COLLECTED PAPERS O F CARL WIEMAN
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COLLECTED PAPERS
OF CARL WIEMAN Carl E. Wieman Director of the Carl Wieman Science Education Initiative University of British Columbia Distinguished Professor of Physics Director of the University of Colorado Science Education Initiative University of Colorado
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ISBN- 13 978-98 1-270-4 15-3 ISBN- 10 98 1-270-4 15-9 ISBN-13 978-981-270-416-0 (pbk) ISBN- 10 98 1-270-4 16-7 (pbk)
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CONTENTS Introduction
1
Precision Measurement and Parity Nonconservation
5
T. W. Hansch, S. A. Lee, R. Wallenstein and C. Wieman, “Doppler-free two-photon spectroscopy of hydrogen ls-2~:’Phys. Rev. Lett. 3 , 3 0 7 (1975).
9
C. E. Wieman and T. W. Hansch, “Doppler-free laser polarization spectroscopy,” Phys. Rev. Lett. 36,1170 (1976).
12
R. Feinberg, T. Hansch, A. Schawlow, R. Teets and C. Wieman, “Laser polarization spectroscopy of atoms and molecules,” Opt. Comm. l8,227 (1976).
16
C. E. Wieman and T. W. Hansch, “Precision measurement on the 1s Lamb shift and of the ls-2s isotope shift of H and D,” Phys. Rev. A 22, 192 (1980).
17
C. E. Wieman and S. L. Gilbert, “Laser frequency stabilization using mode interference from a reflecting reference interferometer,” Opt. Lett. 1,480 (1982).
31
S. L. Gilbert, R. Watts and C. E. Wieman, “Hyperfine structure measurement of the 7s state of cesium,” Phys. Rev. A 21,581 (1983).
34
R. N. Watts, S. L. Gilbert and C. E. Wieman, “Precision measurement of the Stark shift of the 6s-7s transition in atomic cesium,” Phys. Rev. A 21, 2769 (1983).
36
S. L. Gilbert, R. N. Watts and C. E. Wieman, “Measurement of the 6s --+7s M1 transition in cesium with the use of crossed electric and magnetic fields,” Phys. Rev. A 29, 137 (1984).
40
S. L. Gilbert, M. C. Noecker and C. E. Wieman, “Absolute measurement of the photoionization cross section of the excited 7s state of cesium,” Phys. Rev. A 29, 3150 (1984).
47
S. L. Gilbert, M. C. Noecker, R. N. Watts and C. E. Wieman, “Measurement of parity nonconservation in atomic cesium,” Phys. Rev. Lett. 2680 (1985).
51
R. N. Watts and C. E. Wieman, “The production of a highly polarized atomic cesium beam,” Opt. Comm. s , 4 5 (1986).
55
S. L. Gilbert and C. E. Wieman, “Atomic-beam measurement of parity nonconservation in cesium,” Phys. Rev. A 34,792 (1986).
59
S. L. Gilbert, B. P. Masterson, M. C. Noecker and C. E. Wieman, “Precision measurement of the off-diagonal hyperfine interaction,” Phys. Rev. A 3 , 3 5 0 9 (1986).
71
C. E. Wieman, M. C. Noecker, B. P. Masterson and J. Cooper, “Asymmetric line shapes for weak transitions in strong standing wave fields,” Phys. Rev. Lett. 54, 1738 (1987).
75
C. E. Tanner and C. E. Wieman, “Precision measurement of the Stark shift in the 6 s 1/2 + 6P3/, cesium transition using a frequency-stabilized laser diode,” Phys. Rev. A 3 4 162 (1988).
79
C. E. Tanner and C. E. Wieman, “Precision measurement of the hyperfine structure of the 133Cs 6P3,, state,” Phys. Rev. A 38, 1616 (1988).
83
z,
V
vi
M. C. Noecker, B. P. Masterson and C. E. Wieman, “Precision measurement of parity nonconservation in atomic cesium: A low energy test of the electroweak theory,” Phys. Rev. Lett. 61,310 (1988).
85
B. P. Masterson, C. Tanner, H. Patrick and C. E. Wieman, “High brightness, high purity spin polarized cesium beam,” Phys. Rev. A 47,2139 (1993).
89
C. E. Wieman, “Parity nonconservation in atoms; past work and trapped atom future,” in Proceedings of the Workshop on Traps for Antimatter and Radioactive Nuclei, J. Hyperfine Int.
8l,27 (1993).
96
C. E. Wieman, S. Gilbert, C. Noecker, P. Masterson, C. Tanner, C. Wood, C. Cho and M. Stephens, “Measurement of parity nonconservation in atoms,” in Proceedings of the 1992 ‘Enrico Fermi’ Summer School, Varenna, Italy, Course CXX Frontiers of Laser Spectroscopy, (eds.) T. W. Hansch and M. Inguscio (North Holland, 1994), 240.
104
L. Young, W. T. Hill 111, S. Sibener, S. D. Price, C. E. Tanner, C. E. Wieman and S. R. Leone, “Precision lifetime measurements of Cs 6p2P and 6p2P,/, levels by single-photon counting,” 112 Phys. Rev. A 50,2174 (1994).
184
D. Cho, C. S. Wood, S. C. Bennett, J. L. Roberts and C. E. Wieman, “Precision Measurement of the Ratio of Scalar to Tensor Transition Polarizabilities for the Cesium 6s-7s Transition,” Phys. Rev. A 55, 1007 (1997).
192
C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner and C. E. Wieman, “Measurement of parity nonconservation and an anapole moment in cesium.” Science 275, 1759 (1997).
197
S. C. Bennett, J. L. Roberts and C. E. Wieman, “Measurement of the dc Stark shift of the 6s -+ 7 s transition in atomic cesium,” Phys. Rev. A 59, R16 (1999).
202
S. C. Bennett and C. E. Wieman, “Measurement of the 6 s --t 7 s transition polarizability in atomic cesium and an improved test of the standard model,” Phys. Rev. Lett. a 2 4 8 4 (1999).
205
W. C. Haxton and C. E. Wieman, “Atomic parity nonconservation and nuclear anapole moments,” Ann. Rev. Nucl. Part. Sci. 2, 261 (2001).
209
C. E. Wieman, “Pursuing Fundamental Physics with Novel Laser Technology,” in Laser Physics at the Limits (Springer Verlag, 2002), H. Figger, D. Meschede and C. Zimmermann (eds.)
242
Laser Cooling and Trapping
251
R. N. Watts and C. E. Wieman, “Stopping atoms with diode lasers,” Laser Spectroscopy VII, Proceedings of the Seventh International Conference, Hawaii, June 24-28, 1985, eds. T. W. Hansch and Y. R. Shen (Springer-Verlag, 1985), 20.
255
R. N. Watts and C. E. Wieman, “Manipulating atomic velocities using diode lasers,” Opt. Lett. 11,291 (1986).
257
D. E. Pritchard, E. L. Raab, V. Bagnato, R. N. Watts and C. E. Wieman, “Light traps using spontaneous forces,” Phys. Rev. Lett. 57, 310 (1986).
260
C. E. Tanner, B. P. Masterson and C. E. Wieman, “Atomic beam collimation using a laser diode with a self-locking power-buildup cavity,” Opt. Lett. g, 357 (1988).
264
vii
D. Sesko, C. G. Fan and C. E. Wieman, “Production of a cold atomic vapor using diode-laser cooling,” J. Opt. SOC.Am. B 5, 1225 (1988).
267
D. W. Sesko and C. E. Wieman, “Observation of the cesium clock transition in laser cooled atoms,” Opt. Lett. 14,269 (1989).
270
D. Sesko, T. Walker, C. Monroe, A. Gallagher and C. Wieman, “Collisional losses from a light force atom trap,” Phys. Rev. Lett. 63, 961 (1989).
273
T. Walker, D. Sesko and C. Wieman, “Collective behavior of optically trapped neutral atoms,” Phys. Rev. Lett. 64,408 (1990).
277
D. Sesko, T. G. Walker and C. Wieman, “Behavior of neutral atoms in a spontaneous force trap,” J. Opt. SOC.Am. B 8,946 (1991).
28 1
C. Monroe, W. Swann, H. Robinson and C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571 (1990).
294
C. Monroe, H. Robinson and C. Wieman, “Observation of the cesium clock transition using laser-cooled atoms in a vapor cell,” Opt. Lett. 16,50 (1991).
298
E. A. Cornell, C. Monroe and C. E. Wieman, “Multiply-loaded, ac magnetic trap for neutral atoms,” Phys. Rev. Lett. 67, 2439 (1991).
301
C. E. Wieman, C. Monroe and E. Cornell, “Fundamental physics with optically trapped atoms,” in Laser Spectroscopy X, (ed.) M. Ducloy (World Scientific, 1992), 77.
305
K. Lindquist, M. Stephens and C. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46,4082 (1992).
311
C. Monroe, E. Cornell and C. Wieman, “The low (temperature) road toward Bose-Einstein condensation in optically and magnetically trapped cesium atoms,” in Proceedings of the International School of Physics “Enrico Fermi”, Course CXVIII, Laser Manipulation of Atoms and Zons, (eds.) E. Arimondo, W. D. Phillips and F. Strumia (North Holland, 1992), 361.
320
C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt and C. E. Wieman, “Measurement of Cs-Cs elastic scattering at T = 30 pK,” Phys. Rev. Lett. 70,414 (1993).
343
C. J. Myatt, N. R. Newbury and C. E. Wieman, “Simplified atom trap using direct microwave modulation of a diode laser,” Opt. Lett. 47,649 (1993).
347
S. L. Gilbert and C. E. Wieman, “Laser cooling and trapping for the masses,” Opt. Photonics News 4, 8 (1993).
350
M. Stephens, K. Lindquist and C. Wieman, “Optimizing the capture process in optical traps,” J. Hyperfine Int. 203 (1993).
356
M. Stephens and C. E. Wieman, “High collection efficiency in a laser trap,” Phys. Rev. Lett. 72,3787 (1994).
369
M. Stephens, R. Rhodes and C. Wieman, “Study of wall coatings for vapor-cell laser traps,” J. Appl. Phys. 76,3479 (1994).
373
C. Wieman, G. Flowers and S. Gilbert, “Inexpensive laser cooling and trapping experiment for undergraduate laboratories,” Am. J. Phys. 317 (1995).
383
a,
a,
viii
N. R. Newbury, C. J. Myatt, E. A. Cornell and C. E. Wieman, “Gravitational sisyphus cooling of 87Rbin a magnetic trap,” Phys. Rev. Lett. 74,2196 (1995).
397
N. R. Newbury, C. J. Myatt and C. E. Wieman, “S-wave elastic collisions between cold ground state 87Rbatoms,” Phys. Rev. A 51, R2680 (1995).
40 1
M. J. Renn, D. Montgomery, 0. Vdovin, D. Z. Anderson, C. E. Wieman and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75, 3253 (1995).
406
C. J. Myatt, N. R. Newbury, R. W. Ghrist, S. Loutzenhiser and C. E. Wieman, “Multiply loaded magneto-optical trap,” Opt. Lett. 21, 290 (1996).
410
N. R. Newbury and C. E. Wieman, “Resource Letter TNA-1: Trapping of neutral atoms,” Am. J. Phys. 64, 18 (1996).
413
M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman and D. Z. Anderson, “Evanescent-wave guiding of atoms in hollow optical fibers,” Phys. Rev. A R648 (1996).
416
Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell and C. E. Wieman, “Low-velocity intense source of atoms from a magneto-optical trap,” Phys. Rev. Lett. 77, 3331 (1996).
420
Z.-T. Lu, K. L. Convin, K. R. Vogel and C. E. Wieman, “Efficient collection of ’”Fr into a vapor cell magneto-optical trap,” Phys. Rev. Lett. 79,994 (1997).
424
K. L. Corwin, S. J. M. Kuppens, D. Cho and C. E. Wieman, “Spin-polarized atoms in a circularly polarized optical dipole trap,” Phys. Rev. Lett. 1311 (1999).
428
S. J. Kuppens, K. L. Corwin, K. W. Miller, T. E. Chupp and C. E. Wieman, “Loading an optical dipole trap,” Phys. Rev. A 62,013406-1 (1999).
432
S. Duerr, K. W. Miller and C. E. Wieman, “Improved loading of an optical dipole trap by 01 1401-1 (2001). suppression of radiative escape,” Phys. Rev. A
445
Bose-Einstein Condensation
449
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198 (1995).
453
D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, “Collective excitations of a Bose-Einstein condensate in a dilute gas,” Phys. Rev. Lett. -7 420 (1996).
457
J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman and E. A. Cornell, “Bose-Einstein condensation in a dilute gas: Measurement of energy and ground-state occupation,” Phys. Rev. Lett. 77,4984 (1996).
46 1
C. E. Wieman, “The Richtmyer memorial lecture: Bose-Einstein condensation in an ultracold gas,” Am. J. Phys. 64, 847 (1996).
465
C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell and C. E. Wieman, “Production of two overlapping Bose-Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. 78, 587 (1997).
489
D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wieman and E. A. Cornell, “Temperature-dependent damping and frequency shifts in collective excitations of a dilute Bose-Einstein condensate,” Phys. Rev. Lett. 73,764 (1997).
493
s,
a,
a,
ix
E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell and C. E. Wieman, “Coherence, correlations, and collisions: What one learns about Bose-Einstein condensates from their decay,” Phys. Rev. Lett. 79,337 (1997).
497
J. Williams, R. Walser, C. Wieman, J. Cooper and M. Holland, “Achieving steady state Bose-Einstein condensation,” Phys. Rev. A 57, 2030 (1998).
501
C. E. Wieman and E. A. Cornell, ‘The Bose-Einstein condensate,” Sci. Am. 2 x 4 0 (1998).
508
D. S . Hall, M. R. Mattews, J. R. Ensher, C. E. Wieman and E. A. Cornell, “Dynamics of component separation in a binary mixture of Bose-Einstein condensates,” Phys. Rev. Lett. 8 1 1539 (1998).
515
D. S. Hall, M. R. Matthews, C. E. Wieman and E. A. Cornell, “Measurements of relative phase in two-component Bose-Einstein condensates,” Phys. Rev. Lett. sl, 1543 (1998).
519
M. R. Matthews, D. S . Hall, D. S. Jin, J. R. Ensher, C. E. Wieman and E. A. Cornell, “Dynamical response of a Bose-Einstein condensate to a discontinuous change in internal state,” Phys. Rev. Lett. 243 (1998).
523
J. L. Roberts, N. R. Claussen, J. P. Burke, Jr., C, H. Greene, E. A. Cornell and C. E. Wieman, “Resonant magnetic field control of elastic scattering in cold 85Rb,”Phys. Rev. Lett. 81,5109 (1998).
528
E. A. Cornell, J. R. Ensher and C. E. Wieman, “Experiments in dilute atomic Bose-Einstein condensation,” in Bose-Einstein Condensation in Atomic Gases, Proceedings of the International School of Physics “Enrico Fermi” Course CXL, Italian Physical Society, M. Inguscio, S. Stringari and C. E. Wieman, (eds.) (October 1999).
533
M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S . Hall, M. J. Holland, J. E. Williams, C. E. Wieman and E. A. Cornell, “Watching a superfluid untwist itself Recurrence of Rabi oscillations in a Bose-Einstein condensate,” Phys. Rev. Lett. 83, 3358 (1999).
585
M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S . Hall, C. E. Wieman and E. A. Cornell, “Vortices in a Bose-Einstein condensate,” Phys. Rev. Lett. g,2498 (1999).
589
S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell and C. E. Wieman, “Stable 85Rb Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. Lett. 851795 (2000).
593
B. P. Anderson, P. C. Halijan, C. E. Wieman and E. A. Cornell, “Vortex precession in Bose-Einstein condensates: observations with filled and empty cores,” Phys. Rev. Lett. 85,2857 (2000).
597
J. L. Roberts, N. R. Claussen, S. L. Cornish and C. E. Wieman, “Magnetic field dependence of ultracold inelastic collisions near a Feshbach resonance,” Phys. Rev. Lett. -58 728 (2000).
601
J. L. Roberts, N. R. Claussen, S. L. Cornish, E. A. Donley, E. A. Cornell and C. E. Wieman, “Controlled collapse of a Bose-Einstein condensate,” Phys. Rev. Lett. -68 4211 (2001).
605
J. L. Roberts, J. P. Burke, Jr., N. R. Claussen, S. L. Cornish, E. A. Donley and C. E. Wieman, “Improved characterization of elastic scattering near a Feshbach resonance in ”Rb,” Phys. Rev. A 64,024702-1 (2001).
609
a,
X
E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell and C. E. Wieman, “Dynamics of collapsing and exploding Bose-Einstein condensates,” Nature 412, 295 (2001).
612
N. R. Claussen, E. A. Donley, S. T. Thompson and C. E. Wieman, “Microscopic dynamics in a strongly interacting Bose-Einstein condensate,” Phys. Rev. Lett. -98 010401 (2002).
617
E. A. Donley, N. R. Claussen, S. T. Thompson and C. E. Wieman, “Atom-molecule coherence in a Bose-Einstein condensate,” Nature 417,529 (2002).
62 1
E. A. Cornell and C. E. Wieman, “Nobel Lectures: Bose-Einstein condersation in a dilute gas: The first 70 years and some recent experiments,” Rev. Mod. Phys. 74(3), 875 (2002).
626
N. R. Claussen, S. J. J. M. F. Kokkelmans, S. T. Thompson, E. A. Donley, E. Hodby and C. E. Wieman, “Very-high-precision bound state-spectroscopy near a 85RbFeshbach resonance,” Phys. Rev. A 67,060701 (2003).
645
S. T. Thompson, E. Hodby and C. E. Wieman, “Spontaneous dissociation of 85RbFeshbach molecules,” Phys. Rev. Lett. 3,020401-1 (2005).
649
E. Hodby, S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson, D. S. Jin, E. A. Cornell and C. E. Wieman, “Production efficiency of ultracold Feshbach molecules in Bosonic and Fennionic systems,” Phys. Rev. Lett. 94, 120402-1 (2005).
653
S. T. Thompson, E. Hodby and C. Wieman, Ultracold molecule production via a resonant oscillating magnetic field, Phys. Rev. Lett. 95, 190401-1 (2005).
657
Science Education
661
K. B. MacAdam, A. Steinbach and C. Wieman, “A narrow band tunable diode laser system with grating feedback, and a saturated absorption spectrometer for Cs and Rb,” Am. J. Phys. 6 3 1098 (1992).
665
C. Wieman, G. Flowers and S. Gilbert, “Inexpensive laser cooling and trapping experiment for undergraduate laboratories,” Am. J. Phys. 63,3 17 (1995).
679
N. R. Newbury and C. E. Wieman, “Resource Letter TNA-1: Trapping of neutral atoms,” Am. J. Phys. 64, 18 (1996).
693
C. Wieman, “Good science and business practices also yield positive educational results,” Laser Focus World, Comment (April 2004).
696
C. Wieman, “Firming up physics,” AAPT Announcer 34,6 (Summer 2004).
698
K. Perkins and C. Wieman, “Webwatch free on-line resource conncects real-life phenomena with science,” Phys. Edu. 93(January 2005).
699
K. Perkins, W. Adams, M. Dubson, N. Finkelstein, S. Reid, C. Wieman and R. LeMaster, “PhET Interactive simulations for teaching and learning physics,” Phys. Teach. (in press, 2005).
702
W. K. Adams, K. K. Perkins, M. Dubson, N. D. Finkelstein and C. E. Wieman, “The design and validation of the Colorado learning attitudes about science survey,” PERC Proc. (in press, 2004).
710
K. K. Perkins and C. E. Wieman, “The surprising impact of seat location on student performance,” Phys. Teach. 43,30 (2005).
714
xi
K. K. Perkins, W. K. Adams, N. D. Finkelstein, S. J. Pollock and C. E. Wieman, “Correlating student beliefs with student learning using the Colorado learning attitudes about science survey,” PERC Proc. (2004).
718
C. Wieman, “Minimize your mistakes by learning from those of others,” Phys. Teach. 43,252 (2005).
722
C. E. Wieman, “Engaging students with active thinking,” Peer Rev. (Winter 2005).
724
C. Wieman and K. Perkins, “Transforming physics education,” Phys. Today (November 2005), 36.
725
S. B. McKagan and C. E. Wieman, “Exploring Student Understanding of Energy through the Quantum Mechanics Conceptual Survey,” PERC Proc. (accepted).
739
W. K. Adams, K. K. Perkins, N. Podolefsky, M. Dubson, N. D. Finkelstein and C. E. Wieman, “New instrument for measuring student beliefs about physics and learning physics: The Colorado learning attitudes about science survey, Phys. Rev. Spec. Top. PER (submitted), 2005.
743
Development of Research Technology
757
B. R. Brown, G. R. Henry, R. W. Keopcke and C. E. Wieman, “High-resolution measurement of the response of an isolated bubble domain to pulsed magnetic fields,” IEEE Trans. Magnetics 11, 1391 (1975).
761
S. L. Gilbert and C. E. Wieman, “Easily constructed high vacuum valve,” Rev. Sci. Instr. 53, 1627 (1982).
764
D. W. Sesko and C. E. Wieman, “High frequency Fabry-Perot phase modulator,” Appl. Opt. 2 h 1663 (1987).
766
G. J. Dixon, C. E. Tanner and C. E. Wieman, “432-nm source based on efficient second-harmonic generation of GaA 1 As diode-laser radation in self-locking external resonant cavity,” Opt. Lett. Irl, 73 1 (1989).
769
C. Wieman and L. Hollberg, ‘‘Using diode lasers for atomic physics,” (invited review) Rev. Sci. Instrum. 62,l(1991).
772
H. Patrick and C. E. Wieman, “Frequency stabilization of a diode laser using simultaneous optical feedback from a diffraction grating and a narrowband Fabry-Perot cavity,” Rev. Sci. Instrum. 62,2593 (1991).
792
C. Sackett, E. Cornell, C. Monroe and C. Wieman, “A magnetic suspension system for atoms and bar magnets,” Am. J. Phys. 6l, 304 (1993).
795
P. A. Roos, M. Stephens and C. E. Wieman, “Laser vibrometer optical based on feedback-induced frequency modulation for a single-mode laser diode,” Appl. Opt. 6754 (1996).
801
K. L. Corwin, Z.-T. Lu, C. F. Hand, R. J. Epstein and C. E. Wieman, “Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor,” Appl. Opt. 37( 15), 3295 (1998).
809
z,
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Introduction
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The simplest way to characterize my physics career is that I blast atoms with laser light. I actually have been doing this from my early undergraduate days in Dan Kleppner’s lab at MIT. However I like to think that my work has been guided by somewhat more elevated intellectual theme and as a result my research career has followed a somewhat meandering but iairly continuous path that is reflected in my papers. The general theme is that I looked for how I could use the technical capabilities of laser light to study atoms in new ways and in new regimes, with a preference for tackling relatively fundamental physics questions. This meant that I was not only carrying out new physics experiments, but I was also working to advance the technology itself and experimental techniques. As a graduate student with Ted Hansch, I was interested in utilizing the new (at that time) capabilities of narrowband tunable lasers to do high resolution laser spectroscopy in hydrogen. This simplest of all atoms could be quantitatively understood in terms of basic quantum electrodynamics, and thus lasers allowed more precise measures of the energy splittings and correspondingly new tests of QED. By the end of my graduate school days, however, I had become convinced that QED had been so well established that testing it was unlikely to lead to any interesting surprises. The theory of neutral currents and electroweak unification was just coming into vogue at that time, and as such it was the natural extension of my interest. Here was an area where one could do high precision spectroscopy in atoms to measure parity nonconservation (PNC), and thereby explore territory that was far less trod than QED. Of course the downside was that the PNC effects were incredibly tiny and so the experiments were very difficult. This challenge however appealed to my inclination to see just how far one could push experimental capabilities. By pushing the state of the art in several aspects of laser spectroscopy, my group was able to see, and then ultimately measure parity violation in atomic cesium to quite high precision. These were very difficult experiments requiring large amounts of technology development followed by many years of tedious careful measurements and vast amounts of checking and double-checking for possible experimental errors. This was the work that first established my reputation as a scientist, and in retrospect, there were a lot of easier paths that I might have followed. In the midst of doing the PNC experiments, we used portions of the apparatus to carry out numerous other precision measurements on the cesium atom to test the atomic theory that was needed to interpret our parity violation results in terms of fundamental physics. This work also honed our experimental skills and provided publications in the long intervals between each new PNC result. Ultimately, we achieved a much more precise measurement of PNC in atoms, and a correspondingly more precise low energy test of the Standard model of elementary particle physics that, as of this date, is still significantly more precise than any other work. Rather to our surprise, in our second generation measurement we observed that there was the strong suggestion of a small part of the atom PNC that depended on the nuclear spin. After long agonizing over this unexpected and undesired effect in the experiment, we discovered that in fact there were theoretical predictions that such an effect might arise from a strange and never before seen entity called an “anapole moment”. Our third and last PNC experiment, which took more than ten years to complete was sufficiently precise that it provided a reasonably good measurement of the anapole moment. Although this result attracted significant interest and raised a number of significant questions about parity nonconservation in the nuclear forces, it is a bit remarkable that as of this writing, no experimental group other than ours has been able to achieve sufficient sensitivity to observe an anapole moment in any system, now some
3
twenty years after our original observation. To carry out the third generation of atomic parity nonconservation experiments required four tunable very narrowband lasers. At that time the only laser technology available were dye lasers and although, or perhaps because, we were experts in that technology, we knew that the cost and complexity of four suitable dye lasers would be unreasonable. So my graduate student Rich Watts and I began to explore the capabilities of inexpensive, but rather badly behaved, diode lasers. This led to many years of development of diode laser technology and the application of diode lasers in atomic physics research. One of our first and most recognized applications was to show that diode lasers could be used to slow atoms, thereby reducing the cost of the laser system required for atom slowing by nearly a factor of one thousand compared to what had been used. Although the original work was intended merely to show off the capabilities of the inexpensive new laser technology, this work led me to a long and profitable involvement of exploring new capabilities to use light to cool and trap atoms and the study of the light-atom and atom-atom interactions in this new ultracold atom regime. After some years of that work, we had progressed to the point that I thought my group understood enough about the atomic physics and we had developed enough technological capabilities for cooling and trapping atoms in various ways that it would be worth taking the gamble of pursuing the “holy grail” of Bose-Einstein condensation. Eric Cornell joined me in that quest and after five years of work we were successful. After making BEC, there were so many new and exciting experiments that could be done with BEC, and those experiments were so easy compared to the PNC work, that BEC soon became the dominant focus of my research. Throughout most of my career I have been interested in physics education. I always had many undergraduates working in my research labs, and I regularly worked on innovations for teaching undergraduates. I was always struck by the way that students seem to learn little or nothing in classes towards becoming an actual functioning physicist. I could see this in the courses that I taught and in seeing them starting to work in my lab as undergraduate and graduate students. This was in dramatic contrast to the way that a few years in the research lab routinely transformed them into highly competent physicists; something that 16+ years of schooling seemed incapable of doing. From this, it was clear to me that there was some sort of intellectual process present in the research lab that was sorely missing from the traditional education process. As my physics research career was reaching the point where it seemed like there were few if any new heights to reach, I became increasingly interested in science education as a research activity and an area where I might be able to make a substantial impact. I began to see how doing careful research on how people learn physics, and how guiding one’s teaching by the results of that research could work as well in education as it did in the physics lab. This convinced me that science education could be dramatically improved if teachers could be persuaded to break with tradition and follow a new more scientific approach. After receiving the Nobel Prize and realizing the potential for using the stature that comes with the Prize to advance this idea, I have devoted an increasingly large amount of my time and effort to education. This involves both carrying out research in physics education and serving as a public advocate for improved science education and how to achieve it.
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Precision Measurement and Parity Nonconservation
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This work begins with spectroscopy in hydrogen. It involved lots of development of laser to achieve new capabilities. Hidden in the results of the spectroscopy is the development of the method of high power pulsed amplification of a CW laser, as well as the development of the first single frequency tunable blue laser. I also still have fond memories of the excitement at coming up with the idea for polarization spectroscopy as a young graduate student. It was the first time I invented a new experimental technique that was better than what was in use, and so, in my own mind at least, made me feel like a real physicist. As an assistant professor, I staked my career on measuring parity violation in atoms. Fortunately, it did not occur to me for a long time how dangerous a path this was, because I was so wildly and naively optimistic about the difficulty of the experiments. This totally unrealistic optimism in the face of difficulties has stood me in good stead throughout my career. It was getting severely strained though by the time we successfully completed our first PNC measurement. The final PNC measurement was also enormously difficult as a host of unanticipated and unanticipatable problems arose as we pushed far beyond the limits that we or anyone else had previously reached. I had an enormous sense of relief and satisfaction with the completion of that work. Although it did not receive nearly the attention of BEC, I think that it some sense PNC was my best work. It certainly advanced the state of the art and surpassed the work of the competition by a much larger factor.
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VOLUME 34, N U M B E R 6
PHYSICAL REVIEW LETTERS
10 FEBRUARY 1975
Doppler-Free Two-Photon Spectroscopy of Hydrogen I S-2S:g T. W. Hansch,? S. A . Lee, R. Wallenstein,$ and C. WiemanD Department of P h y s i c s , Stanford University, Stanford, Califonzia 94,705 (Received 23 December 1974) We have observed t h e 1.5-2stransition in atomic hydrogen and deuterium by Dopplerf r e e two-photon spectroscopy, using a frequency-doubled pulsed dye laser at 2430 .&. Simultaneous recording of the absorption spectrum of t h e Balmer-p line at 4860 A, using t h e fundamental dye-laser output, allowed u s to precisely compare the energy int e r v a l s 1s-2s and 2 S , P - 4 S , P , D and to determine the Lamb shift of the 1s ground s t a t e to be 8 . 3 i 0 . 3 GHz (D) and 8 . 6 1 0 . 8 GI-Iz (11).
We have observed transitions from the 1s ground state of atomic hydrogen and deuterium to the metastable 2s state, using Doppler-free two-photon s p e c t r o ~ c o p y . " ~ The atoms a r e excited by absorption of two photons of wavelength 2430 A, provided by a frequency-doubled pulsed dye laser, and the excitation is monitored by observing the subsequent collision-induced 2P-lS fluorescence a t the L, wavelength 1215 A. Linewidths smaller than 2% of the Doppler width were achieved with two counter-propagating light beams, whose Doppler shifts cancel. The fundamental dye-laser wavelength a t resonance 4860 coincides with the visible Balmer-P line, and simultaneous recording of the absorption profile of this line permits a precise comparison of the energy intervals 1s-2s and 2 S , P - 4 S , P , D. From our first preliminary measurements we have determined the Lamb shift of the 1s ground state to be 8.3-tO.3 GHz (D) and 8.6*0.8 GHz (H), in good agreement with theory. The only previous measurement of the Lamb shift of the 1s state of deuterium, 7.9*1.1 GHz, has been reported by Herzberg,4 who used a difficult absolute-wavelength measurement of the La line. The hydrogen-1S Lamb shift has never been measured before. Numerous have pointed out that it would be very desirable to observe the 1s-2s transition in hydrogen by Doppler-free two-photon spectroscopy. The +-see lifetime of the 25 state promises ultimately an extremely narrow resonance width. The resolution obtained in the present experiment is already better than that achieved in our recent study of the Balmer-a line by saturation spectroscopy,6 and the implications for a future even more precise measurement of the Rydberg constant a r e obvious. We utilized a dye-laser system, consisting of a pressure-tuned dye-laser oscillator with optional confocal-filter interferometer7 and two subsequent dye-laser amplifier stages, pumped
by the same 1-MW nitrogen l a s e r (Molectron UV 1000) a t 1 5 pulses/sec. This l a s e r generates 102sec-long pulses of 30-50-kW peak power at 4860 A with a bandwidth of about 120 MHz (1-2 GHz without confocal filter). A 1-cm-long crystal of lithium formate monohydrate (Lasermetrics) generates the second harmonic with a peak power of about 600 W. A detailed description of this l a s e r system will be published elsewhere. The ground-state hydrogen atoms are produced by dissociation of H, o r D, gas in a Wood-type discharge tube (1 m long, 8 mm diam, typically 0.1-0.5 T o r r , 1 5 mA). The atoms a r e carried by gas flow and diffusion through a folded transf e r tube about 25 cm in length into the Pyrex observation chamber (Fig. 1). This chamber has two side a r m s with quartz Brewster windows to transmit the uv l a s e r light and a MgF, (originally LiF) window for the observation of the emitted L, photons. A thin coating of syrupy phosphoric acid is applied to all Pyrex walls to reduce the catalytic recombination of the atoms. The uv l a s e r light is focused into the chamber
WOOD DISCHARGE
PRESSURE-TUNED DYE LASER
?f
LITHIUM FORMATE FREQUENCY DOUBLER S O L A R BLIND PHOTOMULTIPLIER
FIG. 1. Experimental setup.
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-1 'F
I S L A M B SHIFT
2430
(b)
.k,
--c
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1 1 1 1 1 1 1 l I I I I I I l I I I I I I I I l l l l l i l l l l
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FIG. 2. (a) Absorption profile of the deuterium Balmer-p line with theoretical fine structure; ( b ) sirnultaneously recorded two-photon resonance of deuterium
1s-2.5.
to a spot s i z e of about 0.5 mm diam. The t r a n s mitted beam is refocused into the cell by a spherical m i r r o r to provide a standing-wave field. The separation between illuminated region and MgF, window is kept s m a l l (1-2 mm) to reduce the l o s s of L, photons due to resonance trapping. The transmitted L, photons a r e detected by a solar-blind photomultiplier (EMR 541 J). An int e r f e r e n c e filter with 6% t r a n s m i s s i o n at 1215 reduces the off-resonance background signal to l e s s than one r e g i s t e r e d photon p e r s e v e r a l hundred laser pulses. The multiplier output i s processed by a gated integrator (Molectron LSDS) with a n effective gate opening t i m e of 200 y s e c and is electronically divided by a signal proportional to the s q u a r e of the uv l a s e r intensity. Figure 2(b) shows a two-photon spectrum of deuterium 1S-2S recorded with moderate resolution (no confocal-filter interferometer). The low Doppler-broadened pedestal is caused by twophoton excitation by each of the linearly polarized uv b e a m s individually and could be eliminated by the use of c i r c u l a r l y polarized light.3 The signal at resonance corresponds to about 10-20 r e g i s tered La photons p e r pulse, and r e m a i n s within the s a m e o r d e r of magnitude when the H, o r D, gas is diluted by He up to a ratio of 1OOO:l. The expected d e c r e a s e in the number of excited meta-
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stable a t o m s i s apparently largely compensated over a wide r a n g e by a concomitant reduction of the loss of L, photons due to resonance trapping and quenching. Despite the lack of a n e a r - r e s o nant intermediate state, the two-photon fluorescence is comfortably strong, and it should easily be possible to o b s e r v e the signal a t a considerably lower total gas p r e s s u r e (loD4T o r r ) , where p r e s s u r e broadening and shifts would become unimportant if the 2s-2P transitions w e r e induced by an applied rf field. F i g u r e 2(a) shows the absorption spectrum of the deuterium Balmer-P line which w a s simultaneously recorded by sending a s m a l l fraction of the blue dye-laser light through a 15-cm-long c e n t e r section of the positive column of a WoodThe posidischarge tube (0.2 T o r r D,, 25 &). tions of the indicated theoretical fine-structure components w e r e located by a computer fit of the line profile. We have m e a s u r e d the separation of the 1s-2s resonance f r o m the strongest component (2P,,,-4D5,,) in the B a l m e r - 8 spectrum to be 3.38 f 0.08 GHz f o r deuterium and 3.3 0.2 GHz f o r hydrogen (in t e r m s of the blue-light f r e quency). The corresponding theoretical s e p a r a tion$ a r e 3.420 and 3.422 GHz, respectively. These separations would be l a r g e r if the 1s s t a t e were not r a i s e d above its D i r a c value by the Lamb shift (theoretically 8.172 GHz for D and 8.149 GHz for H) and our measurement can be interpreted as a determination of the groundstate L a m b shift. A considerable improvement in a c c u r a c y c a n be expected when a high-resolution saturation spectrum6 of the Balmer-P line is used f o r the comparison. A two-photon s p e c t r u m of hydrogen 1s-2s with the laser operating in its high-resolution mode is shown in Fig. 3 (scan t i m e about 2 min). The linewidth is limited by the laser bandwidth of about 120 MHz (in the blue). The s p e c t r u m reveals.two hyperfine components, separated by the difference of the hyperfine splittings of lower and upper s t a t e s , as expected f r o m the selection r u l e AF =O.' It is not difficult to c o m p a r e t h e observed signal strength with theoretical estimates, using Eq. (7) of Ref. 2, derived f o r steady-state conditions. In the p r e s e n t experiment the a t o m s are excited by light pulses whose t i m e duration 7 is s h o r t compared to the inverse linewidth re of the twophoton transition, and which have a n e a r - F o u r i e r transform-limited bandwidth A o = T-'. One can show with the help of time-dependent perturbation theoryg that Eq. (7) of Ref. 2 in this c a s e still
*
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2430.7
tl 0
F:
1*1
0-0
I
4
I
I
I
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I
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D Y E LASER F R E Q U E N C Y D E T U N I N G ( G H z I
FIG. 3. High-resolution two-photon spectrum of hydrogen IS-2s with resolved hyperfine splitting.
correctly predicts the effective two-photon transition r a t e if one replaces the linewidth reby the laser bandwidth A o . W e have numerically evaluated this expression in a calculation similar to that of Ref. 5, but with inclusion of the continuum in the summation over intermediate states, and obtain a two-photon transition r a t e per atom of I?= 7 x 1 0 - 4 1 2 / A ~where , the light intensity I is measured in W/cm2, the laser bandwidth Am in MHz, and r in sec". F o r a comparison with the experiment we consider a H, partial p r e s s u r e of 2 X10'* Torr and 10%dissociation, i,e,, a density of 1.4X10l3 1s atoms/cm3, s o that the loss by resonance trapping is not important. Our estimate then predicts about 3 X l o 5 excited metastable atoms per pulse over a 1-cm path length, With a detection solid angle of 0.02~4.rrsr, a filter transmission of 6%, and a multiplier quantum efficiency of lo%, we expect about thirty registered photons per l a s e r pulse, in reasonable agreement with the present observations. We have also calculated the shift of the 1S-2.S two-photon resonance frequency caused by the intense uv radiation (ac Stark effect). By numerically evaluating Eq. (1) of Liao and Bjorkholm," we estimate a n intensity shift of about 5.5 Hz/ (W/cmz) o r l e s s than 2 MHz under the present operating conditions. The light intensity and
10 FEBRUARY 1975
hence the shift can, in principle, be reduced without loss of resonance signal, by decreasing the l a s e r bandwidth and increasing the interaction time with the atoms. A hundredfold improvement in resolution should be obtainable with the present pulsed dye-laser system, if the hydrogen cell is placed inside a narrow-band confocalfilter interferometer. W e are grateful to Professor T.. Fairchild for lending u s a superb La interference filter of his design. And we thank Professor N. Fortson for stimulating discussions and Professor A. Schawlow for his continuous stimulating interest i n this research.
*Work supported by the National Science Foundation under Grant No. MPS74-14786A01, by the U . S. Office of Naval Research under Contract No. ONR-0071, and by a Grant from the National Bureau of Standards. tAIfred P. Sloan Fellow 1973-1975. $NATO Postdoctoral Fellow. 8 Hertz Foundation Predoctoral Fellow. 'L. S . Vasilenko, V . P. Chebotaev, and A . V . Shishaev, P i s ' m a Zh. Eksp. T e o r . Fiz. 12,161 (1970) [JETP Lett. 12,113 (1970)). *B. Cagnac, G. Grynberg, and F. Biraben, J. Phys. (Paris) 3, 845 (1973). 3 F . Biraben, B. Cagnac, and G. Grynberg, Phys. Rev. Lett. 3, 643 (1974); M. D . Levenson and N. Bloembergen, Phys. Rev. Lett. 32, 645 (1974); T . W. HLnsch, K. C . Harvey, G. Meisel, and A . L. Schawlow, Opt. Commun. 50 (1974). 4G. Herzberg, P r o c . Roy. SOC., S e r . A 34, 516 (1956). k.V . Baklanov and V . P. Chebotaev, Opt. Coinmun. 1 2 , 312 (1974). G T .W. HXnsch, M. H. Nayfeh, S. A . L e e , S. M. C u r r y , and I. S. Shahin, Phys. Rev. Lett. 32, 1336 (1974). 'R. Wallenstein and T. W. Hhhsch, Appl. Opt. 13, 1625 (1974). *J. D. Garcia and J . E . Mack, J . Opt. SOC.Amer. E, 654 (1965). 'A. Gold, in Quantum Optics, Proceedings of the International School of Physics '%nrico Fermi," Course 4 2 , edited by R. J. Glauber (Academic, New York, 1969). lop.F. Liao and J . E . Bjorkholm, Phys. Rev. Lett. 34, 1 (1975).
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VOLUME 36, NUMBER 20
P H Y S I C A L REVIEW LETTERS
17 MAY 1976
Doppler-Free Laser Polarization Spectroscopy* C. Wiemant and T. W. Hsnsch Department of Physics, Stanford U n i v e r s i t y , Stanford, California 94305 (Received 25 March 1976)
We have demonstrated a sensitive new method of Doppler-free spectroscopy, monitoring the nonlinear interaction of two monochromatic laser beams in an absorbing gas via changes in light polarization. The signal-to-background ratio can greatly surpass that of saturated absorption. Polarization spectra of the hydrogen Balmer-P line, recorded with a cw dye laser, reveal the Stark splitting of single fine-structure components in a Wood discharge. We report on a sensitive new technique of highresolution laser spectroscopy based on light-induced birefringence and dichroism of an absorbing gas. Polarization spectroscopy i s related to saturated-absorption1 o r saturated-dispersion' spectroscopy, but offers a considerably better signal-to-background ratio. It is of particular interest for studies of optically thin samples or weak lines and permits measurements even with weak o r fluctuating l a s e r sources. We have studied the Falmer-@line of atomic H and D near 4860 A by the new method, using a single-frequency cw dye laser. The spectra reveal for the f i r s t time the Stark splitting in the weak axial electric field of a Wood-type gas discharge. For saturation spectroscopy it is well known that the signal magnitude depends on the relative polarization of saturating beam and The possibility and advantage of using the resulting optical anisotropy in a sensitive polarization detection scheme seem to have gone unexplored, however. On the other hand, the phenomena of light-induced birefringence and dichroism a r e quite common in optical-pumping experiments with incoherent light source^.^ Recent related experiment^^'^ encourage us to expect that highresolution polarization spectroscopy will also prove useful f o r studies of two-photon absorption and stimulated Raman scattering in gases. The scheme of a polarization spectrometer is shown in Fig. 1. A linearly polarized probe beam from a monochromatic tunable l a s e r is sent through a gas sample, which is shielded from external magnetic fields to avoid Faraday rotation. Only a small fraction of this beam reaches a photodetector after passing through a nearly crossed linear polarizer. Any optical anisotropy which changes the probe polarization will alter the light flux through the polarizer and can be detected with high sensitivity. Such an anisotropy can be induced by sending a second. circularly polarized, l a s e r beam in nearly the oppo-
site direction through the sample. In the simplest case both beams have the same frequency w and a r e generated by the same laser. A s in conventional saturation spectroscopy, a resonant probe signal is expected only near the center of a Doppler-broadened absorption line where both beams a r e interacting with the same atoms, those with essentially zero axial velocity. For a quantitative description we can decompose the probe into two circularly polarized beams, rotating in the s a m e (+) and in the opposite (-) sense as the polarizing beam. As long a s the probe is weak these two components can be considered separately. The polarizing beam in general induces different saturation, i.e., changes in absorption coefficient, Aa' and Aa-, and in refractive index, An' and An-, for these components. A difference Aa'-Acu- describes a circular dichroism which will make the probe light elliptically polarized, and a difference An' -Art- describes a gyrotropic birefringence which will rotate the axis of polarization. A s long as these polarization changes a r e small, the complex field amplitude behind the blocking polarizer is given by
where E , is the probe amplitude, 6 i s some small angle by which the polarizer is rotated from the
r X/4 P L A T E
FIG. 1. Scheme of laser polarization spectrometer,
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perfectly perpendicular position, and 1 is the absorption path length. For low intensities, within third-order perturbation theory* and in the limit of large Doppler widths, the interaction of two counterpropagating circularly polarized beams with two levels of angular momentum J and J' can be described simply in t e r m s of a velocity-selective hole burning' for the different degenerate sublevels of orientational quantum number m , where the axis of quantization is chosen along the direction of propagation. As in conventional saturation spectroscopy' the absorption change as a function of the l a s e r frequency is a Lorentzian function with the natural linewidth yab:
ha+= Acr-/d = - + c Y , , I / I ~ ~1 ~+ X( '). (1 - 5 / ( 4 P d5A~~ACY+=
(2)
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Here, a, is the unsaturated background absorption, I is the intensity of the polarizing beam, Isatis the saturation parameter, and x =(w-wOb)/ yabdescribes t h e l a s e r detuning f r o m resonance. The absorption change corresponds t o the imaginary p a r t of a complex third-order susceptibility, whose r e a l part results in a concomitant change in refractive index, An*= -$Aaixc/w, in agreement with the Kramers-Kronig relation. The magnitude of the anisotropy is described by the parameter d and depends on the angular momenta and the decay rates yJ and y J # of the involved states. If spontaneous re-emission into the lower state is ignored, the steady-state signal contributions of the different orientational sublevels can be added with the help of simple sum rules4 to yield
+ 4 ~ 2)+ for J = J',
j ( 2 3 + 5 r J + 3)/(122- 2) for J = J'+ 1. where Y = ( y j - y j t ) / ( y j + yp). By inserting these results into Eq. (l), we obtain the light flux a t the probe detector
1=I,,[8z + O$sx/( 1 +x') +(is)'/(
1+xZ)],
( 4)
where I, is the unattenuated probe power and s = -i(1 -d)aoZI/Isat gives the maximum relative intensity difference between the two counter-rotating probe components. In practice we have to add an amount I,[ to account f o r the finite extinction ratio t of the polarizer. For a perfectly crossed polarizer, i.e., 6 = 0 , the combined effects of dichroism and birefringence lead to a Lorentzian resonance signal. The signal magnitude is proportional to s2, i.e., it drops rapidly for small s. Small signals can be detected with higher sensitivity a t some small bias rotation 6>>s. The last t e r m in Eq. (4) can then be neglected and the birefringent polarization rotation produces a dispersion-shaped signal on a constant background. If, a s in many practical situations, laser intensity fluctuations a r e the primary source of noise, a figure of merit for the sensitivity is the signal-to-background ratio. Compared t o saturated-absorption spectroscopy, this ratio is improved by a factor (1 -d)O/4(Q2 + 5) which reaches its maximum (1- d ) / 845 for 8 = dt. In addition to the improved sensitivity, such a dispersion-shaped signal is of obvious interest for the locking of the l a s e r frequency to some resonance line. It is also noteworthy that its
(3)
f i r s t derivative h a s a linewidth s m a l l e r than half the natural width, which can greatly facilitate the spectroscopic resolution of closely spaced line components. For the alternative scheme of a linearly polarized saturating beam, rotated 45" with respect to the probe polarization, it can be shown in analogous fashion that the signal always remains Lorentzian, independent of the polarizer angle 6 . As in saturated-absorption spectroscopy, crossover signals a r e expected halfway in between two resonance lines which s h a r e a common upper o r lower level. A third-order nonlinear susceptibility tensor, applicable t o this situation, has actually been calculated p r e v i ~ u s l y .It~ predicts that the ratio A~i-/ha'= d for certain angular momentum states can exceed 1, unlike the expression (3), and hence give r i s e to signals with inverted polarization rotation. For the experimental study of the hydrogen Balmer-P line we used a cw jet-stream dye laser (Spectra-Physics Model No. 375) with 7-diethylamino-4-methyl-coumarin in ethylene glycol, pumped by an uv argon l a s e r (Spectra-Physics Model No. 171). Single-frequency operation was achieved with an air-spaced intracavity etalon (free spectral range 30 GHz; m i r r o r reflectivity 30%) and two additional fixed uncoated quartz etalons (thicknesses 0.1 and 0.5 mm). At 1 W pump power, the l a s e r provid$s single-mode output of about 10 mW near 4860 A with a linewidth of
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~
about 20 MHz. The l a s e r can be scanned continuously over - 4 GHz by applying linear ramp voltages to the piezotransducers of the cavity end m i r r o r and the air-spaced etalon. As in the previous saturated-absorption experiment,g the hydrogen atoms were excited to the absorbing n = 2 state in a Wood-type discharge tube (1 m long, 8 mm in diameter, 0.2 T o r r , 3 mA dc current). The laser light i s sent through a 40-cm-long center section of the positive column. Probe and polarizing beam a r e each about 1 mm in diameter and have powers of 0.1 and 1 mW, respectively, A mica sheet s e r v e s as a h / 4 retarder for the polarizing beam. To avoid residual Doppler broadening due to a finite crossing angle we operated with collinear beams, replacing m i r r o r M , in Fig. 1 with an 80% beam splitter which transmits part of the probe. Standard Glan - Thomp s on prism polarize r s (Karl Lambrecht) were employed for the probe beam. Birefringence in the quartz windows of the gas cell due to internal strain was reduced by squeezing the windows gently with adjustable clamps. We achieve extinction ratios of or better in this way, and the possible improvement over saturated-absorption spectroscopy in signalto-background ratio is on the order of 100-1000. The probe light which passes the blocking polari z e r is sent through a spatial filter to eliminate incoherent light emitted by the gas discharge and scattered light from the polarizing beam, Its intensity is monitored with a photomultiplier. Figure 2 shows a portion of the Balmer-8 spectrum plotted versus time during a l a s e r scan of about 5 min duration. To record the derivative of dispersion-shaped resonances, the dye l a s e r was frequency modulated by adding a s m a l l audiofrequency voltage to the cavity-mirror tuning ramp, and the resulting signal modulation was detected with a phase-sensitive amplifier. The three strongest theoretical fine-structure transitions in this region a r e shown on top for comparison. Hyperfine splitting is ignored. The positions of possible crossover lines due to a common upper o r lower level are indicated by arrows. Obviously, the polarization spectrum reveals many more components. These have to be a s cribed to the Stark splitting in the axial electric field of the positive discharge column, We have calculated the theoretical Stark pattern for an axial field of 10 V/cm by diagonalizing the Hamiltonian. lo The positions of the strongest Stark components and their respective crossover lines are indicated in Fig. 2 and agree well with
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10 vcm-'
I
0
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L A S E R FREOUENCY DETUNING
GHZ ---c
FIG. 2. Polarization spectrum of a portion of the deuterium Balmer-P line. The three strongest fine-structure components and the positions of the strongest Stark components for an axial electric field of 10 V/cm are shown on top for comparison. Crossover lines due to a common upper ( t ) or lower ( + ) level are indicated by arrows.
the observed spectrum. The splitting of the upp e r 4P,D,/, level is essentially a linear function of the electric field, and i t s magnitude can easily be determined to within a few percent from the observed spectrum. The theoretically expected line strengths and the signs of the crossover lines are also in satisfactory agreement with the experiment. The polarization spectrum of light hydrogen looks almost identical to that of deuterium and shows no indication for the 170-MHz hyperfine splitting of the 2.9 state; the components originating in the F = 0 state are missing, because atoms in this level cannot be oriented. The observed Stark pattern changes quite drastically if the l a s e r beams a r e displaced from the tube axis, indicating the presence of additional radial electric fields due to space and surface charges of cylindrical symmetry. Polarization spectroscopy of the hydrogen Ralmer lines thus opens new possibilities for sensitive plasma diagnostics. The spectrum in Fig. 2 also clearly reveals the different natural linewidths of components originating in the short-living 2P state and the
15 VOLUME36, NUMBER 20
PHYSICAL REVIEW LETTERS
l o n g e r - l i v i n g 2s state. T h e narrowest o b s e r v e d components have a width of about 40 MHz, c o r r e sponding t o a r e s o l u t i o n of about 6 p a r t s i n lo*, i.e., they exhibit m o r e than an o r d e r of m a g n i tude i m p r o v e m e n t o v e r o u r earlier p u l s e d - l a s e r saturation ~ p e c t r a .A~ substantial improvement in t h e a c c u r a c y of t h e 1s L a m b s h i f t is e x p e c t e d , i f s u c h a polarization s p e c t r u m is u s e d as a refe r e n c e f o r t h e 1s-2s two-photon s p e ~ t r u m .A~ still f u r t h e r i m p r o v e m e n t i n r e s o l u t i o n should b e p o s s i b l e i f t h e l a s e r linewidth is r e d u c e d by frequency stabilization. A t low e l e c t r i c f i e l d s t h e n a t u r a l linewidth of t h e q u a s i - f o r b i d d e n 2s-4s t r a n s i t i o n is only about 1 MHz. A m e a s u r e m e n t of t h e H-D isotope shift t o better t h a n 0.1 MHz would c o n f i r m o r i m p r o v e t h e i m p o r t a n t r a t i o of e l e c t r o n mass to proton mass, a n d a n a b s o l u t e wavelength o r f r e q u e n c y m e a s u r e m e n t t o better than 6 MHz would yield a new i m p r o v e d value for t h e Rydberg constant. We are p r e s e n t l y e x p l o r i n g t h e s e a n d o t h e r p o s s i b i l i t i e s for new p r e c i s i o n
measurements. We are indebted to P r o f e s s o r A. L. Schawlow for h i s s t i m u l a t i n g interest in this work, and we
17 MAYI976
thank J, E c k s t e i n f o r h i s help i n c a l c u l a t i n g Eq.
(3). *Work supported by the National Science Foundation under Grant No. NSF 14786, and by the U. S. Office of Naval Research under Contract No. N00014-75-C-0841. tHertz Foundation Predoctoral Fellow. 'P. W. Smith and T. W. H b s c h , Phys. Rev. Lett. -26 9 740 (1971). 'C, Borde, G , Carny, B. Decomps, and L. Pottier, C. R. Acad. Sci., Ser. B E , 381 (1973). 'T. W. H k s c h and P. Toschek, 2 . Phys. 266, 213 (1970). '111. Sargent, ID, M. 0. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, London, 1974). %V. Happer, Prog, Quantum Electron. 1, 53 (1970). 'P. F. Liao and G. C. Bjorklund, Phys. Rev. Lett. 36, 584 (1976). 'D. Heiman, R. W. Hellwarth, M. D. Levenson, and G. Martin, Phys. Rev. Lett. 36, 189 (1976). 'M. Dumont, thesis, University of Paris, 1971 (unpublished). 'S. A, Lee, R. Wallenstein, and T. W. Hgnsch, Phys. Rev. Lett. 35, 1262 (1975). 10 J. A. Blackman and G . W. Series. J. P h v a R 6, 1090 (1973).
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U7
traveling in opposite directions, a resonant signal is expected only near the center of a Doppler-broadened absorption line, where both beams are interacting with the same atoms, those with essentially zero axial velocity. With perfectly crossed polarizers, the combined effects of dichroism and buefringence lead to a Lorentzian signal of essentially the natural line width. If the analyzing polarizer is slightly rotated from the perpendicular position, the birefringent polarization rotation produces a dispersion-shaped signal on a small background. With an extinction ratio of lo-', as can be readily achieved with commercial prism polarizers, the ratio of signal to background can surpass that of saturated absorption spectroscopy by 2 or 3 orders of magnitude. To study the hydrogen Balmer-P line, we employed a singlefrequency cw dye laser near 4860 A. The achieved resolution exceeds that of earlier saturation spectra by more than an order of magnitude. The resolved Stark splitting permits the quantitative mapping of axial and radial electric fields in the discharge plasma. To study molecular sodium in an oven at 300°C by lower level labeling, we used two nitrogen-laser-pumped pulsed dye lasers. One provided a broadband probe beam (Ax p. 300 A, centered at 4880 A), the other a monochromatic (Av = 1 GHz) circularly polarized saturating beam, tuned to a molecular X-B transition. The probe light passing through the crossed polarizers is analyzed by a grating spectrograph. Light is transmitted at all wavelengths corresponding to A J = f 1 transitions with the same lower state as the pumped transition. It will be demonstrated how this method can greatly simplify the analysis of unknown atomic or molecular spectra.
LASER POLARIZATION SPECTROSCOPY OF ATOMS AND MOLECULES3 R. FEINBERG, T.W. HANSCH, A.L. SCHAWLOW, R.E. TEETS and C. WlEMAN Department of Physics, Stonford University, Stonford, Gzlifornw 94305, USA
We report on a sensitive new technique of nonlinear laser spectroscopy, based on light induced birefringence and dichroism of an absorbing gas. The technique is related to saturated absorption spectroscopy, but can offer a substantially better ratio of signal to background. Doppler-free polarization spectra of the hydrogen Balmer-line reveal, for the fust time, the Stark splitting of single fine-structure components in the 10 V/cm axial electric field of a Wood-type gaspischarge, and open numerous interesting possibilities for new precision measurements and for sensitive plasma diagnostics [ 11. We have also applied polarization spectroscopy to lower-level labeling [2] of Naz, to provide a useful technique for unraveling the complexities of molecular spectra. The resuiting spectra resemble those of laserexcited fluorescence, but they provide direct information about the spectroscopic constants and quantum numbers of the upper state rather than the lower state. Experimentally, the new technique is rather simple: a probe laser beam is sent t h r o u a a gas sample between crossed polarizers. The light flux reaching a photodetector is a sensitive indicator for any optical anisotropy of the sample. Such an anisotropy can be induced by sending a second, circularly polarized, laser beam nearly collinearly through the probed gas region. By differentially changing the population of various angular momentum sublevels via saturation and optical pumping, this second produces a circular dichrohm and gyrotropic birefringence. For two monochromaticlaser beams of the same frequency,
References (1 ] C. Wieman and T.W. Hansch, Phys. Rev. Letters (1976), accepted for publication. [2] M.E. Kaminsky, R.T. Hawkins, F.V. Kowalski and A.L. Schawlow, Phys. Rev. Letters 36 (1976) 671.
3 Work supported by the National Science Foundation under Grant NSF-14786,and by the U.S. Office of Naval Research under Contract N00014-75C-0841.
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P H Y S I C A L REVIEW A
VOLUME 2 2 , NUMBER 1
JULY 1980
Precision measurement of the 1S Lamb shift and of the 1s-2sisotope shift of hydrogen and deuterium C. Wieman* and T. W. Hansch Department of Physics, Stanford Uniuersity, Stanford, California 94305 (Received 14 September 1979) A precision measurement of the IS Lamb shift of atomic hydrogen and deuterium, using high-resolution laser spectroscopy is reported. The 1S -2stransition was observed by Doppler-free two-photon spectroscopy, using a single-frequency cw dye laser near 4860 A with a nitrogen-pumped pulsed dye amplifier and a lithium formate frequency doubler. T h e n = 2-1 Balmer-P line was simultaneously recorded with the fundamental cw dye-laser output in a lowpressure glow discharge, using sensitive laser polarization spectroscopy. From a comparison of the two energy intervals a ground-state Lamb shift of 8151 k 3 0 MHz has been determined for hydrogen and 8177 30 M H z for deuterium, in agreement with theory. The same experiments yield a tenfold improved value of the IS-2S isotope shift 670992.3 f6.3 MHz and provide the first experimental confirmation of the rclativistic nuclear recoil contribution to hydrogenic energy levels.
-+
result gives the f i r s t experimental confirmation of the relativistic nuclear recoil correction to hydrogenic energy levels. Traditionally, Lamb shifts of S states have been measured by exciting radiofrequency trinsitions to a nearby P state. This technique cannot be used for the ground state, because t h e r e is no P state in the n = 1 level, and a measurement of the 1s shift is consequently considerably more difficult. Herzberg5 attempted the direct approach of measuring the absolute wavelength of the Lyman-a, line to sufficient precision, but such an experiment i s beset by many problems. First, the Lyman-a line (1215 A ) is in the vacuum-ultraviolet region of the spectrum where precision wavelength measurements a r e difficult. Secondly, traditional emission spectroscopy is complicated because emission sources a r e strongly self-reversed, while absorption spectroscopy is plagued by spurious background lines in continuum sources. Thirdly, the Doppler width of the Lyman-a line is about 40 GHz at room temperature, four times l a r g e r than the expected Lamb shift. And finally, if all these difficulties could be surmounted t o obtain a precise value for the Lyman-a! energy, the present uncertainty of the Rydberg constant‘ prevents one from determining the 1s Lamb shift t o better than one part in 1000. Recent advances in high-resolution laser spectroscopy together with improvements in dye-laser technology have made it possible to determine the 1s Lamb shift in a different manner, however, which avoids all these difficulties. This approach, as first described in Ref. 2, u s e s l a s e r spectroscopy t o precisely compare the Balmer-P (n = 2 to 4) transition energy with 2 of the Lyman-a energy. If the Bohr formula were c o r r e c t these two intervals would be exactly the same, & of the
I. INTRODUCTION
The measurement of Lamb shifts in hydrogenic atoms has played a vital role in the development of quantum electrodynamics (QED). Since the f i r s t measurement of the splitting between the 2S,,, and PI,, levels by Lamb and Retherford,’ Lamb shifts have been measured for many hydrogenic states. Some of these measurements a r e among the most precise tests of quantum electrodynamics. One measurement which was notably missing, however, was that of the shift of the 1s ground state. This paper reports on the last of a s e r i e s of three increasingly precise measurements of t h i s quantity performed at Stanford University. Unlike previous Lamb-shift measurements, these experiments a r e based on high-resolution l a s e r spectroscopy. A first and rather preliminary experimental value of the 1s Lamb shift was obtained by m n s c h et al.‘ at the time of the first observation of 1S-2S two-photon excitation in hydrogen. This was followed by the m o r e careful measurement of Lee et aZ.,3combining the techniques of Dopplerf r e e two-photon spectroscopy and saturated absorption spectroscopy. In the present measurements we have made a number of important technical improvements which have enabled us to further reduce the uncertainty to -+30MHz. Among these improvements have been the development of a new, highly sensitive technique of Doppler-free laser spectroscopy, “polarization s p e c t r ~ s c o p y , ”and ~ the construction of a cw dye-laser oscillator with pulsed dye-laser amplifier which offers substantially better power and bandwidth than the previously used pulsed dye-laser system. The new l a s e r has also enabled US to measure the Lyman-a! isotope shift for hydrogen and deuterium to within k6.3 MHz. This 22 -
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@ 1980 The American Physical Society
18 P R E C I S I O N M E A S U R E M E N T O F T H E 1s L A M B S H I F T A N D O F . . .
22
Rydberg energy. Actually they differ by t h e ground-state Lamb shift plus small, well-measured QED and fine-structure corrections t o the excited-state energies. Thus a n accurate comparison of the two intervals allows one to determine the 1s Lamb shift. The comparison of the two energies is made using a powerful and highly monochromatic dye l a s e r with a wavelength near 2860 as illustrated in Fig. 1. The frequency-doubled output of this l a s e r is used to excite two-photon transitions from the IS ground state to the metastable 2s state. Simultaneously, the fine-structure spectrum of the Balmer-P line i s observed using the fundamental l a s e r output, and the position of some component of this line is measured relative to the two-photon reso-
EOHR
.......
4
DlRAC
193
nance. A s can be seen in Fig. 1, a major contribution t o this separation originates from the 1s Lamb shift. Both wavelengths a r e n e a r 4860 A, and one avoids the problems of vacuum-ultraviolet spectroscopy. Furthermore, Doppler broadening of the 15-2s transition can easily be avoided by excitation with counterpropagating beams.’ And because the measurement only involves the comparison of two hydrogen transitions, the uncertainty of the Rydberg constant is unimportant. 11. BACKGROUND
A. Hydrogen energy levels and 1.9 Iamb shift
The energy levels of hydrogen can be written in the form7
QED
-
3
2 2p3/2
/2s1/2 ‘2PI/Z
i
4
_ -
~
-4-
IS-2s TWO-PHOTON EXCITATION WITH
H g FINE STRUCTURE
(b)
where E D is the energy predicted by the Dirac equation using the reduced m a s s Rydberg constant, E R is the additional nuclear recoil correction predicted by the Breit equation for a relativistic two-body system, E N is the correction due t o nuclear s i z e and nuclear s t r u c t u r e effects, and E L , a sum of many t e r m s , gives the QED corrections. In this paper we will be somewhat loose with the t e r m “ground-state Lamb shift,” and use it to r e f e r to the total deviation of the 1s energy from the Dirac energy, i.e., the s u m E , (IS)+ E,(lS) + E,(lS). A detailed list of all the known contributions t o this shift with their most recently computed numerical values is given in Ref. 7. The relativistic nuclear recoil t e r m is given t o lowest o r d e r by
k-n-4
IS-2s TWO-PHOTON RESONANCE (C)
i
I
P I S L A M E SHIFT ! x 114
w
-5 o ~(GHZ) L A S E R FREOUENCY DETUNING-
FIG. 1. Top: Simplified diagram of hydrogen energy levels and transitions. The Dirac fine structure and QED corrections for n =1, 2, and 4 are shown on an enlarged scale. Hyperfine structure and Stark effect are ignored, except for showing the weakly Stark-allowed transition 2S112- 4S1/2. Bottom: Fine-structure spectrum of the Balmer-P line and relative position of the 1s-2s two-photon resonance as recorded with the second harmonic frequency. The dashed line and the dashed arrows above give the hypothetical position of the 1s-2s resonance if there were no 1s Lamb shift. An experimental value of the 1s Lamb shift has been determined from the frequency interval Av between the two-photon resonance and a crossover resonance observed in a polarization spectrum of Balmer-,9 halfway between the fine-structure components 2Si/z4Pi/2 and 2Si/2-*1/2.
where m is the electron m a s s and M the nuclear mass. This recoil shift, which has never been experimentally verified, amounts to -23.81 MHz for hydrogen 1s and -11.92 MHz f o r deuterium 1s. The respective 1.9 shifts due to nuclear structure a r e 1.00*0.05 and 6.78k0.09 MHz. Including these corrections, the 1s Lamb shift has a theoretical value of 8149.43 *0.08 MHz for hydrogen and 8172.23*0.2 MHz for deuterium. The previous measurements of the 1s Lamb shift and the nonlinear spectroscopic techniques involved in this work have been discussed elsewhere in various degrees of completeness.’* 3 1 8 i 9 However, for the sake of clarity we shall give a brief review. B. Doppler-free two-photon spectroscopy of IS-2.9
The technique of Doppler-free two-photon spec-
19
c.
194
W I E M A N A N D 1’.
troscopy, reviewed i n Ref. 10, h a s been cen t r al t o all m e a s u r e m e n t s of t h e 1s L amb shift. F i r s t suggested by Vasilenko et nl. ,11 two-photon excitation of g a s a t o m s with two counterpropagating l a s e r b e a m s can provide n a r r o w resonance s i g n al s f r e e of f i r s t - o r d e r Doppler broadening, b ecau s e t h e a t o m s s e e the two b e a m s with opposite, and thus canceling, Doppler shifts. Two-photon excitation of hydrogen 1.7-2sr e q u i r e s ultraviolet radiation n e a r 2430 of relati.vely high intensity, because t h e r e is no n ear - r es o n an t i n te rmediate level. The t r an s i t i o n r a t e f o r cw excitation h a s been computed numerically by s e v e r a l authors.”’ l 3 According t o Gontier and Trahin,l2 the two-photon absorption c r o s s s ect i o n i s of the order
A
u(Aj=2,75X10-171gcm2,
(3)
w h e r e 1 is t h e light intensity in W/cm2 and g t h e in v e r se atomic transition linewidth i n sec. The expected n at u r al linewidth of t h e transition i n t h e a b s e n ce of collisions i s only about 1 Hz, limited by t h e 4 - s e c l i f et i me of t h e 2 s s t a t e . While c u r r e n t l y available lasers a r e much too broadband to r e a ch t h i s limit, t h ey a r e m o r e than sufficient t o o b s e r v e t h e transition. For a pulsed laser we can easily es t i mat e t h e excitation probability with the help of time-dependent secondo r d e r perturbation theory.14 Assuming, f o r s i m plicity, a s q u a r e excitation pulse of t i m e duration T, intensity I , frequency o, and t r a n s f o r m l i m ite d bandwidth A w =a/T we obtain a n excitation probability p e r a t o m
w h e r e wIs-Dsdenotes t h e at o mi c resonance frequency. At exact resonance, t h e r i g h t mo s t f r a c tion simplifies t o T2; i.e., t h e effective absorption c r o s s section, compared t o t h e s t ead y - s t at e result (Eq. 3), is reduced by a f a c t o r T / 2 g . With a pulse length of 7 n s e c an d a n intensity I = 2 X 10G W/cm2 (approximately t h e experimental conditions), we find a n excitation probability p e r at o m of about 2 X 1 0 - 3 . Since t h e density of 1s a t o m s c a n e a si l y b e as l a r g e as 1014/cm3, t h e signal c a n be substantial even if t h e detection efficiency is poor. C. Previous measurements of the 1.5 lamb shift and limitations
In both previous meas u r emen t s of t h e 1s Lamb shift,2s3t h e frequency-doubled output of a pulsed d y e - l a se r oscillator-amplifier system15 was used t o excite t h e 1s-2s transition, and t h e excitation was detected by observing t h e Lyman-o! radiation emitted i n t h e collision-induced 2.5-1s decay.
w.
22 -
HANSCH
In the f i r s t experiment,‘ the 1s-2s interval w a s compared with t h e n = 2-4 inte rva l by sim ply recording a Doppler-broadened absorption of the Ba.lmer-/3 line in a glow disc ha rge plasma, using the fundamental dye -la se r output. The second measurement3 achieved ;Isubstantial improvement in a c c u r a c y by using the technique of sa tura te d a bsorption spe c trosc opy to obtain better resolution of the Ealnier-P line, again with a portion of the funda m e nta l dye -la se r beam. But despite sub-Doppler linewidths, the fine s t r u c t u r e of t h i s line re m a ine d pa rtly unresolved, and t h e quoted a c c u r a c y of the 1s La m b shift was s t i l l a lm ost e ntire ly limited by the inadequate resolution of this re fe re nc e line. I>. Laser polarization spectroscopy
Searching f o r ways to im prove the resolution of the Ba lm e r-P line, we developed t h e technique of laser polarization ~ p e c t r o s c o p y a, ~method of Doppler-free spe c trosc opy which offers considerably higher sensitivity than conventional sa tura te d absorption spectroscopy. This techniqlie enabled us to obse rve the Ralmer-P line with a low-power cw dye l a s e r in a mild glow discharge. Single Stark components of fine-structure line s could b e readily resolved in t h e s m a l l axial c l e c t r i c field of the disc ha rge plasma. This s u c c e s s opened the way f o r the pre se nt improved m e a sure m e nt of the 1.5 La m b shift. The b a s i c concept of polarization spectroscopy is r a t h e r simple: A line a rly polarized probe laser be a m is s e n t through a g a s s a m p l e and p a s s e s through a ne a rly c r o s s e d l i n e a r pola riz e r, be fore reaching a detector. Any optical anisotropy in the s a m p l e which changes t h e probe polarization can thus b e detected with high sensitivity. Such a n anisotropy is introduced by a second laser be a m of the s a m e frequency, which is counterpropagating and c irc ula rly polarized. In t h i s case, a s s u m i n g low intensities, one de te c ts a Doppler-free signa l as given by Eq. (4) of Ref. 4:
[
1 11, (I>” llxz]’
I=I, 62f6 --+
-
(5)
where I is t h e detected intensity, I , t h e incident probe intensity, and 8 is t h e rotation angle of the analyzing pola riz e r f r o m t h e perfectly c r o s s e d orientation. The frequency detuning f r o m the center of the Doppler-broadened line is de sc ribe d b y the normalized p a r a m e t e r x = (w w a b ) / y a b ,w h e r e w is the l a s e r frequency, wab is t h e tra nsition f r e quency, and yab is the na tura l linewidth of the transition. The p a r a m e t e r s gives t h e maximum relative intensity difference between t h e rightand left-circularly-polarized components of the probe and is defined by s =-$(1- d ) a. I I/Isatr w he re 2 i s t h e s a m p l e length, a,, is t h e unsaturated
-
20 22
P R E C I S I O N M E A S U R E M E N T O F T H E 1s L A M B S H I F T A N D O F ...
absorption coefficient at line center, Isatis the transition saturation parameter, as defined in Ref. 15, and d is the ratio of the light-induced changes in the absorption coefficient for right- and leftcircularly-polarized light. In the chosen mode of operation the analyzing polarizer is rotated s o that 4 8 s i s about 10 times larger than (s/4)', and a derivative signal is obtained by giving the lassr a small frequency modulation and recording the corresponding signal modulation with a phase sensitive amplifier. The resulting line shape, the derivative of x/(l + x ) , appears bell shaped with inverted wings, and its width is less than $ the natural linewidth.
LASER
INTERFEROMETER SHIFTER
/
,-
POLARIZER
I
'0w 7%
PMT
PULSED
nI. APPARATUS
A schematic overview of the apparatus is given by Fig. 2. The output of a single-frequency cw
dye laser (upper left) is split into four beams. One of these is sent into a frequency-marker interferometer (top) for the purpose of calibration. Two of the beams become the polarizing and probe beam of the polarization spectrometer (center). Finally, the fourth beam provides the input for a pulsed dye-laser amplifier system. The amplified beam is sent through a frequency-doubling crystal which generates the ultraviolet radiation for the two-photon spectrometer (bottom right). Because of its complexity, this entire setup is spread over three separate optical tables. A. cw dye-laser oscillator
The cw laser is a Spectra Physics model 375 folded cavity jet s t r e a m dye l a s e r which has been modified to provide tunable single-frequency operation at blue wavelengths, using a solution of 7-diethylamino-4-methylcoumarinin ethylene glycol. The standard tuning elements, a dielect r i c tuning wedge filter and a 0.1-mm uncoated quartz etalon a r e augmented by two additional intracavity etalons. The first is an angle-tuned solid uncoated quartz etalon 0.5 mm thick. The second is an a i r spaced etalon with a gap of 5 mm (free spectral range 30 GHz) and 30% m i r r o r reflectivity. The spacing can be varied with a piezotransducer, and the etalon is contained in a temperature stabilized oven. For cavity fine tuning the laser output m i r r o r is mounted on a piezotransducer. To decrease the linewidth the entire l a s e r is completely enclosed in a large, nearly airtight, wooden box. The dye laser is pumped by a Spectra Physics model 171 argon-ion laser. The pump threshold for single-frequency operation at 4860 is about 600 mW. An output of 20 mW i s obtained with 2-W pump power. During the experiment the dye l a s e r
195
AMPLIFIERS FREOUENCY DOUBLER
1- ABSORPTION TWO-PHOTON CELL
FIG. 2. Schematic of experiment. The Balmer-P line is recorded by polarization spectroscopy (center), while the 1s-2s transition is observed by two-photon excitation with the second harmonic of the dye-laser frequency (bottom).
operated typically at 7.5-mW output power. The dye-laser frequency can be continuously scanned over 4 GHz by applying tuning ramp voltages to the piezotransducers of the cavity end mirr o r and the air spaced etalon. During the experiment the linewidth was on the order of 5 to 10 MHz [full width at half maximum (FWHM)] , although the width decreased to about 2 MHz when the pneumatic vibration isolation of the optical table was activated. The kequency modulation (about 10-MHz sweep amplitude) needed to record a derivative signal is obtained by adding an audio frequency a c voltage (4 V, 2 kHz) to the ramp voltage of the cavity mirror. The dye-laser output beam is sent into an acousto-optic modulator. Diffraction from a traveling acoustic wave in a block of fused quartz generates three beams, one upshifted in frequency (+37.60 MHz), one downshifted (-37.60 MHz), and one with its frequency unchanged. The primary function of this device in our experiment is that of a beam splitter, but the frequency shift improves the signal-to-noise ratio of the polarization spectrometer, as will be discussed later. The upshifted beam, containing about 0.75 mW of power, is sent into the frequency-marker interferometer. The downshifted beam (also 0.75 mW) becomes the probe beam of the polarization spectrometer, while the unshifted beam is used as the polarizing beam. About 30% of this unshifted beam is split off with a partly reflecting m i r r o r and sent into the pulsed amplifier system.
21 196
C . WIEM.43' A N D T. W . HANSCII
22
B. Freqilencp-marker interferometer
D. Polarization spectrometer
The frequency-marker interferometer i s a s e m i confocal interferometer, consisting of two dielect r i c m i r r o r s cemented onto the ends of a quartz tube. It has a finesse of 1 5 , and i t s transmission peaks a r e separated by 113.142 33(15) MHz. This separation was used for frequency calibration in all reported measurements. The thermal drift of this frequency marker was 0.2 to 1.5 MHz/min and typically remained constant to better than 10% o v e r 5 h. The m a r k e r separation was determined to within 2 p a r t s in lo4, by mechanically measuring the length of the spacer. To find a more accurate value, the dye l a s e r was tuned to the center of a transmission peak, and the l a s e r frequency was measured with a precise fringe-counting digital wave meter.lG Then the l a s e r was tuned to another peak, about lo3 fringes away, and the frequency was measured again. From the frequency difference and the known marker spacing the number of fringes between the two peaks can be determined exactly, which in turn yields an improved m a r k e r separation. The procedure can then be repeated with a new, l a r g e r frequency interval. Taking several iterative steps in this manner, the separation between adjacent o r d e r s could be quickly determined to the quoted accuracy.
A simpiified scheme of the polarization spectrometer is included in Fig. 2. The polarizing beam is sent through a quarter wave plate to change its polarization from linear to circular. Two lenses of focal length f =7.8 cm form a 1 : 1 telescope (not shown in Fig. 2) which focuses the beam (original diameter 3 mm) to a waist diame t e r of about 0.2 mm at the center of the hydrogen discharge tube a t 75-cm distance. The probe beam, coming from the opposite direction, passes through an identical telescope to nearly match the confocal parameters of the polarizing beam. To ensure nearly perfect linear polarization the probe is sent through a prism polarizer before entering the discharge tube. It c r o s s e s the polarizing beam near the center of the discharge tube at an angle of 2 mrad. After emerging from the Wood tube, the probe is sent into a second, "analyzing" polarizer whose axis is rotated nearly 90 degrees relative to the first one. Both polarizers a r e cemented Glan Thomson prism polarizers (Karl Lambrecht) with a rated extinction ratio of By using small (about 2mm diameter) and fairly well collimated beams, however, we typically achieve extinction ratios better than During the experiment, the angle of the analyzing polarizer was adjusted s o that the 0s contribution to the signal'was about 5 to 10 times l a r g e r than the s2 contribution. This angle depended on the experimental conditions and varied between 4X and 3 X rad. Probe light which passes through the analyzing polarizer is sent through a spatial filter at 1.5-m distance from the discharge tube (lens of f=7.8 cm, pinhole of 50-pm diameter), and its intensity is monitored by a photomultiplier (RCA 1P28) followed by a lock-in amplifier (PAR model JB5). The internal sine wave reference of this amplifier provides the signal for the 2-kHz dye-laser f r e quency modulation. Although this apparatus can be assembled quite easily, certain pains must be taken t o attain the l a r g e signal-to-noise ratio offered by polarization spectroscopy. The first requirement i s a good extinction ratio for the probe beam. Considering the small rotation angle of the analyzing polarizer it i s obvious that a n extinction ratio much worse than would have been a serious limitation in the reported experiments. Several different meas u r e s a r e taken to a s s u r e this ratio: First, fairly good polarizers a r e used and the beams through them a r e kept small and collimated; second, t h e r e a r e a s few optical components as possible in between the polarizers, and for those which a r e unavoidable (the discharge windows) the birefringence
C. Hydrogen discharge tube
In o r d e r to observe the Balmer-P line in absorption, hydrogen atoms a r e excited in a Wood-type glow discharge tube, similar to those described in Refs. 6 and 9. The tube i s 138-cm long with an inner diameter of 1.5 cni, and the walls a r e coated with orthophosphoric acid to prevent catalytic r e combination of the atoms. Wet molecular hydrogen from a n electrolytic generator provides a continuous gas flow through the tube. A l a r g e a r e a cold aluminum cathode ensures a stable discharge. The l a s e r beams pass through a 60-em-long center section of the positive discharge column. The windows a r e formed by pieces of quartz microscope slides, cemented onto tube extensions with T o r r Seal adhesive. By gently squeezing these windows with small transverse clamps their birefringence can be reduced until extinction ratios better than a r e observed between crossed polarizers. During the experiment the tube was operated at p r e s s u r e s between 0.1 and 1.0 t o r r and at currents between 5 and 20 mA, and the absorption at resonance ranged from as high as 10% to l e s s than l%, depending on p r e s s u r e and current.
22 22
-
P R E C I S I O N M E A S U R E M E N T O F T H E 1s L A M B S H I F T A N D O F
is minimized; finally, all surfaces a r e kept clean to avoid depolarizing scattering. It is also important to minimize any light other than the probe light which reaches the detector. The two primary sources of such background in the present experiment a r e the gas discharge and backscattering of the polarizing beam. The spatial filter reduces the light from both these sources to below the residual probe intensity (10-710). Originally we were facing the problem, however, that the small amount of scattered polarizing light which still reached the detector was coherent with the probe light, causing the detector to respond to an interference between the two electric fields. Slow phase fluctuations resulted in considerably increased noise. The use of the acousto-optic frequency shifter eliminates this problem by causing the interference term to oscillate at a frequency too high for the detector to respond to it. We found that vibrating one of the m i r r o r s reflecting the polarizing beam, for instance, by mounting it on a radio speaker, also eliminates this noise source. The frequency shifter i s preferred, however, because it also reduces amplitude fluctuations of the dye laser due to feedback into the cavity, a problem to which this dye l a s e r is particularly sensitive. E. Pulsed dye-laser amplifier
The pulsed laser amplifier is shown in Fig. 3. The system worked at first t r y more than adequately for this experiment, and little effort was made to optimize the geometry. The pump source is a Molectron W 1000 nitrogen laser operating at 15 pulses per s e c and producing 10-nsec-long pulses of about 650-kW peak power. Approximately 10% of t h i s light is used to pump the first dye amplifier stage, 30% is sent to the second, and 60% to the third stage. The dye cells and the transverse focusing of the pump light into them a r e identical to those used in the pulsed oscillatoramplifier forerunner of this With 1-mW cw input power a gain of about lo5 in peak power is readily achieved in the first stage, but the pulse width is only about 3-4 nsec (FWHM). The power of the amplified beam after the first stage is about 5 to 10 times larger than the amplified spontaneous fluorescence into the same solid angle. The second stage gives a (saturated) gain of about 45 and some pulse stretching, while the third stage gives a gain of 22 and stretches the pulse to about 7 nsec (FWHM). The third stage is saturated to such an extent that it produces the same peak power under almost any alignment condition. Optimizing the alignment broadens the roughly Gaussian-shaped pulse in time, however, and reduces the bandwidth accordingly, as we
TO
...
197
DOUBLING CRYSTAL
f
IMW NITROGEN LASER
25cm
\DIAPHRAGM AMPLIFIER
I
FIG. 3. Pulsed dye-laser amplifier.
found empirically. Pulse energies of about 5 mJ with less than 5% fluctuations are easily obtained, and the spatial beam profile can show a much closer resemblance to a Gaussian TEM,, mode than was normally attained with the earlier pulsed 0s cillato r -amplifi e r system. In hindsight the most serious problem in the present experiment is the fact that the pulsed amplifiers not only broaden but also shift and distort the spectrum. To monitor this shift, a portion of the amplified output beam is sent through a confocal interferometer (2-GHz f r e e spectral range, finesse 150) as indicated in Fig. 4. The transmitted light is measured by two photodiodes; one of these records the pulsed light while the other registers only the simultaneously present collinear cw light. Thus, as the l a s e r is scanned over the transmission peak of the interferometer, the pulsed and cw spectra are obtained simultaneously and can be compared. F. Two-photon spectrometer
Also shown in Fig. 4 are details of the twophoton spectrometer which is almost identical to that used in the earlier experiments.2s3 The output of the pulsed amplifier is focused into a lithium formate frequency-doubling crystal (Lasermetrics) using a lens of f = 1 8 cm. The crystal is 1 cm long and phase matching near 4860 is achieved through angle tuning. With this geometry
23 22
C . W I E M A N A N D 1'. W . H A N S C H
198
'
I 55cm
I
1
\ L I T~. HIUM
FORM ATE FREQUENCY DOUBLER
-
expect a 2s lifetime of about 10 nsec a t a p r e s s u r e of 0.1 t o m , which s e e m s t o a g r e e with our (imprecise) observations. However, t h e r e i s substantial radiation trapping and l o s s of photons through nonradiative decay mechanisms which a r e not well understood. To minimize this loss of signal the ultraviolet beams a r e kept as close to the side window a s possible (distance l e s s than 1 mm). The emerging Lyman-or photons a r e detected with a s o l a r blind photomultiplier (EMR 5415). Scattered l a s e r light is reduced to a negligible level by a Lyman-a interference filter in front of the detector (Matra Seavom Co., 15% peak transmission, 90 I% bandwidth). The output current from the photomultiplier is sent into a gated integrator. G. Data processing
BEAM FROM PULSED A M P L I F I E R SYSTEM
FIG. 4. Details of two-photon spectrometer.
A l l data from the experiment, a r e converted into digital form (12 bits accuracy) and stored on magnetic disk for processing with a minicomputer (Hewlett - Packard 2 1OOA). IV. EXPERIMENTAL PROCEDURE AND ANALYSIS
the doubling efficiency approached 2%, and most data were taken at pulse energies of about 7 pJ. The crystal showed signs of burning after about 15 000 pulses at such a power level. (With tighter focusing it was possible t o obtain over 7% conversion efficiency, but the crystal burned much m o r e rapidly.) The generated ultraviolet beam i s separated f r o m the fundamental beam by a Brewster-angle quartz p r i s m and then collimated with a Suprasil lens (f=1000 mm). The ultraviolet intensity is monitored by observing the light reflected off this lens with a photomultiplier with attenuating diffuser. A s it p a s s e s through the hydrogen absorption cell the beam has a nearly rectangular c r o s s section of 0.2 by 0.4 mm. A flat m i r r o r immediately a f t e r the cell is used t o reflect the beam back onto itself t o provide the required standing-wave field. The two-photon absorption cell is the s a m e as described in Refs. 2 and 3. The ground-state hydrogen atoms a r e produced in a Wood-type discharge and flow and diffuse into the absorption cell through a 25-cm-long folded transfer tube, coated with phosphoric acid. The discharge was run with a current of 20 mA and at p r e s s u r e s between 1.0 and 0.05 t o r r of wet hydrogen. The absorption cell is about 10 c m long with quartz windows f o r the ultraviolet beams on the ends s e t at the Biewster angle. The 1s-2s excitation is monitored by observing the Lyman-a! radiation emitted through a magnesium fluoride side window. The vacuum-ultraviolet radiation i s due t o collisional mixing of the 2.5 and 2P states. F r o m measurements of the collision cross sections" we
A major portion of our efforts were devoted to the investigation of systematic line shifts, as summarized in Table I. The experimental procedure was divided into five distinct portions; the investigation of systematic shifts of the hydrogen two-photon line, the measurement of the s e p a r ation of hydrogen and deuterium two-photon lines, the study of shifts of the Balmer-P reference line, the measurement of the separation between the hydrogen two-photon line and the Balmer-/3 reference line, and finally, the measurement of the hydrogen-deuterium separation of the Balmer-/3 line. The result of the first two p a r t s gives the H-D 1s-2s isotope shift while the first, third, and fourth part yield the hydrogen 1s Lamb shift. The deuterium 1s Lamb shift is obtained by combining results of all five portions. A. Systematic shifts of the 1s-2sline
The only systematic shift of the two-photon signal studied experimentally was that due t o the p r e s s u r e in the two-photon absorption cell. First the pressure was set at 0.05 t o r r and the l a s e r was scanned -5 times over a n -I-GHz range containing the resonance line. During these scans the computer sampled the inputs to six channels on the analog t o digital converter 100 t i m e s p e r sec. Each sweep lasted -25 s e c so the sampling points were about 0.25 MHz apart. The six data inputs were the two-photon signals, the intensity of frequencydoubled (2430-A) light, the frequency-marker signal, the pulsed signal f r o m the spectrum analyzer monitoring the amplified beam, the cw signal from
24 P R E C I S I O N M E A S U R E M E N T O F T H E 1s L A M B S H I F T A N D O F
22 -
...
199
TABLE I. Systematic line shifts considered. a
-
1s-2s two-photon resonance
pressure a c Stark effect laser line shape Balmer-j3 line (4S112-2S1 /,-4P1 polar iz er angle polarizer angle pressure, electric field zero-pressure Stark shift and unresolved hyperfine structure ac Stark effect discharge current frequency-marker thermal drift
meas calc meas/calc
MHz/torr MHZ/MW cm-' of linewidth
0 +5 +0.6
-13
*9%
15 12 -28
*4
crossover) meas caIc meas
MHz/sB" MHz/sB-' MHz/torr
+6
-4.5 * 2 -0.28
calc calc meas meas
MHz MHZ/W cm-2 MHz/niA AfHz/h
-0.1 i: 0.2 60 t l
aShift v (observed) - v (true) in terms of fundamental dye-laser frequency. bGood approximation for 1 sO-'l< 0.2, as in all measurements. See Eq. (5). 'These results a r e not independent. See text for discussion. dTypical value for a given run.
that analyzer, and last, the scanning voltage applied to the cw laser end m i r r o r piezodrive. All pulsed signals were processed by gated integrators. The output proportional to the pulse a r e a was held constant in between laser pulses. After recording these scans the pressure was changed to 1.0 t o r r and 5 more scans w e r e made with the same recording procedure. This was followed by 5 scans again at 0.05 t o r r and then 5 at 1.0 torr. The entire procedure took about 30 min. The first step in the analysis of these data was t o set the frequency scale for each scan by finding the centers of the frequency-marker peaks. For each peak t h e computer generated a smoothed curve where each point of the smoothed curve was the average of 10 adjacent raw data points. For example, the 95th point on the smoothed curve was the average of the 91st through 100th raw data points. The line center for the smoothed curve was found at the Q, and $ peak-height points and the average of these points was taken t o be the frequency-marker center. Although there was essentially no uncertainty in finding the center of a given peak, imperfections in the laser piezodrive caused a 2-3% random fluctuation in the peak-to-peak spacing causing a corresponding uncertainty of the frequency scale. All subsequent analysis in this and the remaining part of the experiment was done in terms of frequency scales determined using this procedure. The two-photon spectra were first normalized by dividing the signal by the square of the 2430-A intensity. This caused no observable shift of the lines, but did reduce the pulse to pulse amplitude fluctuations slightly. Next the spectra were
a,
smoothed in the same manner as used with the frequency marker except that a 40-point average was used. While such a smoothing routine could conceivably cause the peak frequency to shift, for our lineshapes the shift would be less than 0.4 MHz and hence negligible compared to the statistical uncertainty. Examples of such spectra are shown in Figs. 5 and 6(b). After smoothing, the centroid of the line including both hyperfine components was calculated. The values for each set of 5 runs were averaged and the resulting four points (at 0.05, 1, 0.05, and 1 torr) were plotted as a function of the time of acquisition t o determine the thermal drift of the frequency marker. The drift rates shown by the high- and low-press u r e points were the same, and an appropriate correction was made. B. Hydrogendeuterium IS-2s isotope shift
The next stage of the experiment was the measurement of the 1s-2s isotope shift. F o r this, the
tI
1
0
1
1
0.2
1
1
1
1
'
!
-
0.4 0.6 168.0 DYE-LASER FREQUENCY TUNING
,
I
168.2
*
(GHz)
FIG. 5. Two-photon spectrum of 1s-2s transition in hydrogen (left) and deuterium (right).
C . WIEMAN AN D T. W. HANSCH
200
I
4800
I
I
4600
I
I
4400
I
-
4M)
t
I
iS - 2 5 POLARIZATION
SPECTRUM
I
I
200 0 FREQUENCY I M H z )
TWO-PHOTON SPECTRUM
FIG. 6. (a) Portion of the polarization spectrum of the hydrogen Balmer-fi line. @) Two-photon spectrum of hydrogen 1.9-2s transition. The frequency-marker signal at the top has been recorded simultaneously.
two-photon absorption cell was filled with a mixture of H and D at 0.05 t o r r . A crude preliminary measurement using a digital wave meter w a s made to determine the separation of the hydrogen and deuterium lines to within one free spectral range of the frequency marker. The following procedure was then used to determine the additional fraction. Several (4 to 5) scans of the hydrogen two-photon line were made and the spectra stored as previousl y described and then the l a s e r frequency was shifted t o the deuterium line and the procedure repeated. The laser wavelength was returned to the hydrogen line and the sequence repeated until a total of 52 spectra were obtained. The data were analyzed by finding the frequency scale and normalizing and smoothing the twophoton spectra just as before. At t h i s stage in the analysis however, it was necessary to confirm that the deuterium and hjjdrogen line shapes were not systematically different (except for the difference in hyperfine splitting, of course). Because the two-photon line shape was not simple and could change noticeably over periods on the order of a day such a confirmation that it did not change with wavelength was necessary. This was verified by two tests. First, spectra of the fundamental pulsed l a s e r taken at the wavelength of the hydrogen line, the deuterium line, and -100 A on either side were compared and found to be identical. While it is impossible to make a direct correspondence between this spectrum and the twophoton line shape, as will be discussed later, it was observed that changes in one were always correlated with changes in the other. The second test was to compare the two two-photon line shapes directly, a procedure which was complicated by the deuterium hyperfine splitting being
22 -
only partially resolved. This problem was s u r mounted by taking the line shape observed for a single hydrogen hyperfine component and using it to generate a predicted deuterium spectrum using the known hyperfine splitting and intensity ratio for deuterium. The generated line shape was then compared with the observed deuterium spectra obtained at similar times using only the overall amplitude as a free parameter. These were found to match to within the statistical fluctuations, confirming that the hydrogenand deuterium line shapes were the same. Now the centroids of the lines were calculated and these values plotted as a function of time. The drift of the frequency marker was then calculated and appropriate corrections made. C. Systematic shifts of Balmer-0 line
The next and longest portion of t h e experiment was the investigation of systematic shifts of the Balmer-6 signals. The basic procedure was to repeatedly scan the cw dye laser to obtain a polarization spectrum containing the 2S,,4P,,, line, the 2S,, 4S,, line (Stark allowed), and their crossover, l9 while changing various parame t e r s [see Fig 6(a)]. We restricted ourselves to these lines because they a r e narrow and much simpler to interpret than the other components. During each scan the computer sampled and stored the inputs to three channels on the analog-todigital converter (ADC) at 100 times/sec. The three inputs were the l a s e r scanning voltage, t h e frequency-marker signal, and the output of the lock-in amplifier monitoring t h e polarization spectrometer phototube. The first systematic effect studied was the shift of the polarization spectra line centers due to the addition of the small derivative-shaped S0 t e r m to the larger bell-shaped S2term. To measure t h i s shift several scans were made with the polarizer in the + 0 orientation then it was rotated to - 8 , which causes a shift in the opposite direction, and several more scans were recorded. This was r e peated at several times to determine the frequency-marker drift, and at a variety of angles. The following analysis procedure of the polarization spectra was used for this and all subsequent portions of the experiment. First t h e spect r a were smoothed in the standard way. Between one and ten points were used in the smoothing average, depending on experimental conditions, with tests made to ensure this introduced no line shifts, The line center was then determined at regular height intervals (every 5% of the 2S,, - 4 P y , line and every 10% of the crossover) and the average of these centers found. The result of the tests of the 0 dependence
-
26 22 -
PRECISION MEASUREMENT OF THE
showed that f o r all values of 0 used in the subsequent measurements the shift was l e s s than 3 MHz, i n agreement with the theoretical estimates. A s part of all subsequent measurements the data were taken with the polarizer in both + and - orientations and the results averaged, thereby canceling the shift to within a negligible uncertainty. The effects of the discharge environment (cur rent, pressure, electric field) on the Balmer-P reference line were also investigated. First the effect of the current was studied by recording a s e r i e s of spectra at discharge currents of 20 and 11 mA, alternating repeatedly between the two settings. Next the effects of the pressure and the electric field in the discharge were measured. Because only the pressure could be independently varied and this in turn affected the electric field it was necessary to obtain a composite correction. In these measurements the same data-acquisition procedure was used as for the current shift measurements, except that the current was set at 11 mA and the pressure varied. Two data runs were used; one which took data at 0.16, 0.41, and 0.70 t o r r and the second at 0.39, 0.27, and 0.72 torr. The pressure readings were made using a thermocouple gauge. A total of 80 spectra were recorded. D. Comparison of 1s-2s transition with Balmer-0 line
The fourth stage of the experiment was the measurement of the separation between the hydrogen two-photon peak and the crossover line between the 2SII2- 4pl12and the 2SlI,- 4S,,, transitions. This crossover w a s chosen as the reference line for the Lamb-shift measurement because it was l e s s sensitive to the electric field perturbation than the other lines. In this stage seven data channels were read into the computer; the combination of the three cw and four pulsed data lines previously described. The two-photon absorption cell was operated at 0.05 t o r r and the polarization spectrometer discharge ran at 0.27 t o r r and 11 mA. I-GHz scans with the same sampling rate as before were used. First, 5 scans were made over the twophoton line; then as rapidly a s possible (-1 min) the l a s e r wavelength was reset and the Balmer-0 crossover line was scanned several times. Examples of such spectra a r e shown in Fig. 6. The l a s e r was then reset to the two-photon line and the procedure repeated until a total of 45 twophoton spectra and 51 Balmer-P spectra were obtained. The Balmer-P spectra were analyzed using the same procedure a s before, but it was necessary to use a different procedure in the analysis of the two-photon lines because of their asymmetric line shape. The difficulties caused by t h i s line shape
1s
L A M B S H I F T A N D O F ...
20 1
will be discussed in Sec. V. After normalizing and smoothing the two-photon spectra as before, the line centers were found at equal intervals in the region between 30 and 90% of the peak height, This region was used because at lower than 30% the peak is dramatically skewed towards lower frequency, while above 90% the amplitude fluctuations make a center meaningless. The 10 cent e r points could usually be fit with a straight line to within statistical fluctuations. This line w a s extrapolated to the pe& maximum and that point taken to be the “peak” frequency. This value w a s typically about 5 MHz higher in frequency than the line center at the 50% point. Once these centers were found the data were averaged and graphed a s before. Since this data run lasted several hours the frequency-marker drift was fit with a secondorder polynomial (the second-order correction was about the same size a s the statistical spread), and the data corrected. E. Hydrogen-deuterium Balmer-0 isotope shift
The last quantity measured was the isotope shift between the hydrogen and deuterium Balmer -p lines. For this the polarization spectrometer discharge was operated with a 50-50 mixture of hydrogen and deuterium. Polarization spectra of the 2S,,4SS1/,, 2SlI,- 4P,,, , and crossover region were recorded in the usual way alternating between the hydrogen and deuterium lines. This was done at several discharge pressures to ensure that the two had the same systematic shifts. The spectra were analyzed in the standard manner. V. RESULTS A. Systematic shifts of the 1s-2s line
In this and all subsequent sections, numerical results are given in t e r m s of the fundamental dyel a s e r frequency unless otherwise stated. All systematic line shifts a r e given as v (measured)-u (true). The 1s-2s pressure shift was determined to be less than 5 MHz/torr, and since the isotope and Lamb-shift measurements were made at 0.5 t o r r this caused a negligible correction and uncertainty to the result. The ac Stark effect“ was estimated by measuring the 2430-A beam size (0.02X 0.04 cm’) and peak power (0.9-1.1 kw) and using t h e result in Ref. 21 of 0.6 Hz/W/cm shift. This gave a shift of 0.8 f 0.2 MHz. The largest uncertainty in the results is caused by t h e difficulty to relate the line shape of the twophoton signal to the t r u e atomic transition frequency. The atomic linewidth of t h e 1s-2s transi-
27 20 2
c.
WIEMAN A N D T.
tion can certainly be neglected compared to the l a s e r bandwidth. In linear spectroscopy the observed line shape would then simply be given by the laser intensity spectrum, which can be readily meascred with a spectrum analyzer. For twophoton excitation, however, the calculation of the line shape, P (d), involves the convolution of the field spectrum,
where E ( o ) is the component of the electric field of the l a s e r at frequency w. A serious problem arises now, because this integral depends on the relative phases as well a s the amplitudes of the field components, whereas an optical spectrum analyzer can determine only the intensity o r amplitude. In the present experiment the situation is aggravated because there a r e two nonlinear processes, frequency doubling and two-photon excitation, which can each introduce uncertain line shifts and distortions between the measured l a s e r intensity spectrum and the observed two-photon signal. If the relative phases in the amplitude spectrum were truly arbitrary one could create almost any signal line shape. But fortunately the possible phase variations in the present experiment are quite limited by the origin of the l a s e r pulse. We have used various models for these phase v a r i ations, together with the measured laser intensity spectrum, to numerically calculate possible twophoton line shapes. By comparing the calculated and observed line shapes we have determined which model gives t h e best agreement and have used this model to estimate a correction to the observed transition frequency. We have also considered which models correspond to realistic extremes for the possible phase variations, and have used the corresponding corrections to set our uncertainty. The interferometer in Fig. 4 measures the pulsed l a s e r intensity spectrum and relates it to the cw laser frequency. For an ideal laser amplifier t h i s spectrum would be Gaussian, with a Fouriertransform-limited linewidth determined by the pulse length. In this case, there would be no phase variation and t h e peak frequency would be the same as that of the cw laser. The observed spectrum, however, appeared downshifted in its peak frequency by 17 MHz, the linewidth was almost twice as large as the transform limit, and the line shape appeared slightly asymmetric with a small but long tail on the low frequency side. This would imply a shift of the measured 1s-2s frequency of -68 MHz. If the phases varied rapidly and randomly from one frequency interval to the next, the square of
w.
HANSCH
22 -
the convolution integral6 could be replaced by a convolution integral of the intensity spectrum. In this limit there would be no shift between the atomic frequency and the peak of the observed signal. The line profile calculated for this case, however, was 20% narrower and more symmetric than our observed line. Next we investigated the (also quite unrealistic) limit of zero phase variations across the amplitude spectrum, given by the square root of the intensity spectrum. In this case the calculated line profile was drastically skewed, due to the long tail in the spectrum, and had little resemblance to the observed line. Since the long tail is, in all likelihood, due to some brief, rapid frequency chirping, it seemed more realistic to remove t h i s tail and to assume constant phase only for the remainder of the amplitude spectrum before compting the convolution integral. This approximation reproduced the observed line shape f a i r l y well, although the calculated Iinewidth w a s about 10% too large. The predicted resonance peak had a lower frequency. A s a case between these two limits we continued to assume a constant phase in the integrand of the convolution integral, but restricted the integration limits. The rationalization for this is that f r e quency components separated by much more than the Fourier-transform-limited width should be uncorrelated. The calculated peak shape and position is fairly insensitive to what is used as the central frequency of the integration. It matches t h e observed shape quite well when our limits a r e about 2.5 Fourier-transform-limited linewidths on either side of the frequency of maximum amplitude. In t h i s case the predicted shift of the 1s-2s f r e quency is -40 MHz. Because our approach cannot be entirely rigorous we have cautiously chosen e r r o r bars which more than cover the discussed limiting cases, assuming a shift of -40 f 28 MHz. The uncertainty of this shift is much larger than the &3-MHz statistical limit. Of course, the corrections of the observed fundamental frequency a r e one quarter as large. B. Systematic shifts of the Balmer-5 line
The largest correction to the position of the Balmer-P reference line is that due to pressure and electric-field effects. The positions of the %-, 4 P I I ,line and the 4S,I,-2S,,,-4P,, crossover both changed linearly with the pressure to within the statistical accuracy (-1 MHz), although the shifts were due to both pressure and electric field. Taking advantage of that behavior we have extrapolated their positions to zero pressure. The shift from 0 t o 0.27 torr is -17.0i 3.4 MHz for t h e 2S,,-4P,, peak and -7.6-+1.6 MHz for the
28 22 -
P R E C I S I O N M E A S U R E M E N T O F T H E 1s L A M B S H I F T A N D O F ...
crossover, with the uncertainty primarily due to the inaccuracy of the pressure readings. The residual “zero -pressure” electric field can be found by comparing the extrapolated separation between the two peaks with theoretical calculations. Knowing this field w e can determine the residual Stark shift of the crossover line. Because the separation between the two peaks i s approximately 4 times more sensitive to the electric field than the crossover position, the correction of the latter should have very little uncertainty. One difficulty arises, though, because both the 4Sl1, state and the 4Pl,, state possess unresolved hyperfine structure, and we do not know with certainty to what extent the hyperfine interaction is decoupled by collisions in the discharge plasma. Weber and Goldsmith” have found in later investigations of the hydrogen Balmer-a line that the hyperfine splitting of the 3Pl1, state is almost completely destroyed a t helium pressures above 0.1 torr. Unresolved hyperfine splitting is of little consequence in linear spectroscopy where it does not shift the center of gravity of a spectral line. In polarization spectroscopy, however, we expect, in general, some line displacement due to unresolved hyperfine structure, and collisional decoupling can lead to measurable line shifts. In order to consider two limits we have calculated the Stark shift n = 4 energy levels as a function of electric field by numerically diagonalizing the HamiltonianZ3without and with inclusion of hyperfine interaction. A comparison of calculated polarization spectra with the extrapolated observed separation between the two peaks yields a zero-pressure electric field of 4.3 *O.l V/cm if hyperfine interactions can be ignored. The corresponding Stark shift of the crossover position
TABLE 11. Separation A v = v
amounts to -2.6k0.2 MHz. If the hyperfine splitting were fully present the calculated zero-press u r e field would equal 4.6 0.2 V/cm and the combined Stark and hyperfine s h i f t of the crossover would be -6.5+0.2 MHz. The e r r o r bars in the final analysis a r e chosen large enough to span both limits. Including the pressure shifts discussed earlier we then a r r i v e at a total shift of - 1 2 . 2 i 2 . 6 MHz for the position of the crossover line. The a c Stark effect at a beam intensity of 5 + 2 W/cm’ causes a calculated additional shift of -1.4 0.5 MHz of the crossover position. No shift of the line with discharge current between 11 and 20 mA was observed within the statistical e r r o r limits.
*
*
C. Hydrogen 1s lamb shift
The hydrogen ground-state Lamb shift is found from the separation between the Balmer-0 c r o s s ) and t h e over line (4S1,,F=,,1-2S11ZF=1-~11zF=o,1 ISF = = two-photon line, recorded at the fundamental dye -laser frequency. Under the experimental conditions described in Sec. IV D we measure a separation A v = u (HB crossover) - v(lS-2S)/4 of 4760.4* 1.2 MHz. After applying the systematic corrections summarized in Table 11, we a r r i v e a t a true separation of 4764.8 i 7.6 MHz. The theoretical separation, including all t e r m s (7)except for the 1s Lamb shift, is 2’727.0 MHz. From a comparison, we find a 1s Lamb sh i f t of 8151 *30 MHz, which is in excellent agreement with the theoretical value‘ of 8149.43*0.08 MHz. The experimental uncertainty is dominated by the frequency shift introduced by the pulsed dye-laser amplifiers
.
(H,crossover) - v (1S-2S)/4.
Raw measurement (statistical average from 45 two-photon spectra at 0.05 torr, 51 Balmer-P spectra at 0.27 torr, 11 mA)
4760.4 f 1.2 MHz
Systematic corrections (a) 1.9-2s line laser line shape ac Stark effect (b) Balmer-P line pressure, electric field, hyperfine structure ac Stark effect
total correction corrected separation AV
203
-10.0 +0.8
* 7.0 MHz * 0.1
12.2 f 2.6 MHz 1.4 * 0.5 4.4 * 7.5 MHz 4764.8 i 7.6 MHz
29 22 -
C . W I E M A N A N D T . W . HANSCH
204
D . Hydrogen-deuterium IS-2sshift
The IS-2s isotope shift could be measured with much l e s s systematic uncertainty because the twophoton line shape was the same for hydrogen and deuterium. We found an experimental separation of 670992.3 6.3 MHz between the two line centroids. This result is in good agreement with the theoretical prediction7 of 670 994.96 i 0.81 MHz and is about ten times more precise than that of the best previous experimental value.3 This r e sult is of interest in, its own right because it gives the first confirmation of the predicted relativistic nuclear recoil corrections [Eq. ( 2 ) ] .
*
E. Balmer-fl isotope shift and deuterium IS lamb shift
The isotope shif t between the 2 S U Z F= 4F’1,2F - 41 component of the Balmer-0 line was measured to be 167 783.9 i 1.2 MHz. No systematic dependence on pressure o r voltage could be detected, since both isotopes a r e affected in nearly the same way. A correction of +0.7*0.7 MHz has been included, though, to allow for a difference in unresolved 4 P hyperfine splitting, as discussed in Sec. IV B. The result agrees with the theoretical value? of 167 783.7+0.2 MHz. The Balmer-P isotope shift, together with the 1s-2s isotope shift and the hydrogen 1s Lamb shift has been used to derive a n experimental deuterium 1s Lamb shift of 8177 30 MHz in agreement with the theoretical result? of 8172.23i0.12 MHz
*
before been tested. The techniques used offer the possibility of considerable improvements in both these measurements. In traditional excited-state Lamb-shift measurements, the broad P state linewidth has always limited the possible precision. Our approach does not have t h i s limitation. With purely technical improvements, such as better l a s e r s and hydrogen atomic beams it could far surpass the precision of present measurements of any Lamb shift.
It should, for instance, be possible t o achieve a Balmer-j3 reference linewidth as narrow as 1
MHz by observing the 2s-4s dipole-forbidden line component in a two-step process (one optical photon +one radiofrequency photon),24o r simply by single-photon transitions in the presence of a weak electric field. The two-photon linewidth could be reduced by using a l a s e r amplifier of longer pulse duration, o r by using the technique of multiple pulse excitation.Z5*2G It would be particularly interesting to improve the 1s-2s isotope s h i f t measurement because the present theoretical uncertainty is limited by the electron-to-proton-mass ratio, and thus an improved experiment could be used to obtain a better value for this ratio. ACKNOWLEDGMENTS
Using laser spectroscopic techniques we have measured t h e 1s Lamb shift in hydrogen and deuterium with improved precision and we find good agreement with theory. We have also improved the measurement of the 1s-2shydrogen-deuterium isotope shift and find it too agrees with theory; it agrees, in particular, with t h e predicted relativistic nuclear recoil correction which h a s never
W e are indebted to Dr. John E. M. Goldsmith for a computation of the hydrogen Stark effect, taking into account hyperfine interactions. We are also grateful t o Professor Arthur L. Schawlow for his stimulating interest in this work, and we thank Frans Alkemade and Kenneth Sherwin for skilled technical assistance. Finally we would like to thank Dr. Dirk J. Kuizenga for lending us a superb acousto-optic frequency shifter of his design. This work was supported by the National Science Foundation under Grant No. NSF 9687, and by the Office of Naval Research under Cont r a c t No. ONR N00014-78-C-0403.
* P r e s e n t a d d r e s s : Randall Laboratory of Ph y sics, Univ e r s i t y of Michigan, Ann Ar b o r , Michigan 48105. lW. E. Lamb, Jr. and R . C . R eth er f o r d , Phys. Rev. 79, 549 (1950). *T. W. Hznsch, S. A . L e e , R. Wallenstein, and C. Wieman, P hys . Rev. Lett. 34, 307 (1975). 3S. A . Lee, R. Wallenstein, and T. W. Hlnsch, Phys. Rev. Lett. 35, 1262 (1975). 4C. W i e m a n a n d T. W. Hansch, Phys. Rev. Lett. 36, 1170 (1976). 5G. H e rz be rg. Proc. R. Soc. London, Ser. A S , 516 (1956).
6T. W. Hznsch, M. H. Nayfeh, S . A. Lee, S. M. C u r r y , and I. S . Shahin, Phys. Rev. Lett.32, 1336 (1974). ‘G.W. Erickson, J. Phys. Chem. Ref. Data j, 831 (1977). ‘T. W. Hansch, in Tunable L a s e r s and Applications, edited b y A . Mooradian et a1 Sp r i n g er Series in Optical Sciences (Springer, New York. 1976), Vol. 3, p. 326. ’T. W. H h s c h , Phys. T o d a y z , 34 (1977). ‘ON.Bloembergen and M. D. Levenson, in High Resolution L a s e r Spectroscopy, Topics in Applied Ph y si cs (Springer, New York, 1976), Vol. 13, p. 315. “L. S. Vasilenko, V. P. Chebotaev, and A. V. Shtshaev,
.
VI. CONCLUSIONS AND FUTURE IMPROVEMENT
.,
30 22 -
PRECISION MEASUREMENT OF THE
P i s ' m a Zh. Eksp. Teor. Fiz. 12,161 (1970) [ J E T P Lett, 1 1 3 (1970)). "Y. Gontier and M . T r a h i n , Ph y s. Lett. H,463 (1971). 13F. Ba s s a ni, J. J. Fo r n ey , and A . Qu attr o p an i, Phys. Rev. L e tt. 1070 (1977). 14T. W. H&sch, in Proceedings of the Intrmational SchooE of Physics "Enrico Fermi,'' Course L X I V on Nonlinear Spectroscopy, Varenna. Italy, 1975 (NorthHolland, New York, 1977), p. 17. "T. W. Hgnsch, M. D. Levenson, and A . L. Schawlow, Phys. Rev. L e t t . 2 , 946 (1971). 16F. V. Kowalski, R. T. Hawkins, and A . L. Schawlow, J. Opt. SOC.Am. E, 965 (1976). IrR. Wallenstein and T . W. H&sch, Opt. Commun. 14, 353 (1975). '*W. L. F i t e s , R . T. Braclunan, D. G. H u m m e r , and 363 (1959); R . F. Stebbings, Phys. Rev. 9,
12,
2,
124,
1s
LAMB SHIFT A N D OF
...
205
2051 (1961). ''T. W. Hgnsch, I. S. Shahin, and A . L . Schawlow, Phys. Rev. Lett. 7 ,707 (1971). ''A. M. Bonch-Bruevich and V. A . Khodovoi, Usp. Fiz. N a u k E , 71 (1967) [Sov. Phys: Usp. 1(1, 637 (1968)l. 'IS. A . L e e , Ph.D. t h e s i s , Stanford U n i v er si t y , M. L. R e p o r t No. 2460 (1975). "E. W. Weber and J. E. M. Goldsmith, Ph y s. Lett. 95 (1979). 23G. U r d e r s , Ann. Phys. (Leipzig) 2,308 (1951). 24E. W. Weber and J. E. M. Goldsmith, Phys. Rev. Lett. 41, 940 (1978). Teets, J. N. Eck st ei n , and T. W. H h s c h , Ph ys . Rev. Lett. 3, 760 (1977). 26J. N. Eck st ei n , A . I. Fer g u so n , and T. W. H s s c h , Ph y s. Rev. Lett. 40, 847 (1978).
B,
's
480
OPTICS LETTERS / Vol. 7, No. 10 / October 1982
Laser-frequency stabilization using mode interference from a reflecting reference interferometer C. E. Wieman and S. L. Gilbert Department of Physics, University of Michigan, Ann Arbor, Michigan 48109 Received May 17,1982 We present a new method for locking the frequency of a laser to a reference-interferometer cavity. For a nonmode-matched input beam, the light reflected off a cavity contains an interference between the wave fronts corresponding to the various cavity modes. A detector placed a t the proper position on the interference pattern provides a signal proportional to the imaginary component of the reflected field. As a function of laser frequency, this signal is dispersion shaped and can be used as the error signal for electronic frequency stabilization.
The frequency stability of a single-mode laser can generally be improved by using an electronic servo system to lock it to the resonance of a passive reference The most important component of this procedure is the character of the error signal that is fed back to the laser. In this Letter we present a simple new method for obtaining an error signal that is proportional to the 90" -phase-shifted or relatively imaginary component of the field reflected off the cavity. This signal is obtained by detecting the interference between the wave fronts in the reflected beam corresponding to resonant and off-resonance modes of the reference cavity. Traditional locking techniques have used the amplitude of the signal transmitted through the interferometer. One popular approach' is to lock on the side of the transmission peak, but this method is severely limited by the problem that a frequency jump of only half of the peak width can throw the system out of lock. A second traditional approach2is to modulate the frequency of the cavity or the laser and do phase-sensitive detection. One obtains the derivative of the Airyfunction transmission peak and uses it to lock to the center of the peak. This method, as well as requiring modulation and phase-sensitive detection, inherently limits the response time of the feedback. It is also poor a t recovering from large frequency jumps because the signal decreases rapidly as the frequency moves more than a linewidth away from the resonance. Many years ago Pound,3 in working with stabilization of microwave oscillators, realized that the amplitude of the imaginary component of the field reflected off a cavity has nearly ideal characteristics for a feedback signal. As the input frequency varies, this function passes through zero rapidly and has long wings, which aid in recovering from sudden large changes. Recently, this fact was rediscovered in the context of laser stabilization. Two methods for observing this imaginary component and using it for laser stabilization have been demonstrated. Hall and Drever et d 4used the method of putting modulation sidebands on the laser and looking at the reflected beam with phase-sensitive de-
tection at the sideband frequency. This technique, although quite successful, has the practical disadvantage that the need for high-frequency sidebands on the laser and high-frequency detection and signal processing add considerable technical complexity. Hansch and Couillaud5have presented an alternative technique in which a polarizing element is inserted in the reference cavity and the ellipticity of the polarization of the reflected beam is measured. This approach works well but also has certain practical complications because of the need for a polarization-sensitive reference cavity. In general, the most desirable features of a reference cavity are stability and high finesse, but it is normally quite difficult to insert a polarizing element into a high-finesse cavity without appreciably degrading both of these characteristics. Our approach has the same advantages as these methods but is technically simple. Consider the field ERCreflected by a mode-matched interferometric cavity. The standard result6 is
E; is the incident field, 6 = 4rL/X,and R1and TI are the reflection and transmission coefficients, respectively, of the input mirror. R is the amplitude ratio for successive round trips, so the finesse is rdR/( 1 - R ) . 6, the phase difference between the fields on successive round trips in the cavity, is equivalent to frequency. In Fig. 1A we plot Im(ERc)/E; versus 6 to illustrate the desirable dispersion-shaped resonance mentioned earlier. The underlying concept of the approach that we use to isolate this term is shown schematically in Fig. 2. Detector 2 looks at a portion of the pattern formed by the interference of a reference beam Eo and the beam reflected off the cavity, ERC. The detected intensity
Reprinted from Optics Letters, Vol. 7, page 480, October 1982 Copyright 0 1982 by the Optical Society of America and reprinted by permission of the copyright owner.
31
32 October 1982 / Vol. 7, No. 10 / OPTICS LETTERS Mirror
is 12
481
+
+
= ~ E R CE012 = I0 I R c 2 Re(ERc)Re(Eo) 2 Im(ERc)Im(Eo).
+
Writing EOas
Eo =
+
aces 8 + iJ&sin
(2)
= 10
8,
(3) Detector 1
+ IRC+ 2Jz COS 0 Re(ERc)
+ 2Jz sin 0 Im(ERc).
1- 1 7 (4)
The phase angle 8 will be just the difference in the path lengths for the two beams divided by the wavelength. If detector 2 is now positioned so that it sees only the portion of the interference pattern near 0 = go",
+ I R C + 2J&
(5) Detector 1sees the entire interference pattern, so, to a good approximation, 12
= 10
I1 = I0
h(ERC).
+ IRC,
(6)
and the difference, I Z - I1 = 2& Im(ERc). One point that cannot be neglected, however, is the effect of changes in 8. In general, [Im(ERc) Sin 8 + Re(ERc) COS 01. (7) This function has the important property that, for all values of 8 except close to 0" and 180", the frequency dependence of Re(ERc) can be neglected and I2 - I1 can be approximated by
Iz - 11= 2&
The accuracy of this approximation is shown in Fig. 2B. The solid curve is the exact value of ( 1 2 - I 1 ) / 2 a as given by Eq. (7) for 8 = lo", and the dashed curve is the value given by formula (8). The similarity between the two curves rapidly improves as 8 increases toward 90". For comparison purposes we note that the horizontal scale is the same as in Fig. l A , whereas the ver-
Fig. 1. A, Im(ERc)/E; versus 6. B, solid curve, 12 - I1/(24&
Ei);dashed curve, approximation from formula (8). Vertical scale is expanded by a factor of 5.76 from A.
Reference Interferometer Attenuatpr Variable
we have 12
I
Differential Amulifier
12-11
h Diaphragm
Detector 2
Fig. 2, Schematic showing laser-stabilizer concept.
tical scale is expanded by 5.76 (= l/sin 10") and its origin offset by 5.67 (=l/tan 10"). The implications of this dependence on 8 are that, over a wide range of 8, the feedback signal will effectively have the same shape but will have a 0-dependent zero offset. Such an offset, if it can vary with time, is undesirable in a feedback signal. With a setup such as that shown in Fig. 2, extreme care would be necessary to achieve the adequate path-length (8) stability of a fraction of a wavelength. An alternative approach that avoids this problem is the use of the beam reflected off a non-mode-matched cavity. This beam can be described as an appropriate superposition of the modes of the cavity, and the detection of the interference between these modes can provide the desired signal. Relative to a plane wave, the phase of the wave front corresponding to each mode is given in Ref. 7 as +'m,n
= (m
+ n + l)arctan(Xz/awi),
(9)
where m and n are the transverse mode numbers, z is the propagation distance from the waist, and 00 is the diameter of the beam waist. For simplicity let us consider an input beam composed of two modes, one having a resonant frequency near the laser frequency and a second with a resonant frequency far away. From Eq. (1)it is evident that the amplitude and the phase of this latter mode will have very little frequency dependence and hence can serve as a reference signal. The interference of the two modes is then described by Eq. (4). The phase, 8 = +'m,n - +p,,,,,, can be varied by changing the distance between the interferometer and the detector. The relevant length scale for this variation is wi.rr/X,which, for a confocal resonator, is one half of the mirror separation. The transverse variations in the amplitudes of the two modes ( z fixed) are described by the usual products of Hermite and Gaussian function^.^ By evaluating these one can calculate the x and y coordinates where the two mode amplitudes have the same ratio as their spatially integrated amplitudes measured by detector 1. The optimum detector position is then the x , y coordinate where the amplitudes are largest and in this ratio and where the z coordinate is such that 8 = 90". In general, the reflected beam will be a sum of many
33 482
OPTICS LETTERS / Vol. 7. No. 10 / October 1982
modes with different amplitudes and phases a t any given point in space. Calculating the ideal detector position in that case is a rather messy problem. However, finding an acceptable, though not ideal, detector position is a much simpler task because of the b' dependence discussed previously. As is shown by formula (81, the required frequency dependence is obtained for nearly any value of 19and therefore of z . Thus the sole requirement is finding an x , y position where only the Im(ERC) term of formula (8) is left after the two detector signals are subtracted. Such a position is easily found by using the simple empirical procedure given below. We investigated this approach experimentally using a Spectra-Physics Model 380A ring dye laser and a Trope1 Model 240 spectrum analyzer as a reference cavity. The dye laser provided a single-mode output at -540 nm; rhodamine 110 dye was used. The Model 240 spectrum analyzer is a confocal spherical FabryPerot interferometer with a 1.5-GHz free spectral range and a finesse of -150. The detectors were silicon photodiodes. In these experiments we scanned the reference cavity with the laser free running and examined the signals obtained. With the scan turned off, we then used the signal to lock the laser to the reference cavity. We first constructed a rather crude version of the setup shown in Fig. 2 to verify the basic concept of formulas (1)-(8). We obtained signals much like that shown in Fig. 1. Within a few minutes, however, the zero level drifted as much as the height of the peaks. This is hardly surprising since the temperature of our laboratory fluctuates by several degrees centigrade. As was mentioned previously, this is rather unsuitable for laser stabilization. Then to observe mode interference we used the same setup but removed the top mirror so only a single beam went to the detectors. To obtain a dispersion-shaped signal the following procedure was used: Detector 2 was moved across the beam profile while the attenuator in front of detector 1was adjusted to provide a zero error signal when the laser frequency was far off resonance. This approach provided good error signals for a considerable range of input beam diameters and curvatures and detector-interferometer separation. Both the wave-front diameter and curvature were varied by roughly 2 orders of magnitude with the values for the TEMoo mode of the interferometer being at approximately the geometric center of these ranges. The detector-interferometer separation has been as much as 20 times the interferometer length. In nearly all these cases, the signal-to-noise ratio was limited to the same value by imperfectly subtracted laser-amplitude fluctuations. In principle, of course, the optimum signal would be obtained by using an input beam composed of only two modes and positioning the detector as calculated in the manner discussed above. We wish to
emphasize, however, that in practice no such effort is necessary to obtain high-quality signals. After setting the detector position, we turned off the scan of the reference cavity and locked the laser frequency to it. This was done by amplifying the detector-difference signal and feeding it to the galvanometer-driven plates and piezoelectrically movable mirror, which are the standard tuning elements of the dye laser. The frequency stability was measured by using a second Model 240 spectrum analyzer. This feedback reduced the peak-to-peak frequency jitter from -50 MHz in a 10-sec period to under 0.5 MHz. We have also used this approach to lock the laser to a homemade semiconfocal interferometer, which has a 1.5-MHz linewidth. With -95% of the input beam coupled into the TEMoo mode, good error signals and locking were still obtained. We have presented a new method of stabilizing the frequency of a laser by locking it to the resonance of a passive reference cavity. By using mode interference, an error signal is obtained that is proportional to the imaginary component of the reflected field. This signal has nearly ideal characteristics for feedback stabilization. The method works with any reference cavity and requires the simplest of optics and electronics. This technique will apply equally well to the problem of locking the center of a cavity resonance to a laser frequency. This is particularly useful for obtaining intracavity power buildup for second-harmonic generation. This work is supported in part by a Precision Measurements Grant from the National Bureau of Standards, in part by National Science Foundation grant no. PHY-8111118, and in part by a grant from Research Corporation. S. L. Gilbert would like to acknowledge support from the following sources: Laporte Fellowship, AWIS Meyer-Schutzmeister Fellowship, and University of Michigan Rackham Predoctoral Fellowship. We thank R. Watts and J. Ward for useful discussions. References 1. R. L. Barger, M. S. Sorem, and J. L. Hall, Appl. Phys. Lett.
22,573 (1973). 2. A. D. White, IEEE J. Quantum Electron. QE-1, 349 (1965). 3. R. V. Pound, Rev. Sci. Instrum. 17,490 (1946). 4. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, Joint Institute for Laboratory Astrophysics, Boulder, Colorado 80309 (personal communication). 5 . T. W. Hansch and B. Couillaud, Opt. Commun. 35, 441 (1980). 6. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 7.6; Ref. 5. 7. H. Kogelnik and T. Li, Appl. Opt. 5,1550 (1966).
PHYSICAL REVIEW A
VOLUME 27, NUMBER 1
JANUARY 1983
Hyperfine-structure measurement of the 7s state of cesium S. L. Gilbert, R. N. Watts, and C. E. Wieman Physics Department, Universiw of Michigan. Ann Arbor, Michigan 48109 (Received 23 September 1982) The hyperfine structure of the 7Sl/2 state of 133Cshas been measured with the use of laser spectroscopy on a cesium atomic beam. We find the magnetic dipole coupling constant A =545.90(09) MHz. From the amplitude of the hyperfine components we find the ratio of the scalar to tensor polarizabilities ( I a / p I ) for the 6s -7s transition to be 9.80(12).
The measurement of hyperfine structure has long been an important probe of atomic structure. For this reason, the hyperfine structure of the ground and excited states of alkali atoms has been studied extensively.' Information on the 7s1/2 state of cesium is particularly important at the present time because of its role in the study of parity violation in atoms.2 In this paper we present a measurement of the hyperfine structure of the 7s,/2 state of 133Csobtained by studying the laser excited 6S1/2-7S1/2 transition in the presence of an electric field. By measuring the separation of the hyperfine components we determine the hyperfine splitting of the 7S,12state to considerably higher precision than all other measurements of alkali excited-state hyperfine splittings. In addition, by comparing the amplitudes of the components we obtain the ratio between the scalar and tensor polarizabilities of the 6S1/2-7S1/2 Stark dipole matrix element. A measurement of this ratio is of interest because the previous two measurements of this quantity are in serious d i ~ a g r e e m e n t . ~ . ~ The SIR levels of cesium are split into hyperfine doublets with total angular momentum F = 3 and 4. Thus, in a dc electric field ijf, a linearly polarized laser will excite four transitions having the intensities and frequency separations shown below: 6 s ~ '4I S F ~ , 143=21f12E:, IAvl = A 7 S h f r
6S~,3'7s,r,3,
133'7fl'E:
6s~4'7sF-4,
1.,.,'15f12E:
+28a2Ei, 1Avz = A6S h i - A7s h i
+36a2Ei,
neglected. Since A6S hf is the primary standard of frequency, measuring the ratio of either Avl or Av3 to AVZ Av, yields hl. Although the individual transition energies are electric field dependent, the separations, to a very good approximation, are not.s The ratio alp is obtained from the ratios 1 4 4 1 1 3 4 and I 3 3 / 1 4 3 . The apparatus used is shown schematically in Fig. 1. A collimated beam of atomic cesium is produced in a manner similar to that given in Ref. 6. A secondary collimator further reduces the divergence to about 0.025 rad. This beam intersects a standing wave laser field at right angles in a region of static electric field. The static field is obtained by applying 5000 V to optically transparent-electrically conducting coated glass plates (not shown in figure) which sit 0.5 cm above and below the laser beam. The standing wave field is provided by a spherical mirror FabryPerot interferometer (power enhancement interferometer). The output from a single frequency ring dye laser (Spectra Physics model No. 380) with homemade frequency stabilization7 is circularly polarized and modematched into the interferometer. One of the interferometer mirrors is mounted on a piezoelectric transducer, thereby allowing the interferometer resonance to be electronically locked to the laser frequency. The dye laser output is typically 0.1 to 0.2 W at 540 nm and the interferometer increases the electromagnetic field strength by a factor of 12.5. The short term laser frequency jitter is about 0.5 MHz peak to peak. Five percent of the dye laser beam is sent into an
+
IAv.3 = A7s hf
CESIUM
6s~-4+7&-3, 134=21f12E:.
(1) The subscripts I and II are with respect to the direction of laser polarization, a and /3 are the scalar and tensor p~larizabilities,~ respectively, and AbS hf and h[ are the hyperfine splittings of the 6 S and 7 s states. In addition to the electric dipole intensities shown, each transition has a magnetic dipole amplitude. For the electric fields we used ( 5000 V/cm), these contribute less than 1 part in lo4 and can be
i1.V
'
FREQUENCY MARKER INTERFEROMETER !
-
I
POWER ENHANCEMENT lNTERFERoMETER DETECTOR
DETECTOR
VACUUM CHAMBER
0
FIG. 1. Schematic of apparatus.
27
58 1
34
01983 The American Physical Society
35 582
S. L. GILBERT, R. N . WATTS, AND C. E. WIEMAN
additional fixed length Fabry-Perot interferometer which is thermally insulated and hermetically sealed (frequency marker interferometer). The resonances of this cavity provide a frequency scale. The 6s -7s transition is detected by a silicon photodiode which sits under the lower electric field plate. This diode monitors the 850- and 890-nm light which is emitted in the 6P1/2,3/2- 6 s step of the cascade decay of the 7s state. A colored glass cutoff filter prevents the scattered 540-r1m light from reaching the detector. The photodiode output goes into a preamp followed by a precision voltage divider. The linearity of the photodiode preamp combination is better than 1 part in lo4. The voltage divider is used to reduce the 3 -3 and 4 -4 transitions signals by a factor of 155.44, making them approximately the same size as the undivided 4 -3 and 3 -4 transitions. Other photodiodes measure the light transmitted through the frequency marker and power enhancement interferometers. These three signals and the voltage ramp which scans the laser frequency are digitized and stored by a PDP-ll computer for off-line analysis. Spectra were obtained by scanning the laser frequency over each of the hyperfine transitions. For each transition the scan lasted about 1 min and covered about 1 GHz. For each of the three separations a set of about 15 scans were made by alternating between the two transitions of interest. Typical scans are shown in Fig. 2. The 20-MHz linewidth is primarily due to the cesium beam divergence. The marker interferometer is intentionally adjusted to obtain the rather complex signal shown. This structure corresponds to exciting a number of nondegenerate modes and is useful for checking for and correcting scan nonlinearities. Analysis and results. To find the hyperfine splitting hf, the spectra were first normalized to the instantaneous laser intensity. The line centers were then found in terms of the frequency marker scale. This was complicated by the thermal drift of the marker interferometer. To correct for the drift, the position of each transition relative to the frequency marker was found as a function of time. Because the thermal time constant of the interferometer was long, the drift was quite linear with time during each set and varied between 0 and 0.3 MHz per minute for the different sets. Having thus determined the drift, the results were corrected appropriately. Once the line separations were measured in terms of the frequency 'E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys. 49, 31 (1977). 2M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 35, 899 (1974). )M. A. Bouchiat and L. Pottier, J. Phys. (Paris) Lett. 36, L189 (1975). 4J. Hoffnagle et al., Phys. Lett. 143 (1981).
m,
27 -
FIG. 2. Typical spectra.
marker, the separation between the F = 4 -4 and F = 3 -4 lines was used to determine the exact marker spacing. It is amusing to note that the interferometer length is thus known in terms of the fundamental frequency standard. Using this scale, Avl and Av3 were found to be in excellent agreement and their combined average was 2,183.59(36) MHz = 7 s , / 2 hyperfine splitting. This gives the magnetic dipole coupling constant A =545.90(09) MHz, which is in good agreement with the previous result of 546.3(30) M H Z . ~The uncertainty is primarily due to uncertainties in scan linearity and the correction for the drift of the frequency marker. To find 1 a/PI, the area under the four peaks were measured and the ratios Z33/143 and Z44/134 determined. The ratio la/PI was then found using Eq. (1). The light transmitted through the enhancement interferometer was found to be imperfectly circularly polarized, resulting in a correction of 0.6%. From 133/Z43we obtained la/pI =9.91(08), and from Z 4 / Z 3 4 , la//3I =9.68(08). Because these results did not seem consistent with a simple statistical variation we took the combined value to be 9.80(12), where the uncertainty covers both values. This is in good agreement with the value 9.91(1 1) of Ref. 4, but disagrees with the value 8.8(4) given in Ref. 3. This work was supported by the National Science Foundation and the Research Corporation. S. L. Gilbert and R. N. Watts would like to acknowledge support from the University of Michigan Rackham fellowship program. S. L. Gilbert is also pleased to acknowledge receipt of an Association for Women in Science Meyer-Schutzmeister fellowship. SM. Mizushima, Quantum Mechanics of Atomic Spectra and Atomic Structure (Benjamin, New York, 1970), Sec. 10-4. 6P. F. Wainwright, M. J. Algaurd, G . Baum, and M. S. Lubell, Rev. Sci. Instrum. 9, 571 (1978). 'C. E. Wieman and S. L. Gilbert, Opt. Lett. 1,480 (1982). ER.Gupta, W. Happer, L. K. Lam, and S. Svanberg, Phys. Rev. A &, 2792 (1973).
MAY 1983
VOLUME 27, NUMBER 5
PHYSICAL REVIEW A
Rapid Communications ~~
The Ropid Communications section is intended for the accelerated publication of important new results. Manuscripts submitted to this section are given priority in handling in the editorial office and in produclion. A Rapid Cornrnunicalion may be no longer than 3% printed pages and must be accompanied by an absiroci. Page proofs are sent 10 authors. but, because of rhe rapid publication schedule, publication is not delayed for receipt of corrections unless requested by the author.
Precision m e a s u r e m e n t of t h e S t a r k shift of the i n atomic cesium
6s-7stransition
R. N. Watts, S. L. Gilbert, and C. E. Wieman Randall Physics Laborarory, University of Michigan. Ann Arbor, Michigan 48109 (Received 29 November 1982)
We have measured the Stark shift of the 6 s - 7 s transition in a beam of atomic cesium with the use of laser spectroscopy. We find this shift to be 0.7103(24) Hz (V/cm)-*. From this value we determine the static polarizability of the 7 s state, a 7 S = [ 6 1 1 1 ( 2 1 ) l a i ,and the 7P-7S oscillator strength, f 7 P , 7 S = 1.540(7). Finally, we derive a new empirical value for the 6 s - 7 s Starkinduced electric dipole transition probability
INTRODUCTION
cesium and find the frequency shift as a function of electric field. From the literature value for (Y6S and Eq. (21, we determine a l s .
While alkali atom ground- and first excited-state polarizabilities have been studied extensively, little information is available on the polarizabilities of the excited S states.' In this paper, we present a highresolution measurement of the Stark shift of the 6S7s transition of cesium in a dc electric field. From this, we derive the static electric dipole polarizability of the 7s state and obtain a precise value for the oscillator strength f 7 p . l ~ . W e then use f 7 p . 7 ~to make an improved empirical determination of the 6s-7S Stark-induced electric dipole transition amplitude. This amplitude plays an important role in determining the size of the parity-violating interaction recently observed in cesium.2 In a static electric field E, the energy of an atomic S state shifts by an amount I
hens=- i a n s E 2
I
APPARATUS
The experimental apparatus shown in Figs. 1 and 2 is the same as that described in Ref. 4 with the addition of an iodine saturated-absorption spectrometer. A dense highly collimated beam of atomic cesium intersects a standing wave-laser field at right angles in a region of static electric field. The standing wave is produced by a frequency-stabilized cw ring dye laser mode matched into an electronically tunable, spherical-mirror Fabry-Perot interferometer. The laser has a typical output of 150 m W and a shortterm frequency jitter of 0.5 MHz peak to peak. With
(1)
where a d is the static polarizability for that state. For shifts much less than the fine-structure splitting, ad does not depend upon the total angular momen~ electric field tum quantum numbers F o r M F . The makes it possible to drive electric dipole transitions between parity-mixed S states of different n. This transition frequency, expressed in atomic units, is displaced by
1 DYE LASER
', n
5-
FREQUENCY' FREQUENCY
-
CESIUM OVEN
x 14
n
I
1
iy \
POWEk POWER ENHANCEMENT MARKER INTERFEROMETER INTERFEROMETER :, ODETECTOR DETECTOR
In our experiment, we drive the
6s-7s transition in
CHAMBER
FIG. 1. Schematic of apparatus.
21
2769
36
01983 The American Physical Society
37
2110
R . N . WATTS, S. L. GILBERT, AND C. E. WIEMAN
FIG. 2. Detail of interaction region.
I
I
I
I
150
100
50
0
1
50
100
AV (MHZ)
the resonant frequency of the interferometer locked to the laser frequency, the electromagnetic field strength inside the interferometer is 12.5 times that of the incident laser beam. The 6 s - 7 s transition driven by the laser is monitored by observing the 850- and 890-nm light that is emitted in the 6P-6s leg of the 7S-6P-6S cascade decay. This light is detected by a cooled silicon photodiode that sits below the electric field plates. Glass color filters not shown in Fig. 2 prevent scattered 540-nm laser light from reaching the detector. The static electric field is produced by applying 1 to 13 kV to a pair of 5 x 7.5-cm2-coated glass plates 0.9208(5) cm apart. To assure field uniformity, the laser and cesium beams intersect in a 3-cm-long line in the middle of the electric field region. The top field plate is coated with evaporated gold. The lower field plate is coated with a thin layer of chromium having an optical transmission of 75% and a surface resistivity of 7 kCl/o. The voltage applied to the field plates is monitored with a voltage divider and digital voltmeter. Five percent of the laser beam is sent into a thermally isolated, hermetically sealed, fixed-length Fabry-Perot interferometer (frequency marker interferometer). The resonances of this cavity set the frequency scale but thermal drifts make it unsuitable for use as a frequency reference. For this, 30% of the laser beam is sent into a Doppler-free saturatedabsorption iodine spectrometer identical to that described in Ref. 5. Iodine is used because the X ' Z : ( v " - O ) -B311&(v'= 29) J = 81 R-branch transition overlaps the 6&-4-7s~-4 cesium hyperfine transition.6 The iodine hyperfine lines that result as the laser is scanned over the cesium transition serve as stable frequency references. Signals from the frequency marker interferometer, the silicon photodiode, and the iodine spectrometer are digitized and stored by a Digital Equipment Corporation PDP11/23 computer for off-line analysis. Spectra were obtained by scanning the laser over the cesium transition for electric fields of 1.1, 5.7,
FIG. 3. Combination of frequency scans at 1.1 and 11.8 kV/cm. (a) Representative frequency marker interferometer spectrum. (b) Iodine saturated-absorption spectrum. (c) Cesium peaks (amplitude of 11.8-kV/cm peak reduced by factor of 100).
-
9.0, 11.8, and 14.0 kV/cm. Each scan took approximately 30 sec and several scans were made for each value of the electric field. A combination of two such scans appears in Fig. 3. As discussed in Ref. 4 the frequency marker interferometer had been adjusted to give the rather complex structure displayed. The iodine peaks of interest have been arbitrarily labeled (I through e. We believe that the small dip in the center of the cesium peaks at higher (> 6 kV/cm) electric fields is associated with saturation of the transition and has no effect o n the line center. ANALYSIS A N D RESULTS
To find the frequency shifts as a function of electric field, we first chose iodine peak d as an arbitrary zero. Using the frequency marker resonances, which were calibrated to 5 parts in los in Ref. 4, we determined the separations of peaks u through e. Then, using the iodine peaks, we found the displacement from the arbitrary zero of each cesium peak. Uncertainty in determining a peak center was typically 0.1 5 MHz. The data along with the straight-line fit to Eq. (2) is shown in Fig. 4. In this drawing, the error bars are smaller than the data points. The slope d(Av)/ d ( E 2 ) is 0.7103(24) Hz (V/cm)-' with a correlation coefficient of 0.999 98. The statistical uncertainty in the slope is 0.0010 Hz (V/cm)-*. The systematic error is dominated by the uncertainty in the electric field. The separation between field plates was measured to l part in 1800. The voltage divider was checked with the assistance of M. Misakian at NBS to the 0.1% level and the digital voltmeter is good to 0.1%. This combines to give a 0.15% uncertainty in
38
21
PRECISION MEASUREMENT OF THE STARK SHIFT OF T H E . . .
2771
quency tables of Moore,l3 we find 140
f77=1.540(7) .
/
/
/
-2 40
1
E~ ( lo6 V2/crnz)
FIG. 4. Stark-frequency shift vs square of electric field Error bars are smaller than data points.
the electric field and an uncertainty of 0.0021 Hz (V/cm)-2 in the slope. Using Eq. (2) and Q6S= [402(8)Iad from Ref. 7 , we find
This value includes a 0.15% correction due to the inclusion of fine structure in Eq. ( 3 ) . The quoted uncertainty consists of 0.006 from the combined experimental uncertainties and 0.003 from a 25% theoretical uncertianty in A a l s . A 25% uncertainty is reasonable: The calculations of Hofsaess for cesium are usually within 10% and, in the worst case, within 25% of the measured principal series oscillator strengths f n 6 , and the photoionization cross sections. To our knowledge, this value for f 7 7 is one of the most accurately determined oscillator strengths. It is in fairly good agreement with a theoretical value of f 7 7 = 1.574 based on the work of Hofsaess. In a manner analogous to the definition of a7s, the Bouchiats have defined a quantity a that is a measure of the transition probability between the 6 s and 7 s states of cesium in the presence of a electric field.14 This quantity may also be written in terms of experimentally determined oscillator strengths
a7~=[6111(21)I~O3 . This value for (27s can be compared with [6673(386)1ad as determined by Hoffnagle, Telegdi, and W e k 8 The difference in the uncertainties is explained by noting that the second measurement was made in a heated vapor cell. This gave linewidths 40 times larger than those of our experiment and limited the size of the electric field that could be applied. This, in turn, forced Hoffnagle et a/. to make precise measurements of shifts that were at most only 1% of the linewidth.
-
DISCUSSION
The 7s state polarizability can also be expressed in a straightforward manner in terms of oscillator strengths. Following Refs. 8 and 9, we ignore fine structure, expand Eq. (11, and find f67
f77
a 7 s = 1 + 7 - + A a 7 . y w76 W77
,
(3)
where w n n , wnr.n,pand f n t n =f n l p , m . The term A a 7 s contains the contributions from discrete states n’ 2 8, the continuum, and autoionizing states. This quantity has not been measured experimentally but, from the theoretical work of Hofsaess”, ” and Safinya,12 has been estimated by Hoffnagle to be 590d. Using Hoffnagle’s value of f 67 = - 0.573 ( 13 ) and the fre-
IT. M. Miller and B. Bederson, Adv. At. Mol. Phys. 13, I (1977); R . Marrus, D . McColm, and J. Yellin, Phys. Rev. 147, 55 (1966). 2M-A. Bouchiat, J . Guena, L. Hunter, and L. Pottier, Phys.
As in Eq. (31, A a contains the experimentally unknown contributions from the discrete states n’ 3 8, the continuum, and the autoionizing states. Using the theoretical inputs of Hofsaess and Froese-Fischer,” Hoffnagle estimates P a to be - 4.400). Taking Hoffnagle’s value for f 6 7 , f 6 6 = 1.068(21) from Ref. 16, and our value for f 7 7 , Eq. (4) gives a = - [263.7(27)1ad. A 1% correction due to finestructure splitting has been included. T h e experimental uncertainties contribute 2.500) and, as before, we have assumed a 25% (1.10;) uncertainty in ACZ. This value for a is in excellent agreement with a value of - 262.6 calculated by Hoffnagle using the theoretical input of Hofsaess. This work was supported by the National Science Foundation and the Research Corporation. S.L.G. and R.N.W. would like to acknowledge support from the University of Michigan Rackham Fellowship Program. S.L.G. is also pleased to acknowledge receipt of an Association of Women in Science MeyerSchutzmeister Fellowship. We thank M . Misakian for his help in checking our voltage divider and J. C. Zorn for many helpful discussions.
Lett.
w,358 (1982).
)A. Khadjavi and A. Lurio, Phys. Rev.
167,128 (1968).
4S. L. Gilbert, R. N . Watts, and C . E. Wieman, Phys. Rev. A 27, 581 (1983).
39
2112
R. N. WATTS, S. L. GILBERT,A N D C. E. WIEMAN
SW. Derntroder, Laser Spectroscopy (Springer, New York, 19811, p. 489. 6J. D . Simmons and J . T. Hougen, J . Res. Nat. Bur. Stand. Sect. A 7 l , 25 (1977). 'R. W.Molof ef a/., Phys. Rev. A lo,1131 (1974). 8J. Hoffnagle, V. L. Telegdi, and A. Weis, Phys. Lett. 457 (1981). 9J. Hoffnagle, dissertation (Swiss Federal Institute of Technology, Zurich, 1982) (unpublished). 1°D. Hofsaess, Z. Phys. A 281, 1 (1977).
m,
21
"D. Hofsaess, At. Data Nucl. Data Tables 24, 285 (1979). I2K. A. Safinya, T. F. Gallagher, and W. Sander, Phys. Rev. A 22, 2672 (1980). 13C.E. Moore, Atomic Energy Levels, NBS Circular No. 467 ( U S . GPO, Washington, D.C., 1949). 14M. A. Bouchiat and C. Bouchiat, J . Phys. (Paris) 3.493 (1975). ISC. Froese-Fischer, At. Data 4, 301 (1972). 16L. N. Shabanova ef a/.,Opt. Spectrosc. (USSR) 47, 1 (1979).
VOLUME 29, NUMBER 1
PHYSICAL REVIEW A
JANUARY 1984
Measurement of the 6s +7S M1 transition in cesium with the use of crossed electric and magnetic fields S . L. Gilbert, R. N. Watts, and C. E. Wieman Department of Physics, University of Michigan, A n n Arbor, Michigan 48109 (Received 21 June 1983; revised manuscript received 4 August 1983) Th e forbidden 6S+7S magnetic dipole transition amplitude in cesium has been measured by laser spectroscopy of an atomic beam in crossed electric and weak magnetic fields. Th e M I amplitude was determined by observing the change in the transition rate caused by intcrfcrence with a Stark-induced El amplitude. Th e result for the nuclear-spin-independent amplitude is -42.10(80) X 1O-'pB; the result for the nuclear-spin-dependent amplitude is 7.59(55)X 1 0 - 6 p ~ . These values disagree with earlier measurements but they ar e in good agreement with theory. Th e experimental approach is well suited to measuring parity-violating neutral-current interactions.
INTRODUCTION
The 6s-to-7S magnetic dipole ( M 1) transition in cesium has received considerable attention recently because of its role in the study of parity violation in atoms. However, the mechanism responsible for this very small transition amplitude ( -lo5 times smaller than an allowed M I transition) has remained unclear. For some time the dominant mechanism was thoughtIs2 to be the fourth-order product of the interconfiguration and spin-orbit interactions. Recent more accurate calculations334of this product, however, have shown that it is at least an order of magnitude smaller than the experimental value measured by Bouchiat and Pottier' and later by Hoffnagle et a1.6 Stimulated by this, Flambaum et u I . ~proposed that a third-order contribution to the M 1 amplitude would exist and calculated its size. However, this value was nearly a factor of 2 larger than the experimental value, as discussed in Appendix A. Several author^'^^^^ have pointed out that, in addition to this nuclear-spin-independent component of the amplitude, the off-diagonal hyperfine interaction would give rise to a smaller nuclear-spindependent component. There was even poorer agreement between theoretical and experimental values for this component! This was particularly puzzling because the calculation of this quantity is straightforward and can be directly related to well-known hyperfine splittings. We report here the measurement of both components of the M1 amplitude using a new technique which yields higher sensitivity than previously possible and avoids a number of sources of systematic error which may have affected earlier work. Our results resolve the previous disagreements between theoretical predictions and experimental values. The experimental technique used has considerable promise for studying panty-violating neutralcurrent effects in atoms. We conclude with a brief discussion of this future application. The experimental approach employs the interference between the M 1 amplitude and a larger electric dipole ( E 1 ) amplitude. The measurement of a small amplitude by observing its interference with a larger known amplitude has been applied to a variety of problems but was first dis-
cussed in this context by the Bouchiats.' Neglecting parity-violating effects, only M 1 or higher multipole transitions can take place between states of the same parity in an unperturbed atom. The application of a weak dc electric field, however, creates a small admixture of states of opposite parity which gives rise to a "Stark-induced'' E 1 transition amplitude. They pointed out that this amplitude can interfere with both the M 1 amplitude and a panty-violating E 1 amplitude arising from neutralcurrent interactions. Thus the measurement of these two amplitudes via this interference involves very similar experimental considerations. In the absence of a magnetic field, both of these interference terms can create a polarization of the excited state but cannot affect the transition rate.' A number of impressive measurements of small M1 (see Refs. 5,7,8) and parity-violating E 1 amplitudes'*" have been made by determining this polarization. However, such measurements suffer from a loss in sensitivity because the state polarization cannot be measured directly. It must be inferred from the degree of polarization of light absorbed or emitted in a transition from the excited state. In general this will be less than the atomic polarization and can be a source of systematic error in determining the amplitude of interest. Our method avoids these difficulties because the interference is manifested as a direct contribution to the transition rate. This requires that the Zeeman sublevels be resolved. We achieve this by using narrowband laser light to excite transitions in an atomic beam in the presence of a weak magnetic field. The concept of such interference terms affecting the transition rate when a magnetic field is present has been discussed a number of times, though never experimentally demonstrated. The idea was implicit in the experiment to search for panty violation in hydrogen proposed by Lewis and Williams." Following that, it was discussed explicitly by several authors in the context of possible techniques for the measurement of parity violation in heavy atom^.^'"^ Recently Bouchiat and PoirierI4 have extended that discussion to the closely related problem of measuring weak M 1 amplitudes. All these proposed methods require magnetic fields on the order of 1 kG resulting in a complex relationship between the data and the amplitudes 137
29 -
40
@ 1984 The
American Physical Society
41 138
S. L. GILBERT, R. N. WATTS, AND C. E. WIEMAN
-
29 -
LASER BEAM
FIG. 2. Schematic of excitation region. Magnetic field coils and the wire mesh E field plates below and above the plane of intersection of the two beams are not shown.
&, are the same. Thus only a single line is observed and FIG. 1. Cesium energy-level diagram showing hyperfine and weak field Zeeman structure of 6s and 7s states.
of interest. A significant difference between these proposals and our experiment is that we require only tens of Gauss. This small field makes the interpretation of the data quite simple, as we will discuss, and substantially reduces a number of possible systematic errors.
THEORY
The use of crossed atomic and laser beams provides inherently narrow transition linewidths with little background. A n additional experimental consideration however is the configuration of the magnetic (g),the static (g), and oscillating ( E ' ) electric fields. The magnetic field, if weak, causes the Zeeman sublevels to split according to A E / h =mgFpBB,where m is the quantum number for the z component of the total angular momentum F. For transitions between states with the same values of and 5 the resulting spectrum is dramatically different depending on whether is parallel or perpendicular to T. For the parallel case the selection rules for the Stark-induced transitions are A F = O and Am =O. In the weak-magnetic-field limit the energies of all the Am = O transitions remain the same because the initial and final values of F, and hence
the interference terms do not contribute to the transition rate. Only in intermediate and high magnetic fields do the ground- and excited-state Zeeman levels shift differently allowing transitions to different Zeeman sublevels to be resolved. For the case 2 perpendicular to F t h e selection rules are A F = O , f 1, and Am =0, f1. This gives rise to a multiplet structure even in the weak-field regime. Because the different lines in the multiplet involve transitions between different Zeeman sublevels, the transition rate for any given line will contain a contribution from the interference term. Thus the necessary resolution is achieved in quite a weak (tens of Gauss) magnetic field which has the advantages previously mentioned as well as simplifying the experimental apparatus. We shall illustrate the other features of the experimental technique by considering in more detail the 6S-7S hyperfine transitions in cesium which were studied. These can be seen on the cesium energy level diagram in Fig. 1. Although other possible field configurations could be + used, in particular, E along the laser propagation direction, in the interest of brevity we shall limit our discussion to the configuration shown in Fig. 2. The static electric and magnetic fields are perpendicular and the transition is excited by laser light propagating perpendicularly to "E and g, with Pparallel to 5. The Stark-induced E 1 amplitude for a transition from an initial state 6SFmto a final state 7SF.m.is given by
where C,$Am'is proportional to the usual Clebsch-Gordan coefficient and is tabulated in Appendix B. The vector transition polarizability p, introduced in Ref. 2, is given by
42 MEASUREMENT OF THE 6S+7S M1 TRANSITION IN . . .
29
139
I
The M 1 amplitude for this transition is given by f
d ~ ~ ' ( M 1 ) = ( 7 S F , , , I ~i6S,cm) ..~ .
(3)
The selection rules are AF=O,&l and Am = +1. This amplitude can be written as the sum of two terms proportional to M and (F-F'IMhf, respectively, where M is the nuclear-spin-independent component and Mhf is the nuclear-spin-dependent component. This gives
The notable feature of this equation is that the transition rate contains a pure Stark-induced E 1 term, fi2E2,plus an E I-M 1 interference term-the sign of which depends on E and Am. Using Eq. (6) and the fact g ~ = 4 =- g F Z 3 , we can now obtain the spectrum for the three Zeeman multiplets of interest. 6SF=4+7SF=3. The spectrum contains eight lines where each line strength R (i) is given by
d ; ~ ' ( M l ) = [ M + ( F - - F ' M l , f ] ( C ~ , ~ - , )(Am = + l ) and
(4)
.d;Z'(M 1 ) = [ M + ( F - F ) ) M h f ] ( -cj?,,mil) (Am = - 1 ) . For each transition between particular Zeeman sublevels, 6SFm+7sFm., the transition rate I , is the square of the sum of the E 1 and M 1 amplitudes,
I d ( E l ) + d ( M l )1 ' =
I d ( E 1 ) l 2 + 2 ~ ( E l ) ~ ( M 1/ ). d + ( M 1 ) I 2 .( 5 )
For the case we are interested in 1 . d ( M 1 ) 1 << 1 d ( E 1 ) I so we can neglect the I d ( M 1 ) I term. Then from Eqs. ( 1 ) and (4) we can write
The two outermost lines of the multiplet each involve only a single transition ( m=4+3 and m = -4+ -3, respectively) while each of the other six lines is the sum o f a Am = + 1 and a Am = - 1 transition. The calculated spectrum is shown in Fig. 3(a). The actual rate for each line in the figure is the sum of the pure E 1 contribution (solid line) and the E 1-M 1 interference (offset dashed line on an expanded vertical scale). 6SF=3+7SF=4. This has a nearly identical spectrum to the F = 4 + 3 transition. The only difference is that the interference terms are now proportional to M -Mhf instead of M Mhf. 6SF=4+7SF=4. For this transition all the A m = + ] transitions are shifted up in frequency by (0.35 MHz/G)B while all the Am = - 1 transitions are shifted down by this amount. The spectrum thus contains only two lines with their rates given by
+
R(+)=
I
(a1
2
rn = -4-
14.m + I 4,m
+
= +Z(B~EZ 2 ~ 1 3 4 ,)
+3
(8)
R(-)=
2
I>~-'=~&P2E2-2PEM).
rn=-3-+4
FIG. 3. (a) Theoretical spectrum. The solid lines are the
I &'(El) I * contributions while
the dashed lines are the
2 d ( E 1) d ( M1) contributions on an expanded vertical scale.
The dashed lines have been shifted slightly to the right for ease of viewing. Actually both lines occur at the same frequency and the observed intensity will be the sum of the two contributions. (b) Scan of 6 s ~ = 4 + 7 s ~ =transition 3 with B = 100 G.
This analysis used the weak-field limit for the Zeeman effect which assumes no mixing of different hyperfine states by the magnetic field. In the field regime of interest this mixing can be accurately calculated using secondorder perturbation theory. We find its effect is to change the relative transition strengths slightly from the weakfield limit. For a typical field of 40 G the changes range between 0% and 3.4% for the various lines of the multiplets discussed. However, for the spectral lines measured in this paper, the size of the change is unimportant because the corresponding E 1 and M 1 amplitudes are affected by exactly the same amount. Thus the ratio of the two amplitudes, which is the quantity of interest, is independent of the mixing of the hyperfine states. For all three hyperfine transitions the interference t e r n s are odd under reversal of 2, 5,and the changing of the excitation frequency to the opposite multiplet component, providing a simple experimental way to isolate and mea-
43 29 -
S. L. GILBERT, R. N. WATTS, AND C. E. WIEMAN
140
sure them. The experiment consists of carrying out these reversals and detecting the resulting fractional modulation in the transition rate. In principle any single reversal would be sufficient but in practice the extra reversals are valuable for eliminating systematic errors. EXPERIMENT
We have used this technique to measure the ratios of the magnetic dipole amplitude to the Stark-induced amplitude 8,for the 6 s ~ = 3 + 7 s ~ = 4the , ~ S F = , + ~ S F = and ~, 6SF=4-f7SF=4 transitions in cesium. The F = 3 + 4 and F = 4 4 3 transitions were measured, then the apparatus was modified slightly and all three transitions measured. In a previous work15 we determined the absolute value of
8. The experiment is shown schematically in Fig. 2. A narrow-band dye-laser beam intersects a collimated beam of atomic cesium in a region of perpendicular electric and magnetic fields. The 6S+7S transition rate is monitored by observing the 850- and 890-nm light emitted in the 6P3/2,1/2*6S1/2 step of the 7 s decay. The dye laser is a Spectra-Physics model No. 380 dye laser with homemade frequency and amplitude stabilizers. The output power is typically 500 m W with a linewidth of -100 KHz. The cesium beam, which is produced by a two-stage oven to reduce the dirner fraction, is collimated to 0.015 rad. The 2.5-cm-long region of intersection of the two beams is imaged onto a silicon photodiode by a 5-cm-long gold-coated cylindrical mirror with flat ends. The scattered 540-nm light is blocked by a colored glass cutoff filter in front of the detector. This filter is coated with an optically transparent conductive coating. Fine wire mesh was placed above and below the interaction region and the electric field was produced by applying voltage (typically 1.6 KV) to the upper mesh and grounding the lower. For the first run the mesh spacing was 0.5 cm while for the second a new collector was used with a 0.6-cm spacing. A 40-G magnetic field is provided by a 25-cm-diam Helmholtz pair. Data was obtained by locking the laser frequency to the peak of a particular line of the multiplet and reversing the electric field every 0.25 sec. During each half cycle of electric field the detector output was integrated, digitized, and stored by a Digital Equipment Corporation model No. PDP-11 computer. For each of the three hyperfine transitions data was taken on the extreme high-and lowfrequency lines of the Zeeman multiplets as these provide the largest signals. This was done for both directions of magnetic field providing a total of four data sets for each hyperfine line. A t 1-2-min intervals during these measurements the electric field was set t o zero t o determine the baseline for the transition. To test for any asymmetry in the reversal of 3, data was also taken at zero magnetic field for laser frequencies both on and well off the transitions. To test for frequency-dependent background signals the laser frequency was scanned over the transitions both with and without magnetic field and the spectra recorded. One such scan with a rather large magnetic field is shown in Fig. 3(b). The 14 M H z linewidth is due to the residual
-
-
I
I
I
0
0.5
1.0 V (GHI)
1
1.5
I
2.0
FIG. 4. Scan of transition with B = O
Doppler shift from the cesium beam divergence. In order to obtain an absolute value for the M 1 amplitude it is necessary to know the dc electric field. We determined this by measuring the Stark shift of the 6SF=4+7S,7=4 transition as described in Ref. 15. Once the Stark shift was known the field was found using the polarizability given in that reference. DATA ANALYSIS A N D RESULTS
The data analysis primarily consisted of taking the difference between the positive electric field and negative electric field transition rates and dividing it by the average rate. From Eqs. (6)-(8), this is simply (9)
where Meff is M + M , f , M - M h f , and M , for the three transitions studied. However, considerable additional effort was devoted to determining and correcting for possible systematic errors. First we considered systematic errors which could have been introduced by various background signals. The cesium oven and, to a lesser extent, the scattered laser light gave appreciable background signals which were independent of laser frequency. The only effect of these signals on our measurement was to cause a slow uniform drift in the detector zero level, typically corresponding to 1% of the transition rate per minute. This drift was determined from the E = O data and did not introduce a significant uncertainty. The scans of laser frequency over the transition showed a broad frequency-dependent background signal centered on the atomic transition frequency. This can be seen in Fig. 4. We ascribe this signal to a n isotropic background gas of cesium in the interaction region. The line shape was accurately determined from the spectra taken with B=O. The background pedestal could be fitted well by a 575-MHz-wide Gaussian curve. The height varied between 0.040(2) and 0.067(2) times the height of the 14-
44 29 -
MEASUREMENT OF T H E 6S--t7S M1 TRANSITION I N
MHz-wide peak for the different runs. This pedestal contributes to both the average ( B 2 E 2 )transition rate and the E 1-M 1 modulation. The contribution to the modulation is considerably diluted over that given in Eq. (9) because the linewidth is much broader than the Zeeman splitting. Using the measured line shape we calculated the correction needed because of the presence of the pedestal (11-26%). We do not observe any other frequencydependent background-in particular, that due to molecular cesium which was significant in the work discussed in Refs. 5 and 6. A number of possible systematic errors associated with the field reversals were also checked. The use of three independent mechanisms to reverse the sign of the interference term greatly reduces such errors but a number of independent experimental tests were also made. These included tests of the follo_wing:_(1) detector sensitivity depending on direction of E or B, (2) imperfect B reversal, (3) imperfect reversal, (4) error due to ??not perfectly linearly polarized perpendicular to E, ( 5 ) misalignment of B with respect to E and Z,and (6) pickup of '*E fieldswitching transients. None of these sources were found to be significant at the level of the experimental uncertainty except the "E field switching transients. These were determined from the modulation signal measured when no laser light was present. It was found to be -9% of the interference term. Since this spurious signal is always the same sign, its effect averages to zero when the data for different directions of magnetic field and different lines of the multiplet are averaged. In order to obtain an absolute value for the M 1 amplitude from the ratio given by Eq. (9) it is necessary to know E. From the Stark shift of the 6SFS4+7SFE4 transition we obtained 2934U5) V/cm for the first data run. For the second run, values of E between 2661(10) and 2997(10) V/cm were used. The values obtained from the Stark shift agreed well with less accurate ( 5 % ) values calculated by dividing the applied voltage by the mesh separation. The dependence of the transition line shape on electric field also provides a test of the field uniformity. We find that the 14-MHz linewidth is changed by less than 1% in going from 625 to 5000 V/cm while the line center shifts 17.5 M H z [Av=O.71 Hz(V/cm)-2 from Ref. 151. This implies that any spatial variation of the field strength is substantially less than 1%. In Fig. 5 we present the fractional changes in the transition rate observed for the different lines. We have put in corrections for the pedestal and the electric field transients. For display purposes we have normalized all the data to a field of 2997 V/cm. Comparing across each row one can see the expected sign changes. Also the agreement in the magnitudes provides further confirmation that all the reversals are working as planned. From this data and Eq. (9) we obtain from the first run in V/cm
-
-
141
1 1 1 1
''1
1
T
FIG. 5. Fractional changes in transition rates. Direction of and - while the H and the magnetic field is indicated by the L indicate the multiplet line. The highest fi-equency line is labeled H and the lowest L . The AR's in the first and third columns were negative, as indicated. Each dotted line is the average of the four points in that row. All the data was normalized to 2997 V/cm.
+
M-Mhf
= - 34.59( 80)
B -=_
, M f M h f =-24.10(70) , I3
29.43(62),
P where this value for M / B is based on only the F = 4 + 4 data. In Table I we combine these results and compare with earIier measurements. With the exception of the value for ( M -kfhf)/fl our results disagree with the earlier measurements. These earlier measurements of M / P were made using A F = O transitions while for our first data run we derived this quantity from A F = + l and -1 transitions. The disagreement between these led us to speculate that perhaps, contrary to Eq. (4), there was some unsuspected M 1 component which contributed equally to the AF#O lines and did not contribute to the A F = O transitions. However, the agreement between the value of M / f l we obtained from the A F = 1 and - 1 transitions [-29.83(40) V/cm] and the value we subsequently obtained from the F = 4 + F = 4 transition rules out such an idea. Using /3=26.6(4)ai as discussed in Appendix A we obtain
+
M
= -42.10(
80) X 10-6p8 ,
Mhf=7.59(55)X 10-6p,
.
Considering the uncertainty in the calculation, the value for M is in agreement with the recent value of -63 X TABLE I. Measurements of M 1 Amplitudes
Ref. 5 M (V/cm) R
Mhf -
and for the second run in V/cm
. ..
M
- 23.2U.3)
Ref. 6
This work
-26.2(1.7)
-29.73(341
- 34.3(2.1)
- 35.21(56)
-0.3 l(3)
-0.180(13)
45 142
S . L. GILBERT, R. N . WATTS, A N D C. E. WIEMAN
calculated by Flarnbaum, Khriplovich, and Sushkov4 for the contribution due to the product of first-order interconfiguration interaction and second-order off-diagonal spinorbit interaction. M h f arises from the mixing of states of different n by the hyperfine interaction. Hoffnagle6 has shown it can simply be expressed as (theory)
where A w and ~ A ~ W , ~are the accurately known hyperfine splittings of the 6s and 7s states.I6 This value is expected to be quite accurate, and is in good agreement with our measured value. EXTENSIONS OF THE TECHNIQUE
The experimental approach of crossed beams in a weak magnetic field can also be used to measure the parityviolating E 1 amplitude [ d ( ElPv)] arising from neutralcurrent mixing of the § . and P states. As mentioned earlier this interferes with the Stark-induced amplitude in a manner similar but not identical to the M1 amplitude. For the field configuration we have discussed, one significant difference, which was mentioned in the Bouchiats early work,2 is that the parity-violating mixing matrix element is imaginary relative to the Stark mixing matrix element. Thus no interference will be present for linearly polarized light. This can be remedied by using elliptically polarized light, Z=C’l+iC’l,, where the I and / I are with respect to E. NOW iFI1 creates a parity-violating amplitude which has the same phase and hence interferes with the Stark-induced amplitude. Carrying through a similar analysis as before (including the mixing of hyperfine states) for the 6SF=4-+7SF=3 transition one obtains the same Clebsch-Gordan coefficients and the same spectrum as for the M 1 case shown in Fig. 3(a). Th e only difference is that now the vertical scale for the interference terms is proportional to ( e l E f i ) [ E @ ( E lPv)]. Like the M 1 case the interference terms change sign with a reversal of E, 5,and multiplet component. However, unlike the M 1 case they do not change sign if the laser light propagation direction is reversed” and they have the additional signature that they change sign when the “handedness” of the light is reversed (ze,l+-iell). It might also be noted that the ratio between interference and pure Stark-induced contributions t o the transition rate can be enhanced by making elI /el > 1. The apparatus previously described is thus well suited to measure this parity-violating term. Besides the use of elliptically polarized light, the only significant change needed is the addition of a power buildup interferometer cavity of the type we have previously This provides 150 times more laser power i n the interaction region. Because the signal to noise ratio is limited entirely by detector noise this should give a corresponding improvement in the signal-to-noise ratio while suppressing the E 1-M 1 interference. For a panty-violating amplitude of the size measured by Bouchiat et al. lo ( l o p 4 times the M l ) , such a signal-to-noise ratio would allow a one standard-deviation measurement of E I,, with an integrat-
-
-
29 -
ing time on the order of 10 min. The apparatus has potential for considerable future improvement. For the measurement of M 1 amplitudes the addition of a traveling-wave ring buildup cavity would improve the signal-to-noise ratio by 100. Also the use of an optically pumped atomic beam would provide a 16fold increase in signal for measurements of both the d ( M 1 ) and .d(E lpv)amplitudes and eliminate the need for a magnetic field. Note added in prooj We have learned that Bouchiat, Cuena, and Pottier have recently remeasured M / / 3 and measured Mhf/P, and they now obtain results in excellent agreement with ours.
-
ACKNOWLEDGMENTS
We are pleased to acknowledge the assistance of M. C. Noecker on this experiment and D. Kleppner for a critical reading of the manuscript. This work was supported by the National Science Foundation and in part by Research Corporation. One of us (R.N.W.) was supported by a University of Michigan Rackham Fellowship. APPENDIX A: THE VALUE OF /S
A certain amount of confusion has been caused by the value of the vector transition polarizability /3 used in Ref. 5 . The quantity actually measured in that work was the value of M / f i given in Eq. (13) of that reference and listed in our Table I. By using a value of 8.8(4) for the ratio of scalar to vector transition polarizability, 1 a//3 1 , and a theoretical value of ( - 30%~:) for a , they quoted an absolute value for M of 4.24(34)X 1 p B 1 . Subsequent measurements by Hoffnagle et d 6and ourselves,16 and a remeasurement by Bouchiat et al. have obtained values for I a / P [ of 9.91(11), 9.80(12),and 9.90(10), respectively. We have experimentally determined15 a to be -263.7(27)ai. From the average of these results we take l a / P I =9.9(1) and arrive at a value of /3=26.6(4)ai. Using this value for P, the M / P given in Ref. 5 gives M = 3 . 3 ( 2 ) X 10-’pB which is farther from the calculated value of 6.3 x 10-’pB than the number originally quoted. In the text we have taken care to only compare the measured ratios M , f f / P . To derive an absolute value for M and M h f from our data we used 8 = 2 6 . 6 ( 4 ) a i which was obtained as described.
’*
APPENDIX B: CFAm’COEFFICIENTS
We find the C;km’ coefficients to be as follows: c4 4;mr-1 m‘ =++[(5-m ’)(4+m ’)]l” ,
c $ $ +=-+[(5+rn’)(4-m’)1’/~ ~ ~;;l-~
,
, l+$[(4+m = ’)(5+m ’)]1/2 ,
= ++[(4-m’)(5-m‘)11/’
~ 3,m’ 4 , ~ ’ +
c ~ Z : -=-+[(3+m ~ ’)(4+m ’)]1/2
,
= - +[(3-m’)(4-m’)]1/2
,
~ 43 ,.mm, +’ l
c:;Z~.-~=-+[(4-rn’)(3+m’)l1/’
,
.
c 3 3 :m m’ , + l = + + [ ( 4 + r n ’ ) ( 3 - ~ ~ ’ ) ] ~ / ~
46 MEASUREMENT O F T H E 6S+7S M1 TRANSITION I N . . .
29
1M. Phillips, Phys. Rev. 88,202 (1952). *M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 35, 899 (1974); 36, 493 (1975). 3D7V. Neuffer and E. D. Commins, Phys. Rev. A 16,1760 (1977). 4V. V. Flambaum, I. B. Khriplovich, and 0. P. Sushkov, Phys. Lett. 177 (1978). 5M. A. Bouchiat and L. Pottier, J. Phys. (Paris) Lett. 37, L79 (1976). 65. Hoffnagle, L. Ph. Roesch, V. Telegdi, A. Weis, and A. Zehnder, Phys. Lett. @A,143 (1981); J. Hoffnagle, Ph.D. thesis, Swiss Federal Institute of Technology, Zurich, 1982 (unpublished). 'S. Chu, E. D. Commins, and R . Conti, Phys. Lett. 96 (1977). *W. Itano, Phys. Rev. A 22, 1558 (1980). 9P. Bucksbaum, E. Commins, and L. Hunter, Phys. Rev. Lett. 48, 607 (1982). loM. A. Bouchiat, J. Guena, L. Hunter, and L. Pottier, Phys. Lett. 358 (1982). "R. R. Lewis and W. L. Williams, Phys. Lett. B 59, 70 (1975).
m,
a,
m,
143
12M. A. Bouchiat and L. Pottier, in Proceedings of the International Workshop on Neutral Current Interactions in Atoms, (Cargese, 19791, edited by W. L. Williams (unpublished); M. A. Bouchiat, M. Poirer, and C. Bouchiat, J. Phys. (Paris) 40, 1127 (1979). 13P. Bucksbaum, in Proceedings of the International Workshop on Neutral Current Interactions in Atoms (Cargese, 19791, edited by W. L. Williams (unpublished). We are aware that an experiment along the lines discussed in this reference is being worked on by P. Drell and E. D. Commins (private communication). I4M. A. Bouchiat and M. Pokier, J. Phys. (Paris) 43, 729 (1982). lSR. N. Watts, S. L. Gilbert, and C. E. Wieman, Phys. Rev. A 27, 2769 (1983). 1%. L. Gilbert, R. N. Watts, and C. E. Wieman, Phys. Rev. A 27, 581 (1983). I'M. A. Bouchiat and L. Pottier, Lnser Spectroscopy III, edited by J. L. Hall and J. L. Carlsten (Springer, Berlin, 1977). 18M. A. Bouchiat, J. Guena, L. Hunter, and L. Pottier, Opt. Commun. e , 3 5 (1983).
PHYSICAL REVIEW A
VOLUME 29, NUMBER 6
JUNE 1984
Absolute measurement of the photoionization cross section of the excited 7s state of cesium S. L. Gilbert, M. C. Noecker, C. E. Wieman Department of Physics, University of Michigan, Ann Arbor, Michigan 48109 (Received 5 December 1983) We report the first measurement of the absolute cross section for photoionization of the 7 s state of cesium. The measurement employed a new technique in which the density of the excited-state atoms was determined by the amount of fluorescence. The cross section for photoionization by 540-nm light is 1.14( 1O)X cm2. We also propose a second new technique for the absolute measurement of photoionization cross sections which is based on modulated fluorescence.
I. INTRODUCTION
The photoionization behavior of alkali-metal atoms has been a subject of considerable theoretical and experimental study. Cesium has been of particular interest lately because the effects of spin-orbit interaction and core polarization are more pronounced than in the lighter alkali metals. This has been quite significant because the spin-orbit interaction gives rise to the Fano effect which is used for producing polarized electrons. However, these effects make the theoretical calculation of the photoionization cross section more difficult, particularly for S states. Simple quantum-defect theories which are adequate for light alkali metals give rather poor agreement with experimental results for cesium. Weisheit' and Norcross' have carried out semiempirical calculations in which they included both core polarization and the spin-orbit interaction, while ab initio calculations by Chang and Kelly,3 Johnson and Soff; and Huang and Starace' treated the spin-orbit interaction but neglected core polarization. Lahiri and Manson6 have done a simple Hartree-Slater calculation for a number of low-lying states. Attempts to check the accuracy of these theoretical approaches have been limited by the lack of accurate and consistent experimental data on absolute cross sections. In modem times there have been only three absolute measurements of photoionization cross sections for S states of cesium, all of which were for the 6 S ground state. Marr and Creek' obtained results for the ultraviolet region of the spectrum which were about a factor of 2 larger than the values calculated by either Weisheit or Norcross. However, Cook et a1.' repeated this measurement and obtained values in agreement with those calculations but with uncertainties of ?30%. Grattan er aL9 have also measured this cross section at a wavelength in the vacuum ultraviolet with similar results. We report here the absolute measurement of the cross section for the 7 s state accurate to +9%. This measurement used the new technique of "fluorescence normaiization" in which the excited-state atomic density was determined from the amount of fluorescence. This technique avoids the uncertainties in determining the molecular background and the ground-state atomic density which have limited these previous absolute measurements. 29 -
We shall conclude with the discussion of another technique for the absolute measurement of photoionization cross sections. The technique proposed is an extension of fluorescence normalization but under some conditions has substantial advantages over fluorescence normalization and other techniques which have previously been used.
11. EXPERIMENTAL METHOD A N D APPARATUS
For this measurement a beam of cesium atoms in an electric field was excited to the 7s state by a cw dye laser. This normally forbidden transition is allowed due to the small (5 parts in lo') mixing of S and P states by the static electric field. The number density of 7s atoms was determined from the amount of fluorescence emitted as the atoms spontaneously decayed. The laser radiation also caused a small fraction of the 7 s atoms to be photoionized, and the resulting ion current was measured. Strictly speaking, the process observed is two-photon photoionization with a resonant intermediate state. It is an extremely good approximation to treat this as two single-photon processes, however, because both the 6S-7.S transition and the 7S-+continuumtransition were far from saturation. The density of excited atoms was lo-* times the density of ground-state atoms, and only 1 part in lo4 of the excited atoms was photoionized. This allows the 7 s photoionization cross section to be determined from the fluorescence signal, the laser power, and the photoionization current. The apparatus is shown schematically in Fig. 1. A highly collimated beam of atomic cesium intersected a standing-wave laser field at right angles in a region of static-electric field. The cesium beam was produced by a two-stage oven to reduce the dimer fraction. The output nozzle was a microchannel plate which produced a beam with a cross section 0.5X2.5 cm2 and a half-angle divergence of -0.05 rad. This passed through a multislit collimator which reduced the divergence in the direction of the laser beam to 0.013 rad. A final 2-cm-wide aperture just before the intersection with the laser beam provided a precisely defined intersection geometry 2.0 cm long with a diameter equal to that of the laser beam (0.05 cm). The cesium beam density (-5X 109/cm3) was uniform to 3150
47
@ 1984 The
American Physical Society
48
29 -
ABSOLUTE MEASUREMENT OF THE PHOTOIONIZATION CROSS INTERFEROMETER MIRROR
...
3151
fl
il
FIG. 1 . Schematic of apparatus.
within a few percent over the intersection region because the total distance from nozzle to intersection (6 cm) was much shorter than the distance scale for beam-density redistribution along the direction of the laser (100 cm). This uniformity was confirmed using a hot wire detector. The standing-wave laser field was produced by an amplitude- and frequency-stabilized cw ring dye laser mode-matched into an electronically tunable, semiconfocal Fabry-Perot interferometer buildup cavity. The laser had a typical output of 250 mW and a short-term frequency jitter of 0.5 MHz peak to peak. With the resonant frequency of the interferometer locked to the laser frequency, the electromagnetic field strength inside the interferometer corresponded to two linearly polarized traveling waves, each with 116 times the power incident on the buildup cavity. A calibrated photodiode measured the light transmitted through the cavity. The intensity profile for the laser beam in the cavity is that of the lowest-order eigenmode of a semiconfocal cavity with a 25-cm focal-length curved mirror." This is a Gaussian profile with a spot-size radius of 0.021 cm at the input mirror and 0.029 cm at the output mirror. Less than 1% of the power was contained in higher-order modes. The 7s-state population was monitored by observing the 850- and 890-nm light that was emitted in the 6P-6S step in the 7S-6P-6S cascade decay. This light was detected by a cooled silicon photodiode 0.5X 5.5 cm2 that sat 9 mm below the interaction region. Glass color filters not shown in Fig. 1 prevented scattered 540-nm laser light from reaching the detector. The static-electric field was produced by applying voltage to the top field plate and grounding the lower. The laser and cesium beams intersected in the middle of the 5 x 7.5 cm2 electric field region. The top field plate was coated with evaporated gold and the lower field plate was coated with an electrically conducting optically transparent (84%) coating. The photoionization current was measured using an ammeter in the line providing voltage to the top field plate. Data were obtained by scanning the laser over the 6SF=4+7SF=4 Stark-induced transition and simultaneously recording the fluorescence signal and the photoionization current. Typical data are shown in Fig. 2. Scans
100 200 300 LASER FREQUENCY ( M H t )
FIG. 2.
(a)
Photoionization current vs laser frequency. (b)
Fluorescence detector current vs laser frequency. Zero of the laser frequency scale was chosen arbitrarily.
were made at several voltages between 1500 and 3000 V/cm and with both positive and negative voltage. The fluorescence line shape is composed of a narrow (14 MHz) resonance peak superimposed on a low broad (575 MHz) pedestal. The pedestal arises from a diffuse background vapor of cesium in the interaction region and is discussed in more detail in Ref. 1 1 . The photoionization current has an identical line shape but it has an additional background component which is independent of laser frequenCY.
The 7 s number density was calculated from the 7sstate lifetime and the total emitted fluorescence. The determination of the total fluorescence is inherently the least accurate and aesthetically the most unpleasant part of the experiment, as it involves finding the detector quantum efficiency and the detection solid angle. The detector size, uniformity, and angular dependence of the response were measured. The quantum efficiency was given as 0.80(4) by the manufacturer.12 The detection solid angle was calculated by numerical integration using the geometry of the apparatus and the measured transmission of the lower field plate and filters at various angles of incidence. This included the light which was reflected off the upper field plate, the reflectivity of which was measured. There was a negligible contribution due to reflection off other surfaces in the apparatus since these were all far from the interaction region and painted flat black. The laser power in the buildup cavity was determined by dividing the transmitted power by the transmission coefficient for the output mirror. The power measurement was made using a photodiode calibrated against a Coherent Inc. power meter recently calibrated to a NBS standard.
49 3152
S. L. GILBERT, M. C. NOECKER, A N D C. E. WIEMAN 111. RESULTS
their value for the Cooper minimum would explain the discrepancy. They believe the uncertainty in the calculation of this minimum is considerably larger than 0.001 eV.
The total photoionization current is given by I=e J-s(F)~(F)~~F,
29 -
(1)
r
where S(i)is the laser photon flux, n (F) is the density of 7 s atoms, and u is the photoionization cross section of interest. The fluorescence signal is
IV. EXTENSIONS OF PRESENT WORK
The present work has provided the first measurement of a photoionization cross section for an excited S state of cesium. This provides a good test of the different theoretical treatments of photoionization of cesium, a better test than has been possible using the less accurate and disparate values measured for the ground state. However, it is obviously desirable to measure the dependence of this cross section on wavelength. This could be done using the 21gaw2 fluorescence normalization technique if a second laser was U= (3) used to do the photoionization. The obvious choice for a FSor ’ second laser would be a relatively high-power pulsed laser. where So is the total number of laser photons per second However, such a laser would allow the use of a new techand w is the Gaussian beam radius at the interaction renique for measuring the cross section which is something gion. Our measurements give the cross section in cm2 as of a hybrid between fluorescence normalization and the popular saturation te~hnique,’~ but can have significant (T= 1.14( 1O)xl o p i 9. advantages over both approaches. The significant contributions to the 9% uncertainty are This technique, henceforth called modulated fluores5% for detector quantum efficiency, 6% for laser power, cence, could be used quite generally for determining 3% in measurement of photoionization current, 3% for excited-state photoionization cross sections. In the modusolid angle, and 2% for the 7s lifetime. lated fluorescence technique one would excite the atoms to There are several possible sources of additional systhe state of interest with one laser and monitor the change tematic error which we have considered. First, there was in the fluorescence when the photoionizing laser (or lamp) a background current which we attribute primarily to is pulsed. The photoionization rate would then be simply photoemission from surfaces; however, this background the fractional change in the fluorescence signal divided by was eliminated in the analysis by using only the 14 MHzthe lifetime of the state. The only quantities which must wide components of the measured currents shown in Fig. be determined absolutely are the lifetime of the state and 2. A second possible source of error was multiplication of the photoionizing laser intensity. In this respect it is simithe photoionization products through collisions with neular to the saturation technique and superior to fluorestral atoms or secondary surface emission. This can be cence normalization. However, it has a significant advanruled out by the observation that the measured photoionitage over the saturation approach in that it requires conzation cross section was independent of the strength of the siderably lower photoionization rates and hence lower applied field. At the highest field there was a substantial laser power. In the saturation method it is necessary to increase in the noise on the photoionization current, howachieve ionization rates which are at least comparable and ever, suggesting some multiplication or arcing phenomepreferably several times larger than the spontaneous decay na. Therefore only the values obtained at the lowest elecrate. However, the modulated fluorescence technique will tric field (1500 V/cm) were used in obtaining u. Another work with photoionization rates which are a small fracpossible source of systematic error is anisotropic radiation tion of this. The necessary fraction is ultimately limited trapping. Calculations indicate this should be quite small. by the signal-to-noise ratio of the fluorescence measureHowever, we also checked this empirically by measuring ment, but this ratio is characteristically quite high. There the ratio of fluorescence signal to beam density as a funcare many cases where the necessary photoionization rate, tion of beam density. When the density was increased RpI could be 10-2-10-3 of the decay rate. from 5 to 40 times lo9 atoms/cm3 this ratio decreased by The three techniques mentioned are thus complementaless than 20%. This indicates that radiation trapping is ry to each other since each has a range of experimental not a significant effect for the density ( 5 x 109/cm3) at conditions where it is generally superior. If unlimited which the photoionization measurement was made. laser power is available ( RpI > 1/71 the saturation techThe cross section we obtain can be compared with the nique would usually be best because it is usually easier to theoretical value obtained by Lahiri and Mansom6 Their measure photoionization current than fluorescence. For calculation predicts a value about one half of what we an intermediate rate [ loL3(1/ r )< RPI < 1/ r ] the modulatmeasure. However, 540 nm is in a region where their caled fluorescence technique would be superior. Finally, for culated cross section has a very strong dependence on eneven lower photoionization rates, such as in the experiergy (wavelength) so that an error of only 0.001 eV in ment reported, the problem of obtaining a sufficiently where 4 is the detection efficiency and T the 7s lifetime.” As mentioned previously, the 6s -7s excitation rate and 7 s photoionization rate are much smaller than 1/ r . This allows us to assume that n (it) has the same Gaussian form as S(F). With this substitution, Eqs. ( 1 ) and (2) can easily be solved to yield
2
ABSOLUTE MEASUREMENT OF T H E PHOTOIONIZATION CROSS . . .
high signal-to-noise ratio to observe the modulated fluorescence becomes worse than the additional difficulty of determining the absolute fluorescence detection efficiency. In this case the fluorescence normalization technique would be the best choice.
'J. Weisheit, Phys. Rev. A 5, 1621 (1972). Norcross, Phys. Rev. A 7, 606 (1973). 3J. Chang and H. Kelly, Phys. Rev. A 5, 1713 (1972). 4W. R.Johnson and G. Soff, Phys. Rev. Lett. So, 1361 (1983). *K. N. Huang and A. F. Starace, Phys. Rev. A 19,2335 (1979); 22, 318 (1980). 65. Lahiri and S. Manson (private communication). 7G. V. Marr and D. M. Creek, Proc. R. Soc. London, Ser. A 304, 233 (1968). *T. B. Cook, F. B. Dunning, G. W. Foltz, and R. F. Stebbings, Phys. Rev. A 15,1526 (1977). 9K. Grattan, M. Hutchinson, and E. Theocharous, J. Phys. B 13, 2931 (1980). 2D.
3153
ACKNOWLEDGMENTS
We are pleased to acknowledge the assistance in this work of R. N. Watts and T. Miller and the encouragement of D. Norcross. This work was supported by the National Science Foundation.
IOA. E. Seigman, Introduction to Lasers and Masers (McGrawHill, New York, 1971), Chap. 8. "S. L. Gilbert, R. N. Watts, and C. E. Wieman, Phys. Rev. A 29, 137 (1984). I2This is a custom-made photodiode supplied and calibrated by Hughes Aircraft Co., Industrial Products Division. I3J. Hoffnagle, V. L. Telegdi, and A. Weis, Phys. Lett. 457 (1981). I4R. V. Arnbartzumian, N. P. Furzikov, V. S. Letokhov, and A. A. Puretsky, Appl. Phys. 2, 335 (1976); U. Heinzmann, D. Schinkowski, and H. D. Zeman, ibid. l2, 113 (1977); A. V. Smith, D. E. Nitz, J. E. M. Goldsmith, and S. J. Smith, Phys. Rev. A 22, 577 (1980).
m,
VOLUME55, NUMBER 24
P H Y S I C A L R E V I E W LETTERS
9 DECEMBER 1985
Measurement of Parity Nonconservation in Atomic Cesium S. L. Gilbert, M. C. Noecker, R. N. Watts, and C. E. Wieman(’) Joint Institute for Laboratory Astrophysics, Universi@of Colorado and National Bureau af Standards, Boulder. Colorado 80309 (Received 1 1 September 1989 A new measurement of parity nonconservation in cesium is reported. The experimental technique involves measurement of the 6s- 7s transition rate by use of crossed atomic and laser beams in a region of perpendicular electric and magnetic fields. Our results are I m l PNC//3= - 1.65 f 0.13 mV/crn and C t - 2 zk 2. These results are in agreement with previous measurements in cesium and the predictions of the electroweak standard model. This experimental technique will allow future measurements of significantly higher precision.
-
PACS numbers: 35.10.Wb. 11.30.Er.12.30.C~
The standard model for the electroweak interaction predicts a parity-nonconserving (PNC) neutral-current interaction between electrons and nucleons. In 1974 the Bouchiats’ proposed that this effect might be observable in large-Z atoms, thus generating a decade of experimental effort. Measurements of PNC have now been made in bismuth and lead by observation of the optical rotation of light,’ and in thallium3s4 and cesiums by use of the technique of Stark interference. Here we report a new measurement of parity nonconservation in cesium which is more precise than the previous measurements in atoms and is approaching the precision of the best high-energy test of the electroweak theory. This result is in good agreement with previous measurements in cesium. By comparing the PNC observed on two different hyperfine transitions we also set a much lower limit for the proton-axialvector PNC contribution. Parity-nonconservation measurements in atoms are valuable because they test the electroweak theory in a different regime from that probed by high-energy experiments. As well as being sensitive to a very different energy scale for the exchange of virtual particles, atomic experiments also involve a nearly orthogonal set of electron-quark couplings. Because of this, atomic PNC measurements, when combined with high-energy results, can provide useful tests of the electroweak radiative corrections and alternatives to the standard model. Deriving information about the basic neutral-current interaction from atomic PNC measurements requires knowledge of the atomic wave functions. Cesium is a particularly good atom in this respect because of its single-electron character. This enables a more direct and accurate calculation of the wave function than is possible for other heavy atoms. The basic experimental concept has been discussed previously6 but we will review the essential points. The PNC interaction in an atom mixes the S and P eigenstates, allowing a small electric dipole ( E 1) transition amplitude between states of the same parity. In all atomic PNC experiments, this parity-nonconserving 2680
amplitude (APNC) is measured by observation of its interference with a much larger parity-conserving amplitude. In our experiment, the parity-conserving amplitude is a “Stark-induced” E 1 amplitude (A st) created by the application of a dc electric field to mix S and P states. The early Stark interference experim e n t ~ ~measured ,’ the spin polarization of the excited state due to this PNC interference. In our experiment, we are able to observe the interference directly in the transition rate by applying a magnetic field to break the degeneracy of the Zeeman levels. The idea of using electric and magnetic fields was proposed by a number of authors’ and used in the recent thallium PNC measurement of Drell and C o m m i n ~ .A~ similar technique was also demonstrated in our measurement6 of the Cs 6S 7s magnetic dipole amplitude ( A M l ) . A unique feature of our approach is the use of crossed laser and atomic beams. This yields narrow transition linewidths which have the important experimental advantage of allowing the use of a small ( < 100 G) magnetic field. Other desirable features of an atomic-beam experiment include the reduction of collisions, radiation trapping, and molecular backgrounds. We measure the PNC interference term on both the 6 & = 4 - + 7SF,_, and 6 S F E 3 - +7SF,-4 hyperfine lines shown in Fig. 1. The basic field configuration for the experiment is shown in Fig. 2. A standing-wave laser beam along the 9 axis excites transitions in a region with an electric field (E)in the k direction and a magnetic field (B) along the 2 axis. The laser field has polarization d = eZ2 i e x % , where ex and ez are real. For any transition between particular Zeeman sublevels ( m and m“)the transition probability is +
+
I-I~S~+AMMI+A~NCI~,
(1)
where each A is a function of F, F , m, and m’. Using the results of Gilbert* we substitute for the amplitudes in Eq. (1) and obtain for F Z F ,
r;y=
[pZE2€,2 T 2pE€, I m 8 PNCEx1 F’m’ 2 X ( C p m ) 6m,mtkl
@ 1985 The American Physical Society 51
(2)
52
PHYSICAL R E V I E W LETTERS
VOLUME55, NUMBER 24
9 DECEMBER 1985 INTERFEROMETER MIRROR
1.36~ 1.47~
‘LASER
BEAM
0
FIG. 2. Interaction region and field configuration.
FIG. 1. Cesium energy-level diagram showing hyperfine and weak-field Zeeman structure of 6s and 7s states.
plus negligibly small terms involving only 2? PNC and A M 1 . The first term in the brackets is the pure Starkinduced transition rate where p is the vector transition polarizability defined in Ref. 1. The second term is the interference between Ast and the much smaller amplitude A p N c . The quantity i g p N C is the PNC electric-dipole reduced matrix element. The coefficients C are related to Clebsch-Gordan coefficients and are tabulated in Ref. 6. In the low-magnetic-field limit, the spectrum of the F = 4- 3 transition is composed of eight lines with strengths, R ( i ) , given by where i = - 3 to +4. The two outermost lines of the multiplet involve only a single transition ( m = 4- 3 and m = - 4 to - 3, respectively) while the other lines are each the sum of a A m = 1 and a A m = - 1 transition. This spectrum (identical to the F = 3 4 spectrum) is shown in Fig. 3, where the transition rate for each line is the sum of spectra (a) and (b). As we discussed in Ref. 6, the magnetic-field-induced mixing of hyperfine states is small and has no effect on the experiment, although it does cause the slight asymmetry seen in Fig. 3 (c). From Fig. 3 and Eqs. (2) and (3) it is now easy to understand the essence of the experiment. The laser is set to one of the end lines of the multiplet and, by reversing various fields, we change the sign of the PNC interference term without affecting the larger pure Stark-induced rate. Thus we isolate the PNC
term by observing the modulation in the transition rate with reversals of the E field, the B field, and the sign of cX (handedness of polarization). An additional reversal (“m” reversal) of the interference term is achieved by a change of the laser frequency to the other end of the multiplet. The use of four independent reversals is extremely helpful in the suppression of possible systematic errors. The experimental setup is similar to that used in
+
+
FIG. 3. 6&-47SF-3 transition. (a) Theoretical pure Stark-induced spectrum. (b) Theoretical parity-nonconserving interference spectrum on expanded scale. (c) Experimental scan of the transition with B = 70 G .
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53 VOLUME55, NUMBER 24
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Ref. 6. About 500 mW of laser light is produced by a ring dye laser at the 540-nm transition frequency. Servocontrol systems provide a high degree of frequency and intensity stabilization. A Pockels cell with k X/4 voltage applied selects the handedness of the laser polarization. Following the Pockels cell the laser beam enters a Fabry-Perot power-buildup cavity, the length of which is controlled to keep it in resonance with the dye-laser frequency. This produces a standing wave in the cavity with a field which is nearly 20 times larger than the incident field. An intense, well-collimated cesium beam intersects this standing wave at right angles in a line 2.7 cm long. The atomic beam is produced in a two-temperature oven with a capillary-array nozzle followed by a multislit collimator. At the intersection the cesium flux is 1x 10" atoms cm-2 s-'. The interaction region is shown in Fig. 2. Situated 2 mm above and below the line of intersection are flat glass plates with transparent electrically conductive coatings. The dc electric field is produced by application of positive or negative voltage to the top plate and grounding of the lower. Data were taken with use of values of the electric field ranging from 2.0 to 3.2 kV/cm. A 70-G magnetic field parallel to the atomic beam is produced by Helmholtz coils. 7s transition rate is monitored by obserThe 6 s vation of the light which is produced by the ~ P , J ~ , ~6sJ branch ~ - + of the 7s decay. This light is detected by a cooled silicon photodiode below the lower field plate. A gold cylindrical mirror above the top field plate images the interaction region onto the detector. Colored-glass filters in front of the detector block the scattered green laser light. The detector is carefully shielded to reduce electrical pickup, and is quite insensitive to the magnetic field. The output of the photodiode is amplified and sent into a gated integrator controlled by a PDP-11/23 computer. This computer also controls the B, E, and P (polarization) reversals. The reversal rates are 0.02, 0.2, and 2 Hz, respectively, with regular 180" phase shifts introduced in the switching cycles. After a brief dead time to avoid transient effects, the detector current is integrated, digitized, and stored for each half cycle of P. The m reversal is done manually every 30 min. A data run consisted of about 8 h of data accumula3 and tion divided equally between the F -4F = 3 4 transitions. Systematic error checks (discussed below) were made at the beginning and end of each run. Typical experimental conditions were E =2500(15) V/cm and ex/rz==0.94(l), giving a detector current of 3 x lo-'' A and a parityFor the data nonconserving fraction of 1.3 x used here the noise was 2 to 3 times worse than the statistical shot-noise limit, primarily because of noise from laser-light-induced fluorescence in the optics. +
This resulted in an integration time of 20 to 30 min for a ? 100% measurement of the PNC contribution. The data were analyzed by finding the fraction of the , modulated with P,E, B, and transition rate, A p ~ c that m. From Eq. (2) it can be seen that ApNC = 2 ( e x / e , ) (Im %'&E,6). A small calibration correction, 5.0(5)%, was made to account for the incomplete resolution of the lines in the multiplet. Systematic errors, namely, contributions to the signal which mimic the parity nonconservation under all reversals, were a fundamental concern in the design and execution of the experiment. Our approach to the identification and measurement of these contributions was similar to that used in earlier Stark-interference The transition rate was derived for the general case, allowing for all possible componenfs of E, B, Q , and the oscillating magnetic field, r x k . Each of these components was given a reversing and a nonreversing (stray) part. With use of empirically determined limits, all terms which could contribute false signals amounting to greater than 1% of the true PNC were then identified and a set of auxiliary expefiments was designed to measure them. These terms and most of the auxiliary experiments have direct counterparts in the work discussed in Refs. 3-5. One test, however, which is unique to this experiment is the in situ measurement of the birefringence of the buildup-cavity mirror coatings. Table I shows the results from a typical data run along with all the significant corrections due to false PNC signals. The average of all the data runs has a systematic correction of 14(1)%. As a result of space limitations a detailed discussion of these correction terms will be given in a subsequent paper. We made a number of other tests to confirm that there are no additional sources of systematic error. Among these were the introduction of known nonreversing fields, misalignments, and birefringences. All of these produced false PNC signals which agreed with the sizes predicted by the calculation discussed TABLE I. Raw data and corrections to a typical run. AEz and AE, are stray electric fields. 6 represents the birefringence of the buildup-cavity mirror coatings and M is the M1 matrix element.
-
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9 DECEMBER 1985
Fractional modulation ( X lo6) A.PN= (raw data) 4-3 3-4
Corrections A E, EJ ( E , )~ A EyB,/ (E X 5 ,) Mt'/ (PEx)
- 1.82(40)
- 1.49(45) -0.02(1) +0.23(4) O.Ol(2)
-
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VOLUME55, NUMBER 24
P H Y S I C A L R E V I E W LETTERS
above. Also, analysis of the data showed that on all time scales, from minutes to days, the distribution of values for I m I PNC/Pwas completely statistical. This included data taken with two different sets of buildupcavity mirrors, different electric-field plates, several complete realignments of the experiment, and different dc electric fields. Our results are
I
- 1.51 k0.18 mV/cm I m O PNC/P= - 1.80 -t 0.19 mV/cm - 1.65 k 0.13 mV/cm
-
(F=4(F=3
31, 41,
(average),
where the uncertainty includes all sources of error. This is in good agreement with the value of - 1.56 rt 0.17 k 0.12 mV/cm reported by Bouchiat el a/.’ for the average of measurements made o n the 4 - 4 and 3 - 4 hyperfine lines. With p - 2 7 . 3 ( 4 ) a d (see Ref. 91, we have
9 DECEMBER 1985
Because the uncertainty is almost entirely statistical we believe that we can achieve significantly higher precision with future refinements to this experiment. The most immediate gain will come from improved optics which will increase the signal size while decreasing the noise. A more substantial change is the use of a spin-polarized atomic beam. This would provide an increase in signal and allow a number of interesting options on the experimental design. W e have developed a polarized cesium beam of the necessary purity and intensity’‘ and will be exploring these options in the future. W e are pleased to acknowledge the donation of equipment by Jens Zorn and valuable discussions with Dr. J. Ward, Dr. A. Gallagher, Dr. J. Sapirstein, and Dr. W.Johnson. This work was supported by the National Science Foundation. One of us (C.E.W) is an A. P. Sloan Foundation fellow.
x 10”eao. I m g P P N C -0.88(7) =
Using a calculated value for the atomic matrix element we can compare our measurement with the predictions of the standard model for the weak charge, Q,. Since a discussion of the atomic-physics calculation is beyond the scope of this paper, we will simply take the range of reasonable values to be g P ~ ~ (0.85 = = to 0.97) x 10-”ieao(Qw/N) and refer the reader elsewhere” for a review of this subject. When combined with our measured I PNC we obtain - Qw in the range (71 to 81) f 6, where 6 is the experimental uncertainty. This is in agreement with the value of 70 k 4 predicted by the standard model. This agreement provides an improved test of the electroweak theory at low energies and has several specific implications as discussed by Robinett and Rosner” and Bouchiat and Piketty.12 These include improved limits on the radiative corrections to the electroweak theory and on the masses and couplings for additional neutral bosons. From the comparison of the PNC measurements made on the two hyperfine lines13 we find that the proton-axial-vector-electron-vector coupling constant is C,= - 2 ? 2. This is in agreement with the predicted value of 0.1 and is a substantial improvement over the previous experimental limit of IC2,,l < 100 (Ref. 5 ) . To obtain a more precise value for Qw from future experiments the uncertainty in the atomic theory must be reduced. There are presently a number of groups working on this problem and improvements may well be forthcoming in the near future. It is worth noting that this challenge is leading to new ideas and insights into atomic-structure calculations.
(a)Also at Department of Physics, University of Colorado, Boulder, Colo. 80309. ‘M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 35, 899 (1974), and 36,493 (1975). 2A review of experiments in this field is provided by E. N. Fortson and L. L. Lewis, Phys. Rep. 113. 289 (1984). 3P. H. Bucksbaum, E. D. Commins, and L. R. Hunter, Phys. Rev. D 24, 1134 (1981). 4P. S. Drell and E. D. Commins, Phys. Rev. Lett. 53, 968 (1984). 5M. A. Bouchiat, J. Guena, L. Hunter, and L. Pottier, Phys. Lett. 117B, 358 (19821, and 134B, 463 (1984). %. L. Gilbert, R. N. Watts, and C. E. Wieman, Phys. Rev. A 29, 137 (1984). ’R. R. Lewis and W. L. Williams, Phys. Lett. B 59, 70 (1975); M. A. Bouchiat, M. Poirer, and C. Bouchiat, J . Phys. (Paris) 40, 1127 (1979); P. H . Bucksbaum, in Proceedings of the Workshop on Parity Violation in Atoms, Cargbse, Corsica, 1979 (unpublished). 83. L. Gilbert, Ph.D. thesis, University of Michigan, 1984 (unpublished); also see Ref. 7 for a similar derivation. 9This is a semiempirical value of p obtained with use of the approach discussed in Ref. 6 and references therein. The value given here utilizes updated experimental and theoretical inputs which will be discussed in a subsequent paper. low. R. Johnson, D. S. Guo, M. Idrees, and J . Sapirstein, Phys. Rev. A 32, 2093 ( 1 9 8 9 , and a subsequent paper including first-order corrections (to be published). IIR. Robinett and J. Rosner, Phys. Rev. D 25, 3036
(1982).
12C.Bouchiat and C. Piketty, Phys. Lett. 128B, 73 (1983). 13V.N. Novikov et u / . , Zh. Eksp. Teor. Fiz. 73, 802 (1977) [Sov. Phys. JETP 46, 420 (1977)l. 14R.N. Watts and C. E. Wieman, to be published.
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Volume 57, number 1
OPTICS COMMUNICATIONS
1 February 1986
THE PRODUCTION OF A HIGHLY POLARIZED ATOMIC CESIUM BEAM R.N. WATTS and C.E. WIEMAN
'
Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, CO 80309, USA Received 23 October 1985
We have produced a cesium beam of l o i 4 atoms s - ' with more than 98% of the atoms in a single spin state. The beam was polarized by optically pumping it with a single laser diode whose frequency was switched between the 6s( F = 4)-6p,,,( F = 5) and 6 s ( F = 3)-6p3,,(F= 4) transitions at 100 kHz. To d o this, it was necessary to determine the frequency modulation characteristics of the laser. The same laser was used to simultaneously probe the atomic population distribution. We find this to be a remarkably simple and inexpensive way to produce a highly spin polarized beam.
There has long been an interest in producing spinpolarized beams of alkali atoms. Because such beams are in a single well-defined quantum state, they are invaluable tools for studying atomic collisions with other atoms, electrons, or surfaces [I]. In addition, they are useful in a variety of spectroscopic measurements. For example, we plan to use a spin polarized beam in an improved measurement of atomic parity violation [ 2 ] ,Finally, by transferring the electron polarization of one species to the nucleus of another, spin polarized atoms are used to produce polarized nuclei for targets and ion sources [3]. In this paper, we present a new method to spin polarize a beam using optical pumping. We cause the output frequency of a single diode laser to jump rapidly between two different 6 s - 6 ~ ~ 1hyperfine 2 transitions in cesium. These transitions are chosen to deplete one ground state hyperfine level and to polarize the other. We find that, using this technique, we can put essentially 100% of the beam into the 6s(F= 4,Mf = 4) spin state. Polarized beams were originally produced by state selection with an inhomogeneous magnetic field. However, this technique severely limited both the beam intensity and the degree of polarization. The invention
'
of tunable dye lasers dramatically improved this situa. tion by making it possible to optically pump atomic beams. In optical pumping, an atom repeatedly absorbs circularly polarized light and decays spontaneously. After several such cycles the atom is spin polarized along the direction of the laser. The difficulty in this technique lies in the large ground state hyperfine splitting present in all alkalis compared to the narrow linewidth of the pumping laser. Generally, only one of the two hyperfine levels can be in resonance with the laser and so only those atoms will be polarized. Several clever schemes have been devised for overcoming this problem. In one method, an atomic beam is state selected with a hexapole magnet to remove the atoms in one hyperfine level. The atoms in the remaining level are then polarized [4]. For small hyperfine splittings, acoustooptic frequency modulators [5,6] or a broad band multimode laser [6] have been used to pump both hyperfine levels. A fourth approach has been to mix the two hyperfine levels by applying large microwave fields [7]. All of these methods are relatively complicated, In addition, the best polarizations achieved have ranged from 70 to 90%. Diode lasers have some obvious advantages for spin polarization. First, they are far less expensive than the dye lasers which have been used previously. In addition, they are much easier to use, being simple to operate and virtually maintenance free. Finally, they pos-
Also Department of Physics, University of Colorado, Boulder, CO 80309, USA. Sloan Foundation Fellow.
0 030-4018/86/$03.50 0Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 55
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1 February 1986
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sess remarkable frequency modulation characteristics. We have found that it is possible to sweep the frequency of a diode laser by as much as 15 GHz within a microsecond. This makes them uniquely suited to polarizing an atomic beam: by jumping quickly between two transitions, one laser can pump both hyperfme levels. As an added bonus, these modulation characteristics make it possible to use the pumping laser to simultaneously determine the degree of beam polarization, In our experiment, we polarize cesium into the 6s(F = 4,Mf = 4) spin state shown in fig. la. This is done in two steps. First, the circularly polarized laser is tuned to the 6s(F = 3)-6p3p(F = 4) transition. This excited state decays to both the F = 3 and F = 4 ground state levels, so after a few transitions per atom the F = 3 population is completely pumped into the F = 4 state. The laser frequency is then tuned to the 6s(F = 4)-6p3p(F = 5) transition. This excited state can only decay to the F = 4 ground state and so after approximately ten absorptions and emissions the atom is completely polarized. For this process to work the transit time of the atom through the pumping region must be longer than the period of the frequency switching cycle, and must be much longer than the inverse of the transition rate, We find that diode lasers can easily meet both these criteria. To determine the resulting polarization of the cesium beam we measure both the depletion of the F = 3 state and the polarization of the F = 4 state. We make both of these measurements using a small fraction of the beam from the pumping laser. The F = 3 state de-
b’
pletion is obtained by measuring the absorption (or lack thereof) by the optically pumped atomic beam during the time the laser is tuned to the F = 3-4 transition. The F = 4 state polarization is derived from the measurement of the atomic beam circular dichroism (difference in the absorption of right versus left circularly polarized light) while the laser is in resonance with the F = 4-5 transition. Having this information and knowing the matrix elements that connect the various Zeeman levels, we can determine the degree of beam polarization. The success of our technique depends critically on the fast frequency response of a diode laser. Because diode lasers are routinely used in the communications industry, their intensity modulation characteristics have been well studied into the Gbit s-1 regime, In contrast, relatively little work has been done on their frequency modulation characteristics, especially in the 10 kHz to 1 MHz region. It is known that the lasing wavelength of a diode laser depends on the index of refraction of the active region which, in turn, depends on the diode temperature. Changes in injection current affect the temperature of the lasing junction [8] and thus change the wavelength. We have investigated the speed at which this happens using a Hitachi HLP1400 laser and the setup shown in fig. 2. The injection current to the laser consisted of two parts: a dc current of about 100 mA and a 5 kHz square wave. We measured the time delay between the edge of the square wave and when the laser frequency matched the resonance of the Fabry-Perot interferometer. By changing the interferometer resonant frequency by known amounts and determining the time delay for each change, we thus produced a graph of laser fre-
--
- 5 -4
6201 MHz
3 2
LASER
V V POLARIZER
PD
85218
I
_I -2- 3
4
MF
Fig. 1. Cesium energy level diagram showing the transitions used to spin polarize the beam. The u’s give the relative tranxition probabilities out of the F = 4, Mf= 4 level.
46
Fig. 2. Schematic of apparatus used to determine diode laser frequency modulation response.
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OPTICS COM?vIUNICATIONS
I
3 k
'0
10
20
30
40
50 60 t (psec)
70
80
90
100
Fig. 3. Frequency shift versus time for a 5.6 mA injection current step.
1 February 1986
changes could be made even faster. It is somewhat remarkable that the laser frequency settles to 1 part in lo3 of such a large jump after only 1.5 ps. We also note that this frequency change is accompanied by about a 15% change in intensity. This has no effect on the pumping process, however, since for the laser power used, both transitions are well saturated. A schematic for the optical pumping experiment is shown in fig. 4. The cesium bean! is produced in a twotemperature oven heated to about 100°C. The atomic beam is collimated to 25 mrad divergence by a glass capillary array nozzle on the oven followed by a multislit collimator [ 9 ] .The beam typically has an intensity of 1014 s-l and a density of about 1010/cm3 as determined by a tungsten hot wire detector. The laser is mounted on a copper heat sink and is temperature stabilized to a few mK using a thermoelectric cooler The dc current is stabilized to 1 part in lo6 and added to it is the 100 kHz frequency switching component discussed above. The laser has a bandwidth of 30 MHz
*.
quency against time as shown in fig. 3. For times between 0.1 and 100 ps, we found that the curve of frequency versus time could be fitted as the sum of two exponential response curves with time constants of about 2 ps and 17 ps. We were subsequently able t o determine the shorter time constant somewhat more accurately by looking at the response of the laser t o a 0.2 ps current spike. The time constants were found to vary only slightly when the amplitude of the square wave was changed from 1 to 6 mA. It is interesting to note that these results are consistent with those given in ref. [8] for a different laser and larger modulations. This reference explains the shorter time constant as the time required for the chip to reach internal thermal equilibrium. The larger time constant is the time for the chip to come into equilibrium with the heat sink on which it is mounted. Knowing the response function of the laser, we could then make the laser jump rapidly between the two transitions used to polarize our beam. For slow (less than 10 kHz) frequency switching one simply adds a square wave component to the injection current. We need to switch at 100 kHz or faster, however, and hence must compensate for the exponential character of the frequency change. This was done by sending the square wave through two RC highpass filters. We found that by using time constants of 1.5ps and 32 ps, and by adjusting the relative amplitudes of the two filters appropriately, we could force the laser to jump the necessary 8.2 GHz in 1.5 ps and then change its frequency by less than 10 MHz over the remaining 3.5 /JS of a 100 kHz half cycle. Smaller frequency
* These lasers were wavelength selected to lase at 852 * 5 nm at 25°C and 10 mW output power. This is necessary to insure that they can be tuned to the cesium transition but does not affect the& modulation characteristics.
BEAM SPLITTER
PROBE BEAM
-1T b
P O L A R I ZERS
X/4
PLATES
REMOVABLE
ABSORPTION DETECTOR
t
X I 2 PLATE
FLUORESCENCE D E T E CTO R
Fig. 4. Schematic of apparatus used to spin polarize and probe the cesium beam.
47
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Volume 57, number 1
OPTICS COMMUNICATIONS
and a long-term drift of no more than 200 MHz in 5 hours. The output of the laser is collimated to a 0.5 cm diameter beam, then split into a saturating pump beam (the transition saturates at 1 mW/cm2) and a weak probe beam. The light is circularly polarized using a linear polarizer and a quarter wave plate. In addition, a half wave plate can be inserted into either the pump or the probe beams to reverse their respective polarizations. The power in the pump beam is about 5 mW at the interaction region while the probe beam has an intensity of about 0.1 mW. The laser beams intersect the cesium beam at right angles. Detectors monitor the absorption of the probe beam and the fluorescence of the pump beam. From the fluorescence signal we obtain an error signal which we feed back to the injection current to lock the center laser frequency to the 10 MHz wide cesium transition. The quantization axis is defined by a 1 G magnetic field directed along the laser beams, An atom with the most probable velocity spends about 20 ps in the pumping beam which is only two full periods of the laser switching cycle. However this is time for nearly 350 excitations and spontaneous decays. Thus an atom must only spend slightly more than one half cycle (5 ps) in the pump region to be polarized. This explains why even the atoms in the high velocity tail of the Doppler distribution are sufficiently pumped. To confirm this we modulated the laser at 200 kHz and obtained the same polarization. As mentioned previously the atomic beam polarization is obtained by monitoring the absorption of the probe laser beam. While the laser is tuned to the F = 3-4 trmsition we measure an absorption of 2.6% when the pump beam is blocked and an absorption of 0 0.1%when the pump is unblocked. This implies a 100 f 5% depletion of the F = 3 hyperfine level. From the calculated transition rates and interaction times we estimate that this level is at least 99% depleted, which agrees with the measured value. Next we measure the difference in the absorption of right and left circularly polarized light while the laser is tuned to the F = 4-5 transition. We find the ratio of the different absorptions is 33 f 5. From the ClebschGordan coefficients shown in fig. l b , this ratio would be 45 for an atom in the P = 4,Mf = 4 ground state. Thus our result can be interpreted in one of two ways (or some linear combination of the two). First, all the atoms are in the Mf = 4 state and the polarization of the probe light is imperfect. An addition of 0.8% of
*
48
1 February 1986
the wrong circular polarization would give the ratio observed. Given the quality of the optics used, we think that this is a very likely possibility. Alternatively, in the unlikely event that the laser polarization is perfect, some fraction of the population must be in the other Mf levels. Although the fraction implied by the results is quite dependent on the distribution among the Mf levels, if we assume that it is distributed uniformly, the measured ratio implies that theMf = 4 level contains 98% of all the atoms. Because the laser linewidth is twice the residual Doppler width, the entire atomic beam is being sampled by the probe laser beam. Thus we believe that essentially all the atoms are in the F = 4,Mf = 4 state. Simply reversing the laser polarization, of course, puts the atoms in the M f = -4 state. We have produced an intense, very highly polarized beam of atomic cesium in a simple, inexpensive, and compact manner. This technique will make possible an improved measurement of parity violation in cesium and has many other applications. With presently available diode lasers, this technique can easily be applied to rubidium and potassium as well, We would like to thank Dana Anderson, Alan Gallagher and Linden Lewis for helpful discussions. This work is supported by the National Science Foundation.
References [l] H. Kleinpoppen, Adv. At. Mol. Phys. 15 (1979) 423, and references therein. [2] S.L. Gilbert, M.C. Noecker, R.N. Watts and C.E. Wieman, to be published. [3] F.P. Calaprice, W. Happer, D.F. Schreiber, M.M. Lowry, E. Miron and X. Zeng, Phys. Rev. Lett. 54 (1985) 174. [4] D. Hih, W. Jitschin and H. Kleinpoppen, Appl. Phys. 25 (1981) 39; W. Dreves, W. Kamke, W. Broermann and D. Fick, Z. Phys. A303 (1981) 203. [ 5 ] G . Baum, C.D. Caldwell and W. Schroder, Appl. Phys. 21 (1980) 121. [6] J.T.Gusma and L.W. Anderson, Phys. Rev. A28 (1983) 1195. [7] W. Dreves, H. Jansch, E. Koch and D. Fick, Phys. Rev. Lett. 50 (1983) 1759. [8] P. Melman and W.J. Carlsen, Appl. Optics 20 (1981) 2694. [9] S.L. Gilbert,M.C. Noecker andC.E. Wieman, Phys. Rev. A29 (1984) 3150.
PHYSICAL REVIEW A
VOLUME 34, NUMBER 2
AUGUST 1986
Atomic-beam measurement of parity nonconservation in cesium S. L.Gilbert* and C. E. Wieman Joint Institute f o r Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Campus Box 440, Boulder, Colorado 80309-0440 and Physics Department, Uniuersity of Colorado, Boulder, Colorado 80309-0440 (Received 4 April 1986) We present a new measurement of parity nonconservation in cesium. In this experiment, a laser excited the 6s-7s transition in an atomic beam in a region of static electric and magnetic fields. The quantity measured was the component of the transition rate arising from the interference between the parity nonconserving amplitude, LfP,,,, and the Stark amplitude, P E . Our results where C2p is the proton-axialare Im%",,,/~= - 1.65+0.13 mV/cm and C2,= - 2 i 2 , vector-electron-vector neutral-current coupling constant. These results are in agreement with previous less precise measurements in cesium and with the predictions of the electroweak standard model. We give a detailed discussion of the experiment with particular emphasis on the treatment and elimination of systematic errors. This experimental technique will allow future measurements of significantly higher precision.
a wealth of precise experimental data on the various properties of cesium ground and excited states which can be used for testing and refining calculations of its wave functions. In a previous paper6 we briefly presented the results of our first measurement using this new technique. Although we expect considerable future improvement, this measurement is already more precise than previous measurements of atomic PNC and is approaching the pre-
I. INTRODUCTION
The standard model of electroweak unification has stimulated considerable interest in atomic parity nonconservation (PNC) over the last decade. This theory predicted a PNC neutral-current interaction between electrons and nucleons which would mix the parity eigenstates of an atom. Although the standard model has now been tested with moderate precision in a variety of experiments using high-energy accelerators, atomic P N C data can provide unique and complementary information about this interaction. This is because the atomic case probes a very different energy scale and is sensitive to a different set of electron-quark coupling constants. Thus precise atomic data would allow one to measure the radiative corrections' to the electroweak theory and to explore the possible alternatives to the standard model over a larger parameter space. In pursuit of this goal, measurements of panty nonconservation have now been carried out on bismuth,* lead,3 thallium> and cesium.536 The approximately Z3 dependence of the PNC mixing is the reason for the emphasis on high-Z atoms. Aside from some ambiguity in the early bismuth results, all of the data are now in agreement with the predictions of the standard model. While this work has provided significant new information, its importance has been limited by two factors. The first has been the level of precision of the experimental results, and the second is the difficulty in relating the observations to the fundamental electron-nucleon interaction because of the complexities of the atomic structure. To overcome these limitations we have developed a new experimental technique which will allow precise measurements on cesium. Cesium has the virtue that it is the simplest heavy atom, having one S-state electron outside a filled inner core. Thus it is highly single-electron in character and calculations of its structure are more direct and accurate than for other heavy atoms. In addition, there is
1.36~ 1.47U
FIG. 1. Cesium-energy-level diagram showing hyperfine and weak-field Zeeman structure of the 6 s and 7s states.
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59
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@ 1986 The American Physical Society
60 34 -
ATOMIC-BEAM MEASUREMENT OF PAPHTY . . .
cision of the best high-energy tests of the electroweak theory. Here we give a detailed discussion of this experiment with particular emphasis on the critical issue of the treatment and elimination of systematic errors. The PNC interaction in an atom mixes S and P eigenstates, allowing a small electric dipole ( E 1 ) transition amplitude between states of the same parity. Similar to all atomic PNC experiments, we measure this small parity nonconserving amplitude ( A,,, by observing its interference with a larger parity conserving amplitude. In our experiment, the parity conserving amplitude is a "Stark induced" E 1 amplitude ( AST) created by applying a dc electric field to mix S and P states. The use of this amplitude was first suggested by the Bouchiats' and it was used in the cell experiments discussed in Refs. 4 and 5 . In our experiment we use a laser to excite the transition of interest in an atomic beam. The use of an atomic beam nearly eliminates the Doppler broadening and hence we obtain very narrow transition lindwidths. This enables us to observe the PNC interference directly in the transition rate by applying a small magnetic field (70 G) to break the degeneracy of the Zeeman levek8 Other advantages of an atomic-beam experiment include the reduction of collisions, radiation trapping, and molecular backgrounds. A final important feature of our approach is that the transitions can be detected with high efficiency. In this paper we will first discuss the theory of the experiment in Sec. 11. In Sec. I11 we will discuss the apparatus and experimental procedure and in Sec. IV we cover the treatment of systematic errors. In Sec. V we
793
present the results and in Sec. VI discuss the future improvements to the experiment. 11. THEORY
In this section we present the basic theory needed to understand and interpret the experiment. We are interested in the excitation of the 6s state of cesium to the 7 S state by a resonant electromagnetic field in the presence of static electric and magnetic fields. This problem has been previously discussed for the case of large magnetic fields and broad transition linewidths.' Here we consider the case of narrow linewidths and a weak magnetic field which has a relatively simple analytic solution. There are three relevant transition amplitudes which can give rise to this excitation; a Stark induced electric dipole, a magnetic dipole, and the PNC electric dipole. We will first consider the total transition amplitude between a particular ground state (6S,F,m)and excited state (7S,F',m')level, where m ( m ' ) is the z component of the total angular momentum F (F').This derivation applies to the general case of arbitrary dc electric and laser field orientations. Combining this result for the amplitude with knowledge of how the Zeeman levels shift in a weak magnetic field, we will then derive the transition spectrum for the particular field configuration we have chosen. Figure 1 is the cesium-energy-level diagram for the transitions of interest in this experiment. With a static electric field, E, and an oscillating (laser) electric field, E, the Stark-induced transition amplitude between the perturbed 6S(F,m)and 7 S ( F ' , m ' )states is
AST(F,m;F',m')=(7S,F',m'1 -d.e 16S,F,m )
where d is the electric dipole operator. The coefficients CFA"' are proportional to the usual Clebsch-Gordan coefficients and are tabulated in Appendix A. The quantities a and P are the scalar and vector transition polarizabilities respectively and are given by7
Similarly, the magnetic dipole transition amplitude between these states is h
AMI ( F ,m ;F',m ') = f ( k x E ),S,,,,
+ [ L ( k X +i ( k X h
h
E )x
E ) y lSm,rn'? I
] MC;Am'
(3)
where k is the laser propagation vector. M is the highly forbidden magnetic dipole ( M 1 ) matrix element defined as
M = ( 7 S Ip,/c 1 6 s ) ,
(4)
where p, is the z component of the magnetic dipole operator. The panty nonconserving potential, VpN,-, mixes S and P states and gives rise to a transition amplitude given by
61 S. L. GILBERT AND C. E. WIEMAN
794
Combining all three transition amplitudes, the transition probability between particular sublevels (rn and m') is then
I = I AST+AMl+APNC I
2
(7)
where each A is a function of F, F ' , m , and m ' . We chose the experimental design to maximize the interference between AS* and A p N C while minimizing the AsT-A,, interference and other effects which may mimic the P N C signal. It should be noted that the PNC interference term is dependent on m. If the rn levels are degenerate and equally populated this term will sum to zero in the total rate for the F+F' transition. A magnetic field is introduced to break the degeneracy of the m levels and hence avoid this cancellation. The field configuration used" is shown in Fig. 2. A standing wave laser beam along the ^y axis excites transitions in a region of dc electric field in the ^x direction and dc magnetic field ( B ) along the 2 axis. The laser field is elliptically polarized with E = E Z h Z + i E x 2 where E~ and E, are real. Using this field configuration and substituting the amplitudes from Eqs. (l), (31, and ( 5 ) into Eq. (71, we obtain for F#F',
plus negligibly small terms involving only g p and ~ M. The quantities ~ , k + and E:- represent the hz components of the laser field for the ^k=+? and k = - 9 laser beam propagation directions resuectivelv, and E,= 2 I E,k++E,k- I . The first term in Eq. (8)is the pure %ark-induced transition rate and the second term is the interference between AST and the much smaller amplitude A p N C . This corresponds to the pseudoscalar 6G.EX B _
1:
where 6 represents the handedness of the laser polarization. The third term is the interference between As, and A, For our standing-wave laser field, ~ , k = E: - , which leads to a cancellation of the As, - A , I interference. The problem of imperfect cancellation of this term and other effects due to imperfect E, E, and B fields will be addressed in Sec. IV. In the weak magnetic field limit, each Zeeman sublevel is shifted in frequency by an amount Av=mgFpBB,where g F x 4 =- & = 3 = for s states. Applying this to the field configuration of our experiment, we find the spectrum of the 6 S ( F =4)-t7S(F = 3 ) transition to be composed of eight lines with strengths, R (i), given by
,.
+
The two outermost lines of the rnultiplet involve only a single Zeeman transition ( m=4+3 and rn = - 4 - + - 3 , respectively) while the other lines are each the sum of a Am = + 1 and a Am = - 1 transition. The F =4+3 spectrum is shown in Fig. 3 where the transition rate for each line is the sum of contributions (a) and (b). In the weakfield limit, the spectrum for the F = 3-4 transition is identical. These spectra are modified slightly due to magnetic
~
I
A = 0.35MHzIG
17L
FIG.2. Field configuration for experiment.
FIG. 3. 6SF=4+7SF=, transition. (a) Theoretical pure Stark-induced spectrum. (b)Theoretical panty nonconserving interference spectrum on expanded scale. (c) Experimental scan of the transition with B=70 G.
62 34 -
ATOMIC-BEAM MEASUREMENT OF PARITY . . .
field induced mixing of hyperfine states. This causes the right-left asymmetry of the peak heights in Fig. 3(c), for example. However, the mixing is quite small and can be accurately calculated using first-order perturbation theory. We calculate that a 7 0 G magnetic field causes the extreme left- and right-hand peak heights of the F =4+3 transition to differ by about 5%. The asymmetry is smaller on the F = 3 + 4 transition by the ratio of the 7 s to 6 s hyperfine splittings (-0.25). The magnetic field induced mixing, however, is not important in our experiment because we only make measurements on the two outermost lines of the multiplet where the A s , and A p ~ c contributions are affected equally. Thus the ratio ApNC/As*, which is the quantity of interest, is independent of this mixing. 111. EXPERIMENT
A laser, tuned to one of the end lines of the multiplet shown in Fig. 3, excited the 6 S - 7 S transition in cesium. We monitored the transition rate by measuring the amount of 850- and 890-nm light emitted in the 6P,/2,3/2-+6S step of the 7 S - + 6 P + 6 S decay sequence. The essence of the experiment can be understood from Eq. (8) and Fig. 3. The parity nonconserving interference term in Eq. (8) changes sign under the reversal of the E field, the B field, and the sign of E, (handedness of laser polarization). This causes a slight change in the overall transition rate, and hence provides a means of isolating the PNC interference term from the much larger pure Stark-induced term. A n additional reversal ("m" reversal) was achieved by changing the laser frequency to the other end of the multiplet. We have used this technique to measure the ratio of the P N C amplitude to the Stark-induced 6S(F=4)-+7S(F=3) and amplitude for the 6 S ( F = 3 ) - 7 S ( F = 4 ) transitions. A. Apparatus
The basic experimental setup is shown in Fig. 4. Laser light at 540 nm was produced by a dye laser and the beam passed through several optical elements before entering a vacuum chamber. Inside the vacuum chamber the laser beam was coupled into a Fabry-Perot interferometer, referred to as the power-buildup cavity (PBC) in Fig. 4 . The PBC was maintained in resonance with the laser which resulted in a large standing wave field inside the cavity. This field induced transitions in a cesium beam in
FIG. 4. Schematic of apparatus. PC no. 1 and PC no. 2 are the intensity stabilization and polarization control Pockels cells, respectively. D1 is the transition detector (actually situated below the cesium beam) and 0 2 is the PBC transmission detector.
195
a region of static electric and magnetic fields. A silicon photodiode detected the fluorescence emitted by the decay of the excited state. In the following paragraphs, we will discuss each of the key elements of the apparatus. A Spectra-Physics Model No. 380 ring dye laser produced approximately 500 m W of light at 540 nrn. We found it necessary to reduce the frequency fluctuations of the laser. To accomplish this, a few percent of the laser output power was sent to a stable reference interferometer cavity and an error signal was derived from the cavity reflection using the Hansch-Couillaud method." The error signal was used to control the galvanometer driven plates and a piezoelectric transducer (PZT) mounted mirror in the dye-laser cavity. This reduced the laser linewidth to about 100 kHz. A Brewster angle galvanometer driven plate within the reference interferometer allowed cavity optical length adjustment. This in turn produced laser frequency tuning when the laser was locked to the reference cavity. As will be discussed below, the long-term stability of the reference cavity was insured by locking it to the cesium transition frequency. The main laser beam passed through many optical elements. The first element was an electro-optic modulator (EOM) which produced small 4-MH-z frequencymodulation (FM) sidebands on the laser. As discussed below, this was necessary for the scheme which we used to hold the power-buildup cavity in resonance. Following the electro-optic modulator, two lenses modematched the laser into the lowest-order spatial mode of the PBC. The next element was a Pockels cell which, in combination with a linear polarizer, enabled stabilization of the laser intensity with active feedback. Following the intensity stabilization Pockels cell, the laser passed through a Faraday rotation optical isolator which isolated the laser from reflected beams. At the output of the optical isolator, the laser light was linearly polarized at 45"with respect to the ^x axis defined in Fig. 2. The next component in the laser beam path was the polarization control element which set the ellipticity and handedness of the laser polarization. The polarization control element was made up of a h / 2 retardation plate followed by a longitudinal single-crystal [potassium dihydrogen phosphate (KDP)] Pockels cell. When 1.85 kV was applied, the Pockels cell produced a 1 / 4 (90") phase retardation on the x component of the laser field. We adjusted the ellipticity of the light by rotating the (linear) laser polarization at the input of the Pockels cell with the h / 2 plate. The handedness of the laser polarization was then changed by reversing the voltage applied to the Pockels cell (+h/4-+ -h/4 retardation). We reversed this voltage using a double-pole double-throw vacuum relay. The Pockels cell had a 1-cm aperture and provided quite uniform birefringence across the 0.05-cm-diameter laser beam. It was necessary to temperature stabilize the Pockels cell and isolate it from air currents to minimize variations in the resultant laser polarization. With these precautions, the birefringence of the Pockels cell was stable to a few parts in lo5 over the course of an 8-h data run. Following the polarization control element, the laser beam entered the vacuum chamber (pressure 3 X lo-' Torr) and was coupled into the power-buildup cavity.
63 S. L. GILBERT A N D C. E. WIEMAN
196
The PBC was a spherical mirror Fabry-Perot interferometer with a mirror separation of 23 cm. The flat, partially transmitting ( R=98.5% and T=1.3%) input mirror was mounted on a piezo-electric transducer. The second mirror had a reflectivity of 99.8% and a 50-cm radius of curvature. The power-buildup cavity was maintained in resonance with the laser using the FM sideband stabilization technique.12 We implemented this by using a fast photodiode to detect the laser beam which was reflected off the PBC. The output of the photodiode went to a phase-sensitive demodulator operating at the EOM 4MHz drive frequency. This produced an error signal which was then fed to the PZT mounted input mirror of the power-buildup cavity. When locked on resonance, the laser field within the power-buildup cavity was 20 times that of the incident laser beam. We monitored the power in the PBC by detecting the light which was transmitted through the second PBC mirror. The intensity at this detector was held constant to better than one part in lo5 per sec”’ by a feedback loop which controlled the intensity stabilization Pockels cell. We found that our signal-to-noise ratio was improved by about a factor of 2 by inserting a linear polarizer with its axis along the z direction in front of this detector. This means that we were only stabilizing the intensity of the field component ( E, ) which drove the pure Stark-induced term in Eq. (8). The standing-wave laser field in the PBC was crossed by an intense, collimated beam of cesium atoms. The cesium beam was produced in a two-stage oven where the nozzle region was maintained hotter than the main body to reduce the Cs2 dimer fraction. The exit of the oven was a Galileo Electro-optics glass capillary array of 10 p m by 0.05-cm channels with an area of 0.5 cmX2.5 cm. The cesium beam was further collimated in the y direction by passing through a stainless-steel multislit collimator. Several liquid-nitrogen cooled copper plates were placed upstream and downstream of the collimator to pump away cesium and thereby reduce the background cesium pressure. A small amount of background gas still remained, however, showing up as a broad pedestal amounting to a few percent of the transition signal size. At the intersection with the laser beam, the cesium intensity was lOI5 atomscm-2 s-’ with a full angle divergence of 0.04 rad in the y direction. The laser cesium beam interaction region is shown in more detail in Fig. 5. Not shown in the figure is a 30cm-diameter Helmholz pair which produced the 70-G magnetic field. The intersection of the beams was a cylinder 0.05 cm in diameter by 2.7-cm long. Two millimeters above and below this line were optically transparent, electrically conductive coated (InSn02) flat glass plates which had dimensions 2.5 cm x 5.0 cm. A dc electric field of k2.5 kV/cm was produced by applying a positive or negative voltage to the top plate and grounding the lower plate. As with the polarization (P)reversal, this voltage was reversed using a high-voltage double-pole double-throw vacuum relay. As will be discussed in Sec. IV, the plates were heated to avoid stray electric fields. We supplied f W to each plate by running ac (17 kHz) 100-R conductive coatings. This current through the
-
34 I N T E R FE R O M E T E R
MIRROR
‘LASER
\
,
BEAM
FIG. 5 . Detail of the interaction region.
current was sent through isolation transformers to reject any dc component and to isolate the heater supply from the high voltage. The 850- and 890-nm light from the decay of the 7 5 state was detected by a liquid-nitrogen cooled rectangular silicon photodiode (active area 0.5 X 5.6 cm) situated below the lower field plate. A gold-coated cylindrical mirror above the top plate imaged the interaction region onto this detector. Colored glass filters in front of the detector blocked the scattered green laser light while passing the infrared. The output of the photodiode went to a low-noise preamplifier. The detector-preamp combination had a frequency response of -400 Hz (fjde) with a noise equivalent power of 8x lo-” W/Hz”’. The signal from the preamp was sent to a gated integrator controlled by a Digital Equipment Corporation PDPll/23 computer which also stored the integrated data. A more detailed discussion of the data acquisition scheme will be given in the following section. An additional frequency stabilization loop was necessary to remove the effects of thermal drift of the reference cavity. To accomplish this we dithered the laser frequency at 330 Hz by feeding a sine wave to the galvonometer driven Brewster angle plate in the reference cavity. The -2-MHz amplitude of this dither gave rise to a slight modulation on the cesium transition signal. The transition signal, along with the 330-Hz reference, was sent to a lock-in amplifier which provided a very low-frequency correction signal for the reference cavity. We were careful to make sure that this modulation did not produce any signals at the parity reversal frequencies. B. Data acquisition and analysis A typical data run consisted of 8 h of data accumulation divided equally between the F=4-+3 and F = 3 - + 4 transitions. For each hyperfine transition, the laser frequency was locked to the extreme high- or low-frequency line of the multiplet shown in Fig. 3(c). A PDP11/23
64 34
ATOMIC-BEAM MEASUREMENT OF PARITY. . .
computer produced the 'ITL (transistor-transistor logic) signals which controlled the P, E, and B reversals. The reversal rates were 2, 0.2, and 0.02 Hz,respectively, with regular 180" phase shifts introduced in the switching cycles. The transition detector signal was integrated, digitized, and stored for each half cycle of P . Brief deadtimes after each field reversal were necessary to avoid transient effects. The deadtimes used were 25 ms for the P reversal and an entire P cycle (450 ms) for the E and B reversals. After 30 min of data acquisition, the laser was tuned to the other end of the hyperfine multiplet and data acquisition was continued. For normalization purposes, the average signal size was measured using a digital voltmeter and recorded for each data file. The integrator output was also calibrated using this voltmeter. Tests for systematic errors, discussed in the following section, were made before and after each 8 h run. We analyzed each data file to determine the fraction of the total signal which modulated with the P, E and B reversals. From Eq. (8) this fraction, disregarding systematic effects, is (101 The m reversal was implemented by subtracting the APNC for the low-frequency line from that obtained for the high-frequency line of the Zeeman multiplet. In our measurements E, /E, was close to l and E fields between 1750 and 3000 V/cm were used. With a typical voltage of 2500 V/cm we obtained a detector current of 3X A and a parity nonconserving fraction of 1.3 X
-
-
IV. CONTRIBUTIONS DUE TO SYSTEMATIC EFFECTS Systematic errors were of fundamental concern in the design and execution of this experiment. This led to the use of four independent reversals to identify the P N C signal. The quality of each of the reversals was better than one part in lo4. Thus, in principle, we would only require two reversals to cleanly resolve the PNC signal. The two extra reversals provide redundancy which greatly reduces the potential systematic error, since nearly all of the factors which affect the transition rate are at most correlated with only one reversal. The primary concern then becomes the small imperfections in the various field orientations and field reversals which can give signals that mimic the parity nonconserving signal under every reversal. As discussed below, we have identified and measured all such possible errors. Our approach to the identification of these contributions was similar to that used in earlier Stark interference experiments. Using Eqs. (1) and (3) we derived the transition rate for the general case, allowing for all possible components of E,B,E, and the oscillating magnetic field k X E. Each of these components was given a reversing and nonreversing part to characterize its behavior under the P, E, B, and m parity reversals. In this analysis, the 9 axis was taken to be along the laser beam. This means that E~ is absent by definition. The ^x axis of the system was defined to be along the component of the applied E field (i.e., the reversing E field) which was perpendicular to the 3 axis. With this definition, the
797
2 component of the reversing part of E ( E, is absent but there can be a nonreversing stray field part, AE,. These definitions give the following behavior for the E field upon reversal:
The P reversal ( E,
+
- E,
)
can be characterized as
where E, and E, are real. We considered all the combinations of these field components which contribute to the 6S-7S transition rate. Using rough empirical limits for these components, all terms which could be greater than 0.1% of the true PNC were identified. Effectively, this limit means that we needed only to consider terms which changed sign with all four reversals and involved no more than two components which were either stray or misaligned fields. The three terms which satisfy these criteria, along with their typical values, are listed in Table I. Counterparts of all of these terms were considered in earlier P N C experiments, as discussed in Ref. 13. The first term in the table arises from an electric field misalignment (E,) and a stray E field (AE,).The second term is due to the stray E field in the y direction (My ) and a misalignment ( B , ) of the magnetic field. The component B, causes a mixing of states within a particular hyperfine level, and, to a smaller extent mixing of different hyperfine levels with the same principal quantum number. We calculated the size of this mixing using first-order perturbation theory. The third term in Table I is due to the A s T - A w l interference shown in Eq. (8). As mentioned previously, this interference is suppressed by its change in sign under reversal of the laser propagation direction, ^k. For our standing-wave field, this suppression factor, relative to the P N C interference, is about lo3 (PBC output mirror R-99.8%). A second suppression comes from the fact that, though the M1 interference mimics the P N C interference under the E, B, and m reversals, it does not change sign under the P reversal. However, imperfections in the P reversal, such as those indicated in Eq. (121, can cause a significant amount of the M1 interference to leak
TABLE I. Terms which mimic the PNC signal, given as a ratio to the pure Stark induced transition rate of Eq.(8). Term
Average size relative to PNC term 0.01
0.04 0.17
65
34 -
S. L. GILBERT AND C. E. WIEMAN
798
-
through since M / ( I m P p N c ) lo4. Of particular concern is the birefringence in the power-buildup-cavity outputmirror coating as discussed in Ref. 14. This can give rise to an asymmetry between the fields of the counterpropagating laser beams which changes sign with the P reversal. There is an additional contribution to this term which involves the birefringence of the input mirror and other optics. However, these birefringences also cause a modulation in the transition signal size when the polarization is reversed which is about lo5 times larger than the PNC systematic. We observed this modulation and added birefringence to cancel it. With this cancellation the PNC systematic involving these birefringences is negligible leaving only that due to the output mirror. For a more detailed discussion of this point see Ref. 14 or Ref. 15. We have designed a set of auxiliary experiments to measure the fields which contribute to terms 1-3 in Table I. Our basic philosophy was that we should be able to monitor all the possible systematics while we were taking P N C data. T o do this we used the atoms themselves to measure the fields which give rise to the three terms in Table I. This was done without changing the basic experimental configuration. As can be seen below, the procedures used were somewhat excessive for the level of precision of the present measurement. However, much of this effort was preparation for the more precise measurements we plan to make using this technique. The following is a description of each of these auxiliary experiments. (a) AE, / E x measurement. Conditions: linearly polarized laser light E = E , R + E ~ ~ ; IB I =O. The F = 4 - + 4 Stark-induced transition rate from Eq. (1) is given by
I : = 9 a ’ ( E e ~ ) ~%p2 + I E X EI
(17)
This measurement was made simultaneously with measurement (b) and the stored data were analyzed to obtain E,, / E x and AE,, / E x . (d) B x / B z measurement. Conditions: same as in (b) but with B; =O. These were the same conditions as used in the P N C measurement. The fractional modulation of the transition rate with the P,B, and m reversals is (&E’)P,B,m
I:;’
= 2 - -E,, - - - -Bx . Ex 4
Ex
(18)
Ez
This measurement takes no additional time as it was derived from the raw parity nonconservation data. The desired quantity, B x / B z , was obtained from Eq. (18) since E y / E , was known from (b) and E , / E , was measured for each data run. (e) (6~,k+- & - ) measurement. Conditions: circular laser polarization; I B I =O; cesium-beam collimator tilted such that the Cs beam is no longer perpendicular to the laser beam. In this situation, it is simplest to think of the power-buildup-cavity laser field as being made up of two superimposed traveling waves with opposite propagation vectors, ^k = +9 and k - = -9. Due to the tilted collimator, a particular cesium transition is now split into two peaks; one corresponding to the Doppler-shifted resonance with ^k component of the laser field, and the second corresponding to the Doppler-shifted resonance with the ^k component. Using this method we were able to clearly resolve these peaks, as shown in Fig. 6. For the F =4-+3 transition, the transition rates for these two peaks are h
+
= 9 d E ; ~ t+; ~ E , E , E , E) ,
(13)
since a / p - 10 and Ex >>Ey,Ez. The modulation (amplitude) of this transition rate with the E reversal is [ h l : ] =9a2( ~ ExAE,E; + E x AE,E,E,)
The fields B, and B; were measured using a gaussmeter and again E, / E , = 1. With this information and the measurement represented by Eq. (16), we solved for E,, / E x . (c) A E y / E x measurement. Conditions: same as in (b). The measurement is identical to that outlined in (b) with the addition of the E reversal
.
(14)
The above measurement was made and the laser polanzation was then rotated to E’=E,S-E~~. This causes the second term in Q. (14) to change sign. The difference between these two measurements, divided by the overall transition rate yields
+
(19) ~ i
i l
ll
ll
l
ll
ll
l
I--
Knowing E, / E , = 1.O, we obtained AEz / E x . (b) Ey / E x measurement. Conditions: circularly polarized light ( E = E z 2 + i E x B ) ; B,=70 G an additional magnetic field in the 9 direction, B;=O.15B2. In this measurement we monitored the transition rate on the two outermost lines for both the F =4-3 and F =3+4 Zeeman rnultiplets. The effect of the additional magnetic field, B;, is the replacement of Bx with ( B , +Bx,) in term 2 of Table I. The fractional modulation of the transition rate with the P,B;, and m reversals is then (16)
LASER FREQUENCY C H A N G E ( M H z )
FIG. 6 . Scan of the 6SF=4-7S~=4 transition with the cesium-beam collimator tilted so that the ^k and ^k - peaks are resolved ( B = O G ) . +
66 -
~~
ATOMIC-BEAM MEASUREMENT OF PARITY . . .
34
The difference in fractional modulation with the P reversal for the two peaks is then
(20)
wherewehaveused I ~ , k + l = l ~ , k - l = ( l / f i ) I E , l . The expression in Eq. (20) is identical to the coefficient of M / ( P E , ) in term 3 of Table I. The ratio M / ( O E ) has been measured previous~y.’~ Measurements (a), (b), and (c) were made at the beginning and end of each data run to guard against the possibility of AE, and AE, changing with time. These measurements took about l h to complete and, when combined with measurement (d), resulted in an uncertainty for terms 1 and 2 in Table I which was typically an order of magnitude smaller than the statistical uncertainty in the PNC measurement. As discussed below, measurement (el was only made if the PBC output mirror had been moved or rotated. A number of additional tests were made to verify that these field imperfections gave the false signals predicted by Table I, both in magnitude and sign. In each test, a particular term was enhanced and the systematic tests (a)-(e) were made. Parity nonconservation data were then taken and the predictions of the systematic tests were compared with the measured false P N C signals. Nonreversing E fields AEy and AE, of 3 volts/cm (AEy,,/Ex were produced by applying dc voltages across each field plate. The large reversing Bx field ( B X / B , = 3 x lo-’) was produced with an external coil that was reversed with the B, coil. A mirror with a large coating birefringence was put into the PBC to enhance term 3 in Table I. For each of these tests, the prediction of the systematic tests agreed with the measured false P N C to within the uncertainties of the measurements ( 10-20 %). We have carried out extensive studies of the effects which give rise to the terms of Table I. These studies achieved two goals: they led to modifications in the apparatus which reduced the size of the necessary corrections, and reduced the time variation of the systematic errors. This latter point is by far the most important. The measurements described above allow us to measure all the corrections to a high degree of accuracy relative to the P N C rate in a short time. This means it is not particularly important how big these corrections are but it is crucial to know if they vary during the time we then spend taking P N C data. We found that the stray electric fields in particular could be highly time dependent and quite large if preventative measures were not taken. Especially troublesome was the fact that every material we tested tended to acquire stray electric fields after it was exposed to the cesium beam for some time. We tried many different kinds of field plates before arriving at the heated conductive coatings we presently use. Purely empirically we have determined that if these field plates are kept somewhat above room temperature (but not too far above) they have
-
799
very desirable characteristics. When first put into the apparatus they had some modest initial 9 and 2 stray fields, perhaps 0.25 V/cm. After brief exposure to the cesium beam the stray fields would drop to less than 0.05 V/cm; one to two parts in lo5 of the total applied E field. We saw only very slow subsequent drifts in these stray fields. The misaligned fields ( Ey and B, ) were quite stable as expected since all the components of the apparatus were rigidly mounted. The Ey field depends on the alignment of the electric field plates with the laser beam. We found that we could set this alignment to make Ey / E , = lop4. However, a nominal alignment of Ey / E x ~r lo-’ was used as a compromise between enhancing the signal for measurement (d) and minimizing term l in Table I. Magnetic field coils in the f , 9, and 2 directions were used to shim out stray (nonreversing) and misaligned (reversing) B fields. Although most of these fields do not produce false PNC signals, they can give rise to signal modulations with the P reversal which complicated the systematic tests. Thus we found i t worthwhile to eliminate them. The appropriate nonreversing shim field settings were determined by measuring these fields with a gaussmeter. These fields were reduced to about 10 mG. The reversing field shims were set by monitoring the signal modulation with the P, B, and m reversals. After this adjustment, B , / B , was typically 3 x lop3. The possibility of E and B field inhomogeneity along the 1-in line of intersection of the cesium and laser beams was also investigated. The measurements (a)-(d) are only sensitive to the average value of the quantities A E y / E x , AE,/ E x , E, / E x , and B, /B,. These average values were then combined to calculate terms 1 and 2 in Table I. This approach is not strictly correct, however, if both components which make up a single term have spatial inhomogeneity. We tested for this possibility by taking measurements (a)-(d) under the normal conditions and then repeated them with about half the length of the cesium beam blocked. As we expected, these data showed that there were indeed spatial variations in the stray E fields (AE, and AEz) of roughly 50%. However, Ey and B, were found to be homogeneous to better than 10%. This confirms that there is only one inhomogeneous field in each of the terms, and therefore our measurements are valid. The corrections we measure before and after each data run support the conclusion that the stray electric fields vary little with time. The second term in Table I was always found to be the same before and after to within the statistical uncertainty. The average size of this correction for a data run was 3.5% of the P N C with an uncertainty of about the same size. The first term varied by more than the statistical uncertainty on about half the runs. This was hardly grounds for concern, however, since the average value for this correction was 0.4% of the P N C and the typical statistical uncertainty was half of that. When there was variation, we used the average of the two corrections. The error bars were then taken to cover both values, which at worst differed by 0.4% of the PNC. Since the statistical uncertainty in the P N C measurement was about 20% for each data run it is clear that it was not really necessary to check the stray fields before and after
67 ~~~~
S. L. GILBERT AND C. E.WIEMAN
800
each run. Our previous experience, however, made us wary of relying on their constancy until a considerable amount of supporting evidence had been obtained. The only apparatus dependence to term 3 of Table I is the coating birefringence of the PBC output mirror, or to be more precise, the product of the coating birefringence times the angle between the birefringence axis and the x axis. Measurement (e) listed above is a very sensitive way to determine the actual correction due to this term. However, we found the following procedure was a simpler way to study the general characteristics of the birefringence. With the vacuum chamber up to air, the transmission through a linear polarizer following the PBC was monitored while the circular polarization of the laser was reversed. The modulation of this signal showed a periodic dependence with the rotation of the output mirror (PBC on resonance), due to the combination of the output mirror coating and substrate birefringences. The generally smaller contribution due to the substrate birefringence was determined by the same procedure but with the input mirror removed. Using this approach the axis and amount of the output-mirror coating birefringence could be determined. We investigated a number of mirrors from different manufacturers in this manner. In agreement with Ref. 14 we found that the coating birefringence was a general property which varied from mirror to mirror, but for a particular mirror it was largely the same across the entire surface. We did find that there were occasional, usually very local, regions where the birefringence could be significantly different, however. This is in contrast to the results reported in Ref. 14 but we believe this is because the method used in that reference was insensitive to local variations. Based on a limited number of test samples we now believe the birefringence is predominantly determined by the geometry of the coating facilities when the mirror is made. We saw no temporal variations in the coating birefringence. Before taking P N C data we rotated the output mirror to reduce term 3 in Table I. If the angle between the birefringence axis and the x axis is zero, this term vanishes. Because of the spatial variation in the coating and the substrate we could only set this angle to within a few degrees, however. The residual birefringence correction to the P N C data was then determined as described in measurement (e). Because earlier tests had shown there was no time variation to this correction term, measurement (e) was repeated only if the mirror was moved or the laser alignment was changed. We found that laser alignment had very little effect, but rotation of the mirror made a substantial difference, as shown below. For about one third of the data the correction due to 49(4)% of the PNC, for the second third it term 3 was was -58(4)%, and for the remainder it was -0.4(1.0)%. Although these corrections are relatively large, they can be accurately determined and hence are not a serious problem here. However, we have now obtained mirrors with far lower birefringence which will be used in future work. Another conceivable systematic effect we considered was one due to a dependence of the detector sensitivity on the direction of E and/or B. We tested for this effect by
+
34 -
monitoring the detector signal while reversing both E and B. A stable light level was provided by a light-emitting diode. We found that there was no change in detector sensitivity at the part in 10’ level. The existence of the other reversals makes any residual effect from this source negligible. We have also considered the effects of motional E and B fields which arise because the atoms are moving through magnetic and electric fields and we also find these to be negligible. We believe that there are no significant contributions that mimic the P N C signal which have not been taken into account. It should be noted that the uncertainty in determining these corrections is predominantly statistical. Thus improved signal to noise will not only reduce the statistical uncertainty on the P N C results but will also reduce the uncertainty in the corrections. The only remaining source of systematic error is in the calibration of the experiment. This calibration involves measuring the dc electric field, the ratio E, /E=, and determining the contributions to the observed detector current which are not represented in Eq. (9). Such contributions have often been called “dilutions” in earlier papers on this subject. The electric field was determined from the applied voltage and the measurement of the separation of the field plates. The 0.5% uncertainty to this calibration came entirely from the separation measurement. We have previously shown that more accurate field measurements can be made by observing the Stark shift of the cesium atoms, but that was unnecessary for this experiment. We by measuring the polarization of the determined L,/E, light which was transmitted by the power-buildup cavity using a linear polarizer and a photodiode. To measure the background signals we periodically set the static electric field to zero and observed the detector current. This was then subtracted off the signal observed with the E field on. The principal source of background was laser induced fluorescence of the optics which was typically 0.15 times the cesium signal. Though this background did not introduce any systematic uncertainty, it did increase the overall noise by about a factor of 2 and thus increased the statistical uncertainty in our results. The only additional background we observed came from cesium molecules and was about a factor of 5 smaller. We tested for any E field dependent background by tuning the laser frequency well off the transition and measuring the signal for E on and off. This set an upper limit of times the atomic transition signal for such background. A small calibration correction was needed because of the incomplete resolution of the lines in the &man multiplet, as seen in the experimental spectrum of Fig. 3. This correction was obtained in the following way. First, we scanned the laser to obtain the transition spectra for both I B I = O G and 1 B 1 =70 G. The 70 G spectrum, such as that shown in Fig. 3(c), was then fitted as the sum of eight individual lines where each line was assumed to have the 0 G line shape. From this fit we found the contribution of the overlapping lines and, using Eq. (9) we calculated the appropriate correction. This was done for each data run and the correction was typically 4% with negligible uncertainty.
68
ATOMIC-BEAM MEASUREMENT OF PARITY . . .
34
V. RESULTS AND CONCLUSIONS Ten data runs were made in the manner described in Sec. III. The signal-to-noise ratio was typically two or three times worse than that expected in the shot noise limited case. This extra noise was due primarily to the scattered-light-induced fluorescence background mentioned previously. This resulted in an integration time of 20—30 min for a 100% measurement of the PNC term. Our combined results for the ten data runs are -1.5110.18 mV/cm ( J F=4->3), -1.8010.19 mV/cm (F = 3~»4) , — 1.6510.13 mV/cm (average), where the quoted uncertainty includes all sources of error. As discussed earlier, the uncertainty is dominated by the purely statistical contribution. Our value is in good agreement with the value of —1.56±0.17±0.12 mV/cm reported by Bouchiat et al. for the average of measurements made on the F=4—»4 and 3—»4 hyperfine transitions.5 Using /3=27.3(4)do as discussed in Appendix B, we obtain Im^ P N C =-0.88(7)XlO-"ea 0 . To relate this to the weak charge Qw, or equivalently sin2^, it is necessary to know the value of the matrix element in Eq. (6). As we mentioned in Ref. 6 there is some uncertainty in the theoretical evaluation of this quantity. The most extensive calculation has been carried out by Druba et al.11 and their result of § > PNC =0.88(3) X \Q~lliea0(Qw/N) is probably the best value to take for this quantity and its uncertainty. However, a very conservative view would allow a range from 0.85 to 0.97 as can be seen in Ref. 18. It is likely that new results will be forthcoming in the near future which will clarify and hopefully improve this situation. Using the value of Ref. 17 our experimental results give &, =-78+6+3 for the cesium experiment. Where the first uncertainty is due to our experimental uncertainty and the second is due to the theoretical uncertainty. This is in good agreement with the standard model value'7 using sin2^ obtained from the mass of the W boson, e io =-71.0±l. 713.0 for the standard model prediction. The experimental value of Qw can also be used to obtain the weak mixing angle. Using the renormalized weak charges' for the proton and neutron this gives sin2^ =0.257+0.02810.014 for the cesium experiment. A comparison of the PNC measurements for the two hyperfine lines provides information on the nucleon spindependent coupling constants. Novikov et a/.19 have calculated the difference in the PNC between the F=4—>3 and 3—»4 lines using a shell model for the nucleus. They
801
find that the difference is the flip of one proton spin with an estimated uncertainty of 30%. Using this result and our measurements of the two hyperfine lines we find
C2p=~2±2 where C2p is the proton-axial-vector—electron-vector neutral-current coupling constant. This is in agreement with the predicted value of 0.1 and is a substantial improvement over the previous experimental limit of C 2 p <100(Ref. 5). The agreement between our measurements and the predictions of the standard model has implications for a variety of alternative models. It limits the possible values for masses of additional bosons and the strengths of coupling constants in superstring theories,20 supersymmetric theories,21 and others. However, a discussion of this topic is beyond the scope of this paper. VI. FUTURE We believe the experimental technique we have presented here is still in a rather youthful state and that future measurements will provide substantially higher precision. The systematic uncertainties do not appear to be a limitation until a precision of parts in 103 of the PNC is reached. Thus the primary question, which we are actively exploring, is how much the statistical uncertainty can be improved. It is already clear that significant improvements can be achieved with better optics which will provide less scattered light and higher standing-wave fields in the PBC. Another obvious improvement is the use of a spin-polarized cesium beam. Presently only -^ of all the cesium atoms are in the spin state which we exite. We have developed22 a diode laser optically pumped cesium beam which has essentially all the atoms in a single spin state and so will provide much larger PNC signals. With these improvements we believe that the PNC interaction in cesium will be measured to well under 1% as this technique matures. Precise measurements in rubidium will also be possible using the same approach. These data will be a major contribution to our understanding of the neutral-current interaction. ACKNOWLEDGMENTS We would like to acknowledge M. C. Noecker and R. N. Watts for their help in carrying out the experiment. This work was supported by the National Science Foundation. One of us (C.E.W.) acknowledges support by the Alfred P. Sloan Foundation. APPENDIX A: c£,m COEFFICIENTS The coefficients c£mm are proportional to the usual Clebsch-Gordon coefficients and are tabulated in the following: m '4,m' 4
69 S. L. GILBERT AND C. E. WIEMAN
802
34 -
for these quantities have become available. We take /3 to be 27.3(4)ui. The experimental inputs which we use to obtain this number are the ratio a/p= -9.9(1) from Refs. 24-26, the 7 s state lifetime from Ref. 27, the 7s state polarizability from Ref. 28, the oscillator strengths f66 and f76 from Ref. 29, and the measured energy differences of the states involved. As discussed in Ref. 28, the primary theoretical input is the calculation of the contribution t o a from the states with principal quantum numbers greater than or equal t o 8. From the recent work of Johnson and co-workers3' we take this to be
3 m' +(1 6 - ~ 1 ' ~ ) ' ' ~ c4;m, = 4
ha= - 4 . 5 ( 6 ) ~ : .
The tensor transition polarizability, 0, is found by combining several experimental and theoretical results as first discussed in Ref. 23. T h e value which has been quoted in the literature has varied slightly with time as new inputs
T h e value of /3 we give here differs from our previously quoted value of 26.6(4)a: primarily because of a change in the measured value of the 7 s lifetime. Here we are using the newer more precise value obtained by Bouchiat et aLZ7 instead of the value from Ref. 23 which we had used earlier. O u r present value of 0 is also slightly different from the value of 26.8(5)ui first given in Ref. 27 and repeated in many subsequent publications by the same group (in some of the later references the uncertainty was increased from 5 to 8). T h e 2 6 . 8 ~ :result was obtained by using the value of ha calculated in Ref. 31. As discussed in Ref. 30, that value for ha is incorrect.
'Present address: Time and Frequency Division, National Bureau of Standards, Boulder, CO 80303. IW. J. Marciano and A. Sirlin, Phys. Rev. D 27, 552 (1983). 2H. Hollister et al., Phys. Rev. Lett. 46,643 (1981), and references therein. 3T. P. Emmons, J. M. Reeves, and E. N. Fortson, Phys. Rev. Lett. 51, 2089 (1983). 4P. H. Bucksbaum, E. D. Commins, and L. R. Hunter, Phys. Rev. D 24,1134 (1981); P. S. Drell and E. D. Commins, Phys. Rev. Lett. 53, 968 (1984). sM. A. Bouchiat, J. Guena, L. Hunter, and L. Pottier, Phys. Lett. 117B, 358 (1982); lMB, 463 (1984). %. L. Gilbert, M. C. Noecker, R. N. Watts, and C. E. Wieman, Phys. Rev. Lett. 55, 2680 (1985). 7M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 35, 899 (1974);36,493 (1975). *A somewhat different method using high magnetic fields ( 1 kG) in cell experiments was proposed independently by M. A. Bouchiat, M. Poirier, and C. Bouchiat, I. Phys. (Paris) 40, 1127 (1979); and by E. Commins [see P. H. Bucksbaum, in Proceedings of the Workshop on Panty Violation in Atoms, Cargbse, Corsica, 1979 (unpublished)]. This method was used in the 1984 thallium measurement (Ref. 4). gSee Ref. 8, M.A. Bouchiat et al. 'OAnother field configuration is possible using linearly polarized laser light. This configuration was used in the 1984 thallium measurement (Drell and Commins, Ref. 4). For our particular experiment, the advantages and disadvantages of these two field configurations are discussed in Ref. 15. !IT.W. Hiinsch and B. Couillaud, Opt. Commun. 35, 441 (1980). I2R. W. P. Drever, I. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, Appl. Phys. B 31,97 (1983). I3P. S. Drell and E. D. Commins, Phys. Rev. A 32,2196 (1985).
I4M. A. Bouchiat, A. Coblentz, J. Guena, and L. Pottier, J. Phys. (Paris) 42,985 (1981). lsS. L. Gilbert, Ph.D. thesis, University of Michigan, 1984 (unpublished). I6S. L. Gilbert, R. N. Watts, and C. E. Wieman, Phys. Rev. A 29, 137 (1984). The M1 amplitude, M, has a slight dependence on F and F'. This has been taken into account in our adjustments due to the M1 systematic error. 17V. A. Dzuba, V. V. Flambaum, P. G. Silvestrov, and 0. P. Sushkov, J. Phys. B 18,597 (1985). 18W. R. Johnson, D. S. Guo, M. Idrees, and J. Sapirstein, Phys. Rev. A 34, 1043 (1986). 19V. N. Novikov, 0. P. Sushkov, V. V. Flambaum, and I. B. Khriplovich, Zh. Eksp. Teor. Fiz. 73, 802 (1977) [Sov. Phys.-IETP 46,420 (1977)l. It has been pointed out to us that these data could also be used to provide a limit on the size of the nuclear anapole moment, as discussed by V. V. Flambaum, I. B. Khriplovich, and 0. P. Sushkov, Phys. Lett. B 146,367 (1984). zq. Barger, N. W. Deshpande, and K. Whisnant, Phys. Rev. Lett. 56,30 (1986). 21P. Fayet, Phys. Lett. 96B,83 (1980). 22R. N. Watts and C. E. Wieman, Opt. Commun. 57,45 (1986). 23J.Hoffnagle, V. L. Telegdi, and A. Weis, Phys. Lett. 86A,457 (1981); J. Hoffnagle, Ph.D. thesis, Swiss Federal Institute of Technology, Zurich, 1982 (unpublished). 24J. Hoffnagle, L. Ph. Roesch, V. Teledgi, A. Weis, and A. Zehnder, Phys. Lett. 85A,143 (1981). *%. L. Gilbert, R. N. Watts, and C. E. Wieman, Phys. Rev. A 27,581 (1983). *6M. A. Bouchiat, J. Guena, L. Hunter, and L. Pottier, Opt. Commun. 45,35 (1983). 27M.A. Bouchiat, J. Guena, and L. Pottier, J. Phys. Lett. (Paris) 45,L523 (1984).
APPENDIX B: THE VALUE O F B
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ATOMIC-BEAM MEASUREMENT OF PARITY . . .
**R.N. Watts, S. L. Gilbert, and C. E. Wieman, Phys. Rev. A 27, 2769 (1983). 29L. N. Shabanova et a!., Opt. Spektrosk. 47, 3 (1979) [Opt.
803
Spectrosc. (U.S.S.R.)47, 1(1979)]. R. Johnson et al. (unpublished). 31C. Bouchiat and C. A. Piketty, Phys. Lett. B 128, 73 (1983).
3%’.
PHYSICAL REVIEW A
VOLUME 34, NUMBER 4
OCTOBER 1986
Rapid Communications The Rapid Communications section is intended f o r the accelerated publication of important new results. Since manuscripts submitted to this section are giuen priority treatment both in the editorial ofice and in production, authors should explain in their submittal letter why the work justifies this special handling. A Rapid Communication should be no longer than 3% printed pages and must be accompanied by an abstract. Page proofs are sent to authors, but, because of the accelerated schedule, publication is not delayed f o r receipt of corrections unless requested by the author or noted by the editor.
Precision measurement of the off-diagonal hyperfine interaction S. L. Gilbert,* B. P. Masterson, M. C . Noecker, and C. E. Wieman Joint Institute f o r Laboratory Astrophysics, University of Colorado and National Bureau of Standards. Campus Box 440. Boulder, Colorado 80309-0440 and Physics Department, University of Colorado, Boulder, Colorado 80309-0440 (Received 30 May 1986) We have measured the hyperfine mixing of the 6s and 7s states of cesium using a new highprecision experimental technique. By comparing the diagonal and off-diagonal hyperfine interaction for these states, we find that a single-particle description of the states is accurate to better than 2%.
The hyperfine splitting of atomic eigenstates has been studied for many years and is now a major subfield of both atomic and nuclear physics. However, the corresponding off-diagonal terms of this interaction, namely the mixing of states with different principal quantum numbers but the same angular momentum, has only recently been observed.”’ Knowledge of both the diagonal and offdiagonal hyperfine interaction for a valence state allows one to determine the accuracy of a single-particle description of that state.’ This information would allow a better theoretical understanding of the hyperfine structure as well as the related problem of parity nonconservation in the atom. In this paper we present a technique for precisely measuring the off-diagonal mixing. We have used this to measure the hyperfine mixing of the 6s and 7s states in cesium to 2%. We find that the single-particle description of the wave functions for these states is a substantially better approximation than previous calculations have indicated. This is particularly interesting because of the role of these states in the study of atomic parity nonconservation.’ It is hardly surprising that there is little data on the offdiagonal hyperfine mixing when one considers the magnitude of the experimental problem. First, the mixing is very small, typically less than a few parts in lo6, and second, the wave functions that are mixed have the same angular momentum and thus are only distinguished by having different principal quantum numbers. However, these problems have been overcome by exploiting a selection rule for magnetic dipole ( M 1 transitions. The selection rule is that in the nonrelativistic limit the M 1 amplitude is zero between states of the same orbital angular momentum L and different principal quantum number n. But the hyperfine interaction, by mixing states of different n and the ~ measursame L, gives rise to a small M 1 a m p l i t ~ d e .By
ing this amplitude one can obtain the value of the mixing. In practice this idea is complicated by relativistic effects that also allow a magnetic dipole transition amplitude between states of the same L and different n. This is a highorder effect, however, and leads to an amplitude that is only slightly larger than the hyperfine induced amplitude. The two terms can be separated by comparing the amplitudes for different hyperfine components of the transition. Using these ideas the off-diagonal hyperfine mixing of the 6 s and 7 s states of cesium has been measured by several groups. The first reported measurement was by Hoffnagle et aL5 in 1982, although subsequent measurements have shown that this result was in error by nearly a factor of 2. Gilbert, Watts, and Wieman’ and Bouchiat, Guina, and Pottier’ obtained measurements which were in good agreement with each other and had uncertainties of slightly less than 10%. The Bouchiat group has carried out another measurement of this quantity using a different experimental technique and achieved a similar resuk6 In this paper we present a substantially more accurate measurement using the same general ideas, but an improved experimental approach. Before discussing the experimental technique we shall briefly review the relevant theory. The hyperfine mixing of the 6 s and 7s states is given by
where Hhf is the hyperfine-interaction Hamiltonian, and F ( F ’ ) is the total angular momentum quantum number. From Eq. (1) the M 1 matrix element for the 6s- 7s
34
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01986 The American Physical Society
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34
GILBERT, MASTERSON, NOECKER, A N D WIEMAN
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transition is
(2) We now define the M 1 amplitude as A,wi--M+(F-F')Mhr
,
(3)
where M corresponds to the first term in Eq. (2) and arises from relativistic effects. The quantity ( F - F')Mhf corresponds to the second term in Eq. (2) and arises from the hyperfine interaction mixing. From Eq. (3) it is clear that the difference between the M 1 amplitude for a AF = 1 transition and that for a AF = - 1 transition is just 2Mhf. As first shown by H ~ f f n a g l e and , ~ discussed in more detail by Bouchiat et al.,* in the single-particle approximation, Mhf can be written in terms of the well-known hyperfine splittings ( A W ~ Sand A W ~ Sof) the 6s and 7Sstates,
+
noise. Our new technique uses the same approach but we have added an optical power build-up cavity which provided 200 times larger signals and hence a nearly equivalent gain in the signal-to-noise ratio. The new experimental setup was identical to that used in our measurement of parity nonconservation and is discussed in detail in Ref. 8. Here we will only briefly list the key elements. The output of a conventional narrow-band dye laser was sent into a resonant Fabry-Perot interferometer, the power build-up cavity. This interferometer was kept in resonance with the laser frequency and thus produced a standing-wave laser field. For the purposes of this discussion it is more useful to think of this as two traveling waves, each with about 200 times higher power than the incident laser beam. An intense, collimated cesium beam intersected the standing wave field. A uniform electric field was produced in the intersection region by applying voltage to parallel electric field plates that sat above and below the region. The field plates were flat glass plates with optically transparent 7 s transition electrically conductive coatings. The 6 S rate was monitored by detecting the near-infrared fluorescence that was given off during the 6 P - 6 s step of the 7 S 6P 6S decay. This light was detected by a silicon photodiode that was below the interaction region. As pointed out in Ref. 9 the sign of the Ast-AMI interference term in the transition rate depends on the direction of the laser propagation. Thus if the laser and cesium beams are at right angles, the atoms are simultaneously in resonance with both traveling waves and the term we are interested in cancels out. The key feature of this experiment was that the cesium beam did not intersect the laser beam at right angles. We tilted the cesium beam away from perpendicular so the Doppler shift caused the atoms to see different resonant frequencies for the forward going and backward going traveling waves. This caused all the spectral lines to be split into doublets, as shown in Figs. 1 and 2. Thus for a particular Zeeman transition (6sFm 7s F'm') the A s t - A ~ linterference does not cancel out on either line of the doublet. By tuning the laser frequency to one of the lines of the doublet, we were able to make a measurement identical in concept to that of
-
- -
By measuring Mhf and comparing it with the value given in Eq. (4) we have a test of how well these states can be described by a single-particle wave function. The experimental technique used to measure Mhf was a modification of the approach we used in Ref. 1. This modification, however, has improved the signal-to-noise ratio by two orders of magnitude. Before discussing the modified version we will provide a brief review of the technique of Ref. 1. A cesium atomic beam was excited by a 7 s transition. narrow-band dye laser tuned to the 6S Static electric (Eland magnetic ( B ) fields were present in the excitation region. The electric field mixes S and P states in the atom which allows a "Stark-induced" electric dipole transition amplitude (Ast) between these two S states. For the transitions of interest, Ast is proportional to PE, where P is the 6s 7 s vector transition polarizability. This amplitude interferes with the smaller M 1 amplitude, and it was this interference term that was actually detected. A detailed calculation of the spectrum is given in Ref. 1. Here we will simply point out the important features of the transition rate:
-
-
interference ~ depends on The size and sign of the A s ~ - A M the initial and final Zeeman ( m ) levels. If all the m levels are degenerate and equally populated the interference term will sum to zero in the total transition rate. However, with a magnetic field and narrow transition linewidths we can resolve transitions between particular m levels and thus see an A s t - A ~ contribution l to the transition rate. This contribution can be isolated from the larger I Ast 1 contribution by using the fact that only the interference term changes sign with the reversal of E and B fields. It can also be identified by its change of sign when the laser is tuned to excite the opposite m level. In the work reported in Ref. 1, we used this technique to determine M and Mhf. The uncertainty in the measurement of the latter quantity was entirely due to the detector
-
I
I
I
I
I
I
I
tZUY
W
t
urn
ClY
w a
L A S E R FREOUENCY CHANGE ( M H r )
FIG. I . Scan of the 6s F -47s F -4 transition with the cesium beam mjlimator tilted so that the tfansitions due to the
foward going (k') resolved ( B -0 GI.
and backward going (k-) laser beams are
73
PRECISION MEASUREMENT OF THE OFF-DIAGONAL . . .
200
400
600
800
100
LASER FREQUENCY CHANGE ( M H z )
FIG. 2. Scan of the 6s F -47s F - 3 transition with the cesium beam collimator tilted and B -105 G. The two outermost lines are single Zeeman and single laser direction lines. All of the others are the- superposition of a k+ component of one Zeernan line and the k- component of t h e adjacent Zeeman line. The right-left asymmetry of the spectrum is due to the magnetic field induced mixing of hyperfine states. The F -3- F - 4 transition is the same except the asymmetry is smaller.
Ref. 1, but we have increased the signal size by the 200fold enhancement of the power build-up cavity. We took data on the 6s F -4 to 7 s F - 3 and 6 s F -3 to 7 s F -4 transitions. As mentioned earlier the offdiagonal hyperfine mixing was obtained from the difference in the M 1 matrix elements for these two transitions. The spectrum of these transitions is shown in Fig. 2. We measured the fractional modulation of the transition rate on each of the hyperfine transitions as the E and B fields were reversed. In principle the fractional modulation on 4 and F -43 lines is proportional to the F -3(M-Mhf)//3E and ( M +Mhf)//3E, respectively, and therefore, knowing /3 and E, M and Mhf can be easily obtained. III practice the data analysis is more complicated because the lines in the Zeeman-tilted-beam multiplet were only partially resolved, and the different lines had different fractional modulations.' Thus to obtain both M and Mhf it is necessary to determine the contribution of each line at a particular laser frequency. We felt that errors in this determination would cause significant additional uncertainty in the results. To avoid this difficulty we only used the present data to obtain the ratio MhffM. As discussed below this ratio can be determined in a way that is independent of the partial overlap of adjacent lines. To get Mhf we then used the absolute value of M that was measured previously. Data were taken on the extreme high- and lowfrequency lines of the two multiplets. The fractional modulation for the high- and low-frequency lines of a single multiplet differed slightly (<1%) due to magneticfield-induced mixing of hyperfine states in combination with the partial overlap of different lines. The average of the two lines, however, is independent of the magnetic field mixing. lo We obtained Mhr/M from the average modulations as shown in Eq. ( 6 ) , hfhf
-I
M
((Ma) E , B ) - ( ( M j ) E , B )
( ( A I ~ ) E , B ) + ( ( M $ ) E ', B )
(6)
351 1
where ((AIj?)E,B) is the signal modulation with the electric and magnetic field reversals, averaged over the highand low-frequency lines of the F - F' multiplet. Equation (6) is valid independent of the degree of overlap of adjacent lines in the multiplets. We also took data at two different magnetic fields (70 and 105 G ) to empirically confirm this. The uncertainty in our measurement was primarily systematic and was associated with the fact that we were making fractional measurements at the part in lo4 level. The largest uncertainty came from the determination of the signal size in the presence of nonresonant background. We studied the background signals and found there was no nonresonant background associated with the electric field at the part in lo4 level. There was a background (about 10%of the signal size) that was due primarily to scattered laser light. We determined the size of this background by setting the electric field to zero and observing the detector output. This background caused some uncertainty in determining the total signal size, and hence the fractional modulation, because it varied in time. We found that more frequent measurements of the background reduced the variation in our results. Since we could not monitor it continuously, however, we were left with variations that were two to three times larger than the statistical uncertainty in the absolute modulation. Because of this we have chosen error bars to include all of the results obtained. Our results are
1
-0.18316) ( B - 7 0 G ) , -0.1828(6) (B-105 G ) , -0.1830(4) (average). The good agreement between the values obtained with the two different magnetic fields supports our assertion that the results are independent of the magnetic field mixing and the amount of overlap of the lines (both of which are field dependent). In Table I we compare our value with the previous measurements. Our result is an order of magnitude more precise than the other measurements and is in good agreement with measurement (a), (b), and a recent measurement by Herrmann ef al. (d). It is, however, in poor agreement with measurement (c). Using M//3--29.73(34) V/cm from Ref. 1 and /3-27.3(4)ad from Ref. 8, we obtain the off-diagonal hyperfine mixing term Mhr'7.91(15) X 1 0 - 6 p ~ .
Mhf/h!f-
We can now compare this with the value predicted by Eq. (4) using the measured hyperfine splittings of the 6s and 7 s states. The two values agree to within the 2% uncerTABLE I. Measurements of M d M .
Mhr/M
Previous work
This work
-0.180(1 3)a -0.1 85(8Ib -0.191 (4)'
-0.1830(4)
-0. I 84(6)d
"Reference 1. bReference 2.
'Reference 6. dReference I 1.
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GILBERT, MASTERSON, NOECKER, AND WIEMAN
34 -
tation of parity nonconservation measurements in cesium. It is possible that a detailed understanding of the accuracy of Eq. (4) would enable a better determination of p using the measured value of Mhf/P. We have precisely measured the off-diagonal hyperfine interaction in cesium. Our results indicate that the semiempirical calculation of this interaction using the single-particle approximation is remarkably accurate. This measurement was made using a new technique that will allow similar measurements to be made for many atoms. Such measurements will provide unique new information on the nature of the states involved. Note added. After submission of this work we learned that C . Bouchiat'' has calculated the first-order core corrections to Eq. (4) and finds that they are extremely small. This appears to provide a theoretical explanation of our experimental results.
tainty of our result. This is surprising since Eq. (4)is derived using the single-particle approximation. Calculations of the hyperfine splittings of these states using Dirac-Fock and other model potentials have ascribed on the order of 20-30% of the splitting to core electron k e . , multielectron) contributions. Our results suggest that the single-particle approximation is much better than one would have expected, or that the accuracy of Eq. (4)goes beyond this approximation. This could have significant ramifications for both the theoretical treatment of the hyperfine interaction and parity nonconservation in heavy atoms. It would be quite interesting to reduce the present uncertainty and see if Eq. (4)still holds. One possible experimental improvement is the use of a spin polarized cesium beam. This would give a tremendously simplified spectrum and the ratio Mhf/p could be obtained directly with a precision of parts in lo4. Unfortunately, the value for Mhf is already primarily limited by the uncertainty in p and it appears difficult to substantially improve on the present determination of this quantity. On the other hand, the determination of p will also soon be a limit on the interpre-
This work was supported by the National Science Foundation. One of us (C.E.W.) is supported by the Alfred P. Sloan Foundation.
'Present address: Time and Frequency Division, National Bureau of Standards, Boulder, CO 80303. IS. L. Gilbert, R. N. Watts, and C . E. Wieman, Phys. Rev. A 29, 137 (1984). 2M. A. Bouchiat, J. Guina, and L. Pottier, J. Phys. (Paris) Lett. 45, L61 (1984); M. A. Bouchiat, J. Gukna, L. Hunter, and L. Pottier, in Laser Spectroscopy VI, edited by H. P. Weber and W. Luthy (Springer, New York, 1983). 3S. L. Gilbert, M. C . Noecker, R. N. Watts, and C . E. Wieman, Phys. Rev. Lett. 55, 2680 (1985). 4M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 35, 899 (1974). 5J. Hoffnagle, L. Ph. Roesch, V. L. Telegdi, A. Weis, and A. Zehnder, Phys. Lett. 85A,143 (1981).
6M. A. Bouchiat, J. Guina, and L. Pottier, Opt. Commun. 51, 243 (1984). '3. Hoffnagle, Ph.D. thesis, Swiss Federal Institute of Technology, Zurich, 1982 (unpublished). *S. L. Gilbert and C. E. Wieman, Phys. Rev. A 34, 792 (1986). 9M.A. Bouchiat and L. Pottier, J. Phys. (Paris) Lett. 37, L79 (1976). 'OS. L. Gilbert, Ph.D. thesis, University of Michigan, 1984 (unpublished). "P. P. Herrmann, J. Hoffnagle, N. Schlumpf, V. L. Telegdi, and A. Weis, J. Phys. B 19, 1607 (1986). The first measurement of M d M (Ref. 5 ) was omitted from Table I since subsequent results have shown it to be in error. I2C. Bouchiat (unpublished).
VOLUME5 8 , NUMBER17
PHYSICAL REVIEW LETTERS
27 A P R I L1987
Asymmetric Line Shapes for Weak Transitions in Strong Standing-Wave Fields C. E. Wieman, M. C. Noecker, B. P. Masterson, and J. Cooper Joint I n s l i t i r t r f o r Laborurory Asrrophysics. Unirersity of Colorado and National Bureau of Standards. Boulder, Colorado 80309. and Phj,.rirs Deparrmenr. Unil,er.iily of Colorado, Boulder, Colorado 80309 (Received 31 December 1986)
We have observed the resonance line shape for a very weak atomic transition excited when an atomic beam intersects a strong standing-wave laser field. The line shape has a dramatic intensity-dependent distortion which is Doppler free and independent of the excitation rate. We have calculated the line shape predicted by optical Bloch equations that include a spatially varying a c Stark shift, and tind good agreement with o u r experimental results. PACS numbers: 32.70.Jr, 3 2 . 8 - 1
An isolated atom which I S weakly excited in a standing-wave field is one of the most basic systems one can study in spectroscopy. It has been commonly assumed that this simple case would have a symmetric, easily understood line shape. W e have observed that this is not the case, and in fact very weak transitions in a strong standing-wave field can show striking asymmetric distortions of the resonance line shape. This is significant because these are the conditions encountered in the ongoing precision measurement of parity nonconservation in cesium,' and this measurement relies on a thorough understanding of the resonance line shape. Also this is relevant to precision wavelength standards for which weak transitions have been sought because of their narrow linewidths. W e find that the intensitydependent distortion has two rather interesting and surprising characteristics. First, the frequency dependence is characterized by the natural linewidth, and, although the distortion depends on the laser intensity, it is independent of the excitation rate. T h u s we have the unusual situation that on a Doppler-broadened singlephoton resonance line there is a distinct Doppler-free structure for arbitrarily small excitation rates. Second, the frequency dependence of the distortion for electric dipole ( E l ) transitions is the mirror image of that for magnetic dipole ( M I ) transitions. W e have studied this distortion of the line shape for a variety of experimental conditions, and find our results agree very well with the line shape we calculate using optical Bloch equations that include the spatially varying ac Stark effect. Prentiss and Ezekiel have previously studied the intensity dependence of the line shape of the sodium D line. They observed an asymmetry at moderate to high excitation rates which they explained in terms of the induced dipole force changing the atomic trajectories. The asymmetry that we observe is different, though, in that it is independent of the excitation rate. To our knowledge this is the first observation of this type of line-shape distortion. In a rather different context, Roso et uL3 have predicted spectral features due to a spatially inhomo-
geneous ac Stark shift. Their predictions concerned the nonlinear absorption on a probe transition which had a common level with a strongly driven transition. For our study of the line shape, a beam of atomic cesium is excited in an intense standing-wave field. W e observe the 6s to 7s transition, which is extraordinarily weak, and this is not power broadened even in very high fields. Before presenting the experimental results. let us discuss the 6s-7s transition in cesium, because i t has some features that are different from normal transitions. A detailed discussion of this is given in Ref. 1; here we shall simply summarize the results. As shown in Fig. I , both the 6.9 and 7s states are split into two hyperfine states with total angular momenta of F = 3 and 4. I n the absence of an electric field there are four possible M 1 transitions, with matrix elements that
-7P
*
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-1
, /
/ ,/
,
852nm 894nm
-6S
FIG. I . Cesium energy-level diagram
0 1987 The American Physical Society 75
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are a few times 10-5pB. I n the presence of a dc field, these transitions acquire Stark-induced E ] transilion amplitudes given by
As,=aE.d,,A-pIExc/,
(1)
where 6 is the oscillating electric field, E is the static held, and a and /3 are the scalar and vector transition polarizabilities. Since a//3= 10, the AF=O transition rates can be 100 times larger than those of the A F = 1 or - 1 lines. By choice of a large enough E the Starkinduced E l amplitude can be made much larger than the M I amplitude. Although in general these two amplitudes can interfere, this does not happen when they are excited by a standing-wave field. The basic experimental setup is shown i n Fig. 2. Except for minor modifications, it is identical to that discussed in Ref. I . The output of a continuous-wave dye laser goes into a resonant Fabry-Perot interferometer. This acts as a power-buildup cavity and provides an intense standing-wave field. A collimated beam of atomic cesium intersects this field at right angles, and a static electric field can be applied to the intersection region. The 6 s - 7 s transition rate is monitored by observation of the 852- and 894-nm light emitted on the 6 P - 6 s portion of the 7s cascade decay. The dye laser produces up to 300 m W of light a t 540 nm. The laser frequency is locked to the resonant frequency of the power-buildup cavity by electronic feedback, and has a residual jitter on the order of 25 kHz. The laser light is matched into the lowest-order mode of the power-buildup cavity. By use of a piezoelectric transducer the cavity length is adjusted to tune the resonant frequency over the atomic transition. T h e intracavity standing wave is composed of two traveling waves, each with 1600 times the power of the incident laser beam. This is about a factor of 10 higher power than that used in Ref. I . The beam in the cavity has a Gauss-
+
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27 A P R I L1987
ian wave front with a diameter of 0.050 c m and a divergence of about 2 X l o p 3over the 2-cm length where it intersects the cesium beam. The cesium beam is produced by a large-area microchannel plate nozzle followed by a multiple-slit collimator. This collimation reduces the residual Doppler broadening of the 6 s - 7 s transition to 24 MHz. By tilting the collimator, we are able to adjust the intersection angle between the atomic beam and the laser beam around its nominal value of 90 deg. The fluorescence signal is obtained by use of a cylindrical mirror to image the interaction region onto a silicon photodiode. In Fig. 3(a) we show the line shape observed a t relatively low intracavity power. The width is determined by the residual Doppler broadening. In Fig. 3(b) we show a
40
20
LA
‘LASER
BEAM
FIG. 2. Schematic of interaction region
FIG. 3. (a) Line shape of E l transition excited by 13.7 W of power in each of the two traveling waves in the standing wave. (b) Same transition with 277 W per traveling wave and a different vertical scale. (c) The curve corresponds to the curve i n (a) times 277/13.7 minus the curve in (b). The crosses are t h e points given by the theoretical calculation.
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high-power line shape and in Fig. 3(c) we show the difference that is obtained by our scaling the curve in Fig. 3(a) by the laser power ratio and subtracting i t from the curve i n Fig. 3(b). There is no absolute frequency scale. and so the origins of the two frequency scales were set so as to superimpose the undistorted wings of the lines. At moderate power (less than 100 W ) the difference curve passes through Lero a t essentially the undistorted line center, and the width, taken to be the separation between the positive and negative peaks. is always 3 . 5 ( 5 ) MHz. This matches the 3.3-MHz natural linewidth of the 7s state. As the power is increased above 100 W the curve broadens and shifts slightly. as can be seen in Fig. 3(c). For the highest power we used (350 W ) , a width of 5 M H z and a shift of about 0.5 M H L were observed. This power corresponds to a peak intensity of 3.5 x 10' W/cm2 at the center of the Gaussian wave fronts of each of the traveling waves in the cavity. I n Fig. 4 we show the dependence of the distortion on the laser power. W e are characterizing the asymmetry i n terms of the fractional distortion, D,which we define as the ratio of the size of the distortion ( d ) to the peak height ( h ) the line would have i f there were no distortion. This value initially rises linearly, but at higher power it begins to saturate. The line shape for E I transitions was found to be independent of laser polarization, static electric field, and hyperfine transition ( A F = -t 1 and 0 ) . While the line shape remained the same as these conditions were changed, the transition rate per atom varied between I s - ' and l o 4 s-I. When we observed an M I transition by setting the d c electric field to zero, the line shape was the mirror image of that of the E 1 transition. T h e rate was enhanced on the high-frequency side and suppressed on the low-frequency side of the line. W e also found
*
0 4 1
0 3 t X
D X
0 2 1
X
I
100
TYPICAL EXPERIMENTAL UNCERTAINTY
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300
POWER IN TRAVELING W A V E ( W )
FIG. 4. The fractional distortion D v s traveling-wave power. The crosses are t h e theoretical results and the dots are experimental points.
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that when the angle between the cesium beam and the atomic beam differed from perpendicular by a value greater than the divergence of the cesium beam, the line shape began to split into a doublet corresponding to exci[ation by each of the Doppler-shifted traveling waves. and the intensity-dependent distortion went away. Although the explanation is somewhat subtle, these results can be accounted for by consideration of how the ac Stark shift in a standing wave affects the line shape. This Stark shift arises because the field couples both the 6s and 7s states with the P states of the atom. Since the coupling to all of the P states is far off resonance, the shift is essentially constant over our frequency range. Also it is independent of the polarization of the laser. Considering only the coupling to the P states with principal quantum numbers 6, 7, and 8, we calculate thc ac Stark shift of the transition frequency to be 6 . 8 ~ Hz(V/cm) - 2 c 2 , or 21 HL c m 2 / W times the intensity i n one traveling wave. There is an uncertainty in this result due to the contribution from higher states and errors i n the calculated matrix elements, but we estimate that the result is accurate to within a few percent. This shift was incorporated into the usual optical Bloch equations for a two-level system by our simply replacing the energy difference between the states, El - E l , by the Stark shifted difference, El - E l + 6 ( f 1. The ac Stark shift, 6 ( r ) , is proportional to c2cos[kz(t)l, where z ( t ) = Z O +r,/ with zo being an initial position and L'; representing an atomic velocity transverse to the laser wave fronts. Also c still contains the Gaussian x and y dependence of the wave front. We can neglect transitions to other states in the system because the photoionization rate is much smaller than the 7 s - 6 P decay rate.4 W e have solved the Bloch equations numerically to find the time-integrated population in the 7s state for an atom in the laser beam, a s a function of laser frequency. The observed fluorescence is proportional to this population for constant velocity across the beam. The solution involves averaging over the distribution of transverse velocities and initial positions z o in the standing wave, while taking into account the Gaussian field distribution of d x , y ) in the x - y plane. To simplify the solution, we assume that c ( x , y ) is constant over distances i n the x - y plane corresponding to that moved by an atom in an atomic lifetime, and that the atoms follow straight-line trajectories. These assumptions are reasonably good since a typical atom takes more than 20 lifetimes to cross the laser beam and is only deflected by about rad by the dipole force. The calculated line shapes, particularly the intensity-dependent distortions, agree very well with all our observations. Examples of this can be seen i n Figs. 3(c) and 4, which have been calculated with no free parameters. The basic shape of the distortion matches surprisingly well with what we observe and the height matches to within the uncertainty of the calculated ac Stark effect. Even such subtle features as the
Voi.LbiL j X ,
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PHYSICAL REVIEW LETTERS
slight power-dependent shift in the frequency at which the dimerence curve crosses through zero and the small diRerence i n height between the positive and negative peaks are reproduced in the calculated line shapes. W e can understand these results qualitatively by considering the relationship between the Doppler shiit of an atom and the ac Stark shift i t will have. I f an atom has a large 'I it will move rapidly across the wave fronts of the standing wave. I n one lifetime it will sarnple and thus average over many periods of the field, and hence of the ac Stark shift. T h e perturbed resonant frequency of this atom will then be the unperturbed energy difference shiltcd by the spatial average of the Stark shift. T h e spectrum for a group of such atoms will be two Lorentzian peaks which have the natural linewidth and are Doppler-shifted symmetrically above and below the perturbed resonant frequency. All large velocities show similar behaviors; thus the wings of the line, which are due to these velocities, will be symmetric, and the shape of the wings, though not the central frequency, will be independent of the intensity. For very low-velocity atoms the spectrum is quite different, however. A slow atom's position is nearly constant during one lifetime [ z ( [ )= 201 so that its resonant frequency will he shifted by the 6 ( f ) determined by the field a t that particular point in space. T h e spectrum one obtains for a spatial average of such atoms is a single peak skewed toward the maximum value for S ( f ) , because the high-field regions have the largest transition rate as well as the largest ac Stark shifts. The combination of these two very different spectra for high- and low-velocity atoms results in the line shape we observe. The velocity characterizing the distinction between high- and low-velocity regimes is that a t which an atom goes one wavelength in a lifetime. At this velocity an atom will have a Doppler shift equal to the natural linewidth; this is why the characteristic
27 A P R I L 1987
frequency scale is the natural linewidth. From this picture i t is also quite easy to understand the frequency dependence o f the M I transitions. I n a standing wave the antinodes of the oscillating magnetic field occur at the position of tht: electric-field nodes. This means that the MI transitions for low-velocity atoms will occur primarily in regions o f low electric field and hence be skewed toward small ac Stark shifts. Since this is just the oppositc of the E l case, the distortion of the line shape is reversed. I n conclusion, we have found that the simple case of a weak transition excited in a standing-wave field does not have a symmetric line shape. T h e line shape has a n intensity-dependent asymmetry with characteristics that are unlike previously observed distortions of spectral lines. However, these characteristics can be fully explained if one carefully considers the combined effects of the Doppler and ac Stark effects. We are happy to acknowledge useful discussions with Dr. S. Gilbert. Dr. M. Prentiss, Dr. C . Tanner, and assistance in the computations from Dr. D. Kelleher. This work was supported by the National Science Foundation and one of us (C.E.W.) received support from the Alfred P. Sloan Foundation.
IS. I-. Gilbert, M . C. Noecker, R. N . Watts, and C. E. Wieman, Phys. Rev. Lett. 55, 2680 (1985): S. L. Gilbert and C. E. Wieman. Phys. Rev. A 34. 792 (1986). * M . G. Prentiss and S. Ezekiel, Phys. Rev. Lett. 56, 46 (1986).
3L. Roso er al.. Appl. Phys. B 31, I15 (19831, and Opt. Commun. 38, 113 (1981). 4S. L. Gilbert, M. C . Noecker, and C. E. Wieman. Phys. Rev. A 29. 31 50 (1984).
PHYSICAL REVIEW A
VOLUME 38, NUMBER 1
JULY 1, 1988
Precision measurement of the Stark shift in the 6SIl2- + 6 P 3 / 2 cesium transition using a frequency-stabilized laser diode
-
Carol E. Tanner and Carl Wieman
Joint Institute for Laboratory Astrophysics, Uniuersity of Colorado and National Bureau of Standards, Boulder, Colorado 80309-0440 and Physics Department, University of Colorado, Boulder, Colorado 80309 (Received 21 December 1987)
We employ crossed-beam laser spectroscopy with a frequency-stabilized laser diode to measure the Stark shift of the 6P3,, state relative to the ground state of atomic cesium. The scalar and tensor polarizabilities are determined from the measured Stark shifts in the transitions 6S,,, F = 4 - 6 P 3 , , F ' = 5 , M F = 5 and 4. Our results are ao(6P3,2)+a2(6P3/2)-aao(6S,,,)= 977.8( 19)ai and a 2 ( 6 P 3 / , ) =-262.4( 15)ai.
The low-lying electronic states of Cs are being studied extensively in our laboratory because of their connection to the investigation of parity nonconservation (PNC) in the 6 S , / , + 7 S I , , transition.' This paper describes the precise measurement of the Stark shift in the 6S,,, + 6 P , / , transition from which we determine the electric dipole polarizability of the 6 P 3 / , state. Interpretation of PNC measurements involves atomic-structure calculations that include sums of radial integrals. The Stark effect provides an important test of these calculations since it is the only precisely measurable quantity that depends on these matrix elements. The Stark effect in the Cs 6P3/2 state was first measured by Marrus et al. in an atomic-beam rf resonance experiment,2 but they did not achieve the precision necessary to test adequately present theoretical methods of atomic-structure calculaet al. have recently improved upon these t i o n ~ .Hunter ~ measurements by using laser spectroscopy to observe the shifts of the Doppler-broadened 6S,/2 +6P3/2 transitions in a ceL4 We set out to make a careful study of the Stark effect in the Cs 6P3,, state in order to improve the precision and because of a possible discrepancy between theory and experiment (now resolved) in the tensor polari~ability.~ The Hamiltonian describing the interaction between an atom and external electric field, to lowest order, is6
bility of the Cs 6P3/2 state, one must accurately fit the resonance line shape by including all M F levels to extract a. and a2 independently. This requires precise knowledge of the population of each sublevel but many things, especially optical pumping, can lead to nonthermal populations. We avoid these difficulties by resolving the MF levels. This is achieved by performing laser spectroscopy on a Cs atomic beam with a resolution near that of the natural linewidth of the 6P3/2 state ( 5 MHz) in fields up to 45 kV/cm. Figure 1 is a frequency scan of the 6P3/, F'=5 (plus a small admixture of F ' = 4 ) state showing the MF substructure at 40 kV/cm. The 1 MF I = 5 , 4, and 3 levels are clearly resolved, while I MF I =0, 1, and 2 still overlap. The energy shifts of the resolved levels are only a function of E and are not affected by optical pumping. We use a fixed-frequency laser with part of its output shifted in frequency by an acousto-optic modulator to find the shifts in the I MF I = 5 and 4 transition line centers. This technique has the advantage of measuring only frequency differences, without the many uncertainties involved in precise wavelength measurements. The apparatus is shown schematically in Fig. 2. A -I
a
z
H=-+aE2,
a =a o +a z Q ~ , p MF;
2! 9
cn
(1)
where a is the polarizability describing the induced electric dipole moment of the atom. The polarizability of P3/* states contains both scalar a. and tensor a, parts. The scalar part shifts all hyperfine and magnetic sublevels P3,2 F' MF, equally. The tensor part mixes states of different F' through the matrix Q (see Ref. 6 ) , but MF remains a good quantum number. This causes each magnetic sublevel 1 M F I to shift by a different amount.'^^ There are significant experimental difficulties involved in separating the scalar and tensor parts of the interaction because a2 is small. If the M F levels are not resolved, as in the previous measurements of the polariza38 -
79
w V
z
W V
cn W a 0 3 -I LL
lMFl=O I
2
3
4
5
FREQUENCY
FIG. 1. Laser frequency scan showing the MI. substructure of the 6S1,2 F=4-6P,,Z F ' = 5 transition at an electric field of 40 kV/cm. The I M FI = 5 and 4 lines are 38 MHz apart and 11 MHz wide. 162
01988 T h e American Physical Society
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PRECISION MEASUREMENT OF THE STARK SHIFT I N T H E . . .
SATURATION SPECTROMETER BEAM SPLITTERS Cs O V E N
FIG. 2. Schematic of the apparatus.
beam of atomic Cs effuses from a 0.05-cm slit in an oven and is collimated with a second 0.05-cm slit 10 cm downstream. The collimated atomic beam intersects a circularly polarized laser beam at right angles in a region of high-static electric field. The laser propagation direction is aligned parallel to the static electric field. The field is produced by applying between 13 and 18 kV to a pair of coated glass plates 0.3950(2) cm apart. One plate is 5 X 7.5 cm2 and has a transparent conductive coating through which the laser beam passes; the second plate is a 4.5 X 5. 5-cm2 gold-coated glass mirror. To assure field uniformity, the laser beam is only 0.2 cm in diameter and enters the electric field region over 2 crn (five times the plate spacing) away from the edges of the plates. As discussed in Ref. 1, the stray-field effects for these coatings are negligible. The voltage applied to the plates is measured using a high-voltage divider and a Keithley digital voltmeter. Because the divider drifts slightly with temperature, it is calibrated periodically during data collection. The fractional uncertainty in the voltage calibration is 6x lop4. The laser radiation is produced by a frequencystabilized laser diode (Hitachi HLP1400 frequency selected for 852 nm) which is locked to a hyperfine transition in Cs. The unstabilized laser has a typical linewidth of 30 M H z and an output power of 5 mW. The frequency is stabilized by passively locking the laser to a Fabry-Perot (FP)interferometer with optical feedback from the cavity (a property unique to laser diodes). This requires only 0.8 mW of laser power and reduces the laser linewidth to much less than the 5-MHz natural width of the 6S,/,+6P3/2 transition. Although we do not have the capability to determine the laser linewidth accurately, Hollberg et al. have measured laser diode linewidths to be on the order of 20 kHz under similar conditions.' The FP cavity gives short-term stability to the laser frequency, and long-term stability is ensured by locking the cavity to a Cs resonance line. This is accomplished by sending a portion of the laser diode beam into a Cs saturated-absorption spectrometer." The saturatedabsorption spectrum is produced when two counterpropagating laser beams pass through a 4-cm-long Cs vapor cell at room temperature. One beam is intense (8 mW/cm2) and saturates the Cs transition. The second beam is a low-intensity (0.5 mW/cm2) probe whose ab-
163
sorption is measured and provides Doppler-free saturation peaks superimposed on the normal Dopplerbroadened absorption profile. We use another lowintensity probe to subtract off the normal Dopplerbroadened profile to obtain the Doppler-free spectrum shown in Fig. 3. The FP cavity length and the laser frequency which follows it are modulated with an amplitude of 1 MHz at a frequency of 18 kHz. Phase-sensitive detection is used to obtain the first derivative of the saturated-absorption signal, and this signal is fed back to lock the cavity resonance frequency to the peak of the 6S,/, F=4+6P3,, F'= 5 saturated-absorption line. Because of the narrow lines (9 MHz) and very good signalto-noise ratio, the laser stays locked within less than 0.1 MHz of the line center. The main portion of the laser beam is shifted in frequency with one or two acoustooptic modulators (AOM). The rf driving frequency of the AOM is measured precisely with a radio-frequency counter. After the AOM, the frequency-shifted laser beam is sent through a single-mode optical fiber to ensure that the final laser beam direction remains fixed. The total laser power leaving the fiber and reaching the interaction region is about 30 pW. The fluorescence from the interaction region is detected with a cooled silicon photodiode, and the derivative of the 11-MHz wide fluorescence peak is obtained by phase-sensitive detection at the 18-kHz dither frequency. To determine a. and a2 independently, we measured two experimental parameters. One was the Stark shift of the 6S,/, F=4+6P3/, F ' = 5 1 MF I = 5 transition as a function of electric field, and the second was the splitting between the I M F I = 5 to 1 M F I = 4 sublevels of the F ' = 5 state measured at fixed voltages. The first was measured by shifting the laser frequency by known amounts between 140 and 250 MHz with the AOM. At each value of frequency shift, the voltage applied to the plates was adjusted to shift the transition into resonance.
I F'=3
I
4
L 5
FREQUENCY
FIG. 3. Saturated absorption spectrum of the 6S1,* F=4+6P,,2 F' transitions. The 9-MHz wide F ' = 5 line is used to stabilize the laser frequency. This spectrum, obtained from a single 100-ms oscilloscope sweep, shows the very large signal-to-noise ratio.
81 ~
CAROL E. TANNER AND CARL WIEMAN
164
The transition line center was found by observing the zero crossing of the derivative signal. The second parameter, the 1 M F 1 = 5 to 4 splitting, was measured by fixing the voltage and then adjusting the AOM driving frequency to tune the frequency-shifted laser beam from one transition to the other. This measurement was repeated several times for each of three different voltage settings. Before data were collected, we limited some possible systematic effects. First, the Cs beam was aligned perpendicular to the laser beam by adjusting the Cs beam direction while observing the zero crossing of the derivative signal at zero voltage. This reduced the Doppler shifts to less than 0.1 MHz and provided the "measurement" of zero shift at zero field. We also checked that there was no change in the measured shifts when the intensity was varied over a factor of 4. This limited possible optical-pumping effects and ac Stark shifts to less than 0.1 MHz. The total Stark shift of the 6S,,, F=4+6P3/, F ' = 5 1 M F 1 = 5 transition as a function of electric field is
I
I 2oo
n a
?
I-
t
50/
0
38 -
/ I 50
I 100
1 150
I
200 V2 (kVz)
I 250
1
1
300
350
FIG. 4. Stark frequency shift as a function of voltage squared. The shift in M F= 5 fits a straight line. The uncertaint y in each point is much less than the size of the dots on the figure.
MHz from Ref. 12, we find where the Stark shift in the 6S,/, state'' must be subtracted from the shift in the excited state. Figure 4 is a plot of this frequency shift as a function of the square of the applied voltage. Although the I M F 1 = 5 and 4 sublevels were well resolved (Fig. I), the nearby lines pull the line centers slightly. A correction of order +0.10(5)% has been applied to account for this, but it had a negligible effect on the slope (0.01%). The frequency splitting between 6P3,, F ' = 5 1 MF 1 = 5 and 1 M F 1 = 4 sublevels was measured at three different voltage settings (also plotted in Fig. 4). For these measurements, a correction of order +O. 5(2)% was added to the splittings to compensate for the pulling of the nearby lines. The data for v5 fit a straight line to within the statistical uncertainties of the points, and the slope is -0.7796(9) MHz/(kV)*. We combine this slope with the plate spacing and Eq. (2) to obtain
[a,(6P3i2)+a,( 6P3,, )-a,( 6Sl /,
)I =977.8(19 )a: , (3)
where the uncertainty in this result is dominated by the uncertainty in the electric field. Diagonalizing the Stark Hamiltonian (1) in the presence of hyperfine structure and solving for az in terms of the energy splitting, one obtains
(4)
Inserting our measurements of vS4, and vhf=25 l.OO(2 )
a2(6P3,2)= -262.4( 1 5 ) ~ :
(5)
Combining (3) and ( 5 )gives
.
[a,( 6 P 3 / ,) -ao( 6SII2) ] = 1240.2(24)a
(6)
Our results are in good agreement with the much less precise results of Marrus. Hunter et aL4 have obtained the much more recent results a,(6P3,,)= - 2 6 2 ( 7 ) ~ : , [ a,( 6P3,, 1- a,( 6SI,, ) ] = 1264( 13)(I
(7)
.
(8)
Their uncertainty is larger than ours because of the broader linewidths of the transitions measured. Their value for a, is in good agreement with our results, but the value in (8) is different from our value (6) by slightly greater than their uncertainty. The primary significance of our work is as a test of atomic-structure calculations. Early calculations given in Ref. 13 are in reasonable agreement with our results. Much better agreement is obtained with the recent calculations of Zhou and N o r c r o ~ s . ' ~ Using a semiempirical model p ~ t e n t i a l , they ' ~ obtain a polarizability that agrees with our measurements to better than 1%. It will be interesting to compare our measurements to calculations using the relativistic many-body perturbation techniques that are presently being developed for high-precision atomic-structure caiculation~.~ This work was supported by National Science Foundation Grant No. PHY86-04504. We would like to acknowledge the assistance of B. P. Masterson and L. Hollberg. We would also like to thank L. R. Hunter for communicating his results to us prior to publication.
82 PRECISION MEASUREMENT OF T H E STARK SHIFT I N T H E . . .
38
IS. L. Gilbert and C. E. Wieman, Phys. Rev. A 34,792 (1986). *R. Marrus, D. McColm, and J . Yellin, Phys. Rev. A 147, 55 (1966).
3W. R. Johnson, D. S . Gou, M. Idrees, and J. Sapirstein, Phys. Rev. A 32, 2093 (1985); 34, 1043 (1986). 4L. R . Hunter, D. Krause, Jr., S. Murthy, and T. W. Sung, Phys. Rev. A 37, 3283 (1988). 5L.R. Hunter, D. Krause, Jr., S. Murthy, and T. W. Sung (unpublished). 6R. W. Schmieder, Am. J. Phys. 40,297 (1972). 'R. W. Schmieder, A. Lurio, and W. Happer, Phys. Rev. A 3, 1209 (1971).
*T. M . Miller and B. Bederson, Adv. At. Mol. Phys. 13, 1 (1977). This is an excellent review of experimental and
165
theoretical determinations of atomic and molecular polarizabilities. 9B. Dahmani, L. Hollberg, and R. Drullinger, Opt. Lett. 12, 876 (1987).
'OW. Demtroder, in Laser Spectroscopy, edited by Fritz Peter Schafer (Springer, New York, 1981), p. 489. "R. W. Molof, H. L. Schwartz, T. M. Miller, and B. Bederson, Phys. Rev. A 10, 1131 (1974). The polarizability of the Cs 6SI,, ground state reported in this reference is ao(6S,,2 )=402(8)ai.
12C. E. Tanner and C. Wieman, Phys. Rev. A (to be publsihed). 13P.M. Stone,Phys. Rev. 127, 1151 (1962). I4H. L. Zhou and D. Norcross (unpublished). lSD.Norcross, Phys. Rev. A 7, 606 (1973).
PHYSICAL REVIEW A
AUGUST 1, 1988
VOLUME 38, NUMBER 3
Precision measurement of the hyperfine structure of the 133Cs6P312 state Carol E. Tanner and Carl Wieman Joint Institute f o r Laboratory Astrophysics, University of Colorado and National Bureau of Standards, P.O. Box 0440, Boulder, Colorado 80309-0440 and Physics Department, University of Colorado, Boulder, Colorado 80309 (Received 8 March 1988) We report measurements of the hyperfine structure of the 6P,,, state of atomic "'Cdl = 7 / 2 ) . A frequency-stabilized laser diode is used to perform crossed-beam laser spectroscopy of the Cs 6 S , , 2 ( F= 3,4)-6P,,,(F') transitions. From the measured hyperfine splittings, we determine the coefficients of the magnetic dipole ( A ) and electric quadrupole ( B ) contributions to the hyperfine structure. Ou r results are A = 50.273 3 ) MHz and B = -0.53(2) MHz.
The comparison of experimental and theoretical parameters of hyperfine structure is one of the most stringent tests of atomic wave functions near the nucleus. In particular, it is one of the best ways to explore the relativistic, core polarization, and correlation effects of the electrons which are sources of dificulty in accurate atomic structure calculations. At present, such calculations are of particular interest to the study of parity nonconservation (PNC) in atoms because the weak force is a shortrange electron-nucleon interaction. The interpretation of P N C measurements in the 6 S , / ,+7SIl2 transition of atomic cesium requires precise knowledge of atomic wave functions near the nucleus. The experimental determination of hyperfine structure in alkali-metal atoms is reviewed in Ref. 1. Here, we report precise measurements of the hyperfine splittings in the 6 P 3 / , state of '33Cs. Figure 1 shows the hyperfine structure of this state. From these measurements, we determine the coefficients for the magnetic dipole and electric quadrupole contributions to the hyperfine structure. In the approximation that J is a good quantum number, the hyperfine energy is given by
for states where J = L +tf. A and E are, respectively, the coefficients of the magnetic dipole and electric quadrupole contributions t o the hyperfine structure. Measurements of A and B are most conveniently compared to relativistic hyperfine structure calculations through the effective operator formalism developed by Sandars and Beck where A and B are related to various radial matrix
beam is linearly polarized to prevent optical pumping effects. The laser intensity is 13 pW/cm2 so each atom absorbs, on the average, one photon as it traverses the 0.7-cm-diam. laser beam. This low transition rate further reduces the probability of optically pumping the atoms, which would cause a frequency shift of the resonance in the presence of stray magnetic fields. A frequency scan of the Cs beam fluorescence is shown in Fig. 2. We measured the three frequency intervals shown between adjacent hyperfine levels. To avoid the uncertainties encountered in determining changes in laser frequency, the following procedure was used to measure each interval. First, the laser frequency was locked to one hyperfine transition [for example, 6 S , / , ( F =4) - 6 P , / , ( F ' = 5 ) ] in a saturated absorption cell. Then, a portion of the laser output was shifted in frequency, using an acousto-optic modulator (AOM). This shifted light intersected the atomic beam after passing through a singlemode optical fiber which fixes its position. The modular frequency was adjusted to bring the light into resonance, with first one, and then the other of the two allowed hyperfine transitions ( F = 4 and F ' = 3 ) in the atomic beam. The center of each resonance was determined as in Ref. 4. The measured frequency splitting between the ~ ) the difference in the two AOMresonances ( A v ~was drive frequencies needed to excite the two transitions. This approach made the exact laser frequency unimportant, and the quantity of interest, the interval between the 5 251.00 ( 2 ) M H z
element^.^^^ The hyperfine structure of the Cs 6P3/, state has been previously measured by a number of groups using optical double resonance, atomic beam magnetic resonance, and level crossing techniques. We have been able t o improve upon the precision of these measurements using the method described in Ref. 4 for measuring the Stark shift of the 6 P 3 / , state. A highly collimated beam of atomic Cs intersects a linearly polarized laser beam at right angles. The laser radiation is produced by a frequencystabilized laser diode with a 20-kHz line width. The laser 38 -
83
201.24 ( 2 ) M H z
151.21 ( 2 )M H z 6P3,?(F')= 2
FIG. 1. Hyperfine structure of '%s 6P,,, and our measured
splittings. 1616
@ 1988 The
American Physical Society
84 BRIEF REPORTS
1617
5 A +$B =251.00(2) MHz , 4 A -+B = 2 0 1 . 2 4 ( 2 ) MHz ,
(2)
3 A - 4 B = 1 5 1 . 2 1 ( 2 )M H z .
F = 3+ F': 2
3
4
c
4
T h e precision in the frequency intervals is limited by the measured drift in the lock point of the laser. The measurements are independent of laser polarization. The systematic errors introduced by stray magnetic fields and a c Stark shifts are negligible being less than 100 Hz. Using the method of least squares, we determine the coefficients:
5
A = 50.275(3 1 MHz FREQUENCY 6 S I /2(F)-+ 6 P3/2(F'I
B =-0.53(2)
,
MHz .
(31
(4)
FIG. 2. Fluorescence spectra of the 6 S , , , ( F ) 4 6 P 3 , , ( F ' )
transitions.
resonances, was the difference between two easily measured radio frequencies. By locking the laser to different transitions in the saturated absorption cell, we were able to use this method to measure all three intervals. O u r measured hyperfine splittings are shown in Fig. 1. Using Eq. ( l ) , the measured hyperfine splittings in terms of the magnetic dipole ( A ) and electric quadrupole ( B )coefficients are
'E. Armondo, M. Ingusico, and P. Violino, Rev. Mod. Phys. 49, 31 (1977).
'P. G. H. Sandars and J. Beck, Proc. R. SOC.London, Ser. A 289, 97 (1965).
These results are in good agreement with the results reviewed in Ref. 1, but are, respectively, a factor of 20 and 13 more precise. Our results are consistent in that A and B determined from all pair combinations in (2) agree to within the uncertainties. This work is supported by National Science Foundation Grant No. PHY-86-04504.We would like t o acknowledge the technical assistance of B. P. Masterson and L. Hollberg.
3L. Armstrong, Jr., Theory of the Hyperfinr Structure of Free Atoms (Interscience, New York, 1971). 4C. E. Tanner and C. Wieman, Phys. Rev. A 38, 162 (1988).
VOLUME^^, N U M B E R3
PHYSICAL REVIEW LETTERS
18 JULY1988
Precision Measurement of Parity Nonconservation in Atomic Cesium: A Low-Energy Test of the Electroweak Theory M . C. Noecker, B. P. Masterson, and C. E. Wieman Joint Institute f o r Laboratory Astrophysics, National Bureau of Standards and Unirersity of Colorodo,
and Phj’sics Departmenr, Uniwrsity of Colorado, Boulder, Colorado 80309 (Received 2 May 1988)
We have made an improved measurement of the parity-nonconserving electric-dipole transition amplitude between the 6s and ?S states of atomic cesium. We obtain Im(EpNc)/p- - 1.5?6(34) mV/cm. which is in good agreement with the predictions of the standard model and earlier less precise measurements. This places more stringent constraints on alternatives to the standard model. We also see the first evidence of a nuclear-spin-dependent contribution to atomic parity nonconservation. The nuclearspin dependence observed is in agreement with that predicted to arise from a nuclear anapole moment. PACS numbers: 3 5 . 1 0 W b . 11.30.Er. I2.15.-y, 2 I . I O . H w
The measurement of parity nonconservation (PNC) in atoms has the potential to provide tests of the standard model of the electroweak interactions. I Such tests complement high-energy tests because they are sensitive to a different combination of the neutral-current electronquark coupling constants and are at very low energy. Parity nonconservation has been observed in several atoms,2 but two factors have limited the usefulness of P N C measurements for the testing of the fundamental theory. The first was the limited precision of the experiments, and the second was the limited knowledge of the atomic structure, which is needed to compare the experiments with the electroweak theory. In recent years there has been substantial progress in both areas. Cesium is a particularly attractive atom because of its simple atomic structure. It has a single valence electron outside a tightly bound inner core, which makes the calculations of its structure more tractable. Until now, the accuracy of these calculations has exceeded the precision of the experiments. Previous P N C measurements in cesium have already provided a significant test of the standard model, 1,3,4 and have placed unique constraints on alternatives to the standard model which cannot be obtained from any other data. Here we present a substantially improved measurement of P N C in cesium. These improved measurements have also provided the first evidence of a nuclear-spin-dependent parity-nonconserving interaction in an atom. It has been predicted that the major source of such an interaction would be the anapole moment of the nucleus, which arises from charged weak interactions between nucleons. The existence of the nuclear anapole moment was predicted many years ago, but it has never been observed. The experimental technique we use to measure P N C in cesium is the same as we used previously, although we have significantly improved the apparatus. This technique has evolved out of the work of several others in this field and is discussed in detail elsewhere.‘ Here we will only review the basic features. The P N C interaction 310
mixes a small amount of the P states into the 6S (ground) and 7 s states of cesium shown in Fig. 1 . This gives rise to an electric dipole ( E l 1 transition amplitude, E P N C between , these S states. We measure this amplitude by observing its interference with a larger electricfield-induced (“Stark”) transition amplitude, PE. In the presence of a magnetic field, the interference between these two amplitudes gives a small P N C contribution to the laser-driven 6s- 7s transition rate. We separate this P N C interference term from the much larger pure Stark-induced rate by determining the fraction of the rate which changes sign with a reversal of the electric
6s
-
F=3
(a)
FIG. I . (a) Cesium energy-level diagram showing relevant transitions. W e do not show the small Zeeman splitting caused by the magnetic field. The 540-nm laser light is tuned to one of four different transitions between the 6.5 and 7s states. In terms of F and m quantum numbers, these are (4, +-4 to 3, ? 3 ) and (3, ? 3 to 4, +- 4). The transitions are detected by observation of the 852- and 890-nm fluorescence. (b) Orientation of the dc electric and magnetic fields and the laser helicity. 6,in the transition region. To observe the PNC modulation we reverse each of these vectors.
0 1988 The American Physical Society 85
(b)
86 VOI IJMF
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PHYSICAL REVIEW LETTERS
field, the magnetic field, the handedness of the laser polarization, and the rn quantum number of the level being excited. The parity-conserving rate is unaffected by any of these changes, but the PNC term changes sign with all four. The 6s- 7 s rate is measured by observation of the fluorescence emitted in the decay of the 7s state. The general apparatus is quite similar to that described in Ref. 6; we shall omit a detailed discussion of elements which are described there. An intense, highly collimated beam of atomic cesium intersects a standing wave of circularly polarized light at right angles, in the presence of perpendicular electric and magnetic fields. The standing wave is produced by our sending the light from a tunable dye laser into a Fabry-Perot interferometer power-buildup cavity. The major change from our previous experiment is the use of higher-quality mirrors in this interferometer; their higher reflectivities give a larger signal as a result of the greater power buildup, and their very low birefringence gives much smaller systematic corrections. We have about 800 W circulating power in the cavity- 1300 times the power of the laser beam incident on the cavity. Using a rather elaborate servo system, we lock the laser frequency to the cavity resonance, and then the cavity to the chosen cesium resonance frequency. The remainder of the apparatus is identical to that described in Ref. 6, except for a number of small changes that allow more precise alignment of the fields and of the atomic beam. The data-acquisition procedure is similar to that described in Ref. 6. From each day’s run we obtain measurements of the fractional P N C rate on the 6S(F=4)- 7 S ( F ’ = 3 ) and 6 S ( F = 3 ) 7 S ( F ‘- 4 ) transitions, with a typical fractional uncertainty of 9%. We also make several other measurements to determine misaligned and nonreversing field components that could cause systematic errors. Investigation of potential systematic errors occupied the great majority of the time spent on the measurement. We have found five different effects that can shift the results by more than 0.1% of the P N C rate, and have corrected the raw data to account for these. The correc-
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I8 JULY1988
tions are given in Table I. The first correction accounts for the dilution of the P N C fraction due to overlap from adjacent transition peaks involving other rn levels. The other four account for parity-conserving contributions that mimic the P N C rate, changing sign with all four reversals. As shown, these signals arise from misaligned and stray fields, or from interference between the Starkinduced and magnetic dipole (MI ) amplitudes. The two terms involving stray and misaligned dc fields are very small and are treated in the same manner as b e f ~ r e . ~ ” The first MI term, which has also been discussed previo ~ s l y , depends ~.~ on the birefringence upon reflection from the cavity output mirror. Although the reflective coating of this mirror has extraordinarily low intrinsic birefringence, we find it can slowly change because of thin layers of contamination or local deterioration of the coating due to the intense laser fields. To prevent this from introducing significant uncertainty we monitor the birefringence quite closely using the tilted atomic-beam technique discussed in Ref. 6. The high laser intensities used in this measurement also led to a false signal associated with the M1 amplitude on adjacent Am=O transitions; this has not been discussed previously. The high intensities cause a distortion in the resonant line shape of the transition.’ This shifts the maximum of the atomic resonance signal (to which the laser frequency is locked) to a frequency that is in resonance with atoms that have a nonzero Doppler shift. It can be shown from the discussion by Gilbert et al. l o that this leads to false P N C signal which is proportional to the misalignment of the laser and atomic beams, times the imperfection in the reversal of the laser polarization. We determine the size of this contribution by measuring these two factors, and we minimize it by keeping both quite small. As Table I shows, all the corrections were small and the uncertainties in the corrections, which are also statistical, are much smaller than the statistical uncertainty in a one-day P N C measurement. We have carried out extensive theoretical analysis and
T A B L E I . Systematic corrections to the data as a percentage of the PNC signaLa
Systematic contribution
Range
Average all da ta
Daily uncertainty
4.5%
1.4%-5.6% -0.3%-+1.1% -1.3%- +0.4%
-0.1%
0.3% 0.4% 0.4%
+
-0.8%-+4.8% -1.1%- +6.8%
+ I .I% +2.4%
0.6% 0.9%
-+
-0.3%- +0.6% -1.6%- +O. 1%
+0.04% -0.23%
0.04%
( 1 ) Dilution factor
(2) ( A E J E ) ( B * / B ) ( 3 ) (AEz/E)(Ey/E) (4) ( E I ) ( M I ) ( A m - ? I ) A F - - 1 line AF = 1 line ( 5 ) ( E I )(M1 ) (Am = O ) A F - - 1 line AF 1 line
0.3%
0.06%
“AEV and AEz are nonreversing field components, while Ex and Ey are misaligned components. The range column shows the largest and smallest daily corrections.
31 1
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VOLUME^^, N U M B E R3
PHYSICAL REVIEW LETTERS
many experimental tests in an effort to find and eliminate all possible systematic errors, but when one is measuring such small quantities, there is always the concern that something has been overlooked. To test for this possibility we have carefully analyzed the statistics of the fluctuations in the data. We found that the fluctuations on all time scales showed very good agreement with predictions for purely statistical variations. For example, the set of eighteen daily measurements gave a reduced x 2 of 1.0 despite several major changes in the apparatus, such as changing the electric-field plates, running at different electric fields, and moving and realigning the interferometer mirrors. Our final results for measurements of the ratio of P N C to Stark-induced amplitudes are
[ - 1.576(34)(08) mV/cm
82
1
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I I
-
I
PARIS P COMBINED {k",4P,sE~ 86
COLORADO '85
PRESENT WORK
HI
FIG. 2. Comparison of P N C measurements in cesium. From top to bottom, the results shown are from Refs. 1 1 , 12, 12, 13, and 6, respectively.
(average).
The first uncertainty is the statistical uncertainty in the results, and the second is the nonstatistical systematic uncertainty, which is dominated by the uncertainty in the electric-field calibration. We shall combine the two in quadrature in the subsequent discussion. In Fig. 2 we show the various measurements of P N C in atomic cesium. The consistency among the different measurements and the improvement in the precision are clear. Both the difference and the average of the two measured P N C amplitudes (1) are significant. From the different amplitudes of the two hyperfine lines we find that the nuclear-spin-dependent P N C contribution is +0.126(68) mV/cm. There is a 97% probability that this is larger than zero. Khriplovich and co-workers have discussed the different mechanisms that would lead to a nuclear-spin-dependent parity-nonconserving interaction, which they characterize by the dimensionless coupling constant K,. I 4 , I 5 From our data and the matrix elements given in Ref. 15 we find ~ , = + 0 . 7 2 ( 3 9 ) . They have predicted l 4 that the nuclear anapole moment would give a value of K, in cesium between +0.25 and 4-0.33. This is the dominant contribution, but there is also a contribution of about +-0.05 (=Czp) expected from the neutral-current electron-nucleon interaction. l 6 The sum of these two gives a predicted K~ =0.30 to 0.38, in agreement with our measurement. The uncertainty in the atomic structure is unimportant compared with the large fractional uncertainty in the measurement. The average P N C amplitude on the two lines, which is almost entirely due to the electron-nucleon neutralcurrent interaction, was measured with much smaller fractional uncertainty. Thus it can be used to make a quantitative test of the standard model at the few percent level. Our measured P N C amplitude is the product of the weak charge Qw and an atomic structure factor. To find the value of Q,, the quantity of interest, one must know the atomic structure factor. There has been 312
PARIS
18 JULY 1988
'
considerable recent effort devoted to the improved calculation of this factor for the cesium atom. However, the present uncertainty is still about 4 5 % , " or perhaps slightly less. Although there have been many calculations published, we shall use the average of the results of three recent and rather extensive (and presumably accurate) These span a 5% range. We use the most conservative error estimate ( f 5 % ) of the three, and p =27.0ad. 2o This gives Qw=-69.4?
1.5k3.8.
The first uncertainty is experimental; the second comes from the uncertainty in the atomic structure calculations. Marciano and Sirlin" give the renormalized weak charges for the neutron and proton in the standard model in terms of sin2@,. With these, our value of Qw gives sin2@,-0.219 f0.007 f0.018. This result is in agreement with the world average of sin2& =0.23010.005 (Ref. 11, which means that the weak charges in Ref. 21 are correct to within the uncertainties. The confirmation that the standard model is valid in an atomic system at this level of precision implies certain constraints on alternatives to the standard model. These are discussed in detail in Ref. 1. Here we will simply point out one particular aspect of that discussion to emphasize the utility of atomic PNC. This measurement provides the most precise experimental value for the electron axial-vector, down-quark vector coupling constant (Cld). The value of Cld is a sensitive function of the mass of a second neutral boson in many alternative models. A particular class of such models, which has been receiving considerable attention recently, is grand unified theories containing an Eg group. These results
88 VOLUME61, NUMBER3
PHYSICAL REVIEW LETTERS
set the best available limits on the masses of such additional neutral bosons. We have carried out a precision measurement of the PNC transition amplitude in atomic cesium. This confirms the standard model of electroweak interactions with an uncertainty of a few percent, and will allow even more precise comparisons of experiment and theory when the calculation of the atomic structure of cesium is improved. These measurements also provide the first evidence of an anapole moment of the nucleus. Work is presently under way to achieve higher experimental precision with use of an optically pumped atomic beam along with other improvements in the experiment. This will allow us to measure the nuclear-spin-dependent PNC effects more precisely, and make precise measurements of parity nonconservation in rubidium. This work was supported by the National Science Foundation. We are pleased to acknowledge the assistance of Dr. S. Gilbert in the early stages of this work, and valuable conversations with Dr. C . Tanner, Dr. J. Hall, Dr. W. Marciano, Dr. W. Johnson, and Dr. J. Sapirstein.
'U. Amaldi et al., Phys. Rev. D 36, 1385 (1987). 2E. N. Fortson and L. L. Lewis, Phys. Rep. 113, 289 (1984). 3W. Marciano, in Proceedings of the Salt Lake City Meering, edited by Carleton DeTar and James Ball (World Scientific, Singapore, 1987), p. 319. 4S. Capstick and S. Godfrey, Phys. Rev. D 37, 2466 (1988), and references therein. 5Ya. Zel'dovich, Zh. Eksp. Teor. Fiz. 33, 1531 (1958) ISov. Phys. JETP 6, 1184 (1957)l. %. L. Gilbert and C. E. Wieman, Phys. Rev. A 34, 792 (1986). 'P. S. Drell and E. D. Commins, Phys. Rev. A 32, 2196 (1985).
18 J U L Y 1988
*M. A. Bouchiat, A. Coblentz, J . Guena, and L. Pottier, J . Phys. (Paris) 42, 985 (1981). 9C. E. Wieman, M. C . Noecker, B. P. Masterson, and J. Cooper, Phys. Rev. Lett. 58, 1738 (1987). loS. L. Gilbert, B. P. Masterson, M. C. Noecker, and C. E. Wieman, Phys. Rev. A 34, 3509 (1986). IlM. A. Bouchiat el ol., Phys. Lett. 117B. 358 (1982). 12M. A. Bouchiat el a l ., Phys. Lett. 134B. 463 (1984). I3M. A. Bouchiat et al., J . Phys. (Paris) 47, 1709 (1986). I4V. Flambaum, I . Kriplovich, and 0. Sushkov, Phys. Lett. 146B, 367 (1984). 15P. A. Frantsuzov and I. B. Khriplovich, Z. Phys. D 7, 297 (1988). '6V.N. Novikov er al., Sov. Phys. J E T P 46, 420 (1977). 17W. Johnson, S. Blundell, 2. Liu, and J. Sapirstein, Phys. Rev. A 37, 1395 (1988). The value is 0.95(5) in units of ieao( - Qw/78)x 10 - ' I . I8V. A. Dzuba er al., Phys. Scr. 36, 69 (1987). I n the same units as above, this value is 0.90(2). I9C. Bouchiat and C. A. Piketty, Europhys. Lett. 2, 51 1 (1986). In the same units as above, this value is 0.935(2)(3). Reference 17 and 18 are ab initio calculations; this is a semiempirical result. *Owe are breaking with the tradition of using a semiempirical value for p, and instead are using a theoretical value. There are two ways to obtain a reliable theoretical value: using theoretical oscillator strengths from V. Dzuba et al., J. Phys. B 18, 597 (1985). or taking the calculated value for p given in a paper by 2. W. Liu and W. Johnson, presented at the Annual Meeting of the Division of Atomic, Molecular, and Optical Physics of the American Physical Society, Cambridge, Massachusetts, 18-20 May 1987 (unpublished). The two values obtained in this way agree with each other and the semiempirical result to within I%, which is less than the semiempirical uncertainty. We have used the average of the two theoretical results. Qw now is obtained from the quotient of two atomic properties calculated in a similar way. Since both of these quantities tend to behave the same way as higher-order corrections are added, this treatment should minimize the error due to the atomic structure calculation. 21W. J. Marciano and A. Sirlin, Phys. Rev. D 27, 552 ( I 983).
313
VOLUME 47, NUMBER 3
PHYSICAL REVIEW A
MARCH 1993
High-brightness, high-purity spin-polarized cesium beam B. P. Masterson,, C. Tanner,+ H. Patrick, and C . E. Wieman Joint Institute f o r Laboratory Astrophysics and the Department of Physics, University of Colorado, Boulder, Colorado 80309-0440 (Received 19 June 1992)
1 polarized using light atoms sC1,6X 10l6 atomssr - I s ~ ' spin We describe a cesium atomic beam from two diode lasers. Of the atoms, 95% may be placed into either the 6S,,,(F,rnr)=(3,3) or (4,4) state, with less than 2 X of the atoms left in the depleted hyperfine level. The latter fraction rises linearly with atomic beam intensity because of reabsorption of light scattered during the optical pumping process. This and other effects limiting complete hyperfine pumping are discussed. PACS number(s1: 32.80.Bx, 35.80.+s, 42.50.Wm I. INTRODUCTION
Spin-polarized atomic beams are useful in a number of studies. The use of atoms in specific quantum states allows more information to be extracted from atomic collision studies [ 11, and polarized heavy-ion beams likewise allow cleaner nuclear physics experiments [2]. Polarized atomic beams are also employed in microwave frequency standards [3,4]and in spectroscopic studies [ 5 ] . This paper describes a polarized cesium source developed for a third-generation parity-nonconservation (PNC) experiment using a shelving approach [6-81: 6S+7S transitions are detected not in the interaction region through observation of the 7S+6P--t6S cascade fluorescence but instead by detecting those atoms which return to the depleted hyperfine level after a 6 s - 7 s transition. Since more than 100 6S+6P transitions may be detected for each atom that returns t o the depleted hyperfine level when exciting on the cycling 6S transitions, the F =4-+6P3,,F'=5 and F = 3 + F ' = 2 signal size is vastly increased, and photodetector and scattered light noise become negligible by comparison. The removal of the photodetector from the interaction region also allows better control of stray electric and magnetic fields in this region. This approach requires, however, that the fraction of atoms left in the depleted hyperfine state after optical pumping be small compared to the fraction ( of atoms undergoing 6S+7S transitions. Moreover, a good signal-to-noise ratio requires that the polarized beam be as intense as possible. High beam intensity (above 3 X l O I 3 s-') [9] and thorough depletion of one hyperfine state (to the lop4 level) are not entirely compatible because of reabsorption of optical pumping fluorescence [1,10]. This problem and others associated with complete hyperfine pumping are discussed below. Approaches to obtain nearly complete ( > 90%) population of a single F , m , ground state in an alkali-metal atom involve two processes: transfer of angular momentum to the atomic beam t o put atoms into specific magnetic substates, and transfer of atoms out of the unwanted hyperfine level. To d o this, Hils, Jitschin, and Kleinpoppen [ 111 used a hyperfine-level selecting hexapole magnet followed by circularly polarized dye-laser excitation to populate the 3S,,,(F,mF)=(2,2) level in sodium.
Bechtel and Fick [2] employed laser-rf double resonance to put 95% of all sodium atoms into the (2,2) level. Schinn, Han, and Gallagher [ 11 obtained similar results using an electro-optic modulator to excite atoms out of both hyperfine ground levels at once. Our approach has grown out of the work of Watts and Wieman [6]. Like Avila et al. [ 3 ] ,we use two diode lasers to excite atoms out of both ground-state hyperfine levels. Appropriate choice of the lasers' polarizations can then put nearly all atoms into a single (F,m,) state. To populate fully a single ground level, atoms are excited out of all ( F ,m F )ground levels but the desired one; eventually, all atoms decay into the desired level and stop absorbing light. The cesium D, transitions used are indicated in Fig. 1. One laser (the "hyperfine pump") is linearly polarized and excites the
6 S , ,,F 14-
6P3,,F' = 3 F'
-
47 -
89
'
I
I
(o++u-) 852.1 nrn
T
I I I I I I
(a++a-)
2
la+
I I -- I 1 I I
I
I
I 1 I I I
I
I
2139
a+
-
ZEEMAN PU MP HYPERFINE PU MP
F =
t
4
9193 MHz
3
01993 The American Physical Society
90 2140
47 -
MASTERSON, TANNER, PATRICK, AND WIEMAN
transition to drive atoms out of the F = 4 level, while the second (the "Zeeman pump") is tuned to the
6 S ,/,F
= 3+6P3/,F'=
3
transition and is circularly polarized to transfer all atoms into the (3,3) state. By tuning the hyperfine pump laser to the 3-4 transition and the Zeeman pump to the 4-4 transition, all atoms are put into the (4,4) state. The choice of optical pumping transitions is dictated by two requirements: Atoms must be excited out of all ( F ,mF levels but the desired one and the fluorescence from the optical pumping region must be minimized in order to preserve spin polarization downstream. For pumping into the (3,3) level, then, the hyperfine pump laser cannot drive the 6s F = 4 + 6 P F'=5 transition because the AF = 1,0 selection rule prevents depletion of the F = 4 level, while the 4-3 transition is preferred over the 4-4 transition because fluorescence from the optical pumping region is twice as large with 4+4 pumping. Similarly, the 3-2 transition is unsuitable for Zeeman pumping since atoms are excited out of neither the (3,2) nor the (3,3) ground levels, while 3-4 excitation leads to ten times the fluorescence of 3-3 pumping since atoms in the desired (3,3) state continue to be excited. Analogous considerations lead to the transitions chosen for (4,4) pumping.
*
11. EXPERIMENT
The experimental setup is depicted in Fig. 2. A cesium beam of cross section 0.5 X 2.5 cm2 effuses out of a twostage oven through a glass microchannel plate [12]. This ribbon-shaped beam has a divergence [full width at half maximum (FWHM)] near 5" and a flux of 7 X cm-2 s - 1 . After passing through a Iiquid-nitrogencooled aperture, the beam passes through a collimator composed of a set of vertical vanes 2.5 cm long and spaced 0.05 cm apart and then through another cooled aperture. This results in an atomic beam flux of cm-2 s - 1 at the optical pumping region with a horizontal divergence of 25 mrad and a vertical divergence of 70 mrad. The cross section of the cesium beam allows for a long laser-beam interaction length, necessary for studies of weak transitions.
In the optical pumping region, the hyperfine pump and Zeeman pump lasers overlap and make two double passes through the atomic beam, each 0.5 cm wide (yielding a 20-,us interaction time) and separated by 1 cm. Two separate passes through the atom beam are necessary to achieve complete polarization in spite of atomic fluorescence [ I ] . Atoms which enter the desired (F,m,) level and stop absorbing laser light may still absorb the unpolarized fluorescent light emitted by other atoms during the optical pumping process and subsequently decay into other states. In the first pass, then, most of the optical pumping occurs but is not complete because of this reabsorption of fluorescence light. In the second pass 1 cm downstream, the intensity of fluorescent light is much reduced, and the atoms need absorb only one or two photons each to enter the desired nonabsorbing level. The magnetic field in the optical pumping region is 1.7 G and within 10 mrad of being parallel to the pumping laser beams. The hyperfine pump laser power is 3 mW. These laser powers provide far greater than the ten photons required to polarize each atom and suffice to saturate the pumping process (see Fig. 3). This is important since the 1.7-G field produces some splitting of (F,m,)-(F',rn;) transitions (8 MHz for the outmost magnetic sublevels). Moreover, an atomic beam whose populations d o not fluctuate as the pump lasers' frequencies and amplitudes change is essential for the parity experiment. The diode lasers are Sharp Model No. LT015 lasers with high-reflection coatings on the rear surfaces and antireflection coatings on the front surfaces, selected for lasing near 840 nm at room temperature. With heating to 35 "C and optical feedback from blazed diffraction gratings, the wavelengths are pulled to the cesium D , transitions at 852 nm and the laser linewidths narrowed to i 0 . 5 MHz. The lasers are ditherlocked to the appropriate transition peaks in saturated absorption spectrometers [I31 and will remain locked t o within 0.1 M H z for many hours under favorable conditions. These lasers oscillate in a single external cavity mode a t the lop3 level but have a non-negligible amount of power in a pedestal which extends several hundred megahertz out from the lasing frequency.
0.15 ATOMIC
BEAM
1
OPTICAL
2 0.10 LL
0
w
IW J
a W 0.05 n
z 3 CLEAN U P '
I PROBE LASER 6s -SPJ,~
FIG. 2. Experimental arrangement
LNZ-COOlED BEAM D U M P
I
I
I
2
4
6
I 8 HYPERFINE PUMP LASER INTENSITY (rnW/cm2)
FIG. 3. Fraction in wrong hyperfine level vs hyperfine pump laser intensity (no cleanup optical pumping).
91
47
HIGH-BRIGHTNESS, HIGH-PURITY SPIN-POLARIZED . . ,
After the optical pumping region, the atoms pass through a set of soot-coated vertical vanes which stop some of the optical pumping fluorescence from passing downstream. In this region the polarized atoms adiabatically follow the magnetic field as it rotates by 90” and increases to 7 G in the interaction and detection regions. Before entering the interaction and detection regions, the atoms pass through a final hyperfine pump laser beam to deplete the unwanted hyperfine level of atoms put there by absorption of optical pumping fluorescence downstream. At the interaction region the density is 1 X lo9 cm-’ and the intensity is 2 X 10’’ s-I. In the detection region the atoms pass through a 2.5cm wide probe laser beam usually tuned to either the 4-5 o r the 3-2 transition. Fluorescence from the detection region is collected by a large-area silicon photodiode (collection efficiency 30%) and allows sensitive detection of atoms in either hyperfine level. 111. PROBING THE POPULATIONS
The degree of spin polarization and the depletion of the unwanted hyperfine level are tested separately. We consider here population of the (3,3) state but comparable results are obtained when pumping into the (4,4)state. To test the depletion of the F = 4 hyperfine level, the probe laser light is horizontally polarized (parallel to the magnetic field) so that it can excite only Am=O transitions and is tuned to the cycling 4 + 5 transition, where more than 200 photons may be detected for each atom in the F = 4 level. The fluorescence of the atomic beam is then measured with and without optical pumping. Since approximately 30% of the probe laser light is absorbed by the atomic beam in the absence of optical pumping, the fluorescence value is multiplied by 1 . 2 f 0 . 0 5 to obtain a modified fluorescence which remains proportional to atomic beam density. No such correction is required in the optically pumped case since the relevant beam density is then four orders of magnitude smaller. With this correction, the ratio of fluorescence with and without optical pumping is equal to the ratio of the F = 4 populations. Optical pumping by the probe laser occurs in two ways and it is important t o know how this optical pumping affects the measured population ratio. First, m,-state redistribution by the probe laser occurs in ten excitations. Since more than 500 absorption cycles take place in the probe region, this redistribution occurs rapidly and the measured fluorescence should be independent of the initial m,-state populations. Hence our population ratio should not be strongly affected by the fact that the atoms are uniformly distributed among the F = 4 Zeeman substates in the absence of optical pumping but are nearly all in the m,=4 substate with optical pumping. To test this assumption, the magnetic field in the probe region was cancelled to give an average field of less than 0.5 G but with nonuniformities of up to 2 G. With the mF states thus scrambled, the fluorescence ratio was remeasured and found to agree with the ratio obtained with a uniform 5-G field in the probe region. A second source of optical pumping by the probe laser
2141
occurs because of leakage out of the F = 4 level due to off-resonant 4-4 transitions. At our probe intensity (slightly above the saturation intensity 1 mW/cm2), the radiative width of the 4 + 4 transition leads to one 4-4 transition and decay to the F = 3 level in lo4 4-5 excitations and is therefore negligible. The diode laser power spectrum contains large wings, however, and is the primary cause of these off-resonant transitions. Fortunately, such leakage has the same effect on the monitored fluorescence no matter how many atoms occupy the F = 4 level so that the fluorescence ratio is unaffected. A more serious problem occurs because of off-resonant excitation of F = 3 atoms and subsequent repopulation of the F = 4 level due to the probe laser’s spectral impurity, as discussed in Sec. IV. If the probe laser intensity increases above the saturation intensity, then the rate of F = 4 repopulation proceeds more rapidly than the rate of F = 4 depopulation (since the transitions leading to this process are saturated) and the measured hyperfine pumping will deteriorate. It is observed experimentally that the unpumped fraction increases slowly with probe laser intensity above 1 mW/cm2, which is chosen as our operating point. To determine the relative populations of the various rn, substates, microwave excitation on the cesium clock transition at 9.19 G H z is used to drive atoms out of specific m F states in a “flop-in” experiment. With the microwave cavity situated so that the microwave magnetic field is parallel to the 7-G dc magnetic field in the interaction region, only AmF =O magnetic dipole transitions can occur. When the microwave frequency comes into resonance with a given (3,rnF)+(4,mF) transition, ) are transferred to some of the atoms in the ( 3 , m F state the depleted ( 4 , m F ) state where they are efficiently detected by the probe diode laser 10 cm downstream. The 7-G magnetic field completely resolves the microwave spectrum (see Fig. 4) and allows simple determination of the relative populations by comparing microwave spectra with full optical pumping with spectra with hyperfine pumping alone. If one assumes a uniform RESONANCE SIGNAL (arbitrary units)
>
FIG. 4. Microwave spectrum in 7-G magnetic field. Lower:
hyperfine pumping only. Upper: full optical pumping.
92
47
MASTERSON, TANNER, PATRICK, A N D WIEMAN
2142
TABLE I . Populations of the 6S,,*F,mF level with optical pumping. -
State ( F , r n , ) (4,4)pumping: 4,4 4,3 42
4,i 4,0 All others ( 3 , 3 ) pumping:
Fraction of beam 3->4 ( a + + a - - ihyperfine pump, 4-4 0.978 0.018 0.0034 0.0003 0.00008 < 0.0005 4-3
(u4+ C I
hyperfine pump, 3-3
3,3 3,2 3,1 --
3,0 All others
-
-
distribution of atoms among magnetic substates with hyperfine pumping alone, then the relative populations of the rnF states with full optical pumping are equal to the ratios of the line intensities with full optical pumping to the intensities with hyperfine pumping alone. The populations in Table I were derived in this way. The assumption of uniform distribution among the magnetic substates with hyperfine pumping alone requires justification. Symmetry forbids an orientation (the number in the rn state different from the number in the - m state) of the atoms by the hyperfine pump laser but not an alignment. Such an alignment is miniscule, however, since on average only two absorption cycles occur before the atoms enter the desired hyperfine level and become transparent to the hyperfine pumping light. Moreover, one expects a somewhat different alignment to occur when pumping on the 3-4 transition to deplete the F = 3 level compared with 4-3 pumping for F = 4 depletion [4], while no difference is seen in the 4-3 and 3-4 microwave spectra. The relative intensities in the microwave spectra differ significantly from theoretical expectations. Transitions between large - lmF1 states are small compared to O-+O transitions. This may be explained by magnetic-field inhomogeneity in the microwave excitation region. A 10mG inhomogeneity would lead to a 7-kHz broadening of the 1 - 1 transition, a 14-kHz broadening of the 2-2 transition, etc. Since we measure only peak intensities of microwave lines rather than line strengths integrated over the entire resonances, the broadened lines look smaller. The small asymmetry in the spectrum between the m F - m F and the - m F - ' - m F transitions is due to a detuning of the cavity resonance from the 0-0 transition and a subsequent difference in microwave power on different transitions. Since we determine the relative m F populations by measuring only the ratios of microwave line strengths with and without spin polarization, these effects do not influence our results.
+
IV. LIMITS TO HYPERFINE-LEVEL DEPLETION
A high-density atomic cesium beam with less than of the atoms in one hyperfine level is difficult to
(T+
Zeeman pump.
a + Zeeman p u m p .
0.965 0.026 0.007 0.0015 < 0.0005
achieve. If an atomic beam could be produced with no atoms in one level, processes such as reabsorption of fluorescent light and collisional depolarization are likely to put atoms back into the depleted level. I n this section we consider these processes and ways to minimize them. The reabsorption of fluorescent light from the optical pumping region is the dominant source of atoms in the wrong hyperfine level in our atomic beam. Starting with a perfectly polarized cesium beam, we may estimate the effect of fluorescence reabsorption as follows. The number of atoms excited out of the desired F = 3 level per second in a slice dz of the atom beam a distance z downstream from the optical pumping region is
where F is the total beam intensity, K is the number of fluorescent photons emitted per optically pumped atom (roughly ten), ( u ) is the thermal velocity of 2.7X lo4 cm s- and ( (T, ) is the effective cross section for absorption of a fluorescent photon followed by a decay into the wrong hyperfine level. We may crudely estimate ( ur ) from uo, the 6S-6P absorption cross section on resonance. Since the longitudinal Doppler broadening in the cesium beam is 300 M H z while the natural width is 5 MHz, we expect a factor of 5 MHz/300 M H z = & to account roughly for the Doppler detuning of the reabsorbed photons. Another factor of arises since there is a 25% probability that an atom excited out of the (3,3) state will decay into the F = 4 manifold. Hence, ( o r) =u0/240. What is measured is not D but the total fraction of atoms excited into the depleted hyperfine level by the time the atoms reach the detection region:
',
+
where z,,, =O. 5 cm is on the order of the size of the optical pumping region. The fraction in the wrong hyperfine level is seen to be proportional to the beam density in accordance with the trend in the top curve in Fig. 5. At an intensity of l O I 4 s-' at the optical pumping region (or a
93 HIGH-BRIGHTNESS, HIGH-PURITY SPIN-POLARIZED . . .
47
X
X
2
0
1
. .
.A
w > w
J d 1 I LL
f
N O SPIN POLARIZATION (
z
0 t0
4
a:
X
U.
xio-4)
x
WITH SPIN POLARIZATION ( x 1 ~ - 4 )
0
W I T H SPIN P O L A R I Z A T I O N ; “CLEANUPS” BLOCKED ( x l O . * )
X 0
X
i
0
0
0
0 0
0.5
1.0
DENSITY
109
(~rn-~)
FIG. 5. Depletion of F = 4 ground level vs beam density at the detection region.
density of lo9 cm-3 at the detection region), the fraction left in the F =4 state is 0.025. Inserting this into Eq. (2) along with the 20-cm distance between optical pumping and detection regions gives
( u , ) = 4 . 4 ~ 1 0 -cm’, ’~ which is & times the resonant cross section lop9 cm2. This is reasonable because of the relevant Doppler shifts. The fraction of atoms excited into the depleted level by optical pumping fluorescence as a function of distance downstream was tested by sending in a second hyperfine pump laser beam at various positions downstream of the optical pumping region. This “cleanup” pump could deplete all atoms put into the wrong hyperfine level upstream of itself. As expected, the fraction of atoms in the depleted level at the detection region was inversely proportional to the distance from the optical pumping region to the cleanup pump. To overcome the problem of fluorescence reabsorption, we take four steps. The first and most effective is t o send in the cleanup hyperfine pump laser beam as far downstream as is practical. This depletes the unwanted level of all atoms except those excited between the cleanup pump and the detection region and reduces by a factor of 140 the number of atoms in the wrong level, as Fig. 5 indicates. The second step is to improve the cesium-beam collimation. This allows a reduction of the beam flux without a loss in signal on resonance, which is the impor-
2143
tant quantity for the parity experiment. In our case, only those atoms whose Doppler shifts are less than the 6 S + 7 S natural width ( 3 MHz) contribute significantly to the 6S-7S signal size. The vertical-vane collimator described in Sec. 11, by rejecting those atoms with large transverse velocities ( > 8 m s C 1 1, reduces the beam flux by a factor of 3.5 but reduces the residual Doppler width by nearly the same factor (from 3 1 to 9 MHz) so that the signal size is unchanged. Since the residual Doppler width remains larger than the natural width, further collimation is desirable but is difficult to obtain without a reduction in signal. The rather tight atom beam collimation allows us to take a third step since now the atoms moving downstream are collimated to within 25 mrad, while the offending fluorescent light emerges isotropically from the optical pumping region. By installing a second, sootcoated vertical-vane collimator between the optical pumping region and the detection region, we can stop some of the optical pumping fluorescence from coming downstream. This reduces the fraction of atoms in the unwanted hyperfine level by a factor of 2 for a given beam flux in the detection region. The final step taken is to choose those optical pumping transitions which minimize the amount of fluorescent light, as discussed in Sec. I. Comparison of the atomic beam flux at the optical pumping region with the total fluorescence from this region implies that less than ten photons are scattered for each polarized atom. Other sources of light capable of exciting atoms into the depleted hyperfine level must be considered. The hyperfine pump and probe lasers are both tuned to excite atoms in the depleted hyperfine level. Because of spectral impurity due to relaxation oscillation sidebands, however, there may be a small amount of power 9.2 G H z away from the nominal laser wavelength t o excite atoms out of the populated level and into the depleted level. This unwanted optical pumping will proceed until the fraction of atoms in the depleted level is equal to the fraction of laser power available to excite atoms out of the populated level. This problem can be magnified in external cavity lasers which may oscillate in several external cavity modes. To avoid this problem, we shorten the external cavities to 2.5 cm. This forces the lasers to run mainly single mode [I41 and puts the mode spacing at 6 G H z so that any residual sidemodes cannot excite the hyperfine transition 9.2 G H z away. As a final precaution, flat Fabry-Ptkot filter cavities with 1-GHz linewidths and 20-GHz free spectral ranges are placed in the hyperfine pump and probe laser beams t o minimize power a t the unwanted hyperfine transition. Finally, absorption of Zeeman pumping laser light scattered off windows and other optics also degrades the hyperfine depletion. The intensity required t o limit delevel is less than 10 nW/cmz. Scatpletion to the tered light is the most likely reason the depletions with and without spin polarization d o not agree in the limit of low atom beam density. Extensive baffling of stray light both inside and outside the vacuum chamber reduces this source of background atoms to less than 10% of that due to photon reabsorption a t the densities of interest t o us.
94 MASTERSON, TANNER, PATRICK, AND WIEMAN
2144
Collisions are the last potential source of atomic beam depolarization. Cs-Cs, Cs-surface and Cs-background gas collisions need t o be considered. The largest Cs-Cs collisional depolarization mechanism is spin exchange in elastic collisions. The spin-exchange cross section for cesium is [15] C T ~ ~ . 1~0 -~l ~ = cm2 ~ X
and leads to a spin-exchange rate:
This leads to a thermalization of hyperfine populations:
where g , is the degeneracy of the F level. If we start with a fully polarized beam in the F state and assume a low depolarization rate, so that n,/n = 1, then (5)
where doP.detis the distance between the optical pumping and detection regions. When this is 20 cm, we therefore expect
nF
This assumes, however, that all atoms put into the depleted hyperfine level by spin-exchange collisions are detected downstream. Since atoms undergoing these collisions may have their trajectories significantly altered, only a small fraction of them contribute t o the residue detected downstream. In fact, only about 0.01 of the depolarized atoms enter the detection region with small enough Doppler shifts to be detected as part of the atomic beam. [Spin-exchange depopulation is, moreover, easy to detect: Since electron spin polarization is preserved in these collisions, depolarizing collisions cannot occur between two atoms in the (4,4) state. Hence, spin-exchange depolarization would show up unequivocally as a difference between (3,3) pumping and (4,4) pumping, which is not observed experimentally.] Collisions with surfaces lead t o complete depolarization and need t o be avoided. In our case, the use of liquid-nitrogen-cooled apertures and a cooled beam stop downstream of the detection region prevents atoms from striking surfaces and then wandering into the detection region. Finally, collisions with background gases must be considered. Nitrogen is fairly innocuous, with a depolarizing cross section of 5 X cm2 [16], but molecular oxygen has a much larger cross section and since oxygen is much lighter than cesium, 0,-Cs collisions may result in depolarization without significantly changing the Cs atoms’ directions. We find that a pressure less than 1.OX Torr makes this background source small relative t o other sources.
;
47 -
V. RESULTS
Figure 5 plots the hyperfine depletion as a function of atomic beam density at the interaction region. Without spin polarization, all but fewer than 7 X l o p 5 of the atoms are removed from the F =4 level. Contributions to this residue of unpumped atoms have been discussed above. With spin polarization the residue is much larger and rises with atom beam density, indicating that reabsorption of fluorescent libht from the optical pumping remains the dominant source of atoms in the wrong hyperfine level. Since both the hyperfine pump and Zeeman pump intensities in the optical pumping region are greater than the saturation intensity and the laser linewidths are much less than the radiative linewidth of the 6S-6P transition, Zeeman coherences are possible which leave atoms in nonabsorbing states before they reach the desired ( F , r n F )level [17]. Since the optical pumping takes place in a 1.7-G magnetic field, however, Am,=2 coherences oscillate at more than 3 Mrads-’ and higher-order coherences oscillate more rapidly. With an optical pumping time on the order of a few microseconds, then, such coherences are destroyed before atoms evolve into nonabsorbing states. Figure 6 plots hyperfine depletion as a function of magnetic field, showing the destruction of coherence effects by fields greater than 0.4 G . The ramifications of this spin-polarized cesium beam for the atomic parity-nonconservation experiment are threefold. First, since all atoms in the atomic beam are placed into the correct spin state to undergo 6 s - 7 s transitions, there is a sixteenfold increase in useable beam density for a given total density. Because the overall den-
10-1 W
>
-
w
WITH SPIN POLARIZATION
-1
X WITHOUT SPIN POLARIZATION
W
5 U [r
i
0
a 3
cc
U
g
10-4
*-*
--
0
x x x x MAGNETIC x 0.2 FIELD 0 . 4 (G)
FIG. 6 . Hyperfine depletion vs magnetic field: polarization; X , without spin polarization.
0.6
0 , with
spin
95
47
HIGH-BRIGHTNESS, HIGH-PURITY SPIN-POLARIZED . . .
2145
sity is reduced by the extra collimation required for spin polarization, the useable beam density is only increased by a factor of 2.3. Second, the elimination of neighboring transitions out of different spin states removes dilutions d u e t o neighboring lines a n d allows the resolving magnetic field t o be reduced from 7 0 t o 7 G, thereby making hyperfine mixing effects negligible. T h i r d , the use of the shelving approach allows the detection efficiency t o be increased from 25% t o 70% a n d simplifies the design of a 6 s - 7 s excitation region free of stray fields. Since t h e background fluorescence due to unpumped atoms is shot-noise limited over the range of densities explored here, the P N C signal-to-noise ratio does not change a s the density changes, so the largest practical density is used in the P N C experiment in spite of t h e increase in t h e
size of the background. To summarize, a spin-polarized cesium beam with thorough depletion of o n e of the two ground-state hyperfine levels has been demonstrated. I t will prove useful in a parity-nonconservation experiment a n d is of interest for other possible shelving experiments.
'Present address: Melles Griot Electro-Optics, Boulder, CO 80301. tPresent address: Department of Physics, University of Notre Dame, Notre Dame, IN 46556. [l] G. W. Schinn, X. L. Han, and A. C. Gallagher, J . Opt. SOC.Am B 8, 169 (1991). [2] H. Bechtel and D. Fick, Nucl. Instrum. Methods A 257, 77 11987). [3] G. Avila et al., Phys. Rev. A 36, 3719 (1987). [4] R. E. Drullinger et a/.(unpublished). [5] G. Avila et ol., Metrologia 22, 111 (1986). [ 6 ] R. N. Watts and C. E. Wieman, Opt. Commun. 57, 45 (1986). [7] M. C . Noecker, B. P. Masterson, and C. E. Wieman, Phys. Rev. Lett. 61, 310 (1988). [8]D.-H. Yang and Y.-Q. Wang, Opt. Commun. 73, 285 (1989).
[9] Our definition of intensity follows the convention of N. F.
ACKNOWLEDGMENTS This work was supported by National Science Foundation G r a n t No. PHY90-12244. W e a r e indebted t o W. Swann for the design a n d construction of the laser diodes used in this work a n d t o T . Walker for helpful discussions.
Ramsey, in Molecdar Beams (Oxford University Press, Oxford, 1956). [lo] D. Peterson and L. W. Anderson, Phys. Rev. A 43, 4883 (1991). [ I l l D. Hils, W. Jitschin, and H . Kleinpoppen, Appl. Phys. Lett. 25, 39 (1981). [I21 S. Gilbert and C. Wieman, Phys. Rev. A 34,792 (1986). [13] C. E. Tanner and C. Wieman, Phys. Rev. A 38, 162 (1988). [14] L. Hollberg informs us that both shortening and lengthening the external cavity (to make the cavity mode spacing very different from the relaxation oscillation frequency) can force single-mode oscillation. [15] W. Happer, Rev. Mod. Phys. 44,169 (1972). [16] N . Beverini et a [ . , Phys. Rev. A 4, 550 (1971). [17] G. Theobald et al., Opt. Commun. 71, 256 (1989)
Hyperfine Interactions 81(1993)27-34
27
Parity nonconservation in atoms; past work and trapped atom future Carl E. Wieman Joint Institute for Laboratory Astrophysics, University of Colorado and National Institute of Standards and Technology, and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440, USA
Parity nonconservation has now been measured in atomic cesium with a fractional uncertainty of 2%.This was done by observing the 6s-7s laser excited transition rate in a "handed" apparatus. When combined with recent precise calculations of the cesium atomic structure, this provides an important test of the Standard Model. Efforts are under way to achieve a more sensitive test by measuring parity nonconservation in a series of radioactive cesium isotopes which have been trapped using laser light.
1.
Introduction
The basic process we are studying is the exchange of a Zo boson between the electrons and the neutrons or protons in an atom. This gives rise to a parity nonconserving (PNC) neutral current interaction, and the goal of this research is to measure the strength of this interaction and thereby test the Standard Model. Because it is parity violating, this interaction mixes the parity eigenstates of an atom, more specifically, the S states have a very small amount of P state mixed in with them [l]. It is this mixing that is measured in the experiment. The amount of P state &Nc is equal to the weak charge times an atomic matrix element ( y5), which is found by calculating the electronic structure of the atom. The weak charge is the quantity we are interested in, since it is related directly to the fundamental interactions. This weak charge [2] is Q, = 2[(2z
+ N)C,U+ (Z + ~ N ) c+, small ~ I c;, C:
terms.
(1)
The quantities C," and C,"are the fundamental electron quark-neutral-current coupling constants and characterize the quark vector current couplings. There are additional terms involving the axial vector currents for the quarks, as noted, which are much smaller and will be neglected in this discussion. Neglecting radiative corrections, in the Standard Model [23, we have C," = 1/2 - (4/3)sin2OWand C," = -1/2 + (2/3)sin20w.The primary goal of this work is to see if Q,, and hence these constants, are as predicted by the Standard Model, or whether there are small 0 J.C. Baltzer AG, Science Publishers 96
97
28
C.E. Wieman / Parity noncomentation in atoms
corrections to them due to the existence of new physics not included in the Standard Model. The remainder of this paper will discuss how one determines Qw in atoms. It should be emphasized that these experiments are measuring very small quantities and therefore are quite difficult. The mixing of S and P states is typically only about one part in 10”. Nevertheless, these experiments have now succeeded in measuring this mixing to a few percent and should reach a few parts in lo3 in the future. 2.
General experimental approaches
This section will discuss the general experimental approaches which are used to measure the quantity &NC. It is determined by measuring an electric dipole transition between two S states in an atom (or two P states). Such electric dipole transitions are absolutely forbidden by the parity selection rule, and thus when one measures such an E l amplitude it directly indicates the amount of P states which are mixed in with the S states. The “brute force” method that first occurs to most people is to measure such an E l amplitude by looking directly at a “pure” PNC induced rate by simply driving a forbidden S to S transition. The rate in this case is given by the square of the transition amplitude
The transition rate is proportional to (&Nc)~and therefore would be about times a typical allowed electric dipole transition rate. This is far too small to ever be observed and therefore this experimental approach cannot reach the necessary level of sensitivity. The method actually used in all successful atomic PNC experiments is to look for an interference between the PNC amplitude and a parity allowed electromagnetic amplitude. In this case, the transition rate is given by R = I Ao f ApNCI2 = & f 2AoAp~c+ A&c.
(3)
Here, A. is an allowed electromagnetic transition amplitude and A ~ N is C the parity nonconserving amplitude we are interested in measuring. The important point is that the term 2AoAp~cis linear in the panty nonconserving amplitude and therefore can be large enough to measure. The signal of interest is the fractional modulation in the transition rate ARIR when the parity of the experiment (and hence the sign of the interference term) is reversed. In a real experiment, this means carrying out a minor reflection of the experiment and seeing how the transition rate changes. In the Colorado experiment, we interfere ApNCwith an electric field induced E l amplitude by applying a dc field to the atoms. This field produces a parity conserving mixing of S and P states
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C.E. Wieman / Parity nonconservatwn in atoms
29
Here, S, is the electric field or “Stark” mixing term. We observe the interference between the electric field induced mixing and the neutral current induced mixing of P states into the S states, This “Stark interference” method was first used to measure PNC in thallium by the group at Berkeley and in cesium by the group at ENS in Paris. This approach has the advantage that there are a large number of independent reversals which modulate the signal and this makes the systematic errors relatively small and tractable. We have chosen to study cesium because it is a high 2 alkali atom. The weak charge is approximately proportional to the number of neutrons, and the relevant atomic matric element is proportional to the square of the number of protons, so &NC increases approximately as Z3. Being an alkali atom, the atomic structure of cesium is relatively simple, so (75) can be accurately calculated. 3.
Colorado experiment
We will now discuss the details of the Colorado experiment [3].We use a tunable dye laser whose green output excites the 6 s to 7s transition in cesium as shown in fig. 1. The 7s state then decays to the 6P state and subsequently to the 6 s state. We observe the fluorescence on the 6P to 6 s transition to monitor the 6 s to
excite
‘
i
6P
I
I
detect I
. .
: * .
I
*
. ..I
Fig. 1. Relevant energy levels for the cesium atom.
7s rate. The heart of the experimental apparatus is shown in fig. 2. An intensecollimated-cesium beam intersects a standing wave laser field. The laser field is produced by sending light into a Fabry-Perot cavity which is resonant at the frequency of the laser light. This produces a very intense field in the cavity. This
99
C.E. Wieman Parity nonconservation in atoms
30
\m
INTERFEROMETER MIRROR
‘LASER
BEAM
Fig. 2. Interaction region of Colorado PNC experiment.
field excites the laser beam in a region in which there is a “handedness” produced by having a coordinate system defined by the electric field, magnetic field, and photon angular momentum vectors. We measure a fractional change in the 6s to 7s rate ARIR of about 2 x when the handedness of this system is reversed. This reversal is done in four independent ways; E goes to - E , B to - B , CT to -0(this is done by changing the laser light from right to left circular polarization), and m to -m.This m reversal is carried out by changing the laser frequency to excite the opposite m or Zeeman quantum level. Having four independent reversals puts in a tremendous amount of redundancy since, in principle, only one reversal is necessary to observe the parity violation. This redundancy greatly suppresses systematic errors. 4.
Results
In fig. 3, we show the results of all the experimental measurements of parity nonconservation in atomic cesium which is the most thoroughly measured atom. At the top are the two measurements of the Paris group [4], and at the bottom is our 1985 result and our 1988 result [3]. In order to obtain useful elementary particle physics from the experimental numbers, however, it is necessary to know the atomic structure. This comes about because of the fact
Q,
=
8PNC (Y5)nucleus ’
100
C.E. Wieman / Parity noncomemation in atoms
31
COLORADO ‘85 i--O--i COLORAD0’88
W
(2%)
I 0
0.5
1 .o
1.5
2.0
-1m ( E P N C ) / P [mV/cm] Fig. 3. Results of PNC measurements in cesium. This shows the parity nonconserving electric dipole transition amplitude between the 6s and 7s states in units of the amount of dc electric field which would be needed to give a Stark induced amplitude of the same size.
where hNC is the experimentally measured quantity and (y5)s)nucleus is an atomic matrix element which can only come from atomic theory calculations. The most recent and most accurate calculation of (‘ys) has been done by the group at Notre Dame, and they have achieved a 1% uncertainty [ 5 ] . Combining the Notre Dame theory and the Colorado experiment gives Q, = 71.0 f 2% f 1%,
where the first uncertainty is experimental and the second theoretical. Alternatively, this can be expressed in terms of the value of sin2&. This is sin2 8, = 0.223 f 0.007 f 0.003.
If we now compare this value with values obtained from high-energy experiments, we have a precise test of the Standard Model. As shown in fig. 4, this test has a unique sensitivity to a variety of new physics. This figure shows that there is very little experimental data on the C,“coupling constant other than atomic PNC. Thus, any new physics which primarily affects this coupling constant can only be tested by comparing atomic PNC and high energy results. A variety of models (particularly ones involving neutral bosom) have been put forth which would affect this coupling constant. These results set the best constraints on the possible masses or coupling constants in most of these models. 5.
Future
Since the mass of the z is known to better than one part in lo3,any improvement in the determination of Q, will provide a more sensitive test for “new” physics. In
101
C.E. Wieman / Parity nonconservation in atoms
32
c: 1 .o
1 .o
su,
x
u,
-1.0
Fig. 4. Plot of quark axial vector-electron vector coupling constants for the up and down quarks with the electron. The cross hatched area is the region allowed by the SLAC deep inelastic electron scattering results. The solid line is the region allowed by the cesium PNC results. The round spot is the Standard Model prediction. while the arrows indicate how the values would change for three different alternative models.
the near term, we will have an improved experiment at Colorado which uses somewhat fancier laser technology and an optically pumped atomic beam. This experiment is now running with a signal-to-noise ratio which is significantly better than was obtained in the 1988 experiment. Since the previous result was limited entirely by statistical uncertainty, we expect to improve our measurement uncertainty to several parts in lo3 within this year. However, as the experiments get better, the principal limitation will become the uncertainty in the calculated value of (y5). There have been credible speculations that it will be possible to do the theory in cesium to a part in lo3. However, it is not clear when these calculations will be completed and the question of how to check their accuracy becomes a major issue. We have begun a longer term experimental project to try to deal with the atomic theory question. The basic idea is to compare precise measurements of atomic PNC for different isotopes of cesium. The weak charge is sensitive to the number of neutrons and hence will change for different isotopes, but the atomic
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C.E. Wieman / Parity nonconservation in atoms
33
matrix element depends on the electronic structure and hence is almost independent of the number of neutrons. We believe these experiments will be possible because of the new technology of laser trapping which will allow us to collect and hold the isotopes to be measured [6]. If one then looks at appropriate combinations of experimental results, for example
the atomic matrix element will drop out, leaving a ratio of weak charges as shown, which can be directly compared with Standard Model prediction. To achieve adequate sensitivity in a PNC experiment, it is necessary to have a relatively high density of atoms with a low velocity spread. Because of the small number of short-lived (as short as 1 minute) isotopes which can be produced, this experiment is only feasible if these atoms can be efficiently collected and held. We have been developing laser trapping technology which we believe will accomplish this. The laser trap we use [7] is based on the force exerted on the atoms by laser photons scattering off them. An inhomogeneous magnetic field acts to regulate the scattering rate, and hence this force, in a spatially dependent manner. The result is that the atoms are pushed to one point in space and held there by the light pressure. Using this approach, we have achieved atomic densities of 10'0/cm3 and, with minor modifications, have been able to cool these atoms to as low as 1 pK [6]. The proposed trapped atom PNC experiment will work as follows. First, the cesium isotopes will be produced at a facility which is currently under construction at LAMPF. These isotopes will be inserted into a small cell which is filled with the laser beams which make up the optical trap. The efficient trapping of atoms in such a cell is a complicated problem and is discussed in detail by Michelle Stephens in another paper in this volume. After the atoms have been collected in the laser trap and cooled to 4 1 mK, they will be transferred into a second trap for the PNC measurement. The second trap will probably be either a purely magnetic trap or a far off resonance dipole force optical trap. These traps are rather shallow but they avoid the perturbations on the atoms due to frequent excitation and decay by trapping light. In the PNC measurement trap, the 6s + 7s transition will be excited in the presence of electric and magnetic fields, similar to our past experiments. Using this approach, we expect to measure the fractional change in Q, between lZsCs and 139Csto about 2 parts in lo3. This will provide a stringent test of the Standard Model, and may well reveal new physics beyond it.
Acknowledgement This work was supported by the National Science Foundation (US). The author is pleased to acknowledge all the people who have done the work discussed
103 C.E. Wieman / Parity nonconservation in atoms
34
here: S. Gilbert, M.C. Noecker, P. Masterson, C. Tanner, D. Cho, C. Wood, K. Lindquist, and M. Stephens. References M.A. Bouchiat and C. Bouchiat, Phys. Lett. 48(1974)111; I. de Phys. 35(1974)899; I. de Phys. 36(1975)493. E. Commins and P. Bucksbaum, Weak Inferactions of Leptons and Quarks (Cambridge University Press, 1983). M.C. Noecker, B.P. Masterson and C.E. Wieman, Phys. Rev. Lett. 61(1988)310, and references therein. M.A. Bouchiat, I. Gum%L. Hunter and L. Pottier, Phys. Lett. B 117(1982)358;I. de Phys. 47(1986)1709. S.A. Blundell, W.R. Johnson and I. Sapirstein, Phys. Rev. Lett. 65(1990)1411. C. Monroe, W. Swam, H. Robinson and C. Wieman, Phys. Rev. Lett. 65(1990)1571. E. Raab et al., Phys. Rev. Lett. 59(1987)2631.
Proceedings of the 1992 Fermi Summer School on “Frontiers of Laser S p e ~ t r ~ ~ ~Varenna, ~py”, I t a l y (T.W. Hansch and M. I n g u s c i o , Eds.1994) Course CXX, pp. 240
-
Measurement of Parity Nonconservation in Atoms
Carl E. Wieman, Sarah Gilbert,’ Charles Noecker,b Pat Masterson,c Carol Tanner,d Chris Wood, Donghyun Cho and Michelle Stephens
Joint Institute for Laboratory Astrophysics University of Colorado and National Institute of Standards and Technology and Department of Physics, University of Colorado Boulder, Colorado 80309-0440
a) Present address: National Institute of Standards and Technology, Boulder, CO b) Present address: Harvard Smithsonian Institute for Astrophysics, Cambridge, MA c) Present address: Melles Griot Inc., Boulder, CO d) Present address: Dept. of Physics, Univ. of Notre Dame, South Bend, IN 104
105
TABLE OF CONTENTS
. ............................................. 2. PNC Neutral Currents in Atoms ................................. 1 Introduction
..................................... 4 . Design Concepts of Colorado Experiments .......................... 5 . DetailsofApparatus ........................................ 5.1. Signal and Noise Analysis ................................. 5.2. AtomicBeam ......................................... 5.3. Laser Power Buildup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Dewtion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Technical Noise ...................................... 5.6. Servo Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7. Field Reversals and Signal Processing .......................... 6. SystematicErrors .......................................... 7. Implications ............................................. 3. Experimental Approaches
8. Future Improvements ....................................... 8.1. NwTerm .......................................... 8.2. Long Term ..........................................
1
5 9
15 18 18 21
23 25 28 30 35
37 44
49 49
54
106
1 1.
Introduction
This paper is an introduction to the subject of parity nonconservation in atoms. It is intended for the student or scientist who is not familiar with the field. The Colorado cesium experiment is described in detail, and we have attempted to present many of the technical details and considerations that led to the find experimental design. The basic phenomenon discussed in this paper is the parity nonconserving (PNC)weak neutral current interaction in an atom. As shown in Fig. 1, this interaction arises from the exchange of a Z,boson between the electrons and the protons and neutrons in an atom. It
can be contrasted with the much larger Coulomb interaction that arises from the exchange of a virtual photon between the electrons and the protons and predominantly determines the
structure of the atom. Study of PNC in atoms began in earnest with the Weinberg Salam Glashow (WSG) electroweak theory which unified the weak and electromagnetic interactions and predicted a parity violating neutral current interaction.
In an atom, the parity nonconserving Hamiltonian can be written as the sum of two Parts'
107
2
where GFis the Fezmi constant and 73 is the fifth Dirac gamma matrix and operates on the electron. The largest part of the PNC Hamiltonian arises from the so-called weak charge,
Q,. This involves the axial-vectorquark current where the contributions from all quarks simply add in similar manner to the way the electromagnetic charges add. The weak charge,
involves the number of protons (Z) and neutrons (N), and the coupling constants C1, and C,,.? These are two of the four fundamental coupling constants which characterize the neutral current interaction between electrons and quarks. They are multiplied by the appropxiate factors involving the number of up and down quarks respectively. The total Q,
is nearly proportional to the number of neutrons contained in the nucleus. The H2PNC term, arising from the nuclear spin dependent or "weak isospin" contribution, is much ~maller.~ This contribution is about 1%of the total PNC, and because the spins of the quarks in the nucleus tend to add up, they cancel. Actually, this nuclear spin dependent part also has two parts, the smallest of which is the weak neutral current contribution involving the Z, exchange. The much larger part e x l o ) is the nuclear anapole moment contrib~tion.~ This contribution comes from the more traditional weak interactions that take place in the nucleus which then couple to the electrons. We will discuss this in
more detail in Section 7 below.
108
3 The primary interest in studying these interactions is to test the "Standard Model" of the elementary particle interactions. This Standard Model is made up of the WSG electroweak theory plus quantum chromodynamics to account for the strong interactions. h the Standard Model (neglecting radiative corrections), the constants C,, and C,, are given by2
c,, = *A- 4/3 sin2ew
The goal of atomic parity nonconservation work is to determine if these constants are accurately predicted by these formulas or if there are some corrections to these values which arise from "new" physics. By "new" physics, we mean physics that is not included in the
Standard Model.'
To be more specific, we test the Standard Model and hence, look for this new physics using the following prescription. 1) Obtain the value of sin2ewfrom the mass of the 2,
boson, which is now measured very precisely at CERN. 2) Use this value to calculate Q, as in Eq. (3) plus the addition of small radiative corrections we neglected. 3) Compare the calculated value of Q, with what one obtains from atomic parity nonconservation. Before beginning a discussion of the very lengthy and difficult experiments and atomic theory which have been undertaken to determine Q,, the obvious question is: "Why bother?" What is the motivation for doing this? The answer to this is that the Standard Model is
almost certainly not the final answer. It has many undesirable features; in particular, there
109
4
are a number of important quantities that have to be put in on a somewhat artificial ad hoc basis. Notable among these are the coupling constants, the masses of all the particles and
certain basic symmetries such as the handedness of the neutrinos. None of these different quantities arise naturally out of the model, as one would expect from the ultimate theory. In addition, the StandardeModel runs into trouble at high energies. As the energies become large compared to the mass of the 2, boson, the theory becomes divergent.
These features indicate that there must be new physics beyond the Standard Model. We note that many aspects of the Standard Model have only been tested to about 2%. This
is not like quantum electrodynamics where the theory has been tested to nine decimal places and one is asking whether there might be some deviation in the tenth. Because the Standard Model has only been tested relatively crudely, there is still a reasonable likelihood that small improvements in precision may lead to fundamental insight into new physics. Finally, the Standard Model does not occupy an unchallenged position. Many extensions and alternatives have been proposed, and atomic PNC measurements are uniquely sensitive to the physics that
many of these would produce. The reason for this unique sensitivity is that the two coupling constants, Cldand Clu, are measured very poorly, or not at all, by the high energy experiments. In addition, there are a number radiative corrections to the Standard Model, which have energy dependencies. Since the atomic PNC is at low energy, the comparison of atomic parity nonconservation with high energy results is sensitive to such energy dependent
terms. All these features have combined to make atomic parity nonconservation perhaps the
only mainstream particle physics that is currently being done on a table top scale.
110
5 2.
PNC Neutral Currents in Atoms Many of the early concepts of the effects of neutral currents in atoms were introduced
by Curtis-Michel in the 1960's.' However, activity in the field expanded rapidly after the papers of M. A. Bouchiat and C. Bouchiat in 1974lS in which they considered this in the context of the WSG model. The principal effect of this parity Violating interaction in an atom is to mix the S and P parity eigenstates so that the S state is no longer a pure S state, but has a very small amount
(bNJof P state mixed into it,
* (5xlO-'*) (ZC,) (Z2)
-
lO-" !
This quantity SpNc involves the Fermi constant of the weak interactions GF,the weak charge, Qw, mentioned before, and an atomic matrix element which is simply the matrix element of the ys evaluated over the nucleus, The evaluation is only over the nucleus
because the 2, boson is a massive particle and thus this is a short range interaction. It is straightforward to estimate the approximate size of the mixing
m.01.
We have already
indicated the weak charge is proportional to 2 multiplied by constants which are on the order
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6
of one. The T~ matrix element, as pointed out by the Bouchiats, is proportional to Z2. For a relatively heavy atom like cesium, the mixing then works out to be about lo-*'! It is this very tiny s a l e that makes the experiment so difficult and fraught with the possibility of error. To put this in perspective, a ratio of one part in 10" is the same as the ratio of the diameter of a human hair to the diameter of the earth.
In these experiments, the miXing of S and P states is observed as an electric dipole transition amplitude between S states with different principal quantum number n. Or alternatively, between P states with different n. In introductory quantum mechanics one
learns that such electric dipole transition amplitudes are absolutely forbidden by the parity selection rule and therefore, if one measures this electric dipole transition amplitude, it is a measure of how much P state is mixed into the S state. The most direct approach to measuring this transition amplitude would be to simply take a laser, set its frequency in resonance with some S * S transition in an atom, and look for the excitation rate, trying to pick out the part that was due just to the electric dipole contribution. In fact, in the days before the Bouchiats' paper, this approach provided the
best test of the conservation of parity in atoms and set the limits at that time. However, the
parity nonconserving rate is proportional to the square of bNc,
Effectively, this means the oscillator strength for such a transition is about lo-= and therefore, the transition rate is 22 orders of magnitude smaller than a normal allowed electromagnetic transition. With any conceivable experiment, this is an impossibly small
112
7
transition rate, and will always be lost in the noise. Therefore, this approach is clearly unsuitable for achieving the level of sensitivity needed for investigating the weak neutral
current effects. As discussed by Michels and the Bouchiats, a much more intelligent approach is to use
a 'heterodyne" or interference approach. In this case, the transition rate, RH,is equal to the square of the sum of two transition amplitudes, one being the parity nonconserving amplitude
and the other being a much larger parity conserving transition amplitude between the S
states: RH = l A O f A p N C 1 2
2 a
4i2AOApNC+APNc
Here, ApNC is the quantity we are interested in, and amplitude. Notice that there is now a term 2A, A,,,
'
(9)
is the parity conserving
which is linear in the parity
nonconserving amplitude, and therefore can be large enough to measure. It is this interference term that we are interested in determining. This basic idea of mixing and detecting a small amplitude by having it mixed with a large amplitude is, of course, a very
common one in physics and goes back to at least the early days of radio, if not before. It is necessary to give some thought as to the phases of these amplitudes to make sure that such a parity violating interference term can exist in any given experiment. One can work through a l l the mathematics and selection rules to find the appropriate conditions, but the basic point is that this is a parity nonconserving term, and therefore one has to design an experiment which has a handedness built in. If the experiment has no such handedness, it will not be sensitive to a PNC effect. The second important question is, assuming the
interference term exists, how does one isolate it from the much larger b2contribution to the
113
8
rate? This is done by observing the modulation in the transition rate when the handedness is changed, or in other words, a mirror reflection of the experiment is carried out. In that case
there is a change ARH in the transition rate due to the parity violating 2A, APNCkrm reversing sign. Thus, all the experiments we are going to describe determine a fractional change in the transition rate A R H / R H . A key point is that the transition rate itself, RH, is a
very weak transition by n o d standards, and the fractional changes we are interested in are very small. For example, in our experiment RH corresponds to an oscillator strength of lo-’* and the modulation ARH is on the order of one part in 106. Thus, the basic scale of the
experiment is set by the fact that one has a very weak transition, and, in order to see small modulations on that transition, there must be a very high signal-to-noise ratio. It is this issue which makes these experiments quite difficult, and is behind most of the design considerations that we will discuss. There is also a continuing tradeoff between a large 4, which makes ARH larger, but ARH/RH smaller, and a small pb, which does the opposite.
114
9
3.
Experimental Approaches The various experimental approaches that have been tried can be categorized according
to what they use as an interference amplitude, &. The first choice was to use an allowed
magnetic dipole (Ml) amplitude for &. In this case, one drives a transition between two P
states with the Same principal quantum number. This experimental approach has been used to measure parity violation in bismuth, lead and thallium. In this work, the actual observable
is the optical rotation of the plane of linear polarized light. This corresponds to looking at a difference between the index of refraction for left versus right circularly polarized light. Obviously, such a difference reflects a handedness to the system. The basic set-up for such experiments is shown in Fig. 2. The laser beam passes through a linear polarizer to insure that its polarization is very clean. It then passes through a vapor cell which contains the atom of interest, and then finally goes into a second crossed polarizer. This second polarizer blocks out all the laser light unless its polarization has been rotated in the vapor, in which case some light passes through and can be seen at the detector. Of course, the actual
experiments are somewhat more involved, but this is the basic idea. One then tunes the laser over the atomic transition and observes a small rotation in the polarization as an increase in
the light at the detector when tuned on the transition. Several steps have been taken to improve the signal-&noise ratio and to test for potential systematic errors, which have always dominated the uncertainty. First, to improve the signal-to-noise ratio, the polarizers are rotated slightly from perfectly orthogonal, and the incident polarization is modulated using a Faraday rotator. This latter step reduces the noise by shifting the detection bandwidth away from dc. To eliminate sources of pgtential
115
10
systematic errors, some or all of the following steps have been taken in the various experiments: 1) alternating between an oven containing atomic vapor and an identical oven with no vapor, 2) reversing the direction of the light through the vapor, and 3) careful fitting
to the atomic line shape. The parity nonconsemation signal is dispersion shaped and thus has quite a different dependence on laser frequency from the absorption. The second approach is to use a Stark induced
amplitude for the interference. In
this approach, one simply applies a dc electric field to the atom. This electric field mixes S and P states by the Stark effect, in a parity conserving way as shown by
One now obsewes an interference between the & and bpNc mixing terms. This approach has been used to measure parity violation in cesium by the groups at Paris and Boulder, and in thallium at Berkeley. Each of these approaches (allowed M1 and Stark induced) has certain advantages and disadvantages. While these often get quite technical, we can summarize some of the more
notable features. First, let us consider the M1 approach. It can be characterized by the fact is quite large, which makes AR/R small. However, since the transition rate itself is large, the statistical signal-to-noise can be quite good. Another advantage of these
experiments is that they are relatively simple. The drawback to this approach is that because the fractional modulation is quite small (- 1 part in los) and there are relatively few reversals to isolate the PNC component, the systematic errors are major problems.
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11
This can be contrasted with the Stark interference approach. Here
is small and the
problems generally are statistical signal-to-noise. There are many reversals to isolate the
parity nonconserving effects and suppress systematic errors and the fractional modulation is relatively large. Therefore systematic errors are much less of an issue. Because these experiments involve extra applied fields and more reversals, they tend to be rather more complicated, however. The other important experimental issue is the choice of atom. Here there are a number of considerations that come into play. First, one wants a heavy atom because of the Z3 dependence IEq. (7)] of the mixing. Second, one needs an atomic transition between states with the appropriate quantum numbers. If one is using the magnetic dipole interference this
needs to be a nP to nP' transition. If one is using the Stark induced approach, either an
nS+n'S or nP+n'P will work. In this case it is desirable to have different initial and final
n's to suppress the magnetic dipole amplitude, which can introduce systematic errors and background noise. Also, in all cases it is desirable to have the transition at a wavelength where a good tunable laser is available to excite the atom. If one is using fluorescence detection (as in Stark induced experiments), it is important that the fluorescing light be at a wavelength that is significantly different from the excitation light. Otherwise, the scattered excitation light will give an overwhelming amount of background. Another consideration is that the atoms must remain isolated rather than forming molecules, where the PNC effects
are obscured. The final important issue, and at this stage of the field clearly the most important, is the accuracy with which the 'ys matrix element for the atom can be calculated.
This is important because one is now worried about precision tests of the Standard Model
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12 rather than simply detecting parity nonconservation. We will discuss this issue in more detail later. Here we will simply summarize that for alkali atoms such as cesium, francium and rubidium, these calculations can be done with an accuracy on the order of 1%. In thallium calculations the accuracy is on the order of 5 % , while for systems such as bismuth and lead, with many valence electrons, the accuracy of calculations of the ys matrix element is
relatively low. Before discussing the experiments in more detail, we might mention a few other possibilities for PNC experiments which have been proposed but not carried out. Among other interference amplitudes that can be used, there really are very few that have been seriously discussed. There have been proposals by Anderson6 to interfere a two photon allowed transition amplitude with the one photon PNC amplitude. In this case, the interference is sensitive to relative optical phases. This can be both an advantage and a disadvantage, however, and this approach has not been seriously pursued up to this time.
A number of other experiments have been proposed, and a few attempted, involving other atomic species. Probably the most notable is the use of atomic hydrogen. In this experiment, the idea was to observe the mixing of the 2s and 2P states of hydrogen. Although Z is 1, and therefore the PNC matrix element is quite small, this is largely offset by the mixing energy denominator, which is nearly zero for these two states. Hence, the actual bpNc for n=2 hydrogen is nearly the same as that for heavy atoms. Because of this
there were a number of experimental programs initiated to study PNC in hydrogen. However, the great problem with the hydrogen case is systematic errors. Stray electric fields, which can also cause mixing of the 2s and 2P states, are amplified by the same near-
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13 zero energy dominator. As a result, the systematic errors, relative to the PNC signal, are enhanced compared to heavy atom by a Z3factor. Effectively, this means that instead of
needing to worry about millivolt/cm stray fields, one has to worry about nanovolt/cm fields.
This is a nearly impossible problem and therefore, to our knowledge all the experiments on hydrogen PNC have now been abandoned. Another set of experiments which have been proposed and pursued to some degree involve the use of ions or muonic atoms for studying PNC. In these cases,the overlap of the electrons at the nucleus is much larger than for a normal atom, and therefore SpNc can be relatively large. However, this is more than offset by the fact that the sample size is very small. At the present time, the technology is not available to produce large enough samples
to allow meaningful PNC measurements. However, this is very much a function of technology, which is likely to change. We would now like to discuss the first generation of Stark interference experiments which were carried out in Paris' and Berkeley.' The basic schematic for these experiments
is shown in Fig. 3. A circularly polarized laser beam is sent into an atomic vapor cell and excites the transitions of interest in the presence of a dc electric field. The transition rate is monitored by observing the fluorescence as the atom decays back to the ground state. Since the Zeeman transitions are not resolved, the parity nonconserving interference term can not
be observed directly in the total atomic transition rate, but it does cause a polarization of the excited state. This polarization is detected by looking at the degree of circular polarization
of the fluorescence light. These experiments were successful at detecting a small circular
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14
polarization and hence a parity violation. It was measured with a fractional uncertainty at a level of 10-20%. Before beginning a lengthy discussion of the design considerations of the Colorado experiments, we would like to briefly review the lessons we have learned from these pioneering first generation optical rotation and Stark interference experiments. The two main lessons from the optical rotation experiments are that the systematic errors are very serious
and must be considered in great detail, and the atomic structure issue is crucial if one is
interested in a precision test of the Standard Model. These considerations made us decide to go with the Stark interference approach and use cesium, since Cs is a heavy alkali atom.
However, there were also several important lessons from the Stark interference experiments. The first was that the signal-to-noise ratio was disappointingly low, predominantly due to problems from background signals. These signals can arise from sources such as blackbody radiation, molecules, collision induced transitions, and scattered light. Also, in these cell
experiments there were a very large number of non-resonant atoms, atoms that were not
being excited by the laser beam and did not contribute to the PNC signal. These atoms muld give rise to background noise or systematic errors, which is clearly undesirable. Finally, a non-obvious design feature is the issue of experimental flexibility. In such
cell experiments, it is very difficult to build a new cell and therefore one is quite limited in how easily one can change the experiment in response to new data or ideas.
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15 4.
Design Concepts of Colorado Experiments After reviewing the lessons from the work described in the previous section, we had
three major concerns. The first was to have flexibility in the apparatus so that the experiment could be easily modified. The second concern was to increase the statistical
signal-to-noiseratio (SIN), and the third was to control potential systematic errors. To improve the S/N, we set out to achieve high detection efficiency and minimal background. One way to achieve this is to have a modulation directly in the transition rate rather than the polarization of the excited state. An applied magnetic field would allow one b do this by resolving the different m levels. It is particularly useful in conjunction with a
collimated atomic beam because much weaker magnetic fields are required. There are also several other desirable features to an atomic beam. First, it has rather few non-resonant atoms and molecules and allows high detection efficiency. Second, the
beam can be turned off and the system opened up and changed quickly. The major drawback is that the actual number of resonant atoms one has is significantly lower than in a cell. However, we felt that the other factors would more than offset this, which has proved to be
the case. The primary feature for controlling potential systematic errors is to design an experiment with many mirror reversals. However, in addition, it is important to have ways to quickly meaSure potential systematic errors with high precision.
It is quite easy to understand how the addition of a magnetic field helps in these experiments. The ApNCamplitude is proportional to the magnetic quantum number m, while the Stark interference amplitude AE is proportional to the absolute value, I m 1. Therefore,
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16
the sum over all m of the product ApNc AE equals zero. This arises because of the requirement of time reversal conservation. However, if one only looks at transitions between individual m levels, then the product AmC A, is not equal to zero, and will contribute to the transition rate. There are two ways one can experimentally observe transitions between individual m levels. The first way is to simply apply a large enough magnetic field and use the Zeeman effectto resolve the different m levels. Then the laser will only excite transitions from a single m. This approach was used in the Colorado 1985 and 1988 e~periments,~*'~ and in the Drell and Commins thallium experiment" at Berkeley. The
second option is to simply optically pump all the atoms into a single m level. This has the advantage of utilizing the atoms much more efficiently, but it does make the experiment more complicated. This approach is being used in the current experiment at Colorado. The relevant energy levels of the cesium atom are shown in Fig. 4. The 6s ground
state has two hyperfine states with total angular momentum quantum numbers F=3 and F=4. The 7s excited state also has F=3 and F=4 hyperfine states. In the presence of a magnetic field, each of these levels then splits up into the different m levels, and one excites
6s 4 7s transitions between these independent m levels with 540 nm laser light. The excitation of the 7s state is monitored by observing the fluorescence produced primarily at 852 and 894 nanometers when the 7s state decays by an allowed transition to the 6P and then to the 6s states. In Fig. 5 , we show the theoretical spectrum of the 6S,F=4
+
7S,F'=3
transition in a weak magnetic field. Figure 5(a) represents the pure Stark induced rate that is proportional to A:; Fig. 5@) represents the AEApNC interference terms. This illustrates how
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17
the Parity nonconserving contribution adds to the rate for the +m levels, while subtracting for the -m levels.
-
A general layout of the experiment is shown in Fig. 6. A collimated cesium beam is
intersected by a laser beam which excites the 6s
7s transition. In the intersection region,
there are three perpendicular vectors defining a coordinate system. These are a dc electric
-
field,
c, a dc magnetic field, s, and the angular momentum u of the laser photons.
The 6s
7s excitation rate is monitored by observing the subsequent fluorescence. In Fig. 7, we
show a typical spectrum observed when we sweep the laser over one of the hyperfhe transitions. This shows eight lines of the Zeeman multiplet corresponding to the excitation of different m levels. It is quite obvious that there is a rather peculiar line shape. The explanation of this line shape is somewhat complicated. It involves the off-resonance ac Stark shift in combination with the small Doppler shifts. (This has been analyzed and explained in Ref. 12.) The values of laser power and cesium beam divergence for this figure
make the line shape look its most peculiar. For larger or smaller power, or less beam divergence, the lines become smoother and more systematic. In any case, this strange line
shape is a minor anomaly and has no real effect on the parity nonconservation experiments. To summarize the basic experiment, we set the laser frequency on one of the peaks of the Zeeman multiplet and look for a change, typically a part in lo6, in the transition rate when the parity or handedness of the experiment is reversed. There are many different ways
to reverse the handedness; most of them can be seen simply by considering the coordinate
system defined by g,'If and u. Anything that reverses the handedness of this coordinate system will change the sign of the parity nonconserving term. Equivalently, this can be
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18
described as making a mirror reversal with the mirror oriented in three different planes.
Thus the three reversals are -. -2,
-.
-s,
circular polarized light), and finally, m -W -m.
Q
-W
-Q
(right circular polarized light to left
This m reversal is carried out by simply
changing the frequency of the laser so that it moves from one Zeeman peak to the symmetric
&man peak on the other side of the multiplet. The presence of four independent reversals is crucial for suppressing systematic errors. Having many reversals provides a redundancy which tests that the signal observed truly is
parity violating. It greatly suppresses potential systematic errors which might mimic PNC under one or two reversals, but are unlikely to pass the test of all four. While the experiment is rather simple in principle, in fact a tremendous amount of time and effort has gone into optimizing the apparatus to achieve the necessary signal-to-noise ratio. The next section will discuss in detail these signal-to-noise issues, and how they have
led to the construction of the apparatus in its present form.
5.
Details of Apparatus
5.1. Signal and Noise Analysis
The majority of the time spent on this experiment has been used to achieve the
necessary signal-to-noise ratio. The total detector current is given by
R or ((number of atoms)(laser power)(detection efficiency)} [E2&2EbpNJ +background S = AR a {
)[4E bNJand is 10” to
of R in this experiment.
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The main signal depends on the number of atoms, the laser power, and the detection efficiency multiplied by the term in brackets involving the electric field. Of course, the largest electric field term is the E2pure Stark component. Then there is the much smaller PNC interference term which is linearly proportional to the electric field. The actual signal of interest, S, is the change in the detector current when the fields are reversed. This is proportional to (the number of atoms)
X
(laser power)
X
(detection efficiency)
X
4
ESPNc.
The noise, however, is made up of several parts which we will characterize according to their dependence on the electric field E as given by
N =N ,
(detector noise
+ fluctuations in background)
-4
+ Nlh
(shot noiseon R, a
+ Ntcch
(technicalnoiseonR, a {
)
)E%
.
First there is the background noise, which is independent of E. This can be the fluctuations
in the signal due to the fundamental detector noise or any backgrounds due to scattered light,
m m lights, etc. The second term comes from the shot noise fluctuations on the total detector current. Since this noise is proportional to the square root of the current, it is linearly proportional to E. Finally, the third term,what we call the technical noise, is proportional to E2times some fluctuation fraction f. Here, f can be due to changes in a large number of things, for example, laser power, laser frequency, and atomic beam intensity.
Because of this dependence of the noise on E, one has an interesting dependence of the signal-to-noise ratio of S on the electric field. This is shown in Fig. 8.
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20
For a~ ideal case,where ideal is rather low technical and background noise, one would have the signal-to-noiseratio rising linearly with electric field in the region where one is dominated simply by background noise. At higher field, shot noise, which is proportional to
E, begins to dominate. At this point, the signal-to-noise ratio is independent of E since both the signal and noise are proportional to E. Finally, at even larger E fields, the technical
noise with its E2dependence will dominate. When this occurs, the signal-to-noise ratio then falls as 1E. In this ideal case one could then set the electric field anywhere in the region where the shot noise dominated and achieve the same signal-to-noise ratio. In fact, this is not the case in the real world.
In these experiments, we have always had a situation more like that shown in the lower curve of Fig. 8 where the background and technical noise are larger. Here, the signal-tonoise ratio rises with the electric field until it reaches some peak, above which the technical noise dominates. There is no actual flat region where the shot noise dominates, and therefore there is always some optimum electric field. We work hard to push this peak up until it is fairly close to the shot noise limit, and then there is little to be gained by further improvements. At this field, typically the background noise and the technical noise are equal. The signal-to-noise ratio works out to be roughly proportional to the square root of
the number of atoms times the laser power times the detection efficiency. In order to reach
the shot noise signal-&noise-ratio, however, it is necessary to make the background noise small and to minimize the technical noise by making the quantity f small. The former means
keeping the detector noise and scattered light noise low, while the latter requires a number of parameters be highly stabilized. The primary feature that makes these experiments so
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21
difficult is the fact that achieving this to the level necessary to make high precision PNC measurements requires pushing many different technologies to the state-of-the-art or beyond the state-of-the-art. This is different from many experiments where most of the experiment
is based on well- developed technology and there are only one or at most two different technologies that one has to work on very hard. Since the signal is proportional to number of illuminated resonant atoms, the laser power and the detection efficiency, it is clearly important to make all three of these large. We will discuss efforts to maximize each of these factors separately.
5.2. Atomic Beam
In this experiment for an atom in the beam to be "useful" it has to meet slightly different requirements from most atomic beams, where the only requirement is either flux or density. In this case a long thin laser beam is exciting a narrow (2.8 MHz) transition. Thus atoms which have a divergence such that their velocity parallel to the h e r beam gives a Doppler shift larger than 2.8 MHz are out of resonance with the laser and will not contribute
to the signal. The divergence in the perpendicular direction affects the density but does not cause a Doppler shift.
Thus the three basic considerations are: the density of atoms in the
beam, the length of atomic beam intercepted by the laser, and the velocity of the atoms
parallel to the laser beam (the divergence in one direction).
If one starts with an atomic vapor and wants to make a collimated beam out of it, the simplest approach is to let it effuse out of a tube of length L and radius R. This tube is the output nozzle on an oven. In this case the intensity in the forward direction will increase as
~~
22
the pressure of the vapor is increased until the mean free path of the atoms in the vapor becomes less than L. At this pint, the atoms coming out of the oven simply form a cloud, the divergence of the beam increases, and the number of atoms in the acceptable divergence angle will stay constant or decrease. To improve on this, we have used a technology that
had been successful for helium, but which had never been successfully used for cesium.
This technology is a glass capillary m y nozzle. Such arrays are made up of thousands of very tiny tubes, typically 10 pm in diameter and a millimeter long. In order to increase the width of the beam we use a section of array which is 0.4 cm by 2 cm so that it provides a collimated ribbon of Cs atoms. This provides a more intense beam with the appropriate divergence angle than provided by a single tube or slit. We have found that operating a glass capillary array at the pressure where the mean
free path is equal to L does not give the most intense possible beam. We have improved on this by using a glass capi.Uary array as a n o d e on the oven which is operated at a vapor pressure which substantially degrades the collimation. About 2 cm from the array we place a multislit collimator made up of a large number of thin metal veins which provides collimation only along the laser beam direction (Fig. 9). In between these two collimators, we have liquid nitrogen cooled panels which pump away the background Cs vapor. To break up the Cs, dimers, which cause a very noisy background signal we keep the glass capillary
array about 100°C hotter than the rest of the oven. While we have been able to produce a state-of-the-art atomic beam in this manner, the beam intensity was lower than what we hoped for. The intensities we expected were based on results that had been achieved with
helium, but we have achieved less than 1/10 of this g d .
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23 5.3. Laser Power Buildup
Fortunately, we have been able to compensate for lower than anticipated beam intensity by having more laser power than we had originally anticipated. The 540 nm laser light which excites the atoms comes from a cw dye laser. The transition rate is very small, so
only a small fraction (< lo-") of the light is absorbed as it passes through the atomic beam.
This means that one can gain tremendously by reusing the light. There are two common ways to reuse light when looking at weak transitions. The first technique, which has been by
far the mostly widely used, is to have a multipass system as shown in Fig. 10. Here the light comes in and is reflected back and forth between two mirrors and then finally leaves without ever overlapping itself. This technique was used in the Bouchiat cesium experiment, and we initially tried to use it for our experiment, but we discovered it had some disadvantages. First, the number of bounces one can get is limited to the size of the mirrors divided by the size of the laser beam, since the beam occupies a different spot for each bounce. With a reasonably sized mirror, one has a limit of a few hundred bounces. Also, the interaction region becomes quite extended, which is a very serious limitation if one wants efficient detection. This is because imaging fluorescence from an extended region is always much less efficient than if the fluorescence is very localized. Finally, there is also a somewhat subtle effect involving systemic errors due to birefringence of the mirrors which is difficult to deal with in this configuration because one can only observe the average effect of
many reflections.
As a result of these difficulties, we switched to a different configuration, shown in Fig. 11. This is known as a "build-up cavity" because a resonant Fabry-Perot interferometer
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24
builds up the laser power between the two mirrors. The most important issues in a build-up
cavity are the losses and transmissions of the two mirrors. To optimize the buildup, the transmission of the output mirror should be as low as possible. The transmission of the input
&or
should be between one and two times the round trip loss in the cavity. Of course any
optics book will provide the equations you need to calculate the exact optimum. Physically, what is happening is that if the input transmission is much lower than the losses, the light
will die out inside the cavity faster than it builds up. On the other hand, if the transmission
is much larger than the losses, then the light will leak back out the way it came in before it can reflect back and forth many times. From this argument it is easy to see why the optimum transmission should be comparable or slightly more than the total losses due to absorption and scattering. In this case, one achieves a traveling wave power inside the
cavity which is equal to the incident power times the buildup factor B = (1-TJ1,where T,
is the input transmission. This makes it clear that the buildup one obtains is limited entirely by the mirror losses. Over the course of this work, as we have obtained improved mirror
coatings, we have progressed from a buildup factor of 100 to 1,000and, in our latest work, to 15,000. With 0.3 W incident power, we have 4.5 kW of circulating power inside the cavity. Unfortunately, this tremendous enhancement in the laser power, and thus the signal, does not come without a price: laser stabilization. This is the primary disadvantage of the
buildup cavity relative to a multipass cavity where there is no resonance. For a buildup
cavity, the laser must be stabilized to the cavity resonance, which is only about 8 kHz wide for a buildup factor of 15,000. However, it is not sufficient to simply stabilize the laser to 8
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25
IrHZ. If the laser varied by 8 kHz, our 6s -. 7s signal would be varying by nearly 100%, while the desired measurement accuracy is about 1 part in 10’. Therefore, it is necessary to have the laser stabilized to a small fraction of this 8 kHz cavity resonance width. Since the
typical free running dye laser has short term frequency variation of about 10 MHz, this puts severe demands on laser stabilization. In addition, even if the laser is locked to the cavity, should the cavity move relative to the atoms’ transition frequency, there will also be large changes in transition rate. Hence, the cavity must be stabilized to the atomic resonance to within a small fraction of the atom’s natural linewidth. This requires considerable attention
to the mechanical and thermal stability of the cavity. Ultimately all this stabilization is done with sew0 loops; we will discuss some of the concerns of servo systems used in this
experiment in more detail in Section 5.6.
5.4. Detection
The third important factor that determines the signal size is the detection efficiency of the 6P 4 6s fluorescence which is produced as a result of the 7s excitation. As mentioned earlier, the more localized the source, the more efficiently one can collect the emitted light. Although we do not have a point source, we do nearly as well by having a line source of fluorescence, which is defined by the region where the thin laser beam intersects the ribbon
cesium beam (Fig. 12). We use a cylindrical &or
to focus the light from this intersection
region onto a long, namow detector. Essentially we have a two-dimensional system, which allows us to obtain a rather high collection solid angle of nearly 2.r sr.
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26
A major difficulty is that the fluorescence photons must be collected from the same region where we must apply very Carefully controlled electric fields. The only practical way to accomplish this is with a parallel field plates, and hence the photons must pass through the
electric fields plates. This required the development of appropriate transparent, highly conducting Coatings for glass. The development and stability of these coatings has been a major headache in this experiment. Silicon photodiodes operating at liquid nitrogen temperature are used for the detection. Although it is generally believed that a photomultiplier is best for very sensitive detection,
this is not true for relatively large signals. Silicon photodiodes have nearly 100%quantum efficiency at 850 nm, but a photomultiplier detection efficiency is only about 10%or less. Of cowse, a PIN photodiode has no gain, whereas a photomultiplier has gain which
introduces very little noise. However, if the signaI is Iarge enough that the photon shot noise is comparable to the detector noise, then the photodiode is superior due to its quantum efficiency. Thus it is quite important to have the photodiode noise small. One way to accomplish this is to have a small diode, but this is incompatible with having a large detection solid angle. However, even with a large (1 cm2) photodiode, we have been able to achieve a detector noise comparable to about to lo4photonds'n by cooling it to liquid nitrogen temperature. It is also necessary to use a very low noise opamp with a large feedback resistor in the current-to-voltage amplifier so that amplifier noise is not a significant limitation. This means that the photodiode actually gives the best signal-to-noise for signals larger than about lo7 photonds. Clearly, if one is interested in detecting effects of a part in
lo6, it is necessary to have far more that lo7photonds.
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27 Having efficient detection with low detector noise is only half the detection problem. The other half concerns the noise arising from background signals. The major source of background is the scattered laser light reaching the detector. This is a large problem because there are l p green laser photons in the power buildup cavity for every 109 infrared photons produced. Thus, an isolation factor of loi3is required! This isolation is achieved partly by geometry; simply putting in light baffles which absorb any green scattered light and shield
the detector. Of course, this is never perfect, so we also use colored glass filters to block
out the green light but pass the infrared photons. Colored glass filters are not as selective as interference filters, but they have a large acceptance angle, which allows much better detection efficiency. When we first looked at filter specifications, we thought there would
be no problem obtaining a filter that suppressed green light and transmitted near IR light. However, this was not the case. The specifications do not mention that green photons cause
IR fluorescence in the filter at a part in
Id or lo6, which is then transmitted through the
filter. We spent considerable time studying this process and investigating many different filters. We found the conversion efficiency varies widely, but eventually obtained filters that worked quite well. It is interesting to note that the best filter was not one of the standard colored glass filters, but in fact was the red plastic in the back cover of the Schott filter book.
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28
5.5. Technical Noise
These various signal and background issues led to the final design of the 1985 and '88 experiments shown in Fig. 13. Light from a cw dye laser passes through optics, which control polarization very precisely, and then enters the power buildup cavity that is inside a
vacuum chamber. It is neceSSaty to have the cavity mirrors inside the vacuum chamber
because there are no windows with low enough birefringence and loss to be put inside the cavity. The dye laser is frequency stabilized to the PBC with an electronic servo loop and its intensity is stabilized With another servo loop. The light inside the power buildup cavity
intersects the atomic beam in a region of dc electric and magnetic fields. The details of the interaction region are shown in Fig. 14. Once this apparatus is assembled, the prime concern is reducing the technical noise and
eventually reducing the size of the systematic errors. The technical noise must be reduced to a level where the modulation fraction, f, is less than or equal to lo-'&
(Ntcc,, = fxR,
where R is the signal size). There are an astonishing number of processes that can produce
technical noise at this level, including several that would not be encountered in a more typical experiment. For example, we have seen fluctuations in the atomic beam at this level. We found that the standard atomic beam is actually very quiet, but fluctuations of the pressure in the vacuum chamber of 2 lo'* ton cause noise on our signal. We found that
such pressure fluctuations come from the release of small bubbles of gas. These can be eliminated with proper design of the vacuum and pumping system. A more obvious source of technical noise is the laser power fluctuations inside the
power buildup cavity. These can come from fluctuations in the laser power itself or
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29
fluctuations in the laser frequency. The latter is the more difficult to correct; as we mentioned earlier, an 8 kHz change in the laser frequency, relative to the cavity resonant
frequency, leads to f=l. Therefore, considerable effort must be made to stabilize both the frequency and the intensity of the laser. This is done with several servo loops. First, the laser is locked to the buildup cavity using the Pound-Drever frequency modulation technique.
This lock has good signal-to-noise and a rather fast response time so that one can correct the errors in a time as short as 1 ps. However, the length and hence the resonant frequency of the buildup cavity can fluctuate because of inherent instabilities in the mechanical design and vibrations of the environment. To avoid this we lock the buildup cavity to a very stable reference cavity. Because this cavity does not have an experiment inside it, it can be much more rigid. The cavity-tcwwity lock also has good signal-@noise, but has a relatively slow response (400 Hz). This response is limited by the fact that we cannot move the buildup cavity mirrors very fast without introducing strains that cause birefringence and lead to
systematic errors. Although the reference cavity has good short-term stability, it is subject to slow drifts and therefore it must be locked to the atomic transition itself. This lock has
relatively poor signal-@noise because it relies on the atomic transition rate, which is low in
this experiment. Fortunately, the drift-rate of the reference cavity is small, so one does not require the high signal-@noise ratio necessary for a very fast servo response. The signal-tonoise limitation on this lock is the reason we use the reference cavity rather than locking the frequency of the buildup cavity directly to the atoms.
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30 5.6. Servo Systems
Servocontrol systems are very important to this and many other "frontier" experiments of the sort discussed elsewhere in this volume. Because they are often not part of a general physics education, we will provide a brief introduction to servo theory, which will allow one to grasp the key issues involved in the stabilization of systems. A basic servo loop is shown
in Fig. 15. The laser has some noise, for example, the table vibrations which move the
mirrors and thus cause the output frequency to vary. One can compare this output frequency
with some reference, such as a stable optical cavity, and from this obtain an error signal that indicates how far the frequency has moved away from what is desired. The basic error signal is then modified by electronics which provide gain and some type of compensation and the result is a feedback signal that is sent back to the laser. This feedback is negative, so it cancels the errors introduced by the table vibration and shifts the frequency back to where it
is supposed to be. With this feedback, the error signal, & becomes & = E/(gain) in the limit of gain
> > 1 where E is the error signal with no feedback. Thus one wants to make
this gain large in order to push the error signal as close to zero as possible, which is equivalent to forcing the laser frequency to be the Same as the reference fiequency. The key p i n t in designkg any servo loop is that there are always time delays in the system, and much of the design is based on dealing with these time delays. This can be best understood by considering the laser mirror. The mirror is attached to a piezo electric transducer (PZT)which will stretch it when voltage is applied. Although the PZT is a hard
ceramic and the rest of the mount is a solid metal piece, on the scale of frequency errors and
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31 corresponding distances relevant here (atomic diameters or less), there is no such thing as a
stiff mount. In fact, at these atomic scales the stiffest PZT is incredibly spongy and can compress many wavelengths. Therefore, a PZT can be best visualized as a spring with the
mass of a mirror mounted on it, as shown in Fig. 16. We then move the back end of the piezo to try to correct the position of the mirror and compensate for random vibrations. If
one considers how a mass that is attached to a wall by a spring responds to sinusoidal motions of the wall as a function of frequency, this is just a driven harmonic oscillator. For frequencies below the resonant frequency, the response is in phase with the drive. However, above the resonant frequency of the mass-spring system, there is a 1800 phase lag and the mirror moves in the opposite direction of the driving force. This shows the primary problem encountered in designing any semo system; if the amount of feedback is the Same amplitude and phase relative to the error signal for all frequencies, the servo works fine for correcting for errors that fluctuate at frequencies lower than the resonant frequency. However, above
the resonance frequency, this 1800 phase shift causes positive feedback and the system becomes an oscillator if the feedback gain is greater than 1.
This is obviously unacceptable. A straightfoward solution is to make the gain smaller than 1 for frequencies above the resonant frequency and make it larger than 1 for lower frequencies. This gain is produced simply by having the compensation electronics include a simple low-pass filter. The normal way to operate a servo system containing such a filter is
to turn up the gain until it just starts to oscillate at the 1800 phase shift point, and then reduce the gain slightly so that it stops oscillating. This provides a stable sew0 system.
1.17
32 Notice that the gain at low frequencies is set by the response @base lag) of the system at high frequencies. This is characteristic of any servo system. Here we have presented the simplest possible compensation. In a more advanced servo design, one would put in more elaborate compensation involving electronic circuits that change the phase shifts and gain with frequency. With such systems one can optimize the gain at particular frequencies where the noise might be especially large or where one wants
to have the system be particularIy stable, such as at the frequency where the data are being
acquired. Other reasons for more elaborate compensation are to improve the transient
response so the system can recover more rapidly from a sudden shock. In the remainder of
this section we provide a few more examples of relatively straightforward compensation and how one can deal with various kinds of phase lags in systems. Several methods have been developed for designing servo systems - frequency response methods, the root locus method, and state space methods. The method that one
chooses depends on the requirements of the sew0 design, such as transient response and steady state enor. We will describe frequency response methods from an experimentalist’s point of view. We have chosen this method because it is very easy to measure a system’s frequency response with modem signal analyzers. The basis of the fiequency response method is the Bode plot. The Bode plot shows a system’s gain and phase as a function of frequency. Both the system to be controlled and the compensation have characteristic Bode plots. When designing a servo, one first mmures the frequency response of the system to be controlled and then designs a compensation circuit
that tailors the open loop frequency response to provide the desired control. It is important
138
33 to keep in mind thbr high gain at frequencies where the phase is less than 1800 provides good
control, but low gain is required at frequencies where the phase is greater than or equal to 180".
As an example of how to tailor the frequency response of a system with compensation, consider a simple harmonic oscillator. Many physical systems can be modelled as damped harmonic oscillators. The Bode plot of a harmonic oscillator is shown is Fig. 17. There are
two different goals one can have in trying to control this system: (i) to minimize the steadystate error, i.e., to have a large dc gain; or (ii) to maximize the bandwidth of the servo, i.e., to provide damping of the resonant peak with feedback.
As we mentioned previously, the simplest way to prevent oscillation is to make the gain smaller than 1 at frequencies at which the phase shift is greater than 180"; a Bode plot
of such compensation (an integrator, or a low-pass filter) is shown in Fig. 18a. Figure 18b shows the resultant frequency response for the oscillatorcompensation system. Note that the gain at low frequencies has increased, but the phase shift has reached 180" at a lower frequency. We now have a larger dc gain but a smaller bandwidth. The controlled oscillator will lock to the reference signal well at dc but will have a slow transient response and more
noise at higher frequencies. Suppose instead one compensates by adding a "phase lead" (e.g., a differentiator) to the compensation.so that the phase shift of the oscillatorampensation system has not yet reached 180" at the resonance. This allows the resonance to be artificially damped. Figure 19 shows the resultant frequency response for a harmonic oscillator compensated with a phase lead. Note that compared to Fig. 18, the gain near the resonant frequency is large,
34 and the dc gain has decreased. ' f i s system will have a faster transient response, but more
error in locking to the reference signal at dc. There is in general a trade-off between bandwidth (fast transient response) of a servo and dc gain (small steady-state error) of a servo. One way to think of it is in terms of
integrators and differentiators. An integrator will generally provide less steady-state error because it has "memory" to make accurate adjustments at low frequencies. On the other hand, fluctuations that are fast compared to the integration time will be "washed out," and,
as a result, the bandwidth of the system will be reduced. Differentiators predict the future performance of the system by looking at the slope of the error signal, and therefore increase the bandwidth. However, because a differentiator is compensating for future fluctuations it can slightly over or under compensate, leading to less steady-state accuracy.
Systems with more complicated frequency responses than that of a simple harmonic oscillator can be controlled by extending these ideas. A compensation circuit's phase and gain characteristics are tailored to provide the best compromise of bandwidth versus dc
response. We now return to discuss how this stabilization is applied to the actual PNC experiment. The power buildup cavity is stabilized by moving the mirrors with piezoelectric transducers. Then laser frequency is locked to the power buildup cavity by a combination of
elements, most of which are standard in cw dye lasers. First, we have a rotating plate on a galvanometer that changes the optical length of the laser cavity. This has a rather slow
response, but a large dynamic range. Second, one of the mirrors is mounted on a PZT. This
can change the cavity length with a frequency response extending to about 50 kHz. Third,
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35 the fastest feedback is provided by an electrwptic modulator. This has a unity gain frequency of about 2 MHz, but can only correct for rather small errors in the frequency. In addition to the frequency stabilization, to stabilize the optical power inside the power buildup cavity, we sense the light transmitted by the output mirror and hold it constant using acoustooptic or electmoptic modulators to control the incident laser power. One significant difficulty in this experiment is the fact that the power transmitted by this mirror does not
seem to be exactly proportional to the power inside the cavity at the parts in 106 level. This discrepancy has been an ongoing problem which we do not yet fully understand.
5.7. Field Reversals and Signal Processing
With all the necessary frequencies, intensities, and lengths stabilized, one then has to
be concerned about reversing the various fields as precisely as possible, without upsetting the semwontrol systems. The electric field flip is accomplished by reversing the voltage applied to the electric field plates. Initially we tried a sinusoidal reversal, but this gave unacceptably large electrical pickup on the detector. We then switched to a square wave modulation with a few milliseconds of dead time after each reversal before taking data, to
allow the transients to die away. The primary problem in obtaining a perfect electric field reversal is the stray fields. There is considerable black magic we have learned for the preparation and handling of the plates which keeps the stray fields to a minimum, typically, a few tens of mV/cm. For the actual voltage reversal, we have experimented with various solid state and mechanical switches and have obtained the cleanest reversals when we use high voltage
141
36 relays. These have the minor annoyance that they are somewhat slow (less than -40 Hz), but the reversal is much more exact than with any solid state devices we have found. Mercury relays are faster but are limited in the voltage they can handle. To reverse the polarization, we use the same high voltage relays to flip the voltage applied to the Pockels
cell that provides a quarter wave retardation. In this reversal, the major problem is the birefringence of the Pockels cell which drifts with temperature. However, with careful temperature stabilization this can be reduced to a reasonable level. The magnetic field reversal is the easiest; we simply reverse the currents flowing through mils using solid state switches. One of the major concerns when doing any of these reversals is to avoid upsetting any of the servo loops. This takes considerable care and involves the use of various sample-and-holds circuits and gates with precise timing to isolate the servos from the transients. We have succeeded in keeping everything stable enough that the noise while flipping the various fields is as low as when there are no reversals. The signal processing is the final part of the apparatus, and it is hirly simple. The current from the photodiode is sent into a very low noise current-@voltage converter, as mentioned above. The output voltage of this amplifier is monitored with two different systems. The first is relatively crude (1 part in l@),and simply monitors the overall dc signal level for normalization. The second system detects the small changes in the signal due to the PNC modulation. First, the signal passes through a low noise amplifier which
subtracts off a constant voltage so that the dc output is close to zero. This near-zero signal is sent into a gated integrator and the signal is integrated during the time between each reversal. At the end of each interval the output from the integrator is digitized and stored in a
142
37 computer. Each of these numbers is stored with its appropriate label as to the state of E, B,
m, and polarization. Then the computer carries out the next reversal, resets the gated integrator, and the sequence is repeated. Using the offset and gated integrator in this
manner, we avoid the dynamic range problem encountered in trying to measure a very small modulation on top of a large signal.
6.
Systematic Errors Most of the time spent taking data in these experiments is devoted to the study and
reduction of potential systematic errors. Our approach to dealing with systematic errors
follows the same general analysis used in the earlier cesium and thallium experiments. This procedure starts by considering the most general possible case of both dc and ac electric and magnetic fields which have components in the X, Y,and Z directions. Thus we have 12 possible field components. Next we look at all the combinations of these fields that can give
rise to a 6s + 7s transition, either electric dipole or magnetic dipole. We allow each of these 12 field components to have both flipping and non-flipping (henceforth known as
"stray" parts).
We then go through the exhaustive list of combinations that produce terms
that mimic the parity nonconservation by reversing with all of the possible various reversals.
Then we measure the size of these 24 different field components and, in the process, try to make the stray and misaligned fields as small as possible. We have been able to reduce them to typically 10'' to
of the main applied field. Using the measured sizes of the different
components, we look at all the vast number of combinations that mimic PNC,and see which
38
ones are simcant. The 10" to 10-~values effectively mean that any terms involving
more than two stray or misaligned components are negligibly small compared to the true (10'
- 10-6) PNC. At the end of this exercise we found there are three terms that contain
two small components, and these are listed in Table 1. The first of these terms involves a stray electric field in the Y direction times the (misaligned) magnetic field in the X direction, The second term involves a stray electric field in the 2 direction times a misaligned
component of electric field in the Y direction. And the third term is a product of El and M1 transition amplitudes times a mirror birefringence factor. We measure each of the fields and the birefringence involved in these terms while the experiment is running and subtract off their contributions. To do this we run a set of
-
auxiliary experiments simultaneously, or interleaved with the PNC data acquisition. These auxiliary experiments involve observing the effects on the 6s
7s atomic transition rate of
different hyperfine transitions, different laser polarizations, and application of additional E or
B fields. Two points should be emphasized about dealing with systematic errors in this manner. First, it is important to use the atoms themselves so that the same region of space
is sampled at nearly the same time as the PNC experiment. Second, the auxiliary experiments must be designed to allow systematic corrections to be measured with an
uncertainty that is much less than the statistical uncertainty in the parity nonconservation experiment. It is highly desirable to have a measurement time much shorter than that
required to take the parity violation data. If one fails to achieve this, then the uncertainty of an experiment increases because much of the running time is spent in taking data on systematic errors and little on the measurement itself.
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39
In the experiments we have designed, achieving the necessary unceainty requires a small fraction of the PNC integration time. In Table 1, we show the different sizes of the
systematic uncertainties for our 1988 experiment, how much they vary from one run to another, and the average correction and uncertainty. It can be seen that the typical corrections are a few percent or less, and most importantly, the uncertainties in all of these cOrrections in a given day are less than 196, and thus much smaUer than the statistical uncertainty. An obvious question is, "Is this analysis foolproof, or did we miss something?" In fact
we did miss something; there is no analysis that is absolutely foolproof. We overlooked a
small correction the first time through, although we caught it well before we were ready to publish a result. However, it is educational to discuss the statistical analysis procedure we
used to discover this systematic error. This Same type of analysis can (and probably should) be used in any precision measurement. It involves using a 2 test in a particular way to track down systematic errors. The Allan variance used to characterize frequency standards is
related to this approach. Our data consist of a large set of numbers, each number corresponding to a current which was integrated for 0.1 s. In the entire data set there are roughly lo' such numbers
stored in the computer for analysis. The fist step is to find the scatter in the numbers which is due purely to the statistics and has no contribution from any systematic source. This is
accomplished by looking at the fluctuations on the shortest possible time scale where the statistical fluctuations are large. This gives us a standard deviation, Q, which is most likely
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40
to be purely statistical. In our case,we are doubly sure that it is truly statistical because it
corresponds to the shot noise limit for the signal. Having found u, we collect the data into various bin sizes, for example, the frrst million data points would be one bin, the second million would be the second bin, and so on for all the data. This produces 10 bins of data, and we can now predict how the average
values in each bin should distribute based on u, and the number of points in the bin
(cave=c
m).This hypothetical distribution is then compared with the actual distribution. Specifically we find the value of 2 for the distribution Using
Q
to obtain an uncertainty for
the value in each bin, then we look up the probability for having that value of
2. If the
resulting probability is 0.5 or larger, we are confident there was no systematic error that
varied on a time scale of the length of a bin that would be significant relative to the statistical uncertainty. Now, by choosing different bin Sizes we probe for variations on different time scales,
This is quite important because if the bin size is much larger or much smaller than the time scale for the variations in some systematic error, the 2 will probably look reasonable. However, when one chooses a bin size that corresponds to the time scale of the variations, suddenly, the probability will be very low, indicating the presence of some unknown systematic error. This approach does not have to be limited to binning the data by time. It
is equally useful to bin it according to any other factor that may lead to some systematic errors. For example, one might also bin the data according to room temperature to search for temperaturedependent systematic errors.
146
41 Of course this approach only works if the systematic errors vary
- if they are always
constant you will never see them. However, one can make them vary by changing everything about the experiment that might be important, such as realigning all the optics or
replacing critical components. Again, we binned the data corresponding to the different configurations and performed the 9 test. This is a remarkably sensitive test for potential systematic errors, and, although it is not generally taught, it is important to keep in mind in
any precision experiment. In our case it revealed that we had neglected to consider the EIMl interference correction associated with the off-resonance excitation of a m level. This excitation is forbidden at zero magnetic field but could occur because of the second-order
Zeeman effect. Having carried out all the detailed studies of systematic errors and
2 tests we finally
achieved the result
ImbNc
B
-1.639(47)(08) mV/cm
F=4-F'=3
-1.513(49)(08) mV/cm
F=3-F=4
-1.576(34)(08) mV/cm
(average)
.
The size of the parity nonconservhg mixing is given in terms of the equivalent amount of dc
electric field that would be neceSSary to give the Same miXing of S and P states. As shown, we have measured this mixing for two different hyperfine transitions, the 6S, F = 4 -+ 7S, F' = 3 and the 6$, F = 3 + 7S, F' = 4. In both cases, the amount of mixing corresponds to about 1.5 mV/cm. The average of these two is the most important quantity, as we will discuss below. We have measured this to an uncertainty of 2 %, which is dominated by the
0.034 mV/cm statistical uncertainty. The systematic uncertainty is about 1/4 of the size of
147 ~~~
~
42
the statistical uncertainty. It should be noted that this systematic uncertainty is different from many systematic uncertainties, in that it is actually a true statistical uncertainty in the evaluation of the systematic correction. Therefore if the statistical signal-to-noise in the experiment is improved, this uncertainty will be reduced.
In Fig. 20, we show a comparison of the different experimental measurements of parity nonconsewation in cesium, the most thoroughly measured atom. On top are the two experimental results of the Paris group in '82 and '84, below is our 1985 result, and our 1988 result, with its 2% uncertainty. There is good agreement among all of these numbers.
This gives one a Certain amount of confidence that no tremendous systematic errors are being overlooked.
In Table 2, we show a summary of the results from all atomic parity nonconsewation experiments. In the first section are the optical rotation experiments which looked at the 648
nm line of bismuth. These results are somewhat controversial in that the results from the
three groups showed substantial discrepancy, as did the theoretical calculations. In retrospect, the former was probably due to systematic mors that were not sufficiently controlled. More recent optical rotation experiments have shown better consistency, and the uncertainties are mostly in the 15 - 30% range. The one exception is the recent Oxford measurement in bismuth which has an uncertainty of only 2%. The Stark-induced interference experiments are given at the bottom of this table. Most of these are the cesium measurements we have already mentioned, plus there is the one thallium result from Berkeley with an uncertainty of 28 %.
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43
Before we can consider what these measurements tell us about elementary particle physics, we much return to the atomic structure issue. The quantity that is experimentally measured is the GpNc mixing, which is equal to Q, times the T~ matrix element. This matrix element is found by calculating the atomic structure. Thus, in order to obtain Q, to 1%, both the experiment and the matrix element calculation must be accurate to better than 1%.
As mentioned earlier, the calculation of -y5 varies considerably in accuracy from one atom to another. In Table 2 we have given the accuracy quoted for the best calculations for each atom. Here we will limit our discussion to the cesium atom for which there have been the most abundant and most accurate calculations. Two basic approaches have been employed for these calculations. The first is the semi-empirical method which has been used in Paris, Oxford, Colorado and elsewhere. This approach uses experimental data to determine wave functions which are then used to find the matrix element. This technique is relatively easy. However, it is difficult to make a rigorous evaluation of the accuracy of the calculation, since all the relevant experimental data have already been incorporated into the calculation. The estimates for the uncertainty in these calculations are as small as 2%. The second approach is to use ub initio relativistic many-body perturbation theory. "he need for accurate cesium PNC calculations has spurred major advances in this field, although the calculations are very long and difficult. The most recent and most accurate results have
come from the Novosobirsk group of Flambaum, Suskov, et al., who have achieved a 2 % uncertainty, and the Notre Dame group of Blundel, Saperstein and Johnson who have now reached 1%uncertainty. The advantage of this calculational approach is that there is a fairly
149 ~
~
44 clear prescription for evaluating the accuracy of the calculations. The most direct way is to simply use the same calculational technique to determine many properties of the atoms and compare these with experimental data. In this case, this means calculating hyperfine splittings, oscillators strengths between many transitions, energy levels, and fine structure splittings for cesium and other alkali atoms. Fortunately, a tremendous mount of experimental data is available for comparison. In all cases, the agreement between the calculations and the experiments has been within 1%. Another technique for estimating the
uncertainty in these calculations is to estimate the size of the uncalculated higher order terms in the perturbation series expansion. This approach also gives an uncertainty of about 1%.
7.
Implications In this section, we will consider the implications of the Colorado measurement of PNC
in cesium. We will first discuss the significance of the comparison of the two different hyperfine transitions. This difference between the two numbers, A = 0.126 (68) mV/cm, is probably not zero. More specifically, this value indicates a 97% probability that A is greater
than zero. When we made this measurement, we did not anticipate a non-zero result at this level and therefore spent a considerable amount of time trying to determine what was wrong
with the data. The result, however, stubbornly persisted. Only later did we discover that an effect of nearly this size had been predicted. The primary difference beiween these two transitions is that the nuclear spin is reversed relative to the electron spin. Thus, A is a measure of the nuclear spindependent contribution to the PNC signal. Two processes have been discussed which would cause a
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45
Rxlear spin dependent parity nonconservation. The first is simply the electron-quark portion
of the weak neutral current which depends on the spin of the quarks. This interaction is characterid by the C;, and C,coefficients. As we mentioned earlier, because of the size
of these coefficients and the fact that the effect is proportional to the total nuclear spin (and not proportional to the number of quarks), this contribution is much smaller than the weakcharge contribution. However, it has also been pointed out3 that there is a substantially larger contribution, called the nuclear anaple moment, which arises from weak interactions
within the nucleus. The effect of these weak interactions (both charged and neutral) are to mix the parity eigenstates of the nucleus, leading to a parity nonconseming electromagnetic
current in the nucleus. This current takes the form of a toroidal helix, and therefore has no long-range electric or magnetic fields. Thus, it gained the name "anapole moment." This phenomenon was first proposed by Zel'dovich in 1957 in the general context of parity violation in charged systems."
It is not well hown because people shortly thereafter
decided such an effect could never be measured. However, because the cesium electrons penetrate the nucleus, they spend some time inside the toroidal helix and thereby detect its existence. The coupling to the electrons is purely electromagnetic, but because the underlying nuclear currents are parity violating, it leads to parity violation in the electronic transition. There has been a significant amount of interest in this nuclear anapole moment by the nuclear physics community and several authors have calculated the expected size. The first calculations were by Khriplovich and Flambaum3 and their estimates are consistent with our observations. &ton
et aI.26have also made similar calculations, but have treated the
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46
nuclear physiLs rather differently. Finally, Bouchiat and Piketty have done a calculation which is not consistent with our result.27 We have been told by V. Flambaum that the differences in these calculations are not due to any fundamental difference in the theory, but are a problem of the basic interpretation of nuclear PNC from other experiments. Depending
on how one chooses to interpret the other experiments, it is possible to obtain very different constants which characterize PNC interactions in the nucleus. This emphasizes the need for more accurate data in this field. There is hope that these nuclear anapole moment measurements can provide these data. The nuclear anapole moment is unique in that it is a PNC distortion of the nuclear ground state. The previous measurements on nuclear PNC
have observed parity mixing of excited, and often rather distorted, nuclear states where there
is considerable uncertainty about the nuclear wave functions. Thus, it is clear that future improvements in atomic PNC precision should substantially improve the understanding of nuclear parity nonconsewation. Obviously, the uncertainty due to the nuclear physics is a serious issue in the interpretation of atomic parity nonconservation. If we had measured only a single transition, it would seriously compromise our ability to test the Standard Model. Fortunately, if we take the average of the measurements on the two hyperfine transitions, as opposed to the difference, the nuclear spin dependent part cancels out. In this way, we also cancel out any questions involving the nuclear structure, which is critical in allowing a precision test of the Standard Model.
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47
From this average a d the Notre Dame matrix element calculation, we obtain a weak charge, Q, = 71.0 f 2% f 1%. If one assumes the Standard Model is correct, one can then from this extract a value of sin2& which is equal to sin20, = 0.223 i 0.007 (expeshental) t 0.003 (theoretical).
(14)
This value of sin2eWcan now be compared with values obtained from other experiments such
as the measurement of the 2, mass or the neutrino scattering results, as shown in Table 3. In addition to these two measurements, there are many other measurements from high energy experiments which can be used, but which have lower precision or involve other properties
of the &. We have omitted the latter group because, while these are reputed to be independent measurements, the variations in the values of sin%, obtained are much smaller than the quoted uncertainties. This leads to unrealistic 2 probabilities and suggests that
these measurements are not truly independent. The comparison of the values of sin2b provides a precise test of the Standard Model. It is worth noting that in other tests of the Standard Model, particularly those involving the comparison of neutrino scattering and Z,data, the uncertainty in the mass of the top quark introduces an uncertainty of 0.003 in the relative values of
However, the
comparison of the atomic and the Z,mass values are unique in that dependence on the top quark mass is essentially identical in the two cases. Because the atom is sensitive to a different set of electron quark couplings and a different energy scale, this comparison, however, is very sensitive to new physics which is not contained in the Standard Model.
Proposed examples of such new physics include technicolor and extra 2 bosons, which occur in many models.
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48
Figure 21 shows the values of
Gdversus C,,”as determined by a model-independent
analysis of the experiments, along with the Standard Model prediction. The crossed-hatch
area is the constraint from the SLAC deep inelastic electron scattering experiments. In contrast, the constraint set by our cesium experiment and the Notre Dame theory is the narrow solid line which is nearly orthogonal and, in particular, constrains the value of C1,
much more severely. The crossed line shows the values allowed by the SU,
X
U,
Wineberg/Salam/Glashow theory. The point on that line is determined by the value chosen for sin2Bw. This figure shows quite clearly that any new physics that would cause a change
in the value of C,, but not affect the other coupling constants, would only be revealed by the atomic parity nonmnservation measurements. The arrows on this figure show how a few popular proposed models would shift the values of these two coupling constants to a different place in the plane. Because the current atomic physics line passes through the Standard Model point, there is no indication of the existence of new physics. However, this does put
constraints, in some cases quite severe, on the parameters of models that propose such new physics. One example which has drawn considerable attention in the last few years, is how atomic PNC results constrain the proposed mechanism known as technicolor, or more generally, dynamical symmetry-breaking involving heavy particles. This type of new physics has been characterized in terms of S and T parameters which enter directly into QW.**
Generic technimlor models predict the S parameters should be around +2 or somewhat larger.% From the comparison of results just mentioned, one finds that the atomic PNC yields a value of S which is -2.7 f 2 2 1.1 as given in Ref. 29. Thus, one finds that
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49
technicolor is on somewhat shaky ground, although the atomic PNC experimental results are
not good enough to completely rule it out. Finally, atomic PNC provides the best constraints on many models that involve additional neutral Z bosons. While there are many papers on this subject (see references in They consider 11 Ref. 2), we note particularly the paper by Mahanthappa and M0ha~at.m.~~ different models with additional Z’s which have been proposed, and they find that in 8 of
these 11 cases, atomic PNC provides the most severe constraints. Thus, it is clear that the cesium PNC results are providing information on elementary particle interactions that is not available from any other source at the present time.
8.
Future Improvements
8.1. Near Term While atomic PNC experiments are providing useful information, it is clear that more precise results would be desirable and useful. The mass of the 2, is now known to around 1 part in
Id. If atomic PNC results could be improved to that level, we would have a 10-fold
improvement in the test of the Standard Model and correspondingly improved sensitivity to possible new physics. With this in mind, we would like to discuss our efforts to improve the cesium PNC results. Work is also under way to improve atomic parity nonconservation measurements by several other groups: In Paris,the Bouchiat group is building a new experiment that involves stimulated emission probing of the excited state in cesium. At Oxford and Washington, experiments are under way to obtain more precise optical rotation measurements in thallium. At Berkeley, efforts continue to obtain a more precise Stark
155
50
inference measurement in thallium. All of these experiments h v e been under development for a number of years, and we hope to have results in the not too distant future.
In our efforts to improve the Colorado 1988 experiment, our primary focus has been
on improving the signal-to-noise ratio. This is clearly the major limitation of our experiment since the statistical uncertainty was much larger than the systematic uncertainties. We also
took into account the fact that the scattered laser light was a major nuisance requiring frequent realignment of the optics, and the transparent conducting coatings were a substantial problem. The coatings deteriorated under exposure to the cesium vapor, and lead to frequent interruptions in the experiment while they were replaced. With these issues in mind we built a new apparatus that uses an optically pumped atomic beam which, in principle, should provide 16 times more atoms since there are 16 possible m levels of the 6s state. We now have better mirrors for our power buildup cavity; these increase the buildup by about a factor
of 10, resulting in a total buildup of 15,000. A third improvement is using downstream detection of the 6S+7S excitation. This concept is illustrated in Fig. 22, which shows the schematic of the new apparatus. After leaving the oven, the cesium atomic beam is optically pumped into a single F and
mF level by light from two diode lasers which drive two hyperfine transitions of the 6S+6P3, transition. The atoms in the single m-level then propagate down the atomic beam and intersect the power buildup cavity beam where they are excited to the 7s state. They then have a 70% probability of decaying back down into the 6s hyperfine level which was previously depleted. The atoms continue down the optical beam in this state until they reach the probe region. In this region, light from another diode laser again excites the 6S+6P3,2
156
51 transition. However, here we excite a cycling transition (F=4
+
F'=5 or F=3 -. F'=2).
On a cycling transition the atom returns only to the Same initial state and hence can be excited many times. Typically lo00 IR photons are scattered for each 6S-.7S excitation. We detect this fluorescence to determine the 6-7s excitation rate. This detection scheme provides a substantial amount of amplification, yielding a detection of about 200 photons per S+7S transition, instead of the 0.3 detected in the previous apparatus.
In addition, since the detection takes place at a different region from the excitation region, we can now construct our electric field plates out of any material. This greatly simplifies their construction and increases their longevity. Also, scattered light from the green laser light is now negligible, as is detector noise, because the signal size is much larger. All of these improvements would suggest that the experiment should be much easier.
In fact, there have been major headaches and delays with this approach, and it is educational to consider what has gone wrong and what lessons a n be learned about doing experiments at
the frontier of laser spectroscopy. We will now discuss the unexpected problems we encountered in making this "improved" experiment work. The fist problem was noise in the optical pumping and resonance fluorescence detection regions due to the fact that we were using diode lasers. Diode lasers have very rapid (ns) fluctuations in the optical phase. Through a somewhat obscure process, this leads to very low frequency fluctuations in the atomic transition rate.
This was quite puzzling when we first observed it, and has now been explained in a series of papers by Zoller and collaborators.3o While this has become interesting atomic/optical physics to a number of people, to us it is a major experimental problem. To avoid the
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52
problem one must have a feedback system capable of providing gigahertz bandwidth correction signals in order to eliminate the noise at 10-20 Hz,where we detect our signals.
In the first attempt we used optical feedback from narrow-band resonators, as demonstrated by Hollberg and coworkers.31 This approach gave low noise signals, but the locking was not reliable enough to allow three diode lasers to operate for reasonable periods of time. After considerable additional work, we settled on using optical feedback from diffraction gratings.32 These gave much more reliable performance, but the noise levels were still unacceptably high. We solved this problem by finding a laser manufacturer whose
instruments gave superior performance when operated with grating feedback. Thus we have finally succeeded in producing a very reliable source of diode laser light which provides very good signal-to-noise in excitation of narrowband atomic transitions. Specifically, we can
now achieve a noise-to-signal ratio of 3 x
/ Hz'" when exciting a 10 MHz wide
atomic transition. The necessafy laser has a combination of optical, mechanical, and current feedback as shown in Fig. 23. The grating which provides the optical feedback is mounted
on a piezoelectric transducer which allows mechanical adjustment of the grating position.
This holds constant the length, and hence the frequency, of the optical cavity. For faster corrections to the cavity resonant frequency we servo the laser current to achieve the highest level of stabilization. The second major problem in the new experiment was background atoms in the supposedly empty F state. It is relatively easy to deplete one F level of a low intensity atomic beam very well (< 10-4) . However, with a more intense beam there are a number of mechanisms that can repopulate the empty level. For example, collisions between atoms in
158
53 the beam and surfaces or other atoms (parhcularly oxygen) is the first mechanism. We have eliminated this s o u ~ c eby improving the vacuum and carefully positioning the collimating surfaces. A second contribution to the background, which appears to be from atoms in the
wrong state, actually comes from the excitation of the other hyperfine line by the tail of the spectral distribution of light in the probe beam. We have eliminated this source by sending
the probe light through an interferometer filter cavity which blocks out the tails of the spectral distribution. The third and most serious source of atoms in the wrong F state has
been the multiple scattering of the optical pumping light. The optical pumping process produces fluorescence that can travel down the atomic beam and re-excite the atoms out of the desired state. The number of atoms pumped back into the empty state scales as the square of the atomic beam intensity. We have found several ways to reduce this background: multiple pumping beams (the "clean-up" beam in Fig. 22), picking the optical pumping transition which minimizes the scattered fluorescence, and using the photon blocking collimator. This is a collimator with very thin black vanes which allows only the highly collimated photons to pass through. Finally, even with all these steps, the background was still too large, and we ultimately had to reduce our atomic beam intensity. In spite of all
these setbacks and delays, this new, improved experiment is now operational and we are taking data with a signal-@-noise ratio several times better than that of the 1988 experiment. A very painful lesson has been brought home to us in carrying out this improved
experiment. When one is probing a region of technology and physics which is unexplored, it
is important to step warily, and to keep all your options open. In terms of an experiment,
this means you should keep the apparatus flexible and be ready to adapt, as we mentioned
54
earlier. In this experiment we were somewhat seduced by the fact that this approach seemed to solve all our old problems, and we committed ourselves to a design that turned out to be
filled with major unexpected difficulties.
8.2. Long Term As the experimental accuracy improves beyond 196, the principal limitation on the
usefulness of atomic PNC will become the atomic theory. There have been credible speculations that it will be possible to calculate the theory in wium to a part in
Id.
However it is not clear when these calculations will be completed, and the question of how to check their accuracy becomes a major issue.
We have begun a longer term experimental project that avoids the atomic theory question. The basic idea is to compare precise measurements of atomic PNC for different
isotopes of cesium. The weak charge is sensitive to the number of neutrons, and hence will change for different isotopes. The atomic matrix element however, depends on the electronic structure and is almost independent of the number of neutrons. If one then looks at appropriate combinations and ratios of experimental results, for example
the atomic matrix element will drop out, leaving a ratio of weak charge which can be directly compared with Standard Model predictions. In this manner we hope to achieve measurements that can be compared with the Standard Model predictions at the part in I d
160
55
level. There are two major obstacles to carrying out these experiments. First is the need for even better signal-@noise ratios. Second, and most critical, is the need to carry out PNC measurements with small atomic samples, rather than the many grams used in the atomic
beam measurements. This requirement is neceSSary because all the other isotopes of cesium are radioactive and can only be obtained and used in small quantities. We propose to overcome both of these obstacles by using the new technology of laser trapping. This will improve the signal-to-noise ratio because it is possible, even easy, to obtain optical thicknesses in trapped atom samples 10 or 100 times larger than can be achieved in our atomic beam. It is more difficult to show that optical trapping will allow the experiments to be done with very small atomic samples (10'' atoms). We are currently working on this problem. The approach we are using starts with a very small sample of a given isotope (short-lived
isotopes will be produced at an accelerator, while longer lived isotopes can be brought to our
lab), whichis injected into a special cell where the atoms will be efficiently captured by a laser trap (Fig. 24). We have Carried out detailed studies on capturing atoms from a vapor
and we are currently developing wall Coatings which will allow the cesium atoms to bounce
around inside the cell without sticking until they are captured. Preliminary work with silane
coatings has been quite encouraging. Once the atoms are captured, PNC measurements can then be carried out in the cold
dense samples. If all goes according to plan, the next decade will see high precision measurements of PNC in a number of cesium isotopes. This will provide detailed information on the nuclear anapole moment and a very precise test of the Standard Model.
161
56
Acknowledgments This work has been supported by the National Science Foundation.
162
57 References
1. M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 3 (1974)899. 2. P. Langacker, M.-X.Luo, and A. Mann, Rev. Mod.Phys. @ (1992) 87.
3. V. V. Flambaum and I. B. Khriplovich, JE"P
(1980) 835;V. V. Flambaum, I. B.
Khriplovich, and 0. P. Sushkov, Phys. Lett. B
(1984)367.
4. F. Curtis-Michel, Phys. Rev. 138B (1965)408. 5. M.A. Bouchiat and C. C. Bouchiat, Phys. Lett.
m,(1974) 111.
6. D. Z.Anderson, University of Colorado, private communication. 7. M.A. Bouchiat, J. Guena, L. Hunter, and L. Pottier, Phys. Lett. 1178 (1982)358.
8. R. Conti, P. Bucksbaum, S. Chu, E. Commins, and L. Hunter, Phys. Rev. Lett. 42 (1979)343;E. Commins, P. Bucksbaum, and L. Hunter, Phys. Rev. Lett
(1981)
640; P. H. Bucksbaum, E. D. Commins, and L. R. Hunter, Phys. Rev. D 24 (1981)
1134. 9.
S. L. Gilbert, M.C. Noecker, R. N. Watts, and C. E. Wieman, Phys. Rev. Lett. 3 (1985)2680.
10. M.C. Noecker et al., Phys. Rev. Lett. 61 (1988)310. 11. P. S. Drell and E. D. Commins, Phys. Rev. Lett. 3 (1984)968.
12. C. Wieman et al., Phys. Rev. Lett. 58 (1987)1738.
13. P. E. G. Baird et al., Phys. Rev. Lett. 3 (1977)798. 14. L. L. Lewis et al., Phys. Rev. Lett. 3 (1977)795. 15. L.M.Barkov and M.S. Zolorotev, JETP 22 (1978)357. 16. T.P. Emmons et al., Phys. Rev. Lett.
a (1983)2089.
163
58 17.
V. A. Dzuba, V. A. Flambaum, P. G. Silvestrov, and 0. P. Sushkov, Europhys. Lett.
2 (1988) 413. 18. J. H.Hollister et al., Phys. Rev.
Lett 46 (1981) 642. (1989) 147.
19.
V. A. Dzbua, V. A. Flambaum, and 0. P. Sushkov, Phys. Lett A
20.
M. J. D. MacPherson et al., Phys. Rev. Lett.61 (1991) 2784.
21.
T. Wolfenden, B. Baird, and P. Sanders, Europhys. J.
22.
V. A. Dzuba, V. A. Flambaum, P. G. Silvestrov, and 0. P. Sushkov, J. Phys. B 2 p
15 (1991) 731.
(1987) 3297. 23.
M. A. Bouchiat et al., J. Phys. (Paris) 47 (1986) 1709.
24.
S. A. Blundell, W. R. Johnson, and J. SapirSteh, Phys. Rev.
25.
Ya.B. Zel'dovich, Zh.Eksp. Teor. Fiz. 3 (1958) 1531 [Sov. Phys. JETP 2 (1957)
Lett. 65 (1990) 141.
11841.
a (1989) 949.
26.
W. C. Haxton, E. M.Henley, and M. J. Musolf,Phys. Rev. Lett.
27.
C. Bouchiat and C. Piketty, 2. Phys. C
28.
W. Marciano and D. Rosner, Phys. Rev. Lett. 65 (1990) 2963.
29.
K. T. Mahanthappa and P. K. Mohapatra,Phys. Rev. D
30.
T. Haslwanter, H.Ritsch, J. Cooper, and P. Zoller, Phys. Rev. A 3 (1988) 5652.
31.
B. Dahmani, L. Hollberg, and R. Drullinger, Opt. Lett.
(1987) 876.
32.
K. MacAdam, A. Steinbach, and C. Wieman, Am. J. Phys.
a,1098 (1992).
(1991) 91.
a (1991) 3093.
164
59
Table 1. Potential Systematic Errors Systematic Contributiona
(El Mlf)(Am= f1)
(Am=O)
Rangeb
Average
Daily
All Data
Uncertainty
-0.3 964+ 1.1 96
+0.3 96
0.4 %
-1.3%++0.4%
-0.1%
0.4%
-0.8 %++4.8 A
+1.7%
0.6% (AF=-l)
-1.1 %++6.8%
+2.4%
0.9% (AF=+l)
-0.3 %++0.6%
+0.04%
0.04% (AF=-1)
-1.6%++0.1%
-0.23%
0.06% (AF=
+ 1)
L\Er and aE,are nonreversing electric held components, Bxand E,, are misaligned magnetic
t
and electric field components, and f represents the birefringence of the coating on the output
mirror. 'The range shows largest and smallest daily corrections.
165
60 Table 2. Summary of PNC resuLs Ancient (controversial) history Bismuth
Experiment 0xfordl3 Univ. of Wash14 Novosobirskls
variations
Pb
(Wash. '83)16
f28 %
Bi
(Wash. '8 1)'* (Oxford '9 l)')
*18% *2%
(Oxford '9 l)*'
f15%
*3
293 nm (F3erkeley '85)"
*28%
f6%?=
Modern civilized
Wide
wide variations
(?) e a :
Tl 1 . 3 ~
*10%'0(?)17 f 15 % (?)19 II
% (?)22
Stark induced interference: mall
cs (Paris '84-'86)23
(Col.'85)9 (Col. '88)"
*12% *12% *2%
all agree with Standard Model
*1
ll
w
%24
166
61 Table 3. Sin2 0, values.
W mass
-
Neutrino scattering
0.2320f0.0007
4
0.233 f 0.003 f 0.005
167
62 FIGURE CAPTIONS FIG. 1. Parity violating neutral current interaction between electrons and nucleons predicted by the Weinberg-Salaam-Glashow electroweak
theory contrasted with the normal electromagnetic interaction. FIG. 2. Basic experimental set-up for optical rotation PNC experiments. FIG. 3. Basic experimental set-up for the first generation Stark interference PNC experiments.
FIG. 4. Cesium energy level diagram. FIG. 5. Theoretical spectrum of the 6S,F=3 + 7S, F'=4 transition of cesium in a weak magnetic field. (a) The pure Stark induced rate (A;). @) The AE APNCinterference terms multiplied by 106.
FIG. 6. General layout of the Boulder cesium PNC experiment. The coordinate system in the interaction region is defined by a dc electric field, E, a dc magnetic field, g, and the angular momentum, u, of the laser photons.
FIG. 7. Experimental spectrum observed when the laser is scanned over the 6S,
F=3 4 7S, F'=4 transition of cesium in a 70 G magnetic field. FIG. 8. Dependence of the signal-to-noise ratio on the electric field. FIG. 9. Cesium beam wllimation. FIG. 10. Multipass system for reusing laser light. FIG. 11. Laser power buildup cavity. TIis the transmission of the input mirror, T2is the transmission of the second mirror.
FIG. 12. Fluorescence collection and detection.
168
63 FIG. 13. Overall schematic of the apparatus for the 1985 and 1988 Boulder cesium PNC experiment.
FIG. 14. Detail of the interaction region. FIG. 15. Basic sew0 loop. FIG. 16. Spring analogy for PZT. FIG. 17. Bode plot for damped harmonic oscillator. FIG. 18. (a) Bode plot for low pass filter. (b) Bode plot for harmonic oscillator plus low
pass filter compensation. FIG. 19. Bode plot for harmonic oscillator plus phase lead compensation. FIG. 20. Comparison of the experimental measurements of PNC in cesium. FIG. 21. Constraints on the Cld and C,, coupling constants by experimental measurements. The cross-hatched region is from SLAC deep inelastic scattering data, while the solid line is from atomic PNC. The SU, x U,line is the Standard Model value
as a function of sin2 8,. FIG. 22. Schematic of the new Boulder cesium PNC apparatus. FIG. 23. Schematic of the diode laser control system. FIG. 24. Laser trap cell.
169
e
N
e
N PNC Weak Neutral
Figure 1
Figure 2
e
Coulomb
P
170
polarized state
circular polarizer detector
Figure 3
171
540nm
v
m +4
-4
-3
+3 Figure 4
172
-
m=3 tom=4
m = -3 tom=-4
I
I
Figure 5
cs beam
Figure 6
t
173
100 MHr
H
Figure 7
Nshot
G
c 4
en
0 U
E fieId Figure 8
174
capiIlary array
multislit colI imator
I<
Figure 9
Figure 10
ccyo panels
175
e
I?in -P,/(I-Tl)
I TI- 2 x IOSS
T2- 0
Figure 11
cs filter
photodiode Figure 12
176
REFERENCE CAVITY I
EOM
PC #I
POLARIZATION CONTROL
/
OPTICAL X/2 PC ISOLATOR #2
1
DYE LASER CAVITY
Figure 1 3
INTERFEROMETER
&c
+
DETECTOR
'LASER
Figure 14
BEAM
D
CESIUM OVEN
1
177
v out
gnal
laser 1
gain& > compensation electronics
A
negative feedback V
Figure 15
piezo
m
M <-->
Figure 16
178
1
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0-
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Frequency (rad/sec)
Frequency (rad/sec)
Figure 17
I
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179
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. 010'
.. .. . . ., ..
102
103
Frequency (radsec)
10'
102
Frequency (radsec) Figure 18a
103
180
50
! ' . . I." ' .! . I I I 1 I l l l l .. .. .. .. .. .. .. .. .... .... .... .... .... .... ............ .... ... ... ... ... ... ... ...... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... ... ... ...... .. .. .. .. .. .. .. .. .. ---0 L,,+--C--..,cc ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... . . . . . . . . . .. .. .. .. .. .. .. .. .. ... ... ... ... ... ... ... ... ... . .. .. .. .. .. .. .. .. .. . . . . . . . . . .. .. .. .. .. .. .. .. . . . . .. .. ...... ... ... ... ... ... ... ... ... ... . .. .. .. .. .. .. .. .. .. .... ... ... ... .... .... .... .... .... :. . :. .: .: .: :.: :.k - ; ; ; . . ..:.:.:.:A. . ...........:....... .. t......t ..:.t.:.:z:.. .. .. .. .. .. .......A......*... . . .L.&tL . . . &. -50 .......2.......... t.........:..:. .. :...L.t.Lt .. .. .. .. ..J...I:...........t.. .... . .:...:..:.t&:t...:.s+.:.:...&. .... : . .:.:.: .: .. .. ... ... ...... ...... ...... ...... ...... ... .... .... .... ..... ........................ .... ..... ..... ..... ..... ..... ..... ..... ..... ... .... .... .... .... ..... ..... .......... ..: +.;..,.-;:::: ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ... .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .............*::: . ..._ . .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . . . . . .. .. ... ... ... ... ... ... ... ... ... .. .. .. .. .. ... ... ... IIILI. !
I
-
-100
I I I 1 1 1
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.. .. .. .. ... ...! ...!...!...I!!!! . .. ..
...
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1
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.. .. .. .. .. .. .. .. .. .........
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... ... ... ... ... ... ... ... ...
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10'
102
I
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1
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100
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.. .. .. .. .. .. .. .. ..
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-------. . . . . ---... I
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I
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PARIS {I984 i : COMBINED REVISED '86 r---n---l
COLORADO 185 rn
COLORAD0'88 HI (2%) 1.o
0.5
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(EpNC)/P
1.5
[mV/cm]
I
1
I I I I L
105
104
103
P A R I S '84 t
0
1
.. .. .. .. ..,. ... .,,I......
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P A R I S '82 1
I
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181
*
t
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1.0
kk
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Figure 21
182
ATOMIC BEAM cni
I IMATnR
-.,
OPTICAL PUMPING
-
B
sz
1.7G
/
..,............ . ...... . ... .. .... . \. I’:’”””””::~~~:::~:~ ~
‘CLEAN-UP’ PUMP
h’
PROBE LASER 6s -6P312
Figure 2 2
V
LN 2 -COOLED BEAM DUMP
183
to experiment
Figure 23
Cs, Fr
Figure 24
PHYSICAL REVIEW A
VOLUME 50, NUMBER 3
SEPTEMBER 1994
Precision lifetime measurements of Cs 6p 2P1/2and 6p 2P3,2levels by single-photon counting L. Young,* W. T. Hill IIl,+S. J. Sibener,: Stephen D . Price,§ C. E. Tanner," C. E. Wieman,y and Stephen R. Leone** Joint Institute for Laboratory Astrophysics, National Institute of Standards and Technology and Uniuersity of Colorado, Boulder, Colorado 80309-0440 (Received 28 March 1994)
Time-correlated single-photon counting is used to measure the lifetimes of the 6 p 'P,,,and 6p 'P,,, levels in atomic Cs with accuracies =O. 2-0.3 70.A high-repetition-rate, femtosecond, self-mode-locked Ti:sapphire laser is used to excite Cs produced in a well-collimated atomic beam. The time interval between the excitation pulse and the arrival of a fluorescence photon is measured repetitively until the desired statistics are obtained. The lifetime results are 34.75(7)and 30.41(10) ns for the 6p zP,,z and 6p 2P,,, levels, respectively. These lifetimes fall between those extracted from ab initio many-body perturbation-theory calculations by Blundell, Johnson, and Sapirstein [Phys. Rev. A 43, 3407 (1991)land V. A. Dzuba et al. [Phys. Lett. A 142, 373 (198911 and are in all cases within 0.9% of the calculated values. The measurement errors are dominated by systematic effects, and methods to alleviate these and to approach an accuracy of 0.1% are discussed. The technique is a viable alternative to the fast-beam laser approach for measuring lifetimes with extreme accuracy. PACS number(s):.32.70.Fw,32.70.Q 42.55.R~ I. INTRODUCTION
There is renewed interest in accurate lifetime measurements of excited states in alkali-metal atoms [1,2]. The interest stems from the need to test ab initio theory [3,4], which is used to interpret parity nonconservation (PNC) measurements in atomic cesium [5]. Ideally, it is desirable t o establish the accuracy of these many-body perturbation-theory (MBPT) calculations in heavy atoms at an =O. 1-0.2 % level in order t o eliminate the theoretical contribution [6] to the error in the extraction of the weak charge Q, from P N C experiments in Cs atoms. The MBPT calculations can be tested by comparison with binding energies, hyperfine structure (hfs) constants, and dipole matrix elements. While experimental values
'Permanent address: Physics Division, Argonne National Laboratory, Argonne, 1L 60439. tPermanent address: Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742. IPermanent address: James Franck Institute, University of Chicago, Chicago, IL 60637. §Permanent address: Department of Chemistry, University of College London, London WClH OAJ, England. IIPermanent address: Department of Physics, University of Notre Dame, Notre Dame, IN 46556. !Also at Department of Physics, University of Colorado, Boulder, CO 80309-0440. Also at Quantum Physics Division, National Institute of Standards and Technology, and Departments of Chemistry and Physics, University of Colorado, Boulder, CO 80309-0440.
..
1050-2947/94/50(3)/2 174(8)/$06.00
50 -
for the first two quantities are far more precise than theory (theoretical errors are estimated to be 0.5%, I % , and 0.5% for energies, hfs, and dipole matrix elements, respectively), the dipole matrix element comparison is limited by the precision of lifetime data [1,7]. This last comparison is particularly critical for the interpretation of the P N C experiments because the measurement obtains the ratio of the parity nonconserving electric dipole ( E l ) to the Stark-induced amplitude for the 6s-7s transition. Previous precision measurements of the Stark effect [8,9], which involves sums of dipole matrix elements, also provide an important test of these calculations. More generally, it would be comforting t o verify the accuracy of the calculations for the behavior of the wave function both near the nucleus (hfs) a n d a t large r (lifetimes). The reliability of the M B P T calculations of lifetimes at the sub-1% level (dipole matrix elements) for heavy alkali-metal atoms has been cast into some doubt by the discrepancies between experiment and theory in the light alkali-metal atoms. Specifically, in Li and Na experimental lifetimes [lo] for the first excitedp states are -0.6% and 0.9% longer than the "all-order" MBPT theoretical lifetimes [ 11,121. This difference is significant since the estimated errors in Li are 0.15% experimental [lo] and 0.05% theoretical [ l 11. However, other somewhat less precise measurements, such as the (0.5-0.7 %) singlephoton counting measurements [ 13,141, show no significant deviation ( < 2 u ) from theory in either Li or Na, emphasizing the need for extreme experimental accuracy. The situation in Cs is somewhat different: both experimental and theoretical lifetimes for the first excited p states are less accurate than for the lighter alkali-metal atoms. The current MBPT calculations of the 6s-6p dipole matrix element have an estimated error of -0.5% [3]. However, there are credible expectations that improvements in the calculations will yield an increase in 2174
184
@ 1994 The American Physical Society
185 PRECISION LIFETIME MEASUREMENTS OF Cs 6p 2 P , , z . . .
50
theoretical precision of an order of magnitude 13,151. The two previous most precise experimental lifetimes for the Cs 6P3/, level, obtained with the fast-beam laser [ I ] and level-crossing [7] methods, have accuracies of 0.9% and 0.7%, respectively, but differ from each other by -2.1%. Given the expected improvement in theory and the fact that the experimental errors are dominated by systematics, an alternative, high-precision method for measuring lifetimes is necessary. In this paper we use such an alternative method, time-correlated singlephoton counting [16], to measure of the lifetimes of the 6p 2P,,2 and 6p 2P,,2 states in Cs. (Hereafter these states are referred to as 6P,,,and 6P312.I T o date, the lifetime measurement method which has claimed the highest accuracy (0.15%) [lo] has been the fast-beam laser method, where a fast-beam of atoms ( u /c = 0.1 %) is selectively excited with a perpendicularly crossed laser beam to provide a t = 0 point. The decay of the fluorescence is monitored as a function of distance downstream. The measured decay length is then transformed into a decay time by determination of the atom beam velocity. In contrast, in the single-photon counting technique, a fast pulse from a laser selectively excites the state of interest at t =O, starting a clock which is stopped by the arrival of a fluorescence photon. The measured time interval is binned and the sequence is repeated until the required statistics are attained. Because of the very high repetition rates now available from mode-locked laser systems, excellent statistics can be acquired rapidly. Thus, as in the fast-beam laser method, the final accuracy is determined by systematics. Here we demonstrate the capability of the single-photon counting technique to measure nanosecond lifetimes with accuracies at the 0.2-0.3 % level. Improvements to the method should enable lifetime measurements with accuracies =o. 1%. 11. EXPERIMENT
The experiment was performed using the apparatus shown in Fig. 1. Briefly a short, linearly polarized laser
FL ITER-
POLARIZER
-
.-).-:,<'. .
pulse, resonant with the transition of interest 6S,/,-6P,/, (6P,/,) at 895 nm (852 nm), excites Cs atoms in a wellcollimated thermal atomic beam. In contrast to the fastbeam laser method, flight-from-view problems are negligible at these low velocities, since a Cs atom travels only 0.1 mm in 10 lifetimes, corresponding to a change in the solid angle collected of -4X10p7. Fluorescence is detected at right angles to both the electric-field polarization axis of the laser and the laser propagation direction by a red-sensitive photomultiplier tube. Fluorescence photons pass through a polarizer, lens, filter, and slit before reaching the photomultiplier tube. The combined collection/detection efficiency is estimated to be =1X As is usual for high repetition rate sources, the counting electronics are operated in time-reversed mode in order to eliminate dead time due to reset of the time-to-amplitude converter (TAC). Therefore, the start pulse was provided by the arrival of a fluorescence photon that was processed through a constant-fraction discriminator. The stop pulse was provided by a fast photodiode which sampled a portion of the excitation pulse. The output of the T A C was binned by a multichannel analyzer (MCA) in conjunction with a personal computer. Since we expected the experiment to be dominated by systematic errors, the atomic beam apparatus was designed to minimize these effects. In particular, radiation trapping was a major concern. In order to minimize this effect (and other density-dependent effects, i.e., collisions) the Cs atomic beam was well collimated at the interaction region and an extensive system of liquidnitrogen-cooled baffles (not shown in Fig. 1) was employed to scavenge background Cs. In addition, Ruorescence path limiters were placed inside the atomic beam chamber along the detection line of sight. The Cs beam effuses from a 0.17-mm-diam aperture, passes through a cooled aperture whose outer diameter was intentionally designed to capture and thereby eliminate most of the off-axis Cs beam, and is collimated with a 2-mm aperture placed just before the laser interaction region. A cryogenically cooled beam catcher was placed after the in-
-
FIG. 1. A schematic of the experimental arrangement. EO is an electro-optic modulator, CFD is a constant-fraction discriminator, TAC is a time-to-amplitude converter, PMT is a photomultiplier tube, and MCA is a multichannel analyzer. See text for a detailed description.
LENS
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186 50 -
L. YOUNG et al.
2176
teraction region in order to further reduce the Cs backdistribution had spatial variations due to the finite spot ground. Based upon temperature measurements of the size in the EO. However, the pin scattering did serve as oven ( T,,,, =340 K ) and nozzle, the Cs number density an excellent monitor for EO stability during the course of in the interaction region was nominally = 1.5 X l o 6 cm - 3 . the experiment as well as a method for time calibration. The true test of radiation trapping is the comparison of The second method used to map the instrument function measured lifetimes at low and high density. These meaproduced a true spatial average of the laser profile by surements were made with high statistics for the nominal Rayleigh scattering from air and the smoke generated low density cited above and a density approximately five from a dry ice-water mixture. It should be noted that times greater. The chamber background pressure was both these methods of characterizing the afterpulse dismaintained at 2 X l o p 7 Torr. Zeeman quantum beats tribution have much greater dynamic range than simply were also of concern. The magnetic field in the interacmeasuring the output of the fast photodiode on an osciltion region was measured with a Hall probe to be ~ 0 . 6 loscope, where no afterpulses are visible. In addition, the use of a squirrel-cage style (side-on) photomultiplier tube G with an orientation -9” from the electric-field polarization axis of the laser. (PMT) has the advantage that the spurious P M T afterA commercial Ar ’--pumped, self-mode-locked pulse observed in end-on tubes is absent [16,17]. Here we Tisapphire laser was used to provide excitation pulses note that the full width a t half maximum of the P M T ( A t = 150 fs) a t a repetition rate of - 7 6 MHz. Since this response to a &function input is = 1.5 ns, well below the nominal lifetime of 35 n s (30 ns) for the 6P,,, (6P,,, I gives a period between pulses of 13.2 ns and the lifetimes state. of interest are 30 ns, pulse selection was required. The Ti:sapphire laser was passively mode locked; therefore The time calibration was mapped directly on the MCA the pulse selection was accomplished external to the laser by the afterpulses. The laser repetition rate was meacavity by double passing [17] the pulses through an sured with a commercial frequency counter, which has an electro-optic modulator (EO) as shown in Fig. 1. The double-pass geometry was necessary to reduce the leakage of unwanted pulses (hereafter called afterpulses) to a n acceptable level. The amplitude of the afterpulses i n a , 1 single-pass geometry was measured t o be 1% and JIL -0.2% in double pass. For measurement of the 6P,,: (6P,,,) lifetimes 1 out of every 17 (15) pulses was selected, yielding -20 mW of laser power in the atomic beam chamber. %” The characteristics of the Tisapphire laser necessitated careful experimental design. The spectral bandwidth of the laser pulses ( = 10 nm) is so large that > 99.9% of the light is not resonant with the transition, leading to accentuated scattered light problems. Scattered light was decreased t o the background level by the use of (1) long baffle arms with graded apertures, (2) light traps with windows a t Brewster’s angle on both the entrance and exit, and (3) a blackened atomic beam chamber. In addition, the large spectral bandwidth induces the coherent excitation of several hyperfine levels ( F ’ = 2 , 3 , 4 , 5 1 in the excited 6p ’P,,, state leading t o the observation of hyperfine quantum beats [18]. In order to eliminate the 20 000 hyperfine quantum beats, a linear polarizer was placed in “MAGIC ANGLE” the fluorescence detection stack at the “magic angle” 54.7” 15 000 (54.7”) with respect to the laser polarization direction to null the out-of-phase beating of the r and 0 components of the radiation [ 191, as shown in Fig. 2. l 5000 o o o ~ ~ ‘ \ , A thorough understanding of the instrument function, i.e., the response of the detection system to a 6-function input, is also critical for high-precision results since the 0 actual observed wave form is the convolution of the in50 100 150 0 strument function with an exponential decay. In this TIME (ns) case the instrument function assumes greater importance because the extinction of the afterpulses is not complete. FIG. 2. Elimination of hyperfine quantum beats from fluoresThe instrument function was measured by two methods. cence from the Cs 6P,,, state. Top, middle, and lower panels The first involved rotating a pin point to the interaction show fluorescence from the 6P3,z state with the detection polarregion in situ k e . , under the experimental running condiizer oriented at 90’, 0’, and 54.7”, respectively, with respect to tions) in order to scatter laser light into the detection syst h e electric-field polarization vector of the laser. 54.7” corresponds to the “magic angle” where 3 coszO- 1 =O. tem. This was an imperfect method since the afterpulse
-
-
1
r:
1,I
187
50 -
PRECISION LIFETIME MEASUREMENTS OF Cs 6p ’PI/,. . .
accuracy of lo-’. Therefore, any nonlinearities in the time scale were measured directly on the entire TACMCA system. Nonlinearities in the y scale of the TACMCA system were determined by counting random events (induced by scattering the output of a flashlight into the detection system) while maintaining the stop rate from the photodiode. Nonlinearities in the y scale could be induced by rf pickup from, say, the voltage pulse applied to the EO. Extensive shielding of the EO, EO power supply, and a low-pass filter placed directly at the high voltage input to the P M T substantially reduced these effects. A typical experimental run of acquiring a Cs tluorescence decay at a signal rate of 2.5 kHz (4.0 kHz) with a laser repetition rate of 4.4 M H z (5.0 MHz) for the 6P,,, ( 6 P 3 / , ) levels to a peak of 1OOOO or 20000 counts in a particular channel required -5-10 min. The background rate, due entirely to dark counts, was typically 400 Hz, distributed uniformly among all channels. The decays were recorded out to -6.r. Interspersed between accumulations of Cs fluorescence decay data, we measured (1) laser-off resonance signals (to monitor scattered light), ( 2 ) pin scattering signals (to monitor afterpulse amplitude), and (3) “flashlight” scattering (to monitor rf pickup). For the 6P,/, level, a total of 68 runs were taken at “low” density and about half the statistical precision was acquired in 20 runs at “high” density. For the 6 P , / , level, 30 runs were taken at “low” density and 13 runs at “high” density. The linear polarizer in the detection stack was removed and replaced with an aperture of equal size for the 6P,/,runs since no hyperfine quantum beats are expected in this case.
-
2177
tively. In this case, the instrument function was obtained by pin scattering and therefore does not represent a true spatially averaged response. The hyperfine quantum beats have been eliminated from the decay by orientation of the polarizer in the detection stack at 54.7” with respect to the laser polarization axis. The reduced ,y2 of the individual fits was typically 1.1, well within the range expected for good single-photon counting data with slightly non-Poissonian distributed noise (0.8- 1.2) [ 161. In Fig. 4 a similar set of data is shown for the 6P,,, state. In this case, the instrument function was obtained by Rayleigh scattering. Note that the amplitude of the afterpulses is much more constant in this spatially averaged case than in the pin scattered data of Fig. 3. The average amplitude of the afterpulses was =0.2%, consistent with residuals resulting from a fit to the overall summed data. For the 6 P , / , and 6P3,, levels the mean fitted lifetimes at low density were 34.75 and 30.39 ns, respectively. The standard deviations of the mean for all the low-density measurements were 0.1% (0.14%) for 6 P , / , (6P3/,). The
20000
15000
t
I
1
11
I
I
I d
INSTRUMENT FUNCTION
4
111. RESULTS AND ANALYSIS
The observed wave forms were fit to a single exponential decay plus background using a nonlinear leastsquares algorithm. Although this was not the optimal method of analysis, which would have been a deconvolution of the instrument function from the observed decay, we decided on this approach because the instrumental response measured during the experiment was not the appropriate spatially averaged response. However, we note that an analysis by forward convolution of the proper instrument function (acquired the day after the Cs decay curves using Rayleigh scattering as previously described) with an exponential decay yielded the same value as the simple analysis of the decay with a starting point ~ 7 0 % of the peak for the 6 P , / , data. Furthermore, by modeling the convolution of an instrumental response that has a train of equal fractional amplitude A afterpulses with a pure exponential decay, it was found that the simple analysis yielded the lifetime correct to within A / 1 0 . Thus, for our average fractional amplitude of afterpulses, 0.2%, the simple analysis would yield a lifetime in error by only 0.02%, a much smaller uncertainty than contributed by other factors. I n Fig. 3 an example of an instrument function, decay of the 6P3/2state, and residuals with statistical error bars from a fit to a single exponential decay plus background are shown in the top, middle, and bottom panels, respec-
I? a 3
400
9
0
g
-400
0
0
100
150
TIME ( n s )
FIG. 3. Top panel shows an instrument function taken by pin scattering with the laser tuned to the 6SI/2-6P3/2 transition at 852 nm. Middle panel shows fluorescence from the 6P3/2state with the hfs quantum beats nulled as shown in Fig. 2, as well as the fit to a single exponential plus background. Lower panel shows the residuals from the fit.
188 2178
L. YOUNG
values above correspond to a fit of the decay starting at -70% of the peak, or 1 1 ns after the peak (well after any tail of a P M T response to a &function input). For any specific starting point of the data analysis, a fit of the summed data yielded a lifetime consistent with the mean of the individual fits to 0.02%. The statistical uncertainties are small relative to the systematic uncertainties, which are shown in Table I and will be discussed next. The absolute scale for the time calibration was set by counting the repetition rate of the laser excitation pulses as described above. Nonlinearities in the time scale were then determined by fitting the peak positions of the afterpulses to a second-order polynomial. The nominal afterpulse amplitude of 0.2% was easily enhanced for time calibration by detuning the EO voltage from its half-wave value to allow greater leakage of afterpulses. The fit revealed no nonlinear term in the time scale at a level of lo-'. The linear term varied no more than 0.02% over many days, and we take this as the uncertainty in the time calibration. The nonlinearities in the y scale were checked by counting random events as described above. The accu-
-
-
20000
I
i
15000r
/
I
I
i
1'
INSTRUMENT FUNCTION
'
1'
I
0
I
t-
I
2,. 10000/-
1
50
al.
et
-
TABLE I. Single-photon counting error budget for the lifetimes of the 6p 'PI,,and 6p 'P,,,states in Cs.
____
~~~
__
Systematic
01
23 0
(0
I
I
I
I
~
~~
.~~ ~~~
~
~~~
~
6P,/*
~~
~
~
6P3,L -~~
~
Time calibration TAC-MCA nonlinearity Pulse pileup Afterpulse amplitude Radiation trapping Hyperfine quantum beats Zeeman quantum beats Truncation error
*0.02 k0.10 k0.03 k0.02
Total systematic
t0.19 i-0.10
1 0 .1330.27
Statistical Sum in quadrature
T0.22
+O. 1310.31 __
10.02 kO.10 IO.03
!.0.02
1-0.06
t0.06 *~0.07 f 0 . 13i0.10 p0.2 1
t0.15
rO.14
~
~
mulated result should be a constant over all channels if there is no systematic trend that varies the effective bin width of the MCA-TAC combination. Although the accumulated counts showed some nonstatistical deviations (most likely originating from residual rf pickup) at the 0.5% level, the data are fit by a constant with linear and quadratic terms that d o not differ from zero at the level. Analyzing data grouped in 10 and 20 2X channel bins again reveals scatter, but no systematic trends with channel position. The standard deviation in the mean value of the constant is 0.05% and we take twice this value to the largest possible systematic uncertainty. Pulse pileup in the MCA results from the statistics generated by the counting process and can be analytically corrected [16]. That is, due t o the fact that the TAC is stopped only once during an excitation cycle, counts at early stop times are preferentially counted leading to a systematic correction in the fitted decay. The effect is obviously more pronounced when the detection rate is a significant fraction of the excitation rate. We therefore kept the fraction of detected events below & of the excitation events. The analytical correction for the true number of counts N , in the ith channel is given by
N, = -
___.
Error ( % I
Nl
1 l--x
'
1-1
N,
N E ,=I
400
0 -400
0
50
100
150
TIME ( n s ) FIG. 4. Top panel shows an instrument function taken by Rayleigh scattering from air with the laser tuned to the 6S1,26 P , , , transition at 895 nm. Middle panel shows fluorescence from the 6 P , , , state, as well as the fit to a single exponential plus background. Lower panel shows the residuals from the fit.
where N E is the number of excitation cycles and Niis the observed number of counts in channel i. With this correction, we find that, for data analysis starting from 70% of the peak height, the fitted lifetime decreases by 0.03%. The extent of radiation trapping (and other collisional effects) were evaluated by varying the Cs beam density by a factor of - 5 for both the 6 P l / , and 6P,,, levels. This was accomplished by varying the Cs oven temperature and not by any changes in experimental geometry. The change in the lifetime between high and low density was
189 50
PRECISION LIFETIME MEASUREMENTS OF Cs 6p 'PI
-0.29% for both levels. Dividing by 5 (a conservative estimate) for a linear extrapolation to zero density yields a systematic error of 0.06% for radiation trapping-Cs-Cs collisions, which is statistically insignificant. This is consistent with the estimated reabsorption probability for the density and column length along the detection path ( 1 . 5 X lo6 Cs cmP3, 0.24 cm, nabs=lo-'' cm2). Collisions with background gas are not expected to be important at these pressures. The quenching cross sections for the lowest excited p states in Rb by N , [20] are =50 A'. For Cs-N2 collisions with a relative velocity of - 4 X lo4 cm/s and a background density f: N , of 7X 109/cm3, a quenching cross section of lo7 A' would be required to change the lifetime by 1%. Thus these effects are considered negligible. Hyperfine quantum beats are present only in the decay of the 6P3,' level. The magnitude of their effect was estimated as follows. For decays with the detection polarizer oriented at 0" or 90", the quantum beats due to hfs are clearly apparent, as shown in Fig. 2. It is also clear that the lifetime will depend on the range of data included in the fitting procedure, i.e., whether one begins fitting on a peak or a valley. These variations can be quite large, with a peak-to-valley amplitude change of -3-5 %. Therefore, it was considered best to compare a weighted average of the fitted lifetimes over a large fit range with the same quantity for the decays free from hfs quantum beats. In this comparison, it was found that the ( T ( T - ) radiative lifetime differed from the quantum beat free lifetime by -0.1-0.2 %. Unfortunately, the statistical significance of this result is at the 1.2% level due to the small amount of data taken at the 0" and 90" detection angles. Since these lifetime variations are due to the maximum quantum beat size and they are attenuated by at least 50 times due to magic angle orientation of the detection polarizer ( -5500:l extinction at 852 nm), a value of 0.07% is assigned to the systematic uncertainty due to hfs quantum beats [corresponding both to ( 3X 1.2 )%/50 and the peak-to-valley change 4%/50]. The systematic effects of Zeeman quantum beats in the decays can be calculated for the Cs system [2 1,221. These are particularly insidiuous because at the low fields (0.6 G) present in the experiment, there are no observable beats. For an orientation of the B field coincident with the laser electric-field polarization axis, Am =O selection rules rigorously prohibit the generation of Zeeman quantum beats. However, the B field in this experiment was actually tipped at --9" from the laser polarization axis toward the detection axis. This enabled a small admixture (2.7%) of 0 excitation and resulting quantum beats to be present in the decay. A simulation of this was done, including convolution with an instrument function, to yield a decay which could be fitted by the single exponential plus background. The fitted lifetime for the 6P,,, state is found to be 0.13% shorter than the decay free from quantum beats with an estimated uncertainty of 0.10% arising from the combination of an allowed *lo" variation in the orientation and a +15% variation in the B-field strength. We make this 0.13% correction for this state as shown in Table I. For the 6 P , / , state, the Zeeman quantum beat effects are negligible.
-
-
...
2179
Finally, the largest systematic error found is the variation of the lifetime with the starting point of the fit, termed truncation error. By assessing effects through the convolution of instrument functions with model decays, it was determined that this systematic was due to the uariation in the afterpulse amplitudes. As discussed earlier, the actual magnitude of the afterpulses results in only a small distortion of the fitted lifetimes (change of 0.02% for 0.2% afterpulse). However, it is the variation in the afterpulse amplitude that causes this much larger error. These variations in afterpulse amplitude could be caused by, e.g., electrical reflections in the EO transmission line, resulting in some variation in the voltage applied to the EO, and hence in the transmission of the afterpulse. The magnitude of the afterpulse variation is estimated to be at most 4070, which in turn leads to an error in the fitted lifetime of 0.23% when the starting point of the fit is at -70% of the peak. (A more likely value for the magnitude of the afterpulse variation is 20%, based upon the Rayleigh scattering instrument function data, as shown in the top panel of Fig. 4.) As one would expect, the effect of the variation in afterpulse magnitude increases as the starting point is moved out in time, whereas the tail of the PMT response is enhanced at early start times. For the 6P,/' data the observed variation in lifetime as a function of fit range was +O. 15%, for starting points ranging between 80% and 50% of the maximum (9-28 ns after the pulse). The weighted average of these fitted lifetimes is reported for the 6P1/2 level. (The 6P,,, data were also analyzed using the Rayleigh scattering instrument function in a forward convolution with a single exponential decay. In this analysis, there was no systematic variation of the fitted lifetime starting from 90% to 50% of the peak.) For the 6P3/2 data, the observed changes in lifetime over the same fit range were +O. 31%, +O. 12%, and *0.09% for cumulative runs on three different days, with EO realignments interspersed. As in the previous case, for the 6P3/2 data the weighted average of data with starting points between -80% and 50% of the maximum is reported as the actual lifetime and a truncation error of 0.2 1% was assigned.
-
IV. COMPARISON WITH MBPT
The values obtained for the 6 P , / , and 6P3,, lifetimes are shown in Table 11, along with the values derived from TABLE 11. Comparison of selected experimental and theoretical lifetimes for 6Pl/z and 6P3/, levels in Cs.
Lifetime (ns) Experimental method Single-photon counting (this work) Fast-beam laser [26] Fast-beam laser [l] Level crossing [7]
6p1/2
6p3/2
34.75(7) 30.41(10) 34.934(94) 30.499(70) 30.55(27) 29.9(2)
Theoretical MBPT calculations Blundell, Johnson, and Sapirstein [3J 34.51 Dzuba et nl. [4] 34.92
30.f3 30.49
190 50 -
L. YOUNG e t a ! .
2180
the two ab initio MBPT calculations performed by Blundell, Johnson, and Sapirstein [3] and Dzuba et al. [4]. For the theoretical values, ab initio matrix elements and experimental energies are used to compute the lifetimes. In addition, we include the previous most accurate results ( < 1%) derived from fast-beam laser and level-crossing experiments for comparison. A more complete survey of other experimental results can be found in Ref. [I]. As can be seen from the table, the lifetimes measured in this work are consistently longer than the values obtained by Blundell, Johnson, and Sapirstein by -0.7-0.9%. This leads to a disagreement between experiment and Blundell, Johnson, and Sapirstein’s theoretical dipole matrix elements of -0.3-0.5 %, corresponding to the estimated level of uncertainty in the calculation 0.5%. Although the significance is diminished by the theoretical uncertainty, it should be noted that the same trend is found in the lighter alkali-metal atoms, where the measured lifetimes are consistently longer than those calculated [2]. It is unfortunate that only Blundell, Johnson, Sapirstein, and co-workers have performed MBPT calculations on both the light [11,12] and heavy [3] alkali-metal atoms, so that comparisons between few- and many-electron systems are limited to their results. In contrast to those from Blundell, Johnson, and Sapirstein, the calculations of Dzuba et al. are longer than the present results and are in agreement at the few tenths of a percent level. A comparison to other sub-1% experimental results is shown also in Table 11. The present single-photon counting lifetimes appear to be consistently faster than the fast-beam laser results and slower than the level crossing results, but not by an amount that is statistically worrisome, i.e., 5 2ocombined. Not shown in Table I1 is the comparison of relative oscillator strengths obtained from these measurements. is 2.075(8). Our value for the ratio f(6P3,,)/f(6P,/,) This is in good agreement with the previous most accurate study of Shabanova, Monakov, and Khlyustalov [23], who obtain a value of 2.078(12), as well as an earlier semiempirical calculation by Norcross [24], who obtains 2.08.
function (in our case, the spatially averaged Rayleigh scattering) would also eliminate the truncation error and allow data treatment by convolution. This would have the added advantage of permitting the analysis of fast decays in which the lifetime approaches the instrument function width. A more careful alignment or nulling of the magnetic field that provides the quantization axis can essentially eliminate effects due to Zeeman quantum beats. In addition, performing the experiment on a single ion in a trap can eliminate the effects due to radiation trapping and hyperfine quantum beats (with judicious choice of isotope). The feasibility of such experiments has been demonstrated in the alkali-metal-like Ba+ ion [ 2 5 ] . With these improvements, one an expect lifetimes to be measured at the -0.1% level in the near future using single-photon counting. The single-photon counting technique can also be easily extended to measure higher-lying excited states in various alkali-metal atoms. In Cs it is these higher-lying levels, i.e., 7P,,, and 7P,,,, that exhibit much greater discrepancy with MBPT theory. Since the single-photon counting technique uses high-repetition-rate pulsed lasers, frequency doubling to reach the appropriate wavelengths is not difficult. In summary, the lifetimes of the 6P,,, and 6 P j I 2states in Cs have been measured by the single-photon counting technique with accuracies at the 0.2-0.3 % level. The lifetimes are in agreement with those derived from ab initio MBPT calculations of the dipole matrix elements [3,4] in combination with the experimental values for the transition energies [27] at the sub-1% level. This confirms the accuracy of the ab initio calculations of the dipole matrix elements at the 0.5% level, in agreement with the estimated theoretical uncertainty. The trends observed in the lighter alkali-metal atoms, i.e., measured lifetimes longer than those from Blundell, Johnson, and Sapirstein’s MBPT calculations, is also observed for Cs, at approximately the same level.
V. DISCUSSION AND CONCLUSIONS
L.Y., W.T.H., and S.J.S. thank the Joint Institute for Laboratory Astrophysics (JILA) for the support of the Visiting Fellow Program, which made this experiment possible. We also thank J. Hall, A. Phelps, and the JILA electronics shop for the loan of equipment and useful suggestions. H. Green and P. Smith were instrumental in the construction of the atomic beam apparatus. S.J.S. thanks D. Cho for helpful discussions concerning Cs beam operation. L.Y. thanks G. R. Fleming, S. Rosenthal, and C. Kunasz for helpful discussions. C.E.T. was supported by the Luce Foundation and the University of Notre Dame. This research was supported by the National Science Foundation and the U.S. Department of Energy, Office of Basic Energy Science under Contract NO.W-31-109-ENG-38.
We have demonstrated the capability of the singlephoton counting technique to measure lifetimes in the nanosecond regime to an accuracy of a few tenths of a percent. These values are the most accurate lifetime measurements of the first excited p states in Cs. The accuracy of our present results is not limited by factors which are intrinsic to the technique, but rather due to the particular implementation. Complete removal of the afterpulse train (and hence variations in afterpulse amplitude that lead to the largest systematic uncertainty, truncation error) can be accomplished by cavity dumping the laser rather than using an external EO modulator. Alternatively, an in situ measurement of the proper instrument
ACKNOWLEDGMENTS
191
50
PRECISION LIFETIME MEASUREMENTS OF Cs 6p *P,,*. . .
(11 C. E. Tanner, A. E. Livingston, R. J. Rafac, F. G. Serpa, K. W. Kukla, H. G. Berry, L. Young, and C. A. Kurtz, Phys. Rev. Lett. 69, 2765 (1992). [2] J. Jin and D. A. Church, Phys. Rev. Lett. 70, 3213 (1993). [3] S . A. Blundell, W. R. Johnson, and J. Sapirstein, Phys. Rev. A 43,3407 (1991). [4] V. A. Dzuba, V. V. Flambaum, A. Ya. Kraftmakher, and 0. P. Sushkov, Phys. Lett. A 142, 373 (1989). [5]M. C. Noecker, B. P. Masterson, and C. E. Wieman, Phys. Rev. Lett. 61, 310 (1988). [ 6 ]S. A. Blundell, W. R. Johnson, and J. Sapirstein, Phys. Rev. Lett. 65, 1411 (1990). [7] S. Rydberg and S. Svanberg, Phys. Scr. 5 , 209 (1972); S. Svanberg and S. Rydberg, 2. Phys. 227,216 (1969). [8] C . E. Tanner and C. E. Wieman, Phys. Rev. A 38, 162 (1988). [9] L. R. Hunter, D. Krause, Jr., K. E. Miller, D. J. Berkeland, and M. G. Boshier, Opt. Commun. 94, 210 (1992). [lo] A. Gaupp, P. Kuske, and H. J. Andra, Phys. Rev. A 26, 3351 (1982). [ l l ] S. A. Blundell, W. R. Johnson, Z . W. Liu, and J. Sapirstein, Phys. Rev. A 40,2233 (1989). [12] C. Guet, S . A. Blundell, and W. R . Johnson, Phys. Lett. A 143, 384 (1990). [13] J. Carlsson, Z . Phys. D 9, 147 (1988).
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[14] J. Carlsson and L. Sturesson, Z. Phys. D 14, 281 (1989). [15] S . A. Blundell, J. Sapirstein, and W. R. Johnson, Phys. Rev. D 45, 1602 (1992). [I61 D. V. O’Connor and D. Phillips, Time Correlated Single Photon Counting (Academic, London, 1984). [17] D. Bebelaar, Rev. Sci. Instrum. 57, 1116 (1986). [I81 S. Haroche, J. A. Paisner, and A. L. Schawlow, Phys. Rev. Lett. 30, 948 (1973). [19] J. S . Deech, R. Luypaert, and G. W. Series, J. Phys. B 8, 1406 (1975). [20] W. Happer, Rev. Mod. Phys. 44, 169 (1972). [21] M. P. Silverman, S. Haroche, and M. Gross, Phys. Rev. A 18, 1517 (1978). [22] P. J. Brucat and R. N. Zare, J. Chem. Phys. 78, 100 (1983). [23] L. N. Shabanova, Yu. N. Monakov, and A. N. Khlyustalov, Opt. Spektrosk. 47, 3 (1979) [Opt. Spectrosc. (USSR) 47, 1 (1979)l. [24] D. W. Norcross, Phys. Rev. A 7,606 (1973). [25] R. Devoe (private communication). [26] R. Rafac et al. (unpublished). [27] C. E. Moore, Atomic Energy Levels as Derived from the Analyses of Optical Spectra, Natl. Bur. Stand. Ref. Data Circ. No. 35 (US.GPO, Ser., Natl. Bur. Stand. (US.) Washington, DC, 1971), Vol. 111.
1’1-IYSICAI. REVIEW A
VOLUME 55. NUMBER 2
FEBRUARY 1997
Precision measurement of the ratio of scalar to tensor transition polarizabilities for the cesium 6 s - 7 s transition D. Cho I l e p i ~ l i i i e i i tq /
Ph,vsics, Korea Cliziversitv, Seoul, Koi-eii 136-701
C. S. Wood, S. C. Bennett, J. L. Roberts, and C. E. Wieman JIL.4 niid
Uepnrtiiiciit
oj’Pliysics, Uiiivemih, of Colornclo-Boiilcler. Uo7ildei-. Colos~ido80309 (Received 20 September 1996)
The ratio of scalar t o tensor transition polarirabilities (dulp)for the cesium 6S,.,b’= 3 to 7S,,2F’ = 3 transilion has been mcasurcd using a spin-polarized atomic beam and an interference technique. Wc use an electric- and magnctic-ficld configuration where the Stark-induced electric dipole transition amplitudes arising from the scalar and tensor polarirobilities interfere. giving rise to large changes in the 6 s - 7 s transition rate. The ineas~iredratio is c~lp=-9.905-+0.0I 1 . This result represents a tenfold improvement in precision. and agrees well with previous nieasurcmcnts and calculations. The ratio LYIPis of critical importance for ccsitini atomic parity violation esperiments. [S l050-2947(97)08202-4] PACS nuniber(3): 32.10.Dk. 32.60.+i. 32.5O.Hz. 1 I .30.Er
1. INTRODUCTION
The 6s-7s transition in cesium has been thoroughly studied both experimentally [1,2] and theoretically [3-51 for its role in parity nonconservation (PNC) measurements. Our 1988 experiment used the interference of the Stark- and PNC-induced electric dipole (El) amplitudes for this transition to measure PNC with an uncertainty of 2%, and an ongoing experiment seeks to reduce this error further. In these experiments the ratio of the PNC amplitude ElpNCto the tensor (or vector) transition polarizability G, is measured. In order to extract ElpNCthe quantity which contains information about the weak interaction, p needs to be known accurately. Generally the scalar transition polarizability a for the 6 s - 7 s transition is known with better accuracy [6],because it is ten times larger than p. In practice, the calculated value for a is used in conjunction with the measured ratio alp in order to obtain p. A direct measurement of a is difficult with our apparatus, which is more suited to measuring ratios of transition amplitudes. Precise measurements of a by a different research group are underway 171, so the accuracy of /3 may soon be limited by the measurement of a/p. Also, interpretations of atomic PNC experiments rely on atomic theory, and the ratio a/p is one of many important experimental quantities which is used to test the accuracy of these elaborate calculations [4,5]. Finally, our spin-polarized cesium beam will allow measurement of the Stark-PNC interference on the AF=O components of the 6 s - 7 s transition, yielding El P N C I ~ Comparisons . of the three measurements E l p N ~ / a ElpNCIP, , and dulp form a crucial consistency check for the PNC measurement. The ratio alp was measured by three groups in the 1980s. Two of the experiments measured the Stark-induced E l transition strengths of the pure scalar (a,AF=O) and tensor (p,AF= ? 1, F is the hypefine quantum number) parts separately and compared them [8,9]. The third group used hyperfine mixing from a magnetic field to induce an interference of (Y and p in the 6 s - 7 s transition rate [ l o ] .All three groups l050-2947/97/55(2)/1007(5)/$10.00
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measured the ratio with an accuracy around I%, and the results were consistent. The authors of Ref. [4] calculated this ratio, and found a/p=-9.93(14), which agrees well with the experiments. In this paper we report a measurement of a/p using a spin-polarized cesium atomic beam. An interference technique is used which is analogous to that used in the StarkPNC interference measurement. Atomic-spin polarization allows the use of an experimental configuration where there is a strong interference of the Stark-induced a and p transition amplitudes. Many improvements in signal-to-noise ratio and calibration accuracy of our polarized atomic beam apparatus-which was built specifically to measure the much smaller Stark-PNC interference-have allowed us to measure alp easily with a precision of a part per thousand. 11. THEORY
A. Principle of the measurement
When E is the static electric field and E is the laser polarization, the Stark-induced E 1 transition amplitude between and 17S112,F’,ml;,)states can be written the 16S112,F,mF)
The expressions for a and p are given in Ref. [3],and (r is the Pauli spin operator. In our experiment E=Ex, and E = c z i + i E , i is the elliptical polarization of the laser driving the 6 s - 7 s transition. A magnetic field B=B$ is also applied which, in conjunction with the atomic beam, gives the field configuration shown in Fig. 1. For a A F = O and Am,=O transition, both terms in Eq. (1) contribute, and the transition rate becomes
1007
0 1997 The American Physical Society
193 CHO. WOOD, BENNETT. ROBERTS, AND WIEMAN
1008
I
Y
P(‘
Atomic beam
Laser
Ez V
X
FIG. I . The ficld configuration used for the (YIPiiieasurcmcnl
where gi;= - i is the gyromagnetic ratio for F = 3 , and gi. = f for F=4. The ap interference term changes sign when the laser polarization switches from left elliptical to right elliptical ( E = e 2 ; 2 i c / i ) , or when the atoms are optically pumped into the opposite 177, sublevel. We measure the fractional modulation of the transition rate synchronous with these “reversals” to extract the interference term and determine the ratio a/P. In this measurement the use of an optically pumped atomic beam was crucial, because without spin polarization this interference term averages to zero, due to opposing contributions from symmetric, equally populated mi: sublevels. Using interference to measure the ratio greatly simplifies the experiment. With circularly polarized light and highly spin-polarized atoms in the F=3, mF=3 state, the fractional modulation was 15%, and could be made even larger with elliptical light. This is a larger modulation than earlier interference measurements achieved using magnetic-field mixing [lo]. Here the modulation size is nearly independent of the magnetic-field strength (only misalignments contribute). Further, with this field configuration the magnetic dipole ( M I ) transition amplitude for the 6 s - 7 s transition vanishes. This allows a clean measurement because this small A41 amplitude is typically a source of many systematic effects. The size of the fractional modulation depends on the ellipticity of the laser polarization and the degree of atomic spin polarization. As discussed below, we used sensitive laser polarimeters and stimulated Raman spectroscopy to measure these parameters accurately. B. Systematic effect from the ac Stark shift
The AF=O a transitions are the largest of the Starkinduced 6 s - 7 s transitions, yet their oscillator strengths are only lo-” at E=500 V/cm. In order to compensate for this weak oscillator strength, we use a power buildup cavity in a Fabry-Pkrot configuration [ 11. A typical laser intensity inside the cavity is 1 MW/cm2 at the antinodes. The high intensity allows us to observe these weak transitions, but with a serious side effect. The light also induces ac Stark shifts on the 6S112 and 7 S I l 2states. The shift for the InSlnF,mF) state can be written
A f ac( nSu2 F , m F) 1
=
where a; and
E*
+ ip;(
E*)
. ( F , ~ ~ (TI I ~
, m ~ ( )3 ),
are the ac scalar and tensor polarizabilities
55
for the n S l i 2state. This expression is the ac analog of Eq. ( I ) , and the a,: term can be identified as the usual ac Stark shift. Interestingly, the p,: term is a tensor contribution to the energy shift of the InSIIZ,F,rn,..)state. Using tabulated cesium oscillator strengths, we estimated ah = 178, ph = - 4, and a;=43, p;= -71 in atomic units. The 6 s - 7 s resonance frequency can be shifted as much as 20 MHz at our peak intensity due to the ac Stark effect. Furthermore, a,{;, is spatially modulated by the standing wave, leading to inhomogeneous broadening of the transition and an asymmetric line shape [ 1 I]. In our experimental configuration the shift in the 6 s - 7 s transition frequency is A,fa,(6S-7S)= ~ ~ ‘ / + 2 p ‘ e ~ e , g. ~Here . m ~a‘, and p’ are the differential ac polarizabilities of the 6 s and 7s states, and 7 7 . / = E J + E; I S the laser intensity. Note that the tensor part changes sign as the laser ellipticity or the atomic-spin polarization is reversed. This leads to the same modulation pattern as the CYPinterference tenn in the transition rate, Eq. (2). The complicated line shape for the 6 s - 7 s transition in our apparatus is almost completely determined by ac Stark broadening. Modulation of the ac Stark shift causes a change in this line shape, and gives rise to a modulation in the transition rate. This exactly mimics the true ap interference, and was the dominant systematic effect in our measurement. This effect was avoided by taking data at low laser intensities, and extrapolating the results to zero intensity. The dc electric field was adjusted to keep the transition rate suficiently large as the intensity was reduced. For a given laser intensity the systematic effect is largest when the light is circularly polarized ( E J E ~ = l ) , in contrast to the true signal which is proportional to eZ/ e l . To maximize the signal with respect to this systematic error data were taken at E ~ / E { Z =1. The validity of the extrapolation procedure was confirmed by taking data with three different polarization ratios, and comparing the results. 111. EXPERIMENT
A. Atomic beam and 6 s - 7 s excitation
The beam machine was originally built for a thirdgeneration cesium PNC experiment [ 121, but switching between the normal PNC field configuration and the a/p measurement configuration was easy. Three large pairs of coils wound around the vacuum chamber provided the magnetic fields for the a/p experiment. One pair produced a 1 -G magnetic field along they direction, while the other two canceled the ambient field so that the applied field was parallel to the 6 s - 7 s excitation laser beam. A cesium oven produces an intense, highly collimated atomic beam whose flux at the 6 s - 7 s excitation region is l O I 3 s-’. The first step in the experiment is to optically pump the atomic beam using the B field in the y direction as the quantization axis. Linearly polarized light tuned to the 6S1/2F=4+6P3/2F’= 3 transition depletes the F=4 state, and circularly polarized light at the 6SIl2F=3+6P3/*F’ = 3 transition pumps more than 98% of the atoms into the desired 16S112,F= 3,mF=3) state. Both laser beams propagate in the y direction. Fewer than 0.05% of the atoms remained in the F=4 state. The laser beams for the optical pumping were provided by two external-cavity diode lasers
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PRECISION MEASUREMENT OF THE RATIO O F , , ,
at 852 n m [13], and additional details ofthe optical pumping are described in Ref. [12]. By switching between right and left circularly polarized 6 S , / ? F =3 +6P712F’ = 3 light, the atoms were pumped to either r7ii = 3 or -3. This switching, accomplished with a rotating quarter-wave plate, is called the M reversal. The 6 s - 7 S excitation region consists of the power buildup cavity for the light driving the 6 s - 7 s transition (providing E ) and a set of electric field plates (providing E). The field plates are separated by 1 cm and a typical electric field is 300 Vicm. The atomic beam ( 2 ) and the standing wave 6) intersect at the center of the gap between the plates ( E = E o x ) . A ring dye laser system produces 540-nm light for the 6 s - 7 s excitation. The dye laser’s frequency, intensity, transverse mode, and polarization were all precisely controlled. The dye laser frequency was tightly locked to the Fabry-Perot power buildup cavity resonance, which was in turn locked to the peak of the 6 s - 7 s transidon via a stable reference cavity. Detection occurs on the 6S,,,F=4 state further downstream (after the 6 s - 7 S excitation region), where the beam from a third external-cavity diode laser intersects the atomic beam. Atoms in the F = 4 state scatter light tuned to the 6SI1,F=4+6P 3/2F’= 5 “cycling” transition. Approximately 200 scattered photons per atom were collected with a large area photodiode as the atoms pass through this probe region. This large amplification of the fluorescence makes the noise from scattered light, the photodiode, and amplifiers insignificant. B. Measurement of ( m F ) ,Raman spectroscopy
For the duip measurement we need to measure the average magnetic quantum number (rnF)=2fn,mF of the spinpolarized atomic beam accurately. Here f,,, is the fractional population of the mF‘sublevel. Stimulated Raman transitions [14] between the ground hyperfine levels ( ( 6 S F =3 , m F ) to 16SF=4,mF)) were used to probe the m F sublevel populations. Appropriate laser frequencies for the Raman spectroscopy were derived from an external-cavity diode laser whose injection current was modulated by a microwave source [ 151. This allowed us to probe the mF population under the same conditions as the duip measurement, at the same spatial location where the 6 s - 7 s transition was driven, and use the same detection scheme without introducing cumbersome microwave cavities inside the vacuum chamber. We used IJ+ polarization to drive A F = 1 , AmF=O Raman transitions, and integrated the signal from each of the seven Raman resonances to infer the fractional populations. The high degree of spin polarization allowed us to measure ( m F )with 0.1% accuracy, while knowing the relative line strengths for the Raman transitions with only 10% accuracy. Details of the spinpolarization measurement are discussed elsewhere [ 161. C. Generating and measuring
We used a Pockels celI to control the polarization of the dye laser beam. By adjusting the applied high voltage, we could control and reverse the phase relationship between the 2 and i polarization components. For example, a ?90” phase shift results in left or right circular polarizations. We call this the P reversal. The ellipticity was controlled by rotating a
1009
half-wave plate placed just before the Pockels cell. The effect of this wave plate is to change the ratio of major to minor axes of the ellipse without changing the phase relationship, which is determined by the Pockels cell high voltage. In this way we could produce well-defined polarization ellipses, E = ~ , ii ei, i , for the experiment [17]. The power buildup cavity created a unique problem for the polarization measurement. The injected polarization state is distorted due to birefringence associated with the mirror coatings. The mirrors are very high quality, and are mounted carefully, yet they show a birefringence of about 2 X rad per reflection. Because the cavity has very high finesse (= 100 000), the accumulated effect results in a few percent change in E. This birefringence introduces a modulation on the 6 s - 7 s transition rate synchronous with the P reversal. We use a pair of nearly crossed, independently adjustable, temperature-stabilized 1/30 wave plates placed in front of the cavity to compensate for the mirror coating birefringence. This device has the exact opposite effect on the polarization state that the power buildup cavity has, so the combined effect of the wave plates and power buildup cavity is to leave the polarization state unchanged. In order to measure the relevant polarization ratio E ~ / E two ~ , different techniques were employed [IS]. We measured the transmission through either a rotating linear polarizer or a combination of a rotating quarter-wave plate and a fixed linear polarizer. Analysis of the Fourier components in the measured intensity gives the desired information about the polarization state. We used two types of zero-order quarter-wave plates, one made of mica and another utilizing a thin polymer layer. The polarizer was a Clan-Thompson prism. Care was taken to avoid errors from saturation of the photodiode, scattering centers in the polarizer and wave plates, etalon effects from flat surfaces, and unwanted polarizing effects from attenuators, the photodiode window, or the photodiode itself. Repeated measurements of e Z / e Ifor a given polarimeter showed a random distribution with a fractional standard deviation of 0.2%. Average measurements from the different polarimeters agreed to 0.1%. We need the polarization ratio E~ / E, inside the cavity, but can only measure it outside the cavity. The light leaving the cavity is transmitted through the output mirror substrate and a (carefully mounted) vacuum chamber window, so these optics could introduce errors if they are birefringent. Several measurements of LYIP were repeated with the mirror and the output window rotated by 90” to confirm that these optics do not introduce polarization errors. D. Measurement of the Cup interference
The interference causes a large modulation of the transition rate which is easily identified due to the sign change synchronous with the P reversal and the M reversal. When eZ/ E , = 2 , the transition rate modulates by nearly 50%. This provided a large signal-to-noise ratio, but the large and rapid change in the signal, along with the synchronous ac stark shift systematic, interfered with the locking of the laser frequency to the 6 S - 7 S peak. There were several other problems caused by this large modulation, such as linearity of the photodiode and subsequent electronics over such a wide range, and saturation of the transition changing the apparent modulation size.
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CI 10. WOOD. BENNETT. ROBERTS. A N D WIEMAN
A l l of these problems were solved by introducing a complernentaiy modulation of the electric-field strength synchronous with the P and M reversals. The resulting transition rate remained constant. We switched between two high voltage supplies at I/', and V,, and adjusted the voltages until there was no modulation on the signal. In that case ( V , - V z ) / (V , + V 2 ) = ( P ~ _ / ~ ~ , ) g i . ( nand 7 / . )the , only quantities that needed to be measured to determine a//3 were I/, , V ? , E , / E , , and ( m , . ) . This method also eliminated the need to determine the background accurately. In addition to the P and M reversals, we included reversals of E and B i n the measurement procedure. The direction of these fields did not affect the a/3 interference, but did help us eliminate potential systematic errors. For example, patch effects or other surface effects on the field plates can change the estimate of ( V , I/?)/( V , + V?). Such an effect averages out when the E reversal is incorporated. The P , B , E , and M reversals were carried out by computer, with the P reversal most frequent (25 Hz). For each of the 16 possible field configurations the 6 s - 7 s transition signal was integrated and digitized. On-line analysis presented us with fractional modulations under all combinations of the reversals at the end of a 2 min measurement cycle. ~
IV. RESULTS
Measurements of CYIP on the 6S, F = 3 to 7S, F ' = 3 transition were made at E; / e l = 0.995, 1.377, and 1.869. At each value of E, / e l we measured culp at intracavity laser intensities of 25, 50, and 75 kW/cm2. The linearity of the intensitydependent systematic error was quite good out to intensities around 300 kW/cm', and began to saturate above that. At a typical electric field of 350 V/cm, an intracavity intensity of 75 kW/cm2, and circularly polarized light, 1% of the atoms in the beam participated in the 6 s - 7 s transition with a signal-to-background ratio of 5. Our measured signal to noise ratio on the peak of the 6S-7S transition is greater than 50 000 in 1 s. Results of the Cyip measurements and a linear fit to the data are shown in Fig. 2. The data are plotted on a scaled axis, where the intracavity intensity has been divided by E; / e l , which properly accounts for the polarization dependence of the systematic error. This procedure effectively maps three lines with different slopes onto a single line. For each data point we measured the laser polarization before and after the measurement, and if ez/el drifted by more than 0.3% over the run (10 min) the data were rejected. Less than 20% of the data was rejected due to such drifts. Each data point has an uncertainty of 0.1% due to e Z / e I ,but this error is sensitive to small changes in positions and orientations of the analysis optics, and is slightly different for each measurement. We therefore expect these errors in E ; / E ~to look random in nature. This is confirmed by the distribution of residuals from the linear fit, which is consistent with a random uncertainty of 0.1%. The atomic-spin polarization was more stable at ( ml;) = 2.975? 0.003, and was measured less frequently, but this error is the same for all nine points and is treated differently. The whole line could be shifted up or down by 0.1% due to this uncertainty. Linear regression gives Culp=-9.905+0.0043 for the value at zero intensity, where the error is from the random character of the e z / e ,
10.04
1 0 . o ~-
10
a
-_P
t
= syst. error
--
9.98--
9.96 -9.94 -9.92~-
0
9
A
9.9 -9.88
k
__i
FIG. 2. Measurements of a/p vs the scaled intensity inside the po\%er buildup cavity. Thrce sets of data with different values of E , / E ~ are shown. as well as the linear fit Lo the data. e z / ~ , = 0 . 9 9 5 (squares). E , / E / = 1.377 (circles). and E ~ / E / =1.869 (triangles). The systematic error which applies to all points is shown at top left. From the fit. the value of CYIp at zero intensity i s -9.905, with a randoin error from the scatter equal to 0.0043.
measurements. We add this random error in quadrature with the 0.1% systematic error from the (mi.) measurement, and arrive at
alp= -9.905t0.011
(4)
for the ratio of scalar to tensor transition polarizabilities for the cesium 6S-7S transition. This result is compared with the previous measurements and calculation in Fig. 3. From our studies of systematic errors for the PNC experiment we know there are sources of modulation synchronous with the P and M reversals other than Lvip interference. All of them are smaller than the ap term by more than four orders of magnitude, however. We also tried different electric- and magnetic-field strengths, and obtained the same measured value of alp. Misalignments in the magnetic field, however, can reduce the size of the modulation by cos0. Here B is the angle between the magnetic field and the laser direction. Careful alignment ., ensured that this error was negligible. 10.2 T
9.7 ..
1
FIG. 3. Previous measurements and calculation of d/3, and our result.
196
55
PRECISION MEASUREMENT OF THE RATIO O F . . . V. CONCLUSIONS
This cvip result has sufficient accuracy for use in future P N C measurements, in addition to posing an important challenge to atomic theorists. In order to achieve this result we have demonstrated measurements of laser elliptical polarization states with 0. I % accuracy, and atomic-spin polarizations with the same accuracy. It is worth pointing out that this measurement of (yip has additional ramifications for the PNC measurement besides confirmation of atomic theory calculations and interpretation of the final PNC result. Several of the subsystems used here are critical to the measurement of the Stark-PNC interference with this machine. Specifically, measurement of E , / E , and the spin polarization of the atomic beam with an accuracy of 0.1% are very difficult tasks, and it is comforting to confirm
[ I ] M. C. Noecker, B. P. Masterson, and C. E. Wieman. Phys. Rev. Lett. 61: 310 (1988). [2] M. A. Bouchiat. J . Guena, L. Pottier. and L. Hunter. Phys. Lett. 134B, 463 (1984). [3] M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 36. 493 (1975). [4] S. A. Blundell. J. Sapirstein, and W. R. Johnson, Phys. Rev. D 45, 1602 (1992). [5] V. A. Dzuha, V. V. Flamhaum, and 0. P. Sushkov; Phys. Lett. A 141, 147 (1989). [6] 2. Liu and W. R. Johnson, Phys. Lett. A 119, 407 (1987). [7] R. J . Rafac and C. E. Tanner, Bull. Am. Phys. Soc. 40, 1345 (1995). [XI J. Hoffnagle et a/., Phys. Lett. 85A, 143 (1981). [9] S. L. Gilbert, R. N. Watts, and C. E. Wieman, Phys. Rev. A 27, 581 (1983). [lo] M. A. Bouchiat, J. Guena, L. Hunter, and L. Pottier, Opt. Commun. 46, 185 (1 983). [ 111 C. E. Wieman, M. C. Noecker, B. P. Masterson, and J. Cooper, Phys. Rev. Lett 58, 1738 (1987).
101 I
this accuracy prior to measurement o f the more important Stark-PNC interference. Also, this technique which we used for the interference measurement is nearly identical to the technique used for the Stark-PNC interference, and represents a somewhat simpler version of that complex measurement.
ACKNOWLEDGMENTS
The contributions made by B. P. Masterson for the optical pumping work are greatly appreciated. This work was supported by the National Science Foundation, and D.C. acknowledges support from the Korea Science and Engineering Foundation. J.L.R. was supported by the National Science Foundation Graduate Fellowship Program.
[I21 B. P. Masterson. C. 'l'anner: H. Patrick, and C. E. Wiernan. Phys. Rev. A 47. 2139 (1993). 1131 K. B. MacAdarn, A. Steinhach. and C. E. Wieman. Am. J. Phys. 60, 1098 (19921, and references therein. 1141 M. Kasevich, D. S. Weiss. E. Riis; K. Moler, S. Kasapi, and S. Chu, Phys. Rev. Lett. 66. 2297 (1991); P. R. Nemrner, M. S. Shahriar, H. Lamela-Rivera. S. P. Smith, B. E. Bernacki. and S. Ezekiel, J. Opt. Soc. Am. B 10, 1326 (1993). [I51 C. J. Myatt, N. R. Newhury. and C. E. Wieman, Opt. Lett. 18, 649 (1993); P. Feng and T. Walker, Am. J. Phys. 63, 905 (1995); H. R. Noh, J. 0. Kim, D. S. Nam, and W. Jhe, Rev. Sci. Instrum. 67, 1431 (1996). [I61 C. S. Wood et nl. (unpublished). [ 171 Actually, c= c z i + ( cR+ i E,)X is the most general fully polarized state to consider, and this is in fact what we measure, hut cR is kept very small and does not contribute significantly to the alp result. We measure no significant unpolarized component. [I81 J. L. Roberts et al. (unpublished).
Measurement of Parity Nonconservation and an Anapole Moment in Cesium C. S. Wood, S. C. Bennett, D. Cho,* B. P. Masterson,? J. L. Roberts, C. E. Tanner,$ C. E. Wiemans The amplitude of the parity-nonconserving transition between the 6 s and 7 s states of cesium was precisely measured with the use of a spin-polarized atomic beam. This measurement gives lm(ElPnc)/p= -1.5935(56) millivolts per centimeter and provides an improved test of the standard model at low energy, including a value for the S parameter of -1 .3(3)exp (1 The nuclear spin-dependent contribution was 0.077(11) millivolts per centimeter; this contribution is a manifestation of parity violation in atomic nuclei and is a measurement of the long-sought anapole moment.
It
has been recognized for more than 20 years that electroweak unification leads to parity nonconservation (PNC) in atoms ( I ). This phenomenon is the lack of mirrorreflection symmetry and is displayed by any object with a left or right handedness. Perhaps the most well-known example of a PNC effect is the asymmetry in nuclear beta decay first observed in 1957 by W u and collaborators (2). Precise measurements of PNC in a number of different atoms have provided important tests of the standard model of elementary particle physics at low energy (3). Atomic PNC is uniquely sensitive to a variety of “new physics” (beyond the standard model) because it measures a set of model-independent electron-quark electroweak coupling constants that are different from those that are probed by highenergy experiments. Specifically, the standard model is tested by comparing a measured value of atomic PNC with the corresponding theoretical value predicted by the standard model. This prediction requires, as input, the mass of the 2 boson and the electronic structure of the atom in question. T h e 2 mass is now known to 77 parts per million (4), hut the uncertainties in the atomic structure are 1 to lo%, depending o n the atom. In recent years, PNC measurements in several atoms have achieved uncertainties of a few percent (5, 6). Of these atoms, the structure of cesium is the most accurately known (1%) because it is a n alkali atom with a single valence electron outside of a tightly hound inner core. Thus, Theauthorsarewith JllAandtheDepartrnentof Physics, University of Colorado, Boulder, CO 80309. USA. *Present address: Department of Physics, University of Korea, Seoul, Korea. tPresent address: Melles Griot, Boulder, CO 80301, USA. $Present address: Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA §To whom correspondence should be addressed
higher precision measurements of P N C in cesium provide a sensitive probe of physics beyond the standard model. In addition to exploring the physics of the standard model, high-precision atomic PNC experiments also offer a different approach for studying the effects of parity violation in atomic nuclei. In 1957, it was predicted that the combination of parity violation and electric charges would lead to the existence of a so-called anapole moment (7),hut up until now, such a moment has not been measured. Fifteen years ago, it was pointed out that a n anapole moment in the nucleus would lead to small nuclearspin-dependent contributions to atomic PNC that could be observed as a difference in the values of PNC measured o n different atomic transitions (8). W i t h the determination of the anapole moment, the measurement of this difference thus provides a valuable probe of the relatively poorly understood PNC in nuclei. Here, we report a factor of 7 improvement in the measurement of PNC in atomic cesium. This work provides a n improved test of the standard model and a definitive observation and measurement of a n anapole moment. This experiment is our third-generation measurement of P N C in atomic cesium. Conceptually, the experiment is similar to our previous two (6, 9). As a beam of atomic cesium passes through a region of perpendicular electric, magnetic, and laser fields, we excite the highly forbidden 6s to 7s transition. T h e handedness of this region is reversed by reversing each of the field directions. T h e parity violation is apparent as a small modulation in t h e 6s-7s excitation rate that is synchronous with all of these reversals. There are numerous experimental differences from our earlier work, however, including the use of a spin-polar-
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ized atomic beam and a more efficient detection method. This paper describes the basic concept of the experiment, the apparatus, the data analysis, the extensive studies that have been done on possible systematic errors, and finally, the results and some of their implications. Because this experiment has involved 7 years of apparatus development and 5 years studying potential systematic errors, we provide only a relatively brief summary of the work here. Further details o n both the technology and the systematic errors will be presented in subsequent, longer publications. Experimental concept. In the absence of electric fields and weak neutral currents, an electric dipole (El) transition between the 6s and 7s states of the cesium atom (Fig. 1) is forbidden by the parity selection rule. T h e weak neutral current interaction violates parity and mixes a small amount (-lo-”) of the P state into the 6s and 7s states, characterized by the quantity 1m(ElpN,) (Im selects the imaginary portion of a complex number). This mixing results in a parityviolating El transition amplitude A,, between these two states. T o obtain a n observable that is first order in this amplitude, we apply a dc electric field E that also mixes S and P states. This field gives rise to a “Starkinduced El transition amplitude A, that is typically lo5 times larger than A, and can interfere with it. A complete analysis of the relevant transition rates is given in (9). T o get a nonzero interference between A, and A,,, we extransition with a n ellipticite the 6s to cally polarized laser field of the form E,Z
7s
+
Fig. 1. Partial cesium energy-level diagram including the splitting of S states by the magnetic field. The case of 540-nm light exciting the F = 3, m = 3 ievel is shown. Diode lasers 1 and 2 optically pump all of the atoms into the (3,3)level, and laser 3 drives the 6SF=, (Fdefi to 6PF=, transition to detect the 7 s excitation. PNC is also measured for excitation from the (3, -3), (4, 41, and (4, -4) 6s levels. The diode lasers excite different transitions for the iatter two cases.
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198 dancy ineans that although no single reversal is perfect, the product of all the imperfections is far smaller than the PNC signal Apparatus A siinplified schematic of the appaiatus is shown in Fig 2 An effusive beam of atomic cesium is produced by a heated ovcii with a multichannel capillary array nozzle The beam 1s optically pumped into the desiied (F, m) state by light from diode lasers 1 and 2 (9, 10) The 2-cm-wide beam of polarized atoms intersects the 540nin atanding wave (Gaussian diameter, 0 8 min) that IS iiiside a high-finesse (100,000) Fabry-Peiot powci buildup cavity (PBC:) The PBC not only enhances the transition rate, but the standing wave geometry also 2PE,&; 2~lin(~,)Im(El~~~)[C,(F,rn,F’,m‘)] greatly suppresses the troublesome inodula(1) tion arising from A,-A,, interfeience (9) where p is the tensor transition polarizabil- The 540-nm light originates fiom a dye ity, Ex = E is the dc electric field, and C, lase] whose frequency is tightly locked to and C, are combinations of Clebsch-Cor- the resonant frequency of tlie PBC by a clan coefficients [they depend on the initial hi&-speed servorystein (1 1) Before enteiand final values of F and m; C,(m) = iiig the PBC, the dye laser light passe5 through an intensity Ttabilizer, an optical C,(-m), whereas C,(rn) = -C,(-m)l. isolator, and a polarization control system Here we have neglected the tiny (A,,,)’ term and the small 6S-7S magnetic dipole inade lip of a half-wave plate, Pockels cell, transition amplitude A,, (as discussed lat- and adjustable bircfiingence coinpensatoi er, A,, can be a source of many systematic plate We contiol tlie ellipticity of tht light errors). We find the contribution of the by rotating the half-wave plate and reverse parity-violating interference term relative the handedness of the ellipse by reversing to the total transition rate by detcrmin- the h/4 (quarter wavelength) voltage apiiig tlie fraction of the rate AR/R = plied to the Pockets cell The intensity sta2 1 1 n ( ~ , ) l m ( E l ~ ~ , ) / ( ~ ~that P E ) modulates bilizer holds tlie amount of light tidnsinitwith the reversals of E, m,and p. In the ted though the cavity constant and hence experiment, there are actually five “parity” stabilizes the field inside of the cavity This reversals because we reverse m in three field corresponds to dbout 2 5 kW of circiidifferent ways: reversing the polarization of lating power The PBC I esonant fiequency the optical pumping light of laser 2, revers- is held at the fiequency of the desired atoiniiig the optical pumping magnetic field rel- ic transition by a sei vosyrtem that tianslates ative to the puinping light, and revcrsiiig the input mirror The dc electric field in the interaction the magnetic field in the 6S-7S excitation region. These five reversals provide a great region i s produced by applying a voltage deal of redundancy for the PNC signal as between two parallel 5 cm by 9 cin conwell as additional information about tlie ducting plates 0 Y8577(25) cm apart (the experimental conditions, both of which are numbers in parentheses are the error in essential for the detection and elimination the last digits) The plates are inade of flat of potential systematic errors. The rcdiui- pieces of Pyrex glass coated with 100 nm
piIm(E,)x, where the handedness of the polarization p = il , E, is the coinponent of the oscillating electric field that is parallel to E, and E, is the oscillating field perpendicular to E. We ineastire ApNcjAEfnr the 6S,=3 to IS,,, and 6S,=, to IS,=, transitions (F, total angular momentum). In both cases, we only populate and excite out of the states with extreme values of the magnetic quantum number m: 23 and 24, respectively. For these cases a i d the configuration of electric and magnetic fields shown in Fig. 2, the transition rate is
Fig. 2. Schematic of the apparatus. In the interaction 540-nm dye laser beam defines they axis.
. --
,/
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0
mirror
of molybdenum. Both field plates were divided into five electrically separate segments by removing the molybdenum in thin parallel lines. This division allows us to apply sinall uniform and gradient electric fields along the y axis for auxiliary diagnostic experiments. For the PNC measurement, a uniform electric field in the x direction is created by the application of typically 500 V between the two plates. The entire buildup cavity and field plate mounting system is rather elaborate to ensure precise alignment and extreme mechanical stability. After being excited out of the populated 6S F level rip to the 7s state, an atom will decay by way of the 6P states to the previously empty 6s F level (Fdet)more than 60% of the time. We detect this repoptilation of FL,t,,10 cin dowiistream of the interaction region. Light fioin diode laser 3 excites each atom in Fdcl to the 6P,,, state inany times. The resulting scattered photons are detected by a 5-cin’ silicon photodiode that sits just below the atomic beain. When tlie 6S,=, to IS,,, line is measured, the detection laser drives the 6S,.=4 to 6P,,,,,=, cycling transition (Fig. 1). About 240 photons per FCieL atoin are detected. For the hS,_,-IS, -iline, the detection transition is the 6S, .i to 6P3,Z,F=2 cycling transition. This cycle gives about 100 detected photons per F<,,, atom. The signal-to-noise ratio for this transition is about 20% lower. During each half cycle of the most rapid field reversal (E) and after the switching transient has passed, the detector photocurrent is integrated, digitized, and stored. For each stored value, the coinputer also records the field and spin orientations. The signal-to-noise ratio needed for this experiineiit puts extreme requirements on laser stability. A fluctuation in the intcnsity, frequency, or direction of the light from any of the four lasers will introduce noise in the detected atomic fluorescence. The most extensive control is needed for the dye laser (1 1 ), but the requirements on the three diode lasers used for optical pumping and detection are also severe. These requirements have motivated substantial dcvelopmeiit of diode-laser stabilization technology (1.2). T o summarize, both optical and electronic feedback are used to lock the frequency of each diode laser to the desired atomic transition using saturated absorption spectrometers. Extensive and precise control of magnetic fields are required in this experiment. In the optical pumping region, there is a uniform 2.5-G field that must point in the i y direction (parallel to the pumping laser beams). In the interaction region, a 6.4-C field must point precisely in either the +z or the -? direction. Between the two re-
VOL 275 * 21 MARCH 1997 * http://www.sciencemag.org
199 gions, the magnetic field must rotate gently enough that the atomic spins follow it adiabatically. Finally, the field must be near zero in the detection region, and it is necessary to precisely reverse the fields in the optical pumping and interaction regions independently without significantly perturbing the fields in the other two regions. This setup has required the use of 23 magnetic field coils of various shapes to provide the necessary fields and gradients. Most of these coils are driven with both reversing and nonreversing components of current. Many additional elements are required to achieve sufficiently precise alignment and control of all aspects of the apparatus. These include 3 1 different servosystems to ensure optical, mechanical, and thermal stability. Data and results. About 350 hours of PNC data were acquired in five runs distributed over an 8-month period. Each of the five runs followed the same basic procedure. First, a set of auxiliary experiments was carried out. These experiments were used to measure and set numerous quantities: ( i ) all three components of the average E and B fields and their y gradients in the interaction region, (ii) the magnitude and orientation of the birefringence of the PBC output-mirror coating, (iii) the polarization-dependent power modulation of the green laser light, and (iv) the populations of the m levels of the atomic beam as it entered the interaction region. After these measurements were completed, we locked the four laser frequencies to the desired hyperfine transitions and proceeded to take data. T h e data were acquired in blocks of about 1.5 hours each. During this time, the electric field was reversed at 27 Hz; the magnetic field in the optical pumping region, at 0.29 Hz; the laser polarization, at 0.07 Hz; the magnetic field in the interaction region, at 0.018 Hz; and the circular polarization of the optical pumping light, at 0.004 Hz. T h e relative phases of the various reversals were regularly shifted by one half cycle. Before and after each of these 1.5hour blocks, the polarization of the 540-nm standing-wave field was measured and set. A t regular intervals, the PBC output mirror was rotated by 90.0(5)", and at irregular intervals, the four lasers were reset to measure PNC on the other 6s-7.5 hyperfine line. A t the end of the data run (20 to 30 blocks of data), the initial auxiliary experiments were repeated to check that the quantities listed above had not changed significantly. T h e typical size of the 6s-7.5 signal from the photodiode was 200 nA o n the 3-4line and 85 nA on the 4-3 line. These measurements corresponded to about 0.5% of the atomic beam undergoing a 6s-7s transition.
These signals were on top of background signals, -25% as large, arising from unpumped atoms and laser light scattered off various surfaces. T h e signal-to-noise ratio for measuring the 6s-7s transition rate was typically 55,00O/Hz"*. Because the technical noise was small and we detected many photons from each atom that had undergone a 6s-7s excitation, the noise was dominated by the shot-noise fluctuations in the number of atoms making the 6s-7s transition. W e took data with polarization ellipof both 1 and 2 and over a ticities I@,l range of electric fields from 400 to 900 V/cm, with 500 V/cm being the most common. T h e size of the PNC modulation depended o n both field and polarization, but at 500 V/cm and I EJE, 1 = 1, it was about six parts in lo6. T h e signal in this experiment was about 50 times larger than that of our previous experiment (5), which gave a factor of 7 improvement in the PNC signalto-noise ratio. Data analysis and uncertainties. T h e data analysis used to find the PNC modulation for each block of data was relatively simple. W e took the appropriate combination of fractional differences in the signal sizes for each of the 32 different field configurations to find the fraction of the rate that modulated with all five reversals. From the known values of E and EJE,, this fractional modulation was then converted into Im(ElpNcJ/@. T o express this value in terms of a transition from a single m state, a correction is required because the optical pumping is not perfect and the transitions from adjacent m levels are not completely resolved. W e determine this correction from the measurement of the populations of the m levels and the transition line shape. T h e populations were determined by selectively exciting transitions between (F, m) and (F', m) states of the 6s level and comparing the signals o n the respective lines. This work was similar to ( I U ) , except that here, the AF transitions were not excited by microwave magnetic fields, hut rather as two-photon optical Raman transitions (13).
Typically, 97% of the atoms were pumped into the desired m level, and there was less than 0.1% of the population in any but the adjacent m level. This population distribution leads to a +6.0(1)% correction on the PNC value for the 3-4 line and n o correction o n the 4-3. T h e applied electric field was determined from the measurement of the spacing of the electric field plates and the applied voltage. T h e value of &,I&,was obtained from the measurements of the polarization of the light transmitted by the PBC output mirror (14). Our major concern in this experiment was systematic errors (Table 1 ) arising from spurious signals that modulate under all five parity reversals, thus mimicking PNC. Roughly 20 times more data were taken in the investigation and elimination of such errors than in the actual PNC measurement. Several small errors associated with stray and misaligned fields were encountered as in previous PNC measurements (15); we dealt with these as before. W e carried out an exhaustive analysis of all possible combinations of static and oscillating electric and magnetic fields that could mimic a PNC signal. All of the stray (defined as nonreversing) and misaligned dc electric and magnetic fields and their gradients, and many of the laser field components, can he determined by looking a t appropriate modulations in the 6s-7s rate under various conditions. Many of these quantities were extracted from the 31 different modulation combinations we observed in real time while taking PNC data, and the remaining components were determined by the auxiliary experiments that were interspersed with the PNC runs. These auxiliary experiments involved looking a t modulations o n AF = 0 as well as the AF = t l 6s-7s transitions with a variety of applied fields, field gradients, and green laser polarizations, and many are similar to those used in previous work (6, 9 ) . Many tests were also performed to ensure the necessary stability of the relevant fields. Typical misaligned and stray field components were
Table 1. Major contributions and uncertainties of signals that mimic PNC. D,, D,, and D, are line-shape distortion factors, and Apower/Apol is the laser power modulation synchronous with the polarization
reversal. Typical size per block (% of PNC)
Source
1. Misaligned fields, stray fields (imperfect reversals) 2. A,A,, x gradient of stray By x D, 3. AEA,, x mirror birefringence 4. AEA,, x Apower/Apol x D, 5. Re(&,)/E, X D,
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Finat average size (% of PNC)
3-4
4-3
3-4
4-3
O.l(l)
O.l(l)
O.OO(4)
O.OO(4)
0.3(1) 0.5(3) 0.1(2) 0
0
0.0(1) O.OO(5) O.OO(5) 0
0
0.3(3) 0.1(2) 2.0(3)
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200 1X to 7 X 10-' of the main fields. T h e fractional shift in the P N C signal resulting from combinations of such stray and misaligned fields was <4 X Although this procedure was similar in concept to our previous work, here it was more difficult and time consuming because of the higher accuracy required. This requirement made it necessary to consider not only the average fields, but also their gradients across the interaction region. T h e study of gradient effects led to the discovery of another error, which arises from the gradient in the stray B, (Table 1, number 2). This field gradient combines with the velocity gradient across the atomic beam to break the symmetry of the standing wave field in a polarization-sensitive manner and thereby gives a n error proportional to A,A,,. This error can be eliminated by carefully minimizing the stray B, gradient. T h e birefringence of the PBC outputradians per reflecmirror coating ( 2 X tion) will also convert the A,A,, interference into a PNC error (9, 16). We have reduced this error to a negligible level (<0.05% of PNC) through a combination of steps. W e obtained low-birefringence mirror coatings ( 1 7) and carefully mounted and temperature stabilized the mirrors to minimize additional birefringence. Also, by rotating the output mirror we could measure and orient the birefringence before and during the data runs. By orienting the bire-
fringence axis to within 5" of the z or x direction, we reduced the fractional error to 0.5% of the PNC signal in each block. T h e periodic 90.0(5)"rotations of the mirror during the data runs reduced the average fractional error to <0.05%. A third error proportional to A,A,,,,] comes from the distortion in the 6s-7s line shape due to ac Stark shifts produced by the green laser field (6, 18). Because of this distortion, a modulation in the laser power inside the PBC that is synchronous with the polarization reversal results in a PNC error. T o eliminate this error, we measured the polarization-synchronous power rnodulations to 1 part in 10' of the total power. This measurement was done in a n auxiliary experiment that detected power changes in a polarization-insensitive manner by observing the resulting ac Stark shifts o n the 6s-7s transition frequency. W e also tested for any unanticipated errors that might arise from the AF.AMl interference by taking PNC data with polarization ratios / E , / E , / l and 2. T h e ratio A,,,/A,, differs by a factor of 2 for these two cases; the fact that we obtain the same value of I m ( E l p N c ) / ~for both polarizations indicates that there are n o significant systematic errors proportional to AMl. In addition to the tests above, we applied large electric and magnetic fields and gradients in the x , y , and z directions and real and imaginary optical fields in the x and 7 directions. We confirmed that certain applied fields produced the false PNC signals we expected, and others produced the correct changes in the 44 other modulation combinations that we observed during the PNC data runs and the auxiliary measurements. These studies revealed another potential systematic error (Table l , number 5), which arises from imperfections in the polarization of the green light. This error is
related to the distortion in the 6s-7s line shape combined with a nonzero Re(&,)/&,, just as a previously discussed error (Table I, number 4) was related to the line-shape distortion combined with an intracavity power modulation. T o keep this error small, we measured and minimized Re(&,)/&, before each block. I t was adjusted so that the error was typically less than one-half of the PNC statistical uncertainty; we then applied a correction to the results. W e intentionally acquired nearly equal numbers of blocks with positive and negative values of Re(&,)/&, in each run, so the average correction was very small. In such a complex and precise experiment, there is always the worry that there could be some undiscovered systematic error still lurking in the darkness. W e have made numerous checks to reduce that possibility. W e have repeatedly changed many aspects of the experiment (for example, alignments, field plates, PBC mirrors, laser power, atomic beam, laser control systems, optics, and parity-reversal electronics and timing) to ensure these did not cause any unexplained changes in the PNC signal or the many other modulation signals. W e reduced all sources of technical noise until every observed fluctuation in the PNC data was consistent with the independently measured short-term statistical noise o n the 6Srate, and this noise was dominated by the shot-noise fluctuations. Finally, it cannot be overstated how important it is to have the 31 other modulation signals that are obtained from the P N C data. These signals provide a wealth of real-time information about operating conditions in the experiment, including the accuracy of all individual reversals. Results. T h e data, after inclusion of the appropriate calibration factors and corrections listed above, match well to a Gaussian distribution (Fig. 3). This agreement is conprobabilities, which are firmed by the 25% for the 4-3 line and 76% for the 3-4 line. Our final result is
7s
x2
-5 E
P >
165
-'m(E1pNc)'p
E
CI
6
155
145
T 1.40
1.55
1.70
1.85
-Im(El PNC)Ip (mVIcm)
Fig. 3. Histograms of 1.5 hour blocks of PNC
results for the 6SF=3to 7S,=, and the 6S,_,to 7S,=, transitions. The solid bars are the data, and the open bars are the theoretical distributions expected for random samples with standard deviations matching the independently measured short-term noise in the data. 1762
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Fig. 4. Historical comparison of cesium PNC results.Thesquares arevalues for the4-3 transition, the open circles are the 3-4 transition, and the solid circles are averages over the hyperiine transitions. The band is the standard-model prediction for the average, including radiative corrections. The ? l u width shown is dominated by the uncertainty of the atomic structure.
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=
1.6349(80) { 1.5576(77)
mV/cm mV/cm
for the 6S,=, to 7S,,,, and 6S,=, to 7S,,, transitions, respectively. T h e difference was 0.077(11) mV/cm, and the nuclear spinindependent average was 1.5935(56) mV/ cm. T h e statistical uncertainties for the two transitions, 0.0078 and 0.0073 mV/cm, respectively, dominate the error. T h e systematic uncertainties are based on statistical uncertainties in the determination of various calibration factors and systematic shifts, and therefore, it is appropriate to add them in quadrature. T h e final results are in good agreement with previous measurements in cesium (Fig. 4) and are much more precise.
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201 T h e difference between the two lines has important implications for the understanding of PNC in nuclei. A small fraction of the difference (about 15%) is predicted to result from the combination of the hadronic axial-vector neutral-current interaction and the perturbation of the hadronic vector neutral-current interaction by the hyperfine interaction (8). T h e remainder is due to the nuclear anapole moment. Classically, an anapole moment can be visualized as the magnetic moment produced by a toroidal current distribution. Because this moment does not give rise to any longrange magnetic field, there is only a contact interaction between the electron and the nuclear anapole moment. Theoretical predictions for the size of the nuclear anapole moment differ by a factor of -2.5 (8). Given the approximations in the nuclear theory, our measured value is in reasonable agreement (25% larger) with the largest predicted value, but not the smallest. T h e theoretical differences primarily arise from how the strength of the parity-violating force in nuclei is derived from other experimental data and nuclear models. This example illustrates that the measurement of the nuclear anapole moment provides a valuable probe of parityviolating forces in atomic nuclei. T h e weighted average, 0.465[-Im(E1pNc)/b14.3 + 0.535[-Im(El,,c)/pl,.4 (19, 211, contains no nuclear spin-dependent contribution and is solely due to the electron axial-vector weak neutral-current interaction between the quarks and electrons. We obtained a corresponding theoretical value using the standard model (3) and the calculated values of the relevant atomic matrix elements (20, 21). This value agrees with our measured value (Fig. 4), and the 1% uncertainty in the comparison is dominated by the atomic theory calculation. In order to obtain this agreement, the theoretical value
must include radiative corrections, which are about 5%. We find the weak charge (3) Q, to be -72.11(27),,, (89),h,,v, and the S parameter (22) that is used to characterize certain types of physics beyond the standard Assuming model is - 1.3(3)exp ( ll)rheoni. that the standard model is correct, this value of Q, is equivalent to sin20, = 0.2261( 12)exp (41)rhr0ry.These results also set tighter constraints on most models that contain more than one neutral vector boson (3, 23). T h e exact constraints are modeldependent hut usually mean that the second boson must he higher in mass or couple more weakly. Time-consuming but straightforward extensions of the atomic theory calculations are expected to reduce their uncertainties substantially (24), which will either reveal new physics or tighten all of the Constraints discussed. REFERENCES AND NOTES 1 M. A. Bouchiat and C. Bouchiat. J. Phys. (Paris) 35, 699 (19741; ibid. 36, 493 (1975). 2. C. S. W u e t a l , Phys. Rev. 105, 1413 (1957). 3. P Langacker, M. Luo, A. K. Mann, Rev. Mod. Phys. 64, 87 (1992); J. Rosner, Phys. Rev. D 53, 2724 (1995). 4. 2 mass LEP Collaboration. Phys. Lett B 307, 167 (1993). 5. Uncertainties in experimental value: 1.2% in Pb [D. M. Meekhof e l a/. , Phys. Rev. Lett. 71,3442 (199311, 1.2% in Ti [P. A Vetter ef a/., ibid. 74, 2658 (199511, 3.0% in TI [N.H. Edwards e l a/. , ibid., p 26541, and 2% in Bi [M J. D. Macpherson eta/., ibid. 67, 2784 (I 991)l. 6. Further uncertainties in experimental value 2.2% in Cs [M. C. Noecker, B. P. Masterson, C. E. Wieman, Phys. Rev. Lett. 61, 310 (198811. 7. Ya. B. Zei’Dovich, Sov Phys. J R P 6, 1184 (1958) 8. V. V. Flambaum and I. B. Khriplovtch, ibid. 52, 835 (1980);-and 0. P. Sushkov, Phys. Lett. 8146, 367 (1984); W. C. Haxton, E. M. Henley, M. J. Musolf, Phys. Rev. Lett. 63,949 (1989); C. Bouchiat and C. A. Piketty,Z. Phys. C49.91 (1991); V. F. Dmitriev, I. B. Khriplovich, V. B Teiitsin, Nuci Phys. A 577,691 (19941; C. Bouchiat and C. A. Piketty, Phys. Lett. B 269, 195 (1991); M. G. Kozlov. Phys. Lett. A 130, 426 (1988). 9. S.L. Gilbert, M. C. Noecker, R. N. Watts, C. E. Wieman, Phys. Rev. Lett. 55, 2680 (1985); S. L.
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Gilbert and C. E. Wieman, Phys. Rev. A 34, 792 (1986). 10. B. P. Masterson, C. Tanner, H. Patrick, C. E. Wieman, Phys. Rev. A 47. 2139 (1993). 11. C. E. Wiernan et a/. , in Frontiers in Laser Spectroscopy, T. Hansch and M. Inguscio, Eds. (North-Hol1and.Amsterdam. 19941: R W. P. DreveretaiLAooi . Phys. B 31, 97 (1983). 12. C. E. Wiernan and L. Hollberg, Rev. Sci instrum. 62, 1119911. 13. This experiment required a fourth diode laser that had 4.6-GHr side bands and excited Raman transitions similar to that used by P. R. Hemmer eta/. [J. Opt. Soc. Am. 10, 1326 (1993)l. This setup ailowed us to determine them level populations in the actual interaction region without changing anything in the atomic beam apparatus. Cho et a/. (26) discuss the measurements of m level populations in more detail. 14. We have done extensive tests to determine that the average error in the determination of the polarization was much less than 1 X lo-”. See J. L. Roberts et a/., University of Colorado preprint. and (26). 15. See (9)and references therein 16. M. A. Bouchiat, A Coblentr, J. Guena, L. Pottier, J. Phys. (Paris) 42, 985 (1981) 17. Research Electrooptics, Boulder, CO 18. C. E. Wieman, M. C. Noecker, B. P. Masterson, J. Cooper, Phys. Rev. Lett. 58, 1738 (19871. The details of the related PNC systematic are slightly different in this case because of the optically pumped beam and higher laser powers 19. Ya. Kraftmakher, Phys. Lett. A 132, 167 (1968);P. A. Frantsusov and i 5.Khriplovtch, Z. Phys. D 7, 297 (1968). 20. V. A. Dzuba, V V Flambaum, 0. P Sushkov, Phys. Lett. A 141, 147 (1969). 21. The band shown in Fig. 4 uses the value of ElpNc and p from S. A. Blundell. J. Sapirstein, W. R. Johnson, Phys. Rev. D 45, 1602 (1 992); E l PNC is also calculated in (20).The two values agree and have similar 1% error bars. 22. W. Marciano and J. Rosner. Phvs. Rev. Lett. 65. 2963 (1990). 23 K. T . Mahanthapa arid P K. Mohapatra, Phys. Rev. D 43. 3093 11991). 24. J. Saptrstetn’, in Aiomic Physics 14: Proceedings of the 74th international Conference on Atomic Physics. D. Wineland. C Wiernan. S. Smith. Eds iAlP Press, New York, 1995). 25. M. A. Bouchiat et a/., J. Phys. (Paris) 47, 1709 (19861. 26. D. Cho, C. S. Wood, S. C Bennett, J. L. Roberts, C. E. Wiernan, Phys. Rev. A 55, 1007 (1997). 27. This work was supported by NSF grant PHY9512150, and J.L.R. acknowiedgessupport ofan NSF predoctoral fellowship. /
I
18 December 1996, accepted 6 February 1997
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PHYSICAL REVIEW A
VOLUME 59, NUMBER 1
JANUARY 1999
Measurement of the dc Stark shift of the 6 S 4 7 S transition in atomic cesium S. C. Bennett, J. L. Roberts, and C. E. Wicman JILA, Nti/ioiinl Iii,y/i/iite ojS‘/niidnr.d~ arid Technology mid Uviver.si/.v of’ Coloi.ndo. BoiiIdei., Coloi~nclo80309 aiid Deptrr/ii7eii/ of Pliuics, l/tiii~ersi[vof Coloi.ndo. Boiildei., Colomdo 80309 (Received 16 September 1998)
We have measured the dc Stark shift of the 6S-7S transition i n atomic cesiuni using laser spectroscopy. The result of our expcrinicnt is 0.7262( 8) Hz (Vicm-’. This valuc disagrees with a previous experiment but is within 0.3% of the valuc predicted by trh initio calculations. This measurement removes the largcst oiitstanding disagreement bctween cxperiment and ah iiiirio theory of low-lying states i n atomic cesium. [S1050-2947(99)51101-3] PACS numbcr(s): 32.60.+i. 32.lO.Dk, 3 9 . 3 0 . + ~
I. INTRODUCTION
(3) Measurements of atomic parity nonconservation (PNC) can bc used to test the standard model of electroweak interactions, as suggested by Bouchiat and Bouchiat [I]. A recent measurement of PNC in cesium has achieved a precision of 0.35% [2], but to interpret the PNC measurement in terms of standard model parameters, one needs to use atomicsti-ticture calculations of the PNC matrix elements. A h initio calculations of the Cs atomic structure by Dzuba et a/. [3] and Blundell et a/. [4] agree with measurements of many properties [5] to better than 1%, but there remains a 2% disagreement with a measurement of the dc Stark shift of the 6 S i 7 S transition in cesium [6]. In this paper, we present an improved measurement of the Stark shift that agrees with the theory at the 0.3% level. In a static electric field E , the energy shift of an Ins) state in atomic units is given by
We do this for a variety of values of Eiiigl. 11. APPARATUS AND PROCEDURE
The experiincntal apparatus shown in Fig. 1 is similar to that described in Ref. [2]. A collimated beam of cesium atoms is optically pumped by the “hyperfine pumping” diode laser into the 6S,,2F=3 ground state. The optically pumped atomic beam intersects a standing-wave laser field in a region with a dc electric field. The electric field is created by applying a voltage to two 2 X 5 X 0.5-cm molybdenum plates separated by 0.489 94(25) cm. The standing-wave field is produced by coupling 540-nm dye laser light into a highfinesse ( lo5) Fabry-Perot etalon (FPE). The frequency of the dye laser is locked to the FPE and stabilized using the
where a!s is the scalar polarizability of that state. Here we have omitted the tensor polarizability a:, because for S I i 2 states its only contribution is from higher-order effects [7]. In fact, for the 6S,,, state, a& is seven orders of magnitude smaller than mis [8] and is completely negligible. Thus, for the electric fields used in this experiment, does not depend on the quantum numbers F and m F [9]. The dc electric field also mixes states of opposite parity, making it possible to drive a normally forbidden Stark-induced electric dipole transition between different S states. The total dc Stark shift of the 6S-7S transition is given by
To determine the constant k , we scan across the 6S1-7.S transition at high and low electric fields, measure the line centers of the two scans, and calculate the frequency separation, A vStark,of the two centers. The value of k is then given by 1050-2947/99159(1)/16(3)/$15.00
PRA
2 202
2
FIG. 1. Schematic of the apparatus. The dye laser provides 540nm light. The hypefine-pumping (HFP) and probe diode lasers provide 852-nm light. SMOF is the single-mode optical fiber, AOM is an acousto-optic modulator, RefCav is the reference cavity, PD1 is a photodiode used to detect the light transmitted through the RefCav, OP is the optical pumping region, FPE is the Fabry-Perot etalon, PD2 is another photodiode used to stabilize the intensity in the FPE, D is the detection region, and LAPD is the large-area photodiode. R16
01999 The American Physical Society
203 PRA
2
MEASUREMENT OF T H E dc STARK SHIFT OF T H E . . .
Drever-Hall technique [ 101 so that its short-term stability is A ulaacl<5kHz. The 540-nm light drives the 6s F = 3 to 75’ F = 3 transition [ l l ] . When the excited 7s atoms relax, 75% of them repopulate the previously depleted 6S1!?F = 4 statc. The atomic beam then enters a region where the “probc” diode laser is tuned to the 6S,!2 F = 4 to 6 P 3 , 2F = 5 (852 nm) cycling transition. We collect the 852-nm photons scattered by the atoms i n thc F = 4 state on a large-area photodiode. The photocurrent from the photodiode is proportional to the number of atoms making the 65’- 7s transition. To measure A J I ~ , . , , ,with ~ the desired accuracy, the frequency of the dye laser must bc accurately known and rcproducible as it scans over the 6s- 7s transition. The stability needed to provide this reproducibility is much greater than the intrinsic stability ofthc W E . To add this needed stability, 10% of the dye laser light is coupled into a stable rcfcrcncc cavity, and wc lock the frequency of the FPE to the resonant frcqucncy uref of the refercncc cavity. I n order to minimizc the drift of u , , ~the , reference cavity is constructed of Invar, hermetically sealed, and temperature stabilized. To achieve an accurate scan of the frequency of thc dye laser relative to u , . , ~the , light going to thc reference cavity is first double passed through an acousto-optic modulator (AOM). The AOM is driven around 280 MHz by a frequency synthesizer that has an accuracy of better than 1 part i n 10’. Since the frequency of the light coupled into the reference cavity is shifted by twice the AOM drive frequency u h O M , locking the dye laser to the FPE and the FPE to the reference cavity forces
The dye laser is scanned by sweeping uAOM. Any beam motion in the light incident on the reference cavity leads to instability in the lock to the reference cavity [12]. By double passing the AOM we eliminate most, but not all, of the beam motion caused by changing u AO M .To eliminate all beam motion, we couple the light into the reference cavity via a single-mode optical fiber. As vAOMchanges, the output beam from the fiber does not move, so the coupling into the reference cavity remains constant. The laser beam still moves on the front face of the fiber causing 10% changes in the intensity transmitted through the fiber, but these changes do not introduce significant errors. Data are acquired in the following manner. We lock the dye laser, the FPE, and the reference cavity together and tune the laser frequency close to the 6S+7S transition. We then scan over the transition by changing uAOMwhile the electric field is set to a low value. The photocurrent vs uAoMfor the 24-s scan is then stored on disk. We then set the electric field to a higher value, take another scan, and store that data on disk. We repeat the process five times, alternating between the same low and high electric fields to collect a total of ten scans. The voltages on the electric field plates are measured before and after each set of ten scans, using a calibrated precision voltage divider that has an accuracy of 6 X The low field is always 1 kV/cm, but we change the highfield value between sets. The high field values are 5, 6, 7, 8, 9, and 10 kVicm. Typical data for scans at four different electric-field values are shown in Fig. 2.
-20
0
R17
20
60
40
80
100
Dye laser detuning from vref(MHz) FIG. 2. FOLW diffci-ent scaiis of the 6s- 7s transition with dc electric fields of (a) 1 kV/cin. (b) 6 kV/cm, ( c ) 8 kV/cm, and (d) I0 kVicin. 111. ANALYSIS AND RESULTS
The quantity of interest, A ustork, is the frequency difference between the line centers of two scans, one with a low clcctric field and one with a high electric field. The line shapes cannot be fit by a simple functional form and are slightly asymmetric [13], so it is not straightforward to determine the line center. However, the line shape does not change significantly with E , so we can determine an e&tive line center for each scan, and AvSrarkis the difference between two “centers.” A “center” is found by taking the two frequencies at which the fluorescence is n/10 times its peak value, and averaging the pair of frequencies for each n , where n = 1,2, . . . ,9. The average of these nine values is the effective line center. The reproducibility of determining the effective line centers is 0.02 MHz. Saturation can distort the line shape and shift the effective center in an E-field-dependent fashion; we operate at sufficiently low laser power so that saturation effects are negligible. At the low intensity, there is still an ac Stark shift of 0.21 MHz. However, because we stabilize the FPE intracav-
0
20
40
60
80
100
FIG. 3. Measured values of AvStarkplotted as a function of E 2 . Error bars are smaller than data points. The solid line is a fit to the data assuming A v S t a r k ~ E 2 .
204 S. C. BENNETT, J. L. ROBERTS, AND C. E. WIEMAN
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ity power to 1 part in lo5, the ac Stark shift is constant for all scans. The construction of the reference cavity and the tempera~ than 0.3 MHz/min, ture stabilization keep the drift of v , . ,less or about 100 kHz between scans. W e eliminate the effects of this residual drift on the determination of A vStarkby fitting the time dependence of the positions of the line “centers.” A different fit is done for each set of ten scans for a given Ehigll. W e find that it is not adequate to fit the positions to a simple linear time dependence, so we have to consider a third-order polynomial dependence on time to get good fits. Ten high- and low-field line “centers” are fit to a single third-order polynomial plus a constant value of A vSt;l,-k.Our final uncertainty in 4 vStark,which is dominated by the L U certainty i n the line centers, is 0.04% for each ten-scan set. Data for different values o f high electric field are shown in Fig. 3. The solid line is a fit assuming an E 2 dependence. The reduced for the fit is 0.9882, indicating that there is a 61% probability that these data came from a random distribution. The contributions to the uncertainty in our final result for k are 0.1% from the measurement of the field plate sepa-
ration [ 141, 0.04% from determinations of A vSfark,and 0.01% from the measurement of the applied voltage. This yields a total fractional uncertainty of 0.11YO. Our final result of 0.7262( 8) H z (V/cm)-* disagrees with the previous result of 0.7103(24) Hz ( V / c m - 2 [ 6 ] , but is in good agreement with theoretical predictions of 0.7237 Hz ( V i m - * [3] and 0.7257 Hz (V/cin)-2 [4]. This measurement reduces the differencc between the theoretically and experimentally determined values of the dc Stark shift from 2% to about 0.3%. This reduction eliminates the only significant disagreement between ah initio calculations and experimental measurements of the properties o f atomic cesium. I n conjunction with other measurements, we are now able to improve upon the test of the standard model reported in Ref. [2] (see Ref. [ 151).
[ l ] M. A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 35, 899 (1974); 36, 493 (1975). [2] C. S. Wood et al., Science 275, 1759 (1997). [3] V. A. Dzuba, V. V. Flambaum, and 0. P. Sushkov, Phys. Lett. A 141, 147 (1989). [4] S. A. Blundell, J. Sapirstein, and W. R. Johnson, Phys. Rev. D 45, 1602 (1992). [ S ] R. J. Rafac and C. E. Tanner, Phys. Rev. A 58, 1087 (1998); B. Hoeling et al., Opt. Lett. 21, 74 (1996); L. Young et al., Phys. Rev. A 50, 2174 (1994); C. E. Tanner et al., Phys. Rev. Lett. 69, 2765 (1992). [6] R. N. Watts, S. L. Gilbert, and C. E. Wieman, Phys. Rev. A 27,2769 (1983). We have attempted to determine the source of disagreement between the present work and that of Watts, Gilbert, and Wieman. Unfortunately, Watts, who carried out the primary data analysis, is deceased and the records of his analysis are no longer available. [7] J. R. P. Angel and P. G. H. Sandars, Proc. R. SOC.London, Ser. A 305, 125 (1968). [8] H. Gould, Phys. Rev. A 14,922 (1976). [9] A. Khadjavi and A. Lurio, Phys. Rev. 167, 128 (1968). [lo] R. W. P. Drever et al., Appl. Phys. B: Photophys. Laser Chem.
31, 97 (1983). [ 1 11 Because of the limitations imposed by the need to minimize ac Stark shifts and saturation of the 6 s - 7 s transition, we use the 6S1,2 F = 3 to 7SIl2F = 3 transition. It has a much larger signal-to-noise ratio than the other three hyperfine transitions. [I21 Beam motion changes the coupling efficiency into adjacent modes of the reference cavity. These modes are not completely degenerate because the cavity is not perfectly confocal. Since the frequency at which we lock the FPE depends on which modes the input beam i s coupled into, changing the coupling changes uref. [13] C. E. Wieman et al., Phys. Rev. Lett. 58, 1738 (1987). [ 141 We observe a variation in our measurement of k that depends on whether or not a cold plate near the electric field plates is cooled to LN, temperatures. Since we measure the plate separation at room temperature, our final data were taken without cooling the nearby cold plate. This variation with temperature and the strict requirements of the design of the apparatus limit us from performing a more precise measurement of the field plate separation. [15] S. C. Bennett and C. E. Wieman (unpublished).
x’
ACKNOWLEDGMENTS
This work is supported by the National Science Foundation. W e would like to acknowledge helpfd discussions with
D. Cho, V. A. Dzuba, and V. V. Flambaum.
VOLUME82, NUMBER 12
PHYSICAL REVIEW LETTERS
22
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1999
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Measurement of the 6s 7 s Transition Polarizability in Atomic Cesium and an Improved Test of the Standard Model S. C. Bennett and C. E. Wieman JILA, National Institiite of Staiidai-ds a i d Technology . a ~ dUpiiversity of Colorado. Boulder, Colorado 80309 and Department of Plivrics. University of Colorado, Boulder; Colorado 80309-0440 (Received 20 October 1998) The ratio of the off-diagonal hyperfine amplitude to the tensor transition polarizability ( M l , f / p )for 7s transition in cesium has been measured. The value of p = 27.024(43),,l,t(67)t~,cnr~1~ the 6s is then obtained using an accurate seniienipirical value of MI,^. This is combined with a previous measurement of parity nonconservation in atomic cesium and previous atomic structure calculations lo determine the value of the weak charge. The uncertainties i n the atomic structure calculations are updated (and reduced) in light of new experimental tests. The result Q,,, = -72.06(28),,l,,(34)1h2,~1 differs from the prediction of the standard model of elementary particle physics by 2 . 5 ~ . [SO03 1-9007(99)08690-11 --+
PACS numbers: 32.80.Ys, I1.30.Er, 12.15.Ji, 32.10.Dk
Electroweak experiments have now reached high precision in testing the standard model and in searching for new physics beyond it [ 1,2]. These experiments include measurements of parity nonconservation (PNC) in atoms as first proposed in Ref. [3]. Atomic PNC measurements are uniquely sensitive to a variety of new physics, such as the existence of additional Z bosons, because of the different energy scale and because they probe a different set of model-independent electron-quark coupling constants than those measured by high-energy experiments [2]. The most precise atomic PNC experiment [4] examines the mixing of S and P states in atomic cesium. Specifically, it compares the mixing due to the PNC neutral weak current interaction to the S-P mixing caused by an applied electric field (“Stark mixing”). In previous work [4], this measurement was combined with theoretical calculations of the structure of the cesium atom to obtain the weak charge Qw, which characterizes the strength of the neutral weak interaction and can be compared to the value predicted by the standard model. The atomic structure calculations were used to obtain two pieces of information: the amount of Stark mixing and the relevant PNC electronic matrix elements. The 1.2% uncertainty in the determination of Qw was dominated by the uncertainties in those two calculated quantities. In this paper we report a reduced uncertainty in Qw that is obtained by (1) measuring the Stark mixing and (2) incorporating new experimental data into the evaluation of the uncertainty in the calculation of the PNC matrix elements. These new data indicate that the calculations are more accurate than was indicated by the less precise (and in some cases incorrect) data available at the time the calculations were published. Theory.-The 6 s ground state and 7 s excited state of atomic cesium both have two hyperfine levels: F = 3 and F = 4. In the presence of a dc electric field a magnetic field, and a standing-wave laser field with propagation vector k and polarization i , the A F = 2 1
g,
-.
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0031 -9007/99/82(12)/2484(4)$15.00
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6s 7s amplitudes, used in both Ref. [4] and the present work, are given by [5] A6s-7~ =
[iP(iX
Z)
+ MI(i X
Z)
(F’nzklGIFm~),
(1)
where M1 = M -t Mhf8FFtilis the magnetic dipole amplitude (M is from relativistic and spin-orbit effects, Mhf is from the off-diagonal hyperfine interaction) and G is the Pauli spin matrix. The tensor transition polarizability [3] ,B characterizes the size of the Stark mixing-induced electric dipole amplitude, and E 1PNC is the PNC matrix element given by
Here, Ins) is an Ins) state into which the PNC Hamiltonian has mixed a small amount of InP) states, D is the electric dipole operator, N is the number of neutrons, and kpNC is the calculation of the sum of relevant matrix elements between S and P states given by
+ ( ~ ~ I H P N CWIfiIW ~~P) E7S -
EnP (3)
Since H p ~ c= G , ~ y ~ Q w p ~ ( r ) / Jeach 8 , of the terms in Eq. (3) is the product of a dipole matrix element times a y 5 matrix element evaluated in the nucleus. Ninetyeight percent of the sum comes from the 6P1/2 and 7P1/2 states [6]. In Ref. [4], Im(ElpNc)/P is measured. The value Qw is obtained by multiplying this ratio by P N / k p N C . This paper concerns the improved determination of ,f3 and kpNC, and thus Q W.
0 1999 The American Physical Society 205
+ EIPNCZ]
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VOLUME82, NUMBER 12
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PHYSICAL REVlEW LETTERS ~
To determine ,B, we measure M h f / P and take advantage of the fact that Mhf can be accurately determined semiempirically [7]. The amplitude M h f is due to the hyperfine interaction and thus can be expressed in t e r m of well-measured hyperfine splittings. In this experiment we observe the 6s 7s rate driven with a standing-wave laser beam with polarization 2 = € 2 and a field geometry ( E along 2) such that the transition rate is
teracting with the atomic beam is vlaser= v,,f - 2vAoM. Thus, we can change the frequency of the dye laser in a very controlled manner by changing the frequency of 75’transition. the AOM. The dye laser drives the 6s Approximately half of the atoms excited to the 7s state relax to the previously depleted hyperfine ground state ( F = 3 or F = 4). Further downstream, the atoms in the repopulated hyperfine level scatter photons from a diode laser probe beam tuned to an appropriate 6S1/?-6P3/? IA~,~+-~sI’ = ,B’E’E’ + ( M 2 M ~ ~ ~ , L - F ~ ~(I4))’ E cycling ’, transition. We collect the scattered photons on ,a large-area photodiode, and its photocurrent is proportional where sinall interference terms have been omitted. The to the number of atoms making the 65’ 75’transition. ,B-PNC and M 1-PNC interference terms are negligible, To measure the ratio h ’ j ’ 4 (or R:”)), we scan the laser and the P-M 1 interference terms cancel almost identically over the 6s 7s A F = + 1 (or A F = 1) transition in because of their dependence and the standing0.3-MHz steps. After each step we integrate the photocurwave geometry of the experiment. We determine M h f / P rent for 16.67 ins and store that data point on disk. We by measuring the total rate on the two A F = 51 hyperalternate between scans with E = 707.63(68) V/cm and fine transitions with large E , where IA~s-~.~I’ = ,B2zL.-:, and with E = 0, where IA~,T+,SI’ 2 ( M 2 M l , f B ~ ~ t * i ) - . E = 0 V/cm. backThere is a 540-nm-laser-frequency-independent We combine the ratios of the high and low E rates 011 both ground signal from atoms in the wrong hyperfine state transitions to determine Ml,f/P. that is 100 times larger than the desired M I signal for A complication arises because the locations of the E = 0 V/cm. We measure this background before and antinodes of the oscillating electric ( E , ~ ) and magnetic after each data point by detuning the laser -50 MHz from (&) fields are separated by h / 4 in the standing wave. line center and measuring the photocurrent. These backBecause of this separation, photoionization (which is driven by E ~ is~ larger ) for 7s atoms excited by eac ground points are measured alternately above and below the line center to cancel any linear frequency dependence (El atoms) than it is for 7 s atoms excited by O,, ( M I of the background. We subtract the average background atoms). The result is that the detection efficiency for E l from the data points to leave only the contribution from excitations is slightly smaller (- 1% for typical intensities) atoms making the 6s 7s transition. The sum of all than for M 1 excitations. This difference gives a potential the data points (the area under the spectral line) is proporsystematic error that is intensity dependent. The ratios tional to the total transition rate. of the signals, measured at a laser intensity I , for the We looked for but did not observe any frequency deA F = + 1 and A F = - 1 transitions, respectively, are pendence to the background. Also, all likely mechanisms, then such as molecular transitions or light scattering off the mirrors, should have very broad spectral features and, hence, will be eliminated by the background subtraction. The uncertainty in our results due to possible frequency and dependent backgrounds is less than 0.05%. Sample background-subtracted scans are shown in Fig. 1. The two line shapes are asymmetric and slightly offset from one another because of their differing senwhere T is a parameter that describes the difference in the sitivity to ac Stark shifts as discussed in Ref. [9]. The photoionization fraction. different line shapes do not affect our measurement of Experiment. -The apparatus used in the present experithe total transition rate because the atoms’ total transition amplitude is unchanged, even though the resonant frement is very similar to that in Refs. [4,8]. A collimated quency of each atom is shifted according to the local eac beam of cesium is optically pumped into either the F = 3 field. Therefore, by integrating the areas under the entire or F = 4 hyperfine level of the 6S1/2 ground state. The beam of atoms then travels roughly along the 2 axis into a broadened lines we can determine the desired relative and R14’3. ratios R:’4 region with mutually orthogonal dc electric (along A) and magnetic (along 2) fields and intersects a 540-nm standingResults. -The detection efficiency and signal-to-noise wave laser field (along 9 ) at right angles. The laser field is ratio are significantly higher for R3‘4; we measure that produced by a tunable dye laser that is frequency locked to ratio at five different intensities from 0.6 to 2.8 kW and a finesse = lo5 Fabry-Perot etalon. The etalon is, in turn, determine 7 to 1.5 parts in lo3 using a least squares locked to a stable reference cavity. The light going to the fit. We find the ratios Ri-4 = 2.4636(8) X lop3 and reference cavity is double passed through an acousto-optic = 1.1357(6) X where the two uncertainties modulator (AOM), so the frequency of the laser light inhave a common contribution from the extrapolation to
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h
v)
:.
1.5
3
5
m g1.0 0) ._
1 U
f 0.5 3 (I) c I
5 0.0 520 530 540 550 560 570 580 590 Laser Detuning from vreierence(MHz)
FIG. 1. Sample data comparing scans with and without an applied electric field. Open circles are with E = 707 V/cm and the scale on the right. Closed circles are with E = 0 V/cm and the scale on the left. The two lines are offset from one another and have different widths because of the different sensitivitics to ac Stark shifts for the M 1 and E I transitions.
zero intensity. Combining these results using Eq. ( 5 ) we find M h f / P = -5.6195(91) V/cm [ l o From Ref. [7] ?. . . we take Mhf = -151.86(38) (V/cm)nb, which IS based on measured hyperfine splittings with a 0.3 5 0.3% theory correction due to many body effects. This gives
This value is in excellent agreement with the semiempirical values /? = 27.17(35)ai [7] and p = 27.15(13)ni [ l l ] and the Calculated value p = 27.00~1; [12]. Using our measured values for p and Im(ElpNc)/,B, and the calculated value of k p ~ c we , can now extract Q w . The key issue is the uncertainty in the value of k p ~ c .The authors of Refs. [6,12-141 discuss this issue at considerable length. Here we only summarize the conclusion of both groups that the most reliable measure is to use the same ab initio calculations of the electronic structure that are used to find k p ~ to c calculate dipole matrix elements and hyperfine splittings for the 6S112, 7S1/2, 6P112, and 7P1/2 states. The differences between these calculated values and the experimental determinations provide a reliable quantitative indication of the uncertainties in the calculations of k p N C . The authors considered how well these errors in the hyperfine splittings and dipole matrix elements reflect errors in ~ P N Cby rescaling their calculations in a variety of ways and comparing the relative sensitivities of the different quantities. They found that k p ~ has c comparable or smaller sensitivity than the other quantities [15]. From comparing calculated and measured quantities, both groups arrived at uncertainties of about 1% for their value of k p ~ c . Since the time that Refs. [6,12-141 were published, there have been a number of new and more precise measurements of the quantities of interest. In all cases, the new measurements show better agreement with the calculations than earlier measurements and also show that the 2486
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largest previous disagreements were likely due to experimental errors. In Table 1 we have collected the results of the most precise measurements of relevant quantities in cesium. We list the quantities measured, the primary aspect of the electronic wave functions that is being tested in each comparison, and the difference between theory and experiment. Particularly notable are the top three lines of the table, which show that the agreement has dramatically improved from the 1%-2% disagreements of the older experiments. In addition to the data in this table, there have been new experiments that revealed errors in earlier lifetime measurements in sodium and lithium. These new data eliminate what had appeared to be troubling 1% errors in equivalent calculations for those atoms. The standard deviation of the fractional differences between theory and experiment in Table I is 4.0 X lop3. We believe this to be the most valid number to use to represent the 68% confidence level for ~ P N C . Using the average of k p ~ c= 0.905 X 10pl'ieno [12] and k p N C = 0.908 X lO-"ieao [13], this gives a value of k p ~ = c 0.9065(36) X 10p"ieao When combined with our new value for P and the experimental PNC measurement, this gives (7)
The standard model value including radiative corrections is Q w = -73.20(13) [16]. Adding the uncertainties in quadrafure, these values differ by 2 . 5 ~ . Assuming that this difference is not due to an experimental error or a statistical fluctuation, it suggests several possibilities. The first possibility is that the calculated value of the y 5 matrix element is in error by the requisite 1.58%. In light of Table I, such an error would require a wave function with a somewhat peculiar and insidious shape. Although none of the measured quantities depends on the shape of the wave function in a manner identical to that of y 5 , the different comparisons in Table I do probe the value of the wave function in all regions: short, intermediate, and long distances. The largest single difference of the 16 comparisons is only 0.79%, and the standard deviation is only 0.40%. The second possibility is that there are contributions or corrections to atomic PNC within the standard model that have been overlooked. We see no justification for either of these two possibilities, but they clearly need to be explored further. The first offers a formidable but not overwhelming challenge to both theoretical and experimental atomic physicists. The final possibility is that this discrepancy is indicating the presence of some new physics not contained in the standard model. Physics that would be characterized by the S parameter [ 171 is not a likely candidate because the size of the contribution needed [S = - 1.4(6)] would be in conflict with other data [l]. However, there are other types of new physics, such as an additional Z boson, that would be consistent with all other current data.
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VOLUME 82, NUMBER12
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22 MARCH1999
TABLE 1. Fractional differences ( X 10’) between measured and calculated values of quantities relevant for testing PNC calculations in atomic cesium. We only list the most precise experiments. The second column lists the most rclevant aspects of the wave functions that are being tested. ( f / r 3 ) j jisp the average of I / r 3 ovcr the wave function of the electronic state n P . Where the experiment has improved or changed significantly since the publication of Ref. [12], the difference from the old experiment is listed in brackets. Quantity measured 6s
7 s dc Stark shift“ 6P1/2 lifetime‘ 6P3/2 lifetime
+
LYI
ps 6s hfs“ 7.5 hfs‘ 6 P 112 hfs’ 7P1/2 hfsh
Calculation tested (7PllD + s) (6SllD PI/?) (6SIID t P7/2) (7SllD t PI/z),and (7SllD t P3/2) same as LY
*
C//hS(T =
0)
h4- = 0) (1/r3)hr il/r3)7r
Dzuba et al. ‘L‘ -3.4[19] -4.21-81 -2.6[ -411 ... ...
1.8 -6.0 -6.1 -7. I
Difference ( X I 03) Blundell et ul.
(J?,,,l
-0.7[22] 4.3[ I] 7.9[-3 I ]
1.0[4] 1.0[43] 2.3[22]
I .4 -0.8 -3.1
3.2 3 .0
-3.4 2.6 -1.5
0.2 0.2 0.5
-
...
“The value for k p ~ cof Dzuba e t a / . is obtained using “encrgy rcscaling” so we have used the corresponding “rescaled” valucs in the table for consistency. Blundell et al. do not rescale k p ~ cand so we use their pure ah iiiito values in the table. bRefs. [13,14]. ‘Refs. [6,12]. dRef. [8]. ‘Ref. [18]. ‘Using present work’s value of p and a l p from Ref. [19]. “resent work. ”Defined. ‘Ref. [20]. ’Ref. [21]. ‘Ref. [22].
We are happy to acknowledge support from the NSF, assistance in the experiments by J.L. Roberts, and valuable discussions with V. V. Flambaum, P. G. H. Sandars, C. E. Tanner, and W. R. Johnson. V. A. Dzuba graciously sent us his tabulation of calculated matrix elements as well as providing other valuable comments.
A review of the most recent high-energy experiments can be found in M. E. Peskin, Science 281, 1153 (1998). P. Langacker, M. Luo, and A. K. Mann, Rev. Mod. Phys. 64, 87 (1992). M.A. Bouchiat and C. Bouchiat, J. Phys. (Paris) 35, 899 (1974); 36, 493 (1975). C.S. Wood et al., Science 275, 1759 (1997). S.L. Gilbert and C.E. Wieman, Phys. Rev. A 34, 792 (1986). S. A. Blundell, J. Sapirstein, and W. R. Johnson, Phys. Rev. A 43, 3407 (1991); S.A. Blundell, W . R . Johnson, and J. Sapirstein, ibid. 42, 3751 (1990); Phys. Rev. Lett. 65, 1411 (1990). M.A. Bouchiat and J. Guena, J. Phys. (Paris) 49, 2037 (1988). S. C. Bennett, J. L. Roberts, and C. E. Wieman, Phys. Rev. A 59, R16 (1999). C. E. Wieman et al., Phys. Rev. Lett. 58, 1738 (1987). We have also included a -2.3 X correction to Mhf/P to account for the effects of the electric quadrupole interaction, E2, discussed in Refs. [7,23]. This correction depends on the differences in the r n p sublevel populations induced by the optical pumping into a single hyperfine state. The present work determines the value M h r / M = -0.1906(3). By comparing this to the result
in Ref. [24], we find E2/Mh+ = 53(3) X lo-?. These agree with the less precise values M h f / M = -0.1886( 17) and E2/Mhf = 42(13) X found in Ref. [7]. A detailed discussion of these issues can be found in Ref. [25]. [ 111 V. A. Dzuba, V. V. Flambaum. and 0.P. Sushkov., Phvs. , Rev. A 56, R4357 (1997). S. A. Blundell, J. Sapirstein, and W. R. Johnson, Phys. Rev. D 45, 1602 (1992). V.A. Dzuba, V.V. Flambaum, and 0.P. Sushkov, Phys. Lett. A 141, 147 (1989). V.A. Dzuba, V.V. Flambaum, and 0.P. Sushkov, Phys. Lett. A 140, 493 (1989); V. A. Dzuba et al., ibid. 142, 373 (1989). This seems reasonable since the individual dipole and h y p e r h e matrix elements will likely be more sensitive to local errors in the wave function (at long and short ranges, respectively) than will the products of matrix elements that enter into Eq. (3). W. J. Marciano and J. L. Rosner, Phys. Rev. Lett. 65, 2963 (1990); 68, 898(E) (1992). M.E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990). R. J. Rafac et al. (to be published). D. Cho et al., Phys. Rev. A 55, 1007 (1 997). S.L. Gilbert, R. N. Watts, and C. E. Wieman, Phys. Rev. A. 27, 581 (1983). R. J. Rafac and C. E. Tanner, Phys. Rev. A 56, 1027 (1997). E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys. 49, 3 1 (1 977). M.A. Bouchiat and L. Pottier, J. Phys. (Paris) 49, 1851 (1988). S. L. Gilbert et al., Phys. Rev. A 34, 3509 (1986). S. C. Bennett, Ph.D. thesis, University of Colorado, 1998 (unpublished).
2487
Annu. Rev. Nucl. Part. SCI.2001. 53:263-93 Copyright @ 2001 by Annual Reviews. All rights reserved
ATOMIC PARITY NONCONSERVATION AND NUCLEAR ANAPOLE MOMENTS W. C. Haxton' and C. E. Wieman2 'Institutef o r Nuclear Theory, Box 351550, and Department of Physics, University of Washington, Seattle, Washington 98195, and Department of Physics, University of California, Berkeley, California 94720; e-mail:
[email protected] 2Joint Institute for Laboratory Astrophysics and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440; e-mail:
[email protected]
Key Words radiative corrections, hadronic weak interactions Abstract Anapole moments are parity-odd, time-reversal-even moments of the E l projection of the electromagnetic current. Although it was recognized, soon after the discovery of parity violation in the weak interaction, that elementary particles and composite systems such as nuclei must have anapole moments, it proved difficult to isolate this weak radiative correction. The first successful measurement, an extraction of the nuclear anapole moment of 133Csfrom the hyperfine dependence of the atomic parity violation, was obtained only recently. An important anapole moment bound in thallium also exists. We discuss these measurements and their significance as tests of the hadronic weak interaction, focusing on the mechanisms that operate within the nucleus to generate the anapole moment. The atomic results place new constraints on weak meson-nucleon couplings, constraints we compare to existing bounds from a variety of p' p and nuclear tests of parity nonconservation.
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CONTENTS 1. INTRODUCTION . . . . . . . . . . . . . . . . . . 2. ANAPOLE MOMENTS AND ATOMS 2.1. Electromagnetic Moments . . . . . . . 2.2. Atoms and the Generalized Siegert's 2.3. The Spin-Dependent Atomic Interact 3. EXPERIMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 3.1. The Cesium Experiment 3.2. The Thallium Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. ANAPOLE MOMENTS AND HADRONIC PARITY NONCONSERVATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 4.1. Extracting the Anapole Moment . . . . . . . . . . . . . . . 4.2. The Hadronic Weak Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 4.3. Weak Meson-Nucleon Couplings 5. WEAK COUPLING CONSTRAINTS . . . . . . . . . . . . . . . . . . . . . 288 6. OUTLOOK .................................................... 289
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1. INTRODUCTION Until 1957, physicists assumed that the fundamental laws of nature did not distinguish between left and right. However, in that year, following a suggestion by Lee & Yang (l),experimenters discovered that the weak force governing processes such as muon decay and nuclear p-decay violated mirror symmetry maximally (to the accuracy of the measurements) (2). Very soon afterwards, Vaks and Zeldovich (3) noted that weak interactions would then modify the electromagnetic couplings to elementary particles (as well as composite systems), allowing them to have new, parity-violating moments, called anapole moments, in addition to their familiar parity-conserving ones (e.g., the charge and magnetic moments). This new moment has a number of curious properties. It vanishes when probed by real photons, and thus must be tested in processes where a virtual photon is exchanged. For example, the anapole moment of a nucleus can be probed in electron scattering and can influence the energies of bound electrons in an atom, but it cannot be measured through direct in_teractionswith an electric field (unlike magnetic moment interactions in a static B field). The resulting electron-nucleus interaction in an atom is pointlike, thus mimicking the short-range tree-level weak interaction induced by Z o exchange between atomic electrons and the nucleus. The atomic cloud feels the nuclear anapole moment only to the extent that the orbiting electrons penetrate the nucleus. The anapole moment is an electric dipole coupling that is nuclear-spin-dependent; it is this spin dependence, as we will see, that allows anapole effects to be separated from tree-level weak interactions. Finally, the anapole moment is in general one of a larger class of weak radiative corrections. The anapole interaction (Figure la shows one contribution) is thus accompanied by other radiative corrections that do not correspond to virtual photon exchange, such as Figure lb. It is the sum of all such diagrams that contributes to physical observables. It follows that the anapole moment is not a measurable, that is, not separately gauge-invanant. (However, we will see that the dominant contribution to nuclear anapole moments is well-defined and separately gauge-invariant.) Our primary focus in this review is the nuclear anapole moment contribution to atomic parity nonconservation (PNC). Exquisitely precise (sub- 1%) measurements
Figure 1 Weak radiative corrections to electron-proton scattering include anapole contributions ( a ) as well as contributions that do not correspond to a single virtual photon exchange (b).
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263
of atomic PNC have become possible in the past few years (4). The primary focus of these studies has been to obtain precise values of the strength of direct Z o exchange between electrons and the nucleus. The PNC effects are dominated by the exchange involving an axial Z o coupling to the electrons and a vector coupling to the nucleus. The nuclear coupling is thus coherent, proportional to the nuclear weak vector charge (approximately the neutron number), and independent of the nuclear spin direction. It is widely recognized that these atomic measurements are important tests of the standard electroweak model and its possible extensions, complementing what has been learned at high-energy accelerators that directly probe physics near the Z o pole (5-7). The comparison between precision measurements at atomic energies and accelerator energies could reveal the subtle influence of new interactions beyond the standard model. At low energies one expects weak radiative corrections, including the anapole contribution, to interfere with the dominant tree-level weak amplitude, producing corrections to observables of relative size (Y I%, where (Y is the fine structure constant. Therefore, identifying the anapole contribution in weak processes seems a daunting task. The possibility that nuclear-spin-dependent PNC measurements in heavy atoms might prove an exception was first pointed out two decades ago by Flambaum & Khriplovich (8). Such spin-dependent PNC effects involve a vector coupling to the electrons and an axial coupling to the nucleus. In this case, the contribution of Z o exchange is considerably reduced because the nuclear interaction is no longer coherent; instead it depends, in a naive picture, only on the spin of the last, unpaired nucleon within an odd-A nucleus. Additional suppression comes from the small Z o vector coupling to the electron, "(4 sin2 OH, - 1)/2 -0.05. In contrast, the anapole moment is enhanced in heavy nuclei, growing as All3, where A is the atomic number, and thus increasing in proportion to the nuclear surface area. The net result is the surprising conclusion that the anapole contribution-the weak radiative "correction"-will in fact dominate over the tree-level direct Z o spin-dependent contribution for nuclei heavier than A 20. The principal motivation for this review was the recent successful determination of the nuclear-spin-dependent contribution to atomic PNC in '"Cs (9). The extraction of this contribution from the much larger coherent PNC signal was accomplished by studying the dependence of the PNC signal on the choice of hyperfine level. Because the hyperfine differences are quite small, this extraction requires detailed control of possible sources of systematic error. Over a decade ago, the Colorado group (10) succeeded in measuring PNC effects in '"Cs to 2.2%, providing a tentative identification of the anapole moment. Six years ago, the Seattle (1 1) and Oxford (12) groups achieved an accuracy of 1.2% and 2.9%, respectively, in PNC measurements in natural thallium, resulting in bounds on the anapole moment in this nucleus. Finally, in 1997, a definite mea-drement was made: The Colorado group of Wood et al. (9) reported PNC measurements in "'Cs at the 0.35% level, from which a 7a nuclear-spin-dependent PNC signal was extracted. This spin dependence is clearly well above the expected signal from Z o exchange alone but is compatible with the enhancement expected from anapole effects.
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As we describe in this review, a nucleus generates an anapole moment through the weak NN interaction, which operates at long range in the nucleus by meson exchange, where one meson-nucleon vertex is strong and the other weak (13,14). Such interactions admix odd-parity amplitudes into the nuclear ground state and also induce new PNC nuclear currents. Thus, the exquisite precision now achieved in atomic PNC studies has opened a new window on the hadronic weak interaction. This interaction has proven more elusive than the weak interactions involving leptons. Whereas the charge-changing hadronic weak interactions can be studied in strangeness- or charm-changing decays, the standard model predicts that neutralcurrent interactions do not change flavor. Thus, hadronic interactions mediated by the Z o can only be studied in NN interactions, where PNC must be exploited to separate this contribution from much larger strong and electromagnetic effects. This is a difficult task. Only a few hadronic experiments have achieved the requisite precision and even fewer (notably those done with p’ p scattering or in certain light nuclei with special properties) can be interpreted, reasonably free of nuclear structure uncertainties. The importance of atomic anapole moment measurements is not only that they supplement the hadronic data but also that they are sensitive to long-range pion-exchange PNC interactions where the effects of the neutral current can be isolated. We will see, however, that the extraction of weak coupling constant constraints from anapole moments requires some nontrivial nuclear physics analysis. The plan of this review is as follows. In Section 2, we discuss general properties of anapole moments, the associated current distribution, and the spin-dependent interaction generated by nuclear anapole moments in atoms. In Sections 3 and 4, we summarize the experimental status of atomic anapole moment measurements and theoretical progress in estimating the size of nuclear anapole moments from an underlying model of the hadronic weak interaction. In Section 5, we compare weak meson-nucleon coupling constraints obtained from the 133Csanapole result and thallium limits with similar constraints obtained from PNC nuclear observables such as p’ p scattering and the circular polarization of y-rays emitted from “F. It becomes clear that, although all measurements are consistent with the broad “reasonable ranges” for PNC meson-nucleon couplings that theorists have defined (13), there is some disagreement among the experiments when a global fit is performed. We conclude by discussing prospects for improving our experimental and theoretical knowledge of anapole moments.
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2. ANAPOLE MOMENTS AND ATOMS In this section, we discuss general properties of nuclear anapole moments and the interactions they induce in atoms such as 133Csand thallium.
2.1. Electromagnetic Moments A standard multipole decomposition (15) of the electromagnetic current groups interactions according to their multipolarity and symmetry properties into charge
ANAPOLE MOMENTS
(cJ),
265
(?Fg)
transverse electric (?:I), and transverse magnetic operators, where J denotes the multipole rank. Static moments correspond to diagonal matrix elements of these multipole operators. From the transformation properties of the ordinary electromagnetic current operator under parity ( P ) and time reversal (T),
QJI”(2,t ) P = J@(-;,t ) fJI”(s;,
t ) F
= JI”(2,- t ) ,
1.
and the constraint of hermiticity, it is readily verified that the nonzero moments arise from matrix elements of the even-Jprojections of CJ and the odd-Jprojections of (15). More possibilities arise if one turns on the weak interaction, which introduces both parity- and CPIT-violating terms. It is interesting to classify possible static moments more generally, without the assumption that the underlying Hamiltonian conserves electromagnetic symmetries. The results in Table 1 show that, in addition to the ordinary monopole charge and magnetic dipole moments, two new dipole moments then arise. One of these, corresponding to the expectation value of the i.1 multipole, requires both parity and CPiT violation. This is the electric dipole moment, which, though allowed in the standard model because of CP violation in the quark mass matrix and in the theta term (16), has yet to be detected experimentally. The second, an E l moment, requires only parity violation. This is the anapole moment. These monopole and dipole moments must therefore arise in the most general expression for the matrix element of a conserved four-current for a spin-; particle,
?Fg
The two vector terms define the charge F1(4*) and magnetic F2(q2)form factors. The axial terms that follow are the anapole and electric dipole terms, respectively.
TABLE 1 Transformation properties of electromagnetic moments
0
P-even, T-even
1
P-odd, T-odd
P-odd, T-even
P-even, T-even
2
P-even, T-even
P-even, T-odd
P-odd, T-odd
3
P-odd, T-odd
P-odd, T-even
P-even, T-odd
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The anapole term reduces in the nonrelativistic limit to
3. showing that the current is transverse and spin-dependent. We define the anapole operator as
AIL = a(0)alL.
4.
Weak radiative corrections that modify physical processes involving elementary fermions thus include anapole moment contributions. One interesting example is the Majorana neutrino, where the identity under particle-antiparticle conjugation requires the magnetic and electric dipole moments to vanish but permits an anapole moment (17). Yet the interest in anapole moments would probably have remained largely theoretical were it not for the realization that they might play a significant role in atomic PNC experiments.
2.2. Atoms and the Generalized Siegert’s Theorem A generalization of the anapole operator is helpful for the case of a composite system such as a nucleus, where the current operator and wave functions are modified by the interactions among the constituents, inducing odd-parity admixtures in the ground state as well as PNC multibody currents. In the standard multipole decomposition, the El projection of the current is expressed in terms of the multipole operator f:;, 5.
where
6. Current conservation places constraints on the matrix elements of f;”j;(q). A familiar example is the long-wavelength limit of f:l(q) generated from the ordinary, parity-conserving electromagnetic current. The operator then is ? / M , which is of order u / c , where u is the nucleon velocity. It can be shown that the exchange-current contributions to the vector three-current are also of order u/c. For realistic models of the nucleus that account for the interactions among the nucleons, in general we lack a prescription for constructing interactions and currents consistently-and for renormalizing them appropriately to take into account the limited Hilbert spaces employed in nuclear models. As a result, there will be errors in evaluations of ? ,;:
215 ~~
ANAPOLE MOMENTS
~
267
owing to the imperfect construction of the current, that are necessarily of leading order in the velocity, u/c. Siegert (18) showed that the situation could be greatly improved by exploiting the continuity equation +
v
+ '
J ( 2 ) = -i[H,
/I(;)]
7.
TF',
to rewrite in the long-wavelength limit, entirely in terms of the charge operator. This generates the familiar dipole form off;', proportional to wr', where w is the energy transfer. The importance of this rewriting is that the charge operator, which is of order (u/c)O, has exchange-current corrections only of order ( u / c ) ~or, of relative size -1% Thus, the Siegert's form of ?;I, in which the constraints of current conservation are fully exploited, is a far more controlled operator for use in nuclear calculations. An analogous situation arises for the anapole moment. Although a variety of forms of the anapole operator are formally identical up to terms that vanish by current conservation, these forms are not equivalent operationally in realistic calculations because of model violations of current conservation. The simple, longwavelength form of Siegert's theorem does not address the question of moments because it generates an operator proportional to w , which then vanishes for a diagonal matrix element. Fortunately, the generalization of Siegert 's theorem ( 19) exists: At arbitrary q, one can write f,r'(q)= 3 ~ ( q )k,,( q ) ,where all components of the electromagnetic current that are constrained by current conservation have been isolated in 3~ and expressed as a commutator of the charge operator with the nuclear Hamiltonian. All such terms then vanish for a static moment. The resulting generalized Siegert's form of f;' appropriate for diagonal matrix elements is
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From Equations 5 and 8 we have the appropriate threshold form of the current operator. The anapole operator is defined relative to the current operator as in Equations 3 and 4, yielding immediately (20)
9. Other forms of the anapole operator are more commonly used, for example (21)
A,,= --n
/ d3rr24,(r').
10.
J
Apart from the normalization, which is a matter of convention, this form is equivalent to Equation 9 only if the nuclear model is sufficiently simple that exact current operators can be constructed. This would be the case for a simple central potential model, such as an independent-particle harmonic oscillator, for which the appropriate current operator is that of a free particle. This would also be the
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t ___-
- - _ _ .
Figure 2 A toroidal current winding generates a nonzero anapole operator in Equation 9.
case for an independcpparticle model with spin-orbit and orbital potentials of the form . or . l , provided that the additional contributions to the current obtained by minimal substitution, p’ - e i in each appearance of 2 = r‘ x ,; are included in the calculation (22). But in other, more realistic treatments of the nuclear physics, Equation 9 is the unique form that fully enforces the constraints of current conservation, regardless of the complexity of the current operators. Figure 2 gives a classical picture of the anapole moment as a current winding within the nucleus. Although the currents on the inner and outer sides of the torus oppose one another, there is a net contribution to Equation 9 because of r2weighting of the current, leading to an anapole moment that points upward. (Note the sign in Equation 9.) The current distribution drawn in Figure 2 is odd under reversal of parity, as is the ordinary J:?. Thus, it is easy to see that the corresponding anapole moment is a parity-odd operator. If, however, the current has a chirality-a small “pitch” corresponding to a left- or right-handed winding that would signal PNC-a parity-even contribution to the operator would be induced. This is the analog of evaluating Equation 9 with the axial-vector current induced by the weak interaction, leading to an operator component that would have a nonzero expectation value for a parity-conserving ground state. Similarly, the anapole moment associated with JeF would have a nonzero expectation value if the nuclear ground state contains parity-odd components induced by the weak PNC component of the NN interaction.
2 2
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2.3. The Spin-Dependent Atomic Interaction The potential felt by atomic electrons owing to the nuclear anapole moment is generated in the standard way by the dipole current-dipole current interaction. As the (?2 appearing in the nuclear current cancels the photon propagator for a static interaction, this weak interaction has the contact form
11.
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269
where 7 and p(;) are the nuclear spin and density, and where is the Dirac matrix acting on the electrons. The superscript '3.d." denotes that this is the nuclearspin-dependent contribution to the atomic PNC interaction. The portion of this interaction generated by the nuclear anapole moment is then
12. where I I denotes a matrix element reduced in angular momentum and u is the fine structure constant. The reduced matrix element of 1 is 2/Z(Z 1)(21 l).' We have already noted that ~ , , , ~ is~ only l ~ one of the contributions to K . Isolating this anapole contribution would clearly be easier if ~ , , , , , ~ l ~ were the dominant such contribution. It was the important observation of Flambaum & Khriplovich (8) that the A2/3growth of nuclear anapole moments, already implicit in the r2 weighting of the current in Equation 9, would lead to such dominance in heavy atoms. It is also clear that, if ~~~~~~l~ could be extracted from atomic measurements, relating this quantity to the underlying sources of hadronic PNC involves specifying the PNC currents that contribute to Equation 9, as well as the PNC admixtures in the nuclear ground state wave function through which the ordinary electromagnetic current operator will generate nonvanishing matrix elements of We defer this question to Section 4, focusing now on the experimental progress that has produced values and limits for K .
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3. EXPERIMENTS The idea of the anapole moment languished for many years after the early work of Zeldovich and Vaks ( 3 ) . In 1973, Henley et al. (23)-who were unaware of the Zeldovich paper-noted that the anapole moment would contribute to PNC observables in high-energy electron scattering off the proton. Then a revival of interest in the anapole moment occurred in 1980 when its possible relevance to ongoing experiments on atomic PNC was noticed (8). The experimenters were engaged in efforts to measure the tiny PNC mixing of S and P states induced by the coherent, nuclear-spin-independent Z o coupling to the nucleus (24). Because both the coherence and the increased electronic overlap with the nucleus lead to larger PNC effects in heavy atoms, the experimenters focused on atoms with A 2 100. The A2l3enhancement of the anapole moment makes the spin-dependent contribution to atomic PNC dominant over other vector (electron)-axial (nucleus) sources of PNC in such heavy atoms. Spin-dependent effects of the expected sizea few percent of the spin-independent signal-could, in principle, be extracted from 'Elsewhere in the literature, another definition of K is more commonly used, one that associates the nuclear ground state with a single-particle level of orbital angular momentum e and spin I . For example, the K used by Flambaum & Khriplovich (8)is obtained by dividing ours by the factor (-l)'+'/*+'(I 1 / 2 ) / [ I ( I I)].
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the experiments by precisely comparing the amount of mixing for two different hyperfine components, i.e., electronic transitions that differ only in the orientation of the nuclear spin. Nevertheless, the spin-dependent PNC effects are painfully small, corresponding to state mixings on the order of parts in 10”. That level of precision was eventually reached in a cesium PNC experiment and nearly reached in experiments with thallium. The isolation of a definite spin-dependent contribution in cesium provided the first confirmation that anapole moments exist,
3.1. The Cesium Experiment The experiment in cesium is described in detail elsewhere (2.5). The central idea is to exploit the highly forbidden 6s-to-7S transition in atomic cesium. The strength of the transition depends on the handedness within the excitation region. That handedness is defined by various applied electric, magnetic, and laser fields and reverses with appropriate reversals of those applied fields. This reversal forces any PNC contribution to the excitation rate to change sign, thereby altering the excitation rate for the transition. Because the fractional change in the excitation rate is very small, a great deal of work must be done to achieve a signal-to-noise ratio sufficient to see any effect. Then even more work must be invested to verify that the detected effect is truly a violation of parity and not a spurious signal arising from systematic errors such as imperfect reversals or alignments of the fields that define the handedness of the experiment.
3.1.1. TECHNIQUES In the absence of electric fields or parity-violating interactions, the electric dipole (El) transition between the 6 s and 7s states of the cesium atom (Figure 3) is forbidden. Because the nuclear spin of 133Csis I = 712, these S1/2 levels combine with the nuclear ground state to form hyperfine states of total angular momentum F = 4 and F = 3. A PNC interaction mixes a small amount (-lo-’ ) of the neighboring 6P1/2 and 7P1/2 states into the 6s and 7 s states; the P112 hyperfine levels have F = 3-4, so that PNC mixing with both the F = 3 and F = 4 hyperfine S states takes place. The induced 7 s tf 6s PNC El amplitude is thus proportional to the product of the El and PNC mixing matrix elements coupling the S and P hyperfine levels (and inversely proportional to the energy differences between the S and P levels). To obtain a measurable observable that is first-order in this small amplitude, a DC electric field,!? is applied that also mixes S and P states. This field generates an interfering “Stark-induced’’ El transition amplitude AEthat is typically lo5 times larger than the PNC-induced El amplitude APNC. A complete analysis of the relevant transition rates between the various hyperfine magnetic states ( F , m ~is )given elsewhere (2.5). These rates involve a straightfoward evaluation of the hyperfine matrix elements for the Stark amplitude, the nuclearspin-dependent PNC Hamiltonian given in Equation 11, and the spin-independent
21 9
ANAPOLE MOMENTS
6p312
ZZEi
Dye Laser
271
F=5 4 3 2
(540 nm)
6SIl2
Figure 3 Partial cesium energy-level diagram including the splitting of the S states by the magnetic field. The case of 540-nm light exciting the ( F = 3, mF = 3) level is shown. Diode lasers 1 and 2 optically pump all the atoms into the (3,3) level, and laser 3 drives the 6SF=4(Fdet)-to-6P~=s transition to detect the 7s excitation. PNC is also measured for excitation from the (3,-3), (4,4), and (4,-4) 6 s levels. The diode lasers excite different transitions for the latter two cases.
PNC Hamiltonian for the much stronger coherent interaction with the nucleus,
13. Here y5 is the axial coupling to the electrons. The tree-level standard-model result for the vector weak charge of a point nucleus is
Q w = Z(1-4sin2Qw)
-
N
-
-N.
14.
Here we omit the somewhat tedious angular momentum algebra of the rate calcuC AE, which differ by lation. To generate a nonzero interference between A ~ N and a relative phase of i, the 6s-to-7S transition must be excited with an elliptically polarized laser field of the form cz2 pi Im(c,) 2 , where the handedness of the polarization p is f 1,f.x is the component of the oscillating electric field parallel to the DC Stark field E , and cZ is the oscillating field component in the direction perpendicular t o i . The PNC contribution to the transition rate is measured on both the 6S~=3-to-7S~=4 and 6S~=4-to-7S~=3 transitions. To a good approximation, the only difference between the two transitions is the reversal of the nuclear spin. Thus, simply taking the difference between the PNC contributions to the hyperfine transition rates isolates the nuclear-spin-dependent PNC coupling K of Equation 11. For both transitions, the atoms are initially populated and excited only out of states with extreme values of mF, ( F = 3, mF = f 3 ) and ( F = 4, mF = *4). For
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/l
oDtical
//
/
mirror interaction region
...... ....,........................
lasers
dye 7pBc laser
mirror
A# Probe diode &#$“ laser
-
Figure 4 Schematic of the cesium apparatus. In the interactionregion,i is along the P axis,f? is along the 2 axis, and the 540-nm dye laser beam defines the j axis.
these cases and the configuration of electric and magnetic fields shown in Figure 4, the transition rate is
R = IAE
2 2 + APNCI’ - ,62 E,E,Cl(F,
mF; F’, ml,)
+ ~,~E,E,~I~(E,)I~(E~PNc)C~(F, mF; F’, m;),
15.
where ,6 is the tensor transition polarizability, E, = ?I!, I is the DC electric field strength, and C1 and C2 are combinations of Clebsch-Gordan coefficients whose detailed form we have suppressed (25). They depend on the initial- and final-state hyperfine labels (F,mF)and (F’, m/,) and transform as Cl(mF)-+ C1(-mF) and C z ( m ~+ ) - C 2 ( - m ~ ) underreversal of the magnetic labels. The tiny contribution proportional to AgNc has been neglected as well as the small 6s-to-7S parityconserving magnetic dipole transition amplitude AM.(As discussed below, AMcan be a source of many systematic errors that must be addressed carefully.) The experimental quantity of interest is the fractional PNC modulation in the transition rate
16. that modulates with the reversals of E, m, and p . In the experiment, there are actually five “parity” reversals because mF is reversed in three different ways (by reversing the polarization of the optical pumping light, reversing the optical
221 ANAPOLE MOMENTS
273
pumping magnetic field relative to the pumping light, and reversing the magnetic field in the 6s-to-7s excitation region). The use of this large number (five) of independent reversals is essential for the detection and elimination of systematic errors, providing a great deal of redundancy for the PNC signal. This redundancy means that although no single reversal is perfect, the product of all the imperfections is far smaller than the PNC signal. Furthermore, the signal modulations accompanying various combinations of field reversals provide additional information about the field reversal imperfections and orientations that help identify potential systematic errors. 3.1.2.THEAPPARATUS A simplified schematic of the apparatus used by Wood el al. (9) is shown in Figure 4. An effusive beam of atomic cesium is produced by a
heated oven and collimated with a multichannel capillary array nozzle. The beam is optically pumped into the desired ( F , r n ~state ) by light from diode lasers 1 and 2 (25,26). The 2-cm-wide beam of polarized atoms intersects the 540-nm standingwave (0.8mm Gaussian diameter) that is produced inside a high-finesse (100,000) Fabry-Perot power buildup cavity (PBC). The PBC enhances the transition rate, and the standing-wave geometry greatly suppresses the troublesome modulation arising from A E A Minterference (25).The 540-nm light originates from a dye laser whose frequency is tightly locked to the resonant frequency of the PBC by a highspeed servosystem (27). Before entering the PBC, the dye laser light passes through an intensity stabilizer, an optical isolator, and a polarization control system made up of a half-wave plate, Pockels cell, and adjustable birefringence compensator plate. The ellipticity of the light is controlled by rotating the half-wave plate, and the handedness of the ellipse is reversed by switching the sign of the 1/4 (quarter wavelength) voltage that is applied to the Pockels cell. The intensity stabilizer holds the amount of light transmitted through the cavity constant and hence stabilizes the field inside the cavity. This field corresponds to about 2.5 kW of circulating power. The PBC resonant frequency is held at the frequency of the desired atomic transition by a servosystem that translates the input mirror. The DC electric field in the interaction region is produced by applying -500 volts between two parallel 5-cm by 9-cm conducting plates, separated by about 1 cm. The plates are made of flat pieces of Pyrex glass coated with 100 nm of molybdenum. Both field plates are divided into five electrically separate segments. This division makes it possible to apply small uniform and gradient electric fields along the y-axis for auxiliary diagnostic experiments. The PBC and field-plate mounting system are rather elaborate to ensure precise alignment and extreme mechanical stability. After being excited out of the populated 6 s hyperfine level up to the 7 s state, atoms decay by way of the 6P states to the previously empty 6 s hyperfine level ( F d e t ) more than 60% of the time. The population of Fdet is detected 10 cm downstream of the interaction region using laser fluorescence. Light from diode laser 3 excites each atom in F d e t to the 6P3p state many times. The resulting scattered photons are detected by a silicon photodiode that sits just below the atomic beam.
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When the 6S~=3-to-7S~=d line is measured, the detection laser drives the 6 S ~ = 4 t0-6P3,~,~=5 cycling transition (Figure 3). About 240 photons per Fdet atom are detected. For the 6S~=d-t0-7S~=3 line, the detection transition is the 6S~=3-to6 P 3 / 2 , ~ cycling =~ transition. This cycle gives about 100 detected photons per Fdet atom. The signal-to-noise ratio for this transition is about 20% lower. During each half cycle of the most rapid field reversal (E) and after the switching transient has passed, the detector photocurrent is integrated, digitized, and stored. For each stored value, the computer also records the field and spin orientations. The signal-to-noise ratio needed for this experiment puts extreme requirements on laser stability. A fluctuation in the intensity, frequency, or direction of the light from any of the four lasers will introduce noise in the detected atomic fluorescence. The most extensive control is needed for the dye laser (27) but the requirements on the three diode lasers used for optical pumping and detection are also severe. These requirements motivated substantial development of diode-laser stabilization technology (28). Both optical and electronic feedback are used to lock the frequency of each diode laser to the desired atomic transition using saturated absorption spectrometers. The cesium experiment required extensive and precise control of magnetic fields. In the optical pumping region, there is a uniform 2.5-G field that must point in the 9 direction (parallel to the pumping laser beams). In the interaction region, a 6.4-G field must point precisely along either k?.Between the two regions, the magnetic field must rotate gently enough that the atomic spins follow it adiabatically. Finally, the field must be near zero in the detection region, and it is necessary to precisely reverse the fields in the optical pumping and interaction regions independently without significantly perturbing the fields in the other two regions. The setup included 23 magnetic field coils of various shapes to provide the necessary fields and gradients. Most of these coils were driven with both reversing and nonreversing components of current. Many additional elements were required to achieve sufficiently precise alignment and control of all aspects of the apparatus. These included 31 different servosystems to ensure optical, mechanical, and thermal stability. 3.1.3. DATA AND RESULTS Approximately 350 h of data on the PNC observables were acquired in five runs distributed over an eight-month period. Each of the five runs followed the same basic procedure. First, a set of auxiliary experiments was carried out to-measufe and set the following quantities: (a)all three components of the average E and B fields and their 9 gradients in the interaction region; (b) the magnitude and orientation of the birefringence of the PBC output-mirror coating; (c) the polarization-dependent power modulation of the green laser light; and (d) the populations of the mF levels of the atomic beam as it entered the interaction region. After these measurements were completed, the four laser frequencies were locked to the desired hyperfine transitions and data were acquired in 1.5-h blocks. During this time, five parity reversals-the electric field, laser polarization, and three ways of reversing the mF state being excited-were carried out at different
223 ANAPOLE MOMENTS
275
rates. The electric field was reversed at 27 Hz and the others at various lesser rates. The relative phases of the various reversals were regularly shifted by one half cycle. Before and after each of the 1.5-h blocks, the polarization of the 540-nm standingwave field was measured and set. At regular intervals, the PBC output mirror was rotated by 90.0”, and at irregular intervals, the frequencies of the four lasers were changed to measure PNC on the other 6s-to-7S hyperfine line. At the end of each data run (20 to 30 1.5-h blocks of data), the initial auxiliary experiments were repeated to verify that the quantities described above had not changed significantly. The typical size of the 6s-10-7s signal from the photodiode was 200 nA on the 3 + 4 line and 85 nA on the 4 + 3 line. These measurements corresponded to about 0.5% of the atomic beam undergoing a 6s-to-7S transition. The signal-to-noise ratio for measuring the 6s-to-7S transition rate was typically 55,000/1/Hz, and the PNC modulation was typically about six parts in lo6. Because the technical noise was small and many photons were detected from each atom that had undergone a 6Sto-7S excitation, the noise was dominated by the shot-noise fluctuations in the number of atoms making the 6s-to-7S transition. Data were taken over a range of polarization ellipticities and electric fields. The data analysis used to find the PNC modulation for each block of data was relatively simple. Appropriate combinations of fractional differences in the signal sizes were calculated for each of the 32 different field configurations to find the fraction of the rate that modulated with all five reversals. From the measured values of electric field and laser polarization, this fractional modulation was then converted into Im(E 1p N ~ ) / p . Various other small calibration corrections were required. The major concern in this experiment was possible systematic errors arising from spurious signals that modulate under all five parity reversals, thus mimicking PNC. Roughly 20 times more data were taken in the investigation and elimination of such errors than in the actual PNC measurement. Several small errors associated with stray and misaligned fields were encountered, as in previous PNC measurements (25), and were treated as before. All possible combinations of static and oscillating electric and magnetic fields that could mimic a PNC signal were exhaustively analyzed. All of the stray (defined as nonreversing) and misaligned DC electric and magnetic fields and their gradients, as well as many of the laser field components, could be determined by looking at appropriate modulations in the 6s-to-7S rate under various conditions. Many of these quantities were extracted from the 3 1 different modulation combinations observed in real time while taking PNC data, and the remaining components were determined by the auxiliary experiments that were interspersed with the PNC runs. Many tests were also performed to ensure the necessary stability of the relevant fields. Although the procedures were similar in concept to previous work, the Wood et al. (9) measurements were more difficult and time-consuming because of the higher accuracy required. It proved necessary to consider not only the average fields but also their gradients across the interaction region. Several small spurious signals were identified and removed. The absence of any systematic effects
224
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from the troublesome A E A Minterference was confirmed by the independence of results for various laser polarizations sensitive to this interference. Other crosschecks included enhancing sources of error to confirm the predicted response and performing data analyses to verify that transition-rate variations were consistent with fundamental shot-noise fluctuations. The final results are -Im(E l p ~ c ) / P= 1.6349 f 0.0080 mV/cm = 1.5576 f 0.0077 mV/cm
6 S ~ = 4e 7S~,3 6S~,3 + 7S~=4,
17.
yielding for the nuclear-spin-dependent difference of interest for the anapole moment I m ( E l ~ & ) /= ~ 0.077 f 0.011 mV/cm,
18.
a 7 0 effect. The statistical uncertainties for the two transitions, 0.0078 and 0.0073 mV/cm, respectively, dominate the error. The systematic uncertainties are based on statistical uncertainties in the determination of various calibration factors and systematic shifts, and therefore, it is appropriate to add them in quadrature. The final results are in good agreement with previous measurements in cesium but are much more precise. From the available atomic calculations (29,30) one concludes K(’~’CS) = 0.112 f 0.016,
19.
a result we will find is dominated by the anapole moment.
3.2. The Thallium Experiments Experiments measuring parity violation in atomic thallium (70.5% 205Tl,29.5% 203Tl)have not yet detected a nuclear-spin-dependent/anapolemoment contribution, but they have achieved accuracies very near the level where anapole effects are expected. The resulting limits are interesting from the perspective of hadronic PNC. An interference of parity-allowed and PNC contributions to an atomic transition is observed, just as in cesium. However, in thallium the transition is an allowed magnetic dipole transition, 6 P I p +-+ 6P3p. The PNC interference is between the allowed M1 transition amplitude and the PNC El amplitude arising from weak interaction effects that mix S states into the Pl/z ground state. Because thallium has an I = 1/2 nuclear ground state, the P1p state has F = 0 and F = 1 hyperfine sublevels, which will be affected differently by nuclear-spin-dependent sources of PNC. Thus, the search for spin-dependent PNC effects requires a comparison of the strength of PNC transitions out of the F = 0 and F = 1 states. Groups at the University of Washington (1 1) and Oxford (12) have made such comparisons. The optical rotation of linearly polarized light that is produced by the interference of El and M1 transitions is measured in a large vapor cell of atomic thallium, as shown schematically in Figure 5. A beam of linearly polarized laser light passes
225
ANAPOLE MOMENTS
vapor cell
laser
i
polarizer
1 Figure 5
277
detector
i pol.
empty ceii
i
Schematic of the thallium vapor cell apparatus.
through the vapor and then through a nearly crossed linear polarizer followed by a sensitive detector. A Faraday modulator rotates the plane of polarization back and forth through the perfectly crossed position. This allows phase-sensitive detection that minimizes sensitivity to drifts in the signal baseline. The laser is scanned repeatedly across the transition and the detected light signal is then fitted to the predicted lineshape. The PNC rotation reverses sign as the laser is tuned through line center. The amplitude of the rotation can be determined from the fit to the dispersion-like line shape. To distinguish the signal of interest from spurious frequency-dependent rotations in optics, the vapor cell is interchanged with a “dummy” cell that has identical optics but no atomic vapor. An example of the data obtained in this fashion is shown in Figure 6, taken from Vetter et al. (11). The agreement between predicted and observed signals is excellent. As in the cesium experiment, hundreds of hours of data were taken and considerable effort was made to eliminate possible systematic errors. The thallium oven was carefully constructed and shielded to avoid magnetic fields that would cause spurious Faraday rotation. Data were acquired over a wide range of pressures of thallium and a variety of tests of the calibration were carried out. The final results of the experiments are expressed as the ratio of PNC E l to magnetic-dipole M1 transition amplitudes, R = Im(E l p ~ c / M1). The nuclearspin-independent PNC signals obtained are RS.’.= (-14.68 f 0.17) x lo-’ (11) (normalized to 205Tl)and (-15.68 f 0.45) x lo-’ (12), corresponding to 1.2% and 2.9% accuracy. The nuclear-spin-dependent effects are consistent with zero, R”’.(Tl) = (0.15 f 0.20) x lo-’ Seattle = (-0.04
f 0.20) x lo-’ Oxford.
20.
The resulting constraints on K are K ( T ~= ) 0.29 f 0.40 Seattle = -0.08
f 0.40 Oxford.
21.
226
278
HAXTON
WIEMAN I
I
I
I
I
The 133Csand thallium spin-dependent measurements involve odd-proton nuclei, which leads to the prediction that they test a similar combination of isospin components of the weak hadronic PNC potential, as we see below. Note that an anapole measurement for an odd-neutron nucleus would test a roughly orthogonal combination of isoscalar and isovector contributions to the hadronic PNC potential. Several atomic PNC efforts are under way that could produce such constraints (31,32).
4. ANAPOLE MOMENTS AND HADRONIC PARITY
NONCONSERVATION In this section, we discuss the extraction of ~ , , from ~ ~ K~ , the l ~ relation between ~~~~~~l~ and the weak hadronic interaction, and the mechanisms by which a nucleus generates an anapole moment.
4.1. Extracting the Anapole Moment We have seen that atomic PNC measurements place the following constraints on the strength of nuclear-spin-dependent electron-nucleus interaction: K(133CS)= 0.112 f 0.016 K ( T ~=) 0.29 f 0.40,
22.
227 ANAPOLE MOMENTS
279
where we have used the Seattle thallium result above because, due to its central value, it proves to be the more restrictive in the weak meson-nucleon parameter region favored by nuclear PNC experiments. The three principal contributions to K , K
= Kanapole f
KZO
+
KQ,,
23.
,
arise from the nuclear anapole moment, the vector (electron)-axial (nucleus) treelevel Z o exchange, and a term generated by the combined effects of the coherent Z o and magnetic hyperfine interactions between the electrons and the nucleus. In the theory discussions below, we treat the thallium constraint as one on the principal isotope, 205Tl(70.5%). The other isotope, 203Tl(29.5%), differs in structure only by a pair of neutrons, and thus should have very similar properties. The tree-level vector (electron)-axial (nucleus) Z o interaction generates a contribution 24. where g A = 1.26 is the axial-vector coupling and sin20w = 0.223. To get a rough feel for this contribution, we can evaluate the nuclear matrix element in the extreme single-particle limit. I3'Cs would then be described as an unpaired lg7/2 proton outside a closed core, and 205Tlcorresponds to an unpaired 3s1p proton. In this limit, single particle - -(-l)/-e-m,
gA
2e
KZO
+ 1 (1
-
4 sin2 ow),
25.
where l is the single-particle orbital angular momentum and m, is the z-component of isospin (with m, = 1/2 denoting a proton). Thus the single-particle estimates for 133Csand 205Tlare 0.0151 and -0.136, respectively. Nuclear models of various types have been employed to try to estimate the effects of strong correlations in quenching the Gamow-Teller matrix element in Equation 24 from its single-particle value (20,33-35). Here we use the results of a recent large-basis shell-model study (33), which yielded KgF('33cS)
= 0.0140
K ~ F ( ~ '= ~T -0.127. ~)
26.
(We describe these calculations in more detail later.) In addition, one-loop standardmodel electroweak radiative corrections, which we do not include in the numerical results below, modify the tree-level expression in Equation 24 somewhat, reducing the isovector contribution and inducing a small isoscalar amplitude (36). A second contribution to K is generated by the combined effects of the coherent vector Z o coupling to the nucleus, proportional to the weak charge Qw, and the magnetic hyperfine interaction (37). From the measured nuclear weak charge and
228
280
HAXTON
WIEMAN
magnetic moment, Bouchiat & Piketty find (38) = 0.0078 KQ,,,('~~CS)
27.
K ~ , ( ~ ~ '= T 0.044. ~)
Thus, the experimental values for the anapole contributions to K are obtained by subtracting the results of Equations 26 and 27 from Equation 22, yielding K , , , ~ ~ ~ ~ (=~ 0.090 ~ ~ Cf S 0.016 ) K
~
~
~= 0.376 ~ ~f 0.400. ~ ~
(
~
~
~
T28. ~
4.2. The Hadronic Weak Interaction The various mechanisms operating within the nucleus to generate ~ , , , ~ arise ~ l ~ from the hadronic weak interaction. As we noted in the introduction, the only practical strategy for studying the effects of Z o exchange between quarks is the investigation of the PNC NN interaction. Because anapole moments are now measureable in high-precision atomic PNC experiments, they have become a part of that strategy. The low-energy hadronic weak interaction can be described by a phenomenological current-current Lagrangian (14) 29. where Jw and Jz are the charged and neutral weak currents, respectively. If one considers only the light-quark (u, d, s) contributions to these currents, then JW has two components JW = c o s B c J i + s i n B c J & , 30.
-
where sin BC 0.22. The current J& drives the u + d transition and transforms as AI = 1, A S = 0, while J & drives the ii + s transition and transforms as AZ = 1/2, A S = -1. (Here I denotes isospin and S strangeness.) The neutral current also has two components, J$ and J J , which transform as AZ = 0, AS = 0 and AI = 1, AS = 0, respectively. The A S = 0 NN interaction is then governed by the following piece of Equation 29
An important aspect of this equation is its isospin content. The symmetric product of two J$ ( A I = 1) currents transforms as AZ = 0 and 2, while the symmetric product of two J& ( A I = 1/2) currents transforms as AI = 1. Thus, the AI = 1 component of the charged-current weak NN interaction is suppressed by tan2 BC relative to the AZ = 0 , 2 components, whereas the neutral-current AI = 1 contribution is unsuppressed. It follows that the neutral current should dominate this isospin channel.
)
229
ANAPOLE MOMENTS
281
The goals of hadronic PNC studies include isolating this neutral current and understanding the mechanism by which the weak force is communicated over the relatively long distances characterizing Iow-energy NN scattering or NN interactions within the nucleus. Although the standard model specifies the elementary couplings of the weak bosons to quarks, these vertices are dressed by the strong interaction to form couplings between physical particles, such as mesons to nucleons. We know, from the A 1 = 112 rule in strangeness-changing hadronic weak interactions (the strong enhancement of the ratio of AZ = 112 to 312 amplitudes), that strong-interaction effects can substantially alter couplings from their underlying bare values. The low-energy NN weak interaction is conventionally described in a onemeson-exchange model, where one meson-nucleon vertex is weak and the other strong. This long-distance mechanism dominates at nuclear densities. Six weak couplings, f i r , h i , h;, h i , h i , and h k , characterize the strengths of the isovector n,isoscalarJisovectorJisotensor p , and isoscalarJisovector w weak meson-nucleon couplings (13). This model is not as restrictive as it may first appear. First, CP invariance forbids any coupling between neutral J = 0 mesons and on-shell nucleons, eliminating a number of candidate interactions (39). Second, the most general low-energy PNC interaction contains only five independent S t, P amplitudes. From this perspective, the description in terms of n,p , and w exchange can be viewed as an effective theory, valid at momentum scales much below the inverse range of the vector mesons. Because the detailed short-range behavior of the PNC interaction is not resolvable at low momenta, one can characterize the short-range weak interaction by five contact interactions corresponding to the independent S t, P amplitudes, supplemented by long-range n exchange. The six meson-nucleon weak couplings allow one to mimic these six degrees of freedom. If we denote the isoscalar strong meson-nucleon couplings by gir", g, and g,, the resulting NN PNC potential is (14)
230
282
HAXTON
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[c,
{G,
where r‘ = - ;2, ii = e-”‘/4nr], ? = e-m’./4nr}, a n d c = 6, -G2, with ?1 and the coordinate and momentum of nucleon 1. This expression is usually evaluated assuming vector dominance, which fixes the strong scalar and vector magnetic moments, ,uy= -0.12 and pu = 3.70. “Best values” and broad “reasonable ranges” for the weak meson-nucleon couplings were defined by Desplanques, Donoghue & Holstein (13) (DDH), who deduced standard-model estimates for these vertices by such techniques as factorization, the quark model, and current-algebra and sum-rule methods. The broad “reasonable ranges” reflect the large degree of uncertainty implicit in such approximate tools, as well as the potential consequences of missing physics, such as strange-quark amplitudes (40). Nevertheless, the DDH results in Table 2 have provided experimentalists with benchmarks for PNC experiments. In an ideal world, one would determine the low-energy NN S ++ P amplitudes, or equivalently the six weak meson-nucleon couplings, by a series of NN scattering experiments. Such experiments require measurements of asymmetries of -lo-’, which is the natural scale for the ratio of weak and strong amplitudes, 4 n G ~ r n ;/g:”. Only a SingleNNrneasurement, the longitudinal analyzing power A7 for fi p , has been successful (41-43). (Experiments have been done at 13.6, 45, and 221 MeV.) These results have been supplemented by a number of PNC measurements in nuclear systems, where accidental degeneracies between pairs of opposite-parity states can produce, in some cases, large enhancements in the PNC signal. The experiments include A; for 5 01 at 46 MeV (44), the circular polarization Py of the y-ray emitted from the 108I-keV state in 18F(451, and A , for the decay of the 110-keV state in polarized 19F(46). It is widely agreed that the interpretation of these experiments is relatively free of nuclear structure uncertainties; either the systems are few-body, where quasi-exact structure calculations can be done, or they involve special nuclei in which the PNC mixing matrix elements can be calibrated from axial-charge p-decay (47). An anaysis of these results, which have been in hand for some time, suggests that the isoscalar PNC NN
el
+
+
TABLE 2 Weak meson-nucleon coupling “best values” and “reasonable ranges” (in units of from the standard-model calculations of Desplanques et al. (13) Coupling fJr
“Best Value” 0.46
“Reasonable Range” 0.00-
1.14
1.14 --+ -3.08
h:
-1.14
hb
-0.019
h:
-0.95
0.76 --f 1.10
-0.19
0.57 --f 1.03
hl,
-0.11
-0.038 + 0.00
-0.19
-
0.08
23 1 ANAPOLE MOMENTS
283
interaction-which is dominated by p and w exchange-is comparable to or slightly stronger than the DDH best value, whereas the isovector interactiondominated by n exchange-is significantly weaker (21/3) (14). Because the isovector channel is expected to be enhanced by neutral currents, there has been great interest in confirming this result. The 133Csanapole result thus provides a possible cross-check on this tentative conclusion that an isospin anomaly, superficially like the AI = 1/2 mle in flavor-changing decays, may exist in the A S = 0 weak NN interaction.
4.3. Weak Meson-Nucleon Couplings and the Anapole Moment We noted in the introduction that an anapole moment-unlike the magnetic moment-of elementary fermions is not a gauge-invariant quantity. Musolf’s (22) discussion of this point ends with a very straightforward explanation: Because PNC corrections to the electromagnetic current couple only to virtual photons, the amplitude for PNC photon emission is not a physical amplitude and thus need not be gauge-independent. However, the long-distance contributions to the anapole moment in the nucleus-the meson-cloud contributions to the nucleon anapole moment and the many-body contributions due to the PNC NN interaction and associated exchange currents-are both the dominant contribution to the nuclear anapole moment and separately gauge-invariant (20). These contributions, associated with the weak meson-nucleon couplings, are discussed next. To evaluate the nuclear anapole moment in the context of some model of hadronic PNC, one must take the ground-state expectation value of the operator in Equation 9. If one works to first order in the weak interaction, two types of terms contribute to the current matrix element. First are terms corresponding to the PNC weak contributions to JIA. In the context of the weak meson-nucleon couplings, these include both one-body terms-mesonic loop corrections to the ordinary electromagnetic current involving one weak and one strong vertex, as illustrated in Figure 7a-and exchange currents (Figure 7b), where the weak and strong vertices attach to different nucleons. We stress that if the anapole operator of Equation 9 is employed, the exchange currents are model-dependent in the sense that all constraints imposed by current conservation are already explicitly enforced. The argument that minimal substitution in simpler, independent-particle treatments somehow accounts for the exchange currents is incorrect; once Equation 9 is employed, the surviving exchange currents depend on aspects of the NN interaction not constrained by current conservation. The second class of contributions comes from evaluating Equation 9 for the ordinary, parity-conserving JeF, which contributes through PNC admixtures in the nuclear wave function, as illustrated in Figure 7c. The polarization term, which depends on the matrix elements of VPNCbetween the ground state and a complete set of excited opposite-parity states, can be enhanced if opposite-parity states exist very near the ground state; the admixing is clearly inversely proportional to the energy difference (48). However, in the absence of such “accidental” degeneracies,
232
284
HAXTON
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Figure 7 The one-body (a),exchange current (b),and nuclear polarization ( c ) contributions to the nuclear anapole moment.
one expects the mixing to be dominated by the giant resonances, the collective states at -15-20 MeV in heavy nuclei that account for most of the El and other first-forbidden nuclear response. We discuss each of these contributions below, depending rather heavily on recent work (33) in which the various contributions to the nuclear anapole moment were estimated using the formalism that has become standard in other studies of hadronic PNC: the DDH weak-meson couplings, the two-body VPNC based on those couplings, large-basis shell-model nuclear wave functions, and a standard correlation function to modify the short-distance behavior of the shellmodel two-nucleon density. This treatment allows us, in Section 5, to compare anapole, NN, and nuclear constraints on hadronic PNC on an equal footing. The one-body PNC electromagnetic current derived from meson loop corrections (e.g., Figure 7a) yields in the nonrelativistic limit 4.3.1. THE NUCLEON ANAPOLE MOMENT
33.
233 ANAPOLE MOMENTS
285
TABLE 3 Shell-model estimates of the anapole matrix element (IIIA, l l I ) / e , expressed as coefficients multiplied by the indicated weak couplings
"'CS
205~1
nucleonic ex. cur. polariz. total nucleonic ex. cur. polariz. total
0.59 8.58 51.57 60.74 -0.63 -3.54 -13.86 -18.03
0.87 0.02 - 16.67 -15.78
0.90 0.11 -4.88 -3.87
0.36 0.06 -0.06 0.36
0.28 -0.57 -9.79 -10.09
0.29 -0.57 -4.59 -4.87
-0.86 -0.01 4.63 3.76
-0.96 -0.06 1.34 0.33
-0.35 -0.03 0.08 -0.30
-0.29 0.28 2.77 2.76
-0.29 0.28 1.27 1.26
That is, this term is the sum of the anapole moments of the individual nucleons. Because the contributions of spin-paired core nucleons cancel, one expects this term to depend on the unpaired valence nucleon in the odd-A nuclei of interest. Thus the one-body contribution should be roughly independent of A, although-as in the closely related example of magnetic moments-its value will depend on the shells occupied by the valence nucleon. The pionic contribution to a,(O) and a,(O) was evaluated some time ago (20), yielding results proportional to efrgr". The isoscalar coupling a,(O) proved to be about four times larger than a,,(O). Recent extensions of this work have included the full set of one-loop contributions involving the DDH vector meson PNC couplings, using the framework of heavy-baryon chiral perturbation theory and retaining contributions through O( l / A i ) , where A, = 4nFr 1 GeV is the scale of chiral symmetry breaking (36). The contributions due to f r are consistent with the earlier work-the new a,(O) is about 1.3 times larger, and a,(O) is zero to this order in the heavy-baryon expansion. The addition (36) of the heavy meson contributions greatly enhances a,(O). An evaluation with DDH best-value couplings yields a,(O) 7 ~ , ~ ( 0Thus ) . the inclusion of heavy-meson contributions substantially enhances the one-body anapole terms and alters the isospin character, generating opposite signs for the proton and neutron anapole moments. -onebody This then determines A,, . The shell-model studies of Reference 33, which wedescribeinmoredetailbelow,givematrixelements (Ill a(i)llf)= -2.37 and 2.53 and ( Z l l a(i)t3(i)llf) = -2.30 and 2.28 for 133Csand 205Tl,respectively. These lead to the nucleonic anapole contributions shown in Table 3.
-
-
cf=,
cf=,
4.3.2. EXCHANGE CURRENTS Two-body PNC currents arise from meson exchange diagrams where the photon couples to the meson "in flight" (transition currents), or where the photon and either the PC or PNC meson-nucleon vertex creates and annihilates an N N pair (pair currents), as illustrated in Figure 7b. The explicit forms of the pionic exchange currents, which have the longest range, are known but are somewhat complicated. Rather than quoting the result (20), we instead give
234
286
HAXTON
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WIEMAN
the one-body reduction of the pair-current contribution to that operator, which illustrates the underlying physics much more clearly. If one views a nucleus as a single particle outside a closed core, a two-body operator can be replaced by an equivalent one-body effective operator
34. where the sum is taken over the nuclear core. Completing this sum in a Fermi gas model, assuming a spin-symmetric but isospin-asymmetric core, yields the pionic pair-current effective operator
35. where p ( r , ) is the nuclear density operator and m: ( w z ) a proton (neutron) Fermigas response function that depends on k, / kF , the nucleon momentum as a fraction of the Fermi momentum. The ws vary only gently, ranging from 0.33 to 0.19 as k , / k increases ~ from 0 to 1. Thus a suitable average value is -0.25. The overall strength of A::" is given in terms of the pionic contribution to the single-particle anapole moment
~'(0)
- -1.6 efn
gnNN
36.
%fin2
to allow an easy comparison with the one-body current. Using a nuclear density of 0.195/fm3, one finds that the pair isoscalar n exchange current then scales as -0.9A2/3~,: (0). Unlike the nucleonic contribution, the exchange-current contribution grows as A*/'. Clearly, it will increasingly dominate over the nucleonic contribution as A increases. The isovector effective operator is smaller by a factor of 2(Z - N)/3A, reflecting a cancellation between contributions from core protons and neutrons. The calculations of Haxton et al. (20) employed the full two-body form for the pionic currents, evaluating these from the shell-model two-body density matrices. A short-range correlation function was introduced to mimic the effects of hardcore correlations on the density matrix. The extension to include the p and w PNC couplings is a formidable task requiring evaluation of the p and w pair currents and the ppy and p r y currents. This calculation was only recently completed (33). The ppy and p n y transition currents and the component of the w pair current where the photon and PNC w couplings are on different nucleon legs were found to be negligible, well below 1% of the dominant n currents; the remaining heavy-meson currents are more important but, as can be seen in Table 3, still are suppressed relative to the pionic currents. The tabulated results were obtained from the shell-model calculations described below.
235
ANAPOLE MOMENTS
287
The nuclear polarization contribution (Figure 7c) to the anapole moment is given by
4.3.3. NUCLEAR POLARIZATION CONTRIBUTION
37.
a;"'
where is generated from inserting the ordinary electromagnetic current operator into Equation 9, I Z ) is a ground state of good parity, VPNc is the PNC NN interaction of Equation 32, and the sum extends over a complete set of nuclear states n of angular momentum Z and opposite parity. The extended Siegert's theorem again determines the form of A;'"(20). Dmitriev et al. (49) have provided a series of estimates of the polarization contribution in the context of single-particle models, using a one-body effective VPNC. The models include one of uniform density and others based on the harmonic oscillator and Woods-Saxon potentials. The former yields an anapole operator that is explicitly proportional to the nuclear density-a quantity approximately constant in heavy nuclei because of nuclear saturation-and grows as A213,like the exchange-current contribution. Numerically, however, the polarization contribution is about a factor of five larger than the exchange currents; it is the dominant contribution to axial PNC in heavy nuclei. For the present analysis, we need a treatment of the polarization contribution that begins with the DDH form of VPNc. This presents two nuclear structure challenges. The first is the construction of a reasonable model for the ground state. Shell-model calculations (20,33)for '33Cshave been done in the canonical space between magic shells 50 and 82, lg7/2 - 2d512 - 1h - 3.~112- 2d3/2, simplified by the restriction of protons to the first two of these shells and neutron holes to the last three. This produces an m-scheme basis of about 200,000. Similarly, *OST1can be described as a proton hole in the orbits immediately below the Z = 82 closed shell ( 3 s l p - 2d3p - 2d5,~)coupled to two neutron holes in valence neutron space betweenmagicnumbers 126and82(3p1/2-2f5/?-3p3/2- li13,?-2,f7/2-lh0/2). The second challenge is the completion of the intermediate state sum. Direct summations and methods based on moments (20) are impractical because of the dimensions of the spaces. However, because no nonzero E 1 transitions exist among the valence orbitals, an alternative of completing the sum by closure, after replacing 1/ E,, by an average value ( 1/ E ),is quite attractive; the resulting product of the onebody operator A$,, and two-body V""' contracts to a two-body operator, so that only the two-body ground state density matrix is needed. The closure aproximation can be considered an identity if one knows the correct (1/E).In practical terms, this means demonstrating that it can be related reliably to some known quantity, with the position of the El giant resonance being the obvious candidate. A series of shell-model calculations has been completed for a series of light nuclei, where both the 1/E- and non-energy-weighted sums could be done (33). A consistent relationship was obtained between the closure energy and the mean energy of the giant dipole resonance, although the three isospin contributions to VPNCmust be
236
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HAXTON m WIEMAN
treated separately. The average excitation energies found (as fractions of the dipole energy) are 0.604&0.056(h:, h i ) , 0.899 f 0.090 (fir), and 1.28 +0.14(h;). The larger (1/E) for h: and h$ enhances these contributions to the anapole polarizability. The assumption that these relations also hold (48) in the heavier nuclei 133Cs and 'Tl then fixes the appropriate average excitation energies for these nuclei. From the shell-model two-body density matrix, the contracted two-body effective operator that results from the closure approximation (this includes the effects of a short-range correlation function on the two-nucleon matrix elements of VPNC), and these excitation energies, one obtains the polarization results in Table 3. Some situations4hance ground-state panty doublets or nuclear octupole deformation---can lead to enhanced polarizabilities (48). Such enhancement is unexpected in 133Csbecause the nucleus is approximately spherical.
5. WEAK COUPLING CONSTRAINTS Table 3 gives the matrix element of dl, as an expansion in terms of the DDH weak couplings. Equation 12 provides a corresponding expression for ~ , , , ~ ~ which l ~ , can be compared to the experimental constraints (Equation 28). The constraints from 133Csand thallium supplement those available from experiments in 5 p scattering and in nuclear systems. Rather precise measurements of the longitudinal analyzing power A, for p' p have been made at 13.6 and 45 MeV, and a preliminary result at 221 MeV (where only p exchange contributes) is now available. AZhas also been measured for @ a at 46 MeV. There are also two important constraints from nuclei in which observables associated with nearly degenerate parity doublets have been measured. In each case, the nuclear matrix element involved in the mixing has been determined from axial-charge B-decay (14,47), so that little nuclear structure uncertainty remains. The observables are the circular polarization P , of the y-ray emitted in the decay of the 1081-keV state in I8F and angular asymmetry A, for the decay of the 110-keV state in polarized I9F. Table 4 and Figure 8 summarize the PNC constraints. Although the PNC parameter space is six-dimensional, two coupling constant combinations, f r r - 0.12hL 0.18hk and h: 0.7hi, dominate the observables, as Table 4 illustrates. The la error bands of Figure 8 are generated from the experimental uncertainties, broadened somewhat by allowing uncorrelated variations in the parameters in the last four columns of Table 4 over the DDH broad reasonable ranges. Note that only a fraction of the region allowed by the Seattle thallium constraint is shown: The total width of the thallium band is an order of magnitude broader than the width of the cesium allowed band, with most of the thallium allowed region lying outside the DDH reasonable ranges (i.e., in the region of negative frr and positive h: 0.7hi). The corresponding Oxford thallium band, which is not illustrated, includes almost all of the parameter space in Figure 8, as well as a substantial region (to the lower left of Figure 8) outside the bounds of the figure.
+
+
+
+
+
237
289
ANAPOLE MOMENTS
TABLE 4 PNC observables and corresponding theoretical predictions, decomposed into the designated weak-coupling combinations, with fx = fx - 0.12hL - 0.18hk and Lo = h: + 0.7h: Observable
f,
Exp. ( x 10')
h0
h:,
h:
h:,
h:
* 0.21
0.043
0.043
0.017
0.009
0.039
-1.57 & 0.23
0.079
0.079
0.032
0.018
0.073
-0.93
prelim.
-0.030
-3.34 & 0.93 1200 & 3860 -740
zk
190
800 f 140 370
* 390
-0.340
0.140
60.7 -18.0
-0.012
0.021
0.006
-0.039
34.1 -15.8 3.8
-1.1 3.4 -1.8
-0,002 -44
34
4385 -94.2
-0.030
-4.5 I .o
0.4
-0.3
0.1
-0. I 6.1 -2.0
The weak coupling ranges covered by Figure 8 correspond roughly to the DDH broad reasonable ranges. Thus, the anapole constraints are not inconsistent with the most general theory constraints. However, in detail, the pattern is disconcerting. Before the anapole results are included, the indicated solution is a small f n and an isoscalar coupling that is somewhat larger than, but consistent with, the DDH best value, -(h: 0.7hL)cf,H -12.7. The anapole results agree poorly with the indicated solution, as well as with each other. Although the Seattle thallium measurement is consistent with zero, it favors a positive anapole moment, whereas the theory prediction is decidedly negative, given existing PNC constraints. The cesium result tests a combination of PNC couplings quite similar to those measured in A,(I9F) and in A"! but favors larger values. The results plotted in Figure 8 for 133Csare consistent with those of Flambaum & Murray (29), who extracted from the anapole moment an f n 9 about twice the DDH best value, f::: 4.6, and pointed out that theory can accommodate this. However, this ignores (The DDH reasonable range is 0.0-1 1.4, in units of P,("F), a measurement that has been performed by five groups. The resulting constraint is almost devoid of theoretical uncertainty:
+
-
-
-0.6
fn - 0 . l l h ; - 0.19hL
5 1.2.
38.
Allowing h; and hk to vary throughout their DDH reasonable ranges, one finds -1 .O 5 fn 5 1.1, clearly ruling out fir 9. Figure 8 illustrates this, as well as the additional tension between cesium, p a,andA,(I9F).
-
+
6. OUTLOOK The first conclusion one would draw from Figure 8 is that additional experimental constraints would be helpful. In the case of nuclear experiments, the situation has been essentially static for the past 15 years, apart from the @ + p AZmeasurement at
238
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WIEMAN
HAXTON.
30 25 20 0 -
s315 r’: 0
+
0
Q
10
”, 5
0 -5 u
-2
0
2
4
6
8 1
12
10
f, - 0.12 h, - 0.18 h,
14
1
Figure 8 Constraints on the PNC meson couplings ( x 10’) that follow from the results in Table 4. The error bands are one standard deviation. The illustrated region contains all of the DDH reasonable ranges for the indicated parameters.
221 MeV. However, in the next few years, results are expected from experiments on the PNC spin rotation of polarized slow neutrons in liquid helium (B. Heckel, pnvate communication) and on A , in n p -+ d + y (50). Thus, we could soon have the information to test whether the small-f, solution favored by the 3 p and nuclear experiments is correct. This is the solution satisfying the nuclear constraints. The anapole situation is less satisfactory: Even apart from the question of consistency with the nuclear constraints, there is some tension between the 133Cs and Seattle thallium allowed regions, though Figure 8 (which omits the bulk of the thallium allowed region) tends to exaggerate this disagreement. (If the thallium band were enlarged to 2a,it would encompass all of the illustrated region, including the cesium band.) While the Seattle thallium result favors a sign opposite theory, its broad error bar allows either sign. Taken together, the Seattle and Oxford results are not incompatible with any choice of coupling constants within
+
+
239
ANAPOLE MOMENTS
291
the “reasonable ranges.” An improved measurement in thallium would clearly be helpful, as would any new anapole measurement involving an odd-neutron nucleus, which would produce a band in Figure 8 roughly perpendicular to that for cesium. There is an atomic PNC effort under way using dysprosium, which has two abundant odd-neutron isotopes, 161Dyand 163Dy(3 1). Another possibility may come from a new atomic PNC technique using a single trapped Baf ion; the much larger coherence times and field intensities possible with this method compensate for the sensitivity loss stemming from the use of a single atom. (The statistical accuracy of traditional methods goes as f i ,where N is the number of atoms.) Barium has two stable odd-neutron isotopes, ‘35Ba and ‘37Ba. As with cesium, these attempts will have to reach the sub-1% level of precision in order to permit an extraction of the differential effects due to nuclear spin dependence. Alternatively, other methods, such as proposals to measure the anapole moment directly through El/Ml interference in hyperfine transitions ( 5l), could be developed. The underlying issues are not limited to atomic PNC measurements. Our understanding of V ( e )- A ( N ) interactions affects the interpretation of electron-nucleus PNC scattering experiments such as SAMPLE (52), where a similar discrepancy between theory and experiment may originate from an incomplete treatment of PNC effects in the nuclear target. Figure 8 shows that the jump in precision achieved in the 133Csexperiment has now provided a constraint on hadronic PNC that is comparable in accuracy to the best of the nuclear constraints. This is the reason the lack of overlap in the resulting bands is a concern. The nuclear physics required to analyze the 133Csresult is nontrivial (53) and could yet prove to be the source of the discrepancies apparent in Figure 8. The shell-model calculations described in Section 4 depend on a relation between the closure energy and the giant dipole energy that has been tested only in light nuclei. It is possible that the systematics for neutron-rich nuclei might be different. It is also generally appreciated that spin-dependent operators tend to be quenched as model spaces are enlarged to encompass more realistic correlations. Indeed, phenomenological single-particle treatments of anapole moments have previously invoked phenomenological quenching factors (38). The shell-model results are quenched relative to single-particle estimates and would probably be further weakened if those calculations were enlarged. Likewise, core polarization results from the RPA study of Dmitriev & Telitsin exhibit substantial quenching relative to single-particle estimates (34). All of these results argue that more realistic treatments of correlations will tend to reduce matrix element values, thus requiring still larger values of the weak coupling constants to fit the experimental results. Despite this possibility, it is clear that the degree of effort required to complete the ‘33Csexperiment obligates theorists to invest similar effort in improving their calculations. It is unfortunate that the results in Table 3 come from model calculations. With the exception of recent effective field theory treatments, which so far are limited to the deuteron (53),it is difficult to come up with any strategy for quantifying possible errors other than the comparisons that become possible when still more ambitious anapole moment calculations are eventually performed.
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ACKNOWLEDGMENTS This work was supported in part by the US Department of Energy and by the National Science Foundation. WH thanks the Miller and Guggenheim Foundations for support they provided. Visit the Annual Reviews home page at www.AnnualReviews.org
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16. Pospelov M, Ritz A. Phys. Rev. D 63: 073015 (2001) 17. Kayser B. Phys. Rev. D 26:1662 (1982) and 30:1023 (1984); Nieves JF. Phys. Rev. D 26:3152 (1982); Dubovik VM, Kuznetzov VE. Int. J. Mod. Phys. A 135257 (1998) 18. Siegert AJF. Phys. Rev. 52:787 (1937) 19. Friar JL, Fallieros S . Phys. Rev. C 29: 1654 (1984); Friar JL, Haxton WC. Phys. Rev. C 31:2027 (1985) 20. Haxton WC, Henley EM, Musolf MJ. Phys. Rev. Lett. 63:949 (1989); Haxton WC. Science 275:1753 (1997) 21. Khriplovich IB. Parity Nonconservation in Atomic Phenomena. Gordon & Breach: Philadelphia (199 1) 22. Musolf MJ. Electrooweak Corrections to Low-Energy Parity-Violating Neutral Current Interactions. PhD thesis, Princeton Univ. (1989) 23 Henley EM, Huffman A, Yu D. Phys. Rev. D 7:943 (1973) 24 Bouchiat M-A, Bouchiat C. Phys. Lett. 48B: 111 (1974); Barkov LM, Zolotorev M. JETP Lett. 27:357 (1978); Conti R, et al. Phys. Rev. Lett. 42:343 (1979) 25. Gilbert SL, Noecker MC, Watts RN, Wieman CE. Phys. Rev. Lett. 55:2680 (1985); Gilbert SL, Wieman CE. Phys. Rev. A 34:792 (1986) 26. Masterson BP, Tanner C, Patrick H, Wieman CE. Phys. Rev. A 47:2139 (1993) 27. Wieman CE, et al. Frontiers in Laser Spectroscopy, ed. THansch, M Inguscio, p. 243. Amsterdam: North-Holland (1994); Drever RWP, et al. Appl. Phys. B31:97 (1983)
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ANAPOLE MOMENTS 28. Wieman CE, Holberg L. Rev. Sci. Instrum. 62:l (1991) 29. Flambaum VV, Murray DW. Phys. Rev. C 56:1641 (1997) 30. Dzuba VA, Flambaum VV, Silvestrov PG, Sushkov OP. J . Phys. B20:3297 (1987); Kraftmakher AYa. Phys. Lett. A132:167 (1988) 31. Budker D. Physics Beyond the Standard Model, ed. P Herczeg, CM Hoffman, HV Klapdor-Kleingrothaus, p. 41 8. Singapore: World Sci. (1998) 32. Fortson EN. Phys. Rev. Lett. 70:2383 (1993) 33. Haxton WC, Liu CP, Ramsey-Musolf MJ. nucl-th/0101018 and to be published in Phys. Rev. Lett. (2001) 34. Dmitriev VF, Telitsin VB. Nucl. Phys. A674: 168 (2000) 35. Auerbach N, Brown BA. nuclLth/9903032 36. Musolf MJ, Holstein BR. Phys. Lett. B242:461 (1990) andPhys. Rev. D 43:2956 (1991); Zhu S-L, Puglia SJ, Holstein BR, Ramsey-Musolf MJ. Phys. Rev. D 62:033008 (2000); Maekawa CM, Veiga JS, van Kolck U. Phys. Lett. B488:167 (2000) 37. Flambaum VV, Khriplovich IB. Sov.Phys. JETP 62:872 (1985); Kozlov MG. Phys. Lett. A130:426 (1988) 38. Bouchiat C, Piketty CA. Z. Phys. C 49:91 (1991) and Phys. Lett. B269:195 (1991) 39. Barton G. Nuovo Cim.19512 (1961) 40. Dai J, Savage MJ, Liu J, Springer R. Phys. Lett. B271:403 (1991) 41. Balzer R, el al. Phys. Rev. C 30: 1409 (1984) and Phys. Rev. Lett. 44:699 (1980) 42. Potter JM, et al. Phys. Rev. Lett. 33:1307
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Pursuing Fundamental Physics with Novel Laser Technology Carl E. Wieman
One theme that has run through Theodor Hansch’s career has been his development of new laser technology that he then uses to explore interesting fundamental physics. In this paper I would like to discuss a little about how I got started following this approach to doing physics as Theodor’s student, and then how it led me to go on to measure parity violation in atoms to test the current theories of elementary particle physics. As an undergraduate at MIT, I read this paper about a new design of laser that produced intense very narrowband tunable light. Someone named Hansch, whom I had never heard of, had written it. I still remember being quite excited by the thought that now one could get enough photons of any color in a nasrow bandwidth to saturate virtually any visible atomic transition. Even as an undergraduate I realized that this represented a tremendous advance, and that there must be lots of interesting physics that could be done with such a tool. This sentiment was a major factor in my ending up at Stanford a few years later with Theodor Hansch as my Ph.D. advisor. Although I did not realize it at the time, I was one of his first graduate students (the second after Siu Au Lee?), and I started working with him on the effort to observe the 1s-2s two-photon transition in hydrogen. Theodor Hiinsch had already used his narrowband laser to demonstrate Doppler-free saturated absorption spectroscopy in hydrogen, and thereby greatly improved the resolution of hydrogen spectroscopy and the determination of the Rydberg constant. Now he was working on a much more challenging and audacious goal: the observation of the very exotic (at that time) process of a two-photon transition between the simplest two levels in hydrogen. This was the only atom where spectroscopy allowed one to readily test the fundamental theory of quantum electrodynamics. So it was exciting to be a part of this effort. In the same spirit as most of his other work, the success of this experiment was entirely dependent on the invention of new and better laser technology. So in the early days we were working on pulsed oscillators followed by multiple amplifier stages and frequency doubling in very aggravating unreliable crystals. Like many things Theodor Hansch developed, most of this multistage amplifier technology is now standard stuff, but then it had not yet been invented and there was lots of work and trial and error necessary to get it all to function properly. Although I assume that we must have actually worked during the day at some times, all my memories of that period are only of working with Theodor very
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Carl E. Wieman
late at night. He always seemed to be at his happiest and most enthusiastic when he was working in the lab, particularly if it was about midnight. Eventually, this new narrowband high-power laser technology was working well, and the doubling crystal survived barely long enough for us to see the 1s-2s transition and get what, at the time, we thought were impressively narrow lines. Over the next few years, we then extended this work to higher precision by developing more laser technology - a CW single-mode blue dye laser that went through the pulsed amplifiers. The CW blue dye laser may have been a technology that was a bit ahead of its time. I had to mix up the dye in five gallon batches because it wore out so quickly, and we had to replace the large-frame UV argon-ion laser tubes two or three times a month for a year while they perfected the window technology. Fortunately the SpectraPhysics factory was only about a mile away and all the tubes were free. It must have cost the company a fortune, but they were very anxious to have all those visitors to Hansch’s lab see that he was using a Spectra-Physics laser (even if it did not work very well). By the time I finished graduate school, we had obtained hydrogen spectra that were good enough that they were able to confirm a previously untested QED contribution to the hydrogen energy levels. For me, this really proved that the Hbsch approach of building better laser tools to do fundamental physics was both a lot of fun, and really did work. Of course, Theodor has pursued this up to the present time in developing ever-better laser technology in order to push the precision of the hydrogen 1s-2s transition to the almost ridiculous levels of precision that he has now obtained. This has reached resolutions and has tested QED to a level that, when I was working on these experiments, I would never have believed possible. After graduate school, I continued in this spirit and went looking for problems where advances in laser technology could allow one to study basic physics questions. The unification of the weak and electromagnetic interactions that had been predicted shortly before seemed exactly such a problem. The Bouchiats [l]had pointed out that if this theory of unification was correct, it would result in a tiny amount of parity violation (PV) in atoms that, in principle, could be detected using atomic spectroscopy. Rather than trying to further test QED, this offered the exciting potential of testing the structure of atoms “beyond QED”. Having learned from Theodor Hansch that building better lasers and niftier laser technology could solve any atomic spectroscopy problem, I applied that philosophy to this challenging problem. Ultimately it ended up proving successful, although the lessons I had learned as a student about lots of late nights being necessary to get hard experiments to work also turned out to be very important! In parity violation measurements in atoms, one is doing sensitive measurements of atomic line strengths, rather than precision measurements of line centers as in the hydrogen work. However, the same fundamental feature of laser light is important for the success of both, namely the ability to get a great many photons that one can control to be at
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Pursuing Fundamental Physics with Novel Laser Technology
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exactly the desired frequency. Another similarity to the measurement of the 1s-2s transition (although not quite as bad) was that in both cases many years elapsed in the pursuit of ever-higher precision. The parity violation work [2,3] lasted nearly two decades and involved numerous students and postdocs. The primary motivation of these parity violation experiments was to test what is now known as the standard model of electroweak unification. Initially, the interest was just to see whether parity was violated in atoms, but later it became clear that it was interesting to measure this violation as precisely as possible. Although we learned over the years that the standard model worked very well, there have always been compelling reasons to think that it remains an incomplete description of nature. So there have been many extensions or “improvements” to the standard model proposed. Much of elementary particle physics research over the past two decades has been aimed at finding evidence of a deviation from the predictions of the standard model that matches the predictions of one of these extensions. In most cases the “new” physics would manifest itself as a small change in the experimental observable, and so the more precise the measurement in either atoms or at high energy, the more sensitive the test. Atomic parity violation measurements are uniquely sensitive to several types of possible new physics such as extra Z bosons, leptoquarks, or compositeness to the Fermions. A secondary motivation for this work grew out of a discovery that was made in our second-generation experiment that managed to measure atomic P V to a few percent uncertainty. We found that there was a small component to atomic parity violation that depends on the nuclear spin: a sort of hyperfine structure to atomic PV. This component of atomic parity violation is called the “anapole moment” contribution. Although we were unaware of it at the time, the parity-violating interactions between hadrons had been predicted to give such an effect. (Because we were unaware of this, we tried very hard to make it go away, but it stubbornly persisted.) Our later more precise measurements of the anapole moment effect gave a unique and important probe of parity violation in purely hadronic nuclear interactions, for which there are very few good measurements. Although the details are complicated, the basic concepts behind PV in atoms and its measurement are quite simple. The neutral current interaction arising from the exchange of a Z boson between the electrons and the nucleons in an atom leads to a mixing of the S and P states in the atoms. The amount of mixing scales with the atomic number (2)as Z 3 , but even for large 2 it is only about one part in lo1’, which is painfully small! This dependence forced me to abandon my beloved hydrogen, the atom that I had come to know and love as a graduate student, and turn to cesium. (After a serious 20 year relationship with cesium, I recently dumped it to devote myself to yet a third atom, rubidium. I feel downright fickle when compared with Theodor Hansch, who has never wavered in his long attachment to hydrogen!) This parity-violating mixing is detected by observing the electric dipole transition
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amplitude between two S (or two P ) states in an atom. In the case of cesium it is the 6 s to 7s transition at 540 nm. Such an E l amplitude can only exist if parity is violated. Because the amplitude is so small, it can only be observed by using interference techniques. In such a technique the nS to n’S transition rate is given by
where A,, is the desired PV amplitude and is proportional to the S-P mixing, and A0 is a larger parity conserving amplitude. Because the interference term is linear in A,, it can be large enough to measure; however, it must be distinguished from the large background due to the A; contribution. For there to be a nonzero interference term, the experiment must have an inherent “handedness”, and if the handedness is reversed, the interference term will change sign, and can thereby be distinguished. In our experiments the A0 is a “Stark-induced” E l amplitude due to an applied DC electric field. This field mixes S and P states giving rise to an El amplitude that can interfere with A,, under the proper conditions. The handedness of the experiment arises from the combination of orthogonal electric, magnetic, and laser fields that define a handed coordinate system of the apparatus. The Stark-induced 6s-7s transition is excited in an intense cesium atomic beam by a 540 nm beam from a dye laser that intersects the atomic beam at right angles. The PV interference term changes sign, causing a modulation in the excitation rate, as this coordinate system is reversed back and forth between right- and left-handed. There are four such reversals: changing the sign of each of the E and B fields, reversing the laser light from left to right circular polarization, and reversing the sign of the rn level that is being excited. This use of multiple redundant reversals greatly suppresses possible systematic errors. Although it is simple in principle, there are many technical challenges in achieving an adequate signal-to-noise ratio for the measurements, and the elimination of potential systematic errors is even more difficult. It is hard to convey how difficult and time-consuming this and similar PV experiments are to those who have not worked on such projects. To provide a little perspective, I can compare the third-generation PV experiment that ended in 1998 with the progress in my laser cooling and trapping program. The laser trapping and cooling program grew out of our efforts to develop diode laser technology that was the beginning of our third-generation PV experiment, and from then on both programs proceeded in parallel. Both of these involved roughly the same number and quality of people and monetary support (although the parity-violation group tended to work somewhat harder, and for the last several years the cooling and trapping group had Eric Cornell). Over the subsequent 1.5 decades, the cooling and trapping group produced 18 Physics Review Letters articles, as well as many other publications on a variety of topics including the attainment of BoseEinstein condensation (BEC), while the parity violation group was carrying
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out a single measurement! This somewhat illustrates the difference between PV measurements and “normal” atomic physics. The 6s and 7 s levels in cesium are each split into F = 3 and F = 4 hyperfine levels. We drive transitions from a single m state of either the F = 3 ground state up to a single m state of the F = 4 excited state, or the 6S4 to 753 transition. In the first two generations of the experiment we applied a magnetic field that was sufficient to allow us to resolve the individual m states, and then used the frequency of the dye laser to select which level was excited. In the third generation, we improved the signal size by optically pumping all of the atoms into the m state that we were exciting, rather than using the 1/16 of the atoms that normally populates a single m state. This optical pumping also allowed us to more efficiently detect the 7 s excitation using a technique similar to the ‘‘atom shelving” used in ion trapping. About half of the time the 7 s atoms decayed back down into the empty 6 s F level. We excited atoms that refilled this “empty” F level on a cycling transition to the P state, and therefore scattered and detected many (200) photons for each 6s-7s excitation. These changes improved the signal size and detection efficiency, but at a cost of requiring four additional high-performance lasers. Three of these were used in the optical pumping and detection and the fourth was needed to measure the exact degree of optical pumping for calibration. There are a number of aspects of this apparatus that have involved extensive development of laser technology, as I alluded to at the beginning of this article. First, it requires five lasers (four diode, one dye) that must be stabilized to nearly state-of-the-art performance. This means frequency stabilities on the order of one part in 1014/s1/2 and intensity stabilities of one part in ~ O ‘ / S ~ / ~ Getting . this level of performance in a dye laser operating in the green at high power (400 mW) was difficult, but was helped considerably by the expert help of Jan Hall just down the hall. More surprises and more work were involved in getting all those diode lasers to work. This required quite a lot of development work, as we discovered that simply controlling the central frequency of the diode laser to this kind of stability was not nearly good enough; we also had to narrow up the linewidth. That required a fairly extensive program to figure out how to combine optical and electronic feedback to get well-behaved narrowband very stable diode lasers. That technology provided a much cheaper and simpler laser source for cooling and trapping atoms. That was what got me interested in that field and ultimately resulted in our obtaining BEC many years later. One of the biggest technological advances that made the PV experiment possible traces its origins back to my student days. This is the power-buildup cavity, a resonant Fabry-Perot cavity that is used to build up the optical power in order to enhance the excitation of the weak 6s-7s transition. In my work with Theodor Hansch, we used a power-buildup cavity to make the power a few times higher. In the PV work we were exciting an extremely weak transition and so needed as much narrowband power at 540 nm as possible.
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That led us to vigorously pursue power buildup technology. Over the years, we progressed from buildup factors of a few hundred up to nearly 100 000. However, we did eventually find that it was possible to get too much power, because it started to ionize the atoms and distort the mirror coatings. So we backed off a little (to 30 000) for most of the data. With each increase in buildup, concomitant increases in laser-frequency stability and cavity-mirror stability were required. Our final cavity had the spacing between the mirrors held constant to better than 1/100 of an atomic diameter per sl/*. Actually, that was not the hardest part. The hardest part was that this stability had to be achieved while holding the mirrors extremely gently to avoid inducing tiny amounts of birefringence that could give a systematic error. So quite a few years were involved developing the dye laser, diode lasers, power-buildup cavity, and a number of other technologies needed for this experiment. Meeting any one of this list of technical requirements is not easy, but by far the most difficult aspect of the technology for this experiment was that all these things had to work at the same time, and they had to do that for hours, days, and months in order to acquire the necessary statistics. After achieving this necessary level of performance from the apparatus, we then looked at the signal representing the 6s-7s excitation rate as we performed the five parity reversals ( E , B , polarization, and there are two ways to reverse m in the optical pumping process). The quantity of primary interest is the fraction (about 6 parts in lo6) of the signal that modulates with all five reversals. However, there are many other modulation channels that we monitored to measure the modulation with different subsets of these five reversals. These channels provided a great deal of information about the experiment; for example, the size of many stray nonreversing and misaligned field components. This information is essential for dealing with possible systematic errors. Although a lot of time was required to develop the laser technology, probably as much time and more suffering was involved in the study and elimination of systematic errors. This got worse with each generation of the experiment as the target precision improved. For the final generation that achieved a fractional uncertainty of 0.3%, dealing with the systematics required about 7000 hours of data, or roughly 95%, of that acquired. The approximately five years spent on this was a lot less fun than developing the technology! The general approach that was used was that first we calculated all the possible ways that combinations of stray (nonreversing) and misaligned electric and magnetic fields (either oscillating or DC fields) could mimic the PV signal. This involved looking at all field components. Then we found a way to measure and control all the relevant field components to the necessary level. One of the painful aspects of this experiment was that it was necessary to consider all the possible combinations of gradients of these fields as well as the fields themselves. The measurement and control of all these fields and gradients was done by the combination of appropriate construction
248
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of the apparatus, monitoring 32 different modulation channels, and using 31 servosystems t o stabilize everything about the experiment. Many of the modulation channels needed to come directly out of the same data as was used for the parity measurement, but there were several others that came from a set of auxiliary experiments that were interspersed with the acquisition of the P V data. The second stage in eliminating systematic errors was based on extensive statistical tests. Basically, these were quantitative ways t o decide whether anything about the experiment was changing on any timescale by more than what one would expect just due t o the shot noise. Ultimately we reduced the systematic uncertainty t o the 0.1% level. The final result for the parity-violating amplitude on the two different hyperfine lines is: Im(El,,/P)
= 1.6349(80) mV/cm
1.5576(77) mV/cm
for F = 4 to F’ = 3 for F = 3 to F’ = 4 .
(2)
Since we are detecting the P V amplitude as a fractional modulation in an electric field-induced rate, it is given in units of the equivalent electric field required to give the same mixing of S and P states. It was a relief to see that the results of all of our three generations of this experiment agreed with each other, as well as with the earlier less-precise measurement of Bouchiat. It was also exciting t o see that the tantalizing hint of a dependence of the P V on hyperfine transition (the nuclear spin dependence) that was seen in our 1988 result and hotly debated was confirmed by the later work. Our final result is that this difference is 0.077 mV/cm, with a fractional uncertainty of 14%. This is primarily due to the nuclear anapole moment. This is the first (and still only) measurement of an anapole moment, some forty years after its existence was first proposed by Zeldovich. It has provided new information on parity violation in hadronic interactions that does not agree very well with the previous theoretical analysis and very limited experimental data. This has resulted in a re-examination of this entire subject and has stimulated considerable new work. If we take an appropriately weighted average of the two lines, all of the nuclear-dependent parts cancel out, and we are left with a quantity with a fractional uncertainty of 0.3% that can be used to compare t o the predictions of the standard model. That comparison is complicated by the need for atomic structure calculations that relate the measured P V value to the fundamental electron-quark coupling constants of the standard model. When our experimental result was first published, the best calculations, made nearly a decade earlier, indicated that we were lower than the Standard-model (SM) prediction by about the uncertainty of those atomic calculations, which was stated to be 1%. However, this uncertainty was based on the comparison of the calculated and measured values for many other properties of the lowlying states of cesium, such as hyperfine splittings and oscillator strengths. The calculations of those quantities are quite similar to that of the parity
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Carl E. Wieman
violation and were done using the same highly sophisticated techniques. After the completion of the 1998 PV measurement, we went back and checked some of those experimental numbers and carried out an updated analysis [3] of the theoretical uncertainty using the improved experimental numbers obtained by ourselves and others. Remarkably enough, it turned out that in every case where there was an improved experimental measurement, the agreement with these sophisticated atomic structure calculations also improved. So the best estimate of the atomic calculation error based on this updated comparison of measured and calculated observables was 0.4%, rather than 1% [3]. At that level there was then a 2.5 sigma discrepancy between the atomic PV results and the standard-model prediction. Naturally, this attracted considerable attention. On the elementary-particle side it was pointed out how this could be consistent with the existence of an additional high-mass Z boson of a certain type that would also fix up a smaller standard-model difference in the results of a high-energy experiment. On the atomic side it resulted in renewed attention to the atomic calculations. A number of possible problems with the calculations were checked and eliminated. Then a bright young theorist discovered that a correction that had previously been thought to be insignificant, the second-order Breit correction, was surprisingly large and reduced the discrepancy with the standard model down to a modest 1.5 sigma, if one used the same uncertainty [4].However, this same Breit correction also considerably improved the comparison between other calculated and measured properties of cesium, and hence arguably should have reduced the uncertainty. In the process of a couple of other groups checking and confirming the accuracy of this calculation of the Breit correction, it was realized that there is also a nonnegligible vacuum polarization correction that was neglected. This has very recently been calculated and found to shift the result by 0.4% away from the standard-model value, so that there is again a larger than two sigma discrepancy. It is a tantalizing situation that whispers of a possible problem with the standard model, but cries out for further work on the atomic theory and its experimental confirmation at the 0.1% level. In any case, at worst this is confirming the standard model at the fraction of a percent level in atoms, and thereby setting the best available constraints on a variety of possible new physics such as extra Z bosons, leptoquarks, and composite fermions. At best, it is providing us with indications of new physics beyond the standard model. This work has epitomized the Hansch approach to physics of developing novel laser technology that allows one to probe fundamental physics through atomic spectroscopy.
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References 1. M.A. Bouchiat, C. Bouchiat, Phys. Lett. 48B,111 (1974a), J. Physique 35,899 (1974b), J. Physique 36,493 (1975) 2. S.L. Gilbert, M.C. Noecker, R.N. Watts, C.E. Wieman, Phys. Rev. Lett. 55, 2680 (1985); M.C. Noecker, B.P. Masterson, C.E. Wieman, Phys. Rev. Lett. 61, 310 (1988); C.S. Wood, S.C. Bennett, D. Cho, B.P. Masterson, J.L. Roberto, C.E. Tanner, C.E. Wieman, Science 275, 1759 (1997) 3. S.C. Bennett, C.E. Wieman, Phys. Rev. Lett. 82,2484 (1999) 4. A. Derevianko, Phys. Rev. Lett. 85, 1618 (2000)
Laser Cooling and Trapping
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This began as a simple experiment to show that one could slow atoms using very cheap diode lasers. It led to many years of work studying the behavior of very cold atoms and exploring how far one could go in controlling their speeds and positions. Thus it was an interwoven mixture of science and technology. Science in understanding the novel physics involved in the behavior of atoms at these very low temperatures, technology in finding out new, simpler, cheaper, and better ways to trap and cool the atoms, often guided by the physics we were learning. In some respects, I think the development of the diode laser for atom trapping and cooling and the vapor cell MOT, may have been my most influential work, in that these developments so dramatically reduced the cost and complexity required to carry out state of the art laser cooling and trapping. It opened the field up to many more people and allowed it to expand in far more directions than would otherwise have been possible.
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Stopping Atoms with Diode Lasers" R.N. H'a t t s and C.E.Wieman t Joint lnstit ute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, CO 80309, USA
We have succeeded in stopping a beam of cesium atoms usinn freauencychirped diode lasers. We scan over the Doppler profile of a 100°C thermal cesium beam to bring more than 1O1O atomsfs to a temperature of I0K, a limit imposed by the 30 HAz linewidth of our free-running lasers. These results are preliminary.and we expect that further work will provide suhstantial improvements in our final temperature and density. This is an extremely simple and inexpensive way to produce cold atoms.
Two techniques have been devised for using laser light to stop atoms. The NBS group in GaithersburR [ l ] used a single freauency C.W. dye laser combined with a large tapered uolenoid to achieve the necessary condition that the atomic transition and the laser freouency stay in resonance as the atoms slow. Hall and co-workers [2] used an alternative method.in which the frequency of the dye laser was swept (chirped) using state-ofthe-art electro-optic modulators. While both of these approaches have been shown to work well, they involve large investments of money and equipment. We have found that the frequency chirp approach can he implemented using inexpensive diode lasers and simple electronics. The frequency of a diode laser can be smoothly and rapidly varied over many GHz simply by varying the injection current. Thus, by using an appropriate current ramp, we have stopped a beau of cesium atoms. A schematic of the apparatus i s shown in Fig. 1. Cesium atoms In a 100°C oven effuse from a 0.5 w hole and are collimated to 8 mrad. At this oven temperature, the heam haa a mean velocity of 2.7 x 104 cmfs, an intensity of 3 x l0l1/s and a Doppler PWHn of 400 PrAz. The atoms are stopped by bombarding them with resonant counterpropagating 6 s - 6 ~ 3 / 2photons at 8521 A. Each atom-photon collision slowr, the atom hv 0.35 cmfs. Thus, it takes approximately 76,000 photons to bring an atom to a halt. For an excited state lifetime of 30 n s , this process takes 5 ms and requires about 70 cm. Because cesium has two hyperfine iround states SEDUrated by 9192 HAz, two lasers are used. The primary cooling is done hy one laser tuned to the 6s(F-4)-6p3/2(F-5) transition. For circularly polarized light, this transition is a Rood approximation to a leakage-free two-level system. The other laser, tuned to the 6s(F=3)-6p3/2 transition, insures that the P-3 ground state is depleted. The F-4 laser has about 5 mW of power while the F-3 laser has 0.3 mW. Both lasers have a freerunning linevfdth of 30 NXz and are focused to match the atomic beam. Each injection current ramp sweeps the frequencies of the two lasers 1 GHz in 15 ms. The last 10 ma are spent sweeping the full Doppler profile of
*Work supported by the National Science Foundation. 'Sloan Fellow; also with Department of Physics, lhiversity of Colorado.
20
255
256
DlOOE
LASER
-
400YHl
Pig. 1.
Schematic of a p p a r a t u s .
Pig. 2.
Slowed atom scans.
Curves
A-E show v a r i o u s amounts of c o o l i n n . Curve P is t h e Doppler p r o f i l e w i t h no c o o l i n n . velocity.
The arrows mark z e r o
t h e cesium beam. The end of t h e c h i r p may be a d i u s t e d t o h r i n a t h e atoms t o any d e s i r e d speed. The same l a s e r s are used t o monitor t h e r e s u l t i n g v e l o c i t y d i s t r i b u t i o n . After each c h i r p , a p o r t i o n of a much s l o w e r l i n e a r ramu ( c o r r e sponding t o a frequency change of 1.5 GHz i n 6 8 ) i s switched i n t o t h e laser i n j e c t i o n c u r r e n t s . A t t h e same t i m e . a d e t e c t o r I s gated on t o probe t h e slow atom f l u o r e s c e n c e . T h i s g a t e l a s t s f o r 250 US, a f t e r which a new c h i r p i s s t a r t e d . To g i v e a z e r o v e l o c i t y marker, a small f r a c t i o n of t h e l a s e r I s s e n t i n p e r p e n d i c u l a r t o t h e atomic beam. The r e s u l t s are ahown i n F i g . 2. The small bumps on A and R are t h e frequency marker peaks. As t h e f i g u r e shows, by a d i u s t i n g t h e end of t h e frequency sweep t o d i f f e r e n t p o i n t s on t h e Doppler p r o f i l e , we a r e a h l e t o s l o w , s t o p , o r even r e v e r s e a p o r t i o n of t h e atomic beam. The r e s i d u a l Doppler peak, which grows a8 more c o o l i n g I s done, is doe t o uncooled atoms a t t h e probe r e g i o n . The l i n e v i d t h of o u r l a s e r s c u r r e n t l y l i m i t s us t o a 1-K t e m p e r a t u r e i n t h e r e s u l t i n g s t o p p e d d i s t r i b u t i o n . References 1.
2.
J. V. Prodan e t . a l . , Phys. Rev. L e t t . 54 (1985) 992. B l a t t , J.L. hll and M. Zhu, Phys. Rev. L e t t . 54 (1985) 996.
U. E r t m e r , R.
21
Reprinted from Optics Letters, Vol. ! I , page 291, May, 1986. Copyright 0 1986 by t h e Optical Society of America a n d reprinted by permission of t h e copyright owner.
Manipulating atomic velocities using diode lasers R. N. Watts and C. E. Wieman Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colorado 80309 Received January 6.1986: accepted February 19,1986 We have used counterpropagating radiation from a diode laser to cool and stop a beam of cesium atoms. The laser frequency was chirped to keep it in resonance with the slowing atoms. The same laser was used to probe the resulting velocity distributions. We have cooled more than 1O'O atoms/sec to a temperature of 1 K. This is an extremely simple and inexpensive way to manipulate atomic velocities and has a wide range of possible applications.
There is currently considerable interest in cooling and stopping neutral atoms. Such atoms would lend themselves to a variety of interesting experiments involving ultrahigh-resolution spectroscopy and neutral atom traps.l However, the technology for producing cold atoms is still quite young. In the past year, two groups using complementary approaches have stopped beams of sodium atoms using radiation pressure. In these experiments a sodium atom scattered resonant photons from a laser beam directed against the atomic velocity. The transfer of momentum from many (2 X lo4)photons eventually brought the atom to rest. The primary experimental difficulty in implementing this idea was that as the atom slowed, the changing Doppler shift took it out of resonance with the laser light. Phillips and co-workers2 overcame this difficulty by sending the atoms through a spatially varying magnetic field that shifted the atomic transition frequency so that it remained in resonance with the laser. Hall and co-workers3 used the opposite approach of leaving the atomic frequency unchanged and sweeping (chirping) the laser frequency as a function of time to maintain the resonance. Both of these impressive pioneering experiments required substantial investments of money and effort. The Phillips approach used a stabilized single-frequency dye laser and a large tapered solenoid. The Hall approach used a comparable dye laser, a state-of-the-art electro-optic modulator, and sophisticated microwave technology. In addition, both experiments required a second similar dye laser to probe the modified velocity p r ~ f i l e . ~ Here we present a new technology for implementing the frequency chirp technique that provides stopped atoms with a vastly smaller investment of time and money. Using inexpensive commercial diode lasers and simple electronics, we have cooled and stopped a beam of cesium atoms. This is possible because the output frequency of a diode laser is easily controlled by varying the injection current. As well as making it simple to obtain the frequency chirp necessary for cooling the atoms, this allows us to use the same laser to probe the resulting velocity distribution. The probing is accomplished by jumping to frequencies corresponding to different positions on the Doppler profile after the cooling chirp is completed. This is 0146-9592/86/050291-03$2.00/0
done so rapidly that the atoms are essentially fixed in space. The simplicity, low cost, and reliability of this method give an entirely new aspect to the laser manipulation of velocities. Instead of being an exotic technique, it can now be thought of as a laboratory tool with a broad range of possible applications. For example, in most atomic-beam experiments the velocity distribution limits or obscures the results. The technique demonstrated in this Letter provides a means for producing an atomic beam with a large fraction of the atoms having any desired velocity between zero and several times the typical thermal velocity. Also, the simplicity of this technique means that it can be easily applied to many atomic-beam experiments. Stopping cesium atoms in particular is interesting because the cesium clock is the primary frequency standard. At present the velocity distribution of the atoms limits the accuracy of these clocks. The frequency-chirp technique has been discussed in Ref. 3, but we will review the basic idea and give the relevant parameters for cooling cesium atoms. The atomic beam absorbs photons from a counterpropagating laser beam, which is tuned to the 6s(F = 4)-6&(F = 5) resonance line at 852 nm. This is shown in the energy-level diagram in Fig. 1. To excite the moving atoms, the laser must be tuned below the zero-velocity resonant frequency to compensate for the Doppler shift. For a typical thermal cesium beam the most probable velocity is 2.7 X lo4 cm/sec, which gives a Doppler shift of 310 MHz. On the average, each time an atom is excited and isotropically reemits a photon, its velocity is reduced by 0.35 cmlsec along the direction of motion. Thus the atom must make about 76,000 transitions to be brought to a stop. The 31-nsec lifetime of the 6P3/2 state sets the maximum scattering rate at 1.6 X lo7 photonslsec and implies a deceleration of 5.7 X lo6 cm/sec2. As the atom slows, its Doppler shift decreases and the laser frequency must be increased to stay in resonance. The higher frequency then also excites and slows atoms having a lower initial velocity. In velocity space one can picture the laser as a wave that sweeps all atoms ahead of it toward lower velocity. For the velocity and deceleration given, the stopping time and distance are 5 msec 01986, Optical Society of America
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OPTICS LETTERS / Vol. 11,No. 5 / May 1986
much slower probing ramp (1.5 GHz in 6 sec) for 250 Ksec. The switching time is a few hundred nanoseci F 5 onds. The fluorescence detector is gated on for this I 251 MHz 250 psec, and its output goes to an X-Y plotter. By plotting this fluorescence signal as a function of the slow ramp current we produce an image of the atomic velocity distribution as shown in Fig. 4. A 100-psec I 5 1 MHz gate time was also used, and it produced identical 2 results. The frequency sweep changes the output power by -l%/GHz, but this is not a problem since the 85211 cooling transition is well saturated. The cooling is done by the 6-mW laser tuned to the F = 4-5 transition. This is predominantly a two-level system and can be cycled repeatedly. However, there F: is a small amount of leakage to the F = 3 state so that Y 4 most of the atoms are lost before they can absorb the I requisite 76,000 photons. To avoid this, the second 9193 MHz I laser is tuned to the 6S(F = 3 ) ---* 6P3/2(F = 4 ) transi3tion and quickly pumps any atoms that leak to the F = 3 state back to the F = 4. We find that only about 100 Fig. 1. Cesium energy-level diagram. fiW of power is needed to provide the necessary repumping. and 62 cm, respectively. The change in the laser frequency should therefore be 310 MHz in 5 msec, or 67 MHz/msec, in the limit of complete saturation of the transition. This is the maximum useful chirp rate, and the actual rate will be slightly smaller. This chirp is easily achieved with diode lasers. We have determined that our lasers can continuously sweep 15 GHz in 1psec, which is far beyond what is required. A schematic of the experiment is shown in Fig. 2. Cesium atoms effuse from a 0.05-cm hole in a 100°C oven and are collimated to 8-mrad full angle divergence. The intensity is about 1 X loll atoms/sec. The superimposed beams from two diode lasers are FLUORESCENCE circularly polarized and focused to overlap the cesium DETECTOR ZERO VELOCITY MARKER BEAM beam over its entire length. As discussed below, the second laser is needed to depopulate the F = 3 hyperI 6 5 crn fine ground state. A t the vacuum chamber there is 6 mW of power from the first laser and 0.6 mW from the second. A silicon photodiode monitors the fluoresVACUUM cence from the cesium beam a few centimeters from SYSTEM the end of the chamber. A fraction of the first laser beam is sent in perpendicular to the atomic beam in front of the detector. This is used to obtain a fluorescence signal marking the zero-velocity resonant freFig. 2. Schematic of apparatus. quency. To define the quantization axis a magnetic field that varies between 0.5 and 1 G is applied along the beam direction. The results are quite insensitive to the details of this field. The diode lasers are Hitachi Model HLP1400 single-frequency lasers, which have a maximum output power of 15 mW and a free-running linewidth of 35 M H Z . ~They are mounted on temperature-stabilized copper blocks, and coarse tuning of the output frequency is accomplished by varying this temperature. Fine frequency tuning is done with small adjustments of the dc injection current. A graph of the injection current for the diodes is shown in Fig. 3. On top of the TIME dominant dc level is the ramp, about 20 msec in duration, that produces the cooling chirp. Although the dc Fig. 3. Graph of laser injection current. Each 20-msec levels are different, the same ramp goes to both lasers. laser chirp is followed by a 250-psec probe period. For ease At the end of each chirp we electronically switch to a of viewing, a longer probe period is shown.
\
i
1
259 May 1986 / Vol. 11, No. 5 / OPTICS LETTERS - 35 MHz
I c
B RESIDUAL DOPPLER PEAK
\
i , , -,
-+,, 6
4
2
0-2
rn /sec (~100)
Fig. 4. Slowed atom scans. A-E show various amounts of cooling. F is the Doppler profile with no cooling. The arrows mark zero velocity.
The results are shown in Fig. 4. All the cooling chirps start at a frequency well to the left (low-frequency side) of the Doppler profile. The end frequency of the chirp is varied from 100 MHz past the center of the profile in Fig. 4A to -400 MHz past in Fig. 4D. The arrows mark the frequencies corresponding to zero velocity, which were determined by unblocking the perpendicular laser beam to obtain the small additional fluorescence peaks that can be seen in Figs. 4A and 4B. The large narrow spikes are the dramatic signatures of the slowed atoms. By maximizing the slowed atom peaks, we determined the optimum sweep rate to be 54 MHz/msec, in agreement with that predicted above. The final velocity of the atoms can be set to any desired value including the negative velocities shown in Figs. 4D and 4E. The broad Doppler peak, which increases as the cooling sweep is made longer, is due to fresh atoms that come out of the oven after the frequency chirp has passed their velocity range. A significant though not overwhelming fraction of the atoms is lost during the cooling process. One reason for this is that in scattering 7.6 X lo4 photons an atom acquires a random transverse velocity of (7.6 X 104)1/2X (0.35 cm/sec) z 1m/sec. Since the detection region is only 0.5 cm wide and the cooling takes approximately 10 msec this transverse velocity causes many of the atoms to miss the detection region. The additional decrease in peak height as a lower final velocity is produced is due to the length of our stopping region. The machine is only long enough to stop the slower half of the velocity distribution. Thus higher-velocity atoms, which are cooled as the laser
293
sweeps over that portion of the velocity profile and which give rise to the large peaks in Figs. 4A and 4B, run into the end of the chamber before they can be cooled to the lower velocities of Figs. 4C4E. The rapid drop in the peak height as we go to negative velocities occurs simply because the detector is close to the downstream end of the vacuum chamber. The fluorescence linewidth of the cooled atoms is 35 f 5 MHz. This sets an upper limit on the velocity spread of f 1 5 m/sec or a temperature of 1 K. The number of atoms stopped is fundamentally limited by the number of photons and is a few times 10lO/sec. We estimate that the resulting density of stopped atoms is roughly 106/cm3. The linewidth observed just matches that of the diode lasers and is seven times larger than the 5-MHz natural linewidth of the transition. This limitation can be overcome by using optical feedback to narrow the laser linewidth. Thus we believe that diode lasers can be used to cool cesium to the 125-pK limit (0.15 m/sec) set by the natural linewidth.6 We conclude by stressing the simplicity, reliability, and versatility of this method for manipulating atomic velocities. It can be set up with a minimum of time and effort (not to mention expense) and is then largely a matter of turning on the switch in the morning, perhaps adjusting the diode temperature slightly, and turning off the power when finished. It is straightforward to add to existing experiments and provides an easily tuned source of monoenergetic atoms. Although we have worked with cesium, diode lasers are now available that can cool rubidium and potassium as well, Thus it is ideal for use in a wide range of experiments involving spectroscopy or collisions in atomic beams. We are pleased to acknowledge useful discussions with J. Hall. This research was supported by the National Science Foundation. C. E. Wieman is also with the Department of Physics, University of Colorado, Boulder, Colorado 80309, and is a Sloan Fellow. References 1. For a collection of relevant articles see W. D. Phillips, ed., Laser Coaled and Trapped Atoms, U.S. Natl. Bur. Stand. U S . Spec. Publ. 653 (U.S. Government Printing Office, Washington, D.C., 1983). 2. J. Prodan, A. Migdall, W. D. Phillips, I. So, H. Metcalf, and J. Dalibard, Phys. Rev. Lett. 54,992 (1985). 3. W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54,996 (1985). 4. An excellent review of neutral atom cooling and trapping is W. D. Phillips, J. V. Prodan, and H. J. Metcalf, J. Opt. SOC.Am. B 2,1751 (1985). 5. These lasers are wavelength selected to lase in the range 852 f 5 nm a t 25°C and lo-mW output power. 6. S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett. 55,48 (1985).
V O L U M E 57, NUMBER3
PHYSICAL REVIEW LETTERS
21 JULY1986
Light Traps Using Spontaneous Forces D. E. Pritchard, E. L. Raab, and V. Bagnato(a) Physics Deparrment and Research Laboratory of Electronics, Massachusetts Institute of Technologv, Cambridge, Massachusetts 02139
and
C . E. Wieman and R. N. Watts Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Department of Physics, Universiw of Colorado, Boulder, Colorado 80309 (Received 28 March 1986)
We show that the optical Earnshaw theorem does not always apply to atoms and that i t is possible to confine atoms by spontaneous light forces produced by static laser beams. A necessary condition for such traps is that the atomic transition rate cannot depend only on the light intensity. We give several general approaches by which this condition can be met and present a number of specific trap designs illustrating these approaches. These traps have depths on the order of a kelvin and volumes of several cubic centimeters. PACS numbers: 32.80.Pj
Since the optical Earnshaw theorem (OET) was proven,’ it has been widely believed that it is impossible to confine atoms with static configurations of laser beams by use of only light forces associated with spontaneous emission. In this Letter we show that this theorem does not necessarily apply to atom traps and suggest general approaches for making stable “spontaneous-force’’ light traps for atoms. We also present several examples of possible traps which have cubic-centimeter volumes and depths on the order of a kelvin. These numbers are orders of magnitude larger than those predicted for other static-light-force traps and thereby open u p an entirely new range of possible applications. There are two types of radiation forces that can be used to trap neutral particles.* The first is the gradient force arising from the interaction of the induced dipole moment with the field-intensity gradient. The second is the scattering force associated with the transfer of momentum from photons to particles by the scattering of light. This latter force was used to cool beams of thermal atoms3-’ and to viscously damp a collection of already cold atoms.6 Minogin’ and Minogin and Javained proposed that a trap could be constructed using only the scattering force. However, Ashkin and Gordon showed that, in analogy with the Earnshaw theorem of electrostatics, such traps are fundamentally unstable,’ thereby discrediting these proposals and discouraging any others. The current avenues of investigation have therefore been restricted to ac spontaneous-force light traps and the relatively shallow gradient-force traps. The former type was first proposed by Ashkin’ and uses time-varying light intensities and/or frequencies to circumvent the OET in much the same way that rf ion traps overcome the traditional Earnshaw’s theorem. The key idea underlying the OET is that, in the ab310
sence of sources or sinks of radiation, the divergence of the Poynting vector of a static laser beam must be zero. Hence, if the force is proportional to the laser intensity, the force must also be divergenceless, thus ruling out the possibility of having an inward force everywhere on a closed surface. An example is shown in Fig. 1 where the Poynting vector is inward in the x-y plane at z = 0, but outward along the z axis. On axis there is no Poynting vector at z = 0 if the intensities of the right- and left-moving laser beams ( R and L ) are equal. However, the outward Poynting vector, and hence the force, increases in proportion to IzI due to the focusing and consequent increase in intensity of the laser beam traveling away from the origin. Ashkin and Gordon proved the OET for the scattering force on particles with “scalar polarizability” whose “dipole is linearly related to the field.”’ [These conditions assure that the scattering force is proportional to the Poynting vector but the word “dipole” does not imply that the gradient (also known as dipole or induced) force is involved in the present discussion.] They applied the theorem to atoms and atom traps without considering the internal degrees of freedom of the atoms. This was appropriate for traps of X
2
FIG. 1. Basic trap configuration. f/2 optics are used.
@ 1986 The American Physical Society
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the Minogin type. However, the OET does not generally apply to the spontaneous force in atoms because this force is not always proportional to the intensity, and therefore the corollary that “the scattering force by itself cannot form a trap”’ does not rule out atom traps based on spontaneous force. Note our use of the terms “scattering force” to apply to light forces that obey the OET and “spontaneous force” for the analogous forces in atoms (which do not). These two terms have been used interchangeably by previous authors. The basic point of this paper is that the internal degrees of freedom of the atom can change the proportionality constant between force and Poynting vector in a position-dependent way, and thus allow static spontaneous-force traps. Such a change can result, for example, from external fields which shift the resonant frequency or orientation of an atom, or optical pumping which changes the state of the atom. These and other ideas can be exploited to violate strict proportionality and create stable optical traps using spontaneous forces, especially if one uses multibeam arrangements. In the remainder of this paper we give three specific examples of how the sources of disproportionality mehtioned above can be used to produce stable traps. For this purpose we shall restrict out attention to the simple two-beam configuration shown in Fig. 1 . While this is almost certainly not the best practical design for a trap, it will serve to illustrate the general principles. The configuration shown in Fig. 1 is already stable in the x-y plane. For i t to be stable along the z axis and hence form a trap, we must make an atom at z > 0 absorb more strongly from the left-moving laser beam ( L ) than from the right-moving beam ( R ) even though the left-moving beam is less intense. We first show how this can be done by use of a static external field to shift the resonant frequency of the atoms. Consider a two-level atom and imagine that R is tuned below its resonant frequency while L is tuned above it. Assume also that the intensities are adjusted so that there is no force at z=O and that there is a magnetic field gradient in the ?i direction, dB,/dz > 0. As the atom moves toward positive z, its transition frequency will be Zeeman tuned toward the blue, bringing it closer into resonance with L and farther from resonance with R. Obviously if dB,/dz is sufficiently large, the increased absorption of photons from L will more than offset its decreased intensity, resulting in a net spontaneous force back toward z = 0 and consequent stability along 2. Since this configuration is already stable along f and i, it is a spontaneous lightforce trap in violation of the OET. To achieve damping of the velocity (a useful trap must have cooling in addition to stability), R should be detuned slightly below resonance ( r / 2 for maximum damping where r is the natural linewidth) and L should be
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tuned either close to resonance or several I’ above it. The velocity dependence of the force from R, which provides damping, is much greater than that from L and hence dominates and cools. For either tuning, to make the center of the trap an equilibrium point, the powers in the two beams must be different. Although real atoms have more than two levels, a spin-polarized alkali atom, such as sodium in the F,M= 2 , 2 level excited with circularly polarized (m+ light, is a sufficiently good approximation to this two-level case. A plot of acceleration versus longitudinal position and velocity for sodium in such a trap is shown in Fig. 2. Stability results from the fact that the spatial derivative of the force is negative near z = O (and for zero velocity the sign of the force changes). Damping occurs because the higher the velocity of the atom, the larger is the force which is opposing it. In the second example we demonstrate how a static field which changes the orientation of the atoms can be used to produce a stable trap. In this case the laser beams have different linear polarizations, say R polarized along i and L along 9. Application of a magnetic field of constant amplitude but changing direction can cause atoms to interact differently with the two beams provided that the atoms follow the field adiabatically. In particular, for Na atoms in the F , M = 2 , 2 state, transitions to F,M=3,3 are not excited by light polarized parallel to the axis of quantization (the local magnetic field). Consequently, if the B field is a helix that twists toward k for z > 0 and toward i for z < 0, the configuration can be made stable. If the atom moves
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FIG. 2. Accelerations felt by sodium atoms with various velocities in a light trap having frequency-discriminated counter-propagating beams and a magnetic field gradient of 4 G/cm. At z = 0, beam L, tuned r/2 to the red side of resonance, has an intensity IL =0.8/,,; beam R, tuned r/10 to the blue side of resonance, IR -0.151,,. Both L and R are
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sufficiently far in the + i direction, B will be parallel to f and the atom will absorb photons only from the (weaker) L beam. Motion in the - 2 direction will produce a similar restoring force from the R beam. Moreover, both beams can be tuned on the red side of the resonance to provide viscous damping of the velocity. Since the light is not circularly polarized, transitions to F = 3, M = 2 or 1, which destroy the polarization, may also occur unless the magnetic field causes sufficient Zeeman splitting. Approximate modeling shows that a high fraction of population can be maintained in the F,M = 2 , 2 state for appropriate B fields and laser detunings. Our final example demonstrates how optical pumping of the atoms can be used to obtain a stable trap. Assume now that the laser beams have opposite angular momentum (e.g., u + for L and C T - for R ) . Consider a transition where the F quantum number of the excited state is less than that of the ground state but not equal to zero (e.g., Cs, F = 3 I;‘ 2 , which must decay back to F - 3 ) . If atoms are exposed to two beams of different intensity, the atoms become optically pumped so that they absorb more photons from the weaker beam than from the stronger. This can be understood from the transition probabilities shown in Fig. 3. An atom in the M=O state, for z > 0 will be quickly pumped into the M = 2 or 3 sublevel by the stronger u + beam. At that point, the atom can only absorb u- photons from the other beam which is directed toward the center of the trap. In addition, since the matrix elements strongly favor M = 1 decays, the atom will continue to absorb mostly CT- photons, thus generating an average longitudinal restoring force. This peculiar intensity dependence makes it necessary to tune the laser above resonance to provide longitudinal velocity damping. This weakly heats the atoms in the transverse direction. Thus in a real trap it would be necessary to provide additional transverse damping, for example, by use of an asymmetric trap geometry or adding weak “optical molasses’16 beams along and f . Phase-space trajectories for an atom in this trap with such additional cooling beams are shown
21 JULY 1986
in Fig. 4. These show that the atoms quickly are compressed into a region which is a fraction of a millimeter in size and have residual random velocities on the order of several centimeters per second corresponding to T. A weaker version of this trap results from use of orthogonal linear polarizations. We have calculated the performance of the above traps and variations on them. While it is not the purpose of this paper to give detailed results, it seems worthwhile to give the general scale to stimulate consideration of optical traps based on the concepts of this paper. With available dye-laser powers, spontaneous-
5 0 P (cm’secl
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FIG. 3. Relative transition probabilities for the 6 S ( F 3) -6P3,1(F 2 ) transition in cesium.
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(b) FIG. 4. Phase space plots in z and x for a cesium atom of particular initial position and velocity in a light trap having polarization discriminated ( u +and IT- counterpropagating beams along the z axis and weak optical “molasses” in the x-y plane. At z 0 the trap beam characteristics are area 1
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cm2, intensity 0.5 mW/cm2, f12, and detuning r/2. The “optical-molasses” beams are plane waves with an intensity of 0.025 mW/cm2and detuning of - r/2.
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force traps such as those we have described can be constructed with 10-cm dimensions, but 1 cm is more practical with inexpensive optics. The 1-cm size has confinement forces that are a fraction ( < f ) of the unidirectional spontaneous force. Integration of this force over the size of the trap gives a depth of several kelvins. However, a particle of this energy will not necessarily be confined in the trap since Doppler shifts can cause the forces to be significantly smaller for atoms with this much energy. For example, a I - K Cs atom has a Doppler shift of 6r and will interact much more weakly with the radiation than the above calculation of the depth assumes. This trap depth is nearly 100 times deeper and 1OIs times larger than that obtained for gradient-force traps of the type recently demonstrated by Chu et The spontaneous-force optical traps we have proposed are relatively simple ones designed to illustrate general ways in which the optical Earnshaw theorem can be circumvented. More complicated geometries can probably exploit these approaches more fully, particularly ones which provide more direct cooling and confining in the x-y plane. As a simple example, the addition of weak optical molasses along 2 and $, mentioned above, will make any of these traps perform better. Also, additional beams may be needed to ensure that trapped atoms do not escape to untrapped hyperfine ground states such as exist in all alkali atoms. The particular examples we have chosen illustrate three ways the internal degrees of freedom of the atom can be exploited to trap atoms, but there are many additional possibilities. Other types of optical pumping exist, such as pumping between two different hyperfine levels. Also, external static or oscillating fields can be used in a variety of ways to affect how an atom absorbs radiation." Probably all of these can be used to produce light-force traps. Finally, it may be possible to avoid the restrictions of the optical Earnshaw theorem by use of saturation' or absorption. The latter is particularly attractive: An optically dense cloud of Na will experience a maximum spontaneous N/m2 (corresponding to radiative pressure of 5 mW/cm2), enough to contain an atom density of 5 x lOI3/cm3 at T-0.25 mK according to the perfectgas law. Indeed, gas pressure limits the confinement density of a spontaneous-force trap, but this limit may be increased by using a weaker transition with a correspondingly lower ultimate temperature. We have pointed out ways in which spontaneous
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light forces may be used to make traps for atoms or ions. The advantages of this spontaneous-force approach as compared to other proposed neutral-particle light traps include large physical extent, reasonably deep wells, and experimental simplicity. Time-varying laser intensity or frequency is not required and the simple designs proposed are quite forgiving to optical misalignment or intensity mismatch. We hope our suggestions will remove the optical Earnshaw theorem as a practical barrier to the design of spontaneous-force light traps and will open the way to the realization of successful traps of different types. We acknowledge useful conversations with S . L. Gilbert, C. E. Tanner, and D. Z. Anderson. This work was supported by the National Science Foundation and the Office of Naval Research. One of us (C.E.W.) is an A.P. Sloan Foundation fellow.
(a)Permanent address: Instituto de Fisica e Q u h i c a de Sio Carlos da Universidad de SBo Carlos, SHo Carlos, Brazil. IA. Ashkin and J. P. Gordon, Opt. Lett. 8, 511 (1983). *See, for example, the feature issue on the mechanical effects of light, J. Opt. Soc. Am. B 2, 1707-1860 (1985). 3J. Prodan, A. Migdall, W. D. Phillips, I. So, H. Metcalf, and J. Dalibard, Phys. Rev. Lett. 54, 992 (1985). 4W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985). 5R. N. Watts and C. E. Wieman, Opt. Lett. (to be published). %. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett. 5 5 , 48 (1985). 'V. G. Minogin, Kvant. Elektron. (Moscow) 9, 505 (1982) [Sov. J. Quantum Electron. 12, 299 (1982)l. *V. G. Minogin and J. Javainen, Opt. Commun. 43, 119 (1982). 9A. Ashkin, Opt. Lett. 9, 454 (1984). I0S. Chu, J. Bjorkholm, and A. Ashkin, and A. Cable, following Letter [Phys. Rev. Lett. 57, 314 (19861. lwhile these fields do exert forces themselves on the atoms, here we are limiting our discussions to cases, such as the magnetic fields mentioned above, where such forces are many orders of magnitude weaker than the spontaneous force. Hence it is most useful to think of the fields as merely providing a control for the spontaneous force. It is possible to have hybrid traps which are produced by a combination of forces from large static fields and radiation pressure. An interesting possibility for such a trap which employs a large magnetic field for confinement and detuning has been pointed out to us by P. Could, private communication.
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Reprinted from Optics Letters, Vol. 13, page 357, May 1988. Copyright 0 1988 by the Optical Society of America and reprinted by permission of the copyright owner.
Atomic beam collimation using a laser diode with a self-locking power-buildup cavity Carol E. Tanner, Bernard P. Masterson, and Carl E. Wieman Toint Institute for Laboratory Astrophysics, Department of Physics, University of Colorado, Boulder, Colorado 80309-0440 Received January 11,1988;accepted February 18,1988 We have demonstrated a self-locking power-buildup cavity for laser diodes. This device requires only a few simple optical elements and can provide a standing wave containing as much as 1000 times the power emitted by the laser diode. With this device we have obtained an intense standing wave of tunable light that was used to collimate a cesium atomic beam. We have studied the power and frequency dependence of the beam collimation.
Experimental complications are encountered when one considers extending the work of Ref. 5 to a SBC. The first is that the laser will not lock stably to a cavity if there is more than a few percent feedback to the laser. However, one wants to couple all the laser power into the cavity, so the cavity-laser coupling must have a strong directional dependence. A second, slightly more difficult, problem is that for most uses of a power-buildup cavity one would like all the power in the lowest-order (TEMoo) mode of the cavity, but the light that is matched into this mode will have a minimum of reflection at the cavity resonance rather than the maximum needed for locking. If one is interested only in frequency control this is not a problem. As discussed in Ref. 5 , one simply tilts the interferometer relative to the incident beam to excite a higher-order mode; but this solution is unacceptable for a powerbuildup cavity. We solve both of these problems by sending the laser light through a circular polarizer into a slightly birefringent cavity. In this way, one can couple all the light into the lowest-order mode and still obtain the appropriate amount of feedback, since the only light that can return through the circular polarizer is the small amount that has been inside the interferometer and has been transformed into the opposite circular polarization. The simplest implementation of this design is a two-spherical-mirror Fabry-Perot interferometer in which one relies on the nearly universal birefringence of dielectric mirror coatings.6 For the research described below, we use a three-mirror folded cavity because it is more flexible, allowing the birefringence to be adjusted. One can thus vary the intracavity power over a larger range by attenuating the incident light and still maintain sufficient feedback for locking. Also, it is easier to change the beam-waist size in this configuration. The locking of the laser to the cavity depends on the phase as well as on the amplitude of the optical feedback, so the distance between the cavity and the laser must be controlled. If the feedback is optimized, the laser will stay locked for any phase that is not extremely close to 180 deg. Therefore, if one can tolerate very small (megahertz) gaps in the frequency coverage, it is
In the past few years a number of new techniques have been demonstrated for manipulating the position and velocity of atoms by using laser light. These techniques have provided an entirely new level of control in studying the properties of atoms and molecules. We have been working to make these exciting new capabilities widely accessible by showing how they can be implemented with simple inexpensive laser diOne particular technique that was recently demonstrated is the use of the strong dipole forces exerted by an intense standing wave to cool atoms3 and channel them into the field nodes.4 This approach relies on intense fields and thus would not have seemed well suited to the use of low-power laser diodes. However, in this Letter we introduce a new experimental tool, the self-locking power-buildup cavity (SBC), which allows one to use a laser diode to obtain relatively high-intensity standing waves in a simple manner. The SBC has many obvious applications, ranging from particle detection by light scattering to second-harmonic generation and nonlinear spectroscopy. In this Letter we demonstrate the use of a SBC to collimate a beam of atomic cesium, employing the stimulated emission technique in Ref. 3. The SBC exploits the unique sensitivity of laser diodes to external optical feedback. Dahmani et aL5 recently demonstrated that this property can be used to lock the output frequency of a laser diode to a Fabry-Perot interferometer. This dramatically reduces the laser linewidth with no electronic feedback. It is straightforward to extend that work to obtain a high-power SBC. The idea of a power-buildup cavity for a laser is not new. It is well known that the circulating power inside a resonant interferometer cavity can be much larger than the incident power, but in conventional buildup cavities it is necessary to have an electronic servo system to keep the cavity resonance frequency the same as the laser frequency. The higher the buildup desired, the narrower the resonance width, and therefore the more elaborate the electronics. The SBC relies on optical feedback to lock the laser frequency to the cavity resonance and thus requires no electronics. Furthermore, the locking generally improves as the buildup increases. 0146-9592/88/050357-03$2.00/0
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Fig. 1. Schematic of the apparatus. The SBC has not been drawn to scale. It actually has a total length of 24 cm, and the fold angle is approximately 6 deg. LD1 is the prepumping laser that drives the 6S112 F = 3 to 6P3/2 F’ = 4 transition. LD2 is the laser that is locked to the cavity. X/4, quarterwave plate; PZT; piezoelectric transducer.
sufficient that the distance does not rapidly change by more than one-half wavelength. In the research discussed below, this distance could change substantially because the laser was on an optical table and the SBC was in an atomic beam machine. These changes would cause the laser frequency to jump. T o prevent this, we stabilized the laser-SBC distance by using a movable mirror. A simple electronic servo system adjusted the mirror position to maximize the power inside the SBC by holding the separation (or phase, when scanning the frequency) constant. With SBC’s we have achieved a buildup factor (ratio of intracavity traveling wave power to incident power) of approximately 1000. This factor is limited only by the quality of the mirrors and thus could be many thousands with very good mirrors. For a typical lowpower laser diode, even a buildup of 100 means that one has 1 W of tunable power in each traveling wave. The frequency of the light can be tuned over the full range of the laser by changing the resonant frequency of the cavity and the laser current and temperature. We emphasize that this is a simple device to construct and use-it uses only a few optical elements and requires little or no electronics. We have used the standing-wave field in such a cavity to collimate a beam of atomic cesium. A standing-wave field will exert a force on an atom owing to the induced dipole moment, and for intense fields this force can be much larger than the spontaneous force due to the photon momentum. As discussed in Ref. 7 , this force has a velocity-dependent component that is positive (heating) when the light is at a frequency lower than a resonant atomic transition and negative (dampingor cooling) when the light is tuned above the transition. Aspect et a1.3 have demonstrated this effect by using light from a tunable dye laser to collimate a beam of cesium atoms. A schematic diagram of our apparatus is shown in Fig. 1. A thermal beam of atomic cesium is collimated to a full-angle divergence of 15 mrad using small apertures. Initially, all the atoms are put into the 6 S ~ /F2 = 4 hyperfine ground state by optical pumping. Then they pass through the standing-wave field in the SBC and are detected 30 cm beyond the SBC by a movable tungsten hot-wire detector. We measure the flux as a
function of position, which reflects the horizontal velocity distribution of the beam. To produce the standing-wave field, the 852-nm output of a laser diode is sent through a circular polarizer and is mode matched into a folded cavity as previously described. With a SBC, it is easy to reach an intensity high enough that the dipole heating is much larger than the expected cooling. T o avoid this problem, we use a cavity with a flat input mirror of only 85% reflectivity. The other two mirrors, with 5- and 2.5-cm radii, have reflectivities greater than 99%. The separation of the latter mirrors, slightly less than 5 cm, is adjusted so the light beam has a waist diameter of 0.12 cm and is highly collimated where it intersects the cesium beam a t right angles. The maximum power in the cavity is 0.18 W, with a linewidth much less than 1.0 MHz. The standing-wavefield primarily excites the 6S1/2 F= 4 to 6P3/2 F‘ = 5 transition, which is a cycling transition in that the only decay channel is back to the initial state. The F’ = 4 excited state is 250 MHz lower in energy and thus farther off resonance than the Fr = 5 state, but its effect is significant, particularly when the laser is tuned to the red of the F’ = 5 line. We measured the resulting cesium beam velocity profile as a function of laser power and frequency. In Fig. 2 we show the largest collimation (laser tuned to blue) and decollimation (tuned to red) that we achieved. This figure illustrates the dramatic improvement in beam collimation achievable with this method and is similar to the collimation reported in Ref. 3 in which a tunable dye laser and much lower intensity were used. The narrow peak in the center of the curve obtained with red detuning is a reproducible signal t h a t comes from very low-velocity atoms trapped a t the antinodes of the standing wave.3
POSITION RESOLUTION -VELOCITY
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Fig. 2. Signal as a function of hot-wire position. The squares show the beam profile with no laser present, while the crosses show the profile with red detuning and the circles show it for blue detuning. The transverse velocities corresponding to the hot-wire positions are also shown. The intracavity intensity was 21 W/cm2.
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cavities, we have achieved intensities high enough that we actually observed decollimation for all frequencies because the heating was so large. From the data in Fig. 3 we also extract the detuning for optimum collimation as a function of intensity. This is plotted in Fig. 4,and one can see that it scales linearly with the Rabi frequency. The simplicity and low cost of a diode-laser-SBC system makes it attractive for collimating atomic beams in many applications. With minor modifications to the cavity geometry, one could produce a standing wave that intersects itself a t right angles, providing two-dimensional collimation. The velocity-dependent dipole force has two significant advantages for beam collimation over the spontaneous or radiation pressure force that has also been used for this purpose.8 First, the dipole force is much greater so that the collimation can be accomplished in a much shorter distance; second, each spontaneously emitted photon accomplishes far more cooling. This means that for a fixed laser power, a far more intense atomic beam can be collimated. In addition to its use as an atomic lens, a SBC also offers a convenient way to study the behavior of the atoms in intense standingwave fields. For example, one could use two intracavity phase modulators to shift the position of the standing wave back and forth a t a few megahertz and thereby excite the vibrational resonance of the atoms in the potential wells produced by the field. This would provide detailed information on the atom-field interaction.
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Fig. 3. Dependence of the hot-wire signal on laser detuning when the hot wire is at the center of the cesium beam. The four curves represent different values of intracavity power and are characterized by their respective Rabi frequencies at the center of the Gaussian profile of the beam. The maximum intensity is 21 W/cm2 (corresponding to WRabi = 195 r).
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In Fig. 3 we show the frequency dependence of the collimation for various values of intracavity power. The curves in this figure were obtained by placing the hot wire a t the center of the cesium beam and observing the signal while ramping the SBC (and hence the laser) frequency from 160MHz to the red of the transition (30 times the 5.3 MHz natural linewidth = I') to 40 r to the blue. These data show that the maximum collimation achievable at a given intensity increases with intensity up to a peak of about 17 W/cm2, which corresponds to a resonant Rabi frequency of 176 I'. As can be seen from curve 4, if the intensity is increased further the collimation decreases, presumably because of the increased dipole heating. With other
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References '
'
1. R. N. Watts and C. E. Wieman, Opt. Lett. 11,291 (1986). 2. D. Sesko, C. G. Fan, and C. E. Wieman, "Production of a cold atomic vapor using diode-laser cooling," J. Opt. SOC. Am. B (to be published). 3. A. Aspect, J. Dalibard, A. Heidmann, C. Salomon, and C. Cohen-Tannoudji, Phys. Rev. Lett. 57,1688 (1986). 4. C. Salamon, J. Dalibard, A. Aspect, H. Metcalf, and C. Cohen-Tannoudji, Phys. Rev. Lett. 59,1659 (1987). 5. B. Dahmani, L. Hollberg, and R. Drullinger, Opt. Lett. 12, 876 (1987). 6. For a discussion of birefringence in mirror coatings see S. L. Gilbert and C. E. Wieman, Phys. Rev. A 34,792 (1986), and reference therein. 7. J. Dalibard and C. Cohen-Tannoudji, J. Opt. SOC.Am. B 2, 1707 (1985); A. P. Kazantsev, Zh. Eksp. Teor. Fiz. 66, 1599 (1974); V. G. Minogin and 0. T. Serimaa, Opt. Commun. 30, 373 (1979); J. P. Gordon and A. Ashkin, Phys. Rev. A 21,1606 (1980), A. P. Kazantsev, V. S. Smirnov, G. I. Surdutovich, D. 0. Chudesnikov, and V. P. Yakovlev, J. Opt. SOC.Am. B 2,11731 (1985). 8. V. I. Balykin and A. I. Sidorov, Appl. Phys. B 42, 51 (1987).
Reprinted from Journal of the Optical Society of America B, Val. 5, page 1225, June 1988 Copyright 0 1988 by the Optical Society of America and reprinted by permission of the copyright owner.
Production of a cold atomic vapor using diode-laser cooling D. Sesko, C. G.Fan, and C. E. Wieman Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, University of Colorado, Boulder, Colorado 80309-0440 Received November 30,1987; accepted january 25,1988 = 852 nm) to damp the motion of atoms in a cesium vapor. We have been able to contain more than lo7 atoms for 0.2 sec and cool them to a temperature of 100?i:o p K in this viscous photon medium (the so-called optical molasses).
We have used the light from diode lasers (X
In the past few years there has been a flurry of work demonstrating new ways to use laser light to control the positions and velocities of However, nearly all of this work involves rather elaborate and expensive technology. We have been working to achieve these capabilities by using technology that is simple enough to be practical for use in a wide variety of atomic- and molecular-physics experiments. In a previous paper3 we presented a simple and inexpensive way to stop a beam of cesium atoms by using frequencychirped diode lasers. In this paper we present the implementation of optical molasses by using diode lasers and discuss improved results on atom stopping. The cooled125 p K ) and vapor produced in this work was colder (2' significantly denser (-lo8 atoms/cm3) than was previously obtained with sodium in optical molasses2 produced by a dye laser. The idea of cooling an atomic sample by using counterpropagating laser beams was originally proposed by Hansch and Schawlow4 in 1975. They pointed out that if an atom sees counterpropagating laser beams that are tuned slightly below a transition frequency, the Doppler shift will cause the atom to absorb preferentially those photons moving opposite to its velocity. The momentum imparted by these photons will cause the atom to slow down and thus be cooled. In 1985 Chu et aL2demonstrated this idea in a vapor of sodium atoms and coined the name optical molasses. They sent a beam of relatively slowly moving sodium atoms into a region where six laser beams intersected in a cross. These beams cooled the atoms to a temperature, T = hy/2k, which is the theoretically predicted2 minimum temperature one can achieve when exciting a resonance line of width y. Also, they showed that the photons produced a highly viscous medium that confined the atoms for a fraction of a second before they could diffuse out of it. This production of free atoms that are far colder than anything previously obtainable opens up a rich new area for experiments, e.g., using these atoms for ultra-high-resolution spectroscopy or studying the interactions of these cold atoms with surfaces or other atoms. Such experiments are far more practical if diode lasers can be used to cool the atoms. Tunable diode lasers have a number of important advantages over the dye lasers that were used exclusively in the early work on laser cooling.
The most striking feature, of course, is their much lower cost. Other advantages, which are well known, are that they are. simpler to use; because there are no optical elements to get misaligned and dirty, i t is easy to change the output frequency rapidly, and they have good intensity stability. An advantage that is not well known is that it is quite easy to obtain diode-laser linewidths of much less than 1 MHz. A free-running diode laser has a typical linewidth of 30 MHz, but Dahmani and co-workers5 have recently shown that a small amount of optical feedback from a Fabry-Perot interferometer will lock the laser frequency to that of the interferometer and can reduce the linewidth by more than a factor of 1000. In this experiment we utilized this technique; thus the laser linewidth is much smaller than the 5-MHz natural linewidth of the transition. To produce ultracold cesium atoms we first started with a thermal atomic beam with an average velocity of 250 m/sec. These atoms were slowed to a few hundred centimeters per second by a beam of counterpropagating resonant laser light. The slowly moving atoms then drifted into a region where intersecting laser beams, which were tuned slightly below the center of the atomic resonance, formed the optical molasses. This light cooled the atoms and held them for an extended period of time. We studied the atoms by observing their fluorescence while in this region. A schematic of the apparatus for this experiment is shown in Fig. 1. A beam of cesium effused from an oven into one end of a vacuum chamber, and the frequency-chirped laser beam entered from the opposite end. This initial slowing portion of the experiment was identical to that discussed in Ref. 3, except that the stopping laser was locked to an interferometer cavity and the stopping distance was extended to 90 cm. T o lock the laser, a small portion of the output beam was sent into a 5-cm confocal cavity, which was a few centimeters from the laser. The cavity was tilted slightly so that the beam entered a t an angle relative to the cavity axis. Of the order of 1%of the laser power returned to it and caused its frequency to lock to that of the external cavity. If the free-running laser frequency was within approximately 500 MHz of the cavity resonance, the optical feedback would pull the laser frequency to the cavity resonance. This allowed us to tune the laser frequency coarsely by using temperature and current in the usual manner and then to do fine
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J. Opt. Soc. Am. B/Vol. 5, No. 6/June 1988 PUMPING LASER 2 , MOLASSES LASER
V
LASER
C OVEN PHOTODIOD
Fig. 1. Schematic of the apparatus. The third molasses beam, which was perpendicular t o the other two, is not shown. FP1 and FP2 are the Fabry-Perot locking cavities. The saturated absorption spectrometers are labeled SASl and SASS.
tuning by changing the resonant frequency of the cavity by using a piezoelectric transducer to translate one of the mirrors. The stopping laser drove the 6 S ~ = 4 6P3/2,~=5 resonance transition. A second laser, which was not locked to a cavity, was tuned to the 65'~=3 6 P 3 / 2 , ~ = transition 4 to ensure that atoms were not lost to the F = 3 ground state. The frequencies of these two lasers were swept from approximately 500 MHz below the respective transition frequencies to within a few megahertz of the transition. The first half of the chirp does little slowing since the average initial velocity of the atoms is -250 m/sec, but the additional time is needed to allow slower atoms to populate the downstream end of the beam between successive chirps. After a series of chirps, the stopping laser frequency was quickly (=30 gsec) shifted away from the resonant frequency, The frequency of the stopping laser, just before shifting, determined the final velocity of the atoms as they entered the molasses. The F = 3 state depletion laser remained a t the same frequency that it had a t the end of the ramp and thus ensured that the F = 3 state also remained depleted in the molasses. The frequencies of both lasers were monitored using small cesium-saturated absorption spectrometers. The optical molasses was produced by light from a third laser that was also locked to a cavity. The output of this laser passed through an attenuator and an optical isolator and was then split into three linearly polarized beams, each containing approximately 0.5 mW of power, that intersected each other a t right angles. The intersection region was slightly less than 1 cm in diameter and overlapped the cesium beam. The three beams were reflected back onto themselves by dielectric mirrors. A small part of the output from the molasses laser was used to obtain a saturated absorption spectrum in a cesium cell. This spectrum provided an error signal that was fed back to the cavity to hold the laser F = 5 transition. frequency on the red side of the F = 4 These three lasers were sufficient to cool the atoms. However, to obtain @her densities in the molasses by multiple loading, we used a fourth (unstabilized) laser, as described below. A silicon photodiode monitored the fluorescence from the molasses region. The cooling chirp slowed the atoms in the cesium beam to several hundred centimeters per second or less. We found that using a cavity-locked stabilized laser significantly im-
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proved the efficiency of this process. The number of stopped atoms was approximately 2.5 times larger with a stabilized laser than it was with an unstabilized laser, as was used in our previous work. The number of fast background atoms was correspondingly decreased relative to the unstabilized case. Apparently some of the laser-frequency fluctuations were large enough to cause the frequency to change more abruptly than the maximum allowable ramp rate. If this happens, some of the atoms are not decelerated enough to stay in resonance with the laser, and they are no longer slowed. In the present work we obtained approximately 5 X lo6 stopped atoms/cm3, and there were few fast-moving background atoms until a few milliseconds after the stopping chirp was finished. The narrower laser linewidth also permitted much better control over the final velocity of the atoms. Once the atoms were slowed to low velocity they could be caught by the molasses. The fluorescence from the molasses region showed a rapid rise during the stopping laserfrequency chirp, followed by a long decay indicating slow departure of the atoms. The behavior of the atoms in the optical molasses depended quite critically on the velocity they had as they entered the molasses region. To obtain reproducible results, the end of the frequency ramp had to remain constant to within approximately 1 MHz (I MHz corresponds to u = 100 cm/sec). If the atoms were moving too slowly they did not penetrate into the molasses, but instead they piled up on the surface and quickly diffused away. Best results were obtained when they had a final velocity of several hundred centimeters per second so that they penetrated the entire molasses region. This also produced a more dense sample since a larger volume of stopped atoms went into the molasses. In this situation the fluorescence signal from the molasses region would continue to rise for several milliseconds after the stopping laser was switched off. If the atomic velocity was increased further the atoms would fly right through the molasses. Once the atoms were stuck in the molasses they diffused out very slowly. This is illustrated in the plot of the fluorescence as a function of time in Fig. 2. As one would expect, the diffusion time was a sensitive function of loading velocity, alignment, and molasses laser power and frequency. The longest l/e decay time that we observed was 0.2 sec. We found that the conditions for maximum decay time were intensity per beam of approximately one half the 1mW/cm2
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TIME ( s e c 1
Fig. 2. Real-time trace of the fluorescence. The black dots on the left-hand side show the level after each new bunch of atoms was loaded. After eight bunches the loading was stopped, and the righthand side shows the subsequent decay of the fluorescence as the atoms diffuse away.
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Fig. 3. The dots show the fraction of the initial fluorescence that remained after the molasses laser was blocked for the time intervals shown. The solid line is the theoretical fit. Note that the data point a t 41 msec is much lower than the theoretical curve because of gravitational effects.
saturation intensity, frequency detuning between 0.5 and 1 natural linewidth, and loading velocity of a few hundred centimeters per second. These values are in reasonable agreement with what is predicted for these parameters when using the results in Ref. 6, although the intensities are a bit higher than expected. Because the atoms remained in the molasses 20 times longer than the time for a stopping chirp (10 msec), one could obtain much higher densities by repeated loading of the molasses. However, this required a fourth laser to provide the F = 3 state depletion in the molasses while another batch of atoms was being stopped. In the left-hand side of Fig. 2 one can see the effects of loading eight bunches. The dots show the fluorescence level after each additional bunch was loaded. After eight bunches the loading was stopped, and there was a slow decay of the fluorescence. By using multiple loading we were able to increase the number of atoms in the molasses by more than a factor of 10. From the amount of fluorescence, we estimate the maximum number of atoms we contained in the molasses to be approximately 5 X lo7. This was bright enough, when observed with an infrared viewer, that it could be seen in a well-lit room. We determined the temperature of this vapor by measuring the spread of the atoms in the dark, as was done in Ref. 2. After the atoms had been in the molasses for approximately 25 msec, the molasses laser was quickly switched off. A brief time later the light was turned back on, and the decrease in
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the fluorescence was measured. In Fig. 3 we show the fluorescence as a function of the time the light was off. After more than 35 msec with the light off, the subsequent fluorescence was dramatically less (falling off rapidly with longer times without light). We believe that this was due to the gravitational acceleration of the atoms. We fitted the points in Fig. 3 with the calculated curve for the expansion of a uniform sphere of atoms with a Maxwell-Boltzmann velocity distribution. This curve will be changed somewhat by the gravitational acceleration, but this effect appears to be small for times less than 35 m/sec. From this fit we find the temperature to be 100?itop K . There is considerable uncertainty in this measurement because the decay was a strong function of the distribution of the atoms, and we could see that the distribution was only vaguely spherical and could change somewhat from one batch to the next. Nevertheless, this value is in good agreement with the predicted2limit, hr/ 2k = 125 pK for this transition, and is colder than any cooled atoms or ions previously reported. At this temperature the rms velocity for these atoms is only approximately 15 cml sec. We have achieved an extremely cold and moderately dense atomic vapor by cooling atoms with light from inexpensive diode lasers. With the light on, this vapor will remain for a fraction of a second, and with the light off, the atomic motion is largely determined by gravitational acceleration. The technology required to achieve this remarkable behavior is simple enough that such atoms could be produced routinely for use in precision spectroscopy or collision studies.
ACKNOWLEDGMENTS This work was supported by the National Science Foundation and the Office of Naval Research. We are thankful for the assistance of C. Tanner and L. Hollberg in the diodelaser development. The authors are also with the Department of Physics, University of Colorado, Boulder, Colorado 80309-0440.
REFERENCES 1. For a review of this field, see the feature issue on the mechanical effects of light, J. Opt. Soc. Am. B 2,1706 (1985). 2. S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett, 55,48 (1985). 3. R. N. Watts and C. E. Wieman, Opt. Lett. 11,291 (1986). 4. T. W. Hansch and A. L. Schawlow, Opt. Commun. 13,68 (1975). 5. B. Dahmani, L. Hollberg, and R. Drullinger, Opt. Lett. 12, 876 (1987). 6. J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980); R. J. Cook, Phys. Rev. A20,224 (1979);Phys. Rev. Lett. 44,976 (1980).
Reprinted from Optics Letters
Observation of the cesium clock transition in laser-cooled atoms D. W. Sesko and C. E. Wieman Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colorado 80309 Received October 3,1988; accepted December 12, 1988
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We have used the light from diode lasers to produce a nearly stationary (u 15 cm/sec) sample of atomic cesium in optical molasses that is entirely in the F = 3 hyperfine state. In this sample we excite the 9.2-GHz 6s F = 3, m = 0 to F = 4, rn = 0 clock transition. Most of the atoms remain for -20 msec in the 0.4-cm3 observation region. We observe that transitions take place by monitoring the fluorescence when the atoms are illuminated with light tuned to the 6.9 F = 4 to 6P3/2F = 5 transition. Rabi resonance linewidths of less than 50 Hz are obtained.
There has been a great deal of interest recently in the use of laser light to stop and cool atomic samples.*-3 It has been suggested frequently that such atoms would be useful for precision spectroscopy and frequency standards.l However, this potential has yet to be realized. In this Letter we present a step toward this goal by using laser-cooled atoms t o obtain narrow linewidths on the 6 s F = 3, m = 0 to 6 s F = 4, m = 0 transition in atomic cesium. This is the so-called clock transition since it provides the present definition of the second. Although we are interested only in the linewidth that can be obtained and not in the many details that must be addressed in making an actual clock or frequency standard, this research does have two notable features with respect to such devices. First, the interaction volume, which must be extremely well shielded against stray magnetic fields in a real clock, is less than 1 cm3. Second, the cooling of the atoms and the detection of the transition are done using low-power diode lasers. These features allow one to contemplate a relatively compact and portable frequency standard based on this technology. The experimental technique is simple and uses the following time sequence. First, atoms in a cesium atomic beam are slowed and cooled by using the pressure exerted by the laser light. Then the laser light is turned off in such a way that all the cold atoms are left in the F = 3 ground state. During the subsequent dark period microwaves excite the transition to the F = 4 state. The resulting population of the F = 4 state is then determined by switching on a laser that excites this hyperfine state and observing the resulting fluorescence. The relevant transitions are shown in Fig. 1. The primary effort for this experiment is in the laser cooling of the atoms, but we have discussed this in detail previously3 and thus only briefly review it here. Light from a cavity-stablized diode laser is sent counterpropagating to a cesium atomic beam. The output frequency of this laser is swept from 500 MHz below the 6 s F = 4 to 6P3/2 F = 5 transition to 4 MHz below the resonance using a 10-msec ramp. This provides atoms with a velocity of a few meters per second, and the atoms are caught in optical molasses that is created by using the light from a second laser to produce three intersecting orthogonal standing waves. This
light is tuned approximately 10 MHz below the 6 s F = 4 to 6P312 F = 5 transition. In addition there is a third laser, which weakly illuminates the atomic beam and the molasses region and is used t o depopulate the F = 3 ground state. It does this by exciting the 6 s F = 3 to 6P312 F = 4 transition. Its frequency is ramped along with the stopping laser and then held on the transition during the time the atoms are in the molasses. In this manner we produce an atomic sample in the middle of the vacuum chamber with a residual velocity of approximately 15 cm/sec. To carry out microwave spectroscopy on this sample we irradiate it with approximately 7 nW/cm2 of radiation a t 9.2 GHz using a microwave horn driven by a microwave frequency synthesizer. Although the microwave radiation is always present, its effect is unob-
n 4-4 5
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Fig. 1. Energy-level diagram for cesium. 1989 Optical Society of America
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Table 1. Chronology of the Experiment Time
Function
Laser 1 (Stopping and Probe)
Laser 2 (Hyperfine Pump)
Laser 3 (Molasses)
I. 0-10 msec
Stop atoms u, = 400 mlsec uf = 4 mlsec
6s F = 4 6P3iz F = 5 chirp t o compensate for Doppler shift
6s F = 3 76P3iz F = 4 chirp
11. 10-30 msec
Molasses cooling ui = 4 mlsec uf = 15 cm/sec
Off resonance (500 MHz)
6 s F = 3 6P312 F = 4 on resonance
6SF=4+6?3/2F= 5 detuned -10 MHz
111. 30-33 msec
Off-resonant Blocked 6 s F = 4 6P312F = 4 optically pumps cooled atoms into 6s F = 3
Blocked
6s F = 4 6P312F = 5 detuned -10 MHz
Excite clock transition
Blocked
Blocked
Blocked
Blocked
+
6SF=4+6P3/2F= 5 detuned -10 MHz
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V. 53-53.5 msec Probe F = 4 state population
Blocked 6s F = 4 6P3i2 F = 5 on resonance +
VI. Change microwave frequency and repeat sequence
servable when the la& light is present. T o observe the clock transition we follow the time sequence shown in Table 1. After the atoms have been cooled in the molasses for 20 msec (any time from 2 to 80 msec works as well) the F = 3 depopulating laser is switched off using a mechanical shutter, and 3 msec later the main F = 4 to F = 5 molasses beams are also blocked by a shutter. During this 3 msec all the atoms are pumped into the F = 3 ground state owing to the offresonant 6 s F = 4 to 6P312 F = 4 transition. There is then an interval of time during which all lasers are blocked and only microwave radiation is present. All transitions other than the 6 s F = 3, m = 0 to F = 4, m = 0 are shifted far out of resonance by a 0.5 G magnetic field. The microwave power is adjusted to give a ir pulse during this time. At the end of this interval the stopping laser (laser l),which is now tuned to the peak of the 6 s F = 4 to 6P312 F = 5 transition, is unblocked. The fluorescence produced by this laser beam is a direct measure of how many atoms have made transitions to the F = 4 ground state. The fluorescence comes in a brief pulse because the atoms are repumped into the F = 3 state in approximately 0.5 msec. However, during this pulse there are still a few thousand photons scattered for each atom that makes the transition to the F = 4 state, and we detect a few percent of these using a photodiode. The entire time sequence takes 54 msec when the dark period is approximately 20 msec. Because the crude mechanical shutters that we use tend to overheat, we then wait an additional 150 msec before repeating the sequence. If one used acousto-optic shutters to obtain the optimum data rate, it would be possible to obtain a data pulse every 35 msec with a 20-msec microwave transition time. After each cycle the microwave frequency is incremented, and this sequence is repeated. We find that good signals can be obtained for any period of darkness between 1 and 20 msec. For times longer than that the signals drop rapidly owing to the atoms’ leaving the region illuminated by the probe
beam. We believe that the 20-msec interval is primdrily determined by gravitational acceleration and the geometry of the laser beams. The atoms form a somewhat irregular clump in the center of the molasses that is roughly 0.5 cm on a side, and the probe beam is a cylinder with a diameter of 0.5 cm. Thus the atoms simply fall out of the observation region in 20 msec. With a dark period of slightly more than 20 msec we obtain the spectrum shown in Fig. 2. This spectrum represents 5280 55-msec data sequences (each point is the average of 24). It has a linewidth of 44 Hz. If the microwave radiation were pulsed to give a Ramsey separated oscillatory field, i t would give a 23-Hz linewidth. This can be compared with the 26-He linewidths that are currently obtained in the NBS-6 atomic clock that uses a 4.3-m-long transition region.* 1.0
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FREQUENCY CHANGE ( Hz ) Fig. 2. Experimental spectrum and theoretical fit of the 9.2-GHz clock transition in cesium ( 6 S P = 3, m = 0 to 6 s F = 4, m = 0 ) .
272 March 1,1989 / Vol. 14, No. 5 / OPTICS LETTERS
We believe that the signal-to-noise ratio can be improved greatly, since the actual number of photons produced is large. However, instabilities in the frequency of the stopping laser, which is not actively stabilized, cause substantial pulse-to-pulse fluctuations in the number and spatial distribution of the atoms in the molasses. We believe that if this laser were stabilized the signal-to-noise ratio would be larger. It would also be possible to obtain longer interaction times and hence a narrower resonance by changing the geometry or by making an atomic “fountain” as is being built by Ertmer et aL5 The amount of improvement that can be obtained in this fashion is somewhat limited, however, because the gain in time only goes as the square root of the distance. We have demonstrated that it is practical to use laser-cooled atoms t o obtain narrow transition linewidths in small interaction volumes. We hope that this will stimulate the consideration of this technology for high-precision spectroscopy and actual frequency standards.
271
We are pleased to acknowledge the assistance of Chris Monroe and Steve Lundeen and the loan of the microwave equipment from D. Wineland, J. Hall, J. Levine, and D. Anderson. This research was supported by the U.S. Office of Naval Research and the National Science Foundation. The authors are also with the Physics Department, University of Colorado, Boulder, Colorado.
References 1. For a review of this field see the feature issue on the mechanical effects of light, J. Opt. SOC.Am. B 2, 1706 (1985). 2. S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett. 55,48 (1985). 3. D. Sesko, C. G. Fan, and C. E. Wieman, J. Opt. SOC.Am. B 5, 1225 (1988). 4. D. J. Glaze, H. Hellwig, D. W. Allan, and S. Jarvis, Jr., Metrologia 13,17 (1977). 5. W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54,996 (1985).
PHYSICAL R E V I E W LETTERS
V O L U M E63, N U M B E R9
28 AUGUST1989
Collisional Losses from a Light-Force Atom Trap D. Sesko, T. Walker, C. Monroe, A. Gallagher,(a)and C. Wieman Joint Institute for Laboratory Astrophysics, University of Colorado and National Institute of Standards and Technology, and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440 (Received 5 J u n e 1989)
W e have studied the collisional loss rates for very cold cesium atoms held in a spontaneous-force optical trap. In contrast with previous work, we find that collisions involving excitation by the trapping light fields are the dominant loss mechanism. We also find that hyperfine-changing collisions between atoms i n the ground state can be significant under some circumstances. PACS numbers: 32.80.Pj. 34.50.Rk
Spontaneous-force light traps ’,* have provided a way to obtain relatively deep static traps for neutral atoms. These allow one to produce samples containing large numbers of very cold atoms. In this paper we present an experimental study of the collisions which eject atoms from such a trap. These collisions are of considerable interest because the temperatures of the trapped atoms ( l o p 4 K) are far lower than in usual atomic collision experiments. The theory of such low-energy collisions and their novel features have been discussed by several aut h o r ~ . ~Perhaps -~ the most notable feature is that the collision times are very long, and the collision dynamics are dominated by long-range interactions and spontaneous emission. These collisions also have important implications with regard to potential uses of optically trapped atoms. For many applications the maximum density that can be obtained is a critical parameter, and these collisions limit the attainable density. There have been two experimental studies of collisions in optical traps. Gould et al. measured the cross section for associative ionization of sodium. However, there is no evidence that this process is significant in limiting trapped-atom densities, and for some atoms, including cesium, it is energetically forbidden. Prentiss et al. studied the collisional losses which limited the density of sodium atoms which were held in a spontaneous-force trap. Their surprising and unexplained results were a direct stimulus for our work. In particular, they observed no dependence of the loss rate on the intensity of the trapping light. This was quite surprising, because a ground and an excited atom interact at long range via the strong l l r resonant dipole interaction, and a portion of the excited-state energy can be converted into sufficient kinetic energy to allow the atoms to escape from the trap. By comparison, two atoms in their ground states interact only through a much weaker short-range l / r 6 Van der Waals attraction, and even when such collisions occur, they may not produce significant kinetic energy to cause trap loss. This implies that the dominant collisional loss mechanism would involve the excited atomic states, and thus depend on the intensity of the light which causes such excitations.
In this paper we present measurements which show that, in contrast with Ref. 9, the collisional loss rate has a marked dependence on the trap laser intensity. We will present strong circumstantial evidence that the dependence at very low intensities is due to hyperfinechanging collisions between ground-state atoms. We believe that the loss rates at higher intensities are associated with collisions involving excited states, and are the type discussed by Gallagher and P r i t ~ h a r d . ~ As discussed in Ref. 7, these collisions are very different from normal ground-excited-state atomic collisions in which two initially distant atoms, A and A * , approach, collide, and separate in a time much less than the radiative lifetime of the excited state. In contrast, for these very-low-temperature collisions the absorption and emission of radiation in the midst of the collision drastically alter the motion. In particular, if the excitation takes place when the two atoms are far apart ( R > 1000 A) they will reradiate before being pulled into the small-R region where energy transfer occurs. However, if they are sufficiently close when excited, they can be pulled close enough together for substantial potential energy to be transferred into kinetic energy before decaying. The two dominant transfer processes are excited-state fine-structure changes and radiative redistribution. In the first, A* changes its fine-structure state in the collision and the pair acquire a fine-structureinterval worth of kinetic energy. The second process, radiative redistribution, refers to A-A * reemitting a photon which, because of the A-A * attractive potential, has substantially less energy than that of the photon which was initially absorbed. This energy difference is transferred to the subsequent kinetic energy of the ground-state atoms. The trap loss rate depends on the probability of exciting such “close” A-A * pairs, and this probability is determined by the frequency and intensity of the exciting radiation. Light which is tuned to the red of the atomic resonance frequency, VO, excites pairs which are closer together (and shifted in energy) and thus is more effective at causing trap loss than light which is at VO. We tested this hypothesis by examining how the loss
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rate changes when the trapped atoms are illuminated by an additional laser tuned far from resonance. It is particularly straightforward to interpret this second measurement because such a red-detuned “catalysis” laser is very ineffective at exciting isolated atoms and, hence, has a negligible effect on the trap depth. Also, as discussed below, it is considerably simpler to theoretically predict the collisional loss rate for the case of large detuning. Much of the apparatus for this experiment was the same as in our earlier work with cesium in optical molasses, and the arrangement of laser and atomic beams is very similar to that discussed in Ref. 10. A beam of cesium atoms effused from an oven and passed down a I-m tube into a UHV chamber (lo-’’ Torr). In this chamber the atoms were trapped using a Zeeman-shift spontaneous-force trap2 formed by having three perpendicular laser beams which were reflected back on themselves. The beams were all circularly polarized, with the polarizations of the reflected and incident beams being opposite. They had Gaussian profiles with diameters of about 0.5 cm, and were tuned to the 6 S p 4 to 6Py2f-s transition of cesium. Magnetic field coils which were 3 cm in diameter and arranged in an anti-Helmholtz configuration provided a field which was zero a t the middle of the intersection of the six beams, and had a longitudinal gradient of 5.1 G/cm. In addition to the main trap laser beams, there were beams to ensure the depletion of the F - 3 hyperfine ground state in the trap. Thus all the trapped atoms were in the F = 4 ground state. Two other lasers, the “catalysis” and “stopping and probing” lasers, were used on occasion as discussed below. A11 the lasers were diode lasers whose output frequencies were controlled by optical and electronic feedback and had short-term linewidths and long-term stabilities of well under 1 MHz. The fluorescence from the trapped atoms was observed with both a photodiode, which monitored the total fluorescence, and a charged-coupled-device television camera which showed the size and shape of the atomic cloud. The spatial resolution was about 10 pm. Data acquisition involved the following sequence: First, atoms were loaded into the trap by opening a flag which blocked the cesium beam. When the trap laser intensity was very low the stopping laser was used to slow the atoms for loading as in Ref. 10. After a few seconds the cesium and stopping laser beams were again blocked, the video image of the trap was digitized and stored, and the total fluorescence signal was measured as a function of time for the next 205 s. In a separate measurement, we determined the density and the excited-state fraction by loading the trap under the same conditions and observing the absorption of a weak probe beam by the cloud with the trapping light on, and 1 ms after it had been switched off. To measure the temperature of the trapped atoms the probe was moved a few mm to the side of the cloud and the time-of-flight spectrum was observed after the trap light was turned off, as in Ref. 1 1 . The temper962
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ature was found to be between 2.5 and 4 x 10 K. In the initial stages of this work we loaded the trap with as many atoms as possible. This produced a large low-density cloud of trapped atoms which had a loss rate which was proportional to the total number of trapped atoms. The behavior of the cloud in this large-number regime showed a variety of complex cooperative behaviors which will be discussed in a later publication. We subsequently discovered that, when the number of atoms in the trap decreased, the diameter of the cloud diminished, while the density increased. When the number was much lower, the cloud became a small Gaussian sphere with a diameter of about 1.7x l o - * cm. Further reduction in the number of atoms then only reduced the density but not the diameter. In Fig. 1 we show the fluorescence signal from the cloud in this constant-diameter regime, along with a fit which assumes a loss rate of the form d n / d t = -an - p n 2 , where n is the density of atoms in the trap. We also assume that the fluorescence is proportional to the density. The figure shows that this is the appropriate dependence, and the n dependence indicates that collisions between the trapped atoms are causing loss from the trap. The a term is responsible for a simple exponential decay. For all the data presented here, we were careful to ensure that the initial number of atoms in the trap ( 3 l o~4 ) was low enough that the cloud diameter remained constant. This was verified both by directly measuring the cloud diameter and by checking that the fluorescence decay had the expected time dependence. As mentioned earlier, we measured collisional losses under two different conditions. The first approach, similar to that used in Ref. 9, was to simply examine the rate at which atoms were lost from the trap as a function of
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the intensity of the trap laser light. The quantity of interest was the coefficient p, since a depends only on the background pressure in the vacuum chamber. T o find p, we fitted the fluorescence decay curves as in Fig. 1 to get pn0, and from the absorption measurements and the video image we obtained the initial density, no. The digitized image of the cloud showed that the fluorescence was a Gaussian distribution as one would expect. The initial density was between 1 and 4 x l o 9 atoms/cm3, and the probe-beam absorption was 2.5% to 10%. In Fig. 2 we show the dependence of p on the total intensity of all trap laser beams for a trap laser detuning of 5 M H z (I natural linewidth). The scatter in the points provides a reasonable estimate of the uncertainty in the measurements. We should mention that there was far more scatter in the data before considerable effort was expended to improve the quality of the laser beam wave fronts, the alignment, and the measurement of the trap size. W e interpret the data as follows. The large p values seen in Fig. 2 for intensities less than 4 mW/cm2 are due to hyperfine-changing collisions between ground-state atoms. Two atoms acquire velocities of 5 m/s if they collide and one changes from the 6Sj7-4 hyperfine state to the lower F - 3 state. For high trap laser intensities, atoms with this velocity cannot escape, but at lower intensities the trap is too weak to hold them. We also find that if the beams are misaligned slightly, a similarly large+ loss rate is observed for all intensities. W e estimate that hyperfine-changing collisions between ground-state atoms due to the Van de Waals interaction will give a p in the range lo-'' to l o - ' ' cm3/s, which is consistent with the data shown in Fig. 2. Above 4 mW/cm2, p appears to increase linearly with intensity. W e believe that this is due to the energy-transfer collisions of the type discussed above.
28 AUGUST1989
We studied these collisions in more detail by using an additional laser to illuminate the atoms in the trap. As mentioned earlier, measuring the p caused by this laser when it is detuned to the red provides a particularly good way to test the Gallagher-Pritchard model for these collisions. The data acquisition in this case was exactly the same as before. In Fig. 3 we show the results. These data were taken with a trap laser intensity of 13 mW/cm2 and a detuning of 5 MHz. The solid line is the prediction obtained using the Gallagher and Pritchard model.I2 For detuning greater than 100 M H z the curve turns over simply because of the decrease in the number of pairs with such small separations. (A frequency shift of 100 M H z corresponds to a separation of about 500 A.) The agreement between theory and experiment is excellent in view of the estimates used in the calculation, which are discussed below, and the experimental uncertainties. It is much less straightforward to compare the results shown in Fig. 2 with theory because the laser detuning is much smaller. The calculation in Ref. 7 neglects the excited-state hyperfine splitting and the initial atomic velocity which are substantially more important effects in the small-detuning case. When the laser is tuned close to the atomic resonance, the initial interatomic force is weak because the colliding atoms are relatively far apart. As a result, the probability of the atom getting in to small R before radiating has some dependence on its initial velocity. This probability also depends very sensi-
I
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FIG. 2. Dependence of p on the total intensity in all the trap laser beams. The solid line indicates the prediction of the Gallagher-Pritchard model, and would be a straight line if the scale were linear.
FIG. 3. Dependence of p on detuning of the catalysis laser. The zero of detuning corresponds to the center of gravity of the 6P3/2 hyperfine states. The absence of data for a detuning of less than 300 M H z is caused by the need to be well away from all 6S-to-6P3/2 hyperfine transitions to avoid perturbing the trap. These data are for a catalysis laser intensity of 24 mW/cm2. The dotted line shows the value of p which is obtained when the catalysis laser is not present. The solid line indicates the predictions obtained using the Gallagher-Pritchard model.
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tively on the shape of the potential, which, a t these distances, will be strongly affected by the excited-state hyperfine interaction and clearly will not be t h e simple C3/R dependence which was assumed. This could easily change the potential enough to shift the theoretical result by a factor of 5 , and hence would explain the difference between the magnitudes of the theoretical and experimental results. T h e theory does predict that p should increase linearly with intensity which is what we observe. For the larger detunings used with the catalysis laser, the theory is less sensitive to the shape of the potential curves and the atomic velocity, because the probability of reaching small R is approaching 1. However, a factor of 2 error is still not surprising. It is not clear why our results differ from those of Prentiss et a/.,but we do not think it is likely that sodium behaves fundamentally differently from cesium. O n e possibility is that for the range of intensities they used the sum of hyperfine-changing and excited-state collisional losses remained roughly constant. W e have shown that collisions which depend on the trap laser intensity will b e an important factor in determining the densities attainable in optical traps. This will be true of all types of optical traps, but the exact loss rate will depend on the frequency and intensity of the laser light present. Also, ground-state hyperfinechanging collisions can be a n important loss mechanism if the trapped atoms are not in the lowest hyperfine state and the trap is not sufficiently deep. W e are pleased to acknowledge the assistance of W. S w a m with the construction and operation of the stabilized diode lasers. This work was supported by the Office of Naval Research and the National Science
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Foundation (")Staff member, Quantum Physics Division, National Institute of Standards and Technology, Boulder, CO 80309-0440. ID. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986). 2E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987). 3J. Vigue, Phys. Rev. A 34,4476 (1986). 4P. Julienne, Phys. Rev. Lett. 61, 698 (1988). sP. Julienne, in Aduonces in Loser Science III, edited by A. C. Tam, J. L. Gole, and W. C. Stwalley, AIP Conference Proceedings No. 172 (American Institute of Physics, New York, 1988), p. 308. This presents a rough quantummechanical estimate for the radiative redistribution loss, as contrasted with the semiclassical treatment of A. Gallagher and D. Pritchard, preceding Letter, Phys. Rev. Lett. 63, 957
(1989). 6D. Pritchard, in Electron and Atom Collisions, edited by P. Lorents (North-Holland, Amsterdam, 1986). p. 593. 'Gallagher and Pritchard, Ref. 5. 8P. L. Gould, P. D. Lett. P. S. Julienne, W. D. Phillips, H. R. Thorsheim, and J. Weiner, Phys. Rev. Lett. 60, 788 (1988). 9M. Prentiss, A. Cable, J. E. Bjorkholm, S. Chu, E. Raab, and D. Pritchard, Opt. Lett. 13, 452 (1988). 'OD. Sesko, C. G. Fan, and C . Wieman, J. Opt. SOC.Am. B 5 , 1225 (1988). "P. D. Lett et al., Phys. Rev. Lett. 61, 169 (1988). 121n the terminology of Ref. 7, we have used the values C3-49 eV A 3 and qj-0.033 in evaluating the GallagherPritchard model. qj is estimated from a plot of measured fine-
structure-changing collision rates versus the Landau-Zener adiabaticity parameter. We also estimated the probability of sufficient radiative redistribution of energy to cause trap loss to be 0.027, using a trap depth of 1 K.
VOLUME 64, NUMBER 4
PHYSICAL REVIEW LETTERS
22 J A N U A R Y 1990
Collective Behavior of Optically Trapped Neutral Atoms Thad Walker, David Sesko. and Carl Wieman Joint Institute f o r Laboratory Astrophysics, University of Colorado and National Institute of Standards and Technology. and Departrnenr of Physics, University of Colorado, Boulder, Colorado 80309-0440 (Rcccivcd 5 October 1989)
We describe experiments that show collective behavior in clouds of optically trapped neutral atoms. This collective behavior is demonstrated in a variety of observed spatial distributions with abrupt bistable transitions between them. These distributions include stable rings of atoms around a small core and clumps of atoms rotating about the core. The size of the cloud grows rapidly as more atoms are loaded into it, implying a strong long-range repulsive force between the atoms. We show that a force arising from radiation trapping can explain much of this behavior. P A C S numbers: 32.80.Pj.42.50.Vk
ing rate and the collisional loss’ rate. Two other lasers were used to deplete the F - 3 hyperfine ground state in both the trapped atoms and the atomic beam. All the lasers were diode lasers with long-term stabilities and linewidths well under 1 MHz. The trapped atoms were observed by a photodiode, which monitored the total fluorescence, and a chargecoupled-device (CCD) television camera, which showed the size and shape of the cloud of atoms. The optical thickness of the cloud was determined by measuring the absorption of a weak probe beam 1 ms after the trap light was turned off. To measure the temperature of the atoms, the probe beam was moved a few mm to the side of the trap and the time-of-flight spectrum was observed after the trap light was rapidly turned off, as described in Ref. 5. By pushing the atoms with an additional laser, as was done in Ref. 2, we determined that the trapping potential was harmonic out to a radius of 1 mm with a spring constant of 6 K/cm2. In this experiment, care was taken to obtain a symmetric trapping potential. The Earth’s magnetic field was zeroed to 0.01 G, and the trapping laser beams were spatially filtered, collimated, and carefully aligned with respect to the magnetic field zero. Before these precautions were taken we observed a variety of random shapes and large density variations within the trap. Three separate well-defined modes of the trapped atoms were observed, depending on the number of atoms in the trap and on how we aligned the trapping beams. The “ideal-gas” mode occurred when the number of trapped atoms was less than 40000. The atoms formed a small sphere with a constant diameter of approximately 0.2 mm. In this regime, the density of the sphere had a Gaussian distribution, as expected for a damped harmonic potential, and increased linearly in proportion to the number of atoms. The atoms in this case behaved just as one would expect for an ideal gas with the measured temperature and trap potential. When the number of atoms was increased past 40000, the behavior deviated dramatically from an ideal gas, and strong long-range repulsions between the atoms became apparent. In this “static” mode the diameter of
In the past few years there have been major advances in the trapping and cooling of neutral atoms using laser light. It has recently become possible to hold relatively large samples of atoms for minutes at a time and cool them to a fraction of a mK using a spontaneous-force optical trap.’.’ One would expect these atoms to behave like any other sample of neutral atoms, namely they would move independently as in an ideal gas, except when they undergo short-range collisions with other atoms. In this paper we report on experiments which show that contrary to these expectations, optically trapped atoms behave in a highly collective fashion under most conditions. The spatial distribution of the cloud of atoms is profoundly different from that of an ideal gas, and we observe dramatic dynamic behavior in the clouds. We believe that this unusual behavior arises from a longrange repulsive force between the atoms which is the result of multiple scattering of photons by the trapped atoms. This force is sensitive to the modification of the emission and absorption profiles of the atom in the laser field. In the past, this force has been neglected in any system smaller and colder than stars. However, because of the extremely low temperature and relatively high densities of optically trapped atoms, it becomes very important. The apparatus for this experiment has been described e l ~ e w h e r e . ~ ,The ~ trap was a Zeeman-shift spontaneous-force trap,* and was formed by the intersection of three orthogonal retro-reflected laser beams. The beams came from a diode laser tuned 5-10 M H z below the 6Si12 F-4 to 6 P 3 j ~F-5 transition frequency of cesium. The beams were circularly polarized with the retro-reflected beams possessing the opposite circular polarization. A magnetic field was applied that was zero at the center of trap and had a vertical field gradient (5-20 G/cm) 2 times larger than the horizontal field gradient. Bunches of slowed cesium atoms from an atomic beam were loaded into the trap by chirping a counterpropagat~ were ing “stopping” laser at 20 times a ~ e c o n d . We able to load about 2~ lo6 atoms per cooling chirp. The number of atoms in the trap was determined by the load408
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the cloud smoothly increased with increasing numbers of atoms. Instead of a Gaussian distribution, the atoms in the trap were distributed fairly uniformly, as shown in Fig. l(a). To study the growth of the cloud we measured the number of atoms versus the diameter of the cloud using a calibrated CCD camera. The results are shown in Fig. 2. T o fully understand the growth mechanism the temperature as a function of diameter was also measured. For a detuning of 7.5 MHz, the temperature increased from the asymptotic value of 0.3 up to 1 .O mK as the cloud grew to a 3 mm diam. If the cloud were an ideal gas, the temperature increase would need to be orders of magnitude higher to explain this expansion. One distinctive feature of this regime is that the density did not increase as we added more atoms to the trap.
22 JANUARY 1990
When all the trapping beams were reflected exactly back on themselves we obtained a maximum of 3 . 0 l~o * atoms in this distribution. In contrast, if the beams were slightly misaligned in the horizontal plane, we observed unexpected and dramatic changes in the distribution of the atoms. The atoms would collectively and abruptly jump to “orbital” modes when the cloud contained approximately lo8 atoms. The number of atoms needed for a transition depended on the degree of misalignment. The shapes of these orbital modes are illustrated in Fig. 1 11 (b), l k ) , and l ( d ) show the top view, and I(e) shows the view from the side]. We first observed the ring around a center shown in Fig. l ( b ) and then, by strobing the camera, discovered it was actually a clump of atoms orbiting counterclockwise about a central ball
FIG. 1 . Spatial distributions of trapped atoms. (a) Below 10’ atoms the cloud forms a uniform density sphere. (b) Top view of rotating clump of atoms without strobing. (c) Top view of ( b ) with the camera strobed at I10 Hz. (d) Top view of a continuous ring. ( c ) Side view of (d). Horizontal full scale for (a), (d), and (e) is 1 .O cm; for (b) and ( c ) it is 0.8 cm. 409
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FIG. 2. Plot of the diameter (FWHM) of the cloud of atoms as a function of the number of atoms contained in the cloud. For the full figure the magnetic field gradient is 9 G/cm and 16.5 G/cm for the inset. The laser detuning is -7.5 MHz, and the total laser intensity is 12 mW/cm2. The solid lines show the predictions of the model described in the text.
as shown in Fig. 1 (c). Note that there is a tenuous connection between the clump and the center, and that the core has an asymmetry which rotates around with the clump. As more atoms were slowly added to the trap there were several abrupt increases in the radius of these orbits. The clumps orbited at well-defined frequencies between 80 and 130 Hz. W e have also observed a continuous ring encircling a ball of atoms as shown in Fig. 1 (d). In this case we assume that the atoms are still orbiting, but no clumping of the atoms was seen and there was no connection between the ring and the central ball. We observed only one stable radius (2.5 mm) for this ring. The ring is shaped by the intersection geometry of the laser beams with the corners of the ring corresponding to the centers of the beams. The atoms lie in a horizontal plane about 0.5 mm thick [Fig. l(e)l. As mentioned above, the formation of these stable orbital modes depended on the alignment of the trapping beams in the horizontal plane. The beams had a Gaussian width of 6 mm and the return beams were misaligned horizontally by 1-2 mm (4-8 mrad) at the trap region. It was observed that the direction of the rotation corresponded to the torque produced by the misalignment. The degree of the misalignment affected which of the orbital modes would occur. Typically, the atoms would switch into the rotating clump for misalignments of 1-1.5 mm and into the continuous ring for 1.5-2 mm. If the light beams were misaligned even further, the cloud would be in the ring mode for any number of atoms. The rotational frequencies of the atoms were measured by strobing the image viewed by the camera or by observing a sinusoidal modulation of the fluorescence of the atoms on a photodiode. The frequency depended on the detuning of the trapping laser, the magnetic field gradient, and the average intensity of the trapping light. 410
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For each of these, the frequency of the rotation increased as the spring constant of the trap was increased. We also discovered we could induce the atoms to jump into rotating clumps by modulating the magnetic field gradient at a frequency within 8 H z of the rotational frequency. The transition between the static and orbital modes showed pronounced hysteresis. When the number of atoms was slowly increased until it reached lo8, the cloud would jump in < 20 msec from the static mode to the rotating clump and the trap would then lose approximately half its atoms in the next 50-100 msec. This orbital mode would remain stable until there was a small fluctuation in the loading rate. Then the cloud would suddenly switch back to the static mode without losing any atoms. The number of atoms would have to build up to lo8 again before it would jump back into the orbital mode. We have also observed hysteresis between the static and orbital modes when the trap would jump into the continuous ring. The trap started with 10' atoms in the static mode and then 80% of these atoms would jump into a continuous ring with the remaining 20% in the central ball. For certain loading rates the cloud would repeatedly switch back and forth between the two modes. The expansion of the cloud with increasing numbers of atoms and the various collective rotational behaviors clearly show that strong, long-range forces dominate the behavior of the atomic cloud. Since normal interatomic forces are negligible for the atom densities in the trap (lO'o-lO'' ~ m - ~ )other , forces must be at work. In particular, since the optical depth for the trap lasers is on the order of 0.1 for our atom clouds ( 3 for the probe absorption at the resonance peak), forces resulting from attenuation of the lasers or radiation trapping can be important. The attenuation force is due to the intensity gradients produced by absorption of the trapping lasers, and has been discussed by Dalibard in the context of optical molasses.6 This force compresses the atomic cloud and so cannot explain our observations. In the following, we demonstrate that radiation trapping produces a repulsive force between atoms which is larger than this attenuation force and thus causes the atomic cloud to expand. W e will also show how radiation trapping leads to the rotational orbits we observe. The attenuation force, FA,for small absorption, obeys the relation
The atom density is R. the cross section for absorption of the laser light is uL,and the incident intensity of a single laser beam is I,. The absorbed photons must subsequently be reemitted and can then collide with other atoms. Although this was neglected in Ref. 6, this reabsorption of the light (radiation trapping) provides a repulsive force, FR, which is larger than FA. Two atoms separated a dis-
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tance d in a laser field of intcnsity I repel each other with a force
1 FRI
~ a ~ a ~ ~ / ~ r C d ~ ,
(2)
where for simplicity we have assumed isotropic radiation and unpolarized atoms. (The observed fluorescence is unpolarized.) The cross section UR for absorption of the scattered light is in general different than U L due to the differing polarization and frequency properties of the scattered light.7 For a collection of atoms in an optical trap, the radiation trapping force obeys
V.F l p = 6 u ~ c r ~ l , n l c .
(3)
Comparison of (1) and (3) shows that the net force between atoms due to laser attenuation and radiation trapping is repulsive when UR > U L . In the limit that the temperature can be neglected, FR+FA must balance the trapping force - k r . This implies a maximum tichievable density in the trap of n m a x - c k / 2 a ~ ( a R -c~)l-. This allows us to explain the behavior shown in Fig. 1. Once the density reaches nmax.increasing the number of atoms only produces an increase in the size of the atomic cloud. To compare this model to experiment we numerically solve the above equations with the finite temperature taken into account. The only free parameter in the calculation is UR/OL - 1. A value of C R / U L - 1 -0.3 gives the solid curve shown in Fig. 2, which agrees well with experiment. We have estimated U R / U L by convoluting the emission and absorption profiles for a two-level atom' in a ID standing wave field. The intense laser light causes acStark shifts, giving rise to the well-known Mollow triplet emission spectrum. The absorption profile is also modified by the laser light, being strongly peaked near the blue-shifted component of the Mollow triplet, causing the average absorption cross section for the emitted light to be greater than that for the laser light. We calculate alp la^ - I -0.2 for our detuning and intensity ( 1 - - 2 mW/cm2). While this is less than the value of 0.3 needed to match the data in Fig. 2, it is very reasonable that our twenty-level atom in a 3D field should differ from our simple estimate by this amount. The inset of Fig. 2 shows that the expansion of the cloud deviates from this model for sizes greater than 1.5 mm. However, the precise behavior in this region is very sensitive to alignment and is not very reproducible. Other effects not included in the above model, such as magnetic field broadening, optical pumping, multiple levels of the atom, the spatial dependence of the laser fields, and multiple ( > 2) scattering of the light may also be important in this region. We can explain the rings of Fig. 1 by considering the motion of atoms in the x-y plane when the laser beams are misaligned. A first approximation to the force produced by this misalignment is F I = k ' i x r . The atoms are also subject to a harmonic restoring force - k r and a
22 JANUARY 1990
damping force --ydr/dr. If we ncgicct radiation trapping, we find circular orbits can exist with angular frequency o - k ' / y , if k ' / y - ( k / r n ) Since the orbits may have any radius, this clearly does not explain the formation of rings. If, however, we add a force ( a N / r * ) i due to the radiation pressure from a cloud of N atoms within the orbit radius R,we find circular orbits exist only for (4)
if o < ( k / r n ) Thus the effect of the radiation from the inner cloud of atoms is to cause circular orbits at a particular radius 6.e.. rings are formed), and to allow circular orbits for a large range of misalignments. This is in accord with our observations that the orbits always encompass a small ball of atoms (Fig. 1). We have done a more detailed calculation including the Gaussian profiles of the lasers which gives a rotational frequency of 130 Hz and an orbit diameter of 3.5 mm for the conditions of Fig. I (d). While the simple model of the radiation trapping force we have presented explains the expansion of the cloud and the existence of the circular rings, there are several interesting aspects of the observed phenomena which are not explained. These include the formation of rotating clumps of atoms and the dynamics of the transitions between different distributions. We have produced dense cold samples of optically trapped atoms. We find that this unique new physical system shows a fascinating array of unexpected collective behavior at densities orders of magnitude below where such behavior was expected. Much of this collective behavior can be explained by the interaction between the atoms in the trap due to their radiation fields. However, the detailed distributions and the transition dynamics deserve considerable future study. This work was supported by the Office of Naval Research and the National Science Foundation. We are pleased to acknowledge helpful discussions with T. Mossberg and the contributions made by C. Monroe and W. Swann to the experiment.
ID. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986). 2E. L. Raab, M. G . Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987). 3D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63,961 (1989). 4D.Sesko, C. G. Fan, and C. Wieman, J. Opt. SOC.Am. B 5, 1225 (1988).
5P. D. Lett ef 01.. Phys. Rev. Lett. 61, 169 (1988).
6J. Dalibard, Optics Commun. 68, 203
(1988).
7B. R. Mollow, Phys. Rev. 188, 1969 (1969); Phys. Rev. A 5, 2217 (1972); D. A. Holm, M. Sargent, 111, and S. Stenholm, J. Opt. Soc. Am. B 2, 1456 (1985). 41 1
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a reprint from Journal of the Optical Society of America B
Behavior of neutral atoms in a spontaneous force trap D. W. Sesko, T. G. Walker, and C . E. Wieman Joint Institute for Laboratory Astrophysics, University of Colorado, and National Institute of Standards and Technology, Boulder, Colorado 80309-0440 Received March 27, 1990;accepted October 10, 1990 A classical collective behavior is observed in the spatial distributions of a cloud of optically trapped neutral atoms. They include extended uniform-density ellipsoids, rings of atoms around a small central b d , and clumps of atoms orbiting a central core. The distributions depend sensitively on the number of atoms and the alignment of the laser beams. Abrupt bistable transitions between different distributions are seen. This system is studied in detail, and much of this behavior can be explained by the incorporation of long-range interactions between the atoms in the equation of equilibrium. It is shown how attenuation and multiple scattering of the incident photons lead to these interactions.
1. INTRODUCTION
by a balance between the above forces and the trapping force. This paper reports on a detailed study of the behavior of optically trapped atoms and the comparison with a theoretical model that includes effects of radiation trapping and the attenuation of the trapping light. In Section 2 we present the theory, in Section 3 we describe the experimental apparatus, and in Section 4 we compare the results of the theoretical calculations with the experimental data.
Over the past few years rapid progress has been made in the ability to produce large optically trapped samples of cold atoms.'.' There have been orders-of-magnitude improvements in the number of atoms trapped and the densities achieved. The deepest optical traps are the Zeeman-shift spontaneous force traps. We have used these to trap 4 x lo8 atoms with densities of more than 10" ~ m - ~At . these densities and numbers the gas of cesium atoms becomes optically thick, and forces arising from this optically dense vapor can have a profound effect on the behavior of the cloud. In a recent paper3 we described a variety of collective effects that arise from this force between the atoms, and we presented a model to explain this behavior. Here we provide more detailed experimental studies and analysis. The neutral atoms in these traps might naYvely be expected to behave as an ideal gas except when they undergo short-range ~ o l l i s i o n s . Although ~~~ there have been speculations concerning deviations from classical ideal-gas behavior, these centered on Bose-Einstein condensation. This quantum-mechanical phenomenon occurs at temperatures much lower and densities much higher than in the present experiments and is quite different from the behavior discussed here. In this paper we consider collective behavior in which the atomic motion is classical in mature and the behavior is analogous to that observed in plasmas and charged-particle beams. At the densities that we studied, one would not expect such behavior if only conventional interatomic forces between the neutral atoms were involved because the interactions would be dominated by the few nearest neighbors and not involve the cloud as a whole. In this situation the behavior would be much like that of an ideal gas. In contrast, we find that atoms in a spontaneous force trap actually show ideal-gas behavior only for small clouds of atoms. Larger clouds show dramatic dynamic and collective behavior, as Fig. 1 illustrates. We propose that this behavior arises from a strong long-range coupling between the atoms due to the multiple scattering of photons and from the attenuation of the trapping beams as they pass through the cloud. The density and the size of the cloud of atoms are determined 0740-32241911050946-13$05.00
2. THEORY A. Growth of Cloud We consider the force on a trapped atom to be made up of three contributions. The first is the trapping force produced by the laser beams and the Zeeman shifts of the atomic energy levels. The two others are interatomic forces between the trapped atoms. These are the attenuation force, caused by atomic absorption of the laser photons, and the radiation trapping force, arising from the atoms' reradiating the absorbed photons, which are subsequently scattered a second time by other atoms. The first two forces compress the cloud of atoms, while the latter causes it to expand. The three forces are illustrated in Fig. 2 and will be discussed in turn. The trapping force is a damped-harmonic force and was discussed previously.',' The trap is of the Zeeman-shift spontaneous force type, but the following analysis is suitable for any type of spontaneous force trap forming a damped-harmonic potential. The trapping force can be derived if we consider the force on an atom exerted by three pairs of intersecting circularly polarized beams. Each of these beams is reflected back on itself but with the opposite polarization. We define the polarization of the light with respect to the atoms such that r + (r-) polarization drives the Am = +1 (Am = -1) transitions. These beams intersect at the minimum of a magnetic field of magnitude B, = BfZ. By favoring absorption of one polarization over the other, the magnetic field gradient produces a harmonic potential to first order' in each direction such that F = - k r . The Doppler shift provides a damping that makes the atomic motion overdamped by typically a factor of 10. 0 1991 Optical Society of America
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947
Fig. 3 . Spatial distributions of trapped atoms. (a)With fewer than 10' atoms, the cloud forms a uniform-density sphere. (b) Top view rotating around the nucleus in a counof ratating clump without strobing. (c) View of (b) with camera strobed at 110 Hz. terclockwise direction. (d) Top view of rotating clump without strobing but with a than in (b) (the clump is one quarter of the area of the total fluorescence). (e) Top view of continuous ring. (f) Side view of (e). The horizontal full scale for (a), (d), (e), and (f) is 1.0 cm. For (b) and (c) it is 0.8 cm.
are in the trap, the attenubecomes important. The atThe first is a slight reduction of the spring constant of the trap. Much more important, however, is the force resulting from the local intensity imbalance produced by the absorption, as Figs. Z(a) and Z(c) show. This force was discussed in the context of optical molasses by Dalibard' and Kazantsev et a1." Finally, the intensity of the retroreflected beams is reduced by the first pass through the cloud. This causes an uninteresting shift in the pos
The strength of the force associated with the local intensity imbalance is found in the following way. The lasers are initially propagating in opposite directions along the x axis with intensity I , and are attenuated by the trapped atoms. We define an absorption function
AxW = ( 4 I:%ntr)dx,
(1)
where (VL) is the absorption cross section for the incident laser beams and n(r) is the atomic density o f the cloud.
283
948
Sesko et al.
J. Opt. SOC.Am. BIVol. 8, No. 5/May 1991
where AT is the total absorption of light across the cloud and Z+(I-) is the intensity of the beam traveling in the positive (negative) coordinate direction. The two unattenuated beams are assumed to have the same intensity I,. The force due t o the intensity differential between the positive and negative x beams is then
FA,@')= -(+~)IrnAx(F)/c,
(3)
where c is the speed of light. By following the same arguments for they and z directions, we find that the attenuation force F A obeys the relation 0.7
-
V .F A I
0.6
-0.15
-0.09
I
-0.03
x
(4
2.5
-2.5
1
I -0.15
I
I
-0.09
I
I
0.03
0.09
I
I
-0.03
I
I
I
x
0.15
(4
I
I
0.03
0.09
I 0.15
(4
+
Since our maximum absorption of the trapping beams was approximately 20%, we can use the small-absorption approximation (accurate to 2%). Then the spatial dependence of the intensity as a function of x is Ix,(r) = I41 - [AT(xz)
* Ax(r)l/2),
(2)
-6(~~)~I,n/c.
(4)
The negative sign indicates that this attenuation force compresses the cloud or, equivalently, is an effective attractive force between the atoms. This is the force on the atoms due only to the absorption of the trap laser photons. However, for any real atom these photons must be reemitted, and the re-emitted photons also exert a force. This force is illustrated in Figs. 2(b) and 2(c). It is clear that this must be a repulsive force, because in the process of one atom emitting a photon and a second atom absorbing it, the relative momentum of the atoms is increased by 2hk. In a laser field of intensity I, an atom absorbs and subsequently reradiates energy at rate ( U L ) ~ . Thus the intensity of light radiated by one atom at the position of a second atom located a distance d away is
The force on the second atom due to the light emitted by the first is (Irad/c)(uF), where (UF) is the absorption cross section for the scattered photons. A critical point is that the reradiated light is different from the incident light because of such effects as frequency redistribution and depolarization of the scattered light. Thus the average absorption cross section for the scattered fluorescence light, (Q), is different, in general, from the cross section for the trapping photons, (uL). Using Eq. (5), we obtain a repulsive force between the atoms of magnitude
The preceding argument may be extended from two atoms to an arbitrary distribution of atoms, in analogy to Gauss's law of electrostatics, if we assume that an incident photon is unlikely to scatter more than twice. The radiation trapping force thus takes the form
(c) Fig. 2. Forces that arise within an optically thick cloud of atoms. The diameter of the cloud is 0.2 cm. (a) The change in the intensity of the trapping light due to absorption across the cloud of atoms produces the attenuation force Fax a (uL)(I+ - I-). (h) The spontaneous emission of two atoms separated by a distance d produces the repulsive radiation trapping force between the atoms, F Rcc~( n ~ ) d - ' . (c) The three forces in units of kelvins per centimeter. Note that the total force is FT= - k x FA^ + Fnx = 0 within the cloud.
=
FR(r)=
(uF)(uL)z n ( r ' r) m r' d3r', ~
4TC
(7)
and
v . FR = ~ ( U F(UL)I-TZ(r)/C. )
(8)
Here we have replaced the total average intensity I with the average intensity of each of the six trapping beams, I, = I/6,so that Eq. (8)is in the same form as the attenuation force in Eq. (4). Note that the force falls off as l / r 2 for an atom outside the cloud distribution just as it does for a point charge outside a charge distribution. To have the same repulsive force as that due to radiation trapping under our conditions, an atom would need a charge of aptimes the charge of the electron. proximately 5 x
284 Vol. 8, No. 5/May 199115. Opt. Soc. Am. B
Sesko et al.
We may formulate another derivation of the radiation trapping force for a spherical distribution of atoms of uniform density by using the principle of conservation of photons. Here we have that the total outward force on a spherical shell of radius R enclosing N radiating atoms is given by
where the total reradiated intensity in a laser field of intensity I = 61, is
Ir&= (uL)IN/4rR2.
(10)
This result agrees with the solution of Eq. (7) for a spherical cloud of uniform density. It is a good approximation to what we observe for many experimental conditions. The net force between the atoms due to laser attenuation and radiation trapping can be obtained by adding these two forces. Comparing Eqs. (4) and (8), we see that the net force is repulsive if (uF) > (uL). This condition states that if the cross section for absorption of the reradiated light is greater than that for incident laser light, then the repulsive force due t o the re-emission of photons is greater than the attractive force due to the attenuation of the trapping beams. The resulting net repulsive force leads to the expansion of the cloud and thus limits the density of the atoms. In the limit of zero temperature, if the atoms are in equilibrium the attenuation and radiation trapping forces must balance the trapping force - k r . Empirically, we know that for large clouds the temperature is not greatly important, since the observed value of thermal energy is much less than the trap potential energy (ksT << kR 2/2) for a cloud of radius R. If these three forces add to 0, then the divergences of these forces must also add to 0. By adding the divergence of the trapping force, V . ( - k r ) = -3k, to Eqs. (4) and (8), we arrive at an equation of equilibrium that gives the maximum density of atoms in the cloud, nmax= c ~ / ~ ( u ((uF) L ) - (UL))L,
(11)
for (uF) > ( u L ) . If (UF) IQ ) , then the net force is always compressive and Eq. (11) is invalid. Thus, as atoms are added to the trap, the size of the cloud increases but the density is unchanged. We can obtain a more general calculation of the atomic distribution by numerically solving the equation of equilibrium in which a pressure gradient is balanced by the trapping and radiation forces. This is somewhat like the equation of hydrostatic equilibrium in stars, with the trapping force having replaced gravity.8 In our case the pressure gradient across the cloud is V P = TVn + nVT. For simplicity, we assume the temperature gradient across the cloud to be negligible, so for the atoms to be in mechanical equilibrium within the trapped cloud we have TVn(r) = (FA
+ FR- kr)n(r).
(12)
For the special case FA + F R - k r = 0 the density within the cloud is a constant, and we obtain Eq. (11). Taking the divergence of Eq. (12) and using Eqs. (4), (8), and (11) gives TV2 ln[n(r)] = 3k(n/n,.,
- 1).
(13)
949
We may then numerically integrate Eq. (13) for a cloud of
N atoms to find the spatial distribution of the atoms. For the calculation of the spatial distribution of the atoms in the cloud, we assumed strong damping, no convective motions, and no temperature gradients. We also ignored the standing-wave patterns of the lasers (except in the averaging for (UF) and (UL) as discussed below) and the spatial profiles of the lasers. We emphasize that we are not claiming a priori that these and other complications discussed below should (or should not) be significant. We are simply presenting a model that is made tractable by the neglect of them and then showing the successes (and failures) of this model in its explanation of the observations. The next stage of the calculation is to find (UF) and ( u L ) for the atoms in a laser field produced by three intersecting beams. The combination of beams creates a threedimensional standing-wave field
I(x,y, Z ) = 61,
COS'
2TX 2 T y 2rrz -+ - h
h
+
h
+4
where 4 is an arbitrary phase factor. We may visualize this expression for the intensity by first considering the result of two counterpropagating beams of opposite circular polarization. This results in a local linear polarization that traces out a helix in space. Equation (14) follows, then, if we add the light from the two other directions. We obtain the spatially averaged cross section (UL) by averaging the cross section uL = v0[l + Z(x, y, z)/Zs + ~(A/AN)']-~ over a wavelength in an arbitrary direction, where uois the unsaturated, resonant cross section, Is is the saturation intensity, A = W L - oois the trap laser detuning, and A N is the natural linewidth of the transition. We find that
We used this spatially averaged cross section, since this was the approach used in the calculation of (up) in Ref. 9. Note that there are other effects that are, in principle, important in calculating (uF)and (uL), but we neglected them in the spirit of simplification discussed above. These include magnetic field and Doppler shifts, the polarization and coherence properties of the light, local optical pumping of the atoms, and Zeeman precession of the atoms in the magnetic fields. The spatially averaged cross section (UF) for absorption of the reradiated light is different, in general, from that for the incident laser light, (uL). This is due to a change in the frequency properties of the scattered light produced by the ac Stark effectg (in our experiments the average light intensity is 6-12 times the saturation intensity). Calculating (uF)/(uL) exactly is difficult for optically trapped cesium because of the many levels involved, each of which is shifted by the light field. For the purposes of this work we crudely estimated (uF)/(uL), using the emission and absorption spectra for a two-level atom in a monochromatic one-dimensional standing-wave field. These spectra are illustrated in Fig. 3. The scatteredradiation spectrum contains an elastic component at the frequency of the incident photons and an inelastic component, which is the result of the ac Stark shift by the time-
950
Sesko et al.
J. Opt. SOC. Am. B/Vol. 8, No. 5iMay 1991
-ydr/dt. Note that the tangential force k ‘ 2 x r plus the damping force leads to a net orbital drift velocity. The existence of a small ball of atoms within the rings (Fig. 1) suggests that a force (aq,N/r2)? due to radiation pressure from a central cloud of q,N atoms needs to be added as well as a radiation pressure force [ q , N ln(2qrN/7r)/ 2ar2]?on an atom in the ring due to the q,N other atoms Adding these, we obtain in the ring ( a = (uL)(uF)Z/4m). the equation of motion for the trapped atom:
40 -
30 -
m-d ‘r = -kr dt2
- y-dr - k‘2
x r
dt
+ -?. (YN‘
(16)
r2
Here N ’ is the effective number of atoms that provide a repulsive radiation rapping force on the atom and is given by
-1.0
-7.5
-5.0
-2.5
0.0
2.5
5.0
7.5
A/A, Fig. 3. Absorption (solid curve) and emission (dotted-dashed curve) profiles calculated for a two-level atom in a onedimensional standing-wave field. The curves are for an average intensity of 12 mW/cmz and a laser detuning of -1.5A~. The emission profile is in arbitrary units. The zero-width elastic emission peak (which constitutes 50% of the total) is represented by the spike at - 1 . 5 A ~ . dependent fields. This gives the familiar Mollow triplet for the emitted light. The absorption profile is strongly peaked near the blue component of this Mollow triplet. This causes the average absorption cross section for the emitted light, (up), to be greater than that for the incident light, (u~); this critical difference is what causes the combination of attenuation and radiation trapping forces to give a net repulsive force between the atoms. Taking the convolution of the emission and absorption spectra in Fig. 3 to calculate (uF), we find that ( u p ) / ( u L ) = 1.2 for our experimental parameters (a detuning of -1.5AN and a total average intensity 61, = 12 mW/cm2, where AN = 5 MHz is the natural linewidth). Because (uF) > (uL), Eq. (11)applies, and we expect the maximum density to be We emphasize that although a numlimited to n c n-. ber of approximations and simplifications were made in the calculation of ( u ~and ) (uL), the basic physics of the expansion of the cloud relies only on the relation (uF) > (L). As long as the approximations do not change this relation, the cloud expands, subject to a maximum density given by Eq. (11).
B. Formation of Stable Orbits The discussion in Subsection 2.A is valid for any number of atoms as long as the optical thickness is small and the pairs of trapping beams are exactly overlapping. However, a misalignment of the retroreflected trapping beams can put a torque on the cloud of atoms. As Subsection 4.D describes, we observe abrupt transitions t o orbiting rings of atoms surrounding a central nucleus when the number of atoms exceeds a critical value. The formation of the stable orbits can be most easily understood by a consideration of the motion of atoms in the presence of a misalignment of the laser beams in the x-y plane (Fig. 4). Such a misalignment produces a local imbalance between the beams, and thus a force that may be represented phenomenologically by F, = k’& x r. The optical trap also provides a restoring force -kr and a damping force
In order to compare theory and experiment precisely, we must consider the spatial dependence of the trapping beams, which gives an r dependence to k , k’,and ‘y. However, it is illustrative to ignore this spatial dependence for the moment so that Eq. (16)can be solved analytically. By trying the oscillatory solution r = R(32 cos ot + 9 sin w t ) , we find that stable circular orbits exist at a radius
R
=
[a’/(k-
mo2)11/3,
(18)
with orbital frequency o = k’/y Thus we obtain orbital trajectories at a radius R as long as the condition o < (k/m)”’ is satisfied. This condition on the orbital frequency is violated if the torque on the cloud is too high or the damping is too small. In these cases the atoms fly out of the trap rather than orbit. Equation (18) shows that if the radiation pressure is increased (larger a“), the radius of the rings increases. The orbital radius also increases when the spring constant of the trap is decreased
Fig. 4. Geometry of the misalignment in the horizontal plane that gives rise to the orbital modes. The bold boundary and the central ball show the positions of the atoms. The arrow indicates the direction of the orbit. The radiation pressure from the central core is also illustrated.
286 Sesko et al.
Vol. 8, No. 5iMay 199115. Opt. SOC.Am. B
or the frequency of rotation increases. If the radiation pressure term is not included in Eq. (16), we find that there are no stable orbiting solutions. We performed a more general calculation of the orbits than one would obtain by including the r dependence of k , k ' , and y. Rather than using Eq. (16)to calculate this, it is more natural for one to consider the force due t o four separate laser beams. We start by considering the spontaneous force Fxz on an atom for a single beam propagating in the positive or negative x direction, where the Gaussian shape of the beams, the saturation effects, the Zeeman shifts, and the Doppler shifts are included. This force is assumed t o be given by
951
.... 0.2 -
T
g ,,
0.0
-
x
-0.2
-
/ I
-0.4
where Zs is the saturation intensity (IS= 1 mW/cm'), Z(r) is the total local laser intensity, A = O L - w o is the detuning of the laser from line center, oB' is the Zeeman shift in frequency per unit length, and A is the wavelength of the light. This equation for the force is what one obtains for a two-level atom, adding an ad hoc Zeeman shift and assuming that the intensities can be summed to determine the saturation. Although this assumed form is not exactly correct, we believe that it is a good approximation, which contains the relevant physics. The Gaussian intensity profile for the laser beams propagating in the - t x direction is given by
where s is the displacement of the beams with respect to each other and w is the full width a t half-maximum (FWHM) of the Gaussian beam profile. By summing the forces in the forward and reverse directions and adding a radiation trapping force
I
0.0
-0.2
-0.4
I
I
0.2
0.4
(4
x Fig. 5. Trajectories of an atom numerically calculated from our model. The trajectories are shown for ten initial positions and velocities of the atom. The trap parameters were chosen t o be the same as for the conditions in Fig. l(e) (A = - 1 . 5 A ~ ,I/Is = 12, B' = 15 G/cm, w = 6 mm, and s = 1.5 mm). The frequency of the orbit is 82 Hz.
We first numerically solve the equations of motion for a radiation trapping force corresponding to an effective number of atoms, N ' = 5 x lo7. Figure 5 shows the solutions to the equations of motion for several arbitrary starting positions and velocities of the atoms. The model is seen to predict that the atoms orbit in a ring that closely resembles the photographs in Fig. 1. The orbit has a mean radius of 2.5 mm and a rotational frequency of 82 Hz. For N ' = 9 x lo7 the equations of motion predict that the atoms slowly spiral out of the trap, and for larger N ' the atoms are rapidly ejected from the trap. If we examine the case of no radiation trapping force ( N ' = 0), we find that the atoms spiral into the center of the trap. Thus the radiation trapping force is necessary to an explanation of the existence of the orbits.
we obtain a net force
Fx
= FRx
+ ThANz, ~
A Is
{
(g+ - g-) 1 X
Z(r) 4[A2 + ( w B ' x + ux/A)'] ++
Is Z(r) 4[A2 + {I+%+
AN2 (UB'X
+ FyyI,
A.v4
AN'
(23)
where Fr is given by the transposition of x and y and the substitution of uy for ux in Eqs. (21) and (22). These equations of motion for a single atom are solved numerically for the conditions A = -1.5A~,Z(r) = 12 mW/cm', and s = 1.5 mm. The value of W B ' = doB/dr is difficult to calculate, since the atoms are distributed among the various m levels. However, from our measurements of the spring constant and using the correct limits in Eq. (22), we derive a value of W E ' = 10 MHz/cm.
uX/A)
AN'
+ ux/A)'] - 64A'(o~'x + U X / A ) ~'
The totaI force on the atom is found by adding the forces in the x and y directions and is thus
F = FX2
A ( w ~ ' x+
3.
I
'
(22)
EXPERIMENTAL APPARATUS
To produce a cloud of trapped cesium atoms, we started with a thermal atomic beam with an average velocity of 250 m/s. These atoms were slowed to a few meters per second by counterpropagating laser light. The slowly moving atoms then drifted into the spontaneous force Zeeman-shift trap described below and were held there for times on the order of 100 s. The trap cooled the atoms so that their average velocity was approximately 20 cm/s in the nonorbiting mode and compressed the atoms to densities of as much as 10l1~ m - ~We . studied the cloud of trapped atoms by observing the total fluorescence with a
287 952
J. Opt. SOC.Am. BIVol. 8,No. 5/May 1991
Sesko et al.
/I
TRAP LASER
HYPERFINE PUMPING ANTI-HELMHO
Fig. 6. Schematic of the experimental apparatus. The third trapping beam, perpendicular to the page, is not shown.
photodiode and the spatial distribution of the fluorescence with a calibrated charge-coupled device camera. A schematic of the experimental apparatus is shown in Fig. 6. A beam of cesium atoms effused from an oven at one end of the vacuum chamber. A capillary array was used as an output nozzle for the oven to create a large flux of cesium atoms." These atoms were then slowed by a frequency-chirped diode laser entering from the opposite (trap) end of the chamber. The distance from the oven t o the trap was approximately 90 cm. This stopping laser drove the 6S1,2,F=4 + 6P312,p Z 5 resonance transition. A second laser beam, which overlapped the first, excited the 6Sl,z,p=3-+ 6P3,2,~=4 transition to ensure that atoms were not lost to the F = 3 ground state. The frequencies of these two lasers were swept from 500 MHz below the atomic transition to within a few megahertz of the transition in approximately 15 ms. At the end of each chirp the stopping laser was quickly shifted out of resonance, and the F = 3 state depletion laser remained at the same frequency that it had possessed at the end of the ramp. It took atoms 35 ms after the end of the chirp to drift into the trap, after which the 15-ms cooling chirp was repeated. The frequency of the stopping laser at the end of the chirp determined the final drift velocity of the atoms. The number of atoms loaded per chirp was largest when the velocity was approximately 8 m/s. We were able to load approximately 2 x lo6 atoms per chirp into the trap under these conditions. The total number of atoms in the trap was determined by the balance between the loading rate and the collisional loss rate. When no loading was desired, shutters blocked the cesium and stopping laser beams.
The trap was similar to that described in Refs. 1 and 2. The trapping light was produced by a third diode laser tuned below the 6Sliz,F=4 -+ 6P3/2,F = 5 resonance transition. Data were taken over the range of detunings from 5 to 15 MHz. The output from this laser was passed through an optical isolator and was spatially filtered. It was then carefully collimated and split into three circularly polarized beams of 6-mm diameter (Gaussian halfpower widths). These intersected orthogonally at the center of a pair of anti-Helmholtz coils with a 25-mm radius and a 32-mm separation. The beams were then reflected on themselves with the opposite polarization. The magnetic field produced by the coils was 0 at the center of the trap and had a vertical field gradient (5-20 G/cm) two times larger than the horizontal field gradient. The center of the trap was positioned 1 cm below the stopping laser so that the slowing process would not eject the previously trapped atoms. A fourth laser tuned to the 6Sl,z,F=3-+ 6P3/z F=4 transition pumped atoms out of the F = 3 ground state in the trap. All lasers used in this experiment were stabilized diode lasers with long-term stabilities and linewidths of well under 1 MHz. We frequency-stabilized the stopping and trapping lasers by optically locking them to an interferometer cavity." This was done by our sending approximately 10% of the laser light t o a 5-cm Fabry-Perot interferometer located a few centimeters from the laser. The cavity was tilted slightly so that the beam entered at an angle relative to the cavity axis. The cavity returned of the order of 1%of this light to the laser and caused the laser to lock to the resonant frequency of the cavity. The two other lasers, which were used for hyperfine pumping,
288 Sesko et a1
were stabilized by feeding back light from a grating in the Littrow configuration. The front surface of these laser diodes was antireflection coated so that the grating served as a tuning element for the laser frequency and as one of the end mirrors of the laser cavity. The frequencies of all four lasers were monitored with small cesium saturatedabsorption spectrometers. The absorption signals were used to derive error signals, which were fed back electronically to control the laser frequencies by changing the laser currents and/or cavity lengths. In this experiment we observed that the shape and the density of the cloud of atoms depended sensitively on the alignment of the trapping beams. We had also discovered, in a previous experiment5 on collisional loss of atoms from the trap, that the depth of the trap was sensitive to the alignment. For a reproducible experiment, we found that care must be taken to obtain a symmetric trapping potential. We could obtain consistent results only if we carefully centered all the spatially filtered trapping beams on the magnetic field zero while the vacuum chamber was open. In addition, the Earth's magnetic field was zeroed t o 0.01 G so that the field gradient of the trap was radially symmetric. Before these precautions were taken, we observed a large variety of density variations and unpredictable cloud shapes within the trap. We note that motions of atoms in traps with misalignment were briefly described in Ref. 2, but it is difficult to compare our results with those because of the brevity of that discussion.
4.
RESULTS
We now discuss our experimental observations and compare them with our models. First we determined the shape of the trap potential and measured the temperature of the trapped atoms. We then measured the dependence of the cloud shape and size on the number of atoms in the trap. Finally, we studied the atoms' behavior in the orbital mode. This included observing the shape of the orbits, the orbital frequencies, and the transitions between modes. 1.50
-E -E
I
I
I
I
I
I
0.25
0.50
0.75
1.00
1.25
1.50
1.25
1.00
c
C a,
E
0.75
a,
u 0 -
2
0.50
.-
n 0.25
0.00
0.00
953
Vol. 8, No. 51May 199115. Opt. SOC.Am. B
1.75
I (mW/crnz) Fig. 7. Displacement of a small ball of atoms versus intensity of the pushing beam (circles). The solid-line fit shows that the force is harmonic and gives a spring constant of 6 K/cmz. The laser detuning was - M A N , the total intensity was I/Is = 12, and the magnetic field gradient was 15 G/cm. The pushing beam was at the same frequency as the trap laser.
.-. .= u)
24.0
3 3r
$
20.0
.4 L
0
16.0
a,
u
t
C
8
12.0
VI
aJ L
0
2
8.0
LL
4.0
I 0.00
I
0.01
I
0.02
I
0.03
I
0.04
I 0.05
t (sec) Fig. 8. The dots show the TOF spectrum, after the trap light is turned off and the fluorescence is measured, of a probe beam 4 mm to the side of the cloud. The solid curve is a fit to the data for a Maxwell-Boltzmann distribution (T = 265 pK) that heavily weights the rising edge of the TOF spectrum. A. Trap Potential We measured the trap potential by observing the displacement of a small ball of trapped atoms when they were pushed by an additional laser, as was done in Ref. 2. This pushing laser was split off from the trapping laser beam and was linearly polarized. The plot of displacement versus pushing intensity is shown in Fig. 7. The fit indicates that the trap potential was harmonic out to a 1.5-mm radius. The spring constant was 6 K/cmZfor a detuning of -1.5AN and a magnetic field gradient of 15 G/cm. We were unable to measure the trap potential beyond 1.5 mm because at larger radii it was difficult to displace the small ball of atoms in a line.
B. Temperature of Atoms To measure the average temperature of the trapped atoms, we passed a 0.5-mm-diameter probe beam 4 mm from the side of the cloud. We then observed a time-of-flight (TOF) spectrum" after the trap light was rapidly shifted out of resonance. This shift effectively turns off the optical trap quickly, and we accomplished it by switching the current on the trapping laser so that the laser frequency was moved many gigahertz from the atomic transition. The switching time with this technique was well under 10 p s . Approximately 0.5 ms later, all the lasers, except for the probe laser and pumping laser 1, were mechanically shuttered to reduce the scattered light in the vacuum chamber. We then obtained the TOF spectrum from the fluorescence of the probe laser. Figure 8 shows the average of 16 TOF spectra. In calculating the theoretical fit in Fig. 8, we assumed a Maxwell-Boltzmann distribution of velocities and heavily weighted the rising edge of the TOF spectrum. The fit gives a temperature of 265 pK for a cloud with approximately 5 x lo6 atoms. The poor fit shows that either there is a spatially dependent temperature in the cloud or the velocity distribution is non-Maxwellian. The plot of temperature (as defined by the above arbitrary procedure) versus cloud diameter is shown in Fig. 9(a) for two detunings. In both cases the tempera-
289 954
J. Opt. SOC.Am. B/Vol. 8, No. 5/May 1991
Sesko et al.
A=-1.5AN
0
0.0
0
A=-2.OAN
0.8
1.6
3.2
2.4
4.0
FWHM (mm)
(4 2.5
I
I
in the trap and on how the trapping beams were aligned. These modes are described in detail here and in Subsection 4.D. The ideal-gas mode occurred when the number of atoms was fewer than approximately 80,000. The atoms formed a small sphere with a diameter of approximately 0.2 mm. We found the distribution of the atoms by digitizing the image of the cloud and assuming that the observed light intensity was directly proportional t o the number of atoms along the line of sight. The density distribution of the atoms is shown in Fig. lO(a) and clearly matches the Gaussian dependence that one would expect for an ideal gas in a harmonic potential. The density increase was proportional to the number of atoms as atoms were added, but the diameter remained unchanged. Using the measured spring constant, we derived a temperature from the fit to the distribution. We found a temperature of 300 p K for the atoms, which agrees well with the TOF temperature measurement.
I
150.0
I
I
0.1
0.2
I
I
I
0.3
0.4
0.5
-
2.0
v1
125.0
%'
5 P
3
1.5
r
&
3 +
100.0
L
-
+
P W
D
r P
L
2
75.0
a,
:
U
50.0
0.5
0 u
ar L
0.0 -16.0
I
-12.0
I
-8.0
2 25.0 -
I
-4.0
0.0
Detuning (MHz) (b) Fig. 9. Measured temperature of the atoms versw the diameter (FWHM) of the cloud for detunings of - M A N (filled circles) and -2.5AN (open circles). (b) Measured temperature of the atoms versus detuning for a 1.5-mm-diameter cloud. The total laser intensity is 12 mW/cmz, and the magnetic field gradient is 15 G/cm.
LL
0.0
0.0
' (mm) (4
0.6
c n
c
*
ture increases with increasing numbers of trapped atoms. We speculate that this heating arises from radiation trapping. From Fig. 3 we note that since the blue-shifted light is preferentially absorbed, the light that escapes the cloud is red shifted on average from the incident trapping light. The excess energy may then be converted to kinetic energy of the atoms. We did not investigate in detail how this occurs. Figure 9(b) shows the plot of measured temperature versus detuning for clouds of a fixed diameter (1.5 mm). No significant temperature changes were observed when the magnetic field gradient was changed or when the intensities of the lasers were changed. We emphasize that, although the temperature rises with increasing numbers of atoms, the temperature rise would need to be orders of magnitude larger to explain the observed expansion of the cloud. C. Dependence of Cloud Shape on Number of Atoms Three well-defined spatial modes of the trapped atoms were observed. These depended on the number of atoms
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I (mm) ( b) Fig. 10. Top view of fluorescence profile of the cloud. (a) Data for the ideal-gas mode (circles) with a fit (solid curve) for a Gaussian distribution. (b) Data for the static mode (circles) with a fit (solid curve) assuming that we are looking along the minor axis of a ellipsoid of constant density. The fit gives a ratio of 1.5 for the ellipsoidal axes.
290 Sesko et al.
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Number of atoms (X107) Fig. 11. Plot of the diameter (FWHM) of the cloud of atoms as a function of the number of atoms in the cloud (circles). For the main figure the magnetic field gradient is 9 G/cm, and for the inset it is 16.5 G/cm. The laser detuning is - M A N , and the total laser intensity is 12 mW/cmz. The solid curves show the predictions of the mode that are described in the text.
When the number of atoms was increased past 80,000, the cold atoms no longer behaved as an ideal gas and the presence of a strong long-range force was evident. This static mode was characterized by an elliptically symmetric cloud, the diameter of which increased with increasing numbers of atoms. One distinctive feature of this regime was that the density did not increase as we added more atoms to the trap. Instead of following the Gaussian distribution of the ideal-gas mode, the atoms were distributed fairly uniformly, as Fig. 10(b) illustrates. We obtained these data by looking through the cloud along the axis with the greater (ax) magnetic field gradient. The solid curve shows the calculated distribution of fluorescence for an ellipsoidal cloud of atoms with an aspect ratio of 1.5 and constant density. The cloud's minor axis was along the direction of the higher magnetic field gradient. By looking along a major axis, we found that the aspect ratio varied from approximately 1 for a 1-mm diameter (FWHM) cloud to 2 for a 4-mm cloud. We studied the growth of the cloud by measuring the number of atoms versus the major-axis diameter of the cloud. We performed this measurement by filling the trap with as many atoms as possible and then turning off the loading. With a charge-coupled device camera we recorded the evolution of the fluorescence as the atoms were lost from the trap and stored this record on video tape. We calibrated the video signal by also measuring the total fluorescence with a photodiode. To analyze these data, we digitized the frames of the video and stored them for analysis. We determined the total fluorescence in each image to obtain the number of atoms, and we calculated the FWHM diameter of the image. The results are shown in Fig. 11. Note that if the cloud were an ideal gas, the temperature would have t o be of the order of 50 mK to explain the cloud's expansion. This supports our earlier claim that the expansion is not due to a rise in temperature. We obtained the fit (solid curve) to the data (circles) in Fig. 11 by integrating Eq. (13), given N atoms, t o predict the FWHM diameter of the cloud. We assumed the temperature dependence shown in Fig. 9(a) for a detuning of
955
- M A N , but the results are insensitive to this parameter. The only free parameter in this calculation is the ratio (uF)/(uL). A value of (uF)/(uL) = 1.3 gives the solid curve shown in Fig. 11. This is greater than the value of (uF)/(uL) = 1.2 given by the theory, discussed in Section 2, for a two-level atom in a one-dimensional standing-wave field. However, it is not surprising that our cesium atoms with 20 relevant levels in a three-dimensional field differed from the two-level case by this amount. The inset of Fig. 11 shows that our theoretical model for the expansion of the cloud deviates from the data at diameters greater than 1.5 mm. In fact, these data show that the density reached a maximum and then started to decrease as we added more atoms. However, the assumption of a harmonic potential may break down in this regime. Also, the cloud stability at such large sizes was quite sensitive to the alignment and the spatial quality of the trapping beams. Other effects that are not considered in this model but that may be important for clouds of this size are magnetic field broadening, multiple levels of the atom, polarization of the scattered light, and multiple (>2) scattering of the light. Taking these effects into account makes the equation of equilibrium for large clouds quite complex. We were able to obtain as many as 4 x 10' trapped atoms in this static mode if the trapping laser beams were retroreflected ( e l mrad). However, if the return beams were slightly misaligned in the horizontal plane, dramatic and abrupt transitions to orbiting clumps or rings occurred at approximately 10' atoms. D. Rotating Clumps and Rings We now describe the observed spatial distributions and orbital motions for the third mode of the trapped atoms, which we call the orbital mode. This mode is characterized by atoms orbiting a central ball and abrupt transitions between different orbital modes and between static and orbital modes. We studied how the atoms' motion depends on a variety of parameters. Here we compare the data with the numerical model described above. This model explains much of the equilibrium behavior; however, it does not explain the clumping or any of the transition dynamics that are 0b~erved.l~ Photographs of the orbital modes are shown in Fig. 1 [Figs. l(b)-l(e) show the top view, and Fig. l(f) shows the side view]. We first observed a ring surrounding a dense central ball of atoms (henceforth known as the nucleus), as Fig. l(b) shows. By strobing the camera, we then found that the ring was actually a clump of atoms rotating counterclockwise around the nucleus, as Fig. l(c) shows. Note that there is a tenuous connection between the nucleus and the clump and that the nucleus has an asymmetry that rotates with the clump. The orbital radius of these clumps was insensitive to the trapping laser intensity, the detuning, and the magnetic field gradient. The rotational frequency did depend on these parameters, which we discuss below. We also observed clumps rotating at smaller radii, as Fig. l(d) shows. These smaller orbits occurred when fewer atoms were in the trap. The rotating clump had as many as three different radii with abrupt transitions between them as the number of atoms changed. The different orbital radii depended on the number of trapped atoms and varied in size by a factor of 3. Within each of the intermediate stages the radius of the orbit slowly increased as we added more atoms until
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10'
10"
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0.0
30.0
60.0
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TIME (sec) Fig. 12. Fluorescence from a cloud of trapped atoms as a function of time after the loading is terminated. The main figure shows the decay of atoms from the static mode into the ideal-gas mode. The inset shows the decay from an orbitingmode through an abrupt transition (indicated by the arrow) to the static mode.
the cloud suddenly jumped to the next stage. The rotating clump in these intermediate stages was not so extended as it was for the larger radii shown in Fig. l(c), and there was a large number of atoms connecting the clump t o the nucleus. The size of the clump took up approximately one quarter of the area shown in the unstrobed image. Another orbital shape that we observed was a continuous ring surrounding a nucleus [Fig. l(e)]. We assumed that the atoms were still orbiting in this mode but that no clumping of the atoms occurred. Unlike in the rotating clump case, the radius of the continuous ring increased by a factor of 2 as we increased the detuning or decreased the magnetic field gradient. The rectangular shape of the ring was formed by the intersection geometry of the laser beams, with the corners of the ring corresponding to the centers of the beams. The atoms lay in a horizontal plane approximately 0.5 mm thick [Fig. l(f)]. Approximately 80% of the atoms were in the outer ring, with the remaining 20% contained in the nucleus. The formation of these orbital modes depended critically on the alignment of the trapping beams in the horizontal plane. The beams had a Gaussian width of 6 mm, and the return beams were misaligned horizontally by 4-8 mrad (1-2 mm) at the trap region. It was observed that the direction of rotation corresponded to the torque produced by the misalignment (Fig. 4). If the vertical trapping beam was misaligned, the plane of the rotation could be tilted by as much as 20", but no stable orbits were observed in the vertical direction. We believe that this was due t o an asymmetry in the trapping force caused by the vertical magnetic field gradient's being two times larger than that of the horizontal. The shape of the rings depended on the alignment and could be changed by tilting the retroreflection mirrors. The rings had stable orbits at radii from 2 to 3 mm. No rings were observed at radii < 2 mm. At radii > 3 mm the ring would form for only a few tenths of a second, then become unstable, and the atoms would fly out of the trap. The degree of misalignment in the horizontal plane and the number of atoms determined which of the orbital
modes occurred. When enough atoms were loaded into the trap, the cloud typically switched into the rotating clump for misalignments between 4 and 6 mrad and into the continuous ring for those between 6 and 8 mrad. If the light beams were misaligned even further, the atoms continuously loaded into the ring mode for any number of atoms, and for large misalignments we could even form a ring with no visible nucleus. In Ref. 3 we reported that a nucleus of atoms always exists, but we can now make rings without nuclei by a suitable adjustment of the various trap parameters. We also studied how the trap loss rate depended on the mode. This was done by observation of the time evolution of the cloud when no additional atoms were being loaded. Another study of interest was to see how long various cloud modes survived as the number of atoms decreased owing to atomic collisions. These data are shown in Fig. 12. The total fluorescence from the cloud, which we assumed to be proportional to the number of atoms, indicates that there were initially 4 x lo7atoms in the orbital mode (inset of Fig. 12) and that the fluorescence decreased exponentially with a time constant of T = 9 s. With 1.5 x l o 7 atoms remaining, the cloud abruptly jumped into the static mode, and the number of atoms continued to decrease exponentially with time (T = 10 s). This transition is indicated by the arrow in the figure. The main plot in Fig. 12 shows a decay of fluorescence from a different run starting with 4 x lo6 atoms in the static mode. In this run the initial time constant was approximately 17 s rather than the 10 s shown in the inset, owing to a lower vacuum pressure in the chamber. As the cloud started to change from the static mode to the ideal-gas mode after approximately 30 s, the loss rate increased. We believe that this was due to an increase in losses resulting from collisions between the trapped atoms, as Ref. 4 describes. At this point the density had reached its maximum value. Finally, after so many atoms were lost that the density was much lower (=70 s), the fluorescence decayed with a time constant of T = 60 s. This loss rate was due entirely to the collisions with the 2 x 10-l' Torr of background gas in the vacuum chamber. The rotational frequencies of the clumps were monitored either by strobing the image viewed by the camera or by observing the sinusoidal modulation of the fluorescence by a photodiode [Fig. 13(a)]. The peak-to-peak amplitude of this modulation was from 10% to 50% of the total fluorescence. Although it is obvious that the fluorescence from a limited portion of the orbit should vary, it is somewhat surprising that the total fluorescence should show such large modulations. The frequencies of rotation were between 80 and 130 Hz for the larger radii as we changed the trap parameters and were approximately 15% higher for the smaller orbits. As Fig. 14 shows, the orbital frequency depended on the detuning of the trapping laser (A), the magnetic field gradient (I?'), and the intensity of the trapping light ( I ) . Changing any of these parameters so as to increase the spring constant of the trap caused the rotation frequency t o increase. The circles in Fig. 14 show the data points, and the pluses show the results of our model when we use Eq. (22). The experimentally observed orbital radius varied by less than 10% over the range of detunings, magnetic field gradients, and intensities used. In the calculation of the frequency as
292 Vol. 8, No. 5/May 1991/J. Opt. SOC. Am. B
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(b) Fig. 13. (a) Time dependence of total fluorescence for the rotating clump without modulation of the magnetic field. (b) Beat signal of total fluorescence produced after the transition is induced to the rotating clump mode. The orbital frequency was 112 Hz, and the magnetic field was modulated at 119 Hz. The peak-to-peakmodulation was approximately 40% of the total fluorescence for both (a) and (b).
a function of the magnetic field gradient and the laser intensity, the number of atoms composing the nucleus i n our model was adjusted to keep the orbital radius constant because we saw no change i n the experimental radius. The calculated radius had little sensitivity to the laser detuning, so that no adjustment was needed in the calculation of values of orbital frequency versus detuning. The calculated values show excellent agreement with all t h e experimental data, considering the simplicity of this model. One of the notable features that this model does not explain is the observed abrupt transitions between modes. These transitions occurred when the number of atoms exceeded a particular critical value, and they lasted roughly 20 ms. The transition to the largest rotating clump mode involved usually several intermediate transitions to rotating clumps with smaller orbital radii, as we discussed above. In the case of the continuous ring the atoms always jumped directly into this mode from the static mode, with no intermediate stages. We discovered that we could induce the transitions from the static mode to rotating clumps at fewer than the usual number of atoms (=los) by modulating the magnetic field gradient at a frequency close (within 8 Hz) to the rotational frequency. In this case the number of atoms needed
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Fig. 14. Dependence of the rotational frequency of the clump on (a) the laser detuning (B’ = 12.5 G/cm, I = 12 mW/cmz)),(b) the magnetic field gradient (A = -1.5A~,I = 12 mW/cmz),and (c) the total average intensity (A = - 2 . 0 A ~ B’ , = 15 G/cm). The circles are the experimental data, and the pluses give the results of our numerical model.
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for the transition (=lo7) depended on the amplitude and the detuning of the magnetic field modulation from the orbital resonance. Another amusing phenomenon was the following: after the cloud had made a transition, the fluorescence was modulated at the beat frequency between the two frequencies. This is shown in Fig. 13(b). Profound hysteresis effects were observed in the transition between the static and the orbital modes. If the number of atoms was slowly increased to lo', the cloud suddenly jumped from the static mode to the rotating clump, and the trap lost approximately half of its atoms in the succeeding 50-100 msec. This orbital mode remained stable at 5 x lo7 atoms until there was presumably a small fluctuation such as a change in the loading rate. Then the cloud switched back to the static mode without losing any atoms. It was necessary that the number of atoms build back up to 10' before it could again jump into the orbital mode. Thus the orbital modes can be self-sustaining at lower numbers of atoms than those required for a transition. This implies that when the atoms are in a different mode, the cloud obeys a different equation of equilibrium. We occasionally observed a different hysteresis in the transition between the static and continuous ring orbital modes. The cycle started in the static mode with 10' atoms. Subsequently, 80% of these atoms jumped into a continuous ring, while 20% remained in the nucleus. In the succeeding couple of seconds most of the atoms in the ring transferred back into the nucleus. When this occurred, the nucleus grew in size and the radius of the ring decreased. Finally, with approximately 70% of the initial atoms back in the nucleus and 30% in the ring, the cloud collapsed back into the static mode. This transition occurred just before the outer diameter of the nucleus and the inner diameter of the ring merged. During this process the total number of atoms in the trap remained relatively constant. The cloud repeatedly switched between the two modes with an oscillation period of a few seconds. In addition t o the abrupt collective transitions there are a variety of other features of the orbital mode that cannot be described by the simple model. These include the clumping of the orbiting atoms, the number of atoms that appear in the core nucleus versus that in the ring, and the discrete radii of the clumps. The formation of clumps with discrete radii and the interesting transition dynamics may indicate the presence of some nonlinear interaction. A number of effects that we neglected could lead to such nonlinearities. For example, the radiation trapping and attenuation forces can change locally because of the local variation in the density or the polarization of the atoms. It is possible that this dependence of the equation of equilibrium on density could cause a rapid buildup of the number of orbiting atoms and also the forces keeping them there. The possible nonlinear processes inside a dense cloud of low-temperature atoms are not well understood and certainly deserve study.
Sesko et al.
5. CONCLUSION We observed that the motions of atoms in a spontaneous force optical trap have a n unexpectedly rich behavior. They act collectively at surprisingly low densities and make abrupt transitions. This behavior arises from the coupling between the atoms owing to the multiple scattering of photons and is enhanced by the change in the frequency properties of the scattered light. This force is a key element in the equation of equilibrium for optically trapped atoms and limits the obtainable density. A simple model was presented that explains the growth of the cloud and the formation of the circular rings. However, it does not give an explanation of the formation of the clumps or the dynamics of the transitions between different distributions.
ACKNOWLEDGMENTS This work was supported by the U.S. Office of Naval Research and the National Science Foundation. We are pleased to acknowledge the valuable contributions made by C. Monroe and W Swann to this work. The authors are also with the Department of Physics, University of Colorado, Boulder, Colorado 80309-0440.
REFERENCES AND NOTES 1 D. Pritchard, E. Raab, V. Bagnato, C. Wieman, and R. Watts, Phys. Rev. Lett. 57, 310 (1986). 2 . E. L. Raab, M. G. Prentiss, A. E. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987). 3. T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64,408 (1990). 4. A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63, 957 (1989). 5. D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989); M. Prentiss, A. Cable, J. Bjorkholm, S. Chu, E. Roab, and D. Pritchard, Opt. Lett. 13,452 (1988). 6. J. Dalibard, Opt. Commun. 68, 203 (1988). 7. A. P. Kazantsev, G. I. Surdutovich, D. 0. Chudesnikov, and V. P. Yakovlev, J. Opt. SOC.Am. B 6, 2130 (1989). 8. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, Oxford, 1989), p. 6. 9. B. R. Mollow, Phys. Rev. 188, 1969 (1969); Phvs. Rev. A 5, 2217 (1972); D. A. Holm, M. Sargent 111, and S . Stenholm, J. Opt. SOC.Am. B 2, 1456 (1985). 10. S. Gilbert, Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1984). 11. B. Dahamani, L. Hollberg, and R. Drullenger, Opt. Lett. 12, 876 (1988). 12. P. D. Lett, R. N. Watts, C. I. Westbrook, W D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988); P. D. Lett, W D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, J. Opt. SOC.Am. B 6, 2084 (1989). 13. R. Sinclair (Division of Physics, National Science Foundation, Washington, D.C. 20550, personal communication) h a s pointed out that the clumping is similar in character to the negative-mass instability that is observed in electron plasmas. We are now investigating this analogy.
VOLLMF.
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PHYSICAL REVIEW LETTERS
24 S E P T E M B E R 1990
Very Cold Trapped Atoms in a Vapor Cell C. Monroe, W. Swann, H. Robinson,'J' and C. \Vieman Joinr In.ctrrure Jor Laborator) A.rrroph>,sirs, 1'nii.er.Yiiy of' Colorado and ;I'aiional lnsrirure oJ.5randard.r and Terhnologj,, and Deparrnienr of Phj,.tics. L ' n i w r s i r j , of- Colorado. Bortldcr. Colorado 80309-0440 (Reccived 3 I M a y 1990)
We have produced a very cold sample of spin-polarired trapprd atoms. The technique used dramatically simplifies the production of laser-cooled atoms. I n this experiment. I .8 x lo7 neutral cesium atoms were optically captured directly from a low-pressure vapor i n ii small glass cell. We then cooled the < I - m m ' cloud o f trapped atoms and loaded i t into a lowfield magnetic t r a p in the same cell. The magnetically trapped atoms had an effective temperature as low as 1 . 1 I 0 . 2 p K , which is the lowest kinetic temperature ever observed a n d f a r colder t h a n any previous sample of trapped atoms. PACS numbers: 32.8O.Pj. 4 2 . 5 U . V k
I n recent years there has been dramatic progress in the use of laser light to cool' and trap' neutral atoms. We and other^^,^ have previously reported using optical traps to produce cold atomic samples with high densities and optical thicknesses beyond that attainable in most beams. Such samples would be ideal for a variety of experiments which are currently done with atomic beams. However, in previous optical traps, a substantial vacuum apparatus was required to slow an atomic beam prior to trapping. The ability to trap atoms directly and efficiently from a room-temperature vapor means that for many experiments, this apparatus can be replaced with a small cell. Such an optically trapped sample is useful for many applications, but it has some inherent limitations; the atomic spins are randomly oriented, perturbing light fields must be present, and it is difficult to achieve temperatures lower than 300 p K . W e have overcome these limitations by loading the optically trapped atoms into a magnetostatic trap. Because the atoms are very cold when first loaded, we can trap them with relatively small magnetic fields. By properly cooling the atoms before turning on the magnetic trap we have produced a sample which is more than 100 times colder than any previously trapped neutral atoms. In our experiment the atoms are initially captured using the Zeeman-shift spontaneous-force optical trap and used by us in a (ZOT) reported first by R a a b er variety of recent This trap uses the light pressure from six orthogonal intersecting laser beams. A weak magnetic-field gradient acts to regulate the light pressure in conjunction with the laser frequency to produce a damped harmonic potential. In previous work we found that the trap could capture cesium atoms from a cooled beam with speeds up to = 12 m/s and substantial numbers of atoms from an uncooled beam. Similar observations were made by Cable, Prentiss, and Bigelow for s o d i u m 6 This suggested the possibility of capturing atoms directly from a room-temperature vapor. T h e number of atoms contained in a trap which is in a
dilute vapor is determined by the balance between the capture rate into the trap and the loss rate from the trap. Given the maximum capture speed of the trap, [-'., the number of atoms per second entering the trap volume with low enough velocities to be captured is easily calculated.' This rate is R = ~ . S ~ V " ' I ~ , P ( ~ / ~where ~ T ) "n ' , is the density of cesium atoms, m is the mass, T is the temperature, and V is the trapping volume (about 0.1 cm' for our trap). For cesium at a temperature of 300 K, approximately one atom in lo4 is slow enough to be captured. T h e loss rate from the trap, I/r (r=lifetime), is primarily due to collisions with atoms in the vapor. I f we assume the density of noncesium atoms is negligible, then l / r = n a ( 3 k T / m ) " ' , where CT is the cross section for an atom in the vapor to eject a trapped atom. T h e number of atoms in the trap, N, is then given by the solution to the simple rate equation d N l d r = R - N / r . Assuming N(r=O) =0, we obtain N ( t ) =N,(I - e -''r), where the steady-state number N, is given by N , =R r = I
'
( V '/'/a)~,P(rn/2 k T ) .
(I
1
Note that N , is independent of pressure. However, the lifetime r , which is also the time constant for filling the trap, does depend on pressure. Because N , is very sensitive to t'c, we should note which parameters of the ZOT determine r<.. T h e primary stopping force comes from the imbalance in the radiation pressure due to the differential Doppler shift between the two counterpropagating beams. This force decreases rapidly with velocity once the Doppler shift gets so large that the atom is out of resonance with both beams. Since the frequency of the laser is typically I linewidth to the red of the atomic resonance, rc =: 2rh. where T h is the velocity a t which the Doppler shift ( 5 M H z for our transiequals the natural linewidth tion). T h e actual construction of the cell trap was quite simple. Shown in Fig. I , the cell is a vertical cylinder of fused silica I2 cm long with windows on each end and four windows mounted in a cross a t the top of the
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predicted by Eq. (1) with = 15 m/s. A5 we observed in our previous work,’ the density of atoms in the cloud was limited by radiation trapping to about 5 x 10’” ritoms/cm3, and the temperature was about 30 p K . The trapped atom cloud was more than 1000 times brighter than the background fluorescence in the cell at a cesium pressure of 6 x lo-’ Torr. The growth i n the number of atoms with time agreed well with the predicted form. The lifetime of atoms i n the trap was 1 s at a pressure of 6 x lo-’ Torr. The steady-state number of atoms in the trap changed by less than 30% as the cesium pressure was varied between --lo-’ Torr ( r = 0 . 0 6 s) and -1.5x lo-’ Torr ( 5 = 4 s). Lifetimes of 1-2 s were attained using the thermoelectrlc cooler, and lifetimes up to 10 s were reached by immersing the cesium reservoir in a dry ice and alcohol solution. For lower cesium pressures, the number of trapped atoms decreased, presumably because the loss rate from collisions with noncesium atoms became important. At very high pressures the attenuation of the trapping laser beams caused the number of trapped atoms to decrease. From the lifetime and the pressure we find 0 = 2 x cm’. The lifetime at a given pressure is about a factor of 5 shorter than what we previously measured for optically trapped cesium atoms in a stainless-steel UHV chamber. This difference is reasonable since the background gas (mostly H?)in that case was much lighter than cesium. Perhaps the most striking aspect of this trap is its insensitivity to nearly all optical parameters. Although the detailed shape of the cloud changes, the number of atoms is nearly unaffected by changes in alignment, attenuation of the return beams, or quality of the laser wave fronts. Sending the trapping beams directly through the curved glass walls of the cell rather than through windows made little difference in the performance; the trap even worked when a lens tissue was placed in one of the incident beams. Zhu and Hall have trapped sodium in a similar cell, and they have achieved results comparable to ours for cesium.8 With additional laser power it should be possible to increase significantly and thereby dramatically increase N . This could be done by enlarging the diameter of the trapping beams (keeping the intensity near saturation) to allow the stopping force to act over a larger distance. Adding red-shifted sidebands on the laser would give further improvement. We have used these optically trapped atoms to carry out a variety of experiments within the vapor cell. First, the atoms were further cooled by switching off the antiHelmholtz coils to leave the atoms in optical molasses. This technique produces a dense sample of very cold but untrapped atoms, as was demonstrated i n Ref. 9. When 10 cm to the laser light was shut off, these samples fell the bottom of the cell with very little expansion. Alternatively, after turning off the light, we loaded these very cold atoms into a magnetostatic trap or bowl where they 13,
1
c‘
i#
r;
Y
FIG 1 Vapor cell used for optically and magnetically trapping and cooling cesium atoms For the optical trdp, the small coils are operated i n the anti-Helmholtz configuration The six laser beams are indicated by arrows For the magnetic trap, the current to the small coils is turned OK and the large coil turned on The four vertical current bars providing horizontdl confinement are not shown
cylinder. Attached to the main cylinder are two Fmaller tubes. The first is a “cold finger” which contains a reservoir of cesium whose temperature determines the vapor pressure in the cell. It can be cooled to -23°C using a small thermoelectric cooler. The second tube leads to a 1 - 8 s ion pump which removes any residual gas (mostly helium) that may diffuse through the cell walls. The Z O T setup was the same as in our previous work. The light from a diode laser tuned to the 6 S l / ? , F = 4 6 P j / t , F = 5 cycling transition was split into three beams 0.5 cm in diameter with -2 mW per beam. The beams were circularly polarized and aligned to intersect perpendicularly in the center of the cell. After leaving the cell each beam went through a quarterwave plate and was reflected back on itself. Light from a second laser tuned to the 6 S l / ? , F= 3 + 6 P j l ~ , F = 4 transition also illuminated the intersection region and prevented the atoms from accumulating in the F - 3 ground state. A magnetic-field gradient of -10 G/cm was produced by an anti-Helmholtz pair which was wound on the cylinder. The fluorescence from the center of the cell was observed using a charge-coupled-device television camera as well as a calibrated photodiode. When the trapping laser was tuned between 1 and 10 M H z to the red of the transition ( - 6 M H z was optimal), a bright cloud of trapped atoms < 1 mm’ appeared in the center of the cell. By measuring the fluorescence we determined that the cloud contained as ~ atoms, in agreement with the value many as 1 . 8 lo7
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were held by magnetic V ( p . B ) and gravitational forces at the same location as in the optical trap. This produced a very cold, 100% polarized sample which was not subjected to the perturbing fields of the laser. To trap the IF,rnr-) =14,4) weak-field-seeking state, four vertical bars, each carrying a current with direction opposing that of its nearest neighbors, wcre placed around the cell. I n the horizontal (x,.p) plane, they provide a quadrupole field distribution having zero field in the center along the vertical ( z ) axis. I n addition, a horizontal coil below the atoms gives a vertical bias field as well as a vertical field gradient.'" This gradient (aB,/az) supplies the levitating force to support the atom against gravity. At the trap, B; = 100 G and aB, = -24 G/cm. The x and y field gradients, depending on the current through the bars, were 20-80 G/cm. Since B , is much greater than the horizontal components ( B , , B , ) over the trap volume, the trapped atoms see only very small changes in the direction of B. The combination of the magnetic field and gravity produces a very nearly harmonic confining potential within the trap volume in all three dimensions. The vertical 1 , by gravity and our coil spring constant ( ~ ~ fixed geometry, was 0.5 mK/cm' corresponding to an oscillation period of 360 ms. The horizontal spring constants, determined by the corresponding field gradients, were K , =0.5-5 mK/cm' and K~ =0-5 mK/cm2. The following sequence was used to load the atoms into the magnetic trap: ( 1 ) The optical trap was turned on and filled for 3 s. (2) For additional cooling, all magnetic fields were turned off, the laser frequency was shifted 50 M H z to the red, and the intensity was reduced by a factor of 4. After 5 ms, the trapping laser was quickly turned off (20 p s ) . ( 3 ) A 10-G vertical field was applied and a weak circularly polarized vertical laser beam was pulsed on (duration 0-1 ms) to pump the atoms into the 14,4) state. (4) At the end of this pulse, the magnetic fields for the trap were switched on in 20 p s . We observed the distribution and number of magnetically trapped atoms at later times by waiting (0-3 s), then quickly (20 p s ) switching the magnetic fields off and the laser beams back on. A television camera recorded the image of the clouds as the laser came on. With a I-ms optical pumping time in step 3, essentially all the optically trapped atoms I ( 1 - 2 ) x lo')] were loaded into the magnetic trap. However, the scattering of photons during the optical pumping heated the atoms. We avoided this by adjusting the alignment and intensities of the trapping-molasses light beams to enhance the population of the m = 4 state in the molasses. We were able to obtain nearly a factor-of-3 enhancement ( 3 0 9 of the atoms in m = 4 without external optical pumping) without significantly affecting the number of trapped atoms or the final temperature. With this enhancement, the additional optical pumping required to put more than 50% of the atoms into m = 4 produced only slight heating. This will be discussed in a later publication. The
24SEPTEMHER 1990
lifetime in the magnetic trap was == f that of the optical trap. and depended very weakly on the depth of the trap. We studied the temperature of the cooled and magnetically trapped atoms by observing their motion i n the known trapping potentials. For large spring constants the magnetically trapped atoms were confined to much the same volume as they occupied i n the optical trap and quickly reached a static distribution. With weaker spring constants, however, the cloud expanded and contracted as each atom oscillated sinusoidally in three dimensions. We reduced the spring constants until the maximum diameter of the cloud was 3 or more times the minimum (initial) diameter. Under these conditions the initial velocity distribution is mapped into the spatial distribution after of the oscillation period T to within 10%. They are related by L') = ( R , / M ) ' " ~ , . where vi is a component of initial velocity, and r, is the component of displacement at T/4. Thus, we find the velocity distribution by measuring the spatial distribution of the atom cloud at the time of largest expansion. We found the velocity distribution depended on the alignment of the trapping-molasses beams. I n general, the distributions were different in the Y, y , and 2 directions, and could deviate from a thermal Maxwell-Boltzman distribution. For conditions where the initial distribution was thermal to w i t h i n our 10% uncertainty, we can characterize the distribution of velocities in each direction by a tempera. kinetic-energy spread oscillated ture, TI = ~ ~ r , ' / 2 kThis as a function of time between the initial value (a few p K ) and a minimum value which we estimate is about 100 nK. The time-average kinetic energy of the trapped atoms is described by an effective temperature which is the average of these two, or essentially the initial value. With no optical pumping we obtained effective temperatures as low as T,=T,. = 1 . 1 k O . 2 pK, and T: = I .3 -t 0.2 pK. As expected, these effective temperatures are about the temperature recently observed with cesium atoms in optical molasses. " They are more than a factor of 100 lower than has previously been obtained for a sample of trapped atoms. There are many potential uses for samples of magnetically or optically trapped atoms which can be produced and precisely manipulated with such a simple apparatus. They are useful targets for many scattering experiments ahd ideal samples for precision spectroscopy. We have demonstrated one example of the latter by dropping the optically trapped and cooled sample and observing the cesium clock hyperfine transition with good resolution as it fell. ' I Another unique characteristic of these samples is the large optical thickness with negligible Doppler or collision broadening. It should be possible to obtain an optical thickness of 10. Because this is much larger than can be achieved in a collimated atomic beam, it will allow significantly improved measurements of weak optical processes such as parity-nonconserving transitions. l 3 When combined with the ability of the optical trap to I573
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PHYSICAL REVIEW LETTERS
efficiently collect t h e atoms of a very dilute species, rare short-lived radioactive isotopes can be captured and used for such measurements. As an example, this will allow the precise measurement of parity nonconservation i n a large number of isotopes of cesium. This set of measurements will provide an extremely stringent test of the standard model which does not require precise calculations of atomic structure. Trapped polarized samples of other radioactive isotopes collected i n this manner will also allow more precise studies of p decay. I' Finally. by moving the magnetic t r a p into a lower pressure region located above the optical t r a p and propelling many balls of atoms into it, it should be possible t o obtain much higher densities. This would improve all of the above experiments. When combined with additional cooling of the magnetically trapped this approach also offers a potential route toward achieving Bose condensation in an alkali-metal vapor. This work is supported by the Office of Naval Research and the National Science Foundation. W e a r e pleased to acknowledge beneficial discussions with T. Walker and D. Sesko. O n e of us (H.R.) was a Visiting Fellow a t the Joint Institute for Laboratory Astrophysics.
permanent dddress Physics Department, Duke University, Durham, NC 27706
24 SEPTEMBER 1990
[See special issue on laser cooling and trapping of atoms, edited by S. Chu and C. Wieman [J. Opt. Soc. Am. B 6 , No. I I (1989)I. ? D Pritchard, i n Alomrr Physics / I , edited by S. Haroche. J . Gay, and G . Grynberg (World Scientific, Singapore, 1989). 'E. Raab. M Prentiss, A . Cable, S. Chu. and D. Pritchard, Phys. Rev. Lett. 59. 2631 (1987). ". Walker, D. Sesko, and C. Wiernan, Phys. Rev. Lett. 64. 408 ( I990). 'D. Sesko. T Walker, C . Monroe, A . Gallagher, and C Wieman. Phys Rev. Lett. 63, 961 (1989). 6 A . Cable, M. Prentiss. and N . Bigelow. Opt. Lett. 15. 507 (1990). 'F. Reif, Fundanretirals oJ Srarisrical and Thermal Ph,vsics (McGraw-Hill. New York. 1965). 8M. Zhu and J . Hall. JILA, University of Colorado (private communication). 'M. Kasevich, E. Riis, S. Chu, and R. DeVoe. Phys. Rev. Lett. 63, 6 I 2 ( I 989). 'OD. Pritchard, Phys. Rev. Lett. 51, 1336 (1983). "C. Salamon. J . Dalibard, W. Phillips. A. Clarion, and S. Guellati, Europhys. Lett. (to be published). I2C. Monroe, H. Robinson, and C. Wieman. Joint Institute for Laboratory Astrophysics report (to be published). I3M. Noecker, B. Masterson. and C. Wieman, Phys. Rev. Lett. 61, 310 (1988). I4Stuart Freedman, University of Chicago (private communication). I5H. Hess. G. Kochanski. J . Doyle, N. Masuhara, D. Kleppner, and T. Greytak, Phys. Rev. Lett. 59. 672 (1987).
50
a reprint from Optics Letters
Observation of the cesium clock transition using laser-cooled atoms in a vapor cell C. Monroe, H. Robinson,* and C. Wieman roint Institute of Laboratory Astrophysics. University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309-0440 Received June 22,1990; accepted September 17,1990;manuscript in hand October 22,1990
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Cesium atoms in a vapor cell have been trapped and cooled by using light from laser diodes. The 6s F = 4, m = 0 6.9 F = 3, m = 0 hyperfine clock transition was excited as these atoms then fell 2.5 cm in darkness. We observed a linewidth of 8 Hz with good signal-to-noise ratio. This gave a short-term fractional frequency resolution of 6.5 x 10-l*/vG, and there is potential for substantial improvement. The apparatus is extremely simple and compact, consisting of a small cesium vapor cell and two diode lasers.
There has been much interest lately in the use of extremely cold neutral atoms for precision measurements of microwave clock tran~itions.l-~This interest results primarily from the potential for high resolution and small systematic errors. We have reported observation of the clock transition.in laser-cooled cesium atoms,' and Kasevich et aL2 observed the corresponding transition in cooled sodium atoms. Although narrow linewidths were observed, these experiments required large vacuum apparatus and had relatively low signal-to-noise ratios. Here we report the observation of the 6 s F = 4,mF = 0 to 6 S F = 3, mF = 0 clock transition in a sample of laser-cooled (10pK) cesium atoms in a small vapor cell. Since a good signal-to-noise ratio was achieved by using inexpensive diode lasers, this is clearly an attractive technology for use in a high-performance, compact, and portable atomic clock. In this experiment cesium atoms in a low-pressure vapor were optically trapped. The trapped atoms were then cooled further and optically pumped into the 6 s 1 F, m ~=) I4,O) state. Subsequently all light was switched off, permitting the atoms to fall in the dark under the force of gravity. As the atoms fell, two microwave pulses were applied to drive the transition to the 13,O) state. Finally, the light was turned back on to monitor the fraction of the atoms making the transition. This cycle was repeated at different microwave frequencies to observe the Ramsey line shape. The atoms were trapped through a Zeeman-shift spontaneous-force optical trap as in our previous re~earch.~ This trap uses the light pressure from six orthogonal intersecting laser beams. The light pressure is made to vary with the position of the atoms by a weak magnetic field gradient. This, combined with a red detuned laser frequency, produces a damped harmonic potential. Two diode lasers whose frequencies were stabilized by optical feedback from diffraction gratings5 were used for the trapping, cooling, pumping, and probing of the atomic sample. The relevant transitions excited by the lasers are shown in Fig. 1. Light from the trap laser, which was detuned 6 MHz to the red of the 6S112, F = 4 6P312, Ff = 5 cycling
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0146-9592/91/010050-03$5.00/0
transition, was split into three beams 0.6 cm in diameter with 2 mW of power per beam. The beams were aligned to intersect perpendicularly in the center of the cell, then reflect back on themselves. To prevent atoms from accumulating in the F = 3 ground state, light from the pump laser, tuned to the 6S1/2, F = 3 6l'3/2, Ff = 4 transition, also illuminated the intersection region. The magnetic field gradient (=lo G/cm) was produced by an anti-Helmholtz coil pair wound around the cell. As shown in Fig. 2, the cell was a cylinder of fused-silica glass with six windows attached to it. A small cesium reservoir cooled to -2O'C kept the pressure of cesium vapor in the cell at =l X Torr. A 1-L/sec ion pump attached to a small sidearm evacuated any gas (primarily H2 and He) that diffused into the cell. Fluorescence light
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r 5 I
251 MHz
4
4
I
2.1 nrn
HYPERFINE PUMPING
F=
4* 9193 M H r
\
Fig. 1. Cesium energy-level diagram showing the relevant transitions.
01991 Optical Society of America 298
January 1,1991 / Vol. 16, No. 1 / OPTICS LETTERS
51
tion probability. By determining both populations we obtain the probability independent of the number of atoms in the drop. The F = 4 population immediately following the second Ramsey pulse was found by monitoring the atom fluorescence excited by a 1-mW beam of the cycling 4 5’ light from the trap laser. We recorded the photodiode signal (F4) that showed a fluorescence peak that decayed away in 2 msec owing to the F = 4 atoms’ being pushed away from the detector region and accelerated out of resonance owing to the pressure exerted by the exciting light. After 7 msec there was no longer fluorescence from the F = 4 state atoms, and the 4 5’ light was switched off. The F = 3 atoms, which still remained, were then pumped back into the F = 4 ground state by illuminating them for 2 msec with the 3-4’ pump laser light. Finally, the 4-5’ trap light was again applied for 7 msec, and the resulting fluorescence signal (Fa)indicated how many atoms had been left in the F = 3 state. The probability of the I4,O) I3,O) transition, independent of the number of atoms in the sample, is then given by P = F3/(aF4 + F3). In general, the constant cy is not 1 because the 9-msec delay between measurements F4 and F3 causes the falling atoms to be at slightly different spatial locations and have different detection efficiencies. The value of a was determined simply by taking the slope of the plot of F3 versus F4. The entire time sequence between trapping and detection lasted -1.1 sec, including 1.0 sec for the optical trap buildup time. After each cycle the microwave frequency was increased by 1 Hz,and the sequence was repeated. We acquired the Ramsey line shape shown in Fig. 3. The linewidth of 8 Hz is consistent with the pulse separation time. The solid curve is a theoretical fit using the Ramsey two-pulse transition probability with different microwave phases and field strengths for the two spatial locations of the falling atom cloud. The finite fringe contrast is due to the difficulty of providing the proper phase and field strengths at two
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Fig. 2. Vapor cell with fluorescence detector and microwave horn. The arrows show the directions of the trapping laser beams. TEC, thermoelectric cooler.
from the dropped atoms was detected by a 1-cm2silicon photodiode placed against the cell wall 2.5 cm below the optical trap region. The time sequence used to observe the microwave resonance is given in Table 1. Starting with no trapped atoms at t = 0, the number of atoms in the Zeeman-shift spontaneous optical trap would increase as “1 - exp(-t/~)], where N i s 2 X lo7 and T is 0.6 sec. After 1sec the trap contained approximately 1.6 X lo7 atoms at a temperature of -300 pK, in a volume of <1 mm3. The atoms were further cooled to ~ 1 ,uK 0 by turning off the anti-Helmholtz coils to provide the conditions for optical molasses for 5 msec. This temperature was measured by loading the atoms in a magnetostatic trap and measuring the size of the atom cloud as in Ref. 4. Next the sample was pumped into the 14,O) state by blocking the trap laser (but not the pump laser) and turning on both a 100-mG uniform magnetic field and a weak laser beam linearly polarized along the magnetic field direction. This laser field was applied to the atoms for 100 psec and was obtained by taking part of the trap laser output and tuning it to the 6S1/2, F = 4 6P3/2, F‘ = 4 transition by usin an acousto-optic modulator. Because the IF, mF) = 0) IF’, mF’) = Id’, 0’) is forbidden, more than 95% of the atoms were pumped into the 14,0) state. After this sample preparation all light was turned off, leaving a cloud of 1.6 X lo7 atoms that was estimated to be ~ 2 pK 0 in the I4,O) state and occupied a l-mm3 sphere. This cloud then spread slowly as it fell toward the bottom of the cell. As the atoms dropped, we drove the 9.193-GHz magnetic-field-independent I4,O) 13, 0) transition. This transition was excited by two separate6 5-msec pulses of microwave radiation with Brrad polarized along the 100-mG field. The first pulse was applied just as the atoms started to fall; the second pulse was applied 65 msec later, after the atoms had fallen 2.5 cm. A microwave horn driven from a tunable source was used to irradiate the atoms with an intensity of -20 nW/cm2. We next sequentially measured the populations of the 14,O) and 13,O) states to find the microwave transi-
L,
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Table 1. Time Sequence Used to Observe the Microwave Transition” Time
Process
0-1.0 sec Next 1msec Next 5 msec Next 1msec
Buildup of atoms in optical trap Anti-Helmholtz coils turned off Molasses cooling Turning on uniform 100-mG field and turning off 4 5’trap/molasses light 4 4‘ light on for optical pumping Turning off remaining light (3 4’) Apply microwave pulse 1 Atoms fall in the dark Apply microwave pulse 2 4 5’ light on, excites and pushes away F = 4 atoms 3 4‘ light on, transfers F = 3 atoms to F = 4 4 5’ light on, excites and pushes away F = 4 atoms Anti-Helmholtz coils turned on
Next 0.1 msec Next 1msec Next 5 msec Next 65 msec Next 5 msec Next 7 msec Next 2 msec Next 7 msec Next
1msec
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-
-
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a Turning-on or turning-off times represent the switching time; the time duration of all other switching was negligible.
300 52
OPTICS LETTERS / Vol. 16, No. 1 / January 1,1991
0.6
a x ..-Y
0.5
4
0.4
i
I]
0
I]
$2 a
I
01 -40 0
I
I
-20 0
I
20 0
0.0
40 0
Relative Frequency (Hz)
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Fig. 3. Resonance spectrum of the 14,O) 13,O) transition. The solid dots are the data, and the solid curve is a theoretical fit. Most of the dots represent the average of four measurements.
positions in space using a single horn. One striking characteristic of the line shape is the large number of interference fringes, in contrast to the Ramsey line shape observed in beam machine experiments. The uncertainty in the measurement of P was 0.006 per drop. This uncertainty was dominated not by fluctuations in the signal but rather by fluctuations in the background light, which was scattered into the detector by both the cell windows and the background cesium vapor. If we take the maximum frequency resolution ( P = 0.5) to be the linewidth divided by 1.57 times the signal-to-noise ratio (85),we have a fractionwhich al frequency resolution of 66.5 X represents a factor-of-25 improvement over that reported in Ref. 2 and a factor-of-20 improvement over our previous research.’ It should be straightforward to extend the technology demonstrated here to make a compact high-performance atomic clock. The most obvious change is to provide a standing-wave microwave field to eliminate Doppler and phase shifts. A natural way to incorporate such microwave field improvements would be to launch the atoms up through a microwave cavity in a fountain as in Ref. 2. Although we could not use a fountain in this research because of the geometry of our cell, this is a straightforward addition. Another advantage provided by a fountain is the doubling of
the observation time for a given cell size. The final advantage of a fountain is that as the cold atoms fall back down they can be recaptured and used for the next cycle. This would reduce the filling time required for the trap and thus improve the duty cycle of the clock by as much as a factor of 10. It should also be possible to improve significantly the signal-to-noise ratio by reducing the fluctuations in the background light. These were primarily due t o mechanical vibrations of the apparatus, which was necessarily elongated and flimsy because of the cell geometry. A more rigid and compact system, which is also designed to reduce the amount of scattered light, might well approach the shot-noise-limited signal-tonoise ratio. For operation near the optimum value of P = 112, the shot noise will be determined by the \i;E fluctuation in the number of atoms making a transition. For our atomic sample this signal-to-noise ratio would be 3.0 X lo3 per drop, which would yield excellent short-term frequency stability. Although none of our research has dealt with long-term accuracy, there are reasons to expect that a clock based on this technology would be quite good. The low velocities of the atoms reduce the problem of Doppler shifts, and the small interaction volume simplifies the problem of shielding against stray magnetic fields. This research was supported by the U.S. Office of Naval Research and the National Science Foundation. We are pleased to acknowledge valuable discussions with S. Chu, D. Sesko, and T. Walker. The authors are also with the Department of Physics, University of Colorado. H. Robinson is a 1989-90 Joint Institute of Laboratory Astrophysics Visiting Fellow. * Permanent address, Department of Physics, Duke University, Durham, North Carolina 27706.
References 1. D. W. Sesko and C. E. Wieman, Opt. Lett. 14,269 (1989). 2. M. Kasevich, E. Riis, S. Chu, and R. DeVoe, Phys. Rev. Lett. 63,612 (1989). 3. J. Hall, M. Zhu, and P. Buch, J. Opt. SOC.Am. B 6,2194 (1989). 4. C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571 (1990). 5. C. Wieman and L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. (to be published). 6. N. Ramsey, Molecular Beams (Oxford U. Press, London, 1956).
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Multiply Loaded, ac Magnetic Trap for Neutral Atoms Eric A. Cornell, Chris Monroe, and Carl E. Wieman Joint Institute f o r Laboratory Astrophysics and the Department of Physics, LJnii>er.rityof Colorado. Boulder, Colorado 80309 -0440 (Received 5 August 199 I ) We have demonstrated an oscillating-gradient magnetic trap for confining cesium atoms in the lowest-energy spin state. I n this state spin-flip collisions are energetically forbidden and thus the primary density- and temperature-limiting process for magnetic traps is eliminated. The atoms are initially collected in a vapor-cell optical trap, then repetitively tossed into a high-vacuum region, focused in three dimensions, and accumulated in the magnetic trap. An optical pumping scheme inserts each new batch of atoms into the magnetic trap without perturbing the atoms already trapped. PACS numbers: 32.80.Pj. 42.50.Vk
Much of the recent rapid progress [I-31 toward colder and denser atomic vapors has been driven by the intrinsic interest of the processes involved. However, there are several external motivations as well, two of which we find particularly compelling: First, a cold gas of bosonic atoms is by far the cleanest system in which one could hope to observe Bose-Einstein condensation (BEC). The ability, in principle, to cross the phase boundary for BEC at several widely spaced points in the temperature-density plane offers the tantalizing prospect of studying the transition while varying the relative significance of interparticle interactions. Second, cold, dense samples of gas are ideal for the spectroscopy of weak transitions, such as those involved in the measurement of parity nonconservation in atoms. In the past, cesium atoms have been confined in optical traps to a density of 5 x 10’’ atoms per cm’, then cooled in laser molasses to 2 p K [I]. However, several mechanisms limiting temperatures 121 and densities [41 are slowing further progress in optical traps. Magnetic trapping is a way to avoid these limitations. For example, the photon recoil energy, which is the primary limit for optical cooling, is not an obstacle to evaporative cooling in a magnetic trap. Spin-polarized hydrogen in a magnetic trap has been evaporatively cooled (allowing the highestenergy atoms to escape the trap) to a temperature 13 times below the photon-recoil limit I31, and to a density orders of magnitude larger than the present limit for an optical trap. However, traditional magnetic traps have a significant limitation of their own, namely, spin-flip collisions. To date, all magnetic traps have been designed to work only for weak-field seekers, that is, for atoms or neutrons whose magnetic moment is antialigned with the magnetic field. These atoms are confined to a local minimum in the magnitude of a static magnetic field. Unfortunately, because weak-field-seeking states are not the lowestenergy configuration within the confining magnetic field, the atoms are susceptible to two-body collisions that change the hyperfine or Zeeman level of one or both of the colliding atoms. The energy released in these col-
lisions heats the trapped sample and the spin-flipped atoms leave the trap. The loss rate due to these exothermic col:isions scales as the density squared and thus provides an effective ceiling to the attainable density, and indirectly limits temperature as well 151. However, if atoms were trapped in the lowest-energy spin state (magnetic moments aligned with the magnetic field), spin-flip collisions would be endothermic and, hence, greatly suppressed a t low temperatures. Unfortunately, such “strong-field-seeking’’ atoms cannot be confined in a static configuration of magnetic fields because Maxwell’s equations do not allow a local maximum in magnetic-field magnitude. However, there are time-varying field configurations which provide stable confinement. Originally proposed for hydrogen atoms [6l, our trap operates on the same dynamical principle as the Paul trap (71 for ions: Near the center of the trap, the potential is an axially symmetric quadrupole, oscillating at frequency R : ~ = - ~ . B = A ( z 2 - - p 2 / 2 ) c 0 s n t During . each oscillation, the atoms are first confined axially and expelled radially, and then confined radially and expelled axially. For a range of values of the amplitude A, the net force averaged over a cycle of the oscillation is inward; the result is stable confinement in all three dimensions. For easily obtainable oscillating magnetic fields, the trap is extremely shallow (tens of microkefvins). However, using laser cooling we are able to produce atomic samples which are colder than this, and load them into the trap. The loading procedure has a number of steps, as shown in Table I. First, cesium atoms are collected and cooled in an optical trap. The cold atoms are then launched into a differentially pumped vacuum region which contains the ac magnetic trap, as shown in Fig. 1. To reduce the spreading of the atoms they are magnetically focused as they move between the two traps. When they reach the magnetic trap the atoms are optically pumped into the state that is trapped. This approach allows multiple bunches from the optical trap to be transferred to the magnetic trap, thereby increasing the density. The process of preparing room-temperature cesium atoms for magnetic trapping begins with a Zeeman-shift
@ 1991 The American Physical Society 301
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TABLE I . Timing and laser detuning during one cycle of the multiple loading sequence. Relative time (ms) -500 to - 2 -I t o o 0 to 0.4 0.4 to 0.8
0.8 to 130 2 to 4 60 to 62 67 to 70 128 to 130
Laser detuning, function -7 MHz. Atoms accumulate in optical trap. -30 MHz. Molasses cooling to 2 p K . -7 MHz, and offset A ramped from 0 to I .7 M Hz. Atoms accelerate. -30 MHz, and offset A constant at 1.7 MHz. Atoms cool to 4 p K in moving frame. Main beams blocked. Optical trap off. Atoms rise to magnetic trap. -250 MHz. Initial optical pumping. Magnetic focusing pulse. Magnetic focusing pulse. -310 MHz. Final optical pumping as atoms reach magnetic trap.
optical trap (ZOT), which collects the low-energy tail from a vapor of room-temperature cesium atoms. This trap uses light from diode lasers which excites the 6 S 1 / 2 , F = 4 + 6P,/z,F=5 transition. In about 0.5 s, lo7 atoms accumulate in the ZOT to form a ball 0.1 cm i n diameter, with a temperature of a few hundred microkelvin. In the second stage of preparation, the magnetic fields for the ZOT are turned off and the laser frequency is redshifted several linewidths, which allows the more effective molasses cooling mechanism to quickly reduce the temperature to -2 p K . For details of the technique described thus far, see Ref. [I]. To toss the atoms up into the magnetic trap region, we use the technique of moving optical molasses [Sl: We shift the frequency of the downward-angled molasses beams (Fig. 1 ) 1.7 M H z to the red, and the upwardangled beams 1.7 MHz to the blue, relative to the frequency of the horizontal beams. The standing-wave pattern a t the intersection of the six laser beams now becomes a walking-wave pattern. T o achieve the necessary acceleration (200g), while having the lowest possible final velocity spread, we vary the central laser frequency over time during the launch (Table I). After the atoms are accelerated, the molasses beams are shut off abruptly and 0 polarized light optically pumps the a 2-ms pulse of ' atoms to the F = 4 , mf-4 state. The optical pumping beam is tuned to the F = 4 - F ' = 4 transition in order to minimize heating . The tossed atoms begin their ascent with an internal velocity spread of about 3 cm/s rms. I n the 130 ms it takes to reach the site of the magnetic trap, even this small velocity spread would result in a 100-fold decrease in cloud density. We avoid much of this decrease by focusing the atoms with magnetic lenses. Approximately halfway (in time) along their route, the atoms pass through two pulses of magnetic field, each around 3 ms long, separated by 8 ms. These pulses provide an impulse 2440
3a.:
c=- J
....
COILS
\\ A cs
FIG. I . Schematic of the apparatus, showing the upper and lower vacuum chambers, the configuration of key trapping and focusing coils, the paths of the initial optical pumping (IOP) and final optical pumping (FOP) laser beams, and the paths of four of the six laser beams defining the Zeeman-shift optical trap (ZOT) and the molasses. Frequencies of the molasses laser beams during the launch are shown in parentheses. Not shown are the paths of two horizontal laser beams which are perpendicular to the plane of the paper a t the ZOT, the magnetic coils for tuning the optical trap, and a variety of shim magnetic coils.
which reverses all three components of the velocity. During the first focus pulse, the curvature of the magnetic field is such that the atoms are focused axially and defocused radially. During the second focus pulse, the curvature of the lenses is reversed. We adjust the timing and strength of the pulses (that is, the positions and effective focal lengths of the two magnetic lenses), in order to bring the atoms together to a focus axially and radially at the magnetic trap center. As shown in Fig. I , the magnetic trap is created by a combination of ac and dc coils and is located 14 cm above the optical trap. The 60-Hz a c field is generated by two pairs of coils which are arranged to produce maximum field curvature with relatively small oscillating field at trap center. The dc coil produces a field at trap center of 250 G, a gradient of 31 G/cm in the vertical direction, and very little curvature. This static gradient exactly balances the force of gravity, which the magnetodynamic forces are too weak to counteract. The peak amplitude of the ac component is 100 G at the center, with a curvature of 875 G/cm2 in the axial direction and of 440 G/cm2 in the radial direction. T o make contact with the rf-ion-trap literature we note that the forces in our trap correspond to Paul-trap parameters (I, and qz -0.40, where a,
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and q; are stability parameters, defined in Ref. I71. Inserting the atoms into the magnetic trap requires some way of dissipating their energy. This is accomplished by optically pumping the atoms from the F = 4 to the F = 3 ground states. T h e trap will hold atoms in the F = 3 , m f = 3 state, but the atoms rising upward from the optical trap are in the F =4, m/ =4 state. Because this is a weak-field-seeking state, F = 4 atoms approaching the magnetic trap are not only pulled downward by gravity but also repelled downward by the d c field gradient. T h e initial center-of-mass velocity of the atom cloud is adjusted so that this combination of forces brings the cloud to a stop just at the center of the trap. (The large timedependent fringing fields from the trap make it somewhat difficult to achieve a stationary tightly focused cloud a t exactly the correct position and time.) T h e stationary atoms are then illuminated with light linearly polarized along the magnetic field and tuned to the F = 4 - F' =4 transition. This optically pumps the atoms to the trapped, F - 3 , m / = 3 ground state. T h e F = 3 atoms already loaded in the trap are transparent to the light used in this final optical pumping. T h e timing of the final optical pumping relative to the phase of the a c fields is critical. The micromotion (the small-amplitude, highfrequency oscillation induced by the a c forces I711 of the weak-field-seeking atoms just arriving in the trap has an opposite phase from the micromotion of the trapped atoms. Unless the pumping occurs when the micromotion velocity is zero, the atoms are strongly heated. While the atoms are in the magnetic trap, they d o not interact with laser beams. To observe the trapped atoms, we abruptly turn off all the magnetic fields and then illuminate the trap region with a probe beam containing laser light which excites both F = 3 - F ' = 4 and F = 4 F'=S transitions. The resulting fluorescence from the atoms is imaged by a video camera and recorded on tape. Subsequent image processing reveals the spatial distribution and center-of-mass location of the atom cloud. Because the measurement is destructive- the photons from the probe beam blow the atoms away-time evolution of the trapped atoms can be studied only by repeating the load-probe cycle with varying time delays. W e study the magnetic trap by observing the shape and motion of the cloud of trapped atoms. Pulses of magnetic field from supplementary coils give the cloud any desired initial center-of-mass motion. From the evolution of the cloud motion we determine the frequency of the guidingcenter oscillation in the effective potential. For an oscillation amplitude of 0.1 cm or less, we measure an axial frequency of 8..5(4) Hz and a radial frequency of 4.5(1 .O) Hz. These agree with calculated [71 frequencies of 8.5 and 4.25 Hz. T h e overall depth of the trapping potential is difficult to calculate or even to define precisely. The depth of the trap is determined mainly by the behavior of the trapped atoms away from the center of the trap. In those areas the field becomes anharmonic and the approximation that
-
28 OCTOBER 1991
PHYSICAL REVIEW LETTERS
t h e ac fields simply provide an effective d c conservative potential is not valid. The harmonic region is quite small unless there is an axial d c bias field that is much stronger than the radial and axial components of the ac field everywhere in the trapping region. W e confirmed experimentally that increasing the bias field improved the stability of the trap. W e have used a computer simulation to understand the behavior of the trap in the regions where the effective potential approximation is not valid. This simulation calculated the motion of clouds of noninteracting atoms with various temperatures in the trap. For clouds inserted with temperatures of 9 p K or less, nearly all the atoms remain trapped and the final spatial extent of the cloud is consistent with the initial velocity spread and a harmonic potential of the calculated spring constant. For clouds with higher initial temperatures, the final spatial distribution of trapped atoms is largely independent of initial temperature. The high-energy fraction of the atoms leaves the trap immediately, and the atoms remaining in the trap have an rms diameter of about 0.2 cm and an rms velocity of 5 cm/s (equivalent to a temperature of about 12 p K ) . These simulation results match our experimental observations that a fraction of the atoms focused up into the magnetic trap region leave the trap within 0.15 s, and that those remaining form a ball about 0.2 cm in diameter. W e conjecture, then, that the rms velocity of our trapped atoms is about 5 cm/s and that the depth of the trap is somewhat deeper than 12 p K . T h e trap has a I/e lifetime which is typically about 5 s, as shown in Fig. 2. W e believe that this is due entirely to collisions with the Ix torr of residual background gas. When we improved the vacuum by cooling some of the metal surfaces
i/
\',
4.0
3.0
?
t lsec) FIG. 2. Total fluorescence from the magnetically trapped atoms as a function of time. A new bunch was loaded every 0.65 s, for the first 10 s. The dot-dashed line is an exponential decay fit to the data, with 4.9 s I/e time. The y axis has been calibrated to indicate the total number of trapped atoms.
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PHYSICAL REVIEW LETTERS
of the upper vacuum chamber with liquid nitrogen, the lifetime nearly doubled. Th e advantage of multiple loading is apparent in Fig. 2. By accumulating a number of loads, we were able to collect nearly 5 times as many atoms as from a single load. Th e factor-of-5 increase agrees well with our calculated value and is determined by the ratio of magnetictrap lifetime to optical-trap fill time, by perturbations from the fringing fields of the magnetic focus, and by unintentional excitation of the trapped atoms during the initial optical pumping. Without great additional technical effort, this ratio could be a hundred or larger. As the density increases, viscous heating due to the micromotion may become a problem [61. W e have not yet seen any sign of such heating, and a simple estimate "91 of the heating rate leads us to believe that, with a suitable choice of experimental parameters, evaporative cooling can overwhelm the viscous heating. W e have demonstrated an ac magnetic trap for neutral atoms, and we have demonstrated how to load it optically on a quasicontinuous basis. T h e t r a p contains atoms in their lowest spin state, and thus avoids the effects that have previously limited the attainable phase-space densities in magnetic and optical traps. These techniques offer a n appealing way to achieve the phase-space density necessary for Bose condensation in a dilute gas of neutral atoms.
28 OCTOBER1991
This work was supported by the Office of Naval Research and the National Science Foundation. W e acknowledge useful conversations with Dan Kleppner and Mark Kasevich.
[I] C. Monroe, W. Swann, H. Robinson, and C. Wiernan, Phys. Rev. Lett. 65, 1571 (1990). 121 C. Salomon ef a/., Europhys. Lett. 12, 683 (1990). See also the special issue on laser cooling and trapping of atoms, edited by S. Chu and C. Wieman [J. Opt. Soc. Am. B 6 ( I I ) (1989)l. I31 J. Doyle et a / . , Phys. Rev. Lett. 67, 603 (1991). 141 T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64, 408 (1990); D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989). 151 N. Masuhara ef al., Phys. Rev. Lett. 61, 935 (1988). [61 R. V. E. Lovelace, C. Mehanian, T. J . Tommila, and D. M. Lee, Nature (London) 318, 30 (1985); R. V. E. Lovelace and T. J . Tommila, Phys. Rev. A 35, 3597 ( I 987). 171 F. G. Major and H . G. Dehmelt, Phys. Rev. 170, 91 ( I 968). [81 M. Kasevich, D. S. Weiss, E. Riis, K . Moler, S. Kasapi, and S. Chu, Phys. Rev. Lett. 66, 2297 (1991); A. Clarion, C. Salomon, S. Guellati, and W. D. Phillips, Europhys. Lett. 16, 165 (1991). [91 W. Ketterle (private communication).
FUNDAMENTAL PHYSICS WITH OPTICALLY TRAPPED ATOMS
C . WIEMAN, C . MONROE, AND E. CORNELL Joint Institute for Laboratory Astrophysics, and Dept. of Physics, University of Colorado, Boulder, Colorado 80309, USA
ABSTRACT We discuss two experiments which appear feasible because of the progress in laser trapping and cooling of neutral atoms. First, using laser trapping, it should be possible trap a number of radioactive isotopes of cesium and francium. Precise measurements of parity nonconservation could be carried out in these samples. The comparison of the these measurements would be a very sensitive probe for physics beyond the Standard model. In the second experiment we are attempting to achieve Bose-Einstein condensation in a dilute cesium vapor. This involves optically trapping and cooling atoms, and then transporting them into a new type of magnetic trap. This trap avoids the spin-flip-changing collisions which have limited earlier attempts to reach Bose condensation.
In the past year we have demonstrated that it is possible to obtain extremely cold trapped atoms in a simple vapor cell.' By combining optical trapping and cooling with magnetic trapping we have produced atomic samples with as many as 10" atoms/cm3, and temperatures as low as 1 pK. The trapped atoms also had essentially perfect spin polarization. Because this could be done in a vapor cell using diode lasers, the necessary apparatus is remarkably simple and inexpensive. This feature, and the notable characteristics of the trapped atoms, have stimulated us to consider how such atoms might be used for investigating fundamental problems in physics. There are many possible applications, but in this paper we will only discuss two which we are presently pursuing; the measurement of parity nonconservation in radioactive isotopes and the quest for Bose-Einstein condensation in a dilute atomic vapor. These experiments probe physics at opposite extremes of the energy scale. The parity nonconservation (PNC) experiments are investigating physical processes 305
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which are happening at TeV energies, while the Bose-Einstein condensation (BEC) work is sensitive to peV physics, some 24 orders of magnitude lower! Since the latter work has progressed the farthest, it will be the focus of much of the discussion. However, we should emphasize that nearly all of the techniques for trapping and manipulating atoms which we are developing for that work will be directly applicable to the PNC experiments. We will start by briefly discussing the ideas behind the planned PNC experiments. At the present time the so called "Standard model" (SM) of elementary particle physics is consistent with all experimental observations. This model is obtained by combining the electroweak theory of Weinberg, Salaam, and Glashow, which unifies the weak and electromagnetic interactions, with QCD, the theory of the strong interactions. Although this model has been very successful, there are many reasons to think that it cannot be the final answer. First, much of the important physics must be put in on an ad hoc basis. In addition to the various coupling constants and particle masses, this also includes certain basic symmetries, or lack thereof. The most dramatic example of this is the handedness of the weak interactions. Also it is clear that the model will have serious difficulties in describing physics which happens at an energy scale of several TeV. There have been numerous extensions and alternative models put forth, which introduce various types of new physics to avoid these unpleasant aspects of the Standard model. At this time however, there is no experimental evidence which indicates which, if any, of these models is correct. The primary way to probe for the existence of such "new physics" (not included in SM) is by precision measurements of experimental quantities which are related by the SM. By seeing if these experimentally measured quantities are related in the way the model predicts, one probes for the existence of new physics. The three most precise relevant measurements (and their fractional uncertainties) at this time2 are 1) the properties of the Z boson, particularly the mass (0.2%),2) neutrino scattering cross sections (-2.5 %), and 3) the measurement of cesium PNC3*4 (2.5%). In recent months there has been particular interest in the comparison of measurements 1) and 3). This interest has arisen because it has been discovered that this comparison is more sensitive to a variety of new physics than had previously been realized. In particular, it is a sensitive test for the existence of the dynamical symmetry breaking process known as technicol~r,~ and additional neutral 2 bosons.6 Such additional bosons occur in a variety of models which contain such features as left-right symmetry, supersymmetry, grand unification, etc. This sensitivity provides a strong incentive to improve the cesium PNC results. Because the mass of the Z is so well known, if the cesium PNC result could be improved by a factor of 10, we would have a 10-fold improvement in sensitivity to such physics. Our current cesium experiment at JILA presently has a signal-to-noise ratio which is nearly a factor of 10 better than what we obtained in Ref. 3. Thus it is clear that it will be possible to obtain experimental results which are good to a few parts in 103. Unfortunately, this by itself is not enough. The experimentally measured quantity acXp=
Q,, where is an atomic matrix element, and Q, the weak charge, is the quantity that we need to test the SM. The atomic matrix
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element is obtained by calculations: which after rather heroic efforts have achieved an accuracy of 1%. It is not clear if such calculations can reach the 0.1% level, and if they can,how can we test them? Until that is done, it will be impossible to test the SM beyond the 1% level by improving only the present experiment. A potential way around this limitation is to make precise measurements on a variety of isotopes of cesium and take ratios of the 6's and ratios of differences. Since the matrix element involves just the electronic structure, this will be nearly the same for different isotopes and will thus cancel from such ratios. The weak charge, however, is nearly proportional to the number of neutrons and hence will change. Thus the ratios will be well defined quantities which the standard model can relate to the Z mass, without introducing uncertainties from the atomic physics. We should point out that this is a somewhat oversimplified picture. In fact there are small corrections associated with the nuclear size, etc., which must be taken into account, but it appears that it will be possible to do this with sufficient precision. Also it should emphasized that it will always be desirable to have the most accurate atomic calculations possible. There will always be significant information that can only be obtained by having such calculations. The main problem with this idea of comparing isotopes is simply the technical difficulty of doing such an experiment. Our present PNC experiment uses an atomic beam that is many orders of magnitude more intense than one could obtain with radioactive isotopes. We believe that optical trapping of radioactive isotopes will overcome this technical problem. Using a trapped atom sample with the characteristics given in the introduction, it should be possible to obtain PNC signals that are 3 to 10 times larger than in our current experiment. This improvement comes from the much higher density and lower velocity of the trapped atoms compared to the beam. Furthermore, there are only lo8 atoms in the trap. It should be possible to obtain trapped samples of this size for a large number of radioactive isotopes of cesium and a few of francium. For the case of cesium, it should be possible to obtain nearly a 25% change in Q,. The attraction of francium is that it is the next heavier alkali, and its PNC is -10 times larger than in cesium. The comparison of precision measurements on cesium and francium will provide an extremely good test of the calculations of the atomic matrix element. These experiments are only just getting started, and will take many years to complete. However, they are likely to be among the most important elementary particle physics experiments of the next 10 years. Now let us change energy Scales by 24 orders of magnitude and discuss another quite fundamental problem we are presently studying. This is the attempt to obtain a Ebse condensate in a weakly interacting vapor. This would occur if the atoms in the vapor are cooled enough that the average de Broglie wavelength becomes larger than the interparticle spacing. When this happens the atoms in the system should collapse into the lowest energy state of the system, and henceforth behave as a macroscopic quantum state. Thus to achieve BEC one must cool the sample below a critical temperature, T,, which is proportional to the atomic density to the 2/3 power. While superconductivity and superfluid helium are believed to show evidence of Bose condensation, these are both very strongly interacting systems. This makes
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their behavior quite different from an ideal Bose gas. Efforts are currently under way to observe BEC in an exciton gas. This interesting field is discussed elsewhere in this volume. Our work is based entirely on pioneering work in this field by people studying spin polarized hydrogen, and on their attempts to cause it to undergo BEC.' This field started using traditional cryogenics. When this was inadequate to achieve the necessary densities and temperatures, a variety of clever new techniques were developed. The experiment which, to our knowledge, is now the closest to reachin the critical temperature has been Wried out by Kleppner, Greytak, and coworkers. In this experiment they first trapped spin polarized hydrogen in a magnetic trap. The trap depth was then reduced allowing the highest energy atoms to escape, thereby cooling the remainder. Using this "evaporative" cooling they reached a temperature of 100 pK, which was only 3.5 times Tc for their density. They were unable to get any colder because of spin-flip collisions between the atoms which caused heating and loss of atoms from the trap. The atoms were trapped in the so-called "weak-fieldseeking state", which means the magnetic moments of the atoms were parallel to the applied magnetic field. When two atoms collided they could flip their spins and thereby gain energy. This ejected the atoms from the trap and heated up the others in the process. Although they will probably overcome this obstacle eventually, as they have overcome many other obstacles in the past, such collisions are clearly a serious problem. In our work on cesium we are attempting to use laser cooling to allow us to avoid the limitation encountered in the hydrogen work, and hence achieve BEC. We are following a low road to that goal. In particular, relative to the hydrogen work, we are working with low densities (which reduces the atom-atom interactions), very low temperatures, and a low cost apparatus. In the remainder of this paper we will outline the details of the approach we are following and what milestones we have successfully reached along this road. The first step is to obtain a sample of low temperature atoms. This is produced by optically trapping the atoms from a vapor and then cooling them to 2 p K with optical molasses, as we discussed in Ref. 1. These atoms are then thrown upward by radiation pressure so that they pass through a small hole into a second chamber which is at lower pressure. In the upper chamber the atoms are trapped by magnetic fields with no laser light present. In order to move them from the optical to the magnetic trap without a large decrease in density we use pulsed magnets to focus the cloud of atoms in all three dimensions during their flight. We study the magnetically trapped atoms in a destructive manner by illuminating them with light and observing the scattered florescence. To trap the atoms we use a new type of magnetic trap, which we call a magnetodynamic trap. The basic concept for this trap has occurred to numerous people, and has been discussed in detail in Ref. 9. This trap is just the magnetic analogue to the Paul or RF ion trap. The atoms are trapped by a magnetic field, whose magnitude is "saddle shaped" and oscillates sinusoidally in time. To produce the requisite fields we use a large diameter dc coil which is above two smaller closely spaced pairs of coils which are driven at 60 Hz. The smaller coils are arranged so
i
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that at the center their fields and first-order gradients cancel, but the curvatures of the fields add. In such a trap we have held atoms for many seconds. The 6 sec lifetime is limited by the relatively poor pressure we currently have in our upper chamber. This is the first demonstration of a magnetodynamic trap." Although the idea has been around for some time, the reason no one has built such a trap before is that for reasonable fields and frequencies the depth of the trap is quite low. For our case of a curvature of 500 G/cm2 which oscillates at 60 Hz,and a bias field of about 200 G, it has a depth of about 20 pK. Thus such a trap is only practical when used with laser cooled atoms. The important feature of the magnetodynamic trap is that it can hold atoms that have magnetic moments either parallel (weak field seeking) or antiparallel (strongfield-seeking) to the applied field. In our case, we trap the atoms in the F=3, m=3 state. This strong-field-seeking state is the lowest energy spin state, and so it is energetically forbidden for the atoms to flip their spins when they collide. Thus we have eliminated the collisional loss process which plagues the trapped hydrogen, and therefore should be able to reach much lower temperatures. Having demonstrated the magnetodynamic trap, we then wanted to load it with many bunches from the optical trap. This step should allow us to achieve a density in the trap that is the density in the optical trap times the ratio of the pressure in the lower chamber to that in the upper. To load multiple bunches we start by throwing the atoms up in the F=4, m=4 ground state. Because of its g factor, this state is "anti trapped" and thus sees the trap as a hill instead of a well. When gravity is taken into account, the total potential energy of this state is a steep slope as a function of height. If the atoms are given the correct initial velocity they will come to a stop at just the position on this slope which corresponds to the bottom of the trap. Then we iIIuminate them with a pulse of light which excites the atoms to the 6P3,2,F=4 state. This state then decays to the F=3 m=3 ground state, which is trapped by the magnetic field. In this manner we can suddenly turn the trap on around the newly arrived atoms, without perturbing the atoms that are already in the trap. We found that the main difficulty in this procedure was preserving the focusing of the cloud of atoms as it rises up through the wildly varying fringing fields of the magnetodynamic trap. However, we have recently found focusing parameters to accomplish this and have managed to increase the density of the trapped atoms by a factor of 5 by loading multiple bunches of atoms." We expect that with minor improvements in the apparatus we should be able to do much better in the future. Once this is optimized, the final step is to cool the magnetically trapped atoms. We plan to do this using "forced evaporation"." This works exactly like normal evaporation, except the highest energy atoms are selected by a more easily adjusted R F field, rather than the depth of the trap. Since the Zeeman levels of the atoms are shifted by the magnetic field, there is a perfect map between the potential energy of an atom and the transition energy to the m=2 state. By applying RF fields of the proper frequency we can selectively pick out the atoms with potential energy much larger than kT, and drive them from the m=3 to the m = 2 level. These atoms will fall to the bottom of the chamber leaving the remainder colder.
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Reducing the temperature of the atoms will also cause the density to increase since the are contained in a harmonic well. We hope to start with a sample of at least 10t Y atoms/cm3 at a temperature of a few pK. We estimate that by removing 90% of the atoms we can increase the density by 100 and decrease the temperature by a factor of 100. This would be about a factor of 10 below Tc. At present there are major unknowns concerning the formation and nature of the condensate in a cesium vapor. These involve questions concerning the interactions of atoms at temperatures less than 1 pK. We are looking forward to exploring these questions in the near future. To study the behavior of the atoms we will use light scattering in much the same way as we do presently. The scattering from a Bose condensate should be dramatically different from the normal vapor. One of several possible signatures, for example, is that the condensate will form a density spike in the middle of the trap which should be easily observable if one looks at the spatial dependence of the scattered light. In this paper we have discussed how laser trapping and cooling has opened up exciting new opportunities for studying very fundamental properties of nature at both very high and low energies. In the years to come there will no doubt be many other such experiments. This work is supported by the Office of Naval Research, and by the National Science Foundation. One of us (CEW) is a Guggenheim Fellow. C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. B, 1571 (1990). 2. P. Langacker, preprint UPR-435T. 310 3. M. C. Noecker, B. P. Masterson, and C. Wieman, Phys. Rev. Lett. (1988). 4. S. A. Blundell, W. R. Johnson, and J. Sapirstein, Phys. Rev. Lett. 65, 1411 (1990). 5. W. Marciano and J. Rosner, Phys. Rev. Lett. 65,2963 (1990); D. Kennedy and 2967 (1990). P. Langacker, Phys. Rev. Lett. 3093 (1991); G. 6 . K. T. Mahanthapa and P. Mohapatra, Phys. Rev. D Altarelli et al., CERN preprint # th6028/91. 7. Reviews are given by T. J. Greybak and D. Kleppner, in New TrenciS in Atomic Physics, Proceedings of the Les Houches Summer School, Session XXXVIII, eds. G. Greenberg and R. Stora (North-Holland, Amsterdam, 1984); I. Silvera and J. Walraven, in Progress in Low Temperature Physics, ed. D . Brewer (North-Holland, Amsterdam 1986), Vol. 10, Chap. D. 8. J. Doyle et al., MIT preprint. 9. R. Lovelace et al., Nature (London) 318, 31 (1985). 10. E. Corneli, C. Monroe, and C. Wieman, to be published. 11. This idea was proposed by D. Pritchard, K. Helmerson, and A. Martin, in Afomic Physics 11, eds. S . Haroche, J. Gay, and G. Grynberg (World Scientific, Singapore, 1989).
1.
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PHYSICAL REVIEW A
VOLUME 46, NUMBER 7
I OCTOBER 1992
Experimental and theoretical study of the vapor-cell Zeeman optical trap K. Lindquist, M . Stephens, and C. Wieman Joint Institute f o r Laboratory Astrophysics and Physics Department, University of Colorado, Boulder, Colorado 80309-0440 (Received 24 April 1992) We present an experimental study of the number and density of trapped atoms in a vapor-cell Zeeman optical trap. We have investigated how the number (and therefore the capture rate) and density change with the trapping laser’s beam diameter, intensity, and detuning and with the magnetic-field gradient of the trap. We have developed a quasi-one-dimensional numerical model that accurately predicts the number of trapped atoms for all conditions. We also have investigated chirping the laser frequency and trapping with broadband light, neither of which increase the number of trapped atoms.
PACS numberis):32.80.Pj, 42.50.Vk
I. INTRODUCTION
Much recent progress in optical trapping of neutral atoms has been driven by the appeal of large samples of cold, dense atoms [ l ] . Such samples will be useful for high-resolution spectroscopy [2,3], collision measurements [4-71, and time and frequency standards [ 1,8,9]. They will also be useful for studying rare, radioactive isotopes in decay and atomic parity-nonconservation experiments [10,11]. Additionally, optical traps can be used to load magnetic traps in an attempt to see Bose condensation [ 121 and for other fundamental physics experiments. The vapor-cell Zeeman-shift optical trap (ZOT), described below, produces these cold samples in a simple and inexpensive manner. However, despite the growing popularity of the vapor-cell trap, there has never been a detailed study of how to optimize the number and density of trapped atoms. These characteristics are important for most applications. We have determined how the number and density of trapped atoms vary with the laser beam sizes, laser intensities, laser detunings, and magnetic-field gradients. We have developed a simple model of the slowing that agrees well with the measurements of the number of trapped atoms. We have also explored two techniques for increasing the capture rate of the trap: frequency chirping [ 131 and bandwidth broadening [ 141 of the trapping light. Neither technique was successful. In what follows we first explain our experimental setup and measurements in Sec. 11, and then present our data on the number of atoms in Sec. 111. In Sec. IV we describe the model that explains these results, and compare its predictions to our data. In the course of this work, we learned of the results of Gibble, Kasapi, and Chu [15] and have included them in our comparison. In Sec. V we discuss the frequency chirping experiments and calculations and finally, in Sec. VI, we present measurements of the density of trapped atoms. 11. EXPERIMENTAL APPARATUS
Our apparatus is similar to that of Monroe et nl. [16]. The ZOT is a spontaneous-force optical trap consisting of
46 31 1
a spatially varying magnetic field and three orthogonal pairs of counterpropagating laser beams of opposite helicity. The laser beams intersect in a fused-silica cell that is filled with a lO-*-Torr room-temperature cesium vapor. The frequency of the light is slightly below the 852nm cesium D, transition. When Doppler shifts induce absorption imbalances between opposing beams, the resulting difference in radiation pressures slows down atoms in the low-velocity tail of the Maxwell-Boltzmann distribution. A quadrupole magnetic field induces a position-dependent shift in the Zeeman Ievels of the slowed atoms, which causes a position-dependent radiation pressure, thus trapping the slowed atoms [17]. The vapor cell is a 9-cm-long, 2.5-cm-diam cylinder with windows on each end. Two tubes each 4.5 cm long and 2.5 cm in diameter intersect the cylinder at right angles to form a six-way cross. Two smaller tubes are attached to the main cylinder. One is a “cold finger” containing a reservoir of cesium, whose temperature determines the vapor pressure in the cell. The second tube leads to a 2-liter/s ion pump, which removes any residual gas (mostly helium) that diffuses through the walls of the cell [ 161. The trapping beams come from an STC Optical Devices diode laser. We tune the laser frequency and reduce the laser linewidth to much less than 1 MHz with optical feedback from a diffraction grating [18]. The laser frequency is locked to the side of the 6 S , , , , F =4-6P3,,, F=5 cycling transition in cesium using the signal from a saturated-absorption spectrometer. The laser’s output is split into three circularly polarized beams of 5 mW each. These beams enter and exit the cell along three orthogonal axes, pass through quarter-wave plates, and then are retroreflected. The trapping beams are focused t o compensate for intensity losses from the cell windows so that the incident and reflected beams have the same intensity. A 20-mW beam from a second diode laser, frequency locked to the 6S,,,, F =3--t6P3,,, F=4 transition, overlaps one of the trapping beams. This “hyperfine pumping” laser prevents the atoms from accumulating in the F = 3 ground state. The quadrupole magnetic field is provided by a pair of coils with counterpropagating currents. The coil spacing is equal to the 5 cm diam of the coils. This “Maxwell 4082
@ 1992 The American
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312 EXPERIMENTAL AND THEORETICAL STUDY OF THE VAPOR- . . .
46
configuration” provides a very linear field gradient in all three directions, which simplifies the comparison of the model to the experiment. Three additional, orthogonal shim coils cancel stray fields in the region of the trap. We measure the number of trapped atoms by imaging the fluorescence of the cloud onto a photodiode, which gives us the rate at which protons are scattered by the cloud. We assume [5,19] that the number of atoms is proportional to the total fluorescence, with a proportionality constant given by the power-broadened scattering rate, which depends on the laser’s detuning and intensity. The following procedure is used to ensure that the relative uncertainty between measurements is significantly smaller than our uncertainty for the absolute number of atoms. First, we let the trap reach equilibrium at the detuning and magnetic field of interest, then we quickly shift the laser frequency to a standard detuning of - 1.5r and turn off the magnetic field. Here r is the 5-MHz natural linewidth of the transition, and the detuning is defined by A = vlaSer - vatom. We record the fluorescence during the next 10 ms. Turning off the magnetic field eliminates the 10% effects of the magnetic field on the scattering rate. The fluorescence from the cloud takes about 0.4 s to change after the adjustments in detuning and magnetic field, so this provides a n accurate measurement of the number of trapped atoms. Varying the trap parameters to determine how they affect the number of trapped atoms is straightforward. We change the trapping beam intensities by placing neutral density filters in the beam before it is split into three parts. T o change the beam diameter, we replace the lenses in our beam-expansion telescope. The magnetic field can be varied by adjusting the current in the Maxwell coils. Laser detuning is controlled by a variable set point in the frequency locking circuitry. To obtain the density of the cloud of trapped atoms, we measure both the volume of the cloud and the number of atoms in the trap. To measure the volume, we image the cloud from the top and from the side with two chargecoupled device video cameras. The horizontal and vertical axes of the camera images coincide approximately with the semimajor axes (u,b,c)of the ellipsoidal cloud of trapped atoms, allowing us to calculate the volume of the cloud as V =+ubc. The process used to determine the number of pixels covered b y the 1-5-mm clouds is described in Ref. [20]. The resolution is 0.1-0.2 mm, depending on the camera and on the dimension (horizontal or vertical) involved; care must be taken to avoid saturating the cameras. These number and density measurements are quite sensitive to the alignment of the trapping beams, so the beams are carefully aligned to give ellipsoidal cloud shapes. As a check on our alignment, we turn off the magnetic field, then watch the atoms diffuse out of the laser-beam intersection region. Atoms from well-aligned traps diffuse away slowly and isotropically.
I
,
< ,
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OF ATOMS
Figures 1-3 illustrate how the number of trapped atoms depends on various parameters. Only part of our
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15
25
M a g n e r , c Fielc Gradient (G/cm)
FIG. 1. (a) The number of trapped atoms vs the laser detuning at a constant magnetic-field gradient of 11 G/cm. (b) The number of trapped atoms vs the magnetic-fieldgradient, at a detuning of -2.5r. The beam diameter for both (a) and (b) is 1.9 cm and the intensity in each of the six beams is 1.6 mW/cm2.
The asterisks are our data, and the solid line shows the predictions of the model that includes the magnetic field. The dashed line is the prediction of the two-level model. data is shown, to indicate the significant trends. We find that the number of atoms is quite sensitive to alignment. After major realignment of the trap, our numbers are reproducible to within 20%, whereas measurements made when n o realignment is necessary are reproducible t o better than 10%. Figure l(a) shows how the number of atoms depends on detuning a t a fixed magnetic-field gradient; Fig. l(b)
i
/
0
1
2
3
4
5
I (rnw/crn’)
FIG. 2. The number of trapped atoms vs intensity at constant beam diameter: asterisks are our data for a trap with a 1-cm beam diameter, diamonds are for a 1.9-cm beam. Each data point was taken at the optimum detuning and optimum magnetic-field gradient for the given beam size and intensity. The solid lines represent the number of atoms predicted by the model that includes the magnetic field. The dotted line shows the predictions of the two-level model.
313
46
K. LINDQUIST, M. STEPHENS, AND C . WIEMAN
4084
-. . ,
-. 5
“ 3
.5
:‘;
2 -
(c-1)
FIG. 3 . Number of atoms vs l/e beam diameter, L , when the total power in each beam is held as constant as possible. Asterisks indicate data for powers from 0.9 to 1.2 mW per beam, crosses are 2.7-3.2 mW per beam, diamonds are 4-4.5 mW per beam. The power variation in each trace is responsible for some of the scatter in the data. Each data point was taken at the optimum detuning and optimum magnetic-field gradient for the beam size and power. The solid lines indicate the number of
atoms predicted by the model with the magnetic field. The inset shows, on a condensed scale, the predicted number of atoms as a function of beam size for three powers: 2, 4.5, and 7 mW per beam. Solid lines represent predictions made by the model when the magnetic field is included; dotted lines represent predictions made by the two-level model. shows how the number of atoms depends on the magnetic-field gradient at a fixed detuning. For each beam size and beam intensity, there is an optimum detuning and a n optimum magnetic field; as the detuning is increased, the optimum magnetic-field gradient for that detuning increases. Figure 2 shows that as the intensity is increased for a fixed beam diameter, the number of atoms increases approximately linearly. This applies when the detuning and magnetic field are optimized for each intensity, and the intensity is not too high. We also found that, for a fixed power, a larger beam diameter L always resulted in a larger number of trapped atoms (see Fig. 3 ) despite the corresponding decrease in the intensity of the trapping beams to well below the saturation intensity of the optical transition. This was true over the entire range of sizes we were able to study, but, as discussed in Sec. IV, we expect this to fail for very large L. Finally, as L increases while the peak intensity is held constant, the number of atoms increases very rapidly.
This measurement is particularly alignment sensitive; therefore, the uncertainty in our data is quite large. As shown in Table I, for the beam sizes and intensities that are available with this apparatus, the detunings that provide the maximum number of atoms are in a small interval around - 2 . 2 5 r and the optimum magnetic-field gradients along the axis of the coils are between 10 and 15 G/cm. This is in agreement with previous work [15,16]. As can be seen from the table, the optimum values are slowly increasing functions of both intensity and beam diameter. This is reasonable: a larger beam size creates a longer stopping region, which means that faster atoms can be stopped. This results in an increase in the optimum detuning since faster atoms have larger Doppler shifts. To determine whether any important slowing takes place outside the region where the three trapping beams overlap, we varied the volume illuminated by the hyperfine repumping ( F = 3 - + F r = 4 ) laser. We observed that the same number of atoms were trapped whether the repumping light overlapped just one of the trapping beams, or overlapped all three trapping beams. Since repumping is necessary for slowing, we conclude that any slowing that takes place outside the three-beam overlap region does not contribute to the number of atoms trapped. Consequently, the models described in the next section consider the slowing forces only in the overlap region.
LV. ‘rHE~RETLCAL~ O D ~ L S OF THE CAPTURE P ~ O C E § §
In this section we present two different models of the capture process and compare them with the data. The first model is a very simple calculation which treats the atom as a two-level system and ignores the magnetic field and the Gaussian profile of the beam. While it does not match all the observations, it is quite useful in that it does predict most of the general trends quite accurately. This allows one to predict capture rates for most situations reasonably well with just a few lines of code on any personal computer. The second model agrees with all of our observations, as well as those made by Gibble, Kasapi, and Chu 1151 at larger beam sizes and higher intensities than attainable in our experiment. Although this model is somewhat more elaborate, it is still surprisingly simple. Monroe, Robinson, and Wieman [8] have shown that the steady-state number of atoms in the trap is given by
TABLE I. The optimum detunings and magnetic-field gradients predicted by the model are compared to the measured optima. The number of atoms measured in each case and the predicted average capture velocity are also shown. The last line is data of Gibble, Kasapi, and Chu 1151. __~
r)
Field gradient (G/cm)
Beam size
intensity
(cm)
(mW/cm*)
Opt.
Pred.
Opt. -
0.9 0.9
1.9 7.7 4.0
-2.0 -2.5
-2.5
10
-3.0 -3.0 -3.0 -6.5
12
1.o I .9 4.0
A (uniis of
1.6
-2.5 -2.5
22.0
-4.0
13
10 7.7
Pred. 18 18 18 12 9.0
1V
(units of 10’) 0.3 1.2 1.5 2.0 360.0
u,
(average) (m/s)
11 16 14 15 36
314 EXPERIMENTAL AND THEOR5TICAL STUDY O F THE VAPOR-. . .
46
4085
time the atom is being slowed its velocity is high enough that the beam in the negative x direction is Doppler shifted closer to resonance with the atoms than the other beams, and hence dominates the excitation. where R is the capture rate and 1 / ~ ~ ~ ~ is ~ the = n ~ To u find u ~ the ~ maximum ~ capture velocity, the equation of loss rate due to collisions between trapped atoms and motion is solved numerically with the constraint that the atoms in the room-temperature Cs background gas. The atom must be stopped ( u < 10 cm/s) by the time it has cm2, cross section u for these collisions [16] is 2 X traversed the trapping region. The length of the trapping n o = i08/cm3 is the density of the background gas, and region is defined to be the l/e diameter of the trapping u,,, =236 m/s is the root-mean-square velocity. The beam's Gaussian intensity profile (we do not include the capture rate involves the surface area A of the trap Gaussian profile anywhere else in the two-level model). volume, and the ratio uc/uIherrnal, where u, is the maxOnce u, has been found, the number of atoms trapped is imum velocity an atom can have and be captured, and calculated from Eq. (1). To determine the surface area of uthermal=193m/s is the average velocity of the backthe trapping region, we assume that it is spherical, with a ground gas. We assume that all atoms entering the trapdiameter equal to the beam diameter, A =aL2. This ping region at speeds below u, will be trapped. Because T very simple model predicts values for the number of depends only on the characteristics of the background atoms that are approximately 0.3 of those observed (a gas, which are constant, a change in the number of atoms 35% error in u, ). To better illustrate how well the model in the trap is equivalent to a change in the capture rate R . predicts general trends, we have normalized all the preThus, if one is interested in using the trap as a pulsed dicted values by dividing by 0.3. source of cold atoms, the production rate can be calculatIn Figs. 2-4 we compare the normalized predictions ed from our values for N using Eq. (1). This equation with observations. Note that this extremely simple modneglects the contribution of intratrap collisions to the loss el predicts most general trends in the capture rates rerate of the trap [5]. For the densities of background vamarkably well; the two areas where it fails most conspipor and trapped atoms at which we operated, this contricuously are in predicting the optimum detuning, and of bution is relatively small and thus does not affect most of course the dependence on magnetic field. The error in the comparisons we make between our model and our the detuning dependence is reflected in Fig. l(a) and in data. In our discussion, we point out the few cases where the low-intensity part of Fig. 4. The detuning was kept the contribution may be significant. constant at - 4 r throughout the measurements displayed Equation (1) shows that to predict the number of in Fig. 4. This is only slightly higher than the actual optrapped atoms we need to calculate only u, and A . We timum detuning for low intensities, but substantially find u, by computing the one-dimensional slowing forces larger than - 1 . 5 r , which is the optimum predicted by on an atom in the trapping region. These forces are nearthis model. The predicted number of atoms is lower ly the same along the three axes which allows us to conwhen the model detuning is specified to be -4r. If insider only atoms moving in the x direction, but the calculated u, will be valid for all directions. The slowing force for an atom moving in the +x direction is proportional to the difference between the numbers of photons scattered from the two beams propagating along the x axis. In our first model we make the great simplification of treating the atom as a two-level system in one dimension, thus ignoring the magnetic field and the Gaussian intensity profile of the laser beams. The radiation-pressure force can then be written r
ld
(1)
0
l+8+4
I
A ku -+r 2Tr
r ,
(2)
where P=I/I,, is the saturation parameter for one beam, I is the intensity in each beam, and 1,,=2.7 mW/cm2; k=7400 cm-l is the wave number of the trapping light; A=vl,,,,-vYalom is the detuning of the laser, va,,,=3.52X I O l 4 Hz,and T~~~~ is the 31-11s excited-state lifetime. We neglect the saturation effects from the orthogonal trapping beams because during nearly all of the
5
10 '5 I (r-W/crr')
20
25
FIG. 4. The results or our models are compared with the data of Gibble, Kasapi, and Chu [15]. As in Fig. 2, the numbers of atoms trapped are shown as a function of intensity, but rather than optimizing the detuning and the magnetic field at each point (as in Fig. 21, the detuning is kept constant at -4r and the magnetic field is kept constant at 7.7 G/cm, which are the conditions specified in Ref. [15]. The beam diameter is 4 cm. Asterisks are data from Gibble, Kasapi, and Chu, the solid line shows the predictions of the model that includes the magnetic field, the dotted line illustrates predictions of our two-level model. The rolloff at high intensities is due to saturation of the optical transition. For both models the values for the numbers of atoms were normalized to 3.6X 10" atoms at 22 mW/cm2. The normalization factor was 3.2 for the magnetic-field model, and 4.2 for the two-level model.
315 4086
K. LINDQUIST, M. STEPHENS, AND C. WIEMAN
stead of taking the predicted number at a fixed detuning, we take the number at the optimum detuning, as in Fig. 2, then there is much better agreement. In fact, with optimized detuning, this model matches the experimental intensity dependence shown in Fig. 4 quite closely. I n summary, this one-dimensional two-level model makes it possible to easily estimate how capture rates will scale with the laser-beam diameter and intensity, or any of the other factors that are included in Eq. ( 2 ) . Of course it cannot predict the dependence on the magnetic-field gradient, such as shown in Fig. 1, and it is off by about 60% for the absolute number of trapped atoms. To overcome these limitations, we carried out a more elaborate calculation which includes the magnetic field and the multiple m levels of the atom. The quadrupole magnetic field causes position-dependent splitting of the Zeeman levels of the Cs atom. There is a corresponding shift in the frequency of the optical transition:
where B is the magnitude of the magnetic field, gF_,=0.25 and g F# = , = 0 . 4 are Lande g factors, p B is the Bohr magneton, and h is Planck’s constant. This position-dependent frequency shift results in a positiondependent change in the effective laser detuning. Therefore, the slowing force is a function of the position of the atom as well as its velocity. The slowing force in the x direction is a function not only of x, but also of y and z. If the atom that is being slowed enters the trapping region displaced from the x axis (but still with velocity in the +x direction only), it moves through a quadrupole magnetic field that has x, y, and z components (Fig. 5 ) . As a result, the circularly polarized light propagating in the x direction will drive Am = irl and 0 transitions (in the basis defined by the magnetic field). As discussed below, this reduces the effect of the magnetic field on the slowing process. The substantial and changing magnetic field acting upon the
where I(y,z)=Z,exp[ - 4 ( y 2 + z 2 ) / L 2 ] is the intensity of a single beam, and the saturation parameter I,,, depends on the transition and the initial m level. Note that the Gaussian profile of the beam is now explicitly included in the model. Pjlght(Am)is the fraction of the light that drives Am = 1, 0, or - 1 transitions and P,,,,( Am, mF 1 is the mF to mF‘ transition probability. This now models the trapping volume as a cube with side L , rather than as a sphere. With this model, the absolute number of atoms
46
FIG. 5. Spatial variation of the magnetic field in the trap. The grey arrow shows one possible path of an atom through the trap. The dark, black arrows indicate the direction of the magnetic field at several points along the atom’s path.
atom as it moves through the trapping region also tends to equalize the populations of the different m levels. We believe this is the reason our model, which assumes no optical pumping or other ground-state coherences, is successful. Because of this important three-dimensional character of the magnetic field, we must calculate u, in the x direction as a function of the z and y coordinates of the atom when it enters the trapping volume. The (uc/vtherma, in Eq. ( 1 ) must now be replaced by an average over the beam’s cross section, so that Eq. (1) becomes
To find the position-dependent slowing force, the angular momenta of the trapping lasers were rotated into the basis of the magnetic field a t each position. Then the total stopping force was calculated by summing over the force produced by each Am transition:
is predicted more accurately, as one might expect. The normalization factor to best fit all our data with the model is 1.6, which is well within the uncertainty for the value of CT. As mentioned above, we use a different normalization factor when we make comparisons with the data in Gibble, Kasapi, and Chu [15]. In addition to the approximations already noted, this model neglects the effect of saturation on the distribution of atomic populations. For large-m, sublevels of the
46
EXPERIMENTAL AND THEORETICAL STUDY OF THE VAPOR-. . .
ground state, the transition probability for one of the three Am transitions (either A m = + l or -1) is much greater than the transition probability for the other two. If the intensity is high, the strong transition wiII saturate before the others, reducing the ground-state population and therefore reducing the probability that the other Am transitions will be excited. We have tested the importance of this effect by introducing a correction factor in Eq. ( 5 ) t o approximate the effect of the saturation. We found that it did not significantly affect the results. We also neglect the effects of the orthogonal beams, as in the previous model. The results of our normalized model, and comparisons with our data and those of Gibble, Kasapi, and Chu [ 151, can be seen in Figs. 1-4 where the agreement appears to be quite good. One difference from the previous model [see Fig. I(b)] is in the predicted number of atoms, which is larger than the measured values for low fields. The reason for this difference is that the slowing of the atoms is being modeled-not the trapping. When there is no magnetic field, and therefore no ZOT, atoms are still being slowed. There is still a maximum capture velocity; therefore a t zero magnetic field this model will erroneously predict some number of trapped atoms. The only other significant discrepancy is between the predicted and measured optimum detunings (Table I), particularly at the larger beam size and intensity used by Gibble, Kasapi, and Chu [15]. The increase in the loss rate due to light-induced intratrap collisions, which we have neglected, may be large enough to affect the number of trapped atoms for the parameters used by Gibble, Kasapi, and Chu. Also the uncertainty in the detuning data is not clear, so the discrepancy may be smaller than it appears. Our model predicts that as L is increased while the intensity is held constant, the number of trapped atoms increases as L 3 . 6(see Fig. 6). This is consistent with our measurements, and with the exponent of 3.65 measured by Gibble, Kasapi, and Chu [15]. At small beam sizes and high intensities, the number of atoms is not proportional to L3.6,because the number of atoms that can be trapped is limited by the available stopping distance rather than by intensity. The beam size at which the number of atoms begins to follow the relationship N - L 3 . 6 depends on the intensity, as shown. The major difference, besides the normalization, between the predictions made by this model and the twolevel model can be seen in the inset of Fig. 3. This model predicts that there is a large optimum beam size for a given power. This optimum is to be expected: for very large beam sizes the Zeeman shift at the edge of the trap is large enough to interfere with the slowing of the atoms, even with very small magnetic-field gradients. We find that the optimum magnetic field for the slowing is zero. However, since in practice a magnetic field is required to obtain a trap, in the simulations the magnetic-field gradient was not permitted t o drop below 3 G/cm along the axis of the coils. This gradient limits the size over which the advantage of a long stopping region overcomes the disadvantage of lower intensity. A n interesting prediction of this model is that if the intensities of the three trapping beams are equal, the cap-
2.0 L O , ' "
~
" ' '
"
~
' ",'
I
4087
"
"
'
'
"
"
"
"
"
-'
i
1.5 1 .o
0.5 0.0 0
2
I
L~
3
4
(cm)
FIG. 6. Predicted dependence of the number of atoms on the trapping beam size for beams of constant intensities. Predictions are shown for beams with intensities of 2, 5 , and 8 mW/cm*. The solid lines are predictions of the model that in-
cludes the magnetic field; dotted lines shown predictions of the model that does not include the magnetic field (normalized to match predictions of the other model at a 4-cm beam size). The detuning and magnetic field were optimized at each point. The inset shows the same predictions for smaller beam sizes. ture velocity along the longitudinal axis of the magneticfield coils will be slightly smaller than that along the radial axes because the magnetic-field gradient along the axis of the coils is twice that of either radial gradient. For a constant total power in the six beams, our model predicts that a slight (10-15 %) increase in the number of atoms trapped is achieved by increasing the power in the trapping beams along the longitudinal axis by about 20%. This was tested by adjusting the powers in the different beams, which produced an increase in the number of trapped atoms of just the size predicted. Our results clarify a question that has come out of previous calculations. Metcalf [21] had predicted capture velocities for metastable helium that depended strongly on magnetic field. When scaled to cesium, these velocities were far larger than what has been observed. We believe this discrepancy was the result of considering the magnetic field only along the central axis of the laser beams where it is parallel to the wave vector. As we discuss above, most of the atoms enter the trap off axis, where the magnetic field and the wave vector of the laser beam are not parallel. This allows Am = 1, - 1, and 0 transitions to be excited. The net effect is that the magnetic field has far less influence on the slowing than is calculated when one considers only the on-axis case, as was done in Ref. [21].
+
V. CHIRPING We have also explored the possibility of increasing the trapping rate by chirping the laser frequency. Frequency chirping (sweeping the laser frequency repeatedly from large to small detunings) has been used with success to slow atomic beams [22,23], and it has been suggested that the technique would also allow a greater capture rate in a vapor-cell trap [13]. The reasoning is that to increase the
31 7 46 -
K. LINDQUIST, M. STEPHENS, AND C. WIEMAN
4088
capture velocity, one must increase the laser's detuning to adjust for the larger Doppler shifts seen by faster atoms. One then sweeps the laser's frequency to keep it in resonance with the atoms as they slow down. We have studied this idea experimentally and compared our measurements with the predictions of the model described above. The number of atoms in the trap was measured while the laser was being linearly chirped. The data shown in Fig. 7 were taken with a chirp period of 1 ms, which both experiment and theory showed to be the optimum. The chirp period could be varied by about a factor of 2 without substantial change in the number of trapped atoms. The number of trapped atoms is maximized when the starting detuning is about -4r, and never exceeds the number obtained in the unchirped case. In principle, the laser's frequency chirp should compensate not only for Doppler shifts, but also for Zeeman shifts caused by the spatially varying magnetic field. This means that a linear frequency chirp is not the ideal time dependence. However, since the Zeeman shifts are small compared to the Doppler shifts it should still be close to optimum. Our calculations again gave values quite similar to the experimental values, and made it clear why chirping does not work. Since the atoms in the trap are captured from a background gas of cesium in the cell, not all the atoms entering the slowing region will d o so when the laser is at the point of farthest detuning. Some of the atoms will enter the slowing region "out of phase" with the laser's frequency chirp. The number of trapped atoms is therefore given by
(6) While a large capture velocity is obtained for atoms that
i
i
0' -8
I
-7
1
-6 -s -4 Storii-g d e t J n ' l g ( A / r )
-3
-?
FIG. 7. The solid line shows the model's predictions for the number of trapped atoms when the laser's frequency is linearly chirped. The abscissa shows the starting frequency of the chirp cycle. Each chirp cycle stopped at - 1.Or detuning, and the cycles repeated every 1.O ms. The simulation used a beam size of 2.0 cm and an intensity of 1.6 mW/cm2. The dashed line represents the number of atoms predicted for the case of no chirp and the optimum detuning of -3r. The asterisks show our measurements at the same conditions.
enter the trapping volume a t just the right phase of the chirping cycle, this is more than offset by the reduced capture velocities during out-of-phase parts of the chirp cycle. When the chirp is started at a detuning larger than - 4 r , the number of atoms observed is lower than predicted by the model. We believe that this is because hyperfine-changing collisions [S] during the far-detuned parts of the chirp cycle increase the loss rate of the trap. We tested this hypothesis by measuring the trap's characteristic decay times after the trapping laser detuning was rapidly shifted. The rate at which atoms left the trap was more than twice as large at -4r than at - 2 . 5 r , and continued to increase with larger detuning, thus supporting this hypothesis. We tried another proposed [14] method for improving the capture rate: using broadband trapping light, or light with several closely spaced frequency components. This technique also did not work as well as a single fixed frequency. Recent theoretical work by Parkins and Zoller [24,25] provides a plausible explanation for these results. They have shown that low-velocity atoms experience heating rather than cooling under conditions similar to those used in the experiment. Our results are consistent with those of other unsuccessful attempts to increase the number of trapped atoms by frequency chirping [I51 o r trapping with broadband light [15,26], and it is now clear why these techniques fail. VI. DENSITY OF THE TRAPPED ATOMS
As a final study, we have measured the dependence of the density of atoms in the trap on the trap parameters. We have observed a number of trends, but are unable to model the density quantitatively because of its complex dependence on radiation-trapping forces [ 191. An additional complication in the interpretation is that the absolute measurement of the density could vary by as much as 25%. First, we see the density increase with increasing magnetic-field gradient (Fig. 8). Second, the density increases with increasing detuning of the trapping lasers (Fig. 9). Third, the density increases with increasing intensity of the laser beams (this effect tapers off at high intensities). Finally, there is little if any dependence of the density of atoms in the trap on the diameter of the trapping beams (in contrast with our observations of the number of atoms). Some of these trends can be explained qualitatively with the radiation-trapping density model of Ref. [ 191. This model gives a density limit due t o the balance of trapping, absorption, and radiation-trapping forces:
ck
"max
-- ( T r ( U , - - U , ) Z
'
(7)
where Z is the incident intensity, k is the spring constant, (T, and 0,are the cross sections for absorption of incident laser and scattered light, and c is the speed of light. The ( o r - o ), difference depends on the overlap between the reemission spectrum for scattered laser light and the absorption spectrum in a strong standing-wave field. As
318 EXPERIMENTAL A N D THEORETICAL STUDY OF T H E VAPOR-
46
2oT-----
"
"
"
'
'
'
"
'
"
5
,"
1 .?
15
20
25
M c g n e t , c f eld < r o c ent (G/cm)
FIG. 8. The number density of trapped atoms vs magneticfield gradient. Three different trap-laser detunings are shown: - lr (squares), - 1 . 5 r (triangles), and - 2 . 5 r (asterisks). The beam diameter was 1.4 em and the intensity was 3.6 mW/cm2 per beam. The symbols are our experimental data; the connecting lines are only to guide the eye. The field gradients shown on the abscissa are along the symmetry axis of the pair of field coils. Gradients in the other two dimensions are half of this.
discussed in Ref. [19], this difference is quite difficult to calculate accurately for a real cesium atom in three dimensions. The spring constant in Eq. (7)is
dB
AL?
-
k-
[
dz
1+8+
j2j2
(8) '
where dB /dz is the magnetic-field gradient along the axis of the coils. From Eqs. (7) and (8)one would expect the qualitative dependencies on intensity and field gradient
*
0
-I -2 -3 Trap laser detuning (A/r)
-4
FIG. 9. Variation of the number density of trapped atoms with detuning of the trap laser. Three beam sizes were used, all with approximately 5 mW per beam, to give the three curves at 1.6 mW/cmz (squares), 3.6 mW/cm2 (triangles), and 9 mW/cmz (asterisks). The magnetic-field gradient was held constant at 16 G/cm. The symbols show our experimental data; the connecting lines are only t o guide the eye. A characteristic error bar, determined from scatter in the values obtained in repeated trials, is shown for one point. As the detuning was increased beyond - 2 . T , the number of trapped atoms dropped precipitously.
4089
7
0
3
...
5 10 '5 20 N u m b e r of a t o m s (10')
25
FIG. 10. The data of Sesko et al. [I91 (asterisks) for the diameter of the cloud of trapped atoms vs the number of atoms in the trap. The solid line represents the prediction of Eq. (7) ( L - N 1 ' 3 ) . The dotted line shows the same prediction with the added constraint that the optical thickness cannot become so large that an average reemitted photon will scatter more than once in escaping the cloud.
that are observed. The detuning dependence is too complex to predict. On a final note, our data show that, in general, large clouds ( N > 3 X lo' atoms) have lower densities than small clouds. This is consistent with the data of Sesko et al. [19], and inconsistent with Eq. (7). We believe that the source of this discrepancy is the fact that Eq. (7) was derived with the assumption that the photons are scattered no more than twice. However, with more than 3 X lo7 atoms and the density predicted by Eq. (7), the optical thickness would be large enough for a significant number of photons to scatter more than twice in the cloud of trapped atoms before escaping. This would increase the radiation-trapping force, causing the cloud to expand and the density t o drop below the value given by Eq. (7). If the cloud expands too far, however, the optical thickness will begin to decrease, thereby reducing the radiation-trapping force. This leads to a balance of forces that maintains the density of a large cloud at an optical thickness where, on the average, each photon absorbed from the laser beam will, after reemission, scatter no more than once on its way out of the cloud. This means that when the number of atoms is greater than 3 X lo7 the diameter of the cloud will go as the square root of the number of atoms in the cloud rather than as the cube root as Eq. (7)predicts. In Fig. 10 we show that this dependence is very close t o what was reported in Ref. [19]. Our current data are similar but covet a smaller range. VII. CONCLUSION
We have reported results of a study of optical trapping in a vapor-cell Zeeman optical trap. We have developed a model that agrees with our observations of how the number of atoms in the trap varies with the intensity of the trapping beams, the size of the trapping beams, the detuning of the laser, and the strength of the magnetic field. Our model includes the one-dimensional slowing force on an atom moving through the trap, and the three-dimensional effects of the magnetic field. We are
31 9 4090
K. LINDQUIST, M. STEPHENS, AND C. WIEMAN
46
rameters. W e observed small increases in density with magnetic field, laser intensity, a n d detuning.
confident that we can accurately predict numbers of atoms for any reasonable t r a p parameters. In addition, we have shown that for many conditions it is possible t o reliably calculate dependences using a simple onedimensional, two-level model that neglects the magnetic field. This allows one t o estimate the capture rates a n d numbers of trapped atoms quite easily. Attempts to increase the number of trapped atoms by frequency chirping or by bandwidth broadening t h e laser were not successful, a n d we can now explain why these techniques d o not work. Finally, we have described the results of measurements of the t r a p density a s a function of the t r a p pa-
This work was supported by the office of Naval Research and the National Science Foundation. We would like t o thank E. Cornell and C. Monroe for useful discussions. W e are also grateful to K. E. Gibble, S . Kasapi, a n d S. C h u for providing us with their data prior t o publication, and for permission t o reproduce it.
[ I ] J . Opt. SOC.Am. B 6 122) (1989) Special Issue on laser cooling and trapping. [2] D. Grison, B. Lounis, C. Salomon, J. Y . Courtois, and G. Grynberg, Europhys. Lett. 15, 149 (1991). [3] J . W. R. Tabosa, G. Chen, Z . Hu, R . B. Lee, and H. J . Kimble, Phys. Rev. Lett. 66, 3245 (1991). [4] A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63, 957 (1989). [5] D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961 (1989). [6] P. D. Lett, P. S. Jessen, W. D. Phillips, S. L. Rolston, C. I. Westbrook, and P. L. Gould, Phys. Rev. Lett. 67, 2139 (1991). [7] P. S. Julienne and J. Vigue, Phys. Rev. A 44,4464 (1991). [8] C. Monroe, H. Robinson, and C. Wieman, Opt. Lett. 16, 50 (1991). [9] S. L. Rolston and W. D. Phillips, Proc. IEEE 79, 943 (1991). [lo] S. Freedman and K. Coulter (private communication). [ l l ] C. Wieman, C. Monroe, and E. Cornell, in TenthInternational Conference on Laser Spectroscopy, edited by M. Ducloy, E. Giacobino, and G. Camy (World Scientific, Singapore, 19921, p. 77. [I21 E. Cornell, C. Monroe, and C. Wieman, Phys. Rev. Lett. 67,2439 (1991).
[I31 This idea has been suggested by many people. The first, to our knowledge, was D. Pritchard in about 1985. Also see Ref. [15]. [I41 M. Zhu, C. W. Oates, and J. L. Hall, Phys. Rev. Lett. 67, 46 (1991). [ 1 5 ] K . E. Gibble, S. Kasapi, and S. Chu, Opt. Lett. 17, 526 ( 1 992). [I61 C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571 (1990). [17] E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, Phys. Rev. Lett. 59,2631 11987). [18] C. Wieman and L. Hollberg, Rev. Sci. Instrum. 62, 1 (1991). [I91 D. W. Sesko, T. G. Walker, and C. E. Wieman, J . Opt. SOC.Am. B 8,946 11991); T. G. Walker, D. W. Sesko, and C. E. Wieman, Phys. Rev. Lett. 64,408 (1990). [20] E. Cornell, C. Monroe, and C. Wieman (unpublished). [21] H . Metcalf, J. Opt. SOC.Am. B 6,2206 (1989). (221 R. N. Watts and C. E. Wieman, Opt. Lett. 11, 291 (1986). [23] W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985). [24] A. S. Parkins and P. Zoller, Phys. Rev. A 45, R6161 (1992). [25] A. S. Parkins and P. Zoller, Phys. Rev. A 45, 6522 (1992). [26] M. Zhu and J. L. Hall (private communication).
ACKNOWLEDGMENTS
THE LOW (TEMPERATURE) ROAD TOWARD BOSE-EINSTEIN CONDENSATION 1N OPTICALLY AND MAGNETICALLY TRAPPED CESIUM ATOMS
Chris Monroe, Eric Cornell and Carl Wieman Joint Institute for Laboratory Astrophysics University of Colorado and National Institute of Standards and Technology and Department of Physics, University of Colorado Boulder, CO 80309-0440
1. INTRODUCTION There is considerable reason to pursue the extreme limits of low temperature and high density that can be achieved in an atomic vapor. First there is the intrinsic interest in the trapping and cooling of atoms, and the behavior of atoms in new regimes of temperature. Much
of this is discussed extensively elsewhere in this volume. Second, many interesting applications of laser cooled and trapped atoms require very cold high density samples. For example, cooled and trapped atoms are ideal for spectroscopy of weak transitions, such as those involved in
measurement of parity nonconservation in atoms. Clearly, colder and denser samples will enhance any such experiment with trapped atoms. Finally, the ultimate limit of high density and low temperature is of great interest in its own right. This is Bose Einstein Condensation (BEC)
of a trapped vapor sample of bosonic atoms. The ability, in principle, to cross the phase boundary for BEC at several widely spaced points in the temperature-density plane offers the tantalizing prospect of studying the phase transition while varying the relative significance of
interparticle interactions. Although these interactions may well prevent BEC, it will be very interesting to see exactly what will happen when one reaches the necessary condition of the deBroglie wavelength of the atoms exceeding the interparticle spacing. Over the past several years we have been working to obtain higher density and lower 320
32 1
temperature samples of atomic cesium in pursuit of this limit. In the course of this work we have found many interesting new phenomena which have acted as barriers, and we have developed new technologies which have provided ways around these barriers or have generally aided our efforts to cool and trap atoms. In this paper we describe this work and discuss the radiation catalyzed collisions which lead to losses from optical traps, the "radiation trapping repulsion" which limits density in an optical trap, the vapor cell optical trap which allows one to obtain trapped atoms very easily, the loading of optically trapped and cooled samples into magnetic traps, and finally the ac magnetic trap. 11. THE ZEEMAN-SHIFT OPTICAL TRAP
The starting point for all of this work is the Zeeman shift spontaneous force optical trap (ZOT) first demonstrated by Raab et al.'
In our initial studies of collisions and collective
behavior of atoms in an optical trap, a beam of cesium atoms is effused from an oven and passed down a 1 m tube into a UHV chamber (lo-'' TOK). As the atoms speed down the tube they are slowed by a counterpropagating laser beam. In the UHV chamber, the atoms are trapped in a
ZOT formed by having three perpendicular laser beams which are reflected back on themselves. The beams are all circularly polarized, with the polarizations of the reflected and incident beams being opposite. They have Gaussian profiles with diameters of about 0.5 cm, and are tuned to
the 6SF=4to 6P,,*
F=5
transition of cesium. Magnetic field coils, 3 cm in diameter and arranged
in an anti-Helmholtz configuration, provide a field that is zero at the middle of the intersection
of the six beams, and has a longitudinal gradient of about 10 G/cm. In addition to the main trap laser beams, there are beams to insure the depletion of the F=3 hyperfine ground state in the trap. Thus, all the trapped atoms are in the F = 4 ground state. All the laser light in these experiments is provided by diode lasers which are frequency stabilized using optical feedback. The fluorescence from the trapped atoms is observed with both a photodiode, which monitors the total fluorescence, and a charged-coupled-device television camera which shows the size and shape of the trapped atomic cloud. We have discovered that the size and shape (but not the density) of the trapped atom cloud is strongly dependent on the number of atoms.* When the number is less than about 10s atoms, the cloud is a constant diameter ellipsoid with the Gaussian distribution one expects for an ideal
322
gas in a three dimensional harmonic well. Only in this regime can we accurately measure the density -- a necessary condition for collision studies. Therefore, all our collision studies have been carried out with less than 5 X 1 0 4 trapped atoms.
We have studied the low temperature collisions between the trapped atoms by observing the loss rate of the atoms from the optical trap.3 Assuming a loss rate of the form dn/dt = -an
- @n2, where n is the density of atoms in the trap, we find the deviation from a pure exponential decay indicates a trap loss mechanism due to collisions between trapped atoms. Similar observations of collisional trap loss have been obtained previously by Prentiss et aL4 The quantity of interest, 8, is the rate constant for the low temperature collision between trapped atoms; a is the coefficient that describes the traditional collisions with the room temperature background gas in the vacuum chamber. We studied how /3 depends on the intensity of the trapping laser beams.3 Changing the intensity varies the fraction of the atoms that arc in the excited state and also varies the trap depth. In Fig. 1, we show the dependence of /3 on the total intensity of all trap laser beams for
a trap laser detuning one linewidth to the red (5 MHz). As the intensity is decreased from 4 mW/cm2, 13 increases dramatically. We interpret this as due to hyperfine-changing collisions between ground-state atoms. Two atoms acquire velocities of 5 m/s if they collide and one changes from the 6s F-4 hyperfine state to the lower
F=3 state. For high trap laser intensities, atoms with this velocity cannot escape, but at lower intensities the trap is too weak to hold them. We also find that if the beams are misaligned slightly, a similar large-/3 loss rate is observed for all intensities. We estimate that hyperfinechanging collisions between ground-state atoms due to the van der Waals interaction will give a @ in the range lo-'' to lo-'' cm3/s, which is consistent with the observed low intensity value.
Above 4 mW/cm2, 8 appears to increase linearly with intensity. We believe that this is due to the low temperature energy transfer collisions of the type discussed by Gallagher and Pritchard' and by Julienne in this volume. In this mechanism, two atoms, one excited, are accelerated toward each other via the -1/? ground-excited potential. However, the excited atom spontaneously decays during the collision and leaves the trap due to its large accrued kinetic energy. At sufficient intensities or detunings, this loss rate can dominate the loss rate due to background collisions, thereby giving an effective ceiling to the density of trapped atoms.
323
We have studied in some detail the variations in the size and shape of the cloud of trapped atoms with the number of atoms it
contain^.^ We find the atoms behave in a highly collective
manner, and this behavior arises because of a previously overlooked long-range interaction between the atoms. This interaction is negligible at room temperature or for any temperature at which an atomic vapor has previously been studied. However, because of the low temperatures and high densities in a ZOT, it becomes the dominant force in determining the behavior of the
atom cloud, and leads to dramatic "phase-like" transitions. Three separate well-defined distributions or "modes" of the trapped atoms were observed, depending on the number of atoms in the trap and on how we aligned the trapping beams. The "ideal-gas" mode discussed earlier occurred when the number of trapped atoms was less than
105. The atoms formed a small sphere with a constant diameter of approximately 0.2 mm. In this regime, the density distribution of the sphere was a Gaussian, as expected for a damped harmonic potential, and increased linearly in proportion to the number of atoms. The atoms in this case behaved just as one would expect for an ideal gas with the measured temperature and trap potential. When the number of atoms was increased past = 105, the behavior deviated dramatically from that of an ideal gas, and long-range repulsions between the atoms became apparent. In the "static" mode the diameter of the cloud smoothly increased with increasing numbers of atoms. Instead of a Gaussian distribution, the atoms in the trap were distributed fairly uniformly. To study the growth of the cloud, we measured the number of atoms versus the diameter of the cloud using a calibrated CCD camera. The results are shown in Fig. 2. To fully understand the growth mechanism, the temperature as a function of diameter was also measured. For a detuning of 7.5 MHz, the temperature increased from the asymptotic value of 0.3 up to 1.0 mK as the cloud grew to a 3 mm diameter. If the cloud were an ideal gas, the temperature increase would need to be orders of magnitude higher to explain this expansion. One distinctive feature of this
regime is that the density did not increase as we added more atoms to the trap. It remained nearly constant with some decrease at the largest numbers of atoms.
When all the trapping beams were reflected exactly back on themselves, we obtained a maximum of 3.0 x lo8atoms in this "static" distribution. In contrast, if the beams were slightly misaligned in the horizontal plane, we observed unexpected and dramatic changes in the
324
distribution of the atoms. The atoms would collectively and abruptly jump into a third mode when the cloud contained approximately lo8 atoms. We call this the "orbital" mode. The number of atoms needed for a transition depended on the degree of misalignment.
The expansion of the cloud with increasing numbers of atoms and the various collective rotational behaviors clearly show that some long-range interatomic force dominates the behavior of the atomic cloud. Since normal l/r6 or 1/? interatomic forces are negligible for the atom densities in the trap (lO1o-lO" ~ m - ~other ) , forces must be at work. In particular, since the optical depth for the trap lasers is on the order of 0.1 for our atom clouds (3 or more for the probe absorption at the resonance peak), forces resulting from attenuation of the lasers or radiation trapping can be important. The attenuation force is due to the intensity gradients produced by absorption of the trapping lasers, and has been discussed by Dalibard6 in the context of optical molasses. This force tends to compress the atomic cloud and so cannot explain
our observations. We have demonstrated that radiation trapping produces a repulsive force between atoms which is larger than this attenuation force and thus causes the atomic cloud to expand.* It is this force that limits the density of our optical trap to lO''/cm3. 111. THE VAPOR CELL TRAP
During the course of this work, we discovered that the depth of the ZOT is large enough that a significant number of atoms could be captured directly from an uncooled beam. Similar results have also been obtained with sodium.'
We have extended this idea to build such an
optical trap directly in a small vapor ce1L8 We have been able to use this trap to produce clouds
of trapped atoms containing = 2 lo8 ~ atoms with temperatures of a few hundred microkelvins. This now makes it possible to have cooled and trapped atoms with an extremely simple and inexpensive apparatus. This is a significant technological breakthrough because these atoms can then be used for a variety of other experiments. The trap is made with beams from two diode lasers as described in Sec. 11. The only difference is that now the region of intersection of the beams is inside the center of a small glass vapor cell containing approximately lo'* TOKof cesium. The geometric details of the cell and the optical quality of the windows is unimportant. In fact, one can send the beams directly through the curved walls of a glass tube and still obtain very successful atom trapping. Small anti-Helmholtz coils outside the cell provide the necessary
325
magnetic field gradient. The number of atoms contained in such a trap in a cell is determined by the balance between the capture rate into the trap and the loss rate from the trap. Given the maximum capture speed of the trap, v,, the number of atoms per second from the vapor entering the trap volume with low enough velocities to be captured is easily calculated. This rate, R = 1 . 5 ~ */2nV2’3~,4(m/2kT)3/2, where n is the density of cesium atoms, m is the mass, T is the temperature, and V is the trapping volume (about 1 cm3 for our trap). For cesium at a temperature of 300 K, approximately one atom in 7000 is slow enough to be captured. The loss rate from the trap, 1/7 lifetime), is primarily due to collisions with the fast atoms in the vapor. If we assume the density of noncesium atoms is negligible, then 1/7 = na(8T/rm)”* atoms/s where
(T
is the cross section for an atom in the vapor to eject a trapped atom. The
number of atoms in the trap, N, is then given by the solution to the simple rate equation dN/dt = R - NIT. Assuming N(t=O) = 0, we obtain N(t) = N,(l-e-t’q, where the steady-state number, N,, is given by
(-)
N, = 3 vm v:($ 4
.
0
Note that N, is independent of the density of atoms in the cell (i.e. pressure). However, the lifetime 7 , which is also the time constant for filling the trap, does depend on pressure. Because N, is very sensitive to v,, we should note which parameters of the ZOT determine v,. The trap is an overdamped system, so the primary stopping force comes from the damping
force, which arises from the imbalance in the radiation pressure due to the differential Doppler shift between the two counterpropagatingbeams. The force decreases rapidly with velocity once the Doppler shift gets so large that the atom is out of resonance with both beams. Since the frequency of lhc laser is typically one linewidth to the red of the atomic resonance, v, = where
is the velocity at which the Doppler shift equals the natural linewidth
ZrX,
r (5 MHz for
our transition). For cesium, 2rX is 10 m/s. We have constructed several cells which have produced similar results. Cells have been made out of fused silica and of standard stainless steel UHV equipment. It is necessary to have a small (2 Pls) ion pump on the system to keep the non-cesium pressure in the cell less than
326
TOK. After construction the cells are evacuated and baked under vacuum. A fraction of a gram
of cesium is then distilled into the cell and it is sealed off, To maintain the desired cesium pressure of 10-7-10-9 Torr in the cell, a small "cold finger" can be used. This cold finger is maintained at about -25°C to control the vapor pressure of the cesium in the entire cell. Alternatively, the cesium well can be valved off to limit the vapor pressure of cesium in the cell. When the trapping laser was tuned between 1 and 15 MHz to the red of the 6S,,,
to
6P3,2,F=5transition (-8 MHz was optimal), a bright cloud of trapped atoms with a volume < 4 mm3 appeared in the center of the cell. By measuring the fluorescence we determined that the
cloud contained as many as 2 x lo8 atoms, in agreement with the value predicted by Eq. (1) for v, = 15 m/s. As we observed in the work just discussed, the density of atoms in the cloud was
limited by radiation trapping to about 10" atoms/cm3, and the temperature was about 300 pK. The trapped atom cloud was more than lo00 times brighter than the background fluorescence in the cell at a cesium pressure of 6
X
TOK. The growth in the number of atoms with time
agreed well with the predicted form. The lifetime of atoms in the trap was 1 s at a pressure of
6x
TOK. The steady-state number of atoms in the trap changed by less than 30% as the
cesium pressure was varied between
TOK (7 = 0.06 s) and - 5 ~ 1 0 - "Torr (7=10 s).
Similar traps in small vapor cells have been demonstrated in sodium9 as well as rubidium." Results were comparable to those we obtained with cesium. Although the temperature of a large optically trapped sample is typically 300 pK, the sample can be cooled much further by rapidly turning off the ZOT magnetic fields and detuning
the laser further to the red by several linewidths. Three dimensional polarization-gradient cooling forces reduce the temperature to near the recoil temperature (2 pK) in under 1 ms." The temperature is measured by loading the atoms into a magnetic trap, as discussed below. This "sub-Doppler molasses" cooling mechanism is described extensively in this volume. The drawback to optical molasses however, is that the atoms are no longer trapped. Sub-Doppler temperatures in a ZOT have been attained,'* but at the expense of very low densities.
To summarize the performance of optical trapping and cooling, we have produced a sample of 2 X lo8 cesium atoms at a density of 5 X 10'0/cm3 with a temperature of 2 pK. The sample is produced in a small vapor cell, ideal for performing further experiments on the atoms.13
327
IV. THE dc MAGNETIC TRAP
Although optical trapping and cooling represents a very efficient method of collecting atoms and cooling them, optically trapped samples are not polarized and the density and temperature are limited by interactions with the light as discussed above. Furthermore, it is not possible to maintain high densities while obtaining the very low temperatures possible in optical molasses
because there is no confining force in the molasses. Magnetic trapping is a way to avoid these limitations because it provides confinement without heating. Furthermore it allows cooling far below the limits obtainable in optical molasses. The photon recoil energy, which is the primary
limit for optical cooling, is no obstacle to evaporative cooling in a magnetic trap. Spin-polarized hydrogen in a magnetic trap has been evaporatively cooled (allowing the highest energy atoms
a temperature 13 times below the photon recoil limit,14 and to a density orders of magnitude larger than the present limit for an optical trap. to escape the trap) to
To this end, we have used the optical trap to load a magnetostatic trap in the same cell.
To do this, we first turn on the optical trap, then we turn off all the magnetic fields and detune the laser further red for a few milliseconds. At the end of this time, we have a dense, although untrapped sample, which has a temperature of a few microkelvins. We next use the light to pump the atoms to the m F = 4 state and then quickly (500 ps) turn on the magnetic fields to create the magnetic trap. In the magnetostatic trap or bowl, the atoms are held by magnetic
w.3)and gravitational forces at the same location as when they are in the optical trap.
In this
way we have produced a very cold, 100%polarized trapped sample which is not subjected to the perturbing fields of the laser.
Because of the low temperatures, the magnetic fields needed for confinement are quite modest (tens of G k m ) . To trap the I F,m,
>
I
= 4,4 > weak-field-seeking state, four vertical
bars, each carrying a current with direction opposing that of its nearest neighbors, are placed around the cell. In the horizontal (x,y) plane, they provide a quadrupole field distribution having zero field in the center along the vertical (z) axis. In addition, a horizontal coil below the atoms gives a vertical bias field as well as a vertical field gradient."
This vertical field gradient
(aB,/az) supplies the levitating force to support the atom against gravity. Thus, as the atom searches for a local minimum of
13 I , only small changes occur
in the direction of
preventing nonadiabatic spin-flip transitions to other mF states ("Majorana" transitions).
3,
328
To study the magnetically trapped atoms, the magnetic fields were quickly (500 ps) switched off after a variable time delay, and the laser beams quickly turned back on. A television camera recorded the image of the cloud as the laser came back on. By varying the time delay we could observe the shape of the atom cloud and the number of atoms as a function of time in the magnetic trap. Essentially all the optically trapped atoms could be loaded into the magnetic trap, and the density could be as high as in the optical trap. In general, the cloud was not in equilibrium when first loaded into the magnetic bowl. When it is loaded into the bowl off center, secular oscillations or orbits of the cloud are observed with typical periods of 2 - 5 Hz. Even when it is centered, radial oscillations are observed as the atoms slosh back and forth in the bowl. We are presently studying the thermal equilibration of the atoms in the magnetic bowl. This equilibration time depends on the elastic scattering cross section of the atoms at a temperature of a few microkelvins. Obviously, this is a completely unexplored regime and any
data on it will be of interest. Although the atoms are not in thermal equilibrium, we can measure an effective temperature that corresponds to the time averaged kinetic energy. This is done by loading the atoms in a magnetic trap with a small spring constant and observing how far they expand. Taking r to be the radius of the half height point of the spatial distribution at the time of largest spatial extent, the effective temperature is then 0.25 @/K,
where k is the spring constant and
K
is Boltzmann's constant. This technique is valid when the spring constant is small enough that r is several times larger than the initial radius of the cloud. We find that the temperature of the cloud is different along the different axes (which have different spring constants) and depends on the length of time the optical pumping light is on. Neither of these results is surprising since the optical pumping light heats the sample in an anisotropic manner. With no optical pumping light we obtain T, = 1.4 & 0.25 pK and Tx,y= 1.1
0.25 pK. These are the coldest kinetic
temperatures ever observed, and are more than two orders of magnitude colder than any previous sample of trapped atoms. With no optical pumping we have about 1/4 as many atoms in the magnetic trap as we had initially in the optical trap. With enough optical pumping to have 50% of the atoms in the magnetic trap, T, rises to about 2.9 pK and Tx,yis about 1.4 pK. The differences in the temperature in the different directions, and the dependence on the optical pumping is presently being studied.
329
V. THE ac MAGNETIC TRAP Traditional (dc) magnetic traps have a significant limitation themselves, namely spin-flip collisions. To date, all magnetic traps have been designed to work only for weak-field seekers, that is, for atoms or neutrons whose magnetic moment is anti-aligned with the magnetic field. These atoms are confined to a local minimum in the magnitude of a static magnetic field. Unfortunately, because weak-field-seeking states are not the lowest energy configuration within the confining magnetic field, the atoms are susceptible to two-body collisions that change the
hyperfine or Zeeman level of one or both of the colliding atoms. The energy released in these collisions heats the trapped sample and the spin-flipped atoms leave the trap. The loss rate due
to these exothermic collisions scales as the density squared and thus provides an effective ceiling
to the attainable density, and indirectly limits temperature as well? However, if atoms were trapped in the lowest energy spin state (magnetic moments aligned with the magnetic field) spinflip collisions would be endothermic and hence greatly suppressed at low temperatures. Unfortunately, such "strong-field-seeking" atoms can not be confined in a static configuration of magnetic fields, because Maxwell's equations do not allow a local maximum in magnetic field magnitude.
However there are time-varying field configurations which provide stable
confinement. Originally proposed for hydrogen atoms,I7 our trap operates on the same dynamical principle as the Paul trap" for ions: Near the center of the trap, the potential is an axially symmetric quadrupole, oscillating at frequency 0: 4 = -p.B = A(z2-p2/2)cos nt. During each oscillation, the atoms are first confined axially and expelled radially, and then confined radially and expelled axially. For a range of values of the amplitude A, the net force averaged over a
cycle of Lhc: oscillation is inward; the result is stable confinement in all three dimensions. For easily obtainable oscillating magnetic fields, the trap is extremely shallow (tens of microkelvins). However, using laser cooling we are able to produce atomic samples which are colder than this, and load them into the trap."
The loading procedure has a number of steps, as shown in Table I. First, cesium atoms are collected and cooled in an optical trap exactly as done for loading the dc magnetic trap. The cold atoms are then launched into a differentially pumped vacuum region which contains the ac magnetic trap, as shown in Fig. 3. To reduce the spreading of the atoms they are magnetically
330
focused as they move between the two traps. When they reach the magnetic trap the atoms are optically pumped into the state that is trapped. This approach allows multiple bunches from the optical trap to be transferred to the magnetic trap, thereby directly increasing the density.
To toss the atoms up into the magnetic trap region, we use the technique of moving optical molasses2': we shift the frequency of the downward-angled molasses beams (Fig. 1) 1.7 MHz
to the red, and the upward-angled beams 1.7 MHz to the blue, relative to the frequency of the horizontal beams. The standing-wave pattern at the intersection of the six laser beams now becomes a walking-wave pattern. To achieve the necessary acceleration (200 g), while having the lowest possible final velocity spread, we vary the central laser frequency over time during the launch (Table I). After the atoms are accelerated, the molasses beams are shut off abruptly and a 2 ms pulse of a+ polarized light optically pumps the atoms to the F=4, mf=4 state. The optical pumping beam is tuned to the F=4 + F'=4 transition to minimize heating. The tossed atoms begin their ascent with an internal velocity spread of about 3 cm/s rms. In the 130 ms it takes to reach the site of the magnetic trap, even this small velocity spread would result in a 100-fold decrease in cloud density. We avoid much of this decrease by
focusing the atoms with magnetic lenses. Approximately half way (in time) along their route, the atoms pass through two pulses of magnetic field, each around 3 ms long, separated by 8 ms. These pulses provide an impulse that reverses all three components of the velocity. During the first focus pulse, the curvature of the magnetic field is such that the atoms are focused axially and defocused radially. During the second focus pulse, the curvature of the lenses is reversed.
We adjust the timing and strength of the pulses (that is, the positions and effective focal lengths of the two magnetic lenses), in order to bring the atoms together to a focus axially and radially at the magnetic trap center. As shown in Fig. 3, the magnetic trap is created by a combination of ac and dc coils and
is located 14 cm above the optical trap. The 60 Hz ac field is generated by two pairs of coils which are arranged to produce maximum field curvature with relatively small oscillating field at trap center. The dc coil produces a field at trap center of 250 G, a gradient of 31 G/cm in the vertical direction and very little curvature. This static gradient exactly balances the force of gravity, which the magnetodynamic forces are too weak to counteract. The peak amplitude of the ac component is 100 G at the center, with a curvature of 875 G/cm2 in the axial direction
33 1
and of 440 G/cm2 in the radial direction. To make contact with the RF ion trap literature we note that the forces in our trap correspond to Paul trap parameters
= 0 and qz = 0.40, where
a, and qz are stability parameters, defined in Ref. 18. Inserting the atoms into the magnetic trap requires some way of dissipating their energy. This is accomplished by optically pumping the atoms from the F=4 to the F=3 ground states. The trap will hold atoms in the F=3, m,=3 state, but the atoms rising upward from the optical trap are in the F=4, mr=4 state. Because this is a weak-field-seeking state, F=4 atoms approaching the magnetic trap are not only pulled downward by gravity but also repelled downward by the dc field gradient. The initial centersf-mass velocity of the atom cloud is adjusted so that this combination of forces brings the cloud to a stop just at the center of the trap. (The large timedependent fringing fields from the trap make it somewhat difficult to achieve a stationary tightly focused cloud at exactly the correct position and time.) The stationary atoms are then illuminated with light linearly polarized along the magnetic field and tuned to the F=4 4
F'=4 transition. This optically pumps the atoms to the trapped, F=3, mf=3 ground state.
The F=3 atoms already loaded in the trap are transparent to the light used in this final optical pumping. The timing of the final optical pumping relative to the phase of the ac fields is critical.
The micromotion (the small-amplitude, high-frequency oscillation induced by the ac forces") of the weak-field seeking atoms just arriving in the trap has opposite phase from the micromotion
of the trapped atoms. Unless the pumping occurs when the micromotion velocity is zero, the atoms are strongly heated. While the atoms are in the magnetic trap, they do not interact with laser beams. To observe the trapped atoms, we abruptly turn off all the magnetic fields and then illuminate the trap region with a probe beam containing laser light which excites both F=3
-,F'=4
and F=4
--+
F'=5
transitions. The resulting fluorescence from the atoms is imaged by a video camera and recorded on tape. Subsequent image processing reveals the spatial distribution and center-of-mass location of the atom cloud. Because the measurement is destructive -- the photons from the probe beam blow the atoms away -- time evolution of the trapped atoms can be studied only by repeating the load-probe cycle with varying time delays. We study the magnetic trap by observing the shape and motion of the cloud of trapped atoms. Pulses of magnetic field from supplementary coils give the cloud any desired initial
332
center-of-mass motion. From the evolution of the cloud motion we determine the frequency of the guiding-center oscillation in the effective potential. For an oscillation amplitude of 0.1 cm
or less, we masure an axial frequency of 8.5(4) Hz and a radial frequency of 4.5(1.0) Hz.
These agree with calculated'* frequencies of 8.5 and 4.25 Hz. The overall depth of the trapping potential is difficult to calculate or even to define precisely. The depth of the trap is determined mainly by the behavior of the trap@ atoms away from the center of the trap. In those areas the field becomes anharmonic and the approximation that the ac fields simply provide an effective dc conservative potential is not valid. The harmonic region is quite small unless there is an axial dc bias field that is much stronger than the radial and axial components of the ac field everywhere in the trapping region. We confirmed experimentally that increasing the bias field improved the stability of the trap. We have used a computer simulation to understand the behavior of the trap in the regions where the effective potential approximation is not valid. This simulation calculated the motion of clouds of noninteracting atoms with various temperatures in the trap. For clouds inserted with temperatures of 9 p K or less, nearly all the atoms remain trapped and the final spatial extent of the cloud is consistent with the initial velocity spread and a harmonic potential of the calculated spring constant. For clouds with higher initial temperatures, the final spatial distribution of trapped atoms is largely independent of initial temperature. The high-energy fraction of the atoms leaves the trap immediately, and the atoms remaining in the trap have an rms diameter of about 0.2 cm and an rms velocity of 5 cm/s (equivalent to a temperature of about 12 pK). These simulation results match our experimental observations that a fraction of the atoms focused up into the magnetic trap region leave the trap within 0.15 s, and that those remaining form a ball about 0.2 cm in diameter. We conjecture, then, that the rms velocity of our trapped atoms is about 5 cm/s and that the depth of the trap is somewhat deeper than 12 pK. The trap has a I/e lifetime which is typically about 5 s, as shown in Fig. 4. We believe that this is due entirely to collisions with the 1 x
Ton of residual background gas. When we improved
the vacuum by cooling some of the metal surfaces of the upper vacuum chamber with liquid nitrogen, the lifetime nearly doubled. The advantage of multiple loading is apparent in Fig. 4. By accumulating a number of loads, we were able to collect nearly five times as many atoms as from a single load. The factor
333
of 5 increase agrees well with our calculated value and is determined by the ratio of magnetic trap lifetime to optical trap fill time, by perturbations from the fringing fields of the magnetic focus, and by unintentional excitation of the trapped atoms during the initial optical pumping. Without great additional technical effort, this ratio could be a hundred or larger. As the density increases, viscous heating due to the micromotion may become a problem."
We have not yet
seen any sign of such heating, and a simple estimate2' of the heating rate leads us to believe that, with suitable choice of experimental parameters, evaporative cooling can overwhelm the
viscous heating. VI. EVAPORATIVE COOLING AND ADIABATIC COMPRESSION: THE FINISH LINE
To cool magnetically trapped atoms below the recoil limit and finally reach BEC, we plan Because hotter atoms in the to evaporatively cool the sample, as has been done in h~dr0gen.l~ trap experience trajectories through larger magnetic fields, these atoms can be selectively ejected by judicious tuning of RF fields.22
However, effective evaporation requires rapid
thermalization. The high velocity tail of the velocity distribution must be populated by elastic collisions in a time much shorter than the lifetime of the trapped sample. We discuss now a strategy for increasing the rate of desirable, evaporationenabling elastic collisions relative to trapdepleting background collisions. Optical trapping and cooling can
produce a cloud of lo8 cesium atoms with a density of 10" atoms/cm3 and an rms internal velocity of 2 cm/s. There is a large theoretical uncertainty in the low velocity s-wave cross section 0, but for the velocities and densities just mentioned, even the most optimistic estimates do not put the collision rate, nav, very much larger than typical background collision rates of perhaps 0%.
The elastic collision rate can be increased many-fold, however, by adiabatically
compressing the sample in a magnetic trap. The procedure would work as follows: Atoms are optically trapped and cooled and loaded into a magnetic trap as discussed above. The confining magnetic potential is harmonic and nearly spherically symmetric, with the optimal initial trap given simply by the ratio of the atom cloud's initial velocity spread to its initial frequency ainit radial spread (thus preventing entropy-increasing initial collective internal motion in the cloud). The curvature of the magnetic field may then be adiabatically increased. The compression conserves phase space volume (Sr)3(6v)3for the cloud, so that the atoms remain at a constant
334
distance from the BEC transition in phase-space density during the compression. But the coordinate-space density and the rms velocity spread of the atoms increase during the compression, so that the elastic collision rate nav increases by (ofiml/oinil>*. This will likely allow effective forced evaporation to reach the conditions necessary for BEC. VII. CONCLUSION
We have presented a progression of optical and magnetic technology for possibly attaining Rose-Einstein Condensation in a vapor of cesium. The Zeeman-tuned Optical Trap is a very efficient capture device, and is the starting point of the progression. However, optical traps cannot seem to provide the density and temperature necessary for BEC due to the presence of the laser field. Therefore, we have cooled an optically trapped sample of cesium at 10**/cm3 to
2 pK and loaded the atoms into a dc magnetic trap. But exothermic collisional processes are expected to provide a barrier preventing BEC in such a magnetic trap. So we have alternatively loaded the optically trapped and cooled sample into an ac magnetic trap, which traps atoms at their lowest energy, and is immune to any exothermic collisions. To increase the phase space density further in the magnetic trap, we propose to adiabatically squeeze and evaporate the atoms. This "road" offers an appealing way to achieve the phase space density necessary for BoseEinstein condensation in a dilute gas of neutral atoms. This work was supported by the Office of Naval Research and the National Science Foundation. We acknowledge T. Walker and D. Sesko for much of the initial research on limits
of an optical trap.
335
REFERENCES
1.
E. Raab, M. Prentiss, A. Cable, S . Chu, and D. Pritchard, Phys. Rev. Lett.
2,2631
(1987).
2.
T.Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. &, 408 (1990); T. Walker, D. Sesko, and C. Wieman, JOSA B 8, 946 (1991).
3.
D. Sesko, T. Walker, C. Monroe,A. Gallagher, and C. Wiernan, Phys. Rev. Lett. fQ, 961 (1989).
4.
M. Prentiss, A. Cable, J. Bjorkholm, S. Chu, E. Raab, and D. Pritchard, Opt. Lett.
u,
452 (1988).
a,957 (1989).
5.
A. Gallagher and D. Pritchard, Phys. Rev. Lett.
6.
J. Dalibard, Opt. Commun. 68, 203 (1988).
7.
A. Cable, M. Prentiss, and N. Bigelow, Opt. Lett.
8.
C. Monroe, W. Swann, H. Robinson, and C. Wieman,Phys. Rev. Lett.
9.
M. Zhu and J. L. Hall, private communication (1990).
u, 507 (1990). B,1571 (1990).
10. T. G. Walker, private communication (1991). 11.
C. Salamon, J. Dalibard, W. Phillips, A. Clarion, and S. Guellati, Europhys. Lett. 683 (1990).
12. A. Steane and C. Foot, Europhys. Lett.
14,231
(1991).
12,
336
13. C. Monroe, H. Robinson, and C. Wieman, Opt. Lett.,
fi, 50 (1991)
14. J. Doyle, J. Sandberg, I. Yu,C. Cesar, D.Kleppner, and T.Greytak, Phys. Rev. Lett, 67, 603 (1991). 15. D. Pritchard, Phys. Rev. Lett. 51, 1336 (1983). This work demonstrates the use of this trap design to hold neutral sodium atoms.
16. N. Masuhara et al., Phys. Rev. Lett. 61, 935 (1988). 17. R.V.E. Lovelace, C. Mehanian, T.J.Tommila, and D.M. Lee, Nature R.V.E. Lovelace and T.J.Tommila, Phys. Rev. A s, 3597 (1987). 18. F.G. Major and H.G. Dehmelt, Phys. Rev.
m,30 (1985);
m, 91 (1968).
19. E. Cornell, C. Monroe, and C. Wieman, Phys. Rev. Lett. 68, 2439 (1991). 20.
M. Kasevich, D.S.Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, Phys. Rev. Lett. f& 2296 (1991); A. Clarion, C. Salomon, S. Guellati, and W.D. Phillips, Europhys. Lett. 16, 165 (1991).
21.
W. Ketterle, private communication (1991).
22.
D. Pritchard, Proc. 1lth Int'l. Conf. on Atomic Physics, eds. S. Haroche, J. Gay, and G. Grynberg (Singapore: World Scientific, 1989) pp. 179-97.
337
Table 1. Timing and laser detuning during one cycle of the multiple loading sequence.
Relative time (ms)
Laser detuning, function
-500
to -2
-7 MHz. Atoms accumulate in optical trap.
-1
to 0
-30 MHz. Molasses cooling to 2 pK.
0
to 0.4
-7 MHz, and offset A ramped from 0 to 1.7 MHz. Atoms accelerate.
0.4
to 0.8
-30 MHz, and offset A constant at 1.7 MHz. Atoms cool to 4 pK in moving frame.
0.8
to 130
Main beams blocked. Optical trap off. Atoms rise to magnetic trap.
to 4
-250 MHz. Initial optical pumping.
2
60 to 62
Magnetic focusing pulse.
67
to 70
Magnetic focusing pulse.
to 130
-310 MHz. Final optical pumping as atoms reach magnetic trap.
128
338
Figure Captions.
Figure 1.
Dependence of @ on the total intensity in all the trap laser beams. The solid line indicates the prediction of the Gallagher-Pritchard model (Ref. 3, and would be a straight line with linear scales.
Figure 2.
Plot of the diameter (FWHM) of the cloud of atoms as a function of the number
of atoms contained in the cloud. For the full figure the magnetic field gradient is 9 G/cm, and 16.5 G/cm for the inset. The laser detuning is -7.5 MHz, and the total laser intensity is 12 mW/cm2. The solid lines show the predictions of the
model presented in Ref. 2. Figure 3.
Schematic of the apparatus, showing the upper and lower vacuum chambers, the configuration of key trapping and focusing coils, the paths of the initial optical pumping (IOP) and final optical pumping (FOP) laser beams, and the paths of four of the six laser beams defining the Zeeman-shift optical trap (ZOT) and the molasses. Frequencies of the molasses laser beams during the launch are shown
in parentheses. Not shown: the paths of two horizontal laser beams which are perpendicular to the plane of the paper at the ZOT, the magnetic coils for tuning the optical trap, and a variety of shim magnetic coils. Figure 4.
Total fluorescence from the magnetically trapped atoms as a function of time. A new bunch was loaded every 0.65 s, for the first 10 s.
Dashed line is an
exponential decay fit to the data, with 4.9 s l/e time. The y-axis has been calibrated to indicate the total number of trapped atoms.
339
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0
00
o 0
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'
I
I
I
m
0 0 0 . 0
I I I I
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0 ? 7
bt
340
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W
8 0
c4
0
1
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341
cs
Figure 3
342
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nl
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VOLUME70, NUMBER4
PHYSICAL REVIEW LETTERS
25 JANUARY1993
Measurement of Cs-Cs Elastic Scattering at T = 3 0 p K C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, and C. E. Wieman Joint Institute for Laboratory Astrophysics, National Institute of Standards and Technology and Uniuersity of Colorado, and the Department of Physics, Uniuersiry of Colorado. Boulder. Colorado 80309-0440 (Received 9 October 1992)
We have measured the elastic collision cross section for spin polarized atomic cesium. Neutral cesium atoms are optically cooled, then loaded into a dc magnetic trap. We infer t h e scattering rate from the rate a t which anisotropies in the initial energy distribution are observed to relax. The cross section for F - 3 , m F = - 3 on F = 3 , m F = - 3 is 1.5(4)x cm2, and is independent of temperature from 30 to 250 p K , This determination clarifies the technical requirements for attaining Bose-Einstein condensation in a magnetically trapped Cs vapor. We also study heating due to glancing collisions with 300 K background Cs atoms PACS numbers: 34.40.+n, 05.30.Jp, 32.80.Pj. 34.50.--s
Recent advances in optical trapping and cooling techniques have allowed collision studies in an entirely new regime of temperatures. In this regime of mK or even colder temperatures, collisions take on unusual characteristics, and often have cross sections many orders of magnitude different from that observed at previously attainable temperatures. Studies of ultralow-temperature collisions have investigated several inelastic processes which cause loss from optical traps: two-photon quasimolecular photoionization 111, photon-assisted collisions [2,31, and hyperfine-state changing collisions [2,41. All of these processes are inelastic, and while they involve low initial energies for the atoms involved, the final states have kinetic energies of 1 K or greater. It appears that this allows these processes to be treated semiclassically, although the exact accuracy of such treatments is currently a topic of debate. I n the work presented here, we examine elastic collisions in which the initial and final energies are both much less than 1 mK. In order to study such low-energy collisions we have developed techniques for observing collisions in very cold samples of magnetically trapped atoms. From these measurements, we deduce the s-wave collision cross section. The elastic scattering cross section is of particular interest because of its importance in determining the experimental feasibility of achieving Bose-Einstein condensation (BEC) in a cesium vapor. As demonstrated in experiments with spin polarized hydrogen 151, evaporative cooling of magnetically trapped atoms is a promising strategy for achieving the temperature and density necessary for the BEC phase transition, because evaporative cooling does not have the density and temperature limitations encountered in optical cooling and trapping [2,61. To study the very low energy elastic collisions, we first accumulate Cs atoms in a Zeeman-shift optical trap (ZOT) i n a low pressure cell [71. The atoms are then optically cooled and spin polarized in the F = 3 , m p = - 3 state in preparation for magnetic trapping. After all laser light is shut o f f ,magnetic fields are turned on around the atoms in situ 171, thereby trapping them. In this work we use a type of dc magnetic trap which 414
has been previously proposed but, to our knowledge, never used to trap neutral atoms. The field coils consist of wire wound like the seams of a baseball [Sl. The 2.5-cm radius “baseball” coil consists of nine turns of No. 3 Cu capillary tubing, and is cooled by flowing water through the tubing. The field at the center of the trap is aligned with gravity. To counteract the force of gravity, a vertical mapiletic field gradient of -31 G/cm is generated with an additional pair of circular coils which sit above and below the baseball coils. The superposition of the gravity-canceling fields and the baseball coil fields breaks the cylindrical symmetry of the baseball potential. Although the coil geometry is novel, the fields produced near the center of the trap are the same as for the loffe coil configuration; see Refs. 171 and [Sl. The baseball coil geometry has a number of desirable characteristics. It has a nonzero minimum in the magnetic field, allows excellent optical access, and provides strong curvatures for a given electrical power. The trap depth and oscillation frequencies of the atoms in the harmonic magnetic trapping potential are related to the current applied through the coil. With 30 A through the coil, the trap has a central field of 50 G with oscillation frequencies of ( v x , v y , v , ) = ( 4 , 8 , 4 ) Hz for atoms in the F=3, m,==-3 trapped state. At 300 A, the central field is 500 G with oscillation frequencies of (19,20,1 1 ) Hz. When the atoms are first loaded into the magnetic trap from the optical trap the fields are set to give oscillation frequencies ( v , , v y , v , ) = ( 1 0 . 5 , 1 2 . 5 , 6 . 5 ) Hz. The temperature and density of the magnetically trapped cloud can then be increased by ramping up the strength of the trap around the atoms. We typically load 10’ atoms with an initial average density of 7 x 109/cm3 and initial temperature of 30 p K . The temperature is higher than typical optical cooling temperatures due to heating from optical pumping and the addition of magnetic potential energy when the trap is suddenly turned on. As the atoms are compressed in the magnetic trap, the density and temperature can be continuously increased up to 10”/cm3 and 80 p K , respectively. The lifetime of the atoms in the magnetic trap is limited by collisions with room-tem-
@ I993 The American Physical Society
343
344 VOLUME 70, NUMBER 4
PHYSICAL REVIEW LETTERS
25 JANUARY 1993
-
perature background gas. The I/e decay time depends on the pressure and composition of the residual gas, and is typically about 1 5 s. W e have also determined that the trap lifetime is independent of density to within o u r uncertainty, which indicates that the spin-flip relaxation cm3/s. rate is less than 5 x For the higher temperature measurements, the atoms are transferred to a separate chamber [91 where they arc magnetically trapped. I n the double chamber setup, heating i n the transfer process kept attainable temperatures over I00 p K . While the atoms are in the magnetic trap, they do not interact with laser beams and therefore are invisible. T o observe the trapped atoms, we abruptly turn off all magnetic fields and illuminate the atoms with a 0.5-ms pulse of resonant laser light. The resulting fluorescence from the atoms is imaged onto a video camera and a photodiode, revealing both the spatial distribution and total number of atoms. Because the measurement is destructive-the photon pressure in the probe beam blasts away the atoms-time evolution of the trapped atoms is studied by repeating the measurement and changing the amount of time the atoms spend in the magnetic trap before the illumination pulse. Shot-to-shot reproducibility is fairly good- measured size, for instance, reproduces to a few percent. There is always some initial expansion-contraction oscillations and center-of-mass sloshing of the cloud, but these decay away in about 2 s (about twenty oscillations) due to small anharmonicities in the trapping potential. The distribution of atoms in the cloud can be characterized by three temperatures [lo], T,, T,, and T,, where T, is defined in terms of the spring constant ki and the mean-square extent of the cloud in each direction keTi ~ 0 . 5 k i ( r : ) .Because of asymmetries in the loading process, the three temperatures are initially unequal. Perhaps because the oscillation frequencies in the three different directions are well separated, residual anharmonic terms in the trapping potential are not observed to cause the three temperatures to equilibrate, a t least not on the time scale of our experiments. Time evolution of the three temperatures is almost purely due to collisional processes. The time dependence of the individual one-dimensional cloud sizes (Fig. 1) shows two main effects. T h e first, visible only a t high densities of trapped atoms, is that the sizes tend to be mixed. This is due to the effect we are trying to study, low-temperature elastic collisions between trapped cesium atoms bringing the sample into thermal equilibrium. The observed cross-dimensional equilibration rate can, with further analysis, be turned into an elastic collision cross section. Unfortunately, there is a second effect, visible in both high and low density clouds-the sizes in all three directions tend to i n crease uniformly because of elastic collisions occurring between trapped atoms and atoms in the 300-K residual
vapor i n o u r vacuum chamber. Usually, such a collision imparts a substantial amount of energy to the trapped atom, ejecting it cleanly from the trap with no effect on the remaining trapped atoms. But on occasion (in the event of a glancing angle collision), the background atom imparts such a small amount of energy to the trapped atom that the latter remains in the trap, although with increased energy. The glancing collisions with background atoms make our study more difficult, but are of some interest i n their own right [ I I ] . The collisional energies are very high and hundreds of partial waves are involved, yet a classical treatment of the collisions yields incorrect results. I n the extreme glancing collision regime, the momentum transferred can be so small that the associated de Broglie wavelength can be large compared to the classical impact parameter, meaning that diffraction effects become significant. Both trap loss and heating rates vary proportionally with cesium pressure in the background gas, except that, a t very low Cs pressures, the heating rate goes nearly to zero while the loss rate approaches = 0.06 s - ’ . W e attribute this residual loss rate with very little accompanying heating to residual amounts of helium gas i n the system. Because helium is much lighter than cesium, the diffractive regime begins a t a much higher energy transfer, and thus the probability of a He-Cs collision transferring a small enough amount of energy to heat but not to eject an atom is very small [121. In any case, the heating is evidently isotropic and during data analysis can be separated from the interdimensional equilibration arising from intratrap collisions. T o increase the initial temperature anisotropy, and thus to bring out the observable signature of temperature equalizing collisions, we heat one of the dimensions of the trapped atom distribution using a parametric drive. The trapping field is modulated a t twice the y-dimension harmonic frequency. The width of the parametric excitation resonance is about 1 Hz, so this parametric drive does not heat the other two dimensions. After the drive has been turned off, we wait for an interval of time f , and then illuminate the sample and measure the distribution of atoms in the cloud. To simplify the d a t a analysis and interpretation, only very small times f are considered, such that rmixf << 1. rmix is the rate of interdimensional mixing, r,i,=[d/di(T; - Teq)l/Teq,where T,, is the average of T,, T,, and T,. Furthermore, if t < 5 , the t r a p decay lifetime, the atom density does not change significantly in time t . Typical experimental values are Tmi,=0.02/s, t = 5 s, and 5 = I 5 s. T o separate the effects of the background collisions from the intratrap collisions, and also t o correct for any interdimensional mixing caused by anharmonicities, all measurements are taken for both high and low density clouds. This requires the density of the cloud to be decreased without changing its size or shape, which is surprisingly difficult to do. However, after unsuccessfully 41 5
345 ~~
25 J A N U A R Y1993
PHYSICAL REVIEW LETTERS
V O L U M E 70. N U M B E R4
~
trying a number of more obvious approaches we succeeded by irradiating the cloud with about 20 mW/cni’ of blackbody radiation from an incandescent light bulb. After 2 s of this illumination, == 80%)of the atoms are removed uniformly from throughout the cloud by being optically pumped to an untrapped Zeeman level. Figurc I shows the resulting low density ( 1 x 1OY/cm3) growth i n the rms size of the trapped atom cloud i n the three dimensions, along with a comparison with the higher density ( 5 x IOY/cm’) results. The collisions with the background gas cause the cloud to expand in all directions; however, it can be seen that this heating is quite isotropic. From the differences i n the slopes of the high and low density data in Fig. I we extract the mixing rate for energy to be transferred between thc different directions. For the data in Fig. I this rate, is 0.022(6) s - ’ . The elastic collision rate is proportional to this mixing rate, rrtl=armlx. where a is the number of collisions required to spatially mix the different dimensions. Using Monte Carlo methods, we have simulated the collisional dynamicL: of the trapped atom cloud. W e find that a is 2.7.
0
1
2
3
4
5
6
7
8
9
10
TIME ( s ) 10
This value for a is essentially the same for wide variations in spring constants, atom cloud sizes, and the form of cnergy distribution in the cloud. By comparing the observed mixing rate with the mixing rate of a simulated cloud of similar aspect ratio, we determine the cxperimcntal cross section. Several systematic errors can affect our measured density and thus our cross-section value. W e bclicvc the systematic error i n t h e measured total number of atoms, determined from total fluorescence measurements, is no more than 15%. Systematic errors i n size measurements are less than 5% in any one direction, or less than 15% i n volume. The total systematic error i n density measurement is less than about 25%. W e measured cross-dimensional mixing at various temperatures by taking data after varying amounts of adiabatic compression of the atoms. I n Fig. 2, the measured elastic cross section is plotted as a function of mean energy of the cloud. As the figure shows, the cross section appears to be independent of temperature from 30 to 250 p K . The plotted error bars are determined by statistical scatter i n the data. The intratrap elastic collisions can be characterized by a few partial waves, because of the very low temperatures of the atoms. Furthermore, only even partial waves contribute to spin polarized boson-boson scattering. The threshold energy E t h ( l ) for the appearance of a given partial wave / can be approximated by U ( b ) , the potential energy evaluated at the distance of closest approach 6, where b 2 = t i 2 1 ( / +1 ) / 2 r n E , h ( / ) . For the Cs-Cs van der Waals potential, E , h ( / = 2 ) = 150 p K and E , h ( l = 4 ) = I mK. Thus for our conditions of less than 250 pK, the elastic cross section will be predominantly s wave and d wave. I f there are no nearby resonances, o,,will be nearly independent of temperature a t low temperatures, and CTd ( 1 - 2 ) varies as T 3 1131. Thus the most plausible interpretation of t h e lack of temperature dependence in the data is that the d-wave contribution is small and there
0.8
I
-2 E E
-
lo-’oE-
0.6
D
,
-
, , , ,,r
4:
a:
0 4
v)
r
[r
0 2
0 0
1
2
3
4 5 6 TIME ( s )
7
8
9
10
FIG. I . Measured rms radii of cloud vs time spent i n trap for ( a ) an average density of 1 x 109/cm3 and (b) an average densit y of 5 x 109/cm3. The three sets of data are sizes of three orthogonal views ( x , squares; y. circles; z , triangles). The t r a p oscillation frequencies are (v,,v,,v,) =(16.2,17.6.9.8) tlz, corresponding to a mean atom energy of 67 pK. The lines a r e leastsquares fits; each point displayed is the average of four mea-
surements. 416
10.’~
10
102
lo3
TEMPERATURE ( p K )
FIG. 2. Log-log plot of measured elastic collision cross section vs temperature. The line represents the m a x i m u m resonant value for a purely s-wave cross section at a given temperature.
346 V O L U M E7 0 , N U M B E R 4
PHYSICAL REVIEW LETTERS
a r e no nearby s-wave resonances. F r o m this a r g u m e n t we conclude t h a t t h e zero-energy .r-wave elastic cross section = I .5 (4) x 1 0 - I c m o r 5.3 x I 0 4 r for cs is Theoretical calculations of this cross section a r e plagued by t h e difficulty t h a t it is extraordinarily sensitive to t h e exLtct s h a p e of t h e interatomic potential. W e know of two calculations of this cross section by DeVos a n d G r e e n e [I41 a n d Tiesinga el al. [I51 who both obtained values between l o - ’ * a n d l o - ” cm’. However, these a u t h o r s e x a m i n e d t h e sensitivity of t h c cross section t o small c h a n g e s in t h e potential, and found t h a t their results were only valid t o a n order of m a g n i t u d e . In Ref. 1151 it is predicted t h a t there will be large resonances in t h e cross section for particular a m b i e n t m a g n e t i c fields. Each of o u r d a t a points was taken a t a diKerent magnetic tield because in o u r a p p a r a t u s changing t h e t e m p e r a t u r e is conveniently accomplished by c h a n g i n g t h e trapping fields. T h a t we saw no resonances is not surprising since the effect of t h e predicted resonances is expected t o b e reduced for o u r t e m p e r a t u r e s c o m p a r e d t o t h e I p K case considered in Ref. [ 151. T h e primary motivation for this work w a s t o investig a t e t h e feasibility of using evaporative cooling of m a g netically trapped cesium t o reach t h e t e m p e r a t u r e s required for Bose-Einstein condensation. O u r results lend support t o t h e idea t h a t this is a feasible approach. T h i s is in a g r e e m e n t with t h e theoretical work of Ref. [15]. I n t h a t reference, in addition t o calculating t h e elastic collision rate, t h e a u t h o r s also found t h e C s - C s dipole relaxation r a t e for t h e s t a t e we a r e trapping a n d t h e threebody recombination rate. T h e y determined t h e dipole relaxation r a t e t o b e I x l o - ” cm3/s, which is consistent with (although m u c h smaller t h a n ) o u r upper limit of <5 x cm3/s, a n d t h e three-body r a t e t o be ( n 2 ) ( 5 x cm6/s). F r o m o u r M o n t e C a r l o studies we find the criterion for effective evaporative cooling 1161 is t h a t rel/l-~oss must b e g r e a t e r t h a n 150. C o m p a r i n g these t w o loss rates with our measured elastic cross section w e find t h a t evaporation should b e a b l e t o continue t o below t h e B E C transition temperature. Currently t h e limiting loss process for t h e trapped cesium a t o m s is not t h e cold CsC s interactions, b u t rather t h e loss r a t e from collisions with background gas. T h i s technical obstacle c a n b e overcome simply by improving t h e vacuum system a n d increasing t h e density of t h e trapped cloud. T h i s work is supported by t h e N a t i o n a l Science Foundation and t h e Office for N a v a l Research. W e acknowledge very valuable discussions with Boudewijn V e r h a a r , C h r i s Greene, a n d A l a n Gallagher.
’,
[I] P. Could, P. Lett, P. Julienne, W. Phillips, H. Thorsheim,
25 JANUARY 1993
and J . Wiener, Phys. Rev. Lett. 60, 788 (1988); P. Lett, P. Jessen, W. Phillips, S. Rolston. C. Westbrook. and P. Gould, Phys. Rev. Lett. 67, 2 I39 (199 I ); P. Julienne. R. Ileather, and J. Vigue, i n Aloniic Physics 12, edited by J . Zorn and R. Lewis (AIP, New York. 1991 ), pp. I18- 136. [21 D. Sesko, T. Walker. C. Monroe, A . Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 96 I ( I989). I31 D. HoKmann, P. Feng. R. Williamson. and T. Walker, Phys. Rev. Lett. 69, 753 (1992); C. Wallace, T. Dinneen, K. Tan, T. Grove, and P. Gould, Phys. Rev. Lett. 69, 897 (1992); A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63. 957 (1989); P. Julienne and .I. Vigue. Phys. Rev. A 44, 4464 (I99 I ). I41 T. Walker. D. Sesko, C. Monroe, and C. Wicman. in The Phvsics of Electronic and Atomic Collisions. Proceedings of the Sixteenth International Conference, edited by A. Dalgarno, R. Freund, P. Koch, M. Lubell, and T. Lucarto ( A I P , New York, 1990), pp. 593-598. [51 J . Doyle, J. Sandberg, I.Yu, C. Cesar. D. Kleppner, and T. Greytak, Phys. Rev. Lett. 67, 603 (1991); H. Hess, G . Kochanski, J . Doyle, N. Masuhara, D. Kleppner, and T. Greytak, Phys. Rev. Lett. 59. 672 (1987). [61 T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64, 408 (1990); C. Salomon e l a l . , Europhys. Lett. 12, 683 (1990). See also special issue on laser cooling and trapping of atoms, edited by S. Chu and C. Wieman, J. Opt. SOC.Am. B 6, No. I I (1989). [71 C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, I 5 7 1 (1990). [81 T. Bergeman, E. Erez, and H. Metcalf, Phys. Rev. A 35, 1535 (1987). [91 The technique for transferring atoms from an optical trap in one chamber to a magneiic trap in another is described in E. Cornell, C. Monroe, and C. Wieman. Phys. Rev. Lett. 67. 2439 (I99 I ). [I 01 The term “temperature” is not strictly accurate, because the one-dimensional distributions are not purely Gaussian.
2. Phys. 179, 16 (1964); R. Anderson, J. Chem. Phys. 60, 2680 (1974). [ I 21 C. Monroe, Ph.D. thesis, University of Colorado, Boulder, 1992 (unpublished). [ I 3 1 L. Landau and E. Lifshitz, Quantum Mechanics. Nonrelatii.istic Theory (Addison- Wesley, Reading, MA, 1958). p. 405. [I41 S. DeVos and C. Greene, University of Colorado (private communication). [ I 5 1 E. Tiesinga, A. J. Moerdijk, B. J. Verhaar, and H. T. C. Stoof, Phys. Rev. A 46, I167 (1992). [I61 For evaporative cooling to provide many orders of magnitude enhancement of phase space density, the elastic collision rate must not decrease as evaporation proceeds. Evaporation can continue to outpace trap loss only if the initial collision rate is much higher than the background loss rate. [ I I] R. Helbing and H. Pauly,
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Simplified atom trap by using direct microwave modulation of a diode laser C. J. Myatt, N. R. Newbury, and C. E. Wieman Department of Physics, University of Colorado, and Joint Institute for Laboratory Astrophysics, University af Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309-0440 Received November 16, 1992 We demonstrate direct microwave modulation of diode lasers operated with optical feedback from a diffraction grating. We obtain substantial fractions of the laser power (2-30%) in a single sideband a t frequencies as high a s 6.8 GHz with 20 mW of microwave power and simple inefficient microwave coupling. Using a single diode laser modulated a t 6.6 GHz, we trapped 87Rbatoms in a vapor cell. With only 10 mW of microwave power we trapped 85% a s many atoms a s were obtained by using two lasers in the conventional manner.
The use of diode lasers and the vapor-cell trap has greatly simplified laser trapping and cooling of atoms. Such traps have been demonstrated for both cesium and These traps require two diode lasers: one to provide the trapping force by exciting atoms from the upper ground-state hyperfine level, and a second to optically pump the atoms from the lower ground-state hyperfine level to the upper level. The frequency of both lasers must be stabilized by locking to a sub-Doppler absorption feature. The need for the second laser with its associated optics and electronics substantially contributes to the complexity and expense of the trap. However, the laser intensity needed for the hyperfine pumping is typically a small percentage of the intensity of the main trapping laser. As an alternative to the second laser one can put rf sidebands on a single trap laser at the frequency needed for hyperfine pumping. The use of an external modulator to create such sidebands on the output of a dye laser has been the standard approach for trapping sodium. Unfortunately, to obtain sufficient sideband power at the much higher frequencies needed for cesium and rubidium requires expensive (relative to diode lasers) electro-optic modulators and high-power drives. In addition to atom trapping, there are other applications for which one has similar needs for optical frequencies separated by microwave frequency intervals, notably for Raman velocity selection and to excite Raman transitions between cesium hyperfine levels in Raman clocks. In previous experiments, this has been done either by phase locking two diode lasers together with a 9.2-GHz offset3 or by directly modulating a free-running diode laser.4 In this Letter we demonstrate the efficient production of microwave sidebands by direct modulation of the current of a diode laser that is operating with feedback from a diffraction grating. This provides a uniquely simple and inexpensive way to produce the multiple frequencies required for all these applications. It is well known that the use of optical feedback from a diffraction grating makes diode lasers much 0146-9592/93/080649-03$5.00/0
more useful for atomic-physics applications because it reduces the linewidth and provides better control of the output wavelength.5 However, we have found that grating feedback can also improve the modulation efficiency of the laser at multiples of the freespectral range of the external cavity defined by the grating and the back facet of the laser. It is a happy coincidence that the most convenient cavity lengths happen to match well with half the hyperfine splitting of 133Csand 87Rband the full splitting of 85Rb. We have observed large fractions of the laser power in the sidebands at desirable frequencies by using lowpower microwave sources and simple coupling of the microwaves into existing diode-laser mounts. In related research, Hollberg and Ohtsu studied the modulation of a diode laser that was frequency stabilized by using weak optical feedback from a high-finesse cavity.6 For modulation frequencies of 50-1250 MHz, they also observed an enhancement of the modulation at multiples of the free spectral range of the reference ~ a v i t y . ~ The laser we used in this study was a Sharp LT025MDO laser with a grating feedback cavity as described in Ref. 8. With a drive current of 110 mA, the laser produced 30 mW of power, of which 20 mW was coupled out by the grating. At a current of 85 mA, where the modulation is much larger, the power coupled out is 10 mW. The microwaves were coupled into the laser by using 20 cm of RG-174 coaxial cable that was soldered to a -100-pF ceramic capacitor attached to the current lead for the laser. This coaxial cable is rather lossy at high frequencies; however, it is much easier to use than rigid coax, and it transmits little vibration to the laser cavity because it is quite thin and flexible. The microwaves were produced by a Hewlett-Packard HP 8672A frequency synthesizer. Although we do not know exactly how efficiently the microwave power is coupled into the laser, we do know that approximately 50% of the power out of the synthesizer is reflected back to it, and there is approximately a 15% loss in the coaxial cable. It is certainly possible to do much better by following good microwave design practices; however, 0 1993 Optical Society of America
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OPTICS LETTERS / Vol. 18, No. 8 / April 15, 1993
6200
6400
6600
6800
Modulation Frequency (MHz)
Fig. 1. Percent of laser power in a single sideband for both a free-running laser (dashed curve) and a laser with grating feedback (solid curve). The free spectral range of the external cavity formed by the grating and the back facet of the laser was 3.3 GHz. The frequency response of the free-running laser shows many circuit resonances (e.g., at 6.57 GHz) owing to the inefficient coupling of the microwave power.
optimum condition for use in laser trapping where maximum power in the carrier and a small fraction in the sideband is desired. For driving Raman transitions, it is usually optimum to have the maximum power in the sidebands. At a current of 85 mA and 20 mW of microwave power, we can put 25% of the laser output in a single sideband at a frequency of 4.6 GHz (half the cesium hyperfine splitting). We have used a single modulated laser to optically trap 87Rbin a vapor-cell trap that is similar to previously described traps for cesium and rubidium. The 20-mW output from the laser was split into three beams that intersected at right angles in the middle of a cell containing rubidium vapor. The three beams were circularly polarized and reflected back on themselves with the opposite polarization. A small magnetic-field gradient was applied to the cell. We first trapped atoms by using a second diode laser for hyperfine pumping in order to compare to the results from a single modulated laser. This produced approximately 3 x lo7 trapped atoms. We then turned off the hyperfine pumping laser and turned on the microwave modulation. In Fig. 2, we show the relative number of atoms trapped by using the single modulated laser. In this case, the sideband frequency was set to 6.57 GHz to excite the 5SvzF = 1 to 5P3/2F’= 2 transition (5.7-MHz natural linewidth). At our largest modulation, changing the frequency by 2 MHz had little effect on the number of trapped atoms, but a 6-MHz change caused a 5% reduction, and the number rapidly decreased with further detuning. We also set the microwave frequency to 6.4 GHz to excite to the F = 1 to F‘ = 1 transition for repumping. The behavior was similar on this transition except that the modulation needed to obtain the same number was increased by a factor of 3, as one would expect from the branching ratios. It should be noted that in Fig. 2 we quote the microwave power out of the source, but the actual
the point we wish to emphasize here is that one can obtain remarkably good results without expending any such effort on design or construction. We determined the amount of power in the sidebands by sending the light from the diode laser into an optical spectrum analyzer. In Fig. 1,we show the percent of the total power in a single sideband as a function of frequency for both a free-running laser and a laser with grating feedback. The dramatic enhancement at 6.6 GHz for a 3.3-GHz free-spectralrange cavity (4.06 cm long) is evident. We have investigated the amount of modulation versus microwave power and frequency, cavity length, and laser current. We find that the amount of power in the sideband is linear with microwave power, as one would expect, until the carrier becomes substantially depleted. The frequency response of the free-running laser has been well studied p r e v i o ~ s l y . ~ The response rises gradually until it hits the peak of the relaxation oscillations and then falls rapidly as the frequency is increased further. The relaxation oscillation peak shifts to higher frequency and broadens as the laser current is increased. The effect of the grating feedback is to enhance the modulation at multiples of the external-cavity free spectral range and to suppress greatly modulation at other frequencies. This enhancement ranges from unity at a, .-> 0.2 -3 GHz to a factor of -5 at 6.8 GHz (see Fig. 1). 4 We believe that the dependence of the modulation efficiency on laser current is dominated by two 2 0 c4 factors. First, there is the relatively gradual depen15 20 0 5 10 dence caused by the effects on the relaxation oscilMicrowave Power (mW) lations, as just mentioned. Second, the degree of Fig. 2. Number of atoms trapped with a single mimodulation is quite sensitive to how far the current crowave modulated laser as a function of power out of is above threshold. We presume that this is because the microwave source. The number of atoms is plotted the fractional modulation that a given microwave relative to the number obtained in the conventional trap current will produce is proportional to the dc current in which a second laser provides the hyperfine pumping. above threshold. The data shown in Fig. 1 are for At 20 mW of microwave power, there is 2.2% of the laser the maximum laser current of 110 mA. This is the light in the +6.6-GHz sideband.
349 April 15, 1993 / Vol. 18, No. 8 / OPTICS LETTERS power coupled into the laser is no more than 25% of these values. Figure 2 shows that the number of trapped atoms increases with microwave power but quickly approaches saturation. At 10 mW of microwave power the number is 0.85 of that obtained with two lasers, whereas at 20 mW it is 0.87. We believe two roughly equal factors are responsible for the final 13% difference. First, we have not quite reached the infinite hyperfine pumping power limit, and second, the modulation has depleted the power of the trapping laser beams by 4.4%. For most applications, a 10% or 15% decrease in the number of trapped atoms is unimportant, whereas the elimination of a stabilized laser is a major simplification. We have demonstrated that efficient modulation of diode lasers can he obtained for frequencies a s high as 6.8 GHz by enhancing the modulation with a n external cavity of the appropriate length. This makes it possible to laser trap either rubidium isotope by using a single modulated laser. (With the proper cavity length, it is easy to modulate at the 2.9-GHz frequency needed for 85Rh.) Although we used a relatively expensive frequency synthesizer for this study, a 10-mW microwave source within a few megahertz of the desired frequency would be sufficient. Varactor diode sources that meet these requirements are readily available for a few hundred dollars. If more care can be taken in coupling the microwaves into the laser, even less microwave power would be necessary. Such a single-laser vapor-cell trap provides a simple, inexpensive way to obtain laser-cooled and trapped rubidium atoms. We have been unable to observe modulation of our laser at the 8.9-GHz frequency that would be needed to trap cesium with a single diode laser. However, we have demonstrated that it is possible to get nearly complete depletion of the carrier when modulating the laser at half that frequency. Such a single laser could be used to trap cesium atoms by using one sideband for the trapping and the other
651
for hyperfine pumping. There would be at least a factor-of-3 reduction in the number of trapped atoms relative to a two-laser trap, however. Such strongly modulated lasers also provide a convenient source for exciting Raman transitions in either cesium or rubidium. We are pleased to acknowledge valuable suggestions by L. Hollberg regarding how to couple microwave modulation into the diode lasers and what results we should expect. H. Robinson also provided useful suggestions, and C. Monroe and E. Cornell built much of the atom trap apparatus. This research was supported by the Office of Naval Research and the National Science Foundation.
References 1. C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571 (1991). 2 . D. Hoffman, P. Feng, R. Williamson, and T. Walker, Phys. Rev. Lett. 69, 753 (1992). 3. R. D. Steele, Electron. Lett. 19, 69 (1983); S. D. Swartz
and J. L. Hall, Joint Institute for Laboratory Astrophysics, Boulder, Colo. (personalcommunication,1992); K. E. Gibble and D. S. Weiss, Stanford University, Palo Alto, Calif. (personal communication, 1992). 4. S. Ezekiel, Final Technical Report RL-TR-91-413 (Rome Laboratory, AFMC, Griffiss Air Force Base, N.Y., 1993). 5 . C. Wieman and L. Hollberg, Rev. Sci. Instrum. 62, 1 (1991). 6. L. Hollberg and M. Ohtsu, Appl. Phys. Lett. 53, 944 (1988). 7. Hollberg and Ohtsu6 actually observed modulation at frequencies that were rational fractions of the reference-cavity free spectral range. In contrast, we found significant modulation only at integer multiples of the free spectral range of the cavity formed by the grating and the back facet of the laser. 8. K. Macadam, A. Steinbach, and C. Wieman, “A narrowband tunable diode-laser system with grating feedback, and a saturated absorption spectrometer for Cs and Rb,” Am. J. Phys. (to be published). 9. K. Petermann, Laser Diode Modulation and Noise (Kluwer,Dordrecht, The Netherlands, 1988).
LASER COOLING AND
TRAPPING FOR-THE MASSES ATIC A D V A N C E S IN THE TECHNIQUES FOR
cooling and trapping neutral atoms using laser light. In particular, the use of inexpensive diode lasers and the vapor-cell trap have significantly reduced the complexity and cost of trapping atoms, which has opened the field to many additional researchers. Atomic densities of 10" atoms/cm3 have been obtained with temperatures of a small fraction of a millikelvin. In this article, we describe laser trapping techniques and discuss current applications of this technology, ranging from frequency and wavelength standards to basic physics research. The primary force used in laser cooling and trapping is the recoil momentum transferred to an atom when photons scatter from it. This force is analogous to that applied to a bowling ball when it is bombarded by a stream of ping pong balls. The momentum kick that the atom receives from each scattered photon is small; a typical velocity change is about 1 cm/sec (room temperature gas atoms have typical velocities of a few hundred meters per second). However, by exciting a strong atomic transition, it is possible to scatter more than 107 photons per second and produce accelerations on the order of 10sm/sec2 (—10" • g). The fundamental progress in this field has been in finding new ways to harness this acceleration so that it slows ("cools") the atoms in a sample to a particular velocity, usually near zero, and holds ("traps") them at a particular point in space.
Photograph of cesium atoms trapped in a glass vapor-cell magneto-optical trap. The cell is a 6--sided cross with 2.5 cm diameter windows. The copper coils above and below the cell provide the inhomogeneous magnetic field. The trapped atoms are the bright bluish-white cloud at the center of the cell. They are excited to the 85 state to produce the blue fluorescence for the photograph.
LASER COOLING
The cooling is achieved by using the Doppler effect to make the photon-scattering force depend on the velocity of the atom.1 The basic principle is illustrated in Figure 1. If an
e
OPTICS & PHOTONICS NEWS/JULY 1993
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351
BY SARAH L. GILBERT AND CARL E. WIEMAN atom is moving in a laser beam, it will see the laser frequency vliue[ shifted by (V/e^v^^ where V is the component of the atom's velocity that is opposite to the direction of the laser beam. If the laser frequency is below the atomic resonance as a result of this Doppler shift, the atom will scatter photons at a higher rate if it is moving toward the laser beam (V positive) than if it is moving away (V negative). If we then consider the effect of sending laser beams from all six directions, the only remaining force is the velocity dependent part, which opposes the motion of the atom. This provides strong damping of any atomic motion and cools the atomic vapor. Chu et al. first used this configuration of laser fields to obtain very cold atomic samples, and gave it the descriptive name "optical molasses."2 A few years later, Lett ct al? found that, at certain laser frequencies, they could achieve atomic temperatures lower than could be explained by the Doppler cooling just described. This accidental discovery is now understood to arise from some very fortuitous atomic physics. As atoms move through the hills and valleys of potential energy produced by the standing-wave laser fields, the atoms tend to make transitions between states in such a way as to transfer their kinetic energy into that of the scattered photons very efficiently.4
Frequency -
Figure 1. Velocity-dependent photon scattering force. The upper diagram shows the frequency dependence of the atomic excitation rate. The laser frequency v(owr is tuned to the low frequency side of the afomic resonance. When an atom is moving toward the laser beam (middle of figure), it sees the laser frequency Doppler-shifted to higher frequency by rhe amounl &vDa^ff => (V/c) v^, and scoffers many photons. When an atom is moving awoy from the laser beam (bottom), however, it sees the laser frequency Doppler-shifted to lower frequency and scatters very few photons. Thus, the laser photons exert a much larger force if the atom is moving toward the laser beam.
NEUTRAL ATOM TRAPPING
Although optical molasses can cool atoms, the atoms will still diffuse out of the region since there is no position dependence to the optical force. Position dependence can be introduced by using appropriately polarized laser beams and applying an inhomogeneous magnetic field. The magnetic field causes the atoms to be pushed to a particular point in space by regulating the rate at which an atom at a particular position scatters photons from the different beams. As well as holding the atoms, this greatly increases the atomic density since many atoms are pushed to the same point. This atom trap is referred to as the magneto-optical trap (MOT) or the Zeeman-shift optical trap (ZOT) in the literature. Details of how the trapping works are somewhat complex for a real atom in three dimensions, so we will illustrate the basic principle using the simplified case shown in Figure 2. We consider an atom with a J = 0 ground state and a J = 1 excited state, illuminated by circularly polarized beams of light coming from the left and the right. Because of its polarization, the beam from the left can excite only transitions to the m = +1 state, while the beam from the right can excite only transitions to the m = —1 state. The magnetic field is zero in the middle, increases linearly in the positive x direction, and decreases linearly in the negative x direction. This field shifts the energy levels so that the Am = +1 transition shifts to lower frequency as the atom moves to the left of the origin, while the Am = -1 transition shifts to higher frequency. If the laser frequency is below the atomic transition frequencies and the atom is to the left of the origin, many photons are scattered from the a* laser beam because it is close to resonance. The a- laser beam from the right, however, is far from its Am = -1 resonance and scatters few photons. The force from the scattered photons pushes the atom back to the zero of the magnetic field. If the atom
J=1
, B<0 force-
8=0 force = 0
B>0 -force
Figure 2. Magneto-optical trap (MOT) forces in one dimension for an atom with a j = 0 ground state and a J = 1 excited state. A magnetic field gradient of about 0.15 T/tn (15 G/cm| is applied in a region of counterpropagating laser beams with circular polarizations of opposite helicity (IT' and (T }. The upper diagram shows the laser beams and magnetic field and the lower diagrams show the atomic energy levels for negative magnetic field (left diagram, corresponding to an atom to the left of the origin), zero magnetic field (center diagram, atom at the origin], and positive magnetic field (right diagram, atom to the right of the origin). The magnetic field shifts the a' transitions closer to resonance when the atom is to the left of the origin. When the atom is to the right of the origin, the tj" transitions are shifted closer to resonance. This causes a position-dependent force that pushes the atom toward the origin.
352
LASER COOLING AND
TRAPPING
magnetic field coils
Figure 3. Schematic diagram of a three-dimensional MOT.
moves to the right of the origin, exactly the opposite happens, and again the atom is pushed toward the origin. Although it is more complicated to extend the analysis from one to three dimensions, experimentally it is quite simple, as shown in Figure 3. As in optical molasses, laser beams illuminate the atom from all six directions. Two symmetric magnetic field coils with oppositely directed currents create a magnetic field that is zero in the center and changes linearly along the x, y, and z axes. If the circular polarizations of the lasers are set correctly, a linear restoring force is produced along each direction. Damping in the trap is provided by the cooling forces discussed above. It is convenient to characterize the trap's "depth" in terms of the maximum velocity that an atom can have and still be contained in the trap. This maximum velocity is typically a few times the velocity at which the Doppler shift equals the natural linewidth of the trapping transition (V mlx = 20 m/sec). A BRIEF HISTORY
The experimental history of laser cooling and trapping began with laser cooling of trapped ions.5 The first major neutral atom work was the cooling and slowing of atomic beams in one dimension by the groups of Hall6 and Phillips,7 who used laser light that propagated opposite to the atomic beam. The slowed atomic beams provided low velocity atoms that could then be further cooled using optical molasses. Chu, A.shkin, and co-workers then demonstrated the first optical trap by holding the molasses-cooled atoms at the focus of an intense laser beam." The restoring force for this trap came from the interaction of the induced dipole moment of the atom with the light field, rather than the photon-scattering force discussed above. The scattering force has the obvious advantages thai it can extend over a much larger distance, requires lower laser powers, and simultaneously cools the atoms. However, early attempts to use it for trapping were thwarted by the fact that the scattering force, by itself, canno! provide a confining potential in free space.5 Pritchard et at. reali/ed that this problem could be overcome by the use of inhomogeneous external fields to regulate the scattering force.'" This quickly led to the demonstration of the MOT discussed above." A!) of this work used sodium atoms, which started out in atomic beams that were slowed in 1-2 m long atomic beam machines. The slowing and cooling/trapOPTICS & PHOTONICS NEWS/JULY 1993
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ping were accomplished using narrowband cw dye lasers. This work demonstrated that it was possible to produce atomic samples with properties that were unprecedented. Densities on the order of 10" atoms/cm3 were obtained with temperatures of a small fraction of a millikelvin (V = a few centimeters per second). Such gas samples provide large optical absorption without shifts and broadening due to the Doppler effect, since the atomic velocities are so low. A superior signal-to-noise ratio is achieved when probing atomic transitions because background noise due to nonresonant atoms is nearly eliminated. Also, problems caused by perturbations of optically excited levels due to atomic collisions (pressure shifts) are reduced for two reasons. First, the collision rate decreases as the atomic velocity decreases, and second, narrowband excitations can be detected at much lower densities (1/100) due to the absence of Doppler broadening. A final virtue of these samples is that the low velocities allow the interaction times between the atoms and electromagnetic fields to be nearly a second, rather than the fractions of a millisecond available in traditional atomic beams. SIMPLER SYSTEMS
While these experiments demonstrated tantalizing benefits, the size, cost, and complexity of the apparatus limited the potential applications. Recent simplifications in the technology have dramatically changed this situation. The two primary developments are the replacement of dye lasers by narrowband diode lasers (roughly a 100-fold reduction in cost, and even more in electrical power and size) and the vapor-cell MOT. Near infrared diode lasers that can be tuned to the strong resonance lines of cesium and rubidium are readily available. Diode lasers have sufficient power since the intensities required for cooling and trapping are only a few milliwatts per square centimeter for these atoms. The typical linewidth (10 MHz or more) of inexpensive single-mode diode lasers is a problem. However, simple optical feedback schemes12 reduce this linewidth to well below the natural linewidths of the relevant atomic transitions, which is the criterion necessary for trapping and cooling. Diode lasers have been used to slow beams of atomic cesium and rubidium, produce optical molasses, and optically trap these atoms in a MOT.'3 A variety of trapped-atom studies that were carried out with diode lasers revealed new information about very low temperature collisions and elucidated the photon-mediated interatomic repulsion that limited the density of trapped atoms.14 An atomic beam apparatus that included a 1 or 2 m long vacuum system was still required, however. This was subsequently eliminated by the invention of the vapor-cell trap,13 which is simply a MOT that is established inside a small vapor cell. Rather than loading the trap from a slowed atomic beam, the atoms are captured directly from the low velocity tail of the Maxwell-Boltzmann distribution of a dilute room temperature vapor. Vapor-cell traps are now used routinely to produce very cold (as low as a few microkelvins) optically thick samples of cesium, rubidium, and sodium. The cells are constructed from glass or metal with glass windows. Fortunately, if one simply wants to obtain trapped atoms,
353
-.-
I I
LASER COOLING AND
TRAP PI NG the quality of the windows can be poor-a simple round flask is adequate. The pressure in the cell must be fairly low; typically 10-7Pa Torr) is used. This is usually achieved by attaching a very small ion pump to remove the hydrogen and helium that diffuses through the walls, and limiting the pressure of the alkali metal by containing it in a small sidearm that is cooled or isolated by a valve. APPLtCATlONS TO FREQUENCY AND WAVELENGTH STANDARDS
-4bo
-ioo
00
200
-3
Figure 4. 9.2 GHz cesium clock transition spectrum [from Ref. 16) using atoms that were lasercooled and tropped in a diode-laser vapor-cell trap. The oscillafions arise from the use of the Ramsey separated oscillatory fields technique.
I
F=3 -+ F'=3
0
50
100
Fiber Laser Tunina (MHd Figure 5. First derivative of the spectrum of the 1.529 pm 5P,/,, F=3-4D5/,, F'=3 and 2 transitions in rubidium atoms (87Rb)confined in o diode-loser vaporcell trap. The slight asymmetries on the low frequency sides of the lines are due to splitting of the F=3 hyperfine level caused by the trapping laser field.
s
OPTICS & PHOTONICSNEWS/JULY 1993
Diode lasers and vapor-cell traps allow trapped-atom samples to be obtained with a very compact, simple, and inexpensive apparatus, making them well suited for a variety of applications. Either or both of these technologies are now used in the development of improved frequency and wavelength standards, and are being employed in basic research in several areas of physics. The low velocities of the atoms make them useful in improved cesium atomic frequency standards (clocks) that are based on a microwave transition in cesium. The resolution of this "clock transition is limited by the interaction time between the atoms and the microwave field. In the traditional cesium clock, long atomic beam machines (anywhere from 5-0.5 m) are used; these yield linewidths in the range of 25-500Hz. Using a diode-Iaser vapor-cell trap, linewidths as narrow as a few hertz are obtained in a cell only a few centimeters long. An example of a resonance observed in this manner is shown in Figure 4.16In addition to improving the resolution, the use of slow atoms improves the accuracy of an atomic clock, since most of the systematic shifts of the clock frequency are proportional to the velocity of the atoms. There are now several groups pursuing the development of laser trapped atomic clocks, both as research tools and commercial products, and investigating the performance limits of such device^.'^ Wavelength standards for optical communication are another area where laser trapping has proven useful. For future developments in optical fiber communication systems, such as multiple wavelength channels and coherent detection, it is desirable to have a wavelength standard in the 1.5 pm spectral region with an absolute accuracy of about 1MHz. Atomic and molecular references in this spectral region are few and difficult to probe; molecular absorptions are weak overtone or combination bands and there are no atomic absorption lines from the ground state. This lack of absorption features, of course, makes this region attractive for optical communications. To produce a 1.5 pm standard, high resolution spectroscopy of a transition between excited states of atomic rubidium is being carried out in one of our 1abs.l8A diode-laser vapor-cell trap was constructed for rubidium atoms, and the 1.529 pm transition was probed in the trapped atomic sample using a tunable fiber laser. Figure 5 shows a spectrum obtained as the fiber laser's frequency was swept. To demonstrate a stable wavelength reference, the fiber laser's frequency was locked to one of the lines using an electronic feedback circuit. A vapor-cell trap has several advantages over a simple room temperature vapor-cell wavelength reference. Due to the lack of Doppler broadening, the narrow linewidths nec-
354
f
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Optical Properties Database OPTIMATR was developed in conjunction wdh one of the most respected optical properties groups in the country at the Applied Physics Laboratory of The Johns Hopkins University Complex index of refractionis computedfrom several models including one-phonon and free carrier, multiphonon.Urbach Tail and Sellmeier Scatteringcoefficient includes Rayleigh and Mie scattering The database is one of the largest and most comprehensive on-line collections of optical properties in existence All material parameters are listed in the accompanying monual along with references to sources For the working optical scientist or engineer, OPTIMATR is a very convenient optical data resource from looking up a single property of a single material, to reviewinga range of materials and viewing each property graphically
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essary for an accurate standard are easily obtained. Also, as discussed above, a trap significantly reduces pressure shifts of the reference wavelength. Finally, the superior signal-tonoise ratio allows the required sensitivity to be obtained with weaker excitation of the atoms. This reduces lightinduced shifts of the reference wavelength. APPLICATIONS TO B A S I C PHYSICS
Laser traps are also being applied to problems in many areas of basic physics. An obvious application is precision atomic spectroscopy, since most of the requirements for this field are identical to those for wavelength and frequency standards. However, as the advantages of this technology become more apparent, it is being applied in many other types of experiments as well. Here we have space to discuss only two examples with which we are particularly familiar; however, other applications include quantum optics (cavity quantum electrodynamics), squeezed-light generation, and other coherence phenomena), atom interferometers, and studies of very cold atomic collisions. Efforts are underway to use laser traps to collect and hold samples of rare shortlived radioactive isotopes. Experiments planned for these samples include precision studies of nuclear beta decay, searches for electric dipole moments of atoms, and the measurement of parity violation in isotopes of cesium.19Parity violation in atoms is produced by the weak (as opposed to "strong," "electromagnetic," or "gravitational") interactions between the electrons and
quarks in an atom. These interactions are about 10" times weaker than the electromagnetic interaction, but can be detected by observing a small difference in an atomic transition rate depending on whether the experiment is "right-handed" or "left-handed." The handedness of the experiment is defined by the directions of various applied fields. Because of the relatively large optical absorption and low collision-induced background, the signal-to-noise ratio that should be attainable with a trapped-atom sample is much larger than can be achieved in either traditional vapor cells or atomic beams. Furthermore, the comparison of different isotopes will provide an important test of the fundamental theory of elementary particle physics. Another example is the quest to achieve the predicted, but never observed, Bose-Einstein condensation in a dilute vapor. The goal of this work is to produce an extremely cold sample of atoms in which traditional interactions are negligible (due to low density), but the atoms are separated by less than their de Broglie wavelengths. For these conditions, it is predicted that atoms with integral total spin ("bosons") should lose their individual identities and collectively condense into the lowest energy quantum state in the system. This macroscopic quantum state would be quite unlike any known form of matter. Although laser cooling and trapping have not yet achieved the extraordinarily low temperatures required (< 0.1 pK), a cloud of trapped cesium atoms has been reduced to a temperature of 1.1pK.I5 Furthermore, laser cooling and trapping have allowed one of our labs to OPTICS & PHOTONICS NEWS/JULY 1993
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carry out studies of very low energy interactions of atoms, which show how to overcome some particularly serious impediments to achieving Bose-Einstein condensation.”’ Several groups are pursuing different variations on the cooling and trapping techniques discussed here in an attempt to achieve Bose-Einstein condensation. We are observing the rapid growth of an exciting new technology. It is currently applied to a range of research areas, from practical problems such as wavelength standards for optical communication to the study of the fundamental weak interaction between quarks and electrons. In the coming years, there will no doubt be many exciting new results based on this technology and a host of new applications. ACKNOWLEDGMENTS
The work of Carl Wieman is supported by the NSF and ONR, and the work of Sarah Gilbert is supported by NIST and the NCCOSC.This paper represents the work of the U S. Government and is not subject to copyright. S A R A H L. G I L B E R T is n physicist with the Natiorial lrtstitirte of Stnndards and Techology, Boiilder, Cola C A R LE. W I E M A Nis professor of physics, University of Colorado, nrid Clinirriian of ”LA, University of Colorado, Boidder, Cola
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OPTICS & PHOTONICS NEWS/JFJLY 1993
REFERENCES
1. T. W. Hansch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 1975, 68-69; D. J. Wineland and H. Dehmelt, ”Proposed 10“ A”< v laser fluorescence spectroscopy on TI’ mono-ion oscillator Ill,” Bull. Am. Phys. Soc. 20. 1975, 637. 2 . 5. Chu et nl., “Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure,” Phys. Rev. Lett. 55,1985,48-51 3 . P. D. Lett ct nl., “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 1988, 169-172. 4. J. Dalibard and C. Cohen-Tannoudji, ”Laser cooling below the Doppler limit by polarization gradients: Simple theoretical models,” J. Opt. Soc. Am. B 6, 1989, 2023-2045; P.Ungar c’t nl., “Optical molasses and multilevel atoms: Theory,” J. Opt. Soc. Am. B 6, 1989, 2058-2072. 5. D. J. Wineland el nl., “Radiation-t,ressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1978, 1639-1642; W. Neuhauser rt nl., ”Optical-sideband cooling of visible atom cloud confined in parabolic well,” Phys. Rev. Lett. 41, 1978, 233-236. 6. E. Ertmer et nl., “Laser manipulation of atomic beam velocities: Demonstration of stopped atoms and velocity reversal,” Phys. Rev. Lett. 54, 1985,996-999. 7. J. Prodan ct nl., ”Stopping atoms with laser light,” Phys. Rev Lett. 54, 1985,992-995. 8. S. Chu Ct nl., ”Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 1986, 314-317. 9. A. Ashkin and 1. P. Gordon, ”Stability of radiation-pressure traps: An optical Earnshaw theorem,” Opt. Lett. 8, 1983, 511-513. 10. D. E. Pritchard, E.L. Raab, V. Bagnato, C.E. W~emiln,and R.N. Watts, ”Light traps using spontaneous forces,” Phys. Rev. Lett. 57, 1986, 310313. 11. E. L. Raab ct nl., “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 1987, 2631-2634. 12. C. E. Wieman and L. Hnllberg, “Using diode lasers fnr atomic physics,‘‘ Rev. Sci. Instrum. 62,1991,1-20 and references therein; K.B. MacAdam, A. Steinbach, and C. Wieman, ”A narrow band tunable diode laser system with grating feedback, and a saturated absorption spectrometer for Cs and Rb,” Am. J. of Phys. 60,1992,1098-1111. 13. R. N. Watts and C. E. Wieman, “Manipulating atomic velocities using diode lasers,” Opt. Lett. 11, 1986. 291-293; D. Sesko, C.G. Fan, and C. Wieman, “Production of a cold atomic vapor using diode-laser cooling,”J. Opt. SOC.Am. B 5, 1988, 1225-1227; D. Sesko, T. Walker, C Monroe, A. Gallagher, and C. Wieman, “Collisional losses from a lightforce atom trap,” Phys. Rev. Lett. 63, 1989, 961-964. 14. D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, ”Collisional losses from a light-force atom trap,” Phys. Re\,. Lett. 63, 1989, 961-964; D. Hoffman et nl., “Excited-state collisions of trapped “‘Rb atoms,” Phys. Rev. Lett. 69, 753 (1992); T. Walker, D. Sesko, and C. Wieman, “Collective behavior of optically trapped neutral atoms,” Phys. Rev. Lett. 64,1990, 408-411. 15. C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold atoms in a vapor cell,” Phys. Rev. Lett. 65, 1990, 1571-1574; K. Lindquist. M. Stephens, and C. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46, 1992,4082-4090. 16. C. Monroe, H. Robinson, and C. Wieman, “Observation of the cesium clock transition using laser cooled atoms in a vapor cell,“ Opt. Lett. 16, 1991, 50-52. 17. A. Clairon et nl., ”Ramsey resonance in a Zacharias fountain,” Europhys. Lett. 16, 1991, 165-170; K. Gibble and S. Chu, ”Laser-cooled Cs frequency standard and a measurement of the frequency shift due to ultracold collisions,” Phys. Rev. Lett. 70, 1993, 1771-1774; Rincon Research Corp. is developing a trapped-atom clock. 18. S. L. Gilbert, “Frequency stabilization of a fiber laser to rubidium: a high-accuracy 1.53 pm wavelength standard,” Frequency Stabilized Lasers and Their Applications, Proc. SPlE 1837, Boston, Mass, Nov. 1992, paper 1837.20. in press. 19. C. Wieman et nl., ”Fundamental Physics with Optically Trapped Atoms,” Laser Spectroscopy X, ed. M. Ducloy et nl., World Scientific 1992, 77-83; C. Wieman, “Parity violation in atoms; current status and trapped atom future,” J. Hyperfine Interactions (in press). 20. C. Monroe, E. Cornell, C. Sackett, C. Myatt, and C. Wieman, ”Measurement of Cs-Cs elastic scattering at T = 30 pK,” Phys. Rev. Lett. 70,1993, 414-417.
Hyperfine Interactions 81(1993)203-215
203
Optimizing the capture process in optical traps M. Stephens, K. Lindquist and C. Wieman Joint Institute for Laboratory Astrophysics and Physics Department, University of Colorado, Boulder, CO 80309-0440, USA
We present techniques for the most efficient capture of atoms in a modified vaporcell magneto-optical trap. We explain how changing the size and power of the trapping beams affects the capture rate of the trap. We calculate the requirement of wall coatings designed to minimize the interaction of an alkali vapor with the wall and explain how this affects trapping efficiency. Finally, we present measurements of the performance of a "Dryfilm" coated cell.
1.
Introduction
Vapor-cell magneto-optical traps have been shown to be an inexpensive way to obtain cold samples of more than 10'' atoms [ l ] with densities of 5 x 10" atoms/cm3 [2,3]. Although as of yet no one has used these traps to obtain samples of radioactive atoms, trapping radioactive atoms could be useful for parity non-comervation experiments [4], EDM measurements [ 5 ] , P-decay experiments [6], high resolution spectroscopy [7], and cold collision measurements [8]. Some changes in the current methods for trapping from a vapor will have to be made in order to make it possible to trap rare, short-lived isotopes. In particular, one must trap the atoms as efficiently as possible since it may be difficult or impossible to produce large quantities of the desired isotope. We envision producing a small number of the isotopes of interest (around lo8 atoms), releasing the atoms into a closed trapping cell, and cooling and trapping as many of the atoms as possible. To achieve this, we will need to capture the atoms at the maximum possible rate, and we will have to minimize the interactions of the atoms with the walls. How quickly atoms can be trapped depends on parameters such as trapping beam size, power, and detuning, and trapping magnetic field gradient. We have measured and modelled the effects of these parameters on the capture rate and present a summary of those results here (the results have been presented in more detail elsewhere is]). We have also studied how to minimize the interaction of the atom with the walls of the vapor cell. Alkali atoms interact with the wall by chemically bonding with the constituents of the wall (chemisorption) and by sticking to the wall for some time, T,, and then returning to the vapor (physisorption). We have calculated the 356
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effect of both chemisorption and physisorption on trapping efficiency and present minimum requirements for the wall-vapor interaction. We have also begun measuring the properties of some promising wall coatings and present our preliminary results here. In section 2, we describe a typical vapor-cell trap and explain the modifications necessary to trap rare isotopes. In section 3, we briefly present the results of our study on the optimization of the capture rate of a vapor-cell trap, In section 4, we present the requirements for an appropriate wall coating, and in section 5 we present the results of our tests on Dryfilm and Pyrex. 2.
The vapor-cell magneto-optical trap
Our vapor-cell magneto-optical trap has been described in detail elsewhere [9]. Here, we will present a brief description of a standard vapor-cell trap and discuss the changes that must be made to such a trap to allow us to most efficiently trap radioactive isotopes. The magneto-optical trap [ 101 (fig. 1) consists of a spatially varying magnetic field and three orthogonal pairs of counter-propagating laser beams of opposite Torr of room-temperature helicity. The laser beams interest in a cell filled with cesium. The laser beams are tuned to a frequency slightly lower than the 852 nm Cs D2 transition. This provides an imbalance in radiation pressures that slows the atoms in the low-velocity tail of the Maxwell-Boltzmann distribution. A quadrupole magnetic field induces a position-dependent shift in the Zeeman levels of the slowed atoms, which causes a position-dependent radiation pressure that traps the slowed atoms. An ion pump attached to the cell continuously pumps away background gases. We measure the number of trapped atoms by imaging the fluorescence of the cloud orito a photodiode, which gives us the rate at which photons are scattered by the cloud. We assume that the number of atoms is proportional to the total fluorescence, with a proportionality constant given by the power-broadened scattering rate, which depends on the laser’s detuning and intensity. We envision two major changes to our current vapor cell configuration when we begin to trap rare, radioactive isotopes. Both result because we expect to have only small numbers of the isotopes of interest and wish to trap as many of them as possible. First, there will not be a reservoir such as the Cs reservoir in fig. 1. Instead, all of the atoms to be trapped will be released imo the cell at once. Second, because we do not wish to pump away any of the atoms, we will not pump on the trapping cell with the ion pump during the time the atoms are in the cell but not yet trapped. Fig. 2 shows a picture of the changes that will be required. As mentioned in the introduction, there are two areas that require study optimizatioh of the capture rate and minimization of atom- wall interactions. We describe the work we have done on optimizing the capture rate in the following section.
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7
205
-1
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I
I Ion
I
1
pump1
Fig. 1. A standard vapor-cell magneto-optical trap. The fused silica cell has six windows that admit the six slowing and trapping beams. A pair of anti-Helmholtz coils creates a quadrupole magnetic field for the magneto-optical trap. A temperature-controlled Cs reservoir provides a constant source of Cs vapor. A small (2 l/s) ion pump pumps away background gas.
Isotope Source
i/
To ion pump ____)
I V
Fig. 2. A cell for trapping rare isotopes. The Cs reservoir has been replaced with a valve to the isotope source. The ion pump can be valved off so that the rare isotopes are not pumped away during the trapping process. Initially, valve #1 will be closed and valve #2 will be open. Valve #2 will close and valve #I will open and release the isotopes into the cell. Both valves will be closed white the atoms are being trapped. After the maximum number of atoms has been trapped, valve #2 will reopen to pump away the background gasses that have built up.
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M. Stephens
3.
Optimization of the capture rate
We have modelled the effects of beam size, beam intensity, beam detuning, and magnetic field on the capture rate of the vapor-cell magneto-optical trap. We have compared the predictions of the model to measured trends and found good agreement. These results have been reported in detail elsewhere [9] and will only bw summarized here. We find that a simple, one-dimensional model of the slowing that does not include the effects of the magnetic field predicts the capture rate reasonably well. We have also extended that model to a quasi-one-dimensional model that includes the effects of the magnetic field on the slowing and have obtained even better agreement with our measurements. In our simulations, we calculate the number of atoms rather than the capture rate; however, the number of atoms in the trap in steady state is proportional to the capture rate [3]:
N,, = RT, where R is the capture rate and 1/2= (navm,)cs + ( n ~ ~ s ) b a & g r o gas ~ d is the sum of the loss rates due to collisions between trapped atoms and atoms in the room temperature background Cs gas and due to collisions between trapped atoms and atoms in the room temperature non-Cs background gases. The cross section acsis Cm2 and abackgroudgas is around 4 X Cm2, depending on what the 2X background gas is [ l l ] . The capture rate can be written as
where no is the Cs background density, qhermal = 193 m/s is the thermal velocity of the room temperature Cs, and A = 7cL2 is the surface area of the trap ( L is the l/e diameter of the beam). We assume that all atoms entering the trapping region with velocities below vc will be trapped. To predict the capture rate, we need only to calculate v,. We find v, by computing the one-dimensional slowing force on an atom in the trapping region. This force is proportional to the difference between the number of photons scattered from the two beams that are co-propagating and counter-propagating with the atom. If we make the great simplification of considering the atom as a two-level system and ignore the presence of the magnetic field, this can be written as r
1
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M. Stephens et d./ Optimizing capture in optical trap$
207
where /3= Illsatis the saturation parameter for one beam, I is the intensity in each beam, and Isat= 2.7 mW/cm2; k = 74000 cm-’ is the wave number of the light; A is the detuning of the laser, v,,, = 3.52 x 1014 H z and rat,, is the 31 ns excitedstate lifetime. To find the maximum capture velocity v,, the equation of motion is solved numerically with the constraint that the atom must be stopped by the time it has traversed the trapping region. The length of the trapping region is defined to be the lle diameter of the trapping beam’s Gaussian intensity profile. The quasi one-dimensional calculation includes the magnetic field and the multiple rn-levels of the atom. The magnetic field has two effects; first, it causes a position-dependent splitting of the Zeeman levels of the atom. There is a corresponding shift in the frequency of the optical transition:
where B is the magnitude of the magnetic field, gF=4 = 0.25 and g F ’ = S = 0.4 are Land6 g factors, pg is the Bohr magneton, and h is Planck’s constant. Second, the magnetic field defines a quantization axis that is different than that defined by the laser beams, so although the light is circularly polarized, Am = + 1,0, and - 1 transitions are all driven. Furthermore, the direction of the magnetic field changes with position (fig. 3) so the amount that each Am transition is driven changes with position.
Fig. 3. Spatial variation of the magnetic field in the trap. The grey arrow shows one possible path of an atom through the trap. The dark, black arrows indicate the direction of the magneiic field at several points along the atom’s path.
In figs, 4-7, we show the predictions of the model that does include the effects of the magnetic field and the very simple model that does not include the
361
M. Stephcns et al. / Optimizing capture in optical traps
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Fig. 4. (a) The number of trapped atoms versus the laser detuning at a constant magnetic field gradient of 11 G/cm. (b) The number of trapped atoms versus the magnetic field gradient at a detuning of -12.5 MHz. The beam diameter for both (a) and (b) is 1.9 cm and the intensity in each of the six beams is 1.6 mW/cm2. The asterisks are our data, and the solid line shows the predictions of the model that includes the magnetic field. The dashed line is the prediction of the simpler model.
h
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4
5
Fig, 5. The number of trapped atoms versus intensity at constant beam diameter: asterisks are our data €or a trap with a 1 cm beam diameter, diamonds are for a 1.9 cm beam. Each data point was taken at the optimum detuning and optimum magnetic field gradient for the given beam size and intensity, The solid lines represent the number of atoms predicted by the model that includes the magnetic held. The dotted line shows the predictions of the two-level model.
magnetic field, and we compare these predictions with our measurements. We find that the capture rate increases both with beam size and beam power. For this reason, we plan to use a powerful Tixapphire laser rather than a diode laser in isotope traps.
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40 30
20 10
0 0.0
0.5
1.0
1.5
2.0
2.5
Fig. 6. Number of atoms versus lle beam diameter L when the total power in each beam is held as constant as possible. Asterisks indicate data for powers from 0.9 to 1.2 mW per beam, crosses are 2.7-3.2 mW per beam, diamonds are 4-4.5 mW per beam. The power variation in each trace is responsible for some of the scatter in the data. Each data point was taken at the optimum detuning and optimum magnetic field gradient For the beam size and power. The solid tines indicate the number of atoms predicted by the model with the magnetic field. The inset shows, on a condensed scale, the predicted number of atoms as a function OF beam size For three powers: 2, 4.5, and 7 mW per beam. Solid lines represent predictions made by the model when the magnetic field is included; dotted lines represent predictions made by the two-level model.
4
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15
20
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I (mW/cmZ) Fig. 7. The results of our models are compared with the data of Gibble et al. [l]. The detuning is constant at -20 MHz and the magnetic field is constant at 7.7 G/cm. The beam diameter is 4 cm. Asterisks are data from Gibble et al., the solid line shows the predictions of the model that includes the magnetic field, and the dotted line illustrates predictions of our simpler model.
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M. Stephens et al. / Optimizing capture in optical traps
Wall coating requirements
An alkali in a vapor can interact with the walls of the vapor cell through physisorption, in which the atom sticks to the wall for a characteristic time T,, and then returns to the vapor, or through chemisorption, in which the atom chemically bonds to some constituent of the wall, and never returns to the vapor [ 121. Both of these interactions affect the trapping efficiency in different ways. The physisorption reduces the density of the atoms in the vapor and therefore reduces the capture rate. The number of atoms stuck to the wall at any time is the flux of atoms hitting the wall multipled by the average sticking time and the surface area of the cell [13]:
where n is the density of the vapor and Ace,, is the surface area of the cell. The steady-state density is then reduced by the coefficient
where Vcell is the volume of the cell. Since the capture rate is proportional to the density (eq. (3)), the capture rate is also reduced by the coefficient in eq. (6). One catl then calculate the maximum T, that will not significantly reduce the capture rate:
For a room temperature cell with a surface area of 80 cm2 and a volume of 30 cm3, 2, must be less than 80 ps. Physisorption is typically caused by a Van der Waals interaction between the atom in the vapor and the wall. The strength of the interaction is characterized by an adsorption energy E,, which in turn depends on the polarizability of the atom and the wall. The sticking time 2, is related to the adsorption energy by the relationship [ 141 Z,
= 10- 12 e E./kTS*
(9)
Therefore, to find a surface that the atom will only stick to for a short time, we look fot a surface with a low polarizability, i.e. a low dielectric constant. Chemisorption results in the complete loss of trappable atoms, and may also result in an increase of other background gases due to the release of gas during the chemical reaction with the wall. The reaction rate with the wall must therefore be small compared to the capture rate of the trap so that only a few atoms are lost to
364
M. Stephens et al. / Optimizing capture in optical traps
21 1
reactions with the walls during the capture process. The release of gases due to reactions must also be minimized. Since a small number of trappable atoms (around lo8) will be released into a cell with a volume around 30 cm3, the density of the untrapped atoms will be low and the capture rate will be correspondingly low. Because the capture rate is long and we do not wish to pump away the trappable atoms, the ion pump will be “turned off” for a relatively long time and therefore the background gases will build up, increasing the loss rate due to collisions with hot background gas. The choice of the appropriate cell material or wall coating is then a tradeoff between the sticking properties of the wall, the reaction of the Cs with the wall, and the outgassing rate of the wall. The effects of each of these can be calculated by using a modified version of eq. (1). The loss rate l/z is now dominated by background gas and increases with time. The capture rate is reduced by the coefficient in eq. (6), and the total number of atoms available for trapping decreases with a time constant equal to the rate of reaction of the Cs with the wall. Fig. 8 demonstrates how the total trapping efficiency changes for different stick times, reaction rates, and outgassing.
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Fig. 8. Percentage of Cs atoms that have been released into the cell that will be trapped as a function of time. Trace 1 shows the capture efficiency for a wall with a stick time of 1.4 ms, an outgassing of 1.8 lO-’Ton cm3/(cm2 s), and a reaction rate with the Cs of 0.065 cm3/(cm’s). Trace 2 is the same as 1 but with a stick time of only 0.045 ms (this trace represents Dryfilm). Trace 3 is the same as 2 but the stick time if 0 s. Trace 4 is the same as 3 but the reaction rate is 0. Trace 5 is the same as 4 but the outgassing is reduckd by a factor of 60. For all of these, the capture velocity is 32 m/s (corresponding to 200 rtlW/beam and 2 cm beams) and the cell has a surface area of 75 cm2 and a volume bf 30 cm3.
365
M. Stephens
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5.
ef
al. / Optimizing capture in optical traps
The performance of Dryfilm and Pyrex
A number of experimenters have studied wall coatings that preserve spin polarization of optically pumped alkali atoms [ 15- 191. Since the probability that an atom will depolarize at the surface is directly proportional to the amount of time it spends on the surface [15], the surfaces that work well in optical pumping experiments are surfaces with low adsorption energies. In particular, it has been found that paraffin and an organosilicon polymer, often called "Dryfilm", both work well. We chose to test Dryfilm since we expected it to have a lower vapor pressure than paraffin. We have compared Dryfilm-coated Pyrex to uncoated Pyrex. All of our tests have been with 133Cs(the stable isotope). We measured the adsorption energy of Cs on Dryfilm and Cs on Pyrex in two ways. First, we made a direct measurement by measuring the change in Cs vapor as a function of the temperature of the walls (fig. 9). We cooled the cell to liquid
-
+
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Fig. 9. The change in Cs vapor density as a function of wall temperature; n is the measured density at that temperature, nO is the maximum density (defined by the number of atoms initially released into the cell). The slope of the line is the adsorption energy. The asterisks are Cs on Pyrex and the crosses are Cs on Dryfilm. A leastsquares fit provides &, = 0.44 eV for Cs on Dryfilm and E, = 0.53 eV for Cs on Pyrex.
nitrogen temperatures, let in a small amount of Cs which initially completely adsorbed to the wall, and then monitored the Cs vapor density as the cell warmed up. We measured an adsorptioh energy of 0.44 eV for Cs on Dryfilm and 0.53 eV for Cs on Pyrex. Second, we measured the characteristic fill time of the cell. If initially
366
M . Stephens et al. / Optimizing capture in optical traps
213
no Cs is in the cell, and then a valve with a conductance C is opened between a Cs reservoir and the cell, the Cs density in the cell will fill exponentially with a time constant
The stick time, and therefore the adsorption energy, can be inferred from the fill time. The fill time measured for Dryfilm-coated Pyrex was 0.108 f 0.02 s, which s and Ea 5 0.44 eV. The fill time for the uncoated implies z, = 4.5 x f 4.5 x s Pyrex cell was measured to be 1.1 rfi 0.3 s, which implies 2, = 1.4 x f6x and 0.51 < E , < 0.54 eV, in good agreement with the temperature measurements. The primary source of uncertainty for the fill time measurements was the conductance of the valve connecting the cell to the Cs reservoir. The sticking properties of the Dryfilm-coated Pyrex fulfills the requirements for sticking time; however, the Cs does react with the Dryfilm. The reactions have two effects. The first is to cause the permanent loss of trappable Cs, and the second is to increase the outgassing of the Dryfilm. The first effect is not a major difficulty: the Dryfilm can be “cured” by prolonged (several days) exposure to Cs. After the Torr vapor of Cs for a few days, the reaction Dryfilm has been exposed to a rate is 0.065 cm3/(cm2s). This is a reaction time of around 6 s for a cell with a surface area of 80 cm2 and a volume of 30 em3. The capture time for the same size cell (assuming a capture velocity of 32 m/s, i.e. a Ti : sapphire laser and 2 cm beam diameter) is roughly 3 s. This means one would not lose many atoms to the wall reactions during the trapping process. The second effect, however, is large enough to become the dominant effect on trapping efficiency. After exposure to Cs, the Torr cm3/(cm2s). Because of this relatively Dryfilm outgases at around 2 x large outgassing rate, the maximum number of atoms obtained with a Dryfilmcoated cell will be 5-15% of the number released into the cell (see fig. 8). The outgassing is increased by the reactions of the Cs with the Dryfilm [20]. Some possible reactions are shown in fig. 10 [21]. The Cs is probably reacting with the oxygen in the Dryfilm polymer. Although 5-15% of the atoms may be enough for the experiments planned, we will explore other coatings, for example diamond, Langmuir-Blodgett films, and thiolates [22] in the hope of further improving our trapping efficiency.
6.
Conclusions To most efficiently capture atoms in a MOT, one must increase the capture
rate with high laser power and large trapping beams. Wall-coated vapor cells show promise for minimizing atom- wall interactions. We have measured the adsorption energy of Cs on Dryfilm to be 0.44 eV, This corresponds to a 45 ps stick time. Reactions of the Cs with the Dryfilm increase the outgassing of the Dryfilm so that
367
M . Stephens et al. 1 Optimizing capture in oprical traps
214
CH3
I I 0 I CH3-SiI
CH3-Si-
OH
CH3
73Fig. 10. (a) A properly formed Dryfilm polymer. The S i - 0 backbone is shielded from the Cs by methyl groups. (b, c) Some ways in which th epolymer can form improperly. The Cs can react with the oxygen and hydroxide radicals [21].
loss from the trap due to collisions with background gas is now the dominant factor limiting trapping efficiency. We currently believe that with a Dryfilm-coated cell, we will be able to trap 5-15% of the atoms released into the trapping cell.
Acknowledgetnent This work was supported by the Office of Naval Research and the National Science Foundation.
References 111 K.E.Gibble, S. Kasapi and S. Chu., Opt. Lett. 17(1992)526. [2] b.W. Sesko, T.G. Walker and C.E.Wieman, J. Opt. SOC. Am. B8(1991)946. [3] C. Monroe, W. Swann, H. Robinson and C. Wiemafl, Phys. Rev. Lett. 65(1990)1571.
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C. Wieman, C. Monroe and E. Cornell, 10th I n f . Conf. on Laser Spectroscopy, eds. M. Ducloy, E. Giacobino and G. Camy (World Scientific, Singapore, 1992) p. 77. H. Gould, private communication. S . Freedman and K. Coulter, private communication. D. Grison, B. b u n i s , C. Salomon. J.Y. Courtois and G. Grynberg, Europhys. Lett. 15(1991)149; J.W.R. Tabosa, G. Chen, Z. Hu, R.B. Lee and H.J. Kimble, Phys. Rev. Lett. 66(1991)3245. A. Gallagher and D. Pritchard, Phys. Rev. Lett. 63(1989)957; D. Sesko, T. Walker, C. Monroe, A. Gallagher and C. Wieman, Phys. Rev. Lett. 63(1989)961. K. Lindquist. M. Stephens and C. Wieman, Phys. Rev. A46(1992)4082. E.L. Raab, M. hentiss, A. Cable, S. Chu and D. Pritchard, Phys. Rev. Lett. 59(1987)2631. A.M. Steane, M. Chowdhury and C.J. Foot, J. Opt. SOC. Am. B9(1992)2142. J.H. de Boer, in: The Dynamical Character of Adsorption (Clarendon Press, Oxford, 1978) p. 23. F. Reif, in: Fundmntols of Statistical and Thermal Physics (McCraw-Hill, New York, 1965) p. 273. R.G. Brewer, J. Chem. Phys. 38(1963)3015. M.A.Bouchiat and J. Brossel, Phys. Rev. 147(1966)41. J.C. Camparo, 1. Chem. Phys. 86(1987)1533. L. Young, R.J. Holt, M.C. Green and R.S. Kowalczyk, Nucl. Instr. Meth. B24(1987)963. H. Goldenberg, D. Kleppner and N. Ramsay, Phys. Rev. 123(1961)530. D.R. Swenson and L.W. Anderson, Nucl. Instr. Meth. B29(1988)627. J.C. Camparo, R.P. Frueholz and B. Jaduszliwer, I. Appl. Phys. 62(1987)676. D.W. Sindorf and G.E. Maciel, J. Am. Chem. SOC.105(1983)3767. R.G. Nuzzi, L.H. Dubois and D.L. Allara. J. Am. Chem. SOC.112(1990)558.
VOLUME 72, N U M B E R24
PHYSICAL REVIEW LETTERS
13 J U N E1994
High Collection Efficiency in a Laser Trap M. Stephens and C. Wieman Joinr Instirure f o r Laborarory Astrophysics, Uniilersity of Colorado, Boulder, Colorado 80309-0440 (Received 14 October 1993) We present the results of the first study of the fundamental processes governing collection efficiency of neutral atoms in a vapor cell laser trap. The experimental test of our model of the collection process closely agrees with our predictions based on the measured properties of the laser beams and cell. This study has led to the development of special vapor cell wall coatings. We have demonstrated a collection efficiency of 6% for cesium and predict more than 50% could be achieved under optimum conditions. This work develops the concepts for and demonstrates the feasibility of new experiments using shortlived radioactive isotopes. P A C S numbers: 32.80.Pj. 42.50.Vk
There are many important experiments that would be possible if one had a dense vapor of short-lived radioactive isotopes free from perturbations from the container walls. One obvious example is high precision atomic spectroscopy [I]; however, there are a number of more exotic and interesting experiments such as the study of ,i3 decay from polarized samples 121, the search for electric dipole moments [31, and the investigation of atomic parity nonconservation in ways that cannot be done with available isotopes [41. These exotic experiments require dense, polarized samples. Over the years there have been numerous, often heroic, efforts to achieve this goal with very limited success [I]. A large number of experiments have been carried out on alkali isotopes obtained by depositing radioactive ions onto metal foils that are heated to quickly release the volatile neutral atoms 111. However, because of the small numbers of ions initially available, all of these experiments have been limited by low densities and short interaction times ( 1 msec). Here we demonstrate how laser cooling and trapping can be used to efficiently collect small numbers of atoms in a dense (10"/cm3), easily polarizable sample that is held away from the perturbing effects of walls, making a host of new experiments possible. Atoms can be optically trapped from a slowed atomic beam [51 or from an atomic vapor [6,71. Although there are very little data on the collection efficiency from a slowed beam, measured efficiencies have always been low [81. We have chosen to optimize the collection from a vapor. The collection efficiency depends on the geometry of the vapor cell, the properties of the laser beams, and the properties of the cell walls. We have developed and tested a model of how the collection efficiency varies with these properties. Efficient collection of small samples of atoms requires several changes from the usual vapor cell trap. In a standard cell trap [61, atoms from the low-velocity tail of the Maxwell-Boltzman distribution are collected from a dilute vapor that is constantly replenished from a reservoir. Nonalkali background gases are continuously pumped away to prevent them from knocking the trapped atoms
out of the trap. Some of the alkali vapor is also pumped away or sticks to the walls, but i t is replenished from the reservoir. However, when collecting atoms from a limited source, it is necessary to prevent their loss to the pump or to the walls. The cell walls must be a material that the atoms will not stick to, and it is necessary to stop pumping on the cell while trapping the atoms. Without continuous pumping, the density of background gases in the cell builds up over time. This buildup and the corresponding increase in the loss rate from the trap due to collisions with the background gases must be minimized. We have analyzed the situation of suddenly introducing N o atoms into a cell and attempting to trap as many as possible. The number of atoms which are trapped as a function of time depends on the capture rate, R, and the loss rate, 11.5 [71, according to
where N is the number of trapped atoms. Unlike in Ref. 171, R and I/r are time dependent. The capture rate is given by
R(0
---
1
1':
?rL2n(t), (2) 4& .dl where L'th and n are the thermal velocity and the density, respectively, of untrapped atoms, and L', is the maximum velocity an atom can have and be captured. This maximum velocity depends primarily on the laser beam Gaussian width, L, and intensity, I, and can be calculated
171. The untrapped alkali density, unlike the normal vapor pressure, depends on the interactions between the atoms and the walls: (3)
Here A is the surface area of the cell, V is the volume of the cell, r is the effective pumping speed of the wall per cm2 due to permanent chemical reactions between the atoms and the wall, and rs-IO-'Zexp(E,/kT) sec is the
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average time an atom sticks to the cell wall, where E, is the wall-alkali binding energy “91. The loss rate of atoms from the trap depends on the pressure P (in torr). 1/T
=3.3x 10‘6PabL’h,
(4) c
where ‘Tb is the cross section for the nonalkali background gas to knock an atom from the trap, and I‘b is the thermal velocity of the gas. We have measured C s h [ ’ b - 1 0 - ~ crn3/sec for the background gases produced in o u r coated cell. We can neglect collisional loss due to the alkali background because it has a much lower density. The nonalkali pressure will initially be P o = Q A / S , where S is the pumping speed a t the cell and Q is the outgassing of the walls. After the pump is closed off, P rises linearly with time, P = P o + Q A t / V. With Eqs. (1)-(4) we can calculate the number of trapped atoms as a function of time after closing off the pump and releasing the atoms into the cell. The collection efficiency is the ratio of the maximum number of atoms trapped to the number released into the cell. One can see from Eq. ( 3 ) that preventing the sticking time, r S , from impacting the capture rate requires r,<<4V/ AL’th, which is -50 psec for our cell. The sticking time for Cs on Pyrex at room temperature is -2 msec [IOl and is much longer on metals [ I 11. We have investigated several different kinds of wall coatings to reduce T, I l O l . There have been a number of alkali spin relaxation experiments performed that require surfaces with small adsorption energies for alkalis [121. Our requirements for coatings are somewhat different- we can tolerate sticking times many orders of magnitude longer but have much more stringent requirements on the coating’s outgassing and the reaction rate of the alkali with the coating. However, we have found that a coating used in optical pumping experiments also works for efficient trapping. It is a silicon-based hydrocarbon polymer generically called “dry film” [ 1 21. Several different methods for making dry-film coatings have been used (for example, see [131). We tested four methods and found that E, was virtually the same, but they had very different reaction and outgassing properties, the latter of which is the critical property for this application. We believe these variations are due to different polymer lengths and different fractions of incorrectly formed polymers [141. We found that a form of dryfilm on Pyrex made with octadecyltrichlorosilane (OTS) [ I 51 was the best coating. After one day of exposure to Cs, we measured E, =0.38 eV, r=0.08 cm3/seccm2, and an outgassing rate of - 2 . 0 ~ torrcm3/seccm2 [IOI (a factor of -7 greater than 304 stainless steel vacuum baked at 250°C [161). While it is possible to coat stainless steel, the reaction rates and outgassing measured for dry film on stainless were typically 5 to 10 times greater than dry film on Pyrex. We have used our measurement of these and the other relevant parameters to predict the collection efficiency for 3788
FIG. I . The cell is Pyrex with six windows attached to ,I c s lindrical tube. has a surface area of 15.5 c m ’ , and a volume of 75 c m ’
a coated cell a n d have checked the model using 3 trap of stable Cs. A schematic of the apparatus is shown in Fig. I . A magneto-optic trap was established in one end of a coated glass cell. Two glass valves made of a ground ball and socket joint connect the cell on one side to a Cs reser-
voir and on the other side to an I 1 000 cm3/sec ion pump. The glass valves can be opened with magnets and have conductances of roughly 400 cm3/sec. The Pyrex cell and the valves were coated with OTS dry film. There are several reasons one would not want to use such valves if the primary goal were collecting atoms: They have low conductance, they leak, and they introduce substantial undesirable volume and surface area. However, our primary goal here was to test our calculations, and so these flaws were offset by the virtues of opening and closing quickly and providing a coated surface which matched the cell. A vapor-cell magneto-optical trap has been explained in detail elsewhere [61. The beam from a Ti:% laser was expanded to 1.8 cm diameter and split into three beams that were sent through the cell along orthogonal axes and retroreflected with the appropriate polarizations. The laser frequency was tuned below the 6S1/2.F=4 6P3/2, F = 5 transition. The detuning of the laser (15-25 M H z ) and the magnetic field gradient (10-15 G/cm) were set to optimize the capture rate at each beam intensity. Light from a diode laser pumped the atoms out of the F = 3 ground state. A set of Maxwell coils provided the required quadrupolar magnetic field. To measure the trap efficiency we released lo9 atoms into the cell and determined the number of atoms that were trapped from that sample. It is not easy to release a small, known number of atoms into a cell. We used t h e trap itself to prepare the initial sample. The procedure is illustrated in Fig. 2. First about lo9 atoms were trapped from a dilute Cs vapor, then we pumped away the background vapor and closed the valve to the pump so that no other atoms could enter or leave the cell. Next we released the atoms by blocking the laser and turning off the magnetic field, which allowed the atoms to fall to the bottom of the cell and rethermalize. This gave us our initial sample which we then collected by turning the mag-
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371 V O L U M E 12, N U M B E R
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‘. i
.1
f~
“r-i C
S t‘,:
,?,
1: I
,I
FIG. 2. The trap fluorescence as a function of time, in arbitrary units, during a measurement sequence. Initially the valve to the ion pump is open and the valve to the Cs reservoir is closed. At 0 sec the valve to the Cs reservoir is opened, a Cs vapor enters the cell, and the trap fills. At 4.8 sec the valve to the Cs is closed. The fluorescence drops because the background fluorescence changes as the background vapor is pumped away and because atoms are lost from the trap during this time. At 5.8 sec the valve to the ion pump is closed, and 100 msec later the laser is blocked and the magnetic field turned off. After I50 msec the laser is unblocked and the magnetic field is turned back on to begin the capture process. At first the trap fills, then the number of trapped atoms drops as the background gases continue to build up and the trap loss increases.
netic field back on and unblocking the laser. To determine the collection efficiency we monitored the trapped atom fluorescence as the trap refilled and compared it to that of the trap just before we blocked the laser. A leak in the valve between the Cs reservoir and the trapping chamber gave a substantial background signal. This background was measured as a function of time by repeating the above process with the magnetic field initially turned off (no initial trap) until after the laser had been blocked. The time-dependent residual signal was then subtracted from the time-dependent trap signal. Figure 3 shows the signal, with the background subtracted. The number of atoms recaptured rises rapidly, then levels off at a maximum of 6% after 0.9 sec. At this time the non-Cs background pressure has risen to about 5x torr and the collisional loss rate from the trap equals the capture rate. I n an isotope collection experiment, one would reopen the valve to the pump at this point to remove the background gases and preserve this trapped sample. We left the valve closed to study how the trapped sample decreases as the pressure continues to rise. The solid line shows the predictions of our model with no adjustable parameters. Clearly it fits well. We took measurements with the laser blocked for longer times and saw no significant change, indicating the discrepancy at very short times is not due to incomplete rethermalization of the atoms. We measured the collection efficiency for several different intensities (16, 30, 35, and 40 mW/cm2) per beam. The capture efficiency decreased with intensity exactly as predicted by our model. The maximum collection
LL
0
0
1 3 J U N E 1994
L----
----
r
0
1
2
t 7
d
t i r e (sec) FIG. 3. An expanded view of the trap fluorescence after the initial trap has been released with the background subtracted of. This is the fraction of the total sample of atoms collected as a function of time. The smooth solid line represents the predicted collection efficiency for the independently measured parameters of the system: L - 1 . 8 cm, 1-40 mW/cm2, 7,=4 psec, r-0.086 cm’/seccm2, Q - 2 . 2 X lo-* torrcm’/seccm2, V = 7 5 cm’, A - I 5 5 cm’, P 0 - 2 . 0 x torr. The dashed line is the prediction for the collection efficiency if r,=O and r=O. The dash-dotted line is the prediction for an outgassing, Q, a factor of4 smaller ( 5 . 5 ~ torr cm3/seccm2).
efficiency was at 40 mW/cm2. At higher intensities our model predicts higher collection efficiencies. Actually, for traps with more than lo5 atoms, there is an optimum intensity beyond which the collection efficiency decreases, because intensity-dependent intratrap collisions [ I 71, which are not included in our model, increase the loss rate. Figure 3 also illustrates how the efficiency depends on r,, r, and Q. One can see that with this coating only the outgassing is important in limiting the collection efficiency of the cell. Cooling the cell walls would decrease the outgassing and also reduce L’th of the untrapped atoms and therefore increase R [Eq. (2)l. These gains are balanced by the temperature dependence of r , ; however, it would be beneficial to decrease the temperature until rs~lthA/4V- 1 [Eq. (3)1. Optimizing the cell geometry would also improve the collection efficiency. Ideally the volume of the trap region (where all six trapping beams overlap) should be equal to the volume of the cell. Our cell, which was built to study wall coatings, has a volume (75 cm3) substantially greater than the volume of the trap (4.5 cm’). The atoms are out of the trapping region a large fraction of the time, so the collection efficiency is lower. We predict a collection efficiency of 50% for a cell with the same volume as the trapping region, using this coating and 40 mW/cmZ per beam. At higher laser powers, the collection efficiency could be increased by using larger laser beams. Figure 4 shows how we expect the collection efficiency to scale with beam diameter. We predict smaller, but still significant, collection efficiencies for Na and Li, as shown in Fig. 4. We have shown that efficient collection from a very
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beom diameter ( c m ) FIG. 4. Predictions for the collection efficiency as a function of laser beam diameter, L , with a close to optimum cell geometry. The volume of the cell changes with L and is always equal to L The area is 6 L '. The solid lines are predictions for Cs. Na, and Li with a constant intensity of 40 mW/cm2 per beam. The dashed lines are predictions for Cs, Na, and Li with a constant power of 100 mW per beam. We approximate E , for N a and Li by multiplying E. for Cs by the ratio of the respective atomic polarizabilities. We have neglected intensity dependent losses, important at high intensities, as these depend on many factors not discussed here.
'.
small number of atoms is possible by laser trapping in a wall-coated vapor cell. We have measured a 6% capture efficiency and have developed a model that agrees well with our experimental results. The model predicts collection efficiencies greater than 50% for a dry-film coated cell with the optimum geometry. This demonstration opens the door to many new experiments that involve radioactive isotopes. We would like to thank Ruth Rhodes and Kent Lindquist for their help with the experiment. We also thank Harvey Gould. Linda Young, Steve Sibener, Dave Swenson, and Abraham Ulman for helpful suggestions. This work was supported by the National Science Foundation and the Office of Naval Research.
3790
[I] E. W. Otten, Invesrigarion o/ Short-Lired Isotopes b j , Laser Spectroscopy (Harwood Academic Press. Char. 1989), p. 16. 121 Stuart Freedman, LBL (private communication). [31 Harvey Could. LBL (private communication). 141 C. Wieman, C. Monroe, and E. Cornell, in Proceedings u / the 10th Inrernational Conference on Laser Spectroscopj,. edited by M. Ducloy, E. Giacobino. and G. Camy (World Scientific, Singapore, 1992). p. 77. 151 E. J . Raab er al., Phys. Rev. Lett. 59, 2631 (1987). [61 C. Monroe, W. Swann, H. Robinson, and C. Wieman. Phys. Rev. Lett. 65, 1571 (1990). [71 K . Lindquist, M . Stephens, and C. Wieman, Phys. Rev. A 46,4082 (1992). I81 F. Shimizu, K. Shimizu, and H. Takuma, Opt. Lett. 16. 339 (1991), have reported simulations that predict very high capture probabilities from a slowed atomic beam, but the high capture efficiency has not been demonstrated experimentally. We believe these simulations seriously overestimate the efficiency by considering only atoms passing exactly through the trap center. 191 See, for example, J . H. de Boer, The Dynamical Character of Adsorpiion (Clarendon Press, Oxford, 1978). [ l o ] We will explain in detail how we measured stick time, reaction rate, and outgassing for a variety of materials in a future paper. II I] M. D. Scheer, R. Klein, and J . D. McKinley, J . Cheni. Phys. 55, 3577 (1971). [I21 J . Vanier and C . Audoin, in The Quanfum Physics of Atomic Frequency Srandards (Adam Hilger, Bristol, 1989). Vol. 1 , Chap. 3. [I31 D. R. Swenson and L. W . Anderson, Nucl. Instrum. Methods Phys. Res., Sect. B 29, 627 (1988). [I41 J . C. Camparo, R. P. Frueholz, and B. Jaduszliwer. J . Appl. Phys. 62, 676 ( I 987). [ I 51 A. Ulman, An Iniroduction 10 Uliraihin Organic Films-From Langmuir-Blodgett to Self Assembly (Academic, Boston, 1991), p. 294. [I61 J . F. O H a n l o n , A User's Guide to Vacuum Technology (John Wiley and Sons, New York, 1989). p. 444. 1171 D. Sesko et al., Phys. Rev. Lett. 63, 961 (1989).
Study of wall coatings for vapor-cell laser traps M. Stephens, R. Rhodes, and C. Wieman Joint Institute for Laboratory Aswopliysic.s and P1rysic.r Depurtment, Univrrsity of Colorado, Bouldes Colorado 80309-0440
(Received 24 November 1993; accepted for publication 7 June 1994) Efficient collection of atoms into a vapor-cell laser trap requires a special wall material for the cell that minimizes the interactions between the vapor and the wall. Tests of several different wall coatings and materials are reported, and measurements of adsorption enrrgies, outgassing, and chemical reaction rates between the alkali vapor and the walls are described. It is demonstrated that each of these parameters affects the collection efficiency.
1. INTRODUCTION
There are many important experiments that would be possible if one had a dense vapor of short-lived radioactive isotopes free from perturbations due to the container walls. One obvious example is high-precision atomic spectroscopy;' however, there are a number of more exotic and interesting experiments such as the study of p decay from polarized samples,2 the search for electric dipole moments,3 and the investigation of atomic parity noncon~ervation~ that cannot be done with available isotopes. These experiments investigate fundamental interactions such as T violation and the electroweak interaction. They require dense, polarized samples. Because of the small numbers of short-lived atoms initially available, previous experiments have been limited by low densities and short interaction times.' Laser cooling and trapping" is an efficient way to collect small numbers of atoms into a dense (10"/cm'), easily polarizable sample that is held away from the perturbing effects of the walls. Recent efforts have demonstrated that laser cooling and trapping of radioactive isotopes is possible.'.' In this article we study the specialized technical aspects of wall properties relevant to trapping small numbers of neutral atoms. The major technical difficulty in efficient collection is choosing a trapping cell surface that minimizes the loss of trappable atoms to the wall-vapor interactions and also does not compromise the ultrahigh-vacuum environment. Atoms are lost to the wall through both physisorption (the atom sticks to the wall for some characteristic time and then returns to the vapor) and chemisorption (the atom chemically reacts with the wall and is permanently removed from the vapor).' These processes, particularly physisorption, have also been a major difficulty for many resonance cell experiments involving both stable and radioactive isotopes (see, for example, Refs. 8-12) and in optical piston experiments designed to demonstrate light-induced drift (see, for example Refs. 13 and 14). While the solutions developed by these previous experiments guided us in our work we found that our requirements are somewhat different. In this article we explain our requirements, describe our measurements, and discuss the results. In Sec. I1 we model the trapping process and use this model to extract the properties of a cell surface necessary to maximize trapping efticiency. In Sec. 111 we explain our measurements of the adsorption energy and sticking time of alkalis for the wall J. Appl.
Phys. 76 (6), 15 September 1994
coatings and explain what kind of surfaces have a low adsorption energy. In Sec. IV we present our measurements of the outgassing of the coatings and the reaction rates of the coatings with the alkali vapor, and in Sec. V we discuss other issues pertaining to the wall coatings that are important to the trapping process.
II. TRAPPING PROCESS IN A COATED CELL
The trapping process can be modeled by balancing a time-dependent capture rate with a time-dependent loss rate. We can use these equations to calculate the expected capture efficiency for a trap in a cell with particular wall characteristics and geometry and with particular trapping laser-beam characteristics (such as intensity and diameter). Using this model one can see how changes in these characteristics affect the capture efficiency. In a standard vapor-cell trap, atoms from the lowvelocity tail of the Maxwell-Boltzmann distribution are collected from a dilute vapor that is constantly replenished from a re~ervoir.'~ Nonalkali background gases are continuously pumped away to prevent them from knocking the trapped atoms out of the trap. Much of the alkali vapor is also pumped away, chemically reacts with the walls, or sticks to the walls, but the lost atoms are replenished from the reservoir. In an experiment that involves trapping rare, radioactive isotopes we cannot expect to have a reservoir supplying a large, steady flux of atoms. Instead, we will release a short burst of atoms into the cell, either by evaporating the atoms from a hot metal foil or by briefly opening the system to an isotope beam. Thus, we face the problem of having only a small, fixed number of atoms, all of which we want to trap. So, we must minimize the loss due to interactions with the walls. To do this we must construct the cell from a material that does not interact strongly with the alkali vapor but is still compatible with an ultrahigh-vacuum environment. The situation of suddenly introducing N , , atoms into a cell and trapping as many as possible can be modeled in the following way.'6 The number of atoms that are trapped as a function of time depends on the capture rate R and the loss rate 1/r,
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374 TABLE I. Dependence of capture rate and loss rate depend on each vapor-cell trap parameter. Contribution to capture rate R
Parameter Laser properties Beam diameter L
Beam intensity I
Surface properties Adsorption energy E , Reaction ratc r Outgassing Q Cell properties Characteristic size D Pumping speed S
Contribution to trap loss rate
R-L'
constant intensity, large L R - L con~tantpower, large 1, R - 1 constant L , low I (saturates at high I)
negligible
negligible (ignoring intensity dependent trap loss)
''
none
none
none l/r-Qt
r,= 10 exp(FJk7) R- l/[l+(r5Au,,,/4V)/ R-exp( - t A ri V )
R - liD' exp( - r r / D )
I f r-D'+tlD 1/r1,=~,-1/S
none
where N is the number of trapped atoms. The capture rate R isI5
where u t h and n are the thermal velocity and the density, respectively, of untrapped atoms, and u , is the maximum velocity an atom can have and still be captured. This maximum velocity depends primarily on the laser beam size L and its intensity 1.17 The untrapped alkali density, unlike the normal vapor pressure, depends on the interactions between the atoms and the walls, (3) Here A is the surface area of the cell, V is the volume of the cell, r is the effective pumping speed of the wall per cm' due to permanent chemical reactions between the atoms and the wall, and T~ is the average time an atom sticks to the cell wall. The second term in the denominator represents the average number of atoms stuck to the wall at a given time.I8 The loss rate of atoms from the trap depends on the pressure P (in Torr),
The collection efficiency can be calculated by solving Eqs. (1)-(4) for the number of trapped atoms as a function of time. The maximum collection efficiency will be the maximum number of trapped atoms divided by the initial number of atoms released. To completely characterize the capture process, seven parameters must be measured: the laser-beam size and intensity, the characteristic cell dimension, the pumping speed at the cell, and the wall coating's adsorption energy, outgassing, and reaction rate. Table I shows these parameters and how each affects the capture rate and the loss rate. These equations can be used to develop requirements for the cell surface. One can see from Eqs. (2) and (3) that the capture rate is significantly reduced when ~ , > 4 V / v , 4 (-50 ps for typical cell parameters). Equation (4) shows that the loss rate increases with the outgassing Q of the wall. Equation (3) shows that atoms lost to chemical reactions cause a significant reduction in the density of trappable atoms when r becomes larger than V / t , , ~ , where t,,, is the time at which the maximum number of atoms has been captured. To determine the requirements for the coating we numerically integrate Eqs. (1)-(4). Figure 1 shows the capture efficiency as a function of time and how it changes with different parameters. One can see that for realistic values of the cell and laser parameters, to obtain significant capture efficiencies, one would like a ~ , C 5 0 ps, r G 0 . 1 cm3/scm2, and Q C 10 Torr cm3/s cm'. If we compare these constraints to those of previous resonance cell experiments that have used special wall coatings we find that the criteria are different. Most resonance cell experiments require much shorter sticking times, T~-10-'" s or shorter.' The outgassing and reaction rate requirements are much more stringent, however. Resonance cell experiments are typically done at pTorr pressures and with large reservoirs of atoms, so that outgassing and atom loss rates are unimportant. A trap, however, must be established in ultrahigh vacuum Torr). Furthermore, with a fixed number of atoms in the cell, any loss to the walls due to chemical reactions is important.
-'
where (T,, is the cross section for the nonalkali background gas to knock an atom from the trap, and u b is the thermal velocity of the gas. We measured ~ ~ v , - l O -cm3/s ~ by measuring the pressure of the background gases and the fill rate of the trap. We can neglect collisional loss due to the alkali background because its density will be a factor of 10100 lower than the nonalkali background. The nonalkali pressure will initially be P,=QAIS, where S is the pumping speed at the cell and Q is the outgassing of the walls. After the isotopes are released, the cell is closed off from the pump to prevent loss of isotopes to the pump, and P rises linearly with time, P= Po + Q A t / V . 3480
J. Appl. Phys., Vol. 76, No. 6, 15 September 1994
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40 7/kt
FIG. 1. The collection efficiency as a function of time for various parameters. We plot the percentage of the total sample of atoms trapped as a function of time. The smooth solid line represents the predicted collection efficiency for the independently measured parameters of the system: L = 1.8 cm', 1 = 4 0 mWicm', 7,=2 ms, r=0.086 cm'isicm', Q = 2 . 2 X l O - ' Torrcm3/s/cm2, V = 7 5 cm', A = 1 5 5 cm', P , , = 2 . 0 X l O - ' Torr. The dashed line is the predicted collection efficiency if T~ = 4 ,us. The dasheddotted line is the prediction for T ~ = O and r = O . The dashed-dotted-dotted line is the prediction for an outgassing Q, a factor of 10 smaller (2.2X10-' Torr cm3/slcm').
Because of our different requirements it was necessary for us to make more quantitative measurements of the outgassing and reaction rates of the coatings than had been made before. We also measured the adsorption energies of Cs on these coatings rather than relying on previous measurements because our cell geometry is somewhat more complicated than a typical resonance cell (our cell has windows and valves attached to it). 111. ADSORPTION ENERGY AND STICKING TIME MEASUREMENTS Physisorption is the process in which an atom collides with a surface, sticks to it for a characteristic period of time r, , and then returns to the vapor. The dominant long-range force that attracts the atom to the wall is the van der Waals force, which is proportional to ( ~ - l ) / ( e + l )D', where 6 is the dielectric constant of the wall and D is the electric dipole operator of the atom.'9 The polarizable atom is attracted to the wall by the image charge formed in the surface of the wall. Choosing a surface that minimizes the van der Waals force (i.e., a surface that is not very polarizable or, in other words, has a low dielectric constant) will make it less likely that the atom will remain on the surface for a substantial period of time. Therefore, we chose to test materials for the cell walls that have a low dielectric constant. Physisorption is characterized by an adsorption energy E , that is related to the sticking time T,, by the following relation: rs= r o e E a l k T
(5)
where ~ , - 1 0 - ' ~s . We ~ employed two methods for studying the atom-wall interaction: One determines E , directly and the other measures 7,. J . Appl. Phys., Vol. 76, No. 6, 15 September 1 9 9 4
45 (eV
-'
50
)
FIG. 2. The ln(n,,/ng,,-l: as a function of l i k T plotted on semilog scale for the different materials and coatings. The slope of each line, determined by least-squares fit, gives the adsorption energy characterizing the interaction between the vapor atoms and the wall surface. The asterisks represent Pyrex, the plus signs represent dryfilm, and the diamonds represent sapphire. The adsorption energies found werc 0.4Oi0.03 eV for dryfilm on Pyrex, 0.4320.1 CV for sapphire, and 0.53+0.03 eV for bare Pyrex.
To measure an adsorption energy we filled a cell made of, or coated with, the material we wished to test with a Cs vapor and measured the change in vapor density with wall temperature. The total number of atoms no in a closed cell is equal to the number in the vapor plus the average number stuck to the wall at a given time,
where u,, is the thermal velocity of the atoms, A is the surface area of the cell, and V is the volume of the cell. Therefore, by measuring the change in the number of Cs atoms in the vapor as we changed the wall temperature we were able to determine the adsorption energy. Because many of the wall coatings we tested chemically reacted with the Cs at room temperature, and because the reaction rate increased at elevated temperatures [Eq. (6) is only valid when the total number of Cs atoms is the cell is constant], we measured the change in vapor density with wall temperature by cooling the cell rather than heating it. We monitored the Cs vapor density in the cell by measuring the absorption of a diode laser beam sent through the cell as it was swept across the Cs 6S,,,, F = 4 + 6 P 3 / , transition. We measured E , for Pyrex, a number of types of dryfilm (a silicon-based hydrocarbon polymer, see the Appendix), and sapphire. Typical data are shown in Fig. 2. A leastsquares fit of the data gave us an adsorption energy for bare Pyrex of 0.53-CO.03 eV, corresponding to a sticking time of 1.6 ms at room temperature. The dryfilm adsorption energies were similar for all types of dryfilm tested. For the best dryfilm [octadecyltricholorosilane (OTS) on Pyrex] our tests yielded an adsorption energy of 0.40L0.03 eV, which implies a sticking time of 9 ps at room temperature. We measured an adsorption energy of 0.43+0.1 eV for sapphire, which implies a sticking time of 21 ps at 300 K. This is smaller than the adsorption energy of 0.75t0.25 e V for Na on sapphire measured by Bonch-Bruevich, Maksimov, and Khromov.'" Stephens, Rhodes, and Wieman
3481
376 .% x
1-7
1.20
ted intensity as a function of time was recorded, and a fill time was measured. Our measurements are limited for rapid fill times by the fact that we could not open or close the valve in less than about 0.1 s. For a 50 cm' uncoated Pyrex cell, the fill time was 1.1 t0.3 s, which corresponds to a sticking time of 1.4k0.6 ms. This is in good agreement with the sticking time at room temperature calculated from our adsorption energy measurement for Pyrex. It is also in reasonable agreement with the measurement for Na on glass at 400 "C of 8.3+2.0
cn
c
a,
0.96
U U a, N .0 ~
0.72 0.48
EL 0.24 0
c
0.00 0
1
2
3 4 time (sec)
5
x10--5S . l 4
FIG. 3. The calculated dependence of normalized Cs density inside the ccll as a function of time for typical test cell parameters. C = 6 2 5 cm3/s, V = 5 0 cm3, and A = 63 cm'. The solid line shows the curve expected if the Cs docs not stick to the wall at all (~1=0.08 s). The dashed line shows how the cell l similar to fills if the atoms stick to the walls for a time of 2 ms ( ~ 2 = l . s), our measurements for Pyrex.
We developed a second method of studying the surface interaction which directly measured the amount of time an atom sticks to the walls. This allowed us to test coatings on cell geometries that were difficult to test by heating and cooling and provided a check on the accuracy of our adsorption energy measurements. The second method provides the sticking time through a measurement of the time it takes for the test cell to fill with Cs. By measuring the density as a function of time as the cell fills, we can find the characteristic fill time, which is related to T ~ A. Cs reservoir and an evacuated cell are connected by a valve with conductance C. Initially, the valve is closed. The reservoir is at constant pressure P , , and the cell is at pressure P,@P,. When the valve is opened, the Cs flow from the reservoir into the cell can be written" (7)
The pressure in the cell is related to the number of atoms present in the vapor at any time by
P2=
kT
y,
where the second term in the parentheses is the average number of atoms stuck to the surface at a given time." Using Eq. (8) to solve Eq. (7) provides an expression for P, as a function of time, p 2 =p , ( 1- e C N v + u , h A 7,141 1, (9) assuming the reservoir pressure P I stays constant. We can then relate 7s to the exponential fill time Tfill (see Fig. 3),
Once again, the Cs density was measured by optical absorption through the cell. For these measurements the laser was locked to the Cs transition line. As the valve between the Cs reservoir and the evacuated cell was opened, the transmit3482
J. Appl. Phys., Vol. 76, No. 6, 15 September 1994
We measured fill times for a Pyrex cell coated with dryfilm and a stainless-steel cell coated with dryfilm. For dryfilm on Pyrex, r,<35 ps. This is consistent with our adsorption energy measurements (the fill time measurement was limited by the speed with which we were able to open the valve). The sticking time for dryfilm-coated metal was the same as for dryfilm on Pyrex. The sticking time of Cs on sapphire was measured in a differently. We coated a metal system with dryfilm and placed eight 1.3-cm-diam flat sapphire pieces inside the system. The surface area of the sapphire was 21 cm2, while the surface area of the dryfilm was 57.4 cm'. In this hybrid system Ttill has two terms, one that depends on the dryfilm properties and one that depends on the sapphire properties. We obtained an upper limit for the sticking time of 1.6X10-4 s which is consistent with our adsorption energy measurement. Table I1 displays our data for sticking times and adsorption energies of Cs on Pyrex, dryfilm, and sapphire. We attempted to measure the fill time of a paraffincoated Pyrex bulb but had limited success. The coating was made by placing a lump of distilled paraffin in the bottom of the bulb, attaching the bulb to the vacuum system, and then heating the bulb (with the valve to the vacuum system closed) to 400 K. At this temperature, paraffin is in a vapor state and coats the cell. It proved difficult to obtain an even coating with this method. We measured an adsorption energy of 0.53 eV (this corresponds to a stick time of 1.6 ms) with the best coating we were able to obtain. This is much higher than has been measured in other experiment^;^"'^^^^'^ reasons for the high adsorption energy we measured are discussed in more detail in Sec. V. We chose not to pursue paraffin because its vapor pressure is too high for operation at room temperature in ultrahigh vacuum. We discuss in Sec. V how it might work at a lower temperature. We did not test Teflon or any bare metals. Measurements by previous researchers have shown that the reaction rate of Teflon with an alkali vapor is very We did not test bare metal since we expect the adsorption energy will always be very high because the metal can easily form an image charge that will attract the atom to the surface through the van der Wdals force. The adsorption energy of Cs on Mo, for example, has been measured to be 2.47 eV.24The expectation of a high adsorption energy on metals is supported by our observations that Cs fills uncoated glass and stainless-steel systems very slowly and pumps away very slowly. Stephens, Rhodes, and Wiernan
377 TABLE 11. Measurements of adsorption energy and sticking time for C s on dryfilm, Pyrex, and sapphire.a Adsorption energy E , (from temperature measurements)
Surface
Stick time r, (from fill time measurements)
Comments ~
Cs on dryfilm
0.40?0.03 e V
Cs on Pyrex Cs on sapphire
0.5320.03 eV 0.43*0.1 c v
< 35
p.5
OTS on Pyrex r, meawremcnt I$ limited by the time took to open the valve
it
1400 ps
<160
FS
r, measurement is limited by the ratio of the surface area of sapphire to the surface area of dryfilm in the tcst chamber
'The adsorption energies were determined by hcating and cooling thc vapor-fillcd cells, while the sticking times listed were determincd by measuring thc cell fill times at room tempcratiire. In every case the sticking timcs determincd from the fill time measuremcnts are consistent with the sticking time calculated from the measured adsorption energies.
IV. REACTION RATE AND OUTGASSING MEASUREMENTS
While finding a surface with a small 7, is important for efficient collection of atoms in an optical trap, both the outgassing of the surface and the reaction rate of the surface with the alkali gas are also crucial. The reaction rate is responsible for a permanent loss of trappable atoms. The outgassing increases the trap loss rate. While it is important for these properties to be minimized individually, we found a correlation between reaction rate and outgassing rate coatings with high reaction rates had high outgassing rates. A. Chemical reactions
To measure the reaction rate, we filled a coated cell with Cs then closed the Cs inlet and measured the decay of the Cs fluorescence as a function of time (see Fig. 4). The decay
x
.< 1.20 cn a,
I
0.96
-0
0.72 a .p -
0.48
U
E 0.24 c
0.00 0
2
4
6
810
time ( s e c ) FIG. 4. The calculated dependence of normalized Cs density as a function of time after closing the cell off from the Cs reservoir. The solid line shows that if the cell walls were not reacting with the Cs vapor the density would not change. Thc dashed line shows how the density changes it the Cs reacts with the walls. The I / E decay time T is related to the reaction rate r = V / (r A ) . J. Appl. Phys., Vol. 76, No. 6, 15 September 1994
rate of the fluorescence can be interpreted as an effective pumping speed of the wall in units of volislarea. To make sure the Cs was permanently removed from the system we heated the walls and found that the Cs did not reappear. This indicates that the Cs was permanently bonded to the walls rather than physisorbed. In fact, the loss rate increased with temperature, a further indication that these were chemical reactions rather than physisorption. Table I11 shows the measured reaction rates at room temperature for all of the coatings and materials we tested. It also includes some reaction rates at higher temperatures. We found that every surface we tested reacted with the Cs vapor, whether the surface was coated Pyrex, coated stainless steel, bare Pyrex, or sapphire. Every dryfilm coating we tested reacts very rapidly with the Cs vapor initially, then it "cures" to a constant reaction rate. This is consistent with other researchers' experience of an initially low alkali vapor pressure that over a period of a few hours to a few days The curing time for climbs to the expected press~re."~.'~ most of the dryfilms was about 1 day at room temperature when exposed to a Cs flux of approximately l O I 3 atoms/s cm'. The best dryfilm (OTS), however, cured within a few hours, as did paraffin and sapphire. An uncoated Pyrex cell took 1-2 days to cure with the same Cs flux. There are several issues raised in Table 111 that should be discussed. First, it was necessary to clean the sapphire tube in chromic acid at 65 "C before we were able reduce the reaction rate sufficiently to maintain a Cs vapor in the sapphire cell. Second, different types of dryfilm have very different reaction rates. Third, dryfilm on metal always has a higher reaction rate than dryfilm on glass. We believe that the differences in the reaction rates of the various dryfilms is caused by differences in the ratios of correctly formed to incorrectly formed polymers, and by the numbers of methyl groups in a correctly formed polymer that can shield the SiO bond in the backbone of the polymer from Stephens, Rhodes, and Wiernan
3483
378 TABLE 111. The outgassing and reaction rates of all of the surfaces tested.&
Surface Pyrex
Sapphire
Reaction rate r (cm’is cm’)
Outgassing Q (Torr cm’is cm’)
0.03 at 25 “C 0.12 at 37 “C 0.43 at 50 “C 9.6 at 98 “C
il0-H
negligible
... ...
-10-6
...
on Pyrex on stainless steel
0.06 >4.0
SC-I7
on Pyrex on stainless steel
0.025 5.0
Oxysilanes
on Pyrex on stainless steel
0.1 0.8
<4x10-’ 3x10-’
OTS
on Pyrex
negligible 0.086
<4X10-’ 2.2X
DCDMS
Comments
<4X10-’ SXlO-’
The increase in outgassing is probably caused by the reactions with the Cs vapor and is not a property of the glass itself Sapphire must be cleaned with chromic acid first Measurenicnt of r limited by time to close valve
2 x 1 ~ ’
...
In simple test cell In more complicated trapping cell
*In some cases the outgassing was not measured or only an upper limit could be determined.
the Cs vapor. Figure 5 shows a correctly formed short polymer and several possible misformed polymer structures that expose oxygen to the Cs The worst reaction rate was for dryfilm made from dichlorodimethylsilane which has only two methyl groups. In contrast, the best dryfilm was made from OTS, which has 18 methyl groups. Intermediate reaction rates were obtained with dryfilm made from methoxysilanes, which is a thicker coating and has a complicated
m3-si - 0I
sli I
-
0-
7 3 si-013 I
W 3 - S i - CH
I
0
I
CH3-Si-
\ / Si
o/
I b)
CH3
W3 CH3
‘0
C)
FIG. 5. (a) A correctly formed short dryfilm polymer and (b), (c) two possible structures for misformed polymers and from Ref. 26. In (a) the methyl groups shield the oxygen from attack by the alkali, while in (b) and (c) the oxygen is exposed to the alkali atom. 3484
J. Appl. Phys., Vol. 76, No. 6, 15 September 1994
cross-linked structure,26 and the coatings made from SC-77, which is a mixture of short chains. The chemical reactions between the Cs and the dryfilm slowly degrade the dryfilm. Although we made few longterm (>1 week) tests of the dryfilm, one Pyrex cell coated with SC-77 was exposed to -lo-’ Torr of Cs for 3 months. At the end of that time, 5;. for that coating had increased by a factor of 5. There was not a significant change in reaction rate or outgassing. An OTS coated cell was exposed to Torr of Cs for 2 months and no change in 7, was measured. This is consistent with the expectation that coatings with smaller reaction rates will degrade more slowly. We believe the reaction rate of the dryfilm-coated steel is always much higher than the dryfilm-coated Pyrex because it is more difficult for the dryfilm to form correctly on steel than on Pyrex. The dryfilm polymer forms by reacting with OH radicals on the surface to be coated.29 While most glasses have many such radicals, and in fact can be hydroxolated, there are far fewer such radicals on stainless-steel surfaces and therefore the dryfilm does not adhere to the steel as well as it does to the glass. We tested two other possible coatings for stainless steel. They both form polymers similar to dryfilm, with long chains of methyl groups; however, the head group that attaches to the surface is different. The first coating is a “thiolate” that has a sulfur head group. This coating requires that the steel be given a clean coating of gold, silver, copper, or platinum. It has been suggested that these coatings should be less resistant to alkali attack.30 The second nondryfilm coating for metal is one made from octadecylphosphonic acid. This was motivated by the suggestion that the phosphorous might attach to the zinc in the stainless steel” and allow a S t e p h e n s , Rhodes, a n d Wieman
379 surface of long methyl chains to form. Both coatings had high reaction rates with alkalis.
B. Outgassing Coatings with high reaction rates tend to have higher outgassing rates. Because the Cs is probably attacking the oxygen and the SiO bonds in the dryfilm, the coatings that have longer methyl chains and protect the oxygens from Cs attack not only have lower reaction rates, they have lower outgassing rates, also. Outgassing rates were measured in the following manner. A coated chamber was attached to an ion pump through a valve. The steady-state pressure of the system was calculated from the ion pump current. Then the valve between the ion pump and the coated cell was closed and the change in the ion pump current was noted. The change in the pressure at the ion pump can be related to the outgassing of the coating:
S Q=AP A -,
(11)
where S is the speed of the ion pump and A is the surface area of the coating. The outgassing measurements are the least accurate of the coating parameter measurements. The uncertainty is large in part because the ion pump is operating at close to its leakage current and sensitivity limit, so there are large uncertainties in the current readings. Furthermore, there is an uncertainty in the pumping speed of the ion pump for the gases released by the coatings. Finally, in many cases only an upper limit could be set because the surface area of the coated cell was small compared to the uncoated surface area of the whole system and therefore did not provide a significant fraction of the gas load. ?'he outgassing measurements shown in Table 111 are correct to within a multiplicative factor of 3. V. DISCUSSION
One can choose the best wall coating for efficient trapping by using the model described in Sec. 11 and making the measurements described in Secs. 111 and IV. We have found that for our purposes the best coating, OTS on Pyrex, works sufficiently well for a high collection efficiency." There are, however, still several surface issues to be discussed. First, a simple explanation demonstrates why dryfilm works. Second, the adsorption energies we measured for paraffin (0.53 eV) and dryfilm (0.40 eV) were significantly higher than those measured by other researchers (-0.1 eV for a paraffin, slightly higher for dryfilm). Third, one could ask if these or other coatings might work better at a temperature different from room temperature. Finally, there may be other coatings that would work well that we have not tested. We mentioned in Sec. 111 that surfaces with low dielectric constants should have low adsorption energies. Dryfilm works because it is a polymer terminated with methyl groups. Surfaces terminated by methyl groups have low dielectric constants because the methyl groups are polarizable enough to interact with their neighbors to form a smooth J. Appl. Phys., Val. 76, No. 6, 15 September 1994
surface, yet they are not so polarizable that they form an image charge for the approaching atom.3" It is important to have enough layers of methyl groups perpendicular to the wall to effectively shield the underlying surface; therefore, surfaces that form long methyl chains tend to work better than surfaces with short chains." Teflon and paraffin, two other low dielectric constant materials, are also both polymers terminated with methyl groups. The difference between the adsorption energies that we measure and those that other researchers have measured is probably due to a relative lack of cleanliness on our part and the more complicated geometry of our cells. The cIeaning procedures we used for the parts to be coated was the same proccdure we used for a standard UHV system. The parts were immersed for 1 min in hot trichloroethane vapor, rinsed with de-ionized water, immersed in an ultrasound filled with hot soapy water for 1 min, rinsed with de-ionized water and then with methanol. Before coating, each piece was dried in air under a heat lamp. We did not use any special glass cleaners, nor did we do a high-temperature bake in vacuum before coating as many other researchers have done. This is in part because one of our goals was to develop a coating that could he made quickly and with no special techniques. It is likely that residual dirt and water or impurities in the coating chemicals resulted in an imperfect formation of the coating, leading to an increase in adsorption energy, outgassing, and reaction rate. Our cell geometry was also more complicated than other resonance cell experiments, and this added surface roughness is likely to increase the adsorption cnergy. Even our simplest test cell had a coated ground glass valve, and our more complicated trapping cells had two valves and six windows attached. The reaction rate for OTS on Pyrex was much larger in our complicated trapping cell than it was in our relatively simpler test cell (see Table III), indicating that the polymer does not form as well in a cell with many corners and edges. A small increase in surface roughness can increase the adsorption energy by as much as 0.1 eV."',33 For our paraffin coatings in particular, we are not certain of the efficiency of the method described in Sec. I11 (see, for example, Refs. 8 and 34 for a description of the more controlled method used by other experimenters). The second question mentioned, whether there are other materials that might work well at temperatures above or below room temperature, has a complicated answer. Increasing the temperature decreases the sticking time, which could increase the capture rate for a surface with a relatively high adsorption energy [Eq. (211; however, it also increases the thermal velocity of the untrapped atoms, which decreases the capture rate [Eq. ( 2 ) ] with a different temperature dependence. Increasing the temperature will also increase the outgassing of the wall, which will in turn increase the loss rate from the trap [Eq. (4)], and it will increase the reaction rate of the wall which will decrease the capture rate of the trap [Eq. (3)]. Whether one heats or cools the walls will depend on what property of the wall is limiting the capture efficiency. Currently, outgassing is the primary limit to our capture efficiency, so cooling the walls is attractive. Cooling the Stephens, Rhodes, and Wiernan
3485
380 walls should decrease the outgassing, the reaction rate, and the thermal velocity, all of which are beneficial. That is balanced, however, by the increase in the sticking time. With current OTS coatings, the sticking time is long enough that the coatings cannot be cooled beyond 30 "C before the increase in the sticking time begins to affect the capture efficiency [Eq. (3)];however, if by improving our coating techniques we are able to produce paraffin coatings with adsorption energies as low as have been reported by others, it should be possible to enhance the capture efficiency significantly by cooling the walls to liquid-nitrogen temperatures. Paradoxically, there may be situations, such as intensely radioactive environments, where it is advantageous to use a cell with heated walls. Although the capture efficiency is reduced because the thermal velocity, reaction rate, and outgassing will increase, it may be necessary to make such a compromise in order to use more robust wall materials such as sapphire or alkali resistant glass. Although the adsorption energies of sapphire and alkali resistant glass are higher than that of dryfilm and both materials are difficuli to work with, they are more likely to withstand highly radioactive environments. Previous experiments that have used paraffin and dryfilm in an environment with high-energy particles have reported damage to the c ~ a t i n g . ' ~ ~ From " our results we believe that warm sapphire (80 "C) is a viable alternative for such a situation. Finally, one could ask whether there are other wall materials or coatings, which we have not tested, that may be useful. One, alkali resistant glass, has already been mentioned; another is diamond. We did not test alkali resistant glass in part because it is brittle and difficult to work, but primarily because we believe we can predict its performance. Its outgassing rate should be similar to Pyrex. It will not react significantly with the alkali vapor, as has been shown by the success of resonance cells made from alkali resistant glass; however, alkali-resistant glass has a dielectric constant of -6,'* while Pyrex has a dielectric constant of -4.4. As explained in Sec. 111, the adsorption energy increases with dielectric constant, therefore we expect the adsorption energy of alkali-resistant glass to be higher than Pyrex. We expect alkali resistant glass will therefore have a higher adsorption energy than Pyrex and will have to be heated to reduce the sticking time. Diamond is a more promising material. The surface of diamond is terminated by methyl groups, as are the paraffin and the dryfilms.3y Diamond is chemically inert and should not outgas significantly. Currently it is difficult to diamond coat a trapping cell geometry. Furthermore, the optical quality of many coatings is not good. It is likely, however, that the technology of diamond coating will continue to rapidly improve, and this may well be an attractive alternative in the future. VI. CONCLUSIONS
We have performed a series of tests to measure adsorption energy, reaction rate, and outgassing properties for various surfaces interacting with a Cs vapor. The simple measurement methods we have developed can be used to determine these parameters for other coatings. We have also 3486
J Appl. Phys., Vol. 76, No. 6, 15 September 1994
defined the requirements for a surface that will be used in a high-efficiency vapor-cell laser trap, and our model shows how to use these parameters to predict capture efficiencies. The best coating we have found for this purpose is OTS dryfilm on Pyrex. Sapphire is also a viable alternative, although it has a number of drawbacks. These coatings have other uses in vapor-cell laser traps in addition to allowing high collection efficiencies of rare species. For example, a cell coated with dryfilm would allow one to rapidly change the density of untrapped Cs. One could, therefore, fill a trap quickly from a high-density background and then pump away the background to greatly increase the lifetime of the trap. This could be useful for experiments that measure loss from optical traps due to intratrap collisions. ACKNOWLEDGMENTS
We would like to thank Kent Lindquist, Harvey Gould, Linda Young, Steve Sibener, Dave Swenson, and Abraham Ulman for their helpful suggestions. We would like to thank Hans Rohner for making the glass test cells. This work was funded by the Office for Naval Research and the National Science Foundation. APPENDIX
Dryfilm is a generic name for a polymer coating with a silicon-oxygen backbone terminated with methyl groups. There are several different methods for making it; here we describe the four that we tested. We also describe how we made similar sulfur-based and phosphorus-based coatings. Each piece was cleaned in the same way before coating, as described in Sec. V. After being coated each piece was baked at 200 "C in a low vacuum (50 mTorr) bakeout chamber. It is important to prebake the pieces. The heavy hydrocarbons released from the coatings will damage ion pumps and ion gauges. Caution: A vapor with HCI acid is a by-product of all of the chlorosilanes. The chemicals should always be handled in a fume hood and with protective equipment. Chlorosilanes are highly corrosive and should be stored in Teflon bottles. Dryfilm can be removed by rinsing the piece with HF or by soaking it in ammonium bifluoride. Coating Pyrex, stainless steel, or copper with dichlorodirnethylsilane or SC-77
We followed a method very similar to that of Swenson and Anderson:"' 5 ml de-ionized water were put into a large glass test tube, then 10 ml of SC-77 or dichlorodimethylsilane (DCDMS) are added. The mixture will bubble violently and release a thick vapor. Exposure to this vapor coats the piece. After adding the SC-77 or DCDMS we put a silicon rubber stopper with a glass tube through the middle of it into the test tube and forced the vapor into our test cells for 1-2 min. This method ensured that the vapor was entering our cells despite the constricting geometry of the valves attached to the cells. It also made a much thinner coating than directly exposing the piece to the vapor. Our coatings were invisible, Stephens, Rhodes, and Wieman
381 ~~
while the coatings made by direct exposure (described in Swenson and Anderson) had a frosted look and attenuated some of the light from our trapping laser. Once the piece is coated, water should bead up and run off the surface; it should not wet the surface. The by-products from the reaction are HCl and water, which can be neutralized with sodium acetate. SC-77 can be obtained from Silar Laboratories, 10 Maple Rd, Scotia, NY 12302, 518-372-5691. DCDMS can be obtained from Aldrich Chemical Co, Inc., 1001 W. St. Paul Ave., Milwaukee, WI 53233, 3 -800-558-9160.
Coating Pyrex, stainless steel, or copper with rnethoxysilanes
Once again, we followed a method very similar to that of Swenson and Anderson. The piece to be coated was cleaned, then immersed in de-ionized water. A 50-50 mixture of dimethyldimethoxysilane and methyltrimethoxysilane was added to the water (the mixture was 20% of the total volume of the water-methoxysilane solution). Three drops of acetic acid per 100 ml of solution were added to act as a catalyst for the reaction. The mixture was covered and allowed to sit for 24 h. (Swenson and Anderson left the piece in for 2-3 days.) Initially the methoxysilane mixture floats on top of the water. After a day the whole solution is a milky color and a thick sludge begins to form in the bottom of the container. We removed the pieces after 24 h; longer exposures make thicker coatings. After removal from the solution the piece is sticky and will remain so until after it has been baked. Water should not wet the piece after it has been baked. We found it difficult to get a smooth coating with this technique. Letting the pieces "drip dry" for several hours before baking helped. Of all the coating methods, this one made the thickest coatings (around 0.002 in. for a 24 h exposure). Methyltrimethoxysilane can be obtained from Aldrich Chemical Co., and dimethyldimethoxysilane is available from Silar Laboratories, Inc.
Coating Pyrex with OTS
OTS needs to be handled in a nitrogen atmosphere, so to make these coatings we used a portable glovebox inside a fume hood. To make the OTS coating, we prepared a 5 mM solution of OTS in 8% CHC13+12%l CC14+80% n-hexadecane (by volume). We then immersed the piece in the solution for approximately 2 min. After removal from the solution the piece can be baked immediately or allowed to air dry and then baked. As with the other coatings, water will not wet a finished coating. OTS can be purchased from Aldrich Chemical Co. J. Appl. Phys., Vol. 76, No. 6, 15 September 1994
Coating stainless steel with octadecylphosphonic acid
Immerse the piece in a 5 mM solution of n-octadecylphosphonic acid in ethanol for about 1 h. After removal from solution rinse with de-ionized water. The water should bead up on the surface. n-octadecylphosphonic acid can be bought from Johnson Matthey, 30 Bond St., Ward Hill, MA 01835, 1-800-3430660. Coating metal with hexadecanethiol
Thiolate coatings will not bond directly to stainless steel. After cleaning our test pieces in the standard way, we electropolished them, then sputtered a gold coating onto one side. Next we immersed the piece overnight in a 1 mM solution of hexadecanethiol (also called hexadecylmercaptan) in ethanol. After they were removed from the solution we rinsed them with de-ionized water. As with the other coatings, the water does not wet this surface. We were also able to coat copper gaskets (without gold coatings) with this solution. Hexadecanethiol can be purchased from Aldrich Chemical c o .
' E. W. Otten, Irivesligation
of Short-Liwd Isotopes by Luser Spectroscop,v (Hawood Academic, London, 1989), p. 16. 'Z.T. Lu, C. J. Bowcrs, S. J . Freedman, B. K. Fujikdwa, J. L. Mortara, S.-Q. Shang. K. P. Caulter, and L. Young, Phys. Rev. Lctt. 72, 3791 (1994).
'H. Could, LBL (private communication). 'C. Wieman, C. Monroe, and E. Corncll, in I U t A International Conference on Laser Spectroscopy, edited by M. Ducloy, E. Giacobino, and G. Camy (World Scientific, Singapore, 1992), p. 77. 'J. Opt. Soc. Am. B 6, 2058 (1989), special issue on laser cooling and trapping. 'G. Gwinner, J. A. Behr, S. B. Cahn, A. Ghosh. L. A. Orozco, G. D. Sprouse, and F. Xu, Phys. Rev. Lett. 72, 3795 (1994). '5. H. de Boer, The Dynumical Character of Adsorption (Clarendon, Oxford, 1978). 'M. A. Bouchiat and J. Brossel, Phys. Rev. 147, 41 (1966). 'X. Zeng, E. Miron, W. A. Van Wijnagaarden, D. Schreiber, and W. Happer, Phys. Lett. A 96. 1'11 (1983). "'D. R. Swcnson and L. W. Anderson, Nucl. Instrum. Methods B 29, 627 (19x8).
"H. M. Goldenburg, D. Kleppner, and N. F. Ramscy, Phys. Rev. 123, 530 (1961). "G. E. Thomas, R. J. Holt, D. Boyer, M. C. Green, R. S . Kowaalczyk, and L. Young, Nucl. Instrum. Methods A 257, 32 (1987). "W. A. Hamel, A. D. Stream, and J. P. Woerdman, Opt. Commun. 63, 32 ( I 987). "X. Gozzini. G. Nienhuis, E. Mariotti, G. Paffuti, C. Gahbanini, and L. Moi, Opt. Commun. 88, 341 (1992). "C. Monroe, W. Swann, H. Robinson, and C . Wieman, Phys. Rev. Lett. 65, 1571 (1990).
'"M. Stephens and C. Wicman. Phys. Rev. Lett. 72, 3787 (1994). "K. Lindquist, M. Stephens, and C. Wiernan, Phys. Rev. A 46,4082 (1992). "F. Reif, in Fundamentals of Statistical and Thermal Physics (McGrdwHill, New York. 1965), p. 273. '"M. Oria, M. Chevrollier. D. Bloch, M. Fichet, and M. Ducloy, Europhys. Lett. 14, 527 (199I ) ; H. Luth, Surfaces arid Interfaces of Solids (Springer, Bcrlin, IYY3), p. 430. "'A. M. Bonch-Bruevich. Y. M. Maksimov, and V. V. Khromov, Opt. Spectrosc. 58. X54 (1985). 'I F. O'Hanlon, A U.wr's Guide to k c u u m Technology (Wiley, New York, 1989), p. 27. '?C. Rahman and H. G. Robinson, IEEE J. Quantum Electron. QE-23, 452 (1987). 'jR. G. Brewer, J. Chem. Phys. 38, 3015 (1963).
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382 24M. D. Scheer, R. Klein, and J. D. McKinley, J. Chem. Phys. 55, 3577 (1971). 25V. Liberman and R. J. Knize, Phys. Rev. A 34, 5115 (1986). 2hD.W. Sindorf and G . E. Maciel, J. Am. Chem. Soc. 105, 3767 (1983). 27S.R. Wasserman, Y. Tdo, and G. M. Whitesides, Langmuir 5, 1074 (1989). 2XK.Albert, B. Pfleidcrer, E. Baycr, and R. Schnabel, J. Colloid Interface Sci. 142, 35 (1991). 2yA. Ulman, A n Introduction to Ultrathin Organic F i l m s q r o m LangmuirElodgetr to SelfAssembly (Academic, Boston, 1991). '"C. D. Bain, E. B. Troughton, Y. Tdo, J. Evall, G . M. Whitesides, and R. G. Nuuzzo, J. Am. Chcm. Soc. 111, 321 (1989). " G . Cao, H. Hong, and T. E. Mallouk, Acc. Chem. Res. 25, 410 (1992).
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I. C. Camparo, J. Chem. Phys. 86, 1533 (1987). "3. H. de Boer, in Advances in Colloid Science, edited by H. Mark and E. J. W. Verwey (Wiley-Interscience, New York, 1950), Vol. 3, Sec. 111. 14G.Singh, P. Dilavore, and C. 0. Alley, Rev. Sci. Instrum. 43, 1388 (1972). 35 U. Kopf, H. J. Besh, E. W. Ottcn, and Ch. von Platen, 2. Phys. 226, 297 (1969). 3hS. G. Redsun, R. J. Knize, G . D. Cates, and W. Happer, Phys. Rev. A 4 2 , 1293 (1990). 37 D. Swenson (private communication). 3X W. Espe, in Murerids of High Vacuum Technology (Pergamon, Oxford, 1968), Vol. 2, p. 119. "D. C. Harris, Nav. Res. Rev. 3, 3 (1992).
Stephens, Rhodes, and Wiernan
Inexpensive laser cooling and trapping experiment for undergraduate laboratories Carl Wieman and Gwenn Flowers Joint Institute for LaboratoryAstrophysics and Department of Physics, University of Colorado, Boulder, Colorado 80309
Sarah Gilbert National Institute of Standards and Technolom, Bouldec Colorado 80303
(Received 14 July 1994; accepted 15 December 1994) We present detailed instructions for the construction and operation of an inexpensive apparatus for laser cooling and trapping of rubidium atoms. This apparatus allows one to use the light from low power diode lasers to produce a magneto-optical trap in a low pressure vapor cell. We present a design which has reduced the cost to less than $3000 and does not require any machining or glassblowing skills in the construction. It has the additional virtues that the alignment of the trapping laser beams is very easy, and the rubidium pressure is conveniently and rapidly controlled. These features make the trap simple and reliable to operate, and the trapped atoms can be easily seen and studied. With a few milliwatts of laser power we are able to trap 4x10' atoms for 3.5 s in this apparatus. A step-by-step procedure is given for construction of the cell, setup of the optical system, and operation of the trap. A list of parts with prices and vendors is given in the Appendix. 0 1995 American Association
df Physics Teachers.
I. INTRODUCTION Laser cooling and trapping of neutral atoms is a rapidly expanding area of physics research which has seen dramatic new developments over the last decade. These include the ability to cool atoms down to unprecedented kinetic temperatures (as low as 1 GK) and to hold samples of a gas isolated in the middle of a vacuum system for many seconds. This unique new level of control of atomic motion is allowing researchers to probe the behavior of atoms in a whole new regime of matter. Undoubtedly one of the distinct appeals of this research is the leisurely and highly visible motion of the 317
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laser cooled and trapped atoms. Because of this visual appeal and the current research excitement in this area, we felt that it was highly desirable to develop an atom trapping apparatus that could be incorporated into undergraduate laboratory classes. This paper presents a detailed discussion of how to build a simple and inexpensive atom trapping apparatus for atomic rubidium (Rb). Our principal goal was to develop an apparatus which could be built and operated reliably with minimal expense and technical support. In most respects, however, this trap's performance is equal or superior to what is 0 1995 American Association of Physics Teachers
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384 achieved with the “traditional” designs used in many research programs, and some innovations have advantages over these designs. Thus portions of this paper are likely to be of interest to the researcher already working (or considering working) in the field of laser trapping. This paper is written in the same style as the previous paper on grating-stabilized diode lasers and saturated absorption spectroscopy.’ It is intended to provide a “cookbook” discussion that will allow a relative novice to construct and operate an optical trap. This paper is essentially a continuation of the work presented in Ref. 1, and thus the end of that paper is used as a starting point. Without further discussion, we assume that one has two diode lasers of the type discussed in Ref. 1 which produce 5 mW or greater of narrow band tunable light. A small fraction (-10%) of each laser beam is split off and sent to its saturated absorption spectrometer of the type discussed in Ref. 1. This allows for precise detection and control of the laser frequencies, which is essential for cooling and trapping. The remainder of this paper will discuss how to use the light from these lasers to trap Rb atoms. Section I1 provides a brief introduction to the relevant physics of the atom trap; Sec. 111 covers laser stabilization; Sec. IV explains the optical layout; Sec. V details the construction of the trapping cell; and Sec. VI discusses the operation of the trap, measurement of the number of trapped atoms, and measurement of the time the atoms remain in the trap. In the Appendix we provide a list of parts needed in the construction of the apparatus with prices and vendors. In its least expensive version the trapping apparatus, not including the lasers, costs less than $3000, with the ion pump responsible for half of the cost.
II. THEORY AND OVERVIEW We will present a brief description of the relevant physics of the vapor cell magneto-optical trap. For more information, a relatively nontechnical discussion is given in Ref. 2, while more detailed discussions of the magneto-optical trap and the vapor cell trap can be found in Refs. 3 and 4,respectively. A. Laser cooling The primary force used in laser cooling and trapping is the recoil when momentum is transferred from photons scattering off an atom. This radiation-pressure force is analogous to that applied to a bowling ball when it is bombarded by a stream of ping pong balls. The momentum kick that the atom receives from each scattered photon is quite small; a typical velocity change is about 1 cm/s. However, by exciting a strong atomic transition, it is possible to scatter more than lo7 photons per second and produce large accelerations (104.g).The radiation-pressure force is controlled in such a way that it brings the atoms in a sample to a velocity near zero (“cooling”), and holds them at a particular point in space (“trapping”). The cooling is achieved by making the photon scattering rate velocity dependent using the Doppler e f f e ~ tThe . ~ basic principle is illustrated in Fig. 1. If an atom is moving in a laser beam, it will see the laser frequency vase[shifted by an amount (-V/c)y,,, where V is the velocity of the atom along the direction of the laser beam. If the laser frequency is below the atomic resonance frequency, the atom, as a result of this Doppler shift, will scatter photons at a higher rate if it is moving toward the laser beam (V negative), than if it is moving away. If laser beams impinge on the atom from all 318
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LL
Frequency +
-9
force Fig. 1. Graph ofthe atomic scattering rate versus laser frequency.As shown, a laser is tuned to a frequency below the peak of the resonance. Due to the Doppler shift, atoms moving in the direction opposite the laser beam will scatter photons at a higher rate than those moving in the same direction as the beam. This leads to a larger force on the counter propagating atoms.
six directions, the only remaining force on the atom is the velocity-dependent part which opposes the motion of the atoms. This provides strong damping of any atomic motion and cools the atomic vapor. This arran ement of laser fields is often known as “optical molasses. J B. Magneto-optical trap Although optical molasses will cool atoms, the atoms will still diffuse out of the region if there is no position dependence to the optical force. Position dependence can be introduced in a variety of ways. Here, we will only discuss how it is done in the “magneto-optical trap” (MOT), also known as the “Zeeman shift optical trap,” or “ZOT.” The positiondependent force is created by using appropriately polarized laser beams and by applying an inhomogeneous magnetic field to the trapping region. Through Zeeman shifts of the atomic energy levels, the magnetic field regulates the rate at which an atom in a particular position scatters photons from the various beams and thereby causes the atoms to be pushed to a particular point in space. In addition to holding the atoms in place, this greatly increases the atomic density since many atoms are pushed to the same position. Details of how the trapping works are rather complex for a real atom in three dimensions, so we will illustrate the basic principle using the simplified case shown in Fig. 2. In this simplified case we consider an atom with a J=O ground state and a J = 1 excited state, illuminated by circularly polarized beams of light coming from the left and the Wieman, Flowers, and Gilbert
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,
”Rb Energy Levels
I
BcO
B=O
B>O
fOrc8-t
fone=O
t force
780 nm hyperfine pumping diode laser
5%
Fig. 2. One dimensional explanation of the MOT.Circularly polarized laser beams with opposite angular momenta impinge on an atom from opposite directions. The lasers excite theJ=O toJ=l transition. The laser beam from the right only excites to the m = - 1 excited state, and the laser from the left only excites to the m = + 1 state., As an atom moves to the right 01 left, these levels are shifted by the magnetic field thereby affecting the respective photon scattering rates. The net result is a position-dependent force which pushes the atoms into the center.
Fig. 4. “Rb energy level diagram showing the trapping and hyperfine pumping transitions. The atoms are observed by detecting the 780 nm fluorescence as they decay back to the ground state.
right. Because of its angular momentum, the beam from the left can only excite transitions to the rn = + 1 state, while the beam from the right can only excite transitions to the m = -1 state. The magnetic field is zero in the center, increases linearly in the positive x direction, and decreases linearly in the negative x direction. This field perturbs the energy levels so that the Am = + 1 transition shifts to lower frequency if the atom moves to the left of the origin, while the Am=-1 transition shifts to higher frequency. If the laser frequency is below all the atomic transition frequencies and the atom is to the left of the origin, many photons are scattered from the crc laser beam, because it is close to resonance. The d laser beam from the right, however, is far from its resonance and scatters,fw photons. Thus the force from the scattered photons pushes the atom back to the zero of the magnetic field. If the atom moves to the right of the origin, exactly the opposite happens, and again the atom is pushed toward the center where the magnetic field is zero. Although it is somewhat more complicated to extend the analysis to three dimensions, experimentally it is simple, as shown in Fig. 3. As in optical molasses, laser beams illuminate the atom from all
six directions. Two symmetric magnetic field coils with oppositely directed currents create a magnetic field which is zero in the center and changes linearly along the x , y , and z axes. If the circular polarizations of the lasers are set correctly, a linear restoring force is produced in each direction. Damping in the trap is provided by the cooling forces discussed in Sec. I1 A. It is best to characterize the trap “depth” in terms of the maximum velocity that an atom can have and still be contained in the trap. This maximum velocity V,, is typically a few times I‘A (rA is the velocity at which the Doppler shift equals the natural linewidth of the trapping transition, where A is the wavelength of the laser light). A much more complicated three-dimensional calculation using the appropriate angular momentum states for a real atom will give results which are qualitatively very similar to those provided by the above analysis if: (1) the atom is excited on a transition where the upper state total angular momentum is larger than that of the lower state (F-tF’ =F+ 1) and (2) V3(hr/m)’”, where m is the mass of the atom. This velocity is often known as the “Doppler limit” velocity? If the atoms are moving more slowly than this, “sub-Doppler” cooling and trapping processes become important, and the simple analysis can no longer be used.7 We will not discuss these processes here, but their primary effect is to increase the cooling and trapping forces for very slow atoms in the case of F+F+ 1 transitions.
r
(T+
C. Rb vapor cell trap
0-
polar laser
Fig. 3. Schematic of the MOT. Lasers beams are incident from all six directions and have angular momenta as shown. Rvo coils with opposite currents produce a magnetic field which is zero in the middle and changes linearly along all three axes. 319
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We will now consider the specific case of Rb (Fig. 4). Essentially all the trapping and cooling is done by one laser which is tuned slightly (1-3 natural linewidths) to the low =3 transition of frequency side of the 5S1,2F=2-+5P3,2F‘ ”Rb. (For simplicity we will only discuss trapping of this isotope. The other stable isotope, ”Rb, can be trapped equally well using its F = 3 --+ F ’ =4 transition.) Unfortunately, about one excitation out of loo0 will cause the atom to decay to the F=l state instead of the F = 2 state. This takes the atom out of resonance with the trapping laser. Another laser (called the “hyperfine pumping laser”) is used to excite the atom from the 5s F = 1 to the 5 P F’ =1or 2 state, from which it can decay back to the 5s F = 2 state where it will again be excited by the trapping laser. Wieman, Flowers, and Gilbert
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386 In a vapor cell trap, the MOT is established in a low pressure cell containing a small amount of Rb vapor: The Rb atoms in the low energy tail o/
-
-
N4 plnlea
. \
where is the time constant for the trap to fill to its steady state value N o and is also the average time an atom will remain in the trap before it is knocked out by a collision. This time is just the inverse of the loss rate from the trap due to collisions. Under certain conditions, collisions between the trapped atoms can be important, but for coiiditions that are usually encountered, the loss rate will be dominated by collisions with the room temperature background gas. These “hot” background atoms and molecules (Rb and contaminants) have more than enough energy to knock atoms out of the trap. The time constant T can be expressed in terms of the cross sections a, densities n, and velocities of Rb and non-Rb components as
The steady-state number of trapped atoms is that value for which the capture and loss rates of the trap are equal. The capture rate is simply given by the number of atoms which enter the trap volume (as defined by the overlap of the laser beams) with speeds less than V,,,. It is straightforward to show that this is proportional to the Rb density, (Vmx)4, and the surface area A of the trap. When the background vapor is predominantly Rb, the loss and capture rates are both proportional to Rb pressure. In this case No is simply4
No = (0,1A/aRb)( Vmax /VavgI49
(3)
where V,, is (2kT/m)”‘, the average velocity of the Rb atoms in tie vapor. If the loss rate due to collisions with non-Rb background gas is significant, Eq. (3) must be multiplied by the factor nRb~RbV8v$(nRb~RbVavg+~non~nonv“on). The densities are proportional to the respective partial pressures. Finally, if the loss rate is dominated by collisions with non-Rb background gas, the number of atoms in the trap will be proportional to the Rb pressure divided by the non-Rb pressure, but will be independent of the Rb pressure. As a final note on the theory of trapping and cooling, we emphasize certain qualitative features that are not initially obvious. This trap is a highly overdamped system; hence damping effects are more important for determining trap performance than is the trapping force. If this is kept in mind it is much easier to gain an intuitive understanding of the trap behavior. Because it is highly overdamped, the critical quantity V, is determined almost entirely by the Doppler slowing which provides the damping. Also, the cross sections for collisional loss are only very weakly dependent on the depth of the trap, and therefore the trap lifetime is usually quite insensitive to everything except background pressure. As a result of these two features, the number of atoms in the trap is very sensitive to laser beam diameter, power, and frequency, all of which affect the Doppler cooling and hence V,, . However, the number of trapped atoms is insensitive to factors which primarily affect the trapping force but not the damping, such as the magnetic field (stray or applied) and the alignment and polarizations of the laser beams. For example, changing the alignment of the laser beams will dra320
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Fig. 5. (a) Overall optical layout for laser trap experiment including both saturated absorption spectrometers. (b) ,Detail of how laser beams are sent through the trapping cell. To simplify the figure, the N4 waveplates are not shown. Beam paths (1) and (2) are in the horizontal plane and beam (3) is angled down and is then reflected up through the bottom of the cell. The retrobeams are tilted slightly to avoid feedback to the diode laser.
matically affect the shape of the cloud of trapped atoms since it changes the shape of the trapping potential. However, these very differently shaped clouds will still have similar numbers of atoms until the alignment is changed enough to affect the volume of the laser beam overlap. When this happens, the damping in three dimensions is changed and the number of trapped atoms will change dramatically. Of course, if the trapping potential is changed enough that there is no potential minimum (for example, the zero of the magnetic field is no longer within the region of overlap of the laser beams), there will be no trapped atoms. However, as long as the damping force remains the same, almost any potential minimum will have about the same number of atoms and trap lifetime.
D. Overview of the trapping apparatus Figure 5(a) shows a general schematic of the trapping apparatus. It consists of two diode lasers, two saturated absorption spectrometers, a trapping cell, and a variety of optics. The optical elements are lenses for expanding the laser beams, mirrors and beamsplitters for splitting and steering the beams, and waveplates for controlling their polarizations. To monitor the laser frequency, a small fraction of the output of each laser is split off and sent to a saturated absorption spectrometer. An electronic error signal from the trapping laser’s saturated absorption spectrometer is fed back to the laser to actively stabilize its frequency. The trapping cell is a small vacuum chamber with an ion vacuum pump, a Rb source, and windows for transmitting the laser light. In the following sections we will discuss the various components of the apparatus and the operation of the trap. Wieman, Flowers, and Gilbert
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III. LASER STABILIZATION Reference 1 describes how to construct a diode laser system. As mentioned above, two lasers are needed for the trap. Since a few milliwatts of laser power are plenty for hyperfine pumping (F= l+F’ = 1 , 2 ) , the stronger should be used as the trapping laser (F = 2+F‘ = 3 ) if the two lasers have different powers. Clearly visible clouds of trapped atoms can be obtained with as little as 1.5 mW of trapping laser power (after the saturated absorption spectrometer pickoff). However, the number of trapped atoms is proportional to the laser power, so setting up and using the trap is much easier with at least 5 mW of laser power. The trapping laser must have an absolute frequency stability of a few megahertz. This normally requires one to actively eliminate fluctuations in the laser frequency, which are usually due to changes in the length of the laser cavity caused by mechanical vibrations or temperature drifts. This is accomplished by using the saturated absorption signal to detect the exact laser frequency and then using a feedback loop to hold the length of the laser cavity constant. Detailed discussions of the saturated absorption spectrometer, the servoloop circuit, and the procedure for locking the laser frequency are given in Ref. 1, along with examples of spectra obtained. After a little practice one can lock the laser frequency to the proper value within a few seconds. Here, we just mention a few potential problems and solutions. Check that the baseline on the saturated absorption spectrum well off resonance is not fluctuating by more than a few percent of the F = 2 + F J = 3 peak height. If there are larger fluctuations, they are likely caused by light feeding back into the laser or by the probe beams vibrating across the surface of the photodiodes. These problems can normally be solved by changing the alignment and making sure that all the optical components are rigidly mounted. If the fluctuations in the unlocked laser frequency are large it will probably be necessary to improve the vibration isolation of the laser and/or table, or rebuild the laser to make it more mechanically stable. If the fluctuations are stable and synchronized with the 60 Hz line voltage, the problem is probably not mechanical, but rather is the diode laser power supply. Under quiet conditions and reasonably constant room temperature, the laser should stay locked for many minutes and sometimes even hours at a time. Bumping the table or the laser will likely knock the laser out of lock. If the laser frequency does not have excessively large fluctuations before it is locked but jumps out of lock very easily, it is likely that the resonant frequency of the servoloop is too low. This is determined by the response of the diffraction grating mount when driven by the PZT.The resonance limits the gain of the servo and reduces its ability to correct for large deviations. The frequency of this resonance can be determined simply by tuming up the lock gain until the servosystem begins to oscillate and measuring the frequency of the sine wave which is observed. We have found that a reasonable value for the resonance frequency is around 1 kHz. If it is much lower than this, the diffraction grating is too massive, the mounting is not stiff enough, or the PZT has not been installed properly. Even under the best conditions, the system we have described is likely to have residual frequency fluctuations of around 1 MHz. The trap will work fine with this level of stability but these fluctuations will cause some noise in the fluorescence from the trapped atoms. For most experiments this is not serious, but it can limit some measurements. If one has a little knowledge of servosystems, it is quite straightfor321
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ward to construct a second feedback loop which adjusts the laser current. The current feedback loop should have a roll-on filter and high gain. (Because the laser current loop can be much faster than the PZT loop, it can have a much higher gain.) This results in the current loop feedback dominating for frequencies above a few hertz. However, the PZT loop gain is largest for very low frequencies, and thus handles the dc drifts. The combination of the two SeNOlOOpS will make the laser frequency much more stable and will also make the lock extremely robust. We have had lasers with combined PZT and current servoloops remain frequency locked for days at a time and resist all but the most violent jarring of the laser. (A circuit diagram for the combined PZT and current servoloop can be obtained by sending a request and a self-addressed stamped envelope to C. Wieman.) This extra servoloop is a complication, however, which may not be desirable for an undergraduate laboratory. The requirements for the frequency stability of the hyperfine pumping laser are much less stringent than those for the trapping laser. For many situations it is adequate to simply set the frequency near the peak of the 5s F = 1+SP F ‘ = 2 transition by hand. Over time it will drift off, but if the room temperature does not vary too much, it will only be necessary to bring the laser back onto the peak by slightly adjusting the PZT offset control on the ramp box every 5 to 10 min. If better control is desired, this frequency can also be locked to the side of the F = 1-+ F’ = 2 saturated absorption peak using the above procedure. However, it is often more convenient to lock the frequency of this laser to the peak of the F = 1 + F ’ = 1 , 2, and 3 Doppler broadened (and hence unresolved) absorption line by modulating the laser frequency and using phase-sensitive detection with a lock-in amplifier. The output of the lock-in amplifier is then fed back to the PZT to keep the laser frequency on the peak. This is an excessively expensive solution, unless lock-in amplifiers which are too old to use for anything else are available. These work very well for this purpose and this relatively crude frequency lock is adequate for the hypefine pumping laser.
IV. OPTICAL SYSTEM With lasers that can be locked to the correct frequency, one is now ready to set up the optical system for the trap. The basic requirement is to send light beams from the trapping laser into the cell in such a way that the radiationpressure force has a component along all six directions. To motivate the discussion of our optical design for this student laboratory trap, we first mention the design used in most of the traps in our research programs. In these research traps, the light from the trapping laser first passes through an optical isolator and then through beam shaping optics that make the elliptical diode laser beam circular and expand it to between 1 and 1.5 cm in diameter. The beam is then split into three equal intensity beams using dielectric beam splitters. These three beams are circularly polarized with quarter-wave plates before they pass through the trapping cell where the intersect at right angles in the center of the cell. After leaving the trapping cell each beam goes through a second quarterwave plate and is then reflected back on itself with a mirror. This accomplishes the goal of having three orthogonal pairs of nearly counterpropagating beams, with the reflected beams having circular polarization opposite to the original beams. Wieman. Flowers, and Gilbert
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A. Geometry Here, we present a modified version of this setup, as shown in Fig. 5. A number of the expensive optical components (optical isolator, dielectric beam splitters, and large aperture high quality quarter-wave plates) have been eliminated from the research design. The light from the trapping diode laser is sent into a simple two-lens telescope which expands it to a 4.5 cmX1.5 cm ellipse. This is then split into three beams which are roughly square (1.5 cmXl.5 cm), simply by clipping off portions of the beam with mirrors which interrupt part of the beam, as shown. The operation of the trap is remarkably insensitive to the relative. amounts of power in each of the beams; a factor of 2 difference causes little change in the number of trapped atoms. However, the beam size as set by the telescope is of some importance. The number of trapped atoms increases quite rapidly as the beam size increases and, as discussed below, the larger the beam, the less critical the alignment. However, if the beams are too large it is difficult to find optical elements to accommodate them and it becomes harder to see the beams due to the reduced intensity. Beams of 1.5 cm diameter work well and fit on standard 2.5 cm optics. With larger optical components it is possible to use larger beams. However, avoid using beams much smaller than 1.5 cm in diameter because of the decrease in the number of trapped atoms and the increased alignment sensitivity. ' h o of the three beams remain in a horizontal plane and are sent into the cell as shown in Fig. 5(b). The third is angled down and reflects up from the bottom of the cell. There is nothing special about the layout of the three beams we have shown here. The only requirement is that they all intersect in the cell at roughly right angles. There is a minor benefit in having all of the beams follow nearly equal path lengths to the cell to keep the sizes matched, if the light is not perfectly collimated. To simplify the adjustment of the polarizations, the light should be kept linearly polarized until it reaches the quarter-wave plates. As discussed in any introductory optics text, this will be the case as long as all the beams have their axes of polarization either parallel (p) or perpendicular (s) to the plane of incidence of each mirror. This is easy to achieve for the beams in the horizontal plane, but difficult for the beam that comes up through the bottom of the cell. However, with minimal effort to be close to this condition, the polarization will remain sufficiently well linearly polarized. It is easy to check the ellipticity of the polarization by using a photodiode and a rotatable linear polarizer. First, make sure that the polarizer works at 780 nm by looking at the extinction of the laser light by crossed polarizers; some plastic sheet polarizers do not work at 780 nm. An extinction ratio of 10 or greater on the polarization ellipse is adequate. After they pass through tht: cell, the beams are reflected approximately (but not exactly!) back on themselves. The reason for having an optical isolator in the research design is that even a small amount of laser light reflected back into the laser will dramatically shift the laser frequency and cause it to jump out of lock. In the absence of an optical isolator, this will always happen if the laser beams are reflected nealy back on themselves. Feedback can be avoided by insuring that the reflected beams are steered away from the incident beams so that they are spatially offset by many (5-10) beam diameters when they amve back at the position of the laser. Fortunately, for operation of the trap the return beam need only overlap most of the incident beam in the cell, but its 322
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exact direction is unimportant. Thus by making the beams large and placing the retromirrors close to the trap (within 10 cm for example) it is possible to have the forward and backward going beams almost entirely overlap even when the angle between them is substantially different from 180". This design eliminates the need for the very expensive ($2500) optical isolator and, as an added benefit, makes the operation of the trap very insensitive to the alignment of the return beams. If the retro mirrors are close enough to the cell, they can be tilted by up to 30" without significantly changing the number of trapped atoms. It is easy to tell if feedback from the return beams is perturbing the laser by watching the signal from the saturated absorption spectrometer on the oscilloscope. If the amplitude of the fluctuations is affected by the alignment of the reflected beam or is reduced when the beam to the trapping cell is blocked, unwanted optical feedback is occurring.
B. Polarization The next task is to set the polarizations of the three incident beams. The orientations of the respective circular polarizations are determined by the orientation of the magnetic field gradient coils. The two transverse beams which propagate through the cell perpendicular to the coil axis should have the same circular polarization, while the beam which propagates along the axis should have the opposite circular polarization. Although in principle it is possible to initially determine and set all three polarizations correctly with respect to the field gradient, in practice it is much simpler to set the three polarizations relative to each other and then try both directions of current through the magnetic field coils to determine which sign of magnetic field gradient makes the trap work. To set the relative polarization of the three beams, first identify the same (fast or slow) axis of the three quarterwave (W4) plates. For the two beams which are to have the same polarization, this axis is set at an angle of 45" clockwise with respect to thelinear polarization axis when looking along the laser beam. For the axial beam, the axis is oriented at 45" counterclockwise with respect to the linear polarization. This orientation need only to be set to within about 210". The orientation of the h/4 plates through which the beams pass after they have gone through the trapping cell ("retro h/4 plates") is arbitrary. No matter what the orientations are, after the beams have passed though them twice, the light's angular momentum will be reversed. We have tried various options for wave plates. The most straightforward option is to simply use six commercial h/4 plates which have a clear aperture large enough for the laser beams. Ideally, these should be antireflection coated to reduce the attenuation of the laser beams. The only disadvantage to this approach is the cost. There are many suppliers of h/4 plates, but a set of six antireflection coated wave plates will cost over $1O00. An alternative to using expensive wave plates is to replace all or some of the commercial h/4 plates with inexpensive plastic sheet retardation plates, which are widely available in optics demonstration kits. We have replaced the 780 nm h/4 plates with plastic sheet which was nominally XI2 at 500 nm (and hence probably about A13 at 780 nm) and observed little change in the number of trapped atoms. One difference in this case is that the trapping is more sensitive to the orientation of the wave plates and, in particular, is now somewhat sensitive to the orientation of the retro wave plates. Another option is to replace the three retro h/4 plates with retroreflecting right angle mirrors? Although this Wieman, Flowers, and Gilbert
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389 combination of mirrors does not provide an ideal X/2 retardance, we found empirically that for five different dielectric mirrors tested, it is quite close. For gold mirrors the retardance is much farther from A/2, but still adequate for good trapping. Putting the two right angle mirrors on separate mirror mounts gives more flexibility of adjustment, but is probably unnecessary. Simply gluing the two mirrors together at an angle of slightly more or less than 90" (to insure that the beams overlap in the cell, but the return beams does not go directly back into the laser) works fine. An added benefit of this approach is that two reflections off a mirror usually result in much less light loss than one reflection and two passes through a A14 plate. Finally, there are other options for obtaining inexpensive retardation plates that are approximately X/4 at 780 nm. Most transparent plastic sheet is birefringent, and often the retardance can be adjusted by stretching. If one simply tries a number of pieces of plastic sheet, it is likely that some can be found which are not too far from a X/4 retarder. The traditional way of making inexpensive A/4 plates is to split sheets of mica. We have not tried this, but it should work if the mica does not absorb too much light at 780 nm.
C. Hyperfine pumping laser optics Minimal optics are needed for the hyperfine pumping laser. Essentially the only requirement for this light is that it cover the region where the trapping laser beams overlap. In its simplest form this means merely sending the laser through a single lens to expand the beam and reflecting it off a mirror into the trapping cell through any window. In practice, it is often worth going to the small additional effort of using two lenses to make a large collimated beam (typically an ellipse with a 2-3 cm major axis) and send it into the cell from a direction which will minimize the scattered light into the detectors that observe the trapped atoms. The trap is insensitive to nearly everything about the hyperfine pumping light, including its polarization. D. Mirrors and mirror mounts If low cost A/4 retarders are used, the cost of the trapping optics will be dominated by the ten mirrors and mirror mounts required. We have become accustomed to using convenient commercial kinematic mirror mounts (see the Appendix). Similar products are available from many manufacturers, but a price between $50 and $100 per mount is standard. However, the control and stability provided by such mounts are actually far superior to what is necessary in this application. Thus if one has severe budget limitations, simple homemade mounts with limited adjustment should be adequate. We use both gold and high reflectance infrared dielectric mirrors (aluminum is lossy at 780 nm). Although gold mirrors are adequate for any of the mirrors in the setup, we prefer to use the dielectric mirrors for the retroreflection, because their higher reflectance makes the trap more symmetric. Also, while gold mirrors are less expensive than the dielectric mirrors, they are more easily damaged. Gold mirrors with protective overcoatings are readily available, but in our limited experience, mirrors with overcoatings which rival the durability of a dielectric mirror end up costing about the same. In an environment where inexperienced students will be handling the mirrors frequently, inexpensive gold mirrors have a rather short lifetime, in contrast to the standard commercial dielectric coated mirrors. In the long run, we think that the least expensive option for mirrors is to buy 323
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large dielectric mirrors and cut them into smaller mirrors with a glass saw. (Put a protective a v e r such as plastic tape on the surface before cutting.) In the short run, however, it is less expensive to use gold mirrors. The surface quality and flatness of any commercial mirror will be more than adequate. Caution should be taken when using dielectric mirrors at an angle; there can be substantial variation of reflectivity with different incidence angles and polarizations.
V. TRAPPING CELL CONSTRUCTION The primary concern in the construction of the trapping cell is that ultrahigh vacuum (UHV) is required. Although trapped atoms can be observed at pressures of lo-' Pa (=lo-' Torr), trap lifetimes long enough for most ex eri to 10- Pa ments of interest require pressures in the range. Unfortunately, UHV often seems to be synonymous with high cost and specialized technical expertise. Consequently, we have put considerable effort into designing a simple, inexpensive system that can be constructed by someone who does not have experience with ultrahigh vacuum techniques. However, we will present a range of possibilities to best cover the needs of a variety of different users. There are three main elements in the trapping cell: (1) a pump to remove unwanted background gas-mostly water, hydrogen, and helium (helium can diffuse through glass); (2) a controllable source of Rb atoms; and (3) windows to transmit the laser light and allow observation of the trapped atoms. We will discuss each of these elements in turn, and then discuss in detail the actual construction of the cell.
r:-
A. Vacuum pump In all of our systems we have used small ion pumps (2,8, or 11 //s). Ion pumps have the virtues that they are small, quiet, and light, use little power, require no cooling, and have low ultimate pressures. These features make them well suited to this application. The 2 //s pumps have been adequate for a clean system which is evacuated once and remains at high vacuum thereafter. However, in systems which have been cycled up to air a few times the performance of the 2 //s pumps has not been satisfactory. After very few pumpdowns, they become increasingly difficult to start and soon will not pump at all. We recommend an 8-12 //s pump if there is any possibility of letting the system up to air more than once. The power supply for a small ion pump is often more expensive than the pump itself. It is possible to avoid this expense by using a dc high voltage supply, if one is available. The pump will typically require a few tenths of 1 jA at 5-6 kV under normal operating conditions, and as much as a few milliamperes at lower voltage for the first few minutes when it is initially turned on. Any power source which will supply sufficient voltage and current will be adequate. It is quite convenient to be able to monitor the pump current as a measure of the pressure. After a few days of exposure to Rb vapor the pump is likely to have some leakage current for which a correction must be made when determing the pressure, but the leakage does not affect pump performance. These pumps have the minor drawback that they require a large magnetic field, which is provided by a permanent magnet. The fringing fields from this magnet can extend into the trap region and will affect the trap to some extent. Although the trap will usually work without it, we put a layer of magnetically permeable iron or steel sheet around the pump to Wiernan, Flowers, and Gilbert
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shield the trap from this field.g For this same reason it is advisable to avoid having magnetic bases very near the trap.
B. Rb sources We will now discuss how to produce the correct pressure of Rb vapor in the cell. The vapor pressure of a room temperature sample of Rb is about 5X1Op5 Pa. This is much higher than the to Pa to lo-’ Torr) of Rb vapor pressure that is optimum for trapping. At higher pressures the trap will still work, but the atoms remain in the trap a very short time, and it is often difficult to see them because of the bright fluorescence from the untrapped background atoms. Also, the absorption of the trapping beams when passing through the cell will be significant. Thus we would like to maintain the Rb at well below its room temperature vapor pressure. Because it is necessary to continuously pump on the system to avoid the buildup of hydrogen and helium vapor, it is necessary to have a constant source of Rb to maintain the correct pressure. Before discussing specific ways to generate Rb in the vapor cell, we will present some back round on the behavior of Rb in the environment of the cell.$0 This behavior is often counterintuitive. Through chemical reactions and physisorption, the walls of the cell usually remove far more Rb than the ion pump does and the rate of pumping by the walls depends on how well they are coated with Rb. If one has an evacuated stainless steel tube and connects a reservoir of room temperature Rb to one end, initially no Rb vapor will be observed at the other end. Over time the vapor will creep down the tube as it saturates the walls, and in a matter of several days, the pressure differential across the length of the tube will significantly decrease. If the Rb reservoir is then removed and the tube is connected to a pump, the vapor will linger for days as it slowly comes off the walls. If the tube is heated, the vapor will disappear much more quickly. A similar behavior occurs with glass, but the chemical reaction rates vary with different types of glass. With fused silica or Pyrex, the behavior is similar to that observed with stainless steel, and a similar saturation of the chemical reaction rate occurs. With other types of glass (microscope slides, for example) the reaction rate and thus the wall pumping remain large even after extensive exposure to Rb vapor. Since this rapidly depletes Rb and causes large pressure differentials in the cell, we do not recommend making a cell from this type of glass. Our recommended technique for producing the Rb vapor for a student lab experiment is to use a commercial “getter.” This technique was developed specifically for our student lab trap, and, to our knowledge, has not been used before in optical trapping cells. However, as we will discuss, it has advantages over other approaches and is likely to be useful in many research lab traps as well. The getter is several milligrams of a Rb compound which is contained in a small (l.OXO.ZX0.2 cm) stainless steel oven. l b o of these ovens are spotwelded onto two pins of a vacuum feedthrough as shown in Fig. 6. When current (3-5 A) is sent through the oven, Rb vapor is produced. The higher the current, the higher the Rb pressure in the chamber. With this system it is unnecessary to coat the entire surface with Rb, and is in fact undesirable since the getter is likely to be exhausted before the surface is entirely saturated. With a glass cell mounted on a 1.33 in. Conflat-type fiveway cross we are able to produce enough Rb pressure to easiIy see the background fluorescence with as little as 3.4 A through the getter oven. If any 324
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Fig. 6. Drawing of trapping cell. The tubes on the cross have been elongated in the drawing for ease of display. The Rb getter is inside a stainless steel vacuum rube; we show a cut-away view in this drawing. The zero length adapter is a 2.75 in. to 1.33 in. Conflat-type adapter.
fluorescence from the background can be seen, there is more than enough Rb for trapping. According to the data sheets, this is a negligible production rate of Rb, and hence the getter should last a long time. We have operated for 100 h with no sign of depletion of the Rb getter. The principal advantages of this approach are the simplicity of construction and the ability to quickly and easily adjust the Rb pressure. This is particularly important for a lab class experiment where the students will be able to work for only a few hours at a time. After the current through the getter has been turned on, the Rb vapor comes to an equilibrium pressure with a time constant of about 5 min. The overall pressure in the system rises slightly when the getter is first heated, but after degassing overnight by running a current just below that which produces observable Rb, the pressure rise is less than Pa, and continues to decrease with further use of the getter. When the current through the getter is turned off, the Rb pressure drops with a time constant of about 4 s if the getter has been on for only a short while. With prolonged use of the getter, this time constant can increase up to a few minutes as Rb builds up on the walls, but it decreases to the original value if the getter is left off for several days. Presumably this recovery would be much faster if the system was heated. The superior Rb pressure control provided by the getter makes it possible to use high pressures to easily observe fluorescence from the background vapor. This allows one both to check whether the laser and cell are operating properly, and to have rapid response in the number of atoms while optimizing trapping parameters. Once these tasks are complete, simply reducing the getter current provides low pressures almost immediately. Low pressures are desirable for many experiments because they yield relatively long trap lifetimes (seconds) and little background light from the fluorescence of untrapped atoms. We and others have used two alternative methods to produce Rb vapor. We mention the first only because it is used for many ordinary vapor cells, but is not a very good choice for this trap. In this approach a fraction of 1 g of Rb is distilled into the cell under vacuum and is condensed into a thermoelectrically cooled “cold finger. ” The vapor pressure in the cell is then adjusted by changing the temperature of the cold finger and allowing the pressure in the remainder of the cell to come to equilibrium. We do not recommend this Wieman, Flowers, and Gilbert
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approach because the pressure can only be changed very slowly, and (most importantly!) we have found that it is easy to accidently turn off or bum out the thermoelectric cooler. When this happens the entire vacuum system is filled to a relatively high Rb pressure. This is hard on the ion pump, and it is usually difficult and tedious to bake all the Rb back into the cold finger. A better method is to have the Rb reservoir behind a small stainless steel valve. This approach has been used in most of our research work. A small sealed glass ampoule containing about 1 g of Rb is installed in a thin-walled stainless steel or copper tube. We usually distill the Rb into the glass ampoules ourselves to insure that the Rb does not contain other dissolved gases, but it is also possible to buy such ampoules commercially. One end of the metal tube is sealed, and the other end is welded or brazed onto a 1.33 in. Conflat-type flange which is attached to a valve connected to the main cell. After the system has been thoroughly pumped out and baked, the tube is squeezed in a locking pliers (vice grips) until the metal wall has compressed enough to break the glass and the Rb is released. To get the Rb through the valve and into the cell, we heat the tube and the valve to 60100 "C. The first time this is done it takes between several hours and several days to see a significant amount of Rb in the cell (depending on the geometry, the surfaces, and the temperature). Once the pressure in the cell reaches the desired level, the valve to the reservoir is closed. At this point, the vapor will slowly begin to leave the cell, and after some time (days to weeks), the heating process will need to be repeated. After the initial Rb coating of the inside of the cell, it only requires several (10-30) minutes to substantially raise the Rb pressure in the cell upon subsequent heating of the reservoir. The length of time between fills depends on the geometry of the system and the properties of the surfaces. For Pyrex and stainless steel systems it will typically be between a day and a few weeks. For systems with more reactive surfaces it can be just a few minutes, in which case one may simply leave the valve open while carrying out experiments.
C. Windows and cell assembly The third important element of the trap cell is the optical access. This can be either through mounted windows, or through the cell walls themselves if they are transparent and flat. Optical access is necessary for sending the laser beams through the cell and for observing the fluorescence from the trapped atoms. Although there are many ways to assemble a cell with windows, we will discuss the three that are most practical along with their respective advantages and disadvantages. A component common to these designs is the Conflat-type vacuum flange. These flanges are a means of making all-metal vacuum seals which are leakfree, have very low outgassing, and can be baked to high temperatures. The stainless steel flanges are bolted together with a soft copper gasket between them. Knife edges on the flanges press into the copper to make the vacuum seal. The first technique for making a trap cell is one which we have developed specifically for this paper. This design is shown in Fig. 6. Five pieces of plate glass are sealed together with epoxy. This unit is epoxied onto a commercial zerolength 2.75 in. to 1.33 in. Conflat-type adaptor flange which is bolted onto a fiveway cross with 1.33 in. Conflat-type flanges. The sixth window (the bottom window) is a commercial 1.33 in. Conflat-type viewing port. The viewing port, 325
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ion pump, pumpout valve, and Rb getter are attached to the other arms of the cross. The advantages of this design are (1) allows very good optical access and viewing due to the geometrical arrangement and the optical quality of the windows, (2) has relatively low cost, (3) it can be readily built by someone without any special skills in working with glass, and, of least importance, (4) allows the use of antireflection coated windows. Its principal disadvantage is that the ultimate pressure is limited by the temperature to which the epoxy can be baked. However, we have achieved pressures below W7Pa in this type of cell using a small ion pump. We will start by offering some advice on ultrahigh vacuum (UHV)techniques, We strongly recommend, however, that those not familiar with the subject also read about UHV techniques in one of the many reference books on vacuum technology. First, brand new commercial components such as the cross require no additional cleaning and should be kept in their protective wrapping until they are to be assembled. If the components are possibly dirty, they should be cleaned. Our standard cleaning procedure is to put the component in an ultrasonic cleaner with hot water and a good lab detergent for a few minutes. Then it is rinsed in turn with hot tap water, distilled water, and high purity alcohol. After drying in air it is wrapped in clean aluminum foil until it is to be used. Any surface which will be at high vacuum should never be touched with unprotected hands after it has been cleaned. If it must be touched, clean gloves should be used. It is also desirable to avoid having things like hair fall into the vacuum chamber, and it is always wise to carefully inspect the system for such items. Clean new copper gaskets should be used when making the Conflat-type seals, and the flanges should be tightened down uniformly. Graphite or some other high temperature antisiezing compound should be put on the bolts to prevent them from seizing up after they are baked. Also, we find that plate nuts make the assembly of Conflattype seals much easier. Finally, to reach UHV it is necessary to bake even the cleanest systems under vacuum to remove adsorbed water vapor from the surface. The glass part of the cell is a parallelepiped with its long axis oriented vertically. Qpical dimensions of the vertical windows are about 2.75 cm by 4.5 cm, and the horizontal window on top is a square with side length 2.9 cm. The glass for the cell should be at least 2 mm thick; thinner glass may not be strong enough to withstand the pressure and much thicker glass can limit optical access. The windows should be cut to size with a standard glass saw. It is useful to cover the glass with masking tape while it is being cut to avoid scratching the surfaces. The precise dimensions of the glass pieces are not important, but it is important to have the cell fit together tightly to minimize the gaps that will be filled in with epoxy. To achieve this, one should make the opposite sides of the cell the same width, all four sides the same length, and all the comers of each piece perpendicular. To insure that the opposite sides are the same size, it is simplest to stack the two pieces and cut the stack as a single piece. After the pieces are cut, they should be washed carefully to remove any residual grit from the glass saw, and then the protective masking tape should be removed. The windows should then be cleaned just as the other vacuum components. The final assembly is done by first bolting the adaptor flange onto the fiveway cross, and then assembling the cell on top of it. It should be done in this order to avoid stressing the epoxy seals while bolting on the flange. Just before assembly, the inside surfaces of the five pieces of glass should Wieman, Flowers, and Gilbert
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be cleaned with lens tissne and pure alcohol or acetone to remove residual dust, which would scatter the laser light. Residual dust can be easily identified by looking for scattering while illuminating the window using a bright lamp. After the glass slides are clean they are pressed together to form a rectangular tube and placed upright on the flange. l b o small rubber bands are placed around the tube to hold the four glass sides firmly pressed together in this position. The square window is then set on top of the piped and a small weight is put on it to hold everything in place while the epoxy is applied. The seams are then sealed with a bead of low-vapor-pressure epoxy. Warming the epoxy slightly makes it somewhat easier to apply, but the glass itself should not be heated. When the epoxy has hardened somewhat (30 min or so) the elastic bands can be cut off without putting any stress on the cell, and epoxy should then be applied to the seams which were covered by the bands. By assembling the cell in this manner, very little epoxy will be exposed to the vacuum system. The epoxy should be allowed to set for 24 h at room temperature. The rest of the vacuum components can now be bolted in place. There are two items which deserve special mention. First, the pump-out valve should be a bakeable all-metal-seal valve to meet the UHV requirements. For systems which are not going to be let up to air very often, this valve can be replaced by a copper pinch-off tube, which acts as an inexpensive “one use” valve. After the system has been rough pumped and the ion pump started, the tube is pinched off leaving a permanent seal. This works well and avoids the expense of a valve. However, if the system is likely to be opened up a few times or if there is a possibility of leaks in the system, a valve is much more convenient. Second, the Rb getter comes as a powder packed in a small stainless steel boat or oven. It is best to spot weld two of these boats in series and then spot weld them between two 10 A feedthroughs on a 1.33 in. Conflat-type flange. They should be positioned as close to the trapping region as possible without blocking the path of the vertical laser beam. Protective gloves should be used when installing the getter to avoid contaminating the system. Once the entire cell is assembled it is a good idea to perform a helium leak test if a leak detector is available. In the process, it is quite important to avoid contaminating the system with backstreaming of pump oil from the leak detector. If the leak detector does not have a liquid nitrogen cooled trap to prevent this, install one in the line connecting the trap cell to the leak detector.
D. System pumpdown and bakeout The next step is to pump the system down to a low enough pressure to start the ion pump. Use whatever pumping options are available; a clean cryogenic sorption pump is usually sufficient and will not contaminate the system. If the rough pumping is done with a mechanical or diffusion pump it must be very well trapped to avoid oil contamination, as with the leak detector. We have found, particularly with the 2 //s ion pumps, that minimizing the gas load when the ion pump is started improves the long term performance of the pump. If the system was not previously helium leak tested, it should be leak tested after the ion pump has been started and the system closed off from the roughing pump. This testing can be easily done by squirting alcohol on all of the epoxied and copper gasket seals while monitoring the ion pump current. If there is a sudden change in the current when alcohol 326
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is sprayed on the seal, it signifies a leak which has just been covered by the alcohol. Although it is important to leak test the system, it is unlikely there will be any leaks if the Conflat seals have been installed properly and care is taken to insure that the epoxy is applied evenly over all the joints. After confirming that there are no leaks, the system is then baked. One of the drawbacks to this design is the fact that it is quite easy to open up a leak in the glass-to-metal epoxy seal while baking. The reasons for this are the different thermal expansion coefficients of the glass and metal, and more importantly, the fact that the glass and metal tend to heat and cool at very different rates. This results in thermally induced stress which can break the rigid epoxy seal. This problem would be eliminated if a flexible sealant were used, but we have been unable to find one which satisfies the requirements of strength and low vapor pressure, and is also bakeable to 100 O C . After trying a number of baking methods which resulted in leaks, we developed the following procedure, which has not caused leaks the three times we used it. First, the entire system is put inside an oven with a bakeable cable connecting the ion pump to the power supply, which remains outside the oven. The ion pump current (and hence the system pressure) should be monitored while the system is baking. A standard kitchen oven is ideal for this bake. Once the system is in the oven, the temperature should be slowly increased. Normally, the rate at which the temperature can be increased is determined by the maximum operating current of the ion pump and the strong temperature dependence of the outgassing rate in the system. It is not unusual for it to take many (6-8) hours before the temperature can be increased up to the maximum baking temperature of 100 “C. If the epoxy has not been N l y cured, it may be advisable to simply leave the system for a day at about 60 “C without attempting to increase the temperature further until the outgassing has dropped. Even if the pressure does not rise to undesirable levels, the temperature should not be changed faster than about 20 “C/h (up or down) to avoid thermal stresses. Once the temperature has been raised to 100 “Cthe system should be baked for at least a day. Then, it should be slowly cooled by progressively reducing the temperature of the oven. The pressure should drop dramatically as the cell cools and be less than Pa at room temperature. If the pressure in the system does not drop dramatically as it cools, or even increases with cooling, it almost always indicates a leak. The most likely place for a leak will be in the glass-tometal epoxy seal. If such a leak has opened, it can often be sealed by applying new epoxy to the offending region without ever letting the system up to air. This method of baking puts a fairly heavy load of gas into the ion pump. This can be reduced by a “prebake” of the system while it is on the roughing pump. With most facilities, however, it is awkward to insure proper uniform control over the temperature at the rough pumping stage. Once the cell has been baked it is ready to use. The ion pump can be turned off for long periods (up to a few days) without the pressure rising enough to cause problems. This is often convenient while the cell is moved and installed for use.
E.Alternative designs Before discussing the operation of the trap we will briefly mention two alternative methods of making trapping cells. The first is to have a glassblower make a six-sided glass cross with six windows on the tubes of the cross, and a glass-to-metal seal that allows the cell to be attached to the Wiernan, Flowers, and Gilbert
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393 ion pump, pump-out valve, and Rb source. The advantages of this system are that it is simple, it can be baked to very high temperatures, and it has very little tendency to leak. The disadvantages are (1) not everyone has a glassblower available, (2) it is very difficult to modify anything about the trap cell, (3) such designs are often rather fragile, (4) the optical access and trap visibility are limited due to the width of the fused glass seams, and ( 5 ) the glassblowing often causes distortions and scattering centers in the windows. Perhaps the more serious aspect of (4) is that this often results in the windows being so far from the center of the trap (depending on the skill and effort of the glassblower) that the retroreflecting mirrors are far away from the middle of the cell. As discussed above, this renders the alignment more difficult and can make it very difficult to have adequate overlap of the forward and backward going beams without introducing an unacceptable amount of feedback into the diode laser. The second alternative is to buy a commercial 2.75 in. or 1.33 in. sixway cross with Conflat-type flanges (or a cube with six 2.75 in. Conflat-type seals), and six viewports (windows mounted on stainless steel Conflat-type flanges). Then one simply bolts five of the windows onto the sixway cross (or cube). A fourway cross is bolted on the sixth side which has the Rb reservoir and pump attached at right angles, and the sixth window attached to the port opposite the sixway cross. The advantages of this system are that all the components are readily available and can simply be bolted together. The principal disadvantages are the cost and the poor viewing and optical access. Also, metal scatters more light, causing a large background which can obscure the signal from the trapped atoms. These problems and the corresponding difficulty in optical alignment are so great in a sixway cross that we do not recommend it for use in an undergraduate lab. A cube is much better in this respect, but cubes with six Conflat-type seals are relatively expensive.
F. Installation on the optical breadboard After the cell has been baked, it is installed on the optical breadboard. This is simple; one or two ringstand clamps can hold the cell and ion pump in place. It is advantageous to align the six trapping beams before the cell is installed, because testing the overlap of the beams is much easier when the cell is not in the way. The cell must be mounted so that the beams overlap roughly in the center of the glass region and there is enough room below the bottom window to allow the vertical beam to be reflected up from below the cell. Remember also to leave room for the X/4 wave plate below the bottom window and room to adjust the orientation of the h/4 plate and the bottom mirror. Since the cell is elevated, it will also be necessary to support the ion pump since it is the heaviest part of the system. With the cell installed and the laser beams aligned, the final ingredients for trapping are the magnetic field coils. A gradient of up to about 0.20 T/m (20 G/cm; normal trap operation is at 10-15 G/cm) is needed. All other details of the coils are unimportant. We use two freestanding coils 1.3 cm in diameter with 25 turns each of 24 gauge magnet wire. This provides the desired gradients with a current of 2-3 A and a coil separation of 3.3 cm. The coils should be mounted on either side of the cell such that the current travels through the loops in opposite directions and the coil axis is collinear with one of the laser beam axes. Our coils are simply attached to the windows or supporting flange with tape or glue. 327
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VI. OPERATION OF THE TRAP AND MEASUREMENTS A. Observation system Although not essential, it is highly desirable to have an inexpensive CCD TV camera and monitor with which to observe the trapping cell. Standard cameras used for security surveillance work well for this purpose. They will show the cloud of trapped atoms as a very bright white glow in the center of the cell. Because of the poor response of the eye at 780 nm, the trapped atoms can be seen by eye only if the room is quite dark. For aligning the trapping laser beams we usually use a standard IR fluorescent card or a piece of white paper if the room is darkened. (Note that although the laser beam appears weak, it is actually intense enough to cause damage if the beam shines directly into the eye.) A 1 cm2 photodiode with a simple current-to-voltage amplifier is used for making quantitative measurements on the trapped atoms. It is placed at any convenient position which is close to the trap, has an unobstructed view, and receives relatively little scattered light from the windows. The photodiode is used to detect the 780 nm fluorescence from the atoms as they spontaneously decay to the ground state from the 5P3,2 level. This measurement can be quantified and used to determine the number of trapped atoms. The same or a similar photodiode can also be used to look at the absorption by the trapped atoms and to monitor the Rb pressure in the cell by measuring the absorption of a probe laser beam. Although the trap fluorescence is large enough to easily detect with the photodiode, it can be obscured by fluorescence from the background Rb vapor or by scattered light from the cell windows. Over a large range of pressures, the fluorescence from the background gas will be smaller than that from the trapped atoms. However, the scatter from the windows is likely to be significant under all conditions. We have found that with minimal effort to reduce it, the scattered light background will simply be a constant offset on the photodiode signal. For studying small numbers of atoms in the trap, however, the noise on this background can become a problem. In this case, one should use a lens to image the trap fluorescence onto a mask which blocks out the unwanted scattered light.
B. Trap operation When the cell and all the optics are in place, the first step is to turn on the getter to put Rb into the cell. Initially, monitor the absorption of a weak probe beam through the cell to determine the Rb pressure. Although the trap will operate over a wide range of pressures, a good starting point is to have about l%/cm absorption on the F = 2 to upper states transition in the region of the trap. At this pressure it is possible to see dim lines of fluorescence where the trapping beams pass through the cell when the trapping laser is tuned to one of the Rb transitions. It is often easier to identify this fluorescence by slowly scanning the laser frequency and looking for a change in the amount of light in the cell. While absorption measurements are valuable for the initial setup and for quantitative measurements, in the standard operation of the trap, use the setting on the getter current and/or the observation of the fluorescence to check that the pressure is reasonable. After an adequate Rb pressure has been detected, the magnetic field gradient is turned on and the lasers are set to the appropriate transitions. If the apparatus is being used for the Wieman, Flowers, and Gilbert
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first time, it will be necessary to try both directions of current through the field coils to determine the correct sign for trapping. The trapped atoms should appear as a small bright cloud, much brighter than the background fluorescence. Pieces of dust on the windows may appear nearly as bright, but they will be more localized and can be easily distinguished by the fact they do not change with the laser frequencies or magnetic field. The cloud may vary in size; it can be anywhere from less than 1 mm in diameter to several millimeters. Blocking any of the beams is also a simple method for distinguishing the trapped atoms from the background light. If the trap does not work (and the direction of magnetic field and the laser polarizations are set correctly) the lasers are probably not set on the correct transitions.
.’
,I 0
:
:
:
:
:
:
:
5
!
I
10
Time (s)
C. Measurements
Fig. 7. Number of trapped atoms versus time after the trap is turned on at
Although many other more complicated measurements could be made with the trap, here we will discuss how to make the two simplest measurements: the number of trapped atoms and the time which atoms remain in the trap. Both of these measurements are made by observing the fluorescence from the trapped atoms with a photodiode. The number of atoms is determined by measuring the amount of light coming from the trapped atoms and dividing by the amount of light scattered per atom, which is calculated from the excited state lifetime. The time the atoms remain in the trap is found by observing the trap filling time and using Eq. (1). To make a reliable measurement of the number of trapped atoms, it is crucial to accurately separate the fluorescence of the trapped atoms from the scattered light and the fluorescence of the background vapor. To do this one must compare the signal difference between having the trap off and on. Therefore, the trap must be disabled in a way that has a negligibly small effect on the background light. We have found that turning off or, even better, reversing the magnetic field is usually the best way to do this. The magnetic field may alter the background fluorescence, but this change is generally much smaller (1/100) than the signal of a typical cloud of trapped atoms. Once one has determined the photocurrent due to the trapped atoms, the total amount of light emitted can be found using the photodiode calibration and calculating the detection solid angle. The rate R at which an individual atom scatters photons is given by” (4)
where I is the sum of the intensities of the six trapping beams, r is the 6 MHz natural linewidth of the transition, A is the detuning of the laser frequency from resonance, and I , is the 4.1 mW/cm2 saturation intensity. The simplest way to find A is to ramp over the saturated absorption spectrum and, when looking at the locking error signal, find the position of the lock point (zero crossing point) relative to the peak of the line. The frequency scale for the ramp can be determined using the known spacing between two hyperfine peaks. A typical number for R is 6X106 photons/(s.atom). One can optimize the number of atoms in the trap by adjusting the position of the magnetic field coils, the size of the gradient, the frequencies of both trapping and hyperfine pumping lasers, the beam alignments, and the polarization of the beams. With 7 mW from the trapping laser, we obtain nearly 4x10’ trapped atoms when the Rb pressure is large enough to dominate the lifetime. 328
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1.2 s.
The filling of the trap can also be observed using the same photodiode signal. This is best done by suddenly turning on the current to the field coils to produce a trap. The fluorescence signal from the photodiode will then follow the dependence given by Eq. (l),as shown in Fig. 7. The value of the trap lifetime 7 (the characteristic time an atom remains trapped) can then be determined from this curve. We have observed lifetimes which ranged from nearly 4 s down to a small fraction of 1 s depending on the Rb pressure. There are many other experiments which can be done with the trapped atoms. The choices are only limited by one’s imagination and/or available equipment, but here we list a few examples. Although optical traps are being used in a number of experiments, many aspects of the trap performance have not been studied. By changing the current through the getter, one can vary the Rb pressure and see how it affects the number and lifetime of the trapped atoms. This can be compared with the predictions of Eqs. (1) and (2) and can be used to determine collision cross sections; to date, very few trap ejection cross sections have been measured. One can also measure the temperature of the trapped atoms, the spring constant of the trap, and how the number of atoms and the lifetime depend on the various parameters. To our knowledge the detailed dependence of these characteristics on the polarizations of the laser beams, the intensity in the various beams, or the magnetic field gradient have not been studied. You can quickly learn that, in fact, the trap will work well with one of the incident beams linearly polarized, and can even work marginally with two linearly polarized beams, but we know of no analysis or studies of these cases in the literature. It is also relatively easy to do various kinds of high resolution spectroscopy on the trapped atoms because the optical thickness of the trapped atom cloud is substantial and the Doppler shifts are essentially zero. Using another diode laser one can precisely measure the 5S-+5P spectra with the hyperfine components clearly resolved. In addition, there are convenient transitions which allow one to use diode lasers to study excitation from the 5P state to higher levels.’* It is also possible to observe very high resolution microwave transitions between the hyperfine states if a suitable source of microwave power is a~ai1able.l~ By turning the laser light off and adding small magnetic fields, it is possible to magnetically trap4 or “bounce” the Wieman, Flowers, and Gilbert
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laser-cooled atoms off of an inhomogeneous magnetic field. This dramatically demonstrates the quantization of angular momentum because the different Zeeman levels have different magnetic moments, and hence feel different magnetic forces. This causes the cloud to separate into a number of smaller clouds as it bounces, each of which represent a different projection of the angular momentum on the magnetic field. We conclude with both an invitation and a warning about many experiments one can do with optical traps: this is a new and rapidly changing field; one is likely to observe phenomena which are not explained in the current research literature and there are no textbooks to provide answers. The student and instructor may find themselves in uncharted territory.
ACKNOWLEDGMENTS This work was supported by the NSF and the University of Colorado. Early design work on this experiment was carried out by K. MacAdam, A. Steinbach, and C. Sackett. We thank M. Stephens for considerable help and advice. G. Flowers acknowledges the support of a tuition scholarship fiom the NIST PREP Fellowship program.
APPENDIX COMPONENTS USED IN TRAP CONSTRUCTION Listed here are the components that we used in construction of the trap and the suppliers of these components. Commercial trade names are identified to make this article useful to the reader. Since we have not made a careful study of other products, we list only those that we actually used; the prices given are current as of 1993 and are meant for guidance only. No endorsement by NIST or the University of Colorado is implied; similar products by other manufacturers may work as well or better. Buyer’s guides such as those published by Physics Today (with the August issue), Photonics Spectra, and Laser Focus World are good sources for listings of vendors. Optical Base # B A S , 16 @ $8 each, Thorlabs, Inc., P. 0. Box 366, Newton, NJ 07860, (201) 579-7227. Post holder # PH3-ST, 16 @ $12 each, Thorlabs, Inc. (see item 1). Posts # TR2,3 @ $5 each; # TR3,4 @ $6 each; # TR4 1 @ $7; # TR6 14 @ $8 each; # TRlO 1 @ $10, Thorlabs, Inc. (see item 1). Kinematic mirror mounts, # KM-B, nine @ $50 each; # KMS, one @ $41, Thorlabs, Inc. (see item 1). Right angle post clamps # RA90, seven @ $12 each, Thorlabs, Inc. (see item 1). Commercial grade quarter-wave plates # RCQ-1.O0780, 25.4 mm diameter, six @ $184 each (additional charge for antireflection coating). Meadowlark Optics, Inc., 7460 Weld County Road # 1, Longmont, CO 80504, (303) 776-4068. As noted in the text, there are other options for wave plates. These wave plates are destroyed by mild heating. Near-infrared diekectric mirror, 50.8 mm diameter, to be cut in quarters for four mirrors. Coating BD.2, Substrate Code 20D04, $243, Newport Corporation, 1791 Deere Ave., Irvine, CA 92714, (800) 222-6440. Flat gold mirror # G41, 801, 76x102 mm to be cut into six 25x25 mm square mirrors and one 25x50 Am. J. Phys., Vol. 63, No. 4, April 1995
mrn mirror, $22, Edmund Scientific Co., 101 East Gloucester Pike, Barrington, NJ 08007-1380, (609) 573-6250. Rb metal getter Rb/NF/3.4/12/FT 10+10 Code 580125, $250, SAES Getters USA, Inc., 1122 E. Cheyenne Mtn. Blvd., Colorado Springs, CO 80906, (719) 576-3200. Eleven //s ion pump with magnet and bakeable cable, $880, Duniway Stockroom Corp., 1600 Shoreline Blvd., Mountain View, CA 94043, (800) 446-8811; 2 //s ion pump with square magnet and bakeable cable, $890, Perkin Elmer Physical Electronics Division, 6509 Flying Cloud Drive, Eden Praire, MN 55344, (612) 828-6100. High voltage ion pump power supply # 222-0242 Ionpak series 240, $550, Perkin Elmer, Physical Electronics Division, 6509 Flying Cloud Drive, Eden Praire, MN, 55344, (612) 828-6100. Reducing Flange, 2.75 in. to 1.33 in. Conflat-type # 150001, $55, MDC Vacuum Products Corp., 23842 Cabot Blvd., Hayward, CA 94545-1651, (800) 4438817. Fiveway cross with 1.33 in. Conflat-type flange #406000, $160, MDC Vacuum Products Corp. (see above). Vacuum 1.33 in. Conflat-type viewport # VP-133-075, $91, Duniway Stockroom Corp., 1600 Shoreline Blvd., Mountain View, CA 94043, (800) 446-8811. Power feedthrough for Rb getter, Moly 10 APin # EFT0024032, $125, Kurt J. Lesker Co.,1515 Worthington Ave., Clairton, FA 15025, (800) 245-1656. Stainless steel valve. Nupro valve (part SS-4BG-TW) which HPS buys and welds 1.33 in. Conflat-type flanges (HPS # 100881023) on each side and sells for $229. HPS, Division of MKS Instruments, 5330 Sterling Drive, Boulder, CO 80301, (303) 449-9861. These valves have a tendency to leak unless considerable torque has been used to tighten the bonnet seal. Copper Adapter #953-0706 (copper pinch-off tube as alternative to valve), $131 for pkg. of 4, Varian Vacuum Products, 121 Hartwell Ave., Lexington, MA 02173-9856, (800) 882-7426. Pyrex plate window #Q720125, 50.8X50.8X3.18 mm, $12 each, ESCO Products, Inc., 171 Oak Ridge Rd., Oak Ridge, NJ 07438-0155, (201) 697-3700. Torr Seal Epoxy (low vapor pressure epoxy) #9530001, $41, Vanan Vacuum Products (see item 17). Transmissive IR Viewing Card Model #Q-ll-T, $79.20, Quantex, 2 Research Ct., Rockville, MD 20850, (301) 258-2701. A transmissive (as opposed to reflective) viewing card is particularly useful for aligning laser beams. Photodiode #PIN-10 DP, $55.00, United Detector Technology, 12525 Chadron Avenue, Hawthorne, CA 90250, (310) 978-0516. ’K. B. MacAdam, A. Steinbach, and C. Wieman, “ A narrow-band tunable
diode laser system with grating feedback and a saturated absorption spectrometer for Cs arid Rb,” Am. J. Phys. 60,1098-1111 (1992). *S. L. Gilbert and C. E. Wieman, “Laser cooling and trapping for the masses,” Optics and Photonics News 4, No. 7, 8-14 (1993). %e initial demonstration and discussion is E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59,2631-2634 (1987). More detailed analysis of the trap is given in A. M. Steane, M. Chowdhury, and C. J. Wieman, Flowers, and Gilbert
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Foot, “Radiation force in the magneto-optical trap,” J. Opt. Soc.Am. B 9, 2142 (1992), while a discussion of many of thc novel aspects of the behavior of atoms in the trap is given in D. Sesko, T. Walker, and C. Wieman, “Behavior of neutral atoms in a spontaneous force trap,” J. Opt. SOC. Am. B 8, 946-958 (1991). 4C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571-1574 (1990); K. Lindquist, M. Stephens, and C. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46,4082-4090 (1992). ’T. W. Hansch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68-69 (1975). %. Chu, L. Hollberg, J. Bjorkholm, A. Cable, and A. Ashkin, “Threedimensional viscous confinement and cooling of atoms by resonance radiation pressure,” Phys. Rev. Lett. 55, 48-51 (1985). ’5. Dalibard and C. Cohen-Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: Simple theoretical models,” J. Opt. SOC. Am. B 6, 2023-2045 (1989); P. Ungar, D. Weiss, E. Riis, and S. Chu, “Optical molasses and multilevel atoms: Theory,” 1. Opt. SOC.Am. B 6, 2058-2072 (1989). ‘L. Orozco, SUNY at Stonybrook (private communication).
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9Prevenr the shielding from coming into direct contact with the magnet. This can be done by placing small cardboard spacers between the shield and the magnet. ’%. Stephens and C. Wieman, “High collection efficiency in a laser trap,’’ Phys. Rev. Lett. 72, 3787-3790 (1994); M. Stephens, R. Rhodes and C. Wieman, “A study of wall coatings for vapor-cell laser traps,” J. Appl. Phys. 76, 3479-3488 (1994). “For a discussion of saturated transition rates, see W.Demtroder, Laser Spectroscopy (Springer, New York, 1982), pp. 105-106. ‘*R. W. Fox, S. L. Gilbert, L. Hollberg, J. H. Marquardt, and H. G.Robinson, “Optical probing of cold trapped atoms,” Opt. Lett. 18, 1456-1458 (1993); A. G. Sinclair, B. D. McDonald, E.Riis, and G.Duxbury, “Double resonance spectroscopy of laser-cooled Rb atoms,’’ Opt. Commun. 106, 207-212 (1594); T. T. Grove, V. Sanchez-Vllicana, B. C. Duncan, S. Maleki, and P. L. Could, “’ho-photon two-color diode laser spectroscopy of the Rb 5D,,2 state,” Phys. Scr. ”C. Monroe, H. Robinson, and C. Wieman, “Observation of the cesium clock transition using laser cooled atoms in a vapor cell,” Opt. Lett. 16, 50-52 (1991). This work was done using trapped cesium atoms.
6 1995 American Association of Physics Teachers
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Gravitational Sisyphus Cooling of 87Rb in a Magnetic Trap N. R. Newbury, C. J. Myatt, E. A. Cornell, and C. E. Wieman Joint Institute f o r Laboratory Astrophysics. Nutional Institute of Standards and Technology and University of Colorado, and Physics Department, University of Colorado, Boulder, Colorado 80309-0440 (Received 17 October 1994)
We describe a method for cooling magnetically trapped 87Rb atoms by irreversibly cycling the atoms between two trapped states. The cooling force is proportional to gravity. The atoms are cooled to 1.5 p K in the vertical dimension. We have extended this cooling method to two dimensions through anharmonic mixing, achieving a factor of 25 increase in the phase space density over an uncooled sample. This cooling method should be an important intermediate step toward achieving a BoseEinstein condensate of Rb atoms PACS numbers: 32.80.P~
There is an ongoing effort in the field of laser cooling and trapping to achieve lower temperatures and higher densities for a variety of purposes, particularly the attainment of Bose-Einstein condensation. In three dimensions, the lowest achieved temperature for a six-beam molasses is about 13T, 11-31, where T , is the photon recoil temperature [4]. Recently, temperatures as low as 3.5Tr have been reached using both a four-beam molasses [ 5 ]and Raman cooling [6]. These temperatures have been achieved in low density untrapped samples. A method of cooling a trapped atomic sample to comparable temperatures has obvious advantages since the cold and dense sample is preserved for subsequent experiments. The standard magneto-optical trap (MOT) [7] both confines and optically cools atoms. However, a variety of fundamental processes limit both the density and temperature of the atoms in a MOT [8-111. We have developed a new method to cool large numbers of magnetostatically trapped atoms to temperatures near the recoil limit T,. An important bonus of cooling magnetically trapped atoms is that the density increases as the atoms are cooled. Currently, the only proven method of optically cooling atoms in a magnetic trap is doppler cooling [12,13], which reaches the relatively high temperatures of a few mK. Here we describe a new method called “gravitational Sisyphus cooling” which works by cycling the atoms between two parabolic potential wells whose minima are offset due to the gravitational force on the atoms and a slight difference in magnetic moments. The atoms are slowed by a force proportional to gravity, and the irreversibility is provided by the spontaneous emission of a photon during optical pumping. For temperatures above the recoil limit, the process is extremely efficient, removing >50% of the atom’s potential energy per cycle. For technical reasons, we have worked in that regime and have cooled Rb in one dimension to the “submolasses” temperature of 1.5 p K = 4T,. In principle, however, this technique will allow one to cool to below the recoil limit in one dimension in a manner completely analogous to Raman cooling [6]. Although this technique is inherently one dimensional, we have extended it to two dimensions
through anharmonic mixing, and it could be extended to three dimensions using either collisional or anharmonic mixing. Unlike optical molasses [ 1 I], this cooling method is insensitive to optical thickness and therefore can cool very large dense samples to within a few recoil energies. Similar ideas for cooling atoms in a magnetic trap have been proposed by Pritchard and co-workers, but never successfully implemented. In the original proposal [14-161, atoms were to be cooled by cycling between two harmonic potential wells with very different curvatures. However, as realized by Pritchard and Ketterle [17], gravity shifts the minima of two such potential wells vertically and for cold atoms completely foils the cooling. They then proposed a modified cooling scheme which, like gravitational Sisyphus cooling, used gravity as a slowing force. There are several technical but important distinctions between that cooling scheme and gravitational Sisyphus cooling. First, we use states with magnetic moments differing by s 3 % , rather than a factor of 2. For a given trap the final temperature will be a function of the difference in the magnetic moments of the two states. Second, while the proposal of Ref. [ 171 relied on a highly nonparabolic asymmetric potential, we use a parabolic magnetic trap which is simple to construct and provides better confinement of the atoms. Third, and most importantly, with gravitational Sisyphus cooling we have demonstrated how to efficiently drive real atoms (87Rb) between two wells in the desired manner. This is the principle challenge to implementing most cyclic cooling schemes. Gravitational Sisyphus cooling is shown schematically in Fig. 1. Initially there is a thermal distribution of cold magnetically trapped atoms in the “weak-field seeking” IF = I , m = -1) ground state of *’Rb ( I = 3/2). To understand the cooling mechanism, we consider the magnetic trap potentials for the [ I , - 1) and 12, + 1) states. The potential for the 11, - 1) state is V I , - , = I p ~ . - l l l B l+ mgz, where g is the gravitational acceleration and B is the total magnetic field. To cancel gravity, a pair of anti-Helmholtz coils supply a magnetic field gradient d B / d z = m g / p l , - l = -31 G/cm. The mag-
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process), but the change in the potential energy is AU
Vertical Position (rnrn)
Atoms are initially loaded into the 11, - 1 ) slate. A two-photon excitation, resonant at z = zo, excites atoms to the upper 12, I ) potential well. After one-half of an oscillation period, an optical pumping pulse drops the atoms back to the I I , - 1) state. After a number of cooling cycles, the cloud is compressed spatially to a rrns size of - A 2 and a rms velocity width of w , A z = 2 ( A k / r n ) . FIG. 1.
netic field of the remaining coils then produce a simple harmonic potential VI,-l = p ~ , - ~+&rnw,2x2/2 + mw,2y2/2 + rnw:z2/2. To first order in B, the difference in the magnetic moment of the 12, +1) and 11, -1) states is A p = p 1 2 , + ~ -) ~ I . - I= 2 ( Q & - p , ~ ) where , Q = 431 Hz/G2, and p~ = 1.4 kHz/G is the nuclear moment. This slight difference in magnetic moments results in two differences between the potentials of the 11, - 1) and the 12, + 1) states. The first effect is a small and negligible difference in the three oscillation frequencies of atoms in the two potentials. The second effect [I81 is to displace the minima of the two wells in the vertical direction by
Gravitational Sisyphus cooling exploits this separation of the two wells to cool the atoms. A direct result of Eq. (1) is an almost linear variation with z in the energy difference between the two potential wells. Therefore, by selecting the frequency of a two-photon transition between these states, we can excite atoms at a specific vertical position from the 11, - 1) to the 12, + 1) state. Because of the bias magnetic field, this excitation is far off resonance for any other transitions. Consider a narrow slice of the distribution initially at z = zo which is excited to the upper well by a short two-photon excitation pulse. Since the excitation is velocity independent, the velocity distribution of the excited atoms is identical to the velocity distribution of atoms in the lower well [19]. After a half oscillation period P / 2 , the excited atoms will have oscillated to the opposite side of the upper parabolic well which is centered at A z . These atoms are then optically pumped back to the lower well with laser light that does not excite atoms already in the 11, - 1) state. After the cooling cycle, there is no change in kinetic energy (ignoring for now any effects of the optical pumping
=
-2rnw:Az(zo
-
Az).
(2)
Because of the dependence on 2 0 , the larger the initial potential energy, the larger the decrease in the potential energy. For 1.2Az < zo < 7 A z , over 50% of the potential energy of the cycled atoms is removed in a single cooling cycle. In order to cool as many atoms as possible, the ideal cooling cycle would involve rapidly sweeping the two-photon excitation frequency so that all atoms at z > Az are excited to the upper well. This extended cloud will have an initial mean displacement which we identify as zo. Equation (2) then describes the energy removed from the common or center-of-mass motion of the cycled atoms. Depending on the experimental implementation, the optical pumping can impart some number n of directed momentum “kicks” (= l i k ) to the atoms. In order to remove the most total (kinetic and potential) energy from the cycled atoms there is an optimum delay time T , between ~ ~ the two-photon excitation and the optical repumping of the atoms to the lower well. If the optical repumping involves n directed momentum kicks upward, one can show the optimum delay time is given by w z ~ o p= t arctan(-nlik/mw,Az), and does not depend on zo. The optimum delay time ranges from P / 2 for m w , A z >> n h k (the situation described above) to P / 4 for rnw7Az << n h k . The general expression for the amount of energy removed is straightforward to derive but slightly more complicated than Eq. (2) and like Eq. (2) is negative, so that the atoms are cooled, only for sufficiently large ZO. Because the atoms are in a harmonic trap, by repeating the cooling cycle at intervals of 3P/4, all atoms regardless of the initial phase of their trajectories are cooled. Equation (2) shows the atoms can be cooled to a size - A z or, equivalently, T7 rn(w,Az)’ for n = 0. Therefore, by either lowering the bias field or raising w 7 the distribution could be cooled to arbitrarily small Az and temperature. Naively, the limiting temperature is set by the recoil heating in the optical pumping and hence will be no lower than T,. However, for low density clouds, temperatures below T,. are possible through a diffusion to low energy states in a manner similar to that discussed in Refs. [6,15,20]. Of course, as in any subrecoil cooling scheme, the diffusion of the distribution toward lower and lower temperatures takes a progressively longer time. In a high density sample, elastic collisions will prevent any non-Boltzmann accumulation of atoms in very low energy states. Our experimental implementation of gravitational Sisyphus cooling uses much of the apparatus and initial cooling techniques described in Ref. [21]. We use diode lasers and a standard vapor-cell MOT [2,7] to collect 107-108 atoms in about I min, as measured from the fluorescence detected by a photodiode. The atom sample is cooled further with an optical molasses [ l ] and then loaded in situ into the
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dc purely magnetic trap [21,22] with w, = ( 2 ~ ) 9 . 3 S0- I , wy = (2~)14.4 s-I, and w z = ( 2 ~ ) 4 . 3 s5- ' . The lifetime of atoms in the magnetic trap was limited to -6 s by the background pressure in the cell. The distribution and number of the trapped atoms can be measured (destructively) at any given time by suddenly turning off the magnetic trap and exciting the atoms with a 1 ms pulse of both the hyperfine repumping light 5s *S1p(F = 1) 5 p ' P3/ 2 ( F' = 2 ) and laser light tuned three linewidths to the red of the 5s 2 S I / 2 ( F= 2) 5 p ' P 3 / 2 ( F ' = 3) transition. To determine the spatial distribution of the trapped atoms, and equivalently the velocity distribution [21J, the fluorescence is imaged with a charge-coupled-device camera and one raster of the image recorded by a PC. The two-photon excitation is implemented by a microwave photon at 6833 MHz and an rf photon at 2 MHz. The microwave frequency is detuned by 15 MHz from the 11, - 1) to l2,O) transition so that the 11, - 1) and (2, + I ) states can be regarded as a simple coupled two level system. The microwaves are generated by an HP8627A synthesizer, amplified using a 30 W traveling wave tube amplifier and directed at the atoms with an open-ended waveguide. The applied rf is linearly polarized resulting in nearly identical energy shifts of the two trapped states. The coupling of the microwave magnetic field to the actual trap region is poor so that the two-photon Rabi frequency of -lo4 s-' is on the order of the motional linewidth. Increasing the rf field to increase the Rabi frequency is undesirable because it distorts the trap potential and shortens the trap lifetime by heating the walls. For this low Rabi frequency, there is nothing to be gained by sweeping the frequency as described above, and we simply fix the microwave and rf frequency to excite atoms at a fixed vertical position ZO. After the cloud has been cooled to a size less than ZO, the microwave frequency is reduced to excite atoms at a lower vertical position. This approach reduces the cooling efficiency by about a factor of 2 over the ideal case discussed above. We optically pump the atoms using two different laser beams. A 0.5 ms pulse of circularly polarized light propagating at -30" with respect to the quantization axis in the i - 2 plane drives the 5s ' S l / z ( F = 2) 5p *P3/2(F' = 2) transition, quickly pumping atoms into the 12, -2) state. This light will also excite those atoms to the IF' = 2 , m = -2) state which decays into the F = 1 ground state, predominantly into the trapped 11, -1) sublevel. To retrieve the -20% of atoms lost to the 11.0) and 11, + I ) states, two circularly polarized 30 ps pulses of light drive the 5s 2 S l p ( F = 1) 5 p ' P 3 / 2 ( F ' = 1) transition at 795 nm during the 0.5 ms optical pumping 4 recoil kicks upward pulse. The atoms receive n during this repumping. Therefore, we choose w A z = ( n / 2 ) l i k / m = 2Akjm. A typical single cooling cycle begins with an 80 ms two-photon excitation pulse (defined by gating the microwave power), followed by a 40 ms pause, and then a --t
--t
20 MARCH1995
0.5 ms optical pumping pulse to return the atoms from the (2, 1) to the ( I , -1) state. To cool the cloud to 1.5 p K = 4T,, we use eight cooling cycles at a twophoton frequency of 6834.91 MHz and then an additional eight cycles at 6834.92 MHz. The entire cooling process takes less than 2 s. Results of one-dimensional gravitational Sisyphus cooling are shown in Figs. 2(a) and 2(c). The temperatures quoted in Fig. 2 are calculated from the full width at the e - 1 / 2 points of the velocity distribution averaged over one oscillation period of the trap. We can further reduce the residual high velocity tails of the distribution by increasing the number of cooling cycles. If the number of absorbed photons were reduced, we could lower A z correspondingly and reach temperatures closer to T,. This number could be reduced to one or two by using a technique for coherently transfemng atoms between the 12, + I ) and 12, -1) states. The attraction of our current scheme is that less than 2% of the excited atoms are lost to untrapped states in a cooling cycle and it requires only a single additional 20 mW laser. Since gravitational Sisyphus cooling is restricted to the vertical dimension, we cool the sample in two dimensions (see Fig. 2) by anharmonically mixing the vertical and P dimensions. First the vertical distribution is cooled. Then, by adiabatically changing the trap parameters, we produce a degeneracy w , = w z = (2n)3.5 s-'. Anharmonicities
Vertical
~,
13 L
,'
v 0 v
F
0 -10
0
10
-
-+
-
2198
'. - 10
0
10
-10
0
10
Velocity ( c m / s e c ) FIG. 2. The results of two-dimensional gravitational Sisyphus cooling. (a), (b) The vertical and 4 velocity distributions before cooling (dashed lines: T, = 19 p K , T , = 12.5 p K ) and after cooling (solid lines: '7 = 20 p K , T , = 1.6 p K ) . After anharmonic mixing of the vertical and 2 directions, the velocity distributions are given by the dashed lines of (c) and (d) (T, = 3.4 p K , T, = 8.3 p K ) . A final gravitational Sisyphus cooling gives the solid curves of (c) and (d) (TI = 6.4 p K , T,
=
1.5 p K ) .
400 VOLUME74, NUMBER12
PHYSICAL REVIEW LETTERS
cause the atom cloud to rotate in the x-z plane, exchanging the 2 and 2 distributions in -0.65 s. After w e adiabatically restore the trap to the initial field conditions, the one-dimensional phase-space density associated with the L direction is equal to the one-dimensional phase-space density of the cooled vertical direction before mixing. Since one-dimensional phase-space density, rather than energy, is exchanged, w e expect a final horizontal temperature after mixing of T,f = (w,/w,)T,, where TI is the initial vertical temperature, which agrees well with Fig. 2(d). We then cool the vertical distribution again. The entire process is completed in -5 s. T h e phase-space density, which goes as (TXT,Ty)-lfor a simple harmonic trap, is increased by a factor of 25, compared to an uncooled sample after 5 s. Cooling in all three dimensions is also possible using anharmonic mixing but w e cannot achieve the required condition w y = w , with our magnet power supplies. The heating of -3T, in the 4 direction during gravitational Sisyphus cooling results primarily from the velocity kick of n sin30" h k / m in the P direction during an optical pumping pulse and could be substantially reduced by the more advanced optical pumping schemes mentioned earlier. In a denser sample, elastic collisions will mix the energy of the three dimensions so that T, = T y = T, in a time less than the trap lifetime. Anharmonic mixing, on the other hand, gives T,/w, = T y / w y = T,/w, so the corresponding increase in phase space density will be greater for collisional mixing since w,, w), > w z . We have demonstrated gravitational Sisyphus cooling for a magnetic trap loaded in situ from a relatively small MOT (-107-108 atoms). To take full advantage of gravitational Sisyphus cooling would require loading a large number ( z l O 1 o )of atoms into a long-lived magnetic trap. A large sample of magnetically trapped atoms loaded from a M O T would have a high initial potential energy, because of the fixed density of a MOT [8,9], and a high kinetic energy due to an elevated molasses temperature in large samples [ 111. However, gravitational Sisyphus cooling and collisional mixing could cool and compress this distribution to -1 p K in all three dimensions. Any heating effects associated with optical thickness of the sample could be greatly mitigated by optical pumping with light far detuned from resonance. After cooling, the thermalization rate of the sample would be very much larger than the trap loss rate and thereby permit efficient evaporative cooling to lower temperatures [23]. This work is supported by the Office of Naval Research and the National Science Foundation. W e are pleased to acknowledge many useful discussions with M. Anderson and W. Petrich.
[I] See the special issue on laser cooling and trapping of atoms edited by S. Chu and C. Wieman [J. Opt. SOC.Am. B 6, No. 11 (1989)]. [2] C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571 (1990).
20 MARCH1995
C. Salomon, J. Dalibard, W. D. Phillips, A. Clairon, and S. Guellati. Europhys. Lett. 12. 683 (1990). According to the equipartition theorem, the recoil temperature is properly defined as ksT, = ( A k ) * / M , where M is the atomic mass and Ak is the laser photon momentum. This corresponds to twice the recoil energy (Akl2/2M.
A. Kastberg, W.D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and P.S. Jessen, Phys. Rev. Lett. 74, 1542 (1995). N. Davidson, H.-J. Lee, M. Kasevich, and S. Chu, Phys. Rev. Lett. 72, 3158 (1994). E.L. Raab, M. Prentiss, A. Cable, S. Chu, and D.E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987). T. Walker, D. Sesko. and C. Wieman, Phys. Rev. Lett. 64, 408 (1990). W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, Phys. Rev. Lett. 70, 2253 ( I 993). A.M. Steane, M. Chowdhurry, and C. J. Foot, J. Opt. SOC. Am. B 9, 2142 (1992). G. Hillenbrand, C.J. Foot, and K. Burnett, Phys. Rev. A50, 1479 (1994); A. Clairon, Ph. Laurent, A. Nadir, M. Drewsen, D. Grison, B. Lounis, and C. Salomon (unpublished). D.E. Pritchard, K. Helmerson, and A.G. Martin, in Aromic Physics 11, edited by S. Haroche, J. C. Gay, and G. Grynberg (World Scientific, Singapore, 1989). p. 179; V.S. Bagnato, G.P. Lafyatis, A.G. Martin, E.L. Raab, R.N. Ahmad-Bitar, and D.E. Pritchard, Phys. Rev. Lett. 58, 2194 (1987). I.D. Setija, H.G. C. Werij, 0 . J. Luiten, M. W. Reynolds, T. W. Hijmans, and J. T. M. Walraven, Phys. Rev. Lett. 70, 2257 (1993). D.E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983). D. E. Pritchard, K. Helmerson, V. S. Bagnato, G. P. Lafyatis, and A. G. Martin, in Laser Spectroscopy VIIl, edited by W. Persson and S. Svanberg, (Springer-Verlag, Berlin, 1987). p. 68. B. Hoeling and R.J. Knize, Optics Commun. 106, 202 (1994). D.E. Pritchard and W. Ketterle, in Laser Manipulation of Atoms and Ions, edited by E. Arimondo, W. D. Phillips, and F. Strumia (North-Holland, Amsterdam, 1992). p. 473. This result holds even if the anti-Helmholtz field gradient does not exactly cancel gravity. We assume the initial mean velocity of the excited cloud is zero which will be true provided there are no correlations between position and velocity of the atoms in the lower well and there is no center-of-mass motion of atoms in the lower well. F. Bardou, B. Saubamea, J. Lawall, K. Shimizu, 0. Emile, C. Westbrook, A. Aspect, and C. Cohen-Tannoudji. C. R. Acad. Sci. Paris 318, 11, 877 (1994). C.R. Monroe, E.A. Comell, C.A. Sackett, C.J. Myatt, and C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993). T. Bergeman, G. Erez, and H. J. Metcalf, Phys. Rev. A 35, 1535 (1987). J.M. Doyle, J.C. Sandberg, I.A. Yu, C.L. Cesar, D. Kleppner, and T.J. Greytak, Phys. Rev. Lett. 67, 603 (1991); H.F. Hess, Phys. Rev. B 34, 3476 ( I 986).
2199
VOLUME 51, NUMBER 4
PHYSICAL REVIEW A
APRIL 1995
s-wave elastic collisions between cold ground-state ”Rb atoms N. R. Newbury, C. J. Myatt, and C. E. Wicman Joint Institute for Luborutov Astruphysics, Nationcil Institule of S l n d i r d s a i d TLchnology and University of Colorado and Physics Department. University of Colorado, Boulder, C o l o r d o 80309-0440 (Received 17 November 1994)
We have measured the elastic-scattering cross section of ”Rb atoms in the IF= 1, mF.= 1) ground state at 25 ,uK. The cross section is almost purely s wave at these temperatures and has a valuc of (5.45 1.3) X lo-’’ cm’. We have searched for the predicted Feshbach-type resonances in thc elastic cross section [Ticsinga ef a[., Phys. Rev. A 4 6 , 1167 (1992)] as a function of magnetic field. There are no resonances with a magnetic-field width 2 2 G over a magnetic-field range of 15-540 G. ~
PACS number(s): 34.50.-s, 32.80.Pj, 05.30.Jp
Many of the interesting applications of laser-cooled atoms, such as precision measurements, atomic clocks, BoseEinstein condensation, or spin waves, require a fundamental understanding of very-low-temperature ground-state collisions. The ground-state potentials of the heavier alkali-metal atoms such as Rb and Cs are not sufficiently well known to predict the collisional cross sections. Because of the low collision rates ( < I Hz for densities of 10” cm 3, and low energy transfers ( eV), cold ground-state collisions are also difficult to observe experimentally. Recent published measurements are restricted to the elastic cross section for the ( 3 , - 3 ) ground state of 133Cs[ 2 ] and the frequency shift of magnetic resonance lines in 133Cs[3]. Here, we present measurements of the s-wave elastic collision cross section for s7Rb atoms in the ] F = l , mF= - 1 ) ground state. For two colliding spin-polarized atoms in the lower hyperfine ground state Tiesinga et al. have predicted Feshbachtype resonances between the incoming atomic states and quasibound molecular states [l]. Since the magnetic moment of the molecular state differs from that of the colliding atoms, the resonances will occur at specific values of the bias magnetic field. The sign of the scattering length changes across a resonance, so that the existence of such a resonance could be very important for the realization of a Bose-Einstein condensate that requires a positive scattering length [4]. For a relative momentum of f i k , the resonant cross section is 87r/k2 or about ten times larger than our measured nonresonant cross section at 25 p K . A n increased elastic cross section would assist current efforts to achieve runaway evaporative cooling [5] of trapped alkali-metal atoms [6,7]. Unfortunately, given the current accuracy of the Rb interatomic potential, it is impossible to predict the position of the resonances. We have searched for a resonance in the elastic cross section over a magnetic-field range of 15-540 G. We did not find a resonance in the cross section with a width 2 2 G up to 540 G. Our technique is quite similar to the cross-dimensional mixing technique introduced in Ref. [2].Atoms are magnetically trapped in a harmonic well which has differelit oscillation frequencies, v,# v,, # v, , along each dimension. In this work, we use gravitational Sisyphus cooling [S] to cool the vertical dimension to an effective temperature T, , much colder than the horizontal temperatures T , and T v : Elastic collisions will drive the sample towards thermal equilibrium. 1050-2947/95/51(4)/2680(4)/$06.00
51 40 1
Starting from the Boltzmann equation, we can derive the rate of change of the total energy in the vertical direction as [9]
where m is the mass of ”Rb, k B is Boltzmann’s constant, TI,= (T,+ T,)/2, and n = Jn2(r)d3r for a normalized density distribution n(r). To derive this equation, we assume an angle-independent and energy-independent cross section v and separable Gaussian distributions of position and velocity in all three dimensions. By cooling the vertical dimension, rather than heating the dimension as in Ref. [2], we improve the signal-to-noise ratio of the measurement since the initial density n is higher and the initial difference between T h and T , is larger. In addition, our average temperature of 25 y K corresponds to a significantly longer de Broglie wavelength than in Ref. [2], implying that the collisions are almost completely s wave in character. To load the magnetic trap, we initially collect 107-108 atoms using diode lasers and a standard vapor-cell magnetooptic trap (MOT) [ l O , l l ] . The atom sample is cooled further with an optical molasses [12] and then loaded in situ into a magnetic trap. The trap is formed by a “baseball” coil, a pair of Helmholtz coils, and a pair of anti-Helmholtz coils that provide a magnetic-field gradient to balance the force of gravity [8,2,13]. Bias magnetic fields in the range of 15 to 540 G can be achieved with our power supplies by varying the current in the baseball and Helmholtz coils. After loading the trap with a bias field of 25 G and oscillation frequencies of vx=14.0, v,=8.8, and v,=4.4 Hz, we cool the vertical dimensions using gravitational Sisyphus cooling [S]. After cooling we change the bias magnetic field in 0.5-1 sec so that the corresponding changes in the trap oscillation frequencies are adiabatic. The subsequent thermalization rate at the given bias field is determined by observing the increase in the width of the vertical distribution (aTy) over time. To determine the spatial distribution, the magnetic trap is suddenly (<1 msec) turned o f fand the atoms excited with a 1-msec pulse of both 5s ’SIl2(F=1) the hyperfine repumping light -5p 2P312(F’= 2) and laser light tuned three linewidths to the red of the 5 s 2S1,2(F=2)-+5p 2P312(F’=3) cycling transition. The resulting fluorescence is imaged with a
i
R2680
0 1995 The American Physical Society
402
s-WAVE ELASTIC COLLISIONS BETWEEN COLD GROUND-. . . L
R2681
-I_-
1
L
i 0
FIG. 1. The vertical energy as a function of time calculated from the variance of the vertical distribution. To avoid systematic errors associated with “breathing” modes of the distribution, each data point is an average of ten points taken at fractions of an oscillation period. In addition, the horizontal spatial profile of the cloud is imaged to determine T , and n . These particular data represent a single measurement of m at a bias field of 240 G.
charge-coupled-device (CCD) camera and one raster of the image digitized and stored. From this image, we calculate the variance of the spatial distribution ( x : ) and the effective temperature associated with that dimension as T i = r n ~ ? ( x ? ) l k gIn. addition, the total number of atoms N is determined by the fluorescence on a photodiode. The lifetime of the trapped atoms is -8 sec. The average density n is time dependent both through the time dependence of N and the change in the spatial profile of the atom cloud as it comes into thermal equilibrium. The derivation of Eq. (1) assumed a Boltzmann energy distribution for each dimension. In the horizontal dimensions, the cloud shapes are indeed very close to Gaussian and can be characterized by temperatures T, and T y which range from 20-40 p K , depending on v, and u y . After gravitational Sisyphus cooling, the vertical dimension is better approximated as the sum of two Gaussians, with the respective effective temperatures of -1.3 and -11 ,uK at vz=4.4 Hz. (For different vertical spring constants these temperatures will roughly scale as u z . ) Typically the narrow Gaussian consists of about 60% of the atoms and the wider Gaussian contains the remaining 40% of the atoms, although the percentages change as the cloud thermalizes. The effective temperature is calculated from the total energy associated with the vertical dimension as T z = E , l k B . Since T,< Th/2 for our data, the correction to the right-hand side of Eq. (1) for the non-Gaussian vertical distribution is negligible [9]. Since Tz increases by at most 5 ,uK-=ZTh,we integrate Eq. (1) including the time dependence of n but holding T h constant. Figure 1 shows the result of a fit of T , ( t ) to the integrated Eq. (1). In Fig. 2, we present the elastic cross section as a function of bias magnetic field. Averaging the cross section over the magnetic field gives a=(5.420.8? I.0)X 10-l‘ cm’, where the first error is the standard deviation of the data in Fig. 2 and the second error is due to normalization uncertainties in n and Ti.The magnitude of the scattering length is then la I = 46? 11 A, compared to a thermally averaged relative de
200
,
__
c
-7
400 500 Bias Mogvetic Field (G)
100
300
600
FIG. 2. The elastic cross section as a function of magnetic field. Each point is the average of at least two measurements of the type illustrated in Fig. 1. The error bars shown are relative and do not include any overall errors in normalization.
A.
Broglie wavelength at 25 ,uK of ( hdeB) = 1050 For comparison, the maximum possible cross section is a ~ = ( 2 h ~ , $ ~ ) = ( 8 ~ l k 2 ) ~ cm’, 1 0 - 1where 0 hk is the relative momentum of the colliding atoms and the angle brackets denote a thermal average. There are several possible systematics affecting the crosssection measurements. First, residual anharmonicities in the trap potential can couple different dimensions, imitating the effects of elastic collisions. At the trap frequencies chosen for our measurements, we have determined that this coupling is negligible by comparing the thermalization rate of a lowdensity and high-density sample. Glancing collisions with hot background atoms can transfer an amount of energy to trapped atoms which is less than the trap depth. These collisions create a distribution of “hot” trapped atoms in addition to the original cold distribution. The fluorescence from this diffuse cloud is below the noise of our CCD images but is included in the total measured fluorescence of the cloud. Since the energy-transfer cross section for glancing collisions is constant across the energy distribution of the sample, there is no distortion of the cloud profile that would imitate thermalization. (In Ref. [2] the atom sample was hotter so that atoms were not removed uniformly from the original distribution, resulting in an apparent heating of the sample and an important systematic.) However, atoms in this second diffuse distribution can transfer energy back to the original cold distribution through elastic collisions. After 6 sec, this hot distribution holds 15% of the atoms for the deepest trap potentials. At lower trap potentials, the number is considerably smaller. Therefore, by comparing the elastic cross section measured at a constant bias field but different trap depths, we are able to show this effect on a to be under 20%. A final possible systematic in our measurements lies in the use of Eq. (l),which assumes an energy and angleindependent cross section. Clearly any non-s-wave contribution to the cross section will have a dependence on both energy and angle. Since the colliding particles are spinpolarized bosom, only even angular momentum partial waves (e.g., s, d , . . . ) will contribute to the cross section. From the Cb coefficient for Rb [14], we can calculate the
-
403
N. R. NEWBURY, C. J. MYAIT, AND C. E. WIEMAN
R2682
, , , , ,
//
,
15
90
the width of Feshbach resonances associated with the Zeeman interaction to be narrower than 1 G for our low magnetic fields. However, Tiesinga et al. report a broad Feshbach resonance in '33Cs associated with the hyperfine interaction with a width of 10 G. By scanning the magnetic field, we hoped to observe the analogous resonance in a gas of "Rb. Despite the uncertainties in the 87Rb interatomic potential, it is useful to estimate the magnetic-field spacing of such resonances at low fields. Normally the exchange interaction greatly exceeds the hyperfine interaction over the classically accessible region of a molecular bound state. In contrast, high-lying quasibound molecular states extend far into the region where the hyperfine coupling is greater than the exchange interaction. As a result it is difficult to assign spin quantum numbers to these states. Nevertheless, we can consider a Feshbach resonance between our incoming state and a state associated with the potential that dissociates to two 12,- 1 ) atoms. Such a state will have a magnetic moment of - + p B , which will be adjusted by the effects of the exchange interaction at small r . At zero magnetic field the incoming state 11,- 1)@11,-1) has a magnetic moment of - p B and an energy of - 4 u h f + E with respect to two dissociating 12,- 1 ) atoms. Increasing the magnetic field raises the energy of the incoming state and lowers the potential curve associated with the quasibound state. Given the C, coefficient, the maximum energy separation between the incoming state and the nearest quasibound state is 7.9 GHz at zero magnetic field [l5]. For a difference in the magnetic moments of the two states of 2fi.8, we would therefore expect a resonance within a bias field of 0-2.8 kG. A similar rough analysis suggests that a resonance with a second quasibound state that dissociates to an outgoing state 11,- l ) @12,- 1 ) is also possible within a bias field of 0-3.3 kG. Therefore, there is a probability of 1/3 that there is a resonance within 540 G of zero field. (An identical analysis in the standard singlet-triplet molecular basis also gives a probability of one-third for a resonance within 540 G.) Current efforts at understanding the Rb-Rb interatomic potential [I61 may lead to a narrow predicted range of the resonance position. The lack of a resonance in the 15-540 G range, while not as informative as the observation of a resonance, does put some constraints on the interatomic potential. Unfortunately, because of the.limited pbwer dissipation in our magnet coils, we cannot currently search for a resonance at fields much higher than 540 G. One possible method to circumvent the experimental difficulties of sweeping a field from 0 to 3 kG is to couple the incoming state with a quasibound molecular state by applying resonant microwave radiation. Depending on the strength of the microwave radiation required, it may be experimentally easier to scan the microwave frequency several GHz than to scan the bias magnetic field over this range. Our measurement should assist the optimization of evaporative cooling of Rb in the new tightly confining magnetic trap recently demonstrated by Petrich, Anderson, Ensher, and Cornell [6]. In Ref. [8], we presented gravitational Sisyphus cooling as a new method for cooling magnetically trapped atoms. Our value of the elastic collision cross section implies that gravitational Sisyphus cooling, coupled with an im-
-
_ I
165 240 315 390 465 540
Bias Magnetic Field (G) FIG. 3. The effective vertical temperature of the cloud after 4 sec of thermalization time as a function of bias magnetic field. The data points are spaced by -4 G, and each represents an average of 3-5 images of the cloud. For a 10-G-wideFeshhach resonance, the atomic sample would be almost completely thermalized with a vertical temperature given by the dashed line. The various slopes of the dashed line are a result of the complicated dependence of v, , vy , and vi on the baseball and Helmholtz coil magnetic fields.
(nonresonant) d-wave contribution to the elastic cross section to be a2-10-4a. An accidental d-wave shape resonance would have a fractional energy width < l o % and would be unobservable. In the absence of a resonance, the s-wave cross section will have an energy dependence to lowest order in k given by a-8.rra2/(1 + a 2 k 2 ) . At 25 p K , ( a k ) * = 0 . 1 4 , so that given our experimental accuracy we can assume CT=STU~ in deriving Eq. (1). Cross-section measurements taken at a fixed bias magnetic field over a temperature range of 20-60 fiK show no temperature dependence to within our signal-tonoise ratio. Finally, the most interesting possible energy dependence of the cross section would arise from a Feshbach resonance. As discussed below, such a resonance will depend on the bias magnetic field. In order to search for possible resonances in the elastic cross section, we measured the variance of the cloud after it had thermalized for 4 sec. The results as a function of magnetic field are shown in Fig. 3. The trapped atoms sample a spread in magnetic field of ( 2 m ) 3 1 G/cm 2 3 G, so we incremented the bias field in steps of 4 G. If a resonance occurred at a magnetic-field value between two adjacent data points, a peak would be observed in the figure with a signal-to-noise ratio 3 for a full width at half maximum of the resonance of 2 G or more. Feshbach resonances can occur when there is an energy degeneracy and coupling between an incoming unbound state and a quasibound molecular state. Since the magnetic moment of the incoming state is in general very different from that of the molecular state, the existence of such a resonance will depend strongly on the bias magnetic field. Tiesinga et al. [l] discuss Feshbach resonances resulting from the competition between the exchange interaction, Ve,(r)S~-S~, and either the Zeeman interaction or the hyperfine interaction, a h f S i . I i , where Si ( I i ) is the electron (nuclear) spin of a single atom. Based on Ref. [l],we expect
-
51 -
404
51
s-WAVE ELASTIC COLLISIONS BETWEEN COLD GROUND-, . .
R2683
proved experimental apparatus to increase the trap lifetime and number of atoms, should increase the thermalization rate to a value where “runaway” evaporative cooling can proceed.
T h i s work is supported by the Office of Naval Research and the National Science Foundation. We are pleased to acknowledge many useful discussions with E. A. Cornell, M. Anderson, J. Cooper, and C . Greene.
[ I ] E. Tiesinga, A. J. Moerdijk, B. J. Verhaar, and H. T. C. Stoof, Phys. Rev. A 46, 1167 (1992). [2] C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, and
191 C J. Myatt, N. R. Newbury, and C. E. Wieman (unpublished). [lo] E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987). [ I l l C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571 (1990). [12] See special issue on laser cooling and trapping of atoms, edited by S. Chu and C. Wieman [J. Opt. SOC.Am. B 6 (11) (198911. [13] T. Bergeman, E. Erez, and H. Metcalf, Phys. Rev. A 35, 1535 (1987). [14] M. Marinescu, H. R. Sadegbpour, and A. Dalgarno, Phys. Rev. A 49, 982 (1994). [15] R. J. LeRoy and R. E. Bernstein, J. Chem. Phys. 52, 3869 (1970). [16] D. Heinzen and B. J. Verhaar (private communication).
C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993). [3] K. Gibble and S. Chu, Phys. Rev. Lett. 70, 1771 (1993). [4] A. L. Fetter and J. D. Walecka, Quantum Theory of ManyParticle Systems (McGraw-Hill, New York, 1971), p. 222; H. T. C. Stoof, Phys. Rev. A 49, 3824 (1994). [5J J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, and T. J . Greytak, Phys. Rev. Lett. 67, 603 (1991). [6] W. Petrich, M. Anderson, J. Ensher, and E. A. Cornell (unpublished). [7] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell (unpublished); K. B. Davis, M.-0. Mewes, M. A. Joffe, and W. Ketterle (unpublished). [8] N. R. Newbury, C. J. Myatt, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. (to be published).
VOI.IJME 55. NUMBER 4
PHYSICAL REVIEW A
A i m L 1997
Erratum: s-wave elastic collisions between cold ground-state s7Rb atoms [Phys. Rev. A 51, R2680 (1995)] N . R . Neubury, C. J. Myatt, and C. E. Wieman [S I OjO-2')47(97)04403-1]
On page R2681 M e mistakenly quoted the ct-ror bar for the scattering length. The magnitude of the scattering Icngth \\as detemiined to be lul = 4 6 t 6 A. We thank J . P. Burke and J. 1,. Bohn for bringing this to our attention.
1050-2947/97/55(4)/3279(1)/$10.00
55
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0 1997 The American Physical Society
PH Y SI C A L R E V I E W L E T TE R S
VOLUME75, NUMBER18
30 OCTOBERI995
Laser-Guided Atoms in Hollow-Core Optical Fibers M. J. Renn, D. Montgomery, 0. Vdoviii, D. Z. Anderson. C. E. Wiernan. and E. A. Cornell JIW
cin tl
Depci rtm en i
of
Phy., ICJ , U nive r s i p of Colo riido, (Received 24 May 1905)
Boii Ide r,
Coloncirio 8030'3-044 0
W e have used optical forces t o guide atoms through hollow-core optical fibers. Laser light IS launched into the hollow region of a glass capillary libeI- and guided by gruing-incidence reflection from the walls. When the laser is detuned 1-30 GHz red of the Rb D2 resonance lines, dipole forces attract atoms to the high-intensity region along the axis and guide them through the fiber. We show that atoms may bc guided around bcnds in the fiber and that in initial experimcnts thc atonis experience up to 18 reflections from the potential walls with ininirnal loss. PACS numbers: 32.XO.Lg. 32.80.Pj. 39.10.+j
An atom placed in an optical beam is attracted to or repelled from regions of high intensity, depending on the polarizability of the atom at the optical frequency. This ponderomotive force is the basis of several atom-trapping techniques [ I ] and is also the basis of Cook and Hill's original proposal [2] for an atom mirror using evanescent optical fields. Out of the light-invoked atom mirror notion grew the concept of atom waveguides. The idea, first proposed by OI'Shanii, Ovchinnikov, and Letokhov [ 3 ] , and extended by Savage and co-workers [4,S], is to guide atoms down a hollow-core optical fiber using light also guided by the fiber. Fiber-guided atoms may facilitate many atomic physics experiments. For example, atoms could be extracted from a low-quality supply chamber and transported to an ultrahigh vacuum analysis chamber. With atom fiber guides, one has substantial control over atom trajectories. The losses from transverse momentum diffusion in evanescent mirror cavities [6] can be eliminated by the transverse confines of a fiber guide. Furthermore, by using optical access through the fiber walls, atoms can be manipulated and probed without the constraints of a cumbersome vacuum enclosure. At sufficiently low temperatures, the atomic de Broglie wavelength becomes comparable to the transverse dimensions of the hollow core. In this regime, atoms propagate in modes much like the optical modes of conventional fibers. This presents the exciting prospect of fiber-atomic interferometry in analogy with fiber-optic interferometry. In this work we report on an experimental demonstration of atom guiding through short, hollow-core optical fibers using laser light. We implement the configuration suggested by Ol'Shanii, Ovchinnikov, and Letokhov which involves a red detuned laser beam launched into the hollow region of a hollow-core fiber. Atoms in the guide propagate in a manner similar to light in a multimode fiber: axial motion is unconstrained and transverse motion consists of a series of lossless reflections from the potential established by the optical fields. Atoms exit the fiber with a numerical aperture that increases with increasing guiding light intensity. We show that atoms can be 003 1-9007/95/75( 18)/3253(4)$06.00
steered through bends of the flexible fiber. We have also transported atoms through a portion of fiber exposed to atmosphere, dcrnonstrating that the glass walls are sufficient to maintain vacuum in the guide. The theory of atom guiding i n hollow fibers is straightforward. Laser light is coupled into modes 171 which propagate along the fiber axis by grazing incidence reflection from the glass walls. In cylindrical coordinates, the intensity profile for the lowest order EH I , mode is approximately I ( p ) = Z"Ji(,yp) in the limit k a >> 1 where k = 2 v / A and a is the radius of the fiber hollow. ( 0 is the peak field intensity and ,y is found by solving the approximate characteristic equation:
X U , given by 2.405 + 0.022i for a 40-pni hollow-core diameter fiber at A = 780 nm, is related to the axial propagation coefficient, p , by x 2 = k 2 - p2. n is the index of refraction of the glass. The imaginary part of p is the attenuation coefficient of the mode amplitude and is given explicitly in the ka >> 1 limit by
Explicitly, the attenuation length, [Im(P)]-', for a 40-pm-diameter fiber is 6.2 cm. For the intensity profile given in Eq. (l), we can define w mode diameter as the diameter at which the intensity falls to C 1of the peak value. This diameter, 22 p m for the E H , , mode, is substantially smaller than the physical diameter of the core, which implies that the guided atoms may be localized in a transverse area much smaller than the physical area of the core. This fact also allows us to ignore atom-wall loss terms such as van der Waals interactions and quantum tunneling, which would be important for small atom-wall distances [ 5 ] . In curved fibers, the real part of p is unchanged to first order, but the imaginary part has an additional term, which is important when the radius of curvature R becomes comparable to the attenuation length of the straight fiber. The intensity
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profile to first order in 1 / R takes on an asymmetric, elliptical shape, with the center shifted toward larger R . The dipole potential depth felt by the atoms at a radial distance p for a given electric field E ( p ) is 181
where 2 R ( p ) = d E ( p ) / h is the atomic Rabi frequency, d is the dipole moment of the atom, A = w - w g - k u i is the laser detuning from resonance, and 2 y is thc spontaneous decay rate of the upper atomic state. Atoms with sufficiently small transverse velocities are confined in this two-dimensional potential. Equating the atoms’ classical kinetic energy to the potential barrier height, w-e obtain the maximum transverse velocity confined, u,,, = [ 2 U ( 0 ) / m ] 1 / 2where , rn is the mass of the atom. Typical experimental parameters, 45 mW of laser light propagating in a 40-pm-diametcr core and a detuning of -6 GHz, yield a potential depth of 71 mK and a transverse capture velocity of 3.7 m/s. In our experiment, atoms are loaded randomly from a thermal vapor surrounding the fiber opening. To obtain a simple estimate of the guided atom flux, we assume that all atoms that impinge on the opening of the fiber with sufficiently small transverse energy are guided. Integrating the Maxwell velocity distribution up to u , in the transverse direction and over all axial velocities, we obtain
where we take the atom density, no = 10“ m-’, near room temperature, and Ah is an effective area that we take to be the area of the EH I mode. In a straight fiber, no constraint exists on the axial velocities of the guided atoms. In bent fibers, on the other hand, atoms may be guided only if the transverse guiding force can provide adequate centripetal force. The tightest bend radius depends on how deeply the atom is bound in the transverse potential and is approximately R,,, = rnrou:/U(O), where u z is the atomic velocity along the fiber, and t-0 is the mode radius. Atoms with velocities faster than uz will stick to the fiber wall for bend radii R 5 R,,, , For typical experimental parameters the value of R,,, for the median uz is 18 cm. Transverse heating of the atoms by spontaneous scattering of photons is one limit to the distance atoms may be guided through fibers. To estimate this upper limit, we assume that the atomic transition is saturated so that atoms scatter photons at a maximum rate, y . The transverse energy accumulated after N = y r scattering events is E, = ( h k ) ’ N / 2 m . Setting this equal to the potential, U(O), we obtain a guidance time for Rb of T = 0.12 s and an average guidance distance of uz,ave T = 40 m. For the parameters used in the present experiment, transverse heating is negligible even at small detunings. However, for cold transverse temperatures this heating mechanism will be important. Since the spontaneous scatter rate falls
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from the saturated value as A P 2 while the potential scales as A-’, the guidance distance can be extended by detuning the lascr farther from resonance. The experimental apparatus, shown i n Fig. 1, consists of two separate vacuum chambers connected by a 3.1-cmlong capillary tiber with an outer diameter of 144 p m and hollow-core diameter of 40 pin. Light from a titanium sapphire standing wave lascr is coupled primarily into the EH I mode with an aspheric lens. We typically observe a 50% coupling efficiency and an attenuation loss of 7% pcr cm of fiber length for the E H , I mode. A single intracavity etaloii narrows the laser bandwidth to 2 GHz and allows tuning over 250 GHz. The pressure in the detection chamber is lo-* Torr. The input chambei- contains Rb metal and is heated to produce a Rb partial prcssurc of 1 0-6 Torr. The guided atoms leaving the fiber are surface ionized on a heated Pt or Re wire in the second chamber. The resulting ions are dctccted with a channcltron electron multiplier and recorded with a pulse counter. Measurements of the detuning dependence of the guided atom flux are presented in Fig. 2 for two laser intensities. The zero point of the detuning, 6, is taken to be the average transition frequency of the 5 ~ 1 / 2 ( F ) - S p 3 / 2 ( F ’ multiplet ) of 85Rb and ”Rb. Qualitatively, the detuning dependence of atom flux is as expected: atoms are guided by red detuned light, but not by blue detuned light. Particularly striking is the sharp turn-on of the guided atom signal. The signal rises to half maximum at 6 = -2 GHz and full maximum at 6 ^I -3 GHz. For detunings larger than a few GHz, the guided atom signal falls off approxirnatcly as 6 - ’ , as expected from an expansion of Eq. (3). With increasing intensity the signal increases and the position of maximum flux shifts to larger negative detunings. To obtain a precise fit to the data, optical pumping effects, the finite laser bandwidth, and the ground-state hyperfine splittings would have to be considered. In particular, optical pumping redistributes the population of the ground-state hyperfine levels and results in a sharper threshold than predicted from a simple two-state model Vapor cell with Rb atoms
Detector chambe1
FIG. 1. Experimental apparatus. Laser light is coupled into the grazing incidence modes of a hollow-core fiber. Atoms with sinall transverse velocities are extracted frorn a vapor cell in the first chamber and guided through the hollow fiber to a
detection chamber.
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Detuning (GHz) FIG. 2. Guided atom signal vcrsus lascr dctuning from rcsonancc. For the upper curve, I0 = 3.6 X lo3 W/cm' in the guiding modes, and for the lower curve, lo = 1.3 X lo3 W/crn2. The data show a sharp increase in the guided atom signal when the laser is tuned slightly to the red of the 0 2 resonance line of Rb. The solid line is included to show the trend of the data.
for the guided atom flux near 6 = 0. Additional optical pumping effects will be apparent in the signal's intensity dependence, shown in Fig. 4. Overall, the guided atom count rate of -lo4 s-' is in reasonable agreement with the expected flux of -lo5 s-'. The discrepancy, also observed with fiber lengths up to 6 cm, is attributed primarily to uncertainties in the detector calibration. Smaller uncertainties arise from the messy optical intensity profile that exists in the first 400 p m or so of the fiber, before higher-order modes have radiated away. Some of these modes produce finite laser intensity at the fiber walls and effectively reduce the potential gradient created by the fundamental EH I mode. In addition, the derivation of Eq. ( 3 ) does not include a number of effects such as optical pumping, and the guiding or funneling of atoms into the fiber. In Fig. 3 we present measurements of the spatial distribution of the exiting atoms with increasing laser powers. Broadening is clcarly evident as the power is increased. The half-widths, when corrected for the detector wire width and fiber diameter, scale as Z'/2, as expected for low intensities. For the 45 mW curve in
100
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Detector Position (pm) FIG. 3. Spatial distribution of flux measured by translating the surface ionizer at 4 mm from the exit of the fiber. The measurements show broadening of the distribution with increasing laser intensity. The half width of the uppcr curve at 5.9 X 103 W/cm2 corresponds to an average of 18 specular reflections from the potential walls. Background counts come from three sources: ballistic atoms passing straight through the fiber, thermal atoms diffusing slowly through the fiber, and hot wire noise. By bending the fibers slightly and reducing the noise on the hot wire, we have achieved background counts of 10 s - ' .
Fig. 3, the exit cone angle of 15 mrad corresponds to an atomic numerical aperature of 0.015. This corresponds to the atoms being reflected on the average 18 times from the potential walls without losses. We present measurements in Fig. 4 of the power dependence of the flux guided through straight and curved fibers. For R = 00, or straight fiber case, we see that at the largest intensities the guided flux begins to saturate as expected from the form of Eq. (3). The thick curve is a plot of Eq. (4) using a detuning of A = -6 GHz measured from the 5 ~ 1 / 2 ( F= 2)-5p3/2(F')transitions of 85Rb. At intensities smaller than 2 X 106 W/m2, the data obey a power dependence determined by a A = -3.0 GHz detuning from the 5 ~ 1 / 2 ( F= 3)-5p3/2(F') transition, as shown by the thin line in Fig. 4. The switchover at higher power agrees qualitatively with optical pumping of atoms in the upper ground-state hyperfine levels to the lower levels. At small intensities the optical pumping rate is small and negligible population is transferred into the far detuned state. However, at higher intensities, most of the atoms are pumped into this state, and the
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409 VOI.UME 7.5, NUMBER18
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Laser Intensity (1 O3 W/cm2) Intensity dependence of the guided atom flux in 40-pm diameter: curved fibers for radii of curvature are indicated in the legend. The thick line for R = 5. is a plot of Eq. (4), normalizcd to thc data, with A = -6 GHz and assuming that the atoms are optically pumped into the lowest ground-state hyperfine level. The thin line assumes that at low intensity the atoms are distributed equally between the groundstate levels. The solid lines for R < are included to show the trend of the data.
atoms experience the weaker potential determined by the A = -6 GHz detuning. Obviously, a primary interest in using fibers to guide atoms is the ability to guide the atoms around curves 141. The data of Fig. 4 show the dependence of the guided atom flux on intensity and minimum bend radius, R. The fiber is bent by translating the output tip of the fiber while holding the input tip fixed. At small intensities the flux turns on as a power law, reflecting the fact that only the slow velocity tail of the Maxwell distribution is guided around the bend. At high intensities the potential saturates and the flux intensity dependence consequently also saturates. At bend radii comparable to or smaller than the mode attenuation length, R = 6 cm, the flux turns on only at the highest intensities. At these radii the optical intensity profile becomes asymmetric [7] and the mode intensity is enhanced at the wall of the fiber. As a result, the optical guiding potential is weaker and atom losses are higher than expected from our simple picture. It is apparent from these data that the tightest bend through which atom
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guiding can be achieved is limited by a critical radius for effective optical guiding and not by a critical radius that depends on atomic properties. In conclusion, we mention a few additional applications o f guided atomb in fibers. As an atom transport system, a curved fiber could act as a velocity filter, allowing only atoms with small longitudinal as well as small transverse velocities to be extracted from a thermal cell and piped to it UHV system. This would allow, for example, large numbers of atoms to be trapped i n a magnctooptic trap by reducing collisional loss from thermal background atoms. We have already guided atoms in fibers with holes as small as 10 ,urn. Guiding in smaller fibers is feasible but more challenging because of the fast attenuation of the guiding light in simple capillary fibers. In future work we hope to launch optically single-mode light into an annular corc that surrounds the hollow and guide the atoms through the fiber with a blue detuned evanescent field [5,6,9]. This should lead to longer guiding distances in the smaller fibers and less spontaneous scattering. With smaller fiber diameters and colder transverse temperatures, the atoms’ de Broglie wavelength becomes comparable to the size of the hole, and the transverse atomic motion will be quantized. In such a situation, atomic modes interfere to produce atomic speckle patterns. Atoms with recently achieved temperatures of 200 nK [lo] launched into a 2-pm-diameter hollow would propagate in a single transverse atomic mode. This shall be another step to realizing an atom fiber interferometer of the Mach-Zehnder type or a Sagnac loop for rotational inertia sensing. We thank C . M . Savage and P. Zoller for helpful discussions, and we are grateful to the Office of Naval Research (Contract No. NOOO14-94-1-0375), NIST, and NSF for financial support of this work.
[ l ] Special Issue on Cooling and Trapping of Atoms, edited by S. Chu and C:. Wieman [J. Opt. SOC.Am. B 2 (1989)l. [2] R. J. Cook and R. K. Hill, Opt. Commun. 43, 258 (1982). [3J M. A. 01’ Shanii, Y. B. Ovchinnikov, and V. S Letokhov, Opt. Commun. 98, 77 (1993). [4] C.M. Savage, S. Marksteiner, and P. Zoller, in Fundamentals of Quantum Optics IZZ (Springer-Verlag, Berlin, New York, 1993). [5] S. Marksteiner et al., Phys. Rev. A 50, 2680 (1994). [6] V. I. Brlykin and V. S. Letokhov, Appl. Phys. B 48, 5 17 (1989). [7] E.A. J. Marcatile and R . A . Schmeltzer, Bell System Tech. J. July, 1783 (1964). [8] A. Ashkin. Phys. Rev. Lett. 40, 729 (1978). 191 H. Ito er al., Opt. Commun. 115,57 (1995). [lo] W. Petrich et a/., Phys. Rev. Lett. 74, 3395 (1995).
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a reprint from Optics Letters
Multiply loaded magneto-optical trap C. J. Myatt, N. R. Newbury, R. W. Ghrist, S. Loutzenhiser, and C. E. Wieman Joint Insitute for Laboratory Astrophysics, National Institute of Standards and Technology and University of Colorado, and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440 Received September 1, 1995 We report a two-chambered, differentially pumped system that permits rapid collection of trapped atoms with a vapor cell magneto-optical trap (MOT) and efficient transfer of these atoms to a second MOT in a lowerpressure chamber. During the transfer the atoms are guided down a long, thin tube by a magnetic potential, with 90(15%) transfer efficiency. By multiply loading, we accumulate and hold as many as 30 times the number collected by the vapor cell MOT. By thus separating the collection and holding functions of a MOT, we can collect as many as 1.5(0.6) X 10'O rubidium atoms and hold them for longer than 100 s, using inexpensive low-power diode lasers. PAC$ numbers: 34.40+n, 32.80.Pj, 05.30.Jp. 0 1996 Optical Society of America
The simple and inexpensive vapor cell magneto-optical by using a moving molasses; we wish to stress the trap' (MOT), particularly when implemented with simplicity of the method presented here. Our apparatus is shown in Fig. 1. The basic procediode lasers, has led to a proliferation of work on laserdure for accumulating atoms in the'lower MOT begins cooled atoms.' In particular, it has played a key role in achieving Bose-Einstein condensation in a vapor with collecting atoms in the vapor cell MOT for 2.2 s. Then the vapor cell trap is switched off, and atoms of magnetically trapped r ~ b i d i u m . ~In such a MOT, are accelerated into the transfer tube by the push the initial capture of atoms from a vapor and subsebeam for -1 ms. As the atoms travel down the tube quent holding of cold atoms for further experiments are (-30 ms), a hexapole field confines them near the axis. combined in one trap. This imposes a limit on the product of the number of atoms in the trap times As these atoms reach the lower chamber they are captured in the lower MOT, and the vapor cell MOT is the confinement time, N,. The loading rate increases turned back on. with vapor pressure, but the confinement time deThe vapor cell MOT is a standard configuration of creases as a result of collisions with thermal backthree retroref lected beams plus hyperfine repumping ground atoms. This is not a limitation for experiments light. The trap and repump beams come from two needing only short holding times or low capture rates, but, for many studies, both large numbers of atoms and long confinement times are d e ~ i r a b l e . ~ Furthermore, Push the loading rate increases rapidly with the diameter of the laser This is a problem when a magnetic trap is loaded in situ, where the physical size of the field coils, and hence the aperture for laser beams, must be kept small for tight ~onfinement.~Finally, studies of short-lifetime radioactive isotopes created at accelerators could benefit from a system whereby atoms can be collected near the target and quickly and efficiently transferred to a less hostile environment.6 A system with two MOT's, one of which is multiply loaded, rectifies many of the limitations of the vapor cell MOT. By separating the collection and holding functions of the MOT, we can increase the product N , by greater than 1000. With magnetic confinement during the transfer, the tube connecting the differentially pumped chambers can be quite long. Previously, a magnetic trap has been multiply l ~ a d e d , ~ but that method suffered from aberration of the focusing magnetic fields, leading to loss of phase-space Fig. 1. Vacuum system, which is composed of two difdensity. Loading into a second MOT eliminates the ferentially pumped conflat crosses connected by a 40-cmneed for precise focusing. Single load transfer belong, 1.1-cm-inner-diametertransfer tube. The vapor cell tween MOT's in order to take advantage of differential is pumped by an 11-L/s ion pump, and the low-pressure pumping has also been demonstrated: but the transfer chamber by a 30-L/s ion pump and a small titanium subliefficiency was only 20%. After completion of this mation pump. Six wires and/or three strips of uniformly research, we learned of recent research on another magnetized rubber lie along the tube to create a hexapole multiply loaded double-MOT system.' That system magnetic field. A window is aligned along the direction of the transfer tube, and through it we direct the push beam. employed high-power lasers and transferred the atoms 0146-9592/96/040290-03$6.00/0
0 1996 Optical Society of America 41 0
41 1 February 15,1996 / Vol. 21, No. 4 / OPTICS LETTERS
grating-stabilized diode lasers." The trapping beams have a FWHM diameter of -2.0 cm and an intensity of -1.2 mW/cm2 per beam. In this trap -5 x 10' "Rb atoms are collected from a vapor, and we transfer as many as -3.5 x lo-' atoms after the upper trap has filled for 2.2 s. The vapor pressure is estimated to be 1X Torr. The push beam is derived from the upper MOT trapping laser. During the push this laser is tuned near the peak of the cycling transitions: 5s ' S ~ I ~=( F 3) 5p2P3,2(F' = 4) for 85Rb. The push light is circularly polarized, and a small axial magnetic field is applied to the atoms while the push light is on. The atoms are optically pumped into the weak-field-seeking IF = 3, m = 3) ground state. As discussed below, the atoms can also be pumped into the IF = 2 , m = -2) weak-field-seeking state. Typically, the mean speed of the atoms after the push is -16 m/s, measured by time of flight down the transfer tube. A hexapole field provides a radially confining harmonic potential for the spin-polarized atoms. This field is produced by a hexapole coil consisting of six symmetrically placed wires running along the tube and carrying as much as 300-A current. The heating of the uncooled wires, when used alone (-30-ms pulses), limits the transfer rate to one load every -2 s. We therefore supplemented the coil with three long strips of uniformly magnetized rubber-essentially sliced up kitchen magnets-that have a surface field of 600 G. The permanent magnets were 23 cm long, compared with the 40-cm-long transfer tube, in order to reduce fringing fields at the locations of the optical traps. These magnets allow a reduction of the current pulses to 5 ms at the beginning and the end of the transfer when the atoms are not confined by the permanent magnets, and thus the transfer rate is increased by a factor of 3. It is safe to assume that similar performance would be attained by use of longer permanent magnets and short coils that are pulsed on for less than 5 ms. Another pair of diode lasers supplies the trap and repump light for the lower MOT. The trap light is split into six independent beams that have a FWHM diameter of -2 cm and an intensity of -0.7 mW/cm2 per beam. The six-beam configuration is necessary because of the large attenuation of the beams by the trapped atoms, which limits the number of trapped atoms in a retroreflected configuration. We observe more than 50% attenuation when -3 X 10' atoms are held in the lower MOT. In a single transfer we put 90 2 15% of the upper MOT atoms into a lower MOT. We have optimized the transfer efficiency with respect to all variables in the experiment. The efficiency is insensitive to the detuning and field gradient of the lower MOT. The dependence on push parameters-the push detuning, length of push, and intensity of push-can be summarized simply as a dependence on velocity of transfer. The optimum velocity is -16 m/s, and the efficiency drops to 50% at 27 m/s, presumably as a result of the trap depth of the lower MOT. The efficiency is a weak function of current when the hexapole coil is used alone, dropping by only a factor of 1.8 when the current is reduced from 300 to 60 A, and then drop-
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ping more rapidly for lower currents. With no magnetic field the efficiency is 15%, and with just the 23cm-long permanent magnets we transfer 50%. The lifetime 7 6 of atoms in the lower MOT is -120 s, a factor-of-50 longer than for the upper vapor cell, which allows us to transfer and accumulate many Torr. loads. The pressure is estimated to be Little effort has been made to increase this lifetime beyond 120 s; it would certainly improve with a better bake-out of the system to reduce the non-rubidium background gas. Two multiple load cycles are illustrated in Fig. 2. The l / e time constants for buildup of atoms are 44 and 95 s for the two traces, while the trap lifetime is 7 6 = 115 s. The difference between buildup times and lifetime is due to undesirable optical excitation of the spin-polarized atoms traveling down the transfer tube. Light scattered from the lower MOT can excite these atoms so that they decay back into spin states that are not confined by the hexapole field. We find good agreement with a model that treats this undesirable excitation as an additional loss process, l/rex, since this loss of atoms during transfer is proportional to light scattered by the N atoms in the lower MOT. The rate equation for the buildup of N has the form
1.o
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t
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'
'
'
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Fig. 2. Fluorescence of 85Rbatoms multiply loaded in the lower MOT as a function of the number of loads. The time between loads is 2.2 s. In the trace marked by triangles the atoms are transferred in the 13, 3 ) state, and the l / e fill time for collecting atoms is 44 s. The trace marked by circles indicates atoms that are transferred in the 12, 2) state with a fill time of 95 s. The difference in the initial slope illustrates the difference in single load transfer efficiency for the two cases.
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where R is the loading rate (as high as -2 X 108/s). The decay of atoms out of the trap has the usual form dIV/dt = -N/rb. Because most of the light scattered from a MOT is nearly resonant for atoms in the upper hyperfine ground state, multiple loading works best when the atoms are in the lower hyperfine ground state IF = 2 , m = -2) during transfer. The net transfer efficiency with this lower hyperfine state is 75 ? 15%. The reduced magnetic moment of the 12, -2) state relative to the 13, 3) state accounts for a 6% reduction in transfer efficiency, and 9% is due to atoms not pumped into the 12, -2) state. The difference between transfer in the upper and lower states is illustrated in Fig. 2. With the 12, -2) state and the hexapole coil alone, l.l(O.5) X 10" rubidium atoms are collected, whereas with the 13, 3) state 7.5(3.0) x lo9 atoms are collected. The difference in fill times for the lower and upper states, 95 and 44 s, respectively, implies almost a n order-of-magnitude lower optical excitation loss when atoms are transferred in the lower 12, -2) state. The difference in the initial rise rates for the two curves is due t o the difference in single load transfer efficiency (90% versus 75%). These data are taken using the hexapole coil only, without the permanent magnets. The permanent magnets allow the rubidium vapor pressure to be raised (by a factor of -3), increasing the load rate as the heating of the hexapole wires is reduced. However, attenuation of the vapor cell MOT beams, a limitation arising from low laser power and long beam path length through the upper vapor cell, limits the number of atoms trapped at higher vapor pressures. Nevertheless, 1.5(6.0) X 1O1O rubidium atoms were accumulated by use of permanent magnets and transferring in the lower state. We have demonstrated a multiply loaded MOT that can collect more than lolo atoms with inexpensive, low-power diode lasers and hold these atoms for long times. The key feature is a separation of the collection and confinement functions of a MOT. With the use of permanent magnets, this method is a relatively simple extension of a standard vapor cell MOT. The improvement in the product of number and lifetime N , over that attained in a single vapor cell MOT is greater than a factor of 1000. This system is ideal for experiments that require a large N , product, such as evaporative cooling to the Bose-Einstein condensation
phase-space boundary. Both the number and the lifetime can be increased beyond what we report here with relatively minor improvements, and the system should permit the collection of several times 1O1O atoms with a relatively simple, inexpensive apparatus. This research has been supported by the U.S. Office of Naval Research and the National Science Foundation (NSF). R. W. Ghrist acknowledges the support of a n NSF Fellowship, and S. Loutzenhiser was supported by the NSF Research Experience for Undergraduates program. We acknowledge useful discussions with M. Anderson, E. Cornell, and the rest of the Bose-Einstein Condensation research group at the Joint Institute for Laboratory Astrophysics. We thank T. Walker for pointing out the possibility of loss of optical excitation during transfer.
References 1. C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571 (1990). 2. M. H. Anderson, W. Petrich, J. R. Ensher, and E. A. Cornell, Phys. Rev. A 50, R3597 (1994);M. G. Peters, D. Hoffman, J. D. Tobiason, and T. Walker, Phys. Rev. A 50, R906 (1994);J. D. Miller, R. A. Cline, and D. J. Heinzen, Phys. Rev. A 47, R4567 (1993). 3. M. H. Anderson, J. R. Ensher, M. R. Mathews, C. E. Wieman, and E. A. Cornell, Science 269,198 (1995). 4. K. Lindquist, M. Stephens, and C. Wieman, Phys. Rev. A 46, 4082 (1992); K. Gibble, S. Kasape, and S. Chu, Opt. Lett. 17,526 (1992). 5. C. R. Monroe, E. A. Cornell, C. A. Sachett, C. J. Myatt, and C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993); W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Phys. Rev. Lett. 74,3352 (1995). 6. Z.-T. Lu, C. J. Bowers, S. J. Freedman, B. K. Fujikawa, J. L. Mortara, S.-Q. Shang, K. P. Coulter, and L. Young, Phys. Rev. Lett. 72, 3791 (1994). 7. E. A. Cornell, C. Monroe, and C. E. Wieman, Phys. Rev. Lett. 67, 2439 (1991). 8. C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, Phys. Rev. A 52, 1423 (1995);A. Steane, P. Szriftgiser, P. Desbiolles, and J. Dalibard, Phys. Rev. Lett. 74,4972 (1995). 9. K. Gibble, S. Chang, and R. Legere, Phys. Rev. Lett. 75, 2666 (1995). 10. K. B. MacAdam, A. Steinbach, and C. Wieman, Am. J. Phys. 60,109 (1992).
RESOURCE LETTER Roger H. Stuewer, Editor School of Physics and Astronomy, 116 Church Street University of Minnesota, Minneapolis, Minnesota 55455 This is one of a series of Resource Letters on different topics intended to guide college physicists, astronomers, and other scientists to some of the literature and other teaching aids that may help improve course content in specified fields. [The letter E after an item indicates elementary level or material of general interest to persons becoming informed in the field. The letter I, for intermediate level, indicates material of somewhat more specialized nature; and the letter A, indicates rather specialized or advanced material.] No Resource letter is meant to be exhaustive and complete; in time there may be more than one letter on some of the main subjects of interest. Comments on these materials as well as suggestions for future topics will be welcomed. Please send such communications to Professor Roger H. Stuewer, Editor, AAPT Resource Letters, School of Physics and Astronomy, 116 Church Street SE, University of Minnesota, Minneapolis, MN 55455.
Resource Letter TNA-1: Trapping of neutral atoms N. R. Newbury and C.Wieman Joint Institute for Laboratory Astrophysics, National Institute of Standards and Technology and University of Colorado, Boulder, Colorado, 80309-0440
(Received 25 April 1995; accepted 23 August 1995) This Resource Letter provides a guide to the literature on trapping of neutral atoms. 0 1996 American Association of Physics Teachers.
In the past few years there has been spectacular progress in the trapping of neutral atoms. These traps use electromagnetic fields in various forms to contain atomic vapors and prevent them from coming into contact with material walls. This Resource Letter is intended to provide a useful list of references on this field of neutral atom trapping. Such trapping has been accomplished by using two types of fields: (1) rapidly oscillating optical fields from lasers and (2) static or slowly varying magnetic fields. Laser traps can be divided into two groups: dipole-force traps and spontaneous-force traps. The dipole-force traps work on the principle of an induced oscillating dipole moment in the atom that interacts with a spatially inhomogeneous laser field. This approach can also trap macroscopic chunks of matter such as a dielectric ball or a microbe and has been applied to a very diverse range of systems. We limit our coverage of dipole traps to the case of independent atoms. The most common type of neutral-atom trap is the spontaneous-force optical trap, which is based on the momentum transferred when laser light scatters off an atom by the process of excitation and reradiation. By far the most common type of spontaneous-force trap is the magneto-optic trap or MOT (also known as the ZOT, for Zeeman shift optical trap) that uses inhomogeneous magnetic fields and laser polarization to control the spontaneous force to produce the desired trapping potential. Since neutral-atom traps are typically very shallow, all traps also employ some method of cooling the atoms to the very cold temperature of 1 K (Kelvin) or less. Except for the case of hydrogen traps, this cooling is accomplished through the use of lasers. We have not included references that deal exclusively with laser cooling, but discussions of such cooling are included in some of the articles on trapping. In particular, in spontaneous-force traps, both the cooling and trapping forces arise from the scattering and reemission of the laser photons and hence are intimately connected. The purely magnetic traps are based on the interaction of 18
Am. J. Phys. 64 (l),January 1996
the magnetic moment of the atom and an inhomogeneous magnetic field. Most magnetic traps use dc magnetic fields, but a few have been built with ac magnetic fields that vary slowly relative to any internal time scale of the atom, but rapidly compared to the external motion. The number of applications of neutral-atom traps is growing rapidly. Probably the two largest areas of use are precision spectroscopy, including atomic clocks, and the study of cold-atom collisions. We have listed only the first few papers that were published in these areas and have not attempted to cover the large current literature in these subjects. Finally, we should note that the field of neutral-atom trapping itself is changing rapidly. We have no doubt overlooked some important papers in the field, and there are likely to be others that have been published since this Resource Letter was written.
I. JOURNALS THAT HAVE ARTICLES ON SUBJECT The first five journals listed contain the most articles on neutral atom traps. However, the remaining journals often contain longer and more general articles on traps. Physical Review Letters Physical Review A Optics Letters Journal of the Optical Society of America Europhysics Letters Science Nature Physics Today Physics Reports Scientific American American Journal of Physics Optical Communication Journal of Physics B 0 1996 American Association of Physics Teachers
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11. BOOKS 1. Proceedings, Enrico Fermi International Summer School on Laser Manipulation of Atoms and Ions, Varenna, Italy, edited by E. Arimondo, W. Phillips, and E Strumia (North Holland, Amsterdam, 1992). This book has several lengthy articles on neutral-atom traps of various types. (1)
111. REVIEW ARTICLES Below we list a number of useful review articles on trapping of alkali-metal atoms. Typically these articles also contain discussions of laser cooling of atoms. 2. “Cooling and Trapping Atoms,” W. D. Phillips and H. F. Metcalf, Sci. Am. 256 (3), 36-42 (1987). (E) 3. “Laser Cooling and Trapping,” S. Stenholm, Eur. J. Phys. 9, 242-249 (1988). (I) 4. “Cooling, Stopping and Trapping Atoms,” W. D. Phillips, P. L. Could, and P. D. Lett., Science 239, 877-883 (1988). (E) 5. “New Mechanisms for Laser Cooling,” C. Cohen-Tannoudji and W. D. Phillips, Phys. Today 33-40 (Oct. 1990). (E) 6. “Laser Manipulation of Atoms and Particles,” S. Chu, Science 253, 861-866 (1991). (E) 7. “Laser Trapping of Neutral Particles,” S. Chu, Sci. Am. 266 (2), 70-76 (1992). (E) 8. “Manipulation of Neutral Atoms. Experiments,” A. Aspect, Phys. Rep. 219, 141-152 (1992). (I) 9. “Laser Cooling and Trapping of Neutral Atoms: Theory,” C. CohenTannoudji, Phys. Rep. 219, 153-164 (1992). (I)
10. “Laser Cooling and Trapping for the Masses,” S. Gilbert and C. Wieman, Opt. Photon. News 4 (71, 8-14 (1993). (E) 11. “Magneto-Optical Trapping of Atoms,” H. Wallis, J. Werner, and W. Ertnier, Comments At. Mol. Phys. 28 (5), 275-300 (1993). (I) 12. “Cooling and Trapping of Neutral Atoms,” H. Metcalf and P. van der Straten, Phys. Rep. 244, 203-286 (1994). (1)
IV. PARTICULAR SUBJECT REFERENCES The standard dipole-force trap is simply a focused laser beam tuned below the nearest atomic transition. The resulting trapping potential can be described equivalently in terms of the ac Stark shift of the atomic-energy levels, the interaction of an induced atomic-dipole moment with the electric field of the laser, or as energy shifts of the dressed atom (see Ref. 13). A. Dipole-force traps 13. “Motion of Atoms in a Radiation Trap,” I. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606-1617 (1980). (A) 14. “Dressed-Atom Approach to Atomic Motion in Laser Light: The Dipole Force Revisited,” J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707-1720 (1985). (l/A) 15. “Experimental Observation of Optically Trapped Atoms,” S . Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, Phys. Rev. Lett. 57, 314-317 (1986). (I) 16. “Trapping of Neutral Atoms With Resonant Microwave Radiation,” C. C. Agosta, 1. F. Silvers, H. T. C. Stoof, and 8 . J. Verhaar, Phys. Rev. Lett. 62, 2361-2364 (1989). (I) 17. “Far-Off-Resonance Optical Trapping of Atoms,” J. D. Miller, R. A. Cline, and D. J. Heinzen, Phys. Rev. A 47, R4567-4570 (1993). (I) 18. “Demonstration of Neutral Atom Trapping With Microwaves,” R. J. C. Spreeuw, C. Gsrz, L. S. Goldner, W. D. Phillips, S. L. Rolston, C. I. Westbrook, M. W. Reynolds, and 1. F. Silvera, Phys. Rev. Lett. 72, 3162-3165 (1994). (I) 19. “Spin Relaxation of Optically Trapped Atoms by Light Scattering,” R. A. Cline, I. D. Miller, M. R. Matthews, and D. J. Heinzen, Opt. Lett. 19, 207-209 (1994). (I) 20. “Quasi-Electrostatic Trap for Neutral Atoms,’’ T. Takekoshi, J. R. Yeh, and R. 1. Knize, Optics Comm. (in press, 1994). (1) 21. “Evaporative Cooling in a Crossed Dipole Trap,” C. S. Adams, H. 1. Lee, N. Davidson, M. Kasevich, and S . Chu, Phys. Rev. Lett. 74, 35773580 (1995). (1) 19
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22. “Long Atomic Coherence Times in an Optical Dipole Trap,” N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, and S. Chu, Phys. Rev. Lett. 74, 1311-1314 (1955). (I)
B. Magneto-optical traps (MOTs) The first magneto-optical trap (Ref. 22) demonstrated that by combining magnetic-field gradients and circularly polarized counter propagating laser beams, atoms could be cooled and trapped. Subsequently, there has been an enormous effort at understanding, simplifying, and improving the basic magneto-optical trap. By now the magneto-optical trap is the initial step in most experiments on laser cooling and trapping of alkali atoms. In Subsection 1 we list general papers describing the construction and behavior of a MOT. Most often MOTs are used with alkali-metal atoms. In Subsection 2 we list papers describing trapping of alkaline-earth and noblegas atoms. Finally in Subsection 3 we reference some interesting variants on the standard MOT. We have not attempted to reference the enormous literature discussing collisions of atoms within a MOT. 1. General 23. “Stability of Radiation-Pressure Particle Traps: An Optical Earnshaw Theorem,”A. Ashkin and J. P. Gordon, Opt. Lett. 8, 511-513 (1983). (1) 24. “Light Traps Using Spontaneous Forces,” D. Pritchard, E. Raab, V. Bagnato, R. Watts, and C. Wieman, Phys. Rev. Lett. 57, 310-313 (1986). (I) 25. “Trapping of Neutral Sodium Atoms With Radiation Prebsure,” E. L. Raah, M. Prentiss, A. Cable, S . Chu, and D. Pritchard, Phys. Rev. Lett. 59, 2631-2634 (1987). ( I ) 26. “Collisional Losses From a Light-Force Atom Trap,” D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, Y61-9h4 (1989). (I) 27. “Very Cold Trapped Atoms in a Vapor Cell,’’ C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571-1574 (1990). (I) 28. “Observations of Sodium Atoms in a Magnetic Molasses Trap Loaded by a Continuous Uncooled Source,” A. Cable, M. Prentiss, and N. P. Bigelow, Opt. Lett. 15, 507-509 (1990). (I) 29. “Collective Behavior of Optically Trapped Neutral Atoms,’’ T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64, 408-411 (1990). (I) 30. “Behavior of Neutral Atoms in a Spontaneous Force Trap,” D. Sesko, T. Walker, and C. Wieman, 1. Opt. Soc. Am. B 8, 946-958 (1991). (I) 31. “Laser Cooling Below the Doppler Limit in a Magneto-Optical Trap,” A. Stcane and C. Foot, Europhys. Lett. 14, 231-236 (1991). (A) 32. “Experimental and Thcoretical Study of the Vapor-Cell Zeeman Optical Trap,” K. Lindquist, M. Stephens, and C. Wieman, Phys. Rev. A 46, 4082-4090 (1992). (I) 33. “Radiation Force in the Magneto-Optical Trap,” A. M. Steane, M. Chowdhury, and C. J. Foot, J. Opt. Soc. Am. B 9, 2142-2158 (1992). (1) 34. “Improved Magneto-Optic Trapping in a Vapor Cell,” K. E. Gibble, S. Kasapi, and S. Chu, Opt. Lett. 17, 526-528 (1992). ( I ) 35. “Spatial Distribution of Atoms in a Magneto-Optical Trap,’’ V. S . Bagnato, L. G. Marcassa, M. Oria, G. 1. Surdutovich, R. Vitlina, and S . C. Zilio, Phys. Rev. A 48, 3771-3775 (1993). (I) 36. “ A Simple Model for Optical Capture of Atoms in Strong Magnetic Quadrupole Fields,” D. Haubrich, A. Hope, and D. Meschede, Opt. Commun. 102, 225-230 (1993). (A) 37. “Neutral Cesium Atoms in Strong Magnetic-Quadrupole Fields at SubDoppler Temperatures,” A. Hope, D. Haubrich, G. Muller, M. G. Kaenders, and D. Meschede, Europhys. Lett. 22, 669-674 (1993). (A) 38. “Simplified Atom Trap by Using Direct Microwave Modulation of a Diode Laser,” C. J. Myatt, N. R. Newbury, and C. E. Wieman, Opt. Lett. 18, 649-651 (1993). 39. “Collective Atomic Dynamics in a Magneto-Optical Trap,” A. Hemmerich, M. Weidemuller, T. Esslinger, and T. W. Hansch, Europhys. Lett. 21,445-450 (1993). (A) 40. “High Collection Efficiency in a Laser Trap,” M. Stephens and C. Wieman, Phys. Rev. Lett. 72, 3787-3790 (1994). (A)
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415 41. “Measurements of Temperature and Spring Constant in a MagnetoOptical Trap,” C. D. Wallace, T. P. Dinneen, K. Y. N. Tan, A. Kumarakrisnan, P. L. Could, and J. Javaninen, J. Opt. SOC.Am. B 11, 703-711 (1994). (I) 42. “Inexpensive Laser Cooling and Trapping Experiment for Undergraduate Laboratories,” C. Wieman, G. Flowers, and S. Gilbert, Am. J. Phys. 63, 317 (1995). (E)
2. Magneto-optical traps of alkaline earths and noble gases 43. “ A High-Intensity Metastable Neon Trap,” Fujio Shimizu et a/., Chem. Phys. 145, 327-331 (1990). (A) 44. “Laser Cooling and Trapping of Calcium and Strontium,” T. Kurosu and F. Shimizu, Jpn. J. Appl. Phys. (Lett.) 29, L2127-2129 (1990). (A) 45. “Laser Cooling and Trapping of Argon and Krypton Using Diode Lasers,” H. Katori and F. Shimizu, Jpn. J. Appl. Phys. (Lett.) 29, L21242126 (1990). (A) 46. “Laser Cooling and Trapping of Neutral Atoms,” F. Shimizu, Hyperfine Interactions 74, 259-267 (1992). (I) 47. “Laser Cooling and Trapping of Alkaline Earth Atoms,” T. Kurosu and F. Shimizu, Jpn. J. Appl. Phys. Part 1 33, 908-912 (1992). (I) 48. “Magneto-Optical Trapping of Metastable Xenon: Isotope-shift measurements,” M. Walhout, H. J. L. Megens, A. Witte, and S. L. Rolston, Phys. Rev. A 48, R879-881 (1993). (A) 49. “Lifetime Measurement of the Is, Metastable State of Argon and Krypton With a Magneto-Optical Trap,” H. Katorl and F. Shimizu, Phys Rev. Lett. 70, 3545-3548 (1993). (A)
3. Variants on MOT 50. “Four-Beam Laser Trap of Neutral Atoms,” F. Shimizu, K. Shimizu, and H. Takuma, Opt. Lett. 16, 339-341 (1991). (I) 51. “Spin-Polarized Spontaneous-Force Atom Trap,’’ T. Walker, P. Feng, D. Hoffmann, and R. S. Williamson 111, Phys. Rev. Lett. 69, 2168-2171 (1992). (I) 52. “ A Vortex-Force Atom Trap,” T. Walker, D. Hoffmann, P. Feng, and R. S. Williamson 111, Phys. Lett. A 163, 309-312 (1992). (I) 53. “High Densities of Cold Atoms in a Dark Spontaneous-Force Optical Trap,” W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, Phys. Rev. Lett. 70, 2253-2256 (1993). (I) 54. “Behavior of Atoms in a Compressed Magneto-Optical Trap,” W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, I. Opt. Soc. Am. B 11, 1332-1335 (1994). (I) 55. “Reduction of Light-Assisted Collisional Loss Rate From a LowPressure Vapor-Cell Trap,” M. H. Anderson, W. Petrich, J. R. Ensher, and E. A. Cornell, Phys. Rev. A 50, R3597-3600 (1994). (I)
58. “Magnetic Confinement of a Neutral Gas,” R. V. E. Lovelace, C. Mehanian, T. J. Tommila, and D. M. Lee, Nature 318, 30-36 (1985). (I) 59. “Evaporative Cooling of Magnetically Trapped and Compressed SpinPolarized Hydrogen,” H. Hess, Phys. Rev. B 34, 3476-3479 (1986). (1) 60. “Magnetostatic Trapping Fields for Neutral Atoms,” T. Bergman, G. Erez, and H. Metcalf, Phys. Rev. A 35, 1535-1546 (1987). (A) 61. “Magnetic Trapping of Spin-Polarized Atomic Hydrogen,” H. Hess et a/., Phys. Rev. Lett. 59, 672-675 (1987). (1) 62. “Continuous Stopping and Trapping of Neutral Atoms,” V. S. Bagnato, G. P. Lafyatis, A. G. Martin, E. L. Raab, R. N. Ahmad-Bitar, and D. E. Pritchard, Phys. Rev. Lett. 58, 2194-2197 (1987). (I) 63. “RF Spectroscopy of Trapped Neutral Atoms,” A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Pafyatis, and D. E. Pritchard, Phys. Rev. Lett. 61, 2431-2434 (1988). (I) 64. “Experiments With Atomic Hydrogen in a Magnetic Trapping Field,” R. van Roijen, J. J. Berkhout, S. Jaakola, and I. T. M. Walraven, Phys. Rev. Lett. 61, 931-934 (1988). (I) 65. “Quantized Motion of Atoms in a Quadrupole Magnetostatic Trap,” Bergman et a l . , J. Opt. SOC.Am. B 6, 2249-2256 (1989). (A) 66. “Energy Distributions of Trapped Atomic Hydrogen,” J. M. Doyle et al., J. Opt. Sac. Am. B 6, 2244-2248 (1989). (I) 67. “Very Cold Trapped Atoms in a Vapor Cell,” C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571-1574 (1990). (I) 68. “Atom Cooling by Time-Dependent Potentials,” W. Ketterle and D. E. Pritchard, Phys. Rev. A 46, 4051-4054 (1992). (A) 69. “Trapping and Focusing Ground State Atoms With Static Fields,” W. Ketterle and D. E. Pritchard, Appl. Phys. B 54, 403-406 (1992). (A) 70. “Laser and RF Spectroscopy of Magnetically Trapped Neutral Atoms,” K. Helmerson, A. Martin, and D. E. Pritchard, J. Opt. SOC.Am. B 9, 483-492 (1992). (I) 71. “Trapping and Cooling of (Anti)Hydrogen,” J. T. M. Walraven, Hyperfine Interactions 76, 205-220 (1993). (I) 72. “Lyman-Alpha Spectroscopy of Magnetically Trapped Atomic Hydrogen,” 0. I. Luiten, H. G. C. Werij, I. D. Setija, M. W. Reynolds, T. W. Hijmans, M. W. Reynolds, and J. T. M. Walraven, Phys. Rev. Lett. 70, 544-547 (1993). (I) 73. “Atomic Hydrogen in Magnetostatic Traps,’’ J. T. M. Walraven and T. W. Hijmans, Phys. Rev. B 197, 417-425 (1994). (I) 74. “Collisionless Motion of Neutral Particles in Magnetostatic Traps,” E. L. Snrkov, J. T. M. Walraven, and G. V. Shlyapnikov, Phys. Rev. A 49, 4778-4786 (1994). (A) 75. “Measurement of Cs-Cs Elastic Scattering at T = 3 0 p K , ” C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, and C. E. Wieman, Phys. Rev. Lett. 70, 414-417 (1993). (I) 76. “Prospects for Bose-Einstein Condensation in Magnetically Trapped Atomic Hydrogen,” T. J. Greytak, in Bose-Einstein Condensation, edited by A. Griffin, D. W. Snoke, and S. Stringari (Cambridge University, Cambridge, 1995), pp. 131-159. (I)
C. Magnetic trapping
2. ac magnetic traps
Conceptually one of the simplest neutral atom traps is the dc magnetic trap. Atoms are confined in a minimum of magnetic field through the interaction of the atomic magnetic moment and the magnetic field. In Subsection 1 we list references describing dc magnetic traps for hydrogen and alkali-metal atoms. ac magnetic traps are also possible and are listed in Subsection 2. 1. dc magnetic traps
77. “Multiply Loaded, AC Magnetic Trap for Neutral Atoms,” E. A. Cornell, C. Monroe, and C. E. Wieman, Phys. Rev. Lett. 67, 2439-2442 (1991). (1) 78. “ A Magnetic Suspension System for Atoms and Bar Magnets,” C. Sackett, E. Cornell, C. Monroe, and C. Wieman, Am. J. Phys. 61, 304309 (1993). (E) 79. “Magnetic Confinement of a Neutral Gas,” R. V. Lovelace, C. Mehanian, T. J. Tommila, and D. M. Lee, Nature 318, 30-36 (1985). (I) 80. “ A Stable, Tightly Confining Magnetic Trap for Evaporative Cooling of Neutral Atoms,” W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Phys. Rev. Lett. 74, 3352 (1995). (I) 81. “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” M. Anderson, J. Ensher, M. Makthews, C. Wieman, and E. Cornell, Science 269, 198-201 (1995). (I)
56. “Cooling Neutral Atoms in a Magnetic Trap for Precision Spectroscopy,” D. E. Pritchard, Phys. Rev. Lett. 51, 1336-1339 (1983). (I) 57. “First Observation of Magnetically Trapped Neutral Atoms,” A. L. Migdall et al., Phys. Rev. Lett. 54, 2596-2599 (1985). (I)
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N. R. Newbury and C. Wieman
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PIlYSlCAL REVIEW A
VOLUME 53. NlJMBER 2
FEBRUARY 1996
Evanescent-wave guiding of atoms in hollow optical fibers Michael J. Renn,‘ Elizabeth A. Donley, Eric A. Cornell,.’ Carl E. Wieman, and Dana Z. Anderson JILA and Uepnrtmeiit of Pizwics, Univei-sil?/ o j Colorado, Boiilciei: C’olorndo 80309-0440 (Received 2 October 1995) We use evanescent laser light to guide atoins through hollowcore optical fibers. The light. detuned to the blue side o f rubidium‘s D 2 rcsonancc lines. is launchcd into the glass region of a hollow capillary tiber and guided through tlie fiber bj, total internal rcflcctiun liom (lie glass \+alls. Atonis interacting 1%it11 tlie evanescent coinponcnt of thc field are repelled liom the \+all and guided through thc fihcr liollo\+. A second laser tuned to tlie red side of resonance is used to initially iri,icct the atoins into the evanescent guide. An optical intensit! threshold for guiding is observed as the evanescent-Geld-itidticc~ldipole Ihrccs cs Ibrces
I’,4CS ntimbcr(s): 32.80.1’j. 32.8O.Lg. 39. I O.+j
I. INTRODUCTION
An atom placed in a near-resonant laser field is either attracted to or repelled from regions of high intensity depending on the sign of the laser’s detuning from atomic resonance. We have demonstrated that the optical forces induced by laser light guided in a fiber may be used to reflect atoms from the inner wall of a hollow-core optical fiber in a recent work [1,2]. In that demonstration, light was coupled to the lowest-order grazing incidence mode [3] and the laser frequency was tuned to the red side, so that atoms were attracted to the high-intensity region at the center of the fiber. Atoms guided in this way undergo a series of lossless oscillations in the transverse plane and unconstrained motion along the axis. Atoms can also be guided by the evanescent light field of the glass surface surrounding a hollow fiber. With a detuning on the blue side of resonance, atoms are expelled from the high-intensity-field region near the fiber wall. The intensity in the evanescent field is significant at a distance of .=X/2.rr into the hollow region. Consequently, the atoms are nearly specularly reflected from the potential walls. Atom propagation through the fiber in this case is similar to the propagation of light in a multimode, step-index fiber. Evanescent guiding has several advantages over guiding by grazing incidence modes: heating of the atoms due to spontaneous scattering of photons is small in the evanescent case because the atoms spend most of the time in a dark region away from the high laser intensity at the wall. In the grazing incidence configuration, atoms are guided in the high-intensity region, and consequently the spontaneous scattering rate is relatively high. Furthermore, in evanescentwave guiding, the optical potential is generated by light traveling in lossless guided modes. Small-diameter atomic guides of very long length may be practical. By contrast, grazing incidence optical modes decay exponentially with distance [3], effectively limiting the guiding distance to a
*Permanent address: Physics Department, Michigan Technological University, Houghton, MI 4993 1 . ‘Also at Quantum Physics Division, National Institute of Standards and Technology, Boulder, CO 80309. 1050-2947/96/53(2)/648(4)/’$06.00
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few attenuation lengths. For smaller fibers the attenuation length, which scales as the radius cubed, is an increasingl~ severe limitation. Evanescent guiding evidently has a greater practical potential for atom guiding applications. In this paper we report an experimental demonstration of atom guiding in hollow fibers using evanescent fields. We find that severe optical mode-matching constraints, resulting from light scattered into the hollow part of the fiber, can be ameliorated with the help of a second escort laser beam detuned to the red side of atomic resonance. We also observe a threshold intensity level for evanescent guiding that is indicative of van der Waals forces: the repulsive barrier formed by the evanescent wave must be sufficient to overcome attractive van der Waals forces. A measure of these attractive forces is relevant to the design of other atom waveguide configurations [4-6]. 11. THEORY
Comprehensive theories of atom guiding using optical forces may be found elsewhere [1,2,4]. Here we present only the fundamental concepts. An atom interacting with a nearresonant inhomogeneous laser field experiences a conservative ponderomotive energy shift given by [7]
where A=w,,-wo-kuZ is the laser detuning from resonance and k=27r/h. f l ( p ) = d E ( p ) / f i is the atomic Rabi frequency in the presence of an oscillating electric field, d is the atomic transition dipole moment, and y is the spontaneous decay rate. We have written the potential as an equivalent temperature, as indicated by Boltmann’s constant k , . For large detunings from resonance and weak fields, Eq. (1) reduces to the familiar form U ( p ) = h f l * ( p ) / A k , . With a positive detuning from resonance the positive potential shift acts to repel atoms from high-intensity regions. Conversely, atoms are attracted to high-intensity regions of light having a negative detuning from resonance. Laser light in the evanescent configuration is coupled into modes of the annular glass region of the fiber. Approximate expressions for the guiding potential are most easily obtained using ray optics to describe the propagation of light in the R648
0 1996 The American Physical Society
41 7 EVANESCCN I-WAVE GUIDING OF ATOMS IN IHOLLOW . . .
53
A
1‘ 1
10 prn
K649
again treat the waveguide as an infinite dielectric slab. In this case, the van der Waals potential is given by [9]
77 pm
where E is the dielectric constant, (gld’lg) i s the matrix element of the square of the dipole operator, and s is the v distance from the wall. The effective potential, consisting of the van der Waals and evanescent potentials, i s everywhere attractive for small intensities. The intensity above which an atom will experience a positive potential we call the threshold intensity. Slightly above threshold the potential i s positive throughout the hollow region except very close to the wall where the attractive van der Waals potential dominates. Just above threshold the potential exceeds zero at a distance s = 3 / 2 k from the fiber wall and occurs when Uev(0) ‘=lo-’ K. The corresponding intensity threshold is ft1, - 10’ Wim’ for a I-GHz detuning from resonance. The uniform intensity and slab waveguide assumption overestimates the actual potential height. First, the slab waveguide approximation ignores the fact that some laser power i s coupled to sagittal, or skew, rays which may have inner radial turning points larger than the core radius and will FIG. 1. (a) Cross section of hollow capillary fiber used to guide not contribute to the evanescent potential. Thus, the effective atoms. (b) Flat topped optical intensity profile assumed when light intensity of the guide is reduced by the fraction of light is in.jected into the glass region of the fiber. The extent of the cvacoupled to these rays. nescent field extending into the vacuum is exaggerated for clarity. Second, the preceding treatment ignores interference be(c) Transverse intensity profilc of the escort laser. The actual profile is I ( p ) = I , J i ( x p ) where y, is a constant. tween optical modes that give rise to optical speckle and, consequently, to modulation of the intensity of the evanescent field. Atoms encountering a “dark” region in the evahollow fiber [8]. As a further simplification we ignore the nescent field are not shielded from the attractive van der curvature of the fiber walls and approximate the waveguides Waals forces and may be lost from the atomic beam. We as a dielectric slab. For light striking the glass-vacuum intercalculate that room-temperature atoms spend insufficient face at an angle 6‘ and polarization perpendicular to the plane time in the dark to be pulled into the wall before the potential of incidence, the evanescent component of the field extendchanges sign. On the other hand, slower atoms may be lost in ing a distance x into the vacuum is given by these regions. Evanescent guiding requires that the blue detuned field be E ( x ) = E o a exp( - K X ) , (2) confined to the glass waveguide. Typically, some of the laser light scatters from the fiber and couples to grazing incidence where E o is the electric-field amplitude incident on the intermodes existing in the hollow region. Some small fraction of face, and the factors a and K are given in terms of the index light in the hollow region can easily dominate the evanescent of refraction n, incidence angle 0, and laser wavelength A by portion and drive the atoms into the walls. We found it difa=2 J-cosO, and ~ = ( 2 ~ I h ) ( n ~ s i n ~ 6 ’ - 1We ) ” ~ . ficult in practice to match the injected laser beam sufficiently assume that the intensity profile is uniform in the glass rewell. A rough estimate of the mode-matching constraints is gion, as shown in Fig. 1. In particular, we are ignoring interobtained by assuming that efficient guiding is possible ference among modes, which gives rise to an optical speckle only when the laser intensity at the wall, f=Zoa2,exceeds pattern. The intensity on the inner wall of the fiber is then the intensity coupled into the E H l l grazing incidence mode approximately the total power in the guide divided by the by, say, a factor of 10. This suggests that the scattered light cross-sectional area of the glass region, I( 0) = PIA . The field must be suppressed to better than 0.05% in a fiber with a intensity is related to the electric-field strength by E 144-pm outside diameter and a 20 p m inside diameter. To circumvent the mode-matching problem a second laser = giving an evanescent intensity profile of I(x) is used to escort atoms through the spatial transient region = f ( 0 ) a 2 e x p ( - 2 ~ ) . We assume an incident angle correand into the region where the evanescent potential is domisponding to the numerical aperture of the coupling lens, 0.1, nant. This escort laser is coupled into the lowest-order giving -0.26. With a typical laser power of 500 mW and a E H , grazing incidence mode which has an intensity profile fiber cross-section area of 1.8X lo-* m2, the evanescent ingiven by f = f o J i ( x p ) with xa-2.40 [1,3] and a the fiber tensity at the wall is I ( 0 ) a 2 =1.9X lo6 W/m2 corresponding to a potential height at the walls of U ( 0 )= 22 mK. hole radius. f o is the peak intensity on the axis and p is the radial distance from the fiber center. With a red detuning In the absence of an optical potential, the atoms are atfrom resonance atoms are attracted to the high-intensity retracted to the glass wall by long-range, van der Wads forces. gion along the axis and initially guided into the fiber. To estimate the effect of the van der Waals potential we
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Guide Laser Detuning (GW FIG. 3. Evanescent-wave guidinz of atoms injected into the guide with a red detuned escort laser. For an escort laser detuning of 1.6 GHz. upper curve. and an evanescent-wave laser power of 500 mW. the guided-atom signal is enhanced by a factor of 4 at a detuning of +3 GHz. For larger escort laser detunings (-9.4 GHz for the lower curve) the number of injected atoms is smaller but the fractional signal enhancement compared to the absence of the evanescent laser is nearly the same. ~
FIG. 2. Optical coupling scheme. The escort and evanescent laser beams are combined on a polarizing beam splitter (PBS) and separately focused onto the fiber hollow and glass regions.
The grazing incidence modes are leaky, having an intensity attenuation length [3], L,=2.4a3/h2=3.8 mm, where a= 10 pm. The potentials formed by scattering of the blue detuned laser and by the red detuned escort laser both decay within a few centimeters of the fiber input. As the escort potential decays, atoms with the highest transverse energy are lost from the atomic beam. However, when the evanescent potential is present, the atoms that would normally be lost from the grazing mode light may be retained in the evanescent potential and guided through the remaining distance. In this way atoms are escorted by the red detuned laser into the region where only an evanescent potential exists.
111. RESULTS
The experimental setup for guiding atoms is similar to that of Ref. [l] and consists of two vacuum chambers that are connected by a short length of hollow core fiber. The chamber on the input side of the hollow fiber contains a thermal cell of Rb. Laser light is coupled through a window in this chamber and into the fiber. Atoms with sufficiently small transverse velocities are loaded at random from the vapor into the fiber guide. The guided atoms emerge in the second chamber and are detected with a hot wire and channeltron. Figure 2 shows the optical scheme used for launching the laser light and atoms: a high-power (500 mW) laser beam blue detuned from resonance is focused into the annular region of the fiber end; it is coupled primarily to the guided modes but also to the leaky and radiative modes. A second and weaker laser beam, tuned to the red of atomic resonance is focused into the follow region where it couples to the E H , , grazing incidence mode. The optical power in the grazing incidence mode is concentrated within a small region in the fiber core, allowing a potential depth comparable to that of the evanescent field case to be created with nearly two orders of magnitude less power. Thus, a relatively weak, -10-mW diode laser is sufficient to escort the atoms. A few centimeters from the fiber input, the E H , , mode amplitude is
sufficiently attenuated so that the evanescent potential begins to dominate. In Fig. 3 we show the enhancement of the atom flux guided through a 6-cm-long fiber with a 20-pm core diameter using this injection scheme. With the red detuned escort laser alone 200 atoms/s are guided through the fiber. Using a mode attenuation length of 4 mm, the intensity and potential decay by a factor of 500 through the fiber, implying that approximately lo5 atoms/s were initially launched but then lost. Launching 500 mW of evanescent laser power into the glass region recovers some of the lost flux when the laser is tuned to the blue side of resonance but not to the red side, as expected. The flux enhancement is at least a factor of 3 at an optimum detuning of +3 GHz. Increasing the escort laser detuning to -9.4 GHz reduces the number of atoms injected into the evanescent guide and thus decreases the overall guided atom signal. The fractional enhancement, determined as the ratio of guided atom flux with and without the presence of the evanescent guide laser, stays roughly the same as the launch laser is detuned out to -20 GHz. Tuning the escort laser to the blue side of resonance inhibits the injection of atoms into the evanescent guide and completely suppresses the guided atom flux. The intensity dependence of evanescent guiding is qualitatively different than for grazing incidence modes. In grazing incidence configuration atoms are confined near the center of the fiber by a harmonic potential and only weakly interact with the van der Waals potential near the walls. As previously observed [ 13 this leads to a linear intensity dependence of the flux for low intensities. For evanescent guiding the atoms interact strongly with the attractive van der Waals potential, which results in an intensity threshold for guiding. Measurements of the flux enhancement with increasing laser power presented in Fig. 4 show the expected threshold phenomena. For intensity levels below 6 MW/m2 no enhancement in the flux is observed. Above 6 MW/m2 the flux increases roughly linearly with laser intensity. Precise
419 EVANESCENT-WAVE GUIDING OF ATOMS IN H O L L O W . . .
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by using an evanescent field to repel atoms from the inner wall. An intensity threshold is observed for evanescent guiding that is attributed to the intensity required to overcome van der Waals attraction of the atoms to the fiber wall. The two-color guiding combination of guide and escort laser beams ameliorates the problem of mode matching. We anticipate tising this method to inject atoms into a I.4-pm-diam guide where they will then be guided up to 1 in in the dark region with evanescent fields. Finally we mention a few applications of evanescentwave atom guides. Fiber-guided atoms are of particular interest for atom-fiber interferometry [4]. In narrow fibers ( 1 0 5 10 15 pin) and at cold temperatures (<1 p K ) the transverse atomic motion will be quantized. Interference between these atomic intensity (MW/$ modes will give rise to atomic speckle patterns, the simplest FIG. 1. Intensit!’ dependcnce 01‘ thc cvanescent-~~\.ave-guidzd of interferometers. At colder temperatures (< 100 nK) the a t o m will propagate in the lowest mode, and with an atomatom signal. A threshold [or guiding atonis. attributed to attractive van der Waals forces. is ohser\#cdat 6 X I O 6 W/m’. Ahovc threshfiber beam splitter an interferometer of the Mach-Zender old the flux increases linearly with intensity. type may be constructed. The evanescent guiding scheme is particularly useful in that heating and associated loss of comeasurements of the intensity threshold are complicated by herence of the atoms due to spontaneous emission i s minithe uncertainties in the number of modes excited and the mized. effect of mode interference on the evanescent potential. The threshold intensity without correction for any of these effects is approximately 6 X lo6 Wim’ and is roughly a factor of 10 larger than predicted. We note that more precise measurements may be facilitated with the use of single-mode hollow ACKNOWLEDGMENTS core fibers [4]. 130
I
IV. CONCLUSION
In conclusion, we have observed a factor of 3 enhancement in guided atom flux through a hollow core optical fiber
[l] M J. Renn et a/., Phys. Rev. Lett. 75, 3253 (1995). [2] M. A. OI’Shanii et a/., Opt. Commun. 98, 77 (1993).
[3] E. A. J. Marcatile and R. A. Schmeltzer, Bell System Tech. J. 43, 1783 (1964). [4] S. Marksteiner et a/., Phys. Rev. A 50, 2680 (1994). [ 5 ] W. Jhe e t a / . , Jpn. J. Appl. Phys. 33, L1680 (1994). [6] H. Ito e t a / . , Opt. Commun. 115, 57 (1995).
We acknowledge C. M. Savage, P. Zoller, and G. Schmidt for helpful discussions and are grateful to the O f i c e of Naval Research (Contract No. N00014-94-1-0375), NIST, and NSF for financial support of this work.
[7] See, for example, J. Dalibard and C. Cohen-Tannoudji, J. Opt. SOC.Am. B 2, 1707 (1985). [8] C. M. Savage et al., Atomic Waveguides and Fundamentals of Quantum Optics III, edited by F. Ehlotsky (Springer, Berlin, 1993). [9] M. Chevmollier et a/., J. Phys. I1 (France) 2, 631 (1992).
VOLUME 77, NUMBER 16
PHYSICAL REVIEW LETTERS
14 OCTOBER 1996
Low-Velocity Intense Source of Atoms from a Magneto-optical Trap Z. T. Lu, K. L. Corwin, M. J . Renn,* M. H. Anderson,’ E. A. Comell, and C. E. Wieman Joint Iiistifiite .fbr Lahorntory Astrophvsics, National Iiistitiite of Standards and Technology iiiid Uiiiwr~ityqf Colorudo, Boiildci; Colordo 80309 and Pli,v,sic,s Depni.trnent, University qf Colorado, Boillder, Colorado 80309-0440 (Rcccived 6 June 1996)
Wc have produced and characterized an intensc, slow, and highly collimatcd atomic bcani extracted from a standard vapor cell magneto-optical trap (MOT). Thc tcchniquc used is dramatically simpler than previous methods for producing very cold atomic beams. We have created a 0.6 nini diameter rubidium atomic beam with a continuous flux of 5 X IO”/s and a pulsed flux 10 times greater. Its longitudinal velocity distribution is centered at 14 m / s with a FWHM of 2.7 m/s. Through an efficient recycling process, 70% of the atonis trapped in the MOT are loaded into the atomic beam. [SO03 1 -9007(96)01434-21 PACS numbers: 32.80.Pj, 32.80.Lg
Cold atomic beams are usefd in a variety of applications: in ultrahigh resolution spectroscopy, as frequency standards, and in studies of cold atomic collisions [l]. An intense beam of cold atoms is valuable for atom interferometers [2], particularly those sensing rotational and gravitational effects. A cold atomic beam coupled into an atom fiber guide [3] will provide much larger guided atom flux. Current experiments studying Bose-Einstein condensation [4] and trapping of radioactive atoms for fundamental symmetry tests [5,6] require a system of two magneto-optical traps (MOTS) with the capturing and the measurement processes separated in space. Our experiment reveals a simple and efficient way to transfer atoms between two MOT’S via a cold atomic beam. Many examples of cold atomic beams have been demonstrated, such as Zeeman slowers [7,8], chirpedcooled beams [9], and beams slowed by broadband light [ 101 or isotropic light [ 111. These beams all experience serious transverse diffusion effects as the atoms are slowed to very low velocities ( 5 1 5 m/s). This causes a loss of atoms and reduced collimation. To counteract these effects, slow atoms are passed through a twodimensional MOT, or atom funnel, to compress and cool them [12-141. To date, the brightest slow beams employ this technique and include a beam of Na atoms traveling 2.7 m/s with a brightness of 3 X 10” atoms/srs [12], and a beam of Ne* traveling 19 m/s with a brightness of 3 X 10” atoms/srs [13]. In contrast, we have created a low-velocity intense source (LVIS) of atoms with a brightness of 5 X 10l2 atoms/sr s. This atomic beam, the brightest beam of atoms moving slow enough to be easily captured by a MOT ( 5 2 0 m/s), offers the advantage of simplicity, for it is made merely by adding a small modification to the simple vapor cell magneto-optical trap (VCMOT) [15]. In this Letter we report a detailed study of the LVIS beam. We observed the longitudinal velocity distribution by making time-of-flight studies; and we obtained trans-
verse velocity distributions and absolute measures of both pulsed and continuous brightness by adding atomic beam fluorescence measurements. Our model of the system explains most of our results quantitatively, and all of them qualitatively. The LVIS system is nearly identical to a standard VCMOT with six orthogonal intersecting laser beams. The only difference is that one of the six trapping laser beams has a narrow dark column in its center. Atoms in the low-velocity tail of the thermal vapor enter the VCMOT trapping volume and slow down. After they diffuse into the trap center, they enter the central column (“extraction column”) and are accelerated out of the trap by the counterpropagating laser beam (“forcing beam”). The velocity of the extracted atoms is determined by the number of photons they scatter from the forcing beam before leaving the trap. A key feature of this scheme is that these extracted atoms are continuously apertured by laser light along the beam. Those diverging atoms that move out of the extraction column are recaptured and returned to the trap center. This mechanism of recycling the diverging atoms provides a very efficient way of transferring trapped atoms into a collimated atomic beam. The atomic beam flux is determined by the capture rate of the VCMOT. In a conventional VCMOT, the equilibrium number of trapped atoms is N = R / r c , where R is the capture rate and rc is the collisional loss rate [15,16]. In LVIS, most of the atoms are “lost” into the atomic beam. It can be shown that the LVIS beam flux F is given by F = R/(1 + r c / r , ) where r , is the rate of transferring atoms into the beam. Typically rc << r t , e.g., l / r , = 1.0 s a n d I/r, = 30 ms, so F = R. The collisional loss also affects atoms in the beam. Since it takes much less energy to knock an atom out of the LVIS beam than out of a VCMOT, the beam collisional loss rate is roughly 5 times that of the VCMOT. The tradeoff between the collection rate of the VCMOT and the collisional loss rate from the LVIS beam limits the
0 1996 The American Physical Society
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flux and determines the optimum thermal vapor pressure (-1 X lo-’ Ton-). The low-energy collisional cross section in Ref. [ 171 agrees well with the dependence of collimation and flux on Rb vapor pressure that we observe with LVIS. A schematic of the ”Rb LVIS apparatus is shown in Fig. 1. A Ti:sapphire ring laser provides about 500 mW of “trapping” light at a typical frequency 30 MHz detuned 5 p 2 P 3 p ( F ’ = 3) transition. from the 5s ‘ S ! / z ( F = 2) A diode laser supplies 20 mW of “repump” light, tuned to 5y 2P3/2(F‘ = 2) transition. As the 5s 2 S 1 / 2 ( F= 1) in a conventional VCMOT, the trapping beam is split into three beams which intersect inside a vacuum chamber containing Rb vapor. Each of the three beams is reflected back in the opposite direction to make six counterpropagating laser beams in a retroreflecting configuration. The VCMOT beams are relatively large (-4 cin diameter) in order to maximize R . To produce the extraction column, a millimeter sized hole is drilled through the center of one of the retroreflecting assemblies which consist of a quarter-wave plate and mirror [18]. The atomic beam is extracted from the trapping region through this hole. A pair of anti-Helmholtz coils generates the quadrupole magnetic field for the trap, with a gradient of -5 G/cm along the atomic beam direction. To position the trap center in the extraction column, the point of zero magnetic field is moved with a set of orthogonal magnetic shim coils. In normal operation, the plug beam is blocked by a mechanical shutter. When making measurements, the plug beam is unblocked so that the atoms are forced out of the extraction column and returned to the center of the trap. This capability of quickly turning the atomic beam on and off allows us to measure the longitudinal velocity distribution, and to run LVIS in a pulsed mode. With a charge-coupled device camera and a photodiode, we monitor the fluorescence emitted when the atoms cross the de-
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FIG. 1 . Schematic of the LVIS system. Large shaded arrows represent the 4 cm diameter trapping laser beams. A repump laser (not shown) illuminates the trapping volume up to the edge of the retro-optic, which has a small hole and is placed inside the vacuum chamber. This hole, a distance z from the trap center, creates an extraction column through the trap center and causes atoms to accelerate out of the VCMOT. A standingwave light field 30 cm downstream forms the detection region. The plug is a thin beam of trapping laser light; when present, it prevents atoms from leaving the trap via the atomic beam.
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tection region. This allows us to measure the flux, spatial distribution, and velocity distribution of the atomic beam. Given the trap parameters described above and in Fig. I , we found that the trapping laser frequency which maximizes the LVlS beam flux is 5r detuned (where r is the natural linewidth) from the cycling transition, while the detuning that maximizes N in the normal VCMOT is 3.2r. This difference presumably occurs because the transverse light beams can heat and, if there are any imbalances, deflect the atomic beam as it exits the trap. At lower detunings, the scattering rates and hence these deleterious effects increase. The longitudinal velocity distribution in the LVIS beam, as shown in Fig. 2, is measured by the time-offlight method. Typically, we observe v 15 m/s, consistent with our simple model based on a calculation of the photon scattering rate from the forcing beam. In our model the acceleration begins when an atom enters the extraction column. As the atom is accelerated, the scattering rate slows due to Doppler shift and Zeeman shift. Scattering and acceleration cease when the atoms finally leave the region of repump light. Figure 3 shows the dependence of the longitudinal velocity on the intensity and frequency detuning of the forcing beam. Both plots indicate that the final velocity increases with the scattering rate in a manner consistent with our model. Note that the range of useful velocities is limited because the flux decreases rapidly when the intensity and detuning are far from those which optimize the trap capturing process. A narrow longitudinal velocity distribution is usually desired in the applications of cold atomic beams. The velocity spread, about 2.7 m/s FWHM, is much larger than the Doppler cooling limit of 0.12 m/s. The velocity spreads due to a random distribution over magnetic sublevels and statistical fluctuations in the number of scattered photons were both estimated to be -0.7 m/s. We believe the dominant contribution to the longitudinal spread arises because the atoms enter the extraction column within the trap at different positions along the beam axis, and are therefore accelerated over different distances. A calculated velocity and spread match the experimentally observed values ( u = 14 m/s, FWHM = 2.7 m/s) if we assume that the acceleration distance
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FIG. 2. A typical longitudinal velocity distribution. In this case, the average velocity is 14 m/s and the FWHM is 2.7 m/s. This curve was made by recording the shape of the time-offlight signal and taking the derivative of atom flux with respect to velocity.
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FIG. 4. The avcrage longitudinal velocity and spread are shown as a function of lhc forcing beam polarization. The quai-ter-wave plate at 45" gives circularly polarized light. Below 10" and above 80" the flux falls rapidly because the trap capture rate decreases, but the flux varies by less than 30% betwcen 10" and 80". Also, the mean velocity has a much weaker dependence on polarization than the fractional spread.
Detuning (MHz) FIG. 3. The average longitudinal velocity and flux as a function of (a) the forcing laser intensity (with detuning at 32 MHz) and ( b ) the detuning (with I = 38 mW/cm'). In both cases the final velocity increases with increasing scattering rate from the forcing bcam, whilc thc fractional spread remains nearly unchangcd.
covers the range 2.2 to 3.4 cm. This is reasonable since in this case z is 2.5 cm (see Fig. I), and the a t o m trapped in the VCMOT with the plug beam unblocked form a cloud - 1 cm in diameter. The fractional FWHM of the longitudinal velocity distribution depends on the forcing beam's polarization. Figure 4 shows that the fractional FWHM can be minimized by making the polarization significantly elliptical. For circularly polarized light, the Zeeman shift causes an atom to feel an acceleration which is much larger on the upstream side of the trap center than on the downstream side. This principle is essential to the operation of a MOT, but in a LVIS, atoms entering the extraction column on the upstream side experience a larger acceleration than those entering on the downstream side. However, an elliptically polarized forcing beam makes the accelerations more nearly equal on the upstream and downstream sides. This results in a smaller final velocity spread. For angles <20" and >70", the increased spatial spread in the trapped atom cloud outweighs this decreased variation in the acceleration. Many factors contribute at some level to the transverse collimation. lnitially we expected transverse cooling and focusing within the extraction column to dominate. Instead, the measurements described below show that the transverse velocity distribution is primarily determined by a simple geometrical collimation mechanism. Although it is similar to the transverse velocity distribution of a conventional atomic beam collimated with physical apertures, the LVlS beam benefits because the apertured atoms are recycled. In LVIS, the collimation length ( z ) extends from the point where atoms enter the extraction column to the mirror. The divergence angle of the atomic beam 8 is given by 19 = d / z , where d is the diameter
of'the extraction column. The spatial profile is consistent with the triangular profile expected from a geometrical collimation mechanism. When the extraction column was produced by placing the retro-optic with a hole at 2.9 cm from the trap center, the observed divergence angle (36 mrad) agrees well with d / z (40 mrad). To fLirther study this collimation mechanism, the retrooptic with a hole was replaced by a standard retro-optic outside the vacuum chamber. We inserted a piece of glass with opaque spots of various sizes into the laser beam in front of the retro-optic to create the extraction column. To vary the collimation length, we varied the distance (z) over which the repump laser illuminated the atomic beam. The divergence scaled with d / z over a wide range of conditions. The angle vs z is shown in Fig. 5 . Note that while 8 scales as l/7,, the measured values are consistently smaller than the opaque spot diameter divided by z. This is presumably due to diffraction of light into the extraction column which effectively makes d smaller than the diameter of the opaque spot. The tightest collimation was achieved with our maximum collimation length (30 cm) and a 1.6 mm diameter opaque spot. We observed a divergence angle of 5 mrad, implying a transverse temperature of 20 pK. This configuration requires careful alignment, and the atomic beam
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measurements, the retro-optic was removed to the outside of the chamber, and the extraction column was generated by a 1.6 mm opaque spot such that the atoms were accelerated vertically.
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must be sent in a vertical direction to prevent gravity from pulling the atoms out of the extraction column. To better understand the beam collimation, we replaced the conventional MOT field gradient with a quasi-twodimensional MOT which had a magnetic field gradient of 7 G/cm in the transverse direction and < I G/cm in the longitudinal direction. We kept all other conditions the same. The atomic beam width changed from 1.1 m m to 0.65 mm when measured 3 cm above the trap center, but did not change when measured 30 cm above. Thus the quasi-two-dimensional configuration produced transverse focusing but not cooling. We measured the absolute atom flux in the LVIS beam and determined the atom transfer efficiency by comparing this flux with the capture rate of the VCMOT. The capture rate of the trap was determined from measurements of N and r , with the plug beam in place. Our measurements indicated that essentially 100% of the atoms were transferred into the atomic beam for typical values of z . However, the fraction extracted through the hole in the retro-optic into a field-free region varied. The highest flux we achieved was 5 X 109/s, with an extraction efficiency of 30% (0 = 30 mrad, d = 0.7 mm, z = 2.7 cm) which was limited by light scattering around the hole edges. By making a cleaner hole through the retro-optic (0 = 36 mrad, d = 0.8 mm, z = 2.0 cm), we increased the efficiency to 70%, but by that time our deteriorating Ar' laser tube allowed us to trap an order of magnitude fewer atoms. However, with this 70% efficiency and -500 m W of Tkapphire laser power, we expect to achieve a beam flux >lO1"/s. When operated at lower power the LVIS flux drops in proportion to the reduced capture rate for the MOT. However, the beam collimation and velocity remain nearly the same so LVIS would still produce a nice beam with low power diode lasers. Finally, w e observed a much higher peak flux when the LVIS system was operated in a pulsed mode. In these geometries, the VCMOT empties in 50 ms, providing nearly the same time-averaged flux but 10 times the peak flux and brightness. The flux and flux density depend on the collimation angle and geometry of the LVIS setup in the manner predicted above. Although we achieved 70% extraction efficiency when 0 = 36 mrad, we only achieved a trans-
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fer efficiency of 20% when N = 5 mrad. Figure 5 shows the tradeoff between flux density and collimation of the atomic beam. Recycling causes the flux density (number s-' cm') at the detection region to increase while the divergence decreases. Total flux decreases beyond 4.5 cm because for a collimation this tight, the atomic beam transverse temperature becomes comparable to the temperature of atoms in the VCMOT. With tight collimations, an atom must make many more attempts to be successfiilly transferred into the atomic beam. This makes r c / r , larger, decreasing the flux as predicted. The simplicity, brightness, and versatility of LVIS will make it useful in a wide range of applications. This work was supported by the National Science Foundation, the Office of Naval Research, and the National Institute of Standards and Technology.
*Permanent address: Physics Department, Michigan Technological University, Houghton, MI 4993 1. iPermanent address: Meadowlark Optics, 7460 Weld County Road 1, Longmont, CO 80504-9470. [I] K. Cibble and S. Chu, Phys. Rev. Lett. 70, 1771 (1993). [ 2 ] D. W. Keith et al., Phys. Rev. Lett. 66, 2693 (1991). [3] M. J . Renn er al., Phys. Rev. A 53, R648 (1996). [4] M. H. Anderson ef a/., Science 269, 198 (1 995). [5] Z-T. Lu et a/., Phys. Rev. Lett. 72, 3791 (1994). [6] M. Stephens and C. Wieman, Phys. Rev. Lett. 72, 3787 (1994). [7] T. E. Barrett et al., Phys. Rev. Lett. 67, 3483 (1991). [8] W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982). [9] W. Ertmer er al., Phys. Rev. Lett. 54, 996 (1985). [lo] M. Zhu, C. W. Oates, and J. S. Hall, Phys. Rev. Lett. 67, 46 (1991). [ l l ] W. Ketterle et al., Phys. Rev. Lett. 69, 2483 (1992). [12] E. Riis et al., Phys. Rev. Lett. 64, 1658 (1990). [13] A. Scholz et nl., Opt. Commun. 111, 155 (1994). [I41 J . Yu et nl., Opt. Commun. 112, 136 (1994). [15] C. Monroe et al., Phys. Rev. Lett. 65, 1571 (1990). [ 161 K. Lindquist, M. Stephens, and C. Wieman, Phys. Rev. A 46, 4082 (1 992). [ 171 C. Monroe, Ph. D. thesis, University of Colorado, Boulder, CO, 1993. [ 181 A hole was drilled in the quarter-wave plate, and then the back surface was coated with a reflecting layer of gold.
VOLUME 79, NUMBER6
PHYSICAL REVIEW LETTERS
1 1 AUGUST 1997
Efficient Collection of 221Frinto a Vapor Cell Magneto-optical Trap Z.-T. Lu, K. L. Corwin, K. R. Vogel, and C. E. Wieman JILA, National Institute of Standards and Technolog,v and University of Coloivdo, Bozilder, Colorado 80309-0440, and Phj)sics Departnieiit, IJnivei-sity qf Colorado. Boulder, Colorudo 80309-0440
T. P. Dinneen, J. Maddi, and Harvey Could Lait~renccBerltelev National Laboratory, Universit,v of California, Mail Stop 71-259. Berkeley, California 94720 (Reccived 23 April 1997) We have efficiently loaded a vapor cell magneto-optical trap from an orthotropic source of '"Fr with a trapping efficiency of 56( lo)'?& A novel detection scheme allowed us to measure 900 trapped atoms with a signal to noise ratio of -60 in 1 sec. We have measured the energies and the hypefine constants of the 7'P1/2 and 7'P3/2 states. [SO03 I-9007(97)03811-81 PACS numbers: 32.80.Pj, 32.10.Fii
the trap volume (region of laser beam overlap) to the cell There has recently been considerable activity in the surface area. To minimize the loss rate from the cell, the field of laser trapping of short-lived radioactive atoms. opening through which the radioactive atoms enter the cell While a wide range of isotopes is being pursued, laser trapping of "Na [I], 37,38K[2], 79Rb [3], and 209,210,211Frmust be small enough to minimize the leak rate, and the loss of atoms via adsorption to the glass walls must be sig[4] atoms has been experimentally realized. Efficient opnificantly reduced. The latter is accomplished by coating tical trapping is essential for creating large samples of rare the glass surfaces with dryfilm coatings made of siliconatoms. Such samples are very appealing for tests of the standard model including atomic parity nonconservation based hydrocarbon polymers, which are then "cured" by exposure to alkali vapor [ 121. (PNC), the electric dipole moment (EDM), and p decay The analysis of the capture process differs from that of [l]. Francium, which has no long-lived isotopes, is particularly interesting for these tests, because calculations a conventional VCMOT. I n a highly efficient VCMOT, predict PNC amplitudes and EDM enhancements to be the number of atoms in the vapor and the number of atoms 10 times larger in Fr than Cs [5,6]. In this Letter we in the trap are so strongly coupled that the vapor density demonstrate efficient trapping of '"Fr. The general apcannot be considered constant. The time evolution of proach should work as well with any alkali isotope, and such a coupled system of N f atoms in the trap and Nu should make tests of the standard model possible in rare atoms in the vapor depends on three rates: L , the loading rate of atoms from the vapor to the trap; C, the loss rate trapped atoms. of atoms from the trap to the vapor due to collisions with Various techniques have been developed to collect vapor atoms; and W , the loss rate of atoms from the vapor atoms into traps [7], but since short-lived radioactive atoms are available only in limited quantities, improving to the cell walls or out of the cell. In the case where a the optical trap collection efficiency is a central issue. constant flux, I , of atoms enter the cell, the dependence can be described by two coupled differential equations: The highest efficiency yet demonstrated used a vapor cell magneto-optical trap (VCMOT) [8] in a glass cell coated with dryfilm. Using coated cells, Stephens et al. [9] and CN, LN,, Guckert et al. [ 101 have demonstrated, respectively, 6% and 20% collection efficiencies of stable cesium. Similar d ( N v ) = C N , - LN, - W N , + I techniques have been applied to trapping radioactive dt K, Rb, and Fr atoms [2-41, but with far lower trap The time evolution of the number of trapped atoms efficiencies. We have created a highly efficient (56%) follows a double-exponential function, '"Fr VCMOT in a coated cell and used it in spectroscopic measurements on *"Fr. A conventional VCMOT [8] traps a very small fraction of the available atoms. To obtain high collection C + W and klk? = W C . N f ( t = where kl + k2 = L efficiency, one must raise the collection rate and lower 0 ) and N,(t = 0) fully determine a1 and a2. We define the vapor loss rate. The dependence and optimization of trap efficiency (7) as the probability of trapping an atom collection rate on various trap parameters has been previthat has entered the cell [ 131, ously investigated [ 111. To have a high collection rate, L CNf a trap should have large, high-power laser beams. The (CN, I ) ' = (L + W ) trapping cell should be designed to maximize the ratio of
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The "'Fr nuclei are produced in the decay chain 229Th(t1/2= 7300 yr)
u
-
775
'--Ra(tl/2
=
15 d)
22sA~(tl/2 10 d)
a 221
y
Fr(tl/?
=
4.9 min).
In order to produce a portable '*'Fr source with shortlived radioactivity, 2 2 5 Awas ~ first extracted out of a longlived 2'9Th sample, and deposited onto a small piece of platinum using one of two methods. Results reported here were obtained from a sample of "'Ac chemically extracted from a "'Th solution [ 141 and electroplated onto a Pt ribbon. In our preliminary measurements, we used 2 2 5 Aimplanted ~ onto Pt via electrostatic collection [15]. The Pt ribbon was then placed in the cavity of an orthotropic oven [15] where '"Fr daughters were continuously produced and distilled into a collimated atomic beam. At the beginning of the experiment, the oven was loaded with 50 p C i 2 2 5 A ~which , produced 2 X lo6 s-' of *"Fr. The full divergence angle of the atomic beam was measured to be 180 mrad. By counting the cy particles from the decay of the ?"Fr, we measured an atomic **'Fr flux of 3.8 X lo4 s-', or about 2% of the "'Fr atom production rate inside the oven. Thirty days later, at the end of the experiment, the 221Frflux was 5.1 X lo3 s-I, reflecting a decrease matching the natural decay of 2 2 5 A ~ . A schematic of the apparatus is shown in Fig. 1. Both the francium oven and the vapor cell were situated inside a chamber where the vacuum was maintained at 2 X lo-* Tom even when the oven was heated to the operating temperature of 1050 "C. The cell was a quartz glass cube (4.4 cm inside dimension) whose top lid could be opened and closed via a mechanical feedthrough. Following the recipe developed by Stephens et al. [12], the cell walls were coated with a short-chain dryfilm called SC-77 (Silar Laboratories), and then cured by opening the cell lid and
maintaining a Rb vapor in the cell at about 2 X 1 O r 7 Torr for 10 h. The Fr atoms from the oven entered the cell through a 2 mm diameter hole at a lower edge of the cell. Over several days, the coating performance deteriorated, reducing the number of trapped Fr atoms by a factor of 3. This damage to the coatings could be attributed to heat or material evaporating from the oven, and was repaired by providing a continuous, low level curing with - 1 X l o p 8 Torr of rubidium in the cell. Up to 1 W of light from a Ti:sapphire ring laser was tuned to the D l line of Fr at 7 18 nm for trapping. The laser frequency was locked to the side of a Doppler-broadened I? absorption peak [ 161 that conveniently covered the frequency of the 7S1/2, F = 3 7P3/2, F = 4 cycling transition (Fig. 2). In addition, an SDL 100 mW diode laser at 8 17 nm was tuned to the D I line to pump atoms out ofthe 7SI/2,F = 2 state. The frequency of the diode laser was tuned to the 7S1/2,F = 2 7P1p,F = 3 transition and locked to a Fabry-Perot cavity, which was in turn actively stabilized to the rubidium 0 2 line. We assessed the performance of the wall coatings and found the correct laser frequencies by observing fluorescence from the room-temperature Fr vapor. We used an optical-optical double resonance technique to separate the fluorescence from light scattered off the cell walls. For this technique, a beam from the Tksapphire laser was sent through the center of the cell, while its frequency scanned over one Doppler width (300 MHz) about the 7S1/2, F = 3 7P312, F = 4 cycling transition. A 60 mW beam from the 817 nm diode laser, chopped at 100 Hz, copropagated and overlapped the Tksapphire beam through the cell. The diode laser frequency was tuned close to the 7S112,F = 2 7Pl/2, F = 3 transition to pump atoms from the 7S1/2, F = 2 to the 7Sl/2, F = 3 state. The population of the 7Sl/2, F = 3 state was therefore fully modulated for the group of atoms in resonance with both laser beams. Because this was a narrow velocity group, the resulting resonance feature was sub-Doppler (50 MHz
-
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-
F=4-
316MHz
18.6GHz/
7Pg/2
751/2
2
FIG. 2. The atomic level diagram of 22'Fr. In our I2'Fr trap, the trap laser (718 nm) was tuned to -30 MHz below the 7S1/2, F = 3 7P3/2rF = 4 cycling transition, and the repump laser (817 nm) was tuned to the 7s1/2, F = 2 7P1/2, F = 3 transition. To detect the trapped atoms, a 718 nm pump beam (2 mm diameter, 50 mW/cm'), tuned to the 7S,/2, F =3 7P3/2, F = 3 transition, was chopped to modulate the 8 17 nm fluorescence from the trap. -+
FIG. 1. A diagram of the 22' Fr trap setup. Both the oven and the cell were situated inside a vacuum chamber with 10 cm diameter windows. The same oven and cell assembly was used for the "'Fr vapor fluorescence measurements.
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FWHM). The resulting modulated fluorescence at 718 nm generated near the center of the cell was imaged onto a low noise photodiode through a 7 18 n m interference filter and demodulated with a lock-in amplifier. T ~ Uwe S were capable of detecting as few as 600 atoms in the entire cell, or equivalently 1 Fr atom in resonance in the viewing region, with a signal to noise ratio (SNR) of l/&. By monitoring the exponential buildup of the number after the cell lid was closed, we determined the loss rate of Fr atoms from the vapor (W) to be 1.59(9) s - ' , and a constant flux of 2.3 X lo3 s-' Fr atoms entering the cell. We then trapped the Fr atoms in a VCMOT using 4 cm diameter laser beams. These beams provided a sixbeam-total intensity of up to 110 mW/cm2 in the cell. In addition, a 60 mW, 4 cm diameter beam from the diode laser at 8 17 nin was sent through the cell four times along two normal axes of the cube cell. A set of anti-Helmholtz coils generated the MOT quadrupole field gradient of -7 G/cm. The laser light scattered from the six cell walls made detection of the 718 n m fluorescence from the trapped Fr atoms impossible. Instead, to further avoid scattered light and thereby increase our sensitivity, we imaged the 817 nm repump fluorescence froin the trapped atoms onto a photodetector. This light was observed through a cell window not illuminated by any repump beams and through a 8 17 nm bandpass interference filter. The 817 nm fluorescence was modulated using a small, frequency-shifted 7 18 nm pump beam that was chopped and retroreflected through the trap center (see Fig. 1). This beam increased the fractional population in the 7s1/2, F = 2 state to 70% without affecting the trap loading rate. This detection scheme allowed us to detect as few as 15 trapped Fr atoms with a SNR of 1 1 6 in the presence of large amounts of scattered light. The trap contained an estimated 900 atoms, but this is a lower limit assuming that both the 718 nm pump and the repump lasers were tuned to resonance. This trapped atom sample could be maintained for many hours. Figure 3 shows a trap loading curve fitted to the doubleexponential function predicted by our model. The fit gives the trap rate constants, from which we calculate the trapping efficiency to be 7 = L / ( L W ) = 56(10)%. As a check, 7 = C N r / ( C N r + I) gives a lower limit of 30% assuming the repump laser is tuned to resonance, but can be as high as 50% when plausible detunings are assumed in calculating N t . The atomic fluxes at various stages of the experiment are listed in Table I, showing a total efficiency of 0.4% from production to trapping. Engineering improvements in the oven and in the coupling into the cell would significantly improve the total efficiency. An orthotropic source has operated with a 15% efficiency, and 50% (limited by diffusion of Fr into the oven walls) is theoretically possible [ 151. We have made spectroscopic measurements on **'Fr using our apparatus. In the thermal vapor, we measured
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4
Time(sec) FIG. 3. The number of atoms loaded into a "'Fr trap vs time. In a trap with a high collection efficiency, the loading signal is expected to be a double-exponential function. Because the signal was filtered by a low-pass nctwork with a time constant T = 100 ins, it was fitted with the function 01[1 - exp(-klt)] + b2[1 exp(-k~t)] ~ [ b l k lt h?k?][I - e x p ( - t / ~ ) ] . The best fitting results were kl = 4.8(1.0) s-' and k2 = 0.57(23) s-I. Combining these rates with the measured result that on average a Fr atom stayed in the vapor for l / W = 0.63(4) s, we derived that, on average, a Fr atom stayed in the trap for l / C = 0.58(27) s and it takes I/L = 0.48(19) s to load a Fr atom from the vapor into the trap. These rates are consistent with those found in Rb traps, and imply a trap efficiency of L / ( L + W ) = 56(10)%. -
the hyperfine splittings of the 7 P 3 p and 7 P 1 p levels. We observed the individual hyperfine transitions by scanning the 718 nm laser, while the 817 nm diode laser was set (to either the F = 2 -, F' = 3 or F = 2 + F' = 2 transition) to select one velocity group of atoms. We obtained the splittings by measuring the frequency differences between each hyperfine line (Table 11) using a high-resolution A meter 1171. By checking against known splittings and wavelengths in rubidium, we found the accuracy of the A meter to be 3 MHz or less when measuring small differences, but 50 MHz when measuring absolute frequency. This is because uncertainties in the refractive index of air and laser beam alignment are largely canceled in a frequency difference measurement. Using the trapped atoms as a frequency reference, we have measured the wave numbers of the 0 1 and 0 2 transitions of **'Fr (Table 11). While the wave number of the D 1transition measured here is in agreement with the number measured by the ISOLDE Collaboration [ 181, our value of the 0 2 transition is lower by 3 ~ T D As . a check on our calibration. we measured the wave number of the TABLE I. Various rates, fluxes, and efficiencies on Dec. 1718, when the source strength was 9 pCi. Total efficiency is 0.4%.
(1)
(2) (3) (4)
Stage
22'Fr Flux (s-')
Produced in oven Exit oven Enter cell Collected into trap
3.3 x 105 7 x lo3 2.3 x 103
"1.3 x 103
Efficiency 2% of (1) 33% of (2) 56% of (3)
'Deduced from flux entering cell and trap efficiency.
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TABLE II. The hyperfine constants and wave numbers of 221Fr. The wave numbers of the previous work were derived from the published wave numbers of '"Fr and isotope shifts [ 181. ISOLDE [IS] A (7f'3/2) MHz B (7P3p) MHz A I PI/?) M T z v (01)cinv (02) c m - '
65.4(2.9) -259(16) 808(12) 12 236.6601(20) 13 923.2 1 1 8(20)
Current work
66.5(0.9) - 260(4.8)
81 l.O(l.3) 12 236.6579( 1 7) 13923.2041(17)
f2 line that is only 0.5 GHz away from the Lll trapping transition and found an excellent agreement (difference = c n - l ) with the I? atlas [16]. l(3) X We have demonstrated and analyzed the dynamics of a highly efficient collection of short-lived radioactive alkali atoms in an optical trap. The techniques used here, including the optimized VCMOT, the orthotropic source, and sensitive detection, can be easily applied to other alkalis as well [19]. Future applications will likely employ immediate, efficient transfer of trapped atoms out of the cell into a much longer-lived trap [20]. By combining a stronger and currently available 221Frsource (flux - 106/s) with a double-MOT system (trap lifetime - I O2 s), a sample of I Ox trapped 22'Fratoms could be prepared for the next generation of high precision spectroscopy measurements to test fundamental symmetries. We thank the following: Lane Bray and Tom Tenforde for supplying 225Ac,Ken Gregorich for electroplating the 2 2 5 A ~Albert , Ghiorso for showing us how to work with Fr, Michelle Stephens for building the lasers and teaching us the art of cell coatings, and Dave Alchenberg for the design and construction of much of our vacuum system. This work was supported by NSF, ONR, and the Office of Energy Research, Office of Basic Energy Sciences, of the U.S. DOE under Contract No. DE-AC03-76SF00098.
[I] [2] [3] [4]
Z.-T. Lu et d., Phys. Rev. Lett. 72, 3791 (1994). J . A . Behr et u / . , Phys. Rev. Lctt. 79, 375 (1997). C . Cwinner er a/.. Phys. Rev. Lett. 72, 3795 (1994). J. E. Sinisariati e f al., Phys. Rev. Lett. 76, 3522 (1996); L. A . Orozco (private communication). [5] V. A . Dzuba, V. V. Flanibaum, and 0.P. Sushko\', Phys. Rev. A 51, 3454 (1995). [6] P. G . H. Sandars, Phys. Lctt. 22, 290 (1966). [7] W.D. Phillips and H. Mctcalf, Phys. Rev. Lett. 48, 596 (1982); W. Erlmer et NI., Phys. Rev. Lett. 54, 996 ( I 9S5). [S] C. Monroe e f al., Phys. Rev. Lctt. 65, 1571 (1990). [9] M. Stephens and C. Wienian, Phys. Rev. Lett. 7 2 , 3787 (1994). [ l o ] R. Guckert et LII., Nucl. Instrum. Methods Phys. Rcs., Sect. B 126, 383 (1997). [ 1 11 K. Lindquisl, M. Stephens, and C. Wicman, Phys. Rev. A 46, 4082 (1992). [I21 M. Stephens, R. Rhodes, and C. Wieman, J. Appl. Phys. 76, 3479 (1994). [I31 Reference [9] defines a recapture efficiency which is in general lower than 7 , but equal to 7 in the limit of a small collisional loss rate (C << L , W ) . [ 141 L. A. Bray (privatc communication). [I51 T. Dinneen, A. Ghiorso, and H. Gould, Rev. Sci. Instrum. 67, 752 (1996). [16] S. Cerstenkorn, J . Verges, and J . Chevillard, Atlas Du Spectre D 'Absorption De la Molecule D 'lode (Laboratoire Aiine Cotton, Orsay, France, 1982). We used line number 380, Part IV. [ 171 J. Hall and S. A. Lee, Appl. Phys. Lctt. 29, 367 (1976). ~. A468, [IS] ISOLDE Collaboration, A . Coc et al., N L I C Phys. 1 (1987); S. V. Andreev, V. 1. Mishin, and V. S. Letokhov, J. Opt. Soc. Am. B 5, 2190 (1988). [I91 Most alkalis would be easier than 22'Fr, which has an unusually small excited state splitting combined with a large ground state splitting. [20] M . A . Kasevich et al., Phys. Rev. Lett., 63, 612 (1989); C. J. Myatt et a/., Opt. Lett. 21, 290 (1996); Z.T. Lu et a/., Phys. Rev. Lett. 77, 3331 (1996).
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VOLUME 83, NUMBER 7
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Spin-Polarized Atoms in a Circularly Polarized Optical Dipole Trap K. L. Corwin, S. J. M. Kuppens, D. Cho,* and C. E. Wieman JILA, Nutiom1 Imtitiite of Standurd~and Technology and Universit,v of Colorudo, Boiilder, Color-ado 80309-0440 and PIiysics Depmtrii ent, University of Colorado , Bo iilder, Colorado 803 09-0440 (Received 18 March 1999) The behavior of atoms in an optical far-off resonance trap is found to depend strongly on the trap polarization. Largc loss rates occur unless the polarization is perfectly linear or perfectly circular, in which case exponential lifetimcs up to 10 s arc observed. Through optimization of trap loading, samples of 4 X 10' atoms are obtained in a circularly polarized trap. Spin polarization of98(1)% is measured. This trap should prove useful for precision mcasurenients such as p-decay asymmetry and should niakc rf cooling possible in optical traps PACS numbers: 32.8O.Pj. 32.60. -i- i, 32.XO.Lg, 3 2 . 8 0 . B ~
Many precision measurements require spin-polarized neutral atoms and would be improved with trapped samples. However, no trap provides the necessary characteristics: atomic spin polarization, tight confinement, ease of control, and a low photon-scattering rate. An imbalanced magneto-optical trap (MOT) has some inodest polarization [ l], and small spin-polarized samples have been maintained with repeated optical pumping cycles in standard dipole traps [2]. Magnetic traps can confine spinpolarized samples, and offer the advantage of rf transitions between nondegenerate Zeeman states that can be used to drive a variety of cooling schemes, including evaporative cooling and gravitational Sisyphus cooling [3]. Unfortunately, magnetic traps have relatively weak spring constants, and strong magnetic fields are often undesirable and difficult to rapidly control. In contrast, an optical far-off resonance trap (FORT) [4] offers superior confinement and rapid control. As noted in Ref. [5], a dipole trap made with circularly polarized laser light offers all the advantages of an optical trap, plus energy splittings between spin states that make cooling schemes possible in an inherently spin-polarizing trap. In this Letter, we report the creation of such a spin-polarizing FORT for Rb atoms using circularly polarized light. A conventional FORT consists of a linearly polarized laser beam focused to a tight waist [4]. Because of a spatially varying ac Stark shift, laser light tuned below an atom's resonant frequency attracts the atom to regions of high intensity. For alkali atoms in linearly polarized light fields, the potential is the same for all internal spin states of the atom in the ground electronic state n (S1/2) [6]. This degeneracy is lifted when the light is circularly polarized; the laser field acts as a "fictitious magnetic f i e l d [7]. When the laser is tuned between the D1 (nSt12 n P I / * ) and 0 2 ( n Sl/2 n P3/2) transitions, the energy splitting between the spin states is largest (Fig. 1). Also, the absorption of the circularly polarized FORT photons optically pumps the atoms into the deepest potential of the F = 3 manifold, shown for u+ polarization in Fig. 2. -+
The potential energy of atoms in an elliptically polarized FORT may depend on the laser polarization vector 2= 1 (2 + i? with E the ellipticity [8]. The potential depth Uo(I, A, F , n z p , E ) is then
/a -f Jz), z
where the natural linewidth y = 277- X 6.1 MHz in Rb, mF is the Zeeman sublevel of the atom, &'F = [ F ( F + 1) f s(s + 1) - I ( [ + 1)]/[F(F f I)], the saturation intensity IS = 2v2!icy/3A3, and the intensity I0 = 2P/.i.rtvi in terms of power P [9]. The detunings 6112 and 6312 (in units of y ) represent the difference between the laser frequency and the D1 and 0 2 transition frequencies, respectively. The potentials are described in terms of the Gaussian beam waist wg by U ( p ) = UOexp(-2p2/w;). Radial cross sections of these potentials for "Rb are plotted in Fig. 2. The experimental apparatus is shown in Fig. 3. The MOT [lo] collects Rb atoms from a 5 X lop1' tom vapor, and contains a maximum of -3 X lo8 atoms in steady state with a filling time constant of -12 s.
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FIG. 1. Schematic level diagram for 85Rb ( I = 5/2). Laser beams are identified in the text as (a) FORT at 784.3 nm, (b) MOT cooling, (c) MOT (e) probe. AD, = 780 nm, A D ,
repump, (d) transfer, 795 nm.
and
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FIG. 2. ac-Stark shift U ( p ) in the focus of a circularly polarized Gaussian laser beam for both hyperfine levcls in the ”Rb 5 ’SI,~ ground state (‘73 = 1 /3) under typical experimental parameters, for P = 240 mW. Solid lines indicate the spin states toward which the atoms are driven by absorption of the trapping light.
The MOT is made with two external-cavity diode lasers stabilized to atomic lines in Rb at 780 nm with a dichroic atomic vapor laser lock [ l l ] , which allows rapid and convenient tuning over >I00 MHz range. One laser, the “cooling” laser, provides 6 mW (divided between three retroreflected beams) and is tuned to 5 2S1/2 F = 3 5 *P3/2 F‘ = 4. A second laser, the “repump” laser, 5 2P3/2 F i = 3. Both the is tuned to 5 2S1p F = 2 cooling and repump lasers pass through acoustoptical modulators (AOMs) to control their power. The dipole trap with adjustable polarization is created from up to 1 W of laser light from a titaniumsapphire laser, typically tuned to 784.3 nm. The laser beam passes through an AOM and the output power in the first diffraction order (70%) is actively stabilized to prevent parametric heating in the trap [12]. This beam passes through a GlanThompson polarizer and into a Pockels cell, which acts as a quarter-wave retarder when the applied voltage is about 2200 V. A mechanical shutter follows that can turn the FORT off in 60 ps. Finally the FORT beam (wg = 4.2 mm) is incident on an aperture and then a cemented doublet (f = 18 cm) that focuses it to w0 = 26 pm. The number and lifetime of atoms in the FORT are characterized using the following timing sequence. The
-
-
MOT is allowed to fill for some time 7 f before the atoms are transferred from the MOT into the FORT. They are then stored in the FORT in the absence of any other light sources or the quadrupole magnetic field for some time T . ~ . To detect the atoms, the FORT is turned off and the MOT quadrupole magnetic field remains off while the MOT cooling and repump lasers are turned on again, tuned closer to resonance (- y / 2 ) . The number of atoms N is then inferred from the measured fluorescence [13] and plotted as a function of TS (Fig. 4). The decay in N is governed by the differential equation N = - T N P N 2 , where r is the exponential time constant and p characterizes the density-dependent losses. The data are fit to the solution [ 141
as shown in Fig. 4. From this fit, the initial number No, r, and P are extracted. The physics involved in the transfer of atoms from the MOT to the FORT is quite complicated and will be discussed elsewhere [ 151. Here, we give a summary of results. For the parameters in Fig. 4, simple “geometrical loading,” in which the FORT laser is turned on immediately after the MOT turns off, traps about 2 X lo5 atoms. No can be increased by leaving the MOT on with different detuning and power parameters while the FORT is loading. This increase indicates that the MOT lasers cool atoms as they fall into the FORT. For the same FORT parameters, the best loading is typically achieved by detuning the cooling laser to -5 y and reducing the intensity of the hyperfine pump to 10 pW/cm2 for 70 ms before the MOT lasers and magnetic fields are switched off. This procedure transfers up to 20% of the MOT atoms into the FORT. Optimum loading into an elliptically polarized FORT ( E < 1) is different from simply loading into the linear FORT. While geometrical loading still gives similar
6
0
1
2
3
4
s
TLs (s)
nsityj
Piob
bcan
FIG. 3. Schematic of the experimental apparatus. A “transfer” laser, not shown, counterpropagates with the FORT laser. (See text.)
1312
FIG. 4. Number of atoms stored in a linearly polarized FORT (Uo = -1.6 mK at 540 mW) as a function of trapping time, after filling the MOT for T~ = 1 s. No data are taken for the first 100 ms while the MOT atoms fall away. The solid line is a fit to Eq. (2), giving No = 2.2 X lo6 atoms, p = 1.3 X (atomss)-’, and l/r = 3.0 s. For T~ = 4 s, 4 X lo6 atoms are initially trapped in the linear FORT.
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numbers, only about half as many atoms can be loaded using MOT cooling. The most efficient way to transfer atoms into an elliptical FORT is to first load the linear FORT, and then use the Pockels cell to quickly change the polarization from linear to the desired polarization. This procedure transfers about 70% of the a t o m originally held in the linear FORT, regardless of the time constant that governs the change in polarization (from 0.2 to 20 ms). Up to 100% of the atoms in the linear FORT are transferred to the circularly polarized FORT (“circular FORT”) by a “transfer” laser beam, circularly polarized and aligned antiparallel to the FORT laser. This beam, tuned to the 52Sl/2 F = 2 5 2 P 3 / 2 F l transition, is applied for 20 ins after the FORT polarization changes to circular. When u+ polarized, this transfer beam assists the FORT in optically pumping the atoms into positive m~ levels, and tends to keep them in the F 3 states. Thus atoms are pumped to the deep F = 3, I ~ = F +3 state whence they are not readily lost. In this fashion, 4 X lo6 atoms were loaded into a circular FORT at A = 782.2 nm with Uo = 2.1 mK. The lifetime r of the FORT depends dramatically on the polarization of the FORT laser. A measurement of the decay rate r = 117- as a function of the ellipticity E of the FORT laser polarization is shown in Fig. 5. r is smallest when the polarization is perfectly linear or perfectly circular. We believe that the large losses at imperfect polarizations are due to ground-state dipole-force fluctuation heating that arises when atoms change internal spin state [ 161. In the linear FORT, all the ground-state potentials are degenerate, so there is no heating associated with changing the ground level mF state. However, when the polarization is elliptical, these levels are no longer degenerate, and changes between the levels cause instantaneous changes in the dipole force acting on the atom. These changes result in heating, which evaporates atoms from the trap. In the
-
circular 0.0
liiicar 0.2
0.4
0.6
0.S
1.0
FOKI’ Laser Ellipticity te)
FIG. 5. Dependence of the FORT exponential loss rate r on the ellipticity E of the FORT laser polarization. Crosses indicate the measured loss rate, which is small for purely linear ( E = 1) and purely circular ( E = 0) polarizations. Because the circular potentials are deeper as indicated in Fig. 2, I‘(E = 0) > r ( E = 1). P = 280 mW, Uo(mF = 0) = 0.77 mK, and Uo(m~ = +3, E = 0) = 1.5 mK. The solid and dashed curves represent models of the ground-state dipole-force fluctuation heating described by Eq. (3), plus a small offset due to collisions with the background vapor.
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circular FORT, however, only AnzF = 1 transitions can be excited, and the atoms are all pumped into the F = 3, rnr = + 3 state in about 30 ms. After that, they no longer change their ground state and this heating process turns off. Agreement between the data and a simple model supports this picture. The dependence of heating on FORT polarization can be modeled with a two-level system using the two trapping potentials of the 5 ’SI,~ F = 3, I I Z F = + 3 and F = 2, 171,. = +2 states, labeled In) and Ib). These states are selected because the U,, for each is very different, and yet they are closely coupled via offresonant excitation by the FORT beam. We approximate the potentials as hannonic. A t o m initially held in l a ) spread out when they are transfened to lo), which has a smaller spring constant, and then heated when they reenter l u ) with a larger spatial extent and therefore more potential energy. Assuming the atoms hop between the two potentials at a constant rate riI,we derive the exponential time constant as a function of FORT ellipticity:
where U,,(E)and U / , ( E )are calculated for states l a ) and Ih) using Eq. (l), and T , and T f are the initial and final temperatures, chosen to be 100 p K and Uo/kB, respectively. As shown by the solid curve in Fig. 5 , Eq. (3) explains the rapid decrease in r as ellipticity approaches linear, but does not fit well near circular. The decrease in r near circular polarization arises from a dependence of the hopping rate on FORT ellipticity. Because the system actually contains many hyperfine and spin states, it is difficult to model exactly. A simplified approach assumes that absorption of a uphoton causes transfer from la) to Ib) and absorption of a u+ photon transfers atoms back to la). This gives a new rate of hopping between potentials r:, = =c2r/, and modifies the expression for the exponential heating rate in the trap to be r ’ ( E ) = e2 r ( E ) . Figure 5 shows fits to both r ( E ) and r ’ ( E ) , with the only free parameter in each case. They yield r h = 13 s-I and r h = 17 s-’, respectively. To calculate an optical scattering rate from these values, we simply divide by the product of the appropriate Clebsch-Gordon coefficients (0.05) for the four-photon heating process (absorption, emission to F = 2, absorption, emission to F = 3). The resulting scattering rates (260 s-’ and 340 s-’) agree well with the estimated average scattering rate in the FORT to within its 50% uncertainty. The inherent spin-polarizing nature of the circular FORT is confirmed by measuring the fraction of atoms that populate the F = 3, m~ = +3 state. Exciting the atoms in the F = 3 state with an additional laser causes loss by pumping them to the weakly trapped F = 2 state. We compare the loss when all mF states are excited to the loss when all but mF = +3 is excited. The probe laser beam (Figs. 1 and 3) is linearly polarized and tuned 1313
431 VOLUME 83. NUMBER 7
PHYSICAL REVIEW LETTERS
u. 116
, ~1IM
.
, ?(Ill
.
, 1011
,
, (1
,
,
11x1
.
I+rqilrncy ( 1 1 0pIic.rl I'unipiiig hmm IMH,)
FIG. 6. Fraction of atoms remaining in the FORT as a function of the frequency of an applied optical pumping beam for two different polarizations: (0 and 0 ) . Signals are normalized to the number of atoms held in the trap without the optical pumping beam (8 X lo5 atoms). Resonances are ac Stark-shifted about 30 MHz with respect to the resonances in free atoms, and separations are consistent with the 5'P?/2 hyperfine splitting. The difference between 0 and 0 for F' = 2 is clear evidence for spin polarization. This shallow trap [ P = 190 niW, Uo(iiiF = 0) = 0.52 mK, and U,(niF = 3) = 1.0 mK] allows resolution of the hyperfine structure, but should only decrease the level of spin polarization.
to excite 2S1/2 F = 3 + 2P3/2F' transitions. When this beam is aligned as shown in Fig. 3, it induces either A m = 0 or Ani = t- 1 transitions, depending on whether the polarization is oriented horizontally or vertically. The probe beam is pulsed on for 20 ms at an intensity of 1 pW/cm2, 45 ms after the MOT is switched off. We then measure the resulting decrease in the number of atoms in the circular FORT. This signal is plotted as a function of frequency in Fig. 6 for both polarizations. F' = 2 transition When the laser is tuned to the F = 3 and the polarization is vertical (O), we find that the loss saturates at 33%. However, when the polarization is horizontal (0),unsaturated loss is only 4%. The difference in fractional loss indicates a large population in ~ +3 state, because horizontal polarization does the r n = not excite atoms in this state, while vertical polarization excites all spin states. After including a factor of 2 for saturation, the ratio of the two losses implies that 7% of the atoms are not in the F = 3, mF = +3 stretched state. Because these atoms are most likely to be in the mF = f 2 state, the spin polarization is 98(1)%. We have demonstrated and characterized a circularly polarized FORT. We have also shown that the trap provides a high degree of spin polarization. The splittings of mF levels in the circular FORT should allow rf cooling techniques in the trap similar to methods employed in magnetic traps. Finally, we have shown that small polarization imperfections in circular or linear FORTS can
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lead to heating and subsequent loss due to ground-state dipole-force fluctuations. We extend special thanks to several people: Timothy Chupp assisted in the studies of loading into the linear FORT, Neil Claussen modeled the early loading data, Zheng-Tian Lu led the first observation of the circular FORT in 1996, and Kurt Miller worked on recent experimental development. Funding was provided by NSF, ONR, and the Korea Science and Engineering Foundation. Note added. -Recently, the authors have transferred 7 X 10' atoms into a linear FORT with MI() = 37 p m , P = 460 mW, A = 782.4 nm, and 110 = 1.6 mK. This represents 37% of the atoms initially trapped in the MOT.
*Permanent address: Department of Physics, Korea University, Seoul, South Korea 136-701. [ l ] T. Walker, P. Feng, D. Hoffmann, and R. S.Williamson 111, Phys. Rev. Lett. 69, 2168 (1992). [2] J. R. Gardner et al., Phys. Rev. Lett. 74, 3764 (1995). [3] W. Ketterle and N . J. Van Druten, Adv. At. Mol. Opt. Phys. 37, 181 (1996); N. R. Newbury, C. J. Myatt, E. A. Comell, and C.E. Wieman, Phys. Rev. Lett. 74, 2196 (1995). [4] S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, Phys. Rev. Lett. 57, 314 (1986); J. D. Miller, R. A. Cline, and D. J. Heinzen, Phys. Rev. A 47, R4567 (1993). [5] D. Cho, J. Korean Phys. SOC.30, 373 (1997). (61 This is true if the laser detuning is much larger than the hyperfine splitting. [7] C. Cohen-Tannoudji and J. Dupont-Roc, Phys. Rev. A 5, 968 (1972). [8] E is measured by rotating an analyzing polarizer in the laser beam and recording the highest and lowest power ( P A and P , ) transmitted through the analyzer. Then E = (Ph - ~ / ) / ( P + / I PI). [9] Equation (1) is obtained by rewriting the expression for UAc in Ref. [5] in terms of experimentally accessible quantities. For E = 1, Eq. (1) agrees with the expressions for U, in D. Boiron et al., Phys. Rev. A 57, R4106 (1998). [ l o ] E . L . Raab et al., Phys. Rev. Lett. 59, 2631 (1987); C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571 (1990). [ l l ] K. Convin et al., Appl. Opt. 37, 3295 (1998). [12] M. E. Gehm, K. M. O'Hara, T.A. Savard, and J. E. Thomas, Phys. Rev. A 58, 3914 (1998). [I31 C.C. Townsend et al., Phys. Rev. A 52, 1423 (1995). [14] T.P. Dinneen et al., Phys. Rev. A 59, 1216 (1999). [15] S. J. M. Kuppens et al. (to be published). [16] J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).
PHYSICAL REVIEW A, VOLUME 62, 013406
Loading an optical dipole trap S. J. M. Kuppens. K, L, Convin. K. W. Miller, T. H. Chupp,* and C. E. Wieman Jll.A. University of Colorado and National Institute for Standards and Technology, Boulder. Colorado $0309-9440 (Received IS October 1999; published 13 June 2000) We present a detailed experimental study of the physics involved in transferring atoms from a magnetooptical trap (MOT) to an optical dipole trap. The loading is a dynamical process determined by a loading rate and a density dependent loss rate. The loading rate depends on cooling and the flux of atoms into the trapping volume, and the loss rate is Jue to t-ia'iteci slate collisions induced by the MOT Jiyht fields. From (his suidy v,v found wiiys to optimise the loiiding o f i h e optical dipole trap. Key mgietfienb, fur niiixivniiiii loading are Ibuiiti !i> he a reduction of the hyperfiiie repitrnp intensity, increased detiminfj of ibe .VI Of light, and a displjccraenr or ihc- optical dipole trap center with respect 10 the MOT. A factor of 2 increase in the number of loaded atoms is demonstrated by using a hypev'lnc repurnp beam with a shadow n it. I'i this way we load 8 x 10' '°Kb atoms into a I rriK deep optical tfooie trap with a wsist of 58 /iin, which is 40':i of the atoms initially trapped in the MOT.
[, INTRODUCTION In the last, decade, many different schemes for preparing and trapping ultraeold and dense samples of atoms have been demonstrated. Of these, the optical dipole trap [1] requires no magnetic fields and relatively few optical excitations to provide a conservative and lightly confining trapping potential. These characteristics make- it an appealing option for various metrology applications such as parity nonconservation and /3-decay asymmetry measurements. It may also be an "option for reaching Bose-Einstein condensation in a purely optical trap. For these applications, large samples of atoms must be transferred into the dipole trap. This is almost always done from a magneto-optical trap (MOT) [2]. However, the processes determining the transfer between the MOT and the optical dipole trap are poorly understood. Here we give a detailed description and explanation of the loading process and suggest ways in which to improve the loading. The simplest optical dipole trap consists of a focused single Gaussian laser beam. Typically, the light is detuned below the atomic resonance, from a few tenths of a nanometer to several tens of nanometers. The latter are called faroff-resonance traps (FORTs) [3]. We will use the abbreviation FORT in discussing optical dipole traps. Conceptually, a FOR!" works as follows: The ae Stark shift induced by the trapping light lowers the ground state energy of the atoms proportionally io the local intensity. The spatial dependence of the atomic potential energy is therefore equivalent to a spatial dependence of the light intensity. The atom has the lowest energy in the focus of the trapping beam and can therefore be trapped there. For very large detuning, typically several nanometers, the photon scattering rate becomes so low that the potential is truly conservative. The first FORTs were running-wave Gaussian laser beams focused to a waist of about 10 /itn [1,3]- By alternating the FORT with an optical molasses that cooled atoms
into the trap [1.3], about 500 to 1300 atoms were loaded. In later work [4-6J, the FORT was loaded by overlapping it with a MOT continuously, which improved the number of atoms that was transferred to K)6. A key stop in the loading of a FORT' from, a 'MOT is a strong redaction of the hyperfme repump power in the last 1.0 30 ms of the overlap between the traps, ft lias been conjectured [4] that this redaction helps because it reduces three density limning processes, namely, radiative repulsion forces, photoassocialive collisions, and ground suite hyperfine changing collisions. However, to our knowledge, there has not yet been an extensive study of the loading process. Therefore, in this paper we present a detailed and comprehensive investigation of the many mechanisms that govern the loading process. We find that loading a FORT from a MOT is an interesting dynamical process rich in physics. As illustrated in Fig. 1, the number of atoms in the FORT first increases rapidly and nearly linearly in time until loss mechanisms; set a limit to the maximum number. The loading rate and ioss processes both depend in complex ways on the laser fields involved. One factor that determines the loading rale is the flux of atoms into the trapping volume. This flux depends on the MOT density and temperature, i.e., average velocity', of the
2.0-
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time (s) * Permanent address: Department of Physics, University of Michigan, Ann Arbor, MI 48109. 1050-2947/2000/62( 1 )/013406( 13)/$ 15.00
FIG. 1. Number of atoms N in the FORT during loading, for a trap depth of 1 mK and a waist of 26 yum.
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atoms in the MOT, Cooling mechanisms niusi also be active in the region where ilie MOT and FORT overlap for the atoms to be trapped in the FORT. Both the flux anil the probability for snipping depend on the trap depth and light shifts inherent to the FORT. Losses from the trap can be caused by heating rneclutriisms and collision;)! processes. Contributions to heating arise frurn spontaneously scattered FORT light photons, background gas collisions [7,8], intensity fluctuations, and trie pointing s t a b i l i t y of (he FORT beam [ 9 j , However, for large number? of atoms, the losses are dominated by eollisionai processes [ 1 0 1 , including plwfoassodanon, spin exchange/ground stale hyperfine changing collisions, and radiative escape, Puoto^socJative collisions can be induced by the FORT light kself and lead to untrapped molecules, During a ground srarc hyperfine changing collision the atoms gain as much as !lic hyperfme energy splitting in kineticenergy (0.14 K for " R b j . which is enough to eject them out of the FORT. In radiative escape, an atom is optically exeiied and recmits during a collision, ami the attractive dipoleinduecd interaction between the excited and nonexcited atoms leads to an increase of kinetic energv that is enough to eject an atom from the trap. The loading and loss rates depend on the shape and depth of the optical potential, as well as the intensity and detuning of the MOT !igh! fields. By studying the loading rate and loss rate separately as a function of these parameters we have obtained a detailed understanding of the FORT loading process. This understanding has allowed us to optimize parameters in order to improve our loading efficiency to high values. Although we studied the loading of 83Rb into a dipole trap with a detuning of a few nanometers, the physical processes and optimization should be generally applicable to other alkali-metal species and FORTs. when the FORT trap depth exceeds the MOT atom temperature. In the opposite regime, O'Hara et ai [11] have shown that a static equilibration model applies in CO2 laser traps. This paper is organized as follows. In Sec. II, a more detailed expression for the depth and shape of the FORT potential is given. In Sec. Ill, our experimental setup is discussed, including the loading of the FORT and the diagnostic tools for measuring the number of atoms and size of the trapped sample. In Sec. IV, we present measurements of the loading rate and loss rate as a function of different MOT and FORT parameters. In Sec. V, the temperature of the atoms in the FORT is given. In Sec. VI, the loss rates of the FORT in the absence of MOT light are discussed, hi Sec. VII..we present a physical model of the loading process that explains the data presented in See. IV. In Sec. VIII, we discuss how our model explains the interdependencies of the MOT and FORT parameters. In Sec. IX, we demonstrate how, based on our understanding of the FORT loading process, the number of trapped atoms can be improved using a shadowed repump beam. Finally. Sec. X contains summarizing remarks and discusses the general applicability of our results to other traps. II. FORT POTENTIAL
The trapping potential is given by the Gaussian shape of the laser beam.
rcpump bi^i
T:S Laser ~ I W
uu
aperture
mtensitv
gravity
CCD
FIG. 2. Experimental setup
with r and r the radial and longitudinal coordinates, wiz) = vi ; (, v'l ' (-/2«)~ the beam radius as a function of longitudinal position, and :K the beam Rayleigb. range [12]. For an alkali-metal atom, the depth of the potential U0 is calculated from [13]
1 f/n-
24/s !. (2)
where y is the natural linewidth, tiiF is the Zeetnan subleve! of the atoms. g,. = [F(F . 1) I S(S+ !)-/(/+ 1}]/[W + D], /g is the saturation intensity, defined as 7S = 2~-ficy/(3\''), and ./() is the peak, intensity 2P/(7rvv ( 2 l ) in terms of the laser power P [12], The detunings 8m and
The experimental setup is schematically shown in Fig. 2. The starting point of all the experiments described here is a MOT, which collects 85Rb atoms from a 5X 10" 10 Torr vapor and contains a maximum of about 3X10 8 atoms in. steady state with a filling time constant of «=12 s. Two extended-cavity diode lasers, both stabilized to atomic lines in 85Rb at 780 nm with a dichroic atomic vapor laser lock [16] are used for the MOT. One laser provides 6 mW for trapping and cooling (divided between, three retro-reflected beams) and is tuned red of the 5 2 5]/2^ 1 ~3 —>5 ^.Pj/-,F' = 4 transition by an amount denoted as A M . We refer to this light as the "primary" MOT light, and denote its total sixbeam intensity by / M . The beams have a Gaussian beam radius of 9.3 mm. The other MOT laser is used for hyperfine
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LOADING AN OPTICAL DIPOLE TRAP rcpuinping and is tuned near the 5 ~S^!F=2 —'•5 ^P^-f •= 3 transition. We refer to this Sight as the "rcpump" light, and denote its detuning as £i r , ( A K - ~ 0 unless otherwise indicated). We use one single retrorefleefed beam of intensity / R ---3 mW/cm* for repumping the MOT. Both laser beams have acousto-opnc modulators i AOVlsl in than for temporal control of the light intensily. The magnetic: field gradient of the quadrupok field aloni; the strong axis is 5 ti/on. and can be switched otf within 3 mis. The FORT tiejii: is generated with a home-built titaniumsapphire laser, with a nominal output power of 1,2 W. After passing through an AOM. the first order is intensily -stabilized, collimated, and expanded. The beam is focused into ihc vacuum chamber with a 20 cm focal length doublet lens to a focus of radius n- 0 = 26 urn ( \/e': intensity), unless stated otherwise. The detuning from ihe D - l i n e is typically 2 to 4 uni to the red. For P = 30() mW, n- i ,,-=26 /j.m, and X = 784 nm the well depth 6 \ ) / / f H ~ - - 1 . 4 tnK. Approximating the center region of the FORT potential as harmonic, the trap oscillation frequencies are 4.6 kHz and 34 Hz in the transverse and longitudinal directions, respectively. Note that the trap as a whole is highly anharmonic at the edges. The peak photon scattering rate for these parameters is 1.3 kHz.
Loading and diagnostics The sequence for loading the FORT from the MOT and measuring the number of atoms is as follows. The MOT fills for typically 3 s at an optimum detuning of A M ~* 2 7. and at maximum repurnp intensity. This results in typically 3 X iO 7 atoms in the MOT. We then switch on the FORT while simultaneously increasing the detuning of the primary MOT light and reducing the repump light intensily. We will refer to this stage in the timing cycle as the "FORT loading stage/* which is typically 20-200 rns long, depending on FORT parameters. The primary MOT light, MOT repump light, and magnetic fields are switched off, and the atoms arc held, in the FORT for a variable length of time, b u t at least 100 ms, before the detection light comes on, so that the MOT cloud can fall out of the detection region. The atoms are ihen released from the FORT and detected by turning on the MOT light (primary and repump) but no magnetic field, with the primary MOT light frequency now closer to resonance A M *= — 7/2. The number of atoms N is determined from the amount of fluorescence [17] collected with a low noise photodiodc. Under some circumstances, the cloud of atoms trapped in the FORT is so large that it extends beyond the field, of view of our photodetector. In this case the number is measured by recapturing them into a MOT. For this case the detection must be delayed for at least 250 ms to give the MOT cloud time to fall out of the MOT beams. The number of atoms in the FORT is measured as a function of storage time. We call this a lifetime curve, A typical example of a lifetime curve is shown in Fig. 3. Note that the number of atoms does not have a simple exponential dependence on time. We find that the loss of atoms is well described by
10°
10°
time (s) FIG. 3. Number of atoms remaining in the FORT as a function of time, for trap parameters w0 — 2(> /j,m, P=290 rnW, X. = 782.5 nm. The trap depth is Un/kB= — 2.3 mK. The solid curve is a fit of Eq. (3) to the data.
dN di
(3)
with. 1' an exponential loss rate and f}' a colHsiona! loss coefficient. We use the analytical solution of Eq. (3J as a fit [unction to the data to find the number N0 initially trapped, as well as values of I" and /?' (see Sec. VI). The prime on ft1 is used to refer to atom number loss instead of density loss. We use absorption imaging lo measure the size and temperature of the MOT cloud as well as tlie cloud of atoms in. the FORT. Our imaging system consists of a two-lens telescope making a one-to-one image onto a charge-coupleddevice (CCD) array. The lenses are 18 cm focal length cemented doublets. The line of view is perpendicular to the FORT beam. This allows us to observe the transverse as well as longitudinal shape of the cloud of atoms trapped in the FORT. IV. DYNAMICS OF THE LOADING PROCESS In Fig. 1 we showed that the transfer of atoms from the MOT to the FORT is a dynamical process, in which the number of atoms loaded into the FORT increases rapidly until a competing process causes the number to reach a maximum and then decrease at later times. Here we investigate the precise shape of the loading curve in more detail. The number of atoms in the FORT for longer loading times is shown in Fig. 4, The shape of the loading curve is explained as follows. Initially, the number of atoms in the FORT increases as N(t) = Rnt, But at larger times the number starts to roll over. This occurs for two reasons. First, the MOT loses atoms due to the reduced repump intensity and different detuning of the primary MOT light. This reduces the loading rate. Second, the trap loss rates become large enough to counteract the loading. Similarly to Eq. (3) we find that the shape of the FORT loading curve is well described by
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FIG. 4, Number of atoms loaded in the FORT as a function of KORT loading stage duration, for trap parameters wri = 26 /j.m, P = 305 mW, and X = 784.5 nm. Primary MOT intensity /M = 8-2 mW/crn 3 and MOT repump intensity 7 R = 4.7 /jW/cm 2 . The solid curve is a fit of Eq. (4) to the data. The loss coefficient /3,' = (1.56±0.22)X 10~ 5 (atoms s ) " ' ; see text. where TMOT 's lne rat'e at which the MOT loses atoms due to the change of MOT light detuning and repump intensity, and I"i and J3[ characteri/.e- density independent and density dependent losses. The subscript "L" expresses the fact that the loss rates during loading are generally different from those during storage in the FORT without any MOT lijiht present. Four parameters determine the loading: R f ! , >\ ' ' L and /8t' . To understand the physics of the loading process \ve must therefore determine these four parameters under a variety of conditions. The initial loading rate R(l can be determined directly from the initial slope of the loading curve. Then TMOT can t>e determined by measuring the rate at which the MOT fluorescence decreases during the FORT loading stage. We confirmed this explicitly by allowing the MOT to dissipate for a variable length of time rti before the start of the FORT loading stage. The initial slope of the loading curve K(T,/) was observed to be R(TJ) = JR0exp(->-v10TT,/), consistent with the MOT fluorescence measurement. The rate constants f$l and I\ can then be determined by fitting the numerical solution of Eq. (4) to the data. The solid curve in Fig. 4 is such a fit, indicating how well the loading process is described by Eq. (4). With R(> and VMOT constrained to the directly measured values, as mentioned above, the extracted values of ftj and I\ show that the --fi'jN" loss term clearly dominates over the — FLA' term. This is most apparent in the slope of the tail of the load curve. Thus, coilisional processes dominate the losses from the FORT during its loading. The losses during the loading of the FORT can be studied independently from the loading rate. This is done as follows. The FORT is first loaded under optimum conditions, and the atoms are stored for 100 ms, after which the MOT lasers are switched back on. with no magnetic field, for the remainder of the FORT storage time. At the end of this cycle, we detect the number of atoms remaining in the FORT. We refer to such a measurement as a loss curve measurement. In this way we can study the effect of the MOT light field parameters on the loss rate; i.e., we have eliminated the loading
FIG. 5. Number of atoms in the FORT vs time, without any MOT light (•) and with MOT light on (A); primary MOT intensity / M =8.2 tnW/cm 2 . and MOT repump intensity / R = 4.7 /iW/cm 2 . for trap parameters w 0 = 2 6 /j.m, P~305 mW, and X. = 784.5 nm. The curves are fits of liq. (3) to the data. The number dependent loss rates arc ft' = (1,42 ±0.05) Xl(r 6 (atoms s)" 1 and /?L = ( 1.4±0.1)X 10~ 5 (atoms s) ', for the atoms stored in the FORT without any MOT and with MOT light, respectively. term in Eq. (4). Wish the MOT lasers cm, the loss rate is much larger than in the absence of any MOT light and completely dominated by density dependent losses /3[ , The result of such a measurement is shown in Fig. 5 ( A t , which also contains the lifetime curve (•) recorded in the absence of MOT light. Comparing corresponding loading and loss curves we find the sarns values of /?,' from the different types of data sets to within a 30% spread. The advantage of measuring a lass curve is that we can independently change all the MOT light field parameters without altering the number of atoms initially loaded into the FORT. The nice agreement between the value of J3( determined from the loading curve in Fig. 4 and the loss curve in Fig. 5. taken for identical MOT and FORT parameters, gives good confidence that F,q. (4) describes the physics involved in loading the FORT. The fact that coilisional losses, f)^, dominate the tail of the loading curve can also be seen in another way. If one neglects the — FLA; term in Eq. (4) and sets the loading rate to be constant at R(>. i.e., assuming that the MOT does not lose atoms during the loading of the FORT, the steady state solution of Eq. (4) is
(5) Substituting the values of K0 and J3[ obtained as discussed above, the calculated A'st agrees to within 10% with the maxima of the loading curve. We did some extra cheeks on the accuracy of filling Eq. (4) to the data. One test was to set all four parameters in Eq. (4) free. This reproduced the independently determined values of R(i and 7MOT' as we" as those of T, and f } \ . with I\ the least well determined. Both yMor anc-' ^i. S've "se Io exponential decay and are therefore strongly coupled in the
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= 2—F' = 3
1 ^
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so
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E o
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0
20
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FIG. 7. Number of atoms in FORT as a function of MOT hyperfine repump detuning for different repump intensities. 1.4 (uW/cm2 (•), 4 /iW/cm 2 ( H ) , and 280 /xW/cm 2 (*). FORT parameters are »>0~26 fj.m, P~200 mW, and X = 798.1 nm. Solid curves are Lorentzians fitted to the data to guide the eye. Detuning is relative to the unperturbed F = 2—*F' = 3 repump transition.
80
FIG. 6. FORT loading rate R0 (a) and loss coefficient J3'L (b) as a function of hypcrfinc repump light intensity, for trap parameters TV U ~ 26 ,tim, P = 300 mW, and X~784.5 nm. The loss rate was measured for / M = S mW/cm' {•), reduced MOT intensity' of 1M ~ ] mW/cm2 (O), and complete absence of primary MOT light (A). The solid curves are fits to the data based on the model described in Sec. VII A. Error bars are statistical and do not reflect systematic uncertainties in / M . least squares fit. Most often, we find that the fit determines a F L within error oKzero and a >'MOT within 20% of the value determined directly from the MOT fluorescence. Assuming that the value of F, should at least be as big as that of the trap in the absence of the MOT light, we set it to that value and see no significant change in the value of {}[ . Fitting the loading curves with these constraints and different combinations of free parameters, we find a spread in the j3{ values of up to 30% for a given data set, which is more than adequate for the analysis given below. We are now in a position to investigate the loading of the FORT in terms of R(t and /?; separately, by measuring either load curves or loss curves, A, Hyperfine repump intensity and detuning A commonly used technique 1o improve the loading of a FORT is to reduce the hyperiine repump intensity. Here we show how the intensity and detuning of the repump light affect the loading rate Rt} as well as the loss rate /3 t . We concentrate first on the dependence of the loading rate and loss rate on the hyperfine repump intensity at resonance (.F = 2 >F' = 3). The data in Fig. 6 show that for the maximum /M there is a critical repump intensity below which the MOT is not sustained during the FORT loading stage and the loading rate goes to zero. For /R above that critical value, the loading rate decreases slightly because the density in the MOT decreases due to radiative repulsion. The optimum level is about /^=5 jotW/cnr'. Note that the loading rute of the FORT is as high as 5 x 10'' atoms s ', which is a factor
2 higher than that of our MOT. The loss rate /?' increases rapidly with repump intensity and starts to saturate at high repump intensity. For lower / M . saturation sets in at lower repump intensities and /3t' is smaller. The optimum rcpump intensity is intimately linked to the rcpump detuning. This interdependence is shown in Fig. 7. where the maximum number of atoms loaded into the FORT is plotted verssis A R for different values of / f > . Increasing / R from 1.4 /.iW/cnr to 4 /iW/cm 2 , we sec that the number in the trap increases, but the optimum repump detuning stays on resonance. Increasing the intensity to 280 /iW/errr causes the optimum repump detuning to redshift. Interestingly, the optimum number of atoms is loaded when the repump scattering rate is the same for both /R=4 /xW/cm2 and / R = 280 ,u,W/cm2 due to the different detuning, indicating that an optimum repump scattering rate exists for loading the FORT. However, the number of atoms drops and the resonance broadens, which shows that the number of atoms in the FORT is not determined by the repump scattering rate alone. This will be explained in more detail in Sec. VIII. In summary, the loading rate Rn is optimal for a very low repump intensity of about 5 /uW/cm 2 and zero detuning. The loss rate /S'L increases with repump intensity and is larger for higher intensity of the primary MOT light. B, Primary MOT light intensity Both Rn and fi{ depend on the primary MOT intensity / M . The data in Fig. 8 show thai /3' rises rapidly and then saturates with increasing / M . The MOT intensity at which saturation sets in is higher For higher repump intensities. Combined with the dependence of f>'( on repump intensity, this suggests that excited state collisions leading to radiative escape arc responsible for the losses during loading, as is confirmed by the model that we develop in Sec. VII. The solid curves in Figs. 6(b) and Sfb) are based on this model and agree very well with our data.
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KUPPKNS, CORWTN7, MILLER, CTIUPP, AND WIEMAN
PHYSICAL REVIEW A 62 013406
3-
(a)
(a)
4o
2-
1-
O-'r
25-
(b)
5
•C 20W3
S 15-
«r 10'o Cl
n-HK)
<
*
• * i -SO
-60
-40
_
; -20
0
MOT detuning ( MHz )
IM (mW/cm ) FIG. 8. FORT loading rate Ra for a repump intensity of 5 /uW/crrr (a) and loss coefficient fi'L (b) as a function of primary MOT light intensity, where /3'L was measured for zero repump intensity (D) and / R = 5 juW/cm2 (•). The solid curve in (a) is derived from the model described in Sec. VII B. The solid curves in (b) are fits to the data of the model described in Sec. VII A. FORT parameters as in Fig, 6.
As shown in Fig. 8. the loading rate R(l increases nearly linearly with / M . As will be argued in Sec. VII. this implies that the loading rate strongly depends on the cooling mechanisms in the MOT. C. Role of MOT detuning Both R0 and /3,' depend also on MOT detuning, as shown in Fig. 9. For these data, 7R is reduced to the value that gives optimum loading into the FORT. A maximum in Ra is observed at about AM^- 30 MHz. At slightly larger detuning the loss rate fi'L has a minimum. The maximum number of atoms is loaded into the FORT at a detuning close to the maximum in the loading rate and simultaneous minimum in the loss rate. We also observe tSiat the optimum detuning of the primary MOT light during the loading stage depends on the depth of the FORT. This is illustrated in Fig. 1.0. For deeper traps, the optimum detuning is smaller. The FORT depth was varied by changing the wavelength and power of the FORT light.
FIG. 9. FORT loading rate R0 (a) and loss coefficient 01 (b) as a function of the primary MOT light detuning A M during loading. The repump intensity is 5 /tW/crn 2 , For trap parameters w 0 = 26 Mm, P = 300 inW. and X = 784.5 nm.
because it shortens the effective loading time. At intermediate gradients, in the range from .1.5 lo 12.0 (i.'cin. the losses from the MOT arc small, with the maximum number loaded near 5.0 G/cm. E. Alignment with respect to the MOT Another ('actor affecting the FORT loading is the relative alignment of Hie FORT with respect to the MOT. We find that the loading rate is optimum with a longitudinal displacement between the center of the FORT and the MOT. The optimum displacement depends on the FORT depth. Absorption imaging is used to determine the separation between the FORT and the MOT. We observe that the displacement for which the loading rate is optimized increases with trap depth. Typically the displacement between the focus of the FORT and the MOT is about one-half a MOT diameter.
-15
-20 |-25
0. Role of the magnetic field gradient In order to determine the influence of the MOT magnetic field gradient, we measured R^ and ,V0 for different currents in the MOT coils during the FORT loading stage. We find 7?0 to be nearly independent of the magnetic field gradient in the range from 1.5 to 17.5 0/cm. At very small or very large gradients, however, atoms are rapidly lost from the MOT during the FORT loading stage, which strongly reduces /Vu
-30 -35 0.0
0.4
O.S
1.2
trap depth (mK ) FIG. 10. MOT detuning for maximum number of atoms loaded into the FORT versus trap depth, wa~26 /U.IIT.
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438 LOADING AN OPTICAL DIPOLE TRAP 81
.
1
1
1—.
1
.
PHYSICAL REVIEW A 62 013406
r-
1.2 E S 0.8
0.4
ot. 0
1
2
3
4
5
0.0
6
0.5
i.O
1.5
2.0
time ( s )
crap depth ( m K ) FIG. 11. Number of atoms loaded into the FORT as a function of trap depth for n' 0 = 26 Aim ( • ) a n d w 0 = 22 /um ( A ) . For the latter data set, the steady state number [see Eq. (5)] is calculated from K 0 and /?[_ (taken from Fig. 12 below) and plotted as O.
In shallower iraps the displaced loading of the atoms causes an initial sloshing of the atoms in the loiigivudinal direction. The sample of atoms can be seen to make erne Kid one-half oscillations before it thermal izies at the center of the FORT; ibis takes about 100 ens. F. Dependence on FORT depth
We measured how [he loading rale, loss rate, and total number of atoms trapped in the FORT depends on the trap depth. This is shown in Figs. 11 and 12. For each data point, the MOT detuning, repump intensity, and alignment of the FORT with respect to the MOT were optimized to give the highest number of atoms in the FORT. The primary MOT light intensity was fixed at the highest available value of 8 mW/cirr, which optimizes the loading rate (see Sec. IV B), The waist of the trapping beam is fixed, and the trap
12- (a)
FIG. 13. Number of atoms in the FORT as a function of loading time for three different elliplicities of the FORT light. In descending order of the curves, F= 0.999, 0.915, and 0.852. The FORT parameters are w 0 = 22 fj,m, P = 600 mW, and \ = 784.5 nm.
depth is varied by changing both the deuming and power of the FORT be;im. The dependence of ,V.:. on trap depth can be explained as
JT"1 6-
(b)I
I
f 4-
T-
>c
l
I
_J CO.
0-
()
1
2
3
4
5
6
trap depth ( mK ) FIG. 12. FORT Loading rate Ra (a) and loss rate p'L (b) as a function of trap depth Ua for w 0 = 26 ftm.
As the data in Fig. 13 show, the polarization of the FORT light has a profound effect on the number of atoms loaded into the FORT. A change of the cllipticity [15] from linear polarization. e= 0.999, to slightly elliptical polarization, e = 0.915, causes the number of atoms to drop by a factor of 7. At e=0.852 the number dropped by more than an order of magnitude. The drop in number is caused by a combination of effects. Analysis of the shape of the load curves in Fig. 13 shows that the loading rate is reduced by an order of magnitude and the exponential loss rate Fj is increased by a factor of 4. The increase of l\ we ascribe to the ground state dipole force fluctuations related to the optical Zeeman splittings [second (enn on the right hand side of Eq. (2)] induced by the elliptieity of the light [181. The optical Zeeman splittings
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PHYSICAL REVIEW A 62 013406
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level that presumably is due to ground state hypertine changing collisions. This background level depends on the FORT light polarization. Tn a circularly polarized FORT [J8j. the atoms are spin polarized in one of the stretched states of the F===3 ground slate, from which Irypcrfme changing collisions are suppressed. In our 1 trap, we see a factor of 2 reduction of p' in between the pholoassodation lines when the polarization is changed froir. imear u,> circular.
0.8 0.6
0.4
0.2
0.0
VH. ANALYSIS 0.0
0.5
1.0
1.5
2.0
A. Anaiysis of density dependent losses
trap depth ( mK ) FIG. 14. Temperature of the atoms in the FORT as a function of trap depth, for iv0 = 26 /j,m. Solid line is a linear regression giving a slope r/IC/o =0.4. can also be expected to interfere with the functioning of the MOT and thus reduce (he loading rate. In the center of the FORT, the splitting corresponds to i.hai of a magnetic field of 13 O. Although our MOT operates in the sub-Dopp!cr regime, these large splittings disrupt even the Doppler cooling ami trapping mechanisms of !hc MOT, which we believe are mainly responsible for loading of the FORT. V. TEMPERATURE We measured the temperature of the atoms in the FORT for different trap depths. The temperature I'of the atoms in the FORT was determined from the rate at which the cloud of trapped atoms expands after release. The density distribution is well described by a Gaussian of width a thai increases upon expansion as \liT^-r(!<^T/m)t~.. For the data shown in Fig. 14 we used the expansion in the transverse direction LO obtain T. The initial aspect ratio of the cloud is 33 lo 3000 jum. We confirmed that within experimental uncertainties the temperatures in the transverse and longitudinal directions are the same. Note that for the deeper traps the temperature is much higher than that of the atoms in the MOT, which is about 30 fj,K. to 120 /iK, depending on the MOT detuning. Over the range of trap depths that we investigated the temperature is a fixed fraction 0,4 of the FORT depth. VI. TRAP LIFETIME
The number of atoms in the trap decreases after the trap is loaded, as shown in Fig. 3. In the absence of any MOT light, the loss of the trapped sample has a number dependent feollisional) portion and a purely exponential part, as described by Eq. (3). Typically, the exponential lifetime 1/T is between 1 s and 10 s, depending on the trap parameters. The lifetime increases with decreasing FORT scattering rate, until a limit (=•10 s) is reached that we believe is set by the background vapor pressure. The collisional loss, characterized by /?', depends on the wavelength and waist size of the FORT beam. We observe clear resonances in /3', consistent: with photoassociation lines. Between these lines, we see a nonzero background
The main features that we have observed in our data are the strong increase ( x 100) of the collisional loss coefficient fl[ when the MOT l i g h t is present and the strong dependence of this loss rate on the MOT parameters. As we will show in this section, density dependent losses during the loading of the FORT arc miiinh due to radiative escape collisions induced by the MOT light. In the unperturbed FORT, ground state hypcrfroc changing collisions are the dominant loss process. 1. Conversion lo density dependent lf»ss
The typical mechanism driving the collisional loss process in the FORT is more easily identified when the measured rate coefficients J3' and fi{ are converted to density related, volume independent rate coefficients /3 and p's . The relation between these is f}"{3' I ' , with F the volume of the sample of atoms. The volume is found by approximating the trapped sample of atoms as a cylinder with radius and length determined by the size of the FORT beam focus and the temperature of the atoms. The volume is then given by
(6) where j?=/c B 77 f/0| is shown in Sec. V to be a constant 0.4, regardless of the FORT parameters. Thus the volume changes only when vi'o changes. 2. Density dependent loss from the FORT without the MOT present Density dependent losses from the FORT when there is no MOT light present are due to both photoassociation and ground state hypertine changing collisions, as mentioned in Sec. VI. Between photoassociation lines, the values of ft' do not depend strongly on the FORT laser power or wavelength, but change strongly with the trap volume V. Note that, because zt^WQ, F'KVI'Q. However, the resulting /3 = /3' K = ( 4 ± 2 ) X 1 0 ~ 1 2 crrr s ~ ' , for waists ranging from 25 yctm to 58 Aim. This value of (3 agrees well with the value of 4 X10" 1 2 cm-V reported by Miller [19]. 3. Density dependent loss from the FORT during loading In the case of loading the FORT, the density dependent losses are due to light-assisted collisions. To demonstrate this, we compare them with measured rates in MOTs. where light-assisted collisions have been extensively studied. This
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LOADING AN OPTICAL DIPOLE TRAP
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comparison is complicated by the difference in trap depth be;ween the MOT and ihc FORT (I K vs 1 mix) and also the difference in fractional occupation of me upper hypertme component of the ground state, hi the case of the MOT. the fraciiona! population of the lower hypci-fine component is uL'.ijh.nUile, and the loss term is written as fiir. where n is the total d e n s i t y of atoms in t h e MOT. In general, ft depends linciiriy on the intcnsiiy of she K.ssi.sting l i g h t . Those light-assisted collisions. as described by Gallagher and Pntchard 20 j. arc due to two mechanisms: fine structure changing collisions and radiative escape. Ft has been shown tiiat those two mechanisms contribute with the same order of dialive escape is predicted to scale almost inversely with trap dep!h (,#-• t/,7"") [22,23] tbi" trap depths •-• i K, but the fractional contribution of fine structure changing collisions should decrease with decreasing trap depth. Therefore, we can net-ice; the contribution from line structure changing collisions io fi in describing the losses that occur during loading of the 1 inK deep FORT. The fractional ground state population affects the loss rate in the following way. For radiative escape to occur, at least one of liie two colliding atoms must be in the upper hyperfine ground state ( F = 3 ) . The MOT light is too far detuned to excite an atom from the lower ground state ( F = 2 ) to a higher lying molecular state during a collision. Therefore, the loss term takes the form of j3i!-(<-iH~•«-,), where n-\ is the density of atoms in the F ~ 3 stale, n , is the density of atoms in the /•'=-= 2 state, and n- '• n^=-n. The fraction of atoms in the F — 3 state depends on the relative optical pumping rates; of the prmary MOT laser (/" - 3 ^ > " ' - ' : 3 — > > ' - • 2) as compared with the repunip laser ( F = 2 •->{•'' = 3 *F=j). We derive the fraction of atoms in the F=3 state from a simple two-level rate equation model, which results in :IT./H = IR/(!R--alv). where a is a constant that reflects the relative optical pumping rates. These rates depend on the ac Stark shifts induced by the FORT, in addition to Clebsch-Gordan coefficients and the frequency of each laser. The average shift in the transition frequency for atoms in the FORT (A!lc) was extracted from Fig. 1 and found to be i a c =2.3y. This results in « = 0.02. The experimentally determined loss rate coefficient is then
K /M " ..... V
/R
17)
where K is a constant related to the density dependent loss rate, independent of optical pumping effects. We again use the volume V given by Eq. (6) to make the translation from the typical density dependent loss rate to the loss coefficient that we measure in our experiments. For n'o = 26 fj,m, we find ('"= S . 3 X I0~ 6 em 3 . The behavior described by F,q. (1) is clearly seen in the measured dependence of J3'} on both /"M and / R as shown in Figs. 6(b) and 8(b). The solid curves in these plots are fits of Eq. (7) to the data, with K the only free parameter. The fits to the different data sets give AT=(I.1±0.5) X 10~ "1 cm3 mW~' s ~ ' , where the spread is due to the different values we find by fitting different data sets taken on
-50-100-
-150-
-200
0
200
400 600 r (angstroms)
800
1000
FIG. 15. Approximate intramolecular potential for Rb, with C3 = 71 A ' eV. The large graph shows the critical radius r.. for U0/kB=-1 K. while the inset depicts the same parameters for L / 0 / / ( „ = - ! mK. different days. We see that Eq. (7) gives a good description 01 the measured MOT and repump intensity dependences of the ios> raic during loading of the FORT. For a comparison of our numbers with those normally found in MOT's we set « ? / ? ; = 1. assuming all atoms arc in tiie F'= 3 ground state as they would he in a MOT. Thus #L-A:/M. For 7 M = 10 mW/cni 2 we find j3L-\ Y 10" J cnr'/s. In contrast, a MOT with comparable powers and detuiiings has a loss rate of (T 4 ) X i O ' " 1 3 em'/s [24 26,23]. which is a factor of 250 to 1000 smaller than our measured value. As shown below, this difference is due to the smaller depth of the FORT. In order to explain the increased loss rate observed during loading of the FORT as compared to a MOT, we will discuss in more detail how the FORT depth affects tSw radiative escape loss rate. In the ease of a MOT, the dependence of/?, due to radiative escape, on the trap depth is predicted to vary as l/0 " ;t> in tiie limit of large trap depths [22,23]- However, some of the assumptions used to derive this dependence break down at smaller trap depths. Therefore, to calculate /3L for the FORT, we perform a numerical calculation of the radiative escape process, using the semiclassical GallagherPritchard (GP) model [20]. Given the uncertainties in our measured j8 L , a more detailed calculation is unwarranted. Therefore, we do not include angular momentum considerations or enhanced survival probability as described by Julienne and Vigue [22] or any corrections for the hyperfine splittings as introduced by Lett et at. [21]. Instead, we approximate all of the involved intramolecular potentials as a single — C-,/f l in the GP model, as formulated by Peters et a!. [27], and integrate numerically. Figure 15 shows an approximate intramolecular potential for Rb. For initial excitation at radius r 0 , atoms are accelerated toward one another for some time before spontaneous emission leaves two ground state atoms with more kinetic energy than they had initially. If the kinetic energy picked up in the collision exceeds the depth of the trap, both atoms will be lost. The radius at which the pair acquires exactly enough energy to be ejected from the trap is therefore called the critical radius rc, and the time required for the atoms to reach this is called /„.
013406-9
441 KUPPENS, CORWIN, M1LI.F.R, CHUPP, AND WTEMAN 10:
FORT
I
MOT
101
10"
10"
10"
10
10°
10'
5
trap depth ( K )
10
15
20
position z ( mm ) FIG. 16. Radiative escape loss rate coefficient /3 calculated as a function of trap depth. When the atoms have zero initial kinetic energy, A correspond to /3 values for A A C + A M = 2 y and • to 8 y. D depict 8y, but with an initial relative velocity v = kBT/m, corresponding lo 7=800 /uK. The dashed line represents /?=<: f/ 0 M> as mentioned in the text. Once the atoms reach r,,, acceleration is very fast, and the atom pair spends a short time t\ accelerating lo r = = 0 . It' spontaneous emission does not occur during this second time interval, the intramolecular separation will oscillate between 0 and rfl, making multiple orbits until spontaneous emission occurs. The probability of a lossy radiative escape event thus depends or: the excitation probability at i-,, and the probability of decay occurring when r
4vr~(ldr
where the photon scattering rate of atoms in the /•'-•= 3 state, C(r0), is a function of the effective detuning A(r,,} = -CVO'o^) 4 ' ^M with respect to the unperturbed atomic transition frequency. The probability of a radiative escape event resulting in trap loss is / ) RE =sinh(7; ! ysinh['y(/ ( , t i { ) \ . which reflects the important contribution of multiple orbits [27]. The result of the numerical integration of Eq. (8) Cor different trap depths is shown in Fig. 16. The dashed line indicates a dependence of (3~ £/0 s/6 as predicted by Refs. [22] and [23]. This clearly tracks our numerical integration at large trap depths, where it is expected to be valid. Examination of the change in ji with U(> shows that for a change in l/o from -- 1 K to -- 1 mK, ft increases by a factor of ~ 50 (for »i /n= 1 ) . This ratio is at least a factor of 5 smaller than the experimentally observed ratio of 250. However, given the large mi certainties in the experimental determination of n and n-Jn, as well as the approximations that go into the calculations and the sleep dependence of the results, the measured and calculated numbers are not inconsistent. Therefore, we believe that the enhanced loss rates we observe during loading are consistent with light-assisted collisions.
FIG. 17. Equipotential contours of the FORT for U c J k B = -100 fj,K, Un/kB=-\, -2, -3, and -4 mK. depth, (a) (d). and vvo— 26 /Am. B. A model for the FOR!1 loading rate The i n i t i a l loadsnu rate is the flux of atoms into the volume of the FORT Ihnes the probability for an atom in this volume ro become trapped:
where «\.IOT ^ s mc density in the center of the MOT', ;; -- vA n /7«? is the root mean square velocity of the atoms in the MOT. A is the effective surface area of me FORT, and P,,ap is the trapping p r o b a b i l i t y . For the discussion below, we will assume that the MOT density is constant, i.e.. « M ( 1 r- is the density at the start of the loading stage. As was shown in Fig. 9, fie has a maximum at A, v} — — 3D MHz, At the same detuning, the measured product MOT nas a maximum. This supports the idea that In addition to « M O T and v, the loading rate R.,-, also depends on A. For a trap with a waist of 26 /urn. the Rayleigh range is 2.7 mm. This is about 5 times larger than the diameter /JMOT °f me cloud of atoms trapped in the MOT. This means that the FORT radius hardly changes over a distance D MOT . Therefore we approximate A by the surface area of a cylinder with length D M O r and an effective radius reff thai depends on the relative longitudinal position of the MOT with respect to the FORT. The radius is set by the position on the FORT potential where it becomes significant compared to the temperature of the atoms in the MOT, U(re[t,z)=UcK'~-/cBT. The effective radius reff{_-) of the FORT, depending on z and L\. . is then
In Fig. 17 this equipotential contour is plotted for different values of L'"0 and (7(> eft -,-)= 100 //K. The bow tie shape of these contours explains why the best loading is achieved when the FORT and MOT are displaced; the radius has a maximum away from the focus. Moreover, for deeper traps.
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PHYSICAL. REVIEW A 62 013406
the -~ at which ihe maximum radius occurs shifts to larger z values, just as in the loading data. A typical initial MOT density during the FORT loading stage is 2 x 1 0 " cm~\ and the temperature ranges from 30 (.'.¥-. to 100 juK, depending on detuning of the primary MOT iight. A reasonable value for the average speed in one dimension is ihus v~5 ctn/s. The MOT diameter is 500 yum (full vvidlh at half maximum). For l J n - k ^ - 1 mK and l7,_.,'k-A= --100 /.iK. (tie area A-', :•'- 10"' cm 2 . Using these numbers, the loading rate is ?..5 .< l(f s" ' if the trapping probability P | ; . a p =l. The loading rate measured for such a trap is J x ][)"' s " 1 , implying P, --•-0.1. The data in Fig. S show thai ihe loading rale increases nearly linearly with MOT intensity. However, it is known that the MOT temperature and density in the sub-Dopplcr regime scale as intensity and the inverse square root of intcnsily. respectively j 17.29]. The product «MOT L ' > s 'hus expected to show no strong dependence on MOT intensity. We have experimentally confirmed this by measuring the density and temperature of the cloud of MOT atoms as a function of MOT intensity. Over the range of MOT intensities as in Pig. 8, we find that <'.'MOT!;'> is nearly constant. Therefore, we conclude that the observed increase of the loading rate with increasing MOT intensity arises from an intensity dependence of the trapping probability /*1:1,p. We estimate P trap by using the following simple model. The atoms come into the trapping volume with a narrow distribution of velocities around an average velocity :>. To be trapped, an atom's energy has to be reduced to below the edge of the trapping potential. The time scale for this to happen must be about one-half the oscillation period r/2 of the FORT in the strong direction, which is 150 /u,s for the above mentioned trap. The scattering of MOT laser photons affects the atomic velocity in two ways. First, the atom scatters photons, which leads to heating, that is, a broadening of its velocity distribution. The spread in velocity due to photon scattering is u~ \j\{fik/m')iT!X.llr/2. Second, the frictional component of the scattering leads to a reduction of the atom's average kinetic energy. The rate at which the atom loses energy is E— — av~, with a die friction coefficient. This damping leads to a change in average velocity of Ay = v yet rim in the characteristic time r/2. The probability of being captured is the integral of the shifted and broadened distribution,
1
-exp
2<7 2 ,
do.
(11)
The integral leads to p
* trap
—
T
-erfl -=--
(12)
For A M = 30 MHz and / M = 6 mWVem2 the MOT scattering rate r a . lU =9.5x 10s s"1, which makes a=6 cm/s.
We obtain a value of a from the formula given in Ref. 123], based on Doppler cooling. The damping rate calculated from this formula increases linearly with MOT intensity for the velocities we are interested in. The values of a and cr also depend on MOT detuning. There is a strong spatial dependence of the effective MOT detuning due to the light shift induced by the FORT. We make the crude approximation that the MOT is shifted csut of resonance by one-half the FORT light shift. This results in u/tr = 2 and aim — 200 s ' . The result of this mode! is plotted as a solid curve in Fig. S a n d shows good agreement with the data. This is a very approximate treatment, but i' clearly supports our general interpretation of the loading process.
VIII. SUMMARY
From our understanding of ihe loss process and the loading rate, we are now in a position to explain the dependencies presented in Sec. IV. We saw that more atoms arc loaded into deeper FORTs because the loading rate increases and the loss rate decreases. The loading rate increases because the effective FORT radius increases with trap depth, which enhances the flux of atoms into the trap. The loss rate decreases because the probability of light-assisted collisional loss decreases when the trap is deeper. In addition, the light shift of ihe trap increases the effective detuning of the MOT light: and repump light for the atoms in the trap, which" reduces their excitation rate, and therefore reduces radiative escape. The data in Fig. 10 show that the MOT detuning at which loading is optimised is smaller for deeper FORTs. A deeper trap means larger light shifts of the atoms. The MOT detuning and the repump detuning arc both blueshifted, so that the detuning of both lasers from the atoms becomes more negative (rod) in both cases. Due to the increased effective detunings, the MOT cooling rate decreases, and the trapping probability thus decreases. By choosing a smaller MOT detuning the loading rate is increased. The dependence on MOT detuning of the @'L that we measured for a given trap depth (see Fig. 9) is also well described by Eq. (7). The detuning dependence enters through the parameter a. At small MOT detunings, the MOT excitation rate is highest, and $[_ is therefore large. By increasing the MOT detuning further, a becomes smaller. Therefore, n/ M becomes negligible compared to / R , and j3'L becomes independent of detuning. In Sec. IV A, it was shown that loading is optimized for repump light of very low intensity that is tuned on resonance. We saw that the number of atoms in the FORT dropped when the repump intensity was increased, but the repump scattering rate was maintained by changing the repump detuning. Increasing the repump intensity increases the loss rate. Altiiough the detuning is increased for the atoms in the MOT sttch that the scattering rate is the same for the MOT atoms, the increased intensity still causes an increased scattering rate for the atoms trapped in the FORT.
013406-11
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weak rcpump laser. Thus, placing addilionai shadows in the main MOT light may improve the loading even more, but this requires a more complicated optical setup. By reducing f>'{ lo the l i m i t of no MOT light, while maintaining the same loac'lint; rase, the steady state number of atoms will be 2 X U)', ucariy ail of the MOT atoms,
O a
X. CONCLUSIONS
FIG. 18. Number of atoms loaded into the FORT as a function of loading time, using normal loading (O) and enhanced loading with a shadowed repump beam (•). The FORT parameters are n'o = 58 /tm, P=580 mW, and X = 783.2 nm. The solid curves arc fits of I2q. (4) to the data. IX, ENHANCKO AND QUASICONTINUOJJS LOADING The previous data and analysis show that !p'L is very small when the repump power is zero, and reducing the primary MOT intensity aiso helps to reduce the loss rate. However, reducing these light intensities also reduces the loading rate. It is possible to improve the ratio of loss to loading rate by changing the beam geometry. The loss rate only has to be eliminated in the volume occupied by the FORT, which can be accomplished by inserting obstructions to block portions of .the MOT and repump beams. To demonstrate this concept, we make a shadow in the repump beam. Tor this purpose an additional repump beam is used that copropagsles with the FORT beam, A 0.6 mm opaque disk in t h i s beam is imaged into the M.OT and FORT region, such that the shadow covers the length of the FORT. The peak intensity in this repump beam is increased from 5 juW/cm 2 to 460 jciW/cnr, which is sufficient to prevent the MOT from being lost while loading the FORT. The MOT is still loaded as before using a repump beam without a shadow. During the loading of the FORT, this repump beam is switched off, and the shadowed beam switched on. In Fig. ]8 we show two measured loading curves. One shows loading using the conventional method without the shadow and with reduced repump power, and the second curve shows the new method with the shadowed beam and increased repump power. The maximum number of loaded atoms doubles from 3.9X 106 to 7.7X H)6. The transfer efficiency from MOT to FORT increased from 21% to 42%. The increase in the number is caused by a decrease of the loss rate from J8,' = (3.5±0.5)X 10 6 s ' to £{ = (1.7 ±0.3)X 10"6 s" ' and a simultaneous increase of the loading rate, from ftn=5.8X 1C7 s^ 1 to R0= 10.2X U)7 s ~ ' . The loading rate is increased, due to the fact that the MOT density increases, by reducing the repump intensity in the center of the MOT. The loss rate /3( is still larger during loading with the shadow than in the absence of all MOT light, where the loss rate /8'= 2.3x 1(T7 s"" 1 . This is due to the fact that the MOT light is not shadowed and the FORT acts as a very
Our Htudv can be summarized as i:b!iows. Deeper traps load more- atoms and the temperature is a fixed fraction of the !rap depth. Optimum loading is achieved for very low repump scattering rate ( / R P = = 5 «W/cnr). and a MOT det u n i n g Thai depends on the FORT depth. Controlling the geometry of die overlap of the MOT and FORT beams gives substantial improvement. To maximize the number of atoms trapped in the FORT, these are the parameters to adjust. However, underneath this recipe lies a lot of interesting physics. Loading the FORT from a MOT is a dynamical process, governed by a loading rate A',, and density dependent losses characterized by J3[ . During the loading, the MOT l i g h t fields increase the FORT loss rate considerably. The main loss mechanism dining loading of the FORT is radiative escape collision induced by the MOT light fields. The loss rate is higher than in a MOT because the FORT is on the order of only a millikelvin deep, whereas a typical MOT is a kelvin deep. In addition, the light shift due to the FORT changes the hyperfine repump rate and the optical excitation by the MOT trapping and cooling light, which is responsible for the radiative escape. Taking these effects into account, we were able to model how rhe loss rate depends on the intensities and detunings of the primary MOT and repusnp lasers, as well as the FORT depth. This model explains the experimentally observed dependencies of the loss rate on these parameters quite well. The loading rate can be described in two parts: a flux of atoms into the volume of the FORT times a probability of being trapped. Doth parts depend on the MOT and FORT parameters, including the size of the FORT. Here we studied the loading of a FORT with a radius =c60 fim, which is much smaller than the radius of the MOT. Even with this mismatch of size between the MOT and the FORT, we were able to transfer more than 40% of the atoms initially trapped in the MOT to the FORT. For FORTs with a radius comparable to that of the MOT. the description of the loading rate changes, and the loss rate may also behave differently; however, we believe that radiative escape processes may still be enhanced as compared to a MOT. Working with a small waist enables one to make deeper traps and achieve tighter confinement. Such conditions may make it possible to reach, for instance, BoseEiustein condensation at high temperatures of tens of mierokelvins. ACKNOWLEDGMENTS This work is supported by NSF and ONR. We thank Stephan Diirr for his comments during prepration of the final manuscript.
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PHYSICAL REVIEW A 62 01 M06
LOADING AN OPTICAL DIPOLl: TRAP j l ] S. Chu. J. E. Bjorkholm, A. Ashkin, and A. Cable. Phys. Rev. 1 ett. 57, 314 (1986). [2] F. L, Raab, M. Premiss. A. Cable, S. Cliu, arid D. E. Pritchard, Phys. Rev. Lett, 59, 2631 ('1987); C. Monroe, W. Swann. H. Robinson, and ( ' . Wieinsn. -hid 65, 1571 (1990). [3] '). D. Miller, R. A. dins, and D. J. Hemzen, Phys. Rev. A 47, R4567 (1993). [4] C. S. Adams, H. J. Lee. N . Davidson, M. kascvich. and S. Chu, Piiys. Rc-\, Lett. 74, :557^ ('1995). [5] T. Takekoslri and R. J. Knize, Opt. Lett. 21, "7 (1996). [6] I). Boiron. A. Micbaud. J. M. Fotiniisr. 1... Siniard. M. S|>reiigcr, G. Grynberg, and C. Salomon, Phys. Rev. A 57. R4HI6 I J D 9 S 1 . [7] S, Bcili, K. M, O'Hara. M. E. Gchm. S. R. Granada f.nd J. EThornas, Phys. Rev. A 60, R29 (1999). [8] I f . C. W. Beijermek, Phys. Rev. A 61. 033606 (2000), [9] M. E. Gehm. K. M. O'Hara, I. A. Savard, and .1. E. Thomas. Phys. Rev. A 58, 3014 (1998>. j!0] T. Walker and P. Feng, Adv. At.. Mo!.. Opt. Phys. 34, 125 (1997). in] K. M. O'Hara, S. R. Granade, M. E. Gehm, T". A. Savard. S. Bali, C. Freed, and J. E. Thomas, Phys, Rev. Left. 82, 4204 (1999). [12] A. II,. Sie«rri
[16] K. L. Corwin, Zheng Tian Lu. C F. Hand. R. J. F.pstcin. and C, E. Wienian, Appl. Opt. 37. 3295 (1998.1. [17] C. G. rowns(ir,ti, N. H. Edwards. C. J. Cooper, K, P. Zetie, C. J. Foot, A. M. S'.eanc, P, Szriftgiser. IT Pen-in, am! J. Dalibard. Phys. Rev. A 52, 1423 (1995). [ I S ] K. I... Corwin. S. .', M. Kuppens, D. Cho, and C. E. \Vieman, Phys. REV. l.ctl. 83. 1 3 1 1 (i999i. I 19] J. D. Miller, Ph.D. thesis. University of Texas, i ( >94. [ 2 0 j A. Gallagher and D. E. Pritcljard, Phys, Rev. Lett. 63, 957 M989), ! 2 1 l (\ D. Lett, K. Molmer, S. 1). Gesssenier. K.-V. N. Tan, A. Kimiurakrishnan, C. D. Wallace, and I'1. L. Gould, J. Plivs B 28. 65 (1995). [22] P. S. Julit-nns and J. Vigue, Phys. Rev. A 44. 4464 (1991). [23] S. I). Gensemer. V. Sanchez-Villieana, K.-Y. N. Tan. T. T, Grove, sue P. E. Gould, Phys. Rev. A 56. 4055 (1997'. [24] D. Sesko. 1". Walker. C. Monroe, A. GnHagher, and C. Wieman. Phys Kev. Lett, 63, 961 (1989), [25] C. D. Wallace, T. P. Dirmeen, K.-'Y. N, Tan, T, T. Grove, and P. L. Gould, Phys. Rev. Lett. 69. 897 (1992). i 26i L. Marcassa, V. Bagnato, Y. Wang, C. Tsao. J. Weiner. G, Duiieu. Y. B. Band, and P. S. Julienne. Phys. Rev. A 47. R4563 (1993). [27] M. G. Peters, D. Hoffmann, ,!. D. Tabisstm. and T. Walker, Phys. Rev. A 50, R906 (1994), [2K] D, Hoffmann, P. Feng, and T. Walker J. Opt, So;:. Am. B J l , 712 (1994;. [29] M. Drewscn, Ph. Laurent, A. Nadir, G. Santard'i. A. Ciairon, Y. Castin, D. Orison, arid C. Salomon, Appl. Phys. B: Lasers Opt. 59, 283 (1994).
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PHYSICAL REVIEW A, VOLUME 63, 011401(R)
Improved loading of an optical dipole trap by suppression of radiative escape Stephan Durr. Kurt W. Miller, and Carl E. Wieman JILA, NIST, and University of Colorudo, Boulder, Colorudo 80309-0440 (Received 27 June 2000; published 4 December 2OOO) We investigate two-body loss in an optical dipole trap for 87Rb atoms. In the presence of additional near-resonant light, such as from a magneto-optical trap during the trap loading, the two-body loss is strongly enhanced by long-range radiative escape. We suppressed this loss by a factor of 15 by adding a sideband to the optical dipole trap laser. This allows more atoms to be loaded into the optical dipole trap.
DOI: 10.1103/PhysRevA.63.011401
PACS number(s): 32.80.Pj, 34.50.Rk
With most methods for cooling and trapping of neutral atoms, the number of atoms that can be trapped is limited by inelastic collisions between the cold atoms. There has been intensive research in the field of cold collisions, which has been reviewed in Refs. [ 1-41. In order to improve cooling and trapping schemes, a possible suppression of collisional trap loss is of particular interest. Several experiments have demonstrated the possibility of suppressing cold collisions (see Ref. [l] and references therein). To obtain an observable effect, these experiments usually first increase the collisional loss from a magneto-optical trap (MOT) by m,aking the MOT very shallow or illuminating the atoms in the MOT with additional laser light that is driving photoassociative loss. Some fraction of this extra loss can then be suppressed by adding another laser frequency, blue detuned from the MOT transition. This light excites colliding atom pairs to repulsive molecular potentials. The resulting repulsive force between the atoms shields them from a very close encounter, where most inelastic collisions occur. This suppression strategy was successfully applied to metastable rare-gas MOTS. Here, Penning ionization and associative ionization intrinsically create large collisional loss and no further enhancement is needed to obtain an observable effect. This loss has been suppressed by as much as a factor of -30 [5,6]. In this paper, we investigate collisional loss of ”Rb atoms from an optical dipole trap (far-off-resonance trap, FORT), while it is being loaded from a MOT. Adding a sideband shifted by =200 MHz to the FORT light, we observed a strong reduction of the two-body loss coefficient. This allows us to load significantly more atoms from the MOT into the FORT. As discussed below, the observed effect can be explained in a way similar to the suppression experiments mentioned above. Our FORT consists of a linearly polarized, focused Gaussian laser beam, which is detuned =3 nm to the red from the D 2 line in *’Rb. The ac-Stark shift produces a conservative potential with a minimum at the focus, where atoms can be trapped. The FORT is loaded with *’Rb atoms from a MOT. We recently investigated the dynamics of this FORT loading process in great detail [ 7 ] . The best transfer of atoms from the MOT to the FORT is achieved if the FORT light is turned on typically 150 ms before the MOT is turned off. Without this 150-ms “FORT loading stage,” only a small fraction of the MOT atoms is transferred into the FORT. During the FORT loading stage, however, many
MOT atoms enter the FORT volume and are cooled sufficiently by the MOT light to be trapped in the FORT. Unfortunately, the presence of the MOT light drastically increases the loss of atoms from the FORT by light-assisted collisions leading to radiative escape [8], thus limiting the loading efficiency. This can be understood from a simple semiclassical picture, as shown in Fig. 1. The atoms are exposed to the MOT light detuned = -25 MHz (dashed line in Fig. 1) from the cycling transition ( S , 1 2 F = 2 t i P 1 , 2 F = 3 at h=780 nm). If two atoms are separated by R-80 nm. the MOT light is resonant to excite them to an attractive molecular potential. (The number of molecular potentials is large and they can be both attractive and repulsive. For simplicity, Fig. 1 displays only a few potentials.) After excitation, the gradient of the molecular potential accelerates the atoms towards each other. The atoms eventually decay spontaneously; however, the kinetic energy gained is typically greater than the FORT depth ( = h X 20 MHz). The atoms thus escape from the FORT. Note that the FORT is very shallow as compared to a MOT (depth- h X 20 GHz). Thus, most atoms lost from the FORT are immediately recaptured in the MOT. The radiative escape processes predominantly occur at very
1050-2947/2000/63(1)/011401(4)/$15.00
FORT
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....
MOT
k..
-384THZ
I
,
I
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60
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,
80
I
1
90
100
internuclear distance [nm] FIG. I . Optical suppression of radiative escape in a “Rb FORT. The MOT laser resonantly excites atoms around R= 80 nm. In the absence of the two FORT frequencies, the gadient of the excited molecular potential accelerates the atoms towards each other. Before spontaneous decay. the atoms typically gain enough kinetic energy to escape from the FORT. However. if two FORT frequencies are present, they can drive a two-photon Raman transition to a repulsive molecular potential. This potential repels the atoms from each other, thus reducing the radiative escape loss.
63 01 1401-1
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02000 The American Physical Society
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STEPHAN DURR. KURT w . MILLER. AND CARL E. WIEMAN
large R, where the excited-state hyperfine energy is large compared to the molecular binding energy. The molecular potentials in this regime are known [9] and well described by a resonant dipole-dipole interaction, E ( R ) = C , / R 3 . Inelastic collisions at such long range give rise to much larger twobody loss coefficients than short-range processes, because the probability of finding two atoms at an internuclear distance R is proportional to R'. The maximum number of atoms that can be loaded into the FORT is seriously limited by this radiative escape. In order to reduce these losses, it is useful to drastically reduce the power of the MOT repump laser during the FORT loading stage [7,10]. In this situation, a significant fraction of the atoms will populate the S1/'F= 1 ground state (not shown in Fig. 1). Atoms in this state are detuned -6 GHz from the MOT laser and are therefore not excited at large R, thus suppressing two-body losses. At very low repump power, however, the atoms quickly leave the MOT and are not cooled into the FORT so that there is a tradeoff between low radiative escape from the FORT and a reasonable loading rate. However, optical suppression can greatly reduce these radiative escape losses. Consider the case of FORT light composed of two frequencies separated by approximately the excited-state hyperfine splitting (267 MHz). Once the colliding atom pair is in the excited state, the combination of the two FORT frequencies can drive a two-photon Raman transition to a repulsive molecular potential, which is asymptotically connected to a lower hyperfine state, as shown in Fig. I. Atoms in this state will repel each other, thus stopping and finally reversing the relative atomic motion. Since the atoms spend a lot of time around the turning point, they are particularly likely to decay at this point, where no kinetic energy is released. Even atoms that do not decay around the turning point and start moving away from each other are less likely to gain enough kinetic energy for radiative escape, because at increasing separation, the acceleration is getting smaller. The near-resonant state for the Raman transition can be the S S ground state, as shown in Fig. 1, or a doubly excited P P molecular state. If the two FORT frequencies are of roughly the same intensity, the two-photon Rabifrequency will be of the same order of magnitude as the ac-Stark shift (-20 MHz), which will give a substantial transition probability within the excited-state lifetime (-26 ns). The optical suppression will depend on the frequency difference between the two FORT frequencies compared to the spacing of the molecular potentials. Best suppression is obtained if the FORT frequency difference is such that the radius for the Raman resonance is slightly smaller than the radius for the MOT excitation resonance. As the frequency difference becomes much smaller, a greater fraction of the atom pairs will decay spontaneously before a Raman transition can occur, resulting in less suppression. If the FORT frequency difference is too large, however, atoms will still be in the ground state when passing the Raman resonance radius. Thus, no Raman transition occurs and there is no suppression. Thus we expect that the suppression will exhibit an asymmetric line shape as a function of the FORT frequency difference with the steeper slope at larger frequency differ-
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PHYSICAL REVIEW A 63 01 I.U)I(R)
ence. Moreover, the optimum frequency difference will depend on the MOT detuning. For larger MOT detuning, the radius for MOT excitation becomes smaller as well as the required FORT frequency difference. We now turn to the experimental study of this suppression process. To collect "Rb atoms from background vapor, we operate a standard MOT using light detuned =-I5 MHz from the cycling transition and a repump laser resonant with the S I I I F =1*PTizF= 2 transition. After loading the MOT, we switch to the FORT loading stage, i.e., we turn on the FORT and leave the MOT on. For better loading, the detuning of the MOT light is changed and the intensity of the repump light is reduced. After 150 ms, the MOT is turned off. This is followed by a variable time of storage in the FORT, after which we turn the FORT off and the MOT back on. The fluorescence from the MOT immediately after turn on reveals the number of atoms that had been in the FORT. The FORT beam typically has a power of 700 mW focused to a beam waist of -40 pm. This light is created in a homebuilt single-mode Tisapphire ring laser. The laser cavity length is 1.5 m, creating a spacing of neighboring longitudinal cavity modes of 21 2 MHz. The detuning from the "Rb D , line is -3 nm. The first observation of the optical suppression was made by chance, when the Tixapphire laser happened to be operating near a mode hop. such that the laser frequency was jumping back and forth between two neighboring cavity modes. Under these conditions, the number of atoms was about 30% higher than usual. Such a mode-hop operation can be created on demand by sweeping back and forth a Brewster plate mounted on a galvo drive inside the laser cavity. We found that the number of atoms loaded was increased only when the laser was hopping between modes at a rate faster than -1 hHz. In this case, a beat node at 212 MHz could be seen on a photodiode placed in the laser beam, indicating that the laser was emitting at least two light frequencies simultaneously. This increased the number of atoms loaded into the FORT by up to a factor of 3 and had little effect on the FORT performance otherwise. We then investigated the optical suppression more quantitatively by studying the dynamics of the FORT loading stage. This allowed us to measure the radiative escape rate and its suppression. The number of FORT atoms, N. during this time is described by the differential equation [7]
-
d -dr- N = - ~ N - ~ N ' I v + R ~ exp(- yMoTt),
(I)
where r and ,8 are the single-body and two-body loss coefcm3 is the effective ficients, respectively, and V= 1 X FORT volume [7]. Ro is the maximum loading rate at which atoms from the MOT are loaded into the FORT at t=O. Due to the drastically reduced repump power, the MOT decays exponentially during the FORT loading stage as does the loading rate. The factor exp( - yMoTf) takes this into account. We determine the number of atoms in the FORT at the end of the loading stage, as described in Ref. [7]. This number is plotted in Fig. 2 as a function of the duration of the
01 1401-2
447 IMPROVED LOADING OF AN OPTICAL DIPOLE TRAP.. .
PHYSICAL REVIEW A 63 011401(R) 5x1O6
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-40 -35 -30 MOT detuning [MHz]
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FIG. 2. Number of atoms loaded into the FORT as a function of the FORT loading stage duration. Solid (open) circles were measured with (without) optical suppression. From fits (solid lines) to the solution of Eq. ( I ) , we infer a suppression of two-body loss by a factor of IS. FORT loading stage. Solid (open) circles were obtained with (without) rapid mode hop during the FORT loading stage. From fits (solid lines in Fig. 2) to the numerical solution of Eq. (I), we obtain p=1.5X lo-” cm3/s and p=2.2 X lo-” cm3/s with and without mode hop, respectively. The other fit parameters vary only slightly between the two data sets. This shows that the presence of the second FORT frequency, separated by 212 MHz, reduces the two-body loss rate by a factor of 15. Note that p is suppressed down to the same level as is obtained during usual storage in the FORT, i.e., with the MOT off. The method for creating two FORT frequencies, as described so far, has an obvious disadvantage: the frequency difference is fixed by the laser cavity length. In order to overcome this limitation, we operated the Tixapphire laser in single-mode conditions and added sidebands to the light outside the laser cavity using acousto- and electro-optic modulators. Adding sidebands in the frequency range of 0-15 MHz as well as at 43, 65. and 460 MHz, we could not observe any increase of atom number. These results suggest that there is a fairly narrow frequency range within which the presence of a second FORT frequency can increase the atom number. In order to add a tunable sideband at -200 MHz. we split the Tisapphire laser output into two beams. One was sent directly to the vacuum chamber to form a FORT, as before, the other beam was sent through a double-pass acousto-optic modulator (AOM) set up and then overlapped with the first beam. With some effort, we were able to ensure good spatial overlap between the two beams. This tunable sideband makes it possible to observe the dependence of the optimum FORT frequency splitting on the MOT detuning. Figure 3(a) shows the number of atoms in the FORT as a function of the frequency shift in the doublepass AOM setup. Each line of data corresponds to a different MOT detuning. In the range of MOT detunings shown here, the MOT decays away faster for larger detuning, thus reducing the FORT loading. At any fixed MOT detuning, however, there is a clear maximum as a function of the AOM frequency, as predicted by our model discussed earlier. The line shape around this maximum is asymmetric, with the steeper slope &higher AOM frequencies, also as expected from our model.
240
180
200
220
240
FIG. 3. (a) Number of atoms as a function of the difference between the two FORT frequencies. Each line connects data points at the same MOT detuning (dotted line, -23.2 MHz; solid lines, -27.6, -32.3, -36.8, and -41.3 MHz from top to bottom). (b) Optimum frequency difference obtained in part (a) as a function of the MOT detuning.
Figure 3(b) shows the optimum AOM frequency shift as a function of the MOT detuning. If only two excited molecular potentials were involved, as shown in Fig. 1, we would expect a linear dependence in Fig. 3(b), with a slope of 1 - C i / C : , where the C3 coefficients refer to the repulsive and attractive potential, respectively. In reality, however, many molecular potentials are involved. The ones that are most important for radiative escape are the low-lying attractive potentials, because they create the largest acceleration and correspond to the largest resonance radii. The suppression, however, works with any repulsive potential. Considering only the lowest attractive potential, the various repulsive potentials yield values of C;/C; between 0 and -0.5 (see Ref. [9]). We therefore expect a slope somewhere between 1 and 1.5. The experimental data in Fig. 3(b) yield a slope of 1.150.3, which agrees with our model. We also investigated the dependence of the suppression on the FORT wavelength. Using the mode-hop modulation in our standard FORT, we varied the overall Thapphire wavelength in the range between 782.1 and 783.0 nm. We found that the suppression factor depends slightly on the wavelength with a maximum at 782.7 nm. We attribute this to the ac-Stark shift that shifts the energy of the SIl2ground state down and the P,,? excited state up. For atoms inside the FORT, the detuning of the MOT light is therefore effectively larger than in free space. Scanning the FORT wavelength. we change the ac-Stark shift and therefore also the effective MOT detuning inside the FORT. There is some FORT depth where the fixed frequency difference of 212 MHz gives the best suppression. Note that the optimum frequency difference observed in Fig. 3 using the AOM setup is higher than 011401-3
448 PHYSICAL REVIEW A 63 01 l401(R)
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212 MHz, because the FORT laser power, and hence depth, was approximately half as large when using the AOM modulation. We expect a similar suppression to exist in ”Rb. However, the two-body loss coefficient during the FORT loading stage for “Rb without suppression is much lower (p=7 X lo-’’ cm’/s) than in 87Rb, so the effect of suppression would be difficult to observe. It is not obvious what causes this big difference in p between the isotopes. We speculate, however, that it is mainly because the excited-state hyperfine splitting in ssRb is only 121 MHz. Thus, the radius for resonant excitation with MOT light (taking into account the acStark shift in the FORT) is roughly in the region where the molecular binding energy is comparable to the excited-state
hyperfine energy. This gives rise to a large number of avoided crossings between molecular potentials, possibly changing the probability for radiative escape. In conclusion, we have observed a strong suppression of radiative escape from a FORT during the loading stage. This makes it possible to load more atoms into the FORT. The suppression is based on a Raman transition in the presence of two FORT frequencies with an appropriate frequency difference. The observed dependence of the optimum frequency difference on the MOT detuning agrees with our model.
[I] J. Weiner. V. S. Bagnato, S. Zilio, and P. S. Julienne, Rev. Mod. Phys. 71, I (1999). [2] D. Heinzen, in Proceedings of the Iriterriatiorid School oj Physics “Enrico Fermi, Course CXL: Bose-Einstein Condensation in Atomic Gases, edited by M. Inguscio, s. Stringxi. and C. E. Wieman (10s Press, Amsterdam, 19991, pp. 351390. [3] P. D. Lett. P. S. Julienne, and W. D. Phillips, Annu. Rev. Phys. Chem. 46, 423 (1995). [4] T. Walker and P. Feng, in Advances in Atomic, Moleculur, al7d Opficul Physics (Academic Press, San Diego. 1994), Vol. 34,
pp. 125-170. [5] H. Katori and F. Shimizu, Phys. Rev. Lett. 73, 2555 (1994). [6] M. Walhout, U. Sterr, C. Orzel, M. Hoogerland, and S. L. Rolston. Phys. Rev. Lett. 71,506 (1995). [7] S. J. M. Kuppens, K. L. Convin. K. W. Miller, T. E. Chupp, and C. E. Wieman. Phys. Rev. A 62. 013406 (2000). [S] A. Gallagher and D. E. Pritchard, Phys. Rev. Lett. 63. 957 (1989). [9] T. Walker and D. Pritchard, Laser Phys. 4. 1085 (1994). [lo] C. S. Adams, H. J. Lee, N. Davidson, M. Kasevich, and S. Chu, Phys. Rev. Lett. 74. 3577 (1995).
”
We thank K. L. Convin and S. J. M. Kuppens for building much of the apparatus used in this work. This work was supported by the NSF, and one of us (K.W.M.) was supported through the NSF IGERT program.
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Bose-Einstein Condensation
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This effort to find ways to better control the speeds and positions of atoms, ultimately led to reaching the ultimate limit of control, namely Bose-Einstein condensation, where the atoms are at the limits of control set by the Heisenberg uncertainty principle. When we first started work on the pursuit of BEC using laser cooled and trapped alkali atoms, I thought it was unlikely that we would be the first to achieve it. The groups working on getting BEC in polarized hydrogen using traditional cryogenics had such a long head start and seemed so close to reaching BEC. However, I was convinced it was worth working on anyway, because if we could get BEC using our approach, our technology would be far superior for actually using and studying the BEC. Although I was wrong in predicting we would not be the first to reach BEC, I was clearly right about the virtues of our approach for the study of BEC. Once we had BoseEinstein condensation it was like being a kid in a candy store. There were so many relatively easy experiments that could be done, and they gave one interesting result after another. Of course, at that time I was also just finishing up nearly twenty years of parity violation experiments, so my standards as to what were “quick and easy” experiments may have been somewhat distorted. Eric Cornell and I have had a very long and amiable collaboration in BEC work. In the early days we collaborated quite closely so that there was virtually no distinction as to who was “in charge” of any given experiment. After achieving BEC, as our programs expanded the “Cornell group” and the “Wieman group” became necessarily more separate entities, with Eric and I having less intimate connection with the work being carried out in the other’s lab. Up until the awarding of the Nobel Prizes however, to avoid any questions about dividing credit, we continued to have some level of collaboration on all BEC experiments, and we were careful to make sure we were both listed as coauthors on all the significant BEC papers. The author list however provides a clue as to whether the work was carried out in Eric’s lab space and hence primarily under Eric’s supervision or in my labs under my supervision. The person who is listed as the final author is the one who was the supervisor.
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0bservation of Bose-Einstein Condensation in a Dilute Atomic Vapor M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman,* E. A. Cornell A Bose-Einstein condensate was produced in a vapor of rubidium-87 atoms that was confined by magnetic fields and evaporatively cooled. The condensate fraction first appeared near a temperature of 170 nanokelvin and a number density of 2.5 X 10” per cubic centimeter and could be preserved for more than 15 seconds. Three primary signatures of Bose-Einsteincondensation were seen. (i) On top of a broad thermal velocity distribution, a narrow peak appeared that was centered at zero velocity. (ii) The fraction of the atoms that were in this low-velocity peak increased abruptly as the sample temperature was lowered. (iii) The peak exhibited a nonthermal, anisotropic velocity distribution expected of the minimum-energy quantum state of the magnetic trap in contrast to the isotropic, thermal velocity distribution observed in the broad uncondensed fraction.
o n the microscopic quantum level, there are profound differences between fermions (particles with half integer spin) and bosons (particles with integer spin). Every statistical mechanics text discusses how these differences should affect the behavior of atomic gas samples. Thus, it is ironic that the quantum statistics of atoms has never made any observable difference to the collective macroscopic properties of real gas samples. Certainly the most striking difference is the prediction, originally by Einstein, that a gas of noninteracting bosonic atoms will, below a certain temperature, suddenly develop a macroscopic population in the lowest energy quantum mechanical state ( 1 , 2). However, this phenomenon of Bose-Einstein condensation (BEC) requires a sample so cold that the thermal deBroglie wavelength, A,, becomes larger than the mean spacing between particles (3). More precisely, the dimensionless phase-space density, pps = n(A,J3, must he greater than 2.612 ( 2 , 4), where n is the number density. Fulfilling this stringent requirement has eluded physicists for decades. Certain well-known physical systems do display characteristics of quantum degeneracy, in particular superfluidity in helium and superconductivity in metals. These systems exhibit counterintuitive behavior associated with macroscopic quantum states and have been the subject of extensive study. However, in these systems the bosons are so closely packed that they can he understood only as strongly interacting systems. These strong interactions have made it difficult to understand M. H. Anderson, J. R. Ensher, M. R. Matthews, C E. Wieman, JIM, National Institute of Standards and Technology (NIST), and University of Colorado, and Department of Physics, University of Colorado, Boulder. CO 80309, USA. E.A. Cornell, Quantum Physics Division, NIST, JIM-NIST, and University of Colorado, and Department of Physics, University of Colorado, Boulder, CO 80309, USA. ‘To whom correspondence should be addressed.
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the detailed properties of the macroscopic quantum state and allow only a small fraction of the particles to occupy the Bose condensed state. Recently, evidence of Bose condensation in a gas of excitons in a semiconductor host has been reported (5). T h e interactions in these systems are weak but poorly understood, and it is difficult to extract information about the exciton gas from the experimental data. Here, we report evidence of BEC in a dilute, and hence weakly interacting, atomic vapor. Because condensation at low densities is achievable only at very low temperatures, we evaporat i d y cooled a dilure, magnetically trapped sample to well below 170 nK. About 15 years ago, several groups began to pursue BEC in a vapor of spinpolarized hydrogen (6). T h e primary motivation was that in such a dilute atomic system one might be able to produce a weakly interacting condensate state that is much closer tn the original concept of Bose and Einstein and would allow the properties of the condensate to bc well understood in terms of basic interatomic interactions. In the course of this work, 1000-fold increases in phase-space density have been demonstrated with the technique of evaporative cooling of a magnetically trapped hydrogen sample (7); recently, the phase-space density has approached BEC levels. Progress has heen slowed, however, by the existence of inelastic interatomic collisions, which cause trap loss and heating, and by the lack of good diagnostics for the cooled samples. T h e search for BEC in a dilute sample of laser-cooled alkali atoms has a somewhat shorter history. Developments in laser trapping and cooling over the past decade made it possible to increase the phase-space density of a vapor of heavy alkali atoms by more than 15 orders of magnitude. However, sevinvolving the scattered phoSCIENCE
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tons were found to limit the achievable temperatures (8) and densities (9), so that the resulting value for p was lo5 to 10‘ times too low for BEC. d e began to pursue BEC in an alkali vapor by using a hybrid approach to overcome these limitations (10, I I ) . This hybrid approach involves loading a laser-cooled and trapped sample into a magnetic trap where it is subsequently cooled by evaporation. This approach is particularly well suited to heavy alkali atoms because they are readily cooled and trapped with laser light, and the elastic scattering cross sections are very large (IZ), which facilitates evaporative cooling. There are three other attractive features of alkali atoms for BEC. ( i ) By exciting the easily accessible resonance lines, one can use light scattering to sensitively characterize the density and energy of a cloud of such atoms as a function of both position and time. This technique provides significantly more detailed information about the sample than is possible from any other macroscopic quantum system. ( i i ) A5 in hydrogen, the atom-atom interactions are weak [the Swave scattering length a, is about cm, whereas at the required densities the interparticle spacing (x) is about cm] and well understood. (iii) These interactions can be varied in a controlled manner through the choice of spin state, density, atomic and isotopic species, and the application of external fields. T h e primary experimental challenge to evaporatively cooling an alkali vapor to BEC has been the achievement of sufficiently high densities in the magnetic trap. T h e evaporative cooling can be maintained to very low temperatures only if the initial density is high enough that the atoms undergo many (-100) elastic collisions during the time they remain in the trap. Using a combination of techniques to enhance the density in the optical trap, and a type of magnetic trap that provides long trap holding times and tight confinement, has allowed us to evaporatively cool to BEC. A schematic of the apparatus is shown in Fig. 1. T h e optical components and magnetic coils are all located outside the ultrahigh-vacuum glass cell, which allows for easy access and modification. Rubidium atoms from the background vapor were optically precooled and trapped, loaded into a magnetic trap, then further cooled by evaporation. T h e TOP (time orbiting potential) magnetic trap (13) we used is a superposition of a large spherical quadrupole field and a small uniform transverse field that rotates at 7.5 kHz. This arrangement results in an effective average potential that is an axially symmetric, three-dimensional (3D) harmonic potential providing tight and stable confinement during evaporation. T h e evaporative cooling works by selectively re.1
454 leasing the higher energy atoms from the trap; the remaining atoms then retherinalize to a colder temperature. We accomplished this release with a railio frequency (rf) magnetic field (14). Because the higher energy atonis sample the
Fig. 1. Schematic of the apparatus. Six laser beams intersect in a glass cell, creating a magneto-opticaltrap (MOT),The cell is 2.5 cm square by 12 cm long, and the beams are 1.5 cm in diameter. The coils generating the fixed quadrupole and rotating transverse components of the TOP trap magnetic fields are shown in green and blue, respectively.The glass ceil hangs down from a steel chamber (not shown) containing a vacuum pump and rubidium source.Aiso not shown are coils for injecting the ri magnetic field for evaporation and the additional laser beams for imaging and optically pumping the trapped atom sample.
trap regions with higher magnetic field, their spin-flip transition frequencies are shifted as a result of the Zeeman effect. We set the frequency of the rf field to selectively drive these atoms into an untrapped spin state. For optimum cooling, the rf frequency was ramped slowlv downward, causing the central density and collision rate to increase and temperature to decrease. The final temperature and phase-space density of the sample depends on the final value of the rf frequency ( v , , , ~ ) . A typical data cycle during which atoms are cooled from 300 K to a few hundred nanokelvin is as follows: (i) For 300 s the optical forces from a magneto-optical trap (15) (MOT) collect atoms from a room torr vapor (10) of temperature, -I@-" "Rb atoms; we used a so-called dark MOT (16) to reduce the loss mechanisms of an ordinary MOT, enabling the collection of a large number ( lo7)of atoms even under our unusually low pressure conditions (17). (ii) The atom cloud is then quickly compressed and cooled to 20 pK by adjustment of the field gradient and laser frequency (18). (iii) A small magnetic bias field is applied, and a short pulse of circularly polarized laser light optically pumps the magnetic moments of all the atoms so they are parallel with the magnetic field (the F = 2, mF = 2 angular momentum srate.) (1Y). (iv) All laser light is removed and a TOP trap is constructed in place around the atoms, the necessary quadrupole and rotating fields being turned on in 1 ms. (v) The quadrupole field component of the TOP trap is then adiabatically ramped up to its maximum value, thereby
increasing the elastic collision rate by a factor of 5. At this point, we had about 4 X lo6 atoms with a temperature of about YO pK in the trap. The trap has an axial oscillation frequency of about 120 Hz and a cylindrically symmetric radial frequency smaller by a factor of fi.The number density, averaged over the entire cloud, is 2 X 10" cm-'. The elastic collision rate (19) is approximately three per second, which is 200 times greater than the one per 70 s loss rate from the trap. The sample was then evaporatively cooled for 70 s, during which time both the rf frequency and the magnitude of the rotating field were ramped down, as described (13, 20). The choice of the value of v,,,~, for the cycle determines the depth of the rf cut and the temperature of the remaining atoms. If veVapis 3.6 MI&, the rf "scalpel" will have cut all the way into the center of the trap and no atoms will remain. At the end of the rf ramp, we allowed the sample to equilibrate for 2 s (21) and then expanded the cloud to measure the velocity distribution. For technical reasons, this expansion was done in two stages. The trap spring constants were first adiabatically reduced by a factor of 75 and then suddenly reduced to nearly zero so that the atoms essentially expanded ballistically. A field gradient remains that supports the atoms against gravity to allow longer expansion times. Although this approach provides small transverse restoring forces, these are easily taken into account in the analysis. After a 60-ms expansion, the spatial distribution of the
c
Fig. 2. False-color images display the velocity distribution of the cloud (A)just
before the appearance of the condensate,(B)just after the appearance of the condensate,and (C)after further evaporation has left a sample of nearly pure condensate.The circular pattern of the noncondensatefraction (mostlyyellow and green) is an indication that the velocity distribution is isotropic,consistent SCIENCE
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with thermal equilibrium. The condensate fraction (mostly blue and white) is elliptical,indicativethat it is a highly nonthermal distribution.The ellipticalpattern is in fact an image of a single, macroscopically occupied quantum wave function, The field of view of each image is 200 Fm by 270 Fm. The observed horizontal width of the condensate is broadened by the experimental resolution.
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455 cloud was determined from the absorption of a ~ O - F S , circularly polarized laser pulse resonant with the SS,,,,F = 2 to SF',/,, F = 3 transition. T h e shadow of the cloud was imaged onto a charge-coupled device array, digitized, and stored for analysis. This shadow image (Fig. 2) contains a large amount of easily interpreted information. Basically, we did a 2D time-offlight measurement of the velocity distribution. A t each point in the image, the optical density we observed is proportional to the column density of atoms at the corresponding part of the expanded cloud. Thus, the recorded image is the initial velocity distribution projected onto the plane of the image. For all harmonic confining potentials, including the TOP trap, the spatial distribution is identical to the velocity distribution, if each axis is linearly scaled by the harmonic oscillator frequency for that dimension (22). Thus, from the single image we obtained both the velocity and coordinate-space distributions, and from these we extracted the temperature and central density, in addi-
tion to characterizing any deviations from thermal equilibrium. T h e measurement process destroys the sample, hut the entire load-evaporate-probe cycle can he repeated. Our data represent a sequence of evaporative cycles performed under identical conditions except for decreasing values of vevap, which gives a corresponding decrease in the sample temperature and an increase in phase-space density. T h e discontinuous behavior of thermodynamic quantities or their derivatives is always a strong indication of a phase transition. In Fig. 3, we see a sharp increase in the peak density at a value of u , , , ~of 4.23 MHz. This increase is expected at the BEC transition. As cooling proceeds below the transition temperature, atoms rapidly accumulate in the lowest energy state of the 3D harmonic trapping potential (23). For an ideal gas, this state would he as near to a singularity in velocity and coordinate space as the uncertainty principle permits. Thus, below the transition we expect a two-component cloud, with a dense central condensate surrounded by a diffuse, noncondensate fraction. This behavior is clearly displayed in sections taken horizontally through the center of the distributions, as shown in Fig. 4. For values of ucvapabove 4 23 MHz, the sections show a single, smooth, Gaussian-like distribution. A t 4.23 MHz, a sharp central peak in the distribution begins to appear. At frequencies below 4.23 MHz, two distinct components to the cloud are visible, the sniooth broad curve and a narrow central peak, which we identify as the noncondensate and condensate
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.. .. .. .. ......................................... . .. .. .. .. .. .. .. I
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Fig. 3.Peak density at the center of the sample a s afunction of the final depth of the evaporative cut, ueVap. As evaporation progresses to smaller values of vwap, the cloud shrinks and cools, causing a modest increase in peak density until uevapreaches 4.23 MHz. The discontinuity at 4.23 MHz indicates the first appearance of the high-density condensate fraction as the cloud undergoes a phase transition. When avalueforv,,,, of 4.1 Mhz is reached, nearly all the remaining atoms are in the condensate fraction. Below 4.1 MHz, the central density decreases, a s the evaporative "rf scalpel" begins to cut into the condensate itself. Each data point is the average of several evaporative cycles, and the error bars shown reflect only the scatter in the data. The temperatureof the cloud is a complicated but monotonic function of veVap.At ve,,,=4.7MHz,T= 1.6pK,andforueVa,=4.25 MHz, T = 180 nK. 200
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300 prn Fig. 4. Horizontal sections taken through the ve-
locity distribution at progressively lower values of show the appearance of the condensate fraction. vevap
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fractions, respectively. (Figs. 2B and 4). As the cooling progresses (Fig. 4), the noncondensate fraction is reduced until, at a value of veVapof 4.1 MHz, little remains hut a pure condensate containing 2000 atoms. T h e condensate first appears at an rf frequency between 4.25 and 4.23 MHz. T h e 4.25 MHz cloud is a sample of 2 X lo4 atoms at a number density of 2.6 X lo'* cm-3 and a temperature of 170 nK. This represents a phase-space density p,, of 0.3, which is well below the expected value of 2.612. T h e phase-space density scales as the sixth power of the linear size of the cloud. Thus, modest errors in our size calibration could explain much of this difference. Below the transition, one can estimate an effective phase-space density by simply dividing the number of atoms by the observed volume they occupy in coordinate and velocity space. T h e result is several hundred, which is much greater than 2.6 and is consistent with a large occupation number of a single state. T h e temperatures and densities quoted here were calculated for the sample in the unexpanded trap. However, after the adiabatic expansion stage, the atoms are still in good thermal equilibrium, but the temperatures and densities are greatly reduced. T h e 170 nK temperature is reduced to 20 nK, and the number density is reduced from 2.6 X lo'* a f 3to 1 X 10" cm-3. There is no obstacle to adiabatically cooling and expanding the cloud further when it is desirable to reduce the atom-atom interactions, as discussed below (24). A striking feature evident in the images shown in Fig. 2 is the differing axial-toradial aspect ratios for the two components of the cloud. In the clouds with n o condensate (ueVap> 4.23 MHz) and in the noncondensate fraction of the colder clouds, the velocity distribution is isotropic (as evidenced by the circular shape of the yellow to green contour lines in Fig. 2, A and B). But the condensate fraction clearly has a larger velocity spread in the axial direction than in the radial direction (Fig. 2, B and C). This difference in aspect ratios is readily explained and in fact is strong evidence in support of the interpretation that the central peak is a Bose-Einstein condensate. T h e noncondensate atoms represent a thermal distribution across many quantum wave functions. In thermal equilibrium, velocity distributions of a gas are always isotropic regardless of the shape of the confining potential. T h e condensate atoms, however, are all described by the same wave function, which will have a n anisotropy reflecting that of the confining potential. T h e velocity spread of the ground-state wave function for a noninteracting Bose gas should he 1.7 (8114) times larger in the axial direction than in the radial direction. Our observations are in qualitative agreement with this
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simple picture. This anisotropy rules out the possibility that the narrow peak we see is a result of the enhanced population of all the very low energy quantum states, rather than the single lowest state. A more quantitative treatment of the observed shape of the condensate shows that the noninteracting gas picture is not c!)mpletely adequate. W e find that the axial width is ahout a factor of 2 larger than that calculated for a noninteracting ground state and the ratio of the axial tn radial velocity spread is at least 50% larger than calculated. However, the real condensate has a self-interaction energy in the mean-field picture of 4mna,,fiz/m, which is comparable to the separation hetween energy levels in the trap. Simple energy arguments indicate that t h x interaction energy will tend to increase both the size and the aspect ratio to values more in line with what we ohserved. Although an atomic vapor of ruhidium can only exist as a inetastahle state at these temperatures, the condensate survives in the unexpanded trap for ahout 15 s, which is h n g enough to carry out a wide variety of experiments. T h e loss rate is probably n result of three-body recombination (25, 26), which could he greatly reduced hy adiabatically expanding the condensate after it has formed. Much of the appeal of our work is that it perinits quantitative calculations of microscopic hehavior, heuristic understanding of macroscopic hehavior, and experimental verification of both. T h e technique and apparatus described here are well suited for a range of experiments. T h e hasic glass cell design provides flexibility in manipulating and prohing the atoms. In addition, it is not difficult to siibatantially improve it in several ways. First, o u r position and velocity resolution can he improved with minor changes in optics and in expansion procedures. Second, a double MOT technique that spatially separates the capture and storage of atoms will increase our number of atcitns by more than 100 (27). Third, with improvement in measurement sensitivity it should he possible to probe the cloud without destroying it, in order tn watch the dynamics in real time. A n ahhreviated list of future experiments includes ( i ) performing optical spectroscopy, including higher order cnrrelation measurements, on the condensate in situ to study how light interacts differently with coherent matter and incohere n t matter (28); (ii) comparing the hehaviors of K7Rbwith "Rh, which is known to have a negative scattering length (19), potentially making the condensate unstable; (iii) stiidying time-dependent behavior of the phase transition including the
stability of the supersaturated state; (iv) exploring the specific heat of the sainple as it gocs through the transition hciundary (2) by measuring how condensate and noncondensate fractions evolve during cooling; (v) studying critical opalescence and other fluctuation-driven hehavior near the transition temperature; and (vi) carrying out cxperiments analogous to many of the classic experiments on superfluid helium (2, 29). There is a prediction that the scattering length of heavy alkalis can be modified, and even he made to change sign, by tuning the amhient magnetic field through a scattering resonance (26). Directly modifying the scattering length would piovide the ultimate control, but whether or not this is practical, one can still study the properties of the condensate as functions of the strength of the residual interactions hecause we now have the ability to cross the phase-transition curve over a large range of densities. Thus, it will he possible to observe, and to compare with theoretical prediction, the emergence of nonideal behavior such as singularities in the specific heat and many other phenomena, including those mentioned ahove. REFERENCES AND NOTES 1 . S. N Bose. Z Phys. 26. 178 (1924); A Einstein. Sitzunysber. Kgi. Preuss Akad Wiss 1924, 261 (1924); ibid. 1925. 3 (1925);A. Griffin, D. W. Snoke. A Stringari, Eds., Bose Elnstein Condensation (Cambridge Univ Press, Cambridge, 1995). 2. K. Huang, Statistical Mechanics 2nd Edition ( Wiley, New York. 1987). whereh is Planck's constant, m 3. A,, = h/(2~rmkT)'/~. is the mass of the atom, k is Boltrmann's constant, and T is the temperature 4. V Bagnato, D. E. Pritchard, D. Kleppner. Phys Rev. A 35,4354 (1987) 5 J -L. Lin and J P Wolfe, Phys Rev. Lett. 71, 1222 (1993). 6. For reviews of the hydrogen work, see T J. Greytak and D Kleppner, in New Trends !n Atomic Physics, Proceedings oi the Les Houches Summer School, Session xxxV///,Les Houches. France, 2 to 28 June 1993, G. Greenberg and R. Stora, Eds. (North-Holland, Amsterdam, Netherlands, 1984), pp 1 1 271158, I. F Silvera and J T M. Walraven, in Progress in Low TemperaturePhysics, D. Brewer, Ed. (NorthHolland, Amsterdam, Netherlands, 1986). vol. 10. pp 139-173. T. J. Greytak, in Bose Einste!n Condensation, A. Griffin. D. W Snoke. A Stringari, Eds (Cambridge Unrv Press, Cambridge, 1995). pp. 131-1 59 7. N. Masuhara eta/., Phys Rev. Lett. 61, 935 (1988); 0 J Luiten eta/.. ibid 70, 544 (1993). H. F. Hess. Phys Rev. 5 34, 3476 (19861. 8. C. Wieman and S Chu. Eds.. J Opt. SOCAm B 6 (no 11) (1989) (special issue on laser cooling and trapping of atoms; in particular, see the "Optical Molasses" section). 9 D. Sesko, T. Walker, C. Monroe, A. Gallagher. C. Wieman. Phys Rev. Lett. 63, 961 (1989). T Walker, D Sesko, C. Wieman. ibid 64,408 (1990). D Sesko. T. Walker, C. Wieman. J. Opt. Soc Am 5 8, 946 (1991) 10 C. Monroe, W Swann, H Robinson, C.Wieman. Phys Rev Lett. 65, 1571 (1990). 1 1 C Monroe, E. Cornell. C. Wieman. in proceedings of the Enrico Fermi International Summer School on Laser Manipulation of Atoms and Ions, Varenna, Italy, 7 to 21 July 1991, E Arimondo, W Phillips. F Strumia. Eds (North-Holland, Amsterdam, Netherlands, 1992), pp. 361-377.
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12. C. Monroe, E. Cornell, C. Sackett, C . Myatt. C. Wieman, Phys. Rev. Lett 70,414 (1993); N. Newbury, C. Myatt, C. Wieman, Phys. Rev. A 51, R2680 (1995). 13. W. Petrich. M H. Anderson, J R. Ensher, E. A. Cornell. Phys.. Rev. Lett. 74, 3352 (1995). 14. D. Pritchard e l a/., in Proceedings of the 1 lth international Conierence on Atomic Physics, s. Haroche, J C. Gay, G. Grynberg, fds. (World Scientific. Singapore, 1989). pp. 619-621. The orbiting zero-field point in the TOP trap supplements the effect of the rf by removing some high-energy atoms by Majorrana transitions ,Two other groups have evdporatively cooled alkali atoms [C. S Adams, H. J. Lee, N. Davidson, M. Kasevich, S.Chu, Phys. Rev Lett. 74. 3577 (1995); K. 6. Davis, M - 0. Mewes, M A Joffe, M R. Andres, W. Ketterle. ibid., p. 52021. 15 E. Raab, M. Prentiss, A. Cabie, S Chu. D E Pritchard, Phys. Rev. Lett. 59, 2631 (1987). 16. W. Ketterle. K. B. Davis, M. A. Joffe, A. Martin, D. E. Pritchard, ibid. 70, 2253 (1993). I7 M. H. Anderson, W Peterich, J. R Ensher, E. A. Cornell, Phys Rev. A 50, R3597 (1994). 13 W. Petrich. M. H. Anderson, J. R. Ensher, E. A. Cornell, J. Opt. Soc. Am. 5 11, 1332 (1 994). 19. J. R. Gardneretd. [Phys. Rev Lett. 74,3764 (1995)l determined the ground-state triplet scattering lengths and found that they are positive for8'Rb and negative for W b . It is believed that a positive scattering length is necessary for the stability of large samples of condensate. The F = 2. inF= 2 state also has the advantage that, of the Rb 5 s states, it is the spin state with the maximum magnetic trapping force. 20 Afterthe rotating field has been reduced to one-third its initial value, which increases the spring constant by a factor of 3, it is held fixed (at 5 G) and the final cooling is done only with the rf ramp. 21 After the sample is cooled to just below the transition temperature, the condensate peak does not appear immediately after the ramp ends but instead grows during this 2-s delay. 22 This exact correspondence between velocity and coordinate-spacedistributionsrequires that the particles bean ideal gas, which is an excellent approximation in our system, except in the condensate itself. It also requires that sinusoidal trajectoriesof the atoms have random initial phase. This is much less restrictivethan requiring thermal equilibrium. 23 Below the transition temperature the fraction of the atoms that go into the condensate is basically set by the requirement that the phase-space density of the noncondensate fraction not exceed 2.612 (for an ideal gas). As the cloud is further cooled or compressed, the excess atoms are squeezed into the condensate (2). 24 The temperature of a classical gas that would correspond to the kinetic energy of the pure condensate cloud (veVap= 4.1 1 MHz), after adiabatic expansion, is only 2 nK, and during the near-ballisticexpansion it becomes substantially lower. 25 E. Tiesinga, A. J. Moerdijk, B. J. Verhaar. H. T. C. Stoof, Phys. Rev. A 46, R1167 (1992) 26 E Tiesinga, B. J Verhaar, H. T C Stoof, ibid. 47, 4114 (1993). 27 C. Myatt, N Newbury, C. Wieman, personal communication. 28 L. You, M. Lewenstein, J. Cooper, Phys. RevA 51, 4712 (1995) and references therein; 0. Morice, Y. Castin, J. Dalibard. ibid., p. 3896; J. Javanainen, Phys. Rev. Lett. 72,2375 (1994); B. V. Svistunovand G. V. Shylapnikov,JETP 71, 71 (1990). 29 P A Ruprecht, M J. Holland, K Burnett, M. Edwards, Phys. Rev A. 51,4704 (1995). 30 During the adiabatic stage of expanston, we already routinely changed the sample density by a factor of 25 31 We thank K. Coakley, J. Cooper, M. Dowell, J. Doyle, S. Gilbert, C. Greene, M. Holland, D. Kleppner. C. Myatt, N Newbury, W. Petrich, and B. Verhaar for valuable discussions. This work was supported by National Science Foundation, National Institute of Standards and Technology, and the Office of Naval Research 26 June 1995, accepted 29 June 1995
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VOLUME77, NUMBER3
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Collective Excitations of a Bose-Einstein Condensate in a Dilute Gas D. S. Jin,” J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell* Joint Institute for Laboratoiy Astrophysics, National Iiistitute ofStandard,r and T e d i t i o l o ~ and University of Colorado, Boulder, Colorado 80309-0441) and Plysics Depai-tnient, University of Colorado, Boulder, Colorado 80309-0440 (Received 23 May 1996) We observe phononlike excitations of a Bose-Einstein condensate (BEC) in a dilute atomic gas. ”Rb atoms are optically trapped and precooled, loaded into a magnetic trap, and then evaporatively coolcd through the BEC phase transition to form a condensate. W e cxcite the condensate by applying an inhoniogcneous oscillatory force with adjustable frequency and symmetry. W e have observed modes with different angular momenta and different encrgies and havc studied how their characteristics depend on ,interaction energy. W e find that the condensate excitations persist longer than their counterparts in uncondensed clouds. [SO03 1 -9007(96)00722-31 PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj
An important consequence of quantum statistics is that, below some critical temperature, bosom are predicted to pile up in the lowest energy state of a system [l]. This macroscopic quantum phenomenon, termed Bose-Einstein condensation (BEC), has recently been observed in a dilute atomic vapor [2,3]. This condensation provides the basis for theoretical understanding of unusual quantum phenomena such as superfluidity in liquid helium and superconductivity in metals. In these systems, however, strong interactions among the constituent particles fundamentally alter the features of the BEC and considerably complicate any theoretical analysis. In dilute atomic vapors the interactions are primarily binary and can be treated theoretically within an s-wave scattering approximation. The exact shape of the interparticle potential is ignored and a single parameter, a, the s-wave scattering length, characterizes the interactions. Thus, even with interactions, the system is amenable to theoretical understanding. We report herein on studies of low energy, collective excitations of a dilute condensate of ”Rb. Characterizing these lowlying excitations is a first step towards understanding the dynamics of this novel quantum fluid. The apparatus and procedures we use for creating BEC are described elsewhere [2,4]. In summary, we optically precool and trap 87Rbatoms, then load them into a purely magnetic trap‘. We use a time-averaged orbiting potential (TOP) trap consisting of a static quadrupole field plus a small rotating transverse bias field [4]. The effective average potential is axially symmetric and harmonic, with a ratio of axial to radial (or equivalently, “transverse”) We further cool the atoms trapping frequencies of in the TOP trap with forced evaporation by applying a radio frequency (rf) magnetic field to selectively induce Zeeman transitions to untrapped spin states [5]. We observe the atom cloud with absorption imaging. A 26 ps pulse of light resonant with the SS1/2, F = 2 to S P 3 / 2 , F = 3 transition illuminates the atoms, while a lens system with a resolution of 6 p m FWHM images
the shadow of the cloud onto a charge-coupled device array. The image is then digitized and stored for analysis. The probe geometry is such that we get both radial and axial sizes from a single image. By allowing the cloud to expand ballistically before taking the picture, we improve the imaging resolution [2]. Improvements in the procedure used in Ref. [2] now allow us to turn off the trap potential rapidly enough to allow free expansions from traps with any radial frequency between 9 and 308 Hz [6]. Our imaging procedure is destructive, but by repeating the cycle of loading, cooling, and imaging, we study time evolution of the cloud. The final evaporation takes place in a trap with a radial frequency of 132 Hz (373 Hz axial). To approach the zero-temperature limit, we evaporate well below the BEC phase transition [7] so that the expanded clouds show no sign of having a thermal component. At higher temperatures, this component appears as a broad, symmetric Gaussian background [2]. Typically we have 4500 ? 300 atoms in the condensate. The standard theory for BEC in a dilute atomic vapor uses a nonlinear Schrodinger equation, the GrossPitaevskii equation, to describe the condensate wave function in the limit of zero temperature [8]. This equation comes from a second-quantized mean-field theory with the interatomic interactions modeled by an s-wave scattering length. The elementary excitations of BEC in a finite, harmonically confined dilute gas have been studied theoretically using this model r9-121. The lowest energy normal modes of the condensate, corresponding to rigidbody center-of-mass motion (“sloshing”), are predicted to occur at the trap frequencies. The frequencies of the next lowest condensate excitation modes, however, are expected to deviate from the spectrum of a cloud of ideal gas, for which the excitation frequencies are simply multiples of trap frequencies. Not surprisingly, the amount of deviation depends on the strength of the interatomic interactions.
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PHYSICAL REVIEW LETTERS
We excite these collective modes of the condensate by applying a small time-dependent perturbation to the transverse trap potential. We generate the perturbations by applying a sinusoidal current to the coils responsible for the rotating field of our TOP trap (in addition to the normal TOP currents). The response of the condensate depends on the symmetry of the driving force as well as the driving frequency ud. By appropriately setting the phases of the currents through the coils, we can generate perturbations with either of two different symmetries. We label these two driving symmetries m = 0 and m = 2, where m , the angular momentum projection onto 2 , is a good quanhim number because of the axial symmetry of our unperturbed magnetic trap. Equipotential contours for the two trap perturbations are shown in Fig. 1. The rn = 0 drive preserves axial symmetry and corresponds to an oscillation in radial size. For the m = 2 drive symmetry, the trap spring constants along 2 and 9 are modulated 90" out of phase. This corresponds to a normal mode resembling a transverse ellipse whose major axis rotates in the x - y plane. The basic spectroscopic approach is as follows: We distort the cloud by applying the perturbative drive for a short time, then allow the cloud to evolve freely in the unperturbed trap for a variable length of time. Finally, we turn off the confining potential suddenly and image the resulting cloud shape after 7 ms of free expansion. Initial studies were made in a 132 Hz trap. The perturbative drive pulse duration was 50 ms, the center frequency was set to match the frequency of the excitation being studied, and the amplitude was 1.5% of the radial spring constant of our trap. We observe two different collective excitations of the condensate. The observables in both cases are the widths of the expanded clouds as a hnction of the free evolution time. In one case we observe a sinusoidal oscillation of the radial width at a frequency of (1.84 -+ O.OI)v,, where v, is the radial trap frequency and the error quoted reflects only
statistical uncertainties. This mode is driven by the 0 trap perturbation and is not axially symmetric rn observed for the drive with nz = 2 symmetry (with the same drive amplitude and frequency). The cloud widths oscillate in both axial and radial directions, with approximately opposite phase. This response is shown in Fig. 2. The observed phase difference between the oscillations of axial and radial widths is not exactly n-; however, the free expansion of the condensate prior to imaging complicates analysis of this phase shift [ 131. The second excitation oscillates freely at (1.43 2 O.Ol)v,., and appears in response to an rn = 2 drive, and not to an nz = 0 drive. In this case, the radial width oscillates, with no observable response in the axial width. The two-dimensional projection of an elliptical cloud whose major axis rotates in the transverse plane would exhibit this behavior. We calibrate the observed excitation frequencies in units of the trap frequencies by making similar measurements on noncondensate clouds. The temperature of these clouds, in units of the BEC transition temperature, is T / T c -- 1.3; consequently, the density and correspondingly the interactions are very small. Here, we see a response that oscillates at 264 Hz and can be driven with either symmetry. A harmonically confined, noninteracting gas pulses at twice the radial trap frequency, so this gives v, = 132 2 1 Hz. We have also checked that the thermal cloud does not respond when driven at I .43 v,.. In the s-wave scattering approximation, the interactions for ground state 87Rb atoms in our trap are repulsive (positive scattering length) [ 141, providing an effective potential energy which favors a lower central density of
E
,
1
,
, ra;i;
~
A
;ial,
1
,
1
14
12
5
10
15
20
time (ms) FIG. 1. In the unperturbed trap, contours of equipotential in
the transverse plane are symmetric (solid line). To drive the m = 0 excitation (a) we apply a weak harmonic modulation with frequency v d to the trap radial spring constant. The m = 2 drive (b) breaks axial symmetry with elliptical contours which rotate at vd/2. The amplitude of perturbation is shown exaggerated for clarity.
FIG. 2. We apply a weak rn = 0 drive to an N = 4500 condensate in a 132 Hz (radial) trap. Afterward, the freely evolving response of the condensate shows radial oscillations. Also observed is a sympathetic response of the axial width, approximately 180" out of phase. The frequency of the excitation is determined from a sine wave fit to the freely oscillating cloud widths. Each data point represents a single destructive condensate measurement.
42 1
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the Condensate compared to the noninteracting case. This interaction energy determines the excitation spectrum of the condensate [15]. We can examine this effect because BEC in trapped neutral atoms offers the advantage of an adjustable interaction energy. In the standard meanfield picture, the strength of the nonlinear interaction term in the Gross-Pitaevskii equation, relative to the harmonic trap’s energy-level spacing, scales with N a J v , [16]. Thus, by varying the trap frequency or the number of atoms N we can change the relative importance of interactions in the condensates. We measure the excitation fi-equencies of the n z = 0 and m = 2 modes of the condensate as a function of both N and v,.. In the 132 Hz trap, we change the relative interaction strength by reducing the number of atoms. To change v,.,we evaporate to BEC in the 132 Hz trap, then adiabatically ramp the trap fields until the condensate is held in a trap with an axial frequency of 43.2 Hz. In this lower frequency trap, we excite the condensate with a 100 ms pulse and a drive amplitude equal to 3% of the radial spring constant. The observed fractional amplitude of the oscillations in the cloud width (approximately 11% of the mean width) for this drive is the same as observed in the 132 Hz trap. We measure the free oscillation frequency of the M = 0 and rn = 2 modes in this trap to be (1.90 f 0 . 0 1 ) ~and ~ (1.51 t O.Ol)v,, respectively, for N = 3000. The measured excitation frequencies as a function of interaction strength are shown in Fig. 3. By using the product N J V , for the dependent variable we combine our different number and trap frequency data into one graph. The solid lines in Fig. 3 show the mean-field the-
.................................................. I ......._--......
ory calculation by Edwards ef a/. [9], using the current best value of the scattering length for ground state ”Rb atoms, n = 1 I O n o [14], where LZO is the Bohr radius. An extension of this calculation by Esry and Greene is shown with dotted lines [lo]. Finally, dashed lines indicate the prediction by Stringari for the “strongly interacting” limit [ I I ] , in which the kinetic energy of the ground state is ignored. Our data agree reasonably well with these mean-field theory results; the measured energies of the low-lying collective excitations of the condensate deviate from the simple harmonic trap spectruin as predicted, with larger deviation for larger interaction strength. Error bars in Fig. 3 indicate statistical error in the determination of the frequencies, but do not include possible systematic errors such as day-to-day variations in the trap magnetic fields, and therefore frequencies (estimated to be less than 0.50/;,). Also, the theoretical curves are strictly valid only in the limit of zero temperature and zero amplitude. In the limit of low energy, the spectrum of lowlying collective excitations corresponds exactly to the Bogoliubov quasiparticle spectrum [9]. The collective condensate response to our trap perturbation, in the limit of low amplitude, is simply a coherent state of these elementary excitations. To explore the question of whether or not our experiments are performed in this limit, we measure the condensate response for different driving force amplitudes. In this test, we drive the m = 2 mode in the 132 Hz trap, with N -- 4500. The results are shown in Fig. 4 where we plot the frequency of the oscillating radial width as a function of the amplitude of that response. The solid line shows a fit with a parabola, a form which describes an oscillator with anharmonic terms. As our measurements of excitation frequency were performed for a response amplitude between 9% and 1494,
220
/ 1.6
210 -
m=2 excitation
................................................. =
1.4
1.21 0
. I ..-............
200 2
4
6
lo4 Nv ~ ’ ’ ~ FIG. 3. We measure the frequency of the m = 0 (triangles) and m = 2 (circles) condensate modes as a function of interaction strength. The relative interaction strength in the condensate varies as the product of number of atoms, N , and the square root of the radial trap frequency, vr. Solid lines show the mean-field calculation by Edwards and co-workers [9], dotted lines show the results of similar calculations by Esry and Greene [lo], and dashed lines show the prediction by Stringari for the strongly interacting limit [ l l ] .
422
20
40
60
Amplitude (%) FIG. 4. The freely oscillating frequency of the condensate is shown as a function of response amplitude. The condensates, consisting of 4500 atoms, are held in a 132 Hz radial frequency trap and driven with m = 2 symmetry. The solid line shows a parabolic fit to the data.
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which causes a shift of only 1% in the frequency, these data suggest that we are in the regime where the measured spectrum corresponds quite closely to the elementary excitations of BEC in a dilute gas. Finally, we examine the damping of a condeiisate excitation. For comparison purposes we first study the damping in a noncondensed thermal cloud (TIT,. -- 1.3) in the 132 Hz trap. We excite the 264 Hz m = 0 mode, because damping in this mode is not influenced by angular momentum conservation. We fit a sine wave with an exponentially decaying amplitude to the observed oscillations in the radial cloud width. This gives an excitation lifetime of 49 2 13 ins. Since the mean free path in these clouds is long compared to the excitation wavelength and the effect of the trap anharmonicity is small, the excitation lifetime should scale inversely as the atom-atom collision rate. This rate in tun1 scales with the product of the density times the velocity of the atoms. For a given harmonic oscillator confining potential, this collision rate is proportional to the optical depth of the cloud. Using this scaling principle we predict that the damping lifetime in a classical cloud with the same optical depth as the condensate would be 28 t 8 ms. But when we perform the same experiment on the 4500 atom condensate we obtain an excitation lifetime of 110 2 25 ms. Thus, the condensate excitation persists nearly four times longer than can be explained in a classical picture. In summary, we have observed low-lying collective excitations of BEC in a dilute atomic vapor. Both rn = 0 and rn = 2 modes were identified, and their frequencies measured as a function of relative interaction strength. The data were taken in a linear regime where the collective modes should correspond to the elementary excitations of BEC in this system, and reasonable agreement was found between the experiment and mean-field theory results. The damping lifetime of the m = 0 excitation was measured and found to be significantly longer than the prediction of a classical model. We believe further study of these elementary excitations, particularly at different temperatures, will help deepen the understanding of the quantum phenomena of Bose-Einstein condensation of a gas. This work is supported by the National Institute of Standards and Technology, the National Science Foundation, and the Office of Naval Research. The authors express their appreciation for many useful discussions with the other members of the JILA BEC Collaboration, and with K. Burnett, M. Edwards, B. Esry, M. Levenson, D. Rokhsar, and S. Stringari. One of us (D.S. J.)
acknowledges Council.
15 JULY 1996
support from the National Research
*Quantum Physics Division, National Institute of Standards and Technology. [I] K. Huang, Statistical Mechanics (Wiley, New York, 1987), 2nd ed. [2] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E. A. Cornell, Science 269, I98 (1995). [3] K. B. Davis et al.. Phys. Rev. Lett. 75, 3969 (1995). [4] W. Petrich, M. H. Anderson, J . R. Ensher, and E.A. Cornell, Phys. Rev. Lett. 74, 3352 (1995). [5] D. Pritchard Ct al., in ProceediMgs of the I l t h International Corlfhrence on Atomic Physics. edited by S . Haroche, J.C. Gay, and G. Grynberg (World Scientific, Singapore, 1989). pp. 619-621; H. F. Hess et al., Phys. Rev. Lett. 59, 672 (1987). [6] The release of the atoms for ballistic expansion is initiated by a sudden increase in the transverse bias field, which quenches the confining forces on a time scale fast compared to any cloud dynamics. Interactions and residual kinetic energy cause the cloud to fly apart. The error in the final velocity determination due to residual magnetic fields is less than 2%. [7] The remaining noncondensate atoms represent less than 20% of the sample. We find good agreement between the calculated and measured BEC transition temperatures; these data will be presented in a future paper. [8] A. L. Fetter and J. D. Walecka, Quantum Theory ofMunyParticle Systems (McGraw-Hill, New York, 197 1). [9] M. Edwards, P.A. Ruprecht, K. Burnett, R.J. Dodd, and C.W. Clark (to be published); P.A. Ruprecht, M. Edwards, K. Burnett, and C.W. Clark, Phys. Rev. A (to be published). 101 B. D. Esry and C.H. Greene (private communication). 111 S. Stringari, Report No. cond-mat/9603126. 121 A. L. Fetter, Czech. J. Phys. (to be published); K. G. Singh and D. S. Rokhsar (to be published). 131 A connection between the phase of the condensate perturbation and the ultimate width of the expanded cloud may be obtained by numerical integration of the nonlinear Schrodinger equation. See, for example, P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995); M. Holland and J. Cooper, Phys. Rev. A 53, 1954 (1996). 141 J.R. Gardner et al., Phys. Rev. Lett. 74, 3764 (1995); H. M. J. M. Boesten, C.C. Tsai, D. J. Heinzen, and B. J. Verhaar (private communication). [I51 N.N. Bogoliubov, J. Phys. USSR 11, 23 (1947). [I61 V. V. Goldman, I. F. Silvera, and A. J. Leggett, Phys. Rev. B 24, 2870 (1981); G. Baym and C. J. Pethick, Phys. Rev. Lett. 76, 6 (1996); F. Dalfovo and S. Stringari, Phys. Rev. A 53, 2477 (I 996).
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Bose-Einstein Condensation in a Dilute Gas: Measurement of Energy and Ground-State Occupation J. R. Ensher, D. S. Jin,* M. R. Matthews, C. E. Wieman, and E. A. Cornell* JILA, National Institute of Standards Technology and University of' Colorado and P11,vsics Department, University of Colorado, Boulder. Colorado 80309-0440 (Received 4 September 1996) We measure the ground-state occupation and energy o f a dilute Bose gas of "Rb atoms as a function of temperature. The ground-state fraction shows good agreement with the predictions for an ideal Bose gas in a 3D harmonic potential. The measured transition temperature is 0.94(5)T,,, where T, is the value for a noninteracting gas in the thennodynamic limit. We determine the energy from a modelindependent analysis of the velocity distribution, after ballistic expansion, of the atom cloud. W e observe a distinct change in slope of the energy-temperature curve near the transition, which indicates a sharp feature in the specific heat. [SO03 1-9007(96)01891-11 PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 5 I .30.+i
The ability to create Bose-Einstein condensation (BEC) in magnetically trapped alkali gases [l-41 provides an opportunity to experimentally study the thermodynamics of bosonic systems in which the interactions are (i) weak, (ii) binary, and (iii) experimentally adjustable [5-71. One goal of experimental and theoretical work in this field is to understand a variety of low-temperature phenomena from both macroscopic and microscopic points of view, with a quantitative reconciliation of these two approaches. The recent experimental studies of collective excitations of zero-temperature condensates [5,8] were a step in this direction. The purpose of the present effort is to explore the nature of the BEC phase transition by performing quantitative measurements of BEC in a different regimenear the critical temperature. In this paper we analyze a series of images of ultracold clouds of rubidium gas to determine the critical temperature and to extract ground-state occupation and mean energy as a function of temperature. Mean energy (or its derivative, specific heat) has a certain historical significance because it was London's comparison [9] of the specific heats of liquid helium and an ideal Bose gas that began the rehabilitation of BEC as a useful physical concept. Moreover, measurements of thennodynamic quantities such as specific heat are essential in studying any phase transition. Ground-state occupation and critical temperature of a Bose gas are interesting because in liquid helium the former is very difficult to measure, while the latter is almost impossible to calculate accurately. The apparatus and procedures we use for creating an ultracold Bose gas and BEC are described elsewhere [1,10,11]. In summary, we optically trap [12] and precool [13] 87Rb atoms, then load [14] them into a purely magnetic trap. We use a time-averaged orbiting potential (TOP) trap consisting of a static quadrupole field plus a small rotating transverse bias field [I I]. The effective potential is axially symmetric and harmonic, with a ratio of axial to radial trapping frequencies of A. We further
cool the atoms in the TOP trap with forced evaporation [I51 by applying a radio frequency (rf) magnetic field which induces Zeeman transitions of the most energetic atoms to untrapped spin states [16]. By ramping down the frequency of the rf field we control the cloud temperature. The final stages of evaporative cooling are performed in a v z = 373 Hz trap. We observe the atom cloud, after a period of free expansion, with resonant absorption imaging [5]. The confining TOP potential is turned off suddenly, and the cloud of atoms is allowed to expand freely for 10 ms. The cloud is then probed by a 26 ,us pulse of light resonant with the 5S112, F = 2 to 5P312, F = 3 transition. The atoms scatter photons, impressing a shadow onto the probe beam. The shadow is imaged onto a CCD array, and the data are digitally processed to extract the optical depth of the cloud at each point. After point-by-point corrections for imperfect polarization and saturation effects, the result is a 2D projection of the velocity distribution in the expanded cloud. These distributions contain a wealth of thermodynamic information. For instance, the integrated area under the distribution is proportional to the total number N of atoms in the sample. The condensate appears as a narrow feature centered on zero velocity [ 11; the number of atoms in the ground state, N o , is then proportional to the integrated area under this feature. From the mean square radius of the expanded cloud and the expansion time, we get the mean square velocity, or average energy, of the cloud. Finally, as discussed below, the temperature T is extracted from the images, even though the temperature is not merely proportional to mean energy in a degenerate Bose cloud. We have gone to some lengths to extract thesc thermodynamic quantities in a model-independent way. For instance, if we were to fit the observed velocity distributions to a Bose-Einstein distribution, we could hardly avoid coming to the conclusion that the specific heat is
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0 1996 The American Physical Society
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discontinuous-the singular behavior is built-in to the assumed functional form. Moreover, such a fit would preclude our being able to observe effects due to interactions, finite N,critical fluctuations, etc. Fortunately, useful thermodynamic information about the sample can be extracted froin direct calculation of various moments of the velocity distribution, without specific reference to the nature ofthe distribution. The total number and energy of the atoms, as mentioned above, are simply proportional to the zeroth and second moments, respectively, calculated directly by summing over the velocity distribution images [ 171. We define the number of atoms in the ground state to be the number of atoms contributing to the narrow, central feature in the optical depth images [l]. To avoid biased and noisy results we provide the least-squares fitting routine with a tightly constrained template to use in its search for a condensate. With an independent set of measurements on condensates near zero temperature, we have found that the condensate shapes are well-fit with 2D Gaussians whose widths, aspect ratios, and peakheights, for a given trap frequency and expansion time, are functions only of the total number of atoms in the feature [7,18]. The width, for instance, is parametrized by u = a,(l + a N , ) 1 / 5 ,where a. is the predicted noninteracting condensate width and a is extracted empirically. The procedure yields robust values of N,, as long as the temperature is high enough that the noncondensate atoms form a distribution that is significantly broader than the sharp condensate feature. At temperatures below TIT, = 0.5, both T and N o measurements become suspect, as it is no longer possible to cleanly separate the condensate and the noncondensate components without recourse to a detailed model, which is contrary to the spirit of this treatment. Our thermometry differs from previously reported methods for ultracold trapped gases [I-3,5581. For an ideal gas far from quantum degeneracy the velocity distribution is a Gaussian whose width is proportional to As the cloud is cooled closer to the BEC phase transition, higher densities and lower temperatures cause a rapid increase in the significance of quantum statistics and of residual atom-atom interactions. Rather than attempt to model these effects, we assume that the high-energy tail of the velocity distribution (i) remains in thermal equilibrium with the rest of the cloud and (ii) can be characterized by a purely ideal Maxwell-Boltzmann (MB) distribution. The latter is plausible because these highest-energy atoms spend most of their trajectories in the low-density, and therefore weakly interacting, outer part of the trapped cloud. Furthermore the occupation numbers of the corresponding energy states are much less than one. Finally, during the free expansion the highenergy atoms undergo on average much less than one collision. Guided by these assumptions, we detemine the temperature by fitting a 2D Gaussian to only the wings of our velocity-distribution images, excluding the central
16 DECEMBER 1996
part of the cloud, where degeneracy, interactions, fluctuations, etc. may be significant. To test the assumption of thermal equilibrium we have checked that the measured temperature of clouds is independent of the size of the exclusion region, outside of the degenerate regime [ 191. The first quantity we examine is the ground-state fraction N , / N as a fiinction of scaled temperature TIT,, (Fig. 1). The temperature scaling removes the trivial shift in the transition temperature which occurs because as we evaporatively cool through the transition we also reduce the total number of atoms N (Fig. 1, inset). We choose our scaling temperature to be T , ( N ) = t i W / k ~ [ N / [ ( 3 ) ] ' / ~ where i 5 is the geometric mean of the trap frequencies and is the Riemann Zeta filnction. T , ( N ) is also the critical temperature, in the thennodynamic limit, for noninteracting bosons in an anisotropic harmonic potential [20,21]. For this case the temperature dependence of the groundstate fraction is N , / N = 1 - ( T / T , ) 3 below T, (solid line, Fig. 1). We emphasize that, in contrast with the recent work of Mewes ef al. [6], this line contains no free parameters and is not fit to the data, and so comparing this line to our data provides a detailed test of theory. From our data we find a critical temperature of T, = 0.94(5)T11. The uncertainty is dominated by the systematic uncertainty in our measurement of the scaled temperature stemming mostly from a 2% uncertainty in the magnification of our imaging system. Our measurements are thus only marginally different from the theory for noninteracting bosons in the thermodynamic limit. Finite number corrections [22] will shift the transition temperature T , ( N ) down about 3%
z"
0.0
0.5
1.o
1.5
T 1 T, (N) FIG. 1. Total number N (inset) and ground-state fraction N , / N as a function of scaled temperature TIT,. The scale temperature T , ( N ) is the predicted critical temperature, in the thermodynamic (infinite N ) limit, for an ideal gas in a
harmonic potential. The solid (dotted) line shows the infinite (finite) N theory curves. At the transition, the cloud consists of 40000 atoms at 280 nK. The dashed line is a leastsquares fit to the form N , / N = 1 which gives T, = 0.94(5)T0. Each point represents the average of three separate images. 4985
463 VOLUME77, NUMBER25
(dotted line, Fig. I ) . Mean-field [21,23] and many-body [24] interaction effects may also shift T , ( N ) a few percent. The second result we present is a measurement of the energy and specific heat. Ballistic expansion, which facilitates quantitative imaging, also provides a way to measure the energy of a Bose gas [6,7,18]. The total energy of the trapped cloud consists of harmonic potential, kinetic, and interaction potential energy contributions, or Epot,Eki,,, and Eint.respectively. As the trapping field is nonadiabatically turned-off to initiate the expansion, Epot suddenly vanishes. During the ensuing expansion, the remaining components of the energy, Ekin and Einr, are then transformed into purely kinetic energy E of the expanding cloud: Ekin + Eint E , where E is the quantity we actually measure. According to the virial theorem, if the particles are ideal (Eint = O), E will equal half the I total energy, i.e., E = zE,'$"'. However, for a system with interparticle interactions the energy per particle due to Eintcan be non-negligible and then E = u E t o t ,where I a is not necessarily 7. The scaled energy per particle, EINkBT,, is plotted versus the scaled temperature T I T , in Fig. 2. E / N is normalized by the characteristic energy of the transition ksT,, ( N )just as the temperature is nonnalized by T o . The data shown are extracted from the same cloud images as those analyzed for the ground-state fraction. Above T , , the data tend to the straight solid line which corresponds to the classical MB limit for the kinetic energy. Most interesting is the behavior of the gas at the transition. By
-
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PHYSlCAL REVIEW LETTERS
examining the deviation A of the data from the classical line we see (Fig. 2, inset) that the energy curve clearly changes slope near the empirical transition temperature 0.94T, obtained from the ground-state fraction analysis discussed above. The specific heat is usually defined as the temperature derivative of the energy per particle, taken with either pressure or volume held constant. In our case the derivative is the slope of the scaled energy vs temperature plot (Fig. 2), with neither pressure nor volume, but rather confining potential held constant. To place our measurement in context, it is instructive to look at the expected behavior of related specific heat vs temperature plots (Fig. 3). The specific heat of an ideal classical gas (MB statistics), displayed as a daslied line, is independent of temperature all the way to zero temperature. ldeal bosons confined in a 3D box have a cusp in their specific heat at the critical temperature (dotted line) [9]. Liquid 'He can be modeled as bosons in a 3D box, but the true behavior is quite different from an ideal gas, as illustrated by the specific heat data [25] (dot-dashed line): The critical (or lambda) temperature is too low, and the gentle ideal gas cusp is replaced by a logarithmic divergence. We can compare our data with the calculated specific heat of ideal bosons in a 3 0 anisotropic simple harmonic oscillator (SHO) potential [20] (solid line). Note that because we do not measure Epot,we must divide the SHO theory values by two to compare with our measured expansion energies. The specific heat of the ideal gas is discontinuous and finite at the transition. In order to extract a specific heat from our noisy data, we assume that, as predicted, there is a discontinuity in the
'
5 m 4 Y
Z \ 3 0 2 I
I -*
0.4
._--
0.0 0.4
"
0.6
"
"
'
1.2
0.8 I
.
'
1.6
"
'
0.8 1.0 1.2 1.4 1.6 1.8 T 1 To(N)
FIG. 2. The scaled energy per particle E / N k n T , of the Bose gas is plotted vs scaled temperature TIT,. The straight, solid line is the energy for a classical, ideal gas, and the dashed line is the predicted energy for a finite number of noninteracting bosons [22]. The solid, curved lines are separate polynomial fits to the data above and below the empirical transition temperature of 0.94T,. (inset) The difference A between the data and the classical energy emphasizes the change in slope of the measured energy-temperature curve near 0.94T0 (vertical dashed line).
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0 ' . 0.6 0.8
1.0
'
1.2
"
1.4
I
FIG. 3. Specific heat, at constant external potential, vs scaled temperature T I T , is plotted for various theories and experiment: theoretical curves for bosons in a anisotropic 3D harmonic oscillator and a 3D square well potential, and the data curve for liquid 4He [ 2 5 ] . The flat dashed line is the specific heat for a classical ideal gas. (inset) The derivative (bold line) of the polynomial fits to our energy data is compared to the predicted specific heat (fine line) for a finite number of ideal bosons in a harmonic potential.
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slope at the empirically determined transition temperature and fit the data to separate polynomials on either side of T , (curved, solid lines in Fig. 2). We extract the specific heat curve shown in Fig. 3 (bold line in inset). The observed step in the specific heat at the critical temperature is considerably smaller than predicted by a finite number. ideal gas theory [22] (Fig. 3, inset, thin line). A more sensible comparison is to avoid taking the modeldependent derivative and instead to compare theory and experiment directly in the energy-temperahire plot (Fig. 2). The major deviation between the data and the SHO ideal gas theory (dotted line) occurs at scaled temperatures of 0.85 and below. The difference is probably due in part to the effects of interactions. Mean-field repulsion will tend to increase the energy at a given temperature. We have measured the critical temperature, groundstate occupation, and energy of a dilute Bose gas of s7Rb atoms. Our analysis is unique in that it does not rely on detailed models of the quantum degenerate cloud shape. We are thus able to examine the thermodynamics of the Bose gas in an unbiased and quantitative way. The measured ground-state fraction and transition temperature agree well with the theory for noninteracting bosons. However, the qualitative features of the energy data are significantly different from the noninteracting theory. In future work we will attempt to elucidate the role interactions play in the phase transition and the specific heat. For example, we can control the interactions by adjusting the magnetic trap spring constants and changing the number of trapped atoms [ 5 ] . In addition, with larger clouds [4] we can reduce our uncertainty in T,, allowing us to investigate finite number and mean-field effects at the 1% level. This work is supported by the National Institute of Standards and Technology, the National Science Foundation, and the Office of Naval Research. The authors express their appreciation for many useful discussions with the other members of the JILA BEC Collaboration, and with M. Levenson, H. Stoof, and S. Stringari.
*Quantum Physics Division, National Institute of Standards and Technology. [I] M. H. Anderson et al., Science 269, 198 (1995). [2] K.B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995). 131 C.C. Bradley, C.A. Sackett, and R.G. Hulet (to be published).
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[4] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman (to be published). [5] D.S. Jin et al., Phys. Rev. Lett. 77, 420 (1996). [6] M.-0. Mewes e t a / . , Phys. Rev. Lett. 77, 416 (1996). [7] D. S. Jin et a/.. Czech. J. Phys. 46, Suppl. S6 (1996). [8] M.-0. Mewes et al., Phys. Rev. Lett. 77, 988 (1996). [9] F. London, Nature (London) 141, 643 (1938). [ 101 H. Rohner, in Proceedings of the 39th ~,V~17pOSiZ/777on the Art of’ Ghssh/owing (American Scientific Glassblowers Society, Wilinington, Delaware, 1994), p. 57. [ l l ] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Phys. Rev. Lett. 74, 3352 (1995). [I21 E. L. Raab et al.. Phys. Rev. Lett. 59, 2631 (1987). [I31 Special issue on laser cooling and trapping of atoms, edited by S. Chu and C. Wieman [J. Opt. Soc. Am. B 6 (1989)l. [I41 C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571 (1990). [I51 H. F. Hess et a/., Phys. Rev. Lett. 59, 672 (1987). [I61 D. Pritchard et al., in Proceedings of the 11th hiternational Conference on Atomic Physics, edited by S. Haroche, J.C. Gay, and G. Grynberg (World Scientific, Singapore, 1989), pp. 619-621. [I71 Some smoothing is performed in the wings of the distribution, where the signal-to-noise ratio is poor, using the same set of assumptions we use for thermometry. [18] M. Holland, D. S. Jin, M. Chiofalo, and J. Cooper (to be published). [ 191 As the excluded central region is enlarged, systematic bias in the inferred temperature vanishes, but so does the signal-to-noise ratio. We found it necessary to fit to a region which unfortunately samples the outer edge of the degenerate portion of the cloud. From numerical studies of the ideal Bose-Einstein distribution, we derive and apply a modest (
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1996 AAPT Richtmyer memorial lecture to be published in Am. J. of Physics
Bose-Einstein Condensation in an Ultracold Gas
M E .Wieman JILA and Department of Physics University of Colorado, Boulder, CO 80309 ABSTRACT The following article is a written version of the Richtmyer award lecture given to the annual meeting of the American Association of Physics Teachers in January, 1996. I discuss the basic idea of Bose-Einstein condensation in a gas and how it has been produced and examined. To cool the atom to the point of condensation we use laser cooling and trapping, followed by magnetic trapping and evaporative cooling. These techniques are explained, along with the signatures of Bose-Einstein condensation that we observe. I also discuss how very similar laser cooling and trapping techniques have been incorporated into undergraduate laboratory experiments.
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I will cover two topics which at first glance would appear to be quite different: 1) the achievement of Bose-Einstein condensation (BEC) in a very cold gas’, and 2) development of simple and inexpensive techniques for cooling and trapping atoms using the forces of laser light. This second topic has led to the development of experiments for undergraduate lab courses, which involve laser spectroscopf, and laser cooling and trapping. What ties the two topics together is that most of the techniques which are used in the undergraduate lab experiments are also used in the research which has produced BEC. Thus, these experiments allow undergraduates in a laboratory course to become involved with physics which is at the forefiont of current research. Many people, at Colorado and elsewhere, have contributed to the work I will discuss. At JILN Univemity of Colorado there is a substantial group‘ of students, postdocs, and Eaculty who have been working on these projects, particularly BEC, for many years, with Eric Cornell and myself as the coleaders. The work I will discuss is primarily the results of these efforts. This talk will be organized as follows: 1). Introduction to BEC in gas; 2) laser cooling and trapping--basic concepts, simple inexpensive embodiments for BEC and undergraduate labs, 3) magnetic trapping and evaporative cooling of laser cooled samples, and 4) BEC results and conclusions.
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to Base--in a & It is well known that there are two types of objects in nature, Femnions, which have half integer spin, and Bosons which have integer spin. The Fermions are unsociable and thus only one Fermionic object, or Fermion, can occupy a single quantum state. In contrast, Bosons are sociable, and not only is it possible for more than one Boson to occupy a single state, but they prefer to do so. The common example of this is the laser, which works because the photons are Bosons. Although all the constituents of atoms (neutrons, protons, and electrons) are Fermions, ifthey are assembled such that the total spin of the atom is an integer, the atom will be a Boson. Although atoms can be either Bosom or Fermions, an examination of the periodic table reveals that when they are in their lowest electronic state, most atoms are Bosons. Atoms of a gas, when held in a container as shown in Fig. 1, can have only particular quantized energies. However, for any n o d macroscopic container the spacing between these energy levels is extremely small (- lo’= ergs = 1 nK). As long as the sample is “reasonably hot”, (a.) say more than a few pK above absolute zero, whether the atoms are Bosans or Fennions has little effect on the macroscopic properties of the sample. In either case the probability of any given energy level being populated is extremely small, and the atoms can be considered to be much like small classical ball bearings bouncing around inside the container. At very low temperatures, however, a profound and dramatic effect takes place, as first pointed out by Einstein in 1924.’ The history of this sutject actually goes back to the work of Bose which preceded reference 5. Bose was t q h g to understand the black body spectm in terms of photon statistics. He realized that he could predict the correct spectrum ifhe made an assumption about when photons did, or did not, have to be counted as separate particles. Essentially this was saying that they obey what we now know as Bose statistics. Einstein learned of this work and took the fbrther step of postulating that atoms as well as photons should obey these statistics, and he went
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on to write down the now familiar Bow-Einstein distribution formula for an ideal gas.’ He noticed, however, that this formula has the peculiar property dlat at very low, but finite temperatures, it predicts that all the atoms will go into the lowest energy level of the container. This is now known as Bose-Einstein condensation and is discussed in every textbook on statistical mechanics. Although this is n o d y discussed in terms of chemical potentials, a more visual way to understand the condition for BEC is to think in tenns of the Debroglie wavelength, I , . As the temperature is reduced,the Dehroglie wavelength of each atom becomes larger. When the sample is so cold that the Debroglie wavelengths are larger than the interparticle spacing, the atoms begin to fidl into the lowest energy state in the container, as illustrated in Fig. lb. Thus the actual condition for BEC is a requirement on the phase space density. The condition usually given in the texts for an infinite homogeneous ideal gas is that (Am )’n > 2.6, where n is the atomic number density. Although this does not apply exactly to our case of a finite inhomogeneous system, it is fairly close, and provides a good indicator for the necessary temperatures (and hence Am) and densities which must be achieved. BEC is a very strange material in 8 number of respects. First,there are a large number of atoms in a single quantum state. As such, the atoms are indistinguishable in every respect, and hence cannot be considered as separate individual atoms. They have lost their identities as independent atoms, and have now hsed into a sort of “superatom”. Second, the transition to BEC is nonintuitive because, before the transition, the atoms are very fir apart compared to their atomic “size”. The average separation is 10,000 times the Bohr radius, and hence the interactions between them, in the usual sense of electrons pushing up against each other, is extraordinarily small by any measure. However, they still know, through some strange quantum statistical sense, that it is time to all jump into the ground state. Finally, BEC represents a macroscopic population of a single quantum state, and hence provides a macroscopic sample of material which is completely nonclassical in its behavior. It is certainly interesting just to see BEC in a gas, but our primary motivation for this work was not simply to observe it, but rather to use it to explore the subtleties of many particle quantum mechanics. There is some very interesting physics which can be studied through the comparisons between BEC and the other macroscopic quantum states which we all know and love‘, particularly superfluid helium. As a liquid this is quite different fiom the ideal gas discussed by Einstein, but is now generally thought to be closely related to the BEC he discussed; at very low temperatures some macroscopic M o n (-1O?!) of the helium atoms are in the lowest quantum state. Because the atoms are very close together in the liquid, or more formally, the interparticle separation is comparable to the scattering length, this is a strongly interacting system. These strong interactions are actually responsible for much of the interesting behavior we associate with superfluid helium, but the interactions also make it much more complicated to understand the macroscopic properties of superfluid helium in tmns of the microscopic interaction between two helium atoms. BEC in a gas is the perfect tool for exploring how the microscopic interactions between atoms lead to the macroscopic properties of the many atom quantum state. Because the atoms in the condensate are far apart compared to their atomic size, the interactions are weak and well understood, and hence easily treated theoretically. Furthermore, we can readily adjust these
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interactions in experiments by changing the density of the gas and other parameters. Finally, as discussed below, we have very good optical diagnostics for looking at the condensate and measuring its properties. This combination of factors make this an excellent system for studying in detail how one goes fiom the microscopic to the macroscopic. Much of the design of our experiment was motivated by the desire to produce BEC in a simple manner which would fkilitate carrying out experiments on it to explore this area of physics. A gaseous BEC also has a number of potential applications. It is the atomic counterpart to laser light and thus shares the primary feature which makes laser light usefbl, namely very high phase space number density. For this reason, it is likely that BEC will find many uses as well. BEC should revolutionize atom interferometry in much the m e way lasers revolutionized optical interferometry, as well as other applications where extremely high phase space density andor large coherence lengths of atoms are important. Having established the motivation for this work, I will now turn to the experimental difficulties in producing BEC in a dilute gas, and how they have been overcome. When you first consider how one might produce BEC, and the implication of the requirement that the interparticle spacing must be smaller than the Debroglie wavelength, you soon realize that there is a formidable challenge. The obvious approach to try first is to start with a moderately cold sample of atoms, say a few K, and then simply increase the density until the atoms condense. However, you quickly realize that this requires the atoms to be at a density approaching that of a solid, rather than the desired gas. This means the atoms will tend to see each other not as just individual fiiendly Bosons, as they do when the separation is large compared to the atomic size, but instead as a collection of standoffish Fermions. Thus this approach is clearly unsuitable. It is natural to then decide, ok, this means I have to keep the atoms far apart to have a dilute gas, so how cold do I have to make them? A simple calculation indicates that the temperature must be very cold, on the order of 100 nK. If you are a very optimistic experimentalist (or a typical theorist), you are undaunted by the mere technical details of getting a gas that cold, but there seems to be a much more fbndamental problem set by the laws of thermodynamics. This problem is that no atom wants to stay a gas at such a low temperature. They want to be solids, and this is particularly true for rubidium, the atom we use, which is a metallic solid at room temperature! For this reason a number of people thought that it would be impossible to ever produce BEC in a gas, and on occasion assured me that to do so we would have to violate thermodynamics. Every physicist is raised to believe that you can never beat thermodynamics; ofken through the maxim that the three laws are “you can’t win, you can’t break even, and you can’t get out of the game.” However, it is important to realize that while you can’t beat thermodynamics if you play by its rules, it is not so hard to win ifyou are willing to cheat, (or at least use a different set of rules), and thereby get around these oppressive laws. This is exactly what we have done to get BEC. The idea of how one can cheat thermodynamics to get BEC in a gas is a subtle and absolutely crucial concept in this work. What we actually do is avoid ever reaching a true equilibrium where thermodynamics applies. Instead we create a vapor sample which quickly equilibrates to its proper thermal distribution as a spin polarized gas, but which takes a very long time to go to its true equilibrium state (a solid). Its equilibrium ground state is definitely going to be a little chunk of rubidium ice, but we produce conditions (low temperature and low density) so that the gas remains in its metastable super-saturated-vapor state for a long time. During this time
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we can produce and study a gaseous Bose condensate. Although this concept of needing to produce a sample with two very different time scales for equilibration is not widely publicized, I believe that it was the critical step for achieving BEC. Once we fully grasped this concept and its implications, the general route to BEC in a gas was clear. The route we follow is a two step process of cooling and trapping. The first stage uses laser light for the cooling and trapping. This is followed by a second stage which uses magnetic fields for trapping and cools by evaporation. In both stages the trapping is as important as the cooling,because it provides a "thermos bottle" which keeps the very cold atoms fiom coming into contact with the vastly hotter environment only 1 cm away. Historically, these technologies were developed during the 1980's as two independent fields of study. One was the development by many groups of the techniques of laser cooling and trapping', with little or no concern with BEC. Simultaneously, there was a concerted effort to achieve BEC in a gas of spin polarized hydrogen'. The first hydrogen BEC efforts used only traditional cryogenics, but when this failed, magnetic trapping and evaporative cooling were developed to continue the pursuit. In about 1986, I had the idea of combining the two technologies to produce BEC in alkali atoms by first laser cooling and trapping them, and then following that with magnetic trapping and evaporative cooling. Before explaining in more detail these techniques and how this approach succeeded, I want to set the basic scale of the apparatus. Most people automatically associate low temperatures with large complex cryostats, dilution refrigerators, etc., but OUT apparatus uses none of that technology, and in fact is remarkably simple. As illustrated in Fig. 2, the heart of the apparatus is a small glass cell with some coils of wire around it. The only thing in the room which is colder than room temperature is the cooled atom cloud itself.
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Now let me explain how such a simple device can be used to produce BEC. The first step in the process is to cool and trap atoms using laser light. Although cooling and trapping of atoms by light is a large subject which many groups have developed over the years', I want to limit myself to discussing only a few basic ideas. The primary force we use is the radiation pressure force which is produced when one shines resonant laser light on an atom. The atom is excited and then decays, thereby scattering the light, as illustrated in Fig. 3. Each time the atom scatters a photon, it feels a tiny kick, due to the transfer of momentum. Of course an atom can scatter many photons per second and so this scattering force can result in a large acceleration. To cool the atoms, one must make this force have a fiictiona! or velocity dependent part. This is done by using the Doppler shift and appropriate detuning of the laser frequency. As shown in Fig. 3, if the laser is tuned to the red side of the atomic resonance line, then ifthe atom is going towards the laser (9), it sees the light Doppler shifted more into resonance, and hence scatters many photons. This results in a large force opposing the atoms motion. However, ifthe atom is moving in the same direction as the laser beam (b), the Doppler shifted light is W e r off resonance, so the atom only feels a small force increasing its speed. Thus if one has laser beams coming fiom all six directionsto strike the atom, the net effect is that no matter what direction the atom is going it will always feel a force opposing its velocity, and it will be slowed down, and thus cooled. Once the atoms are quite cold (S 1 mk), there is another more complicated process, often called
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“Sisyphus” or “subDoppler” cooling,which cools them somewhat hrther than is possible using the Doppler effect. This cooling arises from a fortuitous coincidence between the manner in which atoms make transitions between states, and their potential energy as they move up and down the potential hills produced by the standing wave laser fields. Laser light can be used not only to cool the atoms, but also to hold them away fiom the hot walls. To use the radiation pressure force for this it must be given a spatial dependence. This is done through the Zeeman shift of the atomic levels induced by m inhomogeneous magnetic field. This Zeeman shift controls the light scattering rate (and thus radiation pressure force) in a position dependent fashion. This effectively creates a potential. The atoms sit at the bottom of the potential, held there only by the laser light. The simplest way to implement laser cooling and trapping is shown in Fig. 4. This is a so This is so simple and inexpensive it is now a called magneto-optic (MOT)trap in a vapor standard experiment in the University of Colorado undergraduate physics lab course3, but it is also what we use for the BEC experiment. We start with a small glass cell which is attached to a vacuum purnp and, intermittently, to a rubidium reservoir, so that it contains about torr of rubidium and very little other gas. Then the laser trap is added by sending in laser beams of the appropriate polarization fiom all six directions and applying the magnetic field. Only a few milliwatts of power are needed in each laser beam, so this light is obtained fiom inexpensive diode lasers, much like those found in CD players. The fiequency of the light must be controlled quite precisely; this is done by using a difiaction grating to send light of a particular fiequency back into the laser. A simple servo control system, which adjusts the grating position and the laser current locks the fiequency of the laser to an atomic resonance lie2. The final component of the MOT, a small magnetic field gradient is produced by running currents in opposite directions through coils of wire on each side of the (often called “antihelmholtz coils”). The trap slowly fills up with atoms captured fiom the low velocity tail of the MaxwellBolt~nanndistribution. The fastest atom which can be captured depends on the diameter and intensity of the laser beams, and this in turn determines the rate at which atoms are loaded into the trap and the equilibrium number. For our low power, 1.5 cm diameter beams about lo7atoms are captured into the trap. The time it takes to collect these atoms depends on the rubidium pressure. We use a similar trapping setup for the BEC experiment as is used in the undergraduate lab experiment. However, the undergraduate lab experiment operates with higher rubidium pressure, and therefore typically fills with a time constant of a second or two. In the BEC experiment, it is desirable to operate at much lower pressures. In this case the time constant for the trap to fill is about 1 minute. With such long fill times the light induced collisional loss can reduce the number of atoms collected, so a “dark spot” trap is used to suppress this loss.” The layout of the undergraduate trapping experiment is shown in Fig. 5. In the normal class, it is broken up into two experiments, each of which takes three lab sedsions, each of which lasts for three hours. For the first experiment, students use either of the two diode-laser saturated-absorption setups to cany out Doppler-fiee laser spectroscopy of rubidium.’ In the second experiment, they lock the laser fiequencies to the appropriate saturated absorption lines and send the beams fiom both lasers into the trapping cell in the middle of the table to observe and study laser trapping and co~ling.~ These experiments are discussed in detail in References 2 and 3.
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e v a p p r a f m Returning to BEC, remember that the requirement is to have a large enough phase space density; the laser trapped sample is a major step toward this goal. After the period of atom collection,the optical trap contains about lo’ atoms, and their temperature is about 10 pK. The good news is that this one easy step has increased the phase space density by about 16 orders of magnitude over the original room temperature vapor. The bad news is that this still leaves one about 5 orders of magnitude short of BEC. Actually my original interest in BEC research grew out of my Curiosity about what was limiting the densities and temperatures which could be reached in these laser traps. We studied this for several years and learned that there were several relevant processes, all of which were due to the presence of the photons.” In some respects the photons are rather like house guests or fresh fish. They are very desirable to have initially, but ifthey stay around too long, they lose their appeal, and even become downright unpleasant. In this case, a photon will “overstay its welcome” by scattering repeatedly offthe atoms in the trapped cloud. Because of this, we decided it would be desirable to get rid of the photons, once we had shamelessly used them to produce a nice laser cooled and trapped sample. You can do this just by blocking the laser beams, but, when that is done with atoms at these temperatures, they simply reconfirm Galileo. They fall until they hit the bottom of the cell with a “thud (admittedly, a very faint one). To save the cold atoms fiom this untimely end, we held them in a magnetic ‘‘Safety net”, often known by the less charikble term, “magnetic trap”.” This fairly old technology uses the fact that each atom has a small magnetic moment, p, and thus can be confined by the p-Binteraction in an appropriately configured inhomogeneous magnetic field. To confine a room temperature atom, or even a 1 K atom, in this way is fairly difficult, because it requires large magnetic fields. However, to confine an atom which has already been laser cooled to 10 pK is quite simple. In fiict, it was the ease of creating such cold magnetic trapped samples that encouraged us to think about what more could be done with them. I was aware that shortly before we carried out this magnetic trapping, the MIT hydrogen BEC group had achieved very impressive results with evaporative cooling of magnetically trapped hydrogen atoms.’ Although they had not quite reached BEC they had shown that it was possible to use this technique to get large increases in phase space density. We decided to see ifwe could use this evaporative cooling of our cold alkali atoms to achieve BEC. We had general hunches as to why certain aspects of atomic physics might fkvor this approach to BEC, relative to the previous hydrogen work. Hydrogen was plagued by dipoledipole spin flip collisions which caused atoms to go into a lower energy spin state and be lost fiom the trap (thus in a sense, quenching the spin polarized metastable vapor). We felt it was likely that heavy akalies would have similar rates for these undesirable collisions, but would probably have much larger cross sections than hydrogen for the desirable two body elastic collisions needed to thermalize the gas for evaporative cooling. The arguments are just that the atomic magnet moments of rubidium and hydrogen are similar, while heavy alkalies are much larger flufEer atoms. In terms of the earlier discussion, this comparison of collision rates is saying that the critical difference in the two equilibration time scales, gas thermalization versus quenching metastability, is larger in alkalies than in hydrogen. However, these ideas could only be hunches, because all of the relevant rates for alkali atoms are extremely sensitive to the exact shape of the
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interatomic potential, and hence were completely unknown six years ago. Much of our work over the past six years was spent in determining the good and bad collision rates. From the work of ourselves and others, in the past few years it has become clear that the original hunch was correct, and it also gave us a much better idea as to exactly what conditions were necessary in order to achieve evaporative cooling to BEC. Although it took us several years to learn this, the key issue for s u c c e d evaporative cooling of a laser cooled sample is simply obtaining a high enough elastic collision rate between the magnetically trapped atoms. This can be understood by considering the evaporative cooling process in more detail. As shown in Fig. 6a and demonstrated with hydrogen, the simplest form of evaporative cooling is to confine the atoms in a magnetic bowl, and let the most energetic atoms escape over the side. When they do this, they carry away far more than their share of the energy, and thus the remaining atoms get colder. This is much like what happens when coffee cools. The most energetic coffee molecules leap out of the cup into the room, carrying away lots of energy and thereby cooling the coffee which remains in the cup. In order for evaporative cooling of a magnetically trapped gas to work efficiently, the time for the atoms to reestablish a proper thermal distribution after atoms escape the trap must be much shorter than the lifetime of the cold atoms in the trap. The trap lifetime is primarily determined by collisions with hot background atoms in the cell, and thus it is important to have good vacuum in the cell. The thermalization time is determined by the elastic collision rate which is equal to the density times the cross section times the relative velocity. This is why the large elastic scattering cross section of heavy alkali atoms is very usem. It is still necessary to make the density larger than provided by a simple laser trapped cloud which is transferred to the magnetic trap. We increase the density in the magnetic trap through a variety of optical techniques to make the cloud as dense and cold as possible before it is put into the magnetic trap”, and then finally we squeeze the cloud as much as possible with the magnetic trapping force. The final step which gave us high enough density to make evaporative cooling work sufficiently well to reach BEC was the invention of a new type of magnetic trap“ by Eric Cornell which provides a large amount of “magnetic squeeze” for a given current. This is the so called “time orbiting potential” or “TOP trap. With this final piece of technology in place, we were then able to evaporatively cool the magnetically trapped atoms to extremely low temperatures. As shown in Fig. 6b. we actually do not evaporate simply by allowing the atoms to escape out of the top of the trap as described above. What we actually do is apply an RF field at fiequency v which causes the magnetic spins to flip when the resonant condition, hv = (Am)gpB, is met. Since higher potential energy in the trap also corresponds to higher magnetic field, this can be pictured as putting a leak in the trap at a position which is set by v. This is very convenient since it allows us to use the RF to skim offthe most energetic atoms, then as the remainder cool and settle lower in the trap, we reduce the RF frequency and skim offthe top of this new distniution to continue to optimize the cooling. In an actual cooling cycle, the rf frequency is ramped slowly to a final value which determines the final temperature of the sample. and conclusions. AU the pieces I have described are put together in a series of steps which cool room
temperature rubidium atoms to a Bose-Emein condensate as follows: 1) collect 10’ rubidium
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atoms 6om the room temperature vapor into the MOT; 2) do some tricks with the magnetic field and the laser detuning to optically get the atoms as dense and cold as possible; 3) use a single circularly polarized laser beam to optically pump all the atoms into the F=2, m=2 spin state (the state which is confined in the magnetic trap); 4) turn off all the lasers and turn on the magnet fields for the TOP trap; 5) ramp up the magnetic fields to squeeze the magnetically trapped m p l e to increase its density and elastic collision rate; 6) turn on the RF and ramp down the frequency to evaporatively cool. The initial collection 1) and the evaporative cooling 6) take roughly a minute each, while all the other steps are essentially instantaneous on that time scale. At the end of the cooling cycle we have a nice cold sample, but since it is sitting in the dark we don’t know anything about it. So we turn the light back on and look at it. When the sample is very cold, since it is sitting at the bottom of a harmonic potential, it is also very small and thus hard to see. To make the cloud large enough to see in detail we turn off the magnetic trap and allow the atoms to fly apart. For a number of technical reasons,this is better than just magnifLing the image with lenses. After the atoms have spread out for 0.06 seconds the cloud is much bigger, and we then take a “shadow snapshot” of it. This image is obtained by illuminating the expanded cloud with a very short pulse of laser light which is tuned to the resonant frequency of the atoms. The atoms scatter the light, thereby casting a shadow in the illuminating laser’beam, and the shadow is imaged onto a CCD array (TV camera). This shadow image is the two dimensional projection of the velocity distribution of the original cloud of atoms in the magnetic trap. From the velocity distribution we can extract the temperature and various other properties of the sample. A set of three such pictures are shown in Fig. 7. (This data is fiom the work described in reference 1.) These correspond to three repetitions of the experiment, where the only difference is the M rfcooling frequency. In the left most picture, we have only cooled the atoms down to a balmy 200 dC,and what we see is a round hill, which looks like the familiar MaxwellBoltPnann velocity distribution. At higher temperatures (not shown here), the cloud has the same shape with a larger width. We determine the temperature by extracting it fiom the measured velocities. The middle shows a cloud (-10,000 atoms) where the sample was cooled firther, down to about 100 nK,and this is when things get exciting. On top of the rounded hill, a narrow spire now emerges which is centered at zero velocity. Ifwe cool even €&her (right), we can produce a sample ( -2000 atoms) in which the hill is completely gone, and only the narrow spire remains. You can see how this behavior is exactly what one expects with BEC if you go back to the original concept illustrated in Fig. 1. The normal atoms that are distributed over many energy levels form the Maxwell-Boltanann-like hill. The atoms in the lowest energy state of the potential are the most localized in both position and velocity space, and they are centered at zero velocity. Thus as atoms condense into that state, they form a very narrow peak in the velocity distn’bution, which sits on top of the broader hill of noncondensed atoms. The condensate lives for 15-20 seconds if it is left in the dark. There are other features of these velocity distributions which indicate we are seeing BEC. One is the peak density of the trapped cloud as a fbnction of temperature, as shown in Fig. 8. Although we measure velocity distributions, one of the very nice features of using a harmonic trap is that we can immediately obtain the density distribution fiom this data just by scaling the velocity
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distribution by the appropriate oscillation frequencies in the trap. The figure shows how the peak density is nearly constant with temperature until it goes below 100 nK,whereupon it jumps up very dramatically. This provides a strong indication of a phase transition, just like the condensation of steam into water. aspect of the data is revealed by looking down on the peaks of Fig. 7 Another intfrom above, as shown in Fig. 9. This allows one to examine the isotropy of the velocity distribution. In Fig. 9a, you can see the contour lines of the rounded hill are essentially circular, indicating an isotropic distribution. This is just what one must have for any thermal sample, because of the equipartition theorem which says there are equal amounts of kinetic energy in each direction. However, you can see that Figs. 9b and c show that the spires are not round; instead they are quite elliptical indicating an anisotropic velocity distribution. The explanation for this is that in these figures you are actually seeing a macroscopic quantum wavefunction. The reason for the shape of this wavefunction is that the harmonic potential in our trap is not isotropic. The spring constants in the z direction are eight time larger than they are in the radial (x-y plane) direction. A simple first year quantum mechanics calculation of the wave fbnction of the ground state in such a 3D harmonic potential will show you that the wavefunction is pancake shaped; it is narrower in the z direction than in the radial direction. This means the atoms are more localized along the z direction and therefore, just fiom the uncertainty principle, (or more formally, calculating the momentum space wave function) you can see that an atom in this eigenstate must have a larger velocity spread in the z direction, and this is why the velocity distribution is elliptical. It is even more enlightening to do this calculation of the harmonic oscillator wave function and quantitatively compare it with our data. (Ifyou are teaching introductory Q.M., you might want to consider giving this to your students as a problem.) It turns out they do not quite agree! There is a discrepancy of about 30%. After discovering this, we got our theory fiends, specifically, M. Holland and J. Cooper, to do the calculation correctly by putting in the small known interactions between the This properly calculated wavefunction shows that the interactions distort the shape of the wavefunction into one which matches perfectly with the data! This is a very satisfvrns result because one of the primary motivations for undertaking this work was the goal of understanding in detail how the microscopic interactions affect the macroscopic behavior, and this first small step in that direction has been very successful. To summarize, we see three clear signatures indicating the atom are undergoing BoseEinstein condensation into the ground state of the confining potential. First, the velocity distribution has two distinct components, a broad thermal distribution and a narrow peak centered at zero. Second, the peak density shows a vcry abrupt increase as the temperature is decreased. Third, the velocity distribution is elliptical rather than round. Thus it seems rather convincing that BEC has been observed. However, the best news is that this was done is a relatively simple and inexpensive apparatus which will allow many properties of the condensate to be studied in detail. This is the goal which Eric Cornell and I net out to achieve years ago. There are now many obvious things to study about the condensate and we and many other groups are eagerly setting out to do this. A very abbreviated list includes looking at how the condensate scatters light and comparing this with n o d atoms, investigating the dynamics of the phase tramition, determining the elementary excitation spectrum of the condensate, and many
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others. One particularly wants to study how all these properties depend on the microscopic interactions, and how they change as we vary the interaction energy. This is straightforward to do simply by changing the density by varying the number of atoms in the trap or by adiabatically increasing or decreasing the confining potential. It will also be interesting to study different types of condensates, and it looks like the same basic approach should work to achieve BEC in many different systems. This has already been demonstrated in sodium,and other atoms will no doubt follow. The study of gaseous Bose-Einstein condensates appears likely to be a very f h i t h l area of physics during the next few years. Most of their properties are as yet unknown, but the basic theoretical tools exist to calculate them and we now have experimental tools for measuring them. I am pleased to acknowledge the help of many people in the work discussed here. These include all my students and postdocs over the last decade, and many other helpfil colleagues. Although the full list is too numerous to mention I want to single out the contributions of the JTLA BEC group and especially Eric Cornell with whom I have had a tremendously rewarding collaboration for many years. Gwenn Flowers and Sarah Gilbert were instrumental in helping set up and carefilly documenting the undergraduate laboratory experiments, and Sarah provided constructive criticism in the preparation of this manuscript (as well as all my previous papers). The work I discussed has been supported by NSF,ONR,and NIST.
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1) M. H. Anderson, J. R Ensher, M.R Matthews, C. E. Wieman and E. A Cornell, “Observation ofBose-Einstein condensation in a dilute atomic vapor”, Science 269, 198-201 (1995).
2) K.IkfacAdam, A. Steinbach and C. Wieman, ”A mow-band tunable diode laser system with grating feedback, and a saturated absorption spectrometer for Cs and Rb”, Am. J. Phy. 60,10981111 (1992). 3) C. Wieman, G. Flowers and S. Gilbert, “Inexpensivelaser cooling and trapping experiment for undergraduate laboratories“, Am. 3. Phy. 63,3 17-330 (1995). 4) This list includes C. Monroe, N. Newbury, C. Myatt, C. Sackett, J. Cooper, M.Holland, R. Ghrist, W.Petrich, D. Jin,E. Burt,G. Flowers, M.Anderson, J. Ensher and M. Matthews. The
BEC data presented here was primarily the work of the last three. 5) A. Einstein, Sitzber. Kg. Preuss. Akad. Wiss. (1924), p. 261; (1925), p.3. 6 ) In addition to superfluid helium, two other examples are superconductivity and BEC in
excitons. The latter work is discussed in J.-L. Lin and J. P. Wolfe, “Bose-Einstein condensation of paraexcitons in stressed CU20’, Phys. Rev. Lett. 71, 1222-1229 (1993). For an extensive discussion of BEC in many different systems, see “Bose-Einstein Condensation”, (eds., A. Griffuz D. Snoke and S. Stringari, Cambridge University Press, Cambridge) 1995. 7) See the following volumes for numerous articles and references on the subjects of laser cooling and trapping. N. R Newbury and C. E. Wieman, “Resource Letter TNA-1:Trapping of neutral atoms”, Am. J. Phys. 64, 18-20 (1996)(this article also provides extensive references on magnetic trapping) ;C. Wieman and S. Chu, Eds., Special Issue on Laser Trapping and Cooling, J. Opt. SOC.Am. B,Vol, 6 #11, (1989); Proceedis, Enrico Fermi International Summer School on Laser Manipulation of Atoms and Ions, Varenna, Italy, (E. Arimondo, W.Phillips, and F. Stmmia, Eds., North Holland, Amsterdam 1992) 8) For review of the hydrogen work, see T. Greytak, pg. 13 1 in “Bose-Einstein Condensation”, (eds. A GrifEn, D.Snoke and S. Strhgari, Cambridge University Press, Cambridge (1995).
9) E. Raab, M. Prentiss, A Cable, S.C h , D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure”, Phy. Rev. Lett. 59,2631-2634 (1987). 10) C. Monroe, W. Swann, H.Robinson and C. Wieman, “Verycold trapped atoms in a vapor cell”,Phys. Rev. Lett. 65, 1571-1574 (1990). 11) M. H.Anderson, W.Petrich, J. R. Ensher, and E. A Cornell, "Reduction of light-assisted
477
collisional loss rate fiom a low-pressure vapor cell trap", Phy. Rev. A 50,R3597-3600(1994).
12) D. Sesko, T. Walker, C. Monroe, A Gallagher and C. Wieman, "Collisionallosses fiom a light force atom trap", Phys.Rev.Lett. 63,961-964(1989); T.Walker, D.Sesko and C. Wieman, "Collective behavior of optically trapped neutral atoms', Phys. Rev.Lett. 64,408-41 1 (1990);D. Sesko, T. Walker and C. Wieman, "Behavior of neutral atoms in a spontaneousforce trap", J. Opt. SOC.Am. B 8,946-958(1991). 13) N. Petrich, M.H.Anderson, J. R Ensher and E. A Cornell, "Behavior of atoms in a compressed magneto-optical trap", J. Opt. SOC.Am. B 11, 1332-1335(1994).
14) W. Petrich, M.H. Anderson, J. R Ensher and E. A Cornell, "Stable tightly confining magnetic trap for evaporative cooling of neutral atoms", Phys. Rev. Lett. 74,3352-3355(1995). 15) M.Holland and J. Cooper, "Expansion of a Bose-Einstein condensate in a harmonic potential", Phys. Rev. A Rapid Corn., to be published, April, 1996.
478
1) a. The energy of hot atoms is very large compared to the spacing of the quantized energy levels in a macroscopic container. For either Bosom or F e d o m there is a very small probabiity of any
given level being occupied. b. When Bosom are cooled sufl[icientlythat the Debroglie wavelength,
&,is larger than the spacing between atoms, d, the atoms fall into the lowest energy state in the potential. All the atoms occupying that state are indistinguishable, and thus occupy the same region in space.
2) BEC trapping cell. A rectangular glass cell (2.5 cm square by about 10 cm high) is attached to a vacuum pump and rubidium reservoir (not shown). Laser beams coming fiom all six directions go through the cell. The magnetic fields are produced by the two large coils, which have currents
flowing through them in opposite direction, and the four smaller coils, which have time varying currents, as discussed in Ref 14. 3) The laser beam exerts a force as large as lO’g’s on the atom by scattering photons. At the bottom is shown how the atom scattering (or equivalently, excitation) rate depends on laser fiequency. If the laser fiequency is below the atom resonant fiequency as shown, then a) when the atom is moving opposite the direction, the light is going the scattering rate and force is high because of the Doppler shift. b) When the atom is moving in the same direction as the light the Doppler shift is in the opposite direction and so the rate is low. 4) Schematic of a vapor cell magneto-optic trap. 9. A rectangular glass cell, typically 2.5 cm square is evacuated, and then a small amount of rubidium (or other alkali) vapor is introduced into
it. Six circdarly polarized laser beams with the helicitiedcircular-polarizations shown, pass through the cell. In practice this is usually done with three beams which are reflected back on themselves through quarter waveplates. The two circles are coils of wire which have currents flowing in opposite directions to produce the desired magnetic field gradient. b. The MaxwellBoltzmann distribution of the room temperature rubidium atoms in the cell. Atoms bounce around inside the cell until they happen to come offthe wall with a velocity less than Vup(-15 m/s) and pass through the laser beams, at which point they are caught by the trap. After a short time, approximately 10’ trapped atoms accumulate in the center of the trap, forming a Small cloud. The temperature of these trapped atoms is about 10 pK. 5) Layout of the undergraduate laser spectroscopy and trapping and cooling experiments on a 4 x 6 foot optical breadboard. At each side are setups for Carrying out saturated absorption laser spectroscopy experiments. In the center is the trapping cell,which uses light fiom both the lasers. 6 ) Schematic of evaporative cooling of magnetically trapped atoms. The atoms are contained in a
magnetic bowl. a) In simple evaporation the most energetic atoms escape over the side and the remaining atoms then become colder. b) The applied RF field causes the atoms magnetic moments to flip at the particuiar value of magnetic field which satisfies the resonant condition.
479
This makes a hole at a particular position (magnetic field) in the bowl through which atoms escape. The position of the hole and the resulting atom distribution is shown for three different times during a cooling cycle in which the RF frequency is being ramped down. 7)Two dimensional velocity distriiutions of the trapped cloud for three experimental runs with different amounts of cooling (different final v RF). The axis are the x and z velocities, while the third axis is the number density of atoms per unit velocity-space-volume. This density is extracted from the measured optical thickness of the shadow. The distribution on the left shows a gentle hill and corresponds to a temperature of about 200 nK. The middle picture is about 100 nK and shows the central condensate spire on top of the noncondensed background hill. In the picture on the right, only condensed atoms are visible, indicating that the sample is at absolute zero, to within experimental uncertainty. The grey bands around the peaks are an artifact left over from the conversion of false color contour lines into black and white pictures for this publication. The original color versions can be seen on the JILA WWW home page and the 1996 A P S dendar. 8) A plot of the peak density of the trapped cloud as a finction of the temperature, in units of the final value of the RF frequency used in evaporative cooling.' The sudden increase in density at 4.2 MHz corresponds to a temperature of about 100 nK. The point at the far left is lower because the RF is ramped down so far it has begun to cut away the condensate itself 9) Plot of x and z velocity distributions of same samples shown in Fig. 7.' Images shown are negatives of actual data, so brighter corresponds to more atoms (less transmitted light). The
circular distribution corresponds to a 200 nK isotropic velocity distribution, while the other images show that the spread in velocity in the condensate is larger in the z direction than in x.
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a. hot atoms
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03
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485
a. simple evaporation
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b
time
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2) RF down, T down
486
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488
VOLUME78, NUMBER4
PHYSICAL REVIEW LETTERS
27 JANUARY 1997
roduction o f Two verlapping Bose-Einstein Condensates by Sympat~eticCooling C. J. MyaSt, E. A. Burt, R. W. Christ, E. A. Cornell, and C. E. Wieman JILA and Department of Physics, University of Colorado and NET, Boulder, Colorado 80309 (Received 20 September 1996) A new apparatus featuring a double magneto-optic trap and an Ioffe-type magnetic trap was used to create condensates of 2 X lo6 atoms in either of the IF = 2, rn = 2) or IF = 1, rn = - 1) spin states of 87Rb. Overlapping condensates of the two states were also created using nearly lossless sympathetic cooling of one state via thermal contact with the other evaporatively cooled state. We observed that (i) the scattering length of the 11, - 1) state is positive, (ii) the rate constant for binary inelastic collisions between the two states is 2.2(9) X cm3/s, and (iii) there is a repulsive interaction between the two condensates. Similarities and differences between the behaviors of the two spin states are observed. [SO03 1-9007(96)02208-91 PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 51.30.+i
One of the more notable recent developments in physics has been the cooling of a trapped dilute atomic gas to below the Bose-Einstein transition temperature [ 1-31. This produced a macroscopic quantum state that is both novel and readily observed. Gaseous Bose-Einstein condensation (BEC) was first reported in a cloud of atoms in a single spin state of the ground state of rubidium [1] and later in single spin states of sodium [2] and lithium [3]. To reach the necessary ultralow temperatures, these experiments used laser cooling and trapping followed by magnetic trapping and evaporative cooling. The observation of BEC led to a number of studies of the properties of these two condensates [4]. Here we report the creation of two different condensates in the same trap. The two condensates in this work correspond to two different spin states of rubidium 87, IF = I, rn = -1) and IF = 2, m = 2). We have created large samples of both condensates separately and compared their properties. We have also created mixtures of the two by using a new cooling technique. The cloud of atoms in the II, - 1) state was cooled by lossy evaporative cooling, as in previous work, but the 12,2) state cloud was cooled only by thermal contact with 11, - 1) atoms. Such “sympathetic” cooling of one species by another has been used at much higher temperatures to cool trapped ions [5] that have strong long-range Coulomb interactions, but this is the first time it has been applied to neutral atoms. This sympathetic evaporative cooling technique may allow the creation of degenerate Fermi gases as well as condensates in rare isotopes. We see differences in the lowtemperature behaviors of the atoms in the two spin states in both condensed and uncondensed phases. When the two condensates are overlapped, additional novel features are observed in their interactions. These condensates were created using a new apparatus that incorporates a double magneto-optic trap (MOT) and a magnetic trap. It is based upon the same vapor-cell/diode-laser technology as the original JILA BEC apparatus [I]; however, it produces much larger condensates and is far more tolerant of imperfect experimental conditions 161.
The apparatus is an extension of the double MOT systern that has been described previously 171. The double MOT system has the advantage of allowing a relatively large sample of atoms to be optically trapped in a very low pressure chamber using low power lasers. It is made up of two small differentially pumped vacuum chambers connected by a 40 cm long X I cm diam tube that is used to transfer atoms between them. The upper vacuum chamber contains about 2 X Torr of rubidium vapor. The transfer tube is lined with strips of permanent magnet that create a hexapole magnetic guiding field. The lower chamber, shown in Fig. 1, is made of glass and is pumped by a 60 L/s ion pump and a small titanium sublimation pump to a pressure of less than Torr. Three coils outside of this chamber create a “baseball coil” magnetic trap that is similar to that used in our previous work [S] but different from that used in the recent JILA BEC
586
0 1997 The American Physical Society
003 1-9007/97/ 78(4)/586(4)$10.00 489
FIG. 1. The glass lower vacuum chamber is connected to the upper chamber through a narrow transfer tube and to sublimation and ion pumps as noted. It is surrounded by the three coils that comprise the magnetic trap. Small additional windows (not shown) allow the cloud to be viewed along some of the diagonals. The trapping laser beams go through the six perpendicular 2.5 cm diam windows, four of which are visible in the figure.
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studies [ 1,4]. The first coil is shaped like the seams on a baseball and provides field curvature in addition to a bias field, while the other two form a Helmholtz pair that can cancel all or part of the bias field. The resulting field configuration is essentially that of an Ioffe-type trap 191, The three coils are wired in series with a variable shunt resistance across the Helmholtz pair. The trapping potential is axially symmetric with the axis nominally horizontal. There is a small additional coil for producing the adjustable-frequencyrf magnetic field used in evaporative cooling. Lenses image the trapped atomic cloud onto both a CCD camera and a calibrated photodiode. The trapping and probing light is provided by low power (50 mW) diode lasers stabilized by grating feedback [lo]. Single-speciescondensates in either the 12,2) or 11, - 1) spin states of the 5S ground state of rubidium are created and examined using the following procedure. First, atoms are collected in a MOT in the upper chamber for 1.0 s. This load, typically several times lo7 atoms, is then pushed down the transfer tube using light pressure, and about 80% of them are recaptured in a second MOT in the lower chamber 171. This procedure is repeated many times to fill the lower MOT with about lo9 atoms. Next, this cloud of trapped atoms is compressed by increasing the MOT magnetic field gradient as in Ref. [ 111. Then the MOT fields (optical and dc magnetic) are turned off and a 1 G bias field turned on. The atoms are then optically pumped into the desired spin state by applying 1 ms pulses of light from two laser beams that excite the 5S112, F = 1 to 5P312, F' = 2 and the F = 2 to F' = 2 transitions, respectively. By suitable adjustment of the polarization and relative timing of the two beams, we pump the atoms into a single state, either 11, - 1) or 12,2), with 90% efficiency. Next, the magnetic trap is turned on around the atoms with full current (200 A) flowing through the baseball coil, but with no current in the Helmholtz pair. The current in the Helmholtz pair is then ramped up in 2 s to reduce the bias field to 1 G. For the l2,2) state this increases the radial frequency from 20 to 400 Hz, while leaving the 10 Hz axial frequency nearly unchanged. (The 11, -1) state has one-half of the 12,2) magnetic moment, and so all of the 11, - 1) frequencies are lower by fi.)This ramp compresses the cloud and raises its temperature to 250 pK. The cloud is then cooled by rf evaporation, in which the applied rf magnetic field drives the most energetic atoms to untrapped spin states. For this cooling, the frequency of the applied rf is ramped down over a period of 30 s. The cooled cloud is probed by absorption imaging in a manner similar to that of Ref. [I]. The cloud is released f?om the magnetic trap, and after it expands ballistically for 20 ms it is i l l ~ i n a t e dbriefly by a near resonant 5 S l p F = 2 to 5P312F' = 3 probe laser beam, as well as F = 1 to F' = 2 hyperfine pumping light. The probe laser is normally tuned 1.6 full linewidths off resonance because an expanded condensate cloud is tens of optical depths thick for resonant light. The resulting shadow
27 JANUARY 1997
produced by the cloud in the iliuminati~gbeam is imaged onto the CCD camera. This absorption imaging is used to find the temperature of the cloud and the fraction of atoms in the condensate as in Ref. [l]. Also, a measurement of total fluorescence is used to accurately determine the number of atoms in the cloud. The fluorescence is measured by the calibrated photodiode after recapturing the evaporatively cooled atoms in a MOT. To create mixtures of the two condensates by sympathetic cooling, the procedure is quite similar. If the two clouds are at the same temperature, the 11, - 1> cloud is less tightly confined by the magnetic field than the 12,2) cloud because of the difference in magnetic moments and, hence, will extend to a larger magnetic field. This causes the 11, -1) state to be preferentially removed by the rf field. The 12,2) atoms are then cooled by elastic collisions with the evaporatively cooled 11, - 1) atoms as long as the two clouds overlap. Because the two states have different magnetic forces but the same gravitational force, the 12,2) cloud is centered slightly above the 11, - 1) cloud. However, for the spring constants given above and a horizontal trap axis, one can easily calculate that the relative displacement of the II, - 1) and 12,2) clouds is much less than their widths when they are not condensed. For condensates, the displacement is a significant fraction of the width, but there is still substantial overlap. By starting with the axis of the trap (the direction of the weak spring constant) slightly tilted off perpendicular to the gravitational force, it is possible to form separated condensates, if desired (Fig. 2). One can also separate the two clouds
FIG. 2(color). A false-color absorption image (475 p m by 675 pm) showing condensates of both 12,2) (left) and 11, - I) (right) states that were created simultaneously by sympathetic
cooling. The condensates are separated because the trap axis was tilted 40 mad to produce a component of the gravitational force along the weak spring constant direction. The noncondensed parts of the clouds (purple and dark blue) stilt overlap. The shape of both of the condensates is a function of expansion time, but the difference in their ellipticities reflects the fact that they have different initial confinements and therefore expand at different rates. The inset shows a vertical trace through the cloud on the left. The dotted line is to guide the eye in distinguishing the broad thermal background &om the narrow condensate peak. 587
49 1 VOLUME 78, NUMBER 4
PHYSICAL RE VIEW LETTERS
after cooling by adiabatically lowering the trap spring constants to increase the displacement due to gravity. For the production of two-species condensates by sympathetic cooling, the MOT portion of a cooling cycle was identical to that given above. However, in the optical pumping the polarization and timing of the two laser beams are set to produce the desired ratio of populations in the 11, - 1) and 12,2) states [12]. The evaporative cooling then proceeds exactly as it would for a pure 11, - 1) cloud. The probing is modified slightly to obtain state selective absorption images of the mixtures. After the cloud has ballistically expanded we probe it two different ways. First, we use only the near resonant F = 2 to F' = 3 light to obtain an absorption image of only the 12,2) cloud. We then take a second absorption image with the F = 1 to F' = 2 light also present [ 131. This produces an absorption image of all the atoms independent of their initial state. In Figs. 2 and 3(a) we show pictures of clouds containing two simultaneous condensates. We have compared the production of condensates for samples of either pure )2,2) or pure 11, - 1) atoms, as well as for mixtures. For the pure cases, both the total transfer efficiency from lower MOT to compressed magnetic trap (-50%) and the evaporative cooling results are similar. This is to be expected because the magnitude of the two scattering lengths are similar [14]. For each factor of 5 loss in the number of atoms during evaporation, we decrease the temperature by a factor of 10 and increase the phase space density by 200. This allows us to reach the -500 nK BEC transition temperature with 6 X lo6 atoms. After further cooling down to where we can no longer see any noncondensed atoms, about 2 X lo6 atoms remain. The efficiency of the evaporative cooling is relatively insensitive to initial conditions and the details of the rf ramp. This is not surprising since the initial elastic
27 JANUARY 1997
scattering rate in the magnetic trap is about 50 times larger than the -200 collisions per trap lifetime required for runaway evaporative cooling. For the low densities obtained before evaporative cooling, this lifetime for both states is -140 s [15]. All of our cooled clouds show the three clear indications of BEC reported in Ref. [ 11: a twocomponent velocity distribution, a nonisotropic velocity distribution, and a sudden large increase in peak density as the temperature is decreased. The first two of these are evident in Figs. 2 and 3. The observed transition temperatures for both states agree with the ideal gas value within our "20% uncertainty. Also, we find that the previously unknown sign of the scattering length for the I1, - 1) state must be positive [ 161. We also see differences between the two states. First, when the rf cooling is disabled, we observe a heating rate that is about 10 times larger for the 12,2) state cloud than a comparable cloud in the 11, - 1) state. This is true both above and below the BEC transition. Second, we see differences in how the two states are lost from the magnetic trap. At the highest densities, the lifetime of the 11, - 1) state remains 140 s but the lifetime for the 12,2) is density dependent and less than 10 s. These differences clearly involve some interesting low-temperature atomic physics that are quite relevant for the creation and study of BEC. In Fig. 4 we illustrate the nearly lossless sympathetic cooling of the 12,2) atoms in a two species cloud. Nearly one-half of the 12,2) atoms remain after cooling from 250 p K to just above the BEC transition. In contrast, I of the 11, - 1) atoms remain. There is a only about small loss of 12,2) atoms as the temperature approaches 1 p K (Fig. 4) and a larger loss in going from that point to a pure condensate (not shown). The loss rate depends on the densities of 11. - 1) and 12,2) atoms and is consistent with it being due to binary, inelastic collisions (presumably spin exchange) between the species. By measuring the densities and loss rates we find the total rate constant for
q a
10
c Y
(a)
0
(b)
FIG. 3(color). Two 475 p m by 475 p m false-color absorption images of 12,2) atoms. (a) A cloud of two overlapping condensates illuminated so that only the 12,2) state atoms are visible. The condensate (white, red, and yellow) is shifted upwards relative to the center of the thermal uncondensed cloud (green, blue, and purple) due to interactions with the 11, -1) condensate (I 1, -1) atoms not visible). (b) A cloud of pure 12,2) atoms cooled to a comparable temperature as in (a). The black line is a guide to the eye going through the center of both thermal clouds.
588
B 0 0
1
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100
Temperature [pK] FIG. 4. Number of ( 1 , -1) and 12,2) atoms in a two-species cloud as a function of the temperature during the sympathetic evaporative cooling. The cloud is being cooled from the initial magnetic trap temperature to just above the condensation temperature.
492 VOLUME78, NUMBER4
PHYSICAL REVIEW LETTERS
inelastic processes is 2.2(9) x cm’ls. When we tilt the trap, as in Fig. 2, we reduce the overlap between the clouds and the observed loss rate. We have measured the temperatures of the 11, - 1) and 12,2) clouds and find that they match closely during the evaporation process. The measured temperature of the 12,2) state is consistently 5%- 10%lower than that of the 11, - 1) state; however, this is just at the limit of our resolution and may not indicate a real difference. Finally, we have briefly examined the degree of interaction of the two overlapping condensates by comparing their ballistic expansion with that observed for single-species condensates. As can be seen in Fig. 3(a), the 12,2) condensate is pushed upward from its position in a single species cloud by the interaction with the lower lying 11, - 1) condensate. This indicates that the interaction is repulsive. We plan to make further studies about the overlap and interactions between the two condensates. This work has demonstrated an improved apparatus and a new cooling method for producing BEC in rubidium. The large number of atoms and long magnetic trap lifetime in this setup make it relatively easy to obtain BEC [17]. It will be straightforward to use this approach to further explore the detailed interactions between the two overlapping condensates. Also, the method of sympathetic cooling will allow future experiments involving the cooling of rare andlor fermionic isotopes. It would be very difficult to cool fermionic atoms into a highly degenerate regime using normal evaporative cooling because of the requirement of a large number of elastic collisions per trap lifetime, combined with the vanishing elastic collision rate in low-temperature spin-polarized fermionic gases. However, it should be quite feasible to use bosonic atoms as a working fluid to sympathetically cool a fermionic gas into the interesting [ 181 degenerate regime. This technique will also allow one to cool to BEC the many species for which inelastic processes make it impossible to obtain high enough densities for conventional evaporative cooling. We are pleased to acknowledge assistance in this work from all members of the JILA BEC group, and particularly N. Newbury and N. Claussen. This work was supported by ONR, NIST, and NSF. R.W.G. acknowledges the support of an NSF graduate fellowship. Note added.-Recent theoretical studies of binary mixtures of Bose condensates [19] predict a rich variety of interesting behaviors for two component condensates. [l] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E.A. Cornell, Science 269, 198 (1995). [2] K. B. Davis el al., Phys. Rev. Lett. 75, 3969 (1995). [3] C.C. Bradley, C.A. Sackett, and R.G. Hulet (to be published). [4] D.S. Jin, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Phys. Rev. Lett. 77, 420 (1996); M.-0. Mewes el al., Phys. Rev. Lett. 77, 988 (1996); J.R. Ensher, D.S. Jin, M.R. Matthews, C.E. Wieman, and E. A. Cornell, Phys. Rev. Lett. (to be published).
27 JANUARY1997
[5] D. J. Larson, J. C. Bergquist, J. J. Bollinger, W.M. Itano,
and D. J. Wineland, Phys. Rev. Lett. 57,?0 (1986). Achieving BEC in the single MOT apparatus of Ref. [I] required precise optical alignment of a dark spot MOT and careful optimization of rubidium pressure, laser trapping frequencies, laser cooling and compression parameters, optical pumping, and rf evaporation rates. In the present double MOT apparatus the alignment of the MOTS is not critical and little care is required to optimize any of the other experimental conditions listed above. C. J. Myatt, N.R. Newbury, R. W. Christ, S. Loutzenhiser, and C. E. Wieman, Opt. Lett. 21, 290 (1996). C. Monroe, E. Cornell, C. Sackett, C. Myatt, and C. Wieman, Phys. Rev. Lett. 70,414 (1993); N. Newbury, C. Myatt, E. Cornell, and C. Wieman, Phys. Rev. Lett. 74, 2196 (1995). T. Bergeman, G. Erez, and H. J. Metcalf, Phys. Rev. A 35, 1535 (1987). BEC has also recently been created in a trap that has a different coil configuration than ours but is the same Ioffe field geometry, by M.-0. Mewes et al., Phys. Rev. Lett. 77, 416 (1996). K. MacAdam, A. Steinbach, and C. Wieman, Am. J. Phys. 60, 1098 (1992). W. Petrich, M.H. Anderson, J.R. Ensher, and E.A. Cornell, J. Opt. SOC.Am, B 11, 1332 (1994). This simple pumping scheme typically results in of the atoms ending up in the other six spin states. However, all of these other states quickly leave the magnetic trap. The only state that is magnetically trapped, other than the 11, -1) and 12,2) states, is the 12,l) state. We see this state quickly depleted, presumably by spin-exchange collisions. When there is a large 12,2) cloud, however, we observe a small residual 12,l) population that we believe is due to the balance between 12,2) + 12,2) dipole collisions populating the 12,l) state and spin-exchange collisions depleting it. We take the second image on the subsequent identical cooling cycle. N. Newbury, C. Myatt, and C. Wieman, Phys. Rev. A 51, R2680 (1995) and errata (submitted for publication) reported the absolute value of the 11, - 1) state scattering length to be 87(10)ao. Note that the uncertainty given in the original publication was incorrect. H. M. J. M. Boesten et al., Phys. Rev. A 55, 636 (1997) gave the 12,2) state scattering length to be 110(10)ao. This is only achieved after a quite careful shielding against stray laser light. If the scattering length were negative, it would not be possible theoretically to form such large condensates. For example, see P.A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995). The critical experimental issue is the stability of the magnetic trap. The potential energy at trap center, i.e., the bias field, must not vary by much more than the trap energy level spacing during the final stages of the rf ramp. H.T.C. Stoof, M. Houbiers, C.A. Sackett, and R.G. Hulet, Phys. Rev. Lett. 76, 10 (1996); M. Houbiers and H.T.C. Stoof, Czech. J. Phys. 47 (suppl sl) 551 (1996). T.-L. Ho and V.B. Shenoy, Phys. Rev. Lett. 77, 3276 (1996); B. D. Esry, C. H. Greene, J. P. Burke, and J. L. Bohn (private communication).
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VOLUME78, NUMBER5
PHYSICAL REVIEW LETTERS
3 FEBRUARY 1997
Temperature-Dependent Damping and Frequency Shifts in Collective Excitations of a Dilute Bose-Einstein Condensate D. S. Jin,* M. R. Matthews, J . R. Ensher, C. E. Wieman, and E. A. Cornell* Joint Institute j o r Lahoi-afoi-yAstrophysics, National Instifiile of Standards and Technologv und Unive/xi@of'Colorndo. and Pli,vsics Departnient, University o j Colorado, Boulder, Colorado 80309-0440 (Received 29 October 1996) We extend to finite temperature the study of collective excitations of a Bose-Einstein condensate in a dilute gas of "Rb. Measurements of two modes with different angular momenta show c in expected temperature-dependent frequency shifts, with very different behavior for the two symmetries; in addition there is a sharp feature in the temperature dependence of one mode. The damping of these excitations exhibits dramatic temperature dependence, with condensate modes at temperatures near the transition damping even faster than analogous noncondensatc oscillations. [SO03 1-9007(96)02283- I ] PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj
Bose-Einstein condensation (BEC) in a dilute atomic vapor [l-31 has been the subject of several recent experimental studies characterizing this weakly interacting quantum fluid. In particular. measurements of interparticle interactions and the low-lying collective excitations of the condensate [4-71 show excellent agreement with theoretical predictions based on a mean-field description of the condensate. This ability to make quantitative comparison between experiment and theory is one of the primary advantages of BEC in a dilute atomic gas. Previous excitation measurements were performed in a regime for which there is no detectable noncondensate fraction. This Letter expands the study of low-lying collective excitations of condensates to include higher temperatures, a regime where theoretical predictions do not yet exist [8]. At these temperatures the interplay between condensate and noncondensate components, a potentially dissipative process not included in the usual mean-field theoretical treatment [9], strongly affects the physics. We find large energy shifts and unexpected structure in the condensate excitation spectrum, as well as a damping rate which plummets with decreasing temperature. These dramatic temperature effects highlight the need for new theoretical understanding. The apparatus and techniques used for creating a Bose-condensed sample in a dilute atomic gas of "Rb [1,10,11], accurately determining its temperature [12], and probing the spectrum of low-lying collective excitations [4] are described elsewhere. We first optically trap and cool the atoms, then load them into a purely magnetic TOP (time-averaged orbiting potential) trap [ 101. This trap provides a cylindrically symmetric, harmonic confinement potential. Cooling the atoms in the TOP trap proceeds by forced evaporation, whereby an applied weak radio-frequency (rf) magnetic field induces Zeeman transitions to untrapped spin states [ 131. The final temperature is controlled by the final frequency of the rf field. The atom cloud is observed by absorption imaging, in which the shadow of the atomic cloud is imaged onto a charge-coupled device array. To circumvent limitations 764
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of optical imaging resolution we allow the atom cloud to expand freely for 10 ms before imaging [ 1,4]. The condensates are produced in a trap with radial frequency v r = 129 Hz (365 Hz axial), with the same evaporation parameters as used in our recent quantitative study of the BEC phase transition [ 121. We thus perform finite-temperature studies of condensate excitations in a well-characterized system, complementing our previous low-temperature study of collective excitations which examined the dependence on relative interaction strength. We report our results as a function of reduced temperature T' = T/T,, where To is the predicted BEC transition temperature for a harmonically confined ideal gas [14]. The estimated systematic uncertainty in T' is 5% for TI > 0.6, and 10% for T' < 0.6. At TI < 0.6 the noncondensate component becomes unobservably small and the temperature is inferred from the final frequency of the rf field. The lowest attainable T' is limited by the evaporation parameters and the extended size of the condensate; deeper evaporative cuts quickly reduce the number of atoms in the condensate. While plotting our results as a function of T ' , we emphasize that the evaporative cooling process changes the total number of atoms as well as the temperature. Figure 1 shows various quantities relevant for our particular evaporation parameters. The basic spectroscopic approach for studying collective excitations follows Ref. [4]. First, a small applied sinusoidal, time-dependent perturbation to the transverse trap potential distorts the cloud. We then turn off the perturbation and the cloud freely oscillates in the unperturbed trap. Finally, the cloud is suddenly allowed to expand and the resulting cloud shape imaged. The symmetry of the drive perturbation can be varied in order to match the symmetry of a particular condensate mode. The modes are labeled by their angular momentum projection rn on the trap axis. In this work, we examine the previously observed m = 0 and m = 2 modes [4,7]. Frequency, amplitude, and damping rate of the excitations are determined as shown in Fig. 2.
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The main results of this work are shown in Fig. 3. For these data, the perturbative drive pulse duration was typically 14 ms, with the center frequency set to match the frequency of the excitation being studied. The response amplitudes (defined in caption for Fig. 2) for these data were kept small, between 5% and 15%. The radial trap frequency v,. is calibrated by driving a rn = 1, or “sloshing,” excitation of a cloud at T’ = 1.3. Interatomic interactions should not affect this excitation which consists solely of rigid-body center-of-mass translation. We have also checked that anharmonicities in the confining potential are negligible in our measurements. The normalized response frequencies as a function of T’ are shown in Fig. 3(a). Collective excitations of a noninteracting cloud should occur at twice the trap frequency (dashed line). Indeed, above the BEC phase transition, the rn = 0 and m = 2 excitations are basically
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T‘ FIG. 1. We plot (a) temperature T and (b) total number of atoms N as a function of normalized temperature T’. The relationship between N and T is not fundamental but rather a consequence of our evaporative trajectory. In (c) we plot the number of condensate atoms NBECand in (d) the inferred noncondensate number Nnc. Solid lines are guides to the eye. Because of their different spatial extents, the peak density of the condensate component is an order of magnitude larger than that of the noncondensate component below T‘ = 0.9.
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FIG. 2. In this example, we plot the observed widths, obtained from 2-Gaussian surface fits to absorption images, of a T’ = 0.79 cloud with an m = 0 excitation. The widths of each component, condensate and noncondensate, of the freely oscillating cloud are fit by an exponentially damped sine wave: A exp(-yt) sin(27rvt + 4) + B, from which we obtain the response frequency v, initial fractional amplitude A / B , and decay rate y . For each pair of points (condensate and noncondensate widths) a fresh cloud of atoms is cooled, excited, and allowed to evolve a time t before a single destructive measurement.
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T’ FIG. 3. Temperature-dependent excitation spectrum: (a) Frequencies (normalized by the radial trap frequency) for rn = 0 (triangles) and rn = 2 (circles) collective excitation symmetries are shown as a function of normalized temperature T’. Oscillations of both the condensate (solid symbols) and noncondensate (open symbols) clouds are observed. Short lines extending from the left side of the plot mark the mean-field theoretical predictions in the T = 0 limit (for 6000 atoms in our trap) [9]. (b) For both the rn = 0 and rn = 2 condensate excitations the damping rate y quickly decreases with decreasing temperature.
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degenerate at v / v r =: 2. The noncondensate frequencies are consistently about 1% higher than 2v,, presumably because interactions are not completely negligible even at the relatively low densities of these clouds [15]. The noncondensate excitation frequencies do not vary significantly with temperature, even for T’ < 1. In contrast, the excitation spectrum of the condensate exhibits strong temperature dependence. While some frequency shift might be expected because of the temperature dependence of the number of condensate atoms, and therefore the relative interaction strength, the magnitude of the observed temperature-dependent frequency shifts is several times larger than that observed due to interaction effects [4]. In addition, the frequencies of these two modes, which are degenerate in the limit of zero interactions, actually shift in opposite directions, with only the m = 0 heading to the noninteracting limit. A sharp change is apparent in the slope of the temperature dependence of the m = 0 excitation frequency. Repeated measurements confirm that this distinct feature at T’= 0.62 reproduces. We speculate this might arise from coupling with another mode, perhaps with a strongly temperaturedependent second-sound excitation or with the excitation of the noncondensate component. No corresponding distinct feature is evident in the temperature dependence of the m = 2 condensate excitation frequency. Figure 3(b) presents the decay rate y as a function of temperature. While the frequencies of the two condensate modes behave very differently, their decay rates appear to fall on a single curve, with y quickly decreasing with decreasing T’[16]. Another remarkable feature is that for temperatures where two-component clouds are observed, the condensate excitations damp much faster than their noncondensate counterparts (see also Fig. 2). The strong temperature dependence suggests that finite decay times reported in Refs. [4] and [7] were due to the finite temperature of the samples and that condensate excitations may persist for very long times at lower T’. In the limit of low amplitude response, the measured spectrum corresponds to the elementary excitations of BEC in a dilute gas. At our lowest temperature, T’ = 0.48, we examine the condensate excitations as a function of increasing response amplitude. The rn = 0 mode exhibits a significantly smaller anharmonic frequency shift than the mode with m = 2 symmetry [Fig. 4(a)]. The decay rates y for the rn = 0 and rn = 2 condensate excitations show some dependence on response amplitude [Fig. 4(b)], but in neither case does y approach zero in the limit of small response. These data demonstrate that v and y measured for a typical response amplitude of 10% are fairly close to the zero-amplitude perturbation limit. For comparison with the results of our free-evolution spectroscopy technique, Fig. 5 presents a more conventional resonance measurement. While the central frequency is in good agreement with that measured from observing the freely oscillating widths as discussed above,
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response (“A) FIG. 4. For large response amplitudes at our lowest temperature, T’ = 0.48, we measure the shift in the response frequency Y and damping rate y . In (a) the frequencies, normalized by the small drive limit, show a larger anharmonic shift for the m = 2 condensate excitation than for the m = 0 case. (Some of the m = 2 data is the same as presented in Ref. [4] and is reproduced here to facilitate comparison.) Solid lines are guides to the eye. In (b) the decay rates y for the m = 0 and m = 2 condensate modes shift in opposite directions with increasing response amplitude.
the resonance technique is inefficient in terms of data collection time and moreover vulnerable to systematic errors. In particular, extracting a damping rate from a linewidth measurement requires a drive duration long compared to
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FIG. 5. The response of the rn = 0 condensate excitation at T’ = 0.79 as a function of the center drive frequency. Solid lines show fits to the expected form for the (a) amplitude and (b) phase of a driven harmonic oscillator in the presence of damping. For these measurements a constant amplitude drive is applied with a duration of 150 ms (about 7 Hz FWHM in frequency), after which the resulting excitation is observed for 10 ms to determine amplitude and phase.
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the damping time, but dissipation during the long drive then alters the cloud temperature and density. This paper extends previous shidies of condensate excitations to temperatures for which both condensate and noncondensate a t o m are present in significant numbers. This regime is not yet well understood theoretically, as the standard mean-field theoretical treatment of collective excitations of a harmonically confined dilute BEC [9] is done for the T = 0 limit and does not include dissipation. Both n z = 0 and in = 2 condensate modes exhibit large temperature-dependent frequency shifts. The decay rate of the condensate collective excitations decreases rapidly with temperature, and shows no signs of leveling off. We believe this data, along with further theoretical study of these elementary excitations, will help deepen the understanding of Bose-Einstein condensation and superfluidity of a gas. This work was supported by the National Institute of Standards and Technology, the National Science Foundation, and the Office of Naval Research. The authors would like to express their appreciation for useful discussions with the other members of the JILA BEC Collaboration. *Quantum Physics Division. [ l ] M.H. Anderson, J.R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995). [2] K. B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995). [3] C.C. Bradley, C.A. Sackett, and R.G. Hulet (to be published). [4] D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 77, 420 (1996). [5] D.S. Jin, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E. A. Cornell, in Proceedings of the XYI International Conference on Low Temperature Physics [Czech. J . Phys. 46, S6 (1996)l; M. Holland, D. S. Jin, M. Chiofalo, and J. Cooper (to be published).
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[6] M.-0. Mewes etal.. Phys. Rev. Lett. 77, 416 (1996). [7] M.-0. Mewes etal., Phys. Rev. Lett. 77, 988 (1996). [S] Since submission of this paper, however, we received a preprint of D.A. W. Hutchinson, E. Zaremba, and A. Griffin, cond-niat./9611023. [9] K . G . Singh and D.S. Rokhsar, Phys. Rev. Lett. 77, 1667 (1996); M. Edwards, P.A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, Phys. Rev. Lett. 77, 1671 (1996); S. Stringari, Phys. Rev. Lett. 77, 2360 (1996); B.D. Esry, Phys. Rev. A (to be published); A.L. Fetter, Czech. J. Phys. 46, S1, 547 (1996); L. You, W. Hoston, M. Lewenstein, and K. Huang (to be published); M. Marinescu and A . F. Starace (to be published). [ l o ] W. Petrich, M. H. Anderson, J . R. Ensher, and E. A. Cornell, Phys. Rev. Lett. 74, 3352 (1995). [ l l ] W. Petrich, M. H. Anderson, J. R. Ensher, and E.A. Cornell, J. Opt. SOC. Am. B 11, 1332 (1994); M. H. Anderson, W. Petrich, J. R. Ensher, and E.A. Cornell, Phys. Rev. A 50, R3597 (1994). [ 121 J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 77, 4984 (1996). [13] H. F. Hess et al., Phys. Rev. Lett. 59, 672 (1987); D. Pritchard et al.. in Proceedings of the 11th International Conference on Atomic Physics, edited by S . Haroche, J. C. Gay, and C. Grynberg (World Scientific, Singapore, 1989), pp. 619-621. [ 141 The observed transition temperature is T’ = 0.94(5) [ 121. [15] In our previous study of condensate excitations at T’ = 0.48, we used the 172 = 0 noncondensate mode for T’ = 1.3 to calibrate the trap frequency. The present work indicates that this caused the values of v/u, given in Ref. [4] to be too low by 1%. We thank W. Ketterle for pointing out this possible source of systematic error. [16] To check that any systematic error due to frequency noise is small we have also determined the standard deviation of the measured cloud widths for time intervals corresponding to a couple oscillation periods. The standard deviations exponentially decay with time to our noise floor at a rate consistent with the y determined from a damped sine wave fit.
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Coherence, Correlations, and Collisions: What One Learns about Bose-Einstein Condensates from Their Decay E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman JILA, Nation01 lnsrirwre of‘Standards and Technology, and Department of’ Physics. University of‘ Colorado, Boi11dt.r: Colorado 80309-0440 (Received 2 May 1997) W e have used three-body recombination rates as a sensitive probe of the statistical correlations between a t o m in Bose-Einstein condensates (BEC) and in ultracold noncondensed dilute atomic gases. We infer that density fluctuations are suppressed in the BEC samples. We measured the three-body recombination rate constants for condensates and cold noncondensates from number loss in the F = I , m i = - I hyperfine state of ”Rb. The ratio of these is 7.4(2.6) which agrees with the theoretical factor of 3 ! and demonstrates that condensate atoms are less bunched than noncondensate atoms. [SO03 1-9007(97)036 1 1-91 PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 4 2 . 5 0 . D ~
The onset of Bose-Einstein condensation (BEC) is defined by the sudden accumulation of many bosons in a single quantum state. The symmetry property of bosons is such that if a gas is indeed composed of many identical bosons all occupying the same single-particle state, the gas will exhibit a collection of correlation properties known as coherence. While most early experiments on dilute-gas BEC [ 1-31 have shown good quantitative agreement with the simple physical model of macroscopic occupation of a single state, no dilute-gas experiment explicitly addressed the issue of coherence in the condensate until the striking observation by Andrews et al. [4] of first-order coherence in a sodium condensate. In this paper we describe collision-rate measurements that probe the higher-order coherence properties of thermal and Bose-condensed rubidium atoms [5]. In particular, the coherence of the BEC ground state is contrasted with the chaotic fluctuations of the ultracold noncondensed states. The correlation properties of degenerate samples of ideal bosons have already been extensively studied in the context of quantum optics [6]. In fact, the close analogies between the macroscopically occupied state of a laser beam (characterized as a “coherent state”) and that of a Bose condensate have prompted the use of the term “atom laser” to describe some aspects of BEC [7]. Quantum optics teaches that a laser beam is described by a quantum field that exhibits both (i) “first-order coherence,” meaning that a measurement of the phase of the field at one point in space and time may be used to predict the phase of the field at some other point [8] and (ii) “higher-order coherences,” meaning in essence that the intensity fluctuations in a coherent sample are suppressed relative to those in a thermal sample with the same mean intensity. The analog of intensity fluctuations in a beam of photons is density fluctuations in a gas of atoms. For example, Fig. 1(a) shows the calculated [9] three-body correlation hnction for a gas of thermal (i.e., noncondensed) bosons. Note that there is an enhanced probability for finding three bosons close together. The same
physics accounts for short-time photon bunching in a thermal light beam (the Hanbury-Brown-Twiss effect [lo]), for the two-atom bunching that has been observed in beams of ultracold (but not condensed) atoms [ l l ] , and for three-pion correlations in p p annihilations [ 12,131. To paraphrase Walls and Milburn [14] (who were in fact actually discussing two-photon correlations), the physical origin of such correlations may be understood in terms of a noisy quantum field: There is a high probability that the first boson is found at a high intensity fluctuation, and hence an enhanced probability for finding a second and third atom boson nearby. The correlations in the positions of multiple, identical bosons thus strongly depend on the type of fluctuations that exist in the density. For example, for Gaussian (thermal) fluctuations, the average of the square of the density is a factor of 2 larger than the square of the average density, and this is precisely the factor of 2 observed in two-boson correlation experiments. The atom-bunching effect is expected to vanish in a condensate, precisely as photon bunching does in an ideal laser bean-see Figs. l(b) and l(c). In principle, with sufficiently high spatial and temporal resolution, the density fluctuations in a dilute gas could be imaged directly [ 151, or detected as coincidence counts in a beam experiment [ l l ] . Kagan and Shlyapnikov [16] have pointed out, however, that an easier experimental approach to probing fluctuations is to take advantage of an observable that is directly sensitive to the probability of finding three atoms near each other, that is, the loss rate of atoms due to three-body recombination. They calculated that three-body recombination in a condensate would be a factor of 3! less rapid than in a thermal cloud at the same mean density, and proposed that this change in recombination rate could be a useful signature for detecting the onset of BEC. It is this proposal that motivates our experimental approach, although in the current experiment, the empirical onset of BEC is independently identified by the appearance of a sharp feature in the center of the coordinate and momentum-space atom distribution. We use
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FIG. 1. (a) Calculated third-order correlation function, g‘’) [6], far noncondensed ideal bosons. Given a particle at the origin, and a second particle a distance x from the origin, the z axis gives the relative, conditional probability for finding a third particle a distance y from the origin. The de Broglie wavelength is defined by Ad = h / ( 2 v m k o T ) ’ / * where , rn is the particle’s mass and T is temperature. The calculation assumes a cloud in the dilute limit and is not valid for distances x and y less than or equal to the range of two-body interactions. (b) A sample cut through the surface shown in (a), with x set equal to y for ease of display. Note that in a thermal cloud one is 3! times more likely to find three atoms close together than one would naively assume given the mean density. The factor of 3! vanishes for the Bose-condensed atoms. Thus if one has Bosecondensed and non-Bose-condensed samples at similar density, one would expect a factor of 3! less three-body recombination. (c) A cut through the surface shown in (a) with the third particle held well away from the origin ( x / A d = 2). Here the surface reduces to a familiar two-boson correlation function, showing the factor of 2 at the origin well known in photonbunching experiments. Preliminary measurements of the “atom bunching” in a thermal atom beam have been reported by Yasuda [ 111.
the shape of the mean density distribution to normalize the observed three-body loss rate, and thus extract a rate constant. The central result of this paper is that our comparison of the three-body recombination rate constant in condensed and noncondensed samples provides a quantitative confirmation of the predicted factor of 3! [17], and thus provides very strong evidence for the existence of higher-order coherence in Bose-condensed rubidium. In this experiment, collisional rate constants were inferred from the loss rate of atoms from the trap. It is known that there are three loss processes for ultracold atoms in a magnetic trap: (1) collisions with background gas, (2) dipolar relaxation, and (3) three-body recombination. However, prior to this work the rate constants for these three processes in very cold rubidium clouds, either condensed or noncondensed, were not accurately known.
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We use the fact that these loss processes have different density dependencies to distinguish their respective contributions. Measurement of the loss rate of atoms from the trap as a function of density is most easily done by preparing a sequence of identical samples and then measuring the number and density of the sample as a function of time. The density decreases with tiine because of both the loss of atoms and the rise in the sample temperature due to heating. The experimental apparatus and techniques used to make and study the cold clouds have been described previously [18]. Using a double magneto-optic trap (MOT) we collect lo9 atoms. These are optically pumped into the F = 1 , inf = - 1 state and loaded into a baseball-coil magnetic trap. The bias field i s determined by an additional set of Helmholtz coils. The net bias is 1.6 G with a radial field gradient of 300 G/cm and axial curvature 84 G/cm?. The atoms are then cooled by radio frequency (rf) evaporation. After being evaporatively cooled to the desired temperature, the cloud is held in the magnetic trap for varying intervals of time t . It is then released from the trap and imaged using absorption, or recaptured in a MOT. We find the temperature T of the cloud from the velocity distribution of the noncondensate atoms observed in the absorption image. Although the number of atoms N can also be obtained from this image, we find that we can determine N more precisely by detecting fluorescence on a photodiode from atoms recaptured in a MOT. To study cold noncondensates we evaporatively cool the atoms to a temperature of 800 nK, slightly above the transition temperature of 670 nK. At this point we have 7 X lo6 atoms at a peak density of 5 X 1013 cmp3. To study condensates, we cool the atoms to and have 250 nK and have 2 X lo6 atoms at a peak density of 5 X l O I 4 ~ m - ~ . A typical set of data is shown in Fig. 2. It can be seen that at long times, or equivalently, low densities, the loss rate is exponential, indicating that it has no dependence on the density of the sample. The deviation from the exponential line at short times indicates that at higher densities the loss rate is primarily density dependent. To determine the dependence we must determine the density of each sample. We obtain this from the temperature and number of atoms in the sample. We fit the absorption images with a thermal Bose-Einstein distribution to find the sample temperature. To avoid problems with large optical depths we fit the absorption images only in the wings. The number of atoms in the trap is simply found from the photodiode fluorescence signal using the known scattering rate per atom. For noncondensate clouds we calculate the original density in the unreleased trap for each hold time from the measured temperature and number and the known spring constants for the harmonic potential. We do not measure condensate density directly but instead measure total number N and temperature T and infer density from experimentally well-established properties of alkali condensates. From measured N and T we infer [2] the condensate occupation number NO. We
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Time [sec] FIG. 2. The natural log of the number of atoms as a function of time in a typical set of data (in this case, for noncondensate atoms). At long times the number loss is due to background collisions and is independent of density. When plotted in this way, the data fall on a straight line with slope equal to the negative inverse of the lifetime, r , set by background collisions, or about 250 s. The deviation of the data from this straight line at short times is the density-dependent loss.
model the condensate density profile n ( x ) as an inverted parabola proportional to Uo - U ( x ) , where U ( x ) is the trapping potential and Uo is a function of No. The function U o ( N ) can be determined from the ThomasFermi limit of the Gross-Pitaevski (GP) [I91 equation combined with molecular spectroscopy data [20], but we need not assume the validity of the GP equation nor indeed even of quantum mechanics to justify the inverted parabola shape. The shape is due to balance of forces and will be valid as long as the cloud is dilute and kinetic energy (KE) is a small contribution to the condensate energy [21]. We determine the form of CJo(N) (i.e., UO a and the prefactor for rubidium from published condensate expansion data [3], again assuming only that KE is small in the condensate, and that the condensate self-interacts in a dilute fashion [21]. This gives the density of the clouds as a function of time. The loss due to three-body recombination is modeled by the rate equations dN - = -K, dt or equivalently
are plotted as shown a single three-body loss process appears as a straight line with a slope equal to the negative of the desired rate constant. The different slopes for noncondensate and condensate data give the rate constants, K;‘ = 4.3(1.8) X cm6/s and KS = 5.8(1.9) X cm6/s, respectively. This value for K;‘ is in good agreement with a calculation by Fedichev et a/. [23], even though it is not clear that the experimental system satisfies the claimed range of validity for the theory. The ratio of the noncondensate to the condensate rate constants is 7.4(2.6). This is in agreement with the theoretical value of 3 ! (for noninteracting atoms) and is dramatically different from 1. This proves that, relative to thermal atoms, the density fluctuations are suppressed for condensate atoms (“higher-order coherence”) as in a laser or any macroscopically occupied state of ideal bosom. The rate constants are quite sensitive to uncertainties in the determination of temperature and number. These uncertainties then are the primary cause for the errors in the rate constants. The heating that we observe also has a density dependence, under some conditions, raising concerns that there may be an associated loss that would distort our results. To check that this was not the case, we took data with the rf evaporative field on and set to different frequencies during the hold time. These frequencies are well above that used to determine the initial sample temperature, but have a dramatic effect on the heating rate. As shown in Fig. 3, the loss rates from the samples were the same although the heating rates were very different.
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n m ( x ,t ) d 3 x
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(2) The rate constant for an m-body process, K,, is determined from a fit to Eq. (2), n is the density, and N ( t ) is the number after time t . The condensate clouds contain some noncondensate fraction and, in general, the rate constants for the two parts will be different. We find that all of the density-dependent loss we observe is due to three-body recombination. This can be seen in Fig. 3. Equation ( 2 ) shows that when the data
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-
We could see no indication of a loss rate that was linear in density which would be the signature of the two-body process dipolar relaxation. With our statistical uncertainties, we can set an upper bound on the dipolar relaxation loss rate constant for the F = 1, m f = - 1 state of 87Rb cm3/s. This is in a 1.6 G field of K?nC5 1.6 X consistent with the small values predicted for alkali atoms in the F = 1, mi = - 1 state [24]. This work represents a quantitative demonstration that BEC atoms have the higher-order coherence characteristic, for instance, of laser photons. We have also shown that the dominant loss process for cold noncondensates and condensates in the F = 1 state of s7Rb is three-body recombination. The rate constant that we have determined sets a limit on attainable lifetimes and densities in condensate samples of 87Rb. We are grateful to all members of the JILA BEC group for much useful advice and assistance, particularly J. Cooper and N . Claussen. We had useful conversations with W. Ketterle on the topic of atomic coherence. This work was supported by ONR, N E T , and NSF. R. W. G. acknowledges the support of an NSF graduate fellowship.
M.-0. Mewes et a/., Phys. Rev. Lett. 77, 416 (1996); D. S. Jin et al., Phys. Rev. Lett. 77, 420 (1996); M.-0. Mewes et a/., Phys. Rev. Lett. 77, 988 (1996); C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 78, 985 (1997). J. R. Ensher et al., Phys. Rev. Lett. 77, 4984 (1996). M. J. Holland, D. S. Jin, M.L. Chiofalo, and J. Cooper, Phys. Rev. Lett. 78, 3801 (1997); M.-0. Mewes et al., Phys. Rev. Lett. 77, 416 (1996); Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996). M. R. Andrews et al., Science 275, 637 (1997). It has recently been pointed out that data on condensate expansion energy, combined with spectroscopic scattering length measurements, can be interpreted as showing second-order coherence in alkali condensates; W. Ketterle and H. J. Miesner (to be published). For a discussion of the formal aspects of first-, second-, and third-order coherence, see, for example, D. F. Walls and G. J. Milbum, Quantum Optics (Springer-Verlag, Berlin, 1994). The fact that the primary difference between a maser field and a chaotic field is demonstrated by higher-order correlation functions was first explored by Glauber [R. J . Glauber, Phys. Rev. 130, 2529 (1963)l. M.-0. Mewes et al., Phys. Rev. Lett. 78, 582 (1997). The term atom laser has also been used to refer to continuousprocess evaporation models, [e.g., M. Holland et al., Phys. Rev. A 54, R1757 (1996)l and to optical pumping-based proposals [e.g., R. J. C. Spreeuw, T. Pfau, and M. Wilkens, Europhys. Lett. 32, 469 (1995)l. First-order coherence can be imposed, albeit over a limited spatial range, on a thermal beam of photons by means of a narrow-band filter and a pin hole, and in atoms by spatial filtering and supersonic expansion. First-order coherence
has been probed in atom interferometry experiments [see, e.g., J. Schmiedniayer et al., Adv. At. Mol. Phys., Suppl. 3, 2 (1997)l. First-order coherence is a natural and dramatic consequence of the macroscopic single-particle BEC state, and it should extend over the entire condensed sample. This was beautifully confirmed by Andrews et a/.[4]. [9] The thermal field which gives the noncondensate surface in Fig. 1 obeys Gaussian statistics with zero mean amplitude. Consequently Wick’s theorem may be iteratively applied to reduce all higher-order correlation functions to simple functionals of the first-order correlation function. I R. Hanbury Brown and R. Q. Twiss, Nature (London) 177, 27 (1956). M. Yasuda and F. Shimizu, Phys. Rev. Lett. 77, 3090 (1996); M. Kasevich (private communication) is perfoniiing a similar measurement. G. Goldhaber et al., Phys. Rev. 120, 300 (1960). I . Juricic et al.. Phys. Rev. D 39, 1 (1989). 0.F. Walls and G. J. Milburn, Quantum Optics (Ref. [6]), p. 40. Direct imaging of “density noise” may yet be a practical technique, particularly near the critical temperature; C. Gardiner (private communication). Y. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, JETP Lett. 42, 209 (1985). The calculated value of 3! is for an ideal gas. Interactioninduced three-body correlations [16] lead to a correction that is smaller than the sensitivity of the present experiment. C. J. Myatt et al., Opt. Lett. 21, 290 (1996); C. J. Myatt et a/., Phys. Rev. Lett. 78, 586 (1997). D. A. Huse and E. D. Siggia, J. Low Temp. Phys. 46, 137 (1982). P. S. Julienne, F. H. Mies, E. Tiesinga, and C. J. Williams, Phys. Rev. Lett. 78, 1880 (1997); see also J. Burke, Jr. et a/., Phys. Rev. A 55, R25 11 (1997); H. M. J. M. Boesten et al., Phys. Rev. A 55, 636 (1997). The assumption that KE in a condensate is small is consistent with all empirical observations, including expansion energy measurements. It is amusing to note that the central result of this paper, that fluctuations in density are much smaller in a condensate than in a thermal cloud, does not depend on the small KE assumption: If KE were not negligible, published expansion data would yield only an upper limit on condensate size such that the value we quote for the suppression of three-body correlations, 7.4(2.6), would be only a lower limit. The error bars in Figs. 2 and 3 represent plus or minus one standard error and are derived from the statistics of the data. They do not include systematic errors explicitly, but due to systematic drifts during the experiment some systematic error is implicitly included. Consequently the error bars are larger than they would be if they were purely statistical. P. Fedichev, M. Reynolds, and G.V. Shlyapnikov, Phys. Rev. Lett. 77, 2921 (1996); see also B. D. Esry et a/., J. Phys. B 29, L51 (1996). F.H. Mies, C. J. Williams, E. Tiesinga, and P. S. Julienne (private communication); H. M. J. M. Boesten, A. J. Moerdijk, and B.J. Verhaar, Phys. Rev. A 54, R29 (1996). ,
I
VOLUME 57, NUMBER 3
PHYSICAL REVIBW A
MARCH 1998
Achieving steady-state Bose-Einstein condensation J. Williams, R. Walser, C. Wieman, J. Cooper, and M. Holland JILA and Department of Physics, Univer,rit);of Colorado, Boulder, Colorado 80309-0440 (Received 7 October 1997) We investigate the possibility of obtaining Bose-Einstein condensation (BEC) in a steady state by continuously loading atoms into a magnetic trap while keeping the frequency of the radio frequency field fixed. A steady state is obtained when the gain of atoms due to loading is balanced with the three dominant loss mechanisms due to elastic collisions with hot atoms from the background gas, inelastic threc-body collisions, and evaporation. We describe our model of this system and present results of calculations of the peak phasespace density p o in order to investigate the conditions under which one can reach the regime p022.612 and attain BEC in steady state. [SlO50-2947(98)05503-6] PACS number(s): 03.75.Fi, 05.20.Dd, 32.XO.Pj
I. INTRODUCTION
In the usual method of evaporative cooling used so far in Bose-Einstein condensation (BEC) experiments [ 1-61, a finite number of atoms are collected in a magnetic trap after being laser cooled to a phase space density at least five orders of magnitude below the critical density needed for BEC. The frequency of an external RF radiation field, which spinflips the atoms to an untrapped state, is then lowered continuously. This further cools the gas by removing high energy atoms from the tail of the distribution. This evaporative cooling procedure increases the phase space density above the critical point needed to reach BEC. The success of this method is well established experimentally, allowing many fundamental properties of Bose-Einstein condensation to be investigated [7-121. This standard method of achieving BEC has one critical drawback: once a condensate has been obtained, it has a finite lifetime in the trap determined by various loss mechanisms, such as collisions with hot atoms from the background gas, and inelastic collisions between the trapped atoms. Although the finite lifetime of the condensate does not prevent many crucial properties of the system to be studied, it is still very desirable to achieve a steady-state situation so that a condensate can be sustained for an indefinite period of time. Such a situation is essential for the continuous output of a coherent beam of atoms in an atom laser [ 13- 181. To date, no experiment has demonstrated a steady-state condensation. We address this problem by constructing an intuitive model describing the two aspects to such an experiment: The continuous loading of atoms into the magnetic trap and the classical kinetic evolution of the trapped atoms toward a steady state during the evaporation. Our description of the loading procedure is based on the experimental setup described in [19], where the authors loaded a magnetic trap with atoms which had been cooled in a separate MOT. This allows us to estimate the rate y, that atoms enter the trap below the RF cut, as well as the mean energy ef of the injected atoms. To model the classical kinetic evolution, we assume a truncated Boltzmann distribution for the trapped atoms and obtain rate equations for the total number N ( t ) and energy 1050-2947/98/57(3)/2030(7)/$15.00
57 -
501
E ( t ) of the system [1,3-6]. These rate equations include the loss of atoms due to elastic collisions with the backgroundgas atoms, inelastic three-body collisions, and evaporation, as well as the gain of atoms due to loading. We then numerically calculate the steady-state solution of these equations and show plots of the peak phase space density p o as a function of the various physical parameters of the system. We show that the critical regime poa2.612 may be reached in order to obtain BEC in steady state. 11. DESCRIPTION OF THE MODEL
In constructing a model of steady-state evaporative cooling, there are several experimental schemes one could consider for describing the loading of atoms into the magnetic trap, as well as several layers of approximation in describing the kinetic evolution of the trapped gas toward steady state. However, we consider only one realization of the loading procedure, assuming the atoms are first trapped and cooled in a MOT and then transferred to a separate magnetic trap [19,20,1 I]. Furthermore, we consider a simplified model of evaporative cooling that assumes classical statistics, and is therefore valid only for phase space densities below the critical point p0=2.612; one would have to include quantum statistics in order to properly model the system above this point. These two parts to our model are described in the following subsections. A. Description of the loading procedure
In a real experiment, irreversibility is introduced at each step of the transfer of the atoms from the MOT to the magnetic trap; the atoms are first pushed out of the MOT, they then travel through a magnetically confining tube, and finally must be caught in the magnetic trap and optically pumped into a trapped hyperfine state. In order not to get lost in the details of modeling all of these heating and loss mechanisms, we consider two extreme idealizations of the transfer: an adiabatic transfer which preserves the phase space density po and a sudden, irreversible transfer which decreases po. We assume the atoms feel an isotropic, linear restoring force in both the MOT and the magnetic trap, neglecting the possibility of a radiation pressure in the MOT, which would 2030
0 1998 The American Physical Society
502 ~
ACHIEVING STEADY-STATE BOSE-EINSTEN CONDENSATION
57 -
203 I
distort the effective harmonic trapping potential [21]. Then the free Hamiltoniaii of an atom in either trap can be written
where ni is the mass of the atom, and w , is the effective radial frequency of the trapping potential. The index i = 1 indicates the MOT, while i = 2 indicates the magnetic trap. We model the transfer of atoms in order to obtain a reasonable estimate of the feed rate y1 and the mean energy e I of atoms injected into the trap below the RF cut. We treat this transfer process as a succession of discrete transfers each consisting of a finite number of atoms. We only need to consider a snapshot of this transfer process: a finite number of atoms N, are collected in the MOT at a temperature TI in equilibrinm, they are then either adiabatically or suddenly traiisfered to the magnetic trap. In our model, we allow these N, atoms to come to an equilibrium in the magnetic trap, characterized by a new temperature T 2 . We then place the RF cut ecutand calculate the fraction of atoms af which remain in the magnetic trap below ecut,as well as the mean energy per atom e f of these atoms,
-
e/kgT2de
ef= J 2 u l e 2 e - e / k g T ~ d e
'
(3)
The e2 factor appears due to the density of states for an isotropic harmonic oscillator potential. A schematic diagram in Fig. 1 illustrates the transfer process. This process can be repeated many times each second so that atoms are transfered to the magnetic trap at a rate y,. The rate that atoms enter below the RF threshold ecutis then given by y,= afy, . We estimate an upper limit on the number of these transfers each second to be on the order of 100. The equilibrium temperature T , which the atoms attain after a sudden transfer can be obtained by considering the sudden change in the energy of the atoms after the instantaneous change in trapping frequencies w I + w 2 . Then for a sudden transfer, the temperature T , is related to the temperature T I in the MOT according to
[; + 1;)
T2=-
1
-
Po",
?>
it is clear that po is invariant through an adiabatic transfer, while it decreases after a sudden transfer. Here, A is the de Broglie wavelength and no is the peak spatial density. The two quantities yf and ef depend on the frequency in the lower trap w 2 , as well as the RF field threshold e,,,; as the trap is made looser, more atoms will make it into the trap below the cut so that yf increases. The feed rate is also increased as ecutis raised, however the mean energy e,, of those atoms increases as well.
B. Description of evaporative cooling
(sudden).
(4)
The adiabatic case can be treated as a succession of infinitesimal steps w , +w I+ S w , each treated as a sudden transfer. This yields the relationship w2
FIG. I . This diagram illustrates the transfer process described in Sec. I I A . A finite number of atoms are cooled in the MOT to a temperature T I in equilibrium. We approxiniate the potential in the MOT as an isotropic harmonic oscillator at frequency w I . They are then transferred to the magnetic trap, either suddenly, or adiabatically. We also approximate the magnetic trap as forming an isotropic harmonic oscillator potential, with a different frequency w 2 . In equilibrium, the atoms have a temperature T , in the magnetic trap. Then, the RF energy threshold ecutis applied and only a portion of the original atoms from the MOT remains. This transfer can be repeated many times in order to obtain a piecewise continuous transfer of atoms.
T2= TI - (adiabatic).
(5)
@I
Note that both cases give T2= TI when w2= w l as they must. With the peak phase space density po=n,A3 of the trapped atoms given by
With the feeding rate yf and mean energy per atom e f of the injected atoms given by the above model of the loading procedure, it remains to describe the kinetic evolution of the atoms in the magnetic trap during evaporation. Our model can be constructed on phenomenological considerations, with the goal of characterizing the steady state of the system. We characterize the trapped atoms by a single-particle distribution over energy p( e ) f ( e , t ) instead of retaining the more detailed description in phase space using f ( x , p , r ) [l]. Here p ( e ) is the density of states for an isotropic harmonic potential. We also make an assumption that the nonequilibrium distribution f ( e , t ) of the system can be well approximated by a truncated Boltzmann distribution [ 1,3-61
503 WILLIAMS, WALSER, WIEMAN, COOPER, AND HOLLAND
2032
where v ( f )and P ( t ) = l / k , T ( t ) are functions of time. They are related to the total number N(t ) and total cncrgy E ( t ) of the atoms according to
TABLE 1. This is a table showing the values used for the various physical parameters needed in the model. vRbis the s-wave scattering cross section for "Rb. The explanations for the other parameters are given in the text. 7 . 5 lo-'' ~ m2 200 s 4 . 9 1~0 - l ~ cm'is 20 ,LLK 100 Hz 10' atoms/s
uRh 7hl-
I/Yhl
K3 T(,ret)
I,I et)
12 7r
yyf) . .
With the assumption of the truncated Boltznianii form for f ( e , t ) , the description of the system can be reduced to finding thc equations of motion for the total number and energy. The equations of motion for N ( t ) and E ( t ) will be written in terms of the various gain and loss processes which occur. There are four competing processes which take place during the evaporation: the constant feeding of atoms into the trap at a rate y f with a mean energy per atom e f , the loss of atoms from the trap due to collisions with the atoms from the hot background gas, characterized by a constant rate ybl, the loss of atoms and heating due to three-body inelastic collisions, given by the rate y; , and the rethermalization due to elastic collisions which will eject atoms from the trap which obtain an energy above ecutafter a collision. We can include all of these effects in the kinetic equation for f(e,t),
where the distribution of atoms injected into the trap is g,(e) and the density of states is p(e)= f e 2 / ( h w 2 ) ' .rcol(t) is the collision integral given by [ 11
r col(
t ) = yo
I
de,de ' de: 6(e
+ e,
- e - e:
) p( emin)
where y , = m a / ( ~ ~ hand ~ ) emin=min{e,e,,e',e:} is the minimum energy. By substituting Eq. (7) into Eq. (lo), and using Eq. (8) and Eq. (9), we obtain the following equations of motion for the total number and total energy:
E = 7P.f- YblE-
f y3N2E- r E
>
(13)
where the three-body loss rate for the total number is y3 = 3 ' . ' K , ( r n w : / 2 ~ k ~ T )K~ 3. is an experimentally determined constant to be specified [22]. In ohtaining the threebody loss terms, an approximation has been made that ecut %k,T(t) in order to simplify the terms. Initially during the evolution, this assumption may not hold, but the density is low enough that the three-body loss terms are negligible in any case. By the time the density has increased enough so
57
~~
that three-body losses are significant, the assumption does hold. The factor of 2/3 in Eq. (1 3) signifies that the energy will decrease at a slower rate than the number due to threebody losses, which gives rise to an effective heating. The two terms r.v and r E represent the loss of number and energy due to evaporation and are given by
The fourth atom in these equations is lost from the trap since its energy is always greater than the RF cut e > ecut. Due to energy conservation and the truncated form of f ( e), this means that emin=e:, as indicated in Eq. (14) and Eq. (15). Also, the energy which appears in the term T,(t) is that of the escaping atom e = e + e,- e: . 111. RESULTS
In order to carry out explicit calculations, we choose realistic values of the various physical parameters needed in our model. These are listed in Table I for a gas of "Rb atoms. The parameters w2 and ecutare not listed in the table but are variables to be specified in the following calculations. We have specified a reference point for the MOT parameters which yields a phase space density in the MOT of po= 6.9 X if one assumes that N , = 5 X lo5 at 20 transfers per second [21]. A. Time evolution
We first consider the dynamical evolution of the system toward steady state. In Fig. 2 we show results of a numerical integration of the rate equations in Eq. (12) and Eq. (13) for the total number N ( t ) and energy E ( t ) . Since the magnetic trap frequency w2 is matched to the MOT frequency w , in this calculation, the adiabatic and sudden transfers are equivalent. For case 1 in the figure, we chose the optimum value of ecut to yield the highest phase space density po,
504
ACHIEVING STEADY-STATE BOSE-EINSTEIN CONDENSATION
57
1
t
0.9
10'
1
1
0.6
1oo
0
G 0.5 0
Q
c
lo-'
0.2 0.1
I"
0
200
400 600 time (s)
100
200
300
400 500 time (s)
600
,
B. Steady-state solution 700
800
FIG. 2. This plot shows the time evolution of the total number
N(/)and total energy E ( / ) for the values of the parameters listed in Table I. The magnetic trap frequency is equal to the MOT frequency w 2 = w I in this calciilation. Two values of eCLitwere chosen: 1.1 p K , labeled by I , and 1 1 p K , labeled by 2. Each of the curves is normalized by its final steady-state value. The solid curve is the total number and reaches a steady-state value of NS,=2.0X lo4 for case 1, and NS,=2.8X 10' for case 2. The dashed curve is the total energy and reaches a steady-state value of E,,=(0.33 p K ) N , , for case 1 (case 2 is not shown). The evolution of the peak phase space density po is shown in the inset for the two cases. while in case 2 the value chosen for ecutis ten times higher than that in case 1. There are some interesting features to consider from this plot. It is instructive to take a simple limiting case of Eq. (12) and Eq. (13) in order to learn something about the build-up time for steady state to occur. If we let e,+m and y3=0, then the solution to the rate equations for N ( r ) and E ( r ) is given by Yl
N ( t ) = -(1
In Fig. 2, the build-up time in case 1 is slightly less than is 200 seconds. This indicates that the choice of eCuti n case 1 minimizes three-body losses. In case 2, on the other hand, whcrc ecutis ten times larger than that in case 1, the build-up time is much shorter at roughly 25 seconds. This i s because in case 2, yf is larger, causing the density to build up more quickly which allows three-body losses to dominate. This also stops the evaporative cooling quickly and so one does not obtain as high of a phase space density po as in case 1. It should be noted that whcn we calculated case 2 with y3 = 0, the build-up timc was approximately equal to T ~ , and the steady-statc value of the phase space density was close to being optimized at that value of e c u twith , p o = 3.9 in steady statc. T ~ , which ,
800
0
0
2033
-e-Yhl'),
Ybl
The time scale for steady state to occur in this simple case is just the lifetime of the trap as determined by background losses, Thl. In the case where the RF cut is present and evaporation is occurring, while still neglecting three-body losses, the build-up time for steady state will be on the order of magnitude of T ~ , although , it will be shorter, based on results of numerical calculations. We define this build-up time to be the time at which N ( t ) = ( 1 - e-I)N,, . When three-body losses are included, the build-up time can be very short compared to Th1 if the density is high enough for threebody losses to dominate. So this gives us an upper limit of the build-up time to be Tbl, and if steady state occurs on a much shorter time scale than this, it indicates that three-body losses are dominating the other loss mechanisms.
Now that we have characterized the time scalc for steady state to occur, it is useful to solve Eq. (12) and Eq. (13) dircctly for the steady-state values of N,, and E,, by setting the left-hand sides equal to zcro. We were not able to solve the resulting coupled algebraic equations analytically, since they are traiiscendcntal in form. However, they are straightforward to solve numerically. In the following sections, we present calculations of the steady-state value of p o while varying some of the physical parameters in order to discern what values of the parameters yield p0=2.612 so that BEC can be achieved in steady state.
I. Vurying ecerand
wt
In trying to understand what it takes to reach a steadystate BEC, it is useful to look at how p o varies with w 2 and ecut.In Fig. 3 and Fig. 4, we show shaded contour plots of the steady-state value of p o , for both an adiabatic and a sudden transfer. Also shown are contours of the total number N,, overlaying the shaded contours. Again, we use the reference point of parameters displayed in Table I. The two different idealizations of the transfer process yield quite distinct shapes for the surfaces of p o and N,, . For the adiabatic case shown in Fig. 3, po increases with increasing 0 2keeping , ecutfixed. However, it levels off quite quickly, varying from 1.1 to 1.5 with an order of magnitude increase in w z / w I from 0.1 to 1.0 at ecut=1 p K . Also, with o2fixed, the optimum value of ecutwhich yields the highest po does not depend much on w 2 , but is roughly a straight line at ecut=1 p K . Perhaps the most interesting and crucial feature exhibited in the plot is that N,, decreases very rapidly as w 2 is increased, going from lo7 down to lo4 as 0 2 / o l goes from 0.1 to 1.0. This is because three-body losses increase as the trap is tightened, since the density increases. Therefore, one will gain a lot in number by keeping the magnetic trap shallow, while losing only a small amount in phase space density. The results of a sudden transfer are shown in Fig. 4. The most striking difference between this and the plot shown in Fig. 3 for an adiabatic transfer is a strong peak which occurs at o2/a, = 1 . This can be attributed to the fact that the phase space density always decreases in a sudden transfer, with a peak occurring at w 2 = w I, where the sudden and adiabatic transfers are equivalent. Notice also that po drops off much more rapidly as o2i o , is varied from unity, compared to the
505 WILLIAMS, WALSER, WIEMAN, COOPER, AND HOLLAND
2034
15
57
~
10'
13
11
09
07
lo-' 10-'
FIG. 3. This plot shows two overlaying contoui-s of the steadystate value of the phase space density and the total nuinbcr vs the ratio of trap frequencies and RF cut threshold for an adiabatic traiisfer. The shaded contours represent the steady-state value of po, with the gray-scale bar shown to the right. The numbered lines represent log,,N,, (i.e., a value of 6 for the line i n the center corresponds to N,,= 10'). It is w 2 that is varied in the ratio, while w I is fixed at 2 ~ 1 0 0Hz. The values used for the other parameters are displayed in Table I.
adiabatic case. Another difference between the two cases is that the optimum value for ecutincreases as w 2/ w I is varied from unity. Finally, it can be seen also that one does not gain that much in number as w 2 is decreased, in sharp contrast to the adiabatic case. 2. Varying T, and y,
We now have an understanding of how the steady-state values of p o and N,, vary with eCUI and w 2 . Another useful 15
13
11
09
07
05
03
lo-'
1oo
10'
1o2
01
ecut (P K)
FIG. 4. This plot is the same as described in the caption of Fig. 3 except for a sudden transfer of atoms from the MOT to the magnetic trap, instead of an adiabatic one.
1 oo
r, 1 jre"
10'
FIG 5 This plot show? thc valucs of 7 , and w , one inust achieve i n order to icacli p , , = 2 612 in the c a x of an adiabatic tiansfer Threc difteient values of w ? / w , ale 5liowii w 2 / w , t { O 1,0 5,l}, with w , = ~ T X100 Hz For each line, eititwas chosen so as to maximize p(, The iefeieiice value5 die f l ' " ) = 2 0 p K and $""= 10' atonis/s
calcu!ation is to see how p o depends on the MOT tcmperahire TI and the transfer rate y , . In the plots below, ecutis chosen so as to maximize po, for a given T , , y r , and w?. Then, given y t and w 2 , TI is chosen so as to reach p o =2.612. This is done for 1 0 h ~ y , S 1 0 8as , well as three values of the trap frequency ratio w 2 / w 1t{0.1,0.5,1}, with w 1 = 2 r X 100 Hz. The results of an adiabatic transfer are shown in Fig. 5. Along each of the three lines p o = 2.612. The most important feature of this plot is that the three lines lie nearly on top of each other. This agrees with Fig. 3 in that p o decreases vary little as w 2 is lowered. The plot also shows that po depends more critically on TI than on yt . Starting from the reference point in the center, one has to either decrease T I by 20%, or increase y , by 100% in order to get to the po=2.612 line. The sudden transfer is shown in Fig. 6. In contrast to the adiabatic case, the three lines are separated, so that as w 2 is decreased, one has to try much harder to reach po=2.612, which is also consistent with Fig. 4. The N,, curves corresponding to the po=2.612 lines in Fig. 5 and Fig. 6 are shown in Fig. 7. The results are the same in both the sudden and adiabatic cases (thus there are only three lines instead of six). For the adiabatic case, by loosening the magnetic trap, one does not have to vary TI and y , much at all in order to stay at po=2.612 while increasing the number N,, by orders of magnitude. On the other hand, for the sudden transfer, one has to decrease TI and increase y , a lot in order to stay at po=2.612 as w 2 is decreased. However, one will achieve the same increase in number as in the adiabatic case. Finally, in Fig. 8 we show a plot of the ratio e,,,/T2 corresponding to the po=2.612 lines shown in Figs. 5-7. This ratio of the optimum cut to the temperature T2 of atoms being injected into the trap is the same in both the adiabatic
506 ACHIEVING STEADY-STATE BOSE-EINSTEIN CONDENSATION
51
FIG. 6. This plot is the same as described in the caption of Fig. 5 but for the case of a sudden transfer.
and sudden transfers. As w 2 is decreased, one does not have to exclude as much of the distribution from the trap. Also, as y, is increased, one has to cut further into the injected distribution in order to prevent three-body losses from dominating. IV. CONCLUSION
In this paper we have addressed the problem of achieving a steady-state condensation by continuously feeding atoms into the magnetic trap below a fixed RF threshold. We have included losses due to elastic collisions with atoms from the
1 0-'
10'
FIG. 7. This plot corresponds to the three lines in both Fig. 5 and Fig. 6, showing the total number of atoms in steady state N,, as a fimction of the transfer rate y , . Along each of these curves, p0=2.612. The legend in Fig. 5 and Fig. 6 applies to this plot also.
2035
FIG. 8. These curves correspond to the curves in Figs. 5-7, showing the ratio of the RF cut to the temperature of atoms injected into the trap, ecutlT2,as a function of the transfer rate y , . Along each of these curves, po=2.612. background gas, as well as inelastic three-body collisions. Our model of the loading of atoms into the magnetic trap treats two idealizations of transferring atoms from a separate MOT; either an adiabatic or a sudden transfer. The description of the kinetic evolution to steady state assumes a truncated Boltzmann form for the nonequilibrium distribution f ( e , t ) , reducing the problem to that of solving coupled rate equations for the total number N ( t ) and total energy E( t ) of the gas. Our calculations show that it is possible to achieve a steady-state condensation using optimistic values of the relevant physical parameters. We have shown several results of numerical solutions of the rate equations in Eq. (12) and Eq. (13). First, we addressed the build-up time for steady state to occur and determined that an upper limit on the build-up time is given by the background loss lifetime T ~ , If . three-body losses are dominating due to a high density, then the build-up time will be much shorter than this. We next looked at how the steadystate value of the peak phase space density po depends on the magnetic trap frequency w2 and the RF cut e,,,. We found that in the adiabatic case, one can gain a large increase in the total number in steady state N,, by loosening the magnetic trap, while only losing a small amount in po. This is not true for a sudden transfer. Finally, we looked at how one must vary the transfer rate y, and the MOT temperature T , in order to reach po=2.612. We found that po depends more critically on T , than yI . Also, it was shown that one must try much harder to reach the critical point while achieving a large N,, in the sudden case compared to the adiabatic case. There are several shortcomings of our model which might be improved, however we believe that the present calculations are qualitatively correct and are sufficient for experimental guidance. An obvious extension to our model would be to include the effect of the growth of the condensate which will make the evaporation more efficient but at the same time increasing three-body losses due to the increase in
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ACKNOWLEDGMENTS
density in the center of the trap [4,23-251. Another improvem e n t would b e t o construct a more accurate model of the transfer process b y understanding the relationship between T , , o,, a n d y, , since these cannot b e varied independently in a n experiment. Alternatively, one could construct a model of the loading procedure based on an entirely different experimental method than that described in [ 19,20,11].
J. Williams and M. Holland appreciated the hospitality of the Institute of Theoretical Physics at the University of Innsbnick during the earlier stages of this work. They thank Peter Zoller and Dieter Jaksch for helpful discussions. This work was supported by the National Science Foundation.
[ I ] 0. Luiten, M. Reynolds, and J. Walraven, Phys. Rev. A 53, 382 (1996). [2] W. Ketterle and N. J. V. Dmten, i n Advar7ces i17 Atomic, Molecuhr; ntid Optical Pl7ysic.r, Vol. 3 7 (Acedemic Press, New York, 1996), p. 181. [3] M. Holland, J. Williams, K. Coakley, and J. Cooper, Quantum Semiclassic. Opt. 8, 571 (1996). [4] M. Holland, J. Williams, and J. Cooper, Phys. Rev. A 55, 3670 (1997). [5] C. A. Sackett, C. C. Bradley, and R. G. Hulet, Phys. Rev. A 55, 3797 (1997). [6] K. Berg-Sdrensen, Phys. Rev. A 55, 1281 (1997). [7] M. H. Anderson et al., Science 269, 198 (1995). [8] C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995). [9] K. B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995). [I01 M.-0. Mewes et al., Phys. Rev. Lett. 77, 416 (1996). [I I ] C. J. Myatt et al., Phys. Rev. Lett. 78, 586 (1997). [12] In addition to the above citations, three other groups have reported observations of Bose-Einstein condensation using the
technique of evaporative cooling on a cold atomic vapor. [ 131 M. 0. Mewes et al., Phys. Rev. Lett. 78, 582 (1997). [I41 M. Holland et nl., Phys. Rev. A 54, R1757 (1996). [I51 H. Wiseman, A. Martins, and D. Walls, Qnantuin Semiclassic. Opt. 8, 737 (1996). [I61 R. J. Ballagh, K. Burnett, and T. F. Scott, Phys. Rev. Lett. 78, 1607 (1997). [ 171 J. J. Hope, Phys. Rev. A 55, R253 1 (1997). [I81 W. Ketterle and H.-J. Miesner, Phys. Rev. A 56, 3291 (1997). [I91 E. Cornell, C. Monroe, and C. Wieman, Phys. Rev. Lett. 67, 2439 (1991). [20] C. J. Myatt et a/., Opt. Lett. 21, 290 (1996). [21] C. G. Townsend et al., Phys. Rev. A 52, 1423 (1995). [22] E. A. Burt et a/., Phys. Rev. Lett. 79, 337 (1997). [23] D. Jaksch, C. W. Gardiner, and P. Zoller, Phys. Rev. A 56, 575 (1997). [24] C. W. Gardiner, P. Zoller, R. J. Ballagh, and M. J. Davis, Phys. Rev. Lett. 79, 1793 (1997). [25] C. W. Gardiner and P. Zoller, Phys. Rev. A (to be published).
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The Bose-Einstein Condensate Three years ago in a Colorado laboratory, scientists realized a long-standing dream, bringing the quantum world closer to the one of everyday experience by Eric A. Cornell and Carl E. Wieman
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n June 1995 our research group at the Joint Institute for Laboratory Astrophysics (now called JILA) in Boulder, Colo., succeeded in creating a minuscule but marvelous droplet. By cooling 2,000 rubidium atoms to a temperature less than 100 billionths of a degree above absolute zero (100 billionths of a degree kelvin), we caused the atoms to lose for a full 10 seconds their individual identities and behave as though they were a single “superatom.” The atoms’ physical properties, such as their motions, became identical to one another. This Bose-Einstein condensate (BEC), the first observed in a gas, can be thought of as the matter counterpart of the laser-except that in the condensate it is atoms, rather than photons, that dance in perfect unison. Our short-lived, gelid sample was the experimental realization of a theoretical construct that has intrigued scientists ever since it was predicted some 73 years ago by the work of physicists Albert Einstein and Satyendra Nath Bose. At ordinary temperatures, the atoms of a gas are scattered throughout the container holding them. Some have high energies (high speeds); others have low ones. Expanding on Bose’s work, Einstein showed that if a sample of atoms were cooled sufficiently, a large fraction of them would settle into the single lowest possible energy state in the container. In mathematical terms, their individual wave equations-which describe such physical characteristics of an atom as its position and velocity-would in effect merge, and each atom would become indistinguishable from any other. Progress in creating Bose-Einstein condensates has sparked great interest in the physics community and has even generated coverage in the mainstream press. At first, some of the attention derived from the drama inherent in the decades26
long quest to prove Einstein’s theory. But most of the fascination now stems from the fact that the condensate offers a macroscopic window into the strange world of quantum mechanics, the theory of matter based on the observation that elementary particles, such as electrons, have wave properties. Quantum mechanics, which encompasses the famous Heisenberg uncertainty principle, uses these wavelike properties to describe the structure and interactions of matter. We can rarely observe the effects of quantum mechanics in the behavior of a macroscopic amount of material. In ordinary, so-called bulk matter, the incoherent contributions of the uncountably large number of constituent particles obscure the wave nature of quantum mechanics, and we can only infer its effects. But in Bose condensation, the wave nature of each atom is precisely in phase with that of every other. Quantum-mechanical waves extend across the sample of condensate and can be observed with the naked eye. The submicroscopic thus becomes macroscopic. New Light on Old Paradoxes
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he creation of Bose-Einstein condensates has cast new light on longstanding paradoxes of quantum mechanics. For example, if two or more atoms are in a single quantum-mechanical state, as they are in a condensate, it is fundamentally impossible to distinguish them by any measurement. The two atoms occupy the same volume of space, move at the identical speed, scatter light of the same color and so on. Nothing in our experience, based as it is on familiarity with matter at normal temperatures, helps us comprehend this paradox. That is because at normal temperatures and at the size scales we are all familiar with, it is possible to de-
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510 scribe the position and motion of each and every object in a collection of objects. The numbered Ping-Pong balls bouncing in a rotating drum used to select lottery numbers exemplify the motions describable by classical mechanics. At extremely low temperatures or at small size scales, on the other hand, the usefulness of classical mechanics begins to wane. The crisp analogy of atoms as Ping-Pong balls begins to blur. We cannot know the exact position of each atom, which is better thought of as a blurry spot. This spot-known as a wave packet-is the region of space in which
The Bose-Emstein Condensate
we can expect to find the atom. As a collection of atoms becomes colder, the size of each wave packet grows. As long as each wave packet is spatially separated from the others, it is possible, at least in principle, to tell atoms apart. When the temperature becomes sufficiently low, however, each atom’s wave packet begins to overlap with those of neighboring atoms. When this happens, the atoms “Bose-condense” into the lowest possible energy state, and the wave packets coalesce into a single, macroscopic packet. The atoms undergo a quantum identity crisis: we can no long-
er distinguish one atom from another. The current excitement over these condensates contrasts sharply with the reaction to Einstein’s discovery in 1925 that they could exist. Perhaps because of the impossibility then of reaching the required temperatures-less than a millionth of a degree kelvin-the hypothesized gaseous condensate was considered a curiosity of questionable validity and little physical significance.For perspective, even the coldest depths of intergalactic space are millions of times too hot for Bose condensation. In the intervening decades, however,
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511 Bose condensation came back into fashion. Physicists realized that the concept could explain superfluidity in liquid helium, which occurs at much higher temperatures than gaseous Bose condensation. Below 2.2 kelvins, the viscosity of liquid helium completely disappears-
putting the “super” in superfluidity. than the rate for hydrogen atoms. These Not until the late 1970sdid refrigera- much larger atoms bounce off one antion technology advance to the point other at much higher rates, sharing enthat physicists could entertain the no- ergy among themselves more quickly, tion of creating something like Einstein’s which allows the condensate to form original concept of a BEC in a gas. Lab- before clumping can occur. Also, it looked as if it might be relaoratory workers at M.I.T., the University of Amsterdam, the University of British tively easy and inexpensive to get these Columbia and Cornell University had atoms very cold by combiningingenious to confront a fundamental difficulty. To techniques developed for laser cooling achieve such a BEC, they had to cool and trapping of alkali atoms with the the gas to far below the temperature at techniques for magnetic trapping and which the atoms would normally freeze evaporative cooling developed by the reinto a solid. In other words, they had to searchers working with hydrogen. These create a supersaturated gas. Their ex- ideas were developed in a series of dispectation was that hydrogen would su- cussions with our friend and former persaturate, because the gas was kuown teacher, Daniel Kleppner, the co-leader to resist the atom-by-atom clumping of a group at M.I.T. that is attempting that precedes bulk freezing. to create a condensate with hydrogen. Although these investigators have not Our hypothesis about alkali atoms yet succeeded in creating a Bose-Ein- was ultimately fruitful. Just a few stein condensate with hydrogen, they months after we succeeded with rubididid develop a much better understand- um, Wolfgang Ketterle’s group at M.I.T. ing of the difficulties and found clever produced a Bose condensate with sodiapproaches for attacking them, which um atoms; since that time, Ketterle’s benefited us. In 1989, inspired by the team has succeeded in creating a conhydrogen work and encouraged by our densate with 10 million atoms. At the own research on the use of lasers to trap time of this writing, there are at least and cool alkali atoms, we began to sus- seven teams producing condensates. pect that these atoms, which include ce- Besides our own group, others working sium, rubidium and sodium, would with rubidium are Daniel J. Heinzen of make much better candidates than hy- the University of Texas at Austin, Gerdrogen for producing a Bose conden- hard Rempe of the University of Konsate. Although the clumping properties stanz in Germany and Mark Kasevich of cesium, rubidium and sodium are not of Yale University. In sodium, besides superior to those of hydrogen, the rate Ketterle’s at M.I.T., there is a group led at which those atoms transform them- by Lene Vestergaard Hau of the Rowselves into condensate is much faster land Institute for Science in Cambridge, EVAPORATIVE COOLING occurs in a magnetic trap, which can be thought of as a deep bowl (blue).The most energetic atoms, depicted with the longest green trajectory arrows, escape from the bowl (above, Zefi). Those that remain collide with one another frequently, apportioningout the remaining energy (Zeft).Eventually, the atoms move so slowly and are so closely packed at the bottom of the bowl that their quantum nature becomes more pronounced. So-called wave packets, representing the region where each atom is likely to be found, become less distinct and hegin to overlap (below, left). Ultimately, two atoms collide, and one is left as close to stationary as is dowed hy Heisenberg’s uncertainty principle. This event triggers an avalanche of atoms piling up in the lowest energy state of the trap, merging into the single ground-stateblob that is a BoseEinstein condensate (below, center and right).
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LASER COOLING of an atom makes use of the pressure, or force, exerted by repeated photon impacts. An atom moving against a laser beam encounters a higher frequency than an atom moving with the same beam. In cooling, the frequency of the beam is adjusted so that an atom moving into the beam scatters many more photons than an atom moving away from the beam. The net effect is to reduce the speed and thus cool the atom.
the laser light relative to the frequency of the light absorbed by the atoms [see illustration above]. In this setup, we use laser light not only to cool the atoms but also to “trap” them, keeping them away from the room-temperature walls of the cell. In fact, the two laser applications are similar. With trapping, we use the radiation pressure to oppose the tendency of the atonis to drift away from the center of the cell. A weak magnetic field tunes the resonance of the atom to absorb preferentially from the laser beam that is pointing toward the center of the cell (recall that six laser beams intersect at Laser Cooling and Trapping the center of the cell). The net effect is he heart of our apparatus is a small that all the atoms are pushed toward glass box with some coils of wire one spot and are held there just by the around it (see illustration on pages 26 force of the laser light. These techniques fill our laser trap in and 271. We completely evacuate the cell, producing in effect a superefficient one minute with 10 million atoms capthermos bottle. Next, we let in a tiny tured from the room-temperature ruamount of rubidium gas. Six beams of bidium vapor in the cell. These trapped laser light intersect in the middle of the atoms are at a temperature of about 40 box, converging on the gas. The laser millionths of a degree above absolute light need not be intense, so we obtain zero-an extraordinarily low temperait from inexpensive diode lasers, similar ture by most standards but still 100 to those found in compact-disc players. times too hot to form a BEC. In the presWe adjust the frequency of the laser ence of the laser light, the unavoidable radiation so that the atoms absorb it random jostling the atoms receive from and then reradiate photons. An atom the impact of individual light photons can absorb and reradiate many millions keeps the atoms from getting any coldof photons each second, and with each er or denser. one, the atom receives a minuscule kick To get around the limitations imposed in the direction the absorbed photon i s by those random photon impacts, we moving. These kicks are called radiation turn off the lasers at this point and actipressure. The trick to laser cooling is to vate the second stage of the cooling proget the atom to absorb mainly photons cess. This stage is based on the magnetthat are traveling in the direction oppo- ic-trapping and evaporative-cooling site that of the atom’s motion, thereby technology developed in the quest to slowing the atom down (cooling it, in achieve a condensate with hydrogen other words). We accomplish this feat atoms. A magnetic trap exploits the fact by carefully adjusting the frequency of that each atom acts like a tiny bar mag-
net and thus is subjected to a force when placed in a magnetic field [see illustrution on opposite page].By carefully controlling the shape of the magnetic field and making it relatively strong, we can use the field to hold the atoms, which move around inside the field much like balls rolling about inside a deep bowl. In evaporative cooling, the most energetic atoms escape from this magnetic bowl. When they do, they carry away more than their share of the energy, leaving the remaining atoms colder. The analogy here is to cooling coffee. The most energetic water molecules leap out of the cup into the room (as steam), thereby reducing the average energy of the liquid that is left in the cup. Meanwhile countless collisions among the remaining molecules in the cup apportion out the remaining energy among all those molecules. Our cloud of magnetically trapped atoms is at a much lower density than water molecules in a cup. So the primary experimental challenge we faced for five years was how to get the atoms to collide with one another enough times to share the energy before they were knocked out of the trap by a collision with one of the untrapped, room-temperature atoms remaining in our glass cell. Many small improvements, rather than a single breakthrough, solved this problem. For instance, before assembling the cell and its connected vacuum pump, we took extreme care in cleaning each part, because any remaining residues from our hands on an inside surface would emit vapors that would degrade the vacuum. Also, we made sure that the tiny amount of rubidium vapor remaining in the cell was as small as it could be while providing a sufficient number of atoms to fill the optical trap. Incremental steps such as these helped but still left us well shy of the density needed to get the evaporative cooling under way. The basic problem was the effectiveness of the magnetic trap. Although the magnetic fields that make
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SCIENTIFIC AMERICAN March 1998
Mass. At Rice University Randall G. Hulet has succeeded in creating a condensate with lithium. All these teams are using the same basic apparatus. As with any kind of refrigeration, the chilling of atoms requires a method of removing heat and also of insulating the chilled sample from its surroundings. Both functions are accomplished in each of two steps. In the first, the force of laser light on the atoms both cools and insulates them. In the second, we use magnetic fields to insulate, and we cool by evaporation.
29
513 up the confining magnetic “bowl” can be quite strong, the little “bar magnet” inside each individual atom is weak. This characteristic makes it difficult to push the atom around with a magnetic field, even if the atom is moving quite slowly (as are our laser-cooled atoms). In 1994 we finally confronted the need to build a magnetic trap with a narrower, deeper bowl. Our quickly built, narrow-and-deep magnetic trap proved to be the final piece needed to cool evaporatively the rubidium atoms into a condensate. As it turns out, our particular trap design was hardly a unique solution. Currently there are almost as many different magnetic trap configurations as there are groups studying these condensates. Shadow Snapshot of a “Superatom”
H
ow do we know that we have in fact produced a Bose-Einstein condensate? To observe the cloud of cooled atoms, we take a so-called shadow snapshot with a flash of laser light. Because the atoms sink to the bottom of the magnetic bowl as they cool, the cold cloud is too small to see easily. To make it larger, we turn off the confining magnetic fields, allowing the atoms to fly out freely in all directions. After about 0.1 second, we illuminate the now expanded cloud with a flash of laser light. The
atoms scatter this light out of the beam, casting a shadow that we observe with a video camera. From this shadow, we can determine the distribution of velocities of the atoms in the original trapped cloud. The velocity measurement also gives us the temperature of the sample. In the plot of the velocity distribution [see illustration on opposite page], the condensate appears as a dorsal-finshaped peak. The condensate atoms have the smallest possible velocity and thus remain in a dense cluster in the center of the cloud after it has expanded. This photograph of a condensate is further proof that there is something wrong with classical mechanics. The condensate forms with the lowest possible energy. In classical mechanics, “lowest energy” means that the atoms should be at the center of the trap and motionless, which would appear as an infinitely narrow and tall peak in our image. The peak differs from this classical conception because of quantum effects that can be summed up in three words: Heisenberg’s uncertainty principle. The uncertainty principle puts limits on what is knowable about anything, including atoms. The more precisely you know an atom’s location, the less well you can know its velocity, and vice versa. That is why the condensate peak is not infinitely narrow. If it were, we would know that the atoms were in the
exact center of the trap and had exactly zero energy. According to the uncertainty principle, we cannot know both these things simultaneously. Einstein’s theory requires that the atoms in a condensate have energy that is as low as possible, whereas Heisenberg’s uncertainty principle forbids them from being at the very bottom of the trap. Quantum mechanics resolves this conflict by postulating that the energy of an atom in any container, including our trap, can only be one of a set of discrete, allowed values-and the lowest of these values is not quite zero. This lowest allowed energy is called the zeropoint energy, because even atoms whose temperature is exactly zero have this minimum energy. Atoms with this energy move around slowly near-but not quite at-the center of the trap. The uncertainty principle and the other laws of quantum mechanics are normally seen only in the behavior of submicroscopic objects such as a single atom or smaller. The Bose-Einstein condensate therefore is a rare example of the uncertainty principle in action in the macroscopic world. Bose-Einstein condensation of atoms is too new, and too different, for us to say if its usefulness will eventually extend beyond lecture demonstrations for quantum mechanics. Any discussion of practical applications for condensates must necessarily be speculative. Never-
CONTINENT-WIDE THERMOMETER is 4,100 kilometers (2,548 miles) long and shows how low the temperature must be before a condensate can form. With zero degrees kelvin in Times Square in New York City and 300 degrees assigned to City Hall in Los Angeles, room temperature corresponds to San Bernardino, Calif., and the temperature of air, frozen solid, to Terre Haute, Ind. The temperature of a nearly pure condensate is a mere 0.683 millimeter from the thermometer’s zero point. 300 KELVIN
50 NANOKELVIN
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514 theless, our musings can be guided by a striking physical analogy: the atoms that make up a Bose condensate are in many ways the analogue to the photons that make up a laser beam. The Ultimate in Precise Control?
SHADOW IMAGE of a forming Bose-Einstein condensate was processed by a computer to show more clearly the distribution of velocities of atoms in the cold cloud. Top and bottom images show the same data but from different angles. In the upper set, where the surface appears highest corresponds to where the atoms are the most closely packed and are barely moving. Before the condensate appears (left),the cloud, at about 200 billionths of a degree kelvin, is a single, relatively smooth velocity distribution. After further cooling to 100 billionths of a degree, the condensate amears as a region of almost sta” tionary atoms in the center of the distribution (centev).After still more cooling, only condensate remains (right). I I
E
very photon in a laser beam travels in exactly the same direction and has the same frequency and phase of oscillation. This property makes laser light very easy to control precisely and leads to its utility in compact-disc players, laser printers and other appliances. Similarly, Bose condensation represents the ultimate in precise control-but for atoms rather than photons. The matter waves of a Bose condensate can be reflected, focused, diffracted and modulated in frequency and amplitude. This kind of control will very likely lead to improved timekeeping; the world’s best clocks are already based on the oscillations of laser-cooled atoms. Applications may also turn up in other areas. In a flight of fancy, it is possible to imagine a beam of atoms focused to a spot only a millionth of a meter across, “airbrushing” a transistor directly onto an integrated circuit. But for now, many of the properties of the Bose-Einstein condensate remain unknown. Of particular interest is the condensate’s viscosity. The speculation now is that the viscosity will be vanishingly small, making the condensate a kind of “supergas,” in which ripples and swirls, once excited, will never damp down. Another area of curiosity centers on a basic difference between laser light and a condensate. Laser beams are noninteracting-they can cross without affecting one another at all. A condensate, on the other hand, has some resistance to compression and some springinessit is, in short, a fluid. A material that is
both a fluid and a coherent wave is going to exhibit behavio; that is rich, which is a physicist’s way of saying that it is going to take a long time to figure out. Meanwhile many groups have begun a variety of measurements on the condensates. In a lovely experiment, Ketterle’s group has already shown that when two separate clouds of Bose condensate overlap, the result is a fringe pattern of alternating constructive and destructive interference, just as occurs with intersecting laser radiation. In the atom cloud, these regions appear respectively as
The Authors ERIC A. CORNELL and CARL E. WIEMAN are both fellows of JILA, the former Joint Institute for Laboratory Astrophysics, which is staffed by the National Institute of Standards and Technology (NIST) and the University of Colorado. Cornell, a physicist at NIST and a professor adjoint at the university, was co-leader, with Wieman, of the team at JILA that produced the first Bose-Einstein condensate in a gas. Wieman, a professor of physics at the university, is also known for his studies of the breakdown of symmetry in the interactions of elementary particles. The authors would like to thank their colleagues Michael Anderson, Michael Matthews and Jason Ensher for their work on the condensate project.
The Bose-Einstein Condensate
stripes of high density and low density. Our group has looked at how the interactions between the atoms distort the shape of the atom cloud and the manner in which it quivers after we have “poked” it gently with magnetic fields. A number of other teams are now devising their own experiments to join in this work. As the results begin to accrue from these and other experiments over the next several years, we will improve our understanding of this singular state of matter. As we do, the strange, fascinating quantum-mechanical world will come a little bit closer to our own.
Further Reading NEWMECHANISMS FOR LASER COOLING. William D. Phillips and Claude Cohen-Tannoudjiin Physics Today, Val. 43, pages 3 3 4 0 ; October 1990. OBSERVATION OF BOSE-EINSTEIN CONDENSATION IN A DILUTE ATOMIC VAPOR.M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell in Science, Vol. 269, pages 198-201; July 14, 1995. BOSE-EINSTEIN CONDENSATION. Edited by A. Griffin, D. W. Snake and S. Stringari. Cambridge University Press, 1995. OBSERVATION OF INTERFERENCE BETWEEN Two BOSE CONDENSATES. M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn and W. Ketterle in Science, Val. 275, pages 637-641; January 31,1997. BOSE-EINSTEIN CONDENSATION. World Wide Web site available at http:// www.colorado.edu/physics/2OOO/bec
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PHYSICAL REVIEW LETTERS VOLUME81
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Dynamics of Component Separation in a Binary Mixture of Bose-Einstein Condensates D. S. Hall, M.R. Matthews, J. R. Ensher, C.E. Wieman, and E.A. Cornell* JILA, National Institute of Standards and Technology and Department of Physics, Universiry of Colorado, Boulder, Colorado 80309-0440 (Received 2 April 1998)
We describe the first experiments that study in a controlled way the dynamics of distinguishable and interpenetrating bosonic quantum fluids. We work with a two-component system of Bose-Einstein condensates in the IF = 1,mf = -1) and 12,l) spin states of s7Rb. The two condensates are created with complete spatial overlap, and in subsequent evolution they undergo complex relative motions that tend to preserve the total density profile. The motions quickly damp out, leaving the condensates in a steady state with a non-negligible (and adjustable) overlap region. [SO03 1-9007(98)06974-91 PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 51.30.+i
Since its realization in dilute atomic gases [ 1-31, BoseEinstein condensation (BEC) has afforded an intriguing glimpse into the macroscopic quantum world. Attention has recently broadened to include exploration of systems of two or more condensates, as realized in a magnetic trap in rubidium [4] and subsequently in an optical trap in sodium [5]. Theoretical treatment of such systems began in the context of superfluid helium mixtures [6] and spinpolarized hydrogen [7], and has now been extended to BEC in the alkalis [8-121. Despite this long-standing and continued theoretical interest, the present paper is the first to explore directly the dynamics of interacting Bose condensates [ 131. The complex structures and motional damping that we observe provide new challenges to the theoretical analysis of this problem. The first experiments involving the interactions between multiple-species BEC were performed with atoms evaporatively cooled in the IF = 2, rnj = 2) and 11, - 1) spin states of 87Rb [4]. These experiments demonstrated the possibility of producing long-lived multiple condensate systems, and that the condensate wave function is dramatically affected by the presence of interspecies interactions. In this Letter, we report results from initial studies of simultaneously trapped BECs in the 12,l) and 11, - 1) states of 87Rb (denoted hereafter as 12) and I l), respectively). The condensates begin with a well-defined 003 1-9007/98/ 81(8)/ 1539(4)$15.00 515
relative phase, spatial extent, and “sag”-the position at which the magnetic trapping forces balance gravity for each state. The fine experimental control of this doublecondensate system permits us to study its subsequent time evolution under a variety of interesting conditions, most notably those in which there remains substantial spatial overlap between the two states. The apparatus and general procedure we use to attain BEC in Rb are identical to those of our previous work [ 141 and will be reviewed here but briefly. We use a double magneto-optical trap system to load roughly lo9 11) atoms into a time-averaged, orbiting potential (TOP) magnetic trap. The atoms are magnetically compressed and evaporatively cooled for 30 s until they form a condensate of approximately 5 X lo5 atoms with no noticeable noncondensate fraction (we estimate that >75% of the entire gas is in the condensate). After completion of the evaporation cycle, the magnetic trap is ramped adiabatically to various bias fields and spring constants for the subsequent experiments. The double-condensate system is prepared from the single 11) condensate by driving a twc-photon transition [ 141 consisting of a microwave photon near 6.8 GHz and a radio frequency photon of 1-4 MHz, depending on the Zeeman splitting. As in [14], we are able to transfer quickly any desired fraction of the atoms to the 12) state by
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selecting the length and amplitude of the two-photon pulse. The two condensates [ 151 are created with identical density distributions, after which they evolve and redistribute themselves for some time T . We then turn off the magnetic trap and allow the atoms to expand for 22 ms for imaging. We selectively image the densities of either of the two states (nl and n2) or the combined density distribution ( n r ) by changing the sequence of laser beams applied to the condensates for probing [14]. Since the expansion and imaging are destructive processes, each image is taken with a different condensate; the excellent reproducibility of the condensates permits us to study the time evolution of the system by changing the time T . The images of the condensates do not always appear in the same absolute locations on the charge-coupled device (CCD) array detector, however, and we compensate for this shot-to-shot jitter by reconstructing the relative positions of the condensates from the images of n l , n2, and nr at each time T . The evolution of the double-condensate system, including the release from the trap and subsequent expansion [ I , 181, is governed by a pair of coupled Gross-Pitaevskii equations for condensate amplitudes QL:
+
where i,j = 1,2 (i j ) , Vi is the magnetic trapping potential for state i, the mean-field potentials are Ui = 4.rrh2aiI@ilz/m and Uij = 4.rrA2ai;l@j12/m, m is the mass of the Rb atom, and the intraspecies and interspecies scattering lengths are ai and aij, respectively. In the Thomas-Fermi limit, the condensate density distributions are dominated by the potential energy terms of Eq. (1). Consequently, the expanded density distributions retain their spatial information and emerge with their gross features (such as the relative position of the condensates) intact. The similarity in scattering lengths a l , a2, and a12 implies that the total density n~ will not change significantly from its initial configuration even though the two components may redistribute themselves dramatically during the evolution time T . In ”Rb, the scattering lengths are known at the 1% level to be in the proportion al:a12:a2::1.03:l:0.97, with the average of the three being 55(3) A [14,19]. The near preservation of the total density nr can be approached theoretically by deriving from Eq. (1) the hydrodynamic equations of motion [20] for n~ and evaluating them in the limit that the fractional differences between the scattering lengths are small. The pressures that tend to redistribute nT must also be small. A similar argument pertains if the minima of the trapping potentials V1 and Vz are displaced from each other (see below) by a distance that is small compared to the size of the total condensate; once again, the effects on the equilibrium distribution of the individual components may be profound but the total density should remain largely unperturbed [21]. 1540
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The rotating magnetic field of the TOP trap gives rise to a subtle behavior that permits us to displace the minima of the trapping potentials V1 and V2 with respect to each other [22,23]. In the rotating frame, the two states see two different magnetic fields as a function of the bias field rotation frequency and sense of rotation (as well as the strengths of the bias and quadrupole fields). By adjusting these parameters, we can change the sign of the relative sag or cause it to vanish [24] while preserving (to first order) the same radial (v,) and axial (v, = v,) trap oscillation frequencies. In the first experiment, we choose a trap that has zero relative sag (v, = 47 Hz) and transfer 50% of the atoms to the 12) state with a -400 ,us pulse. When T = 30 ms, we observe a “crater” in the image of the 11) atoms (Fig. la). The crater corresponds to a region occupied by the 12) atoms (Fig. lb), indicating that the 11) atoms have formed a shell about the 12) atoms. This is consistent with the theoretical prediction that it is energetically favorable for the atoms with the larger scattering length (I 1)) to form a lower-density shell about the atoms with the smaller scattering length (12)) [ll]. At later times the 12) atoms break radial symmetry and drift transversely away from the center of the cloud [25]. In order to explore the boundary between the two condensates, we perform a series of experiments in a trap in which we displace the trapping potentials such that the minimum of V2 is 0.4 p m lower than that of V1, or approximately 3% of the (total) extent of the combined density distribution in the vertical direction. The subsequent time evolution of the system is shown in Figs. 2 and 3.
FIG. l(co1or). (a) The image of the 11) condensate exhibits a crater, corresponding to a shell in which the 12) atoms (b) reside. For this trap, Y, = 47 Hz with zero relative sag. By changing the strength of the magnetic quadrupole field, we can introduce a nonzero relative sag, which shifts the location of the crater (c). (Each square in this postexpansion image is 136 p m on a side.)
517
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3I
I
I *I€
-201, 0
"
20
"
"
40 60 time (ms)
100
80
FIG. 3. The relative motion of the centers of mass of the two condensates under the same conditions as those in Fig. 2.
The two states almost completely separate (Figs. 2a-2c) after 10 ms; they then "bounce" back until, at T = 25 ms, the centers of mass are once more almost exactly superimposed (Fig. 3), although a distinctive (and reproducible) vertical structure has formed (Figs. 2d-20. By T = 65 ms, the system has apparently reached a steady state (Figs. 2g, 2h, and 3) in which the separation of the centers of mass is 20% of the extent of the entire condensate. From these images we observe (i) the fractional steadystate separation of the expanded image is large ( lmpared
2
-150
-100
-50
0
(b) N, / N,,, = 1
% .
50
100
150
50
100
150
,M
1.5
C
4 1.0 .d 0.5 g 0.0
-
I
.
j
.
-150
-150
,
-100
.
-100
0
-50
.
.
-50
.
.
0
.
.
50
.
.
100
.
,
150
vertical position (pm)
FIG. 2(color). Time evolution of the double-condensate system with a relative sag of 0.4p m (3% of the width of the combined distribution prior to expansion) and a trap frequency Y, = 59 Hz.
FIG. 4. Vertical cross sections of the density profiles at T = 65 ms for different relative numbers of atoms in the two states. The combined density distribution (solid line) is shown for comparison to the Thomas-Fermi parabolic fit (dashed line). The trap parameters are the same as those in Fig. 2.
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to the fractional amount of applied symmetry breaking, as we expect for a repulsive interspecies potential; (ii) the placid total density profile (rightmost column of Fig. 2) betrays little hint of the underlying violent rearrangement of the component species; (iii) the component separation is highly damped, although it is not yet certain what mechanism [26] is responsible. With respect to the damping, the excitation is not small and may therefore be poorly modeled by theories that treat the low-lying, smallamplitude excitations [ 121 of double condensates. Finally, we show the optical density as a function of relative number and position on the condensate vertical axis in order to better appreciate the amount of overlap between the two states at T = 65 ms (Fig. 4), which remains substantial despite the underlying separation. Each plot is averaged across a -14 p m wide verticaI cut through the centers of the two condensates. From the overlap shown, one could determine the magnitude of the interspecies scattering length a l 2 by comparison to theoretical calculations conducted within the Thomas-Fermi approximation [S] and numerical solutions of the Gross-Pitaevskii [ 1 11 or Hartree-Fock [9] equations. Such calculations are beyond the scope of the present paper. The overlap region also affords an opportunity to measure the accumulation of relative phase between the two condensates. We explore some of the experimental aspects of the phase evolution of the double-condensate system in the companion article [ 171. We gratefully acknowledge useful conversations with the other members of the JILA BEC Collaboration, in particular, with Chris Greene and John Bohn. This work is supported by the ONR, NSF, and NIST.
*Quantum Physics Division, National Institute of Standards and Technology. [1] M.H. Anderson et al., Science 269, 198 (1995). [2] K. B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995). [3] C. C. Bradley, C.A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 78, 985 (1997). [4] C. J. Myatt et al., Phys. Rev. Lett. 78, 586 (1997). [5] D.M. Stamper-Kurn et al., Phys. Rev. Lett. 80, 2027 (1998); subsequent work [W. Ketterle (private communication)]. [6] I.M. Khalatnikov, Zh. Eksp. Teor. Fiz. 32, 653 (1957) [Sov. Phys. JETP 5, 542 (1957)l; I.M. Khalatnikov, Pis’ma Zh. Eksp. Teor. Fiz. 17, 534 (1973) [JETP Lett. 17, 386 (1973)]; Yu. A. Nepomnyashchi, Zh. Eksp. Teor. Fiz. 70, 1070 (1976) [Sov. Phys. JETP 43, 559 (1976)l. [7] E.D. Siggia and A . E. Ruckenstein, Phys. Rev. Lett. 44, 1423 (1980). [8] T.-L. Ho and V.B. Shenoy, Phys. Rev. Lett. 77, 3276 (1996).
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[9] B.D. Esry, C.H. Greene, J.P. Burke, Jr., and J.L. Bohn, Phys. Rev. Lett. 78, 3594 (1997). [lo] C.K. Law, H. Pu, N.P. Bigelow, and J.H. Eberly, Phys. Rev. Lett. 79, 3105 (1997). [I11 H. Pu andN.P. Bigelow, Phys. Rev. Lett. 80, 1130 (1998). E.V. Goldstein and P. Meystre, Phys. Rev. A 55, 2935 (1997); T. Busch, J.I. Cirac, V.M. PCrez-Garcia, and P. Zoller, Phys. Rev. A 56, 2978 (1997); R. Graham and D. Walls, Phys. Rev. A 57, 484 (1998); B.D. Esry and C.H. Greene, Phys. Rev. A 57, 1265 (1998); H. Pu and N.P. Bigelow, Phys. Rev. Lett. 80, 1134 (1998). The spatial distributions of out-coupled condensates offers indirect evidence of multicomponent dynamics; see M.-0. Mewes e f al., Phys. Rev. Lett. 78, 582 (1997), and, for example, R. J. Ballagh, K. Burnett, and T.F. Scott, Phys. Rev. Lett. 78, 1607 (1997). M. R. Matthews et al., Phys. Rev. Lett. 81, 243 (1998). In the absence of the 6.8 GHz coupling drive, interconversion between the species is negligible (see note [16] below), quantum fluctuations are insignificant, and our imaging system literally distinguishes one species from the other. It is therefore sensible to describe the system as composed of two separate condensates. We discuss the implications of the coherence between the two components in the companion article (Ref. [17]). The 6.8 GHz energy difference is a million times larger than any other relevant energy scale in the system. For example, the energy released by a single atom converting from a 12) state to the 11) state would, if distributed throughout the sample, be enough to drive the entire sample out of the condensate state. D.S. Hall, M.R. Matthews, C.E. Wieman, and E.A. Cornell, following Letter, Phys. Rev. Lett. 81, 1543 (1998). M. Holland and J. Cooper, Phys. Rev. A 53, R1954 ( 1996). J. P. Burke, Jr. (private communication); J. M. Vogels et al., Phys. Rev. A 56, R1067 (1997). E. Zaremba, A. Griffin, and T. Nikuni, Phys. Rev. A 57, 4695 (1998). B. D. Esry (private communication). J. L. Bohn (private communication). D. S. Hall et al., Proc. SPIE 3270 (to be published). The relative sag can also be adjusted in a static magnetic trap by making use of the Breit-Rabi dependence of the magnetic moments on the applied bias field. We believe this effect to be related to asymmetries in the trapping fields. P.A. Ruprecht, M.J. Holland, K. Bumett, and M. Edwards, Phys. Rev. A 51, 4704 (1995); L.P. Pitaevskii and S. Stringari, Phys. Lett. A 235, 398 (1997); Yu. Kagan, E.L. Surkov, and G.V. Shlyapnikov, Phys. Rev. Lett. 79, 2604 (1997); L.P. Pitaevskii, Phys. Lett. A 229, 406 (1997); W.V. Liu, Phys. Rev. Lett. 79, 4056 (1997); P. 0. Fedichev, G. V. Shlyapnikov, and J. T.M. Walraven, Phys. Rev. Lett. 80, 2269 (1998); S. Giorgini, Phys. Rev. A 57, 2949 (1998); T. Nikuni and A. Griffin, cond-mat/ 9711036.
VOLUME81, NUMBER8
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24 AUGUST 1998
Measurements of Relative Phase in Two-Component Bose-Einstein Condensates D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell* JILA. National Insritiite ojStandards and Technolog,v and Department ojPh,vsics, University of Colorado, Boulder Colorado 80309-0440 (Received 21 May 1998) We have measured the relative phase of two Bose-Einstein condensates using a timc-domain, separated-oscillatory-field condensate interferometer. A single two-photon coupling pulse prepares the double-condensate system with a well-defined relative phase; at a later time, a second pulse reads out the phase difference accumulated between the two condensates. We find that the accuinulated phase difference reproduces from realization to realization of the experiment, even after the individual components have separated spatially and their relative center-of-niass motion has damped. [SO03 1 -9007(98)06973-71 PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 4 2 . 5 0 . D ~
The relative quantum phase between two Bose-Einstein condensates is expected to give rise to a variety of interesting behaviors, most notably those analogous to the Josephson effects in superconductors and superfluid 'He [ 11. Experiments with condensates realized in the dilute alkali gases [2-41 have recently drawn considerable theoretical attention, with a number of papers addressing schemes [5-71 by which to measure the relative phase. Two independent condensates are expected to possess [8] (or develop upon measurement [9, lo]) a relative phase which is essentially random in each realization of the experiment. The experimental observation at MIT of a spatially uniform interference pattern formed by condensates released from two independent traps confirms the existence of a single relative phase [ 111. In this Letter, we use an interferometric technique to measure the relative phase (and its subsequent time evolution) between two trapped condensates [ 121 that are created with a particular relative phase. This system permits us to characterize the effects of couplings to the environment on the coherence [13] between the condensates. As in our previous papers [16,17], we create a condensate of approximately 5 X 10' Rb-87 atoms, confined in the IF = 1,mf = -1) (11)) state in a time-averaged, orbiting potential (TOP) magnetic trap. The rotating magnetic field ( V A F = 1800 Hz) is ramped to 3.4 G and the quadrupole gradient to 130 G/cm, resulting in a trap with an axial frequency v z = 59 Hz. The fields are chosen to make the hyperfine transition frequency nearly field independent [18]. We create the second condensate by applying a short (-400 ks) two-photon pulse that transfers 50% of the atoms (f pulse) from the 11) spin state to the IF = 2, mF = 1) (12)) spin state. The coupling drive has an effective frequency of 6834.6774 MHz and is detuned slightly (-100 Hz) from the expected transition frequency in our trap [19]. After an evolution time T and an optional second f pulse, we release the condensates from the trap, allow them to expand, and image either of the two density distributions [16]. The postexpansion images preserve the relative positions and gross 003 1-9007/98/ 81(8)/ 1543(4)$15.00
spatial features of the condensates as they were in the trap [17,20]. The evolution of the double-condensate system, including the coupling drive, is governed by a pair of coupled Gross-Pitaevskii equations for condensate amplitudes D l and @2:
and
where T = -(h2/2m)V2 is the kinetic energy, m is the mass of the Rb atom, Vhf is the magnetic field-dependent hyperfine splitting between the two states in the absence of interactions, condensate mean-field potentials are Ui = 4n-h2ain;/m and Ui, = 4n-h2aijn,/m, n i = ( @ i ( 2 is the condensate density, and the intraspecies and interspecies scattering lengths [16,17] are ai and ai; = a j ; . For the trap parameters given above, the harmonic magnetic trapping potentials V1 and V2 are displaced from one another by 0.4 p m along the axis of the trap [18]. The coupling drive is represented here in the rotating wave approximation and is characterized by the sum of the microwave and rf frequencies wrf, and by an effective Rabi frequency n(t),where
n(t)=
2n-
. 625 Hz,
coupling drive on; coupling drive off.
(3)
Phase-sensitive population transfer between the 11) and 12) states occurs with the drive on, but the two condensates become completely distinguishable once the drive is switched off [17].
0 1998 The American Physical Society 519
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~~
The first f pulse [Fig. l(b)] creates the 12) condensate with a repeatable and well-defined relative phase with respect to the 11) condensate at t = 0. The relative phase between the two condensates subsequently evolves at a rate proportional to the local difference in chemical potentials between the two condensates W ~( 7I , t ) , which in general is a function of both time and space. Couplings to the environment [2I ] can induce an additional (and uncharacterized) diffusive precession of the relative phase, leading to an rms uncertainty in its value Acp,i,+f [22,23]. After an
i
evolution time T , therefore, the condensates have accumuw21(r, t ) d t + Aqd,ft(T). Durlated a relative phase ing the same time, the coupling drive accumulates a phase w,.fT. A second pulse [Fig. l(e)] then recombines the 11) and 12) condensates, comparing the relative phase accumulated by the condensates to the phase accumulated by the coupling drive. The resulting phase-dependent beat note is manifested in a difference in the condensate density between the two states. Immediately after the second pulse the density in the 12) state ( n ; ? fis)
Ji
4
i
In this equation, 1 2 , denote the densities prior to the application of the second pulse. The interference term in Eq. (4) shows that measurement of ( 7 ) in the overlap
5
nl
n2
region is sensitive to the relative phase. Each realization of the experiment (with a freshly prepared condensate) yields a measurement of the relative phase for a particular T ; by varying T , we can measure the evolution of the relative phase. At short times T , for which the overlap between the condensates remains high, varying the moment at which the second pulse is applied causes an oscillation of the total resulting number of atoms in the 12) state. The oscillation occurs at the detuning frequency 6 = w21 - w,f and is completely analogous to that observed in separatedoscillatory-field measurements in thermal atomic beams [ 2 5 ] or in cold (but noncondensed) atoms in a magnetic trap [26]. The fringe contrast, initially loo%, decreases as the condensates separate. After -45 ms, the relative center-of-mass motion damps and comes to equilibrium, leaving the components with a well-defined overlap region at their boundary, as shown in Figs. l(d) and 2(a); see also Fig. 5(b) of Ref. [17]. Application of a second pulse at T 2 45 ms results in a density profile in which the interference occurs only in the overlap region [see Figs. I(f) and 2( b)] . We look at the density of atoms in the 12) state at the center of the overlap region [27] in order to examine the intriguing issue of the reproducibility of the relative phase accumulated by the condensates during the complicated approach to equilibrium. If the phase diffusion term in Eq. (4) is so large that the uncertainty is greater than T , then repeated measurements for the same values of T will yield an incoherent (i.e., random) ensemble of interference patterns. In the opposite extreme (i.e., very little phase diffusion), repeated measurements will give essentially the same interference pattern at T in each experimental run. We plot the optical density in the center of the overlap region as a function of T in Fig. 3, and observe an oscillation at the detuning frequency with a visibility of approximately 50%, corresponding to an rms phase diffusion Aqdiff(T) 5 2;. At longer times the maximum contrast observed in a single realization of the experiment decreases slightly, possibly due to the increasing presence of thermal atoms as the condensates decay.
5
( d)
,
. -----....I .
,.,,-'-
45 ms s t < T ni2-pulse
expansion
.,---tI .____________ 1..._
(f)._.____________
I
t >- T
A schematic of the condensate interferometer [24]. (a) The experiment begins with all of the atoms in condensate 11) at steady state. ( b ) After the first pulse, the condensate has been split into two components with a well-defined initial relative phase. (c) The components begin to separate in a complicated fashion due to mutual repulsion as well as a 0.4 p m vertical offset in the confining potentials (see also Fig. 3 of Ref. [17]). (d) The relative motion between the components eventually damps with the clouds mutually offset but with some residual overlap. Relative phase continues to accumulate between the condensates until (e) at time T a second pulse remixes the components; the two possible paths by which the condensate can arrive in one of the two states in the hatched regions interfere. (f) The cloud is released immediately after the second pulse and allowed to expand for imaging. In the case shown, the relative phase between the two states at the time of the second pulse was such as to lead to destructive interference in the 11) state and a corresponding constructive interference in the 12) state.
FIG. I .
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l.OO,,
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’
I
--
I ’
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50
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vertical position (pm) FIG. 2. (a) The postexpansion density profiles of the condensates in the steady state attained after a single f pulse. These density profiles vary little from shot-to-shot (and day-to-day). (b) The density profiles after the second 7 pulse. The density in the overlap region depends on the relative phase between the two condensates at the time of the pulse; in the case shown, we observe constructive interference in the 12) state and destructive interference in 11). The patterns in (b) are much less stable than those in (a), possibly as a result of unresolved higher-order condensate excitations, issues associated with the expansion, or technical instabilities of the apparatus.
The stable interference patterns show that the condensates retain a clear memory of their initial relative phase despite the complicated rearrangement dynamics of the two states following the first pulse. This is rather surprising, since the center-of-mass motion of the double-condensate system is strongly (and completely) damped, and, in general, decoherence times in entangled states tend to be much shorter than damping times [28-301. The intuition one develops in understanding few-particle quantum mechanics may not apply to experiments involving condensates. The phase between the two condensates seems to possess a robustness which preserves coherence in the face of the “phase-diffusing” couplings to the environment. We have read out the relative phase of two BoseEinstein condensates using a time-domain version of the method of separated oscillatory fields. We observe the persistence of phase memory in this “condensate interferometer,” despite the presence of damping and the complicated rearrangement of the two condensate components. We have established that the time scale for phase diffusion in this system can be longer than 100 ms. The double-
pulse separation (ms)
The value of the condensate density in the ,12), state is extracted at the center of the overlap region (inset) and plotted (a) as a function of T . Each point represents the average of six separate realizations, and the thin bars denote the rms scatter in the measured interference for an individual realization. The thick lines are sinusoidal fits to the data, from which we extract the angular frequency w 2 , - a r t . In (b), the frequency of the coupling drive w,t has been increased by 27r X 150 Hz, leading to the expected reduction in fringe spacing.
condensate methods we have developed will be applicable to other experiments which explore phase diffusion as a function of condensate parameters including temperature, number of atoms [31-331, and collision rates [34]. Collapses and revivals of the “memory” of the relative phase are predicted [ 10,3I] at time scales which may be experimentally accessible should environmentally induced diffusion effects remain small. Our methods will also allow us to examine other phase-related phenomena, such as phase locking and analogs of the superconducting Josephson junctions [35]. We gratefully acknowledge useful conversations with A. J. Leggett, as well as with the other members of the JILA BEC Collaboration. This work is supported by the ONR, NSF, and NIST.
*Quantum Physics Division, National Institute of Standards and Technology. (11 S. Backhaus et al., Science 278, 1435 (1997). [2] M. H. Anderson et al., Science 269, 198 (1 995). [3] K.B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995).
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C.C. Bradley, C . A . Sackett, and R.G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); Phys. Rev. Lett. 79, 1170 (1997). J. Javanainen, Phys. Rcv. A 54, R4629 (1996). A. I m a m o ~ l uand T . A . B . Kennedy, Phys. Rev. A 55, R849 (1997). J . Ruostekoski and D.F. Walls, Phys. Rev. A 56, 2996 (1997). S . M . Barnett, K. Burnett, and J . A . Vaccaro, J . Res. Natl. Inst. Stand. Technol. 101, 593 (1996). J . Javanainen and S . M . Yoo, Phys. Rev. Lett. 76, 161 (1996). Y . Castin and J. Dalibard, Phys. Rev. A 55, 4330 (1997). M. R. Andrews et a/.. Science 275, 637 (1997). Because of technical noise, the intrinsically random nature of the relative phase was not conclusively established. One can equally well describe our experimental system as a single condensate in a coherent superposition of internal states, or as two separate condensates with a particular relative phase. I t would be customary in condensedmatter physics, with a typical coherent sample size of lo2’ particles, to use the latter description. On the other hand, an expert in atom interferometry, with a typical coherent sample size of one particle, would be more likely to use the former. We define “coherence” as the predictability of the relative quantum phase. For interesting discussions of quantum coherence, see Refs. [ 141 and [ 151. A. J. Leggett, in Bose-Einstein Condensation, edited by A. Griffin, D. W. Snoke, and S. Stringari (Cambridge University Press, Cambridge, England, 1995). W.H. Zurek, Phys. Today 44, No. 10, 36 (1991). M. R. Matthews et al., Phys. Rev. Lett. 81, 243 (1998). D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E.A. Cornell, preceding Letter, Phys. Rev. Lett. 81, 1539 (1998). D. S. Hall et al., Proc. SPIE 3270 (to be published). The microwave and rf frequencies are produced by synthesizers locked to global-positioning system signals. The manufacturer of the receiver claims a root Allan variance better than lo-’’. C. H. Greene (private communication). We separate the “environment” into two categories: an “intimate” environment, which includes interactions with thermal atoms as well as with internal modes within the condensate itself, and an “external” environment which includes such experimental factors as uncontrolled fluctuations in the magnetic fields. The former are intrinsic to the physics of the problem, whereas the latter can (in principle) be suppressed. In practice, we
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experience difficulty in keeping the external environment from intruding on our measurements; only in the modes of quietest operation are the oscillations of Fig. 3 observable. When coherence is observed, perturbations due to both the intimate and the external environments must be small; when coherence is washed out, either may be responsible. For these reasons, we have not attempted in this paper to quantify the loss of coherence at longer times. [22] A.J. Leggett and F. Sols, Found. Phys. 21, 353 (1991). [23] Quantum fluctuations introduce an additional uncertainty that grows linearly in time in a process akin to the spreading of a Gaussian wave packet in space (see Refs. [10,31-33]). We estimate the time scale for this process to be much longer than the duration of an individual nicasurement. [24] Atom interferometry between tl?erinal beams is a wellestablished technique. See, for example, D. W. Keith, C. R. Ekstrom, Q. A. Turchette, and D. E. Pritchard, Phys. Rev. Lett. 66, 2693 (1991). An early condensate interferometer is described in H.-J. Miesner and W. Ketterle, Solid State Commun. (to be published). [25] N. F. Ramsey, Molecular Beams (Clarendon Press, Oxford, 1956). [26] E.A. Cornell, D.S. Hall, M . R . Matthews, and C.E. Wieman, J. Low Temp. Phys. (to be published). [27] We take the average of a -14 p m wide (postexpansion) vertical swath down the middle of the condensate density profile and extract the amplitude of the pixel at the center of the condensate image (ix., at the center of the overlap region). [28] A. 0. Caldeira and A. J. Leggett, Phys. Rev. A 31, 1059 (1985). [29] D.F. Walls and G. J. Milburn, Phys. Rev. A 31, 2403 (1985). [30] M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996). [31] E. M. Wright, D. F. Walls, and J. C. Garrison, Phys. Rev. Lett. 77, 2158 (1996). [32] J. Javanainen and M. Wilkens, Phys. Rev. Lett. 78, 4675 (1997). [33] M. Lewenstein and L. You, Phys. Rev. Lett. 77, 3489 (1996); A. Imamoglu, M. Lewenstein, and L. You, Phys. Rev. Lett. 78, 251 1 (1997). [34] T. Wong, M. J. Collett, and D.F. Walls, Phys. Rev. A 54, R3718 (1996). [35] J. Javanainen, Phys. Rev. Lett. 57, 3164 (1986); 1. Zapata, F. Sols, and A. J. Leggett, Phys. Rev. A 57, R28 (1998); A. Smerzi, S. Fantoni, S. Giovanazzi, and S.R. Shenoy, Phys. Rev. Lett. 79, 4950 (1997); J. Williams and M. Holland (private communication).
PHYSICALREVIEW LETTERS VOLUME81
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Dynamical Response of a Bose-Einstein Condensate to a Discontinuous Change in Internal State M. R. Matthews, D. S. Hall, D. S. Jin," J. R. Ensher, C. E. Wieman, and E. A. Cornell* JlLA ar?d Departnient of"Pl7j~sics,Uiiiiwsit,v of Colomdo and Natioiial h7stit~iteof Standal-ds and techno log,^, Boiikkr, co/ol-ado 803 0.9- 044 0
F. Dalfovo, C. Minniti, and S. Stringari Dipal-tiinento di Fisica, Univeisita di Trento, and Istituto Nazionale Fisica della Materia, I-38050 Povo, Italy (Received 1 1 March 1998) A two-photon transition is used to convert an arbitrary fraction of the "Rb atoms in a IF = I , m f = ~ I condensate ) to the IF = 2 , m j = 1) state. Transferring the entire population imposes a discontinuous change on the condensate's mean-field repulsion, which leaves a residual ringing in the condensate width. A calculation based on Gross-Pitaevskii theory agrees well with the observed behavior, and from the comparison we obtain the ratio of the intraspecies scattering lengths for the two ,) 12). [S0031-9007(98)06574-01 states, ~ q , , - , ) / a ~=~ ,1.062( PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 34.20.Cf
The effects of interactions on dilute atomic BoseEinstein condensates (BEC) [ 13 have been studied in several contexts, including density and momentum distributions [2], collective excitations [3], specific heat [4], speed of sound [ 5 ] , and limited condensate number [6]. There has been excellent quantitative agreement between these experiments and theory, made possible by both experimental advances and the fact that the interactions can be modeled relatively simply. Despite complicated interatomic potentials, the mean-field interaction in condensates is well characterized by a single parameter, the s-wave scattering length a. Previous quantitative experiments on interactions have all been done using singlecomponent condensates with a constant value of a, but several authors have proposed using external optical or magnetic fields to shift the mean-field interaction by perturbing the interatomic potential [7-91. We present a method for the creation of condensate mixtures using radiofrequency (rf) and microwave fields. We are able to transfer abruptly a trapped condensate of one hyperfine state into a coherent superposition of two trapped hyperfine states, and then watch the subsequent dynamical behavior. This approach makes possible a variety of twospecies BEC studies. In this paper, we examine quantitatively one special case, in which all condensate atoms are
converted from one state to another. Since these two states have slightly different values of a,the sudden change in self-interaction gives rise to oscillatory spatial behavior of the condensate wave function [ 101. The scattering length ratio can be extracted from a model using analytical equations of motion for the condensate widths [lo-121 based on Gross-Pitaevskii theory. The first demonstration of a binary mixture of condensates by Myatt et al. [13] produced overlapping condensatesofthe5S1/21F = 1,mf = -1)andlF = 2,mf = 2) states of 87Rb. The ratio of the magnetic moments of these states is 1:2, so the condensates experience different potentials in a magnetic trap and are displaced unequally from the trap center by gravity. Because of an accidental coincidence between the singlet and triplet scattering lengths of "Rb, collisional loss is reduced and any mixture of spin states will be relatively long-lived [ 13- 151. Here, we use mixtures of 11, - 1) and 12,l) states, which possess several advantages. First, these two states have essentially identical magnetic moments, and hence feel identical confining potentials. Second, one can conveniently and quickly change atoms from the 11, - 1) state to the 12,l) state by a two-photon transition (microwave plus rf). Finally, we can selectively image the different components using appropriately tuned lasers.
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The apparatus uses a inultiply loaded, double magnetooptical trap (MOT) scheme [16], which consists of two MOT regions connected by a 60 cin long, 1 cin diameter transfer tube. In the first MOT, ”Rb atoms are collected from a background vapor, produced by heating a Rb getter source [17]. The trapping beams are then turned off and a near-resonant laser beam is pulsed on to push the atoms through the transfer tube into the second MOT. To prevent collisions with the walls, the outside of the transfer tube is lined with strips of permanent magnetic material which form a hexapole guiding field. The a t o m exit the hibe and are captured by the second MOT, which has a 70 s lifetime due to differential pumping. This process is repeated twice a second for 10 s until approximately 1O9 atoms have accuinulated in the second MOT. The atoms are further cooled in an optical molasses and then optically pumped into the 11, - 1) state. These atoms are then transferred into a time-averaged, orbiting potential (TOP) magnetic trap [18]. The TOP trap consists of a quadrupole magnetic field with an axial gradient of -250 G/cm and a uniform magnetic field which rotates at 1.8 kHz. The fraction of atoms captured from molasses in the TOP trap depends approximately linearly on the magnitude of the rotating field. We use the maximum rotating field achievable in our apparatus (49 G) to capture up to 50% of the atoms from the MOT, substantially enhancing the initial number as compared to our previous setup [l]. As in [I], the confined cloud is then magnetically compressed and evaporatively cooled to a temperature where there is no noticeable noncondensed fraction remaining, i.e., less than 0.4 of the BEC transition temperature. Evaporative cooling requires about 30 s, and typically produces a condensate of 5 x lo5 atoms. The magnetic moments of the 11, - 1) and 12,l) states are nearly the same to first order in magnetic field. Secondorder dependence, the nuclear magnetic moment, and small effects due to the time varying nature of the TOP trap must be taken into account when calculating the exact trap potential [ 19,201. The quadrupole and rotating magnetic fields are adiabatically changed after evaporation to make the trap potentials the same (to within 0.3%) for the two states. Evaporation takes place in a trap with radial frequency u, = 35 Hz; subsequent work occurs at ur = 17 Hz with a rotating field magnitude of 5.7 G ( u , = u,/& for the TOP trap). The two-photon transition used to change the hyperfine state is shown schematically in Fig. l(a). We apply a pulse of microwave radiation at a frequency slightly less than the ground-state hyperfine splitting of 87Rb (-6.8 GHz) along with a -2 MHz rf magnetic field. This connects the 11, - 1) state to the 12, 1 ) state via an intermediate virtual state with a detuning of 2.2 MHz from the 12,O) state. The two-photon Rabi frequency is 600 Hz, which is much faster than the characteristic frequency 2u, = 96 Hz for the condensate to change shape.
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pulse length (ps) FIG. 1. (a) A diagram of the ground-state hyperfine levels ( F = 1,2) of 87Rb shown with Zeeman splitting due to the presence of a magnetic field. The two-photon transition is driven between the 11, - I ) and 12, I ) states. (b) The Rabi oscillation of population between the 11, - 1) (open circles) and 12, 1) states (solid circles) as a function o f the two-photon drive duration. The lines are fit to the data and show the expected sinusoidal oscillation.
We are able to view either component of a mixture of 11, -1) and 12,l) condensates or the combined density distribution. Observation of the atoms is a destructive measurement in which the TOP trap is suddenly turned off and the condensate is left to expand ballistically for 22 ms. To image the combined density distribution of both states, a short pulse of “repump” light pumps the atoms from the 11, - 1) state into the F = 2 manifold. About 100 ps later, a IT’ circularly polarized probe beam 17 MHz detuned from the 5S1/2 F = 2 to 5P312 F’ = 3 cycling transition is scattered by the atoms and the shadow is imaged onto a charge-coupled device camera. Imaging only the 12,l) condensate uses the same procedure, except that the pulse of repump light is omitted. Atoms in the 11, - 1) state are far (6.8 GHz) from resonance and invisible to the probe beam. Imaging only the 11, - 1) atoms is similar to viewing both species simultaneously, but the 12,l) atoms are first “blown away” from the imaging region by a 2 ms, 60 pW/cm2 pulse of F = 2 to F’ = 3 light applied near the end of the ballistic expansion. This light has
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no effect on the 11, - 1) atoms, to which a subsequent pulse of repump light is applied, pumping them into the F = 2 manifold for probing. In Fig. I(b) we show the population of each state as a function of the two-photon pulse length. The Rabi oscillations have nearly 100% contrast, indicating that we are able to put any desired fraction into the (2, I) state. We have investigated the dynainical behavior of the condensate after more than 99.5% of the atoms are transferred from the 11, - 1) state to the 12, 1) state. After the transfer, the condensate is allowed to evolve in the trap for a time interval T before the cloud is released from the trap and probed. The experiment is repeated for different values of T , and the axial and radial widths of the condensate are measured using a fit to the two-dimensional condensate image [21]. Figure 2 illustrates the time dependence of both the axial and radial sizes of the atom cloud. Qualitatively, the data for both dimensions are consistent with a “compression” oscillation. The condensate shrinks for very early times, indicating a weaker mean-field repulsion in the 12, 1 ) state. In order to describe the response to the change in scattering length, we use the Gross-Pitaevskii (GP) equation for the condensate wave function, with a time-dependent interaction term:
x @(r,t ) ,
(1)
where V(r) is the confining potential, rn is the 87Rbmass, and U ( t ) = 4n-Fi2a(t)/m.This has the form of a nonlinear Schrodifiger equation, with a mean-field characterized by the s-wave scattering length a ( t ) . For t < 0, a ( t ) = a l , the scattering length for 11, - 1) atoms on 11, - 1) atoms.
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FIG. 2. Oscillation in the width of the cloud in both the axial and radial directions due to the instantaneous change in scattering length. The widths are for condensates as a function of free evolution time in units of the radial trap period (w,’ = 9.4 rns), followed by 22 rns of ballistic expansion. Each point is the average of approximately ten measurements. Note that the fractional change in width is quite small. The solid line is a fit of the model to the data, with only the amplitude of the oscillation and the initial size as free parameters.
For t > 0, a ( t ) = a2, the 12,1 ) on 12, 1 ) scattering length. The condensate is formed in its ground state in which the density profile n(r) = I@(r)I2is constant in time, and with the axial and radial widths (w:, w,) determined by V(r), a 1 , and the number of atoms N. After a sudden change in a ( r ) , @ ( r t, ) is projected onto a coherent superposition of its new ground state and collective vibrational modes. Instead of solving the complete GP equation, one can use the Thomas-Feimi (TF) approximation, which corresponds to neglecting the quantum pressure term in the kinetic energy of the condensate. In this case, one can replace the GP equation with a pair of scalar equations of motion for the condensate widths [lO-12]:
Here the widths w, = and w y = are the radial and axial rms radii, and the subscript i is to be replaced with either r or z for the respective widths. The TF approximation holds when N is large [22] and, in this limit, the GP theory coincides with the hydrodynamic theory of superfluids [23]. Time-dependent behavior is predicted by numerically integrating Eqs. (2). The initial conditions ( t < 0) are that the widths are at the values determined by the groundstate solution, &, = &z = 0, and a ( t ) = a [ . After t = 0, n ( t ) = a2 and the integration proceeds for time T . We model the removal of the trap by setting v r = vz = 0 and continue the numerical integration for the 22 ms free expansion. The axial and radial widths subsequently expand with different speeds. In the TF approximation, the ratio between the two expanded widths, averaged over T , can be shown to depend only on v,, uz and the expansion time [ 121. In our case, the theory predicts wz/w, = 1.29, in good agreement with the measured value wz/w, = 1.31. The oscillations of the widths correspond to a superposition of two m = 0 modes. Since no angular momentum is imparted to the condensate by the change of scattering length, these are the only modes excited. The calculated mode frequencies are v/v, = 1.SO and v/v, = 4.99, and turn out to be independent of the amplitude in the range considered here. The two modes contribute to the oscillations in z and r with a different phase and amplitude, the axial motion behaving mainly as the fast mode and the radial motion as the slow one. In each direction, the ratio of the amplitudes of the two modes is predicted to be constant over the range of possible scattering lengths relevant to the experiment. Thus, only the initial size and an overall amplitude are used as adjustable parameters in comparison of theory with experiment. The solid lines in Fig. 2 show the theoretical prediction using the best fit value of the oscillation amplitude, which is related to the scattering length ratio u l / u 2 . The predicted oscillations agree remarkably well with the shape of the data, for both the frequency and the phase of the
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FIG. 3. Dependence of the oscillation amplitude on the rf strength in the two-photon transition. The vertical axis represents the fitted value of the fractional excitation amplitude, with error bars from uncertainty in the fit. The solid line shows extrapolation to zero rf amplitude.
two modes. From the fit amplitude we obtain the ratio of the scattering lengths n I / a ? = 1.062 2 12, which is consistent with the ratio of 1.059’: obtained in a theoretical calculation of binary collision parameters [24]. Analysis of the data is complicated slightly by an observed systematic dependence of the oscillation amplitude on the magnitude of the rf drive during the two-photon pulse. We believe that this effect is partly due to a small dressing of the atoms to the Il,O) or 12,O) state by the rf, and partly due to a coupling of the rf onto electronics controlling the trap potential. Both result in a change in strength of the confining potential during the two-photon pulse which returns to normal when the pulse is complete. After this impulsive perturbation, the BEC can be thought of as freely evolving with an initial “velocity” in the width (wr, Gz # 0). This is in contrast to the discrete change in scattering length, in which the BEC width has an initial offset from the equilibrium width, but no initial velocity. The systematic is manifested as an initial offset in the phase of the resulting oscillation, which is indeed observed for large rf amplitudes. Our result for the ratio was obtained by extrapolating to zero rf amplitude with a quadratic fit as shown in Fig. 3. We have explored the dynamical response of a BoseEinstein condensate to a sudden change in the interaction strength. The agreement with the model is excellent, and presents a new and precise way of measuring a ratio of scattering lengths. This also allows a test of the theory for pressure shifts in atomic clocks [25]. Our method of creating two fully interpenetrating condensates will allow coherent control of the relative population and detailed examination of spatial and phase dynamics in the future. We acknowledge useful conversations with the other members of the JILA BEC Collaboration, and, in particular, with Chris Greene. This work is supported by the ONR, NSF, and N E T . *Quantum Physics Division, National Institute of Standards and Technology.
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[ I ] M.H. Anderson et al., Science 269, 198 (1995); K.B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995); C.C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); C.C. Bradley, C . A . Sackett, J . J . Tollett, and R. G. Hulet, Phys. Rev. Lett. 79, 1170 (1997). [2] M.-0. Mewes et al., Phys. Rev. Lett. 77, 416 (1996); M. J . Holland, D. S. Jin, M. L. Chiafalo, and J. Cooper, Phys. Rev. Lett. 78, 3801 (1997). [3] D. S. Jin et al., Phys. Rev. Lett. 77, 420 (1996); M.-0. Mewes et al., Phys. Rev. Lett. 77, 988 (1996). [4] J. R. Ensher et al., Phys. Rev. Lett. 77, 4984 (1996); see also S. Giorgini, L.P. Pitaevskii, and S. Stringari, Phys. Rev. Lett. 78, 3987 (1997). [5] M. R. Andrews et al., Phys. Rev. Lett. 79, 553 (1997). [6] C. C. Bradley, C . A . Sackett, and R.G. Hulet, Phys. Rev. Lett. 78, 985 (1997). [7] E. Tiesinga, B. J. Verhaar, and H. T.C. Stoof, Phys. Rev. A 47, 41 14 (1993). [8] P.O. Fedichev, Yu. Kagan, G.V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 77, 2913 (1996). [9] John L. Bohn and P.S. Julienne, Phys. Rev. A 56, 1486 (1997). [ l o ] Yu. Kagan, E. L. Surkov, and G.V. Shlyapnikov, Phys. Rev. Lett. 79, 2604 (1997). [ I l l Yu. Kagan, E.L. Surkov, and G.V. Shlyapnikov, Phys. Rev. A 54, R1753 (1996); Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996). [I21 F. Dalfovo, C. Minniti, S. Stringari, and L. Pitaevskii, Phys. Lett. A 227, 259 (1997); F. Dalfovo, C. Minniti, and L. Pitaevskii, Phys. Rev. A 56, 4855 (1997). [13] C. J. Myatt et al., Phys. Rev. Lett. 78, 586 (1997). [I41 P. S. Julienne, F. H. Mies, E. Tiesinga, and C. J. Williams, Phys. Rev. Lett. 78, 1880 (1997). [ 151 James P. Burke, Jr., John L. Bohn, B. D. Esry, and Chris H. Greene, Phys. Rev. A 55, R25 11 (1997). [16] C. J. Myatt et al., Opt. Lett. 21, 290 (1996). [I71 C. Wieman, G. Flowers, and S. Gilbert, Am. J. Phys. 63, 317 (1995). The getter and feedthroughs are in a small glass arm projecting from the glass MOT chamber. The rubidium vapor is readily turned on or off by controlling the current heating. We see no degradation of the trap lifetime compared to apparatus using a traditional (solid metallic) source. After initial seasoning the Rb is not readily consumed by the glass cell walls; a single 1 cm strip of getter material has shown no degradation after a year of nearly daily use. [ 181 Wolfgang Petrich, Michael H. Anderson, Jason R. Ensher, and Eric A. Cornell, Phys. Rev. Lett. 74, 3352 (1995). [19] D. S. Hall et al., Proc. SPIE-Int. SOC.Opt. Eng. 3270, 98 (1998). [20] John L. Bohn (private communication). [21] The expanded condensates are fit using a two-dimensional projection of the Thomas-Fermi form for the density distribution in a harmonic trap. The fit also provides a measure of the number of atoms: we have measured a lifetime of 300 ms for the 12,l) state. This gives an upper limit on the 12,l) inelastic loss rate of 3 X cm3/s. Number, and hence width, is a decaying function of time, so we apply a small correction to the data (Fig. 2) based on this.
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Equations (2) can be indeed generalized to include effects beyond TF approximation. This has been done, for instance, by V. M. Perez-Garcia et al., Phys. Rev. Lett. 77, 5320 (1996). For the condensate used in the present work we have numerically checked that the corresponding changes in the final value of .,/a2 are much smaller than the experimental uncertainty.
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[23] S. Stringari, Phys. Rev. Lett. 7 7 , 2360 (1996) [24] J. P. Burke, Jr. (private communication) based on methods in J . P . Burke, Jr., J . L . Bohn, B . D . Esry, and C . H . Greene, Phys. Rev. Lett. (to be published); J. M. Vogels et a[., Phys. Rev. A 56, R1067 (1997). [25] S. J . J. M. F. Kokkelmans, B. J. Verhaar, K. Gibble, and D. J. Heinzen, Phys. Rev. A, 56, R4389 (1997).
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Resonant Magnetic Field Control of Elastic Scattering in Cold 85Rb J. L. Roberts, N . R. Claussen, James P. Burke, Jr., Chris H. Greene, E. A. Cornell,* and C. E. Wieman JILA arid the Depai-triieiit o j Pjiysics, Universir?jof Co l o rd o , Boulder; Colordo 80309-(1440 (Received 29 July 1998) A magnetic field-dependent Feshbach resonance has been observed in the elastic scattering collision rate between a t o m in the F = 2 , M = -2 statc of “Rb. Changing the magnetic field by several Gauss caused the collision rate to vary by a Factor of lo’, and the sign of the scattering length could be reversed. The resonance peak is at 155.2(4) C; and its width is 11.6(5) G . From these results we extract much improved values for the three quantities that characterize the interaction potential: The van der Waals coefficient C h , the singlct scattering length ci,~, and the triplet scattering length c l T . [SO03 1-9007(98)07904-61 PACS numbcrs: 34.50.-s, 03.75.Fi, 32.80.Pj, 34.20.Cf
Very-low-temperature collision phenomena can be quite sensitive to applied electromagnetic fields. Several groups have altered inelastic collision rates in optical traps by applying laser fields [ I ] . There have also been numerous proposals [2-61 for using laser and static electric or magnetic fields to influence the s-wave scattering length ( a ) , and equivalently, the elastic collision cross sections ( a = 8.rra2) between cold atoms. Particularly notable is the prediction by Verhaar and co-workers [ 2 ] that as a function of magnetic field there should be Feshbach resonances in collisions between cold (-pK) alkali atoms. These resonances were predicted to cause dramatic changes in the cross section and to even allow the sign of the scattering length to be changed. Such resonances are very interesting collision physics, but they also offer a means to manipulate Bose-Einstein condensates (BEC), the properties of which are primarily determined by the scattering length. Some time ago we searched for such resonances in 133Csand 87Rb, without success [7,8]. This was not surprising because there was enormous uncertainty as to the positions and widths of the predicted resonances. However, recent photoassociation spectroscopy has greatly improved the knowledge of the alkali interatomic interaction potentials, allowing these resonance parameters to be predicted with much less uncertainty [9]. As we show below, the fact that the widths and positions of these resonances are so sensitive to the interatomic potentials makes their measurement a good way to determine these potentials. The improved predictions for the resonance positions facilitated their observation. In the past few months Feshbach resonances have been observed in both 23Na [lo] and ”Rb [ l 11. In the sodium work, two magnetic field-dependent processes were observed: (1) A change in the expansion rate of BEC due to a change in the scattering length, and (2) an enormous, and as yet unexplained, increase in the loss rate. This loss rate precluded the study of collisions in the interesting field regime near the center of the resonance where the scattering length changes sign. In the rubidium work
[ I I ] the resonance was detected as a magnetic field dependence of the photoassociation spectra. The resulting resonance width was measured to be far larger than originally predicted by theory [2]. Here we report the study of a Feshbach resonance in the elastic collision cross section between atoms in the F = 2 , M = - 2 state ofX5Rb. By changing the dc magnetic field we are able to change the collision rate by 4 orders of magnihide and explore regions of positive, negative, and essentially zero scattering length. We determine the width and position of the resonance about a factor of 10 more precisely than in Ref. [ 111, and from these data we improve the accuracy of the Rb interaction potential substantially. In contrast to Ref. [lo], we do not observe two or three body loss at resonance because we work at much lower densities (lo9 atoms/cm3). We measured the collision rates using the technique of “cross-dimensional mixing,” as in our earlier work [7]. In this technique, a nonisotropic distribution of energy is created in a magnetically trapped cloud of atoms, and the time for the cloud to reequilibrate by elastic collisions is measured. The apparatus used is identical to that used in our previous work on BEC in 87Rb [12]. It is a double magneto-optic trap (MOT) system in which multiple samples of atoms are trapped in a relatively highpressure chamber and then transferred to a second MOT in a low-pressure chamber. The second MOT is then turned off and a “baseball coil” magnetic trap is turned on around them. The atoms are then cooled by forced evaporation. After the atoms are evaporatively cooled to the desired temperature, the energy in the radial direction (and correspondingly the square of the width of the trapped cloud) is reduced to about 0.6 that of the axial direction. This is done by decreasing the frequency of the rf “knife” used for the evaporative cooling more rapidly than the cloud can equilibrate [ 131. The bias magnetic field is then adiabatically ramped to the selected value, and the cloud is allowed to equilibrate for a fixed time. The shape of the cloud is then measured by absorptive imaging. This is repeated for different equilibration times, and the aspect
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ratios of the cloud vs time are fit to an exponential to determine the equilibration time constant T : ~ . Although the energy dependence of the cross section can lead to small deviations from exponential equilibration, the relative changes in T,', with B are large enough to dominate any error in curve fitting. A s discussed in Ref. [14], T,', is proportional to the inverse of the elastic collision rate [ 151. From the measured trap oscillation frequencies and the measured shape and optical depth, we determine the temperature T and average density ( n ) of the cloud. The magnetic field value at the center of the cloud is determined to ?0.1 G by finding the rf frequency that resonantly drives spin flip transitions for the atoms at that position and using the Breit-Rabi equation. The spread in the magnetic field across the cloud scales as TI/' and is 0.6 G FWHM for a 0.5 pK cloud. The measured equilibration times depend on B and T and vary from 0.15 s to nearly 2000 s. To display data taken at several values of both density and temperature, we convert it to the normalized equilibration rate, rnorm = l/(izu)~,~ Figure . 1 displays the data has units of cross versus mean magnetic field. rnolm section and in fact is proportional to an empirical average over the field- and energy-dependent elastic scattering cross section g ( B , E ) . To calculate this average is problematic because the magnetic field and energy are coupled-a detailed Monte Carlo simulation of particle trajectories [ 161 is probably required. However, in the low-temperature limit the spread in magnetic field across the cloud is small so the maximum and minimum valwill occur at the same magnetic field as the ues of rnorm peak and zero of a ( B ,E = 0). For a rough qualitative comparison, in Fig. 1 we have also plotted the theoretically predicted values of u ( B , E ) for two temperatures. These predictions are calculated using the atomic potentials determined from the observed maximum and minimum values of the relaxation rate, as discussed below. Both the cross section curves and the equilibration data show the same qualitative features. For temperatures below a few p K , there is a slightly asymmetric peak near 155 G, and the width and height of this peak are strong functions of temperature. At 167 G there is a profound drop in the rate. This dip is quite asymmetric, but the shape is relatively insensitive to T , and at the bottom (field value & i n ) the rate is essentially zero. The field value of the peak ( B p e a k ) is customarily defined to be the position of the Feshbach resonance, and the resonance width A is the distance between B p e a k and Bmin [ 111. The scattering length is positive for field values between B p e a k and Bmin and is negative for field values below B p e a k or above & i n . This dependence of the sign is expected from previous theory. The observation that Bpeak is at a lower field than Bmin provides experimental confirmation that the scattering length is negative away from the resonance. 5110
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B ( Gauss) FIG. 1. (a) The data points show the measured equilibration rate divided by average density and velocity (r,,,, = l / ( n u ) ~vs~ magnetic ~) field B. Data are shown for five different temperatures: 0.3 pK (A), 0.5 p K (O), 1.0 pK 3.5 p K ( V ) , and 9.0 p K (M). The lines are calculations for the thermally averaged elastic cross sections (nor equilibration rates) and hence are not expected to fit the data points. The solid line corresponds to 0.5 p K and the dot-dashed line to 9.0 p K . (b),(c) Expanded views of the regions of maximum and minimum cross section.
(o),
Because the functional form of the normalized equilibration rate is not known, the desired quantities B p e a k and Bmincannot be found by a detailed fit to all of the data. We determine them by fitting a simple smooth curve to only the few highest (or lowest) points at each temperature below 5 pK,and assigning correspondingly conservative error bars that more than span the values determined for all temperatures [17]. We find B p e a k = 155.2(4) G and Bmin = 166.8(3) G, giving a resonance width A = 11.6(5) G. The values for B p e a k and A are reasonably consistent with the less precise values of 164(7) G and 8.4(3.7) G measured in Ref. [ll]. In our experiment,
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the accuracy of the peak position is much higher primarily becausz of better field calibration. The better accuracy in the width is largely because the photoassociation measurement is a somewhat less direct way to observe the resonance, and so substantial and somewhat uncertain corrections are required to go from observed signal widths to the actual resonance width. We can now use these measured quantities to determine the singlet and triplet Born-Oppenheimer potentials. The accuracy of predicted cold collision observables hinges on the quality of these potentials used in the radial Schrodinger equation for the nuclear motion. Because the scattering is purely s wave at these temperatures, accurate determination of these potentials in turn relies primarily on three parameters, the long range van der Waals coefficient c6 and the zero-energy singlet and triplet phases. The phases are usually tabulated in terms of singlet as and triplet cly scattering lengths calculated with hyperfine terms omitted from the radial equation. The old "nominal" values of the scattering lengths and c6 (optimized to achieve agreement with previous measurements) predicted the position of this Feshbach resonance reasonably well, but, as in Ref. [ 111 the predicted width was much smaller than observed. This width reflects the coupling of bound and continuum channels and is primarily controlled by the difference between the singlet and triplet scattering lengths. It is inconvenient to tabulate results as a function of singlet and triplet scattering lengths for "Rb, because both are very close to lying on top of a divergent pole. Accordingly, we present the results of our analysis in terms of a bound state phase (fractional bound state number) V D as in Ref. [l I]. We define the corresponding singlet V D S or triplet V D T bound state phase in terms of the scattering length through the relation [18]: V D = (l/.-)cot-'(a/aref - l), where aref = r(5/4)/[& r(3/4)](2mC6/h2)1'4 [19] and m is the reduced mass of the atomic pair. The bound state phase is related to the short range quantum defect psr presented in Ref. [20] by V D = p s r - 1/4. The quantum defect method developed in Ref. [20] was used to generate the theory curves in Fig. 1. We have adjusted the singlet and triplet Rb-Rb potential curves at small distances until the theoretical calculation agrees simultaneously with the present measurements of Bpeak and A, to within the experimental uncertainties. For a given c6, this severely constrains the values of both V D S and V D T (see Fig. 2). However, the position and width are not sufficient to totally constrain all three parameters , VDT. In order to obtain values for all of c6, V D ~ and the parameters, we also require that our triplet potentials predict the g-wave shape resonance in an energy range consistent with the measured value given in Ref. [21]. The results of this analysis are presented in Fig. 2. This illustrates that accurate measurements of Feshbach resonances are an extremely precise method for determining threshold properties of the interatomic potentials.
-
v, -0.02 r
( mod 1 )
[
0.00 l
0.02
t
FIG. 2. Comparison of the allowed V D S , VDT parameter space based on this work and previous measurements for a constant value of Cb. The rectangle i s the allowed range from Ref. [24]. The large diamond is the range from Ref. [ 111. The small filled diamond i s the range from this work. Note that the rectangle and large diamond constraints were established using a different value of C, from the one used in this work (4550 instead of 4700 a.u.). As is discussed in the text, the position of the constrained regions depends linearly upon C g , which i s indicated by the small connected diamonds on either side of the small filled diamond. These diamonds show the effect of 50 a.u. uncertainty in C6. The arrows indicate sensitivity to changes in resonance width and position in this parameter space.
Our measurement has allowed us to reduce the combined c6, U D S , and UDT parameter space by roughly a factor of 80. As discussed in Ref. [11], the position of the resonance depends mostly on c6 and the sum U D S + U D T . The width depends mostly on the difference V D S - V D T . AS Fig. 2 shows, the allowed Y D S , UDT parameter region is an extremely correlated function of c6. In particular, we find that the area of the allowed parameter region is independent of both c6 and c g . The uncertainty in these dispersion coefficients simply shifts the position of the "diamond." Accordingly, we can represent the allowed parameter region in the following manner: U D S + VDT = -0.0442 + 2.14(10-4)(C6 c g ) - 4(10-8)(Cg - cg) 2 0.001 and V D S - V D T = 0.0652 - 6.6(10-5)(C,5 - c6) 2 0.003, where c6 = 4700 and c g = 550600 [22]. Converting the bound state phases into scattering lengths, we find agreement with previous work [23-251. We find for the nominal dispersion coefficients (in a.u.) U T ( "Rb) = -369 t 16, a ~ ( ~ ' R b=) 24003:60°0, c l ~ ( ' ~ R b= ) 106 2 4, u ~ ( ' ~ R b= ) 90 2 1. Our value for c6, 4700(50) a.u., is slightly higher (and with smaller uncertainty) than a previous analysis based solely on the g-wave shape resonance [21]. We have confirmed that several scattering observables predicted by the new Born-Oppeiheimer potentials are consistent with previous measurements. Specifically, the new potentials predict a broad d-wave shape resonance [26] in 87Rb, the scattering length ratio a 2 , ~ / a l , - l [27], and the thermally averaged 12,2) + 11, - 1) inelastic rate constant [28] that are consistent with previous measurements. We also find 10 of the 12 measured d-wave bound states [24] within the 2 a error bars. The new potentials also permit us to predict additional "Rb 5111
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Feshbach resonances [6] at Bpeak = 226.0 -f 4.0 G with a width of -0.01 G , and at Bpeak = 535.5 -f 4.0 G with a width of 2.2 ? 0.2 G. We have shown that precise measurements of the Feshbach resonance in the elastic scattering cross section provide unprecedented knowledge of the Born-Oppenheimer potentials that control Rb-Rb scattering processes at submK temperatures. We have also used this resonance to enhance evaporative cooling of Rb in a low density magnetic trap and change the scattering length froin negative to positive. These capabilities should allow novel studies of BEC in the future. We acknowledge valuable conversations with Dan Heinzen, Boudwijn Verhaar, and John Bohn, as well as the assistance of Richard Christ and Eric Burt in the early stages of the work. NSF, ONR, and NIST supported this work.
*Quantum Physics Division, National Institute o f Standards and Technology, Gaithersburg, MD 20899. [ 11 V. Sanchez-Villicana, S. D. Gensemer, and P. L. Could, Phys. Rev. A 54, R3730 (1996), and references therein. [2] E. Tiesinga et al., Phys. Rev. A 46, R1167 (1992); E. Tiesinga, B . J . Verhaar, and H. Stoof, Phys. Rev. A 47,4114 (1993). [3] M. Marinescu and L. You, Phys. Rev. Lett. 81, 4596 (1 998). [4] P.O. Fedichev et al., Phys. Rev. Lett. 77, 2913 (1996). [5] J. L. Bohn and P. S. Julienne, Phys. Rev. A 56, 1486 (1997). [6] A. J. Moerdijk et al., Phys. Rev. A 51, 4852 (1995); J. M. Vogels et al., Phys. Rev. A 56, R1067 (1997). [7] C. R. Monroe et al., Phys. Rev. Lett. 70, 414 (1993). [8] N. R. Newbury, C. J. Myatt, and C. E. Wieman, Phys. Rev. A 51, R2680 (1995). [9] For a recent review on this subject, see, for instance,
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P. S. Julienne, J. Res. Natl. Inst. Stand. Technol. 101, 487 (1 996). [lo] S. Inouye et al., Nature (London) 392, 151 (1998). [ I I ] Ph. Courteille e/ a/., Phys. Rev. Lett. 81, 69 (1998). [I21 C. J. Myatt et al., Opt. Lett. 21, 290 (1996). [ I31 In our weak trap, the magnetic field lines are approximately parallel across the atomic cloud. Cutting faster than the cloud can equilibrate will remove energy from a single dimension. [I41 C. R. Monroc, Ph.D. thesis, University of Colorado, 1992. [I51 The proportionality constant is about 2.6 when the elastic cross section is temperature independent, which is decidedly not the case here. [ 161 D. Guy-Odelin et al., Opt. Express 2, 323 (1998). [ 171 The three values of Bpr.,h determined for the 0.5, 1, and 0.75 p K data (the latter not shown in Fig. 1 to avoid cluttering it excessively) can be determined to be about the same precision and agree to within 0.4 G. The value o f B,,i, is primarily based on the 3 pK data. [ 181 F. H. Mies et a/., J. Res. Natl. Inst. Stand. Technol. 101, 521 (1996). [ 191 G. F. Gribakin and V. V. Flainba~im,Phys. Rev. A 48, 546 (1993). [20] J.P. Burke, Jr., C.H. Greene, and J. L. Bohn, Phys. Rev. Lett. 81, 3355 (1998). The scattering length is given in terms of the quantum defect by a = -C2 tan(7ipYr)/[1 + G(0) tan(.ir,u")], where C' = 0.957 217(2mC6/h2)'/4 and G(0) = - 1.00260. [21] H.M. J. M. Boesten et al., Phys. Rev. Lett. 77, 5194 (1996). [22] M. Marinescu, H. R. Sadeghpour, and A. Dalgarno, Phys. Rev. A 49, 982 (1994). The uncertainties in the coeflicients are given as ?4%. [23] J. P. Burke, Jr. et al., Phys. Rev. Lett. 80, 2097 (1998). [24] C.C. Tsai et al., Phys. Rev. Lett. 79, 1245 (1997). [25] P.S. Julienne et al., Phys. Rev. Lett. 78, 1880 (1997); J. P. Burke, Jr. et al., Phys. Rev. A 55, R2511 (1997). [26] H.M. J. M. Boesten et al., Phys. Rev. A 55, 636 (1997). [27] M. R. Matthews et al., Phys. Rev. Lett. 81, 243 (1998). [28] C. J. Myatt et al., Phys. Rev. Lett. 78, 586 (1997).
ITALIAN PHYSICAL SOCIETY
PROCEEDINGS OF THE
INTERNATIONAL SCHOOL OF PHYSICS <<ENRICOFER.MI>,
COURSE
CXL
edited by M. INGUSCIO, S. STRINGANand C. E. WIEXrAN
VARENNA ON LAKE COMO VILLA MOX'ASTERO
7-17 July 199s
B ose-E instein Condensat i 0 n in Atomic Gases
1999 Press
AMSTERDAM, OXFORD, TOKYO, WASHINGTON DC
532
Experiments in dilute atomic Bose-Einstein condensation E. A. CORNELL, J. R. ENSHER and C. E. WIEMAN JILA, University of Colorado and National Institute of Standards and Technology and Department of Physics, University of Colorado - Boulder, CO 80309-04d0, USA
1. - I n t r o d u c t i o n
1’1. Why BEC? - In the month we began writing this paper, fourteen papers on the explicit topic of Bose-Einstein condensation (BEC) in a dilute gas appeared in the pages of Physical Review. Both theoretical and experimental activity in BEC has expanded dramatically in the three years since the first observation of BEC in a dilute atomic gas. One is tempted t o ask, why? Why is there so much interest in the field? Is this burst of activity something that could have been foreseen, in, say, 1990? We feel the answer to this last question is yes: while “interesting” is a quality which is impossible to define, there was good reason t o anticipate, even in 1990, that dilutegas BEC was going to be something worth investigating. To see why, one need only perform the following simple test. Ask any physicist of your acquaintance to compile a list of what he or she considers the six most intriguing physical phenomena that occur at length scales greater than a picorneter and smaller than a kilometer. In our experience. such lists almost always include at least two of the following three effects: superfluidity. lasing. and superconductivity. These three topics all share two common features: counterintuitive behavior and macroscopic occupation of a single quantum state. This then would have been our clue-one might have anticipated that dilute-gas BEC was to be interesting because it shares the same underlying mechanism with the widely appreciated topics of lasing. superfluidity, and superconductivity. and yet is quite different from any 1.J
@ Societa Italiana di Fisica
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of them: Dilute-gas BEC is more amenable to iiiicroscopic analysis than superfluidity. more strongl?. self-interacting than laser beanis. and it possesses a range of experimental obseri-alJes n-hicli are conipienientarv to those of superconductii-it:.. This paper will not pretend to proi-ide a thorough suri-ey of the status of dilute-gas BEC experiment. .\iij- attempt to do so at this time n-ould surely be obsolete by the t i n e it appears in print-such is the pace of deyelopnients these days. Rather. n-e will tr;. to coi-er the same pedagogical ground as the set of lectures delivered by one of us (E.1C) at tlie 1998 Enrico Fermi summer school on Bose-Einstein condensation. The lectures n-ere coordinated n-ith other experimenter-lectures at 1-arenna. and thus n-ere not meant to be comprehensive. This paper as well will be selective in its focus. -4n outline of this paper is as follows: 1T’e begin with an introductory section including a revien. of “ideal-gas” BEC theory [l]. The second section presents a history of tlie subject. and the third a forn-ard-looking summary of BEC technology. The fourth section introduces the import ant topic of interactions in the condensate and surveys esperimental n-ork on interactions. The fifth section discusses. mainly by reference to published articles. experiments on condensate excitations. The sixth section is a brief essay on tlie nieaning of tlie **phase“of a condensate. The seventh section surveys what is knon-n about heating processes in trapped atom samples. and examines the implications for trapping very large samples. The eighth section contains some general observations on atomic collisions relevant to evaporation. 1‘2. BEC in an ideal gas. - It is conventional to begin a lecture on ideal Bose statistics ti-ith a ritual chant. ’.Imagine a system of -\-indistinguishable particles . . . *‘ On occasion. however. it is ~ r o r t hpausing to reflect on n-hat an esotic thing it is to imagine a system of indistinguishable particles. One is n-illing enough to admit that yes. two atoms can be so similar one to tlie other as to allow no possibility of telling them apart. Why not? That is not so much to concede. The probleni arises only when one is forced to confront tlie physical implications of the concept of indistinguishable bosons. For example, if there are ten particles to be arranged in tm-o microstates of a system: the statistical weight of the configuration (ten particles in one state and zero in the other) is exactly the same as tlie iveiglit as the configuration (five particles in one state: five in the other). We find this ratio of 1 : 1 disquieting and counter-intuitive. The corresponding ratio for distinguishable particles, “Boltzmannons”, would be 1 : 252. In the second [2] of Einstein’s two papers [2-41 on Bose-Einstein statistics, Einstein acknowledges that in his statistics, “The . .. molecules are not treated as statistically independent . . . ,” and comments that the differences between distinguishable and indistinguishable state counting “express indirectly a certain hypothesis on a mutual influence of the molecules which for the time being is of a quite mysterious nature”. The mysterious statistics of indistinguishable bosons in turn mandates a variety of exotic behavior, e.g., the famous enhanced probability for scattering into occupied states and of course Bose-Einstein condensation. IVhy should the presence or absence of distinguishing tags on the atoms so profoundly change their statistical behavior? For the particular case of photons, the problem may be
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more palatably expressed in terms of fields. rather than particles. When many photons occupy a particular microstate, we talk about the electric-field amplitude in a particular mode. It is relatively easy, at least for the authors of this paper, t o believe that having equal field amplitudes in two adjacent modes is no more entropic than having a large field in one mode and a small field in the other. In some ways, this has not helped much. For as soon as we try t o apply the notion of fields back to atoms, we are left with a new set of troubling notions-can the probability amplitude for a large collection of atoms ever really be like an electric field-coherent, continuous, macroscopically occupied, macroscopically observable? The answer is “absolutely yes!” Troubling or no, this is exactly what a Bose condensate is about. Proceeding conventionally, then, let us imagine a system of h’ indistinguishable particles, distributed among the microstates of a confining potential, such that any occupation number is allo~vable.The mean distribution, the Bose-Einstein distribution (BED), may be derived in several different ways. (See for instance 131.) In the end one may always understand the BED as the most random way t o distribute a certain amount of energy among a certain number of particles in a certain potential. The BED gives the mean number of particles in the i-th state as
where ~i is the energy of a particle in the i-th state, k is Boltzniann’s constant. and T and 11 are identified as the temperature and chemical potential. respectively. In the microcanonical understanding of the BED. p and T are determined from the constraints on total number :Y and total energy E :
(2a) (2b)
For large systems, the constraints may be R-ritten as integrals
where g ( ~ is) the density of states in the confining potential. Equations (1)-(3) contain nearly all the ideal gas physics of BEC. The number of spatial dimensions. and the effects of a confining potential. are all taken care of in the power law of the density of states. g ( E ) . The effects of finite number, and of very asymmetric Potentials [6]. can be determined by using the sums rather than the integrzls to constrain P and T . The critical temperature. the ground-state occupation fraction and the specific
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heat can all be calculated without difficulty. Only number fluctuations, which require a more careful consideration of the underlying ensemble statistics, are left out of this picture. The overall picture is sufficiently easy t o understand that. if the system truly were an ideal gas. there would be little left t o study at this point. -4s it actually has turned out. interactions between particles add immeasurably to the richness of the system, and work in dilute-gas BEC is in its early days.
1'3. Some real systems. - Einstein's original conception of BEC xas in an ideal gas, but the first experiments in BEC were in superfluid helium, a liquid (n-hich is to say a strongly correlated fluid, the opposite limit from an ideal gas). The beautiful and startling experiments on 1-iscosity, vortices, and heat-flow in liquid helium, and the ground-breaking theory those experiments inspired, more or less defined the field of Bose-Einstein condensation for four decades and more ['TI. The BEC concept has been put to use in broader contexts over the years, however, in such diverse topics as Kaons in neutron stars [8], cosmogenesis, and exotic superconductivity 191. Using the term in its broadest meaning ("macroscopic number of bosons in the same state") one need not have a fluid of any sort-lasers and masers produce macroscopically occupied states of optical and microwave photons, respectively. For that matter, a portable telephone. or even a penny-Thistle produce macroscopic occupation numbers of identical bosons (radio-frequency photons in one case, acoustic-frequency phonons, in the other.) In superfluid He-4, the bosons exist independently of the condensate process. The fermionic neutrons. protons, and electrons that make up a H e 4 atom bind to form a composite boson at energies much higher than the superfluid transition temperature. There exists a broad family of physical systems, however. in which the binding of the fermions to form composite bosons, and the condensation of those bosons into a macroscopically occupied state, occur simultaneously. This is of course the famous BCS mechanism. Best-known for providing the microscopic physical mechanism of superconductivity, the Bose-condensed "Cooper pairs" of BCS theory occur as well in superfluid He-3 and may also be relevant t o the dynamics of large nuclei [lo] and of neutron stars. Prior to the observation of BEC in a dilute atomic gas, the laboratory system which most closely realized the original conception of Einstein was excitons in cuprous oxide [ll]. Excitons are formed by pulsed laser excitation in cuprous oxides. There exist meta-stable levels for the excitons which delay recombination long enough to allow the study of a thermally equilibrated Bose gas. The effective mass of the exciton is sufficiently low that the BEC transition at cryogenic temperatures occurs at densities which are dilute in the sense of the mean particle spacing being large compared t o the exciton radius. Recombination events, which can be detected either electrically [12] or by collecting their fluorescence [ll],are the main experimental observable. The most convincing evidence for BEC in this system is an excess of fluorescence from "zero-energy" excitons. In addition, anomalous transport behaviour evocative of superfluidity has been observed 1131. As an experimental and theoretical system, BEC in a dilute gas is nicely complementary t o the variety of BEC-like phenomena described above. In terms of strength of interparticle interaction, atomic-gas BEC is intermediat,e between liquid helium, for
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I
Log Density Fig. 1. - Generic phase-diagram common t o all atoms. The dotted line shows the boundary between non-BEC and BEC. The solid line shows the boundary between allowed and forbidden regions of the temperature-density space. Note that at low and intermediate densities, BEC exists only in the thermodynamically forbidden regime.
which interactions are so strong that they cannot be treated by perturbation theory, and photon lasers, for which “interactions” (or nonlinearity) is small except in effectively tu-odimensional configurations. In terms of the underlying statistical mechanics. atomic-gas BEC is most like H e 4 Unlike in superconductivity, the bosons are formed before the transition occurs, and unlike in lasers. the particle number is conserved. The properties of the phase transition, then, should most closely resemble liquid helium. Finally. in terms of esperimental observables, atomic-gas BEC is in a class entirely by itself. The available laboratory tools for characterizing and manipulating atomic-gas BEC are essentially orthogonal to those available for excitonic systems or for liquid helium. -4lthough dilute-gas atomic BEC provides a nice foil t o more traditional BEC systems. it comes at a price: one is forced t o work deep in the thermodynamically forbidden regime. 1’4. Why BEC an dilute BEC is hard. - Figure 1 is a qualitative phase-diagram u-hich s h o w the general features common to any atomic system. At low density and relatively high temperature, there is a vapor phase. At high density there are various condensedmatter phases. But the intermediate densities are thermodynamically forbidden, except at very high temperatures. The Bose-condensed region of the (n,T)-plane is entirely forbidden, except at such high densities that the equilibrium configuration of nearly all known atoms or molecules is crystalline. (The existence of a crystal lattice rules out a Bose condensate.) The sole exception is helium: m-hich remains a liquid below the BEC transition. Reaching BEC under dilute conditions. say at densities ten or 100 times lon-er than conventional liquid helium. is as forbidden t o helium as it is t o any other atom. All the same, it is possible to do forbidden things. as long as one does not attempt t o do them for very long! While making a dilute-gas BEC in equilibrium is impossible, one
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can exploit metastability to stray temporarily into the forbidden region. In the absence of artificial nucleation sites. the process by n-hich a supersaturated atomic gas finds its n-ay from a gas to a solid begins with molecular recombination. Radiative recombination. by which two atonis collide and emit a photon to stabilize a dimer molecule. has a \-anishingly small rate. -4t any reasonable density the dominant channel is three-body recombination. This then is what makes BEC possible. A gas of atonis can come into kinetic equilibrium via tn-o-bod? collisions. wherea5 it requires three-body collisions t o achieve chemical equilibrium (i. e. to form molecules and thence solids.) -4t sufficiently low densities. the two-body rate n-ill dominate the three-body rate. and a gas n-ill reach kinetic equilibrium. perhaps in a metastable Bose-Einstein condensate. long before the gas finds its way to the ultimately stable solid-state condition. The need t o maintain metastability usually dictates a more stringent upper limit on density than does the desire to create a dilute system. Densities around l O " ~ n i - ~ . for instance, would be a hundred times more dilute than a condensed-matter helium superfluid. But creating such a gas is quite impractical-+\-en at an additional factor of a thousand lox-er density. say 1 0 ' ' ~ r n - ~ .metastability times would be on the order of a few nis; more realistic are densities 011 the order of cmF3. The low densities mandated by the need to maintain long-lived metastability in turn challenge one's ability to achieve the necessary low temperatures for BEC. The critical temperature for ideal-gas BEC in a gas with mass n? is related to density 12 by
(4)
-
where [(z) is the Riemann zeta-function and <(3/2) 2.612. Realistically, one needs to be able to achieve temperatures in the microkelvin scale or still lon-er. which is a challenge indeed. The desire to create a dilute BEC forces one t o work in a forbidden, metastable region of the ( 1 2 , T)-plane. The need to maintain metastability pushes one t o still lower densities. and low densities in turn suppress the phase transition temperature to temperatures that were all but unimaginable two decades ago. Thus, making a dilute-gas BEC n-as an experimentally difficult thing t,o do, and the history of the effort is worth reviewing.
2. - History
2'1.Conceptual beginnings. - The notion of Bose statistics dates back t o a 1924 paper in which Satyendranath Bose used a statistical argument to derive the black-body radiation spectrum 1141. Albert Einstein extended the statistical model to include systems with conserved particle number [2, 31. The result was Bose-Einstein statistics. Particles that obey Bose-Einstein statistics are called bosons, and today it is known t h a t all particles with integer spin (and only those particles) are bosons. Einstein immediately noticed a peculiar feature of the distribution: at low temperature, it saturates. "I maintain that. in this case, a number of molecules steadily growing with increasing density goes over in
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the first quantum state (which has zero kinetic energy) while the remaining molecules separate themselves according to the parameter value -4= 1 (in modern notation, p = 0) . . . . -4separation is effected; one part condenses, the rest remains a “saturated ideal gas” [2,4]. Thus began the concept of Bose-Einstein condensation. BEC has not always been a particularly reputable character. In the decade follo~ving Einstein’s papers, doubts were cast on the reality of the model 1151. Fritz London and L. Tisza [16,17] resurrected the idea in the mid-1930s as a possible mechanism underlying superfluidity in liquid helium 4. London’s view was either disbelieved or else felt to be not particularly illuminating. Certainly the influential helium theory papers of the 50s and 60s make little or no mention of BEC [18-201. The authors of the present reviex are not well-versed in the history of that era, but it is evident that sometime in the intervening years the bulk of expert opinion has shifted. Experiment and theory (neutron scattering [21] and path-integral Monte Carlo simulations [22]. respectively) nom- support the idea that the microscopic physics underlying superfluidity is a zero-momentum BEC. Due to interactions, only about 10% of the helium atoms participate in the condensate. even a t temperatures so low that empirically 100% of the liquid appears to be in the super-fluid state. 2’2. Spin-polarized hydrogen.
Efforts to make a dilute BEC in an atomic gas were spurred by provocative papers by Hecht [23] and Stwalley and Nosanov [24]. They argued on the basis of the quantum theory of corresponding states that spin-polarized hydrogen would remain a gas down to zero temperature, and thus would be a great candidate for making a weakly interacting BEC. -4number of experimental groups [23-281 in the late 70s and early 80s began work in the field. Spin-polarized hydrogen was first stabilized by Silvera and IYalraven in 1980 j2.51. and by the mid-90s spin-polarized hydrogen had been brought within a factor of 50 of condensing I2T). These experiments were performed in a dilution refrigerator. in a cell whose walls n-ere coated with superfluid liquid helium. A radiofrequency (rf) discharge dissociated hydrogen molecules. and a strong magnetic field preserved the polarization of the resulring atoms. Individual hydrogen atoms can thermalize with a superfluid helium surface without becoming depolarized. The atoms were compressed using a (conceptually) simple piston-in-cylinder arrangement [29]. or inside a helium bubble [30].Eventually the helium surface became problematic. howel-er. If the cell is made relatively cold. the surface density of hydrogen atoms becomes so large that they undergo recombination there. If the cell is too hot. the volume density of hydrogen necessary for BEC becomes so high that. before that density can be reached. the rate of three-body recombination becomes too high [31]. -
2‘3.L a s e r cooling a n d t h e ascendancy of alkalis. - Contemperaneous with (but quite independent of) the hydrogen n-ork. an entirely different kind of cold-atom physics n-as evolving. The remarkable story of laser cooling has been reviewed elsexhere [32-35] but we mention some of the highlights in compressed form belon-. The idea that laser light could be used to cool atoms was suggested in early papers from II-ineland and Dehmelt [36]. from Hansch and Schadow [3i]. and from Letokhov‘s group [38]. Earl?
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optical force experiments were performed by Ashkin [39]. Trapped ions were laser-cooled at the University of Washington [10]and at the National Bureau of Standards (now NIST) in Boulder 1413. Atomic beams were deflected and slowed in the early 80s [42-41]. Optical molasses n-as first studied at Bell Labs [dj].lieasured temperatures in the early molasses experiments were consistent with the so-called Doppler limit. which amounts to about 300 niicrokeh-in in most alkalis. Coherent stimulated forces of light were studied [1G.4i].The dipole force of light was used to confine atoms as n-ell [-1S]. In 1987 and 1988 there were tn-o major advances that became central features of the method of creating BEC. First. a practical spontaneous-force trap, the hlagneto-Optical trap (MOT) n-as demonstrated [49]. and second, it was observed that under certain conditions. the temperatures in optical molasses are in fact much colder than the Doppler limit [SO-321. These were heady times in the laser-cooling business. With experiment yielding temperatures mysteriously far below what theory would predict, it was clear that we all lived under the authority of a munificent God. The MOT had generated high densities in trapped gases. Perhaps still colder temperatures and still higher densities n-ere due to arrive shortly. Indeed, it seemed reasonable enough, at the time. to speculate that the methods of laser cooling and trapping might soon lead directly to BEC! 2’4. Cold reality: limits on laser cooling. - It did not happen that n-ay, of course.
By 1990 it was clear that there ir-ere fairlJ- strict limits to both the temperature and density obtainable with laser cooling. Theory caught up with experiment and shored that the sub-Doppler temperatures were due to a combination of light-shifts and optical pumping that became known as Sysiphus cooling. Random momentum fluctuations from the rescattered photons limit the ultimate temperature to about a factor of ten above the recoil limit [53]. The rescattered photons are also responsible for a density limit-the light pressure from the reradiated photons gives rise to an effective inter-atom repulsive force [54]. Momentum fluctuations and trap loss arising from light-assisted collisions limited temperature and density as well [35]. The product of the coldness limit and the density limit works out to a phase-space density of about lo-’, which is five orders of magnitude too low for BEC. Since 1990, advances in laser cooling and trapping have allowed both the temperature [56,5’7]and the density [58]limit to be circumvented. But, it seems that in most cases higher densities have been won a t the cost of high temperatures, and lower temperatures have been achievable only in relatively low-density experiments. The peak phase-space density in laser cooling experiments has increased hardly at all in the decade since 1989 [591* The alkali-atom work of the 1980s included another de~elopment,however, n-hich was to have a large impact on BEC work. Sodium atoms were trapped in purely magnetic traps [60-621. 2‘5. Evaporative cooling and the return of hydrogen. - Harold Hess from the MIT Hydrogen group realized the significance that magnetic trapping had for their BEC effort. Atoms in a magnetic trap have no contact with a physical surface and thus the surface-
541 EXPERIMENTS IN DILUTE ATOMIC
BOSE-EINSTEIN CONDENSATION
23
recombination problems could be circumvented. hloreover, thermally isolated atoms in a magnetic trap were the perfect candidate for evaporative cooling. In a remarkable paper Hess laid out in 1986 most of the important concepts of evaporative cooling of trapped atoms [3l]. Let the highest energy atoms escape from the trap, and the mean energy, and thus the temperature, of the remaining atoms will decrease. In a dilute gas in an inhomogeneous potential, decreasing temperature in turn means decreasing occupied volume. One can actually increase the density of the remaining atoms even though the total number of confined atoms decreases. The Cornell University Hydrogen group also considered evaporative cooling [63]. By 1985 the MIT group had implemented these ideas and had demonstrated that the method was as powerful as anticipated. In their best evaporative run, they obtained, at a temperature of 100pK, a density only a factor of five too low for BEC [64]. Further progress was limited by dipolar relaxation, but perhaps more fundamentally by loss of signal-to-noise, and the need for a more accurate means of characterizing temperature and density in the coldest clouds [65]. Evaporative work was also performed by the Amsterdam group [66]. Both the hIIT and the Amsterdam group began constructing laser sources capable of accessing optical transitions in atomic hydrogen, reasoning that with the powerful tool of laser spectroscopy they would be better positioned to understand and surpass the low-temperature limits [66-681. The Amsterdam group performed some laser cooling on atomic hydrogen [69,70]. The hydrogen evaporation results made a big impression on the JIL-4 alkali group. It seemed to us that a hybrid approach combining laser cooling and evaporation had an excellent chance of working. Evaporation from a magnetic trap seemed like a very appealing way to circumvent the limits of laser cooling. Laser cooling could serve as a pre-cooling technology, replacing the dilution refrigerator of the hydrogen work. I’l’ith convenient lasers in the near-IR. and with the good optical access of a room-temperature glass cell. detection sensitivity could approach single-atom capability. The elastic crosssection. and thus the rate of evaporation, should almost certainly be larger in an alkali atom than it is in hydrogen. Encouraged by these thoughts (and by other collisional considerations, see subsects. 2‘7and 8’1 below) the JIL.4 group set out (see ref. [ill and a presentation a t the 1991 Varenna summer school [72]) to combine the best ideas of alkali and of hydrogen esperiments, in an attempt to see BEC in an alkali gas. 2‘6. Hybridizing MOT and eziaporative cooling techniques. - In a sense efforts to hybridize optical cooling x i t h magnetic trapping are as old as atomic magnetic trapping itself. The original KIST sodium magnetic trap [61] and the first Ioffe-Pritchard (IP) trap [62] were loaded from Zeeman-tuned optical beam-slowers. hlodern alkali BEC apparatuses. however, can trace their conceptual roots through a series of devices built at JIL-4 during an era beginning in late 1989 and continuing into the early 90s [X-Ti]. As things stood at the end of the 19SOs. optical cooling on the one hand and magnetic trapping on the other were both somewhat heroic experiments, to be undertaken only by advanced and well-equipped -4MO laboratories. The prospect of trying to get both worhng. and working well. on the same bench and on the same day was daunting.
542
E.A . CORNELL.J . R. ENSHER and C. E.WIEMAN
24
7 TURNS u+
/+-
\
I?-----
'
20 T U R N S
ION
PUh1P
U TEC -20 O
Fig. 2. - The glass vapor cell and magnetic coils used in early JIL.4 efforts to hybridize laser cooling and magnetic trapping [71]. The glass tubing is 2.5cm in diameter. The Ioffe current bars have been omitted for clarity.
The JILA vapor-cell MOT. with its superimposed IP trap (fig. 2 ) . represented a much-needed technological simplification and introduced a number of ideas that are now in common use in the hybrid trapping business [ i l .i 2 ] : i) vapor-cell (rather than beam) loading; ii) fused-glass rather than n-elded-steel architecture; iii) extensive use of diode lasers; iv) magnetic coils located outside the chamber; overall chamber volume nieasured in cubic centimeters rather than liters; Ti) temperatures measured by imaging an expanded cloud: vii) magnetic-field curvatures calibrated in situ by observing the frequency of dipole and quadrupole (sloshing and pulsing) cloud motion: viii) the basic approach of a RIOT and a magnetic trap which are spatially superimposed (indeed, .n-hich often share some magnetic coils) but temporally sequential; and ix) optional use of additional molasses and optical pumping sequences inserted in time between the MOT and magnetic trapping stages. In the early experiments [71-741 a number of experimental issues came up that continue t o confront all BEC experiments: the importance of aligning the centers of the MOT and the magnetic trap; the density-reducing effects of mode-mismat,ch; the need t.0 account carefully for the (previously ignored) force of gravity; heating (and not merely loss) from background gas collisions; the usefulness of being able to turn off the magnetic fields rapidly; the need to synchronize many changes in laser status and magnetic fields together with image acquisition. At the time the design was quite novel, but by now it is almost standard. It is instructive t o note how much a modern, IP-based BEC device (fig. 3) resembles its ancestor (fig. 2). $7)
EXPERIMENTS IS D I L V T E
ATOMIC
BOSE-EISSTEIN CONDENSATION
25
To Vacuum Pump
Fig. 3. - Glass vapor cell and magnetic-field coils used in mixed-BEC studies (781 and ongoing Feshbach resonance experiments [79].
2'7. Collisional concerns. - From the very beginning. dilute-gas BEC experiments had to confront the topic of cold collisions. Even before evaporation was considered. vhen cooling n a s simply a matter of conduction to the chamber walls. the rate of threebod? decay determined the lifetinie of samples of spin-polarized hydrogen samples. IYith the advent of evaporation. there n-as a demand for understanding several different collisional processes-elastic collisions. dipolar relasat ion. three-body recombination. and (to a lesser estent) spin-exchange. Atomic collisions at very cold temperatures is now a major branch of the discipline of -4110 physics. but in the 1980s there was almost no esperiniental data, and what there was came in fact from the spin-polarized hydrogen experiments [SO]. There wts theoretical rorli on hydrogen from Shlyapnikov and Iiagan [81.82].and from Silvera and Verhaar [13].-4n early paper by Pritchard [84] includes estimates on low-temperature collisional properties for alkalis. His estimates were estrapolations from room-temperature results. but in retrospect several were surprisingly accurate. Esperiments on cold collisions betn-een alkali atoms were initially all performed on light-induced collisions in molasses and 110 1 s [85-57]. The earliest ultra-cold groundstate on ground-state measurements were based on pressure shifts in the cesium clock transition [Ss] and thermalization rates in magnetically trapped Cs [74]. Rb [77] and Xa [89]. S-rave collisions Fere observed directly in cesium [go]. Eventually esperinient and theory of escited-state collisions. in particular photoassociative events. yielded so much information on atom-atom poreritials that ground-state cross-sections could be extracted [91-961. By the tinie BEC was observed in rubidium. sodium. and lithium. there existed. at least at the 20'Z level. reasonabIe estimates for the respective elastic scattering length of each species. Evaporation efforrs on all three species were initially
544
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E . A . CORNELL. J . R. E N S H P R and
c . E.W ~ E M A N
begun, however, under conditions of relatively large uncertainty concerning elastic rates. and near-ignorance on inelastic rates. In 1992 the JILA group came to realize that dipolar relaxation in alkalis should in principle not be a limiting factor. As explained in subsect. 8‘1 belon-. collisional scaling with temperature and magnetic field is such that, except in pathological situations. the problem of good and bad collisions in the evaporative cooling of alkalis is reduced to the ratio of the elastic collision rate to the rate of loss due to imperfect vacuum: dipolar relaxation and three-body recombination can be finessed. It was reassuring to move ahead on efforts to evaporate with the knowledge that, while we were essentially proceeding in the dark, there were not as many monsters in the dark as n-e had originally imagined. Of course, even given successful evaporative cooling. there was still the question of the sign of scattering length, which must be positive to ensure the stability of a large condensate. The JILX group had sufficient laser equipment. however. to trap either s5Rb. s7Rb. or lS3Cs. Given the “modulo” arithmetic that goes into determining a scattering length. it seemed fair to treat the scattering lengths of the three species as statistically independent events.,and the chances then of Nature conspiring to make all three negative were too small to worry about [97]. 2’8. Increasing elastic collisions per background loss. - -4simple 310T, with operating parameters adjusted to maximize collection rate, m-ill confine atoms a t a density limited by reradiated photon pressure to a relatively Ion. value [S], with correspondingly loncollision rate after transfer to the magnetic trap. 2’8.1. M u l t i p l e l o a d i n g . Early efforts to surniouiit the density limit included a multiple-loading scheme pursued at JILA [ r 3 ] . Multiple MOT-loads of atoms were launched in moving molasses, optically pumped into an untrapped Zeeman level, focused into a magnetic trap, then optically repumped into a trapped level. The repumping represented the necessary dissipation, so that multiple loads of atoms could be inserted in a continuously operating magnetic trap. In practice, each step of the process involved some losses and the final result hardly justified all the additional complexity. It is interesting to note, however, that multiple transfers from MOT to MOT [90,’76], rather than from h3OT to magnetic trap, is a technique currently in widespread practice.
2‘8.2. A d i a b a t i c c o m p r e s s i o n . Also in wide, current use is the practice of magnetic compression. After loading into a magnetic trap, the atoms can be compressed by further increasing t*hecurvature of the confining magnetic fields. This is a technique which is available to experiments which use MOTS to load the magnetic trap, but not to experiments which use cryogenic loading (in the latter, the magnetic fields are usually already at their maximum values during loading, in order to maximize capture.) This method is discussed in [72] and was implemented first in early ground-state collisional work [ i d ] .
2‘8.3. E n h a n c i n g MOT d e n s i t y . An important advance came from the MIT Sodium group in 1992, when t,hey developed the Dark-spot MOT [58]. A shadow is
545 EXPERIMESTS IN DILUTE ATOMIC BOSE-EINSTEiN CONDENSATION
27
arranged in the repumping light, such that atoms that have already been captured and concentrated in the middle of the trap are pumped to an internal state that is relatively unperturbed by close neighbors. In effect, the MOT is divided into two spatial regions. one for efficient collection, and one for efficient compression and storage. A related technique is to modulate the MOT parameters temporally, rather than spatially. This approach has been dubbed ChiOT (Compressed MOT) [98]. A very thorough study of related MOT behavior came out of a British-French collaboration [99].
2'8.4.R e d u c i n g b a c k g r o u n d l o s s . Since the most important figure of merit in the evaporation business in the collision rate compared to the trap lifetime, one can do almost as much good improving vacuum as one can improving MOT density. Early machines for cooling atomic beams were relatively dirty, by the standards of modern UHV practice. Except for the Pritchard group's cryogenic trap 1621, confinement times in optical trapping experiments were usually on the order of two or three seconds. Improved atomic-beam design [loo,1011 and the adoption of modern UHV practice has made 100 s lifetimes standard. An experiment at JILA s a x a MOT lifetime of 1000s "151. Cryogenic MOTS have had lifetimes of greater than 3600s [102]. 2'9. Forced evaporation. - Cooling by evaporation is a process found throughout nature. Whether the material being cooled is an atomic nuclei or the Atlantic ocean, the rate of natural evaporation, and the minimum temperature achievable, are limited by the particular fixed value of the work-function of the evaporating substance. In magnetically confined atoms. no such limit esists, because the "work-function" is simply the height of the lowest point in the rim of the confining potential. Hess pointed out [31] that by perturbing the confining magnetic fields. the work-function of a trap can be made arbitrarily low: as long as favorable collisional conditions persist, there is no lower limit to the temperatures attainable in this forced evaporation. Pritchard [lo31 pointed out that evaporation could be performed more conveniently if the rim of the trap were defined by an rf-resonance condition. rather than simply by the topography of the magnetic field: experimentally, his group made first use of position-dependent rf transitions to selectively transfer magnetically trapped sodium atoms between Zeeman levels and thus characterized their temperature [lOi]. From 1993 to 1993. a number of other experimental groups launched their own efforts to create BEC in alkali species using the hybrid lasercooIing/evaporation approach. Progress accelerated and by summer of 1994 three groups had announced the successful use of evaporation t o increase the phase-space density of trapped alkalis [105,106,89]. By the DAhIOP meeting in Toronto. in hfay 199.5. the number of groups with clear evidence of evaporative cooling had increased to four [ l O T ] . The Rice. JIL-4: and MIT alkali groups were all seeing significant increases in phasespace density. The JILA alkali group returned from Toronto B-ith a shared impression that "there is no time left t o fiddle around" [lOS].
2-10.Magnetic trap improvements. - Each of the groups reporting significant evaporation at Toronto had recently implemented a major upgrade in magnetic trap technolo=. In a harmonic trap. the collision rate after adiabatic compression scales as the find con-
546
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E. A . CORSELL.J . R.ETSHER arid C. E.
\\‘IELIAN
fining frequent? squared [Z]. Clearly. there n-as much to be gained by building a more tiglitl?. confining magnetic trap. but the requirement of adequate optical access for the MOT. along n-ith engineering constraints on pon-er dissipation. make the design problem complicated. U’hen constructing a trap for n-eak-field seeking atonis. n-ith the aim of confining the atonis to a spatial size much smaller than the size of the magnets. one n-ould like to use linear gradients. In that case. lion-ever. one is confronted n-ith the prolsleiii of the iiiiniriiuni in the magnitude of the magnetic fields (and thus of the confining potential) occurring at a local zero in the magnetic field. This zero represents a **hole”in the trap. a site at which atoms can undergo Majorana transitions [log] and thus escape from the trap. If one uses the second-order gradients from the magnets to provide the confinement. there is a marked loss of confinement strength. This scaling is discussed in ref. [lOS]. The three esperimental groups each solved the problem differently. The Rice group built a trap with permanent magnets [loll n-hich can be stronger than electromagnets. The MIT and JIL.4 groups each built traps n-ith linear gradients. and then “plugged the hole“. in the MIT case with a beam of blue-detuned (repulsive) light [S9]. and in the JIL-4 case n-ith a time-dependent dither field [103]. -4s it turned out. all three methods n-orked ell enough. In Spring of 1993 (less than three weeks after the D.41IOP meeting), the JIL-4 group saw BEC in rubidium 8i. and by -4utunin of 199.5 all three groups had either observed BEC or else had preliminary evidence for it [110-1121. If one is n-illing to accept tight confinement in two directions only. it turns out that the original IP design. operated n-ith a very lon- bias field. is more than adequate to produce BECs. In 1996, BEC was observed in second-generation traps a t both JIIT [113] and JIL.4 [is].The fields n-ere in the IP configuration: additional coils were used to nearly cancel the central magnetic field. 2’11. Imaging techniques. - The fact that a wealth of quantitative information can be estracted from simple spatial images of the atoni cloud is one of the features that most distinguishes BEC in dilute atomic gas from other BEC experiments, such as escitons and liquid helium. It is u-orth mentioning then some issues in imaging trapped atoins. Early imaging of magnetic all^ trapped atonis was done by collecting fluorescence [711. While acceptable for large, dilute clouds, fluorescence imaging does not work well u-hen the clouds get small and dense because of optical thickness and motional blurring. -4bsorption imaging minimizes motional blurring because the mean spontaneous force from the absorbed light is directly along the line of sight (assuming good optical alignment). The presence of magnetic fields in the trap can make absorption images more difficult to interpret, although the MIT Sodium group has shown that n-ith careful modeling even images taken of atoms in full-strength quadrupole magnetic fields can be correctly understood [89]. The more fundamental problem is that as the clouds get colder, they get both smaller and denser. Two issues arise. First, the cloud can become smaller than the resolution limit of the optics, rvhich makes it easy to misinterpret images [112, 1141. Second, very dense clouds can giye rise to “lensing”. in n-hich the real part of the index of refraction becomes as important as the absorption. In the early experiments, the solution xas to turn off the fields suddenly and let the cloud expand before imaging. The
547 E X P E R f \ l E N T S I N D I L U T E ATOLllC
BOSE-EISSTEIS CONDENS.4TIOS
29
expansion decreases the optical thickness and increases the effective resolution. For the original JILA observation, it was necessary to magnetically support the atoms against gravity in order to allow the atoms to expand sufficiently without dropping out of the imaging field-of-view. Afore recently in situ imaging techniques have been demonstrated to be accurate (see subsect. 3'6 belor). 3.
- Survey of BEC technology, present and future
The list of groups currently observing BEC, and the basic technology each uses (table I) will doubtless be outdated by the time this article appears in print, but, it is interesting to note the variations on a theme. There are many obvious generalizations to make, but most of them have exceptions. In this section we survey the range of existing technology and speculate on what the future may bring.
3'1. Magnetic (and other) traps
1. As a rule, these are of the IP configuation. But TOP traps continue to be built. and the IP traps have yet to zero-in on a standard configuration. As of this writing. the eleven groups producing BEC with IP fields are using nine different coil geometries. Still more exotic trap designs have been proposed [llj]. It seems likely that over the next few years there will be something of a shake-out in coil design. r i t h a small subset of today's designs being found t o be adequate to cover the needs of most experiments.
2. As a rule, the coils are water-cooled. But the NIT hydrogen trappers use superconducting coils. and the Orsay group uses so little power that they can air-cool. 3. It seems likely that evaporative cooling to BEC can eventually be accomplished bx purely optical means. either in FORTS [106]. or their near-dc cousins. the Qt-ESTs [llG.lli].Already. the NIT group has confined condensates in purely optical traps [118]. although the clouds are still precooled in magnetic traps. Still more esotic traps have been proposed. including boses fashioned from blue-detuned evanescent optical waves or from planes of tiny magnetic domains [119]. 3'2. Atomic species. - A majority of groups are currently working in rubidium 87. with most of the remainder using sodium 23. But the Rice group has had success in lithium i and the hydrogen trappers recently have seen condensation [120]. There is considerable experimental effort around the vorld on cesium [121-125]. rubidium 85 I.91. and potassium 39 and 41 [126.12';1 and on the fermionic isotopes lithium 6 [ES] and potassium 40 [129.130]. To the best of our knoxledge. all sodium condensates to date h a w been formed in the F = 1 state. There seems to be some suggestion that the collisional properties of the F = 2. m~ = 2 state are unfayorable for condensation. 117th the recent Successes of buffer-gas loading of molecules and nonalhli atoms [131-1331it seems like1F
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E. A . CORKELL, J . R. ENSHER and C. E . WIEMAN
TABLE I. - Current BEC technology a t a glan.ce-T4-here.
T4:hat and How.
Place
Atom
Collector/Cooler
JILA [110]
s7Rb
MOT/MOT
MI1 [Ill]
2 3 ~ a
Beam/MOT
Rice [112] Ron-land Institute [134] Yale (was Stanford) [135] UTexas [136] Konstanz [13i] Miinchen 11381 NIST Gaithersburg [139] Paris [I401 MIT [120]
" ~ i 23Na 87Rb '"Rb s7Rb "Rb
Orsay [I411 Hannover [142] Otago [143] Sussex [144]
2 3 ~ a
'.IRb
H "Rb "Rb "Rb "Rb
Beam/hlolasses Beam/MOT Vapor/MOT Beam/MOT MOT/MOT MOT/MOT Beam/M OT MOT/MOT Dilution Refrigerator/ He-I1 Surface Beam/MOT Beam/ M 0T MOT/MOT MOT/MOT
Magnetic trap i) T O P ii) IP(basebal1) IP (cloverleaf) (hybrid optical) IP (permanent magnet) I P (four-dee)
TOP TOP
IP (bars) IP (3 coils) T O P (transverse)
IP (3 coils) IP (Superconducting) IP (Pole-Piece) I P (Cloverleaf) TOP IP (Baseball)
that almost any atomic species with a ground-state magnetic moment, and a great many molecular species, can be magnetically trapped. The success of sympathetic cooling [XI lifts most of the requirements on collisional properties. A species with recalcitrant collisional properties can be trapped together with, and cooled sympathetically by, a "good eyaporating" species such as '?Rb; sympathetic cooling requires a far smaller good/bad collision ratio to succeed. Thus the number of different atomic and molecular species that could eventually be cooled to the BEC transition may be in the hundreds. 3'3. Precooling and compression. - Evaporative cooling can work only if the sample is initially cold and dense enough t o permit tight confinement and large collision rates. By far the most popular approach is t o collect, cool, and compress atoms in a h4OT [49], then rapidly turn off the MOT and turn on the fields of the magnetic trap immediately around it [ f l ] .The density of the MOT is often enhanced by a darkspot [58],or by modulating 'the detuning and field gradients of the h.IOT just before transfer t o the magnetic trap [98]. Sometimes a stage of optical cooling is included after the MOT magnetic fields have been turned off and before the magnetic trap has been turned on [71], but for traps with large number, this often provides only marginal benefit. The MOT is a useful but, as it has turned out, hardly an essential tool. The Rice group, which cannot rapidly reconfigure t,heir magnetic fields from RIOT t o IP trap, uses a beam-loaded molasses and dispenses with a hlOT altogether. The hydrogen trapping groups have never needed molasses cooling of any kind t o prepare atoms for magnetic trapping and evaporative cooling-
549
EXPERIMENTS IN DILUTE
ATOMIC
BOSE-EINSTEIN CONDENSATION
31
they rely on thermalization with a dilution-refrigerated He-I1 surface. In the future, buffer-gas loading techniques [131] may rival MOTS as the primary precooling method. Optical cooling may also continue even after the atoms are in their final evaporation trap. Optical cooling has been shown to work in magnetic traps 1145,771 and in optical dipole traps [146,147]. 3.4. MOT loading. - Among the experiments that do use MOTS, there are variants in the methods for loading them. The simplest approach is to collect atoms in the MOT directly from background thermal vapor [71,110], but one encounters a dilemma-should one operate at high vapor pressure, so as to fill the MOT quickly with large numbers of atoms, but then suffer rapid loss during evaporation, or should one operate at low vapor pressure, so as t o have longer time available for evaporation, but then live with slow, meager TOP trap fills? The original JIL-4 apparatus found a workable range of compromise pressures, with the help of a darkspot used t o suppress collisional loss from the MOT during the 200 s fill-time “751. The desire for larger condensates and faster repetition rates has prompted most groups to abandon the simplicity of a single vapor cell in favor of more elaborate loading schemes. Roughly half of the groups fill their MOTS with Zeeman-slowed beams [43]; most of the remainder use a double-MOT scheme-a “dirty“ MOT in a relatively high-pressure region cools atoms which are transferred to a “science MOT” at UHV [go, ’76,1481. In the future, LVIS [149] or funnel-type beam generators [150] may replace the Zeeman-slowed beams. A range of other possibilities exist: one could devise a way t o rapidly modulate the vapor pressure of the desired species. initially high for MOT collection, then low for evaporation. Alternatively, it has been shown also to be possible to spatially separate the MOT from the magnetic trap, and accumulate many MOT loads in a single magnetic trap [73]; perhaps such a technique could eventually find use. In yet another direction, the group at JILA is investigating mechanical means for moving atoms from high-density collection regions to low-density evaporation regions. 3’3. Chamber design. - -4s a rule, the chambers tend to be small glass cells with magnetic coils wound outside of the \-acuum. but the more traditional conflat-based stainless-steel cans with vacuum-compatible coils inside are in use as well, for instance at the Rowland Institute and at the University of Texas. Most likely stainless-steel chambers will continue to find use in certain applications-in cryogenic systems. for instance. or in experiments for which the condensate is t o be deposited on or scattered off of samples. Such experiments may benefit from the greater flexibility of reconfigurable steel chambers.
3‘6. Imaging technique. -Imaging the cloud after ballistic expansion has the advantage of being relatively easy to do. but the technique has its drawbacks. Some traps are difficult to turn off rapidly enough to ensure that no unintentional (and uncharacterized) work is done on the expanding gas. Also, although in most cases the expansion process is relatively easy to model (151.1521. it is an additional and sometimes unwelcome link in the chain of inference that connects an image with the underlying physical behavior. Find]?;,
550
E. A . CORNELL. J . R . ENSHER and C. E.WIEMAN
32
post-expansion imaging is not consistent with acquiring multiple images of the same cloud. The alternative t o post-expansion imaging. imaging in situ. has the prei-iously mentioned dran-backs that the cloud can be m a l l coinpared to the resolution limit. and that the cloud can be so optically thick as to give rise to lensing systematics. If sufficient care is taken. quantitative inforiliation can 1Je extracted from an absorption image even n-hen the object is smaller than the resolution limit [153.154].or when the object is very optically thick (13-11. The problems of optical thickness can be circumvented altogether n-ith imaging methods based on far-detuned probe beams sensitive solely to the real part of index of refraction: polarization rotation [153]. and dark-field and phase-contrast imaging (1531. For n-ork requiring yery high spatial resolution. or high signal-to-noise, expansion follox-ed by absorption imaging will continue to be a useful method. For work in n-hich the (relatively thin) thermal component is of primary interest, in situ ahsorption imaging will probably be preferable. For studies c.f time-dependent behavior (as long as they do not require very high spatial resolution) in situ pliase-contrast methods are probably best. 4.
- Effects
of i n t e r a c t i o n s
As discussed in subsect. 1'3 above. the topic of Bose-Einstein condensation develops its full richness only after interactions are included. 1Iost recent discussions of interactions in a T = 0 condensate begin with the Gross-Pitaevski (GP) equation (5)
Tvhere a is the 2-body scattering length, Vex,is the external potential of the trap and Q is the order parameter. In a dilute gas at zero temperature, the density at any point in space and time is just /912.For an instructive coinparison of several quite distinct routes to arrive at this "starting point" see [156]. For recent surveys of the status of the theory in this area, see [157-161]. We will review here the experimental studies to date on int>eractions. All published experiments on BEC to date have shown signs of these interactions, and these experiments have tested the theoretical predictions t o a greater or lesser degree. For convenience we n-ill discuss excitations in a separate section.
4'1.Energy and site. - In an ideal gas, the statistical mechanics of Bose condensation ensures that the majority of t h e atoms will accumulate in the lowest energy state of the confining potential. In particular, in a harmonic trap, the atoms accumulate in the (0, 0,O) state of the 3-D harmonic oscillator, Ivith a zero-point energy given by (T2/2)(w, + w y + w Z ) per atom. The onset of inter-particle interactions does not change the basic behavior of BEC: atoms still accumulate in the lorn-est energy state. The difference is only that the spatial extent, and total energy, of the interacting ground state is larger than its ideal-gas predecessor.
E X P E R I M E N T S IN D I L U T E ATOMIC‘
BOSE-EINSTEIN CONDENSATION
33
3.5
0.5_ .. . . . : . . . . . : . . : ,
5 : 10 : 15 :Time(p)
:22
.._
Fig. 4. - Comparison of the measured condensate energy ( 0 ) to the prediction from meanfield theory (solid line) as a function of interaction strength. By nonadiabatically releasing the condensate from the magnetic trap, the kinetic and interaction energies of the trapped atoms become the kinetic energy of the released and expanding atom cloud. The kinetic energy of expansion is obtained from a sequence of cloud widths measured at different expansion times. The inset shows experimental widths in the horizontal (0) and vertical ( x ) , compared to the mean-field predictions (dashed and solid lines), for the data point at N v ” ~= 0.53 Hz’l2. Used with permission of M. Holland from ref. [164].
The energy of the condensate is proportional t o the square of the size of its time-offlight image. Interactions perturbed even the earliest images of 2000-atom condensates [110.131]. and are the dominant factor in determining the size of larger condensates [113]. Once the number of atoms in a condensate becomes sufficiently large. the interaction energy and the external potential energy (second and third terms. left side of eq. ( 3 ) above) dominate the quantum kinetic energy (the first term). In solving for the ground state of eq. (Z) it then becomes convenient to ignore the kinetic term. an approach known as the Thomas-Fermi (TF) approximation. Once in the TF regime. the size of the condensate scales as h-1/5 and its energy as N 2 j 5 11621. The energy scaling behavior in the TF limit was confirmed with expanded image data from the MIT group [113]. In the intermediate regime between the ideal-gas limit and the TF limit. all three terms in eq. ( 5 ) are significant, and one espects to see crossover behavior in the energy dependence. This was observed in espanded-image data at JIL.4 [163.164]. The JIL-4 data also verified that the coefficiept in the interaction term of eq. (5) was within 20% of the predicted value. Quantitative checks on the in situ size of the interacting condensate have been performed at M I 1 [165] and the Rowland institute [134]. and are in good agreement with theory.
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In addition to total energy. and in sifu size of the condensate. eq. (3) also yields predictions on the aspect ratio of the expanded cloud and on the detailed shape of the condensate density profile. in the trap or expanded (in the I F limit. the density profile is an inverted paraboloid). See 11571 for a review of this theory. In one high-precision measurement. the aspect ratio of an expanded cloud x a s measured to be in agreement with theory at the 2% level [166]. In the presence of noise and various image distortions. it can be difficult to quantify how well a .‘shape“ agrees Kith theory. but no zero-T ~ major discrepancies with the basic theory measurements n.e are aware of [I321 s h o any
4’2. Two-species condensates. - In thinking about experiments in condensate mixtures, it is easiest to imagine mixtures of two different elements, rubidiurn and sodium. for instance [16T]. -4s it turns out, experiments n-ith tn-o-species condensates have to date mainly been performed in mixtures of two hyperfine states of *’Rb [iS.168-171]. The two-component rubidium work has been recently reviewed [ l i O ] . so we n-ill not discuss it extensively here. X range of tn-o-fluid behavior, including mutual repulsion, component separation, relative-cent er-of-mass oscillation and damping. and residual steady-state overlap have been observed. Qualitatively, the mutual interaction behavior can be accounted for in the framework of two. coupled, G P equations [16T,li2-1i4]. Quantitatively, this model has not been tested to any degree of accuracy. It is worth noting that the rf output-coupler experiment of MIT [IT51 was also sensitive to interactions between different spin-states of the same atomic species. The crescent shape of the out-coupled pulse of atoms arose from the repulsion between the out-coupled F = 1,mF = 0 atoms, and the still-trapped F = 1 . m = ~ -1 atoms [li’G]. Experiments have also been recently performed in a multiple-component system in which spontaneous interconversion of components is an important process in the system‘s dynamics. These are the F = 1 spinor condensates of the MIT sodium group. This work is reviewed in another article in this volume [ 1’Ti].
4‘3.Negative scatterin.g length. - Bose condensates composed of atoms with negative scattering lengths pose a particular challenge to theory. In a homogeneous system, such a condensate cannot exist, because it is unstable t40small density perturbations-for wavelengths larger than a critical value, excitation frequencies become negative. However, if a condensate is confined in a potential such that its size is smaller than the critical wavelength, it is stable (or at least metastable) to density fluctuations. Given a particular atomic species and a particular confining potential, there is a critical number of condensate atoms. If the condensate grows beyond that number, it will implode. Currently the only published experimental data on this phenomenon are from the Rice Lithium group [153]. At present they do not have the stability or sensitivity to watch a single condensate approach the critical value and then collapse. Rather, they collect many images and show that they never observe condensates larger than a certain size, a size consistent with the predicted “implosion limit”. In future work, improved stability and statistical analysis of observed cloud sizes may shed more light on the issue of metastability [178]. Alternatively, experiments utilizing
553 EXPERIMENTS IN DILUTE ATOMIC BOSE-EINSTEIN CONDENSATION
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atomic Feshbach resonances may be able to study implosion dynamics more direct.ly. 4’4. Finite temperature. - The theory of Bose condensation a t nonzero temperature [159-1611 is a more subtle topic than T = 0 BEC. After all, at T = 0 the phenomenon of “Bose condensation”, in the sense of a large fraction of the particles occupying the lowest available orbital, would occur even in Maxwell-Boltzmann statistics [179]. In the decades following the original Einstein papers, it was even believed the phase transition predicted in 1923 [2] would not survive a proper treatment of interactions a t finite temperature [15,179]. Experimentally, we now know that indeed the phase transition does occur [180] close to the (nonzero) temperature predicted in [23, and that the associated coherent behavior [181,148,182] survives the effects of interactions and finite temperature. Beyond that, there has been relatively few critical experimental tests of finite-T theory. There are several published experimental studies on the effect of temperature on excitations, but these data, as discussed in subsect. 5‘4below, are taken in a regime for which theoretical modeling is particularly difficult. There are, however, solid theoretical predictions for the effects of finite-T interactions on several static and thermodynamic properties.
4‘4.1.C r i t i c a l t e m p e r a t u r e . Interactions should affect the temperature a t which the condensate first appears in two different ways. First, the mean-field repulsion arising from the atoms concentrated a t the lowest point in the confining potential will reduce the density there; the net effect is t o reduce the critical temperature relative to what it would be for the same number of noninteracting atoms in the same external potential. See for instance [153]. Second. many-body effects among the confined particles, effects which can be described as a break-down of the dilute limit, are predicted to modestly increase the critical temperature [184.185]. So far there has been no decisive esperimental observation of either effect. The most precise measurement of Tc to date [lSO] quoted an error of about 5%. which was comparable t o the expected effect of interactions. Improvements in the accuracy of measurements of S . X o , and T will probably allow an unambiguous confirmation that interactions affect T,, but a quantitative sorting out of the relative effects of mean-field and many-body interactions will be an enormous experimental challenge [186.l,Si]. 4’4.2. E n e r g y c o n t e n t . Interactions will also affect the specific heat of a confined
Bose gas, or equivalently? the total energy content. which is the temperature integral of the specific heat [188-1901. There has been an esperimental study of the energy content as a function of temperature, based on the analysis of images expanded clouds [180]. The data (fig. 5 ) clearly show a deviation from ideal-gas behat-ior, which can be accounted for by finite-T theory (fig. 6). The experimental procedure is probably not sufficiently accurate t o provide a definitive and quantitative test of the theory.
4’4.3. D e n s i t y d i s t r i b u t i o n s . Interactions of the noncondensed fraction n-ith the condensed fraction, and interactions of the noncondensed fraction with itself. should modify the spatial distribution of the noncondensed fraction. particularly just at the edge of the condensate cloud. There exist some carefully acquired images of this region [I911
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2.0
1.5
0.5
0.01
0.4
"
0.6
'
I
0.8
'
I
1.0
'
TIT
1.2 0
"
1.4
'
I
1.6
'
1
1.8
(N)
Fig. 5 . - The measured scaled energy per particle of a trapped Bose gas is plotted ws. the temperature scaled by the critical temperature. TO.for a noninteracting (ideal) gas of bosons in a 3-D harmonic oscillator. The straight, solid line is the energy for a classical. ideal gas, and the dashed line is the predicted energy for a finite number of ideal bosons [192.193]. The solid, curved lines are separate polynomial fits to the data above and below the empirical transition temperature of 0.94To. The measured energy is actually the kinetic and interaction energies of the trapped atoms and is measured by suddenly releasing the atoms from the magnetic trap and calculating the second moment of the velocity distribution of the released atoms. (Inset) The difference A between the data and the classical energy einphasizes the change in slope of the measured energy-temperature curve near 0.91T0 (vertical dashed line). Figure taken from ref. [lSO].
but a theoretical study of t,he sensitivity of these images to alt.ernat,e models of finite-T interactions has not been undert,aken. paths t,o explore. 4'5. Future directions. - There are a number of exciting e~perirnent~al 4'5.1. F i n i t e T . As discussed just above, our understanding of finite-temperature interactions would benefit from quantitative measurements taken in a regime that facilitates comparison with theory. 4'5.2. F e s h b a c h r e s o n a n c e s . Some years aft,er their significance for BEC was first pointed out [194,195],Feshbach resonances have been observed now in sodium [19G] and in rubidium [197,79]. The interactions between atoms is now in principle an experimentally tunable quantity. Whether the Feshbach resonances will in the end be useful remains
555 E X P E R I X I E N T S I N DILUTE ATOMIC
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0.5
BOSE-EINSTEIN CONDENSATION
1
37
1.5
Fig. 6. - Calculation of the kinetic and finite-T interaction energy of a Bose gas compared with the data of Ensher et al. [180] (diamonds) and with the ideal-gas result (dotted curve). Results are shown for the zero-order solution (full curve), the first-order perturbative treatment (dashed curve) and for the complete numerical solution (circles). The straight line is the classical illaxwell-Boltzmann result. The inset is an enlargement of the region around T'. Used with permission of S. Conti from ref. [188].
an open question. In sodium at least, there appears t o be an enormous increase in the inelastic scattering rate associated with the Feshbach resonance [196]. If this turns out, t o be a general feature of Feshbach resonances, it will make tuned-interaction experiments much less convenient. 4'5.3. V o r t i c e s . Vortices are the heart and soul of superfluidity. Experimental studies of vortices wiII be critical t o our understanding of superfluidity in dilute-gas BEC.
4'5.4.M a n y - b o d y e f f e c t s . Producing experimental evidence for interactions beyond the mean field in dilute-gas BEC will be difficult, but quite worthwhile. 4'6.A perspective.
Equation (3) above looks superficially similar to the LandauGinzburg equation one finds throughout condensed-matter theory. It is worthwhile t o reflect, however, on the important differences. The coefficient 'a' in the nonlinear term not a temperature-dependent quantity, nor is it a factor empirically determined from observed fluid behavior. Rather, 'a' is the scattering length for a tn-0-body collision in free space. For the species which have to date been made t o condense, the value of a h a been determined from spectroscopic studies of pair-wise collisions, entirely independently -
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E. .I\. CORNELL,3. R.ENSHERand C. E.~ Z ' I E M A S
of the measurements of collective behavior discussed in this paper. Equation ( 5 ) . above. and for that matters eqs. (1)-(4). make quantitatii-e predictions for Bose condensation behavior with no adjustable parameters. The ability to predict macroscopic quantities such as critical temperatures. healing lengths. specific heats. etc.. froni calculations based on independent]:. measured microscopic interactions. xvith no adjustable parameters. is a rare occurrence in condensedmatter physics. Power law behavior can be extracted from renormalization group calculations. but the actual \-alues are almost always determined empiricall?. It is common for modern condensed-matter physicists t o dismiss the actual value of a critical temperature. or the actual length of a particular correlation function, as "uninteresting", when perhaps what they really mean is "too difficult". In any case, we find yery appealing the idea that someone Kill soon be able t o begin 1%-iththe elementary two-body collision potential and proceed systematically through to a quantitative prediction for the temperature-dependent time constant of the decay of the spiral motion of a vortex core. The scenario is all the more pleasing because within a year before or after the prediction is published, an experimenter n-ill very likely measure that same quantity. 5 . - Exc i t at i o n s
In this section, by "excitations" we mean '.coherent fluctuations in the density distribution". Excitation experiments in dilute-gas BEC have been motivated by tn-o main considerations. First. BEC is expected to be a superfluid. and a superfluid is defined by its dynamical behavior. Studying excitations is an obvious first step towards understanding dynamical behavior. Second: in experimental physics a precision measurement is almost always a frequency measurement. and the easiest way t o study an effect with precision is to find an observable frequency which is sensitive t o that effect. In the case of dilute-gas BEC, the observed frequency of standing-wave excitations in a condensate is a precise test of our understanding of the effect of interactions. In this section we will review work to date on excitations, concentrating almost exclusively on the experimental side.
5'1. Probing ezcitations. - BEC excitations were first observed in expanded clouds [198]. The clouds were coherently excited (see below), then allowed to evolve in the trap for some particular dwell time, and then rapidly expanded and imaged via absorption imaging. By repeating the procedure many times with varying dwell times, the time evolution of the condensate density profile can be mapped out. From that data, frequencies and damping rates can be extracted. In axially symmetric traps, excitations can be characterized by their projection of angular momentum on the axis. The perturbation on the density distribution caused by the excitation of lowest-lying rn = 0 and nz = 2 modes can be characterized as simple oscillations in the condensate's linear dimensions. Figure 7 shows the widths of an oscillating condensate as a function of dwell time. The use of in situ imaging is particularly useful in studying condensate excitations. In a single measurement, many observations of the width can be made. Simultaneous
557 EXPERIMENTS I N DILUTE ATOMIC
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I
I A
- 26 E
A
2
v
A
A
A
radial
14
A
axial
a
12
39
I
I
5
10
0 I
15 time (ms)
I
20
Fig. 7. - Zero-temperature excitation data from [198]. A weak m = 0 modulation of the magnetic trapping potential is applied to a n K = 4500 condensate in a 132 Hz (radial) trap. Afterward, the freely evolving response of the condensate shows radial oscillations. Also observed is a sympathetic response of the &?cia1 width, approximately 180” out of phase. The frequency of the excitation is determined from a sine wave fit to the freely oscillating cloud widths.
calibration of the trap frequency can be extracted from observing any residual dipolar “slosh” of the condensate (this mode is insensitive to interactions) superimposed on the driven oscillation. Ketterle’s group has developed this technique to the point Fyhere that frequency can be determined to much better than 1% in a single shot [199]. Higher-order modes do not result in oscillations of the overall condensate width but can be observed directly as density waves propagating through the condensate [165]. 5‘2. Driving the excitations. - A frequency-selective method for driving the escitations is to modulate the trapping potential at the frequency of the excitation to be escited 11951.
Experimentally this is accomplished by summing a small ac component onto the current in the trapping magnets. In a TOP trap, it is convenient enough to independentlv modulate the three second-order terms in the transverse potential. By controlling the relative phase of these modulations, one can impose rn = 0, rn = 2 or rn = -2 symmetry on the exitation drive. The frequency selectivity of this method can in instances be a disadvantage-one can miss n-hat one is.not looking for. A still simpler technique for drit-ing the oscillations is to impose a single ‘atepfunction” on the confining porentid [200]. This approach provides an inherently yery broad-band excitation. which has its o m adtantages and disadvantages. Condensates can be manipulated in lossless ways b*:
558
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E.A . CORNELL. J . R.ENSHER and I
72
c I
C.
E.WIEMAN
1
axial
--
I
rad ia I 54
h
E
I
Y
52 UJ
.
0
1 2 3 time (units of wr-’)
4
0
1
,
,
2
3
time (units of
.
1 4
wr:’)
Fig. 8. - Oscillation in the n-idth of a pure BEC cloud in both asial and radial direction due to the instantaneous change in scattering length. The widths are for condensates as a function of free evolution time in units of uT1= 9.1 111s. follon-ed by 22 ms of ballistic expansion. Each point is the average of approximately 10 measurements. The solid line is a fit of a Gross-Pitaevskii model t o the data. with onlj. the amplitude of oscillation and initial size as free parameters. Figure taken from ref. [lGG].
a focused beam of far-detuiied blue light [lll]. This technology has been exploited to excit,e high-order modes in condensate [ 1651. The techniques nientioned above all involve exciting the condensate by manipulating the external potential. Much the same ends can be achieved by manipulating the internal interactions of the condensate. One would ideally like t o modulate the condensate interaction strength arbitrarily and a t will, for instance by the use of Feshbach resonances. The JIL.4 group was able to drive excitations in the condensate by discontinuously transferring the atmomsfrom one internal state to another [lGG]. The different internal states had different interaction strengths and the discontinuous change in niean-field energy resulted in the simultaneous exitation of two m = 0 modes. See fig. 8.
5’3.Connection to theory. - There have been a very large number of theory papers published on excitations; much of this work is reviewed in [157].All the zero-temperature, small-amplitude excitation experiments published to date have been very successfully modeled theoretically. Quantit.ative agreement has been by and large very good; small discrepancies can be accounted for by assuming reasonable experimental imperfections with respect to the T = 0 and small-amplitude requirements of theory. 5’4. Finite T . - The excitation measurements discussed above have also been performed at nonzero temperat.ure [201,199]. The frequency of the condensate excitations
is observed to depend on the temperature, and the damping rates show a strong temperature dependence, This work is important because it represents the only observations
559 EXPERIMENTS IN D I L U T E ATOMIC
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far performed that bear on the finite-temperature physics of interacting condensates. Connection with theory remains somen-hat tentative. The damping rates. which are observed to be roughly linear in temperature, have been explained in the context of Landau damping [202,203]. The frequency shifts are difficult t o understand. in large part because the data so far have been collected in an awkward, intermediate regime: the cloud of noncondensate atoms is neither so thin as to have completely negligible effect on the condensate. nor so thick as t o be deeply in the “hydrodynamic (HD) regime“. In this context, -hydrodynamic regime” means that the classical mean free path in the thermal cloud is much shorter than any of its physical dimensions. In the opposite limit, the “collisionless regime”, there are conceptual difficulties with describing the observed density fluctuations as “collective modes”. The MIT Sodium group has published data [199] on mixed-cloud excitations which are tending to but perhaps not safely within the hydrodynamic limit; the JILA group’s published d a t a [201] describe experiments in which the noncondensate cloud was thin enough to be in the collisionless limit but n-hich still had distinct effects on the condensate excitations. In order to make better contact with finite-T theory, future experiments could be performed on much bigger clouds. so as to make the HD limit more accurate. Alternatively work could be performed in traps with spherical symmetry [204], which would make more tractable the theory of excitations in the collisionless or the intermediate regime. SO
The G P equation is manifestly nonlinear; the observations discussed so far in this section are performed in a small-amplitude, linear limit. Some data exist characterizing the lon-est-order anharmonic behavior [201,200]. but in general there has been little xork on nonlinear dynamics. Theorists have predicted a range of really interesting behavior. Varying the aspect ratio of the confining potential causes the small-amplitude frequencies of various standing-wave modes to move around. One can arrange for accidental degeneracies-for instance at a certain aspect ratio the frequency of the lowest-order ni = 0 mode is exactly half that of the nest higher-order ni = 0 mode. As one approaches the magic-trap parameter. the anharmonic coefficient diverges and one should be able to observe frequency-doubling effects [205]. In the presence of gravity. the aspect ratio of an axially symmetric TOP trap can be varied from fi :: 1 (oblate) down to slightly under 1 :: 2 (prolate) [204]. An IP trap can be varied from extremely prolate (greater than 30 :: 1) to slightly oblate, although in the near-spherical regime. a IP trap’s axial symmetry will be compromised b_v the effects of gravity [-ill. When the condensate is driven very hard. the motion will become chaotic, and energy will couple into many different modes 12061. Dilute-gas BEC may be an instructive model system for experiments to probe the boundary between chaotic and truly thermal behavior. A variety of soliton and soliton-like behavior is also predicted. and is likely to be observable esperimentally. See for example ref. [207]. 5’5. Nonlinear ercitations.
-
5’6. Alany-body effects. - As a condensate becomes more dense. the mean-field G P equation n-ill begin to break down. The first-order corrections t o excitation frequencies
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have been calculated [209], Precision measurements of condensate excitation frequencies may provide sensitive tests of this nontril-ial theory. 5 . i . Perspective. - The two most recent published experiments have in one case measured an excitation frequency that agrees rvith theory to well under l% [199], and in
the other determined amplitude data from which a ratio of interaction strengths was extracted [lGG]. The ratio agreed n-ith the ratio determined from ta-o-body spectroscopic data, also to better than 1%. It is worth pausing t-o think about what the accuracy of these t x o measurements. and the associat,ed theory, tells us about the field of dilute-gas BEC. -4s discussed in subsect. 4‘6 above, this sort of connection between macroscopic interacting behavior and calculations beginning ab initio with two-body potentials is something a little out of the ordinary. 6. - Condensates and quantum phase
Bose condensates are a wonderful environment in which to investigate the subtle concept of quantum phase. Experimental work on phase is really only just now getting started. There are several papers already in print, notably the Ketterle group’s observation of interference between tn-o separate condensates [ l g l ] and a collection of papers on mixed-state Rb-8’7 condensates from the JILX group [169]. rye recently reviewed the mixed-condensate work in a separate publication [ l i O ] , and so n-ill not discuss it here. For the separate-condensate interference experiment, see the original paper [181] or ref. [ l i i ) .K e know of interesting phase experiments going on a t Yale I2091 and a t Gaithersberg [210]. Presumably papers from these groups will have appeared in print by the time this article is published. We postpone to a future paper a thorough review of phase experiments. For the remainder of this section, we would like instead to discuss some of the terminology of the topic of quantum phase in condensates, terminology which has led, in our opinion, to some unfortunate misconceptions. Some may find the content of this section trivial-it is aimed a t the confused nonspecialist. 6’1.A common misconception. - In informal conversat,ions, we have heard it asserted that a precise measurement of the number of at.onis in a condensat,e must “destroy” a condensate, so that it is “not a condensate anymore”. The argument goes like this: i) the defining characteristic of a condensate is its coherence; ii) if you measure the number of atoms in a condensate very precisely, it will be in a number state; iii) if a collection of particles is in a number state, it cannot be in a coherent state; iv) since it is not in a coherent state, it is no longer a condensate! The careful reader will see the flaw in this syllogism, but in oral discussions, the argument can be compelling. The problem of course is that t,he phrase “coherent state” is a technical term from the field of quantum optics [211]. It refers to only a small portion of a broader concept which we could call “having coherence”. It is certainly true that if the atoms have no coherence, then the condensate has been destroyed. And it is equally true that a precision measurement
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of the number destroys a coherent state. The fallaEy is to equate the destruction of the coherent st’ate with the destruction of coherence. The fallacy is perpetuated by a tendency to treat the concept of spontaneous symmetry breaking as a fundament.alist Truth, rather than as a useful metaphor.
6’2. The basic phase thought experiment. - We turn now from the topic of what is wrong to the topic of what is correct. We will couch our discussion in terms of particular thought experiments, rather than in the usuaI language of expectation values of field operators. In our experience the formal notation of the field operators and brackets has sufficient ambiguity that it is as likely t o confuse as t o enlighten. Here, then, is’ the basic experimental module: take a not-too-well determined number of atoms from the vicinity of a particular point in space (XI),and a similar bunch from from the vicinity of another point (z2), and cause both bunches to impinge on an atom counter in such a way as t o ensure t h a t there is no way the counter can tell from which point any given atom originated. The number of atoms detected at the counter will be sensitive t o an interference term between the amplitude of atoms arriving from point one and the amplitude of atoms arriving from point two, and this interference term will be a function of cos 6$(z1, Q), where 6q5(~1, x2) is the difference in local quantum phase between the two points. The interference term will also depend on the additional differential phase which may accumulate while the atoms are being transported to the detector, of course. 6‘3. How is a condensate coherent? - In descriptions of condensates, “coherence” means “predictability of 64.” A condensate is “coherent” because one can predict. for any given time tl = t z . the 60 between any two points z1 and xz within the condensate. For a ground-state condensate at rest in the measurement frame, predicting 64 is particularly easy because b o ( r 1 . n ) = 0 for all q and 2 2 in the condensate. For a condensate moving with a velocity r , one can with confidence predict that @ ( X I , 22) = m ~ i ( tl r z ) / h . For a condensate with a vortex. the dependence of 60 on 21 and xz is more complicated [212]. but again completely predictable. Xote that a measure of the total number of atoms in a condensate, even an arbitrarily precise measurement, need have no effect on the state of the condensate‘s coherence. as defined in this subsection.
6‘4.Relative phase of two condensates. - Although completely irrelevant to the question of whether a condensate is a condensate, the “coherent state” of quantum optics is a useful idea when discussing the diference in phase between two different condensates. In this context, 64 is indeed the conjugate variable t o b:Y, the difference in number between two condensates. and yes, a precise measurement of bN must perforce change unpredictably the 60 between the tn-o condensates, so that there remains no coherence between the two condensates. (Although the tn-o condensates individually can remain ”coherent“ objects!) There is a useful insight due to J. Jawnainen I2131 and to Castin and Dalibard concerning the relative phase of two condensates. They have pointed out that. starting with two condensates in a relative number state. if one proceeds to measure the relative
562 44
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phase, the experiment will yield a perfectly well-defined precise value. w e will not review the formalism of their argunienrs liere, but The gist of it is that the act of sequentially detecting particles builds up correlatioiis in the remaining particles and uncertainties in the reniaiiiiiig relative number. After a relatively small nuniber of particles are detected. the remaining two condensates find themsel\.es in a relative coherent state. The pliase of that state is unknon-able before the nieasurenient begins: in the view of Castin and Dalibard. the act of measurement has projected out. in a quantum-mechanical sense. a particular relative phase. If we return to the "building-block" thought experiments on pliase described above. n-e can see right away what we mean when n-e say *-anieasurenient of the relative phase betxeen two condensates aln-ays yields a particular well-defined value". We take two condensates. called A and B, for simplicity we assunie both are ground-state condensates at rest in the measurement frame. and we take a small nuniber of atoms from near a point s.4 in condensate X and a small number of atonis from near a point SB in condensate B, and beat them on a detector, and from the interference term. n-e determine cos(dqf~(s~. SB)). Immediately thereafter, we pick two new points, xi and zb. again in condensates A and B, respectively. and find that we get exactly the same value for cos(b4). -And in general we find that no matter what point n-e pick in A and n-hat point n-e pick in B: n-e always find the same relative phase as we found in the first measurement. although n-e niay not have been able t o predict ahead of time what phase we would measure in the very first measurement. This then is n-hat we mean by saying that a "measurement of the relative pliase of tn-o condensates always yields a particular welldefined relative value". If we assume a sort of "transitive property of relative phase", ( i e . that S ~ ~ ( . E ~ , . = Z . ~d@(s~,s~) ) ~ @ ( s A , x B )+ d $ ( ~ g , . ~ . ; j )then ) , the fact that a measurement of the relative phase of two condensates always yields a particular, welldefined relative value, is equivalent to the fact that each condensate is independently "coherent" in t.he sense defined in subsect. 6'3 above.
+
6'5. Noncondensate clouds. - One cannot talk about a single relative phase between tn-o conventional, aoncondensed. thermal atomic clouds (unless one has filtered out all but a volume within a single thermal coherence length for each cloud-Lvhich in fact is what is done in noncondensate atom-interferometry experiments.) The significance of the MIT two-condensate interference experiment [181]and of other condensate coherence studies [148,182] is that they demonstrated once and for all that condensates are not simply very cold, relatively dense clouds of atoms.
6'6. Coherence and decoherence. - The relative phase between two condensates can be determined either by measuring it, as described above, or by splitting a single condensate into two with some "gentle" technology [166,169] which does not, too greatly perturb the condensate(s). Immediately after the relative phase has been determined, one can of course predict what will be the result of a new phase measurement. Because the relative phase is known, we say that there is a relative coherence between the condensates. If at some later time we are no longer able to predict the relative phase to
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n-ithin an uncertainty of x radians, we say that the two condensates have undergone decoherence. The decoherence can arise in many ways. For inst,ance, if between our initial deterniiiiation of relative phase and our final measurement of it, someone should perform a precise measurement of the relative number, we know the measurement would cause decoherence. On the other hand, suppose the two condensates are a t slightly different heights, so that they feel slightly different gravitational potentials. Their relative phase n-ill then evolve a t a rate proportional to the difference in the gravitational potential. If the difference in their heights remains very constant over time, we can perform several measurements, determine accurately the rate at which their relative phase evolves, and thereafter be able to predict. what their relative phase will be at some future time. If the difference in their heights varies in time in some uncharacterized way, we can lose track of the relative phase and be unable to make an accurate prediction of it. From an experimenter's point of view, then, decoherence need not (and usually does not) arise from anything particularly fundamental. 6'7. Mixtures and superpositions. - Experimentally, the distinction betxeen "a xnixture of two condensates" and a "single condensate of atoms in a superposition state" is esactly the distinction between decoherence and coherence. A misture is a superposition in xhich we have forgotten the relative phase. ilsuperposition is a misture for which we still remember the relative phase. The distinction between a mixture and a superposition has little to do n-ith the intrinsic nature of the saniple in question, and much to do vith the state of the experimenter's knowledge. 6'8. A final warning on vocabulary. - Much has been written about a particular niechanism for decoherence, one which arises from the nonlinearity of condensate selfinteractions: i) In a finite-sized condensate, the energy per atom depends on the number of atoms. ii) When the phase between two condensates is measured. an uncertainty in the relative number of atoms is always introduced. Therefore. there will be an uncertaintp in the difference in the energy per particle for two systems, and over time an uncertaint.y in the relative phase must develop (2151. The mechanism has been given the deceptive title "quantum phase diffusion." While the mechanism does involve the quantum phase. and while that phase does change unpredictably (which is t o say, it diffuses). "quantum phase diffusion" is only one of many ways in which the phase can become unknowable. For reasonably sized condensates. finite-temperature effects will almost surely be more important [2 161.
7. - Stray heating: a pessimistic note
7'1.Introduction fo the problem. - Ultracold atoms confined in a magnetic trap have a tendency. in the absence of ongoing evaporative cooling. to warm up. Given that the ambient temperature of the trapping environment is typically nine orders of magnitude higher than the temperature of the trapped atonis. the surprise perhaps is not that the atoms heat. but that they heat so slowly. To an optimist. a heating rate of 100 nK/s means that the time constant for the atonis to thermally equilibrate with the 300K
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environment is about one hundred years. Hardly objectionable! To a pessimist (or to a realist) that same heating rate means that the temperature of a sample inirially a t 200nIi n-ill double in only two seconds. which can be a major esperimental inconvenience. The phenomenon of trapped-atom heating is observed. as far as n-e knon-. in all magnetic trapping laboratories. Relatively little has been pubIished on it. lion-elver. In this, the final section of our paper, n-e n-ill surve!. what is knon-n about the various mechanisnis for heating. While the topic has for the niost part not attracted much scientific study, it is worth investigating for two principal reasons. First. this ubiquitous effect, represents a systematic which often complicates the interpretation of condensate studies. such as those 0x1 condensate formation or on collisional loss. Second. heating seems t o scale unfavorably with increasing atom number. and may give rise to an upper limit on the size of condensates one can produce. It is difficult to give a satisfactory summary of the experimental observations. Some quantitative studies are described in [217.21S]. but most of n-hat is knon-n in the con]munity has been passed around anecdotally. by word of mouth. Here are some general observations: -4typical heating rate for a cloud of 10' Rb-ST atoms in a Ioffe-Pritchard trap, in the F = 1:mF = -1 state. a t 11.111;.is perhaps 150nK/s. There is anecdotal evidence. although no careful coniparisons, that indicates that the problem is worse in ~ 2 Rb-8i than it is in S a . The problem seems to be a-orse in the Rb-87 F = 2 . m = state than it is in the F = 1 : n - t ~= -1 state. Heating is observed t o get n-orse n-ith increasing trapped-atom number, with increasing density. with increasing column density. with increasing elastic-collision rate. and with decreasing bias field. Unfortunately. it is difficult to vary each of the above-mentioned quantities independently, therefore it is not clear what really matters. Both the MIT Sodium group and the JILA BEC group have observed that the presence of an ..rf shield" (described in sect. 7'5 below) profoundly affects the heating rate, in most situations. Heating rates as low as 10nK/s have been observed in well-shielded traps [ 1991. Efforts on the part of the JIL-4 group t o come up n-it11 an experimentally validated. comprehensive model for the heating mechanism(s) have not been very successful. Our experimental results have been sonien-hat inconclusive, and our models tend to become too complicated to give simple numerical predictions. The discussion in this section is meant to be taken as a collection of general observations, as suggestions for directions of research, rather than as definitive results. In the remainder of this section we discuss in turn various candidate mechanisms for heating. We believe most of these mechanisms are inadequate by themselves to account for the observed heating. In the final subsection we explore what we feel is the best explanation for the heating-the ultracold t,rapped sampIe is often surrounded by a very dilute haze of much hotter atoms, also trapped, which gradually thermalizes with the ultracold sample. Glancing collisions with background atoms, and inelastic decay products, while inadequate t o directly cause heating, help populate the high-energy haze and thus indirectly contribute significantly t o the heating.
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7’2. Miscellaneous small effects 7’2.1.A n t i - e v a p o r a t i o n . The earliest paper on trapped-atom evaporation [31] disCusses one of the niost ubiquitous mechanisms for heating: inelastic collisions. These can be thought of as giving rise to “anti-evaporation” for the following reason: at the typical temperatures of trapped atoms, the rates for three-body recombination and dipolar relxation are independent of the collision velocities, and the per-volume rates are simple functions of the local density. Since in thermal equilibrium the density is highest where the potential is deepest, inelastic collisions selectively remove atoms with low potential energy, thus increasing the mean energy of the remaining atoms. Heating from anti-evaporation is very common; most experimenters have probably encountered it. In samples below the critical temperature, most of the inelastic collisions occur in the condensate, because its density is much higher than the thermal cloud. If the sample is not too large, the decay products can pass cleanly out of ,the cloud. In this case, the inelastic collisions consume the condensate but do not, strictly speaking, cause heating, because the density and temperature of the thermal cloud is unchanged. But because the overall number of atoms is decreasing. the value of the critical temperature goes down. Anti-evaporation therefore increases the value of T/T,. Anti-evaporation usually arises from inelastic collisions, but more generally anything which preferentially removes atoms from the center of the trap gives rise to antievaporation. For instance, if there is trap loss due to rf magnetic noise driving unwanted Zeeman transitions, and if that noise has a steep frequency dependence (sav, due to Ion-pass filters on the trap-magnet coils). then atoms at the lower magnetic fields at trap center will undergo Zeeman transitions and leave the trap at a faster rate. In many experimental situations, heating is observed that is far too large to be accounted for by anti-evaporation alone. In anti-evaporation’s limiting case, only atoms with zero mean energy would undergo decay. A differential change of number 6-1’would then give rise to a differential increase in temperature 6T such that 6T/T = -6-Y/:Y. This is an upper limit on the amount of heating that can arise from anti-evaporation. When the observed fractional rate of heating is larger than the observed rate of trap loss: one must look elsewhere to identify the dominant source of heating. 7’2.2. F i e l d n o i s e / s h a k i n g t r a p . Atoms in a magnetic trap are by intention highly isolated from the external environment. The one mechanical force that must be present is from the trapping itself. and thus in investigating heating we naturally suspect the trapping fields. In our esperience. however. mechanical noise on the magnetic coils, or electric noise in the coils. is seldom the dominant source of unwanted heating. There are two reasons for this: i) moving the coils. or modulating the current in them. affects all the atoms nearly the same. and thus does not couple well to a heating effect. escept to the extent that it drives macroscopic pulsing or sloshing modes of the cloud. ii) In magnetic traps at least. and at low temperatures. the atoms see a verv harmonic potential. This has the effect of sharply constricting the bandwidth of the noise that can harm the atoms.
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Unless tlie noise is near a trap oscillation frequency. or a harmonic thereof. the energy cannot couple to the atoms. If the noise is a t tlie second har1iioiiic of trap frequency. the motion of the atoms can be driven paranietrjcally. and the resulting .pulsing” of the cloud can convert via collisions to heat. If the noise is directly at a trap frequencJ-. ’.sloshing” can be driT-en. n-hich due to anharnionicity also con\-ertsto heat. Directly driven sloshing is usually only a problem in the vertical direction. for which gravit? breaks symmetry. In an effort to characterize the heating from noise-driven pulsing. n-e have intentionally applied electrical and niechanical noise to coils. If the noise is broad-band. it takes a surprisingly high poR-er density to accomplish any heating. One week at JILX in 199’7. when n-e m-ere having a particularly bad problem with electricai ground loops. our evaporation stopped n-orking n-ell. It turned out that during our evaporation procedure (which includes an adiabatic ramp of the trapping frequencies) n-e were unintentionally pausing for a second at the value of tlie trap parameters which caused one of the trap frequencies to be resonant with a multiple of GOKz (the Xorth -2iiierican power-line frequency). Eventually we fixed tlie power supply. but n-e n-ere able to implement a successful “quick fix“ simply by raniping i-ery quickly over the offending region of trap parameters. In nine years of JIL.4 experience n-ith magnetic traps. this n-as the only occasion in n-hich magnetic coil noise was confirmed to be the dominant source of unn-anted heating. K e have used parametric driving to deliberately heat the atoms (see, e.g.. ref. [74]). but to accomplish this we had to apply a niodulation tone of considerable amplitude compared to background noise. For a thorough. quantitative discussion of heating from noise on the trapping field, see Sai-ard et al. [219]. IT-hile they a n a l p e the same niechanisnis n-e mention above. they come to the opposite qualitative conclusion-they assert that noise on a trap is likely t’o be an important source of heating. The source of the disagreement lies in the technical details-their model system was an optical rather than a magnetic trap, and optical traps tend to have higher confining frequencies. -4s Savard et al. show. the higher the confining frequency. the more susceptible a system is to parametric heating [219]. Moreover, electrical and mechanical stability of electromagnets is likely to be better than intensity and pointing stability in laser beams. Noise-induced heating is likely to be a very important effect in “composite traps”, traps made of magnetic and optical forces [ I l l ] or of more than one laser beam. In those cases noise can lead to relative motion of the various components of the confining potential, and thus heat tlie atonis more directly.
7’2.3. S t r a y rf a n d o p t i c a l f i e l d s . If care is not taken to keep radio and optical frequency phot,ons out from where they do not belong, they can cause trap loss and presumably, under the right conditions, heating as v-ell. In practice it is often necessary to place an opaque box around either the lasers (with their associated absorption cells) or the magnetic trap.
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7'2.4. B l a c k - b o d y r a d i a t i o n . In conventional cryogenic experiments, black-body radiation is a major source of heat transfer, but it is difficult to imagine that it will ever be important for the case of trapped atoms. The cross-section for scattering a photon, at least for atoms, is simply too small. The question may need to be revisited if one were to attempt t o evaporatively cool objects more complicated than individual atoms. objects which niay couple more strongly t o long-wavelength radiation. 7'3. Collisions with background atoms. - Another common source of unmnted heating in conventional cryogenic experiments is imperfect vacuum: residual gas atoms move back and forth between warmer and colder surfaces, transferring heat with each bounce. In trapped-atoms experiments as well the room-temperature residual gas atoms can provide heat to the confined atoms, but the effect cannot be understood as a simple shuttling of heat between a warm surface and a cold sample. Tt'hen a 300K background atom collides with a trapped atom, the most probable result will be a clean ejection-both the incoming and the impacted atoms leave the trap without further collisions. These are the events that give rise to the background loss rate *(bg discussed in subsect. 8'1 below. But there exists also the possibility of glancing collisions which transfer a relatively small amount of energy t o the trapped atom. These events lead to heating rather than to loss. 7'3.1. D i r e c t c r e a t i o n of '.hot" a t o m s . The distinction between heating and loss can be somewhat arbitrary. Imagine a magnetic trap which is 5 m I i deep. confining at its center a sample of atoms with temperature 20pIi. Xow suppose that a trapped atom undergoes a collision that leaves it with 4niK of kinetic energy. Is this heating. or loss? If one were to measure the total number and total energy of the atoms confined in the potential. this particular collision looks like heating. -4more common esperiniental procedure. hon-ever. would be t o image the cloud. and fit its profile to a Gaussian lineshape. If one identifies the energy of the cloud as the mean-square width of the fit lineshape, and the number of atoms in the cloud as the area under the lineshape. then our hypothetical collision looks like loss. Typically, then, "heating" results from background collisions which transfer energy less than three or four times the mean energy of the trapped cloud. -At the opposite extreme. collisions which leave the impacted atom with energy larger than the trap depth clearIy result in loss. Collisions which transfer an intermediate range of energy give rise to a large-area. diffuse cloud of atoms that are still trapped but whose optical density is below detection threshold. These intermediate collisions appear to give rise to loss. rather than heating. at least at short times. The longer-term effects of the diffuse cloud are discussed in subsect. 7'5 below. TOget a rough estimate of the size of these heating effects. we need a simple model for small-angle collisions with background gas. The follon-ing simplified explanation d r a m on thorough treatments by Helbing and Pauly [220] and by Anderson [El]:our results are not meant to be accurate t o n-ithin a factor of two. Small-angle collisions are the result of trajectories with large impact parameters. Therefore only the long-range tail of the inter-atom potential is relevant. i1-e n-rite the inter-atomic potential as L-inr = C6/r6. The value of CS depends on the scattering species. For helium on rubidium. c6 is
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36atomic units (au) [222]. For rubidium on rubidium. the value is 4i00au. Treating the scattering event classically.: in the impact approsimation. and working in the laboratory frame (in which the trapped atom is initially at rest. and the incoming background atom has energy €col). we find the partial cross-section for a glancing collision to transfer energy €t
Xote that the integral of ailass(€l) dE, diverges at Ion. energy; quantum mechanics intervenes to keep the true total cross-section finite. Me define a crossover energy E,: for Et < E,, the deBroglie A-avelength of the transferred momentum becomes comparable to or longer than the classically determined impact parameter, and diffraction effects dominate. E, is given by
where nz is the mass of the trapped atom, 1771, is the mass of the background atom: and it is assumed that ?71b 5 nz. For helium on rubidium, €, is about 40mK. For rubidium on rubidium it is about 3mK. -4s discussed in ref. [221], for Et < E,, ag(Et)shows oscillatory behavior due to diffraction. In our simplified model, we use the following approximation:
where Q = To convert the cross-section into a rate, one needs t o know the density of the residual 300 I< atoms. In the laboratory, it is easier to measure the lifetime of a trapped sample than it is to get an accurate estimate of the local residual vapor pressure at the trap. If we define ybg to be the loss rate measured in the limiting case of a very low-density sample and a very shallow trap, we know that ybg will be proportional to the integrated collision rate and t o the background pressure. Assuming there is one dominant species of background gas, we can normalize eqs. (8) t o get the following partid rate results, for the rate per atom, per differential energy, of energy transfer collisions:
(9)
What then is the resulting heating rate from such collisions? As discussed above, depending on exactly how one measures “heating”, one can estimate a threshold value: call
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it &,, for the transfer energy. For Et > E h , the collisions result in “Ioss”. For E~ < E h collisions result in “heating”. T h e fractional rate of heating is then
is the mean energy per atom in the trapped sample. If we take a plausible where estimate for the E h , say E h X 37?, then ( d z / d t ) / E X 27bgkT/Ec, for a sample with temperature T . This simple model of direct heating from glancing collisions is qualitatively successful in accounting for heating observed in early JILA experiments on trapped Cs 174,2181. By heating the vacuum chamber, we caused the dominant background species to be cesium. In this limit, we observed that the fraction heating rate of a 200 p K cloud scaled linearly with the background loss rate, with a constant of proportionality of about 0.5. For cesium on cesium, Ec is about l m K , and the predicted constant of proportionality between heating and loss is about 0.2. Given the qualitative nature of the model. the agreement is pretty good. When we instead cooled the vacuum chamber, the dominant background species became helium, and (consistent with the large value of Ec for helium on cesium) the heating rate essentially vanished, even though we continued to observe trap loss. In most modern BEC experiments. however, the laboratory conditions are quite dissimilar to the early JILA cs experiments, and the simple model is empirically found to be completely inadequate. In a clean vacuum chamber, the dominant background species is usually a Ion--mass, low-CGspecies such as helium or hydrogen, such that the appropriate value for EC may be 40mK or higher. The observed value of -(bg is typically less than 0.01 s-l. In addition. the sample temperature is typically 1pK or lower. Under these conditions. eq. (10) typically predicts heating rates four orders of magnitude smaller than what is observed.
7’3.2. H e a t i n g f r o m s e c o n d a r y s c a t t e r i n g . As a first attempt to improve the model, we revisit its implicit assumption that collision products can pass unimpeded through the sample, ie. that the sample size I is much less than the mean-free-path .,Xf, If the opposite limit applied. if I >> .,fX, then an atom which has acquired an energy Etfrom a background event would collide with other atoms in the sample many times along its trajectory out of the sample. If the sample is thick enough, the entire energy Et could be distributed among a shower of atoms, all mith energies less than 3r. We can get a rough estimate of the resulting heating rate in this limit from eq. (lo), Kith Eh now defined as the maximum value of energ;\-. an atom in the middle of the sample can have and still be trapped by multiple collisions in the dense sample. E h is non. a function of the column density of the sample, nl. and t o estimate its value we look at the = ( n Q h c ( € ) ) - l , Khere q , c ( E ) is the energy-dependent crossenerg? dependence of ,fX, section for large-angle scattering. h the limit of very large nl. E h d l be sufficientlylarge that %(El-,) m i l l be in the classical limit, for which Q h c ( E ) oc C,‘’3E-’/3. 1Yitl-I each
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collision. the particle loses perhaps 1/2 its energ!.. so that Oh,(€) becomes larger and the corresponding value of Xmfp becomes shorter. The sum over the sequence of successively shorter values of Xmfp converges. so that iiiost of the shower of atoms resulting from the original energ!.-transfer ei.ent should be brought to rest within a distance proportional to E:'3/n. Therefore the nia.;inium energy atom That can be trapped. E h . scales as ( ~ 7 1 ) ~ . Substituting this value for Eh into eq. (10). we see that in the limit of a trapped cloud n-it11 1-eryhigh column density. 111. the heating rate can scale as ( 7 1 1 ) ~ . -4heating rate that scales as the sixth pon-er of the column density is an alarming prospect. but n-hen reasonable numerical values are plugged in. n-e find that our multiplescattering model predicts dramatic effects only for very large trapped samples--n-it11 column densities of perhaps 1013atoms/cm' or greater. We include the above discussion only because it illustrates how. even a-ith perfect rf-shielding (see subsect. 7'5 belon-). heating may el-entually prove to be a forniidable upper limit t o the sizes of condensates one can create. We n-is11to re-elliphasize that the models above are very crude in nature. N e made a number of sinipIif)-ing assumptions n-hich n-e did not explicitly state. For instance, an implicit assumption in the previous paragraph is that. after a particular background scattering event. either all the energx will be captured in the sample by multiple rescattering. or all the energy n-ill escape. In actual fact. most scattering el-ems ivill leave behind as heat some fraction of their initially transferred energy. a fraction that is neither zero nor unity. Such events are not properly accounted for in our model. To conclude this subsection. n-e should emphasize that for the column-densities of the trapped samples in currently ongoing BEC experiments. the amount of heating that can be attributed to the explicit scenario described above (a 300Ii atom scatters from a trapped atom, which in turn rescatters on its way out of the sample) is relatiyely small. On the other hand, n-hile it is true that most of the atoms which are perturbed by background gas will exit the sample without additional scattering, a significant fraction of them may remain within the much larger volume of the magnetic trap. These atoms may cause trouble later on, as m-e discuss in subsect. 7'5 below.
7'4. Collisions with the products of inelastic decay. - When a t atom is lost from the sample due to a collision with a 300K background atom, the chances are 6 / 7 that it will have an energy larger than E, x 40mK (see eq. (9)). This means, in turn, that the large majority of atoms lost due to background collisions will have energies greater than the depth of the confining magnetic trap. With inelastic decay, however, the story is reversed. If we assume most three-body decay populates the least-bound vibrational state, then the decay products will have energies mostly under 1mK. Dipolar decay products may also h a w energy under 1mK. Inelastic loss, then, leads to production of atoms Tit11 energies typically much greater than the trapped sample, but much less than the depth of the confining magnetic potential. Given that lifetimes due to background decay often exceed IOOs, and given that the three-body decay constants for alkalis are in the yicinity of it is often the case that, as the density of a trapped cloud approaches the critical \ d u e for BEC, the dominant loss process is inelastic decay. rather than background collisions. The combined result then is that as a sample approaches the
571
critical density, the dominant mechanism for the production of atoms with “dangerous” energies ( L e . 1pK < E < 2mII;) will be inelastic decay, rather than glancing collisions with background atoms. The decay products will help populate a diffuse cloud of hot atoms. which can cause heating, as discussed below. Depending on the column density of the trapped sample. they niay also contribute to direct heating, exactly as background-scattered atoms can. There is a widely held assumption that the decay products of three-body recombination are so energetic that they have a relatively small cross-section for colliding with a trapped atom. This is completely incorrect. Given that the dominant decay channel is into the highest vibrational state, a simple calculation for Rb-8T shows that the outgoing “witness” atom will have a cross-section for scattering which is about the same as the zero-energy s-wave cross-section [223]. Beyond the specifics of the rubidium potential, general considerations on the relationship between s-wave scattering and the binding energy of the highest vibrational level suggest that the witness atom will usually have an energy low enough that its cross-section will be comparable to the zero-energy value. -4s for the outgoing dimer, the kinetics of three-body recombination is such that the dimer will have per-atom translational energy four times smaller than the witness atom’s energy. There is no reason to believe that the dimer’s cross-section on its atomic counterpart will be small. Xfter all, the loosely bound dimer is presumably no less polarizable than a single atom. 7’5. The Oort cloud paradigm. - The usable volume of a Ioffe-Pritchard magnetic trap can be quite significant, perhaps 2cm3 or more, and the depth of the confiiiing potential is typically on the order of several millikelvin. A typical sample of ultracold atoms. n-ith temperature on the order of a microkelvin, occupies only a tiny fraction of the available volume: the ratio of sample volume to the trap volume may be lo-’ or less. There esists considerable evidence that, towards the end of an evaporative cooling cycle Khen the central sample is nearing degeneracy, the outer region of the trap is filled with a very dilute. very high-energy halo of trapped atoms which we refer to as the ”Oort The millikelvin Oort cloud is of course not in thermal equilibrium with cloud” [??-I]. the microkelvin ultracold sample. and the coupling between them is weak. due to the Oort cloud‘s very low density. Nonet heless occasional collisions may transfer energy (and atoms) between the two trapped components, and we believe this coupling provides the dominant mechanism for sample heating in many experimental situations [225]. The Oort cloud is not readily imaged. Its radius may be ten to 100 times larger than that of the uItracold sample. which means that its cross-sectional area may be 100 to 10‘ times larger. Thus even if the Oort cloud comprises more atoms than the ultracold sample. its optical depth may be so small as to make it easy to overlook. In our experience. a thermal cloud that is allowed to remain in a magnetic trap for several seconds in the absence of evaporative cooling will sometimes develop wings.“ which can not be fit Kith a single Gaussian density profile. IVhile these wings are usually thought of as a high-energy tail to the ultracold distribution, they niay as well be described as a low-energy tail to the Oort cloud.
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Direct evidence for the esistence of the Oort cloud can be found in discrepancies between two distinct methods of measuring the number of trapped atoms in a magnetic trap. The first nietliod is to pass a probe beam through the t r a p and image the tn-odinlensional absorption profile. The number of atoms in the trap is then assumed to be proportional to the integrated signal under the central feature in the shadow cast by the ultracold sample. This method is not sensitive to the presence of an additional. spatially dispersed population of atoms. The second method is suddenly t o turn off the magnetic trap and t o turn on the MOT. and then to collect the total fluorescence emitted from the atoms recaptured in the optical trap. The MOT laser beams are several cni in diameter and sweep up any atoms that had been trapped anywhere n-ithin the magnetic trapping volume. The tn-o methods can be performed on similar clouds of trapped atoms. and the results compared. What is often found is that n-hen a cloud of trapped atoms has just been freshly loaded into the magnetic trap. the tn-o methods give similar estimates of the total number of trapped atoms. Later. after extensive evaporation has been performed and the number of atoms in the sample reduced 100fold, the MOT-recapture method may shon- perhaps five times more atoms than the shadow-image method. In this esaniple. then. only 20% of the trapped atoms are in the ultracold sample. The remainder populate the Oort cloud. The strongest e\-idence that the Oort cloud is responsible for much of the typically observed heating conies from the effect of an *‘rfshield”. Both at J1L.A [217.148.204] and at MIT [113.118] it has been noted that the application of an rf field can affect the lifetime and heating rate of trapped samples. During evaporation, an rf magnetic field is applied to the trapping region. and its frequency is gradually ramped downward as evaporation proceeds. There is a 2-dimensional surface surrounding the center of the trap. a surface over which the magnetic field has the correct magnitude to bring the atoms on resonance with the rf-field. -\toms whose trajectories cross this surface have some probability of being transferred into an untrapped spin state. -4s the frequency of the rf is decreased. the radius of the rf-surface decreases, cutting into the edge of the ultracold sample, and driving the evaporative process. In a typical experiment, the rf surface is smoothly brought inn-ard until the ultracold sample reaches the desired temperature. then moved back out again and left on while experiments are performed on the sample. This post-evaporation configuration is referred to as an “rf shield”. The radius of the rf surface is typically sufficiently large that the rate of evaporation essentially vanishes; no ultracold atoms have trajectories with sufficiently high energy to reach the rf shield. Yet the shield is observed t o have a profound effect-turn the rf field off, and the rates of heating and loss from the ultracold sample can increase by an order of magnitude. Presumably the rf shield depletes the Oort cloud or prevents it from being populated in the first place. Hon7 is the Oort, cloud populated? We speculate that there may be several niechanisnm. First, there is the mechanism of glancing collisions. This is discussed in subsect. 7’3. above. Second, there is the generation of inelastic decay products. This is described in subsect. 7‘4, above. Third, “incomplete evaporation”. In experiments with atoms in the F = 2, ??IF = 2 state, rf evaporation may remove an atom from the mF = 2
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55
state but fail to transfer it into an untrapped state, leaving it instead in the m F = 1 state. Some of these atoms can later cause trouble. Fourth, “primordial remnants.**At any given instant, a small fraction of atoms in an optical molasses have energies which are extremely superthermal. These atoms may end up loaded into the magnetic trap with trajectories that are outside the initial radius of ’the evaporative rf knife. While these atoms may initially represent less than 1%of the total sample, the process of e\*aporation reduces the size of the trapped sample by a factor of 100, which increases proportiona1Iy the relative significance of the high-energy remnants. Fifth, the Oort cloud can populate itself through erosion of the ultracold sample. For example, an Oort atom with 500pE; of energy can plunge into the ultracold sample and undergo a collision, with the result that two, 250pK atoms emerge. The net population of the Oort cloud is increased, and it is left colder and denser. The colder the Oort cloud is, the more tightly it is coupled to ultracold sample, and the more heating it will cause.
How is the Oort cloud depopulated? The presence of the rf shield (or equivalently, ongoing rf evaporation, vihich amounts to the limiting case of a dose-in rf shield) will tend to deplete the Oort cloud. As the atoms pass through the surface in space on which the rf is resonant, they are transferred into untrapped states, and may leave the trap. In a TOP trap, atoms in the Oort cloud may encounter as well the orbiting zero-field point, which can induce hfajorana transitions [log] and detrap atoms. The rf shield is an imperfect device for removing the Oort cloud. If the amplitude of the rf is too low, then the Oort atoms (whose average velocity is high compared to the atoms undergoing evaporation) may pass through the rf‘s resonant surface too rapidly to undergo a transition. The trajectories of such Oort atoms may carry them through the ultracold cloud many times before they are finally ejected by the weak rf field. On the other hand, if the amplitude of the rf is too high. such that there is unity probabilit? for an Oort atom t o undergo a transition as it passes through the surface, the situation is equally unsatisfactory. As the Oort atom passes through the resonant surface. towards the center of the trap, it is flipped into an untrapped state. But its energy will be such that it can still reach the ultracold atoms a t the center of the trap. To leave the trap. it must pass once again through the resonant surface, where with high probability it will be flipped back into a trapped spin-state! See fig. 5b of ref. [226] and accompanying test for a nice description of the relevant physics.
-4quantitative understanding of the interplay between the rf shield and the Oort cloud. and of the consequences for trap loss and heating. may require a fairly elaborate model. Certainly any attempt to create a condensate with say lo9 rubidium atoms will need to consider the problem of heating very carefully. Perhaps optical dipole traps, which are SO shallow that they cannot support an Oort cloud [llS].may play an important role in the creation of extremely large condensates. For a discussion of heating in shallow traps. see ref. 12271.
574
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8.
E. A . CORNELL. J . R. EXSHERand C. E. N'IEMAY
- Collisions
and ev ap o r at i o n : an o p t i m i s t i c note
The crucial issue in planning a successful dilute-gas BEC experiment is collisions. L-nti] a ne\v technology comes along. the last step in a BEC experiment n-ill be el-aporation. and evaporation n-orks only if the collisional properties of one's system are favorable. In practice. this means that planning a BEC experiment requires learning to cope with ignorance. So much is currently knon-n about the elastic and inelastic collisional properites of the atoms in the first column of the periodic table [225] that it is easy t o forget that essentially nothing is knon-n about the low-temperature collisional properties of any other atomic or molecular species. One cannot expect theorists to relieve one's ignorance-inter-atomic potentials derived from room-temperature spectroscopy are not adequate t o allon- theoretical calculations of cold elastic and inelastic collision rates: even at the order-of-magnitude lei-el. Cold-atom spectroscopy can do the job (as n-e have seen u-ith lithium. sodium. and rubidium), but starting an experimental program to investigate the cold collisional properties of a nen- atom is a major and uncertain endeavor. In most cases, the easiest, way to discover if evaporation will n-ork for a particular atom or molecule is simply t o try it. Launching a major new project n-ithout any assurances of success is a daunting prospect: but we believe that, if one works hard enough. the probability that any given species can be evaporatively cooled to the point of BEC is actually quite high. In this section we n-ill revien- the scaling laws behind this optimistic assertion.
8'1. Collisional scaling predicts success! - The estensive literature on evaporation will not be revierx-ed here. See refs. [31,G5] for important early n-ork. The results of the earliest Monte Carlo trajectory simulations are presented in refs. [T4,218]. See ref. [22G] for a useful revien- paper. The literature can be summarized in the following requirement: In order for evaporative cooling to succeed, there must be a great many elastic collisions per atom in trap per lifetime of the atoms in the trap. Exactly what is meant by a "great many" depends on how evaporation is implemented, but a reasonable guess is "200" give or take a factor of four. Since the lifetime of the atoms in the trap is usually limited by collisions [229], the requirement can be restated: the rate of elastic collisions must be about two orders of magnitude higher than the rate of bad collisions. At sufficiently low collision energy, the cross-section for elastic collisions becomes energy-independent, and the rate per atom can be written ~
where n is the mean density, 0 is the zero-energy s-wave cross-section, and z, is the mean relative velocity. Assuming t h at atoms are trapped in a state for which there is no spinexchange, there are three bad collisional processes: background collisions, three-body recombination, and t,wo-body dipolar relaxation, with the per-atom loss rate respectively denoted by ?bg, "y3b and Y2b. Loss due to background collisions is t o first approximation
575 E S P E R I ~ I E N T SIN DILUTE ATOMIC
BOSE-EIXSTEIS COSDESSATION
57
independent of the cloud density and temperature. ( K e examine the breakdon-n of this approsinlation in sect. 7 above.) U'e write the rate as
n-here Q is proportional to an effective total residual pressure in the vacuum chamber. The constant of proportionality depends on the composition of the residual gas. The other tn-o rates are inelastic processes among the trapped atoms. At low energies these rates become independent of collision energy, and we can write
where 5'is the dipolar rate constant and X is the three-body rate constant. The total rate of bad collisional processes is ?bad = -]bg + *f3b + 7 2 b . For good evaporation? the three inequalities
not. >> 0. not.>>
noit
A12?
>> pi1
must be separately satisfied. Moreover. the inequalities must be satisfied not onl- initially (i.e. immediate$ after the atoms are loaded into the trap). but also as evaporation proceeds towards ever greater phase-space density. towards larger n and smaller 2.. K i t h respect to evaporating rubidium 87 and the lower hyperfine level of sodium 23. nature has been kind. The respective values of o are sufficiently high. and X and 3 sufficiently low. that the collisional rate inequalities (13b) and (13c) are routinely met along a reasonable evaporation path through n-2- space to BEC. One need *.only" arrange for the initial trapped cloud t o have sufficiently large n, and design a vacuum chamber with sufficiently small a. and evaporation works. The main point of this section. hon-ever. is that evaporation is likely to be possible even with less favorable values of o,d. and A. We deal n-ith the constraints (13) in order. 8'1.1. B a c k g r o u n d loss. The technical difficulties of ensuring that the elastic rate is much greater than the background loss rate can be formidable. All the same. eq. (13a) does not represent a fundamental collisional limit. for the simple reason that with sufficient laboratory effort Q can be made almost arbitrarily small. Lifetimes of lo4 s are attainable in cryogenic magnetic traps. Cryogenic experiments are notoriously difficult. but if one is willing t o go to the effort and the expense. one should ultimately succeed. The challenge is only t o satisfy (13a) at the beginning of evaporation. Once efficient evaporation is established. nt. increases. while a remains the same (except see sect. 7 above).
576
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E. A . CORSELL,J . R. EYSHERand C. E. VIEM MAY
8'1.2. T h r e e - b o d y r e c o m b i n a t i o n . It is unlikely that X could ever be so large that three-body recombination is a problem when the atoms are first loaded into the trap. -4s evaporation proceeds. however. density increases and velocity decreases. so that inequalit!, (13b) may well fail. if one is working with a species with small c or large A. The solution lies in manipulating the trapping potential. -4diabatically ramping the mean harmonic confinement frequency ut has no effect on the phase-space density-t he ramp takes atoms neither closer to nor further from the BEC transition. The mean density and mean velocity are both affected by the ramp, however, such that 73b/Te = An2/(17cv)x at. Therefore, as long as one can continue t o turn down the confining strength of one's trap. one can ensure that (13b) remains satisfied all the way to the BEC transit ion. 8'1.3. D i p o l a r r e l a x a t ion. -4s with three-body recombination. if dipolar relaxation is to be a problem, it will likely not be until late in the evaporative process when the decrease in velocity may threaten the validity of (13c). -4diabatically turning down the confining potential is not helpful. because the scaling with ut has the opposite sign: 72b/?e 3: ut-lI2.Evidently one can win by turning up the trap. 11-hile this may be a useful tactic in a particular laboratory situation, it is unsatisfactory in terms of our "Esistence Proof" for evaporation. One may always make a trap weaker, but there is likely to be an engineering-based upper limit to trap strength, and if 3 is particularly large, a problem can remain. More fundanlentally. if one is cursed with a species for which both f and X are large, there may be no value of ut for which (13b) and (13c) can be simultaneously satisfied in a cloud near degeneracy. Fortunately, one is not required to accept the value of ,8 nature provides. In fact, all one really has to do is to operate the magnetic trap with a very low bias field. Below a threshold field of perhaps five Gauss, the value of ,8 for the lowest hyperfine level drops rapidly to zero. This behavior is simple to understand. -4t low temperature, the incoming collisional channel must be purely s-wave. Dipolar relaxation changes the projection of spin angular momentum, so to conserve angular momentum the outgoing collisional channel must be at d-wave or higher. The nonzero outgoing angular momentum means that there must be an angular-momentum barrier in the effective molecular potential,. a barrier which rises a few hundred microkelvin above the r = 00 limit. This barrier is irrelevant for atoms trapped in the stretch-state, for instance (F = 2 , 7 7 1 ~= 2 in rubidium 87). Dipolar relaxation releases the (relatively large) hyperfine energy, and the outgoing atoms barely feel the angular-momentum barrier. But if the atoms are ~ -1, in rubidium 87) the outgoing energy is trapped in the lower state ( F = 1 , m = only the Zeeman energy in the trapping fields. For fields below about 5 G, this energy is insufficient to get the atoms back out over the angular-momentum barrier. If relaxation is t o occur, it can happen only at inter-atomic radii larger than the outer turning point of the angular-momentum barrier. For smaller and smaller fields, the barrier gets pushed further out, with correspondingly lower transition rat.es [230]. This effect turns out t o be irrelevant in atomic hydrogen, since the lonver hyperfine state is F = 0, and is not magnetically trappable. In most magnetically trappable atoms and molecules, however,
577 E X P E R I M E S T S IN DILUTE ATOMIC
BOSE-EINSTEIN CONDENSATION
59
the lowest state has F # 0, and one of the Zeeman sublevels will have no spin-exchange. and suppressed dipolar relaxation. To summarize this subsection, given i) a modestly flexible magnetic trap, ii) an arbitrarily good vacuum, iii) a true ground state with F # 0 and S # 0 and iv) nonpathological collisional properties, almost any species can be successfully evaporated to
BEC. 8'2. Except of course . . . . - The assumption of "nonpathological properties'' may actually be fairly constraining. For example: i) Rb-87 has a d-wave resonance [231]. ii) Cs-133 appears t o have a zero-field Feshbach resonance [232]. iii) The elastic crosssection of Rb-85 has a pronounced energy dependence and drops to near-zero at a very inconvenient temperature [197,79]. One is forced to confront the fact that at least in the alkalis, pathology seems almost to be the rule. Also, for technical reasons having to do with low-frequency electronics noise, it can be inconvenient to trap atoms at fields belom- about 0.5 G, which may not always be lon- enough to completely suppress dipolar relasation. Also the scaling arguments discussed above assume T-independent elastic cross-section; optical molasses temperatures are not always low enough to ensure that one is in this regime. Further, we have completely neglected the issue of heating. Therefore we invite you to adopt an appropriately skeptical attitude towards the universality of the arguments outlined in this section. 8'3. But on the other hand . . . . - Requiring that a particular atomic or molecular species be able, on its own, to cool to BEC is unnecessarily restrictive. Sympathetic cooling has been shoivn to work well. -4s far as we know, thorough modeling of the collisional requirements for sympathetic cooling has not been performed. but simple estimates suggest that in the presence of a robustly eyaporating working fluid, the collisional requirements on the species to be cooled: and on interspecies collisions, are d o m bv a factor of ten or more from those outlined in the first paragraph of subsect. 8'1. Once you have cooled to the BEC transition. your ability to accumulate a significant number of atoms in the condensate depends on the scattering length's being positive. Here again. even in total ignorance. your odds are pretty good: see the argument in the footnote [ g i ] .
* * * We acknon-ledge support. from the Sational Science Foundation? the Office of Saval Research. and the National Institute of Standards and Technology. We have benefited enormously from ongoing discussions Kith other members of the JIL-4 BEC collaboration.
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Independently of the details of the short-range pot.entia1, any molecular pot.entia1 possessing a sufficiently strong l / r 6 potential at long range has only 25% probability of having a negative scattering length. Even neglecting the fact that there are two magnetically trappahle st,ate for each of the three species, t,he probability of all three species having negative scattering lengths is only 1/64. See GRIBAKIS G. F. and FLAMBAUSI 1 '. I:., Phys. Rev. A: 48 (1993) 346. -4ct,ually, we also had at JILA in 1991 an unpublished, in-house esti1nat.e that in a harmonic ext.erna1 potential. a species with a negative scatt.ering length could form a stable condensat,e as long as the number of at.oms in the condensate was not much larger than th e ratio of th e absolute value of the scatt.ering length t.o the size of the noninteracting ground state of the c.ondensate. We t.hought such a condensate would still be interesting, and so it has turned out. See ref. [l53] and references therein. PETRICH W., AXDERSON 11. H., ENSHER J. R. and CORXELL E. X., J . Opt. SOC.Am. B, 11 (1994) 1332. TOWNSEND C. G. et al., Ph.ys. Rev. A , 5 2 (1995) 1423. JOFFEM., Ph.D. thesis, hlIT (1991). TOLLETT J. J., BRADLEY C.C., SACKETT C. A . and HULETR. G.,Phys. Rev. A , 51 (1995) R22. WILLEMS P.A . and LIBBRECHT K. G., Phys. Rev. A , 51 (1995) 1403. PRITCHARD D. et al., in Proceedin,gs of the 11th International Conference on Atonzic Ph.ysics, edited by S . HAROCHE, 3. C. GAY and G. GRYKBERG (World Scientific, Singapore) 1989, pp. 619-621. MARTIN A . G . et al., Phys. Rev. Lett., 61 (1988) 2431. PETRICH TI;., ASDERSON31. H., ENSHER J. R.. and CORNELL E. X., Phys. Rev. Lett., 74 (1995) 3352. ADAMS c. e f a!., Ph.ys. Rev. Lett., 74 (1995) 3577. HVLETR. G., Bdl. Am. Ph.ys. SOC.,40 (1995) 1267.
s.
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ANDERSOXM. H., private communication (1995). E., ~ u o u oCzmento, 8 (1931) 107. ANDERSOS M. H. et al., Scieme, 269 (1995) 198. DAVISK. B. et al., Phys. Rev. Lett., 75 (1995) 3969. C. C., SACKETTC. A . , TOLLETT J. J. and HULETR. G., Phys. Rev. Lett., BRADLEY 75 (1687) 199.5; 79 (199T) 1170. [113] MEWEShi.-0. et al., Phys. Rev. Lett., 77 (1996) 416. [114] At the DAMOP meeting in hlay of 1995, the JILA group presented an image taken of a very small, very cold cloud. The image consisted of a distorted diffraction pattern consisting of apparent “negative absorption” and surrounding rings. No attempt was made to present an interpretation of this cloud, beyond identifying it as being smaller than resolution and thus presumably very cold. A month later, after our gravity-compensated expansion imaging had evolved to the point where we could unambiguously identify condensates, we re-examined the data from May. Reviewing the evaporation protocols that we used, we could infer that some of the distorted images we took had been of Bose condensed clouds, and some had not. Inferring the existence of BEC from diffraction patterns is a risky business. [I151 SHAPIRO l’., Phys. Rev. A , 4 7 (1996) R1019. T . , YEH J. R. and KNIZER. J., Opt. Commun., 114 (1995) 421. Ill61 TAKEKOSHI [117] TAKEKOSHI T. and KNIZER. J., Opt. Lett., 21 (1996) 77. [118] STAMPER-KURN D. R l . et al., Phys. Rev. Lett.. 80 (1998) 2027. I. G., Phys. Rev. Lett., 80 (1998) 615. (1191 HIXDSE. A . , BOSHIERM. G. and HUGHES 11201 FRIED D. G . et al., e-print physics/9809017, to be published in Phys. Rev. Lett. (1998). [121] ARNDT&I. et al., Phys. Rev. Lett., 79 (1997) 625. [122] SODIKGJ. et al., Phys. Rev. Lett., 80 (1998) 1869. [123] GU~R\.-ODELIX D., SODISGJ., DESBIOLLES P. and DALIBARD J.. Optics Ezpress. 2 (1998) 3-33. (1241 ARLTJ. et al.. J . Phys. B. 31 (1998) L321. [125) ARIMOSDOE., private communication (1998). [136] FORTC. et al.. Euro. Phys. J . D.3 (1998) 113. [127] PREVEDELLI hf. et al.. Phys. Rev. A , 59 (1999) 886. 3f., STOOFH. T. C.. LICALEXANDER W. I. and HULETR. G., Phys. Reif. (1281 HOUBIERS A . 57 (1998) R1497. [129] CATALIOTTI F. S. et al.. Phys. Rev. A , 57 (1998) 1136. B. and JIS D. S.. Phys. Rev. A , 58 (1998) 426’7. [130] DEMARCO [131] KIM J. H. et al., Phys. Rev. Lett., 78 (199;) 3665. [132] ti-EINSTEIN J. D. et a!., Phys. Rev. A . 57 (199s) R3173. [133] ’CI‘EISSTEIX J. D. et al., Nature, 395 (1998) 148. [134] HAUL. 1-.et al., Phys. Reu. A , 58 (1998) R54. [135] A S D E R S O S B. P. and KASEVICH hI. A . , Bull. Am. Phys. Soc., 43 (1998) 1251. [136] HAS D. J., WYNAR R. H., COURTEILLE P. and HEINZEN D. J., Phys. Rev. A , 57 (199s) R1114. I1371 ERNSTL-. et al., Europhys. Lett.. 41 (1998) 1. [138] ESSLINGER T., BLOCH I. and HANSCHT. W., Phys. Rev. A , 5 8 (1998) R-3661. L. et al., Bull. Am. Phys. Soc.. 43 (1998) 1379. (1391 DENG [140] DALIBARD J., this Volume. p. 321. [I111 ASPECT-I.* this Volume. p. 503. [l12] ERTMERJV.,private communication (1998). [143] WILSOSX., private communication (1998). [I111 BOSHIER hi.. private communication (1998). [lo81 [log] [110] [111] [I121
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mat/9806038. PARKISS .I\. S. and \I-.ILLS D. F.. Phys. Rep.. 303 (1996) 2. GRIFFIX-4.. this lblume. p. 591. FETTER -4.L.. this l'olurne. p. 201. BI'RYETTI<.. this l'olume. p. 265. LOI'ELACER . 1%.E. and TOMMILA T. J.. Phys. Rezi. A . 35 (1987) 359T. J I N D. S. ~t al.. Czech. J . Phys.. 46 (3070) 1996. suppl. S6. [161] HOLLAND 31. J.. J I XD. S.. CHIOF.%LO 11. L. and COOPER J.. Phys. Rev. Lett.. 78 (1997) 3801. [165] .I\NDRE\VS1f. R . et al.. Phya. Rev. Lett.. 79 (1997) 553: 80 (1998) 2967. [166] ~IATTHEIVS 11. R . ~t ~ 1 . .Phys. Re?,. Leff.. 81 (1998) 213. [167] LA\$'C . I<.. PC H . BIGELOIV S . P. and E B E R LJ. ~ ' H.. Phys. Rezi. Letf.. 79 (1997) 3105. [168] HALLD. S. et al.. Phys. Rev. Lett.. 81 (1998) 1539. I ' SR., V-IEXIX C. E. and CORNELL E. .I\.. Phys. Rev. Left.. [169] HALLD. S.. ~ I A T T H E ~11, 81 (1998) 1543. [ I i O ] CORSELLE. -4.. HALLD. S.. MATTHEIVS >I R. . and I?-IEI\IANC. E.. J . Low Temp. Phys.. 113 (1998) 151. [ l T l ] lVhile it has been argued that such systems are not true mixtures, the distinction turns out t o be largely academic. except in special cases. [I721 Ho T.-L. and SHENOI'1'. B., Phys. Rev. Lett., 77 (1996) 32TG. [1i3] ESRYB. D.. GREENEC. H.. BL-RKE J. P. jr. and BOHNJ. L., Phys. Rev. Lett.. 78 (1997) 3594. I1741 SINATRA .I\. et a!., e-print condmat/9809061. [li5] XIE\VESM.-O. ct ~ l . Phys. , Rev. Lett., 78 (199i) 582. [li6] ZHANGlV. and W ~ L LD.S F., Phys. Rev. A , 57 (1998) 1248. [lii] KETTERLE W., this l'olume, p. 137. [1i8] HULETR. G.. private communication (1998). [1i9] HOLZMASN hl., GRUTERP. and LALOEF., e-print cond-mat/9809356. [180] ENSHER J. R. et al., Phys. Rev. Lett., 77 (1996) 4984. 11. R. et al., Sczence, 275 (1997) 63'7. [181] ANDREWS [182] KETTERLEW. and MIESSERH.-J., Phys. Rev. A , 56 (1997) 3291. [183] BAGNATO V. S., I i L E P P N E R D. and PRITCHARD D. E.. Phys. Rev. A , 35 (1987) 4354 lV., Phys. Reu. Lett., 77 (1996) 3695. [184] KRAUTH 1185) B I J L S ~ 11. I A and STOOFH. T. C., Phys. Rev. A . 54 (1996) 5085. [18G] HOVBIERS If.,STOW H.T. C. and CORSELLE. A . , Phys. Rev. A , 56 (1997) 2041. [18i] HOLZMANS If., ICR.4UTH 17'. and NARASCHEWSKI 31.. e-print cond-mat/9806201. [188] h h N G U Z Z 1 -4.) CONTI s. and T O S l h l . P.. J . Phys. Condens. Matter. 9 (199T) L33. [l58] [l59] [lGO] [161] [162] [163]
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ADASIS C‘. S., Phys. Rev. A. 55 (199i) R252i. GIORGIXI S., PITAE\‘SKII L. P. and S T R I N G A R I S., Phys. Rev. Lett., 78 (1997) 3957. HALL. lr..private communication (1998). G ~ o s s a r ~ Ss .r and HOLTHAWhI., Phys. Lett. A , 208 (1995) 188. KETTERLE Q-. and VAN DRVTEN X. J., Phys. Rev. A . 5 4 (1996) 636. TIESISGA E.,hfOERDIJE: A. J., I’ERHAAR B. J. and STOOFH. T. C., Phys. Rev. A . 46 (1992) R1167. [195] TIESIXGA E., I‘ERHAAR B. J . and STOOFH. T. C., Phys. Rev.A , 47 (1993) 4114. [196] ISOVYES. et al., Nature, 392 (1998) 151. [19i] COVRTEILLE P. et al., Phys. Rev. Lett., 81 (1998) 69. [198] J I XD. S. e t al., Phys. Rev. Lett., 77 (1996) 420. I1991 S T A ~ I P E R - K UD. R NhI. et al., Phys. Rev. Lett.. 81 (1998) 500. [200] MEWEShf.-O. et al., Phys. Rev. Lett., 77 (1996) 988. (2011 JIS D. S. et al., Phys. Rev. Lett.. 78 (199i) i64. [202] L I I ~ V . Y., Phys. Rev. Lett., 79 (1997) 4056. [203] F E D I C HP E.~O., ’ SHLYAPXIKOV G . IT.and II-.ALRA\.EN J . T. hl.. Phys. Rev. Lett.. 80 (199s) 2269. [204] ENSHER J. R.. Ph.D. thesis, Vniversity of Colorado, Boulder, Colorado (1998). F., hfISNIT1 C. and PIT.AEVSKII L. P., Phys. Rev. A , 56 (1997) 4555. [205] DALFOVO [206] 1.. CASTISR. D., Phys. Rev. Lett., 79 (1997) 3553. [207] ~ I O R G AS.NA . . BALLAGH R. J . and BCRSETT I<.. Phys. Rev. A. 55 (1997) 4335. [208] PIT;\E\’SKII L. and STRINGARI S.. Phys. Reu. Lett., 81 (1998) 4541. [209] AXDERSOSB.. private communication (1998). I<.. private communication (1991). [210] HELLIERSOS [211] GL.\VBER R . J.. Phys Rezr.. 131 (1963) 2766. [212] Only slightly more complicated! If we assume the vortex has winding number R and that its core lies along the &xis of symmetry of a condensate, and if n-e use cylindrical . cSo(zl..r~) = n(81 - 82). coordinates x L = ( r l . O I . z 1 )then “131 J.\\-.IS.AISESJ . and 1-00 S. 11.. Phys. Re[*.Lett.. 76 (1996) 161. [211] C A S T I S E’. and D.ALIB.IRD J . , Phys. Rev. A . 55 (1997) 4330. [215] LEGGETT-4.J.. in Bose-Einstein Condensafron. edited by A . GRIFFIS.D. I!-. S S O K E and S. STRINGARI (Cambridge 1-niversit? Press. Cambridge) 1995. Chapt. 19. pp. 452462. [216] GR.AHAIIR., Phys. Rev. Lett.. 81 (1998) 5262. [217] >fY.lTT C. J.. Ph.D. thesis. Cniversity of Colorado (1997). [218] MOSROEC. R . . Ph.D. thesis. Pniversity of Colorado (1992). J. E.. Phys. lieu. A . 56 (1997) R109.J. [219] S.-II-.IRDT. A . . O’HARAK. &I. and THOMAS [220] HELBISGR. and PACLY H.. 2. Phys., 179 (1961) 16. [221] ASDERSOSR., J . Chem. Phys., 60 (1974) 2610. [222] I i L E I S E K A T H O F E R t-.et al.. Chem. Phys. Lett.. 249 (1996) 257. [223] BVRKE J.. private communication (1998). [221] Dutch astrophysicist Jan Oort proposed the existence of a vast. extremely disperse. essentially undetectable cloud of comets located far outside of the orbit of Pluto. but still within the solar gravitational well. See OORT J. H.. Bull. Astron. Inst. Pv‘eth.. 11 (1950) 91. [%] The existence of hot. diffuse cloud of trapped atoms has been discussed in refs. [113.11S. 14S.217]. (2261 V.IX DRL-TESX. J. and KETTERLE TI-.. Adr =It. illol. Opt. Phys.. 37 (1996) 381. I2271 BALIS., O’HARAK. M..G E H ~11. I E.. GR.IS.\DE S. R . and THOLI.\S J . E..to be published in Phys. Rev. A .
[I891 [190] [191] [I921 [193] [I911
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[2%] HEINZEN D.: this l'olume, p. 351. [229] Vsually, but not always. L-nless Care is taken in the experimental set-up. lifetimes can be limited by absorption of stray scattered laser light, or by unwanted radio-frequent? magnetic noise. [230] The argument in this paragraph is due to MOSROEC. and CORNELL E. The results have been numerically validated by B. I'ERHAAR'S group. [231] BOESTEN H. 3.1. J. 15. et al.. Phys. Rev. -4.55 (1997) 636. [232] I
VOLUME 83, NUMBER 17
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PHYSICAL REVIEW LETTERS
Watching a Superfluid Untwist Itself: Recurrence of Rabi Oscillations in a Bose-Einstein Condensate M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, M. J . Holland, J. E.Williams, C. E. Wieman, and E. A. Cornell" JILA. National Institute oj' Standards and Technalog,v and Departinerit University of Colorado, Boulder, Colorado 80309-0440 (Received 16 June 1999)
of' Physics,
The order parameter of a condensate with two internal states can continuously distort in such a way as to remove twists that have been imposed along its length. We observe this effect experimentally in the collapse and recurrence of Rabi oscillations in a magnetically trapped, two-component Bose-Einstein condensate of "Rb PACS numbers: 03.75.Fi, 42.50.Md, 67.57.Fg, 67.90.+r
Quantization and persistence of current in superconductors and superfluids can be understood in terms of the topology of the order parameter. Current arises from a gradient in the phase of the order parameter. Quantization of flow around a closed path is a consequence of the requirement that the order parameter be single valued; metastability, or "persistence," arises from the fact that the number of phase windings (the multiple of 277- by which the phase changes) around the path can be changed only by forcing the amplitude of the order parameter to zero at some point. If the energy this requires exceeds that available from thermal excitations, the current will then be immune to viscous damping. This familiar argument relies, however, on the order parameter's belonging to a very simple rotation group. The order parameter in superfluid 4He, for example, is a single complex number. Its phase can be thought of as a point lying somewhere on a circle, and is subject to the topological constraints mentioned above. The order parameter of a more complicated superfluid, on the other hand, will in general be capable of ridding itself of unwanted kinetic energy by moving continuously through a higher-dimensional order-parameter space in such a way as to reduce, even to zero, its winding number. Presumably this ability will reduce a superfluid's critical velocity; in the limit that the order-parameter space is fully symmetric, the critical velocity may even vanish [ 11. In this paper, we discuss experiments on a gas-phase Bose-Einstein condensate with two internal levels [2]. This is equivalent to a spin- 1/2 fluid: the order parameter has SU(2) rotation properties. A differential torque across the sample is applied to the order parameter so that with time it becomes increasingly twisted. Eventually the sample distorts through SU(2) space so that the steadily applied torque now has the effect of untwisting the order parameter, which returns nearly to its unperturbed condition. The pattern of twisting and then untwisting is manifested experimentally as a washing out followed by a recurrence of an extended series of oscillations in the population between the spin states. Related behavior has been previously observed in A phase 3He [3]; a major difference in 3358
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this paper is that we can directly observe components of the order parameter with temporal and spatial resolution. Magnetically confined 87Rbcan exist in a superposition of two internal states, known as 11) and 12) [4]. The two internal states are separated by the relatively large S7Rb hyperfine energy, but in the presence of a near-resonant coupling field the states appear, in the rotating frame, to be nearly degenerate. The condensate can then dynamically convert between internal states. The order parameter for the condensate is the pair of complex field amplitudes and @2 of states 11) and 12). Evolution of these fields is governed by a pair of coupled Gross-Pitaevskii equations which model the coupling drive, the external confining potential, kinetic energy effects, and mean-field interactions [5-71. The SU(2) nature of the order parameter (Q,, @ 2 ) is more evident if we write
and where 0, 4 , n t , and (Y are purely real functions of space and time. 0 and 4 give the relative amplitude and phase of the two internal components, and may be thought of, respectively, as the polar and azimuthal angles of a vector whose tip lies on a sphere in SU(2) space. The total density and mean phase, nt and a , respectively, remain relatively constant [8] during the condensate evolution described in this paper. The apparatus has been previously described [5,6,9]. The starting point for the measurements is a magnetically confined cloud of -8 X lo5 evaporatively cooled, BoseEinstein-condensed 87Rb atoms near zero temperature. The combined gravitational and magnetic potentials [ 101 yield an axially symmetric, harmonic confining potential V1 (V2) for particles in the 11) (12)) state, in which the aspect ratio of the axial oscillation frequency in the trap to the radial frequency w z / w , can be varied from 2.8 to 0.95 [ l l ] . V] and V2 are nearly identical but can optionally be spatially offset a distance zo in the axial direction [12].
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The effect of the coupling drive is to induce a precession of the order parameter at the local effective Rabi frequency = [ ~ ( z +) ~~( z ) ’ ] ’ ’ ’ . In a preliminary experinearly uniment, we chose parameters so as to make form, with w : = 2n- X 63 Hz,w,- = 2n- X 23 Hz, 2~ X 340 Hz, and 6 ( z ) 0. A condensate at near-zero temperature was prepared in the pure 11) state. The coupling drive was then turned on suddenly, inducing an extended series of oscillations of the total population from the 11) to the 12) state (“Rabi oscillations”) [Fig. 11. The robustness of the Rabi oscillations is proof that our imaging does not significantly perturb the quantum phase of the sample [ 161 (population transfer via Rabi oscillations is phase sensitive). If there is an axial gradient to fie,,, then a relative torque is applied to the order parameter across the condensate, which can cause a twist to develop along the axial direction. If we naively model the sample as a collection of individual atoms, each held fixed at its respective location, then the order parameter at each point in space rotates
aeff(:)
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FIG. I . (a) With the trap parameters adjustcd for high spatial unifonnity in fit,,, we drive the coupling transition and rccord a streak-camera image of 60 Rabi oscillations between the 11) (white) and (2) (black) states. The vertical dimension of the figure is 80 pin. (b) The value of S N , the total number of atoms in 12) minus the total in II), is extracted from the image in part (a). The contrast ratio remains near unity; the observed loss of signal is due to overall shrinkage of’ the condensate through collisional decay.
The coupling field has a detuning 6 from the local 11) to 12) resonance. If zo is nonzero, 6 depends on the axial position z, with 6 ( z ) - S ( z = 0) linear in z and in zo. The strength, characterized by the Rabi frequency R , of the coupling field also varies with an axial gradient [ 131. We are able to measure the population of both spin states nondestructively using phase-contrast microscopy [ 14,151. W e tune the probe laser between the resonant optical frequencies for the 11) and 12) states. Since the probe detuning has an opposite sign for the two states, the resulting phase shift imposed on the probe light has an opposite sign, such that the 11) atoms appear white and the 12) atoms appear black against a gray background on the CCD array. We can acquire multiple, nondestructive images of the spatial distribution of the 11) and 12) atoms at various discrete moments in time, or we can acquire a quasicontinuous time record (streak image) of the difference of the populations in the 11) and 12) states, integrated across the spatial extent of the cloud. FIG. 2. (a) We represent the polar-vector order parameter as an arrow in these simulations. The angle 0 from the vertical axis determines the relative population, and the azimuthal angle 4 is the relative phase of states 11) and 12) [see Eqs. (l)]. Each column in the arroy array is at fixed time, and each row at fixed axial location. R is perpendicular to the plane of the page, so that a uniform, on-resonance Rabi oscillation would correspond to all the arrows rotating in unison, in the plane of the image. The tips of all the arrows are (on the relatively fast time scale of Ocff)tracing out circles nearly parallel to the plane of the page (in our rotating-frame representation, small excursions out of the page are a consequence of finite detuning). In the figures, we “strobe” the motion just as the central arrow approaches vertical, to emphasize the more slowly evolving “textural” behavior. (b) The total density of the condensate n, maintains a Thomas-Fermi distribution (integrated through one dimension, as imaged) and changes only slightly during the evolution of the cloud. (c) In a simple model of individual, fixed atoms, a continuous inhomogeneity in will cause the Rabi oscillations in 6 N to wash out. (d) When a condensate is simulated [ 191, the kinetic energy causes the order parameter to precess through the full SU(2) space, coming out of the page to cast off the winding and thus reduce its kinetic energy. (e) The corresponding plot of 6 N shows that, when the arrows are once more aligned, the Rabi oscillations recur.
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independently at the local effective Rabi frequency, !&,(. In Fig. 2(a) we see the implications of the “fixed-atom’’ model for R = 2~ X 700 Hz and 6 ( z ) = 2 7 ~X (100 14:) Hz with z in microns: a twist develops in the order parameter which leads to a washing out of the Rabi oscillations [Fig. 2(c)]. In contrast, the kinetic energy provides stiffness for a tme condensate. Simulation of the condeiisate [ 171 shows that, for early times, the order parameter begins twisting, as in the fixed-atom model, but the twisting process self-limits about 40 ms into the simulation. At this point, there is nearly a full winding across the condensate. Thereafter, although the two ends of the order parameter continue to twist with respect to one another, the order parameter has been sufficiently wrapped around the SU(2) sphere so that the effect of further torque is to return the condensate close to its unperturbed condition [Fig. 2(d)]. The Rabi oscillations exhibit a corresponding revival [Fig. 2(e)]. The factor driving the untwisting process is the increasing kinetic energy cost associated with an increasing twist in the order parameter. For a simple U( 1) order parameter, continuously increasing the winding ultimately results in a “snap,” in which the order parameter is driven to zero and a discontinuous (and presumably dissipative [ 181) process releases the excess windings. The revival in the present case is made possible by the larger rotation space available to a two-component cloud.
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Under experimental conditions similar to those of the siinulations in Fig. 2, we have observed as many as three complete cycles of Rabi-oscillation decay and recurrence. These data appear in Ref. [19]. In this paper we present data which correspond to the case of a more vigorous twisting. We increase the axial dimension of the condensate cloud by a factor of 4; the kinetic energy cost of twisting the condensate is correspondingly lower, so that at the point in time when the condensate is maxiinally distorted there are four windings across the cloud. The parameters of the experiment were as follows: w,. = w y = 2 v X 7.8 Hz and mean fl,ft = 2~ X 225 Hz. There was a gradient in both 6 and R across the 54 pin axial extent of the cloud, resulting in a - 2 X ~ 60 Hz difference in a,,, fi-om the top to the bottom of the condensate. The result of the experiment is seen in Fig. 3. The observed recurrence of the Rabi oscillations at 180 ins [Fig. 3(a)], when corrected for overall decay of the cloud, corresponds to 60% contrast. We find it remarkable that the distorted order-parameter field seen in Fig. 3(b) at times 65 and 75 ms should find its own way back to a nearly unifoim configuration. The simulations qualitatively reproduce the integrated number and state-specific density distributions observed in the experiments. For large inhomogeneity in a,,,, however, the simulations predict the development of smallscale spatial structure not observed in the experiment.
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FIG. 3. A condensate with large axial extent undergoes twisting. (a) The streak-camera data show a rapid decay in the Rabi oscillation in the integrated popidation difference, from full contrast at t = 0 to near zero by t = 20 ms. The oscillations recur at 180 ms. (b) Individual phase-contrast images (at distinct moments in time) of the spatial distribution of (I)-state atoms show that the spatially inhomogeneous Rabi frequency is twisting the order parameter, cranking successively more windings into the condensate until, by -75 ms, four distinct windings are visible. Further evolution results not in more but in fewer windings until, at time 180 ms, the order parameter is once more uniform across the cloud. Each image block is 100 p m on a side, and the probe laser is tuned much closer to the 11) state than to the 12) state. (c) The numerical simulation reproduces the qualitative features of the corresponding experimental plot (a). The simulation used 6 ( z ) = 0, n ( z = 0) = 257 X 225 Hz and a 2 n X 60 Hz spread in 0 across the extent of the condensate.
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The simulations contain no dissipation, whereas finitetemperature damping may occur in the experiment [20] Heuristically, what value do we expect for the recurrence time f,,,,, for the data in Fig. 3? The difference in a,,, from the top to the bottom of the condensate is about 60 Hz. From the data in Fig. 3( b) we see that the recurrence occurs only after four windings have one by one been twisted in and then twisted out of the condensate. A rough estimate then would be rrccur = (4 + 4)/60 Hz = 133 ms, shorter than the observed value of 180 ms, but reasonable given that edge effects have been neglected. An interesting theoretical challenge would be to develop simple arguments that would allow an a yriori prediction of the spacing of the windings at the instant of maximum twist. Also, for particularly strong torques, should not one expect suppression of the total density, rather than continuous evolution through SU(2) space, to be the preferred mode of relieving accumulated stress? We have observed that the presence of the coupling drive need not result in population oscillations. For any given frequency and coupling strength, there are two steady-state solutions which are completely analogous to the dressed-state solutions of the single-atom problem [21]. We have been able experimentally to put the condensate in such states via an adiabatic process: the strength of the drive is increased gradually from zero, and the frequency (initially far detuned) is gradually ramped onto resonance. The resulting “dressed condensate” is extremely stable-after the ramp is complete, the cloud remains motionless in a near-equal superposition of the two bare states. The twisting experiment discussed in this paper can be thought of as the evolution of a highly nonlinear superposition of the two dressed states. It would be interesting to work in the opposite limit and explore the spectrum of small-amplitude excitations on a dressed state. Analogies to (i) the transverse zero sound in the He3 B phase [22] and (ii) spin waves in spin-polarized atomic hydrogen [23] should also be studied [24]. The ability to fundamentally alter the topological properties of a condensate has already proven useful. With the coupling drive off, the SU(2) properties vanish, and states 11) and 12) become distinct species, each forced to live in its separate U(1) space. With a variation [25] on this technique we have created a vortex-state condensate [26]. We acknowledge funding from the ONR and the NSF, and useful conversations with Jason Ho and Seamus Davis.
*Quantum Physics Division, National Institute of Standards and Technology. [ 11 P. Bhattacharyya, T. Ho, and N. Mermin, Phys. Rev. Lett. 39, 1290 (1977); T.-L. Ho, Phys. Rev. Lett. 49, 1837 (1982). [2] For other recent experimental work on multicomponent Bose-Einstein condensate, see J. Stenger et al., Nature
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(London) 396, 345 (1998); H.-J. Miesner et al., Phys. Rev. Lett. 82, 2228 (1999). [3] D. Paulson, M. Krusius, and J . Wheatley, Phys. Rev. Lett. 37, 599 (1976). [4] The states I I) and 12) correspond to the two Zeeman sublevels, F = 1,177 = - 1, and F = 2,111= 1, respectively. For further details on the system, see Ref. [9]. [5] E.A. Cornell, D. S. Hall, M.R. Matthews, and C.E. Wienian, J . Low Temp. Phys. 113, 151 (1998). [6] D.S. Hall, M. R. Matthews, C.E. Wienian, and E.A. Cornell, Phys. Rev. Lett. 81, 1543 (1998). [7] R. DLIIII,J . I. Cirac, M. Lewenstein, and P. Zoller, Phys. Rev. Lett. 80, 2972 (1998); R. J . Ballagh, K. Burnett, and T.F. Scott, Phys. Rev. Lett. 78, 1607 (1997); K.-P. Marzlin, W. Zhang, and E. M. Wright, Phys. Rev. Lett. 79, 4728 (1997); J . Williams et ul., Phys. Rev. A 57, 2030 (1998). [8] D.S. Hall ef a/., Phys. Rev. Lett. 81, 1539 (1998). [9] M. R. Matthews eta/., Phys. Rev. Lett. 81, 243 (1998). [lo] W. Petrich, M.H. Anderson, J . R. Ensher, and E.A. Cornell, Phys. Rev. Lett. 74, 3352 (1995). [ l 11 J . R. Ensher, Ph.D. thesis, University of Colorado, 1998. [ 121 D. S. Hall et a/., cond-niat/9903459. [ 131 The microwave coupling drive is actually accomplished by a two-photon transition. The quoted values of R and 6 refer to the combined effect of a two-frequency drive (see Ref. [9]). Since R depends on the intermediate-state detuning, it can in general have a spatial dependence due to Zeeman shifts. f1 also depends on the (poorly characterized) polarization of the coupling fields, whose projection onto the local quantization axis varies weakly across the cloud. In our comparisons with theory, the gradient in R is an adjustable parameter. [I41 F. Zemike, in Nobel Lectures, Physics (Elsevier, Amsterdam, 1964), p. 239 (text of 1953 Nobel Prize lecture). [15] M. R. Andrews et al., Science 273, 84 (1996). [16] The probe beam does impose a small, reproducible shift on the energy difference between the 11) and 12) states, a measurable shift for which we correct. [17] The simulation is a numerical integration of the coupled Gross-Pitaevski equations in [6]. See [ 191 for more detail. [18] K. Schwab, N. Bruckner, and R. Packard, Nature (London) 386, 585 (1997); 0. Avenel, P. Hakonen, and E. Varoquaux, Phys. Rev. Lett. 78, 3602 (1997). [I91 J. Williams, R. Walser, J. Cooper, E. A. Cornell, and M. Holland, cond-mati9904399. [20] P. 0. Fedichev, G.V. Shlyapnikov, and J . T. M. Walraven, Phys. Rev. Lett. 80, 2269 (1998); L.P. Pitaevskii and S. Stringari, Phys. Lett. A 235, 398 (1997); W.V. Liu, Phys. Rev. Lett. 79, 4056 (1997). [21] P. B. Blakie, R. J. Ballagh, and C. W. Gardiner, cond-mat/ 99021 10. [22] Y. Lee, T.M. Haard, J.B. Kycia, J.A. Sauls, and W.P. Halperin, Nature (London) 400,43 1 (1999). [23] N. Bigelow, J. Freed, and D. Lee, Phys. Rev. Lett. 63, 1609 (1989). [24] Obviously, oscillatory behavior in coupled spin systems is a recurring theme. At present it is unclear to us how deep the analogy runs. [25] J. E. Williams and M. J. Holland, cond-mati9909163. [26] M.R. Matthews et al., Phys. Rev. Lett. 83, 2498 (1999).
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Vortices in a Bose-Einstein Condensate M. R. Matthews, B. P. Anderson,* P. C. Haljan, D. S. Hall,+ C. E. Wieman, and E. A. Cornell* JILA, National Institute of Standards and Technology and Department of Physics, University of Colorado, Boulder, CoIorado 80309-0440 (Received 6 August 1999)
We have created vortices in two-component Bose-Einstein condensates. The vortex state was created through a coherent process involving the spatial and temporal control of interconversion between the two components. Using an interference technique, we map the phase of the vortex state to confirm that it possesses angular momentum. We can create vortices in either of the two components and have observed differences in the dynamics and stability. PACS numbers: 03.75.Fi, 42.50.Md, 67.57.Fg, 67.90.+2
The concept of a vortex is at the center of our understanding of supeduidity. A vortex is a topological feature of a superfluid-in a closed path around the vortex, the phase undergoes a 2 7 ~ winding and the superfluid flow is quantized. Following the experimental realization of a dilute atomic Bose-Einstein condensate (BEC) [ 11, much theoretical effort has been directed towards the formation and behavior of vortices in atomic BEC [2-41. This paper presents the experimental realization and imaging of a vortex in BEC. We use the method proposed by Williams and Holland [5] to create vortices in a two-component BEC. An interference technique is used to obtain phase images of the vortex state and confirm the 277 phase winding required by the quantization condition. We have also carried out preliminary studies of the stability of the vortices. Vortices can be created in superfluid helium by cooling a rotating bucket of helium through the superfluid transition, and a vortex forms for each unit of angular momentum. This does not work for BEC because it is formed in a harmonic magnetic trap. When the condensate first forms it occupies a tiny cross-sectional area at the center of the trap and is too small to support a vortex. Eventually, the condensate grows to a sufficient size so that it can support vortices, but the time scale for vortices to be generated in the vortex-free condensate due to coupling with the rotating environment is unknown, and may well be longer than the lifetime of the condensate. This is the potential difficulty with using an optical “stirring beam” or magnetic field distortion to rotate the cloud during condensation, as has been frequently proposed. Another proposal has been to use optical beams with appropriate topologies to “imprint” a phase on an existing condensate. This technique must drive the local density to zero at some point and then rely on uncertain dissipative processes for the condensate to relax into a vortex state. We have avoided these uncertainties by creating vortices using a coherent process that directly forms the desired vortex wave h c t i o n via transitions between two internal spin states of 87Rb. The two spin states, henceforth 11) and (2), are separated by the ground-state hyperfine splitting and can be simultaneously confined in identical 2498
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and fully overlapping magnetic trap potentials. A twophoton microwave field induces transitions between the states. As we have seen in previous experiments, this coupled two-component condensate is exempt from the topological rules governing single-component superfluids [6]-mles that make it difficult to implant a vortex within an existing condensate in a controlled manner. In the coupled system, we can directly create a 12) (or 11)) state wave function having a wide variety of shapes [5] out of a 11) (or 12)) ground-state wave function by controlling the spatial and temporal dependence of the microwaveinduced conversion of 11) into 12). We control the conversion by shifting the transition frequency using the ac Stark effect. A spatially inhomogeneous and movable optical field (a focused laser beam) provides the desired spatial and temporal control of the ac Stark shift. The vortex state is an axially symmetric ring with a 25- phase winding around the vortex core where the local density is zero. To create a wave function with this spatial symmetry, the laser beam is rotated around the initial condensate as in Fig. la. The desired spatial
FIG. 1. (a) A basic schematic of the technique used to create a vortex. An off-resonant laser provides a rotating gradient in the ac Stark shift across the condensate as a microwave drive of detuning 6 is applied. (b) A level diagram showing the microwave transition to very near the 12) state, and the modulation due to the laser rotation frequency that couples only to the angular momentum 1 = 1 state when w = 6. In the figure, the energy splitting (
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phase dependence is obtained by detuning the microwave frequency from the transition, and rotating the laser beam at the appropriate frequency w to make the coupling resonant. For large microwave detunings 6 , the necessary rotation frequency is simply 6 . For smaller detunings, the rotation frequency must be the effective Rabi frequency of the microwave transition [7]. As shown in Fig. Ib for large detunings, the energy resonance condition now means that atoms can change only the internal state through the coupling of the time-varying perturbation, and are therefore obliged to obey any selection rule that the spatial symmetry of that perturbation might impose. The center of the condensate (at the axis of the beam rotation) feels no time-varying change, while regions near the circumference of the condensate feel a near-sinusoidal variation, with a phase delay equal to the azimuthal angle 0 around the circumference of the cloud. Williams and Holland show that this is precisely the geometry best suited to couple the condensate into a vortex state. It should be emphasized that it is not simply the mechanical forces of the optical field that excite the vortex: a laser beam rotating clockwise can produce clockwise or counterclockwise vortex circulation, depending on the sign of the microwave detuning. In the absence of the microwave coupling field, the two states can be thought of as two distinguishable, interpenetrating superfluids that interact with each other and with themselves via a mean-field repulsion proportional to the local densities. The interaction coefficients differ slightly [8,9], so the 11) fluid has slight positive buoyancy with respect to the 12) fluid [lo]. When the 11) fluid has a net angular momentum, it forms an equatorial ring around the central 12) fluid. The 11) fluid partially penetrates the constant-phase 12) fluid, which creates a central potential barrier. Conversely, a 12) vortex forms a ring that tends to contract down into the 11) fluid. We use the overlap of the 11) and 12) fluids to image the phase profile of the vortex state via the interconversion interference technique that we introduced in [ 111. In the presence of a near-resonant microwave field (and no perturbing optical field), the two states interconvert at a rate sensitive to the local difference in the quantum phases of the two states. Thus the application of a resonant 7r/2 microwave pulse transforms the original two-fluid density distribution into a distribution that reflects the local phase difference, a “phase interferogram.” Looking at the condensate both before and after the interconversion pulse provides images of both the amplitude and phase of the vortex ring. The basic experimental setup for forming condensates and driving them between different spin states is the same as in [8]. Using laser cooling and trapping, followed by trapping in a time-averaged orbiting potential magnetic trap and evaporative cooling, we produce a condensate of typically -8 X lo5 atoms in the 11) state ( F = 1, mF = -1). We then adiabatically convert the trap to a spherically symmetric potential by reducing the
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quadrupole magnetic field gradient [12]. This leaves us with a condensate 54 microns in diameter in a trap with oscillation frequencies of 7.8 +- 0.1 Hz in the radial and axial directions for both spin states. In this trap, a 11) state condensate has a lifetime of 75 s and the 12) state (F = 2, r n F = + 1) about 1 s. oscillating magnetic fields can then be pulsed on to drive the microwave transition between the 11) and 12) states. The power and oscillation frequency of the fields are adjusted to obtain the desired effective Rabi frequency for the 11) to 12) transition. To create a vortex in 12) we add a 10 nW, 780 nm laser beam that has a waist of 180 ,um and is detuned 0.8 GHz blue of resonant excitation of the 12) state. Using piezoelectric actuators we rotate the beam in a -75 ,um radius circle around the condensate at 100 Hz. The following procedure is used to obtain the precise location of the laser beam that is required to create a vortex. We first set the effective Rabi frequency to 100 Hz by adjusting the detuning of the microwave frequency away from resonance in the presence of the light. Typically the resonant Rabi frequency is 35 Hz and the total detuning about 94 Hz from the 11) to 12) transition. By making fine adjustments of the detuning we then optimize the amount of transfer (typically 50%) to the 12) state. We then adjust the center of rotation of the beam to obtain the most symmetric rings. After -70 ms the vortex has been formed, and we turn off the laser beam aiid the microwave drive. We can take multiple images of a vortex both during and after formation using nondestructive state-selective phase contrast imaging [6,13]. Rapid control of the microwave power and frequency allows us to apply various pulse sequences to explore many different options for the creation, manipulation, and observation of a single vortex. For example, we can put the initial condensate into either the 11) or 12) state and then make a vortex in the 12) or 11) state, respectively, and we can obtain phase interferograms or quickly switch the internal state of the vortex at any time after the rotating laser beam is off. We can also watch the evolution of the vortex over time scales from milliseconds to seconds. All of these techniques are nondestructive to both the density and phase. In Fig. 2 we show a detailed picture of the phase profile of a 12) vortex. To obtain this we first take a picture of the vortex (Fig. 2a), then we apply a resonant microwave 7r pulse. Halfway through the 4 ms long T pulse, we take a second image (Fig. 2b), and at the completion of the pulse we take a third image (Fig. 2c) that shows the original density distribution of the interior 11) state. Normalizing by the density distributions of the vortex and interior states we obtain the phase image in Fig. 2d [14]. The figure dramatically shows the variation and continuity of the phase around the ring (Fig. 2e) that are required by the quantization of angular momentum. In Fig. 3 are pictures of the time evolution of vortices to show their dynamics and stability. As expected, the dynamics of the 11) state vortices (Figs. 3a and 3b) are 2499
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FIG. 2 (color). Condensate images (100 pm on a side) in which the imaging laser is tuned such that only 12) is visible. (a) The initial image of the vortex, (b) after 7r/2 of the interconversion pulse, and (c) after completion of the 7r pulse. The vortex is now invisible (in 11)) and the interior fluid is imaged (in 12)). (d) The normalized difference in densities between the local average ring and interior densities [(a) and (c), respectively] and the phase interferogram density (b) for each corresponding point in the images. This is approximately the cosine of the local phase difference q5 between the vortex state and the interior state [14J The values are shown only for those regions where the densities of each state were high enough to give adequate signal to noise for the phase reconstruction. (e) The radial average at each angle 0 around the ring is shown in (d). (The data are repeated after the azimuthal
angle 2rr to better show the continuity around the ring.) different from the 12) state vortices (Fig. 3c) due to the different scattering lengths. The 12) vortex ring sinks in toward the trap center (Fig. 3c), and then rebounds and apparently breaks up. This pattern is repeatedly observed in measurements of the 12) vortices. Conversely, the equilibrium position of a 11) state fluid is obtained by ”floating” outside the 12) state fluid. The inner radius of the 11) state ring shrinks slowly as the interior 12) fluid decays away with a -1 s lifetime. A variety of additional behaviors has been seen for ll) vortices. Initial asymmeby is very sensitive to beam position and condensate slosh. For small differences in the initial vortex state density distribution, asymmetric density distributions sometimes develop and/or “heal” between 0.5 and 1 s. The nonrotating 12) fluid within a 11) vortex is analogous to the defects that pin vortices in superconductors. This ( 2 ) “defect” can be removed quickly, by a properly tuned laser pulse, or allowed to decay slowly (as in Figs. 3a and 3b). Thus this system can be varied between two relevant physical limits. In the limit of a large repulsive central potential (produced by a large amount of 12) fluid) the system most closely resembles quantized flow in a fixed, 3D toroidal potential. The vortex core is pinned and its size is determined by the central potential. Fig2500
ures 3a and 3b are near this limit for t = 0 and 200 ms. In this case the density distribution of the vortex state is vulnerable to instabilities due to the fact that the relative densities of 11) and 12) may evolve with relatively little energy cost so long as the total density remains constant [lo]. In the opposite limit (a small amount of 12) fluid), the central potential is negligible and the size of the vortex core (in equilibrium) is determined entirely by its own centrifugal barrier. Figures 3a and 3b at t = 600 ms are evolving towards this limit. In Fig. 3a, we see that the interior 12) fluid is no longer pinning the vortex at 600 ms, but rather is being dragged around by the precessing vortex. The 12) vortex provides an interesting mixture of the above limits. The 11) fluid “floats” to the outside so there is no pinning of the vortex core, but the tendency towards density instabilities remains. Expanded studies of stability issues are under way. We also expect to be able to observe interesting transitional behavior between these limiting cases. For example, it is a straightforward extension of our method to create an 1 = 2 vortex. In the presence of a strong pinning potential, 1 = 2 vortices should be stable, but in the weak potential limit, I = 2 vortices are predicted to spontaneously bihrcate [4].
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FIG. 3. (a),(b): Two separate instances of the free evolution of a 11) state vortex in the magnetic trap. It is stable over a time long compared to the trap oscillation period (128 ms). (c) The free evolution of a 12) vortex is much more dynamic. It is seen shrinking quickly into the invisible 11) fluid and rebounding into fragments. Each column is from a single run,where time t is referenced to the end of vortex creation ( t is the same for each row). The 11) and 12) state images appear different due to different signs of the probe detuning.
W e gratefully acknowledge useful conversations with
J. Williams and M. Holland. This work is supported by the ONR, NSF, and NIST. *Quantum Physics Division, National Institute of Standards and Technology. 'Present ad+ess: Department of Physics, Amherst College, Amherst, MA 01002.
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M.H. Anderson et al., Science 269, 198 (1995); W. Ketterle et al., cond-mav9904034; E. Cornell et al., cond-mav9903 109. K.-P. Marzlin, W. Zhang, and E.M. Wright, Phys. Rev. Lett. 79, 4728 (1997); R. Dum, J.I. Cirac, M. Lewenstein, and P. Zoller, Phys. Rev. Lett. 80, 2972 (1998); E. L. Bolda and D. F. Walls, Phys. Lett. A 246, 32 (1998). These vortex generation methods involve transitions between different internal states, as in Ref. [5]. F. Dalfovo and S . Stringari, Phys. Rev. A 53,2477 (1996); T.-L. Ho and V. B. Shenoy, Phys. Rev. Lett. 77, 2595 (1996); D.S. Rokhsar, Phys. Rev. Lett. 79, 2164 (1997); R. Dodd, K. Burnett, M. Edwards, and C. Clark, Phys. Rev. A 56, 587 (1997); B. Jackson, J.F. McCann, and C. S. Adams, Phys. Rev. Lett. 80, 3903 (1998); A. Fetter, J. Low. Temp. Phys. 113, 189 (1998); S. Stringari, Phys. Rev. Lett. 82, 4371 (1999); T. Isoshima and K. Machida, Phys. Rev. A 59, 2203 (1999); B. Caradoc-Davies, R. Ballagh, and K. Burnett, Phys. Rev. Lett. 83, 895 (1999); D. Feder, C. Clark, and B. Schneider, Phys. Rev. Lett. 82, 4956 (1999). H. PU,C. Law, J. Eberly, and N. Bigelow, Phys. Rev. A 59, 1533 (1999). J. Williams and M. Holland, Nature (London) (to be published), cond-mat/9909 163. M. R. Matthews et al., Phys. Rev. Lett. (to be published), cond-matl9906288. In the limit of large detuning, 6 is equal to the effective Rabi frequency. In our implementation, w and the effective Rabi frequency are 100 Hz and 6 is 94 Hz. M. R. Matthews et al., Phys. Rev. Lett. 81, 243 (1998). J. P. Burke, Jr., J. L. Bohn, B. D. Esry, and C. H. Greene, Phys. Rev. Lett. 80, 2097 (1998). D. S . Hall et aL, Phys. Rev. Lett. 81, 1539 (1998). D.S. Hall, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Phys. Rev. Lett. 81, 1543 (1998). J. R. Ensher, Ph.D. thesis, University of Colorado, 1998. M. R. Andrews et al., Science 273, 84 (1996). The normalized difference is defined as (24)12) - dlz) d l l ) ) / ( 2 a )where dl,) is the local density for the ith state and dll)12)for the interferogram (Fig. 2b). This is not exactly the cosine of the phase difference 4 between the vortex and interior state due to small effects, such as incomplete overlap between the states along the line of sight and uncertainties in the zero levels.
VOLUME85, NUMBER9
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Stable 85 R b Bose-Einstein Condensates with Widely Tunable Interactions S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell,* and C . E. Wieman JILA, National Institute of Standards and Technology, and the Department of Physics, University of Colorado, Boulder, Colorado 80309-0440 (Received 7 April 2000) Bose-Einstein condensation has been achieved in a magnetically trapped sample of *'Rb atoms. Longlived condensates of up to lo4 atoms have been produced by using a magnetic-field-induced Feshbach resonance to reverse the sign of the scattering length. This system provides new opportunities for the study of condensate physics. The variation of the scattering length near the resonance has been used to magnetically tune the condensate self-interaction energy over a wide range, extending from strong repulsive to large attractive interactions. When the interactions were switched from repulsive to attractive, the condensate shrank to below our resolution limit, and after -5 ms emitted a burst of high-energy atoms, PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 34.50.-s
allowed us to reach novel regimes of condensate physics. These include producing very large repulsive interactions (npklaI3 = lov2), where effects beyond the mean-field approximation should be readily observable. We can also make transitions between repulsive and attractive interactions (or vice versa). This now makes it possible to study condensates in the negative scattering length regime, including the anticipated "collapse" of the condensate [13], with a level of control that has not been possible in other experiments [14]. In fact, this ability to change the sign of the scattering length is essential for the existence of our "Rb condensate. Away from the resonance the large negative scattering length (= - 4 0 0 ~ 0 [4,15]) limits the maximum number of atoms in a condensate to -80 [2]. However, we have produced condensates with up to 104 atoms by operating in a region of the Feshbach resonance where a is positive. The experimental apparatus was similar to the double magneto-optical trap (MOT) system used in our earlier
Atom-atom interactions have a profound influence on most of the properties of Bose-Einstein condensation (BEC) in dilute alkali gases. These interactions are well described in a mean-field model by a self-interaction energy that depends only on the density of the condensate ( n ) and the s-wave scattering length ( a ) [l]. Strong repulsive interactions produce stable condensates with a size and shape determined by the self-interaction energy. In contrast, attractive interactions ( a < 0) lead to a condensate state where the number of atoms is limited to a small critical value determined by the magnitude of a [2]. The scattering length also determines the formation rate, the spectrum of collective excitations, the evolution of the condensate phase, the coupling with the noncondensed atoms, and other important properties. In the vast majority of condensate experiments the scattering length has been fixed at the outset by the choice of atom. However, it was proposed that the scattering length could be controlled by utilizing the strong variation expected in the vicinity of a magnetic-field-induced Feshbach resonance in collisions between cold (-pK) alkali atoms [3]. Recent experiments on cold 85Rb and Cs atoms and Na condensates have demonstrated the variation of the scattering length via this approach [4-71. However, extraordinarily high inelastic losses in the Na condensates were found to severely limit the extent to which the scattering length could be varied and precluded an investigation of the interesting negative scattering length regime [8]. These findings prompted the subsequent proposal of several exotic coherent loss processes that remain untested in other alkali species [9- 113. Here we report the successful use of a Feshbach resonance to readily vary the self-interaction of longlived condensates over a large range. In "Rb there exists a Feshbach resonance in collisions between two atoms in the F = 2, mf = -2 hyperfine ground state at a magnetic field B - 155 gauss(G) [4,12]. Near this resonance the scattering length varies dispersively as a function of magnetic field and, in principle, can have any value between --M and +m (see inset of Fig. 1). This has 0031 -9007/00/85(9)/ 1795(4)$15.00
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Magnetic Field (G) FIG. 1. Scattering length in units of the Bohr radius (a,,) as a function of the magnetic field. The data are derived from the condensate widths. The solid line illustrates the expected shape of the Feshbach resonance using a peak position and resonance width consistent with our previous measurements [4,18]. For reference, the shape of the full resonance has been included in the inset.
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work [16]. Atoms collected in a vapor cell MOT were transferred to a second MOT in a low-pressure chamber. Once sufficient atoms have accumulated in the lowpressure MOT, both the MOT’S were turned off and the atoms were loaded into a purely magnetic, Ioffe-Pritchard “baseball” trap. During the loading sequence the atom cloud was compressed, cooled, and optically pumped, resulting in a typical trapped sample of about 3 X lo8 85Rb atoms in the F = 2,mf = -2 state at a temperature of T = 45 p K . The lifetime of atoms in the magnetic trap due to collisions with background atoms was -450 s. Forced radio-frequency evaporation was employed to cool the sample of atoms. Unfortunately, ”Rb is plagued with pitfalls for the unwary evaporator familiar only with s7Rb or Na. In contrast to those atoms, the elastic collision cross section for ”Rb exhibits strong temperature dependence due to a zero in the s-wave scattering cross section at a collision energy of E / k B 350 p K [17]. This decrease in the elastic collision cross section with temperature means that the standard practice of adiabatic compression to increase the initial elastic collision rate does not work. 85Rb also suffers from unusually high inelastic collision rates. We recently investigated these losses and observed a mixture of two- and three-body processes which varied with B [18]. The overall inelastic collision rate displayed several orders of magnitude variation across the Feshbach resonance, with a dependence on B similar to that of the elastic collision rate. However, the inelastic collision rate increased more rapidly than the elastic collision rate towards the peak of the Feshbach resonance, and was found to be significantly lower in the high field wing of the resonance than on the low field side. This knowledge of the loss rates, together with the known field dependence of the elastic collision cross section [4,15], has enabled us to successfully devise an evaporation path to reach BEC. This begins with evaporative cooling at a field ( B = 250 G) well above the Feshbach resonance. To maintain a relatively low density, and thereby minimize the inelastic losses, a relatively weak trap is used. The low initial elastic collision rate means that about 120 s are needed to reach T = 2 pK, putting a stringent requirement on the trap lifetime. As the atoms cool, the elastic collision rate increases and it becomes advantageous to trade some of this increase for a reduced inelastic collision rate by moving to B = 162.3 G where the magnitude of the scattering length is decreased. The remainder of the evaporation is performed at this field [with a radial (axial) trap frequency of 17.5 Hz (6.8 Hz)]. In contrast to field values away from the Feshbach resonance, the scattering length is positive at this field and stable condensates may therefore be produced. The density distribution of the trapped atom cloud was probed using absorption imaging with a 10 pus laser pulse 1.6 ms after the rapid (= 0.2 ms) turn-off of the magnetic trap. The shadow of the atom cloud was magnified by about a factor of 10 and imaged onto a CCD
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array to determine the spatial size and the number of atoms. The emergence of the BEC transition was observed at T 15 nK. Typically, we were able to produce “pure” condensates of up to lo4 atoms with peak number densities of n p k = 1 X 10l2 cmP3. The lifetime of the condensate at B = 162.3 G was about 10 s [19]. This lifetime is consistent with that expected from the inelastic losses we have measured in cold thermal clouds [18]. It is notable that our evaporation trajectory suffered a nearcatastrophic decline prior to the observation of the BEC transition. We approached the required BEC phase space density at 100 nK with about lo6 atoms, but then lost a factor of about 50 in the atom number before the characteristic two-component density distribution was visible. Over this part of the trajectory, the cooling efficiency has become low (and the phase space density remains approximately constant). This is because the mean-free path is comparable to the cloud size and the high density results in high losses. This situation improves when the number of atoms becomes sufficiently low, and we are then able to obtain a significant fraction of the atoms in a condensate. The number of atoms in the condensate after we reach this favorable low-density regime is obviously a delicate balance between the elastic (cooling) and inelastic (loss) collision processes in the cloud. Both of these are strongly field dependent near the Feshbach resonance. Although we are able to decrease the loss rate by moving to higher fields, the ratio of elastic to inelastic collisions actually decreases and it becomes harder to form condensates. For example, at B = 164.3 G we can produce condensates of only a few thousand atoms. Conversely, moving to lower fields does not help because we reach the favorable low-density cooling regime at smaller numbers of atoms. This restriction together with the larger loss rate means that at B = 160.3 G, for example, we are unable to form condensates. One of the features of the high inelastic loss rates reported in the Na experiments was an anomalously high decay rate when the condensate was swept rapidly through the Feshbach resonance [S]. In light of this work, it was essential to determine to what extent the ”Rb condensate was perturbed in being swept across the Feshbach peak during the trap turn-off. These measurements also provide an additional test of coherent loss mechanisms such as those in Refs. [9- 111. We applied a linear ramp to the current in the baseball coil to sweep the magnetic field experienced by the atoms from B = 162.3 G across the Feshbach peak to B 132 G and then immediately turned off the trap and imaged the atom cloud. From the images we determined the fraction of condensate atoms lost as a function of the inverse ramp speed (Fig. 2). The loss for the fastest ramp, which corresponds to the direct turn-off of the magnetic trap, is less than 9%. This was determined in a separate experiment where the condensate was imaged directly in the magnetic trap both before and after the ramp. For comparison, the experiment was repeated using
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Inverse Ramp Speed (ps/G) FIG. 2. Fraction of atoms lost following a rapid sweep of the magnetic field through the peak of the Feshbach resonance as a function of the inverse speed of the field ramp. Data are shown for a condensate (a) with a peak density of npk = 1 .O X 10'' cm-3 and for a thermal cloud ( 0 ) with a temperature T = 430 nK and a peak density of n p k = 4.5 X 10" cm-3.
a cloud of thermal atoms much hotter than the BEC transition temperature. The results for the thermal atoms are consistent with the known inelastic loss rates in the vicinity of the Feshbach resonance [18]. The strong and poorly characterized temperature dependence of these known loss rates near the Feshbach peak makes it difficult to determine what fraction of the observed condensate loss can be attributed to the usual inelastic loss processes and we cannot, therefore, rule out a coherent aspect to the loss process. There have been several models of coherent loss processes put forward to explain the corresponding sodium results [9-111. However, these calculations are based on the Timmermans theory [9] of coupled atomic and molecular Gross-Pitaevskii equations which is unlikely to be applicable to the conditions of the present experiment [20]. We also changed the self-interaction energy by varying the magnetic field and observed the resulting change in the condensate shape. By applying a linear ramp to the magnetic field we have varied the magnitude of a in the condensate by almost 3 orders of magnitude. The duration of this ramp was sufficiently long (500 ms) to ensure that the condensate responded adiabatically. Figure 3 shows a series of condensate images for various magnetic fields. They illustrate how we are able to easily change a over a very wide range of positive values. Moving towards the Feshbach peak the condensate size increases due to the increased self-interaction energy. The density distribution approaches the parabolic distribution with an aspect ratio of h j - ~= w z / w , expected in the ThomasFermi (TF) regime [21]. Moving in the opposite direction the cloud size becomes smaller than our 7 p m resolution limit shortly before we reach the noninteracting limit where the condensate density distribution is a Gaussian whose dimensions are set by the harmonic oscillator lengths [Zi = ( f i / r n ~ i ) l where /~ i = r,z1 [21]. We took condensate images similar to those shown at many
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Horizontal Position (pm) FIG. 3 (color). False color images and horizontal cross sections of the condensate column density for various magnetic fields. The condensate number was varied to maintain an optical depth (OD) of -1.5. The magnetic field values are (a) B = 165.2 G, (b) B = 162.3 G, ( c ) B = 158.4 G , (d) B = 157.2 G , and (e) B = 156.4 G .
field values between 156 and 166 G. From the images the full widths at half maximum (FWHM) of the column density distributions were determined. The scattering length was then derived by assuming a TF column density distribution with the same FWHM [K In Fig. 1 we plot the scattering length derived in this manner versus the magnetic field. It shows that these values agree with the predicted field dependence of the Feshbach resonance. The ability to tune the atom-atom interactions in a condensate presents several exciting avenues for future research. One is to explore the breakdown of the dilutegas approximation near the Feshbach peak. The lifetime of the condensate decreases with larger a, but for a lifetime of about 100 ms, which is sufficient for many experiments, we have created static condensates with = For such values, effects beyond the mean-field approximation, such as shifts in the frequencies of the collective excitations [22], are about 10%. A second avenue is the behavior of the condensate when the scattering length becomes negative. When we increased the magnetic field beyond B = 166.8 G , where a was expected to change sign, a sudden departure from the smooth behavior in Figs. 1 and 3 was observed. As a was decreased the condensate width decreased, and then about 1797
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5 ms after the change in sign there was a sudden explosion that ejected a large fraction of the condensate. This left a small observable remnant condensate surrounded by a “hot” cloud at a temperature on the order of 100 nK. These preliminary results on switching from repulsive to attractive interactions suggest a violent, but highly reproducible, destruction of the condensate. The ability to control the precise moment of the onset of a < 0 instability is a distinct advantage over existing methods for studying this regime which rely on analyzing ensemble averages of “post-collapse’’ condensates [ 141. The dynamical response of the condensate to a sudden change in the sign of the interactions can now be investigated in a controlled manner, probing the rich physics of this dramatic condensate collapse process. We are pleased to acknowledge useful discussions with Murray Holland, Jim Burke, Josh Milstein, and Marco Prevedelli. This research has been supported by the NSF and ONR. One of us (S. L. C.) acknowledges the support of a Lindemann Fellowship.
*Quantum Physics Division, National Institute of Standards and Technology. See, for example, the review by F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999). P. A. Ruprecht, M. J. Holland, K. Bumett, and M. Edwards, Phys. Rev. A 51, 4704 (1995). W.C. Stwalley, Phys. Rev. Lett. 37, 1628 (1976); E. Tiesinga, B. J. Verhaar, and H.T.C. Stoof, Phys. Rev. A 47,4114 (1993); E. Tiesinga, A. J. Moerdijk, B. J. Verhaar, and H. T.C. Stoof, Phys. Rev. A 46,R1167 (1992). J.L. Roberts et al., Phys. Rev. Lett. 81, 5109 (1998). Ph. Courteille et al., Phys. Rev. Lett. 81, 69 (1998). V. VuletiC, A. J. Kerman, C. Chin, and S. Chu, Phys. Rev. Lett. 82, 1406 (1999). S. Inouye et aZ., Nature (London) 392, 151 (1998). J. Stenger et al., Phys. Rev. Lett. 82, 2422 (1999). E. Timmermans et al., Phys. Rep. 315, 199 (1999).
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[lo] F.A. van Abeelen and B. J. Verhaar, Phys. Rev. Lett. 83, 1550 (1999). [ l l ] V.A. Yurovsky, A. Ben-Reuven, P.S. Julienne, and C. J. Williams, Phys. Rev. A 60, R765 (1999). [12] P. Courteille et al., Phys. Rev. Lett. 81, 69 (1998). [13] Y. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. Lett. 79, 2604 (1997); C.A. Sackett, H.T.C. Stoof, and R.G. Hulet, Phys. Rev. Lett. 80, 2031 (1998); M. Ueda and A.J. Leggett, Phys. Rev. Lett. 80, 1576 (1998); H. Saito and M. Ueda, e-print cond-mat/0002393. [14] C.C. Bradley, C.A. Sackett, J.J. Tollett, and R.G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); C.A. Sackett, J.M. Gerton, M. Welling, and R.G. Hulet, Phys. Rev. Lett. 82, 876 (1999). [15] J.P. Burke, Jr., NIST Gaithersburg (private communication). [I61 C. J. Myatt et al., Opt. Lett. 21, 290 (1996). [17] J. P. Burke, Jr., J. L. Bohn, B. D. Esry, and C. H. Greene, Phys. Rev. Lett. 80, 2097 (1998). [18] J.L. Roberts, N.R. Claussen, S.L. Cornish, and C.E. Wieman, Phys. Rev. Lett. 85, 728 (2000). [19] This is in dramatic contrast to the very short lifetimes obtained with Na condensates anywhere near the vicinity of a Feshbach resonance [S]. We believe that the primary difference is that we have 3 orders of magnitude lower density. [20] Timmermans, in effect, models the Feshbach resonance as an avoided crossing between atoms and molecules, with a density-dependent splitting given by a&, where a is a characteristic strength of the resonance [9]. Implicit in this model, and in related work [10,11], is the assumption that the Gross-Pitaevskii equation mean-field description of the atoms applies when there is small atom-molecule coupling. For this to be valid ? 2 p k a 3 must remain <<1 if the detuning from the Feshbach resonance is large compared to a&. As we rapidly ramp over the resonance this clearly fails, because we reach values of a for which npku3> 1 when the detuning is still much larger than a& and hence well before the model would predict atoms are converted to molecules. [21] G. Baym and C.J. Pethick, Phys. Rev. Lett. 76, 6 (1996). 1221 L. Pjtaevskii and S. Stringari, Phys. Rev. Lett. 81, 4541 (1998); E. Braaten and J. Pearson, Phys. Rev. Lett. 82,255 (1999).
VOLUME 85, NUMBER 14
PHYSICAL REVIEW LETTERS
2 OCTOBER 2000
Vortex Precession in Bose-Einstein Condensates: Observations with Filled and Empty Cores B. P. Anderson," P. C. Haljan, C. E. Wieman, and E. A. Cornell''' J I L A , Nntioiinl 1rz.rtitrrte of' Stuiidcirds m c l Tec~hiiologyarid Depnrtiizeiit of Ph!,sic.~,Uiiiver.rity of' Colorci~lo, Boirldei; Colorcido 80309-0440 (Received 19 May 2000) We have observed and characterized the dynamics of singly quantized vortices in dilute-gas BoseEinstein condensates. Our condensates are produced in a superposition of two internal states o l Y7Rb, with one state supporting a vortex and the other filling the vortex corc. Subsequently, the state filling the core can be partially or completely removed, reducing the radius of the core by as much as a [actor of 13. all the way down to its bare value of the healing length. The corresponding superfluid rotation rates, evaluated at the core radius, vary by a factor of 150, but the prccession fl-eqtiency of the vortex core about the condensate axis changes by only a factor of 2. PACS numbers: 03.75.Fi. 67.90.+z, 67.57.F_:,32.80.Pj
The dynamics of quantized vortices in superfluid helium and superconductors have been fascinating and important research areas in low-temperature physics [ 1,2]. Even at zero temperature, vortex motion within a superfluid is intricately related to the quantization of current around the vortex core. Besides these superfluid systems, studies of the dynamics of optical vortices have also become an active area of research [3]. More recently, demonstrations of the creation of quantized vortices in dilute-gas Bose-Einstein condensates (BEC) [4,5] have emphasized the similarities between the condensed matter, optical, and dilute-gas quantum systems. Because of the observational capabilities and the techniques available to manipulate the quantum wave function of the condensates, dilute-gas BEC experiments provide a unique approach to studies of quantized vortices and their dynamics. This paper reports direct observations and measurements of singly quantized vortex core precession in a BEC. Numerous theoretical papers have explored the expected stability and behavior of vortices in BEC [6- 171. One interesting expected effect is vortex core precession about the condensate axis [7,9-161. Radial motion of the core within the condensate can also occur, and may be understood as being due to energy dissipation and damping processes. Core precession may be described in terms of a Magnus effect-a familiar concept in fluid dynamics and superfluidity [I]. An applied force on a rotating cylinder in a fluid leads to cylinder drift (due to pressure imbalances at the cylinder surface) that is orthogonal to the force. Analogously, a net force on a vortex core in a superfluid results in core motion perpendicular to both the vortex quantization axis and the force. In the condensate vortex case, these forces can be due to density gradients within the condensate, for example, or the drag due to thermal atoms. The density-gradient force may be thought of as one component of an effective buoyancy: just as a bubble in a fluid feels a force antiparallel to the local pressure gradient, a vortex core in a condensate will feel a force towards lower
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condensate densities. The total effective buoyancy, however, is due less to displaced mass (the "bubble") than it is to dynamical effects of the velocity-field asymmetry, a consequence of a radially offset core. Typically, the total buoyancy force is away from the condensate center, and the net effect is an azimuthal precession of the core via the Magnus effect. Drag due to the motionless (on average) thermal atoms opposes core precession, causing the core to spiral outwards towards the condensate surface. In the absence of this drag (for temperature -O), radial drift of the core may be negligible. Our techniques for creating and imaging a vortex in a coupled two-component condensate are described in Refs. [4,18]. The two components are the IF = l , m F = - 1) and IF = 2, mF = 1) internal ground states of 87Rb, henceforth labeled as states 11) and 12), respectively. We start with a condensate of lo6 (2) atoms, confined in a spherical potential with oscillator frequency 7.8 Hz. A near-resonant microwave field causes some of the 12) atoms to convert to 11) atoms. The presence of a rotating, off-resonant laser beam spatially modulates the amplitude and phase of the conversion. The net result is a conversion of about half of the sample into an annular ring of 11) atoms with a continuous quantum phase winding from 0 to 2n- about the circumference-a singly quantized vortex. The balance of the sample remains in the nonrotating 12) state and fills the vortex core. With resonant light pressure we can selectively remove as much of the core material as we desire. In the limit of complete removal, we are left with a single-component, bare vortex state. In this bare-core limit, the core radius is on the order of the condensate healing length [ = ( 8 n - n 0 a ) - ' / ~where , no is the peak condensate density and a is the scattering length. For our conditions, [ = 0.65 p m , well under our imaging resolution limit. The bare core can be observed after ballistic expansion [5] of the condensate, but this is a destructive measurement. On the other hand, if we leave some of the 12)-state atoms filling the core, the pressure
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of the filling material opens up the radius of the 11) vortex core to the point where we can resolve the core in a time series of nondestructive phase-contrast images. Filled-core cLyrzanzics.-We first discuss vortex dynamics in two-component condensates, where 10%-50% of the atoms were in the 12) fluid filling the 1 1 ) vortex core. We took successive images of the 11) atoms in the magnetic trap, with up to 10 images of each vortex. The vortex core is visible as a dark spot in a bright 11) distribution, as shown in Fig. l(a). Instabilities in our vortex creation process usually resulted in the creation of off-center vortex cores, allowing us to observe precession of the cores. We observed precession out to -2 s, after which the 12) fluid had decayed to the point that the vortex core was too small to be observed in the trapped condensate. The recorded profile of each trapped condensate was f i t with a smooth Thomas-Fermi distribution. Each vortex core profile was fit with a Gaussian distribution to determine its radius and position within the condensate. From the fits, we determined the overall radius R , of the trapped condensate (typically 28 ,urn), the HWHM radius r of the filled vortex core, and the displacement d , and angle 8, of the core center with respect to the condensate center. Core angles and radii for the images in Fig. l(a) are shown in Figs. I(c) and l(d). The vortex core is seen precessing in a clockwise direction, which is the same direction as the vortex fluid flow around the core. The angular precession frequency was determined from the time dependence of 8 , [Fig. l(c)]. This and other similar data sets showed no reproducible radial motion of the core over the times and parameters examined. However, consistent decrease in the
size of the core was observed, which we interpret as being due to known decay of the 12) fluid through inelastic atomic collisional processes. For each data set, we determined a mean core radius and displacement. The data cover a range of core radii ( I , = 7( to 13[), displacements ( d , = 0.17R, to 0.48R,), and percentage of atoms in the core ( 1 0% to 50%). Except for a few “rogue vortices” (discussed below), the measured precession frequencies are clustered around l .4 Hz, as shown in Fig. 2, precessing in the same direction as the fluid rotation. The data [Fig. 2(a)] suggest a slight increase in frequency for cores farther from the condensate center. We also see [Fig. 2(b)] a slight decrease in precession frequency for larger cores. These measurements are in qualitative agreement with two-dimensional numerical simulations for two-component condensates [ 161. As indicated in Fig. 2, a few vortex cores exhibited precession opposite to that of the fluid flow (negative frequencies). The quality of the corresponding vortex images was routinely lower than for the positive-frequency precession points, with vortices looking more like crescents and “D” shaped objects rather than like the images of Fig. 1. We only found such occurrences with our two-component (intrap) measurements. We speculate that this “inverse precession” may be due to rare events in which total angular momentum may be distributed in a complicated way among both internal states. Such cases might arise due to position instabilities of the rotating laser beam during the vortex creation process. Recent theoretical attention has addressed the possibly related situation of nonsymmetric configurations of vortices in two-component condensates [ 171. Bare-core dynamics. -To examine the dynamics of bare vortices, our procedure consisted of taking a nondestructive phase-contrast picture of the partially filled 11) vortex
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FIG. I . (a) Seven successive images of a condensate with a vortex and (b) their corresponding fits. The 75 pm-square nondestructive images were taken at the times listed, referenced to the first image. The vortex core is visible as the dark region within the bright condensate image. (c) The azimuthal angle of the core is determined for each image, and is plotted vs time held in the trap. A linear fit to the data indicates a precession frequency of 1.3(1) Hz for this data set. (d) Core radius r in units of healing length 5. The line shown is a linear fit to the data.
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FIG. 2. Compiled data for filled vortex core precession, with each data point extracted from a series (as in Fig. 1) of nondestructive images of a single vortex. Precession frequency is plotted vs (a) core displacement d, in units of condensate radius R , , and (b) core radius r in units of healing length ,$. Circles correspond to positive frequencies and filled squares to negative frequencies. (Positive frequency is defined as core precession having the same handedness as the vortex angular momentum.) The triangle at r = 5 shows for reference the average measured precession frequency of many bare vortices [see text and Fig. 3(b)]. A line is drawn as a guide to the trend in frequency vs core size.
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distribution [Fig. 3(a) inset], as previously discussed, followed by complete removal of the core filling 1191. We then held the bare vortex in the trap for a variable hold time til, after which the condensate was released from the trap. We took a final near-resonance phase-contrast image [201 of the atomic distribution [Fig. 3(a)J after the condensate had ballistically expanded by a factor of -3.5 (211and the empty core had expanded 1221 to a fit radius of -9 p m . Displacements dl and angular positions 0, of the cores for the in-trap images were extracted as described before. The images of the expanded clouds were fit with identical distributions, and the Thomas-Fermi radius R , of the expanded cloud and the vortex core displacement d, and angle 0, were obtained for each image. For each pair of images, we determined the angular difference AOer = 8 , - O r between the cores in the expanded and in-trap images. We also determined the core displacement ratio d,/d,, an indicator of the radial motion of the core during the hold time t h . From the measurements of AOel at different hold times rl, [Fig. 3(b)], we find a bare-core precession frequency of 1.8(1) Hz, slightly faster than the precession of filled cores and consistent with the trend shown in Fig. 2(b) for filled cores. To emphasize that our measurements of filled and empty cores are different limits in a continuum of filling
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FIG. 3. (a) Ballistic expansion image of a vortex after all 12)
atoms have been removed. The dark spot is the hare vortex core. Inset-the corresponding, preceding in-trap nondestructive image of the partially filled core. (b) Angular differences AO,, between vortex cores from the in-trap and expansion images, plotted against hold time t ~in, the magnetic trap. The line is a fit through the data, indicating a bare-core precession frequency of 1.8(1) Hz. (c) Radial core motion is determined by ( d e / R c ) / ( d l / R , ) ,the ratio of the fractional core displacements from the expansion and in-trap images of each data set. The data are shown as open circles, with the average of all data at each given hold time plotted as a filled triangle. (d) Core visibility of an expanded vortex, defined as the conditional probability for observing a vortex in an expanded image given the observation of a vortex in the corresponding, preexpansion in-trap image. Visibility drops dramatically for hold times th > 1 s.
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material, we indicate the measured bare-core precession frequency in Fig. 2(b) with a point at P = 5. From Fig. 2(b) it is apparent that the structure and content of the vortex core have a relatively modest effect on precession frequency. One can calculate, for instance, the fluid rotation rate v,. at the inner core radius. The value of v,-is given by the quantized azimuthal superfluid velocity evaluated at the radius of the core, divided by the circunferential length at that radius. For bare vortices, v,.is about 260 Hz, while for the largest filled cores of Fig. 2(b) (for which nearly half of the sample mass is composed of core filling), v , is only about 1.7 Hz. Thus between vortices whose inner-radius fluid rotation rates vary by a factor of 150, we see only a factor of 2 difference in precession frequency. The slower precession of filled cores can be understood in terms of our buoyancy picture. Because of its slightly smaller scattering length, 12) fluid has negative buoyancy with respect to 11) fluid, and consequently tends to sink inward towards the center of the condensate [23]. With increasing amounts of 12) material in the core, the inward force on the core begins to counteract the outward buoyancy of the vortex velocity field, resulting in a reduced precession velocity. It is predicted that with a filling material of sufficiently negative buoyancy in the core, the core precession may stop or even precess in a direction opposite to the directioli of the fluid flow [16], but our data do not reach this regime. Various theoretical techniques involving two- and threedimensional numerical and analytical analyses have been explored to calculate the precession frequency of a vortex core within a condensate. We briefly compare those most readily applied to our physical parameters, assuming a spherical, single-component condensate with 3 X lo5 atoms ( R , = 22 p m ) in a nonrotating trap. Where relevant, we assume a core displacement of d, = 0.35R,. A two-dimensional hydrodynamic image vortex analysis 1241 has been analytically explored in the homogeneous gas [ 1 11 and two-dimensional harmonic confinement [ 141 limits. The latter of these predicts a bare-core precession frequency of -0.8 Hz. Svidzinsky and Fetter's threedimensional [ 121 solution to the Gross-Pitaevskii equation predicts a precession frequency of 1.58 Hz. Jackson et al. [lo] have obtained results in close agreement with this analytical solution using a numerical solution to the Gross-Pitaevskii equation. Finally, a path-integrals technique by Tempere and Devreese 1131 predicts a 1.24 Hz precession, and a two-dimensional simulation by McGee and Holland [ 161 using a steepest-descents technique predicts a precession frequency of 1.2 Hz. Measurements of & / d , for different hold times t h show the radial motion of the bare cores and are a probe of energy dissipation of the vortex states. The plot of Fig. 3(c) displays no trend of the core towards the condensate surface during t h , indicating that thermal damping is negligible on the 1 s time scale [2.5]. However, we notice a 28.59
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sharply decreasing visibility of expanded bare vortices for hold times greater than t l , 2 1 s, as indicated in Fig. 3(d). The absence of radial core motion suggests that decreased visibility is due to imaging limitations rather than true decay. One hypothesis is that the vortex core may be tilting away from the imaging axis [26], suppressing contrast in optical depth below our signal-to-noise threshold. Such a situation may arise if the trap is not perfectly spherical, and if the vortex is not aligned along a principle axis of the trap. Evidence in support of this hypothesis [27] will be presented in a future paper [28]. Through a combination of destructive and nondestructive imaging techniques we have obtained measurements of vortex dynamics in bare- and filled-core vortices in dilutegas BEC. Vortex precession frequencies show only modest dependence on the radius and content of the vortex core. Further measurements of vortex dynamics in condensates may reveal the rate of loss of angular momentum at finite temperatures, an indication of energy dissipation. Such measurements may suggest interpretations for “persistence of current” in condensates, further strengthening ties between BEC and superfluidity. In order to pursue these goals, we plan to extend the studies reported here to investigate higher-order dynamical behavior and to characterize the dissipative effects of finite temperatures. We gladly acknowledge helpful discussions with Murray Holland and Sarah McGee. This work was supported by funding from NSF, ONR, and NIST.
*Quantum Physics Division, National Institute of Standards and Technology. [I] D. R. Tilley and J. Tilley, Superfluidity and Superconductivity (IOP Publishing Ltd, Bristol, 1990), 3rd ed.; R. Donnelly, Quantized Vortices in Helium I1 (University Press, Cambridge, 1991). [2] D. J. Thouless et al., Int. J. Mod. Phys. B 13, 675 (1999). [3] G.A. Swartzlander, Jr. and C.T. Law, Phys. Rev. Lett. 69, 2503 (1992); Y.S. Kivshar et al., Opt. Commun. 152, 198 (1998). [4] M.R. Matthews et al., Phys. Rev. Lett. 83, 2498 (1999). [5] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000). [6] T. Isoshima and K. Machida, J. Phys. Soc. Jpn. 66, 3502 (1997). [7] D. S. Rokhsar, Phys. Rev. Lett. 79, 2164 (1997). [8] D. A. Butts and D. S. Rokhsar, Nature (London) 397, 327 (1999); H. Pu, C. K. Law, J. H. Eberly, and N. P. Bigelow,
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Phys. Rev. A 59, IS33 ( I 999); T. Isoshima and K. Machida, Phys. Rev. A 59, 2203 (1999); J. J. Garcia-Ripoll and V. M. PCrez-Garcia, Phys. Rev. A 60, 4864 (1999). 191 E . L . Bolda and D.F. Walls. Phys. Rev. Lett. 81, 5477 (1998). [ 101 B. Jackson, J . F. McCann, and C. S. A d a m , Phys. Rev. A 61, 013604 (1999). [ I I ] P.O. Fedichev and G.V. Shlyapnikov, Phys. Rev. A 60, R1779 (1999). [ I21 A. A. Svidzinsky and A. L. Fetter, Phys. Rev. Lett. 84, 59 I9 (2000); (private communication). [I31 J. Tempere and J.T. Dcvrcese, Solid State Commun. 113, 471 (2000). [I41 E. Lundh and P. Ao, Phys. Rev. A 61, 63612 (2000). [ 151 D. V. Skryabin, cond-mad0003041. [I61 S.A. McGee and M. J. Holland, cond-mad0007143. [ 171 J. J. Garcia-Ripoll and V. M. Ptrez-Garcia, Phys. Rev. Lett. 84,4264 (2000); V. M. PCrez-Garcia and J. J. Garcia-Ripoll, cond-mat/99 12308. [I81 J.E. Williams and M. J. Holland, Nature (London) 401, 568 (1999). [I91 The core-filling material is removed slowly enough (100 ms) that the vortex core has time to shrink adiabatically to its final bare size, but rapidly enough that the initial location of the bare core is well correlated with the location of the original filled core. [20] We use phase-contrast imaging for both the in-trap and expansion images to simplify imaging procedures. When imaging the expanded atom cloud, we obtain good signal when the probe is detuned three linewidths from resonance. [21] Because of our low trapping frequencies (7.8 Hz), expansions are correspondingly slow. We get larger final spatial distributions by giving the atoms a 6 ms preliminary “squeeze”at high spring constant, followed by a 50 ms expansion period. [22] F. Dalfovo and M. Modugno, Phys. Rev. A 61, 023605 (2000). [23] Tin-Lun Ho and V.B. Shenoy, Phys. Rev. Lett. 77, 3276 (1996); D.S. Hall et al., Phys. Rev. Lett. 81, 1539 (1998). [24] G.B. Hess, Phys. Rev. 161, 189 (1967). [25] Our measurements were performed at temperatures of T = 23(6) nK, and relative temperatures of T / T c = 0.8(1), where T, is the BEC critical temperature. Temperatures were measured using fits to trapped bare vortices where the core was not resolvable. [26] A. A. Svidzinsky and A. L. Fetter, cond-mat/0007139. [27] P. C. Haljan, B. P. Anderson, C. E. Wieman, and E. A. Cornell, in Proceedings of the Quantum Electronics and Laser Science Conjerence 2000, Sun Francisco, 2000, Abstract No. QPDl (Optical Society of America, Washington, DC, 2000). [28] P. C. Haljan, B. P. Anderson, C. E. Wieman, and E. A. Cornell (to be published).
VOLUME 85, NUMBER4
PHYSICAL REVIEW LETTERS
24 JULY 2000 ~
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Magnetic Field Dependence of Ultracold Inelastic Collisions near a Feshbach Resonance J. L. Roberts, N. R. Claussen, S.L. Cornish, and C. E. Wieman JILA, Natioiial Iiisriture of Stririclard,s arid Tecliriologj c i r d the Uriiversity of Coloi-ritlo, nrid rlir Drpcirfriieirr of PIiysir..r, Uiiiversiry of Colorado, BoLilrler; Color-riclo 80309-0440 (Received 28 February 2000) Inelastic collision rates for ultracold "Rb atoms in the F = 2, r i i f = -2 statc have been measured as a function of magnetic field. At 250 gauss (G). the two- and three-body loss rates wcre measured to be +(I711 K2 = (1.87 -C 0.95 2 0.19) X lo-'' cm'/s and K1 = (4.24-,129 t 0.X5) X cm"/s. respectively. As the magnetic field is decreased from 250 G towards a Feshbach resonance at 155 G . the inelastic rates decrease to a minimum and thcn increase dramatically, peaking at the Feshbach resonance. Both two- and three-body losses are important, and individual contributions have been compared with rhcory. PACS numbers: 34.50.-s. 03.75.Fi. OS.30.Jp. 32.80.Pj
Feshbach resonances have recently been observed in a variety of cold atom interactions, including elastic scattering [l], radiative collisions and enhanced inelastic loss [2], photoassociation [3], and the loss of atoms from a BoseEinstein condensate (BEC) [4]. By changing the magnetic field through the resonance, elastic collision rates can be changed by orders of magnitude and even the sign of the atom-atom interaction can be reversed [ S ] . The work in Ref. 143 has received particular attention because the BEC loss rates were extraordinarily high and several proposals for exotic coherent loss processes have been put forward [6-81. However, even ordinary dipole relaxation and three-body recombination are expected to show dramatic enhancements by the Feshbach resonance [9-141, and the calculations of these ordinary enhancements have never been tested. This has left many outstanding questions as to the nature of inelastic losses near a Feshbach resonance. How large are the dipole relaxation and three-body recombination near the Feshbach resonance and how accurate are the calculations of these quantities? How much of the observed condensate losses in Ref. [4] are due to these more traditional mechanisms and how much arise from processes unique to condensates? How severe are the "severe limitations" [4] that inelastic loss puts on the use of Feshbach resonances to change the s-wave scattering length in a BEC? The nature of the inelastic losses near Feshbach resonances also has important implications for efforts to create BEC in 85Rb, because these losses play a critical role in determining the success or failure of evaporative cooling. Thus, it is imperative to better understand the nature of inelastic collisions between ground state atoms near a Feshbach resonance. In this paper we present the study of the losses of very cold 85Rb atoms from a magnetic trap as a function of density and magnetic field. There are two types of inelastic collisions that induce loss from a magnetic trap. The first is dipolar relaxation where two atoms collide and change spin states. The second process is three-body recombination, where three atoms collide and two of those atoms form a molecule. Measuring the losses as a function of density
has allowed 11s to determine the two-body and three-body inelastic collision rates, while measuring the variation of these losses as a function of magnetic field has allowed us to find out how the Feshbach resonance affects them. In contrast to the work of Ref. [4], we observe a pronounced dip in the inelastic losses near the Feshbach resonance. This will make it possible to create "Rb BECs with a positive scattering length. To study these losses we needed a cold, dense "Rb sample. This was obtained through evaporative cooling with a double magneto-optic trap (MOT) system as described in Ref. [IS]. The first MOT repeatedly collected atoms from a background vapor and those atoms were transferred to another MOT in a low-pressure chamber. Once the desired load size was achieved, the MOTS were turned off and a baseball-type Ioffe-Pritchard magnetic trap was turned on, resulting in a trapped atom sample of about 3 X 10' F = 2, rnf = -2 "Rb atoms at 45 p K . Forced radio-frequency (rf) evaporation was used to increase the density of the atoms while decreasing the temperature. Because of the high ratio of inelastic/elastic collision rates in 85Rb and the dependence of the ratio on magnetic field ( B ) and temperature ( T ) ,trap conditions must be carefully chosen to achieve efficient cooling. We evaporated to the desired temperature and density (typically 3 X 10" cm-3 and 500-700 nK) at a final field of B = 162 gauss (G) [16]. We then adiabatically changed the dc magnetic field to various values and measured the density of the trapped atom cloud as a function of time. We observed the clouds with both nondestructive polarization-rotation imaging (171 and destructive absorption imaging. In both cases, the cloud was imaged onto a CCD array to determine the spatial size and number. The nondestructive method allowed us to observe the time evolution of the number and spatial size of a single sample. The destructive method required us to prepare many samples and observe them after different delay times, but had the advantages of better signal to noise for a single image and larger dynamic range. A set of nondestructive imaging data is shown in Fig. 1. In
728
0 2000 The American Physical Society
0031-9007/00/85(4)/728(4)$15.00 601
602
PHYSlCAL REVIEW LETTERS ~~
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Nuinbcr of atoms versus time ( 0 ) taken by polarization~rotation imaging at I59 G. Fits to both purely two-body (dashcd line) and purely thrcc-body (solid line) inelastic rates are shown. illustrating the difficulty in separating two- and three-body loss processes.
addition to these techniques. we made a redundant check of the number by recapturing the atoms in the MOT and measuring their Fluorescence [ 181. We measured the time dependence of the number of atoms and the spatial size, and from these measurements determined the density. We assigned a 10% systematic error to our density determination, based primarily on the error in our measurement of number. The value of B was measured in the same way as in Ref. [ I ] : the rf frequency at which the atoms in the center of the trap were resonantly spin flipped was measured and the Breit-Rabi equation was then used to determine the magnitude of B . The magnetic field width of the clouds scaled as TI/' and was 0.39 G FWHM at 500 nK. In each data set, we observed the evolution of the sample while a significant fraction (20%-35%) of the atoms was lost. The temperature of the sample also increased as a function of time. This heating rate scaled with the inelastic rates and therefore varied with B . Since the volume and temperature in a magnetic trap are directly related, the change in the volume (typically 50%) with time was fit to a polynomial-usually a straight line was sufficient within our precision. The number ( N ) as a function of time was then fit to the sum of three-body and two-body loss contributions given by
Here, ( n ) = / n 2 ( x )d'x is the density-weighted density and ( n 2 )= $ / 1 z 3 ( x ) d3x. The time evolution of the volume is contained in the computation of the density as a function of time. The background loss rate is given by T. It was independently determined by looking at low-density clouds for very long times. Typically, it was 450 s, with some slight dependence on B (<20%), particularly near the Feshbach resonance. The change in the combined inelastic rates as a function of B around the Feshbach resonance is shown in Fig. 2(a). All of the inelastic data shown in Fig. 2(a) were taken
~~
24 JULY2000
~
with initial temperatures near 600 nK. The points less than 157 G were taken with initial densities within 10% of 1 X 10" cm-3, while the points higher than 157 G were taken with initial densities that were 2.7 X 10" omp3. The decrease in initial density below 157 G was due to ramping through the high inelastic loss region after forming the sample at 162 G. Because of this density variation, we have used a different weighting, parameter p , in the sum of the two- and three-body rate constants for the two regions in Fig. 2(a). Also shown in Fig. 2(a) is the elastic rate determined previously [I]. This elastic rate was measured by forming atom samples similar to the ones in this work (but at much lower density), forcing them out of thermal equilibrium, and then observing their reequilibration time. The shape of the inelastic rate vs B roughly follows that of the elastic rate vs B . In particular, the peak of the inelastic rate occurs at 155.4 t- 0.5 G , identical to the position of the elastic peak at 155.2 2 0.4 G within the error. Also, just as is the case for the elastic rates, the inelastic loss rates around the Feshbach resonance vary by orders of magnitude. The peak in the inelastic rate is much less symmetric, however. Another interesting feature is that the loss rates not only increase near the elastic rate peak but decrease near its minimum. The field where the loss is minimum, 173.8 2 2.5 G, is higher than the minimum of the elastic rate at 166.8 2 0.3 G. Even though the two- and three-body inelastic rates have different density dependencies, it is difficult to separate them. Figure 1 shows how a purely two-body or purely three-body loss curve fits equally well to a typical data set. An excellent signal-to-noise ratio or, equivalently, data from a large range of density are required to determine whether the loss is three-body, two-body, or a mixture of the two. To better determine density dependence, we decreased the initial density by up to a factor of 10 for several B values. For fields with relatively high loss rates, the signal-to-noise ratio was adequate to distinguish between two- and three-body loss. However, where the rates were lower this was not the case. Figure 2(b) shows the two-body inelastic rate determined from the density-varied data and Fig. 2(c) shows the same for the three-body rate. The character of the inelastic loss clearly changes as one goes from higher to lower field (right to left in Fig. 2). On the far right of the graph, at B = 250 G, the inelastic rate is dominated by a three-body process. From B = 250 G down to B = 174 G the inelastic rate decreases to a minimum and then begins to increase again. Near the minimum, we could not determine the loss character, but at 162 G the losses are dominated by a twobody process at this density. At 158 G the two-body process is still dominant and rising rapidly as one goes toward lower field. However, by 157 G it has been overtaken by the three-body recombination that is the dominant process at 157-145 G. At lower fields, both two-body and tbreebody rates contribute significantly to the total loss at these densities.
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PHYSICAL REVIEW LETTERS
Along with varying the density, the initial temperature was varied at B = 145, 156, 160, and 250 G. There was no significant rate change in the loss rate for temperatures between 400 and 1000 nK for the 160 and 250 G points. The combined loss rate increased by a factor of 8 at 156 G from 1 p K to 400 nK, and by a factor of 2 at 145 G. This temperature dependence near the peak is expected in analogy to the temperature dependence of the elasticrates [ 191. The fact that the loss rates near the peak exhibit both a two- and three-body character plus a temperature dependence and considerable heating makes interpreting the data challenging. This introduces additional uncertainty that is included in the error bars in Figs. 2(a)-2(c). A calculated dipolar loss rate is compared with the data in Fig. 2(b). This was calculated by finding the S-matrix between the trapped and untrapped states using carefully determined Rb-Rb molecular potentials [ 121. Keeping in mind the significance of error bars on a log plot, the agreement is reasonably good. In the region between B = 160 and 167 G where the determination of K2 is not complicated by three-body loss and temperature dependence, the agreement is particularly good. Likewise, we show a prediction of the three-body recombination rate in Fig. 2(cj [lo]. It is predicted that the recombination rate scaIes as the s-wave scattering length to the fourth power (a:) for both positive and negative a,$, although with a smaller coefficient for the positive case [9,10,14]. Since a, varies across the Feshbach resonance [see Fig. 2(a), upper plot], the recombination rate is expected to change [20]. Temperature dependence has not been included in the prediction. Qualitatively, the main features of the predicted three-body recombination match the data. The three-body rate decreases as a, decreases,
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FIG. 2. (a) Elastic rate (upper plot and 0 symbol) and s-wave scattering length a , (upper plot and solid line) from Ref. [I], and total loss rates (lower plot) vs magnetic field. The total loss rate is expressed as a sum of the two- and three-body loss as Kz + pK3. Because of initial density differences caused by ramping across the peak, = 1.6 X 10" cm-3 for the points below B = 157 G (filled circles) and 4.0 X 10" cm-3 for the points above (filled triangles). The vertical lines represent the positions of the elastic rate maximum and minimum at B = 155 and 167 G,.respectively. (b) The determination of the two-body inelastic rate for several B fields. The theory prediction from Ref. [I21 is shown as a solid line. The open circles (0) are the two-body rates determined from the total loss from (a) by assuming (not explicitly measuring) that the loss between 162 and 166 G is predominantly two-body. These points are added to aid in comparison with theory. (c) The determination of the three-body inelastic rate. The points with down arrows (1) are to be interpreted as upper limits on the three-body rate. The solid line here is a prediction of the loss rate from Ref. [lo]. The open circles in (c) are similar to those in (b), only in (c) we assume (but again do not measure) that the loss above 175 G is predominantly three-body. The error bars on the 250 G point are relatively small because a large amount of data was taken there. In addition to the statistical errors shown in (a)-(c), there is another 10% systematic uncertainty in K2 and 20% in K3 due to the estimated error in our density determination.
604
VOLUME85, NUMBER4
24 JULY 200U
PHYSICAL REVIEW LETTERS ~p~
and it increases rapidly where ( 1 , diverges at N = 155 G. From 155 to 167 G, a, is positive and the three-body loss is much smaller than it is at B fields with a comparable negative a,?,as expected from theory. This overall level of agreement is reasonable given the difficulties and approximations in calculating three-body rates. The marked dependence on B of both the elastic and inelastic rates has important implications for the optimization of evaporative cooling to achieve BEC in "Rb. As the density increases with evaporative cooling, the absolute loss rate, which is already much greater than in "Rb and Na, becomes larger. However, the important ratio of elastic rate to inelastic rate is both temperature and density dependent and magnetic field dependent. The fact that the losses are dramatically lower at fields above the Feshbach resonance suggests that it should be possible to devise an evaporation path that will lead to the creation of a "Rb BEC [21]. The 85Rb Feshbach resonance has a profound effect on the two- and three-body inelastic rates, changing them by orders of magnitude. Far from the resonance at 250 G, the two-body and three-body loss rates are measured to be K2 = (1.87 t 0.95 2 0.19) X cm'/s and K3 = (4.24!::;: t 0.85) X cm6/s, respectively. Here the first error is the statistical error and the second in the systematic uncertainty due to the number determination. Near the resonance, the two- and three-body rates change by orders of magnitude. The dependence of the inelastic rates on magnetic field is similar in structure to the dependence of the elastic rate: the maxima in both rates occur at the same field, while the minima are close but do not coincide. The total loss is a complicated mixture of both two- and three-body loss processes. They have different dependencies on field so both have field regions in which they dominate. We are pleased to acknowledge useful discussions with Eric Cornell, Jim Burke, Jr., Chris H. Greene, and Carl Williams. We also thank the latter three and their colleagues for providing us with loss predictions. This research has been supported by the NSF and ONR. One of us (S. L. Cornish) acknowledges support from the Lindemann foundation.
.
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111 J . L. Roberts e r n / . , Phys. Rev. Lett. 81, 5109 (1998). [2] V. Vtiletit. et nl., Phys. Rev. Lett. 82, 1406 (1999); V. Vuletii et al., Phys. Rev. Lett. 83, 943 (1999). [ 3 ] Ph. Courtcille et nl.. Phys. Rev. Lett. 81, 69 (1998). 141 S. Inouyc et nl., Nature (London) 392, I5 I (1998); J. Stengcr et cil., Phys. Rev. Lett. 82, 2422 (1999). IS] E. Tiesinga ef a/.,Phys. Rev. A 46, RI I67 (1992); E. Tiesinga. B. J. Verhaar, and H.T.C. Stoof, Phys. Rev. A 47, 41 14 (1993). [6] E. Timmcrmans er a/., Phys. Rev. Lett. 83, 269 I (1999). [7] F. A. van Abcclcn and B. J. Vei-haar, Phys. Rev. Lett. 83, 1550 (1999). 181 V. A . Mii-ovsky Bf a/., Phys. Rev. A 60, R765 (1999). 191 E. Niclsen and J.H. Macek, Phys. Rcv. Lett. 83, 1566 (1999). [lo] B. D. Esry, C. H. Grecnc, and J. P. Burke. Jr., Phys. Rev. Lett. 83, 1751 (1999). [ I I ] J. P. Burke, Jr. er d., Phys. Rcv. Lett. 80, 2097 (1998). 1121 F. H. Mies ef a/., J. Res. Natl. Inst. Stand. Technol. 101, S21 (1996). The Rb-Rb potential has been updated since this literaturc was published in light of additional Rb data [Carl J. Williams, NIST, Gaithersburg (private communication)]. [I31 A.J. Moerdijk, H.M.J.M. Boesten, and B.J. Verhaar, Phys. Rev. A 53, 916 (1996). [ 141 P. 0. Fedichev, M. W. Reynolds. and G. V. Shlyapnikov, Phys. Rev. Lett. 77, 2921 (1996). [l5J C.J. Myatt et nl., Opt. Lett. 21, 290 (1996). [ 161 The choice of 162 G for the final stage of evaporation was made somewhat arbitrarily. 1171 C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 78, 985 (1997). [I81 C.G. Townsend et nl., Phys. Rev. A 52, 1423 (1995). [I91 For instance, see J.R. Taylor, Scatrering Theory: The Quantum Theory of Nonrelativistic Collisions (Geiger, Malabar, Florida, 1987), pp. 89, 352-353. [20] The predicted change in u , ~with field is obtained using the potential determined in Ref. [ I ] by J.P. Burke, Jr. [NIST, Gaithersburg (private communication)]. That potential predicts a , = -422 bohr away from the Feshbach resonance. Private communications with C. J. Williams indicate that n, = -600 bohr away from the resonance may be more consistent with all the available Rb data. [21] In fact, after the completion of this work, a 85Rb BEC with a highly adjustable scattering length was created and studied (to be published).
73 1
VOLUME86, NUMBER19
PHYSICAL REVIEW LETTERS
7 MAY2001
Controlled Collapse of a Bose-Einstein Condensate J. L. Roberts, N. R. Claussen, S . L . Cornish, E. A. Donley, E. A. Cornell,* and C. E. Wieman JILA , Nat ioI in1 lrist it LI te o f Sta litla rcl.c a1id Techriology and f he Urzive rsity of Co lo rado, %oulder; Colorcido 80309-0440 a i i c ! the Departriieiit of Physics, University of Colorcido, Boulder; Colorado 80309-0440 (Received 22 January 2001)
The point of instability of a Bose-Einstein condensate (BEC) due to attractive interactions was studied. Stable 85RbBECs were created and then caused to collapse by slowly changing the atom-atom interaction from repulsive to attractive using a Feshbach resonance. At a critical value, an abrupt transition was observed in which atoms were ejected from the condensate. By measuring the onset of this transition as a function of number and attractive interaction strength, we determined the stability condition to be -~ = 0.459 2 0.012 -C 0.054, slightly lower than the predicted value of 0.574. DOI: 10.1 103/PhysRevLett.86.42I 1
PACS numbers: 03.75.Fi, 05.3O.Jp, 32.80.Pj, 34.50.-s
The creation of Bose-Einstein condensation in 85Rb [ 1 J allows detailed control of the atom-atom interactions via a Feshbach resonance [2,3]. The magnitude and sign of the interaction between atoms can be tuned to any value: large or small, repulsive or attractive. The presence of attractive interactions between the atoms has a profound effect on the stability of a Bose-Einstein condensate (BEC), since a large enough attractive interaction will cause the BEC to become unstable and collapse in some way. In a confining potential, the kinetic energy of the condensate counteracts the attraction and stabilizes the condensate for small enough attractive interactions. However, as the magnitude of the attractive interaction energy is increased, either by increasing the interaction strength or the number of atoms, the attractive interaction will eventually overwhelm the kinetic energy and make the condensate unstable. Our ability to tune the magnitude of the attractive interactions allows us to study the onset of this instability and the subsequent behavior in a highly controlled fashion. In Ref. 141, Ruprecht and co-workers predict when the condensate will become unstable. The prediction can be cast in the following form:
The condensates grow in number until they reach a critical value and then their number decreases. Following the decrease, the number begins to grow again and the growth and collapse cycle is repeated as atoms are fed into the condensate from the large thermal cloud that is always present. These cycles are observed to be variable, and so a statistical treatment of postcollapse ensembles has been used to examine the ’Li growth and collapse process [8]. Here we used the Feshbach resonance in 85Rb to tune the scattering length by changing the magnetic field. After forming stable and nearly thermal-atom-free condensates with positive scattering length (repulsive interactions) and a particular number of atoms, we tuned the scattering length to a selected negative value (attractive interactions) in the vicinity of the expected critical value to study the instability. This allowed us to study the onset of instability and the collapse dynamics in a very controlled manner. From these studies we were able to confirm the functional form of Eq. (1) and measure the stability coefficient k . Also, the transition from stable to unstable condensates was found to be very sharp. Finally, the number of atoms remaining in the condensate after the collapse event was measured as a function of the initial number. The first stage in our experiment was to form a stable ”Rb condensate. We used a double magneto-optic trap, or MOT, system [9] to load a Ioffe-Pritchard-type “baseball” trap with 3.5 X 10’ F = 2 mF = -2 85Rb atoms for evaporative cooling. The frequencies of the trap were 17.24 X 17.47 X 6.80 Hz. Following the somewhat unusual evaporative cooling scheme discussed in Ref. [ 11 we created a BEC of -10000 atoms at a temperature less than 6 nK. The magnetic field was then ramped very slowly (800 ms) to a value where the scattering length was near zero. We then ramped the field [lo] linearly in 200 ms to the desired negative scattering length value and waited at that field for SO ms. For these small scattering lengths, the condensates were smaller than the 7 p m resolution limit of our imaging system and we found that errors in the measurement of number were unavoidable when measuring subresolution limit condensates. To avoid these errors, after
N,,lal
=k , (1) a ho where k is a dimensionless constant, a is the s-wave scattering length that characterizes the strength of the atomatom interactions, N,, is the maximum number of atoms is that will be stable in the condensate, and ah,, = the mean harmonic oscillator length, which characterizes the kinetic energy in the trap (5is the geometric mean of the trap frequencies). Several others [5] also calculate k = 0.574 using a variety of methods. Condensates with attractive interactions have been observed in 7Li [6], and the maximum number has now been observed to be consistent with the predictions in Refs. [4] and [5]. There are several differences between these 7Li experiments and our work, however. In 7Li, condensates are formed from a rapidly cooled thermal cloud at a negative scattering length whose value remains fixed, and condensate formation kinetics has been a central study [7].
0031-9007/01/86(19)/4211(4)$15.00
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the 50 ms wait we expanded the condensate by changing the magnetic field to achieve a large positive scattering length (-80 nm). This caused the condensate to expand outward rapidly due to the mean-field repulsion and become larger than the resolution limit of the imaging system. After the condensate had expanded sufficiently, we turned off the magnetic trap suddenly and imaged the sample using destructive on-resonance absorption imaging to determine both the number of atoms and the shape of the condensate. This was repeated for different values of the magnetic field and numbers of condensate atoms. The major source of difficulty in this experiment was the variation in the number of atoms in the BEC at the end of evaporative cooling. Several steps were necessary to reduce the variation to under 10%: moving all ferromagnetic materials away from the vicinity of the magnetic trap, reducing table and trap coil vibrations, stabilizing the dc magnetic field to better than 6 ppm, and adding another stage of sample preparation. This additional stage consisted of waiting 2-10 sec at the a - 0 field to allow density-dependent inelastic losses [1 I ] to smooth out number variations. The condensates showed no effect from their excursion into the region of negative a as long as la1 was below a critical value. Above that value, two dramatic changes in the condensate were observed. First, there was a substantial decrease in the number of atoms in the condensate [12]. Second, the condensates’ ratio of axial to radial widths was variable and usually different from the initial ratio, indicating that the condensates were oscillating in a highly excited state. We refer to this abrupt change as the condensate “collapse.” The transition from stable to unstable is obviously very sharp (see Fig. 1). We were not able to determine any width to the transition beyond the 4 mG apparent width arising from the shot-to-shot variations in initial condensate number, and we never observed “partial” collapses. The data in Fig. 1 and other sets like it also show the fraction of atoms that remain after the collapse event for an initial number around 6400. That fraction was 0.58(3). Changing the starting number to -2600 atoms gave a fraction remaining of 0.76(3). As indicated by the standard deviation bars (Fig. l), the standard deviation in the observed number of atoms after the collapse was larger than the precollapse variation. For the 6400 atom case the initial number had a fractional standard deviation of 7%, while the fraction remaining varied by 20%. For the 2600 atom case, the precollapse and postcollapse variations were 9% and 20%, respectively. The expanded postcollapse condensates were safely above the resolution limit so we do not believe that this effect is due to systematic errors in our imaging system. To examine the validity of the theory prediction in Eq. (l), we varied the number of atoms that were initially in the condensate and then determined the minimum magnetic field at which that number of atoms would collapse. 4212
7 MAY2001
PHYSICAL REVIEW LETTERS 1.1
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166.00
Magnetic Field (Gauss) FIG. 1. Transition from stable to unstable condensates. This figure shows the fraction of atoms remaining as the magnetic field was ramped to higher magnetic fields ( i c , stronger attractive interactions). The hatched region shown is the expected 4 mG width due to the measured initial number variation. The initial BEC was kept close to 6400 atoms. At the field in the hatched region, the data fell into two groups and the individual points (filled triangles) are displayed rather than the average of those points. The solid bars show the standard deviation of the points to the left of the hatched region (left bar) and those to the right of the hatched region (right bar).
Since the scattering length varies linearly with magnetic field for the negative scattering lengths of interest, Eq. (1) would predict that a plot of vs the collapse field would be a straight line. The slope of this line would then determine the stability coefficient k . The presence of drift and random variation in initial number complicated the measurement by making it impossible to know the exact number of atoms that were in the condensate before the ramp to the negative a . When a collapse event is observed, the initial number of atoms in the precollapse condensate is known only as well as the shot-to-shot reproducibility of the experiment. To monitor the initial number, normalization points, for which the condensates were prepared, handled, and measured in an identical manner except that they were not ramped to negative a , were frequently interspersed with the points with a ramp to negative a . Two days’ negative a data are shown in Figs. 2(a) and 2(b). From data sets such as these we are able to measure the stability coefficient k and test whether or not the onset of instability scales with N u . Figure 2(c) shows a summary of several days worth of data taken in the same fashion. As is evident in Fig. 2 , the onset of instability scales well with the product of Nlul. The all-or-nothing nature of collapse led us to use an unusual approach to determine the stability coefficient k in the presence of variations in the initial number. Instead of fitting a line to a set of random data, we had to determine a boundary between collapse and noncollapse events. Data were taken at fixed high number (-6500) and a selected low number that was varied. Many points were taken at the
k
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I
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0.0001 165.9
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Magnetic Field (Gauss) FIG. 2. Determining the onset of collapse as a function of condensate number and magnetic field. (a) and (b) show data
sets used to determine the stability coefficient via the boundary that divides the data between collapse (0)and noncollapse ( 0 ) events. In (a), the data were concentrated in two field regions to accurately measure the slope of the boundary line. In (b), the initial number was varied over a larger range to illustrate the functional form of the onset of instability. Because of initial number variations, some collapse points appear to be on the wrong side of the stability boundary. ( c ) shows stable condensate numbers known to be on the verge of instability (see text) as determined from several data sets like those in (a) and (b). The fit to these points provides the value of k . critical magnetic fields at which the initial number variation resulted in some condensates collapsing while others remained stable. This ensures that some of the stable condensate points are just on the verge of collapsing. The two stable condensate data points closest to the boundary, one from the low-number set and one from the high-number set, were then used. The magnetic field drifted by tens of m G from day to day due to variations in the ambient field. To allow day-to-day comparisons, the value of the highnumber collapse field was used to measure the field drift. The remaining closest-to-boundary collapse points for various numbers are shown in Fig. 2(c) and are fit to determine a slope. While this is not the most efficient way to determine the slope, the drift in initial number complicates
more sophisticated analyses and this method already gives a statistical error for k that is much smaller than the systematic errors. The uncertainty in the individual slope determination was estimated by using a standard Monte Carlo simulation [ 131 with the observed initial number variation as an input parameter. The uncertainty predicted by the simulation is consistent with the measured day-to-day variation in the slope. The sirnulation also confirms that there was no significant systematic error introduced in this method of analyzing the data. The average of all of the data gives a slope of 0.001 26(3) (atoms G ) - ' for l/Ncr vs magnetic field. The change in magnetic field was determined as in Ref. [2], with a negligible uncertainty of 0.5%. The number of atoms was determined from the optical depth of the cloud taking into account the intensity and frequency of the probe beam, optical pumping effects, and the small Doppler shift arising from scattering miiltiple photons. The intensity and frequency of the probe were systematically varied to check the scaling of these effects, and from these measurements, we estimate the systematic error in determining the number to be 210%. To convert Gauss to scattering length, the scattering length vs magnetic field was determined using the rethermalization time technique as in Ref. [2], but with better control of density, temperature, and reproducibility. Rethermalization data were taken at magnetic fields B = 168, 170, and 251 G. Those rethermalization times were combined with the number calibration of the absorption imaging system and the relation between elastic cross-section and rethermalization time [141 to determine the scattering length at those three fields. These scattering lengths were then fit to the functional form A of the Feshbach resonance a = ab,(l using the previously determined peak ( B p k ) and the width (A) to measure the background scattering length to be -20.1 nm [15]. From this we determined that the change in scattering length vs magnetic field near 166.05 G is -1.770(46) nm/G, or -33.4 bohr/G. Combining all of these calibrations with the measurement of the slope of l/NCr vs magnetic field gives a value of k = 0.459 f 0.012 (statistical) 2 0.054 (systematic). The uncertainty is dominated by the systematic uncertainty in the determination of the number [ 161. This value is 2 standard deviations lower than the predicted value of 0.574, which would indicate that the condensates were collapsing at a slightly lower value of interaction strength than was predicted. We investigated the effects of both finite temperature and condensate dynamics on the condensate stability. There have been predictions for the effect of finite temperature on the condensate stability in 7Li [17,18] and 85Rb [18]. Reference [ 181 predicts that the shift in the stable condensete number between zero temperature and the temperature at which we took our data should be less than 1%. We deliberately created hot clouds that had roughly equal numbers of condensate and thermal atoms and performed
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a collapse measurement. This was roughly a factor of 10 more thermal atoms than were present in the previous measurements. We could detect no change in the collapse point between the hot and cold condensates. To search for a dependence on condensate dynamics, we increased the initial spatial size of the condensate by starting with an initial scattering length of + I 1 nm instead of zero and used ramp times of both 200 ms and 1 ms to magnify any possible effect of dynamics. The previous data were taken using a magnetic field ramp duration (200 ms) that was more than twice the period of the lowest collective excitation of the condensate, to avoid exciting the condensate prior to the collapse. This condition is true except for very near the point of instability where the lowest collective excitation frequency must go to zero [ 191. For the 200 ms ramp there was no detectable change in the collapse point between ramps starting at a = l l nm and a = 0 nm. The 1 ms, initial a = + I 1 nm ramp gave substantial precollapse excitation to the condensate but the apparent value of k decreased by only 8%. We have observed that condensates are stable with sufficiently small attractive interactions. Once the interactions are increased beyond a sharply defined value, the condensates collapse and rapidly lose 25%-40% in atom number. The exact point at which the onset of collapse = occurs has been measured and determined to be 0.459 ? 0.012 ? 0.054, which is 25(12)% lower than the predicted value. The study of the precise nature of the collapse dynamics and the mechanisms by which the atoms are lost will be a future topic of work by this group. The authors acknowledge the assistance of John Obrecht in developing methods to analyze our data. One of us (S. L. Cornish) acknowledges the support of the Lindemann Foundation. This work was supported by the ONR and NSF.
+
%
*Quantum Physics Division, National Institute of Standards and Technology. [I] S . L. Cornish et a/., Phys. Rev. Lett. 85, 1795 (2000). [2] J. L. Roberts et a/., Phys. Rev. Lett. 81, 5109 (1998). [3] W.C. Stwalley, Phys. Rev. Lett. 37, 1628 (1976); E. Tiesinga, A. Moerdijk, B. J. Verhaar, and H. T. C. Stoof, Phys.
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Rev. A 46, RI 167 (1992); Ph. Courteille et al., Phys. Rev. Lett. 81, 69 (1998); S. Inouye etul., Nature (London) 392, 151 (1998). [4] P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, 4704 (1995). [5] F. Dalfovo and S . Stringari, Phys. Rev. A 53, 2477 (1996); M. Houbiers and H.T.C. Stoof, Phys. Rev. A 54, 5055 (1996); R.J. Dodd ef al., Phys. Rev. A 54, 661 (1996); V.M. Perez-Garcia, H. Michinel, and H. Herrero, Phys. Rev. A 57, 3837 (1998); A. Gamrnal, T. Frederico, and Lauro Tornio, Phys. Rev. E 60, 2421 (1 999); L. BergC, T. J. Alexander, and Y.S. Kivshar, Phys. Rev. A 62, 023607 (2000); A. Eleftheriou and K. Huang, Phys. Rev. A 61, 043601 (2000). See also M. Ueda and K. Huang, Phys. Rev. A 60, 3317 (1999). [6] C.C. Bradley, C.A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 78, 985 (1997). [7] J.M. Gerton et nl., Nature (London) 408, 6813 (2000). [8] C.A. Sackett, J. M. Gerton, M. Welling, and R. G. Hulet, Phys. Rev. Lett. 82, 876 (1999). [9] C. J. Myatt et al., Opt. Lett. 21, 290 (1996). [ 101 Changing the bias magnetic field changed the trap frequencies by less than 0.07%. [ I I ] J.L. Roberts et al., Phys. Rev. Lett. 85, 728 (2000). [ 121 When comparing the number of atoms before and after the collapse event, we measured only atoms that remained in a central component. In other words, we were not including the atoms in the burst described in Ref. [I]. 1131 See, for instance, W.H. Press et al., Numerical Recipes: The Art of Scientijic Computing (Cambridge University Press, Cambridge, England, 1989), pp. 529-538. 1141 B. DeMarco, J.L. Bohn, J.P. Burke, Jr., M. Holland, and D. S . Jin, Phys. Rev. Lett. 82, 4208 (1999). 1151 J. L. Roberts, J. P. Burke, Jr., N. R. Claussen, S. L. Cornish, E. A. Donley, and C. E. Wieman, e-print physics/0104053. 1161 A precise independent determination of the scattering length would provide a calibration of number through the rethermalization measurements and therefore lead to a corresponding improvement in the determination of k . [ 171 M. J. Davis, D. A. W. Hutchinson, and E. Zaremba, J. Phys. B 32, 3993 (1999); M. Houbiers and H. T. C. Stoof, Phys. Rev. A 54, 5055 (1996). [18] E.J. Mueller and G. Baym, Phys. Rev. A 62, 053605 (2000). [19] See, for example, the review by E Dalfovo, S . Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
PHYSICAL REVIEW A. VOLUME 64, 024702
Improved characterisation of elastic scattering near a Feshbaeh resonance in
S5
Rb
J, L. Roberts. 1 James P. Burke, Jr.,-' N. R. Claussen,' S, T,, Cornish/'* E. A. Doniey.' and C. E. Wienian 1 *JILA, National Instiim? of Standards and Technology wid the University of Colt.trf.nio, tifU'J !)i.'f>ni-tmen! of Physics, Unive<'$it\< of Colorado. Bouidti; Colorado 80309-044(1 1 Atumic Pliysict Division. National Institute nj Standards and Technology, Gtiithsrshurz. Maryland 20tJ9V-0-il43 (Received I March 2001; published 5 July 2001) We report extensions anil corrections to the measurement of the Feshbaeh resonance in 's"Rb cold atom collisions reported earlier [J. L. Roberts el a!., Phys. Rev. Lett. 81, 5109 (H98'i]. In addition to a better determination of the position of the resonance peak I54.9M) G| and its width [11.0(4) Gj. improvements in our techniques now allow the measurement of the absolute si;"c of the clastic-scaUfring rate, this provides a measure of the ,v-wave scattering length as a function of magnetic field near the Feshbaeh resonance and constrains the Rb-Rb interaction potential. DOI: 10.1103/PhysRevA.64.()247()2 The presence of a Feshbaeh resonance in cold atomic collisions allows the clastic and inelastic collision properties to be dramatically altered as an external field is changed [1], In particular, the .s-wave scattering length (<-/), which completely characterizes elastic scattering in ultraeold collisions, can be tuned over a large range. In 1998, several groups reported the observation of external-magnetic-field Feshbaeh resonances in ultraeold collisions in "Rb [2,3] and in ~ J Na [4]. These tunable interactions have been used to study the stability [5] and dynamics [6j of 9-Tlb Bose-Einstein condensatcs (BECs) with attractive interactions. Interpreting the results of these recent studies and other future studies in S5 Rb requires a more accurate knowledge of a as a function of magnetic field (B) near the Fesbbach resonance than that obtained in Ref. [2], In order to improve the accuracy of the determination of a versus B, we extended and improved the measurements performed in Ref. [2]. The cold collision Rb-Rb interatomic potential and hence the s-wave scattering length vs magnetic field is characterized primarily by three parameters: the triplet scattering length a 7-, the singlet scattering length as, and the Cg van der Waals coefficient [7,8]. In our previous work, we measured two characteristics of the prominent Feshbaeh resonance in N5Rb: the magnetic field at which the magnitude of the s-wave scattering .length was maximum (Bpeak) and the magnetic field at which the .v-wave scattering, length was zero (B2eru). This was done by measuring the relative elasticscattering cross section as the magnetic field was changed. This tightly constrained two of the three parameters, aT and as, for a given C6. However, C6 was much less constrained, even when combined with the results from another experiment [9]. At the time, the temperature of the trapped atom clouds was too hot to allow the determination of the absolute value of the scattering length [2], We have now been able to improve the measurements made in Ref. [2] in two ways. First, the measurement of the stability of BECs with attractive atom-atom interactions (negative a) has allowed a much more precise measurement
*Prcsent address: Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road. Oxford 0X1 3PU, United Kingdom. .1050-294 7/2001 /64(2>/024702<3)/S2Q,00
of B-.,,.,,. This more precise determination showed the presence of some small errors in the previous- measurement. Second, we now have a Far greater range of density and temperature of the magnetically trapped atom clouds available, and much better control over those quantities. The reproducibtlity and calibration were also improved. The absolute value of the scattering length could now be accurately determined. This additional constraint allows us to completely characterize a vs B near the Feshbaeh resonance. The more precise measurement of the position of B.c..f, was a natural outgrowth of our work determining the stability of magnetically trapped BECs with attractive atom-atom interactions [5j. When large enough attractive interactions (defined by Na) are present, the BCCs will become unstable and collapse. The collapse criterion is given by the relation A' f r ja| = /3, where Ncr is the number of atoms in the condensate and /3 = 1.41 /j,m for our magnetic trap. In Ref. [5], A;,.r was determined as a function of magnetic field down to very small a. A simple extrapolation to the point where N.... = ='-• determined the position of B7l,rn. The value of B-cn, determined in this way is 165.85(5) G. a factor of 6 more precise than before. This increase in precision occurs both because the stability measurement is more sensitive and it avoids temperature-dependent shifts. The new determination of Bzero disagrees with the previous measurement [2] B.ero= 166.8(3) G by more than the stated errors. A small error in the magnetic-field calibration in Ref. [2] is responsible for 0.3 G of the difference. The remainder of the difference is likely due to temperature shifts of the elastic cross section and the temperature-dependent asymmetry in the cross section versus the magnetic field near Bzfm DO] that were not adequately included in Ref. [2 j. The new magnetic field calibration also shifts the measured position of B,jeak to 154.9(4) G. The absolute value of the scattering length was determined at several values of the magnetic field, using ''crossdimensional mixing" measurements as in Ref. [2], In this technique, a cloud of cold atoms, created by evaporative cooling as in Ref. [11], is perturbed from its equilibrium energy distribution. The time for the cloud to equilibrate by elastic collisions is then measured. In the zero-temperature limit, the elastic collision cross section (
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610 PHYSICAL REVIEW A 64 024702
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Equilibration Time (sec) FIG. I . Relaxation of the aspect ratio to equilibrium. The data shown here were taken at B= 169.7 G. Each data point (•) represents a single destructive absorption measurement. The solid line shows the fit to the data used to determine the elastic-scattering cross section and hence the scattering length. The deviation of the data points from the fit around 3 4 seconds is due to the oscillations between the two radial directions as described in the main text.
tically reduce the elastic collision rate. By cutting faster with the if "knife" used for evaporative cooling than the cloud could equilibrate, it was possible to remove energy in just the radial direction of the axially symmetric magnetic trap [2]. The width of the cloud in the radial direction was decreased until the radial energy had been reduced to 0.6 of the axial direction, and then the bias magnetic field was ramped adiabatically to a selected value and the cloud was allowed to equilibrate for a fixed time. The size, number, and shape ol' the cloud were measured using on-resonance destructive absorption imaging. This procedure was repeated for different equilibration iimes and the rate of the relaxation of the aspect ratio of the clouds (the ratio of the axial width to the radial width) to equilibrium was recorded. The rate of flow of energy out of any particular direction, axial or radial, is proportional to the difference between the average energy in that direction and the equilibrium value multiplied by the factor r-(2/K)(n}a(v} (I.e., dE-,!dt = -T(Ej-Kal!!!), i = x.y,z]. In the factor F, («} is the average density defined by fn(x')2d?'x/fn(x)d*x, ( v ) is the root-mean-square relative speed \l}6kBTirnm, (T=8ira 2 is the elastic collision cross section, aod K is a numerical factor. The parameter K is weakly dependent on the difference between radial and axial energy, but is always within 2% of the value 2.50. it is calculated using classical transport theory [12] with the assumption that the velocity and density distributions can be described in each direction by a single parameter, an effective temperature that is equal to the average energy divided by the Boltzmann constant k K . A Monte Carlo simulation, agrees with this classical calculation at the 4% level. [1.3], The relaxation of the aspect ratio was fit appropriately to determine F and hence a. F is not strictly constant as a function of equilibration time since the density and temperature of an atom cloud varied during equilibration due to loss, heating, and change in shape. These small (< 5%) variations were included in the fits to the data. The density and temperature of the atom cloud were measured directly along
FIG. 2. The value of a/,,, implied by several cross-dimensional mixing measurements. This value was determined by dividing the ,>-wave scattering length determined at a particular magnetic field by the factor [ 1 - (B,ef,,-8^,^1(8-B^^)], The 4--wave scattering lengths measured at 251.0 G were divided by an additional factor of 0.99 to account for the deviation of the approximate description of a vs B from the full theory calculation. The error bars shown include the statistical errors plus the fit uncertainty due the initial radial energy imbalance. The fit uncertainty is nonstatistical and so the central values are clustered together more than the error bars would suggest.
with the aspect ratio in the absorption imaging. The density was maintained near 2.5XI0 1 ' 1 cm" ' and the temperature was kept near 130 nK. For these cold clouds, the energy dependence of the average elastic cross section is expected to be less than 5%, and the tits to the data included the energy dependence due to the uniLarity limit 141. Data were taken at B= 168.0, 169.7, and 251.0 G. The magnetic field was calibrated in the same way as Ref. [2J: the rf frequency thai: resonantly drove spin-flip transitions for atoms at the center of the cloud was measured and the BreitRabi equation was used to determine the magnetic field. Colder clouds were used, improving the precision. The spread of the magnetic field across the clouds was —-0.2 G full width at half maximum. Figure 1 shows a set of aspect ratio equilibration data and the corresponding fit. Across the range of magnetic fields measured, the equilibration times varied by about a factor of 30. The distance from Bt,eak ensures that the spread in a due to the spread in B across the cloud was not significant and that the precise location of Bpeat is also not a limiting factor in analyzing the results. The uncertainty in the determination of a is dominated, by the determination of number and the lit to obtain F. The' number determination relies on the calibration of our absorption imaging system, which was performed while taking into account the frequency and intensity of our probe laser, optical pumping effects, and the small Doppler shift arising from scattering multiple photons. We estimate the uncertainty to be 10%, which results in a 5% uncertainty in each determination of a. The uncertainty in the fit to obtain F is primarily caused by small oscillations in the aspect ratio due to the vertical and horizontal radial directions not being perfectly degenerate. Although this limited the accuracy of the fit. the size of this imbalance was varied and no significant shifts hi the results were observed. Also, the period of the oscillations does not depend on the value of a and so the agreement between data sets of very different equilibration times also
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BRftT REPORTS indicates that this effect is not distorting the results significantly, The measured value (in atomic units) of a is -342(10), 9 7 ( 6 ) . and -60(4) at 5 = 251.0. 169.7. arid 168.0 G, respectively. The listed uncertainty does not ineiude the common uncertainty due to the atom number determination. Near t h e fcshbach resonance, the change in a with B can be well approximated by ihe relation i i - - a [ , , , [ \ — ( f t . f , . , , - / ? , , , , J / ( / J #, H .,,i)] [15]. Using this, we can determine a!t,, for eacli measured value of u (Fie. 2\. The weighted average of all of these data gives
This work has been supported by are pleased to acknowledge helpful DeMajco and Murray Holland. One edges the support of the Lindemann
[!J W.C. SiwaHey. Phys. Rev. Lett, 37, 1628 (1976); B. Tiesinga, A. Moerdiik. B.J. Yerhaar, and H.TC. Stoof". Phys. Rev. A 46, R1167 U992']. [2] J.L. Roberts et al.. Phys. Rev. Lett. 81, 5109 (1998). [3] Ph. Courteille et ai. Phys. Rev. Lett. 81, 69 (1998). [4] S. Inouye el al, Nature (London) 392. 151 (UWi. [5] /.I.. Roberts et al, Phys. Rev. Lett. 86. 4211 (2001), [6] E.A. Oonety et at., e-print cond-mat;0!05019. [7] J.R Burke, Jr., C.H. Greene, and J.L. Bohn, Phys. Rev, l^ett. 81, 3355 (1998). [.Sj J.M. Vogels. B.J. Verhaar, and R.H. Blok, Phys. Rev. A 57, 4049 (1998). [9] H.M.J.M Boesten et al., Phys. Rev. Lett. 77. 5194 (1996). [10'] J, P. Burke, Jr.. Ph.D. thesis, University of Colorado, 1999. [11] S.L. Cornish et al.. Phys, Rev. Lett. 85, 1795 (2000). [12] See, for instance, F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York. 1965), pp. 516 527, [13] B. DeMavco et al., Phys. Rev. Lett. 82, 4208 (1999). [14] For instance, see J.R, Taylor, Scattering Theory: The Quantum
TheuiT of Nanrelarivi'ttic Collisions (Kreiiier, Malabar. FL, 1987), p. 89. [15] From 155.3 G ur-550 nm) to 175 G. this agrees with the full theory (see 110]) to better than 0.5%. and at 251 G there is a 1% difference. [16] J.P, Burke, Jr.. C.H. Greene, and J.L. Bohn, Phys, Rev. Lett. 81, 3355 (1998). [17j The singlet scattering potential has a bound state very near to the collision threshold ;uid so a,., is close to diverging. This accounts for the large error bars on ag. [18j The quoted values for ar, a$. and C,, are calculated using the values of Cj and C]o predicted in M. Marinescii and L. You. Phys. Rev. Lett. SI, 4596 (1998). Allowing Cf to vary by 10% would increase the uncertainties on a,,^,/,,, ,a,,„,,!,,,, and C^ by about 25%. [19] A. Derevianko et al., Phys. Rev. Lett. »2, 3589 (1999). In Ihis work. C(, was calculated to be 469J±50. [20] J.M, Vogels et al., Phys. Rev. A 61, 043407 (2000), Here, Q was measured to be 4650 ±50.
served vaiues. The calculation of a vs B is performed using a quantum-defect method |16]. Mulching the Rb-Rb potential to the observed scattering lengths implies «;-( S 5 Rb)—- 332 ± 18, a<,( g5 Rb}-36SO:^;f [17], and C G -4660 ±20, where all of the values tire listed in atomic units 18]. This value of the C(, coefficient is consistent both with an ab iniiio calciiia'ion 1 1 9 ] and a recent photoassociation measurement |20], As a result of improvements in our experimental techniques and the measurement of S;Rb BFC stability in the presence of attractive interactions, we have been able to improve our characterization of the prominent Feshbach resonance in "Rb.
024702-3
the ONR and NSF. We discussions with Brian of us (S.L.C.) acknowlFoundation,
Dynamics of collapsing and exploding Bose-Einstein condensates Elizabeth A. Donley*, Neil R. Claussen*, Simon 1. Cornish*, Jacob 1. Roberts*, Eric A. Cornell*t & Carl E. Wieman* *JIM,Campus Box 440, and Department of Physics, Campus Box 390, University of Colorado, Boulder, Colorado 80309, USA
t . . ~ . E " m . P h v S ~ ~ ~ ~ ! ~ ~ ~ . ~ ~ ~ ! . ~ ~ ! ~.................. ~ ~ .................................................... ~ . ~ ~ ~ ~ ~ ~ ~ ~ S ~ E When atoms in a gas are cooled to extremely low temperatures, they will-under the appropriate conditions-condense into a single quantum-mechanical state known as a Bose-Einstein condensate. In such systems, quantum-mechanical behaviour is evident on a macroscopic scale. Here we explore the dynamics of how a Bose-Einstein condensate collapses and subsequently explodes when the balance of forces governing its size and shape is suddenly altered. A condensate'sequilibrium size and shape is strongly affected by the interatomic interactions. Our ability to induce a collapse by switching the interactions from repulsive to attractive by tuning an externally applied magnetic field yields detailed information on the violent collapse process. We observe anisotropic atom bursts that explode from the condensate, atoms leaving the condensate in undetectedforms, spikes appearing in the condensate wavefunction and oscillating remnant condensates that survive the collapse. All these processes have curious dependences on time, on the strength of the interaction and on the number of condensate atoms. Although the system would seem to be simple and well characterized, our measurements reveal many phenomena that challenge theoretical models. Although the density of an atomic Bose-Einstein condensate (BEC) have observed many features of the surprisingly complex collapse is typically five orders of magnitude lower than the density of air, the process, including the energies and energy anisotropies of atoms interatomic interactions greatly affect a wide variety of BEC proper- that burst from the condensate, the timescale for the onset of this ties. These include static properties, such as the size, shape and burst, the rates for losing atoms, spikes in the wavefunction that stability of the condensate, and dynamic properties, like the collec- form during collapse, and the size of the remnant BEC that survives tive excitation spectrum and soliton and vortex behaviour. Because the collapse. The level of control provided by tuning a has allowed us all of these properties are sensitive to the interatomic interactions, to see how all of these quantities depend on the magnitude of a, on they can be dramatically affected by tuning the strength and sign of the initial number and density of condensate atoms, and on the the interactions. initial spatial size and shape of the BEC before the transition to Most of the physical processes in BECs are well described by instability. mean-field theory', in which the strength of the interactions A great deal of theoretical interestl2-'' was generated by BEC depends on the atom density and on one additional parameter experiments in 7Li (ref. 18), for which the scattering length is also called the s-wave scattering length, a. The parameter a is determined negative and collapse events are also The 7Li by the atomic species. When a > 0, the interactions are repulsive. In experiments do not use a Feshbach resonance, so a is fixed. This contrast, when a < 0, the interactions are attractive and a BEC tends restricts experimentation to the regime where the initial number of to contract to minimize its overall energy. In a magnetic trap, the condensate atoms is less than or equal to N,, and the collapse is :ontraction competes with the kinetic zero-point energy, which driven by a stochastic process. In addition, studies of collapse tends to spread out the condensate. For a strong enough attractive dynamics in 7Li are complicated by a large thermal component. mteraction, there is not enough kinetic energy to stabilize the BEC, Our ability to tune the scattering length, and to explore the regime and it is expected to implode. A BEC can avoid implosion only as where the initial condensate is 'pure' (near T = 0 ) and the number ong as the number of atoms in it, No, is less than a critical value No is much larger than N,,allows us to explore the dynamics and given bf compare it with theory in far more detail. Ncr = kaho/lal (1) Nhere the dimensionless constant k is called the stability coefficient. The precise value of k depends on the ratio of the magnetic trap lequencies3.ahois the harmonic oscillator length, which sets the size )f the condensate in the ideal-gas (a = 0) limit. Under most circumstances, a is insensitive to external fields. But n the vicinity of a so-called Feshbach resonance, a can be tuned over i very large range by adjusting the externally applied magnetic ield4,5.This has been demonstrated in recent years with cold "Rb ind Cs atoms6-', and with BECs of Na and "Rb (refs 9, 10). For "Rb itoms, a is usually negative, but a Feshbach resonance at -155G illows us to tune a by orders of magnitude and even change its sign. rhis gives us the ability to create stable BECs of 85Rb(ref. 10) and to tdjust the interatomic interactions. We recently used this flexibility o verify the functional form of equation ( l ) , and to measure the tability coefficient to be k = 0.46(6) (ref. 11). Here we study the dynamical response ('the collapse') of an nitially stable BEC to a sudden shift of the scattering length to a ,slue more negative than the critical value acr= - kah,/No.We
Experimental techniques The procedure for producing stable "Rb condensates has been described in detail elsewhere". A standard double magneto-optical trap (MOT) system2' was used to collect a cold sample of "Rb atoms in a low-pressure chamber. Once sufficient atoms had accumulated in the low-pressure MOT, the atoms were loaded into a cylindrically symmetric cigar-shaped magnetic trap with frequencies vrSdial= 17.5 Hz and vUial = 6.8 Hz. Radio-frequency evaporation was then used to cool the sample to -3nK to form pure condensates containing >90% of the sample atoms. The final stages of evaporation were performed at 162 G, where the scattering length is positive and stable condensates of up to 15,000 atoms could be formed. After evaporative cooling, the magnetic field was increased adiabatically to 166 G (except where noted), where a = 0. This provided a well-defined initial condition, with the BEC taking on the size and shape of the harmonic oscillator ground state. We could then adjust the mean-field interactions within the BEC to a variety of values on timescales as short as 0.1 ms. The obvious manipulation was to jump to some value of a < a,, to trigger a
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613 collapse, but the tunability of a was also a great help in imaging the sample. Usually the condensate size was below the resolution limit of our imaging system (7 p m full-width at half-maximum). However, we could increase the scattering length to large positive values and use the repulsive interatomic interactions to expand the BEC before imaging, thus obtaining information on the pre-expansion condensate shape and number. A typical a ( t ) sequence is shown in Fig. la. We have used a variety of such sequences to explore many aspects of the collapse, and enhance the visibility of particular components of the sample.
Condensate contraction and atom loss When the scattering length is changed rapidly to a value acollapre< acr, a condensate’s lunetic energy no longer provides a sufficient barrier against collapse. As described in ref. 13, during collapse one might expect a BEC to contract until losses from density-dependent inelastic collisions2’ would effectively stop the contraction. This contraction would take place roughly on the timescale of a trap oscillation, and the density would sharply increase after T,,,/4 = 14 ms, where Trad is the radial trap period.
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Figure 1 An example of a ramp applied to the scattering length a, and a plot of the condensate number Nversus time after a jump to a negative scattering length. a, A typical a([)sequence. a, = 0.529A is the Bohr radius. The scattering length is jumped at t = 0 in 0.1 ms from a,,,, to &oIia,,,, where the BEC evolves for a time ~ e y o i y e .The field is carefully controlled so that magnetic-field noise translates into fluctuations in &oilapse on the order of -0.1 a, in magnitude. The collapse is then interrupted with a jump to auUench, and the field is ramped in 5 ms to a large positive scattering length which makes the BEC expand. After 7.5 ms of additional expansion, the trap is turned off in 0.1 ms, and 1.8 ms later the density distribution is probed using destructive absorption imaging with a 40-ks laser pulse (indicated by the vertical bar). The increase in afrom acoilapse to aexpand is far too rapid to allow for the BEC to expand adiabatically. On the contrary, the smaller the BEC before expansion, the larger the cloud at the moment of imaging. Thus we can readily infer the relative size of the bulk of the BEC just before the jump to The density of the expanded BEC is so low that the rapid transit of the Feshbach resonance polez5 during the trap turn-off and the subsequent time spent at magnetic field B = 0 (a = 400a,) both have a negligible effect. b, The number of atoms remaining in the BEC versus T~~~~~~ at a, = - 30a,. We observed a delayed and abrupt onset of loss. The solid line is a fit to an exponential with a best-fit value of t,,i,, = 3.7(5) ms for the delay. ~
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How does this picture compare to what we have seen? A plot of the number of atoms in the condensate, N, versus T~~~~~~ for acollapre= - 30a, and a,, = +7a0 is shown in Fig. Ib. The number remained constant for some time after the jump in a until atom loss suddenly began at tcollapse After the jump the condensate was smaller than our resolution limit, so we could not observe the contraction directly, but we could infer the extent of the contraction by observing the degree of mean-field expansion. We observed that the post-expansion condensate widths changed very little with time ~,,,l~~ before tcollapse. From this we infer that the bulk BEC did not contract dramatically before loss began. The equations in ref. 23 (in the gaussian approximation) predict fairly well the observed expanded shape over a fairly large range of acollapse and t before collapse. So we used these equations to estimate the density before collapse, and find that the predicted contraction only corresponds to a 50% increase in the average density to 2.5 X loL3ern-'. Using the decay constants from ref. 22, this density gives an atom loss rate, 7decaythat is far smaller than what we observe and does not have the observed sudden onset. For the data in Fig. 1b and most other data presented below, a was We changed (in 0.1 ms) to aquench= 0 after a time ~~~~l~~ at acollapse. believe that the loss stopped immediately after this jump in a. This interpretation is based o n the observation that the quantitative details of curves such as that shown in Fig. l b did not depend on whether the collapse was terminated by a jump to aquench = 0 or to aquench = 250a0. We have measured loss curves like that in Fig. I b for many The collapse time versus a,,llapse for different values of acouapse No= 6,000 presented in Fig. 2 shows a strong dependence on acollapse Reducing the initial density by a factor of -4 (with a corresponding increase in volume) by setting a,,,t = +89ao for (-15ao), increased tcollapse by a factor of about one value of acollapse three. The atom loss time constant T~~~~~ depended only weakly o n aco1lapse and No. For the range of aco~apse shown in Fig. 2, Tdecaydid not depend on acollapsc or No outside of the experimental noise (-20%). On average, 7decaywas 2.8(1) ms. For the very negative value of acollapse= - 250a0, however, Tdecay did decrease to 1.8ms for N , = 6,000 and to 1.2 ms for N o = 15,000. Bursts of atoms. Atoms leave the BEC during the collapse (Fig. lb). There are at least two components to the expelled atoms. One component (the ‘missing atoms’) is not detected. The other component emerges as a burst of detectable, spin-polarized atoms with energies much greater than the initial energy of the condensate but much less than the magnetic trap depth. The dependences of burst energy on acollapse and No are complex, but as they will provide a stringent test of collapse theories, we present them in detail.
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614 The angular kinetic-energy distribution with which the burst atoms are expelled from the condensate can most accurately be measured by observing their harmonic oscillations in the trap (Fig. 3a). For example, half of a radial period after the expulsion ( Trad/2),all atoms return to their initial radial positions. Well before or well after this 'radial focus', the burst cloud is too dilute to be observed. Fortunately, at the radial focus, oscillations along the axial trap axis are near their outer turning points, and the axial energy can be found from the length of the 'stripe' of atoms along the axial axis. The radial energycan be found with the same procedure for an axial focus. The sharpness of the focus also provides information on the time extent of the burst.
Figure 3b shows an image of a radial focus. The size scales for thc burst focus and the remnant were well separated because the lattei was not expanded before imaging. Figure 3c shows cross-sections o the burst focus, and fits to the burst and to the thermal cloud. Thc burst energy distributions were well fitted by gaussians character ized by a temperature that was usually different for the two traE directions. The burst energy fluctuated from shot to shot by up to i factor of 2 for a given acouapse. This variation is far larger than thc measurement uncertainty or the variation in initial number (botk -loyo), and its source is unknown. (We also discuss observec structural variability when we present the jet measurements below.: Although the burst energies varied from shot to shot, the averagt 250
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3gure 3 A burst focus. a, Conceptualillustration of a radial burst focus. is the time at which the burst is generatedand is the radial trap period. b,An image of a radial burst ocus taken 33.5 ms after a jump from a,",!= 0 to -30afl for No = 10,000, T& = 28.6 ms, which indicates that the burst occurred 4.9(5)ms after the jump. The utial energy distribution for this burst correspondedto an effective temperature of 62 nK. The image is 60 x 310 pm. c, Radially averaged cross-section of b with a gaussianfit to he burst energy distribution. The central 100 p m were excluded from the fit to avoid listortion in the fit due to the condensate remnant (u = 9 bm) and the thermal cloud u = 17 bm). The latter is present in the pre-collapse sample due to the finite emperature, and appears to be unaffected by the collapse. The dashed line indicates the it to this initial thermal component. We note the offset between the centres of the burst ind the remnant. This offset varies from shot to shot by an amount comparable to the iffset shown.
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{ATUREIVOL 412 119 JULY 2001 Iwww,nature.com
Figure 4 Burst energies and energy anisotropies. a and b, the axial and radial burst energies versus for No= 6,000 and No= 15,000, respectively.On average, ten studied. The burst focuses were measuredfor each trap direction at each value of &,Il,, energies were higher for the larger-& condensates over the full range of &o,lw studied. The vertical and horizontal error bars indicate respectively the standard error of the arising from the magnetic-field calibration. measurementsand the uncertainty in acollapse For several of the points, the uncertainties are smaller than the symbol size and the error bars are not visible. c, The ratio of the radial to the axial energies, E,d€,, which is a measureof the burst anisotropy. For values of 1 just past &,the burst was isotropic 1 , larger-Nflcondensates for both No = 6,000 and No= 15,000. At larger values of lacdlaps gave rise to stronger anisatropies.For No= 6,000, a,,,l/aowas 0 and h was 1.6, and for No= 15,000, a,,,Jao was +7 and h was 2, where h = uJur is the aspect ratio of the initial condensate. When instead we started at a,",,= +100a,, the BEC was initially more anisotropic (h = 2.4),but the burst became more isotropic, with E, going up by -40% and Caddropping by -60% for No= 15,000 and acollapse = - 1OOa,. 297
615 value was well-defined and showed trends far larger than the jump to aquench = 0, the kinetic energy per atom in the resulting jet is variation. The axial and radial burst energies versus acouapse are equal to the confinement energy that the spike had before quenchshown in Fig. 4a and b for N o = 6,000 and 15,000, respectively. The ing the collapse, that is, (112)mv’ = h’/(4muz), where u is the width burst-energy anisotropy shown in Fig. 4c depended on No, acollapse of the spike in the wavefunction, m is the mass, and v is the and ship expansion velocity. The anisotropy of the jets indicates that the When we interrupted the collapse with a jump back to a = 0 as spikes from which they originated were also highly anisotropic, discussed above, we also interrupted the growth of the burst. The being narrower in the radial direction. From the widths and the ‘interrupted’ burst atoms still refocused after sitting at a = 0 for the number of atoms in the jets, we can estimate the density in the requisite half trap period. The energyof the atoms in the interrupted spikes. Plots of the number of jet atoms and the inferred density in bursts appeared to be the same, but the number of atoms was the spikes versus ~~~~l~~ are presented in Fig. 6. The jets exhibited smaller. By changing the time at which the collapse was interrupted, variability in energy and number that was larger than the -10% we could measure the time dependence of the creation of burst measurement noise. atoms. For the conditions of Fig. lb, the number of burst atoms Nburst grew with 7,,,lVe with a time constant of 1.2 ms, starting at Ove~iewof the current theoretical models 3.5 ms and reaching an asymptotic final number of -2,500 for all Several theoretical paper^'^^^'' have considered the problem of varied randomly by -20% for the data in Fig. 4, collapse of a BEC when the number of atoms exceeds the critical times >7 ms. Nburs, but on average the fraction of atoms going into the burst was about number. These treatments all use a mean-field approach, and describe the condensate dynamics using the Gross-Pitaevskii 20% of No and did not depend on acoUapse or No. the loss mechanism is threeRemnant condensate. After a collapse, a ‘remnant’ Condensate (GP) equation. In most casesiz*13s15,16, containing a fraction of the atoms survived with nearly constant body recombination, but Duiiie and Stoof17propose that the loss number for more than 1 second, and oscillated in a highly excited arises from a new elastic scattering process. In both cases the loss is density dependent, and so the loss rate is quite sensitive to the ~ collective state with the two lowest modes ( v 2 ~ , ,and v 2v,,dlal)being predominantly excited. (The measured frequen- dynamics of the shape of the condensate. Because a full threecies were v = 13.6(6) Hz and v = 33.4(3) Hz.) To find the oscilla- dimensional anisotropic time-dependent solution to the GP equation frequencies, the widths of the condensate were measured as a tion is very difficult, these calculations have used various approximations to calculate the time evolution of the condensate shape. function of time spent at amllapae. The number of atoms in the remnant depended on acullapre and No, Kagan, Muryshev and Shlyapnikov” numerically integrate the GP and in general was not limited by the critical number, N,, The equation for the case of an isotropic trap and large values of the stability condition in equation (1) determined the collapse point, three-body recombination coefficient K3. In this regime, the conA fixed fraction of No went into the densate smoothly contracts in a single, collectivecollapse. Saito and but did not constrain Nremnant. remnant independent of No, so that smaller condensates often Uedai5,16perform a similar numerical solution for the isotropic case < N,,, but larger condensates rarely did. but with smaller values of K3, and observelocalized spikes to form in ended up with N,,, The fraction of atoms that went into the remnant decreased with the wavefunction during collapse. Duine and Stoof” model the lacoliapsel, and was -40% for lacoilupre/ < IOa, and -10% for dynamics for the anisotropic case, but use a gaussian approximation laCoUapsel > 100a,. We do not think that surface-wave excitations2* rather than an exact numerical solution. These calculations have all been performed over a certain range of are responsible for stabilizing the remnant because we excite largeamplitude breathing modes. For N , = 6,000 and Iacolinpse/ < 10ao, parameters, but none have been done for the specific range of more atoms were lost than the number required to lower Nremnanl to parameters that correspond to our experimental situations. None of the predictions in these papers match our measurements except for below N,, As Nburst was independent of aco~apse but Nr,m,,,t decreased with la,llapseI, the number of missing atoms increased with /acoiiapsei. The number of missing atoms also increased with No,hut the percentage of missing atoms was equal for No = 6,000 and No = 15,000 with 1 acouapsefixed and was -40% for lacOnapael < 10a, and -70% for lucoliapse/ 2 100~~ The . missing atoms were presumably either 05 expelled from the condensate at such high energies that we could not detect them (>20 FK), or they were transferred to untrapped atomic states or undetected molecules. 0 Jet formation. Under very specific experimental conditions, we observed streams of atoms with highly anisotropic velocities emerging from the collapsing condensate. These ‘jets’ are distinguished from the ‘burst’ in that the jets have much lower kinetic energy (on the order of a few nanokelvin), in that their velocity is nearly purely radial, and in that they appear only when the collapse is interrupted (that is, by jumping to aquench = 0) during the period of number loss. Collapse processes that were allowed to evolve to completion (until N =: N,,,,,) were not observed to emit jets. Examples are shown in Fig. 5 for different ~ , , l ~ for ~ the conditions of Fig. lb. The size and values for the conditionsof Fig. I b. The evolution shape of the jets varied from image to image even when all Figure 5 Jet imagesfor a series of T~~~~~~ conditions were unchanged, and as many as three jets were times were 2, 3,4,6,8 and 10ms (from a to f). Each image is 150 X 255 wm. The bar = +250a, was applied, so the sometimes observed to be emitted from the collapse of a single indicatesthe optical depth scale. An expansion to aemSnd condensate. Also, the jets were not always symmetric about the jets are longer than for the quantitative measurementsexpiainedin the text. The jets were longest (that is, most energetic)and containedthe most atoms at values of for which condensate axis. We believe that these jets are manifestations of local ‘spikes’ in the the slope of the loss curve (Fig. I b) was greatest. A tiny jet is barely visible for = 2 ms (a),which is 1.7 rns before &ollapsp.The images aiso show how the number condensate density that form during the collapse and expand when 7avolue the balance of forces is changed by quenching the collapse. We can of condensate atoms decreaseswith time. The time from the appiicationof aquenchuntii the estimate the size of the spikes using the uncertainty principle. After a acquisition of the images was fixed at 5.2 rns.
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From the experimental point ofview, there are at least two questions to be answered. First, is the burst coherent? It may be possible to answer this by generating a sequence of 'half bursts' and see if they interfere. Second, where do the missing atoms go? If molecules and/ or relatively high-energy atoms are being created, can we detect them? It is clear that adjustable interactions open up new avenues 0 for BEC studies. Received 23 Avrd, accevted 18 Tune 2001
I .
1. Dalfavo, F., Giorgim, S., Pitaevshi, L. P. & Stimgari, S.Theory of Bose-Einstein condensation in trapped gases Rev Mod Phyi 71, 463-512 (1999). Ruprecht, P A , Holland, M. I., Burnett, K. &Edwards, M. Time-dependent solution of the nonlinear Schrodingei equation for Bose-condensed trapped neutral atoms. Phyr. Rev. A 51, 4704-4711 (1995). 3. Gammal, A,, Fredenco, T.& Tomio, L. Critical number of atoms for attractwe Bose-Einstein condensates with cylindrically symmetrical traps. Phyr. Rev A (submitted). 4. Tiesinga, E., Morerdqk, A. I., Verhaar, B 1. k Stoof, H. T. C. Conditmns for Bose-Emstem condensation in magnetically trapped atomic cesium. Phyr. Rev. A 46, R1167-RI 170 (1992) 5. T i e h g a , E., Verhaar, 8.J. & Stoof, H. T. C. Threshold and resonance phenomena m ultracold groundstate collisions Phyi Rev A 47, 4114-4122 (1993). 6. Courteille, Ph., Freeland, R. S.,Hemien, D. 1 , van Abeelen, F. A. & Verhaar, B. 1. Observation of a Feshbach resonance m cold atom scattering. Phyi. Rev, Letr 81,69-72 (1998) Phyr. Rev Lett. 81, 7. Robertr, I. L. eral. Resonant magnetic field con~iolofelasticscatteringincold~~Rb. 5109-5112 (1998). 8 Vuletit, V., Kerman, A. I., Chin, C. & Chu, S. Observation of lowfield Feshbach resonances in collisions ofcesium atoms Phyi. Rev Lett 82, 1406-1409 (1999). 9. Inouye, 5.et 01. Observation of Feshbach resonance^ I" a Bose-Einstein condensate. Nature 392,151154 (1998) 10. Cornish, 5.L., Ciaussen, N. R.,Roberts, I. L., Cornell, E A & Wieman, C. E. Stables5Rb Bose-Emstein condensates with widely tunable mteractmn~.Phyi. Rev. Len. 85, 1795-1798 (2000). 11. Roberts, I L. et a1 Controlled collapse of a Bose-Emstem condensate. Phyi Rev L e a 86,4211-4214
2
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.-
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0 al
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0.0 0
2
4
6 TevC'Ve
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1
0
Figure 6 Quantitativejet measurements. a, The number of atoms in the jets versus T ~ for the conditions of Fig. 1b. b,The spike density inferred from the kinetic energies of the jets.The bars indicate the range of shot-to-shotvariability. For the analysis,we assumed
the jets were disk-shaped,as the magnetic trap is axially symmetric.The images were taken perpendicular to the axial trap axis, viewing the disks edge-on.The jets expanded with velocity I/ = 1 mm s - ' , which correspondsto a kinetic energy of -6 nK and a radial pre-quench gaussian r.m.s. width of -0.5 p,m. As the axial size was below our resolution limit, we could not measure the axial expansion rate. For estimating the spike density,we assumed an axial width equal to the harmonic oscillator length. The atom density in the spikes decreased for larger values of ~ac,ll,,,,~, and was haif as large for acoIlaose = - 100a, as for -30ao. the general feature that atoms are lost from the condensate. Also, we see several phenomena that are not discussed in these papers. Whether this lack of agreement is due to the fact that these calculations do not scale to our experimental situation or do not contain the proper physics remains to be seen. Theoretical challenges. Collapsing "Rb condensates are simple systems with dramatic behaviour. This behaviour might provide a rigorous test of mean-field theory when it is applied to our experimental conditions. Some of our particularly puzzling results are as follows. (1) The decay constant 7decay is independent of both No and (ucollapse/ for (uiollapre/ < IOOu,, and only weakly depends on these quantities for larger lucollspsel. ( 2 ) The burst energy per atom dramatically increases with initial condensate number. (3) The changes. (4) The fraction of burst atoms is constant when acollappc number of cold remnant BEC atoms surviving the collapse varies between much less and much more than N,, depending on Noand acollapse,but the fractions of remnant atoms, burst atoms, and missing atoms are independent of No.
QATUREIVOL 412 119 JULY 2001 (urww.nature.com
~ (2001). ~ , ~ ~ I 2 Kagan, Y, Surkou, E L. & Shlyapmkov, G. liEvolution and global collapse oftrapped Bose condensates under variations of the scattering length. Phyi Rev. Len. 79, 2604-2607 (1997). 13. Kagan, Y,Muryshev, A. E. & Shlyapnikou, G. V. Collapse and Bose-Einstein condensation in atrapped Bose gas with negative scattering length Phyr R e v Lett 81,933-937 (1998). 14. Ucda, M. & H u n g , K Fate of B Bose~Einsteincondensate w t h an attractive mteractmn. Phys Rev. A 60,3317-3320 (1999). 15 Saao, H &Ueda,M Powerlaw\andcollapringd~amics ofa IrappedBose-Einsteincondensatewith attractive interactions. Pltys. Rev. A 63, 043601-8 (2001). 16. Sam, H. & Ueda, M. intermittent implosion and pattern formation of trapped Bose-Emstein condensates with an attractive interaction. Phvi R n , Lett. 86, 1406-1409 (2001). 17. Dune, R. A. & Stoof, H. T C. Explosion o f a collapsing Bose-Einstein condensate. Phyi Rev Lett. 86, 2204-2207 (2001). 18. Bradley C. C , Sackett. C. A. & Hulet, R. G. Bose-Einstein condensation of hthium: Observarian of limited condensate number. Phyi. Rev. Lett. 78,985-989 (1997). 19. Sacketr, C. A,, Gerton, I. M., Welling, M. & Hulet, R. G. Measurements ofcollective collapse ~na BoseEinstein condensate with attractive interactions. Phyi. Rev. Lett 82, 876-879 (1999) 20. Gerton, 1. M., Strekalov, D., Prodan, I. & Hulet, R. G. Direct observatm o f growth and collapse of a Bose-Einstein condensate with attractive interactions. Nature 408, 692-695 (2000). 21. Myatt, C. I., Newbuv, N.R., Ghnst, R.W., Loutzenhiser, S. Wieman, C. E. Multlply loaded magneto-optical trap Opt Lett 21, 290-292 (1996). 22 Roberts, I. L., Claussen, N. R., Cornish, S. L. & Wieman, C E.Magnetic field dependence ofultracold inelastic collisions near a Feshbach resonance Phys. R e v Lee. 85, 728-731 (2000). 23. Perez-Garcia, V M ,Mxchinel,H., Cirac, 1. L., Lewenstem, M. & Zoller, P. Dynamics of Bose-Emstem condensates. Variational solutions of the Gross-Pitaevskii equations. Phyr. Rev. A 56, 1424-1432 (1997) 24. Pattanayak, A. K., Gammal, A., Sackett, C. A & H u l a , R. G. Stabilizing an attractive Bose-Einstein condensate by driving a surface collective mode. Phyi. Rev. A 63,033604-4 (2001). 25. Stenger, I. et 01. Strongly enhanced inelastic collisions ~na Bose-Einstein condensate near Feshbach resonances. Phys. Rm Lett. 82, 2422-2425 (1999).
Acknowledgements We thank S. Thompson for laboratory assistance, and S. Durr, G. Shlyapnikov,H. Stoof, M. Holland, M. Ueda and R. Duine for discussions. This work was supported by the ONR, NSF, ARO-MURI and NET. S.L.C. acknowledgesthe support of a Lindemann Fellowship. Correspondence and requests for materials should be addressed to E.A.D. (e-mail: [email protected]).
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Microscopic Dynamics in a Strongly Interacting Bose-Einstein Condensate N.R. Claussen, E.A. Donley, S.T. Thompson, and C.E. Wieman JILA, National Institute o j Stnizdards arid Techriology arid the Uiiiver.c.itj, of' Colorcrdo, a r? d the Depa r tin ei I t of' Physics, Ur I i ve r.s ity of' Coloi d o , Boulder; Colo rcirlo 80309-0440 (Received 10 January 2002; published 14 June 2002)
An initially stable "Rb Bose-Einstein condensate (BEC) was subjected to a carefully controlled magnetic field pulse near a Feshbach resonance. This pulse probed the strongly interacting regime for the BEC, with the diluteness parameter ( n n ' ) ranging from 0.01 to 0.5. Condensate number loss resulted from the pulse, and for triangular pulses shorter than I ins, decreasing the pulse length actually irio-ecisrtl the loss, ~ i n t i lvery short time scales (-I0 ~ s were ) reached. The obsei-vcd time dependence is very different from that expected i n traditional inelastic loss processes, suggesting the presence of new i i i i croscopic BEC physics DOI: 10.1 103/PhyaRevLett.XC).~10401
PACS numbers: 03.7S.-b, 05.30.-d. 32.80.-t, 34.50.-s
The self-interaction strength that determines most of the properties of a Bose-Einstein condensate (BEC) is characterized simply by the s-wave scattering length, a [I]. Near a Feshbach resonance [2], the scattering length depends strongly on the magnetic field. This property has been used to vary the self-interaction energy in BEC [3-71. In particular, we have used the Feshbach resonance to create stable "Rb condensates [5]. We also varied the scattering length and studied the effect on the condensate, most notably its collapse when the scattering length was made negative [6,7]. Here we discuss BEC behavior in the positive a region near the 85Rb Feshbach resonance, where the selfinteraction is large and repulsive. We used rapid magnetic field variations to probe the strongly interacting regime in the condensate and to investigate the possibility of collisional coupling between pairs of free atoms and bound molecules [8-131. We changed the magnetic field to approach the Feshbach resonance from above, which causes the positive scattering length to increase. As .a3 becomes comparable to or larger than one, the customary mean-field description of the dilute gas BEC, which assumes no correlations between the atoms, breaks down. Correlations between the atoms become increasingly important and a BEC with sufficiently large .a3 will eventually reach a highly correlated state such as that found in a liquid. We are interested in the nature of this transition to a highly correlated state and the time scale for the formation of the condensate correlations. In addition to such interesting physics, our time-dependent experiments also allow us to probe the coupling between atomic and molecular BEC states that are degenerate at the Feshbach resonance. Several authors have proposed mechanisms for transferring atom pairs into molecules using time-dependent magnetic fields near resonance [8-131. To investigate such physics we examined the response of the condensate when we briefly approached the resonance by applying a short magnetic field pulse (much shorter than the period for collective excitations). The most obvious feature of the BEC response was a
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loss of atoms that increased when we made our pulses shorter, until the loss became very large for very short pulses. Bose-Einstein condensates always disappear if one waits long enough. The dominant loss process was found to be three-body recombination into molecules [ 14,151, with a time dependence characterized by a simple rate equation and rate constant. All previous observations of condensates are consistent with a mean-field description that includes such a density-dependent loss process. In the vicinity of a Feshbach resonance the decay rates have been seen to increase dramatically [4,5,16], becoming so large that various novel coherent conversion processes have been put forth to explain their magnitude [ 10- 121. However, the observations in Refs. [4,5,16] always revealed that the more time spent near the Feshbach resonance, the greater the loss. This is still consistent with a basic mean-field model with some inelastic loss term, albeit a very large one. Here we report losses occurring on very fast time scales even when we remain some distance from the resonance, and we see that shorter and more rapid pulses lead to a greater loss than longer, slower pulses. This behavior contrasts dramatically with the conventional picture presented above and indicates the presence of novel BEC physics. To study BEC dynamics we first formed a 85Rbcondensate, following the procedure given in Ref. [5]. A sample of 'jRb atoms in the F = 2 mF = -2 state was evaporatively cooled in a cylindrically symmetric magnetic trap (v,,d,,l = 17.5 Hz, v,,ial = 6.8 Hz). The magnetic field at completion of evaporation was 162.3 G, corresponding to a scattering length of 200ao. Typically, the cooled sample had No = 16 500 BEC atoms and fewer than 1000 noncondensed atoms. The magnetic field was then ramped adiabatically (in 800 ms) to -166 G where the scattering length was positive but nearly zero [17], and the BEC assumed the shape of the harmonic trap ground state. We next applied a short magnetic field pulse (duration <1 ms) so that the field briefly approached a value
0 2002 The American Physical Society 61 7
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moderately close to the Feshbach resonance at -155 G (see Fig. 1). We used destructive absorption imaging to look at the number of atoms remaining in the condensate. This experiment was repeated with a variety of differently shaped magnetic field pulses. We found that the magnetic trap must be turned off and the condensate spatial size must be significantly larger than our resolution limit to obtain the most sensitive and accurate measurements of number [7]. To expand the BEC after the short pulse, we ramped in 5 ins to -157 G (Li = 1900ao), and then held at that field for 7 ins. The mean-field repulsion during the ramp and hold times decreased the density by about a factor of 30; then we rapidly turned off the trap and imaged the cloud. The density decrease avoided density-dependent loss that we have observed during the trap turn-off [5,16]. A significant number loss from the BEC was observed for pulses lasting only a few tens of microseconds. The loss was accompanied by the generation of a "burst" of relatively hot (- 150 nK) atoms that remained in the trap, as in Ref. [ 7 ] . The few thousand burst atoms represented a fairly small fraction of the total number lost from the BEC and so in this Letter we have focused only on the number in the condensate remnant. The burst will be the subject of future work. To study the remarkably short time scales for the loss, we designed a low-inductance, high current auxiliary electromagnetic coil. The coil current was supplied by a capacitor bank that was charged to 580 V, then discharged through the coil at a rate controlled by a transistor. Our goal was to create a perfect trapezoidal magnetic field pulse with adjustable but identical rise and fall times and a hold time during which the field was constant. To compensate
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for observed mutual inductance effects with other coils in the apparatus, we empirically determined the auxiliary current pulse shape needed to produce a total magnetic field pulse that closely approximated our ideal, as shown in Fig. 1. The presence of induced currents limited the maximum ramp speed to clB/clt = 1 G/,us, and the magnetic field uncertainty was 1 part in 10' (0.16 G). We first examined the BEC loss for trapezoidal field pulses as a function of the hold time (see Fig. 2). Using a linear rise and fall time of 12.5 ,us, we observed BEC number loss of 10%-20% when the hold time was set to zero (triangular pulse). The number of BEC atoms then showed a smooth exponential decrease as hold time was increased. Surprisingly, when we reduced the initial BEC density by more than a factor of 2, the time constant was nearly unchanged. For the low (high) density data, the value of the time constant was over 2 (1) orders of magnitude shorter than predicted by our previous inelastic decay measurements with cold thermal clouds [ 181. We next measured how the loss depended on the rise/fall time of the pulse for a variety of pulse amplitudes and hold times (Figs. 3 and 4). We varied the rise time from 12.5 to 250 ,us and changed the hold time at the pulse peak from 1 to 100 ps. In addition, the amplitude was varied to examine fields from 158.0 G (a = 1 100ao)to 156.0 G (a = 4000a~l).Here we list the corresponding scattering lengths that we have observed by slowly adjusting the magnetic field, as in Ref. [5]. The scattering length was calculated from the equation: a ( B ) = ab,[l - A / ( B Bo ) ] ,where A = 11.0(4) G is the width of the resonance, Bo = 154.9(4) G is the resonant magnetic field, and a b g = -450(3)ao is the background scattering length [19]. For the range fields examined here, the initial value - of magnetic of the diluteness parameter varied from nu3 = 0.01 for a = 1100nt~to nu3 = 0.5 for a = 4000at~.
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Hold Time ( us ) FIG. 2. Fraction of BEC remaining vs pulse hold time. The pulse riselfall time was 12.5 ps and the magnetic field during the hold was 156.7 G ( 2 3 0 0 ~ ) .Number decay was measured for two different initial densities: ( n ) = 1.9 X lOI3 cm-3 ( 0 ) and ( n ) = 0.7 X l0l3 cm-3 (0). Fitting the data to exponential functions (solid lines) gave time constants of 13.2(4) and 15.4(14) ps. Thus, reducing the density by a factor of 2.6 caused an increase of only 17(1I)% in the time constant. 010401 -2
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Scaled Rise Time ( ps ) FIG. 3. Dependence of remnant BEC number on pulse rise/fall time for various hold times (see legend) with N O = 16500 ((n) = 1.9 X 1013 cm-3). For the majority of the data points, the symbol is larger than the statistical en-or bar (not shown). The lines are spline fits to guide the eye. The abscissa was multiplied by a factor of 1/4 to show the time required to ramp from 75% to 100% of the pulse amplitude. This scaling reflects the observed fact that most of the loss occurred at fields closest to the Feshbach resonance. The magnetic field during the hold time was 156.7 G (2300no).
The number remaining in the BEC after the pulse vs pulse rise time for a variety of different hold times is shown in Fig. 3. For hold times tho]d 5 15 ps, there is an initial decrease in Nremas the rise time increases. Then the slope changes and fewer atoms are lost for longer rise times. All of the hold time data display this upward slope over some range, but the range is largest for the 100 ps hold time. The increase in remnant number for longer rise time provides clear evidence that the loss is not conventional inelastic decay that is characterized by a rate constant. It is interesting that the short hold time data show a distinct minimum in N,,, vs rise time, which shifts toward shorter rise times as the hold time is increased.
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In Fig. 4 we display the remnant number vs ramp time for triangular magnetic field pulses (1 ps hold). The remnant number is shown for various pulse amplitudes. For all cases, N,,,, is large at the shortest rise time and then decreases with rise time until it reaches a minimum. Then longer rise times cause N,,, to increase over a time scale of tens of microseconds. The rise time that induces maximal loss becomes longer as the pulses come closer to resonance. Conventional condensate loss is characterized by a rate constant for a density-dependent decay process, and thus the loss increases monotonically with time. Near a Feshbach resonance the rate constants have been observed to increase enormously [4,5,16], but, nevertheless, a longer time spent near the peak, or equivalently a slower ramp getting over it, resulted in greater loss. In contrast, we have measured an increase in the loss when the ramp time is decreased, which reveals the existence of previously unexplored BEC physics. Of course, the above interpretation of our data would be modified if the density of the sample was changing due to the rapid increase in the mean-field interaction, but the characteristic time for such readjustments in cloud shape is far longer [of order 1/(2v,,dinl) = 29 ms] than the time scales we considered here. For example, using a simple analytic model based on the Gross-Pitaevskii equation [22], we calculate that for a 250 ps ramp to Bfinal = 156.0 G (4000ao), the change in mean-field energy causes the density to decrease by only 1% from its initial value. Thus, the observed dependencies on rise time must reflect microscopic physics in the BEC and not any macroscopic changes in the shape of the condensate. The response of the BEC remnant to magnetic field ramps, observed in Fig. 4, is qualitatively similar to what one would expect in a Landau-Zener (L-Z) model [23,24] of an avoided crossing of two states with linear Zeeman shifts, if the second state were undetectable. In our case, the obvious candidate for this second state is the Feshbach resonance bound state corresponding to a diatomic molecule. With a L-Z avoided crossing, the behavior when the field approaches and then backs away from the crossing point with a triangular pulse is qualitatively similar to the more familiar and analytically soluble case [25] of a linear field ramp that goes from far above the crossing to far below and then back again. In both cases, and as seen in Fig. 4, the L-Z model predicts a steep rise from zero in the transition probability as the length of the ramp increases from zero (diabatic limit). As the ramp time increases further, the transition probability reaches a maximum-where the time derivative of the relative energy matches the square of the coupling strength- and then slowly decreases to zero as the ramp approaches the adiabatic limit. We found that the L-Z model does a rather poor job at reproducing any more quantitative features of the data, however, even when we allowed the coupling strength and relative magnetic moment 0 10401-3
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to be arbitrary parameters in our numerical simulations [26]. Nevertheless, it seeins likely that some atoms are being converted to another state (possibly molecular) by nonadiabatic mixing, although the process is more complicated than a simple Landau-Zener avoided crossing picture. In future experiments, we plan to further investigate the time response of the BEC loss using asymmetric pulses and double pulses with variable spacing. We will explore the burst production process and attempt to determine the fate of the lost atoms. We acknowledge contributions from the entire JILA BEC/Degenerate Fermi Gas Collaboration and helpful discussions with L. Pitaevskii and E. Cornell. S. T. acknowledges the support of the ARO-MURI Program. This work was also supported by ONR and NSF.
[I] F. Dalfovo et a]., Rev. Mod. Phys. 71, 463 (1999). [2] W.C. Stwalley, Phys. Rev. Lett. 37, 1628 (1976); E. Tiesinga et ul., Phys. Rev. A 46, R1167 (1992). [3] S. Inouye et al., Nature (London) 392, 151 (1998). [4] J. Stenger et ul., Phys. Rev. Lett. 82, 2422 (1999). [5] S.L. Cornish et ul., Phys. Rev. Lett. 85, 1795 (2000). [6] J.L. Roberts et al., Phys. Rev. Lett. 86, 421 1 (2001). [7] E.A. Donley et nl., Nature (London) 412, 295 (2001). [8] E. Timmermans et al., Phys. Rep. 315, 199 (1999); E. Timmermans et al., Phys. Rev. Lett. 83, 2691 (1999).
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191 P. D. Drummond, K. V. Kheruntsyan, and H. He, Phys. Rev. Lett. 81, 3055 (1998). [lo] F.A. van Abeelen and B. J. Verhaar, Phys. Rev. Lett. 83, 1550 (1999). [ I I ] V. A. Yurovsky et nl., Phys. Rev. A 60, R765 (1999). [ 121 F.H. Mies, E. Tiesinga. and P. S. Julienne, Phys. Rcv. A 61, 022721 (2000). [I31 M. Holland, J. Park, and R. Walscr, Phys. Rev. Lett. 86, 1915 (2001). 1141 E. A. Burt et ul., Phys. Rev. Lett. 79, 337 (1997). [ 151 C. J. Myatt, Ph.D. thesis, University of Colorado, 1997. 161 J. L. Roberts, Ph.D. thesis, University of Colorado. 200 I , available at http://jilawww.colorado.edu/www/sro/thesis/. (171 J.L. Roberts rt d., Phys. Rev. A 64, 024702 (2001). [ 181 J. L. Roberts et al., Phys. Rev. Lett. 85, 728 (2000). [I91 The values for A and Bo were taken from our previous work 1171, while u h p came from Ref. [20]. Although this value for nhg disagrees somewhat with our measurements i n Ref. [17], we use it because of the substantially smaller uncertainty stated in Ref. [21]. [20] S. J. J. M. F. Kokkelmans (private communication). [21] E. G . M . van Kempen et al., Phys. Rev. Lett. 88, 093201 (2002). [22] V. M. PCrez-Garcia et al., Phys. Rev. A 56, 1424 (1997). [23] L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932). [24] C. Zener, Proc. R. Soc. London A 137, 696 (1932). [25] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Pergamon, New York, 1965), 2nd ed., pp. 322-330. [26] See, for instance, J. R. Rubbmark et al., Phys. Rev. A 23, 3107 (1981).
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.............................................................. Atom-molecule coherence in a Bose-Einstein condensate Elizabeth A. Donley, Neil R. Claussen, Sarah T. Thompson & Carl E. Wieman JILA, Universityof Colorado and National Institute of Standards and Technologfi Boulder, Colorado 80309-0440, USA .................................... .................................... ................... .......................................................,..,............,,,,.......
Recent advances in the precise control of ultracold atomic systems have led to the realisation of Bose-Einstein condensates (BECs)and degenerate Fermi gases. An important challengeis to extend this level of control to more complicated molecular systems. One route for producing ultracold molecules is to form them from the atoms in a BEC. For example, a two-photon stimulated Raman transition in a 87Rb BEC has been used to produce "RbZ molecules in a single rotational-vibrational state', and ultracold molecules have also been formed' through photoassociation of a sodium BEC. Although the coherenceproperties of such systems have not hitherto been probed, the prospect of creating a superposition of atomic and molecular condensates has initiated much theoretical work&'. Here we make use of a time-varying magnetic field near a Feshbach resonance8-" to produce coherent coupling between atoms and molecules in a *'Rb BEC. A mixture of atomic and molecular states is created and probed by sudden changes in the magnetic field, which lead to oscillations in the number of atoms that remain in the condensate. The oscillation frequency, measured over a large range of magnetic fields, is in excellent agreement with the theoretical molecular binding energy, indicating that we have created a quantum superposition of atoms and diatomic molecules-two chemicallydifferent species. A Feshbach resonance is a scattering resonance for which the total energy of two colliding atoms is equal to the energy of a bound molecular state, and atom-molecule transitions can occur during a A schematic representation of the potentials involved is shown in the inset of Fig. la. For our *'Rb resonance, BEC atoms in the F = 2, rnF = -2 state collide on the open-channel threshold. F and mF are the total spin and spin-projection quantum numbers. The bound state in the closed channel differs in energy by an amount E from the open-channel threshold. The bound molecular wavefunction can be described as a sum of amplitudes of different hyperfine components (F, mF)havingMF = mE1 mF,2 = -4 (ref. 13). Because of their different spin configurations, the atoms and molecules generally have different magnetic moments and the difference depends on magnetic field. Thus E depends on magnetic field and the degree of atom-molecule coupling is magnetically tunable. The energy difference between the free atoms and the bound molecules is plotted in Fig. la. This behaviour of the boundstate energy also causes a resonance in the scattering length, a, which is shown in Fig. lb. The scattering length characterizes the meanfield interaction energy of a BEC. When the magnetic field is tuned to values near the Feshbach resonance, theory predicts coherent oscillations between the atomic and molecular states, but there is significant disagreement on the conversion fraction and the coherence pr~perties'~''. In experiments with a sodium BEC", inelastic losses were greatly enhanced when the magnetic field was ramped across the Feshbach resonance. We observed similar results for 85Rb, but with lower rates2'.22.It is likely that the formation of molecules played a role in the atom loss, but there was no experimental evidence for the presence of molecules and the results followed a loss-rate dependence on time. More recently, we measured the time dependence of the loss in an *'Rb BEC by applying controlled magnetic-field pulses
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622 toward but not across the Feshbach resonance23. We observed the surprising result that under some conditions, shorter, more rapid pulses actually led to more loss than longer, slower pulses that spent more time near the resonance. The time dependence of the loss was suggestive of a nonadiabatic mixing of states, with the only states within a reasonable energy range being the normal atomic BEC state and the nearby bound molecular state. Here we show that much of the loss is probably due to the coherent mixing of atomic and molecular states. To create a superposition and probe its coherence, we applied two short magneticfield pulses toward the Feshbach resonance, separated by a ‘free evolution’ time during which the magnetic field was held at a constant value some distance from the resonance. We measured the number of atoms in the condensate as a function of time between the two pulses for various values of the steady-state magnetic field between the pulses. We observed dramatic oscillations in the number of atoms remaining in the atomic BEC at frequencies corresponding to the energy splitting between the molecular and the atomic states. The a paratus has been described in detail e l ~ e w h e r e ~We ’ . ~first ~. created Rb condensates2’typically containing 16,500 atoms, with fewer than 1,000 uncondensed thermal atoms. The initial number N,,,, fluctuated from shot to shot by about 500 atoms (-3% number noise). After producing the condensate at a field of -162G, we ramped the magnetic field adiabatically to -166G, corresponding to an initial scattering length = lo%, where = 0.053 nm. The spatial distribution of the atoms was gaussian with a peak atom density of no = 5.4 X 1013cm-3, and the trap
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frequencies were (17.5 X 17.2 X 6.8) Hz. After preparing the condensate we applied a selected fast magnetic-field pulse sequence by sending an appropriate time-dependent current through an auxiliary magnetic-field coil2’. A typical pulse sequence is shown in Fig. 2. It is composed of two nearly identical short trapezoidal pulses separated by a region of constant (but adjustable) magnetic field. Upon completion of the fast-pulse sequence in Fig. 2, we ramped the magnetic field from 166 G to 157 G in 5 ms and held at that field for an additional 7 ms to allow the repulsive mean-field energy to expand the condensate. Then we turned off the magnetic trap and used destructive absorption imaging 12.8 ms later to observe the atomic condensate and measure the number of remaining This detection scheme was sensitive neither to atoms with kinetic energies larger than -2 pK nor to atoms in off-resonant molecular states. We determined the value of the magnetic field between the i,, by measuring the resonance frequency for transitions pulses,,,B from the F = 2, mF= -2 to the F = 2, mF = - 1 spin state by applying a 10-ps radio-frequency pulse to a trapped cloud of atoms2’. As we observed for single pulses toward the Feshbach resonancez3, there were two distinct components of atoms present in the absorption images and a third ‘missing’ component that we could not detect. One of the observed components was a cold remnant BEC which was not noticeably heated or excited by the fast-pulse sequence, while the other component was a relatively hot (-150 nK) ‘burst’ of atoms that remained magnetically trapped during the BEC expansion time. Using a variational approachz5to model the mean-field expansion that we applied to the BEC we found that we should remnant to measure its number, Nremnani, impart at most 3 nKworth of energy to the remnant before imaging. This estimate agrees well with the expansion velocity that we observed after the trap turn-off. Thus the remnant BEC was approximately 50 times colder than the burst. The missing component contained atoms that were in the initial sample but were not detected after the trap turn-off. To find the number of atoms in the remnant BEC and the number of burst atoms, we allowed the magnetic trap to ‘focus’ the burst cloud before imaging16. A typical image is shown in Fig. 3. We fitted the focused burst (which had a much larger spatial extent than the remnant) with a two-dimensional gaussian surface, excluding the central region of the image that contained the remnant. This fit yielded the number of burst atoms, Nburst.Subtracting this fit from the image and performing a pixel-by-pixel sum of the central region of the image gave Nremnant. Nremnant versus tevolve is plotted in Fig. 4 for two different values of .B ,,e ,, The number clearly oscillates. Changing the value of Bevolve strongly affected the oscillation frequency (note the change in scale
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Figure 1 Feshbach resonance bound-state energy and scattering length. a, Energy splitting versus magnetic field. The resonance, 5,is centred at -1 55 G. The solid curve is a theoretical estimate of the energy found with a coupled-channels calculation, and the dotted line indicates E = 0. The coupled-channels calculations were provided by S. J. J. M. F. Kokkelmans and C. H. Greene. The calculations used the best estimates of the Feshbach resoname parameters found by combining the results of three highprecision experiments. The determination of the parameters and the construction of the potentials is described in detail in ref 29. The inset schematicaliy shows the collision channels involved in the resonance. E depends on magnetic field because the atoms and molecules have different magnetic moments and thus the potentials have different Zeeman shifts. b, Scattering length versus magnetic field for fields above the Feshbach resonance. 530
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t (PS) Figure 2 Magnetic field pulse shape. Fields shown for pulses 1 and 2 correspond to scattering lengths of -2,500ao, and the free precession field 5,,,,,, corresponds to a scattering length of -57080. The dashed line indicates the position of the Feshbach resonance. In the text, we refer to the free precession time as tevolve. The average riseifall time for ail of the pulses that we used was 14 ps. NATURE 1 VOL 417 I30 MAY 2002 1 hiWW.natlire.Com
623 from Fig. 4a to Fig. 4b). After only pulse 1 and the subsequent constant field but with no pulse 2, NremndLlt showed no variation except for a slow decay consistent with the loss rate expected for a single pulse to that fieldz3. We have taken data similar to those in Fig. 4 for a variety of different LI,,,I,, values. As in Fig. 4, we fitted each curve to the $J) to find the oscilfunction y = yo +Aexp(-t/7d,,,)sin(2.rrvt lation frequency u and decay time constant 7drc.i,,. The measured in Fig. 5 along with theoretical frequencies are plotted versus B,,,, predictions for the bound-state energy relative to the atomic state. In the regime where the scattering length is much larger than the radius of the interatomic potential well, the bound state energy for an arbitrary attractive potential can be approximated by" E = -h2/ma2. h is Plank's constant divided by 27r, rn is the atomic mass, and a is the scattering length. The same equation relates the bound state energy to the effective scattering length, which is calculated from the Feshbach resonance parameters through the relation a = ahg X (1 - A / ( B - Bo)). where abg is the background scattering length, A is the width ofthe Feshbach resonance, and Bo is the resonant magnetic field. Rethermalization measurements in 85 Rb thermal clouds2' yield values o f Bo = 154.9(4) G and A = 11.0(4)G. The best estimate of ahg is an estimate found by mass scaling (S.J.J.M.F. Kokkelmans, personal communication) spectroscopic mea~urements'~of the four highest-lying bound states of 87Rb,which gives abg = -450(3)a0. The quantity 1eI/h is plotted with no adjustable parameters in Fig. 5. The measured oscillation frequencies are in excellent agreement with this simple model over the range of magnetic fields where the model is expected to be valid. The theoretical results found with a much more sophisticated coupled-channels scattering calculation in Fig. 5 are in excellent agreement with the data over the entire range. The fact that the oscillations occurred at exactly the frequency corresponding to the bound-state energy clearly indicates that we are creating a coherent superposition of atoms and molecules with the sudden magnetic-field p ~ I s e s " ~ ~Although ~~'~-~ we~ do . not have a detailed understanding of how the field pulses couple atoms and molecules, by choosing the shapes ofthe perturbing pulses such that a single pulse results in roughly 50% atom loss, we observe highcontrast oscillations in the number of atoms in the atomic BEC. From the amplitude of the oscillations, we can put a lower bound on the number of molecules being created. Take, for example, the data in Fig. 4a. The amplitude of the atom oscillations was 1,800(300) atoms. Assuming the fringes are coming from interference with molecules, there must be at least 1,800/2 = 900(200) molecules on average. Assuming that the missing atoms are molecules that we fail to convert back into atoms gives an upper bound of 3,2OO(100) molecules for the conditions of Fig. 4a.
+
The damping time for the oscillations, T~~~~~ was more difficult to measure with high precision than the oscillation frequency. To within our measurement precision, Tdecay did not depend on Bevolve, but our uncertainties in 7decay were as large as 100% for some fields. We had the highest precision measurements for frequencies around 200 kHz where the oscillation period was long compared to our experimental timing jitter but short compared to 7decayAt fre= 38(8) ps for quencies near 200 kHz, we measured 7drcay no = 1.3 X 10'4cm-3 and Tdecay = 91(33) ps when we decreased no to 1.1 x 1 0 ~ ~ ~ ~ - 3 . N,,urstalso had interesting dependencies. For the conditions under which most of the data were collected, the burst contained Nburstdepended -5,000 atoms on average, which is -30% on density and varied from one-half of the atoms lost from the , = 5.4 x 1013cm-3to condensate for our typical peak density of ,n nearly all of the atoms lost from the condensate for no = 1.1 X 10'3c1n-3. NbUrS,, Nremnant, and total number of atoms detected are plotted in Fig. 6 for Bevolve= 159.84(2) G and no = 1.1 X 1013cm-3 (5 times lower density than was used for the data shown in Fig. 4). All three components oscillated at the same frequency. The burst oscillation lagged behind the remnant oscillation by 155(4)". The relative phase shift was nearly 180", so the oscillation amplitude for the total number was smaller than either the burst or the remnant oscillation amplitudes. The relative phase depended sensitively on the fall time of pulse 2. For example, when we increased the fall time from 11 ps to 159 ps, the burst oscillation then lagged behind the remnant oscillation by 68(7)" and the peakto-peak amplitude of the total number oscillation was 5,600(400). For the conditions of Fig. 6, Nlnit= 17,100 exceeded the timeaveraged total number of atoms counted after the pulse sequence by 8(3)"/0 on average. For the higher-density measurements in Fig. 4,
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:igure 3 An absorption image taken after the fast magnetic fleld pulse sequence and the m a n field expansion The shaded bar indicates the optical density The horizontal and iertical directions coincide with the axial and radial axes of the trap respectively The limensions ot the image are 365 x 52 p m The BEC remnant is the roughly spherical loud at the centre while the burst atoms are tocused into a thin line along the axial iirection We note the dramatic difference between the two spatial distributions owing to he large difference in their mean energies (€,,J = 50 x (E,,,,) iATUREIVOL417130MAY 2 0 0 2 ~ ~ n a t u r e c o m
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Figure 4 N,,,,,,, versus fevolvefor no = 5.4 x 10'3cm-3. a, 5e,,1,,= 159.6914) f (aevolve= 580ao). The data fitted to a damped sine wave with a frequency of 207(2)kHz and a decay time of 46121) ps. The open squares near N,,,,,,, = 5,000 indicate the at 159.69 G. The error number remaining versus time after only pulse 1 and levalve estimates in oarentheses are the standard errors in the units of the last decimal place. b, Bevolve = 157.60(4) G (aevolve= 1 ,400ao).We note the increase in time scale from a. These data fitted to an oscillation frequency of 23.9(12)kHr and a decay time of 82138)p ~ . 531
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Figure 5 Oscillation frequency versus magnetic field. The points are the measured frequencies. The solid line is the energy difference between the atom-atom threshold and the bound molecular state found by S.J.J.M.F. Kokkelmans with a coupled-channeis scattering calculation.The dotted line is a plot of d h . The inset is an expanded view of the lower-frequency data. The maximum frequency that we could measure was limited only by timing jitter and finite resolution in the experiment. The magnetic-field measurements for the points with the smallest horizontal error bars were performed on the same days as the corresponding frequency measurements.The error bars for the points with larger field uncertainties were inflated by 100 mG to account for estimated day-to-day field drifts.
39(4)% of the atoms were missing. Experiments with longer pulses 1 and 2 also had a higher fraction ofmissing atoms. For example, when weused50pspulseswithno = 5.4 X 10'3cm-3,56(3)%weremissing. We have carried out double-pulse measurements with a variety of widths and amplitudes for pulses 1 and 2 and a variety of different densities and initial magnetic fields. Although the oscillation frequency was unchanged, the phase, contrast and damping of the oscillations did vary. The contrast was very sensitive to the pulse length, and was lower for longer pulses that created more missing atoms. Defining the contrast as the oscillation amplitude divided by the time averaged number of remnant atoms detected, we observed an optimum contrast of 0.42(7) for 15 ps pulses to 156.6(1) G. A single such pulse removed about half of the atoms from the BEC. When the pulse length was comparable to ?-decay the contrast was reduced by about a factor of two. Under those conditions, threequarters of the atoms were lost after pulse 1. The phase was shifted, but we did not observe a change in the contrast when we varied the levels of pulses 1 and 2 from B = 156.6(1) G (2,400 ao) to B = 155.1(1)G (24,000 ao). The contrast did depend o n the intermediate level, however, and was reduced for Bwolve values closest to the resonance, for which the magnetic-field jumps between Bevolveand pulses 1 and 2 were shortest. We also looked for a temperature dependence of both the damping and the frequency at -200 kHz and did not see any. The high-temperature data were much noisier than the data for pure condensates, owing to unexplained enhanced noise in the number of thermal atoms after the magnetic-field pulse, but when the initial thermal fraction was increased from <5% to 30%, the data still showed oscillations with frequency, amplitude and damping consistent with what was observed with low-temperature data. Our interpretation of our observations is that the first magneticfield pulse provides a sufficiently rapid perturbation to result in nonadiabatic mixing between atomic and molecular states. The superposition then evolves according to the energy difference between the states, which is determined by the magnetic field during the free evolution stage, Bevolve.The second pulse mixes atom and molecule states again, such that the final state of the system depends on the relative phase of atomic and molecular fields at the time of the second pulse. This is somewhat analogous to Ramsey's method of separated oscillating fields". Over a very limited range of pulse amplitudes (a near 1,7004), we could also abserve Rabi-like oscillations with a single pulse towards the Feshbach resonance. The narrow observational window results 532
10
15
20
25
30
35
40
tWOl",
(G)
Figure 6 Number versus tevOtve for no = 1 1 x 1 O'3cm-3. From bottom to top, the data are plots of NbUISt (open circles),Nlemnant (filled circles],and the total number of observed
atoms (grey squares).Each data set was fitted to a damped sine wave resulting in the displayed fits. N,,,,= 17,100is indicated by the flat dashed line. B,,,,,, = 159.84(2]G and the remnant data fitted to an oscillation frequency of 196(l)kHz and T~~~~~ = 91 (33)~ sNot. all of the data used to determine rdecay are shown. To produce condensates with lower density for these measurements,the initial magnetic field before the fast-pulsesequence was 162.2(1)G and the amplitudes for pulses 1 and 2 were reduced to -7 G.
from the conflicting needs for both strong coupling and a condensate loss time23that is long compared to a Rabi oscillation period. After pulse 2, a fraction of the coherent molecular component is converted into the energetic but still spin-polarized burst atoms through a yet to be determined process. Another mystery is the missing atoms. Are they molecules that are not converted back into atoms and are not detected in the burst or the remnant signals? If so, why do we not see them as atoms after the field is turned off and the corresponding molecular state is no longer bound? Why are there fewer missing atoms for lower-density condensates and quicker pulses towards the Feshbach resonance? What is the actual conversion efficiency from atoms to molecules and how could we maximize it? Very near the Feshbach resonance, the molecular state has a magnetic moment nearly the same as that of the free atoms, and hence will remain magnetically trapped. Another remaining question concerns the nature of the molecules. Could they be considered a molecular BEC? Clearly, there is much to be learned about this curious system. 0 Received 22 Apnl: accepted 30 April 2002. 1. Wynar, R H., Freeland, R. S., Ha", D. I.,
Rp,C. & Hemzen, D 1. Molecules in a Bose-Einstein condensate. Science 287, 10161019 (2000). 2. McKenzie, C. et 01. Photoassociation of sodium in a Bose-Einstein condensate. Phyi Rev. Lett. 88, 120403-1-120403-4 (2001). 3 Anglin, 1. R. & Vardi. A Dynamics of a two-made BowEinstein condensate beyond mean-field theory. Phyr. Rev A 64, 013605-1-013605-9 (2001). 4. Cusack, B. I., Alexander, T. I., Ortrovikaya, E. A. & Kivshar, Y.S. Existence and stability of coupled atomic-molecular Bose-Einstein condensates. Phyi. Rev. A 65,013609-1413609-4 (2001). 5. Calsamiglia, I., Mache, M. & Suominen, K. Superposition of macroscopic numbers of atoms and molecules. Phyr. Rev. Lett 87, 160403-1-160403~4 (2001). 6. Drummond, P.D., Kheruntsyan, K. V., Helnzen, D 1. & Wynar, R. H. Stimulated Raman adiabatic passage from an atomic to a molecular Bore-Einstein condensate. Preprint cond-mat/0110578 at (http://uor.lanl.gov) (2002). 7. Heinzen, D. 1, Wynar, R., Drummond, P. D. & Kheruntsyan, K V. Superchemistry: dynamics of coupled atomic and molecular Bose-Einstein condensates, Phyr. R w Len. 84, 5029-5033 (2000). 8. Tiesmga, E , Moerdijk, A,, Verhaar, B. 1. & Stoof, H. T. C. Conditions for Bose-Emstem condensation tn magnetically trapped atomic cesium. Phyi. Rev A 46, R1167-RI170 (1992). 9 Tminga, E., Verhaar, B. 1.8Stoof, H. T. C. Threshold and resonance phenomena in ultracold groundstate collisions. Phyr. Rev A 47, 41144122 (1993). 10. Moerdijk, A I., Verhaar, B. 1. & Axelson, A. Resonances in ultracold collisions of6Li, 'Li, and "Na Phys Rev A 51,48524861 (1995). 11. van Abeelen, F. A. & Verhaar, B. I. Time-dependent Feshbach re~onancescattering and anomalous decay of a Na Bose-Einstein condensate. Phyi Rev Lett. 83, 155s-1553 (1999). 12. M m , F H., Tiesinga, E. &Julienne, P. S . Manipulation of Feahbach resonances m ultracold atomic collisions using time-dependent magnetic fields Phyr. Rev. A 61,022721-1-022721-17 (2000). 13. van Abeelen, F A,, Heinzen, D. I &Verhaar,8. I. Photoassociation as a probe of Feshbach resonances in cold-atom scattering Pkyi. Rev. A 57, R4102-R4105 (1998). 14. Timmermana, E., Tommasini, P., Hurrein, M. & Kerman, A. Feshbach resonan~esin atomic BoreEinstein condensates. Phyr Rep. 315, 199-230 (1999). NATURE 1 VOL 417 I30 MAY 2002 I m . n a t u r e . c o m
625 15. Timmermans, E., Tammasini, P., Cbtt, R., Husrein, M. & Kerman, A. Rarified liquid properties of
hybrid atomic-molecular Bose-Einstein condensates. Phyr. Rev. Len. 83,2691-2691 (1999). 16. Drummond, P. D., Kheruntsyan, K. V. &He, H. Coherent molecular solitons in Bore-Einstein condensates. Phyr. Rev. Left. 81, 3055-3058 (1998). 17. Holland, M., Park, I. & Walser, R. Formation of pairing fields in resonantly coupled atomic and molecular Bore-Einstein condensates. Phyi. Rev. Lett. 86, 1915-1918 (2001). 18. Ghral, K., Gajda, M. & Rzgiewrki, K. Multimode dynamics of a coupled ultracold atomic-molecular system. Phyr. Rev. Len. 86, 1397-1401 (2001). 19. Vardi,A,, Yurovsky, V. A. & Anglin, I. R. Quantum effects on the dynamics of a two-mode atommolecule Bose-Einsteincondensate. Phys. Rev. A 64,063611-146361 1-5 (2001). 20. Stenger, I. ef ai. Strongly enhanced inelastic collisions in a Bose-Einstein condensate near Feshbach resonances. P h p . Rev. Lett. 82,2422-2425 (1999). 21. Carnish,S. L.,Claussen, N. R.,Roberts,J.L.,Cornell,E. A.&Wkman, C.E.Stablp”RbBose~Einstein condensates with widely tunable interactions. Phyi. Rev Len. 85, 1795-1798 (2000). 22. Claussen,N.R., Cornish, S. L., Roberts, I. L., Cornell, E. A. & Wieman, C. E. in AromicPhyiici 17 (eds Arimondo, E., DeNatale, P. & Inguscio, M.) 325-336 (American Institute of Physics, New York, 2001). 23. Claussen, N. R., Donley, E. A., Thompson, 5. T. & Wieman, C. E. Microscopic dynamics in a strongly interacting Bose-Einstein condensate. Phyr. Rev. Len. (submitted): preprint cond-mati 0201400 at (http:llxu.lanl.gov) (2002). 24. Roberts, 1. L. et 01. Controlled collapseof a Bose-Einsteincondensate. Phyr. Rev. Len. 86,421 1 4 2 1 4 (2001). 25. Ptrez-Garcia,V. M., Michinel, H., Cirac, 1. I., Lewenstein, M. & Zoller, P. Dynamics of Bose-Einstein condensates: variational solutions of the Gross~Pitawskuequations. Phyi. Rev. A 56, 142G1432 (1997). 26. Donley, E. A. et al. Dynamics of collapsing and exploding Bose-Einstein condensates. Nature 412, 295-299 (2001). 27. Sakurai, I. I. Modem Quantum Mechanics 410416 (Addison-Wesley,Reading, Massachusetts. 1994). 28. Roberts, I. L. et ai. Improved characterization of elastic scattering near a Ferhbach resonance in ”Rb. Phyr. Rev. A 64, 024702~1424702-3(2001). 29. van Kempen, E. G. M., Kokkelmans, S. 1. 1. M. F., Heinzen, D. 1. & Verhaar, B. I. Interisotope
determination of ultracold rubidium interartions from three high-precision experiments. Phys. Rev Len. 88, 093201~1-093201-4 (2002). 30. Ramsey, N.F. A molecular beam resonance method with separated oscillating fields. P h F 695-699 (1950).
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Acknowledgements We acknowledge contributions from E. A. Cornell a n d the JILA q u a n t u m gas collaboration. We are grateful t o C. H. Greene and S. J. J. M. E Kokkelmans for providing the coupled-channels scattering calculations presented in Fig. 5 a n d t o L. Pitaevskii for numerous discussions. S.T.T. acknowledges the support of a n ARO-MURI Fellowship. This work was also supported by ONR a n d NSF.
Competing interests statement The authors declare that they have no competing financial interests. Correspondence and requests for materials should be addressed to N.R.C. ( e m a i l : nclausseejilaul.colorado.edu).
‘IATURE1 VOL 417 I30 MAY 2002 1 m . n a t u r e . c o m
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REVIEWS OF MODERN PHYSICS, VOLUME 74, JULY 2002
Nobel Lecture: Bose-Einstein condensation in a dilute gas: the first 70 years and some recent experiments* E. A. Cornell and C. E. Wiernan JIU, University of Colorado and National Institute of Standards and Technology, and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440
(Published 19 August 2002)
Bose-Einstein condensation, or BEC, has a long and rich history dating from the early 1920s. In this article we will trace briefly over this history and some of the developments in physics that made possible our successful pursuit of BEC in a gas. We will then discuss what was involved in this quest. In this discussion we will go beyond the usual technical description to try and address certain questions that we now hear frequently, but are not covered in our past research papers. These are questions along the lines of: How did you get the idea and decide to pursue it? Did you know it was going to work? How long did it take you and why? We will review some our favorites from among the experiments we have camed out with BEC. There will then be a brief encore on why we are optimistic that BEC can be created with nearly any species of magnetically trappable atom. Throughout this article we will try to explain what makes BEC in a dilute gas so interesting, unique, and experimentally
challenging.’ The notion of Bose statistics dates back to a 1924 paper in which Satyendranath Bose used a statistical argument to derive the black-body photon spectrum (Bose, 1924). Unable to publish his work, he sent it to Albert Einstein, who translated it into German and got it published. Einstein then extended the idea of Bose’s counting statistics to the case of noninteracting atoms (Einstein, 1924, 1925). The result was Bose-Einstein statistics. Einstein immediately noticed a peculiar feature of the distribution of the atoms over the quantized energy levels predicted by these statistics. At very low but finite temperature a large fraction of the atoms would go into the lowest energy quantum state. In his words, “A separation is effected; one part condenses, the rest remains a saturated ideal gas”* (Einstein, 1925). This phenomenon we now know as Bose-Einstein condensation. The condition for this to happen is that the phase-space density must be greater than approximately unity, in natural units. Another way to express this is
*The 2001 Nobel Prize in Physics was shared by E. A. Cornell, Wolfgang Ketterle, and C. E. Wieman. ‘This article is our “Nobel Lecture” and as such takes a relatively personal approach to the story of the development of experimental Bose-Einstein condensation. For a somewhat more scholarly treatment of the history, the interested reader is referred to E. A. Cornell, J. R. Ensher, and C. E. Wieman, “Experiments in dilute atomic Bose-Einstein condensation in Bose-Einstein Condensation in Atomic Gases,”Proceedings of the International School of Physics “Enrico Fermi” Course C X L , edited by M. Inguscio, S . Stringari, and C. E. Wieman (Italian Physical Society, 1999), pp. 15-66, which is also available as cond-mat19903109.For a reasonably complete technical
review of the three years of explosive progress that immediately followed the first observation of BEC, we recommend reading the above article in combination with the corresponding review from Ketterle, cond-mat/9904034. ’English translation of Einstein’s quotes and the historical interpretation are from Pais (1982), Subtle is the Lord. . . . 0034-6861/200U74(3)/875( 19)/$35.00
that the de Broglie wavelength, X ~ B ,of each atom must be large enough to overlap with its neighbor, or more precisely, nX&> 2.61. This prediction was not taken tembly seriously, even by Einstein himself, until Fritz London (1938) and Laszlo Tisza (1938) resurrected the idea in the mid 1930s as a possible mechanism underlying superfluidity in liquid helium 4. Their work was the first to bring out the idea of BEC displaying quantum behavior on a macroscopic size scale, the primary reason for much of its current attraction. Although it was a source of debate for decades, it is now recognized that the remarkable properties of superconductivity and superfluidity in both helium 3 and helium 4 are related to BEC, even though these systems are very different from the ideal gas considered by Einstein. The appeal of the exotic behavior of superconductivity and of superfluidity, along with that of laser light, the third common system in which macroscopic quantum behavior is evident, provided much of our motivation in 1990 when we decided to pursue BEC in a gas. These three systems all have fascinating counterintuitive behavior arising from macroscopic occupation of a single quantum state. Any physicist would consider these phenomena among the most remarkable topics in physics. In 1990 we were confident that the addition of a new member to the family would constitute a major contribution to physics. (Only after we succeeded did we realize that the discovery of each of the original Macroscopic Three had been recognized with a Nobel Prize, and we are grateful that this trend has continued!) Although BEC shares the same underlying mechanism with these other systems, it seemed to us that the properties of BEC in a gas would be quite distinct. It is far more dilute and weakly interacting than liquid-helium superfluids, for example, but far more strongly interacting than the noninteracting light in a laser beam. Perhaps BEC’s most distinctive feature (and this was not
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something we sufficiently appreciated, in 1990) is the ease with which its quantum wave function may be directly observed and manipulated. While neither of us was to read C. E. Hecht’s prescient 1959 paper (Hecht, 1959) until well after we had observed BEC, we surely would have taken his concluding paragraph as our marching orders: The suppositions of this note rest on the possibility of securing, say by atomic beam techniques, substantial quantities of electron-spin-oriented H, T and D atoms. Although the experimental difficulties would be great and the relaxation behavior of such spin-oriented atoms essentially unknown, the possibilify of opening a rich new field for the study of supefluid properties in both liquid and gaseous states would seem to demand the expenditure of maximum experimental effort? In any case, by 1990 we were awash in motivation. But this motivation would not have carried us far, had we not been able to take advantage of some key recent advances in science and technology, in particular, the progress in laser cooling and trapping and the extensive achievements of the spin-polarized-hydrogen community. However, before launching into that story, it is perhaps worthwhile to reflect on just how exotic a system of indistinguishable particles truly is, and why BEC in a gas is such a daunting experimental challenge. It i s easy at first to accept that two atoms can be so similar one to the other as to allow no possibility of telling them apart. However, confronting the physical implications of the concept of indistinguishable bosons can be troubling. For example, if there are ten bosonic particles to be arranged in two microstates of a system, the statistical weight of the configuration with ten particles in one state and zero in the other is exactly the same as the weight of the configuration with five particles in one state, five in the other. This 1:l ratio of statistical weights is very counterintuitive and rather disquieting. The corresponding ratio for distinguishable objects, such as socks in drawers, that we observe every day is 1:252, profoundly different from 1:l. In the second of Einstein’s two papers (Einstein, 1925; Pais, 1982) on BoseEinstein statistics, Einstein comments that “The. . . molecules are not treated as statistically independ e n t . . . , and the differences between distinguishable and indistinguishable state counting . . . express indirectly a certain hypothesis on a mutual influence of the molecules which for the time being is of a quite mysterious nature. This mutual influence is no less mysterious today, even though we can readily observe the variety of exotic behavior it causes such as the well-known enhanced probability for scattering into occupied states and, of course, Bose-Einstein condensation.” Not only does the Bose-Einstein phase transition offend our sensibilities as to how particles ought best to distribute themselves, it also runs counter to an unspo-
3Emphasis ours. Rev. Mod. Phys.,Vol. 74, No. 3, July 2002
Log Density
FIG. 1. Generic phase diagram common to all atoms: dotted line, the boundary between non-BEC and BEC solid line, the boundary between allowed and forbidden regions of the temperature-density space. Note that at low and intermediate densities, BEC exists only in the thermodynamically forbidden regime.
ken assumption that a phase transition somehow involves thermodynamic stability. In fact, the regions immediately above and immediately below the transition in dilute-gas experiments are both deep in the thermodynamically forbidden regime. This point is best made by considering a qualitative phase diagram (Fig. 1), which shows the general features common to any atomic system. At low density and high temperature, there is a vapor phase. At high density there are various condensed phases. But the intermediate densities are thermodynamically forbidden, except at very high temperatures. The Bose-condensed region of the n - T plane is utterly forbidden, except at such high densities that (with one exception) all known atoms or molecules would form a crystalline lattice, which would rule out Bose condensation. The single exception, helium, remains a liquid below the BEC transition. However, reaching BEC under dilute conditions (say, at densities 10 or 100 times lower than conventional liquid helium) is as thermodynamically forbidden to helium as it is to any other atom. Of course, forbidden is not the same as impossible; indeed, to paraphrase an old Joseph Heller joke, if it were really impossible, they wouldn’t have bothered to forbid it. It comes down in the end to differing time scales for different sorts of equilibrium. A gas of atoms can come into kinetic equilibrium via two-body collisions, whereas it requires three-body collisions to achieve chemical equilibrium (i.e., to form molecules and thence solids). At sufficiently low densities, the twobody rate will dominate the three-body rate, and a gas will reach kinetic equilibrium, perhaps in a metastable Bose-Einstein condensate, long before the gas finds its way to the ultimately stable solid-state condition. The need to maintain metastability usually dictates a more stringent upper limit on density than does the desire to create a dilute system. Densities around 10’” ~ m - for ~ , instance, would be a hundred times more dilute than a condensed-matter helium superfluid. But creating such a gas i s quite impractical even at an additional factor-of1000 lower density, say lo” ~ m - when ~ , metastability times would be on the order of a few microseconds;
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more realistic are densities on the order of 1014~ m - ~ .temperature is decreasing. The Cornell University hydrogen group also considered evaporative cooling The low densities mandated by the need to maintain (Lovelace et al., 1985). By 1988 the MIT group had demlong-lived metastability in turn make necessary the onstrated these virtues of evaporative cooling of magachievement of still lower temperatures if one is to reach netically trapped spin-polarized hydrogen. By 1991 they BEC. obtained, at a temperature of 100 OK, a density that was Thus the great experimental hurdle that must be overonly a factor of 5 below BEC (Doyle, 1991a). Further come to create BEC in a dilute gas is to form and keep progress was limited by dipolar relaxation, but perhaps a sample that is so deeply forbidden. Since our subsemore fundamentally by loss of signal-to-noise, and the quent discussion will focus only on BEC in dilute gases, difficulty of measuring the characteristics of the coldest we shall refer to this simply as BEC in the sections beand smallest clouds (Doyle, 1991b). Evaporative work low and avoid endlessly repeating “in a dilute gas.” was also performed by the Amsterdam group (Luiten Efforts to make a dilute BEC in an atomic gas were sparked by Stwalley and Nosanow (1976). They argued et al., 1993). that spin-polarized hydrogen had no bound states and At roughly the same time, but independent from the hence would remain a gas down to zero temperature, hydrogen work, an entirely different type of cold-atom and so it would be a good candidate for BEC. This physics and technology was being developed. Laser cooling and trapping has been reviewed elsewhere (Aristimulated a number of experimental groups (Silvera and Walraven, 1980; Hardy et al., 1982; Hess et al., 1983; mondo et al., 1991; Chu, 1998; Cohen-Tannoudji, 1998; Johnson et al., 1984) in the late 1970s and early 1980s to Phillips, 1998), but here we mention some of the highbegin pursuing this idea using traditional cryogenics to lights most relevant to our work. The idea that laser cool a sample of polarized hydrogen. Spin-polarized hylight could be used to cool atoms was suggested in early papers by Wineland and Dehmelt (1975), by Hansch and drogen was first stabilized by Silvera and Walraven in Schawlow (1975), and by Letokhov’s group (Letokhov, 1980, and by the mid 1980s spin-polarized hydrogen had 1968). Early optical force experiments were performed been brought within a factor of 50 of condensing (Hess by Ashkin (Bjorkholm et al., 1978). Trapped ions were et al., 1983). These experiments were performed in a dilution refrigerator, in a cell in which the walls were laser-cooled at the University of Washington (Neucoated with superfluid liquid helium as a nonstick coathauser et aZ., 1978) and at the National Bureau of Staning for the hydrogen. The hydrogen gas was compressed dards (now NIST) in Boulder (Wineland etaZ., 1978). using a piston-in-cylinder arrangement (Bell et al., 1986) Atomic beams were deflected and slowed in the early 1980s (Andreev et al., 1981; Ertmer et al., 1985; Prodan or inside a helium bubble (Sprik et al., 1985). These attempts failed, however, because when the cell was made etal., 1985). Optical molasses, where the atoms are very cold the hydrogen stuck to the helium surface and cooled to very low temperatures by six perpendicular recombined. When one tried to avoid that problem by intersecting laser beams, was first studied at Bell Labs warming the cell sufficiently to prevent sticking, the den(Chu et al., 1985). Measured temperatures in the early sity required to reach BEC was correspondingly inmolasses experiments were consistent with the so-called creased, which led to another problem. The requisite Doppler limit, which amounts to a few hundred midensities could not be reached because the rate of threecrokelvin in most alkalis. Light was first used to hold body recombination of atoms into hydrogen molecules (trap) atoms using the dipole force exerted by a strongly goes up rapidly with density and the resulting loss of focused laser beam (Chu et al., 1986). In 1987 and 1988 there were two major advances that became central feaatoms limited the density (Hess, 1986). Stymied by these problems, Harold Hess (Hess, 1986) tures of the method of creating BEC. First, a practical from the MIT hydrogen group realized that magnetic spontaneous-force trap, the magneto-optical trap trapping of atoms (Migdall et al., 1985; Bagnato et al., (MOT) was demonstrated (Raab et al., 1987); and sec1987) would be an improvement over a cell. Atoms in a ond, it was observed that under certain conditions, the magnetic trap have no contact with a physical surface temperatures in optical molasses are in fact much colder and thus the surface-recombination problem could be than the Doppler limit (Lett et al., 1988; Chu et aZ., 1989; circumvented. Moreover, thermally isolated atoms in a Dalibard et aZ., 1989). The MOT had the essential elemagnetic trap would allow cooling by evaporation to far ments needed for a widely useful optical trap: it required lower temperatures than previously obtained. In a rerelatively modest amounts of laser power, it was much markable paper, Hess (1986) laid out most of the impordeeper than dipole traps, and it could capture and hold tant concepts of evaporative cooling of trapped atoms relatively large numbers of atoms. These were heady for the attainment of BEC. Let the highest-energy atoms times in the laser-cooling business. With experiment escape from the trap, and the mean energy, and thus the yielding temperatures mysteriously far below what temperature, of the remaining atoms will decrease. For a theory would predict, it was clear that we all lived under dilute gas in an inhomogeneous potential, decreasing the the authority of a munificent God. temperature will decrease the occupied volume. One During the mid 1980s one of us (Carl) began investican thus actually increase the density of the remaining gating how useful the technology of laser trapping and atoms by removing atoms from the sample. The all imcooling could become for general use in atomic physics. portant (for BEC) phase-space density is dramatically Originally this took the form of just making it cheaper increased as this happens because density is rising while and simpler by replacing the expensive dye lasers with Rev. Mod. Phys., Vol. 74, No. 3,July 2002
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vastly cheaper semiconductor lasers, and then searching for ways to allow atom trapping with these low-cost but also low-power lasers (Pritchard et al., 1986; Watts and Wieman, 1986). With the demonstration of the MOT and sub-Doppler molasses Carl’s group began eagerly studying what physics was limiting the coldness and denseness of these trapped atoms, with the hope of extending the limits further. They discovered that several atomic processes were responsible for these limits. Light-assisted collisions were found to be the major loss process from the MOT as the density increased (Sesko et al., 1989). However, even before that became a serious problem, the light pressure from reradiated photons limited the density (Walker et al., 1990; Sesko et al., 1991). At about the same time, the sub-Doppler temperatures of molasses found by Phillips, Chu, and Cohen-Tannoudji were shown to be due to a combination of light-shifts and optical pumping that became known as Sysiphus cooling (Dalibard and CohenTannoudji, 1989). Random momentum fluctuations from the scattered photons limit the ultimate temperature to about a factor of 10 above the recoil limit. In larger samples, the minimum temperature was higher yet, because of the multiple scattering of the photons. While carrying out studies on the density limits of MOT’S Carl’s group also continued the effort in technology development. This resulted in the creation of a useful MOT in a simple glass vapor cell (Monroe et al., 1990), thereby eliminating the substantial vacuum chamber required for the slowed atomic beam loading that had previously been used. Seeking to take advantage of the large gains in phasespace density provided by the MOT while avoiding the limitations imposed by the undesirable effects of photons, Carl and his student Chris Monroe decided to try loading the cold MOT atoms into a magnetic trap (Monroe etaL, 1990; see Fig. 2). This worked remarkably well. Because further cooling could be carried out as the atoms were transferred between optical and magnetic trap it was possible to get very cold samples, the coldest that had been produced at that time. More importantly, these were not optical molasses samples that were quickly disappearing but rather magnetically trapped samples that could be held and studied for extended penods. These samples were about a hundred times colder than any previous trapped atom samples, with a correspondingly increased phase-space density. This was a satisfying achievement, but as much as the result itself, it was the relative simplicity of the apparatus required that inspired us (including now Eric Cornell, who joined the project as a postdoc in 1990) to see just how far we could push this marriage of laser cooling and trapping and magnetic trapping. Previous laser traps involved expensive massive laser systems and large vacuum chambers for atomic beam precooling. Previous magnetic traps for atoms were usually (Bagnato et al., 1987; Doyle, 1991) extremely complex and bulky (often with superconducting coils) because of the need to have sufficiently large depths and strong confinement. Laser traps and magnetic traps were Rev. Mod. Phys., Vol. 74, No. 3, July 2002
both somewhat heroic experiments individually, to be undertaken only by a select handful of well-equipped AM0 laboratories. The prospect of trying to get both traps working, and working well, in the same room and on the same day, was daunting. However, in the first JILA magnetic trap experiment our laser sources were simple diode lasers, the vacuum system was a small glass vapor cell, and the magnetic trap was just a few turns of wire wrapped around it. This magnetic field was adequate because of the low temperatures of the lasercooled and trapped samples. Being able to produce such cold and trapped samples in this manner encouraged one to fantasize wildly about possible things to do with such an atom sample. Inspired by the spin-polarized hydrogen work, our fantasizing quickly turned to the idea of evaporative cooling further to reach BEC. It would require us to increase the phase-space density by 5 orders of magnitude, but since we had just gained about 15 orders of magnitude almost for free with the vapor cell MOT, this did not seem so daunting. The JILA vapor-cell MOT (Fig. 3), with its superimposed ion pump trap, introduced a number of ideas that are now in common use in the hybrid trapping business (Monroe et al., 1990; Monroe, 1992): (i) Vapor-cell (rather than beam) loading, (ii) fused-glass rather than welded-steel architecture, (iii) extensive use of diode lasers, (iv) magnetic coils located outside the chamber, (v) overall chamber volume measured in cubic centimeters rather than liters, (vi) temperatures measured by imaging an expanded cloud, (vii) magnetic-field curvatures calibrated in situ by observing the frequency of dipole and quadrupole (sloshing and pulsing) cloud motion, (viii) the basic approach of a MOT and a magnetic trap which are spatially superimposed (indeed, which often share some magnetic coils) but temporally sequential, and (ix) optional use of additional molasses and optical pumping sequences inserted in time between the MOT and magnetic trapping stages. It is instructive to note how a modem, Ioffe-Pritchard-based BEC device (Fig. 4) resembles its ancestor (Fig. 3). As we began to think about applying the technique of evaporative cooling with hydrogen to our very cold alkali atoms we looked carefully at the hydrogen work and its lessons. When viewed from our 1990 perspective the previous decade of work on polarized hydrogen provided a number of important insights. It was clear that the unique absence of any bound states for spinpolarized hydrogen was actually not an important issue (other than its being the catalyst for starting the entire field, of course!). Bound states or not, a very cold sample of spin-polarized hydrogen, like every other gas, has a lower-energy state to which it can go, and its survival depends on the preservation of metastability. For hydrogen the lower-energy state is a solid, although from an experimental point of view the rate-limiting process is the formation of diatomic molecules (with appropriately reoriented spins). Given that all atomic gases are only metastable at the BEC transition point, the real experimental issue becomes: How well can one
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FIG. 4. Modern MOT and magnetic trap apparatus, used by Cornish et al., 2000 [Color].
FIG. 2. Chris Monroe examines an early hybrid MOTmagnetic trap apparatus [Color]. preserve the requisite i~etastabili~y while still cooling sufficiently far to The realization bility was the key experimental challenge one should focus on least as important to the attainment o the experimental techniques we subsequently developed to actually achieve it. The work on hydrogen provided an essential guide for evaluating and tackling this challenge. It provided us with a potential cooling technique (evaporative cooling of magnetically trapped atoms) and mapped out many of the processes by which a magnetically trapped atom can be lost from its metastable state.
FIG. 3. The glass vapor cell and magnetic coils used in early JILA efforts to hybridize laser cooling and magnetic trapping (see Monroe er al., 1990). The glass tubing is 2.5 em in diameter. The Ioffe current bars have been omitted for clarity. Rev. Mod. Phys., Vol. 74, No. 3, July 2002
The hydrogen work made it clear that it was all an issue of good versus bad collisions. The good collisions are elastic collisions that rethermalize the atoms during evaporation. The more collisions there are, the more quickly and efficiently one can cool. The bad collisions are the inelastic collisions that quench the metastability. Hydrogen had already shown that three-body recombination collisions and e spin-flip collisions were the major inelastic culprit ers were fairly close to reachin encouragement. It meant that was not ridicudo a little better lously distant and that one on1 in the proportion of good The more we thought about this, the more we began to suspect that our heavy alkali atoms would likely have more favorable collision properties than hydrogen atoms and thus have a good chance of success. Although knowledge of the relevant collision cross sections was totally nonexistent at that time, we were able to come up with arguments for how the cross sections might scale relative to hydrogen. These are discussed in more detail below in the section discussing why collisional concerns make it likely that BEC can be created in a large number of different species. Here we will just give a brief summary consistent with our views circa 1990. The dipole spin-flip collisions that limited hydrogen involve spin-spin interactions and thus could be expected to be similar for the alkalis and for hydrogen because the magnetic moments are all about the same. The good collision. needed for evaporative cooling, however, should be much larger for heavy alkalis with their fat fluffy electron clouds than for hydrogen. The other villain of the hydrogen effort, three-body recombination, was a total mystery, but because it goes as density cubed while the good elastic collisions go as density squared, it seemed as if we should always be able to find a sufficiently low-density and low-temperature regime to avoid it (see Monroe, 1992). As a minor historical note, we might point out that during these considerations we happily ignored the fact that the temperatures required to achieve BEC in a
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heavy alkali gas are far colder than those needed for the same density of hydrogen. The critical temperature for ideal-gas BEC is inversely proportional to the mass. It was clear that we would need to cool to well under a microkelvin, and a large three-body recombination rate would have required us to go to possibly far lower temperatures. To someone coming from a traditional cryogenics background this would (and probably did) seem like sheer folly. The hydrogen work had been pushing hard for some years at the state of the art in cryogenic technology, and here we proposed to happily jump far beyond that. Fortunately we were coming to this from an A M 0 background in a time when temperatures achieved by laser cooling were dropping through the floor. Optimism was in the air. In fact, we later discovered optimism can take one only so far: There were actually considerable experimental difficulties, and further cooling came at some considerable effort and a five-year delay. Nevertheless, it is remarkable that with evaporative cooling a magnetically trapped sample of atoms, surrounded on all sides by a 300-K glass cell, can be cooled to reach temperatures of only a few nanokelvin, and moreover it looks quite feasible to reach even colder temperatures. General collisional considerations gave us some hope that the evaporative cooling hybrid trap approach with alkali atoms would get us to BEC, or, if not, at least reveal some interesting new physics that would prevent it. Nonetheless, there were powerful arguments against pursuing this. First, our 1990-era arguments in favor of it were based on some very fuzzy intuition; there were no collision data or theories to back it up and there were strong voices in disagreement. Second, the hydrogen experiments seemed to be on the verge of reaching BEC, and in fact we thought it was likely that if BEC could be achieved they would succeed first. However, our belief in the virtues of our technology really carried the day in convincing us to proceed. With convenient lasers in the near-IR, and with the good optical access of a roomtemperature glass cell, detection sensitivity could approach single-atom capability. We could take pictures of only a few thousand trapped atoms and immediately know the energy and density distribution. If we wanted to modify our magnetic trap it only required a few hours winding and installing a new coil of wires. This was a dramatic contrast with the hydrogen experiments that, like all state-of-the-art cryogenics experiments, required an apparatus that was the better part of two stones, and the time to modify it was measured in (large) fractions of a year. Also, atomic hydrogen was much more difficult to detect and so the diagnostics were far more limited. This convinced us that although hydrogen would likely succeed first, our hybrid trap approach with easily observed and manipulated alkali samples would be able to carry out important science and so was well worth pursuing in its own right. From the very beginning in 1990, our work on BEC was heavily involved with cold atomic collisions. This was somewhat ironic since previously both of us had actively avoided the large fraction of A M 0 work on the Rev. Mad. Phys., Vol. 74, No. 3,July 2002
subject of atomic collisions. Atomic collisions at very cold temperatures is now a major branch of the discipline of A M 0 physics, but at the end of the 1980s there were almost no experimental data, and what there was came in fact from the spin-polarized hydrogen experiments (Gillaspy et al., 1989). There was theoretical work on hydrogen from Shlyapnikov and Kagan (Kagan et al., 1981, 1984), and from Silvera and Verhaar (Lagendijk et al., 1986). An early paper by Pritchard (1986) includes estimates on low-temperature collisional properties for alkalis. His estimates were extrapolations from roomtemperature results, but in retrospect, several were surprisingly accurate. As we began to work on evaporative cooling, much of our effort was devoted to determining the sizes of all the relevant good and bad collision cross sections. Our efforts were helped by the theoretical efforts of Boudewijn Verhaar, who was among the first to take our efforts seriously and attempt to calculate the rates in question. Chris Greene also provided us with some useful theoretical estimates. Starting in 1990 we carried out a series of experiments exploring various magnetic traps and measuring the relevant collision cross sections. As this work proceeded we developed a far better understanding of the conditions necessary for evaporative cooling and a much clearer understanding of the relevant collisional issues (Monroe et al., 1993; Newbury et al., 1995). Our experimental concerns evolved accordingly. In the early experiments (Monroe et al., 1990, 1993; Cornell et af., 1991; Monroe, 1992) a number of issues came up that continue to confront all BEC experiments: the importance of aligning the centers of the MOT and the magnetic trap, the density-reducing effects of mode-mismatch, the need to account carefully for the (previously ignored) force of gravity, heating (and not merely loss) from background gas collisions, the usefulness of being able to turn off the magnetic fields rapidly, the need to synchronize many changes in laser status and magnetic fields together with image acquisition, an appreciation for the many issues that can interfere with accurate determinations of density and temperature by optical methods, either florescence or absorption imaging, and careful stabilization of magnetic fields. The mastery of these issues in these early days made it possible for us to proceed relatively quickly to quantitative measurements with the BEC once we had it. In 1992 we came to realize that dipolar relaxation in alkalis should in principle not be a limiting factor. As explained in the final section of this article, collisional scaling with temperature and magnetic field is such that, except in pathological situations, the problem of good and bad collisions in the evaporative cooling of alkalis is reduced to the ratio of the elastic collision rate to the rate of loss due to imperfect vacuum; dipolar relaxation and three-body recombination can be finessed, particularly since our preliminary data showed they were not enormous. It was reassuring to move ahead on efforts to evaporate with the knowledge that, while we were essentially proceeding in the dark, there were not as many monsters in the dark as we had originally imagined.
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It rapidly became clear that the primary concerns would be having sufficient elastic collision rate in the magnetic trap and sufficiently low background pressure to have few background collisions that removed atoms from the trap. To accomplish this it was clear that we needed higher densities in the magnetic trap than we were getting from the MOT. Our first effort to increase the density two years earlier was based on a multipleloading scheme (Cornell et al., 1991). Multiple MOTloads of atoms were launched in moving molasses, optically pumped into an untrapped Zeeman level, focused into a magnetic trap, then optically repumped into a trapped level. The repumping represented the necessary dissipation, so that multiple loads of atoms could be inserted in a continuously operating magnetic trap. In practice, each step of the process involved some losses, and the final result was disappointing. Later, however, as discussed below, we resurrected the idea of multiple loading from one MOT to another to good advantage (Gibble et al., 1995; Myatt et al., 1996). This is now a technique cllrrently in widespread practice. In addition to building up the initial density we realized that the collision rate could be dramatically increased by, after loading into a magnetic trap, compressing the atoms by further increasing the curvature of the confining magnetic fields. In a harmonic trap, the collision rate after adiabatic compression scales as the final confining frequency squared (Monroe, 1992). This method is discussed by Monroe (1992) and was implemented first in early ground-state collisional work (Monroe et al., 1993). In fall of 1992, Eric’s postdoctoral appointment concluded, and, after a tour through the job market, he decided to take the equivalent of an assistant professor position at JILANIST. He decided to use his startup money to build a new experimental apparatus that would be designed to put these ideas together to make sure evaporation worked as we expected. Meanwhile, we continued to pursue the possibility of enhanced collision cross sections in cesium using a Feshbach resonance. At that point our Monte Carlo simulations said that a ratio of about 150 elastic collisions per trap lifetime was required to achieve runaway evaporation. This is the condition where the elastic collision rate would continue to increase as the temperature decreased, and hence evaporation would continue to improve as the temperature was reduced. We also had reasonable determinations of the elastic collision cross sections. So the plan was to build a simple quadrupole trap that would allow very strong squeezing to greatly enhance the collision rate, combined with a good vacuum system in order to make sure evaporative cooling worked as expected. Clearly, there was much to be gained by building a more tightly confining magnetic trap, but the requirement of adequate optical access for the MOT, along with engineering constraints on power dissipation, made the design problem complicated. When constructing a trap for weak-field-seeking atoms, with the aim of confining the atoms to a spatial size much smaller than the size of the magnets, one would Rev. Mod. Phys., Vol. 74, No. 3, July 2002
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like to use linear gradients. In that case, however, one is confronted with the problem of the minimum in the magnitude of the magnetic fields (and thus of the confining potential) occurring at a local zero in the magnetic field. This zero represents a hole in the trap, a site at which atoms can undergo Majorana transitions (Majorana, 1931) and thus escape from the trap. If one uses the second-order gradients from the magnets to provide the confinement, there is a marked loss of confinement strength. This scaling is discussed by Petrich et al. (1995). We knew that once the atoms became cold enough they would leak out the hole in the bottom of the trap, but the plan was to go ahead and get evaporation and worry about the hole later. We also recognized that even with successful evaporative cooling, and presuming we could solve the issue of the hole in the quadrupole trap, there was still the question of the sign of scattering length, which must be positive to ensure the stability of a large condensate. In setting up the new apparatus Eric chose to use rubidium. Given the modulo arithmetic that goes into determining a scattering length, it seemed fair to treat the scattering lengths of different isotopes as statistically independent events, and rubidium with its two stable isotopes offered two rolls of the dice for the same laser system. Eric then purchased a set of diode lasers for the rubidium wavelength, but of course we kept the original cesium-tuned diode lasers. The wavelengths of cesium and of the two rubidium isotopes are sufficiently similar that in most cases one can use the same optics. Thus we preserved the option of converting from one species to another in a matter of weeks. The chances then of Nature’s conspiring to make the scattering length negative, for both hyperfine levels, for all three atoms, seemed very small. Progress in cold collisions, particularly the experiment and theory of photoassociative collisions, had moved forward so rapidly that by the time we had evaporatively cooled rubidium to close to BEC temperatures a couple of years later there existed, at the 20% level, values for several of the elastic scattering lengths. In particular, we knew that it was positive for the 2,2 state of Rb-87 (Thorsheim et al., 1987; Lett et al., 1993; Miller et al., 1993; Abraham et al., 1995; Gardner et al., 1995; McAIexander et al., 1995). Our original idea for the quadrupole trap experiment was to pulse a burst of rubidium into our cell, where we would catch a large sample in the MOT and then hold it as the residual rubidium was quickly pumped away, leaving a long trap lifetime. We, particularly Eric’s postdoc, Mike Anderson, spent many frustrating months discovering how difficult this seemingly simple idea was to actually implement in practice. The manner in which rubidium interacted with glass and stainless-steel surfaces conspired to make this so difficult we finally gave up. We ended up going with a far-from-optimum situation of working with extremely low rubidium pressure and doing our best at maximizing the number of atoms captured in the MOT from this feeble vapor and enhancing the collision rate for those relatively few atoms as much
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as possible. We recognized that this was a major compromise, but we had been trying to evaporate for some time, and we were getting impatient! We had no stomach for building another apparatus just to see evaporation. Fortunately we were able to find two key elements to enhance the MOT loading and density. First was the use of a dark-spot MOT in which there is a hole in the center of the MOT beams so the atoms are not excited. This technique had been demonstrated by Ketterle (Ketterle etal., 1993) as a way to greatly enhance the density of atoms in a MOT under conditions of a very high loading rate. The number of atoms we could load in our vapor cell MOT with very low rubidium vapor was determined by the loading rate over the loss rate. In this case the loss rate was the photoassociative collisions we had long before found to be important for losses from MOT'S. The dark-spot geometry reduced this two-body photoassociative loss in part because in our conditions it reduced the density of atoms in the MOT (Anderson et al., 1994). Using this approach we were able to obtain 10' atoms in the MOT collected out of a very low vapor background (so that magnetic trap lifetime was greater than 100 s). The second key element was the invention of the compressed MOT (CMOT), a technique for substantially enhancing the density of atoms in the MOT on a transient basis. For the CMOT, the MOT was filled and then the field gradient and laser detuning were suddenly changed to greatly suppress the multiple photon scattering. This produced much higher densities and clouds whose shape was a much better match to the desired shape of the cloud in the magnetic trap. This was a very transient effect because the losses from the MOT were much larger under these conditions, but that was not important; the atoms needed only to be held for the milliseconds required before they were transferred to the magnetic trap (Petrich et af., 1994; see Fig. 5). With these improvements and a quadrupole trap that provided substantial squeezing, we were able to finally demonstrate evaporative cooling in rubidium. Cooling by evaporation is a process found throughout Nature. Whether the material being cooled is an atomic nucleus or the Atlantic Ocean, the rate of natural evaporation and the minimum temperature achievable are limited by the particular fixed value of the work function of the evaporating substance. In magnetically confined atoms, no such limit exists, because the work function is simply the height of the lowest point in the rim of the confining potential. Hess (1986) pointed out that, by perturbing the confining magnetic fields, one could make the work function of a trap arbitrarily low; as long as favorable collisional conditions persist, there is no lower limit to the temperatures attainable in this forced evaporation. Pritchard (Pritchard et al., 1989) pointed out that evaporation could be performed more conveniently if the rim of the trap were defined by an rf-resonance condition, rather than simply by the topography of the magnetic field; experimentally, his group made first use of position-dependent rf transitions to selectively transfer Rev. Mod. Phys., Vol. 74,NO.3, July 2002
magnetically trapped sodium atoms between Zeeman levels and thus characterized their temperature (Martin et af., 1988). In our experiment we used Pritchard's technique of an rf field to selectively evaporate. It was a great relief to see evaporative cooling of laser precooled, magnetically trapped atoms finally work, as we had been anticipating it would for so many years. Unfortunately, it worked exactly as well, but no better, than we had anticipated. The atoms were cooled to about 40 p K and then disappeared, at just the temperature we had estimated they would be lost, through the hole in the bottom of the quadrupole trap. Eric came up with an idea that solved this problem. It was a design for a new type of trap that required relatively little modification to the apparatus and so was quickly implemented. This was the Time Orbiting Potential (TOP) trap in which a small rotating magnetic field was added to the quadrupole field (Petrich et al., 1995). This moved the field zero in an orbit faster than the atoms could follow. It was the perfect solution to our problem. Mike Anderson, another postdoc, Wolfgang Petrich, and graduate student Jason Ensher quickly implemented this design. Their efforts were spurred on by the realization that there were several other groups who had now demonstrated or were known to be on the verge of demonstrating evaporative cooling in alkalis in the pursuit of BEC. The TOP design worked well, and the samples were cooled far colder, in fact too cold for us to reliably measure. We had been measuring temperature simply by looking at the spatial size of the cloud in the magnetic trap. As the temperature was reduced the size decreased, but we were now reaching temperatures so low that the size had reached the resolution limit of the optical system. We saw dramatic changes in the shapes of the images as the clouds became very small, but we knew that a variety of diffraction and aberration effects could greatly distort images when the sample size became only a few wavelengths in size, so our reaction to these shapes was muted, and we knew we had to have better diagnostics before we could have meaningful results. Here we were helped by our long experience in studying various trapped clouds over the years. We already knew the value of turning the magnetic trap off to let the cloud expand and then imaging the expanded cloud to get a measure of the momentum distribution in the trap. Since the trap was harmonic, the momentum distribution and the original density distribution were nearly interchangeable. Unfortunately, once the magnetic field was off, the atoms not only expanded but also simply fell under the influence of gravity. We found that the atoms tended to fall out of the field of view of our microscope before they had sufficiently expanded. The final addition to the apparatus was a supplementary magnetic coil, which provided sufficient field gradient to cancel the effects of gravity while minimizing any perturbation to the relative ballistic trajectories of the expanding atoms. Anderson, Ensher, and a new graduate student, Mike Matthews (Fig. 6), worked through a weekend to install the antigravity coil and, after an additional day or two of
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FIG. 5. Wolfgang Petrich working on CMOT [Color]. trial and error, got the new field configuration shimmed up. By June 5 , 1995 the new technology was working well and we began to look at the now greatly expanded clouds. To our delight, the long-awaited two-component distribution was almost immediately apparent (Fig. 7) when the samples were cooled to the regime where BEC was expected. The excitement was tempered by the con-
FIG. 6. From left, Mike Anderson, Debbie Jin, Mike Matthews, and Jason Ensher savor results of early BEC experiment [Color]. Rev. Mod. Phys., Vol. 74, No. 3, July 2002
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cern that after so many years of anticipating two component clouds as a signature of BEC, we might be fooling ourselves. Almost from the beginning of the search for BEC, it was recognized (Lovelace and Tommila, 1987) that as the sample started to condense, there would be a spike in the density and momentum distributions corresponding to the macroscopic population of the ground state. This would show up as a second component on top of the much broader normal thermal distribution of uncondensed atoms. This was the signature we had been hoping to see from our first days of contemplating BEC. The size of the BEC component in these first observations also seemed almost too good to be true. In those days it was known that in the much higher density of the condensate, three-body recombination would be a more dominant effect than in the lower-density uncondensed gas. For hydrogen it was calculated that the condensed component could never be more than a few percent of the sample. The three-body rate constants were totally unknown for alkali atoms at that time, but because of the H results it still seemed reasonable to expect the condensate component might only be a modest fraction of the total sample. But in our first samples we saw it could be nearly loo%! In the light of the prevailing myth of unattainability that had grown up around BEC over the years, our observations seemed too good to be true. We were experienced enough to know that when results in experimental physics seem too good to be true, they almost always are! We worried that in our enthusiasm we might confuse the long-desired BEC with some spurious artifact of our imaging system. However, our worries about the possibility of deluding ourselves were quickly and almost entirely alleviated by the anisotropy of the BEC cloud. This was a very distinctive signature of BEC, the credibility of which was greatly enhanced to us by the fact that it first revealed itself in the experiment, and then we recognized its significance, rather than vice versa. It was a somewhat fortuitous accident that the TOP trap provided a distinctly anisotropic trapping potential, since we did not appreciate its benefits until we saw the BEC data. A normal thermal gas (in the collisionally thin limit) released from an anisotropic potential will spread out isotropically. This is required by the equipartition theorem. However, a Bose-Einstein condensate is a quantum wave and so its expansion is governed by a wave equation. The more tightly confined direction will expand the most rapidly, a manifestation of the uncertainty principle. Seeing the BEC component of our two-component distribution display just this anisotropy, while the broader uncondensed portion of the sample observed at the same time, with the same imaging system, remained perfectly isotropic (as shown in Fig. 8), provided the crucial piece of corroborating evidence that this was the long-awaited BEC. By coincidence we were scheduled to present progress reports on our efforts to achieve BEC at three international conferences in the few weeks following these observations (Anderson et al., 1996). Nearly all the experts in the field were represented at one or more of these
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FIG. 7. Three density distributions of the expanded clouds of rubidium atoms at three different temperatures. The appearance of the condensate is apparent as the narrow feature in the middle image. On the far right, nearly all the atoms in the sample are in the condensate. The original experimental data were two-dimensional black and white shadow images, but these images have been converted to three dimensions and given false color density contours [Color].
conferences, and the data were sufficient to convince the most skeptical of them that we had truly observed BEC. This consensus probably facilitated the rapid refereeing and publication of our results.
In the original TOP-trap apparatus we were able to obtain so-called pure condensates of a few thousand atoms. By pure condensates we meant that nearly all the atoms were in the condensed fraction of the sample.
FIG. 8. Looking down on the three images of Figure 7 (Anderson et al., 1995). The condensate in B and C is clearly elliptical in shape [Color]. Rev. Mod. Phys., Vol. 74, No. 3, July 2002
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Samples of this size were easily large enough to image. Over the few months immediately following the original observation, we undertook the process of a technological shoring up of the machine, until the machine reached the level of reliability necessary to crank out condensate after reproducible condensate. This set the stage for the first generation of experiments characterizing the properties of the condensate, most notably the condensate excitation studies discussed below. Although by 1995 and 1996 we were able to carry out a number of significant BEC experiments with the original TOP-trap machine, even by 1994, well before the original condensates were observed, we had come to realize the limitations of the single-cell design. Our efforts to modulate the vapor pressure were not very successful, which forced us to operate at a steady-state rubidium vapor pressure. Choosing the value of vapor pressure at which to operate represented a compromise between our need to fill the vapor-cell MOT with as many atoms as possible and our need to have the lifetime in the magnetic trap as long as possible. The single-cell design also compelled us to make a second compromise, this time over the size of the glass cell. The laser beams of the MOT enter the cell through the smooth, flat region of the cell; the larger the glass cell, the larger the MOT beams, and the more atoms we could herd into the MOT from the room-temperature background vapor. On the other hand, the smaller the glass cell, the smaller the radii of the magnetic coils wound round the outside of the cell, and the stronger the confinement provided by the magnetic trap. Hans Rohner in the JILA specialty shop had learned how (Rohner, 1994) to create glass cells with the minimum possible wasted area. But even with the dead space between the inner diameter of the magnetic coils and the outer diameter of the clear glass windows made as small as it could be, we were confronted with an unwelcome tradeoff. Thus, in 1994, in parallel with our efforts to push as hard as we could toward BEC in our original, single-cell TOP trap, we began working on a new technology that would avoid this painful tradeoff. This approach was a modified version of our old multiple loading scheme in which many loads from a MOT were transferred to a magnetic trap in a differentially pumped vacuum chamber. That approach had been defeated by the difficulty in transferring atoms from MOT to magnetic trap without losing phase-space density. There was no dissipation in the magnetic trap to compensate for a slightly too hard or too soft push from one trap to the other. This made us recognize the importance of having dissipation in the second trap, and so we went to a system in which atoms were captured in a large-cell MOT in a region of high rubidium pressure, and then transferred through a small tube into a second, small-cell MOT in a low-pressure region. This eliminated the previous disadvantages while preserving the advantages of multiple loading to get much larger numbers of trapped atoms in a low-vacuum region. The approach worked well, particularly when we found that simple strips of plastic refrigerator magnet material around the outside of the transfer tube between Rev. Mod. Phys., Vol. 74, No. 3, July 2002
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the two traps provided an excellent guide to confine the atoms as they were pushed from one trap to the other (Myatt et al., 1996). With this scheme we were still able to use inexpensive low-power diode lasers to obtain about one hundred times more atoms in the magnetic trap than in our single MOT-loaded TOP magnetic trap and with a far longer lifetime; we saw trap lifetimes up to 1000 s in the double MOT magnetic trap. This system started working in 1996 and it marked a profound difference in the ease with which we could make BEC (Myatt et aZ., 1997). In the original BEC experiment everything had to be very well optimized to achieve the conditions necessary for runaway evaporative cooling and thereby BEC. In the double MOT system there were orders of magnitude to spare. Not only did this allow us to routinely obtain million-atom pure condensates, but it also meant that we could dispense with the dark-spot optical configuration with its troublesome alignment. We could be much less precise with many other aspects of the experiment as well. The first magnetic trap we used with the double-MOT BEC machine was not a TOP trap, but instead was our old baseball-style Ioffe-Pritchard trap. The baseball coil trap is rather complementary to the TOP trap in that each has unique capabilities. For example, the geometry of the TOP trap potential can be changed over a wide range, although the range of dc fields is quite limited. In contrast, the geometry of the baseball coil trap potential can be varied only by small amounts, but the dc bias field can be easily varied over hundreds of gauss. Thus in 1996, when we upgraded the original BEC machine to incorporate the double-MOT technology, we preserved the TOP trap coil design. Each is well suited to certain types of experiments, as will be evident in the discussions below. With the double-MOT setups we were able to routinely make million-atom condensates in a highly reliable manner in both TOP and baseball-type magnetic traps. These were used to carry out a large number of experiments with condensates over the period from 1996 to the present. Some of our favorite experiments are briefly discussed below. FAVORITE EXPERIMENTS Collective excitations
In this section, by excitations we mean coherent fluctuations in the density distribution. Excitation experiments in dilute-gas BEC have been motivated by two main considerations. First, a Bose-Einstein condensate is expected to be a superfluid, and a superfluid is defined by its dynamical behavior. Studying excitations is an obvious initial step toward understanding dynamical behavior. Second, in experimental physics a precision measurement is almost always a frequency measurement, and the easiest way to study an effect with precision is to find an observable frequency that is sensitive to that effect. In the case of dilute-gas BEC, the observed fre-
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.-
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time (ms) FIG. 9. Zero-temperature excitation data from Jin et al. (1996). A weak m=O modulation of the magnetic trapping potential is applied to a 4500-atom condensate in a 132-Hz (radial) trap. Afterward, the freely evolving response of the condensate shows radial oscillations. Also observed is a sympathetic response of the axial width, approximately 180" out of phase. The frequency of the excitation is determined from a sine wave fit to the freely oscillating cloud widths.
quency of standing-wave excitations in a condensate is a precise test of our understanding of the effect of interactions. BEC excitations were first observed by Jason Ensher, Mike Matthews, and then-postdoc Debbie Jin, using destructive imaging of expanded clouds (Jin et al., 1996). The nearly zero-temperature clouds were coherently excited (see below), then allowed to evolve in the trap for some particular dwell time, and then rapidly expanded and imaged via absorption imaging. By repeating the procedure many times with varying dwell times, the time-evolution of the condensate density profile can be mapped out. From these data, frequencies and damping rates can be extracted. In axially symmetric traps, excitations can be characterized by their projection of angular momentum on the axis. The perturbation on the density distribution caused by the excitation of lowest-lying m = 0 and m = 2 modes can be characterized as simple oscillations in the condensate's linear dimensions. Figure 9 shows the widths of an oscillating condensate as a function of dwell time. A frequency-selective method for driving the excitations is to modulate the trapping potential at the frequency of the excitation to be excited (Jin et al., 1996). Experimentally this is accomplished by summing a small ac component onto the current in the trapping magnets. In a TOP trap, it is convenient enough to independently modulate the three second-order terms in the transverse potential. By controlling the relative phase of these modulations, one can impose m=O, m=2, or m=-2 symmetry on the excitation drive. There have been a very large number of theory papers published on excitations; much of this work is reviewed by Dalfovo et al. (1999). All the zero-temperature, small-amplitude excitation experiments published to Rev. Mod. Phys., Vol. 74,No. 3, July 2002
date have been very successfully modeled theoretically. Quantitative agreement has been by and large very good; small discrepancies can be accounted for by assuming reasonable experimental imperfections with respect to the T=O and small-amplitude requirements of theory. The excitation measurements discussed above were then revisited at nonzero temperature (Jin et al., 1997). The frequency of the condensate excitations was clearly observed to depend on the temperature, and the damping rates showed a strong temperature dependence. This work is important because it bears on the little-studied finite-temperature physics of interacting condensates. Connection with theory (Hutchinson et al., 1997; Dodd et al., 1998; Fedichev and Shlyapnikov, 1998) remains somewhat tentative. The damping rates, which are observed to be roughly linear in temperature, have been explained in the context of Landau damping (Liu, 1997; Fedichev et al., 1998). The frequency shifts are difficult to understand, in large part because the data so far have been collected in a theoretically awkward, intermediate regime: the cloud of noncondensate atoms is neither so thin as to have completely negligible effect on the condensate, nor so thick as to be deeply in the hydrodynamic (HD) regime. In this context, hydrodynamic regime means that the classical mean free path in the thermal cloud is much shorter than any of its physical dimensions. In the opposite limit, the collisionless regime, there are conceptual difficulties with describing the observed density fluctuations as collective modes. Recent theoretical work suggests that good agreement with experiment may hinge on correctly including the role of the excitation drive (Stoof, 2000; Jackson and Zaremba, 2002). Two-component condensates
As mentioned above, the double-MOT system made it possible to produce condensates even if one were quite sloppy with many of the experimental parameters. One such parameter was the spin state in which the atoms are optically pumped before being loaded into the magnetic trap. As our student Chris Myatt was tinkering around setting up the evaporation one day, he noticed, to his surprise, that there seemed to be two different clouds of condensate in the trap. They were roughly at the locations expected for the 2,2 and 1,- 1 spin states to sit, but that seemed impossible to us because these two states could undergo spin-exchange collisions that would cause them to be lost from the trap, and the spinexchange collision cross sections were thought to be enormous. After extensive further studies to try and identify what strange spurious effect must be responsible for the images of two condensate clouds we came to realize that they had to be those two spin states. By a remarkable coincidence, the triplet and singlet phase shifts are identical and so at ultralow tem eratures the spin-exchange collisions are suppressed in 'Rb by three to four orders of magnitude! This suppression meant that the different spin species could coexist and their
638 E. A. Cornell and C. E. Wiernan: BEC in a dilute gas
OJ
F-1
FIG. 10. Energy-level diagram for ground electronic state of 87Rb. The first condensates were created in the 2,2 state. Mixtures containing the 2,2 and 1,- 1 state were found to coexist. In later studies we created condensates in the 1,- 1 state and then excited it to the 2,l state using a microwave plus rf twophonon transition.
mixtures could be studied. In early work we showed that one could carry out sympathetic cooling to make BEC by evaporating only one species and using it as a cooling fluid to chill the second spin state (Myatt et al., 1997). We also were able to see how the two condensates interacted and pushed each other apart, excluding all but a small overlap in spite of the fact that they were highly dilute gases. These early observations stimulated an extensive program of research on two-component condensates. After Myatt’s original measurements (Myatt et al., 1997), our work in this field, led by postdoc David Hall, concentrated on the 1,- 1 and 2,+ 1 states (see Fig. 10) because they could be coherently kterconverted using twophoton (microwave plus rf) transitions and they had nearly identical magnetic moments and so saw nearly the same trapping potentials (Matthews et al., 1998). When the two-photon radiation field is turned off, the rate of spontaneous interconversion between the two spin species essentially vanishes, and moreover the optical imaging process readily distinguishes one species from the other, as their difference in energy (6.8 GHz) is very large compared to the excited-state linewidth. In this situation, one may model the condensate dynamics as though there were two distinct quantum fluids in the trap. Small differences in scattering length make the two fluids have a marginal tendency to separate spatially, at least in an inhomogeneous potential, but the interspecies healing length is long so that in the equilibrium configuration there is considerable overlap between the two species (Hall etal., 19?8a, 1998b). On the other hand, the presence of a near-resonant two-photon :oupling drive effectively brings the two energy levels quite close to one another: on resonance, the corresponding dressed energy levels are separated only by the effective Rabi frequency for the two-photon drive. In this limit, one may in a certain sense think of the condensate as being described by a two-level, spinor field (Cornell et d., 1998; Matthews et al., 1999b). We got a lot of mileage out of this system and continue to explore its properties today. One of the more Rev. Mod. Phys., Vol. 74, No. 3, 2uly 2002
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dramatic experiments we did in the two-level condensate was the creation, via a sort of wave-function engineering, of a quantized vortex. In this experiment we made use of both aspects of the two-level system-the distinguishable fluids and the spinor gas. Starting with a near-spherical ball of atoms, all in the lower spin state, we applied the two-photon drive for about 100 ms. At the same time, we illuminated the atoms with an offresonant laser beam whose intensity varied both in time and in space. The laser beam was sufficiently far from resonance that by itself it did not cause the condensate to transition from state to state, but the associated ac Stark shift was large enough to affect the resonant properties of the two-photon drive. The overall scheme is described by Matthews et al. (1999a) and Williams and Holland (1999). The net effect was to leave the atoms near the center of the ball of atoms essentially unperturbed, while converting the population in an equatorial belt around the ball into the upper spin state. This conversion process also imposed a winding in the quantum phase, from 0 around to two pi, in such a way that by the time the drive was turned off, the upper-spin-state atoms were in a vortex state, with a single quantum of circulation (see Fig. 11). The central atoms were nonrotating and, like the pimento in a stuffed olive, served only to mark the location of the vortex core. The core atoms could in turn be selectively blasted away, leaving the upper-state atoms in a bare vortex configuration, whose dynamic properties were shown by postdoc Brian Anderson and grad student Paul Haljan to be essentially the same as those of the filled vortex (Anderson et aL, 2000). Coherence and condensate decay
One of our favorite BEC experiments was simply to look at how a condensate goes away (Burt et al., 1997). The attraction of this experiment is its inherent simplicity combined with the far-reaching implications of the results. Although it was well established that condensates lived for a finite period, fractions of a second to many seconds depending on conditions, no one had identified the actual process by which atoms were being lost from the condensate. To do this our co-workers Chris Myatt, Rich Ghrist, and Eric Burt simply made condensates and carefully watched the number of atoms and shape of the condensate as a function of time. From these data we determined that the loss process varied with the cube of the density, and hence must be threebody recombination. This was rather what we had expected, but it was nice to have it confirmed. In the process of this measurement we also determined the threebody rate constant, and this was more interesting. Although three-body rate constants still cannot be accurately calculated, it was predicted long ago (Kagan et al., 1985) that they should depend on the coherence properties of the wave function. In a normal thermal sample there are fluctuations and the three-body recombination predominantly takes place at high-density fluctuations. If there is higher-order coherence, however, as one has in macroscopically occupied quantum states such as a
639 888
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00
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FIG. 11. Condensate images showing the first BEC vortex and the measurement of its phase as a function of azimuthal angle: (a) the density distribution of atoms in the upper hyperfine state after atoms have been put in that state in a way that forms a vortex; (b) the same state after a pi/2 pulse has been applied that mixes upper and lower hyperfine states to give an interferogram reflecting the phase distribution of the upper state; (c) residual condensate in the lower hyperfine state from which the vortex was formed that interferes with a to give the image shown in (b); (d) a color map of the phase difference reflected in (b); (e) radial average at each angle around the ring in (d). The data are repeated after the azimuthal angle 2 v to better show the continuity around the ring. This shows that the cloud shown in (a) has the 27r phase winding expected for a quantum vortex with one unit of angular momentum. From Matthews et al., 1999a [Color].
FIG. 12. Bosenova explosion from Roberts et al. (2001). From top to bottom these images show the evolution of the cloud from 0.2 to 4.8 ms after the interaction was made negative, triggering a collapse. On the left the explosion products are visible as a blue glow expanding out of the center, leaving a small condensate remnant that is unchanged at subsequent times. On the right is the same image amplified by a factor of 3 to better show the 200 nK explosion products [Color].
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single-mode laser, or as was predicted to exist in a dilute gas BEC, there should be no such density fluctuations. On this basis it was predicted that the three-body rate constant in a Bose-Einstein condensate would be 3 factorial or 6 times lower than what it would be for the same atoms in a thermal sample. It is amusing that such a relatively mundane collision process can be used to probe the quantum correlations and coherence in this fashion. After measuring the three-body rate constant in the condensate we then repeated the measurement in a very cold but uncondensed sample. The predicted factor of 6 (actually 7.4%2.6) was observed, thereby confirming the higher-order coherence of BEC (Burt et al., 1997). Feshbach resonance physics
In 1992 Eric Cornell and Chris Monroe realized that dipole collisions at ultralow temperatures might have interesting dependencies on magnetic field, as discussed in the Appendix. With this in mind we approached Boudwijn Verhaar about calculating the magnetic-field dependencies of collisions between atoms in the lower F spin states. When he did this calculation he discovered (Tiesinga et al., 1993) that there were dramatic resonances in all the cross sections as a function of magnetic field that are now known as Feshbach resonances because of their similarity to scattering resonances described by Herman Feshbach in nuclear collisions. From the beginning Verhaar appreciated that these resonances would allow one to tune the s-wave scattering length of the atoms and thereby change both the elastic collision cross sections and the self-interaction in a condensate, although this was several years before condensates had been created. In 1992 we hoped that these Feshbach resonances would give us a way to create enormous elastic collision cross sections that would facilitate evaporative cooling. With this in mind we attempted to find Feshbach resonances in the elastic scattering of first cesium and then, with postdoc Nate Newbury, rubidium. These experiments did provide us with elastic scattering cross sections (Monroe et al., 1993; Newbury et al., 1995), but were unable to locate the few-gauss-wide Feshbach resonances in the thousand-gauss range spanned by then theoretical uncertainty. By 1997 the situation had dramatically changed, however. A large amount of work on cold collisions, BEC properties, and theoretical advances provided accurate values for the interaction potentials, and so we were fairly confident that there was likely to be a reasonably wide Feshbach resonance in rubidium 85 that was within 20 or 30 gauss of 150 G. This was a quite convenient bias field at which to operate our baseball magnetic trap, so we returned to the Feshbach resonance in the hope that we could now use it to make a Bose-Einstein condensate with adjustable interactions. The time was clearly ripe for Feshbach resonance physics. Within a year Ketterle (Inouye et af., 1998) saw a resonance in sodium through enhanced loss of BEC, Dan Heinzen (Courteille et al., 1998) detected a Feshbach resonance in photoassociation in "Rb, we (RobRev. Mod. Phys., Vol. 74, No, 3, July 2002
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erts et af., 1998; notably students Jake Roberts and Neil Claussen) detected the same resonance in the elastic scattering cross section, and Chu (Vuletic etal., 1999) detected Feshbach resonances in cesium. Our expectations that it would be as easy or easier to form BEC in 85Rb as it was in "Rb and then use this resonance to manipulate the condensate were sadly naive, however. Due to enhancement of bad collisions by the Feshbach resonance, it was far more difficult and could only be accomplished by following a complicated and precarious evaporation path. However, by finding the correct ath and cooling to 3 nK we were able to obtain pure 'Rb condensates of 16000 atoms (Roberts et al., 2001). The scattering length of these condensates could then be readily adjusted by varying the magnetic field over a few gauss in the vicinity of the Feshbach resonance (Cornish et al., 2000). This has opened up a wide range of possible experiments, from studying the instability of condensates when the self-interaction is sufficiently attractive (negative a ) to exploring the development of correlations in the wave function as the interactions are made large and repulsive. This regime provides one with a new way to probe such disparate subjects as molecular Bose-Einstein condensates and the quantum behavior of liquids, where there is a high degree of correlation. This work represents some of the most recent BEC experiments, but almost everything we have explored with this system has shown dramatic and unexpected results. Thus it i s clear that we are far from exhausting the full range of interesting experiments that are yet to be carried out with BEC. In the first of these Feshbach resonance experiments our students Jake Roberts, Neil Claussen, and postdoc Simon Cornish suddenly changed the magnetic field to make a negative. We observed that, as expected, the condensate became unstable and collapsed, losing a large number of atoms (Roberts etal., 2001). The dynamics of the collapse process were quite remarkable. The condensate was observed to shrink slightly and then undergo an explosion in which a substantial fraction of the atoms were blown off (Donley, 2001). A large fraction of the atoms also simply vanished, presumably turning into undetectable molecules or very energetic atoms, and finally a small cold stable remnant was left behind after the completion of the collapse. This process is illustrated in Fig. 12. Because of its resemblance (on a vastly lower energy scale) to a core collapse supernova, we have named this the Bosenova. There is now considerable theoretical effort to model this process and progress is being made. However, as yet there is no clear explanation of the energy and anisotropy of the atoms in the exFlosion, the fraction of vanished atoms, and the size of the cold remnant. One of the more puzzling aspects is that the cold remnant can be far larger than the condensate stability condition that determines the collapse point would seem to allow (Donley, 2001). Another very intriguing result of Feshbach resonance studies in 85Rb was observed when our students Neil Claussen and Sarah Thompson and postdoc Elizabeth Donley quickly jumped the magnetic field close to the
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resonance while keeping the scattering length positive. They found that they could observe the sample oscillate back and forth between being an atomic and a molecular condensate as a function of time after the sudden perturbation (Donley et al., 2002). This curious system of a quantum superposition of two chemically distinct species will no doubt be a subject of considerable future study. An optimistic appendix
Until a new technology comes along to replace evaporative cooling, the crucial issue in creating BEC with a new atom is collisions. In practice, this means that planning a BEC experiment with a new atom requires learning to cope with ignorance. It is easy to forget that essentially nothing is known about the ultralowtemperature collisional properties of any atomic or molecular species that is not an atom in the first row of the Periodic Table. One cannot expect theorists to relieve one's ignorance. Interatomic potentials derived from room-temperature spectroscopy are generally not adequate to allow theoretical calculations of cold elastic and inelastic collision rates, even at the order-ofmagnitude level. Although the cold collisional properties of a new atom can be determined, this is a major endeavor, and in most cases it is easier to discover whether evaporation will work by simply trying it. Launching into such a major new project without any assurances of success is a daunting prospect, but we believe that, if one works hard enough, the probability that any given species can be evaporatively cooled to the point of BEC is actually quite high. The scaling arguments presented below in support of this assertion are largely the same as those that originally encouraged us to pursue BEC in alkalis, although with a bit more refinement provided by age and experience. Although there is an extensive literature now on evaporative cooling, the basic requirement is simply that there be on the order of 100 elastic collisions per atom per lifetime of the atoms in the trap. Since the lifetime of the atoms in the trap is usually limited by collisions, the requirement can be restated: the rate of elastic collisions must be about two orders of magnitude higher than the rate of bad collisions. As mentioned above, there are three bad collisional processes, and these each have different dependencies on atomic density in the trap, n: background collisions (independent of n ) , twobody dipolar relaxation ( a n ) , and three-body recombination ( a n 2 ) . The rate for elastic collisions is nuu, where n is the mean density, u is the zero-energy s-wave cross section, and u is the mean relative velocity. The requirement of 100 elastic-to-inelastic collisions must not only be satisfied immediately after the atoms are loaded into the trap, but also as evaporation proceeds toward larger n and smaller u . With respect to evaporating rubidium 87 or the lower hyperfine level of sodium 23, Nature has been kind. One need only arrange for the initial trapped cloud to have sufficiently large n , and design a sufficiently low-pressure vacuum chamber, Rev. Mod. Phys., Vol. 74, NO. 3, July 2002
and evaporation works. The main point of this section, however, is that evaporation is likely to be possible even with less favorable collision properties. Considering the trap loss processes in order, first examine background loss. Trap lifetimes well in excess of what are needed for "Rb and Na have been achieved with standard vacuum technology. For example, we now have magnetic trap lifetimes of nearly 1000 s. (This was a requirement to achieve BEC in 85Rbwith its less favorable collisions.) If one is willing to accept the added complications of a cryogenic vacuum system, essentially infinite lifetimes are possible. If the background trap loss is low enough to allow evaporative cooling to begin, it will never be a problem at later stages of evaporation because nu increases. If dipolar relaxation is to be a problem, it will likely be late in the evaporative process when the density is high and velocity low. There is no easy solution to a large dipolar relaxation rate in terms of changing the spring constant of the trap or the pressure of the vacuum chamber. Fortunately, one is not required to accept the value of dipolar collisions that Nature provides. In fact, all one really has to do is operate the trap with a very low magnetic bias field in a magnetic trap, or if one uses an optical trap very far off-resonance (such as C02 laser), trap the atoms in the lowest spin state, for which there are no dipole collisions. The bias field dependence comes about because below a field of roughly 5 G, the dipolar rate in the lower hyperfine level drops rapidly to zero. This behavior is simple to understand. At low temperature, the incoming collisional channel must be purely s wave. Dipolar relaxation changes the projection of spin angular momentum, so to conserve angular momentum the outgoing collisional channel must be d wave or higher. The nonzero outgoing angular momentum means that there is an angular momentum barrier in the effective molecular potential, a barrier of a few hundred microkelvin. If the atoms are trapped in the lower hyperfine state ( F = l,rnF= - 1, in rubidium 87) the outgoing energy from a dipolar collision is only the Zeeman energy in the trapping fields, and for B less than about 5 G this energy is insufficient to get the atoms back out over the angular momentum barrier. If relaxation is to occur, it can happen only at interatomic radii larger than the outer turning point of the angular momentum barrier. For smaller and smaller fields, the barrier gets pushed further out, with correspondingly lower transition rates. It is unlikely that the three-body recombination rate constant could ever be so large that three-body recombination would be a problem when the atoms are first loaded from a MOT into the evaporation trap. As evaporation proceeds, however, just as for the dipolar collisions, it becomes an increasingly serious concern. Because of its density dependence, however, it can always be avoided by manipulating the trapping potential. Adiabatically reducing the trap confinement has no effect on the phase-space density but it reduces both the density and the atom velocity. The ratio of three-body to elastic collisions scales as llnv. Therefore, as long as
642 E. A. Cornell and C. E. Wiernan: BEC in a dilute gas
one can continue to turn down the confining strength of one s trap, one can ensure that three-body recombination will not prevent evaporative cooling all the way down to the BEC transition. To summarize, given (i) a modestly flexible magnetic trap, (ii) an arbitrarily good vacuum, (iii) a true ground state with F # 0, and (iv) nonpathological collisional propeities, almost any magnetically trappable species can be successfully evaporated to BEC. If one is using a very far off-resonance optical trap (such as a C 0 2 dipole trap) one can extend these arguments to atoms that cannot be magnetically trapped. In that case, however, current technology makes it more difficult to optimize the evaporation conditions than in magnetic traps, and the requirement to turn the trap down sufficiently to avoid a large three-body recombination rate can be more difficult. Nevertheless, one can plausibly look forward to BEC in a wide variety of atoms and molecules in the future. ACKNOWLEDGMENTS
We acknowledge support from the National Science Foundation, the Office of Naval Research, and the National Institute of Standards and Technology. We have benefited enormously from the hard work and intellectual stimulation of our many students and postdocs. They include Brian Anderson, Mike Anderson, Steve Bennett, Eric Burt, Neil Claussen, Ian Coddington, Kristan Convin, Liz Donley, Peter Engels, Jason Ensher, David Hall, Debbie Jin, Tetsuo Kishimoto, Heather Lewandowski, Mike Matthews, Jeff McGuirk, Chris Monroe, Chris Myatt, Nate Newbury, Scott Papp, Cindy Regal, Mike Renn, Jake Roberts, Peter Schwindt, David Sesko, Michelle Stephens, William Swann. Sarah Thompson, Thad Walker, Yingju Wang, Richard Watts, Chris Wood, and Josh Zirbel. We also have had help from many other JILA faculty, including John Bohn, Chris Greene, and Murray Holland. REFERENCES
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644 E. A. Cornell and C. E. Wieman: BEC in a dilute gas
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PHYSICAL REVIEW A 67, O f i i n O i i K ) (2003)
Very-high-precisiou bound-state spectroscopy near a
S5
R.b Feshbaeh resonance
R, Gaussen, S. J. J. M. F. Kokkclmans,* S. T. Thompson, E. A, Donley, F. llodby, and C. LL Wicman JILA, Narimiui Ins'in'h <>; ' Stamiarflu and Techmi/ngv. Univa'sisv of Cf.-loiado. Colorado 8<)M)9-fi44«, USA and Dfjiunnirni ti!'rhvi-if.s, University of CoI/iraJn, Hnuliisi: Colorado f
The phenomenon of a Feshbach resonance in uhracold collisions of alkali-metal atoms lias received much theoretical and experimental interest in recent years, sparking interest in the subjects of resonant Bose-Einstein condensatcs (Bt'Csj and superfluidity in degenerate Fermi gases. Here we focus, on ultracold bosonic gases, in which magnetic-field Feshbach resonances [ l , 2 j have been used both to control elastic and inelastic collisions [3 6] and to tune the selfinteraction in BEC [7-1!]. In a BEC, the magnetic field controls the self-interaction of the uoridonsate by affecting the .v-wave scattering length a. Close to resonance, the scattering length varies with £ field according io
A (1)
where Spe,,k is the resonance position and is defined to be the magnetic field where :.he magnitude of a becomes infinite, aby is the background scattering length, A = S zcm —B pei ,i, is the resonance width, attd 5zero is the field where the scattering length crosses zero. The resonance is due a (quasi)-bound state that can be tuned in close proximity to the zero-energy threshold via a magnetic field. Measurements of Feshbach resonance positions and widths have been used in a variety of alkali-metal atoms to improve the determination of the interatomic potentials, which in turn can be used for calculattne interaction properties relevant for cold atomic gases [12-15], Recently, we applied rapid magnetic-field variations near a Feshbach resonance to create an atom-molecule superposition state in a esRb BEC [16], which has allowed us to precisely determine the Feshbach resonance position and width. Our technique for studying the Feshbach resonance relies on the presence of atom-molecule coherence [17-19]. By inducing periodic oscillations in the number of condensate atoms, we obtain
*Present address: Laboratoire KststltT Brossel. ENS. 24 rue Lhomond, 75005 Paris, France 1050-294 7/2003/67C6J/060701 (4)/S20.00
nance, we t u n e '.lie molecular state very close to threshold ( < ? h | . . , t — ' 0 " ' e m " " ' ) lo our knowledge, this is the most weakly bound diatomic state ever observed in a spcctroseopy experiment. The present method for studying the Feshbach resonance through atom-molecule oscillations offers all of the many inherent advantages of a frequency measurement, including the possibility of high measurement precision, a lack of sensitivity lo errors in the absolute atom number calibration, and a s i m p l e interpretation of the oscillation frequency in terms of the relative energy difference between the atomic and molecular states. When these advantages are combined with an improved method for magnetic-field calibration [20], the present technique for probing the Feshbach resonance is much more precise than previous experiments that examined such Feshbaeh resonance observables as variable rethermalization rates in a trapped eloisd of atoms [4], enhancements of photoassociation rates [3] and inelastic loss rates [6] near the resonance, and variations of the mean-field expansion energy of a BEC [7], To complete our precise characterization of the Feshbaeh resonance, we also made an improved measurement of BK!0> the magnetic field where the scattering length vanishes. This experiment is very similar to our previous work [14.21], where we determined the a = 0 field by measuring the critical number (;Vcrii) for collapse of a BEC, and then wre extrapolated to the magnetic field where A?crjt would be infinite. We have improved the measurement precision by about a factor of 4 by improving our magnetic-field calibration and using a larger number of coadensate atoms to measure the collapse. We find Efm= 165.750(13) G. The procedure used to generate atom-molecule oscillations in 1 are variable quantities. The double pulse sequence is followed by a slow change in the B field to expand the BEC [9], then
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156 157 158 159 160 161 magnetic field (G)
is
'evolve (l- )
162
13000 FIG. 2. Molecular binding energy versus magnetic-field 5 evolve . The points are measured values of the atom-molecule oscillation frequency vu, while the solid line fit is the molecular binding energy from a coupled-channel scattering theory. Only black points were included in the fie; white points were excluded because they experienced a statistically significant mean-field shift. To improve visibility, the points are larger than the error bars. The inset shows the deviation of the lowest frequency data from the fit to the rest of the data.
5 11000 .a O LU CO
9000 7000
10
15
20
25
30
'evolve (MS)
FIG. 1. BEC number versus pulse spacing, / ev0 | ve . (a) BEV,,IV£ = 156.840(25) G. The oscillation frequency is very low [v0 = 9.77(12) kllzj so that the damping and atom loss are significant -here /3=2wX0.58(12) kHz and a = 7.9(4)atoms//iS. Nevertheless, the oscillations remain underdamped, with an effective frequency shift due to damping [see Eq. (2)] of only 0.2%, which is far smaller than the 1 % statistical frequency error from the fit. (b) B evo i ve = 159.527(19) G, Farther from resonance, the damping of the oscillations and atom loss are negligible in the relatively short time window used to determine v0= 157.8(17) kHz.
the trap is switched off (jS—>0) and destructive absorption imaging is used to count the number of atoms remaining in (he condensate. As in Ref. [16], periodic oscillations in the BEC number were observed as a function of/ e v i : l v u , (see Fig. 1). We fit: the BEC number oscillation to a damped harmonic oscillator function with an additional linear loss term:
N( ! } ~ A
r
- al
exp( - 0t) sin{
(2)
where A'.lvg is the average number, A is the oscillation amplitude, a and j3 are the number loss and damping rates, respectively, and ci>i,— 2TT^Vft — [ f i i ( 2 I T ) ] ~ , The quantity of interest here is f c ,, the natural oscillator frequency corresponding to the molecular binding energy, v0~-ebjni/h. We measured the oscillation frequency for values of Scvolvt. from 156.1 G to 161.8 G. Over this range, the frequency varies by over two orders of magnitude (10 1000 kHz), but the linear loss rate of roughly — 5 atoms/ f t s changes very little [20]. The damping rate shows a significant 5-field dependence, increasing from /3=2irX0.8 kHz near 156 G to /3^2w X22 kHz near 162 G, This dependence is most likely related to dephasing of the atom-molecule oscillations due to a finite S-field gradient. To characterize the Feshbach resonance, it is necessary to know both the oscillation frequency and -5 eYO ive- We prc-
cisely measured B::vah., by transferring atoms to an untrappcd spin state by driving A;«= H 1 spin-flip transitions with an applied pulse of rf radiation (pulse length=5 25 ytts). After measuring the rf transition frequency, we inverted the BreitRabi equation to obtain the corresponding 6 field. To ensure that the magnetic field was sufficiently constant during 'evniv-' we mapped out B(t) using rf pulses with lengths short compared to few\vi. Due to interference of the rf radiation with the magnetic-field control circuitry, there was a small systematic shift of the field as a function of the rf power used. The total uncertainty for each magnetic-field determination was —25 mG due to the quadrature sum of the uncertainty from the line shape measurements (~15mG) and the uncertainty in the extrapolation to zero rf power (-20 mG). The measured oscillation frequencies versus magnetic field are plotted in Fig. 2, We use these data and the zerocrossing field S7f.m to completely characterize the scattering length and binding energy as a function of magnetic field near the Feshbach resonance. As a starting point, we use the coupled-channel analysis of van Kempen et al. [15], where several high-precision data for 85Rb and 87Rb were combined to perform an interisotope determination of the rubidium interactions with unprecedented accuracy. The predictive power of this analysis can be seen from Ref. [16], where the initial data on the atom-molecule coherence were already in good agreement with the predicted binding energy of the underlying Feshbach state. Another example of the accuracy of the analysis in Ref. [15] is its agreement with more than 40 Feshbach resonances recently discovered in 87 Rb [22]. Van Kempen et al. used the best known values [14] for the resonant magnetic field B^ and zero crossing Szero, In this work, we ignore the relatively imprecise value of Bp^ from Ref. [ 14], and instead use the measured dependence of binding energy on magnetic field along with the new Bzsfo measurement given above to determine the interaction parameters. We observe that the fitting procedure is mainly
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VERY-HIGH-PRECISTON BOUND-STATE . TABLE I, Sensitivities of the determined interaction parameters vng, !.'/>•/• and (-'•-, 5" fractional uncertainties in C\, (',(,- A1,' , and J. Far instance, the systematic eiTor in C,, due to a 10% uncertainty 'in C\ is 123x0.10- 12.3 a.u. A,//../ 1
ii'ixj - - i . 5 3 * 1 0 ' * Au.::.,. .-4.14X10'" 4 ACV 123
6.HOX 10~
sensitive 10 o n l y three parameters: the van tier Waals dispersion coefficient C,, and the nonintcgral vibrational quantum numbers at dissociation, Cn\' and u/-,-/•, which determine the position of the last bound state in the singlet and trip Lei posentials. respectively. Varying the additional parameters f.\:, C;,~, (h'r (the first-order energy dependence of the phase of the oscillating triple! radial wave function), and ,./. the strength of the exchange interaction, does not improve iho fitting because these changes can be absorbed in small shifts of vDs, VD-,~, and Cf,. Therefore, we take the mean values for these four parameters [23] from the most recent determination in Ref. [221. The best fit to B7aa and the seven highest frequency data points yields a reduced ;^ = 0.3() for 5 degrees of freedom. This value of x1 is improbably low due to the fact that the uncertainty in the data is dominated by the systematic uncertainty in magnetic field related to the magnitude of the if power shift. Figure 2 shows the theoretical f i t to the bindingenergy data as a function of magnetic field. From (he fit, we find substantially improved values for the Feshbach resonance position g peak = 155.04](18) G, width A = 1 0 . 7 I ( 2 ) G . and background scattering length a bli = 443(3ja 0 . These results may be compared to previously obtained results 5pc,)k= 154.9(4) G and A = 11.0(4) G [14], and obp= -450(3)^0 [24], Our best interaction parameter values are C 6 =4707(2) a.u., vns= 0.009 16( 171, and VDT -0.94660(29). Here the error bars do not include systematic errors due to the uncertainties in other interaction parameters that are not constrained by our data. To compare our values with those of Ref. [15], we determined the sensitivity of our three interaction parameters to systematic shifts in the other parameters, as shown in Table I. Using the fractional uncertainties in C 8 , C, 0 , <•/>£, and J from Ref. [15], we find C6 = 4707(13) a.u. ; vos= 0.0092(4). and tJ D7 =0.9466(5). All of these values agree with those given in Ref. [15]: C6 = 4703(9) a.u., t,',JiV=0.009< 1). and u /rr =0.9471{2). Our value for u^s is more precise than that of Ref. [15], while VDT and C6 are slightly less precise. If future experiments allow improvements to the other interaction parameters, then our results will also become more precise since the systematic errors are comparable to or larger than our statistical errors from the fit. To understand the strong parameter constraints that we obtain with our bound state spectroscopy. one must: consider the nonlinear dependence of the binding energy on magnetic field. The magnetic-field dependence of e bjlll j as it approaches the collision threshold depends sensitively on the exact shape of the long-range interatomic potentials, which are mainly
characterized by she van der Waals coefficient C,,, At magnetic fields far from resonance, the bound-state wave function is confined to a short internucleai distance and eb,,.d varies linearly with magnetic field. The linear dependence on E field gives relatively little information about C 6 . As the B tick! approaches resonance, the detuning decreases until the bound-state lies just below threshold. Now the bound-state wave function penetrates much deeper into the classically forbidden region, which causes e h l n d to curve toward threshold as a function of magnetic field. Because the energetically forbidden region stretches oul as C 6 / r h , the observed curvature depends seriesively on (he C,, coefficient. One can show [25] that an analy'.ica! Feshbacb model that includes the correct potential range and background scauering processes [26] can reproduce the binding energy curve over the full range of magnetic field. The coupled-channel theory used in this work applies to two-body scattering; therefore, this theory cannot account for many-body effects in the atom-molecule BEC system, such as a mean-field shift to the observed oscillation frequency [17,27], Any such mean-field shift must be fractionally largest near the Feshbach resonance, where the binding energy approaches zero while the atom-atom scattering length increases to infinily. We searched for a mean-field shitl 10 the oscillation frequency when 5ev.)iv(, was decreased to —• 156 (i. As shown in Fig. 2, the lowest magnetic-field data display a clear frequency shift with respect to the coupledchannel theory prediction. As Rtw]ve approaches resonance, the observed shift increases to 1.7 kHz. which significantly exceeds a simple estimate for the average atom-atom meanfield shift in the BEC: 4-!rh2(n')a/m^0.5 kHz at Bevolvc -•-156,1 G. We are presently investigating new experimental techniques to further study the frequency shifts, including their density dependence. We use a statistical test to exclude the low-frequency data from the (two-body) theory fit. We first fit the dataset including all frequency measurements with u(,^9 kHz, Eliminating the lowest frequency point from this set causes the reduced \2 to decrease from 0.3 to 0.2, and there is no significant change in parameter values. In contrast, adding the next lower frequency point increases the reduced x~ to 1.9, causing a large systematic shift in the parameter values. TSie observed behavior seems sensible since we expect mean-field shifts to increase rapidly as one moves toward resonance. The fact that the two-body theory fits the oscillation data over a frequency range spanning two orders of magnitude, but fails when the B field approaches B^, strongly suggests the influence of many-body physics. We note that a recent theoretical treatment [27] of the atommolecule oscillations that includes mean-field effects shows excellent agreement with our data over the entire range of magnetic field shown in Fig. 2. As a result of our improved determination of the 85 Rb Feshbach resonance parameters, we find that our new value for the off-resonant or background scattering length a bg = 443(3)00. is inconsistent with the value given in Ref. [14], where tfbg= - 380(21)a 0 . The most plausible explanation we can find for disagreement is that the theoretical expression used to relate measured rethermalization rate to
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cross section is i n s u f f i c i e n t for the: requisite level of accuracy. However, the new value for (.'(li, allows us to revise our previous estimate for the stability condition of a DEC w i t h negative scauerma length 121 j. The s t a b i l i t y condition rn;>y he expressed in terms of a stability uoettic'cnt Ay,.nap-.,— N,.:,,\a\iai,:,. Here k.^^.,rv. depends on the critical number of aioiiis, to obtain She linear slope oi scattering length versus /? lield near B=B.,.,, i: . We then combine the value of iWAS = 39.^7{22)t/ ! i /(.T ftiih our previously measured slope
''roiinp-ce determination agrees witli the most recent theoretical value of 0.55 [28]. In conclusion, we present a unique method for exploring a " Rb Feshbach resonance. Ihe observed atom-molecule coherence allows us to study the highly nonlinear dependence of (be molecular binding energy on magnetic field. We find gr,od agreement with an analysis of van ICempen ei al, [15] and \ve also substantially improve the precision of the 8 > Rb Feshbaeh resonance parameters, In addition, we observe, mean-field shifts to the mokvular binding energy, offering the possibility for future studies of many-body effects In this exciting system. Tins research was supported by the Office of Naval Research, the National Science Fotimifction. and the U.S. Army Research Office ( A R C ) Grant No. DAAU19-00-1-01632X
[I] H. Feshbaeh. A n n . Plrys. f N . Y . ) 5, ?57 (19581; ib.'J. 19. 287 U962): Theoretical Nuclffiii- Pky.iif:<, [Wiley, New York. 1992;. [2] V'. Tiesingii el a!., Phys. Re;.'. A 46, R I 1 6 7 (1992). [>] R Courleiiie a a!., Phys. Rev. Lett. 81, 69 (1998). [4] j.L Roberts at al... Phys. Rev. Lett. 81, 5109 (1998). [5] V. Vuletic cru!.. Phys. Rev. Lett. 82. 1406 (19991. [6] C. Chin etui, Phys Rev. Lett. 85. 27!7 i2000)_ [7] S. Inouye i:t ai, Ntdure (London) 392, 151 11948). [ K j S.L. Cornish ,'/ a!., Phys. Rev. Lett, 85, 1745 (2000). [9] B.A, Donley et al. Nature (London) 412, 295 (2001). [10] L. Khaykovich el c:i.. Science 296. 1290 (2002). [il] K.E. Strecker el al., Natiire (London) 417. 150 (2002) [12JF.A. van Abeeleii and Q.I Vcrhaar, Phys. Rev, A 59, 578 (1999). [13] P.J. Leo, C.J. Williams, and P.S. Julienne. Phys. Rev. Lett, 85. 2721 (2000), f!4] J.L. Robert's el
1S040! (2002), 8" M. Miu'lfie, K..-A. Suonihien. and J, Javamiinsn, Phys. Rev. Leu. 89. 180403 (2002), M, Mackic er al., (.--print pljyiics/0210131. [19] T. K.ohler, T. Gasen/er. ntu:l K. Burnett. Phys. Rev. A 67, 013601 (2003 J. N.R. Clausen ,-t ,>i., Phys, Rev, Lett. 89. 010401 (2002). J.L. Roberts- ct ai, Phys. Rev. Lett. 86, 4211 (200Ll. A. Mane el a!., Phys. Rev, Lett. 89. 283202 (2002). For the $£ value, which is not listed in Ref. [22] we used ^f = 0.160(18) K" 1 [E.G.M. van Kempen and B.J. Verhaar (pri vate communication) ], , 2 4 ! The value for a^,,— —450(3) a a was extracted from the Ref. [15] analysis. We refer to this value instead of our earlier experimentally determined value of 0), ? ~~380(2l)« u [14], which is less precise and less accurate. [25] S.J.J.M.K. Kokkeimans et al. (unpublished). [26] SJJ.MF. Kokkeimans et at., Phys, Rev. A65, 053617 (2002). [27] R.A. Duine and II.T.C. Stoof, e-print cond-mat/0303230. |28] A. Gamrnal, T. Frederico, and L. Toinio, Phys. Rev. A 64. 055602 (2001).
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PHYSICAL
REVIEW
LETTERS
2 I JANUARY 2005
Spontaneous Dissociation of 85 R b Feshbach Molecules S. T. Thompson, E. Hodby, and C. E. Wieman JILA, Ncitiorial Iristittire of' Stnridards arid Techriology m d the Uri;vers/t~of' Colorado, arid the Departmerit of' Physics, Uriiver.sity qf' Colorciclo, Boiilrlec Colorciclo 80309-0440. USA (Received 5 August 2004; published I8 January 2005) The spontaneous dissociation of "Rb dimers in the highest lying vibrational lcvcl has been observed in the vicinity of the Feshbach resonance that was used to produce them. Thc molecular lifetime shows a strong dependence on magnetic licld. varying by 3 orders of magnitude between 155.5 G and 162.2 G. Our measurements are i n good agreement with theorctical predictions i n which molecular dissociation is driven by inelastic spin relaxation. Molecule lifetimes of tens of milliseconds can be achievcd within approximately a I G wide region directly above the Feshbach rcsmance. PACS numbers: 03.75.Nt. 34.50.-s. 36.90.+f
DOI: 10. I I03/PhysRevLett.94.020401
Magnetic Feshbach resonances were first used to dramatically alter the strength and sign of interatomic interactions in ultracold atoms 11-61, Several years ago it was predicted that they could also be used to produce molecules [7-1 I]. Today Feshbach resonances have become very useful tools for creating ultracold gases of diatomic molecules. In our initial experiments we saw molecules formed from a x5RbBose-Einstein condensate (BEC) by nonadiabatic mixing of atomic and molecular states when the magnetic field was rapidly pulsed close to the Feshbach resonance [12]. Subsequently, it has been shown that both fermionic [13-15] and bosonic 116-181 atoms can be converted into molecules by adiabatically sweeping the magnetic field through a Feshbach resonance. Molecules formed using these techniques are very weakly bound and very highly vibrationally excited and are of considerable experimental and theoretical 119-21] interest. The lifetime of these molecules has varied widely under different conditions and their decay processes have not been fully established. Several experiments have shown that such molecules can undergo rapid vibrational quenching in which they collide with atoms or other molecules and relax to lower vibrational states 122,231. For the case of molecules created from a Fermi gas, it has been observed that near resonance the molecular lifetime increases by several orders of magnitude [23]. It is speculated that collisional relaxation is greatly suppressed close to the Feshbach resonance due to the Fermi statistics of the atoms [21]. A systematic study of the lifetime of molecules composed of bosons near a Feshbach resonance has not yet been published. However, it is believed that the observed low atommolecule conversion efficiencies for bosonic atoms [ 18,221 are actually the result of very high vibrational quenching rates near the Feshbach resonance. In general, all of these experiments have started with an atom cloud with an initial peak density of no - 1013 to 1014 ~ m - This ~ . collisional quenching mechanism will become much less significant at lower densities. such as the conditions we have used in
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studying the conversion of "Rb atoms to molecules (11" 10'' cni-'1. There were several experiments that raised questions as to the possible decay of these molecules even at densities where collisional quenching was implausible. These experiments emphasized the need for a better understanding of the molecular lifetime. First, there were some unanswered questions regarding the results of our initial experiments in which we observed coherent atom-molecule oscillations [ 121. Comparisons with theoretical calculations [24,25] showed superb agreement in all respects except for the amount of atom loss that was observed. If the molecules were decaying in such a way that they were being ejected from the trap, this would resolve the disagreement. In addition, the damping of this atom-molecule coherence has yet to be explained in part because it was unknown whether molecular decay contributed to this damping, and if so, to what degree. Further questions about the molecular lifetime were raised by our subsequent unsuccessful attempts to observe the spatial separation of atomic and molecular clouds juxtaposed with successful experiments by Grimm et al. [ 161 and later Rempe et al. 1181 demonstrating just such a separation. A possible explanation for this difference was that our molecules were for some reason decaying more quickly than theirs, quicker than the time required to spatially separate them from the remaining atoms. For all of these reasons, in addition to the intrinsic interest of better understanding the properties of these remarkable Feshbach resonance molecules, there has been considerable incentive to determine the fundamental (i.e., noncollisional) decay processes of these molecules and to learn how it might depend on magnetic field, and, assuming there was an intrinsic decay mechanism, what range of lifetimes could be achieved. There is a rather practical use to a relatively short well understood and characterized molecular lifetime. This is to definitively settle the substantial disagreement [24,25] over the fraction of the sample that has been converted to molecules in our initial experiments [ 121. If the molecular lifetime is known, one can simply observe what fraction of
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the sample disappears with this characteristic time and this issue is easily resolved. All of the above issues inspired the theoretical work of Kohler et nl. discussed below and the parallel experimental program presented here. We have systematically investigated the molecular lifetime of "Rb diiners in the highest vibrational state as a function of magnetic field in a 7 G wide region directly above the Feshbach resonance. By starting with an ultracold but uncondensed gas of bosonic 85Rb atoms in a magnetic trap we have been able to study the molecular lifetime at an initial atom density which is 2 to 3 orders of magnitude smaller than in other experiments and thus distinguish collisional destruction of molecules from the intrinsic lifetime of the molecular state. Kohler et nl. have predicted that a novel decay mechanism should dominate under these conditions [26]. In short, they expect inelastic spin relaxation to lead to the spontaneous decay of these molecules. One of the atoms in the molecule experiences a spin flip that is similar to an inelastic spin relaxation collision between two atoms. This causes the molecule to dissociate and releases sufficient kinetic energy for both atoms to be lost from the trap. A high dependence of this dissociation rate on magnetic field is anticipated. Close to resonance, the size of the molecule increases and spin relaxation is suppressed. Our work directly tests this theoretical prediction and determines the range of experimentally accessible molecular lifetimes. To carry out these lifetime measurements we start with what has become a rather standard technique for molecule production, namely, ramping the magnetic field adiabatically through a Feshbach resonance [ 13-1 81. We have used the 10.7 G wide resonance at 155.0 G for this purpose and have observed a 30% atom-molecule conversion efficiency. We found the lifetime of the molecules by holding them for various lengths of time, then converting all remaining molecules back into atoms and measuring the number of atoms remaining versus the duration of the hold. We have repeated this process holding the molecules at several different magnetic fields. The apparatus used in this study has been described in detail elsewhere [2]. We first prepared an ultracold (30 nK) thermal cloud of 100000 85Rbatoms in the F = 2, mF = -2 state in a magnetic trap at a bias field of 162.2 G. The standard deviation of the atom number from shot to shot was -3%. The spatial distribution of the atoms was Gaussian with a peak density of no = 6.6 X 10" cm-3 and the trap frequencies were (17.5 X 17.2 X 6.8) Hz. We then used the trapping coils to apply a magnetic field time sequence as shown in Fig. 1 to produce molecules and subsequently measure their lifetime. Having performed evaporative cooling at 162.2 G where the scattering length is positive, we first ramped the magnetic field to 147.2 G as rapidly as experimentally convenient (an inverse ramp rate of 46 ps/G) simply to get to the correct side of the resonance to begin molecule pro-
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FIG. 1. Magnetic field ramp sequence for producing molecules and measuring their decay rate. The Feshbach resonance is indicated by the dashed line. The field is first swept as quickly as possible from the evaporation field to the opposite side of the resonance. A second ramp back across the resonance converts somc atoms to molecules. The molecules are then held at a constant field above the resonance for a variable amount of time. A third ramp across the resonance then converts any remaining molecules back into atoms and the magnetic trap is turned off.
duction[27]. A second ramp (57 p s / G ) back across the resonance then adiabatically converted 30% of the atoms into molecules. We did not observe any atom or molecule loss during this ramp; if we quickly shut the trap off after the ramp (converting any molecules back into atoms), all of the atoms in the original sample were still present. This field ramp continued to the chosen field &dI, above the resonance. The field was then held constant at & , I d for a variable amount of time thold, during which time a fraction of the molecules could decay. A third ramp across the resonance (65 ps/G) then converted any remaining molecules back into atoms. The trap was then turned off and the atom cloud was allowed to expand for 22 ms before destructive absorption imaging was used to determine the number of atoms in the cloud. By measuring the decrease in the number of atoms as a function of &,Id we were effectively measuring the decay of the molecules. This method, of course, relies on the assumption that the decaying molecules leave the magnetic trap so we do not see them in our absorption images. The observed exponential loss indicates that this must be true for at least a large fraction of them. On theoretical grounds it is likely all leave since it has been predicted that the decay energies associated with the various available decay channels are all on the order of several mK [26] and our trap depth is only -1 mK. Also, we have looked at absorption images at a large range of expansion times and have not seen any evidence for modestly energetic atoms arising from less energetic decay channels. By measuring the atom number as a function of thold and by fitting this to an exponential decay we were able to extract the molecular lifetime at Bhold.
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tional form including molecular decay and three-body loss, Data from such a measurement are shown in Fig. 2(a) for Bhold= 156.6 G. We have investigated a range of BhOld but in practice the uncertainties are less if we simply measure the three-body loss versus time in a sample of from 155.5 G to 162.2 G. The decay we observe fits very nicely to an exponential. We found that the time constant pure atoms (no molecules) at 155 G and 156 G and subtract off this loss from the raw molecular decay data before for the decay depends very strongly on field; it changes by fitting it to a single exponential. A similar technique was 3 orders of magnitude over this 7 G wide region. There is a used by Regal et al. [23] in the measurements of the lifeminor complication in the data analysis for fields less than time of 4"K molecules produced from a Fermi gas near a 156.5 G. At fields this close to the Feshbach resonance the Feshbach resonance. This three-body atomic loss is atoms leave the trap via three-body collisions at a rate that strongly field dependent [28] and was negligible for data is slower but not entirely negligible relative to the molecule loss rate, as we have previously studied [28]. This causes above 156.5 G. A summary of our molecule lifetime measurements is shown in Fig. 3. For B fields between 155.5 G the apparent sloping baseline in Fig. 2(b). In principle we could fit the loss versus time curve with the correct funcand the peak of the Feshbach resonance at 155.0 G, the quantity ngni (where n is the s-wave scattering length) is equal to or greater than one. In this regime the atomic and moleciilar states are not well defined and hence it becomes
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decay .'' We have confirmed that the molecular lifetime is independent of density which illustrates the one-body nature of the decay. For an initial atom density 12" = 6.6 X 10" cm-' we measure a lifetime of 2.1(4) ms at 159 G. Repeating this measurement with no = 2.6 X 10" cm-' yielded a lifetime of 2.5(6) ms which agrees within the error bars. The solid curve in Fig. 3 is the result of a coupled channels calculation done by Kohler et nl. in Ref. [26] in
(a)
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FIG. 2. Measurements Qf molecular lifetime. (a) Number of atoms remaining after holding the system at 156.6 G as a function of the hold time. The decay fits nicely to an exponential and from this we get a lifetime of 10.4(1.7) ms. The error represents the statistical error from the exponential fit to the data. The baseline indicates we have converted 30% of the atoms into molecules. (b) Number of atoms remaining after holding at 155.5 G as a function of the hold time. In this case atoms are being lost from the trap during the hold time due to three-body collisions and we never observe a horizontal baseline. Analyzing the data taking into account this atom loss we get a molecule lifetime of 24.7(6.4) ms.
158
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Magnetic Field (G)
FIG. 3. Molecule lifetime as a function of magnetic field. The experimental data are represented by the closed points. The two lines are the results of theoretical calculations with no free parameters by Kohler et al. (Ref. [19]) in which molecules spontaneously decay due to inelastic spin relaxation. The solid line arises from an exact coupled channels scattering calculation. The dashed line results from a simpler calculation in which the detailed nature of the interatomic potentials is ignored, resulting in an analytic solution for the molecular lifetime. The inset shows the discrepancy between experiment and theory close to the 155.04 G Feshbach resonance.
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decay of these molecules. There is good agreement between experiment and theory for fields greater than - 157 G covering a factor of 100 in lifetime. The discrepancy within - 1 G of the Feshbach resonance is most likely due to many-body effects becoming more important as n,)a3 becomes large and, as mentioned above, the loss processes cannot be simply described in terms of distinct molecular decay and three-body atomic loss. The essentially binary approach of the theoretical model is expected to fail in this regime. The dashed curve is the result a universal calculation which does not depend or1 the detailed nature of interatomic interactions, also by Kohler ct a/. in Ref. 1261. It predicts that the molecular lifetime as a function of magnetic field is given by 4-lrn'(B)/K2(B), where a(B) is the s-wave scattering length and K,(B) is loss rate constant for inelastic spin relaxation collisions. This simple formula also does a good job of predicting the molecular lifetime over the magnetic field range we have investigated and in addition provides good physical insight into the decay mechanism. It has been theoretically shown that the spatial extent of the wave functions of these Feshbach molecules is of the order of the scattering length [19]. Thus, as a(B) becomes large near resonance so does the volume containing the atom pair, and the spontaneous decay of the molecule is suppressed. As pointed out in Ref. [26], if K,(B) is known, such measurements of the molecular lifetime can be used as a direct probe of the size of the molecule. In summary, we have measured the lifetime of 85Rb dimers in the highest lying vibrational level in the vicinity of the Feshbach resonance. We have observed a very strong dependence of this lifetime on magnetic field which is in good agreement with theoretical predictions where molecules decay due to dissociation driven by inelastic spin relaxation. These results show that it is possible to create 85Rb dimers with lifetimes of tens of milliseconds. Therefore, one should be able to carry out further experiments with these molecules if operating within a few gauss of the Feshbach resonance. These results also explain the unexplained atom-molecule loss observed in our previous experiments [7,12,24,25,29] creating coherent superpositions of atomic and molecular BECs of "Rb. Because of our thorough understanding of this novel decay mechanism, we can use it as a tool for further investigation of the molecular production process which has received little experimental attention to date. For example, it is unclear what factors determine the atommolecule conversion efficiency when the magnetic field is swept across the Feshbach resonance. By making molecules under a controlled set of conditions and then subsequently ensuring that they all undergo one-body decay before turning the magnetic trap off and imaging, we can
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get an accurate measure of how many molecules were produced. This is the subject of current work. We thank P. Julienne and T. Kohler for their theoretical assistance and D. Jin and C. Regal for helpful discussions. This work has been supported by ONR and NSF. S.T. Thompson acknowledges support from ARO-MURI and E. Hodby acknowledges the support of the Lindemann Foundation.
[I J 121 [3] [4] [5J [6J [7] 181
[9] [lo] [1I]
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[I41 [I51 [I61 [17] [IS] [I91 [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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S. lnouye et a / . , Nature (London) 392, 151 (1998). S. L. Cornish er nl., Phys. Rev. Lett. 85, 1795 (2000). T. Loftus et a/.. Phys. Rev. Lett. 88, 173201 (2002). K. Dieckinann et a/., Phys. Rev. Lett. 89, 203201 (2002). K.M. O'Hai-a et ul., Science 298, 2179 (2002). T. Bourdel et d., Phys. Rev. Lett. 91, 020402 (2003). E. Tiinmermans et nl., Phys. Rev. Lett. 83, 2691 (1999). E. Timinermans, P. Tommasini, M. Hussein, and A. Kerman, Phys. Rep. 315, 199 (1999). E H . van Abeelen and B. J. Verhaar, Phys. Rev. Lett. 83, 1550 (1999). F.H. Mies, E. Tiesinga, and P. S. Julienne, Phys. Rev. A 61, 022721 (2000). V. A. Yurovsky, A. Ben-Reuven, P. S. Julienne, and C. J. Williams, Phys. Rev. A 62, 043605 (2000). E.A. Donley, N.R. Claussen, S.T. Thompson, and C.E. Wieman, Nature (London) 417, 529 (2002). C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Nature (London) 424, 47 (2003). K. E. Strecker, G. B. Partridge, and R. G. Hulet, Phys. Rev. Lett. 91, 080406 (2003). J. Cubizolles et al., Phys. Rev. Lett. 91, 240401 (2003). J. Herbig et al., Science 301, 1510 (2003). K. Xu et al., Phys. Rev. Lett. 91, 210402 (2003). S. Diirr, T. Volz, A. Marte, and G. Rempe, Phys. Rev. Lett. 92, 020406 (2004). T. Kohler, T. Gasenzer, P.S. Julienne, and K. Burnett, Phys. Rev. Lett. 91, 230401 (2003). P. Soldan e t a / . , Phys. Rev. Lett. 89, 153201 (2002). D. S. Petrov, C. Solomon, and G. V. Shlyapnikov, Phys. Rev. Lett. 93, 090404 (2004). T. Mukaiyama et al., Phys. Rev. Lett. 92, 180402 (2004). C. A. Regal, M. Greiner, and D. S Jin, Phys. Rev. Lett. 92, 083201 (2004). S. J. J.M. F. Kokkelmans and M. J. Holland, Phys. Rev. Lett. 89, 180401 (2002). T. Kohler, T. Gasenzer, and K. Bumett, Phys. Rev. A 67, 013601 (2003). T. Kohler, E. Tiesinga, and P. S. Julienne, Phys. Rev. Lett. 94, 020402 (2005). V.A. Yurovsky and A. Ben-Reuven, Phys. Rev. A 67, 04361 1 (2003). J.L. Roberts, N.R. Claussen, S.L. Cornish, and C.E. Wieman, Phys. Rev. Lett. 85, 728 (2000). K. G6ra1, T. Kohler, and K. Bumett, cond-matlO407627.
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Production Efficiency of Ultracold Feshbach Molecules in Bosonic and Fermionic Systems E. Hodby,' S. T. Thompson,' C. A. Regal,' M. Greiner,' A. C. Wilson,* D. S. Jin,' E. A. Cornell,' and C. E. Wieman' ' J I L A , National Institute of Standards and Technology and the University of' Colorrido, Bo~ilder;Colorcido 80309-0440, USA 'Department of Physics, UniverJity of Otagn, Diinedin, New Zenlnnd (Reccived 17 November 2004; published 30 March 2005) We investigate the production efficiency of ultracold molecules in bosonic X5Rband fermionic 'OK when the magnetic field is swept across a Feshbach resonance. For adiabatic swecps of the magnetic field, our novel model shows that the conversion efficiency of both species is solely determined by the phase space density of the atomic cloud, in contrast with a number of theoretical predictions. In the nonadiabatic regime our measurements of the 8sRb molecule conversion efficiency follow a Landau-Zener model. DOI: 10. I l03/PhysRevLett.94.120402
PACS numbers: 05.30.Jp. 03.75.S~.05.30.Fk. 36.YO.+f
The production of ultracold diatomic molecules is an exciting area of research, with applications ranging from the search for the permanent electric dipole moment [ 11 to providing unique experimental access to the predicted BCS-BEC (Bose-Einstein condensate) crossover physics [2]. A widely used production technique involves the association of ultracold atoms into very weakly bound (- 10 kHz binding energy) diatomic molecules by applying a time varying magnetic field in the vicinity of a Feshbach resonance. By using a slow adiabatic sweep of the field through the resonance into the region where bound molecules exist r3-51, samples of over 10' molecules at temperatures of a few tens of nK have been produced from both quantum degenerate two-component Fermi gases [681 and atomic Bose-Einstein condensates [9-111. Over the past two years, several experiments have probed the unique and exotic properties of these molecules [12-15]. The molecular production technique investigated in this Letter involves only two-body interactions, and hence we define "adiabaticity" in terms of a Landau-Zener system coupled by such interactions. Molecular production by three-body collisions is negligible in this work and is important only when higher densities and slower ramps or even static fields are used [16]. Only under such conditions (unlike those described here) can molecular formation be discussed in terms of thermodynamic equilibrium. Despite the widespread use of the two-body molecular production technique, the process itself has received little experimental attention, and a microscopic description that accurately predicts the conversion efficiency as a function of sweep rate, atom type, density, and temperature has not yet been developed. Theoretical work on bosonic systems always assumes the existence of a condensate and hence 100% molecular conversion for sufficiently slow sweeps. Meanwhile in quantum degenerate fermionic systems, conversion limits of 50% have been suggested for two-body molecular production in certain limiting cases [17j, but these theories do not cover the full range of experimentally accessible parameters [13]. In this Letter we present conversion data in the adiabatic regime from both an ultracold but noncondensed cloud of bosonic 85Rb atoms and an
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ultracold cloud of 4"K fermions, at a range of degeneracies. We show that a siizgle, very general description of the pairing process accurately predicts the relationship between conversion efficiency and phase space density in both cases. This theory demonstrates that the complete conversion of the condensate is just the limiting behavior of a bosonic thermal cloud as the phase space density increases. Outside the adiabatic regime, several predictions have been made for conversion efficiency as a function of magnetic field sweep rate [3j (and references therein) but have been tested only against experimental data over a narrow range of parameters [6,7]. In the first part of this Letter, we present a thorough investigation of the nonadiabatic regime using the 85Rb system. Our results support a general Landau-Zener- type theory for conversion as a function of magnetic sweep rate and density that is discussed in detail in [3,4]. Detailed descriptions of the 85Rb experimental apparatus and the magnetic field sweeps for producing molecules are contained in [ 18,191, respectively. We use evaporative cooling to produce thermal clouds of 50 000-130 000 atoms at temperatures of 26-94 nK. The atoms are held in a purely magnetic "baseball" trap [18], with fixed trapping frequencies of 17.6 X 17.6 X 6.8 Hz. For efficient evaporation, the bias field is held at 162 G, where the scattering length a is positive. For slow magnetic field ramps, Rb2 molecules are produced only when the field is ramped upward through the resonance, which is located at 155 G; hence the first step in molecule production is to rapidly jump the magnetic field from 162 to 147.5 G. We then sweep the field back up to 162 G at a chosen linear rate, producing molecules as we pass through the Feshbach resonance. The cloud is then held at 162 G for 10 ms, during which time all the molecules undergo spontaneous one-body dissociation and the two dissociated atoms rapidly leave the cloud with several mK of kinetic energy [19,20]. Thus the number of molecules formed is just the difference in atom number before and after the molecular conversion sweep divided by 2. Given the unusually low densities of the cloud in our experiments (-10" ~ m - ~ ) ,
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two- and three-body atomic decay and collisional molecular decay rates are negligible and do not affect our measurement of the molecular fraction [19]. In other experiments, operating at higher densities (e.g., [lo]), collisional decay cannot be ignored. First we investigated the molecular conversion efficiency as a function of the magnetic field ramp rate. A typical data set is shown in Fig. 1 (a) and is well fitted by a Landau-Zener formula for the transition probability at a two-level crossing
where N,,, is the asymptotic number of molecules created for a very slow ramp, B is the magnetic field sweep rate, and p is a fitting parameter. ,BIB is often called the Landau-Zener parameter c?,~. The dependence of BLz on the mean density of the cloud n, the magnetic field width of the resonance A , the background scattering length ubg, and B can be derived intuitively up to a constant a by considering the time taken to cross the resonance divided by the mean-field coupling
Inverse ramp rate ( p / G )
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strength, =
anAab,/B.
(2)
Equation (2) shows that for true Landau-Zener behavior, the quantity / 7 / B , / , should be constant, independent of density. (At B , / , , 6 , = ~ I and NnlO1/N,,,,, = 63%.) Similar datasets to Fig. ](a) were taken over a range of initial conditions and n/81/eis displayed versus / I in Fig. I(b). The data support a constant value for / 7 / B , / ( , and hence a Landau-Zener dependence on the sweep rate. The value for a , extracted from the data of Fig. I(b), is 4.5(4) X lo-' in's-'. In Ref. [3], the Landau-Zener behavior is rigorously derived for a zero temperature condensate. The authors predict LY to be roughly 1/8 of the value that we extract from our thermal cloud data. This difference is currently being investigated [21]. Our second experiment used both bosonic and fermionic systems and investigated molecular conversion efficiency at a sweep rate that is slow (adiabatic) with respect to B , l p [e.g., slower than 150 ps/G for the conditions of Fig. ](a)] but still fast compared to the rate of molecule production via three-body inelastic collisions. This regime [22] is important for producing molecules with a limited lifetime. The conversion efficiency for "Rb is shown in Fig. 2 as a function of the peak phase space density of the cloud. Note that for adjacent points on Fig. 2 (i.e., points with very similar phase space density), the number of atoms often varies by over a factor of 2, indicating that the conversion efficiency is related to phase space density but not to number or temperature individually. The inset plot in Fig. 2 shows that the same relationship between conversion efficiency and phase space density may be observed in ultracold fermionic 40K, as discussed later in this Letter. One can argue intuitively that there should be a close relationship between phase space density and conversion
-d 1.5
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Mean density (10"cm~') FIG. 1. (a) Molecular conversion in 85Rb as a function of the inverse magnetic field sweep rate. The initial conditions of the atomic cloud were N = 87000, T = 40.6 nK, and n = 1.3 X 10" ~ m - The ~ . solid line is a fit to Eq. (1). (b) Initial mean density of the atomic cloud ( n ) divided by 1/e sweep rate and plotted against mean density. The dashed line shows the best fit of Eq. (2) to the data. For the four low density points, the vertical error bars are dominated by the uncertainty in fitting B,,=. At higher densities, the cloud experienced significant heating during the ramps across the resonance; hence the density of the final point has significant uncertainty. This heating limited the maximum density used.
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Peak phase space density FIG. 2. Molecule conversion efficiency in 85Rbas a function of peak phase space density (circles). The solid line shows a simulation based on our conversion theory, fitted to the data with a single pairing parameter yo = 0.44(3) [Eq. (3)]. The dashed lines indicate the uncertainty in yb. Inset: molecule conversion efficiency in 40K as a function of peak phase space density (for full data range, see Fig. 3). The best-fit simulation based on our conversion theory is shown as a solid line, with the uncertainty indicated by dotted lines.
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efficiency via the adiabatic sweep process. An adiabatic process smoothly alters the wave function describing atom pairs but does not change the occupation of states in phase space. Thus to form a molecule, a pair of atoms must initially be sufficiently close in phase space (small relative momentum compared to / I divided by separation) that their combined wave function can evolve smoothly into the highest bound molecular state as the resonance is crossed. This theory predicts 100'70 conversion in a condensate because every atom occupies the s m i e state in phase space and so is able to form a molecule with any other atom, although lower values are always observed in experiments due to finite molecular lifetimes. For a noncondensed sample, we are still able to convert a substantial fraction to molecules because there are no constraints on the center of mass velocity of the atom pair; it simply converts to the velocity of the molecule. An analytical relationship between phase space density and conversion efficiency is possible only in the limit where the conversion fraction is very small. For almost any finite efficiency, one cannot ignore that the formation of molecules reduces the pairing options for those atoms that remain, leading to the saturation behavior observed in the experimental data of Fig. 2. To account for this, we must add a monogamy clause to our model; two atoms will form a molecule if they satisfy the phase space proximity condition described above and neither has already paired with another atom to form a molecule. To test this model for the pairing process we used a Monte Carlo-type simulation based on these two principles. For each run, we randomly assigned positions and momenta to typically 10000 atoms 1231 ensuring that the ensemble retained the correct phase space distribution for the specified temperature and trap frequencies. The simulation program searched for a partner for each atom in turn, removing both atoms from future searches if the pairing condition was satisfied. The simulation ended when no more molecules could be formed. Atoms were considered sufficiently close in phase space to form a molecule if they satisfied the relation
<
l ~ ~ , - e l ~ ~ ~ Yr k c l l
(3)
where 8rre, is the separation of the pair, ni is the atomic mass, 8urelis the relative velocity between the two atoms, and y is a constant to be determined by fitting the output of the simulation to experimental data. Finally, for the boson case, the simulated conversion fraction was increased by 2% to allow for two-body correlations that assist molecule formation in a cloud of identical bosons 1241. For each value of y , the simulation was run over a range of phase space densities and compared to the experimental data using a least squares routine. The best fit for the bosonic data was given by Y b = 0.44(3) and is shown as the solid curve on Fig. 2. The error on this result is dominated by a 12% systematic uncertainty in the phase space density (resulting from uncertainties in atom number and temperature). The dotted lines indicate the range of
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values for the pairing parameter y,, that result from this phase space density uncertainty. The excellent agreement between experiment and simulation over a wide range of phase space density provides strong evidence in support of our molecular-conversion model. Our model for the pairing process is based on very general arguments about adiabaticity and atom availability. With appropriate phase space distributions, it is equally applicable to moleciile formation in a fermionic system. This provides another stringent test of its validity. To study the molecule conversion efficiency in a Fermi system, we create an ultracold, two-component Fermi gas as outlined in [2S]. To achieve a wide range of phase space density, we follow cooling of the gas in an optical dipole trap with a recompression of the trap and controlled parametric heating through a modulation of the trap strength. This allows creation of Fermi gases ranging in temperature from <0.05TF to 1.3TF, where T , is the Fermi temperature. The dipole trap that confines the final Fermi gas is characterized by radial frequencies, v,., between 312 and 630 Hz and an aspect ratio of v,/v; = 70 5 20. With this ultracold Fermi gas, we create molecules as described in [6] using a broad s-wave Feshbach resonance between the I f , m f ) = 19/2, -9/2) and 19/2, -7/2) states of 40K located at B , = 202.1 ? 0.1 G [IS]. The magnetic field sweep used to create molecules starts at 202.83 G (where n < 0), ends between 20 1.10 and 201.59 G, and occurs at inverse sweep speeds ranging from 640 to 2900 ,us/G. Within this range the conversion fraction is independent of sweep rate, confirming that the conversion is adiabatic. On the time scale of these magnetic field sweeps, the loss of molecules and atoms is negligible 1261. At the end of the magnetic field sweep, we immediately turn off the dipole trap and allow the gas to ballistically expand. We then use one of two techniques to probe the molecular conversion: (i) We compare the number of atoms remaining on the a > 0 side of the resonance to the number of atoms measured at a < 0 as described in [6]. (ii) We convert the remaining atoms in the 19/2, -7/2) state to the )9/2, -5/2) state by applying a radio frequency T pulse. We then dissociate the molecules by ramping the magnetic field back to the a < 0 side of the Feshbach resonance. Finally, we image the 19/2, -7/2) and )9/2, -5/2) states separately to extract the molecule and atom numbers, respectively. Experimental data for molecule formation efficiency in the fermionic gas are shown in Fig. 3, and a limited range of the same data is shown in the inset of Fig. 2. The data were simulated with the same molecular conversion model as the bosonic data, modified to provide the correct FermiDirac phase space distributions for a 50/50 spin mixture and allow pairing only between atoms with unlike spins. The pairing parameter for the best-fit simulation is y I = 0.38(4), where the error includes a 10% systematic uncertainty in TIT,. This result is in excellent agreement with the value of Yb = 0.44(3) from the bosonic system, especially given that both results are dominated by systematic
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Note added. -It has recently been brought to our attention that a relationship between T I T , and molecular conversion efficiency was mentioned in [ 2 7 ] ,but these results are not in quantitative agreement with our experimental
Ool-
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FIG. 3. Molecule conversion cflicicncy in '"K as a function of T I T F . (Note that T I T , unic]tiely dctci-mincs the phasc space density of a fermionic cloud.) Thc initial clouds have mean densities for each spin state ranging fI.om 1 X 10" to 2 X 10" and 52(2)% of thc atoms are i n the m i = -7/2 state. The data are fitted with thc same conversion model as for "Rb. The best-fit curve has a pairing parameter of0.38(4). The dottcd lines indicate the uncertainty on this result. Note that conversion efficiencies far greater than 50% are measured.
rather than statistical uncertainties. This agreement provides clear evidence that our model and the physics underlying it accurately describes molecular conversion efficiency in both bosonic and fermionic systems. One fundamental difference between our model and many previous theories is that we allow each atom to pair with any other available atom that satisfies the phase space proximity condition. Other theories have considered only one potential partner for each atom, which has led to a ceiling for the fermionic conversion of 50% [ 171, contradicting the experimental data of Fig. 3. In summary, we have performed a detailed experimental investigation of the efficiency of molecular formation when the magnetic field is swept across a Feshbach resonance. In the nonadiabatic regime, we have shown that the conversion efficiency follows a dependence on density and sweep rate that is well approximated by a two-level Landau-Zener model. In the adiabatic regime, using an ultracold but noncondensed gas of bosonic 85Rb atoms and a quantum degenerate gas of fermionic 4"K, we find that the conversion efficiency is monotonically related to the initial phase space density of the atomic cloud. We have shown that the pairing process allows each atom to form a molecule with any other atom in the cloud provided that they are sufficiently close together in phase space. The agreement of experimental data from both bosonic and fermionic systems with a single simulation provides compelling evidence for the validity of this molecule conversion model. We thank J. Stewart for experimental assistance. This work has been supported by ONR, NSF, and NIST. S. T. T. acknowledges support from ARO-MURI and C. A. R. from the Hertz Foundation. 120402-4
J. J. Hudson et nl.. Phys. Rcv. Lett. 89, 23003 (2002). M. Randeria. in Bosc-Eiristeiri Coritlrrisnrioii (Cambridge University Press, Cambridge, U.K.. 1Y95). pp. 355-392. I<. Goral et d., J. Phys. B 37. 3457 (2004). F. H. Mies et nl., Phys. Rev. A 61, 022721 (2000). E. Timniermans et al., Phys. Rep. 315, 199 (1999); E A . van Abeclcn et d., Phys. Rev. Lett. 83, 1550 (1999); V. A. Yurovsky et n1.. Phys. Rev. A 62, 043605 (2000); V.A. Yurovsky et nl., Phys. Rev. A 67, 04361 I (2003); M. Mackie ct d., physics/0210131. C. A. Regal et al., Nature (London) 424, 47 (2003). K. E. Strecker et nl., Phys. Rev. Lett. 91, 080406 (2003). J. Cubizolles et nl., Phys. Rev. Lett. 91, 240401 (2003). J. Herbig et nl., Science 301, 1510 (2003). K. Xu et al., Phys. Rev. Lett. 91, 210402 (2003). S. Durr et al., Phys. Rev. Lett. 92, 020406 (2004). E. A. Donley et al., Nature (London) 417, 529 (2002). M. Greiner et al., Nature (London) 426, 537 (2003). S. Jochim et nl., Science 302, 2101 (2003); M.W. Zwierlein et nl., Phys. Rev. Lett. 91, 250401 (2003); T. Bourdel et nl., Phys. Rev. Lett. 93, 050401 (2004). C. A. Regal ef nl., Phys. Rev. Lett. 92, 040403 (2004). S. Kokkelmans et al., Phys. Rev. A, 69, 031602 (2004); M. W. Zwierlein et al., Phys. Rev. Lett. 92, 120403 (2004); S. Jochim et al., Phys. Rev. Lett. 91, 240402 (2003). E. Pazy et al., Phys. Rev. Lett. 93, 120409 (2004); J. Chwedenczuk et al., Phys. Rev. Lett. 93, 260403 (2004). S. L. Cornish et al., Phys. Rev. Lett., 85, 1795 (2000). S. T. Thompson et al., Phys. Rev. Lett. 94,020401 (2005). T. Koehler et al., Phys. Rev. Lett. 94, 020402 (2005). T. Koehler (private communication). J. E. Williams et al., cond-mat/0403503. The 85Rb simulations used only 10000 atoms for speed, while typical experimental points used 100000 atoms. The 40K simulations used 60 000 atoms while experimental points used between 250000 and 1.5 X lo6 atoms. By running the simulations with 100 to 350000 atoms, we found that in afinite system, the conversion efficiency has a small number dependence that was included in comparing the simulation to experimental data. The factor of 2% is an excellent approximation over the range of phase space density investigated and was determined by adding atoms to a Gaussian distribution at appropriate locations to approximate a Bose distribution with the correct two-body correlations. The conversion efficiency of the correlated and original Gaussian distributions could then be compared. It is only 2% because of the small fraction of pairs within a thermal de Broglie wavelength. C.A. Regal et al., Phys. Rev. Lett. 90, 230404 (2003). C.A. Regal et al., Phys. Rev. Lett. 92, 083201 (2004). J. Javanainen et al., Phys. Rev. Lett. 92, 200402 (2004).
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PHYSICAL REVIEW LETTERS
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Ultracold Molecule Production via a Resonant Oscillating Magnetic Field S . T. Thompson, E. Hodby, and C. E. Wieman JILA, National Institute of Standards and Technology and The University of Colorado, and the Department of Physics, University of Colorado, Bouldez Colorado 80309-0440, USA (Received 10 May 2005; published 2 November 2005)
A novel atom-molecule conversion technique has been investigated. Ultracold ssRb atoms sitting in a dc magnetic field near the 155 G Feshbach resonance are associated by applying a small sinusoidal oscillation to the magnetic field. There is resonant atom to molecule conversion when the modulation frequency closely matches the molecular binding energy. We observe that the atom to molecule conversion efficiency depends strongly on the frequency, amplitude, and duration of the applied modulation and on the phase space density of the sample. This technique offers high conversion efficiencies without the necessity of crossing or closely approaching the Feshbach resonance and allows precise spectroscopic measurements. Efficiencies of 55% have been observed for pure Bose-Einstein condensates. PACS numbers: 03.75.Nt, 34.50,-s, 36.90.+f
DOI: 10.1103/PhysRevLett.95.190404
The conversion of ultracold atoms to ultracold molecules by time varying magnetic fields in the vicinity of a Feshbach resonance is currently a topic of much experimental and theoretical interest. This particular conversion process lends itself well to the formation of molecular BECs [1-41 and atom-molecule superpositions [5]. These Feshbach molecules and their creation process are also important for understanding ultracold fermionic systems in the BCS-BEC crossover regime because they are closely related to the pairing mechanism in a fermionic superfluid that occurs near a Feshbach resonance [6-91. Finally, Feshbach molecules are interesting themselves because they are very weakly bound and very large in spatial extent-comparable to the spacing between atoms in the sample from which they were created [lo]. To date three Feshbach molecule creation techniques have been demonstrated. Atom to Feshbach molecule conversion was first directly observed by applying very rapid (tens of ps) time dependent magnetic fields to a 85Rb Bose-Einstein condensate (BEC) in a Ramsey type manner [5]. The magnetic field was pulsed very close to the Feshbach resonance which created a superposition of free atoms and Feshbach molecules. This technique was plagued by low conversion efficiencies (on the order of a few percent) that were difficult to control. It also lead to heating and loss of atoms from the atomic sample. The most popular atom-molecule conversion scheme to date involves slowly sweeping the magnetic field through a Feshbach resonance. This has been demonstrated for both fermionic [11,12] and bosonic [13-171 atoms. Although high conversion efficiencies have been observed in degenerate Fermi systems, high vibrational quenching rates near the Feshbach resonance have lead to low conversion efficiencies for BECs of bosonic atoms [15,18]. There are also problems caused by density dependent heating processes. As the resonance is crossed, we observe significant heating. "Rb BECs with peak densities of no = 1 X l O I 3 ~ r n were - ~ heated by a factor of roughly 250 (a few pK) which prevented any molecule formation in the 0031-9007/05/95( 19)/190404(4)$23.00
field sweep. This heating was much less, but still significant for thermal clouds that were 2 orders of magnitude less dense. For example, a 30 nK cloud with peak density no = 4 X 10" cm-3 was heated by a factor of 0.5. This heating is likely due to three body recombination collisions. For this reason, we were only able to observe molecule production in thermal clouds. For adiabatic magnetic field sweeps (adiabatic with respect to the two body physics governing molecule formation, not in a thermodynamical sense) we have shown that the atom to molecule conversion efficiency is solely determined by the phase space density of the atomic sample after this heating has been taken into account [17]. Therefore, by limiting the achievable phase space density, this heating is also limiting the conversion efficiency. A third atom-molecule conversion technique has been demonstrated in two experiments with fermionic atoms [19,20]. This technique utilizes the enhanced three body recombination collision rates near a Feshbach resonance to efficiently associate atoms into molecules. Molecules were formed simply by holding a degenerate Fermi cloud of atoms for several seconds on the positive scattering length side of a Feshbach resonance where a weakly bound molecular state exists. Conversion efficiencies as high as 85% have been reported [19]. This technique would not be useful in systems such as ours where the molecular lifetime is comparatively short and cannot exceed tens of ms [21]. Low bosonic atom-molecule conversion efficiencies and the heating observed when using both of these time dependent magnetic field techniques has prompted us to investigate alternative conversion methods. In this Letter we report on a novel atom-molecule conversion method in which atoms are resonantly associated to form molecules by applying a sinusoidallyoscillatingmagnetic field modulation. Note that this molecular formation process is different from typical single photon photoassociation processes where a colliding pair of ground state atoms absorbs a photon to form an excited state molecule 122-241. The photon adds energy to the system, allowing a molecule to
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form. In our case, photons from our oscillating magnetic field have the opposite effect-they cause the system to lose energy by stimulating an atom pair to emit a very low frequency photon and thereby decay to a lower energy bound molecular state. These experiments were motivated in part by work done by Regal et al. where radio frequency photons were used to dissociate molecules formed from a degenerate two component Fermi gas [ll]. This technique circumvents problems of heating and enhanced collision rates because it does not require crossing or closely approaching the Feshbach resonance. The maximum achievable conversion efficiency for a given phase space density is the same as was observed in the slow field sweep experiments [17]. However, since this new technique greatly reduces the heating the cloud experiences in the conversion process, we can achieve higher ultimate phase space densities and hence higher conversion efficiencies. It enables us to convert more than half of the atoms in a BEC to molecules. This new scheme also allows spectroscopy that is comparable or superior to the Ramsey technique discussed in Ref. [5]. A detailed description of the 85Rb experimental apparatus can be found in Ref. [25]. We use evaporative cooling in a purely magnetic trap at a bias field of 162 G to produce either degenerate or nondegenerate atomic samples. BoseEinstein condensates used for these experiments contain 5000-20 000 atoms and range from having a 50%condensate fraction to nearly a 100% condensate fraction, and the nondegenerate clouds used typically have 40 000200000 atoms at temperatures ranging from 20-80 nK (BEC transition temperature is approximately 14 nK). The majority of these experiments were done with uncondensed samples. After producing an ultracold atomic sample, we utilize the 11 G wide Feshbach resonance at 155.0 G to convert a fraction of the sample to molecules. The molecular state exists above the resonance. We first ramp the magnetic field from 162 G to a selected value between 156 and 157 G in 5 ms. We then use the trapping coils to apply a sinusoidal modulation to the magnetic field for 0-50 ms whose peak-to-peak amplitude ranges from 130-280 mG and whose frequency is close to the molecular binding energy as measured in Ref. [26]. We then slowly ramp the field (in 5 ms) back to 162 G where the molecular lifetime is 700 ~s [21]. The field is held here for about 10 ms to ensure that any molecules we have made decay, ejecting their constituent atoms from the trap and our field of view in the process [21]. The trap is then rapidly turned off and we measure the number of atoms remaining using absorption imaging. The loss of free atoms is negligibly small, so any loss of atoms in this process must be due to molecule production. We observe significant molecular formation that is very dependent on modulation frequency, time, and amplitude and on the initial phase space density of the sample. As further confirmation that the observed loss is due to molecular formation and decay, we have done a slightly modified version of this experiment. After applying the
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sinusoidal modulation to create molecules, the trap is immediately turned off instead of ramping the magnetic field back to 162 G where all of the molecules would quickly dissociate. In turning the trap off, we are sweeping the magnetic field through the Feshbach resonance, converting any remaining molecules back into atoms. In this case, most of the original atoms are still present in the absorption image and the slight loss we observe is consistent with the molecular lifetime at the field at which the molecules were created. Figure 1 shows the observed atom loss as a function of modulation frequency for three different modulation durations (coupling times). There is a clear resonant frequency. As the coupling time increases, more atoms are converted into molecules. Over the range of modulation amplitudes and coupling times we have investigated, the width of the loss feature increases linearly with peak loss to within our measurement uncertainty. For 37(2)% conversion at a field of 156.45 G the width is 1.88(16) kHz while for 17(2)% conversion it is 0.82(14) kHz. By fitting the linewidth versus conversion data to a straight line we find the width in the zero percent conversion limit to be 0.2(2) kHz. We would expect the width due to the spontaneous lifetime of the molecules to be 1 / 2 m which, with T = 11 ms [21,27], is 0.01 kHz. We have investigated the dependence of the resonant frequency on the value of the dc magnetic field. One would expect this to provide information about the field dependent molecular binding energy. The results of all of our measurements are shown in Fig. 2. The solid line is the molecular binding energy that resulted from fitting our Ramsey measurements described in Ref. [26] with an exact coupled-channel scattering calculation and the dashed 40
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FIG. 1. Percentage of atoms converted to molecules at 156.45 G as a function of modulation frequency for three different coupling times. The modulation amplitude was 130 mG. The triangles are the data and the solid line a Lorentzian fit for a coupling time of 6 ms. The circles and short dashed line are for a 25 ms coupling time and the squares and long dashed line are for a 38 ms coupling time. The widths are 0.82(14), 1.44(18), and 1.88(16) kHz, respectively. The peak positions for all three curves agree within the 0.04 lcHz fitting uncertainty.
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Magnetic Field (G) FIG. 2. Resonant frequency as a function of magnetic field. The solid line is from Ref. [26] and is the result of fitting our Ramsey measurements with a coupled-channels scattering calculation. The dashed line indicates the uncertainty in that fit. Measurements were made with both condensed and uncondensed (T = 20-80 nK) 85Rbclouds. The open squares indicate BEC measurements and the closed circles represent thermal cloud measurements. The data has been corrected for the temperature dependent shifts discussed in the text. The uncertainty in the measured frequency is represented by the width of the points.
lines represent the uncertainty in that fit. The solid circles are measurements made with uncondensed thermal clouds that have been corrected for the temperature shift discussed below. The open squares are measurements made with much denser, partially condensed clouds. One would expect that the exact transition frequency would depend on the relative energy of the free atoms, and we see this as a temperature dependent shift in the measured resonant frequency as shown in Fig. 3. In order to form molecules, hotter, more energetic atoms require higher frequency photons to carry away their excess energy. We have measured the resonant frequency at 155.44 G for clouds with temperatures of 20, 50, and 80 nK. Over this range the shift in the resonant frequency is linear with a slope of 0.0126(4) kHz/nK which corresponds to 0.60(2) kBT where kB is Boltzmann’s constant and T is the temperature of the sample. From the results of Ref. [17] we expect this to be lower than the 1.5kBT relative energy, but the precise factor requires a theory of the formation process which does not yet exist. In Ref. [ 171 we showed that the pairing is selective and favors atom pairs with a lower relative energy than the sample average. The thermal cloud data points shown in Fig. 2 have been corrected to show the zero temperature resonant frequency. Most of the error bars for these measurements overlap with the error bars on the binding energy curve but it is interesting that they are all consistently shifted to slightly higher frequency. This is consistent with the peak of the Feshbach resonance being about 40 mG lower than the value found in Ref. [26]. The two BEC measurements were carried out with clouds which had 50% condensate fractions. We found
FIG. 3. Measured resonant frequency at 156.44 G as a function of the temperature of the atomic sample. Fitting the data to a straight line gives a frequency shift equal to 0.60(2) kBT where kn is Boltzmann’s constant and T is the temperature.
the results of BEC measurements to show larger statistical fluctuations which may be due to shot-to-shot fluctuations in the phase space density of the partially condensed samples. The sample was not significantly heated during the molecule production process and the condensate fraction remained the same to within our level of uncertainty (approximately 15%). Small corrections (0.03-0.06 kHz) due to the 5-10 nK thermal cloud fractions have been made to the measurements shown in Fig. 2. We assumed that the thermal fraction and condensate fraction have slightly different resonant frequencies so that the frequency we measure is an average of the two. The measurement at 156.54 G lies above the best fit line for measurements made with lower density thermal clouds and hence may show evidence for a mean field shift to the binding energy. A shift consistent with this one was also observed using our Ramsey technique as reported in Ref. [26]. We have thoroughly investigated the resonant conversion efficiency at 156.45 G as a function of coupling time. Figure 4 shows that the conversion efficiency increases with coupling time until a saturated value is reached. Interestingly, for a given phase space density, this value is the same as we would expect if we had instead ramped the magnetic field adiabatically across the Feshbach resonance [17]. In both cases the maximum possible conversion is determined just by the phase space density. However, this new technique alIows much higher phase space densities (and hence higher conversion efficiencies) because the cloud is heated much less. The relevant phase space densities to compare are the ones after the conversion process. We have confirmed that this same phase space limit on conversion efficiency applies at all magnetic field values we have investigated and at this time it is not understood why the efficiencies of these two conversion schemes have the same dependence on phase space density. Figure 4(b) shows the converted fraction in the short coupling time regime. We observe Rabi-like oscillations in
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higher than those achieved by ramping the magnetic field through the Feshbach resonance because it greatly reduces problems associated with working close to resonance such as heating and rapid collisional loss. For example, by applying a resonant modulation with an amplitude of 0.5 G to a pure BEC for 1.6 ms we observe 55% conversion. We have observed Rabi-like oscillations between atomic and molecular populations that damp out on the time scale of the molecular lifetime. This new technique also allows precise spectroscopic measurements that are comparable to the results of our Ramsey measurements in Ref. [26]. We thank C. Regal, D. Jin, E. Cornell, and S. Papp for helpful discussions. This work has been supported by ONR and NSF.
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HG. 4. (a) Percentage of atoms converted to molecules at 156.45 G (using 6.5 lcHz resonant modulation frequency) as a function of coupling time. The number of molecules increases with coupling time until the conversion becomes saturated. (b) Oscillations in the converted fraction as a function of coupling time. The oscillation frequency is approximately 2.5 kHz and the damping time is approximately 4 ms.
the atomic population at a frequency of about 2.5 kHz. Note that the initial peak-to-peak amplitude of these oscillations is only 6% of the total atom number, making this measurement rather difficult. We have mapped out the oscillations for slightly longer coupling times and observe that, as one would expect, they damp out with a time constant of 4 ms which is of the same order as the molecular lifetime. One question that remains is why the initial oscillation amplitude (6%) is much less than the maximum converted fraction of 30%. In a simple two level system one would expect a Rabi oscillation amplitude of 30% because the maximum conversion would occur during the first Rabi cycle. Here, we observe a slow build up of molecules over a time scale much slower than the Rabi frequency. It is unclear what determines the coupling strength relevant to both these Rabi-like oscillations and the time scale associated with reaching the saturated conversion regime. In summary, we have devised a new and very useful technique for atom-molecule coupling in the vicinity of a Feshbach resonance that offers superb control of the conversion process. It allows conversion efficiencies that are
[ l ] M. Greiner, C. A. Regal, and D. S. Jin, Nature (London) 426, 537 (2003). [2] M. W. Zwierlein et al., Phys. Rev. Lett. 91,250401 (2003). [3] S. Jochim et aL, Science 302, 2101 (2003). [4] M. Bartenstein et al., Phys. Rev. Lett. 92, 120401 (2004). [5] E.A. Donley, N. R. Claussen, S.T. Thompson, and C.E. Wieman, Nature (London) 417, 529 (2002). [6] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser, Phys. Rev. Lett. 87, 120406 (2001). [7] E. Timmermans, K. Furuya, P.W. Milloni, and A.K. Kerman, Phys. Lett. A 285, 228 (2001). [8] M. W. Zwierlein etal., Phys. Rev. Lett. 92, 120403 (2004). [9] M. Greiner, C. A. Regal, and D. S. Jin, Phys. Rev. Lett. 94, 070403 (2005). [lo] T. Kohler, T. Gasenzer, P.S. Julienne, and K. Burnett, Phys. Rev. Lett. 91, 230401 (2003). [I 11 C.A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Nature (London) 424, 47 (2003). [12] K.E. Strecker, G.B. Partridge, and R.G. Hulet, Phys. Rev. Lett. 91, 080406 (2003). [13] J. Herbig et al., Science 301, 1510 (2003). [I41 K. Xu et aZ., Phys. Rev. Lett. 91, 210402 (2003). [15] S. Diirr, T. Volz, A. Marte, and G. Rempe, Phys. Rev. Lett. 92, 020406 (2004). 1161 M. Market al., Europhys. Lett. 69,706 (2005). [17] E. Hodby et aL, Phys. Rev. Lett. 94, 120402 (2005). [18] T. Mukaiyama et al., Phys. Rev. Lett. 92, 180402 (2004). [19] J. Cubizolles et al., Phys. Rev. Lett. 91, 240401 (2003). [20] S. Jochim et al., Phys. Rev. Lett. 91, 240402 (2003). [21] S. T. Thompson, E. Hodby, and C. E. Wieman, Phys. Rev. Lett. 94, 020401 (2005). [22] H. R. Thorsheim, J. Weiner, and P. S. Julienne, Phys. Rev. Lett. 58, 2420 (1987). [23] P. Pillet et al., J. Phys. B 30, 2801 (1997). [24] W. C. Stwalley and H. Wang, J. Mol. Spectrosc. 195, 194 (1999). [25] S.L. Cornish et al., Phys. Rev. Lett. 85, 1795 (2000). [26] N. R. Claussen, Phys. Rev. A 67, 060701(R) (2003). [27] T. Kohler, E. Tiesinga, and P. S. Julienne, Phys. Rev. Lett. 94, 020402 (2005).
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I have been interested in physics education for many years. My publications in this area began with trying to find ways to make current topics in atomic physics more accessible to undergraduate students in their teaching labs. This gradually progressed into greater involvement in education both through research into more effective educational practices and as an advocate for improved science education. I now have a substantial research group in physics and chemistry education, and we are ramping up our output of research papers in this field. In my advocacy role, I am often asked to write about various aspects of science education, and the resulting papers are usually commentaries rather than research.
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A narrow-band tunable diode laser system with grating feedback, and a saturated absorption spectrometer for Cs and Rb K. B. MacAdarn,a)A. Steinbach, and C. Wiernan Joint Institute for Laboraiory Astrophysics and the Department of Physics, Uniuersity of Colorado, Boulder, Colorado 80309-0440
(Received 6 February 1992; accepted 26 June 1992) Detailed instructions for the construction and operation of a diode laser system with optical feedback are presented. This system uses feedback from a diffraction grating to provide a narrow-band continuously tuneable source of light at red or near-IR wavelengths. These instructions include machine drawings for the parts to be constructed, electronic circuit diagrams, and prices and vendors of the items to be purchased. It is also explained how to align the system and how to use it to observe saturated absorption spectra of atomic cesium or rubidium.
I. INTRODUCI'ION Tuneable diode lasers are widely used in atomic physics. This is primarily because they are reliable sources of narrow-band ( < 1 MHz) light and are vastly less expensive than dye or Ti-sapphire lasers. However, the frequency tuning characteristics of the light from an "off the shelf" laser diode is far from ideal, and this greatly limits its utility. In particular, the laser output is typically some tens of MHz wide and can be continuously tuned only over certain limited regions. These charactcristics can be greatly improved by the use of optical feedback to control the laser frequency. Reference 1 gives a lengthy technical review of the characteristics of laser diodes, the use of optical feedback techniques to control them, and various applications in atomic physics. An earlier review by Camparo2 also gives much useful information, primarily relating to freerunning diode lasers. The use of a wavelength-dispersive external cavity for diode laser tunning and mode selection was described by Ludeke and Harris,' and the spectral characteristics of external-cavity stabilized diode lasers were investigated in detail by Fleming and M ~ o r a d i a n . ~ During the past several years our laboratory has carried out a large number of experiments in optical cooling and trapping, and general laser spectroscopy of cesium and rubidium using diode lasers. In the course of this work, we have developed a simple inexpensive design for a diode laser system that uses optical feedback from a diffraction grating. This system produces over 10 mW of light with a bandwidth of well under 1 MHz and can be easily tuned over atomic resonance lines. We now have over a dozen such laser systems operating, including two in an undergraduate teaching lab, and the design has reached a reasonable level of refinement. There are many other designs for optical feedback systems' and we make no claims for this one being superior. However, it is a reasonable compromise between several factors which are relevant to 1098
many laboratories: ( 1 ) low cost (about $400not including labor), (2) ease of construction (several of these systems have been built by novice undergraduates), and ( 3 ) reliability. These lasers have achieved several notable successes in experiments on cooling and trapping cesium atoms, and the design has been successfully duplicated in a number of other laboratories. We prepared this article in response to a large number of requests for detailed instructions on how to build and operate such a system. This article provides a detailed and fully comprehensive recipe for construction of the system and its use to observe saturated absorption spectra in a rubidium or cesium vapor cell. We refer the reader to Ref. 1 and the references therein for information about the physics of laser diodes and the factors that motivated this design as well as design alternatives. In this paper, we have attempted to respond to three frequent requests for information we receive. The first is from the undergraduate wanting to do high-resolution laser spectroscopy for a project, without expert local supervision. The second is from the faculty member who wants to construct a teaching laboratory experiment and wants instructions that can be given to a technician or undergraduate with favorable results. The third is from the research scientist who wants to use diode lasers in an experiment and would like to benefit from the accumulated practical experience in another laboratory. We will first discuss the construction or purchase of the basic components and then explain how to put them together, align the laser system, and tune the frequency. Finally, we discuss how to observe saturated absorption spectra and how to use these spectra to evaluate the laser performance or to actively stabilize the laser frequency by locking it to narrow saturated absorption features. 0 1992 American Association of Physics Teachers
Am. J. Phys. 60 (12), December 1992
665
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I’I
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Fig. 1. Assembly top view of laser. The arrow showing the blaze direction on the grating is for the low feedhack-large output case.
11. SYNOPSIS OF COMPONENTS As shown in Fig. 1, the laser system has three basic components, a commercial diode laser, a collimating lens, and a diffraction grating. These components are mounted on a baseplate. The laser and lens are mounted so that the lens can be carefully positioned relative to the laser to insure proper collimation. The diffraction grating is mounted in a Littrow configuration so that the light diffracted into the first order returns to the laser. As such, the grating serves as one end “mirror” of a laser cavity, with the back facet of the diode providing the second mirror. This means the grating must be carefully aligned and very stable. To achieve this we mount the grating on a standard commercial mirror mount which is attached to the baseplate. As with any laser, changes in the length of the cavity cause shifts in the laser frequency. Therefore, to obtain a stable output frequency, undesired changes in the length due to mechanical movement or thermal expansion must be avoided. To reduce movements due to vibration of the cavity we mount it on small soft rubber cushions. To avoid thermal changes, the bascplate is temperature controlled using heaters and/or thermoelectric coolers. In addition to controlling the temperature of the baseplate, we independently control the temperature of the laser diode. Finally, to avoid air currents interfering with the temperature control we enclose the entire laser system in a small insulated metal box. Of course, to finely tune the laser frequency one must have some way to change the length of the cavity in a carefully controlled manner. We do this using a piezoelectric transducer speaker disk which moves the grating in response to an applied voltage. The laser system also requires a small amount of electronics. A stable low-noise current source is needed to run the laser, and temperature control circuits are used to stabilize the diode and baseplate temperatures. This electronics is readily available commercially. However, for those 1099
Am. J. Phys., Vol. 60,No. 12, December 1992
with more time than money, we provide circuit diagrams for the relatively simple circuits that we normally use. This system contains both purchased and “homemade” components. Before discussing the construction aspects, we will provide some information concerning the purchasing of the commercial components. We purchase the diode laser itself, the collimating lens, the fine adjustment screw which controls the lens focus, the diffraction grating, the mirror mount which holds the grating, and the piezoelectric disks. The purchase of most of these items is straightforward. Fine adjustment screws and mirror mounts are available as standard items from most companies that sell optics hardware. Similarly, laser diode collimating lenses and diffraction gratings are available from numerous companies. For the convenience of the reader we list in the Appendix the exact products we use along with the prices and vendors. However, for these items our choice of vendors was primarily determined by expediency, and we have no reason to think that other vendors would not provide equal or superior products. In contrast, in order to obtain satisfactory laser diodes and piezo disks we have tried and rejected a large number of different vendors. Piezo disks are widely sold as electronic speakers and are very inexpensive, but most models are not adequate for this application. The Appendix gives the only suitable product we have found. The purchasing of diode lasers can be filled with frustrations and pitfalls, and we refer the reader to Ref. 1 for a full discussion of the subject. Here, we shall just give a brief summary of what must be specified, and our recommendations for suppliers. The basic requirement for a diode laser which is to be used in this system is that it have a high reflectivity coating on the back facet and a reduced reflectivity on the front, or output facet. Very inexpensive diodes which produce a few milliwatts of power have two uncoated facets, and will not work very well. We have used 20-mW lasers, but their performance is marginal. However, we have found that any laser we have tried that is specified to provide 30 mW or more single mode will have the necessary coatings and will work weL5 It will provide narrowband laser light that is tuneable over 20-30 nm. If one wants this range to cover the 852-nm cesium or 780-nm rubidium resonance lines, the diode laser wavelength must be specified when purchasing. This greatly complicates the purchasing. We have tried numerous suppliers, but have now settled on STC as our supplier of lasers for 852 nm and Sharp as the supplier for 780 nm. The Sharp lasers are far less expensive and can usually be obtained rather quickly since 780 nm is near the center of the distribution of their normal mass-produced product. This is not the case for 852 nm, and thus the lasers must be produced as a custom run. STC has made several such custom runs and hence usually has 852-nm lasers available although they cost 3 to 4 times more than the Sharp lasers. The long wavelength edge of the distribution of Sharp lasers is at about 839 nm, and we have used such lasers to reach the cesium line by heating them. However, it can be difficult to obtain 839-nm lasers and to obtain reliable performance when tuning the laser this far from its free-running wavelength. The heating of the laser also degrades its lifetime. The remaining components of the laser system are homemade. The key components are the laser mounting block which holds the actual diode laser, the holder for the collimating lens, and the baseplate onto which all the comK.MacAdam, A. Steinbach, and C . Wieman
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667 COLLIMATOR LENS MOUNT
FRONT VIEW
0.250
SIDE VIEW
LENS
x 0.275 OPEN TO / WITH, 3/32" END MILL 0.150
4-40 TAP
\
'
0 040 ASSEMBLY (CUT AWAY)
~
,'1
4 40 SCREEN
+WASHER
ALIGNMENT JIG
30" BEVEL FOR BEAM CLEARANCE
0 688
TOP VIEW
0 500
Fig. 2. Laser mounting block, machine drawing. Dimensions are in inches. The hole sizes, spacings, and depths are correct for a Sharp LT025MDO laser and may be modified for other types.
ponents are fastened. In addition, we also make the box that encloses the system and a small jig that is useful for setting the position of the collimating lens. All these components have been designed so that they can be constructed by a novice machinist.
111. INSTRUCTIONS FOR CONSTRUCTION OF LASER COMPONENTS Construction of the diode laser system begins in the machine shop and primarily requires a milling machine and drill press. Detailed machine drawings for the laser mounting block, baseplate, collimating-lens holder, and an alignment jig are given in Figs. 2 and 3. In addition, an enclosure should be fabricated, but its design is not critical. We provide dimensions for mounting the standard Sharp laser package. Small changes may be needed for lasers from other vendors. In view of the setup time required in machining, and the fact that many interesting experiments with diode lasers require more than one of them, it will probably be found economical to make two (or more) systems at once.
Fig. 3. Collimating lens mount and alignment jig machine drawing. Dimensions are in inches.
"bridge" design has been found to make a significant improvement on laser cavity stability. The 0.356-in. diam hole to receive the diode package may be made either by boring on a lathe fitted with a four-jaw chuck or, more easily, by a suitable end mill. Reground 3/8-in. end mills can often be found near this diameter. Some deburring or filing may be necessary to allow the diode to fit snugly into its recess but allow it to be rotated to its proper orientation in the initial step of alignment. A small hole whose diameter is selected to fit the thermistor should be drilled into the back side of the mounting block near the diode recess.
A. Laser mounting block The laser diode is held firmly in a small aluminum block whose details are shown in Fig. 2. The critical dimensions are the 0.500-in. height of the laser center above the baseplate and the depths of holes that ensure that the 9-mm flange of the diode package is gripped by the mounting screws. For stability when the block is screwed down to the baseplate, the bottom surface of the block should be machined as shown with a 0.020-in. relief cut down the middle so that contact is along the edges of the block. This I100
Am. J. Phys., Vol. 60, No. 12, December 1992
I
I
$ Fig. 4. Jig usage in collimation. LD=laser mounting block (Fig. 2), C=collimating lens mount (Fig. 3 ) , J=alignment jig (Fig. 3). SP =spring or rubber pad to provide a restoring force against adjusting screw S2. K. MacAdam, A. Steinbach, and C . Wieman
1 loo
668 this time will allow comparison of laser performance with different cavity lengths without complete disassembly of the collimated laser.
DIODE L A S E R B A S E P L A T E
4 0 656
3 112
PRECISION BUSHING. ' DRILL CLEARANCE
i t 1 5/3-24 b
-
2 5/16+
HOLE(S) FOR SCREW TO ATTACH GRATING MOUNT.
MATERIALS: ALUM. l / Z " THK DIMENSIONS IN INCHES
Fig. 5. Laser baseplate, machine drawing
B. Collimator lens mount Figure 3 shows the aluminum block that holds the flanged collimator lens. The placement of the lens axis at 0.500 in. above the base and the diameter of the hole are again the critical dimensions. The 30" bevel shown on the front side of the block allows clear passage of the output beam off the diffraction grating when very short cavities are used. The figure also shows dimensions of a suggested alignment jig that is used to allow transverse displacement of the lens holder without rotation or longitudinal movement. A 2-56 screw with rounded tip and a small piece of bent spring steel or resilient cushion should be prepared for use with the jig (as shown in Fig. 4).
C. Baseplate The baseplate is shown in Fig. 5. We have found that aluminum is adequate for most purposes and is easy to machine. If greater thermal stability is required, however, the baseplate can be made of invar. The two pairs of 4-40 holes should be carefully positioned to match corresponding holes in the laser and collimator-lens blocks. The single 4-40 tapped hole is used to mount the alignment jig. The most obvious feature of the baseplate is its flex hinge design, which allows smooth variation of the spacing between diode and collimation lens by action of a commercial precision screw mounted to push against the hinge. The slot that forms the hinge can be cut by a bandsaw after all holes are laid out. The hole intended to receive the precision adjusting screw should be reamed to allow a fit without excess clearance. After all machining of the baseplate is complete the screw can be mounted in this hole with adhesive or by a set screw. The web that provides the flexible hinge should be left 1/16 in. or more in width: One can always remove material later if it proves too stiff. One or more holes should be drilled in the baseplate to mount the diffraction grating holder, but the exact position(s) depends on the dimensions of the holder and grating and on the desired cavity length. Several suitable holes drilled at 1101
Am. J. Phys., Vol. 60, No. 12, December 1992
D. Grating and grating mount The baseplate design is intended for use with a 1200 line-per-mm grating. Suitable gratings are readily obtained with 500- and 750-nm blazes and dimensions 1X 1X 3/8 in. thick. When mounted, the grating has its rulings vertical and diffracts its first-order interference maximum back into the laser. The output beam is the zero-order beam or specular-reflection maximum, which passes horizontally beside the collimator block and out of the enclosure. The direction of the blaze is toward the output beam. A laser diode whose free-running wavelength is within about 3 nm of the desired wavelength requires less feedback for stabilized operation than a laser that must be pulled more severely. For this case, lower diffraction efficiency and thus a shorter blaze wavelength (500 nm) is suitable, and this allows more power to be brought out in the zero-order beam. If a laser must be pulled more severely, a longer blaze wavelength (750 nm) is used to provide stronger feedback at the price of lower output power.6 When a grating of suitable blaze has been selected it may be cut down to a small size since only about 0.3 in. parallel to the rulings and 0.5 in. perpendicular is required. Thus several gratings can be had for the price of one, and the others may be used to duplicate the diode laser system or for testing grating properties outside the laser. The cutting may be safely done as follows. Apply a generous coating of clear acetate fingernail polish to the ruled face of the grating. Spread the fluid using a soft camel's hair brush, and avoid physical contact with the grating. After the coating is thoroughly dry, wax the back of the grating to a block of bakelite or phenolic to support the grating while it is sawed. Mark the coated surface of the grating into pieces of the desired size. A 1 x 1 in. grating will yield six suitable pieces. Saw the grating in an abrasive-wheel glass saw by holding the support block on its edge as the saw cuts directly into the face of the grating. Make sure the saw cuts penetrate completely through the grating into the support block without severing the block. Then melt off the cut segments. The nail polish can then be removed by submerging the grating segments in a small beaker of methanol and placing the beaker in an ultrasonic cieaner. Remove the gratings with tweezers, being very careful to avoid any contact with the now-exposed ruling surface, refill the beaker with fresh methanol, and repeat once or twice until the gratings, when drained and dried, appear completely clean. Harsher solvents may attack the plastic substrate of replica gratings, but methanol has been found to be safe and effective. After cleaning, the grating is attached to the movable face of the grating mount in a location where the collimated laser beam will strike near the middle of the grating. Care should be taken to make the rulings vertical. A stiff but readily removable adhesive such as Duco cement is recommended for attaching the grating to the mount. The grating segment can be easily damaged when it is necessary to remove it or shift its position unless it can be detached with little physical force. If necessary, the efficiency of most inexpensive gratings at the 852-nm wavelength for Cs can be improved by 10%-20% by evaporating a gold coating onto the grating before it is installed.' K. MacAdam, A. Steinbach, and C. Wieman
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N
PZT should be inserted between the mounting plate and the ball end of the adjusting screw as shown in Fig. 6. Small pieces of mylar should be inserted to electrically isolate the PZT from the mount. The PZT will provide about f 1 pm of displacement when f 15 V are applied.
GRATING
TO TRIANGLE WAVE GENERATOR
Fig. 6. Piezoelectric disks (not to scale)
Before the grating is attached to the mount, the mount should be modified, if necessary, so that it has the same “bridge” profile on its base as described earlier. In addition, the grating mount should be modified so that its adjustment screws can be turned by a ball-end wrench through holes in the temperature-control enclosure of the assembled laser. A good way to do this is to remove the heads from 1/4 in.-20 socket screws in a lathe and to attach them to the centers of the knobs of the adjustment screws with epoxy cement. E. Piezoelectric disks Piezoelectric (PZT) disks are inserted between the grating mount adjustment screw and the movable face of the mount in order to rotate the grating about a vertical axis and alter the cavity length with electrical control (Fig. 6). Each PZT element consists of a thin brass backing about 1 in. in diameter to which a thin smaller-diameter silverplated piezoelectric slice is attached in the center with adhesive around its edge. When voltage is applied, the piezoelectric stress causes the backing to “dish” on the opposite side. Two such elements can be attached back to back, doubling the displacement of a single one, by lightly soldering the adjacent brass backings at four places around their circumference. If necessary for clearance in the grating mount, some of the excess brass can be clipped away without damaging the piezoelectric center. The double PZT is wired by lightly soldering one connection to the brass and the other to the two silver-plated piezo elements in parallel. For this and all other wiring of the laser, it is best to select a limp insulated wire that will not transmit vibration to the laser structure. Rubber covered No. 24 test prod wire has been found suitable. After the PZT is assembled and wired, and the grating is glued to the mount, the 1102
Am. J. Phys., Vol. 60,No. 12, December 1992
F. Enclosure for the laser An aluminum enclosure should be fabricated to hold the laser. It should have a sufficient thermal mass and conductivity to aid in temperature stabilization. Such a box can be made out of rectangular side plates screwed together, placed on a rectangular baseplate, and capped by a lid, or a single piece of hollow rectangular tubing may be selected to form the walls. Wall thickness should be 1/4 to 1/2 in. Inside dimensions about 3.5 in. wide, 5.5 in. long and 4 in. high are adequate. The floor of the enclosure should stand on some firm support, to bring the laser output beam to a desired height above the table. The lid of the enclosure should be easily removable to allow frequent access to the laser with minimal disruption of the thermal or mechanical stability. After the laser is assembled and satisfactorily aligned, drill holes in the box to allow access to the grating adjustment screws and drill an opening to allow exit of the laser beam. These steps should be delayed until one knows for certain where the holes should be placed. The output aperture is ultimately covered by a microscope slide, and the access holes should be plugged to limit air currents. Tapped holes on opposite edges of the bottom plate of the enclosure allow the laser structure to be anchored onto the vibration isolation pads discussed below, for instance by stretching a rubber band over the laser baseplate and looping it over screw heads in the edges of bottom plate. The bottom edge of one of the side walls of the enclosure should be provided with a notch or channel at both ends of the laser for egress of all wires. Soft rubber placed in the notches can serve to press the wires firmly against the bottom plate, and in this way the movement of wires outside the box will not transmit stress or vibration to the laser structure. Finally, one should make sure the enclosure is electrically grounded.
IV. TEMPERATURE CONTROL Precise control of the temperature of both the baseplate and the diode laser itself is essential for the long term reliable operation of the laser at a particular wavelength. We control these temperatures using identical independent servosystems. The sensing element for the servo is a small thermistor, which is part of a bridge circuit. The amplified and filtered error signal drives a heater or thermoelectric cooler. In this area of thermal control we have made the largest compromises of potential performance in order to simplify the mechanical and electrical designs. Part of the reason we are willing to make this compromise is that we usually sense the output frequency of the laser and lock it directly to atomic transitions to insure long term stability at the sub-MHz level. This is discussed in Sec. X. The temperature of the diode laser mounting block is controlled only by heating, which means that it must be kept 1-2 “C hotter than the baseplate for proper temperature control. The heating is done by a small (0.3 X 1.5 in.) adhesive film heater which is attached to the top or side of the laser mounting block. The sensing thermistor rests in a small hole packed with heat sink compound in the mountK. MacAdam, A. Steinbach, and C. Wieman
1102
The r ~ s l s t o ro n~ S I
+15u ?
a r e each
4.39k 1%
m e t a l film resistors.
.
SET POINT’
~~1399 u1
’THERII. NON’
TL074
H
R I 1 5W f 2 Ohm
*Resistors are I% m e t a l f i l m .
*
6 Fig. 7. Temperature control circuit.
ing block. The temperature control circuit which drives the heater is shown in Fig. 7. Although this circuit is rather crude compared to what is usually used for precision temperature control, we have found it adequate for most purposes. It is simply a bridge, an amplifier and an RC filter which rolls off the gain as Vfrequency, for frequencies between 0.005 and 0.50 Hz. The components have been chosen so that above 0.5 Hz the electrical gain is constant. This frequency response was selected so that the combination of this electrical response and the thermal response of the laser mounting block results in a net servo gain which goes nearly as l/f. The gain is set by the 5-kR potentiometer to be just below the point where the servo loop oscillates. One can readily observe saturated absorption spectra and carry out other atomic spectroscopy experiments with temperature control only on the laser mounting block. However, temperature stabilizing the baseplate greatly reduces the thermal drift of the laser frequency and changes in the cavity alignment. The baseplate is either heated or cooled depending on the requirement. Heating is much simpler since it only requires the attaching of a film heater to the baseplate. The film heater is similar to that used on the laser, except it is larger in area and power output. To keep the baseplate controlled it is necessary that it be at least 1-2°C above the room temperature, and the laser must be an equal amount hotter than the baseplate. This is not difficult if the laser’s free running wavelength at room temperature is shorter than wavelength desired. In that case it is advantageous to heat the laser. If however the laser’s free-running wavelength is significantly to the red, 1103
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the laser should be run near or below room temperature. In this case, the baseplate must be cooled below room temperature using a thermoelectric cooler (TEC). This is somewhat more trouble, and the vibration isolation pads between the baseplate and the bottom of the enclosure are now replaced by a rigid TEC. The TEC is a square 1.5 in. on a side and fits between the baseplate and the aluminum plate which is the bottom of the enclosure. A thin layer of heat sink compound is applied on both sides of the TEC to insure good thermal contact. The bottom plate of the enclosure must have a large enough surface area or be in contact with a thermal reservoir so that it does not heat up enough to cause “thermal runaway” of the TEC. The baseplate temperature is monitored using a thermistor glued onto the middle of the baseplate. Since the thermal time constant for the baseplate is much longer, some adjustment (or removal) of capacitors C1, C2, and C3 from the temperature control circuit may be desirable to improve stability. If the laser is cooled below the dew point condensation may form. This may be avoided by flushing gently with dry N2. An alternative to controlling the baseplate temperature is to control the temperature of the entire enclosure. This is more effort because much more heating or cooling power is needed and the thermal time constant is very long. We find, however, that this technique gives better ultimate stability of the laser alignment. For most purposes, we have found that this is not worth the effort. However, the small additional effort required in putting insulation on the outside of the aluminum enclosure to attenuate room temperature fluctuations is worthwhile. K. MacAdam, A. Steinbach, and C. Wieman
1103
671
RI
A c c u l e x DP-650
RZ
‘CURRENT
meter.
2,
The b l o c K r a n t h e b a r e r o f 01 a n d 02 a r c f e r r i t e b e a d s . 3) R e r r r t o r r e v e 1% m e t a l f i l m .
*
Fig. 8. Laser current control circuit.
V. LASER DRIVE ELECTRONICS The circuit diagram* for the laser current controller is shown in Fig. 8. This is a stable low-noise current source. The output current can be modulated rapidly by sending a voltage into the “RF MOD I N ’ input. If such modulation is not needed and novices may be operating the laser, it is wise to disconnect or cover this input to minimize the possibility of accidentally damaging the laser. The output current of the supply is limited by potentiometer R36 to a value that cannot exceed the maximum allowed for the diode laser. The primary concern when working with the current source is to avoid damaging the laser with an unwanted current or voltage spike. In Ref. 1, we discuss this danger at some length so here we will just provide a few helpful techniques. To avoid accidents we always carefully test a new power supply with resistors and light emitting diodes in place of the laser. We check that it produces the voltage and current desired, and that there are no significant transients when turning it on or off. It is also wise to check that all the appropriate grounding connections have been made so that turning on and off nearby electrical equipment or static discharges do not cause current or voltage spikes that exceed the maximum allowed by the laser diodes. When making these tests it is important to realize that lasers can be destroyed by spikes that last only a fraction of a microsecond. Only after the power supply has passed all these tests is it connected to the laser. The cables from the power supply to the laser should be shielded and there should be no possibility of them being accidentally disconnected, One has to be fairly careful in handling the lasers to avoid static discharges, and it is a good idea to keep the leads shorted together as much as possible. Normally such lasers come with handling instructions that should be followed. These instructions will usually also mention that a 1104
Am. J. Phys., Val. 60, No. 12, December 1992
fast, reverse-biased protection diode should be connected across the laser leads at the laser mounting block. This diode protects against voltages spikes which may exceed the few volts of back bias a diode laser can tolerate. We have found that the lifetime of diode lasers is substantially increased by also connecting several forward-biased diodes across the leads at the same point as shown in Fig. 9. These diodes have a large enough voltage drop that current does not flow through them under normal operation. However, if there is a large forward voltage, these diodes turn on allowing the current to flow through them instead of the laser diode. It may also be helpful to place a 1 0 4 current limiting resistor in series with the supply right at the laser diode (at LD+ in Fig. 9 ) and, if modulation much above 1 MHz is not required, ferrite beats on the supply lead at this point. Switch S1 in Fig. 8 should be used to short the
LD-
7--r--l
I
D2
D2 D2
LD+ 0
0
I
PD+ 0 D1
=
IN5711
D2 = 1N914
Fig. 9. Protection diode wiring to laser.
K. MacAdam, A. Steinbach, and C. Wieman
1104
672 supply to ground before connecting the laser, and the current control R3 should be fully “OR’whenever 5’1 is toggled.
VI. ASSEMBLY AND TESTING A top view of the assembled laser is shown in Fig. 1. A. Diode mounting
The laser diode, with its protection diodes already wired on and its leads temporarily shorted together for safe handling, is mounted in its recess in the laser mounting block with the screws only gently tightened at first. The desired orientation of the laser will produce a vertically polarized output beam and a widely diverging elliptical beam pattern whose major axis is horizontal. This corresponds to the rectangular output facet of the diode chip having its longer dimension vertical. Next, the mounting block is attached to the baseplate by 4-40 screws extending from beneath. The baseplate should then be secured to some temporary stand so that the uncollimated laser beam can be easily observed after it has gone 1.5 m or more from the laser. The current supply is then set to a normal operating current. The output beam at 780 nm, when projected onto white paper attached to the wall, will hardly be visible with the naked eye, but will show up readily in an IR viewer. The 852-nm light can only be observed by the viewer or an IR sensitive card. At this time, interference rings or fringes may be apparent in the projected beam. These are normally caused by dust, fingerprints, etc., on the laser’s output window. The window should be cleaned with an optical tissue dampened in methanol so that the beam pattern is uniform and clear. The orientation of the diode in its recess should be set either by noting the major axis direction or by checking the output polarization. Once the proper orientation is achieved, the mounting screws that hold the laser in its recess should be tightened. Make sure that connections to the diode, including the network of protection diodes, are insulated and arranged so that short circuits will not occur during routine handling. Note carefully the center position of the dispersed beam spot, both its height and lateral position and mark it on the wall. Despite the broad and undifferentiated beam spot, the center can be judged reliably within *2”.
ing point of the laser. Temporarily tighten the screws that hold the lens block in that position and remove the slide. Next, adjust the precision screw to bring the laser beam to a sharp focus on the wall. By a very slight adjustment of the screw the beam should then be brought to collimation in an oblong spot about 5 mm wide. It should be confirmed that no focus occurs between the laser and the wall. This constitutes a preliminary alignment. The beam spot will very likely fall 2” or more from the aiming spot. Horizontal corrections can be made smoothly by use of the alignment jig later, but vertical corrections require shimming first. Note the vertical displacement of the spot from the aiming point. A low spot will require raising the lens mount by about 0.0025 in. per degree of misalignment and a high spot will require raising the diode block by the same amount. Layers of aluminum foil (avoiding crinkles) or shim stock should be selected to shim the preliminary alignment beam height to within 1”of the aiming spot. The alignment jig is installed next by screwing it to the laser baseplate using its oversize hole and a large washer (or stack of washers) so that it snugly touches the lens mount as shown in Fig. 4. It should be positioned with its 2-64 screw and a spring or elastic cushion so that when the screws of the lens mount are released, the mount can be pushed in both directions without losing contact. With the lens-mount screws now loosened the mount may be displaced smoothly to bring the collimated spot horizontally to the aiming point. A rubber band or finger pressure should be used to hold the loose lens mount against the jig. A properly aligned laser will exhibit a symmetrical and elliptical beam spot. The effects of aberration can be observed by purposely misaligning the lens to one side or the other with the jig, and a symmetrical behavior allows one to confirm that the designated aiming spot was initially correct. After a satisfactory alignment and collimation has been obtained, the lens mount screws should be firmly tightened and the jig removed. After the lens mount is tightened in place, the fine adjustment screw should again be adjusted to precisely collimate the beam. Positioning the lens without the jig is also possible for those users with a steady hand, but it is very difficult to avoid random rotations and displacement along the beam when only a transverse adjustment is desired.
C. Power output and threshold current measurements
B. Collimation The next step is collimation of the output beam and shimming, if necessary, of the laser or lens mounting blocks to the correct height. The collimating lens should be firmly fastened into its mounting block with a set screw and the mounting block should be loosely screwed to the laser baseplate with the flat side of the lens toward the diode. The precision adjusting screw that pushes against the baseplate hinge should be advanced so that the hinge is opened enough to allow plus and minus 0.020 in. of motion without losing contact with the ball end of the screw. Next insert a clean microscope slide (about I-mm thickness) between the lens flange and the front face of the diode laser mounting block. While holding the lens block, the slide, and the diode block together with finger pressure, observe the beam spot again with the IR viewer. It should be possible to slide the lens block back and forth to bring a more concentrated intensity maximum near to the original aim1105
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After the laser has been aligned and collimated and before the grating is installed, the power output and threshold characteristics should be recorded and compared with the specifications. The threshold current depends on laser temperature, so it may be desirable to stabilize the temperature of the diode mount at this time. The output power can be measured as a function of drive current by illuminating the face of a wide-aperture photodiode.
D. Mounting and adjusting the diffraction grating The mounting of the diffraction grating has been described earlier. The laser cavity length is determined by the distance from the back of the laser chip to the illuminated spot on the grating and can be as short as about 20 mm in this design. The grating mount should be screwed to the laser baseplate to form a cavity of the desired length so that the collimated beam illuminates the center of the grating at K. MacAdam, A. Steinbach, and C . Wieman
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673 approximately the Littrow angle. Best results have been obtained with the shortest possible cavities, apparently because the corresponding mode spacing (about 8 G H z ) avoids excitation of adjacent cavity modes by the inherent relaxation noise of the diode at around 3 GHz from line center. It is possible, with a carefully aligned 780-nm laser having 20-mm cakity length to tune electrically over 7 GHz without a mode hop using only the PZT. The following procedure is used to align the diffraction grating. A small card cut from stiff white paper or a file folder, about 2 X 1/4 in., is useful as a probe to see that the beam diffracted from the grating returns approximately to the center of the lens. The beam spot at 780 nm is readily visible to the eye on the card, but at 852 nm the IR viewer is required. Before screwing the grating mount firmly to the baseplate in this coarse alignment make sure that the adjustment screws are in midrange. The card should be used next to make a more careful alignment of the grating. If the return beam is, for example, too high, as the card is lowered vertically in front of the lens the outward face of the card will be illuminated along a narrow region at its edge until the beam is completely cut off. The width of this narrow region indicates the degree of vertical misalignment. When an edge of the card is raised from below to cut off the beam, no such region of direct illumination will be visible in this example, although direct light from the lens may weakly filter through the card. Probing from all four directions into the collimated beam will indicate both the horizontal and vertical misalignment of the return beam, and the objective is to adjust the screws of the grating mount so that the width of the illuminated region on the card edge is brought exactly to zero for each direction of approach. A precise vertical alignment of the return beam is made by reducing diode current to just above threshold. Then observe the intensity of the output beam while adjusting the tilt of the grating around a horizontal axis. If the preliminary Littrow alignment was adequate, the output beam should significantly brighten at the exact vertical position that optimizes feedback into the diode. After completing this adjustment the threshold current will be lower than the value recorded earlier for the diode. The laser should now be operating with grating controlled feedback near its free-running wavelength. If more than one vertical setting of the grating appears to enhance the laser output near threshold, or if the output beam projected on a distant surface consists of more than a single collimated spot, the fault may lie with imperfections (chips, scratches, dirt) on the grating, laser window, or lens surfaces.
VII. TUNING THE LASER FREQUENCY A low-resolution ( < 1 nm) grating spectrometer is useful to assess the tuning characteristics of the laser discussed below. After initial alignment of the grating, the output wavelength of the laser will be within about 2 nm of the wavelength specified by the manufacturer, and near the center of the tuning range. Small adjustments of the grating rotation screw (vertical axis) should smoothly shift the laser wavelength. A region of the grating angle adjustment should be identified over which the laser can be tuned. As one nears the end of the tuning range the laser output will be seen to hop back and forth or share power between two very different frequencies. One is the fixed “free-running” 1106
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frequency at which the laser will operate if there is too little or no feedback from the grating, and the other is the angle dependent frequency set by the grating feedback. At a given temperature, tilting the grating should tune the output wavelength over a range 10 to 30 nm, depending on the particular laser and the amount of feedback. Changing the diode temperature shifts the entire range by 0.25 nm/”C. If the grating is misaligned, the output wavelength will either be insensitive to small changes of the grating angle or will move only a small amount and then jump backwards. Although this tuning may appear continuous when observed on a low or medium resolution spectrometer, there can actually be small gaps. These occur because the wavelength-dependent feedback of the grating dominates but does not always totally overwhelm feedback off the AR coated output facet of the chip. If it proves impossible to excite some desired atomic absorption line by tilting the grating, it is necessary to operate at a different temperature and/or current. This is best assessed by means of an atomic absorption cell (discussed below) since the gaps in tuning can be narrow and vary randomly from one laser to another. It is helpful to record the tuning rate vs grating rotation (about 14 nm/turn with an 80 thread-per-in. screw pitch), because one can easily mistune the grating grossly, requiring a retreat to earlier steps in the alignment process. After the grating rotation has been set to produce approximately the correct wavelength, the vertical alignment should be rechecked using the threshold current technique. The simple laser design described here suffers from a defect that may be annoying in wideband usage: Its output beam is deflected horizontally as the wavelength is scanned, approximately at the angular rate,
for grating constant d. This is normally of no consequence, however, for saturated absorption or neutral atom trapping, e.g., in Rb where the 5s,/,- 5p3/2hyperfine multiplets of the two naturally occurring isotopes span a total of less than 0.014 nm. If it is necessary to avoid beam deflection, the simplest technique is to take the output beam off a beam splitter inserted between the collimating lens and the grating.’
VIII. ENCLOSURE AND VIBRATION ISOLATION After all preceding steps of alignment have been completed, the laser should be thermally and vibrationally isolated in its enclosure. We have achieved an adequate degree of mechanical isolation by supporting the laser inside the enclosure on three rubber pads that form a tripod under the solid part of the baseplate (avoiding the hinge). Soft “sorbothane” rubber, 1/8 in. thick, cut into 1/2-in. squares and stacked to a height 3/8-in. forms a springy but well damped support that isolates from vibrations above about 100 Hz. For extra isolation, additional rubber may be placed under the support which holds the enclosure at the desired height. The wires from heaters, thermistors, the diode, and the PZT should be taped down to the laser and/or the enclosure baseplates to decouple them mechanically from the laser cavity. The suggested enclosure design offers additional decoupling by pressing the wires firmly against the bottom plate where they exit from the box. K. MacAdam, A. Steinbach, and C. Wieman
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DIODE L ASER
Rb VAPOR C E L L
BEAM
GAIN
PH OT OD IODES
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AMPLIFIER OFFSET
PO OUT
l00K 1200pF
Fig. 10. Beam layout for saturated absorption.
+ LF356
lQQK
Holes should now be drilled in the sidewalls of the box to allow the beam to exit and to allow manual grating adjustments without having to open the enclosure. Because of the very nonrigid support of the laser, mechanical adjustments, although not often necessary after stabilization, require a delicate touch. The laser structure takes one or more hours to fully stabilize inside its box with temperature-control electronics active. However, preliminary output tests can proceed immediately if steady frequency drift is not an obstacle. Depending on the degree of stability required and the environment, the laser may be operated on anything from an ordinary laboratory bench, to a fully isolated optical table. A room location near a load-bearing wall or in a basement laboratory can often be worth the price of an expensive optical table. Since the laser itself is one of the best vibration detectors obtainable, experience will be the best guide.
system. Often when one purchases ampules of alkali metal they come packed with an inert gas. In this case there will be a burst of gas also released into the system which must be pumped away. The alkali metal can be moved into the cell simply by heating the glass around it and thus distilling it down the glass tubing into the cell. It is only necessary to have a few very small droplets in the cell so one ampule is sufficient to fill many cells. It is desirable to put much less than 1 g of metal into the cell to reduce the tendency of the metal to coat the windows. Once the alkali is in the cell?the sidearm is tipped off and the cell is ready for use.
IX. SATURATED ABSORPTION SPECTROMETER
B. Optical setup
The simplest spectroscopy one can perform with these lasers is to observe the absorption and Doppler-free saturated absorption spectra’ of rubidium or cesium. This can easily be done in small glass vapor cells which are at room temperature. Such spectroscopy experiments also provide the simplest way to determine the short and long term frequency stability and tuning behavior of the laser frequency.
Figure 10 illustrates a typical layout of beams for a simple saturated absorption apparatus. Initially only a single beam passing through the cell is required, which should be the full laser intensity for maximum sensitivity. In this step one tunes the laser to an atomic transition and finds the optimum laser temperature, current, and mechanical arrangement for stable operation. When the cell is viewed through an IR viewer, or a CCD television camera, a strong track of fluorescence should become visible as the laser is tuned within the Doppler profile of an absorption line by mechanically rotating the grating. It is helpful to ramp the PZT at a frequency of 20 Hz over a 15-V range during this search. The diode current should be arbitrarily set between about 75% and 90% of lop. If no fluorescence is apparent at any grating angle with the known tuning range, the temptation to turn the grating farther or to adjust the vertical alignment of the grating should be resisted. Most likely the laser has a tuning discontinuity that encompasses the desired wavelength. The current should be changed several mA and the procedure repeated. If this still fails, the temperature should be changed up or down 0.5 “C to 1 “C and the search for the absorption line should be repeated. If this process is iterated several times without success, it may be desirable to look once again with the grating spectrometer to confirm that the laser is still tuning in the desired range and that the grating has not been grossly misaligned by a random walk. When one finds a grating position which produces fluorescence, the current can be adjusted to maximize the fluorescence.
A. Vapor cells Rubidium and cesium vapor cells can be obtained commercially, but they are usually rather expensive. However, they can be prepared quite easily, if one has a vacuum pump and some basic glassblowing skills. We use Pyrex or quartz tubing, typically 1 in. in diameter and 2 to 4 in. long, and fuse windows onto the ends. The optical quality of the windows is unimportant. The glass cell is connected to a vacuum system through a glass tube about 1/4-in. diameter so that the cell can be evacuated to between lop5 and Torr. After the cell is filled, it will be “tipped off’ by heating this connecting tube until it collapses in on itself. A few grams of alkali metal in a glass ampule are placed in a separate arm on the vacuum system. The system should be pumped down and the cell outgassed briefly by heating it for several minutes with a torch. At the point where it will be tipped off, the glass connecting arm should be repeatedly heated until it just starts to soften but does not collapse in. After the outgassing is completed, the ampule should be broken to release the alkali metal into the 1107
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Fig. 11. I to V amplifier circuit
K. MacAdam, A. Steinbach, and C. Wieman
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+
F')
85Rb (F = 2
+
F')
85Rb (F = 3
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"Rb
+
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87Rb (F
=
W
2
I-
5W n -1
8 5 Rb (F = 3
+
F')
a z
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PZT VOLTAGE
z G F Q
Fig. 12. Single beam saturated absorption in 85Rb,F = 3 - F '
a: 0
v)
m
a D W
Once a proper temperature has been set it should not be necessary to change it. However, when the laser is turned on in the morning it is not uncommon to find that the proper drive current has changed by up to 1 mA, or, at a fixed current, minute readjustment of the grating angle is needed, to hit the absorption line again. This drift may be caused by environmental changes, hysteresis in the electrical tuning characteristics, aging of the diode, or mechanical creep of laser cavity components.
c a a: 3 I-
a
v)
0
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=
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100
300
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500
FREQUENCY (MHz)
(a)
C. Piezoelectric scanning After the laser is mechanically tuned onto an absorption line as observed in the IR viewer, the transmitted (probe) beam should be attenuated so that the intensity is less than 3 mW/cm2 and directed into a photodiode. A preliminary assessment of mechanical, electrical, and thermal stability may be made merely by observing the single-beam absorption line. The photodiode output is converted to a voltage by an I/V amplifier, whose circuit is shown in Fig. 11, and the resulting signal is displayed on an oscilloscope. Make sure the I / V offset is not set to an extreme value that saturates the amplifier at either the positive or negative supply voltage. Next the piezoelectric element should be driven by a triangle wave from an ordinary function generator at 15 to 30 Hz, with peak-to-peak amplitude up to 30 V. The photodiode signal should vary by 5%-50% (depending on the particular cell) as the PZT scans the laser across the absorption line. It is helpful to trigger the scope from the function-generator sync pulse or TTL output, or operate the scope in X-Y mode in order to obtain a stable display as the PZT drive is adjusted. When electrical tuning of the laser over the absorption line has been obtained, it is a good time to reexplore mechanical adjustments of the grating angle and diode drive currents. An absorption line or its neighbors, corresponding to different hyperfine levels of the ground state or different isotopes, recurs several times for nearby currents or grating angles. Also discontinuous steps of photodiode output occur across the oscilloscope trace. These steps correspond to transitions from one longitudinal external cavity mode to another. These mode hops may be as far as 8 GHz apart but nil1 exhibit somewhat random spacings as well as hysteresis. 1108
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I
133Cs (F = 3-
I
(b)
F')
I
I
FREQUENCY (MHz)
Fig. 13. Saturated absorption curves for (a) Rb and (b) Cs. The "Rb F=2-F' peaks are broader than the others because they were made with a different setup. The widths and relative heights are affected by beam alignment, beam intensities, electronic damping constants, and absorption cell pressure. These are only representative results.
K. MacAdam, A. Steinbach, and C. Wieman
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A l l c a ~ a c i t o rv a l u e s a r e I ” u F .
R
f
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i
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+?
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-15U
Fig. 14. Servolock circuit
D. Observing the saturated absorption The full saturated-absorption setup of Fig. 10 is required for a more detailed test of stability and tuning rates and for locking of the laser output frequency at the level of 1 MHz or better. When first observing a saturated absorption signal it is useful to block the nonoverlapped probe beam. The counter propagating saturating beam can easily be aligned to overlap the probe beam at a 1” intersection angle or smaller. The intensities of the beams are not important for initial adjustments, but typically only a small fraction of the laser output, less than a few percent should be used for the saturated absorption. Reflection from a microscope slide provides an ample intensity that will allow further attenuation by neutral density filters or exposed photographic film. When adequate pump and probe beam overlap has been obtained, small saturated absorption dips should become evident near the center of the absorption line (Fig. 12). They may be recognized unambiguously by their disappearance from the Doppler profile if the saturating beam is blocked. The height of the narrow dips may be maximized by adjusting the alignment. The width can be reduced by reducing the angle of intersection of the overlapped beams and by attenuating either or both beams to avoid power broadening. The triangle wave amplitude and dc offset can be adjusted to zoom in on a particular region of the scan. For more detailed observations it is helpful to unblock the second probe beam. This second probe beam is directed into a photodiode identical to the first and wired in parallel with reversed polarity. The two probe beams can easily be obtained by utilizing the reflections off both front and rear surfaces of a piece of 3/8-in.-thick transparent plastic or glass. When the two photodiodes are properly positioned, the differential output signal cancels the large and featureless Doppler profile of the absorption line and allows saturated absorption features from the first probe beam to appear on a nearly flat background. If the Doppler broadened absorption is observed but the saturated absorption 1109
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peaks cannot be seen, it often means that there is too much background gas in the vapor cell.
E. Saturated absorption patterns in Rb and Cs After saturated absorption peaks have been observed, one can compare the patterns to known hyperfine structures of the ground and excited states to assess the electrical tuning range possible without hopping external cavity modes and to establish the tuning direction. The widths and resolution of the saturated absorption peaks for a given resonance line will depend on electronic time constants, the triangle-wave frequency, and possibly on diode current, in addition to alignment and intensity factors noted above. Figure 13 shows several saturated absorption patterns in Rb (780 nm) and Cs (852 nm) vapors photographed from an oscilloscope. These may aid new users in finding their way. Note that the patterns contain both true Doppler-free peaks and crossover peaks,’ which occur at frequencies (vI+ v 2 ) / 2 for each pair of true peaks at frequency Y , and v2. The crossovers are often more intense than the true peaks.
F. Typical tuning rates observed by saturated absorption Tuning rates for the grating-feedback laser, operated with a single longitudinal mode of the external cavity, depend on geometrical, thermal, and electrical properties of the laser components. In particular, tilting the grating changes both the wavelength of light diffracted back to the diode and the length of the cavity. These two effects interact in determining the change of output frequency. Typical tuning rates for a 780-nm laser having a 20-mm cavity on an aluminum baseplate are: ( 1 ) diode drive current: 200 MHz/mA, ( 2 ) temperature change of diode: 4 GHzTC, ( 3 ) temperature change of baseplate (cavity length): 7 GHzPC, ( 4 ) grating angle change (80 pitch screw): 5 X lo6 MHz/turn, and ( 5 ) piezoelectric tuning: 1 GHz/V. K. MacAdam, A. Steinbach, and C. Wieman
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PHOTODIODE
I
I
Fig. 15. Electronic layout schematic. For operation without the servolock box, the ramp is connected directly to the PZT as shown by the dashed line.
excellent indicator of the magnitude and spectral characteristics of the compensated noise, out to the bandwidth of the servolock circuit. The drift rate of the unlocked laser is normally under 5 MHz/min when the system is properly stabilized, and this slow drift is eliminated by locking. The short-term jitter amplitude of the unlocked laser frequency is typically f3 MHz on a 1-s time scale if the laser is on a reasonably stable lab table. The short-term intensity variations are much smaller than 1%. When locked, the laser frequency is stabilized to 1 MHz or better. The locked diode laser described in this paper is well suited for studies of neutral-atom cooling and trapping, for which some elaborations of the servolock circuitry are desirable. A future paper will describe trapping of Rb and Cs atoms from a vapor cell in a user-manual style similar to that used here. ACKNOWLEDGMENTS
X. SERVOLOCKED OPERATION OF THE DIODE LASER For stabilized operation of the laser, it may be locked to either side of any of the sufficiently well-resolved saturated absorption peaks such as those shown in Fig. 12. A simple servolock circuit is given in Fig. 14. Figure 15 indicates how the lock box is connected to the other components. Locking is not difficult after a little practice, provided that the saturated absorption signals are not too noisy and the laser frequency jitter caused by environmental or electrical backgrounds is less than the saturated absorption linewidths. First, the laser is tuned to the desired hyperfine multiplet of saturated absorption peaks and the ramp gain and ramp offset are adjusted, both on the ramp generator and on the lock box, so that one can zoom in to the desired side of a particular peak simply by turning down the ramp gain on the lock box to zero. With feedback and output gain controls set at minimum and the laser tuned to the side of a peak, the error offset is adjusted to a value near 0 V, as observed on an oscilloscope. The feedback and output gains are then gradually increased until the circuit corrects for deviations from the desired lock point and thus holds the frequency on the side of the peak. If the servolock seems to “repel” the saturated absorption peak, the input invert switch is reversed to select the opposite slope. When the laser is properly locked, it should be possible to turn the feedback fully on and the output gain up to a point where the PZT begins to oscillate at about 1 kHz. The best operating point is just below the onset of oscillation. Locking is confirmed by noting that the setpoint, indicated by the level of the now flat saturated absorption signal on the oscilloscope, can be varied by the error offset control within a range from about 10% to 90% of the height of the selected peak without a noticeable change in the monitored error output. Independently, the error output can be varied over a wide range by the ramp offset control without affecting the locked level of the saturated absorption signal. When the laser is locked, environmental noise appears on the error output and error signal monitor instead of on the saturated absorption signal because the error output compensates for laser frequency variations that would otherwise occur. The error signal monitor thus becomes an 1110
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This work was supported by the NSF and ONR. We are indebted to many people who contributed ideas which have been incorporated into the present design. Much of the basic design work was carried out by Bill Swann, Kurt Gibble, and Pat Masterson. Steve Swartz, Jan Hall, and nearly every member of the Wieman group during the past several years have also provided valuable contributions. Melles Griot Inc. loaned us an excellent diode laser current supply which was used for part of this work. APPENDIX: PARTS AND SUPPLIERS 1. Collimating lens #1403-,108, $7S.00, f = 5 mm, numerical aperture 0.5, Rodenstock Precision Optics Inc., 4845 Colt Road, Rockford, IL 61109, (815) 874-8300 2. Sorbothane Pad P/N: C37,M)o, $49.95, Edmund Scientific, 101 E. Gloucester Pike, Barrington, NJ 080071380, (609) 573-6250 3. Photodiode Pin-1OD ( 1 cm2 active area), $55.25, United Detector Technology-Sensors, 12525 Chadron Avenue, Hawthorne, CA 90250, (213) 978-1150 x360 4. Kodak IR detection card R11-236, $49.50, Edmund Scientific, (address as above) 5. Hand held infrared viewer P/N 84499, $1195.00, FJW Optical Systems, Inc., 629 S. Vermont Street, Palatine, IL 60067-6949, (708) 358-2500 A less expensive alternative is to use a CCD surveillance camera. These can be purchased from many sources including home and office security companies, and discount department stores. For an adequate model, prices for a camera, lens, and monitor will range from $500 to over $1OOo. 6. Sharp Diode Laser LT025MD0, $170.85, wavelength 780 nm, Added Value Electronic Distributors, Inc. (local Sharp distributor), 4090 Youngfield Street, Wheatridge, CO 80033, (303) 422-1701 STC LT50A-034 laser diodes (STC was recently purchased by Northern Telecom), wavelength 852 nm. We have purchased these lasers for ~ $ 6 5 0from a German distributor: Laser 200 GMBH, Argelsrieder Feld 14, D8031 Werling, Germany 7. Minco Thermofoil Kapton Heater, Minco 8941 P/N HK5207R12SL12A, $23.50, # 10 PSA (Pressure sensitive K. MacAdam, A. Steinbach, and C. Wieman
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adhesive) sheet, $4.00, Minco Products, Inc., 7300 Commerce Lane, Minneapolis, MN 55432, (612) 571-3121 x3177 8. Diffraction grating 1200 l/mm, 500 nm blaze: P/N C43,005, $72.85, 750 nm blaze: P/N C43,210, $72.85, Edmund Scientific (address as above) 9. Thermistor, P/N 121-503JAJ-Q01, $8.25, Fenwall Electronics, 450 Fortune Blvd., Milford, MA 01757 (also available from electronics distributors) 10. PZT disk, P/N PE-8, $0.75 (Murata/Erie # 7BB27-4), All Electronics Corp., P.O. Box 567, Van Nuys, CA 91408, (818) 904-0524 11. Kinematic Mirror Mount Mod. MML, $52.00, Thorlabs, Inc., P.O. Box 366, Newton, NJ 07860, (201) 579-7227 12. Fine Adjustment Screw Mod., AJS-0.5, $30.00, Newport Corp., P.O. Box 8020, 18235 Mt. Baldy Circle, Fountain Valley, CA 92728-8020, (714) 963-9811 13. Thermoelectric cooler, 3 0 x 30 mm, #CP1.4-71045L, $19.00, MELCOR, 990 Spruce St., Trenton, NJ 08648, (609) 393-4178 14. Cesium and rubidium vapor cells. We have never used these cells, but this company has announced that they will sell low cost vapor cells to educational institutions. Environmental Optical Sensors, Inc., 3704 N. 26th St., Boulder, CO 80302, (303) 440-7786
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a’JILA Visiting Fellow 1991-1992. Permanent address: Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-0055. ’C. Wieman and L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Inst. 62, 1-20 (1991). *. C.I. Camparo, “The diode laser in atomic physics,” Phys. 26, 443477 (1985). ’R. Ludeke and E. P. HarrL, “Tunable GaAs laser in an external dispersive cavity,” Appl. Phys. Lett. 20, 499-500 (1972). 4MM. W. Fleming and A. Mooradian, “Spectral characteristics of external-cavity controlled semiconductor lasers,” IEEE J. Quantum Electron. QE-17, 44-59 (1981). ’Diode lasers that are supplied without an output window, or diodes whose hermetic package has been carefully opened, may be AR coated with SiO by the user who has suitable optical coating apparatus, and improved operation may result. See M. G. Boshier, D. Berkeland, E. A. Hinds, and V. Sandoghdar, “External-cavity frequency-stabilization of visible and infrared semiconductor lasers for high resolution spectroscopy,” Opt. Commun. 85, 355-359 (1991). ‘The user should be aware in evaluating gratings for use that the efficiency is highly sensitive to polarization. See E. G. Loewen, M. NeviAre, and D. Maystre, “Grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 271 1-2721 (1977). ’Steve Chu, Stanford Univ., private communication. ‘Steve Swartz, Univ. of Colorado, private communication. ’For a discussion of saturated absorption spectroscopy, see W. Demtroder, Laser Spectroscopy (Springer-Verlag, New York, 198 1 ).
@ 1992 American Association of Physics Teachers
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Inexpensive laser cooling and trapping experiment for undergraduate laboratories Carl Wieman and Gwenn Flowers Joint Institute for Laboratory Astrophysics and Department of Physics, University of Colorado, Bouldel; Colorado 80309
Sarah Gilbert National Institute of Standards and Technology, Boulder, Colorado 80303
(Received 14 July 1994; accepted 15 December 1994) We present detailed instructions for the construction and operation of an inexpensive apparatus for laser cooling and trapping of rubidium atoms. This apparatus allows one to use the light from low power diode lasers to produce a magneto-optical trap in a low pressure vapor cell. We present a design which has reduced the cost to less than $3000 and does not require any machining or glassblowing skills in the construction. It has the additional virtues that the alignment of the trapping laser beams is very easy, and the rubidium pressure is conveniently and rapidly controlled. These features make the trap simple and reliable to operate, and the trapped atoms can be easily seen and studied. With a few milliwatts of laser power we are able to trap 4x10’ atoms for 3.5 s in this apparatus. A step-by-step procedure is given for construction of the cell, setup of the optical system, and operation of the trap. A list of parts with prices and vendors is given in the Appendix. 0 1995 American Association of Physics Teachers.
I. INTRODUCTION Laser cooling and trapping of neutral atoms is a rapidly expanding area of physics research which has seen dramatic new developments over the last decade. These include the ability to cool atoms down to unprecedented kinetic temperatures (as low as 1 p K ) and to hold samples of a gas isolated in the middle of a vacuum system for many seconds. This unique new level of control of atomic motion is allowing researchers to probe the behavior of atoms in a whole new regime of matter. Undoubtedly one of the distinct appeals of this research is the leisurely and highly visible motion of the 317
laser cooled and trapped atoms. Because of this visual appeal and the current research excitement in this area, we felt that it was highly desirable to develop an atom trapping apparatus that could be incorporated into undergraduate laboratory classes. This paper presents a detailed discussion of how to build a simple and inexpensive atom trapping apparatus for atomic rubidium (Rb). Our principal goal was to develop an apparatus which could be built and operated reliably with minimal expense and technical support. In most respects, however, this trap’s performance is equal or superior to what is
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680 achieved with the “traditional” designs used in many research programs, and some innovations have advantages over these designs. Thus portions of this paper are likely to be of interest to the researcher already working (or considering working) in the field of laser trapping. This paper is written in the same style as the previous paper on grating-stabilized diode lasers and saturated absorption spectroscopy.’ It is intended to provide a “cookbook” discussion that will allow a relative novice to construct and operate an optical trap. This paper is essentially a continuation of the work presented in Ref. l, and thus the end of that paper is used as a starting point. Without further discussion, we assume that one has two diode lasers of the type discussed in Ref. 1 which produce 5 mW or greater of narrow band tunable light. A small fraction (-10%) of each laser beam is split off and sent to its saturated absorption spectrometer of the type discussed in Ref. 1. This allows for precise detection and control of the laser frequencies, which is essential for cooling and trapping. The remainder of this paper will discuss how to use the light from these lasers to trap Rb atoms. Section I1 provides a brief introduction to the relevant physics of the atom trap; Sec. 111 covers laser stabilization; Sec. IV explains the optical layout; Sec. V details the construction of the trapping cell; and Sec. VI discusses the operation of the trap, measurement of the number of trapped atoms, and measurement of the time the atoms remain in the trap. In the Appendix we provide a list of parts needed in the construction of the apparatus with prices and vendors. In its least expensive version the trapping apparatus, not including the lasers, costs less than $3000, with the ion pump responsible for half of the cost. 11. THEORY AND OVERVIEW
We will present a brief description of the relevant physics of the vapor cell magneto-optical trap. For more information, a relatively nontechnical discussion is given in Ref. 2, while more detailed discussions of the magneto-optical trap and the vapor cell trap can be found in Refs. 3 and 4,respectively.
A. Laser cooling The primary force used in laser cooling and trapping is the recoil when momentum is transferred from photons scattering off an atom. This radiation-pressure force is analogous to that applied to a bowling ball when it is bombarded by a stream of ping pong balls. The momentum kick that the atom receives from each scattered photon is quite small; a typical velocity change is about 1 cmis. However, by exciting a strong atomic transition, it is possible to scatter more than lo7 photons per second and produce large accelerations (104.g).The radiation-pressure force is controlled in such a way that it brings the atoms in a sample to a velocity near zero (“cooling”), and holds them at a particular point in space (“trapping”). The cooling is achieved by making the photon scattering rate velocity dependent using the Doppler e f f e ~ tThe . ~ basic principle is illustrated in Fig. 1. If an atom is moving in a laser beam, it will see the laser frequency ulasershifted by an amount (-V/C)~,,,~,where V is the velocity of the atom along the direction of the laser beam. If the laser frequency is below the atomic resonance frequency, the atom, as a result of this Doppler shift, will scatter photons at a higher rate if it is moving toward the laser beam (V negative), than if it is moving away. If laser beams impinge on the atom from all 318
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laser beam
I , force
Fig. 1. Graph of the atomic scattering rate versus laser frequency. As shown, a laser is tuned to a frequency below the peak of the resonance. Due to the Doppler shift, atoms moving in the direction opposite the laser beam will scatter photons at a higher rate than those moving in the same direction as the beam. This leads to a larger force on the counter propagating atoms.
six directions, the only remaining force on the atom is the velocity-dependent part which opposes the motion of the atoms. This provides strong damping of any atomic motion and cools the atomic vapor. This arran ement of laser fields is often known as “optical molasses.
,,k
B. Magneto-optical trap Although optical molasses will cool atoms, the atoms will still diffuse out of the region if there is no position dependence to the optical force. Position dependence can be introduced in a variety of ways. Here, we will only discuss how it is done in the “magneto-optical trap” (MOT), also known as the “Zeeman shift optical trap,” or “ZOT.” The positiondependent force is created by using appropriately polarized laser beams and by applying an inhomogeneous magnetic field to the trapping region. Through Zeeman shifts of the atomic energy levels, the magnetic field regulates the rate at which an atom in a particular position scatters photons from the various beams and thereby causes the atoms to be pushed to a particular point in space. In addition to holding the atoms in place, this greatly increases the atomic density since many atoms are pushed to the same position. Details of how the trapping works are rather complex for a real atom in three dimensions, so we will illustrate the basic principle using the simplified case shown in Fig. 2. In this simplified case we consider an atom with a J=O ground state and a J = l excited state, illuminated by circularly polarized beams of light coming from the left and the Wieman, Flowers, and Gilbert
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”Rb Energy Levels
E
3 2
1
5Pm
0
B
force +
6.0 force 0
780 nm hyperfine pumping diode laser
B>O t force
780 nm trapping diode laser 2
5%
Fig. 2. One dimensional explanation of the MOT. Circularly polarized laser beams with opposite angular momenta impinge on an atom from opposite directions. The lasers excite t h e J = O t o J = l transition. The laser beam from the right only excites to the m =-I excited state, and the laser from the left only excites to the rn = + 1 state. As an atom moves to the right or left, these levels are shifted by the magnetic field thereby affecting the respective photon scattering rates. The net result is a position-dependent force which pushes the atoms into the center.
Fig. 4. ”Rb energy level diagram showing the trapping and hyperfine pumping transitions. The atoms are observed by detecting the 780 nm fluorescence as they decay back to the ground state.
right. Because of its angular momentum, the beam from the left can only excite transitions to the m = + 1 state, while the beam from the right can only excite transitions to the m = - 1 state. The magnetic field is zero in the center, increases linearly in the positive x direction, and decreases linearly in the negative x direction. This field perturbs the energy levels so that the Am = + 1 transition shifts to lower frequency if the atom moves to the left of the origin, while the Am = - 1 transition shifts to higher frequency. If the laser frequency is below all the atomic transition frequencies and the atom is to the left of the origin, many photons are scattered from the (T’ laser beam, because it is close to resonance. The (T- laser beam from the right, however, is far from its resonance and scatters few photons. Thus the force from the scattered photons pushes the atom back to the zero of the magnetic field. If the atom moves to the right of the origin, exactly the opposite happens, and again the atom is pushed toward the center where the magnetic field is zero. Although it is somewhat more complicated to extend the analysis to three dimensions, experimentally it is simple, as shown in Fig. 3. As in optical molasses, laser beams illuminate the atom from all
six directions. Two symmetric magnetic field coils with oppositely directed currents create a magnetic field which is zero in the center and changes linearly along the x , y , and z axes. If the circular polarizations of the lasers are set correctly, a linear restoring force is produced in each direction. Damping in the trap is provided by the cooling forces discussed in Sec. I1 A. It is best to characterize the trap “depth” in terms of the maximum velocity that an atom can have and still be contained in the trap. This maximum velocity V,,, is typically a few times TA (Ti is the velocity at which the Doppler shift equals the natural linewidth r of the trapping transition, where A is the wavelength of the laser light). A much more complicated three-dimensional calculation using the appropriate angular momentum states for a real atom will give results which are qualitatively very similar to those provided by the above analysis i f (1) the atom is excited on a transition where the upper state total angular momentum is larger than that of the lower state (F-F’ = F + 1) and (2) V>(fiT/m)”2,where rn is the mass of the atom. This velocity is often known as the “Doppler limit” velocity.’ If the atoms are moving more slowly than this, “sub-Doppler” cooling and trapping processes become important, and the simple analysis can no longer be used.7 We will not discuss these processes here, but their primary effect is to increase the cooling and trapping forces for very slow atoms in the case of F+F+ 1 transitions.
1
0’
C. Rb vapor cell trap
0‘
0 Fig. 3. Schematic of the MOT. Lasers beams are incident from all six directions and have angular momenta as shown. Two coils with opposite currents produce a magnetic field which is zero in the middle and changes linearly along all three axes. 319
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We will now consider the specific case of Rb (Fig. 4). Essentially all the trapping and cooling is done by one laser which is tuned slightly (1-3 natural linewidths) to the low frequency side of the 5S,,,F= 2+5P,,,F’ =3 transition of ”Rb. (For simplicity we will only discuss trapping of this isotope. The other stable isotope, ”Rb, can be trapped equally well using its F = 3 4 F ‘ = 4 transition.) Unfortunately, about one excitation out of 1000 will cause the atom to decay to the F = l state instead of the F = 2 state. This takes the atom out of resonance with the trapping laser. Another laser (called the “hyperfine pumping laser”) is used to excite the atom from the 5s F = l to the 5 P F ‘ = l or 2 state, from which it can decay back to the 5s F = 2 state where it will again be excited by the trapping laser. Wieman, Flowers, and Gilbert
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682 In a vapor cell trap, the MOT is established in a low pressure cell containing a small amount of Rb vapor! The Rb atoms in the low energy tail (V
N ( ~ ) = N , ( -Ie - * ’ T ) ,
(1)
where T is the time constant for the trap to fill to its steady state value N o and is also the average time an atom will remain in the trap before it is knocked out by a collision. This time is just the inverse of the loss rate from the trap due to collisions. Under certain conditions, collisions between the trapped atoms can be important, but for conditions that are usually encountered, the loss rate will be dominated by collisions with the room temperature background gas. These “hot” background atoms and molecules (Rb and contaminants) have more than enough energy to knock atoms out of the trap. The time constant r can be expressed in terms of the cross sections a, densities n , and velocities of Rb and non-Rb components as 1/7=nRbuRbVRb+nnonunonvnon.
\ 10 t
0
Spectrometer
-
A/4 plates
mirrors
saturated absorption
hypertine pumping laser
(a)
A
v
A
(3)
(2)
The steady-state number of trapped atoms is that value for which the capture and loss rates of the trap are equal. The capture rate is simply given by the number of atoms which enter the trap volume (as defined by the overlap of the laser beams) with speeds less than V,,,. It is straightforward to show that this is proportional to the Rb density, (V,ax)4, and the surface areaA of the trap. When the background vapor is predominantly Rb, the loss and capture rates are both proportional to Rb pressure. In this case N o is simply4
Fig. 5. (a) Overall optical layout for laser trap experiment including both saturated absorption spectrometers. (b) Detail of how laser beams are sent through the trapping cell. To simplify the figure, the hi4 waveplates are not shown. Beam paths (1) and (2) are in the horizontal plane and beam (3) is angled down and is then reflected up through the bottom of the cell. The retrobeams are tilted slightly to avoid feedback to the diode laser.
matically affect the shape of the cloud of trapped atoms since it changes the shape of the trapping potential. However, these very differently shaped clouds will still have similar numbers where Vavg is (2kT/m)”’, the average velocity of the Rb of atoms until the alignment is changed enough to affect the atoms in the vapor. If the loss rate due to collisions with volume of the laser beam overlap. When this happens, the non-Rb background gas is significant, Eq. (3) must be muldamping in three dimensions is changed and the number of tiplied by the factor It RbaRbVavg/(nRbaRbVavg+nnonUnonvnon). trapped atoms will change dramatically. Of course, if the The densities are proportional to the respective partial prestrapping potential is changed enough that there is no potensures. Finally, if the loss rate is dominated by collisions with tial minimum (for example, the zero of the magnetic field is non-Rb background gas, the number of atoms in the trap will no longer within the region of overlap of the laser beams), be proportional to the Rb pressure divided by the non-Rb there will be no trapped atoms. However, as long as the pressure, but 7 will be independent of the Rb pressure. damping force remains the same, almost any potential miniAs a final note on the theory of trapping and cooling, we mum will have about the same number of atoms and trap emphasize certain qualitative features that are not initially lifetime. obvious. This trap is a highly overdamped system; hence damping effects are more important for determining trap perD. Overview of the trapping apparatus formance than is the trapping force. If this is kept in mind it is much easier to gain an intuitive understanding of the trap Figure 5(a) shows a general schematic of the trapping apbehavior. Because it is highly overdamped, the critical quanparatus. It consists of two diode lasers, two saturated absorptity V,, is determined almost entirely by the Doppler slowtion spectrometers, a trapping cell, and a variety of optics. The optical elements are lenses for expanding the laser ing which provides the damping. Also, the cross sections for beams, mirrors and beamsplitters for splitting and steering collisional loss are only very weakly dependent on the depth of the trap, and therefore the trap lifetime is usually quite the beams, and waveplates for controlling their polarizations. insensitive to everything except background pressure. As a To monitor the laser frequency, a small fraction of the output of each laser is split off and sent to a saturated absorption result of these two features, the number of atoms in the trap is very sensitive to laser beam diameter, power, and frespectrometer. An electronic error signal from the trapping quency, all of which affect the Doppler cooling and hence laser’s saturated absorption spectrometer is fed back to the laser to actively stabilize its frequency. The trapping cell is a V,, . However, the number of trapped atoms is insensitive to factors which primarily affect the trapping force but not small vacuum chamber with an ion vacuum pump, a Rb the damping, such as the magnetic field (stray or applied) source, and windows for transmitting the laser light. In the and the alignment and polarizations of the laser beams. For following sections we will discuss the various components of the apparatus and the operation of the trap. example, changing the alignment of the laser beams will dra(3)
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683 111. LASER STABILIZATION Reference 1 describes how to construct a diode laser system. As mentioned above, two lasers are needed for the trap. Since a few milliwatts of laser power are plenty for hypefine pumping (F= 1+F’= 1,2), the stronger should be used as the trapping laser (F= 2 4 F ’ = 3) if the two lasers have different powers. Clearly visible clouds of trapped atoms can be obtained with as little as 1.5 mW of trapping laser power (after the saturated absorption spectrometer pickoff). However, the number of trapped atoms is proporticnal to the laser power, so setting up and using the trap is much easier with at least 5 mW of laser power. The trapping laser must have an absolute frequency stability of a few megahertz. This normally requires one to actively eliminate fluctuations in the laser frequency, which are usually due to changes in the length of the laser cavity caused by mechanical vibrations or temperature drifts. This is accomplished by using the saturated absorption signal to detect the exact laser frequency and then using a feedback loop to hold the length of the laser cavity constant. Detailed discussions of the saturated absorption spectrometer, the servoloop circuit, and the procedure for locking the laser frequency are given in Ref. 1, along with examples of spectra obtained. After a little practice one can lock the laser frequency to the proper value within a few seconds. Here, we just mention a few potential problems and solutions. Check that the baseline on the saturated absorption spectrum well off resonance is not fluctuating by more than a few percent of the F=2+F’=3 peak height. If there are larger fluctuations, they are likely caused by light feeding back into the laser or by the probe beams vibrating across the surface of the photodiodes. These problems can normally be solved by changing the alignment and making sure that all the optical components are rigidly mounted. If the fluctuations in the unlocked laser frequency are large it will probably be necessary to improve the vibration isolation of the laser and/or table, or rebuild the laser to make it more mechanically stable. If the fluctuations are stable and synchronized with the 60 Hz line voltage, the problem is probably not mechanical, but rather is the diode laser power supply. Under quiet conditions and reasonably constant room temperature, the laser should stay locked for many minutes and sometimes even hours at a time. Bumping the table or the laser will likely knock the laser out of lock. If the laser frequency does not have excessively large fluctuations before it is locked but jumps out of lock very easily, it is likely that the resonant frequency of the servoloop is too low. This is determined by the response of the diffraction grating mount when driven by the PZT. The resonance limits the gain of the servo and reduces its ability to correct for large deviations. The frequency of this resonance can be determined simply by turning up the lock gain until the servosystem begins to oscillate and measuring the frequency of the sine wave which is observed. We have found that a reasonable value for the resonance frequency is around 1 kHz. If it is much lower than this, the diffraction grating is too massive, the mounting is not stiff enough, or the PZT has not been installed properly. Even under the best conditions, the system we have described is likely to have residual frequency fluctuations of around 1 MHz. The trap will work fine with this level of stability but these fluctuations will cause some noise in the fluorescence from the trapped atoms. For most experiments this is not serious, but it can limit some measurements. If one has a little knowledge of servosystems, it is quite straightfor321
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ward to construct a second feedback loop which adjusts the laser current. The current feedback loop should have a roll-on filter and high gain. (Because the laser current loop can be much faster than the PZT loop, it can have a much higher gain.) This results in the current loop feedback dominating for frequencies above a few hertz. However, the PZT loop gain is largest for very low frequencies, and thus handles the dc drifts. The combination of the two servoloops will make the laser frequency much more stable and will also make the lock extremely robust. We have had lasers with combined PZT and current servoloops remain frequency locked for days at a time and resist all but the most violent jarring of the laser. (A circuit diagram for the combined PZT and current servoloop can be obtained by sending a request and a self-addressed stamped envelope to C. Wieman.) This extra servoloop is a complication, however, which may not be desirable for an undergraduate laboratory. The requirements for the frequency stability of the hyperfine pumping laser are much less stringent than those for the trapping laser. For many situations it is adequate to simply set the frequency near the peak of the 5s F = 1+5P F’ = 2 transition by hand. Over time it will drift off, but if the room temperature does not vary too much, it will only be necessary to bring the laser back onto the peak by slightly adjusting the PZT offset control on the ramp box every 5 to 10 min. If better control is desired, this frequency can also be locked to the side of the F = 1 +F’ = 2 saturated absorption peak using the above procedure. However, it is often more convenient to lock the frequency of this laser to the peak of the F = 1 +F‘ = 1 , 2, and 3 Doppler broadened (and hence unresolved) absorption line by modulating the laser frequency and using phase-sensitive detection with a lock-in amplifier. The output of the lock-in amplifier is then fed back to the PZT to keep the laser frequency on the peak. This is an excessively expensive solution, unless lock-in amplifiers which are too old to use for anything else are available. These work very well for this purpose and this relatively crude frequency lock is adequate for the hyperfine pumping laser.
IV. OPTICAL SYSTEM With lasers that can be locked to the correct frequency, one is now ready to set up the optical system for the trap. The basic requirement is to send light beams from the trapping laser into the cell in such a way that the radiationpressure force has a component along all six directions. To motivate the discussion of our optical design for this student laboratory trap, we first mention the design used in most of the traps in our research programs. In these research traps, the light from the trapping laser first passes through an optical isolator and then through beam shaping optics that make the elliptical diode laser beam circular and expand it to between 1 and 1.5 cm in diameter. The beam is then split into three equal intensity beams using dielectric beam splitters. These three beams are circularly polarized with quarter-wave plates before they pass through the trapping cell where the intersect at right angles in the center of the cell. After leaving the trapping cell each beam goes through a second quarterwave plate and is then reflected back on itself with a mirror. This accomplishes the goal of having three orthogonal pairs of nearly counterpropagating beams, with the reflected beams having circular polarization opposite to the original beams. Wieman, Flowers, and Gilhert
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684 A. Geometry Here, we present a modified version of this setup, as shown in Fig. 5. A number of the expensive optical components (optical isolator, dielectric beam splitters, and large aperture high quality quarter-wave plates) have been eliminated from the research design. The light from the trapping diode laser is sent into a simple two-lens telescope which expands it to a 4.5 cmXl.5 cm ellipse. This is then split into three beams which are roughly square (1.5 cmX1.5 cm), simply by clipping off portions of the beam with mirrors which interrupt part of the beam, as shown. The operation of the trap is remarkably insensitive to the relative amounts of power in each of the beams; a factor of 2 difference causes little change in the number of trapped atoms. However, the beam size as set by the telescope is of some importance. The number of trapped atoms increases quite rapidly as the beam size increases and, as discussed below, the larger the beam, the less critical the alignment. However, if the beams are too large it is difficult to find optical elements to accommodate them and it becomes harder to see the beams due to the reduced intensity. Beams of 1.5 cm diameter work well and fit on standard 2.5 cm optics. With larger optical components it is possible to use larger beams. However, avoid using beams much smaller than 1.5 cm in diameter because of the decrease in the number of trapped atoms and the increased alignment sensitivity. Two of the three beams remain in a horizontal plane and are sent into the cell as shown in Fig. 5(b). The third is angled down and reflects up from the bottom of the cell. There is nothing special about the layout of the three beams we have shown here. The only requirement is that they all intersect in the cell at roughly right angles. There is a minor benefit in having all of the beams follow nearly equal path lengths to the cell to keep the sizes matched, if the light is not perfectly collimated. To simplify the adjustment of the polarizations, the light should be kept linearly polarized until it reaches the quarter-wave plates. As discussed in any introductory optics text, this will be the case as long as all the beams have their axes of polarization either parallel (p) or perpendicular (s) to the plane of incidence of each mirror. This is easy to achieve for the beams in the horizontal plane, but difficult for the beam that comes up through the bottom of the cell. However, with minimal effort to be close to this condition, the polarization will remain sufficiently well linearly polarized. It is easy to check the ellipticity of the polarization by using a photodiode and a rotatable linear polarizer. First, make sure that the polarizer works at 780 nm by looking at the extinction of the laser light by crossed polarizers; some plastic sheet polarizers do not work at 780 nm. An extinction ratio of 10 or greater on the polarization ellipse is adequate. After they pass through the cell, the beams are reflected approximately (but not exactly!) back on themselves. The reason for having an optical isolator in the research design is that even a small amount of laser light reflected back into the laser will dramatically shift the laser frequency and cause it to jump out of lock. In the absence of an optical isolator, this will always happen if the laser beams are reflected nearly back on themselves. Feedback can be avoided by insuring that the reflected beams are steered away from the incident beams so that they are spatially offset by many (5-10) beam diameters when they arrive back at the position of the laser. Fortunately, for operation of the trap the return beam need only overlap most of the incident beam in the cell, but its 322
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exact direction is unimportant. Thus by making the beams large and placing the retromirrors close to the trap (within 10 cm for example) it is possible to have the forward and backward going beams almost entirely overlap even when the angle between them is substantially different from 180". This design eliminates the need for the very expensive ($2500) optical isolator and, as an added benefit, makes the operation of the trap very insensitive to the alignment of the return beams. If the retro mirrors are close enough to the cell, they can be tilted by up to 30" without significantly changing the number of trapped atoms. It is easy to tell if feedback from the return beams is perturbing the laser by watching the signal from the saturated absorption spectrometer on the oscilloscope. If the amplitude of the fluctuations is affected by the alignment of the reflected beam or is reduced when the beam to the trapping cell is blocked, unwanted optical feedback is occurring.
B. Polarization The next task is to set the polarizations of the three incident beams. The orientations of the respective circular polarizations are determined by the orientation of the magnetic field gradient coils. The two transverse beams which propagate through the cell perpendicular to the coil axis should have the same circular polarization, while the beam which propagates along the axis should have the opposite circular polarization. Although in principle it is possible to initially determine and set all three polarizations correctly with respect to the field gradient, in practice it is much simpler to set the three polarizations relative to each other and then try both directions of current through the magnetic field coils to determine which sign of magnetic field gradient makes the trap work. To set the relative polarization of the three beams, first identify the same (fast or slow) axis of the three quarterwave (h/4) plates. For the two beams which are to have the same polarization, this axis is set at an angle of 45" clockwise with respect to the linear polarization axis when looking along the laser beam. For the axial beam, the axis is oriented at 45" counterclockwise with respect to the linear polarization. This orientation need only to be set to within about 210". The orientation of the hi4 plates through which the beams pass after they have gone through the trapping cell ("retro X/4 plates") is arbitrary. No matter what the orientations are, after the beams have passed though them twice, the light's angular momentum will be reversed. We have tried various options for wave plates. The most straightforward option is to simply use six commercial Xi4 plates which have a clear aperture large enough for the laser beams. Ideally, these should be antireflection coated to reduce the attenuation of the laser beams. The only disadvantage to this approach is the cost. There are many suppliers of h/4 plates, but a set of six antireflection coated wave plates will cost over $1000. An alternative to using expensive wave plates is to replace all or some of the commercial h/4 plates with inexpensive plastic sheet retardation plates, which are widely available in optics demonstration kits. We have replaced the 780 nm Xi4 plates with plastic sheet which was nominally Xi2 at 500 nm (and hence probably about hi3 at 780 nm) and observed little change in the number of trapped atoms. One difference in this case is that the trapping is more sensitive to the orientation of the wave plates and, in particular, is now somewhat sensitive to the orientation of the retro wave plates. Another option is to replace the three retro X/4 plates with retroreflecting right angle mirrors.* Although this Wiernan, Flowers, and Gilbert
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685 combination of mirrors does not provide an ideal X/2 retardance, we found empirically that for five different dielectric mirrors tested, it is quite close. For gold mirrors the retardance is much farther from h/2, but still adequate for good trapping. Putting the two right angle mirrors on separate mirror mounts gives more flexibility of adjustment, but is probably unnecessary. Simply gluing the two mirrors together at an angle of slightly more or less than 90" (to insure that the beams overlap in the cell, but the return beams does not go directly back into the laser) works fine. An added benefit of this approach is that two reflections off a mirror usually result in much less light loss than one reflection and two passes through a hi4 plate. Finally, there are other options for obtaining inexpensive retardation plates that are approximately Xi4 at 780 nm. Most transparent plastic sheet is birefringent, and often the retardance can be adjusted by stretching. If one simply tries a number of pieces of plastic sheet, it is likely that some can be found which are not too far from a hi4 retarder. The traditional way of making inexpensive X/4 plates is to split sheets of mica. We have not tried this, but it should work if the mica does not absorb too much light at 780 nm.
C. Hyperfine pumping laser optics Minimal optics are needed for the hyperfine pumping laser. Essentially the only requirement for this light is that it cover the region where the trapping laser beams overlap. In its simplest form this means merely sending the laser through a single lens to expand the beam and reflecting it off a mirror into the trapping cell through any window. In practice, it is often worth going to the small additional effort of using two lenses to make a large collimated beam (typically an ellipse with a 2-3 cm major axis) and send it into the cell from a direction which will minimize the scattered light into the detectors that observe the trapped atoms. The trap is insensitive to nearly everything about the hyperfine pumping light, including its polarization.
D. Mirrors and mirror mounts If low cost h/4 retarders are used, the cost of the trapping optics will be dominated by the ten mirrors and mirror mounts required. We have become accustomed to using convenient commercial kinematic mirror mounts (see the Appendix). Similar products are available from many manufacturers, but a price between $50 and $100 per mount is standard. However, the control and stability provided by such mounts are actually far superior to what is necessary in this application. Thus if one has severe budget limitations, simple homemade mounts with limited adjustment should be adequate. We use both gold and high reflectance infrared dielectric mirrors (aluminum is lossy at 780 nm). Although gold mirrors are adequate for any of the mirrors in the setup, we prefer to use the dielectric mirrors for the retroreflection, because their higher reflectance makes the trap more symmetric. Also, while gold mirrors are less expensive than the dielectric mirrors, they are more easily damaged. Gold mirrors with protective overcoatings are readily available, but in our limited experience, mirrors with overcoatings which rival the durability of a dielectric mirror end up costing about the same. In an environment where inexperienced students will be handling the mirrors frequently, inexpensive gold mirrors have a rather short lifetime, in contrast to the standard commercial dielectric coated mirrors. In the long run, we think that the least expensive option for mirrors is to buy 323
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large dielectric mirrors and cut them into smaller mirrors with a glass saw. (Put a protective cover such as plastic tape on the surface before cutting.) In the short run, however, it is less expensive to use gold mirrors. The surface quality and flatness of any commercial mirror will be more than adequate. Caution should be taken when using dielectric mirrors at an angle; there can be substantial variation of reflectivity with different incidence angles and polarizations.
V. TRAPPING CELL CONSTRUCTION The primary concern in the construction of the trapping cell is that ultrahigh vacuum (UHV) is required. Although trapped atoms can be observed at pressures of Pa (.=lo-' Torr), trap lifetimes long enough for most ex eri ments of interest require pressures in the to lo-' Pa range. Unfortunately, UHV often seems to be synonymous with high cost and specialized technical expertise. Consequently, we have put considerable effort into designing a simple, inexpensive system that can be constructed by someone who does not have experience with ultrahigh vacuum techniques. However, we will present a range of possibilities to best cover the needs of a variety of different users. There are three main elements in the trapping cell: (1) a pump to remove unwanted background gas-mostly water, hydrogen, and helium (helium can diffuse through glass); (2) a controllable source of Rb atoms; and (3) windows to transmit the laser light and allow observation of the trapped atoms. We will discuss each of these elements in turn, and then discuss in detail the actual construction of the cell.
A. Vacuum pump In all of our systems we have used small ion pumps (2, 8, or 11 //s). Ion pumps have the virtues that they are small, quiet, and light, use little power, require no cooling, and have low ultimate pressures. These features make them well suited to this application. The 2 //s pumps have been adequate for a clean system which is evacuated once and remains at high vacuum thereafter. However, in systems which have been cycled up to air a few times the performance of the 2 //s pumps has not been satisfactory. After very few pumpdowns, they become increasingly difficult to start and soon will not pump at all. We recommend an 8-12 //s pump if there is any possibility of letting the system up to air more than once. The power supply for a small ion pump is often more expensive than the pump itself. It is possible to avoid this expense by using a dc high voltage supply, if one is available. The pump will typically require a few tenths of 1 p A at 5-6 kV under normal operating conditions, and as much as a few milliamperes at lower voltage for the first few minutes when it is initially turned on. Any power source which will supply sufficient voltage and current will be adequate. It is quite convenient to be able to monitor the pump current as a measure of the pressure. After a few days of exposure to Rb vapor the pump is likely to have some leakage current for which a correction must be made when determing the pressure, but the leakage does not affect pump performance. These pumps have the minor drawback that they require a large magnetic field, which is provided by a permanent magnet. The fringing fields from this magnet can extend into the trap region and will affect the trap to some extent. Although the trap will usually work without it, we put a layer of magnetically permeable iron or steel sheet around the pump to Wieman, Flowers, and Gilbert
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686 shield the trap from this field.’ For this same reason it is advisable to avoid having magnetic bases very near the trap.
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B. Rb sources We will now discuss how to produce the correct pressure of Rb vapor in the cell. The vapor pressure of a room temperature sample of Rb is about 5X10-5 Pa. This is much higher than the to Pa to Torr) of Rb vapor pressure that is optimum for trapping. At higher pressures the trap will still work, but the atoms remain in the trap a very short time, and it is often difficult to see them because of the bright fluorescence from the untrapped background atoms. Also, the absorption of the trapping beams when passing through the cell will be significant. Thus we would like to maintain the Rb at well below its room temperature vapor pressure. Because it is necessary to continuously pump on the system to avoid the buildup of hydrogen and helium vapor, it is necessary to have a constant source of Rb to maintain the correct pressure. Before discussing specific ways to generate Rb in the vapor cell, we will present some back round on the behavior of Rb in the environment of the cell. This behavior is often counterintuitive. Through chemical reactions and physisorption, the walls of the cell usually remove far more Rb than the ion pump does and the rate of pumping by the walls depends on how well they are coated with Rb. If one has an evacuated stainless steel tube and connects a reservoir of room temperature Rb to one end, initially no Rb vapor will be observed at the other end. Over time the vapor will creep down the tube as it saturates the walls, and in a matter of several days, the pressure differential across the length of the tube will significantly decrease. If the Rb reservoir is then removed and the tube is connected to a pump, the vapor will linger for days as it slowly comes off the walls. If the tube is heated, the vapor will disappear much more quickly. A similar behavior occurs with glass, but the chemical reaction rates vary with different types of glass. With fused silica or Pyrex, the behavior is similar to that observed with stainless steel, and a similar saturation of the chemical reaction rate occurs. With other types of glass (microscope slides, for example) the reaction rate and thus the wall pumping remain large even after extensive exposure to Rb vapor. Since this rapidly depletes Rb and causes large pressure differentials in the cell, we do not recommend making a cell from this type of glass. Our recommended technique for producing the Rb vapor for a student lab experiment is to use a commercial “getter.” This technique was developed specifically for our student lab trap, and, to our knowledge, has not been used before in optical trapping cells. However, as we will discuss, it has advantages over other approaches and is likely to be useful in many research lab traps as well. The getter is several milligrams of a Rb compound which is contained in a small (l.OX0.2X0.2 cm) stainless steel oven. Two of these ovens are spotwelded onto two pins of a vacuum feedthrough as shown in Fig. 6. When current (3-5 A) is sent through the oven, Rb vapor is produced. The higher the current, the higher the Rb pressure in the chamber. With this system it is unnecessary to coat the entire surface with Rb, and is in fact undesirable since the getter is likely to be exhausted before the surface is entirely saturated. With a glass cell mounted on a 1.33 in. Conflat-type fiveway cross we are able to produce enough Rb pressure to easily see the background fluorescence with as little as 3.4 A through the getter oven. If any
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12 cm
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Fig. 6. Drawing of trapping cell. The tubes on the cross have been elongated in the drawing for ease of display. The Rb getter is inside a stainless steel vacuum tube; we show a cut-away view in this drawing. The zero length adapter is a 2.75 in. to 1.33 in. Conflat-type adapter.
fluorescence from the background can be seen, there is more than enough Rb for trapping. According to the data sheets, this is a negligible production rate of Rb, and hence the getter should last a long time. We have operated for 100 h with no sign of depletion of the Rb getter. The principal advantages of this approach are the simplicity of construction and the ability to quickly and easily adjust the Rb pressure. This is particularly important for a lab class experiment where the students will be able to work for only a few hours at a time. After the current through the getter has been turned on, the Rb vapor comes to an equilibrium pressure with a time constant of about 5 min. The overall pressure in the system rises slightly when the getter is first heated, but after degassing overnight by running a current just below that which produces observable Rb, the pressure rise is less than Pa, and continues to decrease with further use of the getter. When the current through the getter is turned off, the Rb pressure drops with a time constant of about 4 s if the getter has been on for only a short while. With prolonged use of the getter, this time constant can increase up to a few minutes as Rb builds up on the walls, but it decreases to the original value if the getter is left off for several days. Presumably this recovery would be much faster if the system was heated. The superior Rb pressure control provided by the getter makes it possible to use high pressures to easily observe fluorescence from the background vapor. This allows one both to check whether the laser and cell are operating properly, and to have rapid response in the number of atoms while optimizing trapping parameters. Once these tasks are complete, simply reducing the getter current provides low pressures almost immediately. Low pressures are desirable for many experiments because they yield relatively long trap lifetimes (seconds) and little background light from the fluorescence of untrapped atoms. We and others have used two alternative methods to produce Rb vapor. We mention the first only because it is used for many ordinary vapor cells, but is not a very good choice for this trap. In this approach a fraction of 1 g of Rb is distilled into the cell under vacuum and is condensed into a thermoelectrically cooled “cold finger.” The vapor pressure in the cell is then adjusted by changing the temperature of the cold finger and allowing the pressure in the remainder of the cell to come to equilibrium. We do not recommend this Wieman, Flowers, and Gilbert
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approach because the pressure can only be changed very slowly, and (most importantly!) we have found that it is easy to accidently turn off or bum out the thermoelectric cooler. When this happens the entire vacuum system is filled to a relatively high Rb pressure. This is hard on the ion pump, and it is usually difficult and tedious to bake all the Rb back into the cold finger. A better method is to have the Rb reservoir behind a small stainless steel valve. This approach has been used in most of our research work. A small sealed glass ampoule containing about 1 g of Rb is installed in a thin-walled stainless steel or copper tube. We usually distill the Rb into the glass ampoules ourselves to insure that the Rb does not contain other dissolved gases, but it is also possible to buy such ampoules commercially. One end of the metal tube is sealed, and the other end is welded or brazed onto a 1.33 in. Conflat-type flange which is attached to a valve connected to the main cell. After the system has been thoroughly pumped out and baked, the tube is squeezed in a locking pliers (vice grips) until the metal wall has compressed enough to break the glass and the Rb is released. To get the Rb through the valve and into the cell, we heat the tube and the valve to 60100 "C. The first time this is done it takes between several hours and several days to see a significant amount of Rb in the cell (depending on the geometry, the surfaces, and the temperature). Once the pressure in the cell reaches the desired level, the valve to the reservoir is closed. At this point, the vapor will slowly begin to leave the cell, and after some time (days to weeks), the heating process will need to be repeated. After the initial Rb coating of the inside of the cell, it only requires several (10-30) minutes to substantially raise the Rb pressure in the cell upon subsequent heating of the reservoir. The length of time between fills depends on the geometry of the system and the properties of the surfaces. For Pyrex and stainless steel systems it will typically be between a day and a few weeks. For systems with more reactive surfaces it can be just a few minutes, in which case one may simply leave the valve open while carrying out experiments.
C. Windows and cell assembly The third important element of the trap cell is the optical access. This can be either through mounted windows, or through the cell walls themselves if they are transparent and flat. Optical access is necessary for sending the laser beams through the cell and for observing the fluorescence from the trapped atoms. Although there are many ways to assemble a cell with windows, we will discuss the three that are most practical along with their respective advantages and disadvantages. A component common to these designs is the Conflat-type vacuum flange. These flanges are a means of making all-metal vacuum seals which are leakfree, have very low outgassing, and can be baked to high temperatures. The stainless steel flanges are bolted together with a soft copper gasket between them. Knife edges on the flanges press into the copper to make the vacuum seal. The first technique for making a trap cell is one which we have developed specifically for this paper. This design is shown in Fig. 6. Five pieces of plate glass are sealed together with epoxy. This unit is epoxied onto a commercial zerolength 2.75 in. to 1.33 in. Conflat-type adaptor flange which is bolted onto a fiveway cross with 1.33 in. Conflat-type flanges. The sixth window (the bottom window) is a commercial 1.33 in. Conflat-type viewing port. The viewing port, 325
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ion pump, pumpout valve, and Rb getter are attached to the other arms of the cross. The advantages of this design are (1) allows very good optical access and viewing due to the geometrical arrangement and the optical quality of the windows, (2) has relatively low cost, (3) it can be readily built by someone without any special skills in working with glass, and, of least importance, (4) allows the use of antireflection coated windows. Its principal disadvantage is that the ultimate pressure is limited by the temperature to which the epoxy can be baked. However, we have achieved pressures below Pa in this type of cell using a small ion pump. We will start by offering some advice on ultrahigh vacuum (UHV) techniques. We strongly recommend, however, that those not familiar with the subject also read about UHV techniques in one of the many reference books on vacuum technology. First, brand new commercial components such as the cross require no additional cleaning and should be kept in their protective wrapping until they are to be assembled. If the components are possibly dirty, they should be cleaned. Our standard cleaning procedure is to put the component in an ultrasonic cleaner with hot water and a good lab detergent for a few minutes. Then it is rinsed in turn with hot tap water, distilled water, and high purity alcohol. After drying in air it is wrapped in clean aluminum foil until it is to be used. Any surface which will be at high vacuum should never be touched with unprotected hands after it has been cleaned. If it must be touched, clean gloves should be used. It is also desirable to avoid having things like hair fall into the vacuum chamber, and it is always wise to carefully inspect the system for such items. Clean new copper gaskets should be used when making the Conflat-type seals, and the flanges should be tightened down uniformly. Graphite or some other high temperature antisiezing compound should be put on the bolts to prevent them from seizing up after they are baked. Also, we find that plate nuts make the assembly of Conflattype seals much easier. Finally, to reach UHV it is necessary to bake even the cleanest systems under vacuum to remove adsorbed water vapor from the surface. The glass part of the cell is a parallelepiped with its long axis oriented vertically. Typical dimensions of the vertical windows are about 2.75 cm by 4.5 cm, and the horizontal window on top is a square with side length 2.9 cm. The glass for the cell should be at least 2 mm thick; thinner glass may not be strong enough to withstand the pressure and much thicker glass can limit optical access. The windows should be cut to size with a standard glass saw. It is useful to cover the glass with masking tape while it is being cut to avoid scratching the surfaces. The precise dimensions of the glass pieces are not important, but it is important to have the cell fit together tightly to minimize the gaps that will be filled in with epoxy. To achieve this, one should make the opposite sides of the cell the same width, all four sides the same length, and all the corners of each piece perpendicular. To insure that the opposite sides are the same size, it is simplest to stack the two pieces and cut the stack as a single piece. After the pieces are cut, they should be washed carefully to remove any residual grit from the glass saw, and then the protective masking tape should be removed. The windows should then be cleaned just as the other vacuum components. The final assembly is done by first bolting the adaptor flange onto the fiveway cross, and then assembling the cell on top of it. It should be done in this order to avoid stressing the epoxy seals while bolting on the flange. Just before assembly, the inside surfaces of the five pieces of glass should Wieman, Flowers, and Gilbert
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688 be cleaned with lens tissue and pure alcohol or acetone to remove residual dust, which would scatter the laser light. Residual dust can be easily identified by looking for scattering while illuminating the window using a bright lamp. After the glass slides are clean they are pressed together to form a rectangular tube and placed upright on the flange. Two small rubber bands are placed around the tube to hold the four glass sides firmly pressed together in this position. The square window is then set on top of the piped and a small weight is put on it to hold everything in place while the epoxy is applied. The seams are then sealed with a bead of low-vapor-pressure epoxy. Warming the epoxy slightly makes it somewhat easier to apply, but the glass itself should not be heated. When the epoxy has hardened somewhat (30 min or so) the elastic bands can be cut off without putting any stress on the cell, and epoxy should then be applied to the seams which were covered by the bands. By assembling the cell in this manner, very little epoxy will be exposed to the vacuum system. The epoxy should be allowed to set for 24 h at room temperature. The rest of the vacuum components can now be bolted in place. There are two items which deserve special mention. First, the pump-out valve should be a bakeable all-metal-seal valve to meet the UHV requirements. For systems which are not going to be let up to air very often, this valve can be replaced by a copper pinch-off tube, which acts as an inexpensive "one use" valve. After the system has been rough pumped and the ion pump started, the tube is pinched off leaving a permanent seal. This works well and avoids the expense of a valve. However, if the system is likely to be opened up a few times or if there is a possibility of leaks in the system, a valve is much more convenient. Second, the Rb getter comes as a powder packed in a small stainless steel boat or oven. It is best to spot weld two of these boats in series and then spot weld them between two 10 A feedthroughs on a 1.33 in. Conflat-type flange. They should be positioned as close to the trapping region as possible without blocking the path of the vertical laser beam. Protective gloves should be used when installing the getter to avoid contaminating the system. Once the entire cell is assembled it is a good idea to perform a helium leak test if a leak detector is available. In the process, it is quite important to avoid contaminating the system with backstreaming of pump oil from the leak detector. If the leak detector does not have a liquid nitrogen cooled trap to prevent this, install one in the line connecting the trap cell to the leak detector.
D. System pumpdown and bakeout The next step is to pump the system down to a low enough pressure to start the ion pump. Use whatever pumping options are available; a clean cryogenic sorption pump is usually sufficient and will not contaminate the system. If the rough pumping is done with a mechanical or diffusion pump it must be very well trapped to avoid oil contamination, as with the leak detector. We have found, particularly with the 2 //s ion pumps, that minimizing the gas load when the ion pump is started improves the long term performance of the pump. If the system was not previously helium leak tested, it should be leak tested after the ion pump has been started and the system closed off from the roughing pump. This testing can be easily done by squirting alcohol on all of the epoxied and copper gasket seals while monitoring the ion pump current. If there is a sudden change in the current when alcohol 326
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is sprayed on the seal, it signifies a leak which has just been covered by the alcohol. Although it is important to leak test the system, it is unlikely there will be any leaks if the Conflat seals have been installed properly and care is taken to insure that the epoxy is applied evenly over all the joints. After confirming that there are no leaks, the system is then baked. One of the drawbacks to this design is the fact that it is quite easy to open up a leak in the glass-to-metal epoxy seal while baking. The reasons for this are the different thermal expansion coefficients of the glass and metal, and more importantly, the fact that the glass and metal tend to heat and cool at very different rates. This results in thermally induced stress which can break the rigid epoxy seal. This problem would be eliminated if a flexible sealant were used, but we have been unable to find one which satisfies the requirements of strength and low vapor pressure, and is also bakeable to 100 "C. After trying a number of baking methods which resulted in leaks, we developed the following procedure, which has not caused leaks the three times we used it. First, the entire system is put inside an oven with a bakeable cable connecting the ion pump to the power supply, which remains outside the oven. The ion pump current (and hence the system pressure) should be monitored while the system is baking. A standard kitchen oven is ideal for this bake. Once the system is in the oven, the temperature should be slowly increased. Normally, the rate at which the temperature can be increased is determined by the maximum operating current of the ion pump and the strong temperature dependence of the outgassing rate in the system. It is not unusual for it to take many (6-8) hours before the temperature can be increased up to the maximum baking temperature of 100 "C. If the epoxy has not been fully cured, it may be advisable to simply leave the system for a day at about 60 "C without attempting to increase the temperature further until the outgassing has dropped. Even if the pressure does not rise to undesirable levels, the temperature should not be changed faster than about 20 "C/h (up or down) to avoid thermal stresses. Once the temperature has been raised to 100 "C the system should be baked for at least a day. Then, it should be slowly cooled by progressively reducing the temperature of the oven. The pressure should drop dramatically as the cell cools and be less than Pa at room temperature. If the pressure in the system does not drop dramatically as it cools, or even increases with cooling, it almost always indicates a leak. The most likely place for a leak will be in the glass-tometal epoxy seal. If such a leak has opened, it can often be sealed by applying new epoxy to the offending region without ever letting the system up to air. This method of baking puts a fairly heavy load of gas into the ion pump. This can be reduced by a "prebake" of the system while it is on the roughing pump. With most facilities, however, it is awkward to insure proper uniform control over the temperature at the rough pumping stage. Once the cell has been baked it is ready to use. The ion pump can be turned off for long periods (up to a few days) without the pressure rising enough to cause problems. This is often convenient while the cell is moved and installed for use.
E. Alternative designs Before discussing the operation of the trap we will briefly mention two alternative methods of making trapping cells. The first is to have a glassblower make a six-sided glass cross with six windows on the tubes of the cross, and a glass-to-metal seal that allows the cell to be attached to the Wieman, Flowers, and Gilbert
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689 ion pump, pump-out valve, and Rb source. The advantages of this system are that it is simple, it can be baked to very high temperatures, and it has very little tendency to leak. The disadvantages are (1) not everyone has a glassblower available, (2) it is very difficult to modify anything about the trap cell, (3) such designs are often rather fragile, (4) the optical access and trap visibility are limited due to the width of the fused glass seams, and (5) the glassblowing often causes distortions and scattering centers in the windows. Perhaps the more serious aspect of (4) is that this often results in the windows being so far from the center of the trap (depending on the skill and effort of the glassblower) that the retroreflecting mirrors are far away from the middle of the cell. As discussed above, this renders the alignment more difficult and can make it very difficult to have adequate overlap of the forward and backward going beams without introducing an unacceptable amount of feedback into the diode laser. The second alternative is to buy a commercial 2.75 in. or 1.33 in. sixway cross with Conflat-type flanges (or a cube with six 2.75 in. Conflat-type seals), and six viewports (windows mounted on stainless steel Conflat-type flanges). Then one simply bolts five of the windows onto the sixway cross (or cube). A founvay cross is bolted on the sixth side which has the Rb reservoir and pump attached at right angles, and the sixth window attached to the port opposite the sixway cross. The advantages of this system are that all the components are readily available and can simply be bolted together. The principal disadvantages are the cost and the poor viewing and optical access. Also, metal scatters more light, causing a large background which can obscure the signal from the trapped atoms. These problems and the corresponding difficulty in optical alignment are so great in a sixway cross that we do not recommend it for use in an undergraduate lab. A cube is much better in this respect, but cubes with six Conflat-type seals are relatively expensive.
F. Installation on the optical breadboard After the cell has been baked, it is installed on the optical breadboard. This is simple; one or two ringstand clamps can hold the cell and ion pump in place. It is advantageous to align the six trapping beams before the cell is installed, because testing the overlap of the beams is much easier when the cell is not in the way. The cell must be mounted so that the beams overlap roughly in the center of the glass region and there is enough room below the bottom window to allow the vertical beam to be reflected up from below the cell. Remember also to leave room for the X/4 wave plate below the bottom window and room to adjust the orientation of the X/4 plate and the bottom mirror. Since the cell is elevated, it will also be necessary to support the ion pump since it is the heaviest part of the system. With the cell installed and the laser beams aligned, the final ingredients for trapping are the magnetic field coils. A gradient of up to about 0.20 T/m (20 G/cm; normal trap operation is at 10-15 Gicm) is needed. All other details of the coils are unimportant. We use two freestanding coils 1.3 cm in diameter with 25 turns each of 24 gauge magnet wire. This provides the desired gradients with a current of 2-3 A and a coil separation of 3.3 cm. The coils should be mounted on either side of the cell such that the current travels through the loops in opposite directions and the coil axis is collinear with one of the laser beam axes. Our coils are simply attached to the windows or supporting flange with tape or glue. 327
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VI. OPERATION OF THE TRAP AND MEASUREMENTS
A. Observation system Although not essential, it is highly desirable to have an inexpensive CCD TV camera and monitor with which to observe the trapping cell. Standard cameras used for security surveillance work well for this purpose. They will show the cloud of trapped atoms as a very bright white glow in the center of the cell. Because of the poor response of the eye at 780 nm, the trapped atoms can be seen by eye only if the room is quite dark. For aligning the trapping laser beams we usually use a standard IR fluorescent card or a piece of white paper if the room is darkened. (Note that although the laser beam appears weak, it is actually intense enough to cause damage if the beam shines directly into the eye.) A 1 cm2 photodiode with a simple current-to-voltage amplifier is used for making quantitative measurements on the trapped atoms. It is placed at any convenient position which is close to the trap, has an unobstructed view, and receives relatively little scattered light from the windows. The photodiode is used to detect the 780 nm fluorescence from the atoms as they spontaneously decay to the ground state from the 5P,,, level. This measurement can be quantified and used to determine the number of trapped atoms. The same or a similar photodiode can also be used to look at the absorption by the trapped atoms and to monitor the Rb pressure in the cell by measuring the absorption of a probe laser beam. Although the trap fluorescence is large enough to easily detect with the photodiode, it can be obscured by fluorescence from the background Rb vapor or by scattered light from the cell windows. Over a large range of pressures, the fluorescence from the background gas will be smaller than that from the trapped atoms. However, the scatter from the windows is likely to be significant under all conditions. We have found that with minimal effort to reduce it, the scattered light background will simply be a constant offset on the photodiode signal. For studying small numbers of atoms in the trap, however, the noise on this background can become a problem. In this case, one should use a lens to image the trap fluorescence onto a mask which blocks out the unwanted scattered light.
B. Trap operation When the cell and all the optics are in place, the first step is to turn on the getter to put Rb into the cell. Initially, monitor the absorption of a weak probe beam through the cell to determine the Rb pressure. Although the trap will operate over a wide range of pressures, a good starting point is to have about l%/cm absorption on the F = 2 to upper states transition in the region of the trap. At this pressure it is possible to see dim lines of fluorescence where the trapping beams pass through the cell when the trapping laser is tuned to one of the Rb transitions. It is often easier to identify this fluorescence by slowly scanning the laser frequency and looking for a change in the amount of light in the cell. While absorption measurements are valuable for the initial setup and for quantitative measurements, in the standard operation of the trap, use the setting on the getter current and/or the observation of the fluorescence to check that the pressure is reasonable. After an adequate Rb pressure has been detected, the magnetic field gradient is turned on and the lasers are set to the appropriate transitions. If the apparatus is being used for the Wieman, Flowers, and Gilbert
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690 first time, it will be necessary to try both directions of current through the field coils to determine the correct sign for trapping. The trapped atoms should appear as a small bright cloud, much brighter than the background fluorescence. Pieces of dust on the windows may appear nearly as bright, but they will be more localized and can be easily distinguished by the fact they do not change with the laser frequencies or magnetic field. The cloud may vary in size; it can be anywhere from less than 1 mm in diameter to several millimeters. Blocking any of the beams is also a simple method for distinguishing the trapped atoms from the background light. If the trap does not work (and the direction of magnetic field and the laser polarizations are set correctly) the lasers are probably not set on the correct transitions.
0
5
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Time (s)
C. Measurements Although many other more complicated measurements could be made with the trap, here we will discuss how to make the two simplest measurements: the number of trapped atoms and the time which atoms remain in the trap. Both of these measurements are made by observing the fluorescence from the trapped atoms with a photodiode. The number of atoms is determined by measuring the amount of light coming from the trapped atoms and dividing by the amount of light scattered per atom, which is calculated from the excited state lifetime. The time the atoms remain in the trap is found by observing the trap filling time and using Eq. (1). To make a reliable measurement of the number of trapped atoms, it is crucial to accurately separate the fluorescence of the trapped atoms from the scattered light and the fluorescence of the background vapor. To do this one must compare the signal difference between having the trap off and on. Therefore, the trap must be disabled in a way that has a negligibly small effect on the background light. We have found that turning off or, even better, reversing the magnetic field is usually the best way to do this. The magnetic field may alter the background fluorescence, but this change is generally much smaller (1/100) than the signal of a typical cloud of trapped atoms. Once one has determined the photocurrent due to the trapped atoms, the total amount of light emitted can be found using the photodiode calibration and calculating the detection solid angle. The rate R at which an individual atom scatters photons is given by”
(4) where I is the sum of the intensities of the six trapping beams, I- is the 6 MHz natural linewidth of the transition, A is the detuning of the laser frequency from resonance, and I , is the 4.1 mW/cm2 saturation intensity. The simplest way to find A is to ramp over the saturated absorption spectrum and, when looking at the locking error signal, find the position of the lock point (zero crossing point) relative to the peak of the line. The frequency scale for the ramp can be determined using the known spacing between two hyperfine peaks. A typical number for R is 6X106 photons/(s.atom). One can optimize the number of atoms in the trap by adjusting the position of the magnetic field coils, the size of the gradient, the frequencies of both trapping and hyperfine pumping lasers, the beam alignments, and the polarization of the beams. With 7 mW from the trapping laser, we obtain nearly 4 X lo7 trapped atoms when the Rb pressure is large enough to dominate the lifetime. 328
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Fig. 7. Number of trapped atoms versus time after the trap is turned on at 1.2 s.
The filling of the trap can also be observed using the same photodiode signal. This is best done by suddenly turning on the current to the field coils to produce a trap. The fluorescence signal from the photodiode will then follow the dependence given by Eq. (l), as shown in Fig. 7. The value of the trap lifetime 7 (the characteristic time an atom remains trapped) can then be determined from this curve. We have observed lifetimes which ranged from nearly 4 s down to a small fraction of 1 s depending on the Rb pressure. There are many other experiments which can be done with the trapped atoms. The choices are only limited by one’s imagination and/or available equipment, but here we list a few examples. Although optical traps are being used in a number of experiments, many aspects of the trap performance have not been studied. By changing the current through the getter, one can vary the Rb pressure and see how it affects the number and lifetime of the trapped atoms. This can be compared with the predictions of Eqs. (1) and (2) and can be used to determine collision cross sections; to date, very few trap ejection cross sections have been measured. One can also measure the temperature of the trapped atoms, the spring constant of the trap, and how the number of atoms and the lifetime depend on the various parameters. To our knowledge the detailed dependence of these characteristics on the polarizations of the laser beams, the intensity in the various beams, or the magnetic field gradient have not been studied. You can quickly learn that, in fact, the trap will work well with one of the incident beams linearly polarized, and can even work marginally with two linearly polarized beams, but we know of no analysis or studies of these cases in the literature. It is also relatively easy to do various kinds of high resolution spectroscopy on the trapped atoms because the optical thickness of the trapped atom cloud is substantial and the Doppler shifts are essentially zero. Using another diode laser one can precisely measure the 5S-5P spectra with the hyperfine components clearly resolved. In addition, there are convenient transitions which allow one to use diode lasers to study excitation from the 5 P state to higher levels.” It is also possible to observe very high resolution microwave transitions between the hyperfine states if a suitable source of microwave power is a ~ a i l a b l e . ’ ~ By turning the laser light off and adding small magnetic fields, it is possible to magnetically trap4 or “bounce” the Wieman, Flowers, and Gilbert
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69 1 laser-cooled atoms off of an inhomogeneous magnetic field. This dramatically demonstrates the quantization of angular momentum because the different Zeeman levels have different magnetic moments, and hence feel different magnetic forces. This causes the cloud to separate into a number of smaller clouds as it bounces, each of which represent a different projection of the angular momentum on the magnetic field. We conclude with both an invitation and a warning about many experiments one can do with optical traps: this is a new and rapidly changing field; one is likely to observe phenomena which are not explained in the current research literature and there are no textbooks to provide answers. The student and instructor may find themselves in uncharted territory.
ACKNOWLEDGMENTS This work was supported by the NSF and the University of Colorado. Early design work on this experiment was carried out by K. MacAdam, A. Steinbach, and C. Sackett. We thank M. Stephens for considerable help and advice. G. Flowers acknowledges the support of a tuition scholarship from the NIST PREP Fellowship program.
APPENDIX: COMPONENTS USED IN TRAP CONSTRUCTION Listed here are the components that we used in construction of the trap and the suppliers of these components. Commercial trade names are identified to make this article useful to the reader. Since we have not made a careful study of other products, we list only those that we actually used; the prices given are current as of 1993 and are meant for guidance only. No endorsement by NIST or the University of Colorado is implied; similar products by other manufacturers may work as well or better. Buyer’s guides such as those published by Physics Today (with the August issue), Photonics Spectra, and Laser Focus World are good sources for listings of vendors. Optical Base # BAlS, 16 @ $8 each, Thorlabs, Inc.. P. 0. Box 366, Newton, NJ 07860, (201) 579-7227. Post holder # PH3-ST, 16 @ $12 each, Thorlabs, Inc. (see item 1). Posts # TR2, 3 @ $5 each; # TR3,4 (@ $6 each; # TR4 1 @ $7; # TR6 14 @ $8 each; # TR10 1 @ $10, Thorlabs, Inc. (see item 1). Kinematic mirror mounts, # KM-B, nine @ $50 each; # KMS, one @ $41, Thorlabs, Inc. (see item 1). Right angle post clamps # R A Y O , seven @ $12 each, Thorlabs, Inc. (see item 1 ) . Commercial grade quarter-wave plates # RCQ-1 .00780, 25.4 mm diameter, six @ $184 each (additional charge for antireflection coating). Meadowlark Optics, Inc., 7460 Weld County Road # 1, Longmont, CO 80504, (303) 776-4068. As noted in the text, there are other options for wave plates. These wave plates are destroyed by mild heating. Near-infrared dielectric mirror, 50.8 mm diameter, to be cut in quarters for four mirrors. Coating BD.2, Substrate Code 20D04, $243, Newport Corporation, 1791 Deere Ave., Irvine, CA 92714, (800) 222-6440. Flat gold mirror # G41, 801, 76x102 mm to be cut into six 25x25 mm square mirrors and one 25x50 Am. J. Phys., Vol. 63, No. 4, April 1995
mm mirror, $22, Edmund Scientific Co., 101 East Gloucester Pike, Barrington, NJ 08007-1380, (609) 573-6250. Rb metal getter Rb/NF/3.4/12/FT 10+10 Code 580125, $2.50, S A W Getters USA, Inc., 1122 E. Cheyenne Mtn. Blvd., Colorado Springs, CO 80906, (719) 576-3200. Eleven /ils ion pump with magnet and bakeable cable, $880, Duniway Stockroom Corp., 1600 Shoreline Blvd., Mountain View, CA 94043, (800) 446-8811; 2 //s ion pump with square magnet and bakeable cable, $890, Perkin Elmer Physical Electronics Division, 6509 Flying Cloud Drivc, Eden Praire, MN 55344, (612) 828-6100. High voltage ion pump power supply # 222-0242 Ionpak series 240, $550, Perkin Elmer, Physical Electronics Division, 6509 Flying Cloud Drive, Eden Praire, MN, 55344, (612) 828-6100. Reducing Flange, 2.75 in. to 1.33 in. Conflat-type # 150001, $55, MDC Vacuum Products Corp., 23842 Cabot Blvd., Hayward, CA 94545-1651, (800) 4438817. Fiveway cross with 1.33 in. Conflat-type flange #406000, $160, MDC Vacuum Products Corp. (see above). Vacuum 1.33 in. Conflat-type viewport # VP-133-075, $91, Duniway Stockroom Corp., 1600 Shoreline Blvd., Mountain View, CA 94043, (800) 446-8811. Power feedthrough for Rb getter, Moly 10 NPin # EFT0024032, $125, Kurt J. Lesker Co., 1515 Worthington Ave., Clairton, PA 15025, (800) 245-1656. Stainless steel valve. Nupro valve (part SS-4BG-TW) which HPS buys and welds 1.33 in. Conflat-type flanges (HPS # 100881023) on each side and sells for $229. HPS, Division of MKS Instruments, 5330 Sterling Drive, Boulder, CO 80301, (303) 449-9861. These valves have a tendency to leak unless considerable torque has been used to tighten the bonnet seal. Copper Adapter #953-0706 (copper pinch-off tube as alternative to valve), $131 for pkg. of 4, Varian Vacuum Products, 121 Hartwell Ave., Lexington, MA 02173-9856, (800) 882-7426. Pyrex plate window #Q720125, 50.8X50.8X3.18 mm, $12 each, ESCO Products, Inc., 171 Oak Ridge Rd., Oak Ridge, NJ 07438-0155, (201) 697-3700. Torr Seal Epoxy (low vapor pressure epoxy) #9530001, $41, Varian Vacuum Products (see item 17). Transmissive IR Viewing Card Model #Q-11-T, $79.20, Quantex, 2 Research Ct., Rockville, MD 20850, (301) 258-2701. A transmissive (as opposed to reflective) viewing card is particularly useful for aligning laser beams. Photodiode #PIN-10 DP, $55.00, United Detector Technology, 12525 Chadron Avenue, Hawthorne, CA 90250, (310) 978-0516. ‘K. B. MacAdam, A. Steinbach, and C. Wieman, “A narrow-band tunable diode laser system with grating feedback and a saturated absorption spectrometer for Cs and Rb,” Am. J. Phys. 60, 1098-1111 (1992). I S . L. Gilbert and C. E. Wieman, “Laser cooling and trapping for the masses,” Optics and Photonics News 4, No. 7, 8-14 (1993). ‘The initial demonstration and discussion is E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631-2634 (1987). More detailed analysis of the trap is given in A. M. Steane, M. Chowdhury, and C . .I. Wieman, Flowers, and Gilbert
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Foot, “Radiation force in the magneto-optical trap,” J. Opt. SOC.Am. B 9, 2142 (1992), while a discussion of many of the novel aspects of the behavior of atoms in the trap is given in D. Sesko, T. Walker, and C. Wieman, “Behavior of neutral atoms in a spontaneous force trap,” J. Opt. SOC. Am. B 8, 946-958 (1991). 4C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571-1574 (1990); K. Lindquist, M. Stephens, and C. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46, 4082-4090 (1992). 5T. W. Hansch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68-69 (1975). ‘S. Chu, L. Hollberg, J. Bjorkholm, A. Cable, and A. Ashkin, “Threedimensional viscous confinement and cooling of atoms by resonance radiation pressure,” Phys. Rev. Lett. 55, 48-51 (1985). ’J. Dalibard and C. Cohen-Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: Simple theoretical models,” J. Opt. SOC. Am. B 6, 2023-2045 (1989); P. Ungar, D. Weiss, E. Riis, and S. Chu, “Optical molasses and multilevel atoms: Theory,” .I. Opt. SOC.Am. B 6, 2058-2072 (1989). ‘L. Orozco, SUNY at Stonybrook (private communication).
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’Prevent the shielding from coming into direct contact with the magnet. This can be done by placing small cardboard spacers between the shield and the magnet. ‘OM. Stephens and C. Wieman, “High collection efficiency in a laser trap,” Phys. Rev. Lett. 72, 3787-3790 (1994); M. Stephens, R. Rhodes and C. Wieman, “A study of wall coatings for vapor-cell laser traps,” J. Appl. Phys. 76, 3479-3488 (1994). ”For a discussion of saturated transition rates, see W. Demtroder, Laser Spectroscopy (Springer, New York, 1982), pp. 105-106. ”R. W. Fox, S. L. Gilbert, L. Hollberg, J. H. Marquardt, and H. G. Robinson, “Optical probing of cold trapped atoms,” Opt. Lett. 18, 1456-1458 (1993); A. G. Sinclair, B. D. McDonald, E. Riis, and G. Duxbury, “Double resonance spectroscopy of laser-cooled Rb atoms,” Opt. Commun. 106, 207-212 (1994); T. T. Grove, V. Sanchez-Villicana, B. C. Duncan, S. Maleki, and P. L. Gould, “Two-photon two-color diode laser spectroscopy of the Rb 5D,,, state,” Phys. Scr. I3C. Monroe, H. Robinson, and C. Wieman, “Observation of the cesium clack transition using laser cooled atoms in a vapor cell,” Opt. Lett. 16, 50-52 (1991). This work was done using trapped cesium atoms.
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Roger H. Stuewer, Editor School of Physics and Astronomy, 116 Church Street University of Minnesota, Minneapolis, Minnesota 55455 This is one of a series of Resource Letters on different topics intended to guide college physicists. astronomers, and other scientists to some of the literature and other teaching aids that may help improve course content in specified fields. [The letter E after an item indicates elementary level or material of general interest to persons becoming informed in the field. The letter I, for intermediate level, indicates material of somewhat more specialized nature; and the letter A, indicates rather specialized or advanced material.] No Resource letter is meant to be exhaustive and complete; in timc there may be more than one letter on some of the main subjects of interest. Comments on these materials as well as suggestions for future topics will be welcomed. Please send such communications to Professor Roger H. Stuewer, Editor, AAPT Resource Letters, School of Physics and Astronomy, I 1 6 Church Street SE, University of Minnesota, Minneapolis, MN 55455.
Resource Letter TNA-1: Trapping of neutral atoms N. R. Newbury and C. Wieman Joint Institute for Laboratory Astrophysics, National Institute of Standards and Technology and University of Colorado, Boulder; Colorado, 80309-0440
(Received 25 April 1995; accepted 23 August 1995) This Resource Letter provides a guide to the literature on trapping of neutral atoms. 0 1996 American Association of Physics Teachers.
In the past few years there has been spectacular progress in the trapping of neutral atoms. These traps use electromagnetic fields in various forms to contain atomic vapors and prevent them from coming into contact with material walls. This Resource Letter is intended to provide a useful list of references on this field of neutral atom trapping. Such trapping has been accomplished by using two types of fields: (1) rapidly oscillating optical fields from lasers and (2) static or slowly varying magnetic fields. Laser traps can be divided into two groups: dipole-force traps and spontaneous-force traps. The dipole-force traps work on the principle of an induced oscillating dipole moment in the atom that interacts with a spatially inhomogeneous laser field. This approach can also trap macroscopic chunks of matter such as a dielectric ball or a microbe and has been applied to a very diverse range of systems. We limit our coverage of dipole traps to the case of independent atoms. The most common type of neutral-atom trap is the spontaneous-force optical trap, which is based on the momentum transferred when laser light scatters o f f an atom by the process of excitation and reradiation. By far the most common type of spontaneous-force trap is the magneto-optic trap or MOT (also known as the ZOT, for Zeeman shift optical trap) that uses inhomogeneous magnetic fields and laser polarization to control the spontaneous force to produce the desired trapping potential. Since neutral-atom traps are typically very shallow, all traps also employ some method of cooling the atoms to the very cold temperature of 1 K (Kelvin) or less. Except for the case of hydrogen traps, this cooling is accomplished through the use of lasers. We have not included references that deal exclusively with laser cooling, but discussions of such cooling are included in some of the articles on trapping. In particular, in spontaneous-force traps, both the cooling and trapping forces arise from the scattering and reemission of the laser photons and hence are intimately connected, The purely magnetic traps are based on the interaction of 18
the magnetic moment of the atom and an inhomogeneous magnetic field. Most magnetic traps use dc magnetic fields, but a few have been built with ac magnetic fields that vary slowly relative to any internal time scale of the atom, but rapidly compared to the external motion. The number of applications of neutral-atom traps is growing rapidly. Probably the two largest areas of use are precision spectroscopy, including atomic clocks, and the study of cold-atom collisions. We have listed only the first few papers that were published in these areas and have not attempted to cover the large current literature in these subjects. Finally, we should note that the field of neutral-atom trapping itself is changing rapidly. We have no doubt overlooked some important papers in the field, and there are likely to be others that have been published since this Resource Letter was written.
I. JOURNALS THAT HAVE ARTICLES ON SUBJECT The first five journals listed contain the most articles on neutral atom traps. However, the remaining journals often contain longer and more general articles on traps. Physical Review Letters Physical Review A Optics Letters Journal of the Optical Society of America Europhysics Letters Science Nature Physics Today Physics Reports Scientific American American Journal of Physics Optical Communication Journal of Physics B
Am. J. Phys. 64 (l),January 1996
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694 11. BOOKS 1. Proceedings, Enrico Fermi International S u m m e r School on Laser Manipulation of Atoms and Ions, Varcnna, Italy, edited by E. Arimondo, W. Phillips, and F. Strumia (North Holland, Amsterdam, 1992). This book has several lengthy articles on neutral-atom traps of various types. (1)
1x1. REVIEW ARTICLES Below we list a number of useful review articles on trapping of alkali-metal atoms. Typically these articles also contain discussions of laser cooling of atoms. 2. “Cooling and Trapping Atoms,” W. D. Phillips and H. F. Metcalf, Sci. Am. 256 (3), 36-42 (1987). (E) 3. “Laser Cooling and Trapping,” S. Stcnholm, Eur. 1. Phys. 9, 242-249 (1988). (I) 4. “Cooling, Stopping and Trapping Atoms,” W. D. Phillips, P. L. Could, and P. 0 . Lett., Science 239, 877-883 (1988). (E) 5. “New Mechanisms for Laser Cooling,” C. Cohen-Tannoudji and W. D. Phillips, Phys. Today 33-40 (Oct. 1990). (E) 6. “Laser Manipulation of Atoms and Particles.” S. Chu, Science 253, 861-866 (1991). (E) 7. “Laser Trapping of Neutral Particles,” S. Chu, Sci. Am. 266 (2), 70-76 (1992). (E) 8. “Manipulation of Neutral Atoms. Experiments,” A. Aspect, Phys. Rep. 219, 141-152 (1992). (I) 9. “Laser Cooling and Trapping of Neutral Atoms: Theory,” C. CohenTannoudji, Phys. Rep. 219, 153-164 (1992). (I) 10. “Laser Cooling and Trapping for the Masses,” S. Gilbert and C. Wieman, Opt. Photon. News 4 (7), 8-14 (1993). (E) 11. “Magneto-Optical Trapping of Atoms,” H. Wallis, 1. Werner, and W. Ertnier, Comments At. Mol. Phys. 28 (51, 275-300 (1993). ( I ) 12. “Cooling and Trapping of Neutral Atoms,” H. Metcalf and P. van der Straten, Phys. Rep. 244, 203-286 (1994). (I)
IV. PARTICULAR SUBJECT REFERENCES The standard dipole-force trap is simply a focused laser beam tuned below the nearest atomic transition. The resulting trapping potential can be described equivalently in terms of the ac Stark shift of the atomic-energy levels, the interaction of an induced atomic-dipole moment with the electric field of the laser, or as energy shifts of the dressed atom (see Ref. 13).
A. Dipole-force traps 13. “Motion of Atoms in a Radiation Trap,” J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606-1617 (1980). (A) 14. “Dressed-Atom Approach to Atomic Motion in Laser Light: The Dipole Force Revisited,” J. Dalibard and C. Cohen-’I’dnnoudji, J. Opt. Soc. Am. B 2, 1707-1720 (1985). (VA) 15. “Experimental Observation of Optically Trapped Atoms,” S. Chu, J . E. Bjorkholm, A. Ashkin, and A. Cable, Phys. Rev. Lett. 57, 314-317 (1986). (I) 16. “Trapping of Neutral Atoms With Resonant Microwave Radiation,” C. C. Agosta, 1. F. Silvers, H. T. C. Stoof, and B. J. Verhaar, Phys. Rev. Lett. 62, 2361-2364 (1989). (I) 17. “Far-Off-Resonance Optical Trapping of Atoms,” J . D. Miller, R. A. Cline, and D. J. Heinzen, Phys. Rev. A 47, R4567-4570 (1993). (I) 18. “Demonstration of Neutral Atom Trapping With Microwaves,” R. J . C. Spreeuw, C. Gerz, L. S. Goldner, W. D. Phillips, S. L. Rolston, C. I. Westbrook, M. W. Reynolds, and 1. F. Silvera, Phys. Rev. Lett. 72, 3162-3165 (1994). (I) 19. “Spin Relaxation of Optically Trapped Atoms by Light Scattering,” R. A. Cline, J. D. Miller, M. R. Matthews, and D. J. Heinzen, Opt. Lett. 19, 207-209 (1994). (I) 20. “Q~iasi-ElectrostaticTrap for Neutral Atoms,” T. Takekoshi, J. R. Ych, and R. J. Knize, Optics Comm. (in press, 1994). (I) 21. “Evaporative Cooling in a Crossed Dipole Trap,” C. S. Adams, H. J. Lee, N. Davidson, M. Kasevich, and S. Chu, Phys. Rev. Lett. 74, 35773580 (1995). (I) 19
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22. “Long Atomic Coherence Times in an Optical Dipole Trap,” N. Davidson, H. J . Lee, C. S. AddmS, M. Kasevich, and S. Chu, Phys. Rev. Lett. 74, 1311-1314 (1955). (I)
B. Magneto-optical traps (MOTs) The first magneto-optical trap (Ref. 22) demonstrated that by combining magnetic-field gradients and circularly polarized counter propagating laser beams, atoms could be cooled and trapped. Subsequently, there has been an enormous effort at understanding, simplifying, and improving the basic magneto-optical trap. By now the magneto-optical trap is the initial step in most experiments on laser cooling and trapping of alkali atoms. In Subsection 1 we list general papers describing the construction and behavior of a MOT. Most often MOTs are used with alkali-metal atoms. In Subsection 2 we list papers describing trapping of alkaline-earth and noblegas atoms. Finally in Subsection 3 we reference some interesting variants on the standard MOT. We have not attempted to reference the enormous literature discussing collisions of atoms within a MOT.
I . General 23. “Stability of Radiation-Pressure Particle Traps: An Optical Earnshaw Theorem,” A. Ashkin and J. P. Gordon, Opt. Lett. 8, 511-513 (1983). (1) 24. ”Light Traps Using Spontaneous Forces,” D. Pritchard, E. Raab, V. Bagnnto, R. Watts, and C . Wieman, Phys. Rev. Lett. 57, 310-313 (1086). (I) 25. “Trapping of Neutral Sodium Atoms With Radiation Pressure,” E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, Phys. Rev. Lett. 5Y, 2631-2634 (1987). ( I ) 26. “Collisional Losses From a Light-Force Atom Trap,” D. Sesko, T. Walker, C. Monroe, A. Gallagher, and C. Wieman, Phys. Rev. Lett. 63, 961-Yh4 (1Y89). (I) 27. “Very Cold Trapped Atoms in a Vapor Cell,” C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571-1574 (1990). (I) 28. “Observations of Sodium Atoms in a Magnetic Molasses Trap Loaded by a Continuous Uncooled Source,” A. Cable, M. Prentiss, and N. P. Bigelow, Opt. Lett. 15, 507-509 (1990). (I) 29. “Collective Behavior of Optically Trapped Neutral Atoms,” T. Walker, D. Scsko, and C. Wieman, Phys. Rev. Lett. 64,408-411 (1990). (I) 30. “Behavior of Neutral Atoms in a spontaneous Force Trap,” D. Sesko, T. Walker, and C. Wieman, J . Opt. Sac. Am. B 8, 946-958 (1991). (1) 31. “Laser Cooling Below thc Doppler Limit in a Magneto-Optical Trap,” A. Steanc and C. Foot, Europhys. Lett. 14, 231-236 (1991). (A) 32. “Experimental and Theoretical Study of the Vapor-Cell Zeeman Optical Trap,” K. Lindquist, M. Stephens, and C. Wieman, Phys. Rev. A 46, 4082-4090 (1992). (I) 33. “Radiation Force in the Magneto-Ontical Trap,” A. M. Steane, M. Chowdhury, and C. J. Foot, J. Opt. Soc. Am. B 9, 2142-2158 (1992). (1) 34. “Improved Magneto-Optic Trapping in a Vapor Cell,” K. E. Gibble, S. Kasapi, and S. Chu, Opt. Lett. 17, 526-528 (1992). (I) 35. “Spatial Distribution of Atoms in a Magneto-Optical Trap,” V. S. Bagnato, L. G. Marcassa, M. Oria. G. 1. Surdutovich, R. Vitlina, and s. C. Zilio, Phys. Rev. A48, 3771-3775 (1993). (I) 36. “ A Simple Model for Optical Capture of Atoms in Strong Magnetic Quadrupole Fields,” D. Haubrich, A. Hope, and D. Mcschede, Opt. Commun. 102, 225-230 (1993). (A) 37. “Neutral Cesium Atoms in Strong Magnetic-Quadrupole Fields at SubDoppler Temperatures,” A. Hope, D. Haubrich, G. Muller, M. G. Kacnders, and D. Meschede, Europhys. Lett. 22, 669-674 (1993). (A) 38. “Simplified Atom Trap by Using Direct Microwave Modulation of a Diode Laser,” C. J. Myatt, N. R. Newbury, and C. E. Wieman, Opt. Lett. 18, 649-651 (1993). 39. “Collective Atomic Dynamics in a Magneto-Optical Trap,” A. Hemmerich, M. Weidemuller, T. Esslinger, and T. W. Hansch, Europhys. Lett. 21, 445-450 (1993). (A) 40. “High Collection Efficiency in a Laser Trap,” M. Stephens and C. Wieman, Phys. Rev. Lett. 72, 3787-3790 (1994). (A)
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695 41. “Measurements of Temperature and Spring Constant in a MagnetoOptical Trap,” C. D. Wallace, T. P. Dinneen, K. Y. N. Tan, A. Kumarakrisnan, P. L. Could, and J. Javaninen, J. Opt. Soc. Am. B 11, 703-711 (1994). (I) 42. “Inexpensive Laser Cooling and Trapping Experiment for Undergraduate Laboratories,” C. Wieman, G. Flowers, and S. Gilbert, Am. J. Phys. 63, 317 (1995). (E)
2. Magneto-optical traps of alkaline earths and noble gases 43. “A High-Intensity Metastable Neon Trap,” Fujio Shimizu ef al., Chem. Phys. 145, 327-331 (1990). (A) 44. “Laser Cooling and Trapping of Calcium and Strontium,” T. Kurosu and F, Shimizu, Jpn. J. Appl. Phys. (Lett.) 29, L2127-2129 (1990). (A) 45. “Laser Cooling and Trapping of Argon and Krypton Using Diode Lasers,” H. Katori and F. Shimizu, Jpn. J. Appl. Phys. (Lett.) 29, L21242126 (1990). (A) 46. “Laser Cooling and Trapping of Neutral Atoms,” F. Shimizu, Hyperfine Interactions 74, 259-267 (1992). (I) 47. “Laser Cooling and Trapping of Alkaline Earth Atoms,” T. Kurosu and F. Shimizu, Jpn. J. Appl. Phys. Part 1 33, 908-912 (1992). (I) 48. “Magneto-Optical Trapping of Metastable Xenon: Isotope-shift measurements,” M. Walhout, H. J. L. Megens, A. Witte, and S. L. Rolston, Phys. Rev. A 48, R879-881 (1993). (A) 49. “Lifetime Measurement of the Is, Metastable State of Argon and Krypton With a Magneto-Optical Trap,” H. Katori and F. Shimizu, Phys. Rev. Lett. 70, 3545-3548 (1993). (A)
3. Variants on MOT 50. “Four-Beam Laser Trap of Neutral Atoms,” F. Shimizu, K. Shimizu, and H. Takuma, Opt. Lett. 16, 339-341 (1991). (I) 51. “Spin-Polarized Spontaneous-Force Atom Trap,” T. Walker, P. Feng, D. Hoffmann, and R. S. Williamson 111, Phys. Rev. Lett. 69, 2168-2171 (1992). (I) 52. “A Vortex-Force Atom Trap,” T. Walker, D. Hoffmann, P. Feng, and R. S. Williamson Ill, Phys. Lett. X 163, 309-312 (1992). (I) 53. “High Densities of Cold Atoms in a Dark Spontaneous-Force Optical Trap,” W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, Phys. Rev. Lett. 70, 2253-2256 (1993). (I) 54. “Behavior of Atoms in a Compressed Magneto-Optical Trap,” W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, I. Opt. SOC.Am. B 11, 1332-1335 (1994). (I) 55. “Reduction of Light-Assisted Collisional Loss Rate From a LowPressure Vapor-Cell Trap,” M. H. Anderson, W. Petrich, I. R. Ensher, and E. A. Cornell, Phys. Rev. A 50, R3597-3600 (1994). (I)
58. “Magnetic Confinement of a Neutral Gas,” R. V. E. Lobelace, C. Mehanian, T. J. Tommila, and D. M. Lee, Nature 318, 30-36 (1985). (I) 59. “Evaporative Cooling of Magnetically Trapped and Compressed SpinPolarized Hydrogen,” H. Hess, Phys. Rev. B 34, 3476-3479 (1986). (I) 60. “Magnetostatic Trapping Fields for Neutral Atoms,” T. Bergman, G. Erez, and H. Metcalf, Phys. Rev. A 35, 1535-1546 (1987). (A) 61. “Magnetic Trapping of Spin-Polarized Atomic Hydrogen,” H. Hess ef a!., Phys. Rev. Lett. 59, 672-675 (1987). (I) 62. “Continuous Stopping and Trapping of Neutral Atoms,” V. S. Bagnato, G. P. Lafyatis, A. G. Martin, E. L. Raab, R. N. Ahmad-Bitar, and D. E. Pritchard, Phys. Rev. Lett. 58, 2194-2197 (1987). (I) 63. “RF Spectroscopy of Trappcd Neutral Atoms,” A. G. Martin, K. Helmerson, V. S. Bagnato, G. P. Pafyatis, and D. E. Pritchard, Phys. Rev. Lett. 61, 2431-2434 (1988). (I) 64. “Experiments With Atomic Hydrogen in a Magnetic Trapping Field,” R. van Roijen, J. J. Berkhout, S. Jaakola, and J. T. M. Walraven, Phys. Rev. Lett. 61, 931-934 (1988). (I) 65. “Quantized Motion of Atoms in a Quadrupole Magnetostatic Trap,” Bergman ef al., J. Opt. Soc. Am. B 6, 2249-2256 (1989). (A) 66. “Energy Distributions of Trapped Atomic Hydrogen,” I. M. Doyle ef a / . , J. Opt. Soc. Am. B 6, 2244-2248 (1989). (I) 67. “Very Cold Trapped Atoms in a Vapor Cell,” C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571-1574 (1990). (I) 68. “Atom Cooling by Time-Dependent Potentials,” W. Ketterle and D. E. Pritchard, Phys. Rev. A 46, 4051-4054 (1992). (A) 69. “Trapping and Focusing Ground State Atoms With Static Fields,” W. Ketterle and D. E. Pritchard, Appl. Phys. B 54, 403-406 (1992). (A) 70. “Laser and RF Spectroscopy of Magnetically Trapped Neutral Atoms,” K. Helmerson, A. Martin, and D. E. Pritchard, J. Opt. SOC.Am. B 9, 483-492 (1992). (I) 71. “Trapping and Cooling of (AntijHydrogen,” J. T. M. Walraven, Hyperfine Interactions 76, 205-220 (1993). (I) 72. “Lyman-Alpha Spectroscopy of Magnetically Trapped Atomic Hydrogen,” 0. J. Luiten, H. G. C. Werij, 1. D. Setija, M. W. Reynolds, T. W. Hijmans, M. W. Reynolds, and J. T. M. Walraven, Phys. Rev. Lett. 70, 544-547 (1993). (I) 73. “Atomic Hydrogen in Magnetostatic Traps,” J. T. M. Walraven and T. W. Hijmans, Phys. Rev. B 197, 417-425 (1994). (I) 74. “Collisionless Motion of Neutral Particles in Magnetostatic Traps,” E. L. Surkov, J. T. M. Walraven, and G. V. Shlyapnikov, Phys. Rev. A 49, 4778-4786 (1994). (A) 75. “Measurement of Cs-Cs Elastic Scattering at T=30 pK,” C. R. Monroe, E. A. Cornell, C. A. Sackett, C . J. Myatt, and C. E. Wieman, Phys. Rev. Lett. 70, 414-417 (1993). (I) 76. “Prospects for Bose-Einstein Condensation in Magnetically Trapped Atomic Hydrogen,” T. J. Greytak, in Base-Einstein Condensation, cdited by A. Griffin, D. W. Snoke, and S . Stringari (Cambridge University, Cambridge, 1993, pp. 131-159. (I)
C. Magnetic trapping
2. ac magnetic traps
Conceptually one of the simplest neutral atom traps is the dc magnetic trap. Atoms are confined in a minimum of magnetic field through the interaction of the atomic magnetic moment and the magnetic field. In Subsection 1 we list references describing dc magnetic traps for hydrogen and alkali-metal atoms. ac magnetic traps are also possible and are listed in Subsection 2. 1. dc magnetic traps
77. “Multiply Loaded, AC Magnetic Trap for Ncutral Atoms,” E. A. Cornell, C. Monroe, and C. E. Wieman, Phys. Rev. Lett. 67, 2439-2442 (1991). ( I ) 78. “ A Magnetic Suspension System for Atoms and Bar Magnets,” C. Sackett, E. Cornell, C. Monroe, and C. Wieman, Am. J. Phys. 61, 304309 (1993). (E) 79. “Magnetic Confinement of a Neutral Gas,” R. V. Lovelace, C. Mehanian, T. J. Tommila, and D. M. Lee, Nature 318, 30-36 (1985). (I) 80. “ A Stable, Tightly Confining Magnetic Trap for Evaporative Cooling of Neutral Atoms,” W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Phys. Rev. Lett. 74, 3352 (1995). (I) 81. “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” M. Anderson, I. Enshcr, M. Matthews, C. Wieman, and E. Cornell, Science 269, 198-201 (1995). (I)
56. “Cooling Neutral Atoms in a Magnetic Trap for Precision Spectroscopy,” D. E. Pritchard, Phys. Rev. Lett. 51, 1336-1339 (1983). (I) 57. “First Observation of Magnetically Trapped Neutral Atoms,” A. L. Migdall ef al., Phys. Rev. Lett. 54, 2596-2599 (1985). (I)
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Am. J. Phys., Vol. 64, No. 1, January 1996
N. R. Newbury and C. Wieman
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Good science and business practices also yield positive educational results
Carl Wieman The reliance of modem society on science and technology has created a serious and growing need for a large high-technology workforce and a technically literate population. Nowhere is this clearer than in the area of optics. Furthermore, providing an effective science education for a large and diverse segment of the population is something no educational system has ever achieved (sedaser Focus World December 2003, p. 47). The following are some examples of how using familiar practices that have greatly enhanced the progress of optical science can also lead to progress in science education. These practices include using and building upon past research results, utilizing modern technology, and being guided by quantitative objective measurements of results. Although there are numerous useful facets of this, I mention only a couple of representative examples here. 1. When I select material and plan how it is to be presented in a class (or even in a technical talk), I make use of what research on cognition has revealed about the limitations of short-term memory and the problem of "cognitive overload."' Short-term memory can handle seven (give or take two) different items at any one time. If related items can be presented in a sufficiently coherent way, they are "chunked" together in memory and take up a smaller total allocation. Unfortunately, sufficient coherence in a student's understanding is seldom achieved without considerable explicit emphasis.
2. Using results of psychology and advertising research to motivate students to follow desired learning practices, I have been able to get students to spend up to two times longer on homework and to use that time more effectively. First, I spend considerable effort creating, or borrowing whenever possible, problems that both cover the desired physics concepts and are perceived by the students as having obvious significance outside the confines of the classroom. This is particularly easy to do with optics topics, since light and its uses surround us. Sadly, many instructors ignore basic human nature and assume that most students will be motivated to thoughtfully work through a difficult abstract problem, merely because there is the suggestion that at some point in the future this exercise will be worthwhile. Second, my use of advertising strategy has an obvious impact when I survey the students and then announce to the class: "95% of the students who work in a collaborative problem-solving group say that this is very beneficial for their learning. Here is a typical student comment: 'Since I started working with a study group it is taking me less time to do my physics homework and I am learning so much more. I only wish I had started doing it sooner.' " The reason companies spend billions of dollars a year on this sort of advertising is that it works. 3. I also use a personalized electronic-response system (PERS) based on simple, inexpensive TV remotes that transmit a student's identity code and have five buttons (A thru E) that enable students to vote in
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response to multiple-choice questions in class. A computer records every response and provides a histogram of the accumulated responses for display. The psychological impact of the combination of individual accountability (the computer, and thus I, know what each student chose), knowledge of overall class response, and anonymity to peers is particularly valuable and makes this more useful than other real-time forms of student feedback and assessment. The students are more engaged in the material because they are continually responding to questions about it. Also the PERS provides ongoing feedback from the students to me and to the rest of the class as to how well they understand the material. When used properly, PERS can profoundly change the nature of the classroom and the student's expectation about learning the material2 This became startlingly clear to me the first (of many) times a mixture of whoops and curses (depending on their respective PERS predictions) greeted the outcome of an introductory physics demonstration experiment I carried out in a large lecture. The number and quality (and gender balance) of unsolicited questions also goes up dramatically, which makes class much more fun. Now, instead of reciting a lecture to a group of bored andor mindlessly scribbling students, I am having a conversation with them about physics. PERS also can provide ongoing data that is both enlightening and sobering.
I have been able to measure, for example, that information I simply tell the class, even with a visual cue, is retained by only 10% of the students 15 minutes later. In contrast, if that information is given in the context of a PERS question, so that they are invested in the answer, more than 90% of the students retain the information two days later. This is the case even though most of the class initially answered the question incorrectly. PERS also provides a natural mechanism for students to discuss questions and concepts, which can dramatically improve understanding and long-term retentiod The use of PERS and other forms of collecting data on student learning (such as regular open-ended online questions and listening in on student collaborative problem-solving sessions) has made me realize how inadequate my traditional exam and homework questions were at probing students' conceptual understanding and their ability to "think like scientists." Like bad data in any science, these inadequate exam questions led me to faulty conclusions about my teaching effectiveness despite 20 years of experience. I have now found that basing my teaching practices and conclusions on good data and on past research is as effective for advancing teaching as it is for advancing optical sciences. I believe it is our best hope for achieving the science education system we need.
REFERENCE 1. How people learn, NAS Press, Wash. D. C. (2000). 2. M. Dubson and C. Wieman, (unpublished)
www.colorado.edulphysicslEducationIssues/HITTMITTDescription.html. 3. C. Crouch and E Mazur, Am. J. Phys., 69,970 (2001). Nobel laureate CARL WIEMANis a iiieinber of the APS/AlP/AAPT Task Force on Undergraduate Physics, a riieiizber of the National Research Council Board on Science Education, and a professor ojpliysics at the Univeisity of Colorado, 440 UCB. Bouldel; CO 80309-0440; e-niail: cwieinaii@ila. Colorado.edu.
Guest Editorial Firming Up Physics of the university-supported tutors, which were usually junior- or senior-level physics majors. Many of the students who used tutors were coming to me complaining that their tutors were either giving them incorrect answers that disagreed with what they were told in class or in my homework solutions, or telling them the questions were either impossible or dealt with material that was far too advanced for the student, because it was not covered until graduate or very advanced undergraduate physics courses. Then I discovered that the graduate student TAs, who did grading or ran help sessions for the class, were similarly making frequent mistakes in answering questions. During the first couple of years (until I learned better), I would have the Fraduate TAs check my exams and homework problems by working out the solutions. I discovered that the best Z4s did about as well as the best students in the class, and that the weaker TAs did considerably worse. Although these graduate and undergraduate physics students had excelled in many physics courses and could solve textbook problems requiring elaborate mathematical calculations, they were in fact frequently incapable of applying basic physics concepts to simple real world situations. So which is the “watered down” physics course? The course in which students learn to use Green’s functions to solve complex boundary value problems, but have a totally misguided idea as to the physics involved in a microwave oven or the greenhouse effect? Or the course in which the students learn 10 use the concepts of physics to explain and predict phenomena like these in the world around them, even if their predictions are only based on simple algebra and conceptual reasoning? While I agree with Bernie that we need far more physics courses that give a diverse group of students a solid conceptual understanding of physics even if the use of math is quite limited, I would argue that there need be nothing “waterttd down” about those courses!
Many months ago, Bernie Khoury wrote one of his lively and thought-provoking columns’ in which he passionately advocated “watered down physics.” This stimulated a discussion between us that he urged me to write up as a response. Now, sitting on an airplane on my way to the Miami AAPT meeting, I finally have time to do this. Although Bernie’s words are always memorable, I should perhaps remind you of his primary point. He argued that it is neither realistic nor useful to try to have every student take the standard introductory physics course, and that there is great virtue to teaching a less mathematical course that can be of educational value to far more students. While I completely agree with this statement in principle, I object to the idea that such a course is necessarily “watered down” physics. My experience is that such a course for students with poor math and science backgrounds can in fact be “firming up” physics. Such students can come to understand the concepts of physics and apply them to the world around them better than undergraduate physics majors and beginning graduate students who have gone through a traditional physics program. These claims are based on my teaching of the introductory physics course for non-science students (sometimes referred to as “physics for poets”). The math prerequisite is high school algebra. The course focuses on the “physics of everyday life” and uses Bloomfield’s text of the same name. The focus is on conceptual understanding and being able to apply basic concepts of physics to explain and predict a variety of everyday phenomena. This is a large lecture course with no recitations, but to the extent possible in this setting I use numerous proven pedagogical strategies, including peer-instruction, interactive lecture demonstrations, context-rich problems, collaborative leaming, etc. I discovered some unexpected problems with teaching this course that are directly related to my discussions with Bernie as to what constitutes a “watered down” physics course. First, many of these students, naturally feeling nervous about taking a physics course, availed themselves
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ANNOUNCER
Carl Wieinan, Distinguished Professor of Physics aiid Fellow of JILA, University of Colorado iii Boulder 1. B Khoury, ‘ThreeCheers for Watered Down Physics,” Announcer,32 (3) 1 (2002).
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Free online resource connects real-life phenomena with science The University of Colorado’s Physics Education Technology project (PhET) provides an extensive online suite of simulations that covers the majority of high school and introductory college physics; it also illustrates the concepts behind advancedtopics such as semiconductors. The simulationscreate animated gamelike environments where the visual and conceptual models that physicistsuse are made accessible to students. They often animate what is invisible to the eye, such as atoms, electrons, photons and electric fields. You’ll find around 35 simulations at w\yW.C‘(~lorado.edu/physica/pht:t/and because this is an active project, the website is regularly updated with newly developed or improved simulations. User interaction is encouraged by PhET’s engaging graphics and intuitive controls that include click-and-drag manipulation, sliders and radio buttons. By immediately animating the response to m y user interaction, the simulations are particularly good at establishing cause-and-effectand at linking multiple representations. For example, in the Moving Man (figure 1) simulation, you can use the mouse to drag a little man back and forth on a pavement, watching as graphs of position,velocity and acceleration appear. Alternatively, you can ‘play back’ graphs to see the resulting motion of the man. For quantitative exploration, the simulations have measurement instruments available, such as a ruler, stop-watch, voltmeter and thermometer.All the simulations are extensively tested for usability and educational effectiveness, and a rating system (gold star, beta or alpha) is used to indicate what level of testing they have received. The tests involved student January 2005
Figure 1. The Moving Man is thefirst of the Motion simulations listed on the PhETwebsite. Others include Masses and Springs and Projectile Motion. interviews and use of the simulations in a variety of settings, including lectures, group work, homework and lab work. The PhET simulations are easy to use. They are written in Java and Flash, and can be run using a standard Internet browser as long as the latest Flash and Java plug-ins are installed. Instructions on the PhET site make it easy to download these free plug-ins for those who do not already have them. The simulations can be run online, or can be downloaded to your machine, which is particularly convenient for users without high-speed Internet connections. Electricity and circuits When you visit the website, you’ll find the simulations organized under seven partially overlapping topics: Motion, Work, Energy & Power, Sound &Waves, Heat & Thermo, Electricity & Circuits, Light & Radiation,Quantum Phenomena and Math Tools. The PhET simulations range from the quite simple to the highly elaborate.For example, John Travoltage in Electricity & Circuits is a simple sim-
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Figure 2. Balloons and Static Electricity can be found in PhET’s Electricity & Circuits section.
Figure 3. Circuit Construction Kit: a more advanced Electricity & Circuits simulation.
ulation, where the user rubs John’s foot on a rug and sees how he picks up electrons that will create a spark if his hand is moved too near a doorknob - all with accompanying soundeffects.Thisshows therelationship between the size and extent of the spark, the amount of charge accumulated and the distance from the doorknob. Another quite simple but useful simulation, Balloons and Static Electricity, involves the classic demonstrationof rubbing a balloon on a sweater (figure 2), but with the electric charges made visible. After rubbing, the user can observe the Coulomb attraction between the sweater and balloon and the shorter range attraction when the balloon is stuck to a wall, which explicitly shows the electron motion polarizing the wall. Examples of much more elaborate simulations from the Electricity & Circuits category incIude the Circuit Construction Kit (CCK) and Radio Waves simulations. In CCK (figure 3 ) , you construct arbitrarily complex circuits involving light bulbs, switches, batteries, resistors and wires, adjust any component resistances or battery voltages, and simultaneously observe the effect on the motion of electrons through the circuit and the brightness of the bulbs. Realistic looking voltmeters and ammeters measure voltage difference and current. This simulation has been very effective at helping 94
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students to grasp the concepts of electric current and voltage. In Radio Waves (figure 4), the user can move electrons in the broadcasting antenna via the mouse or by setting the frequency and amplitude of oscillation, and simultaneously observe the electromagnetic wave that is produced, how it propagates away from the transmitting antenna, and its effect on the electron in a distant receiver antenna. Other simulations in Electricity & Circuits include Signal Circuit, Electric Field of Dreams, Electric Field Hockey, Battery Voltage,Battery-ResistorCircuit, Ohm’s Law, Resistance in a Wire, Conductivity and Semiconductors.
From motion to maths The Motion topic includes The Moving Man, Masses & Springs, 2D Motion, Maze Game and Projectile Motion (a favourite of instructors,particularly when they realize that they can launch pumpkins, bricks, and more, to see how air resistance affects them). The Work, Energy & Power category includes Conservation of Energy and Ideal Gases & Buoyancy. Heat & Thermodynamics has simulationsof The Greenhouse Effect, Friction, Blackbody Spectrum and Ideal Gases & Buoyancy. The Ideal Gas simulation is rather elaborate. It begins with the usual capabiliJanuary 2005
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Figure 5. Waves on a String is one of two Sound & Waves simulations.
Figure 4. Radio Waves is an engaging Electricity & Circuits simulation that enables you to broadcast radio wavesfrom KPhET (aspictured here), then observe electron positions. ties of being able to put atoms in a box and view behaviour while their volume, number and temperature are altered. However, you can also control gravity, make mixtures of heavy andlight atoms, investigatethe physics of helium and hotair balloons, turn atom-atom interactions on and off and examine diffusion, all while quantitatively measuring relationships between temperature, pressure, velocity and energy distribution. The Light & Radiation section features Colour Vision & Filters, Microwave Oven, Radio Waves, Geometric Optics and Blackbody Spectrum. Sound & Waves has only two simulations: Sound, and Waves on a String (figure 5). Both have been found to be effective for teaching and learning-the latter being engrossing for student and instructor alike. The Quantum Phenomena section takes a number of advanced topics and provides useful visual models of the concepts behind them,making these subjects accessible to the introductory student by hiding the complex mathematics in the depths of the program. These simulations deal with subsections including Nuclear Physics (radioactivedecay and weapons), Lasers, Conductivityand Semiconductors.
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Finally, the Math Tools category containing Vector Addition, Equation Grapher, Ohm's Law and Resistance in a Wire are experimental attempts to help studentsobtain a conceptual feel for mathematical operations and relationships. When used in their simplest form, these PhET simulations enable any teacher with an LCD projector to replace the traditional static pictures they use in lectures with lively animated views of the phenomena. Varying the controls to show the animated response can provide strong visual images of the cause-and-effectrelationships between various physical quantities.Amorepedagogical1ypowerful use of these simulations is for the students to control the simulations themselves in discovery activities, designed with guiding questions from the instructor to allow them to develop a robust conceptual understanding of physics.
Kathy Perkins and Carl Wieman Acknowledgments The success of the PhET project is due to its dedicated and talented team members: Wendy Adams, Krista Beck, Noah Finkelstein, Michael Dubson, Ron LeMaster, Noah Podolefsky and Sam Reid. We gratefully acknowledge the Kavli Operating Institute and the NSF for providing the support to develop the simulations, and to make them freely available to all educators and students.
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PhET: Interactive Simulations for Teaching and Learning Physics Katherine Perkins, Wendy Adams, Michael Dubson, Noah Finkelstein, Sam Reid, and Carl Wieman University of Colorado at Boulder Ron LeMaster Kavli Operating Institute The Physics Education Technology ( P E T ) project is an ongoing effort to provide an extensive suite of simulations for teaching and learning physics and to make these resources both freely available from the PhET website (phet.colorado.edu) and easy to ircorporate into classrooms. The simulations (sims) are animated, interactive, and game-like environments in which students learn through exploration. In these sims, we emphasize the connections between real life phenomena and the underlying science and seek to make the visual and conceptual models of expert physicists accessible to students. We use a research-based approach in our design - incorporating findings from prior research and our own testing - to create sims that support student engagement with and understanding of physics concepts. We currently have about 35 sims posted on our website. Most of these sims cover introductory high school and college physics, but some focus on making traditionally more-advanced topics (e.g. lasers, semiconductors, greenhouse effect, radioactivity and nuclear weapons) accessible to students. On the website, the sims are organized under seven somewhat loose and partially overlapping categories: Motion; Work, Energy & Power; Sound & Waves; Heat & Thermo; Electricity & Circuits; Light & Radiation; Quantum Phenomena; and Math Tools. We update the website regularly with newly developed or improved sims, and we plan to post 10-20 new sims by the end of 2005. In this article, we introduce the P E T sims and their basic design; we describe how to access and run the sims; and we provide suggestions for effectively incorporating the sims into a variety of educational settings.
Creating PhET Sims for Engagement and Learning We have two main goals for the PhET sims: increased student engagement in learning and improved learning. Our target users are today’s physics students - a widely diverse population of young people. Thus, much of our attention focuses on connecting with the student and on creating student-sim interactions that facilitate construction of a robust conceptual understanding of the physics. While we do draw from the research literature on how students learn’, conceptual difficulties in physics2,and educational technology design3,we also make extensive use of student interviews and classroom testing to uncover any usability, interpretation, or learning issues and to develop a set of principles for highly effective designs. We design the sims to present an appealing environment that literally invites the student to interact and explore in an open-style play area with simple, intuitive controls, e.g. click-and-drag manipulation, sliders, and radio buttons. In the Ideal Gas sim (Fig. l), for example, the opening panel greets the user with a wiggling invitation to “Pump the handle!”. We use many connections to everyday life, both to engage the students and to support their learning by providing ties to their own experiences. This emphasis influences both the small details (e.g. using a bicycle pump to add gases) and the larger design questions where the underlying science is often presented in the context of real life phenomena (e.g. learning about buoyancy with hot air and helium balloons in Ideal Gas).
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Figure 1. In the Ideal Gas sim, pump the handle to add heavy or light particles to the box and see them move about, colliding with each other and the walls. Cool the box with “ice” and see the particles motion slow as the thermometer and pressure gauge readings fall. Increase gravity and see a pressure gradient form.
Using dynamic graphics, the PhET sims explicitly animate the visual and conceptual models that expert physicists use in their understanding. In many cases, the sims show what’s not ordinarily visible to the eye, e.g. atoms, electrons, photons, and electric fields. All of the PhET sims directly couple the student’s interaction with the animation. Adjustment of any controls results in an immediate animated response in the visual representations, making these sims particularly good for establishing cause-and-effect relationships and for enhancing students’ abilities to connect multiple representations. For more quantitative explorations, the sims have various measurement instruments available, e.g ruler, stop watch, voltmeter, thennometer, pressure gauge, etc. The Ideal Gas sim described in ~ i 1 illustrates ~ . many of these design features.
Making PhET Sims Accessible These sims are easy to access and free at the PhET website (phet.co1orado.edu). We write the sims in either Java or Flash so that they can be run using a standard web-browser with the recent Flash and Java virtual machine plug-ins. We provide instructions on how to download these free plug-ins for users who do not have them. In addition to being able to run the sims directly from the website, users can download an installer file (currently about 45 MB) which will install the entire website on to any local machine for use offline. This is particularly convenient if either the student computer lab or the lecture halls do not have an internet connection available. We recommend using these sims on PCs. The Flash-based sims work well on Macs, but the Java-based sims will only work on Macs running OSX - the older the version of OSX, the less reliable the Apple Java code. Even with the latest software, the Java-based sims often still run slower on Macs than on PCs. Teaching and Learning with PhET Sims Each P E T sim is created as a stand-alone, open learning tool often with several layers of complexity, giving teachers the freedom to pick and choose which sims to use and how to incorporate these into their class. While our design approach emphasizes improved conceptual understanding, the sims are most effective when students’ use is constrained to be productive, either through instructor guidance in lecture or through the use of a guided activity in homework, lab, or recitation. Here, we want to introduce some of the PhET sims and provide some suggestions for how to effectively incorporate these sims into different learning environments. Many of the examples come from our large lecture course - “The Physics of Everyday Life” - for non-science majors (though we have also used these sims effectively in algebra- and calculus-based introductory courses and advanced level courses such as physical chemistry).
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Figure 3. In the Sweater-Balloon sim, students can rub the balloon on the sweater and then stick it to the wall while seeing the charges move and the effects of Coulomb attraction.
Lecture. These sims are versatile tools for teaching in lecture: serving as powerful visual aids; complementing traditional classroom demos; and providing opportunities for interactive engagement Figure 2. In Wave-on-a-String sim, you can wiggle the through sim-based interactive lecture demos4or end of the string with the mouse or a piston to create a concept tests. wave and explore effects of tension and damping. Here we use the sim (A) to helps students visualize a standing Every physics teacher knows that it is often wave and follow with concept tests (B). Even after very difficult for students to visualize the physics. discussing the first concept test, only 23% of the students We use pictures, words and gestures in an attempt shown the tygon tube demo answered correctly compared to help them share the same visual models that have with 84% of the students shown the sim. worked for us. Unfortunately, this is not always successful - a student’s picture may not be the same or necessarily useful. When using the sims, the students and the teacher see the same objects and motions, allowing the teacher and students to focus their time and attention on creating an understanding of the physics rather than on establishing a common picture. In teaching about the physics of violins, we wanted students to have a good visualization of a standing wave on a string. In 2002, we used the conventional demonstration of shaking a long tygon tube stretched across our lecture hall to create a standing wave. In 2003, we demonstrated the motion of a standing wave with our Wave-on-a-Stringsim shown in Fig. 2A. We followed each demo with the two concept tests in Fig. 2B. As shown, the sim was much more effective at helping the students visualize the string’s motion. We regularly use the sims to show students what is not visible to the eye. When teaching about electrostatics for instance, we follow the traditional balloon demos with the Sweater-Balloon sim in Fig. 3 where the students can now see the electric charges. This sim is relatively simple but effective, animating the Coulomb attraction between oppositely charged objects and the movement of negative charges (electrons) as they are transferred from the sweater to the balloon when rubbed together. Polarization is also represented as the negative charges in the wall shift away from their positive ion cores (nuclei) as a charged balloon approaches.
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a. haw fast this pe&moves to t h e right b. how fast this peak moves up and down. c. could be a or b
Figure 4.In Radio Waves, users create EM waves by moving the electron in the broadcasting antenna either by hand (mouse) or by setting the frequency and amplitude of oscillation. Here we’ve created a concept test to help students distinguish between wave speed and frequency.
Figure 5. In Moving Man, users control the man’s motion either by dragging the man about or using the position, velocity, or acceleration controls. By graphing the motion simultaneously and including a “playback” feature, this sim helps students build connections between actual motions and their graphical representation.
These sims naturally couple with the use of interactive engagement techniques in the classroom. In our classrooms, we use an adaptation of Mazur’s Peer Instruction’ technique with both concept tests and interactive lecture demos. In teaching about electromagnetic waves, students are challenged to understand and conceptualize: how EM waves are created by accelerating charges, how they exert forces on charges, and how their frequency, wavelength, and wave speed are related. We use the Radio Waves sim in guiding students’ understanding of these ideas. In the PowerPoint slide in Fig. 4,we ask the students to discuss and vote on how the speed of the wave is measured. About 1/3 of our students had not yet clearly distinguished the ideas of frequency and speed. By using the sim, we were able to immediately address this confusion; we focused the students’ attention on following the peak as it moved to the right and relating that to the speed of the wave. By varying the frequency, we showed students that speed is independent of frequency. The sims are also useful tools for interactive lecture demos (ILDs). For instance, the Moving Man sim (Fig. 5 ) is ideal for use with Thornton and Sokoloff s force and motion ILD where students predict the graphs of position, velocity, and acceleration for a described m ~ t i o nUsing . ~ the Moving Man sim, the student’s predictions are tested as the instructor reproduces the described motion of the man on the sidewalk and graphs of position, velocity, and acceleration simultaneously appear. This motion can be repeated with the sim’s “playback” feature or the position scale on the sidewalk can be flipped with “invert x-axis” to guide students’ thinking about the meaning of the sign of velocity and acceleration. We have noticed that using sims in lecture often leads to unprompted high quality questions and comments from students e.g. connecting to their own experiences, asking probing “what if’ questions, or extending the discussion to applications or consequences of the physics. With the open design of the sims, we are often able to immediately use the sim to test the students’ ideas or answer their questions.
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Figure 6 . The Masses and Springs sim creates an open lablike environment in which students are free to investigate. Challenge them to measure the mass of the green weight, to measure gravity on Planet X, or to make sense of the energy conversions.
Figure 7. In the CCK sim, students can construct these circuits, close the switch, and immediately see the response - the electrons flow faster from the battery, the ammeter reads higher, the voltage meter reads lower, and one bulb dims while the other bulb glows brighter. Results from a recent study show improved performance on the final from students using CCK in lab.
Lab/Recitation. These sims are specifically designed to allow students to construct their own conceptual understanding of physics through exploration. This makes them useful learning tools for small group activities in lab and recitation. We have found that such activities need to have well-defined learning goals and be designed to guide, but not excessively constrain, the students’ exploration of the sim - promoting lines of inquiry that help students develop their understanding of the important concepts. A number of the P E T sims are particularly well-suited for use in these environments, including Moving Man, Circuit Construction Kit, Masses and Springs, and Ideal Gas. In Masses and Springs (Fig. 6), students can complete traditional laboratory activities, such as hanging objects on springs and measuring spring displacement or oscillation period; however, students can extend their exploration by slowing down time and following the conversion between different forms of energy, by instantaneously porting the whole apparatus to a different planet, or by varying the spring constant continuously. With the lifelike look and feel of the sim, the students’ interactions mimic the real world experience in many ways. Through guided activities that investigate the physics of spring scales or bungee jumpers for example, students can reason about and construct a conceptual understanding of a range of topics including Hooke’s law, damped and undamped harmonic oscillators, conservation of energy, and net force and motion. In several ways the Circuit Construction Kit (CCK) sim (Fig. 7) offers a learning environment similar to a real-life lab. Students connect light bulbs, switches, batteries, resistors, and wires to create arbitrarily complex DC circuits. Realistic looking voltmeters and ammeters are used to measure voltage differences and currents. But to this, the CCK adds an animation of the electrons flowing through the circuit elements and the ability to continuously adjust the resistance of any component (including the light bulbs) or the voltage of the battery. For example, after building the circuit in Fig 3, students can close the switch and continuously change the resistance of the 10 ohm resistor. Simultaneously the students observe the effect on the motion of electrons, the brightness of the bulbs, and the measured voltage difference. These features provide the students with powerful tools in understanding current and investigating the cause-and-effect relationships between voltage, current, resistance, and power.
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In a recent research study, we found that the students who used CCK in lab performed better on conceptual questions about circuits than the students who used real equipment (Fig. 7 inlay).6 While it is reasonable to suspect that the CCK students would have more difficulty building a circuit out of real equipment, the same study demonstrates that the CCK students could actually construct real circuits as well as or better than the students who had used the real equipment in an identical lab experience.6 While it is useful for students to make sense of the non-idealized, real-world situation, these sims use a layered structure to allow students to me/False In this simulation, a darker shade of gray indicates an increase first explore and construct a conceptual in pressure compared to the undisturbed air pressure ruelfalse To increase the volume of a tone, the speaker must oscillate understanding with idealized equipment and then to back and foreh more bmes each second R extend beyond this ideal to make sense of more Figure 8. In the Sound sim, a speaker oscillates back and subtle complexities encountered in real life. For forth producing pressure waves that propagate from the example, in CCK, the advanced features introduce speaker. Students can: adjust frequency and amplitude; see finite resistivity to the wires and an internal changes in the pressure waves and hear changes in pitch and volume; use the ruler and timer on the “Measure” resistance to the batteries.
F
panel to measure speed, frequency, period, and wavelength; and look at and listen to the interference of waves from !NO speakers on the “Interference” panel.
Homework. As with lab or recitation activities, we design homework questions that guide the students’ work with the sim. These questions require the student to interact with the sim to discover, explain, or reason about the important physics concepts. Often we ask the students to explore cause-and-effect relationships, both qualitatively and quantitatively, or make connections to their everyday life experiences. We use a range of question styles - true/false, multiple choice, numeric, and essay response. We prefer essay-style questions where students must explain their ideas and reasoning in words, but we are limited by grading time in our large-enrollment courses. In homework on sound, we ask students to make sense of what the Sound sim (Fig. 8A) is showing with true/false (Fig. 8B) and essay questions: “You hear a Concert A tone from the speaker. Describe the required motion of the speaker and how this motion leads to detection of Concert A by your ear. Include in your explanation the chain of cause-and-effect logic.” In a discovery exercise, students are asked to use the ruler and timer in the “Measure” panel to develop a procedure for measuring the speed of sound and then measure the speed at 200 Hz and at 400 Hz. We ask them: “Does the speed depend on the fiequency? How are your observations consistent or inconsistent with your experience in everyday life? Explain.” They also measure the period and wavelength at these frequencies and explain how this is consistent or inconsistent with their measurement for the speed.
Hearing from the students At the end of the term, we asked the students in our large lecture how useful the sims were for their learning in the course, responding on a 5-point scale from “not useful” to “a great deal”. For the usefulness of sims in lecture, 62% of the students rated the sims as very useful for their learning (4-5) with an additional 22% finding them somewhat useful (3). When asked about the usefulness of the homework questions coupled with sims, 49% rated these as very useful with an additional 24% finding
Submitted to The Physics Teacher (Nov. 2004)
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them somewhat useful. In contrast, only 27% found their text very useful with the majority (52%) rating the text of little use in their learning (1-2). When we receive complaints from the students about the sims, they tend to fall into two categories - trouble getting the sims to run on their home computer (typically because they have a Mac or a slow internet connection) or the discovery of an unknown bug within the sim. The easiest solution to the first issue is to make sure the students have access to a school computer which is ready to run the sims. As for the second, students are our best testers, and we fix the bugs as they find them. In a calculus-based physics course, we recently used the CCK sim in conjunction with the Tutorials-in-Physics7for the circuits tutorial. In a follow-up survey, we asked students to comment on which of the tutorials (out of the 9 they had completed so far) had been particularly effective. Of those who listed specific tutorial(s), about 70% identified the circuits tutorials in which they used CCK. This favorable response is similar to when real batteries and light bulbs are used in tutorial, but two strong themes emerged in the comments on the CCK sim - how helpful it was to be able to easily experiment and adjust the circuits and how the sim helped with visualizing what was going on: ,I
I really like the circuits tutorial where i got to build circuits on the computer and change variables to see an instantaneous reaction. This really helped me conceptualize circuits, resistors, etc... ’’ “I liked the Voltage, Current and Resistance tutorial with the computers. I am a visual learner, and I am strugging with electricity because it is not something you can really see. I mean you can see a light bulb go on, but you can not see what is going on inside. The tutorial with the computers helped me out, because they showed what was really going on inside the circuit. ”
Finally, we asked our students if they “missed” having sims on the topics (MRIs, x-rays, cameras) where they had not been used. Most said they would have used these sims to help their learning. P E T is an ongoing program. We continue to upgrade older sims and add new ones. By the end of 2005, we expect to have added enough new sims to cover nearly all of the core content in introductory college physics. During this time, we will also be expanding our coverage of quantum physics and physical chemistry and developing a shared electronic space for PhET sim educators. This space will provide the community of educators using PhET sims with a forum for sharing guided discovery activities, concept tests, or other curricular materials that they have written and tested with their students. We are pleased to acknowledge support for this work by the NSF, the Kavli Operating Institute, and the University of Colorado. We thank Steve Pollock, Chris Keller, and the other members of the newly formed Physics Education Research Group at Colorado (PER@C) for their valuable contributions to this effort.
References 1. e.g. J.D. Bransford, A L. Brown, and R. R. Cocking, editors How People Learn (Natl. Acad. Press, Washington, DC, 2002). 2. e.g. references within L.C. McDermott and E.F. Redish, “Resource letter on Physics Education Research,” Am. J. Phys. 67,755-772 (1999). 3. e.g. R.C. Clark and R.E. Mayer, e-Learning and the Science of Instruction: Proven Guidelinesfor Consumers and Designers of Multimedia Learning, (Pfeiffer, San Francisco, CA, 2003). 4. D. Sokoloff and R. Thomton, “Using interactive lecture demonstrations to create an active learning environment,” The Phys. Teach. 35,340-346 (1997). 5 . E. Mazur, Peer Instruction: A User’s Manual (Prentice-Hall, New Jersey, 1997). 6. N. Finkelstein, W. Adams, C. Keller, P. Kohl, K.Perkins, N. Podolefsky, S. Reid , and R. LeMaster, “When learning about the real world is better done virtually: a study of substituting computer simulations for laboratory equipment”, submitted to Phys. Rev. -PER, 2004.
Submitted to The Physics Teacher (Nov. 2004)
7
7. L.C. McDermott, P.S. Shaffer, and the Physics Education Group at the University of Washington, Tutorials in Introductoiy Physics (Prentice-Hall, New Jersey, 2002).
The PhET Team Members: Curl Wieman,Distinguished Professor of Physics and a Fellow of JILA, leads the PhET project housed in the Department of Physics at the University of Colorado at Boulder. Other departmental team members include: Katherine Perkins (research associate and lecturer), Wendy Adums (graduate student), Michael Dubson (senior instructor and flash programmer), Noah Finkelstein (assistant professor), Sam Reid (software engineer), and Krista Beck (administrative assistant). Ron LeMaster (software engineer) is supported by the Kavli Operating Institute. Contact: Katherine Perkins, Department of Physics, UCB 390, University of Colorado at Boulder, Boulder, CO 80309; [email protected]
Submitted to The Physics Teacher (Nov. 2004)
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The Design and Validation of the Colorado Learning Attitudes about Science Survey W. K. Adams, K. K. Perkins, M. Dubson, N. D. Finkelstein and C. E. Wieman Department of Physics, University of Colorado, Boulder Abstract. The Colorado Learning Attitudes about Science Survey (CLASS) is a new instrument designed to measure
various facets of student attitudes and beliefs about learning physics. This instrument extends previous work by probing additional facets of student attitudes and beliefs. It has been written to be suitably worded for students in a variety of different courses. This paper introduces the CLASS and its design and validation studies which include analyzing results from over 2400 students, interviews and factor analyses. Methodology used to determine categories and how to analyze the robustness of categories for probing various facets of student learning are also described. This paper serves as the foundation for the results and conclusions from the analysis of our survey data.4%’ prevalent in a typical introductory student’s vocabulary, were avoided. Every effort was made to avoid questions that include two different statements. Finally, one of the most difficult tasks was creating questions that were interpreted in only one way in interviews with both faculty and students.
Over the last decade, researchers in science education have identified a variety of student attitudes and beliefs (ABs) that shape and are shaped by student classroom experience.’ Over the last year at Colorado, we have developed and validated an instrument, the Colorado Learning Attitudes about Science Survey, CLASS2, which builds on existing surveys (MPEX, VASS, EBAPS)3. This survey probes student’s ABs and distinguishes the ABs of experts from novices. The CLASS was written to make the questions as clear and concise as possible and is readily adapted to use in a wide variety of science courses. Students are asked to respond on a Likert-like (5-point agree to disagree) scale to questions such as: “I study physics to learn knowledge that will be useful in life.”, or “After I study a topic in physics and feel that I understand it, I have difficulty solving problems on the same topic.”, or “TO learn physics, I only need to memorize important equations and definitions.” In this paper we will discuss the methods used to validate the survey. We will also discuss the subtleties of choosing categories of questions and list the seven categories we have chosen. The survey has generated some very interesting results which are discussed briefly here and in depth in the companion papers by Perkins et al.4 and Pollock et al.’.
Students (though perhaps not physicists) apply the word physics in at least three ways: a particular course, the scientific discipline, or the physics that describes nature. We designed the survey to embrace a single meaning of the word physics to avoid confusion. We focused the questions on physics that describes nature; noting this sense sometimes overlaps with physics as a discipline. By taking this approach, it made the questions meaningfil even if a student had never taken a physics course. This survey has been administered before (pre) and after (post) instruction to 2400 students in 10 courses over the past year either online or paper and pencil. Scoring is done by determining the percentage of a group of students who agree with the experts’ view. The survey is scored overall and then in the seven categories listed in Table 2. Each category consists of three to eight questions that correlate with one another and target a specific attitude or belief about science.
The CLASS was designed for use with a broad population, takes only ten minutes to complete and covers many areas of student’s ABs about physics. To make it suitable for a variety of courses serving nonscience majors, physics majors or graduate students words such as “domain” or “concepts”, which are not
VALIDITY AND RELIABILITY Validation was done in three steps: First experts were interviewed and then took the survey; second students were interviewed to confirm the clarity and
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meaning of questions; and finally a detailed factor analysis was performed to create and verify existing categories of questions. Three experts underwent a series of interviews once the initial design was complete. These experts were physicists who have extensive experience with teaching introductory courses and worked with thousands of students. Some of these experts are involved with physics education research others are simply practicing physicists interested in teaching. Their comments were used to hone the questions and remove any that could be interpreted more than one way. When this process was complete, seven experts took the survey. Their answers matched on all except three questions, two of which have been reworded. Their answers were used to determine the expert point of view for scoring. Student interviews were carried out by obtaining 34 volunteers from six different courses at a mid-size multipurpose state university (MMSU) and a large state research university (LSRU). Care was taken to acquire a diverse group of interviewees. Interviews consisted of first having the student take the survey with pencil and paper. Then, during the first ten minutes, students were asked about their major, course load, besuworst classes, how they study, class attendance and future aspirations to characterize the student and their interests. After this, the interviewer read the questions to the students while the student looked at a written version. The students were asked to answer each question using the 5-point scale and then talk about whatever thoughts each question elicited. If the student did not say anything, helshe was prompted to explain hisher choice. After the first five or six questions, the students no longer required prompting. If the students asked questions of the interviewer, they
were not answered until the very end of the interview. Interview results showed students and experts had consistent interpretations on nearly all of the questions. A few questions were unclear or misinterpreted by some of the students. Some of these were reworded or removed in the Spring version 2 of the survey and a few remain to be changed. Finally there were questions that elicited unexpected student ideas, which will be used for further refinement of the survey. Statistical analyses were used to test the validity of the sub-groupings of questions into categories. We performed a factor analysis, a data reduction technique that groups similar questions using correlations between question responses. We used the principle components extraction method along with a direct oblimin rotation and performed both an exploratory and a confirmatory factor analysis. For more detail on factor analysis see reference 6. First we did an exploratory factor analysis, which analyzes the results from all questions and then groups questions that were answered similarly by students into independent factors. The exploratory factor analysis was performed with Spring version 2 of the survey on three sets of data from a calculus-based physics I course (N416): pre-test results, post-test results and the shift from pre to post. The results from this exploratory factor analysis were used to indicate potentially bad questions (gave inconsistent results or seemed to be independent of the rest of the questions) and provided a set of independent categories. These categories emerge from the student responses and thus are factors that span the space of student ideas and characterize our students.
Robustness
Solution
Independence Coherence
MF PC
How to Learn Coherence
Exploratory Categories Category i Category 2
Concepts Reality World View Reality Personal View Math
MF S
How to Learn Reality World
Category 3 Category 4
WF
How to Learn Reality World and Personal Metacognition Dropped
S
Reality
Category 5
NS
Dropped
Category 6 Category 7
SF WF
Sense Making Dropped
Original Categories
Effort Skepticisin
Robustness BQ SS*
BQ
Solution
Personal S PC PC
Math Dropped Dropped
712
TABLE 2. Version 2 CLASS Categories Category
Reality Personal View Reality World View Math Sense Making Metacognition How to Learn Coherence
Description
Physics is part of the student’s life - student cares about physics. Physics describes phenomena in the World around us. Mathematical formulae describe physical phenomena. It is important to me to make sense out of things when learning physics. Awareness of what is necessary to learn and understand physics - self reflection. Best learned by memorization of facts and methods without understanding. Physics consists of connected ideas.
Another usefil perspective is to look at specific groups of questions that probe facets of learning that the physics professor can directly address. Following this idea we chose our original categories based on the categories used by the MPEX and the VASS and expanded upon them slightly during the first two phases of validation of the CLASS. These types of categories emphasize both what a physicist believes is a useful breakdown of what is important for a student to learn physics and pedagogical organization (expert perspective) rather than emphasizing the way students think (student perspective). This means that some of these categories may not be independent of one another; however, if questions are properly designed, these categories are still self-consistent and provide useful information. We performed a confirmatory factor analysis next using our originally chosen categories and the exploratory factor analysis categories. With a confirmatory factor analysis, the categories are predetermined by the researcher and the analysis determines how well each question within a factor correlates with that particular factor. Results of this analysis were used in conjunction with correlations between individual questions to create seven very robust (defined below), albeit not completely independent, categories. Table 1 above lists both our original categories and the exploratory categories and their robustness. Robustness is determined by the confirmatory factor analysis and our assessment of the usefulness of that grouping of questions as a research tool. If a category was not robust, we either made it stronger by addinglsubtracting questions or simply dropped the category. The third column lists the fate of each category after completing the analysis. Table 2 lists the categories that resulted from this process. Reliability studies were conducted in Calculusbased physics I at LSRU which is offered every semester with an enrollment over 500 students. During the 2003-2004 school year the course was taught by the same professor, who allowed us to administer the survey to his course pre and post, both Fall and Spring
semester. The pre and post results for the two semesters were not statistically different for questions that were the same on both surveys. See Table 3 for overall scores, Reality World View and Math for both Fall and Spring semesters. TABLE 3. Reliability Data
Category Fall Overall Reality World Math
Pre
Post
Uncertainty
63% 73% 71%
65% 76% 69%
1Yo 2% 2%
64% 73% 69%
66% 76% 68%
1% 1% 1Yo
Spring
Overall Reality World Math
APPLICATIONS There are several useful ways to use the scores from the CLASS. One can look at the pre-test results and their influence on student learning or retention. One can also look at the change in attitudes over a semester, the shifts, to determine what effect instruction had on students’ ABs. In Table 4 we show results for six courses covering a range of introductory physics classes. We see that students’ incoming ‘Reality Personal View’ increases with level of physics course. Thus, students who make larger commitments to studying physics tend to be those who identify physics as being relevant to their own lives. As seen with other surveys, the CLASS shows student ABs deteriorate after instruction; unless, ABs are explicitly addressed by the instructor. We see in Table 4 that in the courses at LSRU, which explicitly attended to ABs, the overall scores did not deteriorate; however, in the courses at MMSU there was a substantial decline in ABs. A companion paper by Perkins et aL4 goes into more detail on these courses and also carefilly looks at correlations of students’ ABs with their learning gains. They show that students with large learning gains have a greater
713
TABLE 4. Correlations between favorable ‘Reality Personal’ and physics course selection Course School Dominant # of students Overall %favorables Type Type/Term student wl CLASS population Pre Post
Non-Sci-I LSRUEa03 non-sci 77 56% LSRU/SpO4 non-sci 34 71% Non-Sci-I1 MMSUEa03 Alg-I pre-meds 60% 36 LSRUEa03 Calc-I engineers 174 63% LSRU/SpO4 engineers Calc-I 416 64% MMSUga03 Calc-I phy& maj 64 ‘Yo 41 I=la semester, II=2”dsemester; typical standard deviation for ‘Overall’ is -16%.
’
This paper describes the philosophy and methods behind the development of the CLASS. In addition we detail the validity and reliability studies for this survey. We also define a process of selecting categories of questions and determining their robustness. This paper serves as the foundation for the results and conclusions from the analysis of our survey data and future applications of the survey. Analysis and refinement of the CLASS is still in process. Over the next year we plan to perform a factor analysis of the results for other courses, a final revision of the current questions and creation of questions to target other categories that were not adequately addressed by the current version of the survey. The survey will also be altered slightly to be appropriate for use in Biology, Math, Astronomy and Chemistry and administered to these courses this Fall. Finally we would like to step beyond simply characterizing groups of students to identifying individual student characteristics.
ACKNOWLEDGMENTS Thank you to Steven Pollock and Courtney Willis for many helpful discussions and the members of the PER@C group. This work is supported in part by NSF.
REFERENCES 1.
Bransford, J.D., Brown, A.L., and Cocking, R.R. (2002). How People Learn Washington D.C.: National AcademyPress. Hammer, D. (2000) Student resources for learning introductory physics, American Journal of Physics, 68, S52-S59. Redish, E.F.,(2003). Teaching Physics with Physics
51%
65% 66% 54%
44% (4%) 61% (5%) 61% (5%) 63% (2%) 64% ( 1%) 71% (5%j
Suite, John Wiley & Sons. Seymour, E. and Hewitt, N.,( 1997). Talking about Leaving, Westview Press.
positive shift in ABs while students with lower learning gains show a deterioration in ABs.
CONCLUSIONS AND FURTHER WORK
57% 73%
Reality Personal %favorable on Pretest (uncertainty)
2.
A copy of the CLASS can be found httu://cosmos.colorado.edu/uhetlsurveviCLASS/
at
3.
Redish, E., Saul, J.M. and Steinberg, R.N. (1998). Student Expectations in Introductory Physics American Journal of Physics, 66, 212-224. www.uhvsics.umd.edu/uereiexDects/index.html Halloun, I. A. “Views About Science and Physics Achievement: The VASS Story.” In The Changing Role of Physics Departments in Modem Universities: Proceedings of the ICUPE, E.F.httu://modeling.asu.edu/ R&E/Research.html Elby, A., Epistemological Beliefs Assessment for Physical Science. httu://www2.~hvsics.uind.edui-elbv/ EBAPS/home.htm
4.
Perkins, K., Adams, W., Finkelstein, N. and Wieman, C. (2004). Correlating student attitudes with student learning using the Colorado Learning Attitudes about Science Survey, submitted to PERC Proceedings 2004.
5.
Pollock, S.(2004). No Single Cause: Learning Gains, Student Attitudes, and the Impacts of Multiple Effective Reforms, submitted to PERC Proceedings 2004.
6.
Crocker, L. and Algina, J., (1986). Introducrion to Classical and Modern Test Theory, Fort Worth Hold, Rinehart and Winston, Inc. Kachigan, S. K., (1986). Statisticaf Analysis, New York: Radius Press. Kim, J. and Mueller, C. W., (1978). Factor Analysis Statistical Methods and Practical Issues, Beverly Hills: Sage Publications
an
I
very physics instructor knows that the mosr engaged and successful students tend to 'sit at the front of the class and the weakest students tend to sit at the back. However, it is normally assumed that this is merely an indication of the respcccive seat location preferences of weaker and stronger students. Here we present evidence suggesting that in fact this may be mixing up the cause and effect. It may be that the seat selection itself contributes to whether the student does well or poorly, rather than the other way around. While a number of studies have looked at the effect of seat location on students, the results are often inconclusivc, and few, if any, have studied the effects in college classrooins with randomly assigned seats.' In this paper, we report on our observations of a large introductory physics course in which we rarzdom(y assigned students to particular seat locations at the beginning of the semester. Seat location during the first half of che semester had a noticeable impact on student success in the course, particularly in the top and bottom parts of the grade distribution. Students sitting in the back of the room for the first half of the term were nearly six times as likely to receive an F as students who started in rhe fiont of the room. A corresponding but less dramatic reversal was evident in the fractions of students receiving As. These effects were in spite of many unusual efforts to engage students a t the back of die class and a front-to-back reversal of seat location halfivay through the term. These results suggest there may be inherent dctrimental effects
30
University of Colorado ar Boulder; Boulder, CO
of large physics lecture halls that need to be further explored.
The Course This study was done in the "Physics of Everyday Life" course we taught at the University of Colorado in Boulder. This is an algebra-based introductory physics course for notiscience, noriengineering majors and uses the textbook by L. Bloomfield with a similar name.2 Out 20 I students were a diverse mix of majors and ages, with 43Yo being first-term freshmen. T h e class included two 75-minute lectures per week, regular gre-class reading assignments, extensive weekly homework assignments, three evening hourly exams, arid a coniprehensive final.
The Lecture The lectures were designed to be highly interactive and engaging for the students via a number of methods. Peer instruction techniques3 and a personal electronic response system (PEKS)4 were used extensively during every class to stimulate student discussion and to provide feedback to both the instructor and the student. During a typical. class, students were asked to consider numerous questions (7to 10). These questions were designed to, for cxample, elicit/revcal students' misconceptions, test fsr conceptual understanding, predict or reflect on demonstration outcomes, or draw on intuition from everyday life. We emphasized student-student discussions that focused on sense-making and reasoning. In order to
110:: 10.1 119iI.lXS')87
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715 Table 1. Students grouped by initial seating assignment. ensure all studencs were part of a well-defined and - . _ _ . s _ _ _ _ . . _--...... .... ._ stable discussion group, students were randomly asSeating Initial seat +of . I Average GPA signed to a seat location and a group at the start ofthe group distance from i styleiits Inut i!iclud;:ig j thi. worst,) (rant o f class j . . term., where each group was composed of three or four . ... --. srudents seated adjacent to each other. We had the srudents debate many of the in-class questions with their group members and then click in their answer using their PERS device. To improve both the quality and amount of student discussion within the group, many of the questions reclu.ired "group consensus answers."5 H a l h a y through the term, seat locations were reversed front to back in the lecture hall, with some reorganization of groups where necessary. A significant eRort was made to keep all studenrs engaged in the course. Two undergraduate teaching assistants and often a second faculty member moved throughout the classroom encouraging enga,metilent and discussion with an explicit god of getting students in the back involved. Although the lecture hall is relatively large with the projection screen and dcinonstration table abour 2 meters from the nearest students a.nd I 3 meters from the most distant, the hall i s sloped s o that all seats have a clear view of the front.'; To ensure good visibility from the back, we used PowerPoint slides projected on a large screen with a good Fig. 1. lnitial seating location vs attendance. The average attenLCD projector. All slid.esused large figures and fonts. dance is plotted for the first (blue) and second (red) half of the term for students grouped by the distance of their initial assigned seat h video camera was used to project a large image of from the front of the classroom. a n y smaller demonstrations. ,
The Grading Studerits earned points for reading quizzes, class participation, homework, and exams. Typically, m70 points per class (in total accounting for -12% of the possible points) were given for responding to the inclass questions (almost always regardless of whether or not the answer was correct). This grading structure encouraged attendance and involvement in the questions and in-class d.iscussion.T h e attendance averaged 82.7% over the first half of the term with 1% of stud.ents virtually never attending. Over the second halfof the term, the attendance averaged 79.6% with 5.4% ofstudents virtually never attending.
The Impact of Seat Location Halfivy through the semester when we switched the seating location of the groups to bring those in the hack to the front and mow the ones i n from to
~
i
^
the back, we found ourselves in a strange situation. Students sitting in the hack ofthe room were atrending more regularly and asking significantly more questions than those sitting in t:he front. Struck by chis behavior, we carried out a more detailed analysis of the impact of the student's seating location. We analyzed the course data by grouping the students Lased on their original seating location as listed in Table I. When the seat locations were reassigned at the midpoint of the term, the group locations were generally reversed (e.g.. group # I students were now near the back). We found the average of die GPAs for each group (not including this course) to be identical (see Table I), suggesting that the student populations were similar. In our analysis, we looked for correlations between group number and a variety oforher variables, and fcxind some striking diffcrcnccs i n the foul- groups.
31
!
, I
.
716 .
”-
.
Fig. 2. Initial Seating location vs course performance (not including attendance). This plot shows the fraction of students within each group who were in the top 20% and in the bottom 10% of the class for total points earned excluding attendance points. The effect is even more pronounced when looking at final grades, where attendance points are included.
Figure 1 shows each group’s average attendance for the first and second half of the term. Two trends are evident: The further the original seating location is from the front of the classroom, then 1) the lower the average attendance and 2) the larger rhe drop-off in attendance between the first and second half of thc term. The trend in the drop-off in attendance is particularly notable because during the second half of the semester, students in group #I were s i t t i n g j n from the front while students in group #4 were sitting close to the front. Even though the students in group #I were now sitting far from the front and skiing season had begun, their average awmdance declined by less than I%! We also looked at the relationship between seating location and grade. We found a difference in average grade for thc four groups that is at the edge of statistical significance; however, the effects on the top and bottoni of the grade distribution are quite pronounced. The fraction of A’s decreased steadily a s the groiip’s migznnlseat location was further ftom the front (2WO in group #1 received A’s compared to 18% in group #4), while the fraction o f F i increased (2041in group #1 to 12% i n group $4). Student performance did not change significantly between the first and second halvcs of the semesrcr; the scudcnts
32
7
Fig. 3. Initial seating location vs students’ beliefs. The fraction of students within each group whose beliefs about physics improve over the term is correlated with the distance of their initial assigned seat from the front of the classroom.
who started in front and doing well continued to do so when they moved to the back. As shown in Fig. 2, even when the attenda.nce contribution to the grade i s removed, there is still a clear effect. Finally, we looked a t die students’ beliefs about physic5 and learning physics for the different groups. These were probed using the Colorado Learning Attitudes about Science Survey (CLASS),? where students are asked to consider statements about physics and respond on a five-poin t, srrongly-agree-to-srronglydisagree scale (e.g., “l(now1edge in physics consists of many disconnected topics” or “I think about the physics I experience in everyday life”). About half of the students in each group were given the CLASS both at the beginning and end of the course to measure these beliefs. In Fig. 3, we show the fraction ofstudents within each group whose beliefs improved (moving froni more novice-like to more expert-like beliefs) or deteriorated. While there is considerable uncertainty because of the limited statistics, we find that a larger fraction ofstudents who starced the semester in front showed improved beliefs compared to those who started the semester in back. It should be noted that i t is typical for the beliefs ofstudents to decline in introductory physics courses that use mostly traditional teaching practices.y
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+Val. 33, ianuary 2005
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In summary, we have seen that- the assigned seat location in a large lecture hall lias a significant effect on students’ attendance, grades, and beliefs about pliysics. This i.s in spite of ma.ny activities that were designed to engage all students in the class and some activities specifically aimed at students at the rear of the room. It would be interesting to see how such seat location effects might vary wich different instructional styles and student populations. Acknowledgments XJe gratefdly acknowledge the NSF and the University of Colorado at Boulder for providing thc snpport for this work. We also thank all the memb a s of the Phpics Education Research Group at Colorado for many valuable discussions.
learning physics, students’ beliefs about physics, and sustainable reform.
Department of Physics, UCB 390, University of Colorado at Boulder, Boulder, CO 80309: Katherine. [email protected] Carl E. Wieman is a Distii?guishedProfessor of Physics and a Fellow of JlLA at the University of Colorado at Boulder. He received his 6,s.from M./.T: in 1973 and his Ph.D. from Stanford University in 1977. He maintains active research programs in physics education and in laser spectroscopy and Bose-Einstein condensation, for which he was awarded the Nobel Prize in physics in 2007. A highlight of his physics education work is the Physics Education Technology Project, which creates online interactive simulations for learning physics (http://phet. colorado.edu).
Department of Physics, UCB 390, University of Colorado at Boulder, Boulder, CO 80309; cwiemanO jila.colorado.edu
References C. Weinstein, “The physical environruent ofthc sclrool: A review ofthe research,” Rev. Educ. Kes. 49, 577-610 (. 1979). 2. L.A. Bloomfield, I-lou) Things Work: The Physics of&rryduy Lifl. (Wiley, New York,200 1). I. .
E. LMazur, Perr tnstrLrction:A Lisrr? Mmaual (Prentice Hall, New Jersey, 1997). 4. H-TI-; see b t t p : / / ~ ~ . l ~ - i t t . c o m / . 5. Having &signed groups and consensus answers had a notable beneficial.impact on che amount and level of 3.
student discussion compared to the more traditional infoorm.al peer instruction we used the previous year.
6.
4. 13artlett, “The Frank C. Walz lecture halls: A new concept in the design of lecture auditoria,” Am. /. Phy. 41,1233-1240 (1973).
7.
W.K. Adams, K.K. Ptrkins, M. Dubson, N.D. Finkelstein, and. C.E. Wieman, “Design and valida.tion of the Colorado Learning About Science Survey,” in review for Priiceedings oftbe PERCzOM, Sacramento, CA; http:// cosmos.colorado.edu/phet/surveylCLASS/.
E.F. Redish, J.M. Saul, and R.N. Steinberg, “Student. expectations in introductory physics,’’Am. J Phys. 66, 212-224 (March 1998). PACS codes: 01.40D, 01.40Gb, 01.40R 8.
Katharine K. Perkins is a research associate and lecturer in the Deparfment of Physics at the University of Colorado at Boulder. She received her B.A. in physics in 1992 and her Ph.D. in atmospheric science in 2000 both from Harvard Universiv. Her current researcli interests include the use of interactive simulations for teaching and
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Correlating Student Beliefs With Student Learning Using The Colorado Learning Attitudes about Science Survey K. K. Perkins, W. K. Adams, S. J. Pollock, N. D. Finkelstein and C. E. Wieman Department of Physics, University of Colorado, Boulder, CO 80309 Abstract. A number of instruments have been designed to probe the variety of attitudes, beliefs, expectations, and epistemological frames taught in our introductory physics courses. Using a newly developed instrument - the Colorado Learning Attitudes about Science Survey (CLASS)[11 - we examine the relationship between students’ beliefs about physics and other educational outcomes, such as conceptual learning and student retention. We report results from surveys of over 750 students in a variety of courses, including several courses modified to promote favorable beliefs about physics. We find positive correlations between particular student beliefs and conceptual learning gains, and between student retention and favorable beliefs in select categories. We also note the influence of teaching practices on student beliefs.
ever, in courses specifically designed to attend to student attitudes and beliefs.[2]
INTRODUCTION In addition to the traditional content within any course, there are extensive sets of attitudes and beliefs about science that are taught to our students. How we conduct our class sends messages about how, why, and by whom science is learned. Such messages are being studied with the goal of developing more expert-like views on the nature and practice of science in our students. [2][3][4]
With these new measures of student beliefs about physics and about learning physics, the question emerges as to how these factors impact and are impacted by students’ pursuit of physics study and their mastery of the content.[lO] In this paper, we begin to examine the relationships among these different aspects of student learning. We look at: 1) the influence of teaching practices on student beliefs; 2) the relationship between students’ beliefs about physics and their decisions about which physics course to take and whether to continue on in physics; and 3) the relationship between student beliefs about physics and their conceptual learning in the physics course.
Over the last decade, physics education researchers have used several survey instruments to measure these attitudes and beliefs and to distinguish the beliefs of experts from the beliefs of novices.[5][6][7][8][9] Experts think about physics like a physicist. For instance, they see physics as being based on a coherent framework of concepts which describe nature and are established by experiment. Novices see physics as being based on isolated pieces of information that are handed down by authority (e.g. teacher) and have no connection to the real world, but must be memorized.
DATA The Colorado Learning Attitudes about Science Survey (CLASS)[l] was used to measure student beliefs at the start @re) and end (post) of several introductory physics courses. The newly-developed CLASS survey builds on the existing attitude surveys (MPEX[5], VASS[6], EBABS[7]). The details of the design and validation of the CLASS are reported by Adams et al.[l] The survey consists of 38 statements to which students respond using a 5-point Likert scale. The ‘Overall’ favorable score is measured as the average percentage of statements to which the students
Data have shown that, traditionally, student beliefs become more novice-like over the course of a semester.[5] Even in courses using reformed classroom practices that are successful at improving student conceptual learning of physics, student beliefs tend not to improve.[4] Some success has been achieved, how-
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answer in the favorable sense, e.g. as an expertphysicist would. The ‘Overall’ unfavorable score is similarly determined. The survey is used to measure specific belief categories by looking at subsets of statements. Here, we include measurements of the following facets: ‘Conceptual Understanding’ (physics is based on a conceptual framework), ‘Math Physics Connection’ (equations represent concepts), ‘Sense Making / Effort’ (I put in the effort to make sense of physics ideas), ‘Real World Connection’ (physics describes the world), and ‘Personal Interest’ (I think about physics in my life).
8.2% for the Calc-I courses taught at MMSU. While the Alg-I course was traditionally taught, the Calc-I course was taught using interactive engagement methods; however, neither course specifically attended to improving student attitudes and beliefs about physics. These declines are consistent with those observed in similar courses.[5] In contrast, we do not see these declines in the LSRU courses. All of these LSRU courses incorporated teaching practices specifically aimed at improving student beliefs. Despite being constrained to a large lecture format, these courses resulted in increases of 1-2% in the ‘Overall’ score.
We look at the influence of teaching and at student’s course selection and retention using data from six courses. These courses range in size (less than 40 to over 600 students), student population (non-science majors; pre-meds; physics, chemistry, and engineering majors), and school setting (from a large state research university (LSRU) to a mid-size multipurpose state university (MMSU)). Table 1 lists the courses as well as the other data available for each course.
Course selection and retention. The courses listed in Table 1 represent a range of commitments to the study of physics. We see that the students’ incoming favorable beliefs on the ‘Personal Interest’ category increases with the level of the physics course in which students enrolled. The non-science majors average only a 54% favorable belief while the average for the course for physics majors is 74%. Thus, students who make larger commitments to studying physics tend to be those who identify physics as being relevant to their own lives - as measured by ‘Personal Interest’.
We look at the correlation with conceptual learning using data from the LSRU’s large calculus-based course in Spring 2004. The large number of the students allows us to examine sub-groups of students and retain good statistics. The structure of this course included multiple reforms designed to improve student learning and student beliefs, including interactive engagement in lecture, tutorial-style recitations, and an emphasis on conceptual understanding. In addition, the development and application of expert-like beliefs and approaches to problem solving were emphasized across the course components. For a detailed description of the reforms and an analysis of the contribution of various reforms to learning see Pollock.[ 1 11
RESULTS AND DISCUSSION Influence of teaching practices. In Table 1, we show the average pre- and post- ‘Overall’ percent favorable score for the six introductory physics courses. In bold, we see a decline of 9.8% for the Alg-I and
In addition, we see that the non-science majors who chose to continue and take the second term had significantly more favorable ‘Overall’ and ‘Personal Interest’ beliefs than those who did not - their scores being 14% and 15% more expert-like, respectively. For the MMSU Alg-I course, a significant number of students (41) dropped the course. The students who completed the course had a substantially higher favorable ‘Personal Interest’ score initially (64%) than those who dropped (49%). These two pieces of data suggest a link between retention of students (both within and across courses) and students’ favorable beliefs.
Student Beliefs and Conceptual Learning. The LSRU’s large calculus-based courses were highly successful at achieving their goal to improve student learning.[ll] On two standard exams for measuring conceptual learning, the students achieved median normalized gains of 0.67 on the FCI[12] (Fall 2003) and 0.76 on the FMCE[13] (Spring 2004). While the
TABLE 1. Evident correlations between favorable ‘Personal Interest’ and physics course selection Dominant # of Normalized ‘Personal Interest’ Course School student students learning ‘Overau’ %favorable on Pre-test Type TypelTerm population wl CLASS gains Pre Post Shift (Std. Err) (Std. Error of Mean) Non-Sci-I LSRUffa03 non-sci 76 57% 58% +1.0%(1.5%) 54% (3%) Non-Sci-I1 LSRU/SpO4 non-sci 36 71% 72% +1.4%(2.2%j 69% (5%) -9.8%(2.8%) 64% (3%) Alg-I MMSUfFa03 pre-meds 35 g-FCFo.13 63% 53% Calc-I LSRUIFa03 engineers 168 g-Fc14.67 65% 67% +IS% (1.2%) 70% (2%) Calc-I LSRUKp04 engineers 398 g FMCE4.76 68% 70% +1.5%(0.7%) 72% (1%) __ Calc-I MMSU/Fa03 physics maj 38 g FCI4.35 65% 57% -8.2%(2.7%) 74% (4%j I=ls’ semester, II=2”dsemester; typical standard deviation for ‘Overall’is -16% -
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TABLE 2. Correlations between beliefs and learning Belief Correlations of category normalized FMCE gain with* Pre-beliefs Post-beliefs (p-value) (p-value) Overall 0.21 (0.0008) 0.26 (0.00002) Conceptual Understanding 0.22 (0.0005) 0.30 (0.00001) Math Physics Connection 0.20 (0.001) 0.20 (0.001) Sense Making / Effort 0.11 (0.09) 0.17 (0.007) Personal Interest 0.03 (0.63) 0.15 (0.01) Real World Connection 0.02 (0.79) 0.08 (0.19)
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* for students in LSRU Calc-I Spring 2004 with FMCE pre-test<60.
median normalized gain was quite high, some students had significantly smaller learning gains and a large number of students had much higher learning gains. With prelpost CLASS and FMCE data on 307 students from Spring 2004, we are able to explore the relationship between students’ beliefs and their learning gains. In Table 2, we show the correlations between students’ beliefs and their normalized learning gain. We limit these to the 90% of the students who had FMCE pre-test scores < 60 (Thomton’s level of conceptual mastery) because we are interested in students who have not already mastered the material (N=256).[13] The correlations in the various belief categories range from 0.02 to 0.22 for pre and 0.08 to 0.30 for post. We see the categories of ‘Conceptual Understanding’ and ‘Math Physics Connection’ have correlations with learning gains that are statistically different fi-om zero; where as, ‘Real World Connection’ and ‘Personal Interest’ do not. It makes sense that expert beliefs in these former two categories are more important to the form of learning measured by the FMCE. We also see larger correlations with post-beliefs than pre-beliefs in almost all categories. While these correlations are small, it is notable that they are larger than the correlations of learning gain with homework, attendance, and a math pre-test, which are all 0.22 or lower.[l 11 In Figure 1, we show students average pre- and post- beliefs as a function of normalized learning gain for ‘Overall’ and ‘Conceptual Understanding’. Two trends emerge: (1) students with higher conceptual gains tend to have more favorable beliefs in these categories and (2) students in the lowest gain category tend to regress in beliefs, while higher performing students tend to post gains in beliefs. It is important to note that the correlations between beliefs and learning gain listed in Table 2 apply to this course with its teaching practices, student population, and curriculum and with its choice of instrument for measuring conceptual learning. Changes to any of these elements can effect the correlations with beliefs. Analysis of the observed affect can then help interpret the meaning of the change and provide some more in-
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(N=23)
(N=51)
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FIGURE 1. Average pre- and post- beliefs vs. normalized conceptual learning gain. sight into the relationship of student beliefs and leaming. For example, the Fall 2003 LSRU course had the same instructor (Pollock) using the same teaching practices and materials, but using a different measure of conceptual learning gain (the FCI). While the data do show the same general trends as observed in Figure 1, they show somewhat stronger correlations between normalized learning gain and students’ beliefs. In addition, some categories which show relatively weak correlations here show stronger correlations with the FCI learning gain, and vise versa. These differences likely reflect the differences in the learning measured by the two instruments.[131 The correlations reported here are between individual belief categories and normalized learning gain. If multiple categories contribute to improved learning, it is likely that a weighted combination of categories would result in higher correlation coefficients and is a topic of future analysis. Higher correlations may also result from the ongoing refinement of the CLASS statements and categories to better probe beliefs.[ 11
As a start to understanding possible cause and effect, we have looked further at the relationship between pre-belief and normalized learning gain and the meaning of the observed correlation of 0.2 1. In Figure 2, we have binned the students by their incoming ‘Overall’ belief and plotted the percent of students within each belief bin who achieve learning gains of greater than 0.8, 0.8 to 0.3, and less than 0.3. We see that over 50% of the students with favorable incoming beliefs of 80- 100% achieve normalized learning gains in excess of 0.8 and that this percentage decreases for each successive bin while the percentage of students with learning gains less than 0.3 increases. A chisquared test shows that the observed combination of distributions of students within each bin are statistic-
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FIGURE 2. Conceptual learning gain vs. Pre‘Overall’ belief. cally different (p=0.006) from the class-average distribution - 14% g<0.3, 43% 0.3O.8. These data are consistent with the idea that students’ beliefs about science when they enter a course influence their conceptual learning. Of course, we cannot rule out the alternative conclusion that other factors are simultaneously influencing both students’ incoming beliefs and their conceptual learning. It is also interesting to consider how learning is impacting or impacted by changes in students’ beliefs over the course of the term. One approach is to select a subgroup of students with nearly identical incoming beliefs and look at the connection between their sh$t in beliefs and their learning gain. We preformed an analysis of this type for the data presented here. The results are suggestive of a positive relationship between occurrence of favorable shifts in beliefs over the course of the term and higher learning gains, but additional data is necessary to investigate this further.
CONCLUSION Our analysis of students’ attitudes and beliefs measured using the CLASS suggests that beliefs play a role both in the physics courses students choose to take and in their retention within those courses and in the discipline. In addition, we observe positive correlations between student beliefs and normalized conceptual learning gains. We see that students who come into a course with more favorable beliefs are more likely to achieve high learning gains. These data are consistent with the idea that beliefs are a factor in student learning; however, we must consider the possibility that other factors are simultaneously influencing students’ beliefs, their learning, and their choices in pursuing physics. In our work, we found that it is possible to create environments, even in large lecture courses, that support improved student beliefs about physics and that this specific attention can avoid the common decline in beliefs commonly observed in introductory physics courses.
The authors thank NSF for providing the support for this work. We are also very grateful to all of the instructors who generously agreed to administer the CLASS in their courses. We also give thanks to all the members of the Physics Education Research at Colorado group for many valuable discussions.
REFERENCES 1 W. Adams et al., “Design and Validation of the Colorado Learning About Science Survey” in review for Proceed-
ings of the PERC 2004, Sacramento, CA. http:l/cosmos.colorado.edu/uhetlsun/ev,‘CLASS/ 2 D. Hammer, Am. J. Phys. 68, S52-S59 (2000). 3 A. Elby, Am. J. Phys. 69, S54-S64 (2001). 4 Redish, Teaching Physics with Physics Suite, Wiley 2003 5 E.F. Redish, J.M. Saul, and R.N. Steinberg, Am. J. Phys. 66,212-224 (1998). 6 I.A. Halloun, “Views About Science and Physics Achievement: The VASS Story.” In The Changing Role of Physics Departments in Modem Universities: Proceedings of the ICUPE. 7 A. Elby, Epistemological Beliefs Assessment for Physical Science h~te:/iww\y2.uhvsics.uind.edu,’-clbviEBAPS/ homchtm 8 D. Hammer, “Misconceptions or p-prims: How may alternative perspectives of cognitive structure influence instructional perceptions and intentions?’ Journal of the Learning Sciences, 5,97-127, (1996). D. Hammer, “More than misconceptions: multiple perspectives on student knowledge and reasoning, and an appropriate role for education research,” Am. J. Phys., 64, 13161325, (1996). D. Hammer, “Discovery learning and discovery teaching,” Cognition and Instruction, 15,485529, (1997). 9 Chi, M.T.H, Feltovich, P.J., and Glaser, R. “Categorization and representation of physics problems by experts and novices,” Cognitive Science, 5, 121-152, (1981). 10 A. Schoenfeld, “What‘s all the fuss about metacognition,“ in A. Schaenfeld, Cognitive science and mathematical education. Philadelphia: Lawrence Erlbaum. 1987. 11 S. Pollock, “No Single Cause: Learning Gains, Student Attitudes, and the Impacts of Multiple Effective Reforms’’ in review for Proceedings of the PERC 2004, Sacramento, CA. 12 R. Hake, Am. J. Phys. 66,64-74, (1998) 13 R. Thornton and D. Sokoloff, Am. J. Phys. 66, 338-352, (1998); also R. Thornton, D. Kuhl, K. Cummings, and J. Man, “Comparing the FMCE and the FCT’, unpublished.
Patricia Blanton, Column Editor Department of Physics and Astronomy Appalachian Stale University, Boone, NC 28608; [email protected]
Minimize Your Mistakes by Learning from Those O f Others, by Carl Weman, University of colorado, Boulder, co 80309-0440 Carl Wieman is a distinguished professor of physics at the University of Colorado at Bouider and has been teaching undergraduate physics for nearly 30 years He was awarded the Nobei Prize in Physics in 2001 and the NSF Director’s Distinguished Teaching Schoiar award the same year He was named CarnegielCASE National Professor of the Year for 2004
ather than repeat much of the ood advice of past columns, I have listed here some general ideas that I keep repeating to myself. I know they are true, but I am still struggling to fully integrate them into my own teaching of physics.
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1. Minimize your mistakes by learningjorn those of others. The best place to start is by learning what research tells us about how people learn in general and how they specifically learn physics. The three references are an excellent starting point. Perhaps the most basic and fiequently neglected idea from this research is the concept of “cognitive load.” The more information and ideas that people have to deal with, the less effectively they can process them. In fact, once a limit of about seven ideas in short-term working memory is reached, processing pretty
much shuts down. This means that every iiew tupic, digression, or technical term that you introduce in class, 10 mattcr how unimportant to the primary goal of the course, will use up that precious limited cognitive capacity of your students’ brains. Incidenrally, this problem of cognitive load applies eqtially well to technical t a l k given to your peers.
2.Stzdent beliej are crucialfor learning. Beliefs have a profound influence on student niotivarion and learning. For all aspeccs of the course you should ask yourself: (1) \Vhy is this iriarerial v,orrh learning?, mtl (2) How can I convince the st:idents of rhar? ~ ~ s u a l lthis y, latter question means figuring out how to relare the material ro their lives in some way. Almost no one is motivated to learn something that is total!y disconnected from hidher interests or existence, par-ticulatly if the only justifica:ion is “because some da:; you will need to know this.”
3. Listen to your students. People learn b>7 building on their previous thinking. YQUi studeni-s are ntccssarily buildirig on different backgrounds and inenmi frarneu.orks than !’ours. No matter how clear
and lucid your understanding of the subject is you can never simply pour that understanding into them. How well students learn (or not) will be determined by how well you listen and understand what they are thinking and then guide that thinking. There are many wavs to “listen in on student thinking” including “Just-inTime Teaching,”i questions and personal response systems in class,-’ and literally listening in on student discussions during peer instruction3 or group problem-solving discussions.
4. Make your students active partners in the learningprocess. Discuss explicitly with them the purpose of the class, what you want them to learn, and how the class is designed to accomplish that learning. The more your students become responsible partners in the learning process, the more they will learn and the more they- (and ;:OLI) will enjoy it.
5. Focus on reasoning and discourse. Keep scientific reasoning and discourse at the forefront in all aspects of your course rather than letring them get buried under facts and formulas. If all you do is recite (and/or rest) facts and problem-solving recipes, most students will conclude that memorizing these is what learning physics is all about (/?OW boring!).
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6. Beflexible. If you follow my earlier advice, you will regularly find yourself in class torn between either completing your carefully prepared lesson plan/lecture or stopping to address either some point of confusion or an interesting and relevant extension of the topic that has been raised by rhe students. Keep reminding yourself that often student learning is best served by gritting your reerh and abandoning much of your exquisitely prepared material to more directly address their current thinking. Finally, remember that reaching is like politics. There will almost always be a few vocal students who dislilie both you and physics no matter what you do and other students who love you. Don" be discouraged by the complaints or distracted by flattery. Concentrate on the learning ofthe large, often silent, majority.
References 1. Commission on Behavioral and Social Sciences and Education, How Peopk Learn (Naciond Academies Press, Washington, D.C., 2000). 2. R. Mayer, Learning arid brmxctioii (Prentice Hall, Upper Saddle Rivet-, NJ, 2002). 3. E. Redish, Teaching Physirs with tbe Physics Sztirr (Wilcy, New, York. 2003).
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Engaging Students with Active Thinking Wieman, Carl E This Peer Review issue focuses on science and engaged learning. As any advertising executive or politician can tell you, engaging people is all about attitudes and beliefs, not abstract tacts. There is a lot we can learn froin these professioiial coininuiiicators about how to effectively engage students. Far too often we, as educators, provide students with the content of science-often in the distilled formal representations that we have lound to be the most concise and general-but fail to address students' owii attitudes and beliefs. (Although heaven forbid that we should totally abandon reason and facts, as is typical in politics and advertising.) What does it mean for studeiits to be meaningfully engaged in learning science? I would argue that it means that students are both actively thinking about the subject and applying scientific ideas to solve problems, in much the same manner as an expert. So how well are our science courses accomplishing this? Recent research (including some froin my own research group) has measured studeiits' attitudes and beliefs about physics and physics problem solving and how introductory physics courses affect these beliefs. W i a t is consistently found is that completing such courses actually shifts students' beliefs to be less expert-like. For example, a large fraction become convinced of such statements as, "To solve a physics problem, I should look for an equation that has the variables given and just plug in the values." Or they believe statements like, "The subject of physics has little relation to what I experience in the real world," or "I cannot leaiii physics if the teacher does not explain it well in class." This shift towards less expert beliefs is seen even for courses that incorporate a number of "interactive engagement" methods associated with good conceptual learning gains. We also see direct evidence that student beliefs are very important: they correlate with learning gains, course retention, and the inclination to pursue (or switch out) of physics as a major. Although most of the data comes from physics, our sampling of courses in other sciences shows similar negative impacts of instruction on student beliefs.
If we are going to seriously engage students in learning science, we must attach as much importance to student beliefs, and how different teaching practices affect those beliefs, as to the content we cover. W e must recognize that when we present material in formal, abstract ways, use unnecessary technical jargon, and assign homework and exam problems that are correspondingly abstract and can be completed by following memorized recipes, we are teaching more than just content. To a student who does not share our experience and expert insight, we also teach that the subject is abstract and disconnected from the real world, that problem solving is basically rote memorization, and that there is no use for solving a science problem other than to pass a course. This needn't be the case. With a little effort, virtually all introductory content can be presented in terms of understanding the behavior of real-world phenomena, with little or no obscuring technical terminology. We can also learn from advertising how to choose those illustrative phenomena that will most attract and interest students in the subject. Problem solutions can require reasoning and have obvious real-world utility. I have found that rather modest efforts of this sort have a substantial impact on the beliefs about physics for science and nonscience majors. Only when we recognize that education is more about changing student minds than transferring information, and guide our teaching and evaluation of learning accordingly, will we be able to truly engage and educate students in science. The opinions expressed in the article do not necessarily reflect the official position of the National Academy of Science. By Carl E. Wieman, distinguished professor of physics, University of Colorado at Boulder; 2004 Council of Advancement and Support of Education and Carnegie Foundation U S . Professor of Year; recipient of 2001 Nobel Prize in physics; and chair, National Academy of Science Board on Science Education Copyright Association of American Colleges and Universities Winter 2005 Provided by ProQuest Information and Learning Company. All rights Reserved
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Transforming Physics Education By using the tools of physics in their teaching, instructors can move students from mindless memorization to understanding and appreciation. Carl Wieman and Katherine Perkins The science community needs to change science education to make it effective and relevant for a much larger fraction of the student population than in the past. This need is the result of significant changes in the environment and society over the past several decades. First, society now faces critical global-scale issues that are fundamentally technical in nature-for example, climate change, genetic modification, and energy supply. Only a far more scientifically and technically literate citizenry can make wise decisions on such issues. Second, modern economies are so heavily based on technology that having a better understanding of science and technology and better technical problem-solving skills will enhance a person's career aspirations almost independent of occupation. Furthermore, a modern economy can thrive only if it has a workforce with high-level technical understanding and skills. As a community, we must now ask ourselves, "How successfully are we educating all students in science?" This objective is very different from in the past, when the goal of science education was primarily to train only the tiny fraction of the population that would become future scientists. The new, broader educational need does not eliminate the need to educate future generations of scientists. However, improving science education for all students is likely to produce more and better-educated scientists and engineers. This claim is supported by data showing that the fraction of students who complete a physical science major in college is determined more by the students' ability to tolerate traditional physical science instruction than by their ability to do science.' For a variety of reasons, the physics community should and can take the lead in providing an effective and relevant science education for all students. Moreover, this is in their enlightened self-interest. A better-educated citizenry would better appreciate the value of supporting physics research. But what specifically do we mean by effective physics instruction? It is instruction that changes the way students think about physics and physics problem solving and causes them to think more like experts-practicing physicists.2 Experts see the content of physics as a coherent structure of general concepts that describe nature and are established by experiment, and they use systematic concept-based problem-solving approaches that are applicable to a wide variety of situations. Most people ("novicesf1)see physics more as isolated pieces of information handed down by some authority and unrelated to the real world. To novices, "learning" physics simply means memorization of information and of problem-solving recipes that apply to highly specific situations2 725
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Research on traditional instruction We now examine how well traditional instruction does at getting the average student to think like an expert. Traditional science instruction is used in the overwhelming majority of college physics courses and has familiar characteristics. Most of the class time involves the teacher lecturing to students; assignments are typically back-of-the-chaptertype homework problems with short quantitative answers, and grades are largely based on exams containing similar problems. Over the past couple of decades, physics education researchers have studied the effectiveness of such practices. (For reviews with useful citations, see references 3-5 and the article by Edward Redish and Richard Steinberg, PHYSICS TODAY, January 1999, page 24). In this section, we present representative examples of research on three quite different but important aspects of learning: conceptual understanding, transfer of information, and basic beliefs about physics. The first aspect of learning, conceptual understanding, has been extensively studiedu and is particularly relevant because the great strength of physics is that a few fundamental concepts can explain a vast range of phenomena. Most studies have looked at students' learning of basic physics concepts in traditional introductory physics courses. The results are remarkably consistent. We will discuss two examples, one from mechanics and one fiom electricity. Physics education researchers have developed several carefully constructed tests that explore student understanding of the basic concepts of force and motion. These tests have been administered at the beginning and end of many, many courses across the country. The oldest and best-known test is the Force Concepts Inventory Figure 1 (FCI).6Figure 1 shows a sample question from the FCI and results compiled by Richard Hake from data on 62 courses (14 traditi~nal).~ As shown in the figure, students receiving traditional instruction master, on average, less than 30% of the concepts that they did not already know at the start of the class. The result is largely independent of lecturer quality, class size, or institution. Eric Mazur, a highly renowned teacher at Harvard University, has studied students' understanding of concepts 1 in electricity. Motivated by FCI results, Mazur gave his students an exam with a series of paired problems8 such as those shown in figure 2. His and similar data show that 1i students are able to correctly answer traditional test questions and complete traditional courses without understanding the basic physics concepts or learning the useful concept-based problem-solving approaches of physicists.
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We next examine a second aspect of learning, simple transfer of information and ideas from teacher to student in a traditional physics lecture. The following example is from data collected in our own introductory physics class for non-science majors. After explaining the physics of sound in our usual incredibly engaging and lucid fashion, we brought a violin into class. We explained how, in accordance with the physics we had just explained, the strings do Figure 3 not move enough air to create the sound from the violin. Rather, the strings cause the back of the violin to move via the soundpost, and thus it is the back of the violin that actually produces the sound that is heard. Fifteen minutes later, we asked the students the multiple choice question shown in figure 3, "The sound you hear from a violin is produced mostly by . . ." As illustrated in the figure, only 10% gave the correct answer. We have seen that this 10% level of retention after 15 minutes is typical for a nonobvious or counterintuitive fact that is presented in a lecture, even when the audience is primarily physics faculty and graduate students. When we have asked physics teachers to predict the student responses to the violin question, nearly all of them greatly overestimate the fraction of students who answer correctly. Many physics faculty go so far as to simply refuse to believe the data. For readers who may share their skepticism, we briefly mention two other studies. Redish had students interviewedjust as they came out of his l e ~ t u r eThe . ~ interviewer simply asked the students, "What was the lecture about?" The students were unable to recall anything beyond the general topic. In a more structured study,' Zdeslav Hrepic and coworkers gave 18 students six elementary questions on the physics of sound. Immediately after attempting to answer the questions, the students were told that they were to get the answers to the six questions from watching a 14-minute commercially produced videotaped presentation given by a nationally renowned physics lecturer. For most of the six questions, no more than one student was able to learn the correct answer from the lecture, even under these highly optimized conditions! When presented with these data, teachers often ask, "Does this mean that all lectures are bad?" The brief answer is no, but to be effective, lectures must be carefully designed according to established, but not widely recognized, cognitive principles about how people learn.lo Our third topic is research on students' general beliefs about physics and problem solving in physics. Research groups including our own have studied these beliefs through extensive interviews and well-tested surveys.l 1 These surveys measure where students' thinking lies on the expert-novice scale discussed above, and how their views are changed by taking a physics course. The surveys have now been given to many thousands of students at the beginning and end of introductory physics courses at many different institutions. After instruction, students, on average, are found to be Iess expert-like in their thinking than before. They see physics as less connected to the real world, less interesting, and more as something to be memorized without understanding. This is true in almost all courses, including those with teaching practices that have substantially
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improved conceptual mastery. If it is any consolation to physics teachers, we have measured similar results from introductory chemistry courses. The examples we have discussed are just a few from a large body of research on the effectiveness of the traditional approach to teaching physics. The definitive conclusion is that no matter how "good" the teacher, typical students in a traditionally taught course are learning by rote, memorizing facts and recipes for problem solving; they are not gaining a true understanding. Equally unfortunate is that in spite of the best efforts of teachers, typical students are also learning that physics is boring and irrelevant to understanding the world around them.
A better approach Is there a way to teach physics that does not produce such dismal results for the typical student? Our answer, and that of many others doing research in physics education, is unequivocally yes. Many of the same methods that have worked so well for advancing physics research also improve physics education. These methods include basing teaching practices and principles on research and data rather than on tradition or anecdote; using new technology tools effectively; and disseminating and copying proven results. Considerable evidence shows that this approach works. Classes using research-based teaching practices have shown dramatic increases in retention of information, doubling of scores on the FCI and other conceptual tests, and elimination of negative shifts in beliefs about physics. Research on learning has provided results that both explain many of the disappointing results of traditional instruction and provide guidance as to how to improve. We present three examples here, chosen in part because they are relatively easy to use throughout the standard curriculum and classroom setting. Numerous other examples, including many about specific physics topics, are given in references 3-5.
Figure 4
Cognitive research shows that the amount of new material presented in a typical class is far more than a typical person can process or learn. People's brains function in a way somewhat analogous to a personal computer with very limited random-access memory. The more things the brain is given to process at the same time-the cognitive load-the less effectively it can process anythingI2 (see figure 4). Any additional cognitive load, no matter what form it takes, will limit people's abilities to mentally process and learn new ideas. This is one of the most well-established and widely violated principles in education, including by many education researchers in their presentations.
Cognitive load has important implications for both classroom teaching and technical talks. To maximize learning, instructors must minimize cognitive load by limiting the amount of material presented, having a clear organizational structure to the presentation,
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linking new material to ideas that the audience already knows, and avoiding unfamiliar technical terminology and interesting little digressions. Expert competence5'12is a primary goal of education and is another area in which research has provided usefil insights. Expert competence has been found to have roughly two parts: factual knowledge and an organizational structure that allows the expert to effectively retrieve and apply those facts. Organizing physics ideas around general concepts is part of building such a structure. If students do not have a suitable organizational structure, simply pouring additional facts on them may actually deter learning.
To move a student toward expert competence, the instructor must focus on the development of the student's mental organizational structure by addressing the "why" and not just the "what" of the subject. These mental structures are a new element of a student's thinking. As such, they must be constructed on the foundation of students' prior thinking and e~perience.~,'~ This prior thinking may be wrong or incorrectly applied, and hence must be explicitly examined and adequately addressed before further progress is possible. The physics education research literature can help instructors recognize and deal with particular widespread and deeply ingrained misconception^.^^^ In summary, expert competence is likely to develop only if the student is actively thinking and the instructor can suitably monitor and guide that thinking. Our final example of useful research concerns students' beliefs. Students' beliefs about physics and how it is learned are important.'." They affect motivation, approaches to learning and problem solving, and, not surprisingly, choice of major. As we noted earlier, teaching practices influence students' beliefs, usually by making them more novice-like. Presenting mechanics in terms of general concepts and the motion of abstract items such as blocks on frictionless ramps can inadvertently teach many students that these principles do not apply to real-world objects. Assigning problems that are graded strictly on a final number, or that can be done by plugging the correct numbers into a given procedure or formula, can teach students that solving physics problems is only about memorization and coming up with a correct number-reasoning and seeing if the answer makes sense are irrelevant. The good news is that courses with rather modest changes to explicitly address student beliefs have avoided the usual negative shifts.'' Those changes include introducing the physics ideas in terms of real-world situations or devices with which the students are familiar; recasting homework and exam problems into a form in which the answer is of some obvious utility rather than an abstract number; and making reasoning, sense-making, and reflecting explicit parts of in-class activities, homework, and exams.
New educational technology Utilizing principles established by educational research can greatly improve physics education. Technology can make it easier to incorporate these principles into instruction. For example, online surveys and student-faculty e-mail are rather simple ways to enhance communication, thereby helping faculty understand and better guide student
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thinking. Here we will discuss a couple of more novel technologies-personal electronic response systems and interactive simulations. These technologies are relatively simple and inexpensive, and we have found them to be pedagogically powerful and easy to incorporate into the standard curriculum. A variety of commercial vendors sell personal electronic response systems, or "clickers" as they're usually known to our students. The various systems are all based on a similar idea. Each student owns a clicker and uses it to answer multiple-choice questions asked during class. A computer records each student's answer. After all the responses are in, the system displays the answers in a histogram like that given in figure 3. Software grades the responses and allows the instructor to later examine each student's answer. A clicker system for a classroom of about 200 seats requires several receivers, a computer, and a projector; the total cost is about $5000. If used properly, clickers can have a profound impact on the students' educational experience. The value of the clicker is that it provides a way to quickly get an answer for which the student is accountable, and that answer is anonymous to the student's peers. While the clickers provide some measure of what students are thinking, it is the specifics of the implementation-the change in the classroom dynamic, the questions posed, and how they are followed up-that determines the learning experience. These specifics need to be guided by an understanding of how people learn. Instructors must also make sure their students understand how and why the clickers are being used. If students perceive clickers merely as a way to give more tests, rather than as a method to improve engagement and communication, the clickers will be resented. We have found that the biggest impact of clickers comes when they are used with a combination of practices that others have developed. We randomly assign students to groups the first day of class (typically three or four students in adjacent seats). Each lecture is designed around a series of about six clicker questions that cover the key learning goals for that day. Although multiple-choice questions may seem limiting, they can be surprisingly good at generating the desired student engagement and guiding student thinking. They work particularly well if the possible answers embody common confusions or difficult ideas. Useful clicker questions and valuable guidance on writing effective questions are now a~ailable.*~'~ It is important to actively encourage students to talk to each other about the questions. We do this, sometimes after they have answered individually, by requiring our groups to come to a consensus answer, enter it with their clickers, and be prepared to offer reasons for their choice. Those peer discussions are the times when most students are doing the primary processing of new ideas and problem-solving approaches. Critiquing each other's ideas to arrive at a consensus answer also enormously improves their ability to carry on scientific discourse. Finally, the discussion helps them to learn to evaluate and test their own understanding. Experts have the ability to monitor and test their own thinking on an ongoing basis by asking questions like "Does this make sense?" and "How can I test this?" However, it is very difficult for students to learn this skill without some amount of social interaction and feedback. The student discussions in our classes are inspired by the
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peer instruction technique popularized by M aw. ' The clickers and consensus groups just provide a way to enhance the process, particularly for the less active or less assertive students.
A major value of clickers is how they can enhance communication in the classroom. The sometimes painful feedback provided to the instructor by histograms like figure 3 is the most obvious. However, there are other, more valuable forms of feedback. By circulating through the classroom and literally listening in on the consensus-group discussions, the instructor can quickly learn particular points of student understanding and confusion. Then in the follow-up lecture or whole-class discussion, the instructor can directly target those specific items of confusion. Perhaps even more important than the feedback to the instructor is the feedback provided to the students through the histograms and peer discussions. Students become much more invested in their own learning. One manifestation of this change is that we now receive many more substantive questions, and they are asked by a much broader distribution of students; 10-1 5 questions per class period is typical. Clickers can also be useful in other ways. For instance, we use them to quickly survey the range of student backgrounds and to quiz students at the start of class to check that they've done the assigned reading. Ensuring background reading considerably facilitates useful in-class discussions.
The reality of virtual physics Interactive simulations that run through a regular Web browser can be highly effective. Using an existing simulation also often takes less preparation time than more traditional materials. Our research group has created and studied the effectiveness of about 45 simulation^.'^ We have explored their use in lectures, as part of homework problems, and as laboratory replacements or enhancements. Figure 5 shows our circuit construction kit simulation. This simulation allows one to build arbitrary circuits Figure 5 involving lifelike resistors, light bulbs, wires, batteries, and switches; measure voltages and currents with realistic meters; and see light bulbs lighting up. It also shows what cannot normally be seen---electrons that flow around the circuit with their velocity proportional to current, immediately responding to any changes in circuit parameters. Our studies14have found this simulation helps students understand the basic concepts of electric current and voltage and, when substituted for an equivalent lab with real components, improves how well students can build and explain real-life circuits. Many physicists find it quite mysterious and somewhat disturbing that carefully developed simulations are more educationally effective than real hardware. Both the efficacy of simulations and the physicists' discomfort can be understood by recognizing the difference between how the beginning student and the expert instructor perceive the
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same situation. These perceptual differences are readily apparent in our testing of simulations and in other research on the effectiveness of lecture demonstrations.l5 A real-life demonstration or lab includes enormous amounts of peripheral information that the expert instructor filters out without even thinking about it. The student has not learned what can be filtered out, and so all this other information produces confusion and a much heavier cognitive load. The student's attention is often on things the instructor doesn't even notice, because they are irrelevant. For example, in a real circuits lab, inexperienced students will often spend considerable time and concern on the significance of the different colors of the plastic insulation on the wires. A carefully designed computer simulation can maintain connections to real life but make the student's perception of what is happening match those of experts. This is done by enhancing certain features, hiding others, adjusting time scales, and so on, until the desired student perception is achieved. Simulations also can provide visual representations that explicitly show the models that experts use to understand phenomena that are not directly visible, such as the motion of electrons. It is likely that both features are important in explaining the observed benefits of simulations. The educational importance of recognizing and dealing with differences between student and expert thinking goes well beyond the use of simulations. An apt metaphor is that of the student and the expert instructor separated by the mental equivalent of a canyon; the function of teaching is to guide the student along the path that leads safely and effectively across the canyon to the nirvana of expert-like thinking. Guidance that ignores the student's starting point or that is interpreted differently than intended usually just sends the student over a cliff. But education research, carell measurement, and new technology make it possible to guide most students safely along the path toward a true understanding and appreciation of physics.
We are pleased to acknowledge the valuable inputfrom all the members of the University of Colorado at Boulder physics education research group.
Carl Wieman, corecipient of the Nobel Prize in Physics in 2001 and the Carnegie-CASE US University Professor of the Year in 2004, is a Distinguished Professor of Physics at the University of Colorado in Boulder. Katherine Perkins is an assistant professor of physics attendant rank at the University of Colorado.
References 1. I . E. Seymour, N. Hewitt, Talking About Leaving: Why Undergraduates Leave the Sciences, Westview Press, Boulder, CO (1997); K. Perkins et al., http://ww . c o l o r a d o . ~ d ~ ~ h ~ ~ i ~ s / ~ d u ~ a t i o n I s s u e s l PFormatted-submitte ERC2005 d.pdf. 2. 2. D. Hammer, Cogn. Instr. 15,485 (1997). 3 . 3. L. McDermott, E. Redish, Am. .I. Phw. 67, 755 (1999) IINSPECj.
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4. 4. E. Redish, Teaching Physics with the Physics Suite, Wiley, Hoboken, NJ (2003). 5. 5. J. Bransford, A. Brown, R. Cocking, eds., How People Learn: Brain, Mind, Experience, and School, National Academy Press, Washington, DC (2000). 6. 6. D. Hestenes, M. Wells, G. Swackhamer, Phys. Teach. 30, 141 (1992) ISPINI; see ref. 4 for a compilation of other useful concept surveys. 7. 7. R. Hake, Am. .I Phys. 66, 64 (1998) rlNSPEC1. 8. 8. E. Mazur, Peer Instruction: A User's Manual, Prentice Hall, Upper Saddle River, NJ (1997). 9. 9. Z. Hrepic, D. Zollman, N. Rebello, in Proc. 2003 Physics Education Research Conference, J. Marx, K. Cummings, S. Franklin, eds., American Institute of Physics, Melville, NY (2004), p. 189. 10. 10. D. Schwartz, J. Bransford, Cogn. Instr. 16,475 (1998). 11. 11. See the discussion of the Maryland Physics Expectations (MPEX) survey in ref. 4, and the articles by W. Adams et al. and by K. Perkins et al. in Proc. 2004 Physics Education Research Conference, J. M a , P. Heron, S. Franklin, eds., American Institute of Physics, Melville, NY (ZOOS), p. 45 and p. 61, respectively 12. 12. R. Mayer, Learning andInstruction, Merrill, Upper Saddle River, NJ (2003). 13. 13. I. Beatty et al., http:l/umperg.physics.umass.edu/libra~/~per~-20050 1 /entirepaper. 14. 14. Those simulations, along with copies of research articles, are available at the Physics Education Technology website htt&het.colorado.edu. 15. 15. C. Crouch et al., Am. J. Phys. 72,835 (2004) [SPIN]; W.-M. Roth et al., JRes. Sci. Teach. 34, 509 (1 997) . 16. 16. C. Crouch, E. Mazur, Am. .I. Phys. 69,970 (2001) J'INSPECI; S. Pollock, in Proc. 2004 Physics Education Research Conference, J. M a , P. Heron, S. Franklin, eds., American Institute of Physics, Melville, NY (2005), p. 137. ._____
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Students master relatively few concepts in physics courses using traditional instruction. The histogram shows, for the Force Concepts Inventory (FCI), the average normalized learning gain-that is, the fraction of the concepts that students learned that they did not already know at the start of the course. Results from 14 traditional courses are in red, and results from 48 courses using a wide variety of interactive-engagement techniques are shown in green. Superimposed on the histogram are data (blue arrows) from two large lecture courses that use well-tested research-based practices,l6 The inset shows a figure accompanying a typical FCI question: Students are asked which path the ball will follow upon exiting the tube. (Adapted from ref. 7.)
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Figure 2. Paired problems compare students' ability to calculate quantitative answers with their conceptual understanding. (a) Students were asked, "For the circuit shown, calculate (a)the current in the 2-52 resistor and ( b ) the potential difference between points P and Q." The average score of 69% on the question indicates that most of them were able to calculate the currents and voltages in this moderately complex DC circuit. (b) Those same students performed much worse (average score of 49%) when asked to explain what happens qualitatively to, for instance, the brightness of these light bulbs and the current drawn from the battery when you close the switch S-questions that seem far simpler to any physicist. The message is that students can answer traditional test questions without really understanding basic physics concepts or mastering conceptbased problem-solving approaches. (Adapted from ref. 8, with permission of the publisher.)
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Figure 3. Counterintuitive facts are not retained by lecture students. Fifteen minutes after being explicitly told that it is the back of the violin that produces the sound, students were given the boxed multiple-choice question. The histogram of their responses shows that only 10% answered correctly.
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“Mr. Osbarne, may I be excused? My brain is full.” Figure 4. Students in lecture are apt to suffer from cognitive overload.
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,The circuit construction kit is a simulation that allows students to build virtual circuits containing a number of different elements. A pedagogically useful feature of the simulation is that it displays motion of the electrons, shown here in blue. Students who work with the interactive simulation are better able to understand and build real-life circuits. The simulation and many others are available at the Physics Education Technology website, ref. 14.
Exploring Student Understanding of Energy through the Quantum Mechanics Conceptual Survey S. B. McKagan* and C. E. Wieman” *Departmentof Physics and JILA, University of Colorado, Bouldel; CO, 80309, USA Abstract. We present a study of student understanding of energy in quantum mechanical tunneling and barrier penetration. This paper will focus on student responses to two questions that were part of a test given in class to two modem physics classes and in individual interviews with 17 students. The test, which we refer to as the Quantum Mechanics Conceptual Survey (QMCS), is being developed to measure student understanding of basic concepts in quantum mechanics. In this paper we explore and clarify the previously reported misconception that reflection fiom a barrier is due to particles having a range of energies rather than wave properties. We also confirm previous studies reporting the student misconception that energy is lost in tunneling, and report a misconception not previously reported, that potential energy diagrams shown in tunneling problems do not represent the potential energy of the particle itself. The present work is part of a much larger study of student understanding of quantum mechanics. Keywords: physics education research, conceptual surveys, quantum mechanics, tunneling, misconceptions PACS: 01.4O.Fk,O1.40.Gm,01.5O.Kw,03.65.-~
INTRODUCTION Quantum mechanics is a fascinating subject because it is so challenging to the intuition. Learning quantum mechanics requires learning to accept such counterintuitive notions as “particles” reflecting off barriers even though they have enough energy to cross them as well as tunneling through barriers that they do not have enough energy to cross. Perhaps even harder than accepting these notions is actually understanding them. Previous research shows that even when students accept strange ideas, they often do not understand them [ 141. In order to change this, we must first gain a clearer understanding of how students actually think about these concepts. We are in the process of developing an instrument called the Quantum Mechanics Conceptual Survey (QMCS) [S] to measure understanding of basic concepts in quantum mechanics. The QMCS is a multiple-choice survey, designed to provide quantitative data to complement and extend the qualitative interview data that already exists on this subject. Through in-class tests and student interviews, we have used the QMCS to elicit and explore student thinking about many concepts in quantum mechanics. Here we focus on two QMCS questions that were developed to hrther explore student misconceptionspresented in a previous study [ 11. We elaborate on the source and extent of these misconceptions and present a new misconception not seen in previous work. We present results from two modem physics classes where the QMCS was given at the end of the Spring 2005 semester and from 17 student interviews. The two classes used the same textbook [6] and covered similar
material, but were taught by different professors with different teaching siyles. One class was intended for engineering majors (ENGspOS, N=68) and one for physics majors (PHYSspO5, N=64). The interview subjects included four students from ENGspOS, nine students from PHYspOS, and four students who took the equivalent of ENGspOS in a previous semester (ENGfaM), taught by a different professor. In interviews, students were asked to work through the QMCS, thinking out loud and explaining why they chose the answers they did. The interviewer (SBM) asked questions to further probe their thinking. There is some controversy in the Physics Education Research community over the definition of the word “misconceptions” and the extent to which students have them [7]. In this paper we will take the perspective that student thinking can take many forms, ranging from fragmented and incoherent ideas that apply only in certain contexts to robust theories that are consistent across all relevant contexts. Here we will use the word “misconception” to mean any incorrect student idea that can be clearly articulated and is seen consistently in numerous students in at least one context.
REFLECTION: A RANGE OF ENERGY? In an extensive study of student understanding of wave properties of light and matter [ 11, Ambrose has reported on the “failure to recognize that reflection occurs at the boundary between regions of different potential or wave speed” and the “mistaken belief that reflection and transmission of a beam of particles is due to a range of energies of the particles in the beam.” Using a survey with an
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open-ended question similar to that shown in Fig. 1, he found that many students did not believe that any electrons would be reflected, using the classical reasoning of answer A. Of those students who knew that some electrons should be reflected, many thought the reason was that the beam contained electrons with a range of energies, as stated in answer B, in spite of the fact that the beam was described as honoenergetic”. We adapted the question in Fig. 1 from Ref. [l] in order to further explore these misconceptions. We wanted to determine the extent to which students hold these misconceptions and why. Further, we wanted to determine to what degree the misconception described by answer B was due to students’ simple misunderstanding of the word “monoenergetic,” and whether it was robust enough to appear even if the contradiction between the problem statement and the answer was more apparent. Our results show that this misconception is quite robust; most students hold on to it even when the contradiction is explicitly pointed out. A beam of electrons that all have tk same energy E are traveling through a conductingwire. At x = 0. tk wire becomes a different kind of metal so that the potential energy ofthc ckctmns increases h m m o to UO.IfE > UO,which statement most accurately describes the transmission and reflectionof elemom?
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A All the ~lectronsare transmittedbecause they all have E > UO. 6. Some ofthe electrons are transmitted and some arc reflected bccause they achlally have a range ofenergies. C. Some of the electmm are transmitted and some are retlectcdbecause they bchave as waves. D. All afthc electmm are reflcacd because they prefer to bc in the region with IOWR potential energy. E. None ofthe above statements are correct.
FIGURE 1. A barrier penetration question from the QMCS. This question is adapted from an open-ended question in Ref. [l], and the distracters are based on student responses reported therein. We have changed the wording from “Monoenergetic electrons” to “A beam of electrons that all have the same energy E ’ , and added the figure and description of the wire to make the question more grounded in physical reality.
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reflection occurred in some cases. In ENGspO5 a nearly identical question had been discussed in class at great length, which probably explains why these students did better on this question than the PHYspO5 students, although the PHYspO5 students did better on most QMCS questions. In spite of instruction, the test results, coupled with the interviews discussed below, show that a significant fraction of students in both classes held the misconceptions described by answers A and B. Out of the 15 students interviewed on this question, eight students initially selected answer B. In all of these cases, the interviewer then asked, “How do you reconcile that answer with the statement in the question that all the electrons have the same energy?’ In response to this question, three students stuck by their answer, giving detailed justifications, two students changed their answers to C (the correct answer), two students changed to A, and one student changed to D. Of the three students who defended answer B, two used the Heisenberg uncertainty principle, arguing that you could never really know the energy, and E was just the average energy. The third student gave a more elaborate explanation, based on a misunderstanding of a type of diagram commonly used quantum mechanics in which wave functions are drawn on top of energy levels: “. .. every picture I’ve ever seen where they tell us what the wave function is and they say it has this energy E , he draws a line down the middle and then draws the wave function around it. And I guess I just internalize that as saying that. ..that’s like their average energy. . .” Of the 15 students interviewed, we argue that seven had a robust misconception that reflection at a barrier is caused by electrons having a range of energies. In addition to the three students who defended answer B, there is strong evidence that the three students who switched to answers A or D, as well as one student who initially selected answer A, also held this misconception. These four students all argued that in this case, in which all the electrons had the same energy, they would all be transmitted (or reflected), but in all the other cases they had discussed in class, in which there was reflection, the electrons must have a range of energies. We view this misconception as an extension of the first misconception discussed by Ambrose, that all electrons with sufficient energy should be transmitted. It is essentially a way of reconciling the first misconception with the remembered fact that sometimes electrons are reflected.
FIGURE 2. Percentageofstudents who selected each answer to the question shown in Fig. 1 on a test given in two classes.
Fig. 2 shows the answers that students selected for the question in Fig. 1 on the in-class exam. It should be pointed out that transmission and reflection through a barrier was discussed in both classes, and that all interview subjects from these two classes remembered that
TUNNELING: ENERGY LOSS AND THE MEANING OF POTENTIAL ENERGY Several previous studies have found that students often believe that particles lose energy in tunneling [ 141.
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While these studies provide extensive interview data on this misconception, there is little quantitative data on the extent to which it is held, as we provide here. Morgan, Wittmann, and Thompson [3] suggest several explanations for why students might believe that energy is lost in tunneling. One explanation is that most textbooks and lecturers draw the energy and the wave function on the same graph, leading many students to confuse the two, believing that the energy, like the wave function, decays exponentially during tunneling. A second explanation is classical intuition about objects physically passing through obstacles, in which energy usually is dissipated. Muller and Sharma [4] propose another explanation: students may be thinking of the energy of an ensemble of particles, rather than the energy of a single particle. In this case, since not all of the particles are transmitted, it is actually correct that an ensemble as a whole loses energy during tunneling. uppose that $the experznt described in the previous question*,you would like to ccnasc the s p e d ofthe elcctmn coming out on the right side. Which ofthe following hangcs to the apimental sef-up would decrase this speed? A lnnrascthcwidthwofthegap: U(X)
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B. lncrcase Uo,the potential energy ofthe gap:
C. Increase the potential energy to the right ofthe gap:
D. Decrease the potential emrgy to the right ofthe gap: C/(X)
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E. More than one ofthc changes above would decrease the s p e d ofthe electron The previous question in the test, which is not discussed in t h s paper. states “An ‘ectronwith energy E is traveling thmugh a conductingwire when it encounten a small ap in the wire ofwidth w. The uotcntial enerev ofthe electron as a function ofoosition &en by the plot [above lei?], where Uo > E.:
FIGURE 3. A tunneling question fiom the QMCS. This question was developed to explore the misconception that energy is lost in tunneling.
Fig. 3 shows a QMCS question designed to elicit the misconception that energy is lost in tunneling. In interviews, all students who selected answers A, B, or E argued that since energy was lost in tunneling, making the bamer wider andlor higher would lead to greater energy loss. The fraction of students who selected one of these three answers (62% in ENGspO5 and 53% in PHYspOS)), shown in Fig. 4, gives us a lower bound on the fraction of
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FIGURE 4. Percentage ofstudents who selected each answer to the question shown in Fig. 3 on a test given in two classes.
students who hold the misconception that energy is lost in tunneling. It is only a lower bound because in interviews, even students who gave the correct answer with the correct reasoning often second-guessed themselves and wondered whether energy might be lost in cases A and B as well. Some students who initially held the misconception eventually chose the correct answer because cases D and E, which were unlike any examples they had seen in class, forced them to consider the energy on the right of the barrier more carefully. In interviews we saw evidence for all three of the explanations for this misconception listed above. We found this question particularly useful in exploring student thinking about energy in tunneling, not only because it elicits the idea that energy is lost, but because understanding the correct answer, C , requires a clear understanding of the relationship of potential, kinetic, and total energy in the context of tunneling. In many cases, this question elicited significant cognitive dissonance, as students struggled to reconcile two contradictory ideas: that energy is lost, and that kinetic plus potential equals total. While it is technically possible to reconcile these ideas if it is the kinetic and total energy that is lost, we found that most students who thought that energy is lost did not have a clear idea of which energy is lost. When asked, they were just as likely to say potential energy as any other kind. Often a single student would use two or even all three types of energy interchangeably within the same explanation. Most of the interview subjects had fragments of both the correct view and the view that energy is lost simultaneously. Of the four students interviewed from ENGspOS, all held a robust misconception not seen in any previous study, and not seen in any of the interview subjects from the other courses. These students thought that the quantity U ( x ) ,plotted here and in nearly every problem involving solutions to the Schrodinger equation, is not the potential energy of the electron itself, but some kind of “external energy.” We discovered this misconception in the first interview conducted in this study, in which a student drew an exponentially decaying curve over the potential energy graph shown in the question, and
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consistently referred to this curve as representing “the potential energy.” The interviewer then asked what the graph shown in the question represented, since it was also referred to as “the potential energy.” The student replied, “I don’t know, that’s just the bump that it goes through. I don’t know what it means. I just see that and I know that it’s some kind of obstacle that it goes through.” When pressed, he said that the “bump” was “the external energy that the electron interacts with” and insisted that it was not the potential energy of the electron itself, in spite of the fact that it was explicitly labeled as such. The interviews with other ENGspOS students were very similar, with all of them referring to the graph as either “the external potential energy” or “the potential energy of the medium,” and quickly dismissing the idea that it was the potential energy of the electron itself. It is unclear why this misconception was held so robustly by all of the interviewees from one course and not present at all in the interviewees from the other two courses. The sample sizes are small and the courses, as well as the student populations, were different. Therefore we do not wish to speculate on which factor caused the discrepancy. However, it seems unlikely that this misconception is confined to students in this particular course, and we hope that other researchers will continue to probe student thinking about this topic in other contexts. We find this misconception interesting because while many studies have shown that students think energy decays in tunneling, none of these studies discuss how students reconcile this idea with the fact that they are often drawing these decaying curves on top of graphs of energy curves that are not decaying. The source of this misconception becomes clearer if we consider that textbooks and lecturers nearly always refer to “a particle in a potential” as if the potential is something external. One physics professor, in a discussion of these results with SBM, stated that the potential is an external thing, caused by an external field of some kind. SBM replied that it may be caused by an external field, but the potential energy is a property of the particle itself. The professor said that he always thinks of it as a potential, rather than a potential energy, in which case it is not a property of the particle itself. We suspect that many physics professors think this way, easily switching between potential and potential energy in their minds, forgetting that the distinction between the two is not a trivial factor of charge q in the minds of their students. In all other fields of physics, the symbol V is used for potential and the symbol U is used for potential energy, but most quantum mechanics textbooks use the symbol V for potential energy (the textbook used in these classes is an exception to this rule) and then use the terms potential and potential energy interchangeably. Few textbooks give any kind of physical explanation for the source of the potential energy term in the Schrodinger equation.
Because we had seen this confusion over the meaning of potential through the semester, throughout the QMCS we used the symbol U ( x )rather than V ( x )and referred to this symbol as “the potential energy function of the electron.” In interviews students simply skimmed over this unfamiliar phrase and focused on the familiar symbols. It is important to note that one should not overinterpret the statistical results given in Fig. 4 as indicating the fraction of students in a class that hold a particular view. First, answer E does not distinguish between students who think that A and B alone are correct and students who think that A, B, and C are correct. Furthermore, even if this ambiguity were resolved, for example by allowing students to mark more than one correct answer, the answers alone would not tell us much about the thinking of those students who held fragments of the correct view and the view that energy is lost. In interviews we found that these students selected a wide range of answers. Some students who held both views decided that A, B, and C were all correct, while others picked only one or two of these options, either at random or because one sounded slightly more plausible. Several students changed their answer after several minutes of thinking through all the implications, which would not have happened under normal test-taking circumstances. This project is the first step in a comprehensive study of student thinking about quantum mechanics. We have found the QMCS to be a usehl tool in probing student thinking, and will extend these results in further studies.
ACKNOWLEDGMENTS We would like to thank Dana Anderson and Mihaly Horanyi for useful feedback on the QMCS and for allowing us to give it to their classes, the physics education research group at the University of Colorado for additional feedback, Brad Ambrose, Michael Wittmann, Jeff Morgan, and Derek Muller for discussions of their studies on these topics, and finally, the students who participated in interviews. This work was supported by the NSF.
REFERENCES 1. B. Ambrose, Ph.D. thesis, University of Washington (1999).
2. L. Bao, Ph.D. thesis, University of Maryland (1999). 3. J. T. Morgan, M. C. Wittmann, and J. R. Thompson, Phys. Educ. Res. Con$ Proceedings (2003). 4 . D. A. MulIer, and M. D. Sharma, in preparation (2005). 5 . http:Ncosmos.colorado.edu/phet/survey/qmcs/ (2005). 6. J. R. Taylor, C. D. Zafiratos, and M. A. Dubson, Modern Physics for Scientists and Engineers, Pearson Prentice Hall, 2004,2 edn. 7. A. Elby, J. Math. Behav., 19,481 (2000).
PHYSICAL RFVJEW SPtCIAL TOPICS •• PHYSICS EDUCATION RESEARCH 2. O i O I O l (2006)
New instrument for measuring student beliefs about physics and learning physics: The Colorado Learning Attitudes about Science Su rvey W. K. Adams, K. K. Perkins, N. S. [''odolefsky, .VI. Dubson, N, D, Finkclstd.n, and C. H, Wicman Depanmem of Physic*, iiurtcmiy >.)j'Colorado, Roiiidt'r, Cutoiud!/ Mt.'Jil'J, f/.sV! I Received 22 Mav yi(X: published 10 January 2!X>6) The Colorado Learning Attitudes about Science Survey (CLASS) is a new instrument designed to measure student beliefs about physics and about learning physics. This instrument extends previous work by probing additional aspects of studrnL beliefs and by IKMUT wording suitjble for students in a wide variety of physics courses. The CLASS has been validated using intervii; 1 ,^;,. r e i i j b i l i i y studies, and extensive statistical itrulyses of responses from over 5000 students, in addition, a new methodology for determining useHil ana stfuisiiwilly robust categories of'stiuteni bel'cts h,is been developed. This piipti-r stives as the f o u n d u l i f / r i !"ur ;ii! extensive study of how student beliefs impact and lire impacted by their edueiitujiuil experience;,. Fur c\u-'np!e, this survey measures [he following: (hat most teaching practices uuusr s n b s u i n l i a l drops in ssuden: sco:es; 1:h,i! a student's likelihood of becoming a physics major uorralutes wills their "Persona! Interest" score; and rhnt, for a majority of student populations, women's scores in som-? ,:;itt'p.orles. i n c l u d i n g "Persona! Interesl" and "Real World Connections/" ;;re s i g n i f i c a n t l y different from men'* scores. DO!: 10.1 K)1/PhvsRevSTPER.2.010101
PACS numherls): 01 .-iO.Fk. 01.40.G---. ni.4U.Di, 01.50.K.W
I. INTRODUCTION
Over the last decade, researchers in science education have ideniificd a variety of siijdcru attitudes and belief's thai shape, and are shaped by student classroom experience. 1 '"' Work by House"- 6 and Sadler andT;ii' indicate that students' expectations are better predictors at college science performance than the amount of high-school science or math they completed. House found that students" achievement expectations and academic self-concept were more significant predictors of chemistry achievement than were students' prior achievement and their prior instructional experience. Sadler and Tai found that professor gender matching student gender was second only to quality of high-school physics course in predicting students" performance in college physics. A n u m ber of surveys have been created to measure various aspects of student's beliefs and expectations. We have developed and validated an instrument, the Colorado Learning Attitudes about Science Survey (CLASS)," 'which builds on work done by existing surveys. This survey probes students' beliefs about physics and learning physics and distinguishes the beliefs of experts from those of novices. The CLASS was written to snake the statements as clear and concise as possible and suitable for use in a wide variety of physics courses. Students are asked to respond on a Likert 13 (live-point, agree lo disagree) scale to 42 statements such as the following. ( 1 ) "[.study physics to learn knowledge that will be useful in rnv life outside of school." (2) "After 1 study a topic in physics and fee! thai 1 understand it, I have difficulty s o l v i n g problems on the same topic." (3) ''If I get stuck on a physics problem my first try, 1 usually try to figure out a different: way that works." The statements are scored overall and in eight categories. What we mean by the term "category" is fundamentally different from what is meant by that label as used in previous belief surveys in physics. Our categories are empirically de1:554-9! "?K,<2006/2( I),'()!0101(14)
termined groupings of statements based on student responses. This is in contrast to n prion groupings of statements by the survey creators based on their belief as to which statements characterize particular aspects of student th.ink.rn0. Some researchers argue that mil all of a student's ideas about learning; physics have become coherent and thus it does not matter whelher or i;o( their responses to statem e n t s w i t h i n a categoiy are correlated. Our empirically determined categories and interviews demonstrate that students do have many consistent ideas about learning physics and problem solving; although we have found certain ideas, such as ihe nature of science, where our interviews and survey results suggest that students do no! have coherent ideas, at least none that we are able to measure. Our empirical approach to category creation identifies', through statistical analysis of student responses, those aspects of student thinking where there is some reasonable degree of coherence. 11 The degree of coherence is itself an empirically determined quantity. The definition oi what aspect of t h i n k i n g such an empirically determined category describes is determined entirely by the statements that our analysis shows the students answer in a correlated fashion. Normally, one can see from looking at the groups of correlated statements that these represent certain identifiable aspects of t h i n k i n g that the teacher can address. Rennie and Parker 1 " provide a particularly powerful example, w h i c h supports the value of empirically determined categories, (hey present an attitude survey designed to focus on the singSe idea of interest in science. The instrument was giver) !.o lour to seven year olds and analyzed as a whole: they found no difference between boys" and gills' interest in science. The researchers believed, based on theory, that the questions could be broken into four types; learning about science, doing experiments, "work with...," and "create or grow...." When analyzed using this categorization scheme, very little difference was seen between boys and girls. Then, a factor analysis was performed on the data and two different categories emerged that showed very dear distinctions be-
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tween boys and girls Girls preferred items relating to plants, animal, and shadows w h i l e boys were more interested in energy. whecH. and earthworms. There were also several items that they wore equally interested in such as weather, This research demons! rates that student ideas may not be cieariy understood u priori hu! can be idcmincd through statistical analysis of responses. Tn (his paper we first describe the design principles used for the CLASS and how these principle's and the instrument itself d i f f e r iron) previous survevs. We then discuss how it. was validated, ami how the eight general categories of s t u dent beliefs that ii measures were determined. A number of subtleties involved in choosing and interpreting these categories are discussed. We also present results of studies eon ducted to confirm Ihe survey's r e l i a b i l i t y and o u t l i n e the important factors that must be considered when interpreting results of this survey, e.g , student gender. Since the survey development and v a l i d a t i o n have gone through three iterations, these sections are necessarily rather interconnected. Finally, we present a few brief examples of the results we arc finding insrn widespread use of this survey. I I . DESIGN Three well-known survevs for probing student beliefs about the physical sciences are the Maryland Physics Expectation survey (MP^X).'- the Views About Science Survey (V'ASS),'' :uid Ihe EpiMemologiea! Beliefs Assessment about Physical Science iEBAPS;. ! X Each of the three has. a particular locus, primarily aspects of epistemoiogy or expectations. Some focus on breadth while others delve into a limited number of ideas in depth. There are also severs.! other natureof-science surveys, such as Views of N a t u r e Of Science (VNOSi. 1 0 Several design principles shaped the CLASS and distinguish i! Iroirs the previous surveys, f j j I E was designed t o address a wider variety of issues that educators consider imporiaju aspects of learning physics. l'2) Die wording of each statement was earefuHy constructed and tested to be clear and concise and subject lo only a s i n g l e interpretation by both a broad population of students and a range of experts. Tins makes the survey >iutab!e for use in many different courses covering a range of levels, and also allows most of the statements to ba readily adapted for use in other sciences such as chemistry. (,3) The statements were written to he m e a n i n g f u l even to students who had n e v e r taken physics. For t h i s reason we chose not to include statements that ask the student to rcllcct an ihc requirements of the course. (4; The '"expert" and "novice" responses to each statement were unambiguous so scoring of the responses was simple and obvious. (5) The amount of time required to thoughtfully complete the survey was kept to ten minutes or less by requiring clear and concise statements and using a simple response format. This also limits the .vjrvey. to less than about 50 stateroeoLS, We believe that a longer survey will encounter significant difficulties with widespread t'acuhy arid student acceptance, i f i l The administration and scoring were designed to be easy, allowing for an online survey and for automated scoring. (7) The grouping of statements into cat-
egories of student beliefs was subject to rigorous statistical analysis and only statistically robust categories were accepted. The resulting categories characterize identifiable and useful aspects of student t h i n k i n g . Our initial starting points for the survey statements were MPEX and VASS statements. We modi lied many of these statements to make them consistent w k h the guidelines above, particularly after evaluating them in interviews with experts and students. We found that the most effective way to successfully modify and create statements was to listen to students and write down statements thai: we heard them sav. These statements then represent student ideas about learning and arc in a vocabulary sluGenis understand. Here we mention some of the issues that arose in these interviews, t i l Words such as "domain" and "concepts" are not prevalent in a typical introductory student's vocabulary, and so need to be avoided to make die survey suitable for a broad range of students. (2) Students {'though perhaps not physicists) apply the word physics in al least three ways, referring to their particular physics course, the scientific discipline, or the. physics that describes nature. The survey is designed for use iii the context of an academic environmenl; however, we believe it is important to ask qucslions not specifically about the course bt'.t rather about the physics that describes nature; noting that this sense sometimes overlaps with physics as a discipline. !!' .statenients do refer to the course, students sometimes have varied responses such as referring to their high-school course or t h e i r college course. We. do not claim that this survey would not elicit different responses for some statements if it were given in a completely different context; it has been designed and validated for this particular context. (3) Statements that include two siatements in one, as do a number of statements on the .MPEX. are often interpreted inconsistently by students, although not by experts. A number of new statements were also created to address certain aspects of learning not covered by the earlier surveys such as personal interest, aspects of problem solving, and the coupled beliefs of sense making and effort. III. SCORING Scoring is done by deteirnining, for each student, the percentage of responses for which the student agrees with the experts' view ("percent favorable") and then averaging these i n d i v i d u a l scores to determine the average percent, favorable. The average percent unfavorable is determined in a comparable manner. The survey is scored "overall" and for the eight categories listed in Table 1. Each category consists of i'our to eight statements that characterize a specific aspees of student thinking, as shown in the Appendix. Together, these categories include 27 of the statements. The overall score includes these statements plus an additional nine statements, all of which pass our v a l i d i t y and reliability tests. The rem a i n i n g ssx, to complete the 42-statement survey, are statements that do not have an "'expert"' response or are statements that are not useful in their current form. Table I shows typical CLASS V.3 restills for a calculusbased Physics I course (A r =397) from a large state research university (LSRIJ). These arc typical results for a first-
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TABLE I. Typical CLASS percent favorable results. The percentage of favorjble responses (students agreeing witli experts!, pre- and postsemester, given by A''-:,*1)? students, taking a reform-oriented course thai led to a 0.58 nunnali/ed gain on the Force and Motion Conceptual E v a l u a t i o n (Ret'. ?(!!. See the Appendix tor category details, .Standard error-, sue in parentheses.
Post Overall Rual world connections Personal interest Sense making/error! Conceprtiitl a>jinee;:ons Applied coiicepTUiii understanding Problem solving genera! Problem solving confidence Ptoble?)! solving sophistication
semester course—-regardless of whether it: is a t rail! t ion til lecture-based course or a course w i t h interactive engagement in which the instructor does not attend to students' a t t i t u d e s add beliefs about physics. The standard deviations vary with class, but they are typically '15-20% for the overall score and 25-30 % for the categories. The post standard deviations are typically slightly larger than the pre. The stunriard deviation and hence uncertainty (standard error) is iniiuenccd in part by the number oi' statements included in the particular category being scored -with fewer statements, the m i n i m u m difference between the individual perecni favorable scores is larger, which will result in a larger standard deviation. There are ?wo common methods for scorine Liken scales/' - 1 One can assume that the characteristics under study can be considered as either an interval scale or an ordinal scale. When assumina an interval scale each possible response receives a value from 1 to 5 (1 for strongly agree and 5 for strongly disagree with the spacing between each of these values bearing equal wsiphl. The responses tor each item can then be summed. The second method, ordinal scale, assumes there is not equal difference between each possible response; therefore, scoria" must be done MS a presentation of percentage of agreement. In our interviews, students expressed a variety of reasons for choosing neutral. incSudim: the following: has no idea how to answer; lias no opinion; has conflicting beliefs arising tVorn different experiences in different physics courses: or is coniTiaed between answering according to what they think they should do versus what they actually do in practice. For these reasons, it is clear that it is preferable to score the CLASS survey responses on an ordinal scale. In scoring, neutrals are scored as neither agree nor disagree with the expert so that an individual .student's percent favorable score (and thus the average for the class) represents only the percentage of responses for which the student agreed with the expert and similarly for percent unfavorable. The difference between 100% and the sum of percent favorable and percent unfavorable represents the percent of neutral responses. The use oi' a live-point Likeri. scale of strongly disagree to strongly agree is important for validity and scoring for two
reasons. First, students' interpretations of au.ree vs strong!;1 agree are not consistent; the same conviction of belief may noi result in the same selection such t!;at <>ue student may respond with strongly agree w h i l e another responds with agree. Thus, in scoring (he survey, we treat strongly agree and agree as the same answer (similarly for .strongly disagree and disagree). This has previously been shown to be important when comparing different populations because t h e i r responses are aftected by differences in how conservative the populations are. 2 ' Collapsing the scaie is also frequently done when scoring small samples.- 1 We Sind in interviews and based on the above results thai by collapsing the scale when scoring, we ni;iv have losl some d e f i n i t i o n but have no reason to believe that we have distorted the results. Interviews also revealed that the use of a live-point scale in the survey as opposed to a three-point scaie was impottant. Students expressed that agree vs strongly agree (and disagree vs strongly disagree! was an important distinction and that without the two levels of agree and disagree thev would have chosen neutral more often. When a student skip:* a statement, the survey is scored as if the statement did not e.xist for that student. A siudem must answer a m i n i m u m number of statements on the survey (3? on! of 36 scored statements) to be included in the overall .-core and a m i n i m u m number of statements for each category to he included in the results for thai category. In our experience, only a very small number o! students skip more than two statements, but from a statistical analysis of the difference between dropping skipped statements or including them as a neutral response, we believe thai effectively droppine them from the scoring gives the most accurate results if there is an anomalous population where a large number of students skip many statements.
IV. ADMINISTRATION
Since fall 2003, we have administered Ihe CLASS survey before (pre) and after (post) instruction lo over 7000 students in 60 physics courses. In addition, faculty members from at
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leas! 45 other universities are using the CLASS in their physics courses. After some experimentation, we have settled on the following approach for maximizing the number of student responses from a given class. We ( I ) announce the survey both in class and by emu:!. (2} give a short (three to seven day) window for taking the survey, (?) provide a f o l l o w - u p email reminder lo students who s t i l l need to take the survey, and (4) oiler a small a m o u n t of course credit for s u b m i t t i n g the survey, although the actual responses are not graded and a student recei\e? ful! eiedi! for s u b m i t t i n g only name ane ID. Some student-- w i l l randomly choose answers. We have added statement 3 i to i d e n t i f y the majority of these students f 'We use this statement to discard (he survey of people who are no! reading the statements. Please select agree (riot strongly agree) for iivls .statement." We f i n d iha! 7-12 '# {inversely related to icve! of course) of the students fail to cOfTectiy answer this statement and an experts answered it correctly, 'in addition to statement 31. we have added a timer ior surveys administered online. If students take less than three mmules to complete the survey, we discard their answers. We typically achieve 90% precourse response rate and 85% poslcourse response rate. Of these responses, approximately 10--!? % are dropped because the students did not answer statement 31 correctly, chose the same answer for essentially all the statements, or .simply did not answer most of the statements. The remaining responses provide useful pre and post data sets. To determine the shifts in beliefs from pre to post, ii is important to include only students who took hmh ore and post surveys. This ensures that any calculated change in beliefs measures shifts in students' t h i n k i n g rather than a difference in student, population pre to post. Thus, an additional data set restricted to students who took both pre and post is also created. This matched data set t y p i c a l l y includes about 65 70 % of t h e students enrolled in the course, V. VALIDITY AND RELIABILITY We have performed a series of rigorous validation and reliability studies that i n v o l v e d several iterations to revise and refine ihc survey statements. The validation process included face validity - interviews with and survey responses from physics f a c u l t y ro establish the expert inteipretauon and response; interviews iviih students to confirm the clarity and meaning of statements: construct validity- administration of the survey to several thousand students ibiiovved by extensive statistical anal 1 ;sis of Hie responses i n c l u d i n g a detailed factor analysis to create and verify categories of statements: predictive validity correlation with students' Incoming beliefs and course performance;"'^ 1 and concurrent validity analysis of responses of the .survey to show that it measures certain expected results such as that physics majors are more expertlike in their beliefs than nouscienee majors. Revisions were made to the survcv based on the results of the interviews and factor analysis and then the above validation studies were repeated w i t h the uevv version of the survey. A. Validation interviews Three experts underwent a series of interviews on the initial draft of CLASS V.I (version !, fall 200.'*). Their com-
ments were used to hone the statements and remove any that could be interpreted more than one way. When this process was complete, '6 experts took the survey. Their answers confirmed the expert pom! of view used in scoring. These experts were physicists who have extensive experience with teaching introductory courses arid worked with thousands of students. Some of these experts are involved with physics education research: others are simply practicing physicists interested in teaching. The above process was repeated for CLASS V.3 (version 3, fall 200-4. shown in the Appendix). The experts provided consistent responses ft; all statements in V.J except to four statements, none of which are included in the f i n a l eight categories. Two are "learning style" statements that we do not expect to have a correct "expert" answer, b u t are i n c l u d e d to provide useful i n f o r m a t i o n about student t h i n k i n g . These stalements arc "it is useful for me to do lots and lots of problems when learning physics" and "1 find carefully analyzing only a few problems in detail is a good way for me to learn physics." The other two statements (nos. 7 and 41) targeted beliefs about the nature of science, and are being revised. So Far. we have been unable to find a set of statements that measure student t h i n k i n g about the nature of science and meet our criteria for statistically valid categories. Student interviews were carried out on V.! by obtaining u total of 34 volunteers from six different physics courses at a midsize multipurpose state university (M. MS 1.0 and a large state research university S'LSRU). Eight additional students from three different: physics courses at the LSRU were interviewed to analy/.e V.3 statements. Care was taken to interview a diverse group of students by selecting from introductory courses catering to the full range of majors, having equal numbers of men and women, and having 20% nonCaucasian students. Interviews consisted oj first having the student take the survey with pencil and paper. Then, during the firs! ten minutes, students were asked about their major, course load, best/worst classes, study habits, class attendance, and future aspirations, in order to characleri/e the student and his or her interests. After this, the interviewer read the statements to tfie students while the student looked at a written version. The students were asked to answer each statement using the five-point scale and then talk about whatever thoughts each statement elicited. If the student did not say anything, he/she was prompted to explain his/her choice. After the first few statements, most students no longer required prompting, if the students asked questions of the interviewer, they were iiOt answered until the very end of !he interview. Interview results showed students and experts had consistent interpretations of nearly all of the statements. A few statements on V.I were unclear or misinterpreted by .some of the students. Some of these were reworded or removed in the spring on V.2 of the survey; the remainder were addressed in the fall with V.3, hi addition, the interviews exposed some unexpected student ideas about physics; some of these were incorporated into V.3, Student interviews on V.3 revealed problems with only three statements. Two of the three arc being revised, The third "It is important for the government to approve new scientific ideas before they can be widely accepted" is interpreted differently by experts and novices,
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NEW INSTRUMENT i'OR MEASURING STUDENT... but in this case, the interpretation itself is consistent and indicates an expert-novice distinction, m a k i n g it a useful statement thai w i l l remain on the survey. Finally, these interviews provided some new insights into s t u d e n t s ' t h i n k i n g , such as the distinction between whether students t h i n k that physics describes the real work! and whether they y c m a i l y care or t h i n k abom she physics they experience in their e v eryday lives. This important distinction was not ivetinni/ed in previous surveys.- 1 - 5 B. Validating categories Statistical analyses were used to test the vaiuiily ol the subgroupings of statements i n t o categories. In i b i s regard, the CLASS is different from previous surveys. There is no published s t a t i s t i c a l analysis of the MPEX categories. Inn we had a substantial number of students take the M P H X s u r v e y and did a statistical a n a l y s i s of their responses. We round some MPbX categories were made up of s t a t e m e n t s for which the studcm responses were very w e a k l y correlated. We later found a brief discussion of this point in l;;e inesis o! Saul's," 1 which suggested that he had similai hr,dii>g:s. We believe that this poor correlation between responses in a category indicates that such a category is not valid for chancteri/ing a facet of student thinking. The VASS and liHAPS use essentially the same categories as the MPEX. and we have been unable in lirul any discussion of s t a t i s t i c a l tesis of the validity oi' the categories for those surveys, it is likely that a statistical analysis would show s i m i l a r results u:> those found for the MPEX. Because of this lack of statistical v a l i d i t y to the categories used in previous surveys, here we present a de.taile-d discussion of the approach we developed to obtain categories i n a t are both useful and itniisl:i:ally valid. t. Categorization philosophy There are t w o different philosophies that can he followed in establishing a set of categories—we w i l l label them as '"raw statistical" and ''predeterminism." The raw statistical approach is where one puts in no prior constraints and allows the categories to emerge purely from the data via exploratory factor analysis. Exploratory factor analysis is a statistical data reduction technique that uses a large set of student responses to all survey statements and groups the statements according to the correlations between statement responses This produces a set of factors that are independent, emergent categories. These provide an oblique basis set that best .spans the space of student responses. This approach has been employed with rntuiy survey instruments and e x a m s in the education community, for more detail on factor analysis see Rets. 22. 23, and 27. In predetermimssri, a set of predetermined categories is chosen based on the expert physicists' or teachers" perspective. The categories rellect the experts' eategon/.afion and definition of useful beliefs for learning physics and 'heir assessment of which statements will probe w h i c h of these belief's. This approach is the one used to establish categories in the other beliefs surveys used in physics. In practice, both of these philosophies have strengths and deficiencies, and so we f i n d the optimum procedure is to use
PHYS, RFV. ST PHYS. EUUC. RES. 2. 010101 (2006i a combination. The strength of the exploratory factor analysis is that it guarantees thai one has statistically valid categories, and it provides new insights i n t o student thinking and how best to characterize that t h i n k i n g . For example, it showed us that there is a high correlation between the responses to statements that involve sense m a k i n g and those that i n v o l v e effort, thus revealing that in the s t u d e n t m i n d these were inexorably linked. This suggests that many students see sense making as an a d d i t i o n a l effort and whether or not they do i! is based, on their cost-beneii! a n a l y s i s ol She effort required. This inievprst.at.ion is supported by our interviews. Statements such as ""There are time'- 1 solve a physics problem more than one way to help my understanding" are quite often answered as disagree; however, students who disagree qualify t h e i r answers d u r i n g interviews with comments such as "1 like to do t h i s when i can," "I know it w i l l help me h u t . . . . " and "I try to go hack and do this before the exam but u s u a l l y don't have time." i-aeior a n a l y s i s also snowed thai statements i n v o l v i n g the connection of p h y s i c s w i t h reality separated inio two distinguishable categories, supporting our findings from student interviews. The i\vo categories distinguish between wbeJhe.r students t h i n k that physics describes the real world a n d whether they actually care or think about the physics they experience in their everyday life. A drawback to the raw statistical categories obtained w u h exploratory factor analysis, however, is that many are. not very useful. 'There is a hierarchy of categories according to level of correlation. Some of the categories that have relat i v e l y low correlations cannot be related to any clearly defined aspect of .Student t h i n k i n g and so cannot be related to particular classroom practices or learning goals. Also, ihe mathematical constraint imposed by factor analysis that all statements must be Ikied into independent categories—can cause an undesirable mismatch between the emergent categories and actual student t h i n k i n g , w h i c h does not follow such rigid mathematical constraints. The strength of predeterminisrn is that the categories are by definition useful in that they are of interest to teachers However, they also have some serious weaknesses. The first deficiency of predetermined categories is that some categories may riot be v a l i d when subjected to a statistical test. Predetermined categories are not statistically vali'd when there is l i t t l e or no correlation between responses, reflecting the fact that student beliefs may be organ i/ed or connected quite differently t h a n was assumed in creating the category, i f statements do no! correlate in the students' minds and hence in their responses, we assert that it is unjusiiriiihlc to claim that there is some definable aspect of student t h i n k i n g that can be labeled and measured by such a category. As Rennie and Parker demonstrated, w i t h t h e i r study of four to seven year olds' interest in science, gender differences were apparent only when appropriate groups of statements were determined by a factor analysis. The second deficiency ss that using predetermined categories precludes learning anything new about how beliefs are organi/ed and related in the students' minds. 2. Pragmatic design approach Our approach is an empirical approach, which embraces elements of both oi' the above philosophies to determine the
748
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Raw Statistical Categories P merge from ra1ory Factor AiLrtly.sis
Predetermined Categories Consist [>f researcher classified statements.
Principle componenl. analysis performed. One poleiUiiil category at \\ lime.
Reduced Basis Factor Analysis
Remove slnienienK vvh'eh do not fit or analy/e as two (or more) separate categories.
Add additional statements thai arc highly correlated wi;h current, set oFstatemmils^
Repeiit until ;i rulrusi (duffned below) group of statements (a minimum ot 3 -preferably more) has been wealed.
optimum set ot categories. We lake advaiitatie of die strengths of" both approaches and moid the weaknesses to obtain statistically robust categories that best characterize student t h i n k i n g in the acudemic context in which this .survey is intended to be used, and address facets ot most use to teachers. In the preliminary stage of this approach, we carried out both exploratory factor analysis and statistical tests of a n u m ber of predetermined categories {including those used in earlier surveys such as MPEX). Guided by those results, we then group the statements into new categories that we judge are likely to be useful and are evaluated to be statistically valid. These auegories are not necessarily independent and not all statements must go into a category. This approach is instilled because different aspecls of student beliefs are noi. necessarily independent and because we are not trying ro describe all of our data; ralher, we are trying to identii'y which portions ot our data are useful For describing particular general aspects of student thought .1. Reduced basis factor analysis
We examine the statistical validity ol" these new categories by carrying out factor analysis, but use a basis set that is limited to those statements we believe should be in the category plus a small number of additional statements that are candidates for the category based on their correlations (Fig, I ) . We use the principal components extraction method along with a direct oblimin rotation* 5 ' when performing both the exploratory arid reduced basis factor analysis. The analyses are performed on three sets of data (pretest responses, posttest responses, and the shifts from pre to post:) from three large f i r s t - t e r m introductory courses (physics for nonscieuce majors, algebra-based physics, and calculus-based physics).
After carrying out a reduced basis factor analysis, we evaluaic the scree plots, correlation coefficients, and Facloi loadings as discussed in the factor robustness section beJov., Multiple iterations of ihts analysis and adding or subtracting statements are used to optimise the categories. After determ i n i n g robust categories iu this fashion, we evaluate the statements not included in any category and search for new categories by looking specifically for correlations w i t h those statements. Whenever there were correlation coefficients of 0.15 or greater, we searched for new categories that would include the correlated statements. This categorization process is illustrated in Table il, which lists the original predetermined categories (based on the type of categorization used by other surveys.) and the emergent categories from the "raw statistical" analysis. The "FA results" columns indicate the results of the reduced basis factor analysis. The ''optimum categories" columns list the fate of each predetermined and emergent category after completing the full process of choosing "optimum'" categories on V.2 in summer 2004. Based on interview and factor analysis results, a major revision of the CLASS was undertaken to create V..H. Table 11! shows the o p t i m u m categories for V.2 and the optimum categories found with another lactor analysis done on V.3 with 800 student responses collected in full 2004. Following each category is a numerical rating of the category's robustness, which is described in Sec. V C below. !t is important to note that there is no such thing as a "perfect" set of categories: these are simply our choices as to the best combination of usefulness and statistical robustness. A subtlety of the factor analysis is that the statistical tests of the categories give the clearest results if students with highly expert or highly novice views are excluded. If students are fully expertlike, for example, their responses will be those of the expert and provide only one dimension-—that
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TABLE II. Reduced Basis Factor Analysis (FA) of Categories—CLASS V.2. SS = strong single factor; BQ=better with one or two different .statements; WF = weak factor; NS = .statements didn't make sense together: V1F=multipIe factors; PC = poorly correlated, Predetermined categories
HA results
I !
Optimum categories
'Independence
Conceptual understanding
Coherence
Conceptual under standing
Concepts Reality world view Reality persona! view Math Effort Skepticism
Conceptual undmtanding Real world connection Personal interest Math physics contieciis.>!i Sense making/effon Dropped
Emergent categories
FA results
! Category !
,SS:!
Real world connection and personal interest
SS'1
Real world connection jtid personal interest
BQ
Conceptual uiKkrstanijinii
i i i
of the expert. Typical students do nul yet have I'uliy coherent ideas about learning physics httl do have coherent ideas about specific: aspects of learning. These specific aspects, which are probed by smaller groups of statements, determine the CLASS categories. For this reason we chose to do the initial factor analysis work for V.3 on students who were not as OKnertlike. We combined the responses from the algebra-based physics course and the physics tor nonseienee majors course and then removed the 27 students who had overall scores of over 80% agreement w i t h experts. leaving an A7 of 400. Only 1% ol' students were more than 80?< novicelike, so we did nol exclude students at that end of the distribution. Once a set of optimum categories was established for this data set. a reduced basis factor analysis was performed on the responses from the more expettlike calculus based Physics 1 students (;V=400,I. This analysis confirmed thai the categories were consistent between the different classes. The result is nine robust categories. One of the nine categories included all of the statements in the "problem solving general" category plus four additional statements. Based on additional analysis, we, concluded that this emended problem solving category provided no additional useful information beyond that provided by the ''problem solving peiseral" category and so have not included it here. Thus, eight categories
Optimum categories
WF
Dropped
XS SS Wi7
Dropped Sense mtkinc/erfort Dropped
resulted from our analysis of V.3. as listed in Table Hi. The statements included in each category are shown in the appendix. 4. Category names Category names are chosen after "'optimum" categories have been determined, The name is simply u label, which attempts to summarize the statements within a category. The name discs not define Ihc beliefs contained within a category. One must read the statements to do this. C. Category robustness Robustness of a category is determined by the reduced basis factor analysis ou thai group of statements. Various indicators of statistical validity are evaluated, including the correlation coefficients between statements, ihc percent of variance explained by the weighted combination of statements represented by the first factor, and the factor loadings for each statement in that first factor. (A factor analysis always produces as many factors as statements in the basis.) An example of these indicators for a very robust factor can be seen in Table IV and Fig. 2. Table IV shows the correlation coefficients between the statements and the factor ioau-
TABLE HI. CLASS V.2 and V.3 category robustness ratings.These ratings were clone on post data for calculus-based Physics 1 students at I . S K U . Robustness Personal interest
Personal interest
Real world connections
7.38
Real world connections
Conceptu a I wi der^ta ndi ng
6.11
Conceptual connections
Sense making/effort
5.89
S^nse making/effort
Math physic? connection
6.51 Problem solving sophistication Problem solvim; confidence Problem solving genera! Applied conceptual understanding
750
0 0
1
2
3
4
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TABLE V. Physics I vs Physics I I ; 20U3-20U4 calrulus-based course at MMSU w i t h same instructor ami "students. 'These, results are using V.2 of the survey so the categories are slightly different from those see,
Pest
Full l.,V-4!i Ovenill Real workl connection Personal inlenesr Conceptual understanding Mail; physics connection Spring (iV=4 i i 57% 549i
Overall Real work! connection
63'X
6S')r
Persona! intiMt-st
d2%
(>K%
Sense-makint/Vflbi't Coiiceptna! understanding
69% Tir-
fij'Ji 1 43%
Math physics connection
62'"A
63%
coefficients i2, 1, and 5) are chosen to give the three lenns in the sum the relative weightings that we believe are most appropriate. The shape of the scree plot contributes approximately 45%. while She average correlation coefficients and factor loadings both contribute about 27.5%. The overall factor of 3 is so that the rating of the best category is nearly 10, for convenience. Table I I I indicates the robustness rating lor each of our categories. Between V.2 and V.3 of the survey, we slightly .revised the wording of many of the statements with the intention of making them clearer and improving t h e i r til: to the categories identified in V.2. It can be seen that this resulted in distinctly more robust categories, For comparison, our tenth best category in V.3 had a robustness value of 4.1.
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D. Making valid interpretations To correctly interpret ihc results of the survey, k is important not to assume that all changes in student beliefs arc due purely to instruction. Here we present a list of other factors thin our data have shown are significant. ( ! ) Physics 1 vs Physics II. There is a fairly consistent difference between responses in Physics 1 and Physics II courses that is largely independent of other factors. An example of this is shown in Table V. which compares Physics I and 11 courses taught at a rnidsi/e multipurpose .state university. Physics I courses [with the notable exception of courses where beliefs are e x p l i c i t l y addressed) typically result in signiiica.nl deterioration in a!! categories of beliefs as illustrated by the results in Table I, while Physics 11 courses have variable results w i t h the exception of the sense- making/effort category, which shows a decrease in expertlike beliefs for ail courses surveyed. (1) The (winter,) break effect. Statistically s i g n i f i c a n t shifts in some student beliefs were measured between the end of Physics I in fall and the beginning of Physics 11 live weeks later. This finding indicates the importance, when comparing different courses, of being sure that, the survey was given at the same time relative to the beginning and end of the course for the results being compared. (3) Student ago. .Statistically sianificant differences (5% or more) were measured on about a quarter of the statements when we compared !H- and !9 years-old students with their 20- to 21-years-oki classmates. Younger students displayed more expertiikc beliefs on statements 2, 3. 5. 15. 25. and .3^ i 14% difference) while older students displayed more expert like beliefs on statements 13, 27, 38, and 39. Jn particular, the younger Ntudents scored higher on all three problem solving categories. Not surprisingly, the 22- to 25-years-okl students scored much higher in both "real world connections" and ''personal interest" categories. (4) Gender. The responses to nearly half the statements show significant gender differences. Comparing responses from risen and women in the same classes, which typically represent the same set of majors, women arc generally less
» SenseslVbkinryBTort : • Ftea! Wbrld C&inections Interest
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(a): Women
Men
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60,00
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20.00 Unfavorable
Q.OO
40.OO
20.00
40,00
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FIG. 3. (Color) Women's, (a) l'A;=88) and men's fb) OV=62) responses to "What would a physicist say?" and "What do YOU think?" 010101-9
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TABLE Vi. Tesi-retesi reliability tall 2004 to ipving 2005. LSRU. CLASS V.3. Agree
Neutral
TABLE VII. Correlations between student shifts with different instructors. LSRC. CLASS V..i: algebra-bused instructors had quite different teaching pliiluiophies while calculus -b;ist:d instructors had verv similar ideas aboaf teaching.
Disagree
0.88 0.88
expert-like on statements in the "real world connections,'' "personal interest." "problem solving confidence." and ''problem solving sopnisiication"' categories and a b i t more e\pe;i!ike on some "sense-making/effon" type statements. The results from the calculus-based courses show smaller gender differences, hut there are sliil significant dilVcvenccs. particularly in the "real world connections" and "persona! interest" categories. Mile Vlli below includes data on the "personal interest" category by gender. Perhaps one of the biggest questions about the validity of this type of survey is whether the students are actually answering as they believe or what they t h i n k the instructor wants.'" Our student interviews indicated that when students take the survey, they sometimes consider both what: they feel is the correct response and what they personally believe. We studied this issue by administering the survey in a siii'hlly different format at the end of an algebra-based Physics I course, We asked the students to give two answers for each statement: ( 1 ) What would a physicist say? and (2) what do YOU think? These "comparative" results are revealing. Figure 3 shows what students (broken out by gender) believe a physicist would say (hollow markers) and what students actually believe (solid markers). From these data it is clear that, by the end of the term, the students were good at identifying die expert response, but their personal beliefs were much more novice. This difference is large lor men and noticeably larger For women. The CLASS was administered the following se.rnef.lcr to the same course, taught by the same instructor, in the 'traditional' format (students were asked to respond to each statement only once, as they believe). Comparison of the "comparative" results to data from the "traditional" administration indicates that typical student responses to the CLASS align w i t h their responses to "What do YOU think'.'" rather than to "What would a physicist say?" We have also administered the survey at the beginning of the semester to a small algebra-based course and. data
E. Reliability
Reliability studies were performed at the LSRU on Phvsics i courses for both calculus-based and algebra-based physics, 'These courses have enrollments over 500 and 400 students, respectively. Studeni incoming survey responses were compared between fall 2004 and spring 2005. Since it is reasonable to assume there is little variation in these large student populations over a few semesters, this provides a good direct measure of the survey reliability. We have compared the average incoming beliefs from one semester to the next. This comparison was done with two different courses, both algebra-based Physics I ana calculus-based Physics i. In both cases we see very consistent slatemerit responses across semesters. The results of lest-re test reliability for the calculus-based arid algebra-based courses are shown in Table VI.- 1u Note that the correlations between neutral responses are not quite as high as those for the agree and disagree responses. As mentioned previously, students choose neutral for a variety of reasons, rrutking it a less reliable measure. VI. APPLICATIONS There are many useful ways to analyze and use CLASS data. One can look at the pre results and their correlation with student learning, course selection, retention, gender. age, major, etc. One can also look at the shift in beliefs over a semester to determine correlations between various teaching practices and students" beliefs. We have found high correlations between students" shifts in beliefs over a semester
TABLE VIII. Relationships between favorable "personal interest." physic* course selection, and gender l\-\ sent favorable shown tor east of display. Overall pre (standard error! Course type
School
Physical science
MMSU
Principles of scientific inquiry
MMSU
Physics T I'Aljz)
LSRU
Physics 1 (Caici
LSRU
Dominant student population Elementary ed. (sophomores) Elementary ed. (seniors) Premeds Engineers
Personal interest pre (standard error]
Women
Men
Men
42
6
43% (11%)
54
5
186
114
104
293
010.101-10
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in moving in the unfavorable direction. Validation studies and further analysis of these data are under way. VII. CONCLUSIONS, CAVEATS. AND fill,RE WORK
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for instructors with very s i m i l a r ideas ahoui teaching and quite low correlations between shifts for student 1 ; who received reform vs traditional instniction (see Table V I ] ; . These data provide an additional demonstration of concurrent validity. In (able V I I I we show examples, of how "'overail" and "personal interest" pre results vary fur four courses covering a ran^e of introductory physics. We sec that students' incoming scores increase with level of physics course. Thus, students who make .larger commitment 1 ; 10 studying physics tend to he those who identify physics as being more relevant to their own lives. Also women have lower "personal interest'' scores than men for all courses surveyed (Fig. 4i. We also have dala showing that the two courses in which the instructors made modest efforts to e x p l i c i t l y address beliefs obtained substantially better results t h a t is, no observed decline in beliefs -than other courses These various results are obviously relevant to the question of now to increase the number and diversity of students going into STEM (science, technology, engineering, and mathematics) disciplines. In a companion paper that is in preparation, we will examine many of these issues in more detail and also examine correlations between students' beliefs and (heir learning gains. The CLASS has also been altered s l i g h t l y to create appropriate versions for chemistry, biology, astronomy, and math and administered to a number of courses in these disciplines. These versions were written in cooperation with experts in each respective field; however, validity and reliability studies have been completed only in physics and are currently being done for chemistry. Approximately 5000 students at the LSRU have taken these nonphysics version;, of the CLASS. Preliminary analysis of chemistry results indicates thai shifts after instniction are similar to, if not worse than, in physics
This paper describes the process by which we have developed a new i n s t r u m e n t to survey student beliefs. The survey can be easily used in mmmous different courses and lias been subjected (o rigorous validation testing. As part of this v a l i d a t i o n process, we have created a method for selecting catesoric-s of beliefs that both are statistically v a l i d and measure categories that are useful to teachers and education researchers. We have also established a quantitative measure of the statistical v a l i d i t y of belief categories that can be applied to any survey. When using the CLASS there are a number of influences on students" beliefs t h a t must he considered while using and interpreting survey results, such as gender, major, age. and time in college. This paper serves as the foundation for the use of this survey instrument to study student beliefs about physics and how they are affected by leaching practices. Because this survey is highly suited for widespread use, it can serve as a valuable tool for research and to improve physics teaching, Ou; preliminary data already show me importance of certain beliefs for success in physics courses nml a student's inclination to continue in or drop out of physics, and it shows thai most teaching practices have a detrimental impact on all of these critical beliefs. It also shows that teaching practices aimed at explicitly addressing student beliefs about physics can have clearly measurable effects. The survey results also show thai (here are large gender differences in beliefs that are undoubtedly relevant to the discussion as to how to attract more women into physics. These preliminary results make i\ clear that the CLASS will allow detailed studies of student beliefs for a variety of different student populations and how these beliefs arc related to t h e i r physics educational experience. The work presented here has only been validated for characterizing student beliefs in the aggregate. Further work is needed to establish whether or not this survey can characterise an i n d i v i d u a l student in a useful way. We do have hints, however, that this may be possible, in addition to our interview results, for several notable students (both good and bad! we have retrospectively looked at their individual survey results, and these were quite consistent with the highlyexpert or highly novice behavior these students indicated in their work and discussions with faculty. Copies of the CLASS V.3 online and in PDF format and the Excel scoring form are available at http:// CLA SS, col orado.edu. A C.KNOW LEDGMENTS We (hank Andy El by for his thorough and thoughtful comments, Steven Pollock and Courtney Willis for their many helpful discussions, Krista Beck for her assistance with the project, and the Physics Education Research Group at Colorado for their support. This work has been supported by the National Science Foundation DTS program and the University of Colorado.
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APPENDIX: CLASS V.3 STATEMENTS AND CATEGORIES Statement 1. A significant problem in learning physics is being able lo memorize all the information 1 need to know. 2. When i am solving a phvsics problem. I try to decide what would he a reasonable value for the answer. 3. I t h i n k about the physics I experience in everyday life. 4. It is useful for me to do lots and lots of problems-when learning physics. 5. After 1 study . L topic in physics and feel that I understand it, I have d i f f i c u l t y solving problems on the same topic. 6. Knowledge in physics consists of many disconnected topics. • l i. As ;. 1 VM ists learn more, most physics ideas we use today are likeiy to be proven wrong, 8. When I solve a physics problem, I. locate an equation that uses the variables given in 'he problem and p!ug in the values. 9. [ find that reading ill? text in detail is a good way for me to learn physics. 10. There is u s u a l l y only one correci approach to solving a physics problem. I I. I am not satisfied until 1 understand why something xvorks the way it docs. 12. I cannot learn physics if the teacher does not explain t h i n g s well in class. 13. 1 do not expect physics equations to help my understanding of the ideas: they arc just for doing calculations. 14. I study phvsics to learn knowledge that will be useful in my life outside of school. 15. If 1 gel stuck on a physics problem on rny first Iry, 1 usually try to ligure oul a different way that works, 16. Nearly everyone is capable of understanding physics if they work at it. J7. Understanding physics basically means being able to recall something you've read or been shown. 18. There could he two different correct values to a physics problem it 1 use two different approaches. 19. To understand physics I discuss it with friends and other students. 20- I do not spent! more than five minutes stuck on a physics problem before giving up or seeking help from someone else. 21. If 1 don't remember a particular equation needed to solve a problem on an exam, there's nothing much 1 can do (legally!) to come up with it. 22. If i want to apply a method used for solving one physics problem to another problem, the problems must involve very similar situations. 23. In doing a physics problem, if my calculation rjives a result very different from what I'd expect, I'd trust the calculation rather than going back through the problem. 24. In physics, it is important for me to make sense out of formulas before 1 can use them correctly 25. i Mijoy solving physics problems. 26. In physics, mathematical formulas express meaningful relationships among measurable quantities, 27. It is important for the government i.o approve new scientific ideas before they can be widely accepted. 28. Learning physics changes my ideas about how the world works. 29. To learn physics. I only need to memorize solutions to sample problems, 30 Reasoning skills used to understand physics can be helpful to me in ny everyday life.
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Statement 31. We use this question to discard the survey of people who are not reading the statements. Please select agree—option 4 (riot strongly agree) to preserve your answers. 32. Spending a lot of time understanding where formulas come from is a waste ol' time. 33. 1 find carefully ana!y/.im: only a lew pri'hienis in detail is a good way for me So learn physics. 34 I can usually figure out a way to solve physics problems. 35. The subject of physics hiis litile relation to what' I experience in the real world. 36. There arc times I solve n physics problem more chnn one way to help my understanding. 37. To understand physics. 1 sometimes Think about my personal experiences and relate them to the topic being antily/ed. 38. It is possible to explain physics ideas without mathematical formulas. 39. When 1 solve a physics problem, 1 explicitly t h i n k about which physics ideas upply to the problem. 40. If I srel stuck on a physics problem, there Is no chance I'll figure it out on my own. It is possible for physicists to carefully perform the same experiment and get two very different results that are both correct. 42. When studying physics. 1 relate the important information to what I already know rather than just memori/ing it the way it is presented.
Categories Real World Connection Personal Interest Sense Making/Effort Conceptual Connections Applied Conceptual Understanding Problem Solving General Problem Solving Confidence Problem Solving Sophistication Not. Scored
Statements comprising category 28. 3(1, 35, 37 3, 11. 14, 25, 28, 30 I I , 23, 24, 32, 36, 39, 42 I. 5, 6, 13, 21. 32 1 , 5 , 6 , 8, 21, 22, 40 13, 15, 16, 25, 26. 34, 40. 42 15, 16, 34,40 5, 21, 22, 25, 34,40 4, 7. 9, 31, 33, 41
'.!. D. Bramford, A. L, Brown, and K. R, Cocking. Hmv t'evpla Leant (National Academy Press. Washington D. C.. 200?.!. -D. Hummer. Am. J. Phys. 68, S52 .2000]. •'E. F. Redish. leaching Physics milt Physii's Suite (John Wiley & Sons, New Yurk, 2003). H R . Seymour and N. Hewitt, Talking abixts f-enviafi (Westview Press, liouldev, CO, 1997). 5 J. D. House. Int. ,T. Inst. Med. 22, 157 (1995). "J. D. House. Int. .1. Inst. Med. 21, 1 (1994). 7 P. M. Saddler and R. H. Tai, Sd. Ednc. 85, 1! 1 (200!). 8 A copy of the CLASS can be found at http://CLASS.cnIoradii.edu
S
\V. K. Adams. K. K. Perkins. M. Dubson, N. D. Finkelsreiti, and C. E. Wieman, in Proceeding! of the 2004 Physics Ldncaiim: Kcseairh Conference, edited by J. Marx, P. Heron, am] S, Franklin. AIP Conf. Proc. No. 790 (AIR Melville. New York. 2005). p. 45. i0 K. Perkins, VV. Adams. N. Finkelstein, and C. VVisman, in Pivceedings of She 200-f Physics Education Kesnarch Conference (Ref. 9). "S. Pollock, in Proceedings of the 2004 Physic.', Education Research Conference {Ref. 9). '-K. K. Perkins, M. M. Grainy. W. K. Adams, N. D. Finktistem,
010101-13
756 PHYS. REV, ST PHYS. EDUC. RES, 2. 010101 (2006}
ADAMS el at. and C. R \Vieinsn, Proceedings of the 2005 Physios Education Research Conference- (U) be published). 13 R. Liken. Arch. Psycriul. (Frankf) 140. 44 (1932). i4 D. Huffman and P. Heller. Phy<. leach. 33, 138 (1995). 15 L. J. Rennie and 1. H Parker, I. Re-,. Sci. Teach. 24, 567 ! !9S?'l. |f 'E. RecJish. .!. M. Saul. <md R. N. Steinberg, Am. .1 Phys. f>6. 2 1 2 (I'W). I; I . A. Ha'louti. i n Proceedings of the Internal ional Conference on UndercrudriiUe Physics Education. Coiieoe Park. Maryland. i996 (impsiblished). A, Plhy. !;Ltp://w\vw2.physics.un)d/eciii/-f-iby/E;BAPS/idea.htm N. (-. ].idenn;sn, F. Abd-El Khalick, R. L. Be!], and R. S. Schwart/. J. Res. Sci. Teach. 39. 497 (2002). R. K. Thornton ana D. R. Sokoloff, Am. .1. Phys. 6
:<
S . K. Kachigan, Statistical Analv-'ii'i (Radius Press, New York, 1986). M I. A. Halloun, Beirat: Phoenix Series/Fducational Research Center, Lebanese University. 2001!. -'We understand that the MPEX2, which is under developmen?.. also recognJK-s this distinction, "CM, Saul, Ph.D. thesis, University of Maiylaiid (nripuMisliedl, http:/Av%vw.lphyi,ic 1 -. amd.edu/ripe/perg/dissertations/Saiii/ - ' J . K.im and C. W. Mueller. Factor Analysis •StMi.-nical Method.-; and Pi'di-iical l-i.tites (Sage Publications. Beverly Hifis, CA, 1978.. •^CompiHsrcude SPSS \.l:i lor Windows (2003). "''T. 1... McCaskey, M. H. Dancy. and A. tilhy. in fi-n:eediitt>s of I/IK 200.! P/j.v.svti Kdiicaiitiii Research Co'iifivncc. edited by M. C'. Wi-tmann and R. E. Schcrr, AIP Coirt. 'Pmc. No. 720 (AEP, Melville, New York, 2004), 31. •"•'It is preferable to use a iesL-reicst reliability nuher l!;an j C'runha.c!) iilpha coefficient fRefs. 22 and 23i.
OIOIOI-M
Development of Research Technology
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As mentioned above, my research has usually been about expanding the limits of experimental technology in order to carry out interesting physics experiments. Most of the technology advances that we achieved are contained as notes in the papers in the other sections where we talk about the resulting science that we were able to carry out. However, there were some technology developments that did not lead to further research, or ended up being published on their own for one reason or another. Those papers are collected here. Certainly the most notable of these is the review that Leo Hollberg and I did on using diode lasers in atomic physics. As I was working on the writing of it, I frequently cursed myself for agreeing to write it, and it remains the one serious review paper that I have ever done. In the end though the effort apparently was worthwhile. It ended up being a very widely used reference with many people, particularly in countries outside the US, telling me how useful it was to them. Of course now it is badly out of date.
759
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IEEE Transactions on Magnetics, vol. Mag-11, no, 5 , September 1975
HIGH-RESOLUTION MEASUREMENT OF THE RESPONSE OF Ah' ISOLATED BUBBLE DOMAIN TO PULSED MAGNETIC FIEU)S B. R. Brown*, G. R. Henry*, R. W. Koepcke*, and C.
E. Pieman§#
ABSTRACT The response of an i s o l a t e d magnetic bubble domain t o pulsed magnetic f i e l d s normal t o t h e f i l m plane is measured w i t h high s p a t i a l and temporal resolution. D i g i t a l s i g n a l averaging techniques are used on a succession of i d e n t i c a l pulses, permitting changes in bubble r a d i u s t o b e measured t o a precision of 2 nanometres, in successive t i m e i n t e r v a l s of 10 nanoseconds. The r a d i a l w a l l velocity, obtained by d i f f e r e n t i a t i o n , is determined with a precision of 0.2 metres p e r second. S i g n i f i c a n t " i n e r t i a l " e f f e c t s and v e l o c i t y n o n - l i n e a r i t i e s are seen a t q u i t e modest drive f i e l d s (a few t e n t h s of kA/m, o r , a few Oe) and w a l l v e l o c i t i e s (a few m / s ) , i n response both t o an applied bias-field s t e p and t o a pulse.
each a t a rate of 600 t r a c e s per second. The e f f e c t of s i g n a l averaging is seen in Fig. 1; t h e t h r e e oscilloscope traces correspond t o t h e magnetic b i a s f i e l d (shaving an applied s t e p of 0.28 U / m ) , many traces of t h e raw PMT s i g n a l ("phosphor-averaged") , and t h e d i g i t a l l y averaged r e s u l t of 65,536 traces.
I. INTRODUCTION We have measured with high s p a t i a l and temporal resolution t h e response of an i s o l a t e d magnetic bubble domain t o pulsed magnetic f i e l d s normal t o t h e f i l m plane. The observations c l e a r l y show s u b s t a n t i a l i n e r t i a l e f f e c t s , as w e l l as non-linearities i n w a l l v e l o c i t y as a function of d r i v e f i e l d . The resolution obtained corresponds t o changes of 2 nanometres i n radius, over time i n t e r v a l s of 10 nanoseconds. In our experiment w e used an as-grown, Y-Eu Ga-substituted i r o n garnet film' , supporting bubble domains of approximately f i v e microns i n diameter, 61.8 with a c o l l a p s e f i e l d of 4.92 kA/m (4ax4.92 Oe). Current flowing in a s i n g l e t u r n loop i n contact with t h e garnet produced t h e desired magnetic f i e l d pulse. The response of t h e bubble was measured by monitoring t h e Faraday magneto-optical s i g n a l with a PMT-equipped p o l a r i z i n g microscope. An argon laser w a s used t o provide a d i s k of i n t e n s e illumination only s l i g h t l y l a r g e r than t h e bubble i t s e l f . The r e s u l t i n g s i g n a l is approximately f i v e orders of magnitude g r e a t e r than t h a t obtained from normal a r c lamp illumination. With t h i s arrangement, v a r i a t i o n s in the s i g n a l are r e l a t e d t o changes in bubble radius. The radius (and i t s t i m e d e r i v a t i v e , t h e w a l l v e l o c i t y ) can then be measured in response t o an applied magnetic f i e l d pulse.
Time (50nddiv)
Fig. 1. Oscilloscope photograph of b i a s - f i e l d with t h e raw and d i g i t a l l y averaged PMT s i g n a l .
A number of possible sources of systematic e r r o r w e r e examined. The CW laser illumination (0.04 mW per square micron) produced some measure of l o c a l heating. W e found t h a t t h e bubble w a s i n f a c t weakly a t t r a c t e d t o t h e c e n t r e of t h e ' n o t spot"; consequently i t shaved no tendency t o wander away from t h e illuminated area when exposed t o b i a s - f i e l d pulses. A t t h e same t i m e , t h e i n t e r n a l p r o p e r t i e s of t h e bubble appeared t o be l i t t l e affected by t h e heating, t h e collapse f i e l d changing by about 0.1 kA/m with and without t h e laser on. Changes i n l a s e r power produced no q u a l i t a t i v e changes i n t h e observed PMT s i g n a l . In the absence of a bubble t h e r e was no observable s i g n a l , demonstrating t h a t t h e r e was no appreciable e l e c t r o n i c cross t a l k and t h a t t h e applied magnetic f i e l d pulses produced no s h i f t i n the optical p r o p e r t i e s of the sample o r system. The PMT was operated w e l l within t h e l i n e a r region, and t h e s i g n a l represented a l i g h t modulation of only a few p e r cent. The time resolution w a s limited by t h e c a p a b i l i t i e s of t h e t r a n s l e n t d i g i t i z e r (Biomation model 8100, with 10-ns sampling i n t e r v a l ) . C i r c u i t and e l e c t r o n i c bandwidths were somewhat b e t t e r ; t h e rise t i m e f o r an applied s t e p i n t h e b i a s f i e l d was measured t o b e A number of o t h e r t e s t s ( p o l a r i t y about 5 ns. changes, etc.) a l s o yielded r e s u l t s expected f o r r a d i a l bubble w a l l motion.
-
11. EXPERIMENTAL TECHNIQUE
The two main problems t o be solved in t h e experiment centered on t h e need f o r s i g n a l averaging t o improve signal t o noise r a t i o s , and systematic t e s t i n g t o ensure t h a t the signal was indeed produced by t h e bubble motion of interest r a t h e r than by some experimental a r t i f a c t . Signal averaging w a s accomplished by d i g i t i z i n g t h e PMT. s i g n a l at a 10 nanosecond samplfng i n t e r v a l and accumulating r e s u l t s of successive i d e n t i c a l pulse experiments in a d i g i t a l computer. In t h i s manner we could average s i g n a l traces of about 1.3 microseconds
111. RESULTS
The f l g u r e s below a r e i l l u s t r a t i v e of r e s u l t s obtained. Figure 2 i n d i c a t e s t h e observed response of t h e bubble t o a slowly varying symmetric sawtooth b i a s f i e l d modulation (3.93 t o 4.73 Ufm, peak t o peak). The o r d i n a t e shows t h e d i g i t a l l y averaged s i g n a l , with smaller numbers corresponding t o smaller bubbles. Over a wide region of t h e collapse to stripe-out range t h e bubble radius has been t h e o r e t i c a l l y predicted2 and experimentally found t o be l i n e a r l y r e l a t e d t o t h e Thus, if our disk of l a s e r applied b i a s f i e l d . illumination were uniform, we would expect t h e curve i n Fig. 2 t o be concave upward, s i n c e changes i n
Manuscript received March 17, 1975.
*
IBM Research Laboratory Monterey & C o t t l e Roads San Jose, California 95193
§
Hertz Foundation Pre-doctoral Fellow.
#
Physics Department Stanford University Stanford, C a l i f o r n i a 94305
step,
1391
76 1
762 bubble area produce the l i g h t modulation. In f a c t t h e curve is approximately l i n e a r (indeed, i t i s s l i g h t l y convex upward), i n d i c a t i n g t h a t t h e l i g h t is more i n t e n s e toward t h e c e n t r e of the disk. A s t h e system c a l i b r a t i o n we used t h e b e s t s t r a i g h t - l i n e f i t t o Fig. 2 i n conjunction with independent measurements of bubble radius as a function o f applied b i a s f i e l d .
The curves f o r the two lowest s t e p s are more o r less what might be expected; t h e bubble shrinks t o its new equilibrium radius with l i t t l e appreciable overshoot, and a doubling of s t e p height roughly doubles t h e maximum w a l l velocity. A r s l i g h t l y l a r g e r drives n o n - l i n e a r i t i e s set in. Under t h e 0.28 kA/m s t e p , t h e v e l o c i t y follows t h a t of t h e 0.2 kA/m s t e p i n i t i a l l y , but then decreases t o an approximately constant "saturation" value. This v e l o c i t y behaviour is c o r r e l a t e d with the onset of t h e s u b s t a n t i a l overshoot now observed. As the s t e p i s f u r t h e r increased, t h e "saturation velocity" remains nearly constant, u n t i l , at a step height of 0.4 kA/m, there is a (Higher "breakthrough" t o s t i l l higher v e l o c i t i e s . s t e p s i n t h i s series would collapse t h e bubble.)
Time (ps) Fig. 2. D i g i t a l l y averaged response t o t r i a n g u l a r b i a s - f i e l d modulation; t h e d i g i t a l "grain s i z e " may be i n f e r r e d from the i n t e g e r numbers on t h e ordinate. The rounding of t h e corners may be due t o coercive e f f e c t s ; i n all of our experiments t h e bubble response f o r small e x c i t a t i o n s is less than would be expected on t h e b a s i s of l i n e a r i t y . Figure 3 shows a family of bubble reponses t o b i a s f i e l d s t e p s of various heights on top of a d.c. b i a s f i e l d of 4.3 kA/m (the Fig. 1 photograph i s of the 0.28 kA/m s t e p ) . The corresponding r a d i a l w a l l v e l o c i t i e s are p l o t t e d in Fig. 4. 2.8
2.6 2.4
--_----
-E 2.2 cn
2.0 LT
1.8 1.6
1.4
0.0
0.1
0.2
0.3
0.4
0.5
Time (ps)
Fig. 3. Response bias-field steps.
of
bubble
radius
to
several
1392
-10 0.0
0.1
0.2
0.3 Time Ips)
0.4
0.5
Fig. 4. Radial w a l l v e l o c i t y i n response to bias-fi.eld s t e p s ( f i r s t d e r i v a t i v e of Fig. 3). In each of the l a t t e r two cases, t h e v e l o c i t y is approximately constant when the bubble passes through its new equilibrium radius. Apparent s a t u r a t i o n is again seen on t h e rebound, being s u b s t a n t i a l l y more pronounced in the 0.4 k A / m case, where the e f f e c t i v e d r i v e f i e l d is greater. It i s i n t e r e s t i n g t h a t t h e "Vo" l i m i t i n g v e l o c i t y 3 f o r our f i l m i s about 4 m / s , i n t h e region of t h e observed "saturation". The e f f e c t seen is i n e r t i a l i n t h e sense t h a t energy s t o r e d during t h e i n i t i a l response l a t e r c a r r i e s the bubble past i t s f i n a l equilibrium point, b u t t h e extreme non-linearity observed makes i t d i f f i c u l t t o a s c r i b e a "mass" t o the w a l l . Proposed mechanisms f o r the "winding up" of Bloch l i n e s in t h e wall4 may o f f e r b e t t e r explanations f o r t h e data. Another view of bubble behaviour i s offered i n Figs. 5 and 6: a b i a s f i e l d pulse of 20 nanoseconds ( f u l l width, h a l f maximum) i s applied, again on top of a d.c. b i a s f i e l d of 4.3 kA/m. A t the smaller e x c i t a t i o n , the response looks f a i r l y l i n e a r as f ( t ) , although t h e r e is some s i g n of overshoot on t h e r e t u r n t o equilibrium. A t the higher drive, t h e v e l o c i t y i n Fig. 6 again shows the c h a r a c t e r i s t i c "saturation" e f f e c t on the rebound. We conjecture t h a t t h e pul$e is too s h o r t t o allow complex w a l l s t r u c t u r e t o evolve during the application of the pulse itself. (In Pigs.
763 In s ~ n r m a q , w e obsercre a number of interesting and complex e f f e c t s in the r a d i a l wall motion of bubbles at q u i t e modest drives and v e l o c i t i t e s . Clearly i t i s the high resolution obtained i n t h i s experiment that
5 and 6 , the pulae extends over only four channels, with FMM of two channels.) These data suggest that measurements based on bubble collapse t e c ~ i q u e s 5should perhaps be viewed in a new l i g h t . In particular, it can e a s i l y be argued t h a t inertial e f f e c t s a r i s i n g from domain w a l l mass do not a f f e c t mobility measurements obtained from bubble collapse, but it is not clear that t h e e f f e c t s reported here do not influence such measurements. For example, i f the "winding up" picture is applicable, i t may be that not a l l of the energy taken from t h e system during "wind up" is returned in the "unwinding" (since the dissipative e f f e c t s i n the winding process are unknown), and i n t h i s case a spurious deviation from t h e steady-state mobility could be inferred.
1
'
1
'
1
'
1
'
'
"
l
2.8
permits observation of the fine details presented i n The resolution is particularly Figs. 3 through 6. crucial i n determination of wall v e l o c i t i e s ; an estimate of precision i s given by the radial resolution divided by the time resolution: 2 nm/lO ns = 0.2 m/s. It would be useful t o improve the time resolution (although the s p a t i a l resolution must of course be improved by the same r a t i o i f the precision of velocity determination is co be maintained). A limitaion of t h i s technique is that transient, one-of-a-kind events cannot be recorded: the power density required t o produce the desired resolution i n a single trace would r e s u l t i n unacceptable thermal effects. While a t present we cannot o f f e r complete explanations for the experimental findings, w e f e e l t h a t the high-resolution techniques discussed here w i l l prove helpful i n understanding a number of magnetic bubble phenomena.
-1
ACKNOWLEDGMENTS It i s a pleasure t o thank B.A. Calhoun, T.-L. A m , L.L. Rosier, and R.L. White f o r numerous helpful discussions. REFERENCES
2.21
0.0
"
0.1
'
'
I
0.2
0.3
'
0.4
'
'
0.5
1. E.A. Giess, B.E. Argyle, B.A. Calhoun, D.C. Cronemeyer, E. Klokholm, T.R. McGuire, and T.S. Plaskett, Mater. Bes. Bull. 5, 1141 (1971) 2. A.A. Thiele, J. Appl. Phys. 4 l , 1139 (1970) 3. B.E. Argyle, J.C. Slonczewski, and A.F. Mayadas, AIP Conf. Proc. 5, 175 (1972) 4. A.P. Malozemoff. J.C. Slonczewski, and J . C . de Luca, paper 1-10, This Conference. 5 . A.B. Bobeck, I. Danylchuk, J.P. R e m e i k a , L.G. Van U i t e r t , and E.M. Walters, presented at I n t . Conf. F e r r i t e s , Kyoto, Japan (1970)
I
Time (M)
Fig. 5 . pulses.
-101
0.0
Radial
'
'
0.1
response
'
I
0.2
to
I
two
20
ns bias-field
I
I
0.3
0.4
t
1
I
0.5
Time (Ms)
Fig. 6. pulses.
Wall v e l o c i t i e s produced by 20 ns bias-field
1393
Easily constructed high-vacuum valve S. L. Gilbert and C. E. Wieman Physics Department, University of Michigan, Ann Arbor, Michigan 48109 (Received 17 May 1982; accepted for publication 23 June 1982)
We present a design for a high-vacuum valve which can be built very quickly and easily. It has the additional feature that it is very thin. PACS numbers: 07.30.Kf
We describe a high-vacuum valve which is extremely simple to construct. It is particularly well suited for use where a thin valve mechanism is needed to close a port in a large chamber. As shown in Fig. 1, the valve itself is simply a %-in.thick stainless-steel rectangle 6 X 2% in. with %-20 threaded holes at each end. For our particular case of a 1% in. port to be sealed, a 2-in. 0 ring was seated in a groove in the face. A 'h-20 bolt is threaded through the 1627
Rev. Sci. Instrum. 53( lo), Oct. 1982
left-hand hole (when facing the 0 ring) and a stop (two counter tightened nuts) is locked on its end. The smooth upper shaft of this bolt passes through a standard commercial '14 in. quick connect feedthrough (ULTRA-TORR Male Connector, Cajon Company, Solon, Ohio) mounted in the chamber wall as shown in Fig. 2. When the valve is closed, a slightly shorter bolt comes through an identical feedthrough, and is screwed into the hole in the other end. A stop could be fastened to this bolt to prevent ac-
0034-6748/82/101627-02801.30
764
0 1982 American Institute of Physics
1627
765
R
FIG. 1 . Drawing of the valve as viewed from within the vacuum chamber. Solid figure is open position with O-ring groove on hidden face.
cidental removal under vacuum, but we have not found this necessary. Both bolts are tightened down, which presses the 0 ring against the smooth chamber wall. To open the valve, the second bolt is disengaged and then the first is rotated counterclockwise. This pulls the 0 ring away from the wall by moving the valve plate out along the threads to the stop. Another half turn pivots the plate away from the opening and then locks it against the
1628
Rev. Sci. Inatrum., VoI. 53, No. 10, October 1982
FIG.2. Rotary feedthrough for the bolts. The 0 ring is lightly greased and the seal into the wall is made with Teflon pipethread tape.
stop. To close the valve the bolt is rotated clockwise which swings the plate back over the opening against a post set in the chamber wall, after which further turning pulls it against the wall and the second bolt is engaged. This is easily done under all conditions if the post has been mounted at the proper position. This valve has worked well for 18 months of frequent use at pressures of to lo-’ Torr. As well as the ease of construction and low profile (the entire mechanism inside the vacuum chamber is less than 5/8 in. wide), we have found this design to avoid two common problems. There is no burst of air let into the system when opening or closing, as is frequently the case with valves that employ a shaft which slides through a feedthrough, and the 0 ring is not worn out rapidly as happens in valves where an 0 ring slides over a surface. One of us (SLG) is happy to acknowledgesupport from the Bendix Corporation Scholarship Fund.
Notes
1628
Reprinted from Applied Optics, Vol. 26, Page 1693, May 1, 1987 Copyright 0 1987 by the Optical Society of America and reprinted by permission of the copyright owner.
High frequency Fabry-Perot phase modulator David W. Sesko and Carl E. Wieman
We have constructed an efficient high frequencyphase modulator using a KDP crystal modulator in a FabryPerot interferometer. We present the theoretical analysis for the performance of such a modulator and compare this with experimentalresults. We have obtained a modulation index of 0.6 at 2.7 GHz with 0.3 W of microwave power. This implies that the use of a Fabry-Perot cavity can reduce the amount of drive power needed to achieve a given modulation index by 3 orders of magnitude or more. Such a device is particularly useful in the ultraviolet region of the spectrum where one must use relatively inefficient modulator crystals.
The technique of FM or frequency modulation spectroscopy has made possible optical absorption measurements of unprecedented sensitivity.lI2 One of the many exciting applications of this technique is the detection of trace species in the atmosphere.3 For this application, it is necessary to have an optical phase modulator which produces a modulation index of the order of one, at a frequency higher than the pressure broadened transition linewidth of the species of interest. This is typically several gigahertz. There has been a great deal of work done on high frequency modulation.68 Recently, impressive advances in modulator technology have attained a high modulation index a t microwave frequencies using LiTaO3 as a modulator material.7 This material works very well in the visible portion of the spectrum. Unfortunately, it is opaque in the ultraviolet where the absorption lines of some particularly important species such as OH are found. The modulator materials which are transparent in this spectral region, such as KDP, are far less efficient and thus would require unacceptably large amounts of drive power to achieve a satisfactory modulation index. A possible solution to this problem is to modulate a visible laser and then double the laser frequency. This involves substantial problems for cw laser sources, however, because the doubling efficiency is low and boosting the efficiency by intracavity dou-
bling can introduce large amounts of undesirable amplitude modulation. We have explored the alternative approach of enhancing the frequency modulation efficiency in the UV by using an optical cavity. The realization that a resonant optical cavity will provide an enhancement of the modulation index is quite old. It was demonstrated qualitatively by Ruscio a t low modulation frequencies: and a number of people including Gallagher et aL8 have proposed it could be done at microwave frequencies. We have demonstrated this by placing a high frequency KDP modulator inside a resonant Fabry-Perot interferometer. In this paper we will provide a theoretical analysis of our microwave modulator and then a qualitative discussion of the experimental performance. Phase modulation is accomplished by orienting the optic axis (2)of a nonlinear crystal along the direction of propagation of a light beam. The light beam is polarized along the X axis of the crystal so that the modulating field along the Y axis produces a phase modulation index of 6 = wn~4JhlEm1/2c,
where w is the optical frequency, no is the refractive index of the ordinary ray, r41 is the electrooptic coefficient, Em is the stxength of the modulation field, and 1 is the length of the crystal. Equation (1)is valid if 1 is short enough so that the optical wave and the modulating field remain in phase throughout the ~ r y s t a l .The ~ optical field, as it leaves the phase modulator, is described by a sum of Bessel functions: E,,, = A(J,(b) cowt + J,(~)[cos(w+ w m ) t - COS(W - w,)t]
~
The authors are with Joint Institute for Laboratory Astrophysics, University of Colorado and US. National Bureau of Standards, Boulder, Colorado 80309-0440. Received 20 October 1986. 0003-6935/87/091693-03$02.00/0. 0 1987 Optical Society of America.
(1)
+ (higher-order terms)),
(2)
where A is the amplitude of the incident field, and wm is the frequency of modulation. Equation (2) shows the 1 May 1987 I Vol. 26,No. 9 / APPLIED OPTICS
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field amplitude of the carrier and the sidebands as a function of the modulation index. We will now calculate the enhancement of power in the sidebands when the crystal is inside the FabryPerot interferometer. As we will show, the resonant frequency of the Fabry-Perot and the modulation frequency must be set so that the optical carrier and the sidebands all fall within the transmission of the interferometer. Also, the position of the KDP crystal in the interferometer must be chosen to insure that the sidebands produced before and after reflection of the optical wave are in phase. We analyze this system by first decomposing the microwave and optical standing waves into forward and backward traveling waves. The traveling microwave field propagating in the same direction as one of the optical waves produces a modulation on this wave. If the modulator is an integral number of microwave half-wavelengths long, there is no modulation due to the counterpropagating microwave field.4 This condition was true for our experiment and will be assumed in the following analysis. To get an explicit expression for the modulation, we consider an input optical field of magnitude A into the interferometer. The steady-state optical field may be obtained by summing the multiple round-trip passes of the field^,^ keeping track of the phases of the carrier and the sidebands in each of the passes through the modulator. To first order in Jl(6)the fields of the first pass, the next round trip, and the nth round trip through the crystal are, respectively, E, = Atl(l - y ) ( J o ( 6 )exp(iw,t)
+ Jl(6)(exp[i(wo+ w,)t]
- exp[i(wo- ~ , ) t I l ) ,
x exp[i(w,
(3)
+ w,)t]
the crystal, L is the optical path length of the interferometer, and x is the optical path length from the center of the crystal to the input mirror. By requiring the phases of the carrier wave terms in Eqs. (3)-(5) to match, we obtain the usual Fabry-Perot condition that 2L
w o - = 2MT. C
Similarly, for the sideband terms to also match in phase,
where N and M are integers. This states that both the carrier and sideband frequencies must be an integral number of the free spectral range of the interferometer. By summing all the components of the carrier from one to an infinite number of passes, we find that the field transmitted by the Fabry-Perot cavity is E, = A t , t z ( l - ~ ) J ~ ( 6 ) / [-1 rlr2(1 - Y)~J@)].
(8)
The field of the sideband may be obtained by summing in a similar manner and taking the real part: At,t,(l - y)Jl(6)[l
E, =
f
rirz(1 - Y)'&6)
C
O
[I - rlrz(l - r)'~(s)IZ
S
T
(9)
The quantity of interest is the ratio of sideband to carrier field amplitudes which we represent by an effective modulation index defined as 2(E,/E,). For 6 << 1,we can make the approximations that Jo(6) = 1and Jl(6) = 6/2 and from Eqs. (8) and (9) we obtain
where F is the finesse of the cavity given by F = r(rlrz)l'z(l - y ) / [ ~- rlrz(l - r)'].
(11)
From Eq. (10) we see the effective modulation index will be largest when the crystal is positioned so that -am 2n =
2 p r orx = PL ,
N
C
where p is an integer less than N . Thus for N = 4, there will be three locations in the interferometer where the crystal can produce maximum modulation. Equations (10) and (11)can be simplified by assuming the plausible conditions that the crystal is positioned properly, 7-1,~are equal and close to one, and y is small. This gives the simple result that 6eff
where t j and r, are the amplitude transmission and reflection coefficients of mirror j , y is the loss through 1694
APPLIED OPTICS / Vol. 26, No. 9 / 1 May 1987
26F
= - )
and F = n/l - R, where R is the mirror reflectivity. One other relevant factor is the maximum transmission through the interferometer which islo Tint = (1 A/1 - R)2,where A is the single pass absorption. Dielectric mirrors typically have losses of <0.1% and
768
KDP crystals can be obtained with insertion losses of <1%. Thus for a mirror reflectivity of 95%,from Eq. (13) we see the modulation index will be increased by a factor of -40 with less than a 30% loss in the total optical power. Equivalently, to obtain a given modulation index this means the microwave power required is reduced by a factor of 1600. We have demonstrated the predicted enhancement of the modulation index at microwave frequencies using the setup in Fig. 1. The output of a polarized HeNe laser was mode matched into an optical resonator which contained a microwave modulator (commercial Sylvania modulator, Q N 1000). The resonator was a Fabry-Perot interferometer with one mirror having a radius of curvature of 15 cm and R = 0.99 and the other being flat with R = 0.995. The curved mirror was mounted on a piezoelectric transducer which allowed it to be translated by a few microns and thereby vary the resonant frequency of the interferometer. The mirror separation was -20 cm. The phase modulation was done by a 3.53-cm long KDP crystal contained in a cylindrical resonant microwave cavity. The TElll mode of the cavity was excited to produce a standing wave with its field perpendicular to the optic axis of the crystal. The resonant frequency of the modulator was 2.7 GHz and could be tuned by changing the length of the cavity. The cavity was excited with 0.3 W of microwave power measured just before entering the modulator. With the modulator in place, the free spectral range of the optical cavity was very close to one-quarter of 2.7 GHz. It should be noted that the purpose of this experiment was only to demonstrate the enhanced modulation. Therefore many of the parameters of this setup were determined by what equipment was readily available and hence are far from optimum for a real modulator. In particular the KDP crystal had antireflection coatings for the wrong wavelength which caused an insertion loss of 12%per pass. This limited the finesse of the cavity, and in combination with the excessively high reflectivity of the cavity mirrors, gave a low optical throughput. The resonant frequency of the optical cavity was held at the frequency of the He-Ne laser using a servo system based on the Hansch-Couillaud approach.ll In this scheme an error signal is derived by analyzing the circular polarization of the light reflected off the optical cavity. This error signal was sent through a high voltage integrating amplifier whose output drove the piezoelectric transducer. We examined the light transmitted through the optical cavity with a scanning Fabry-Perot interferometer optical spectrum analyzer. This allows us to measure the relative power in the carrier and th e modulation sidebands. With the microwave frequency adjusted to satisfy the phase-matching condition the ratio of sideband to carrier height was O.lO(l), or a modulation index of 0.60(4). With only a single pass through the modulator (mirrors removed) the ratio was 1.7(3) X giving a modulation index of 0.026(4). We determined that the finesse of the opti-
BEAMSPLITTER
BREWSTER
VOLTAGE
Fig. 1. Diagram of optical servo system phase modulator and analyzer.
cal cavity was 25(5) by measuring the linewidth and free spectral range. Using this value and Eq. (9), we calculate that the optical cavity should increase the modulation index by a factor of 18(3). This is in reasonable agreement with the measured value of 23(4). Thus we have demonstrated that a resonant optical cavity can give an enormous improvement in the efficiency of a high frequency optical modulator. This allows a large modulation index to be obtained a t microwave frequencies using relatively inefficient materials such as KDP and modest modulation powers. The primary advantage to such a device is that it will work in the ultraviolet region of the spectrum where traditional LiTaOa modulators are opaque. We are happy to acknowledge the loan of the microwave modulator by J. Magyar and valuable discussions with J. Ward. This work was supported by the National Science Foundation. One of us (CEW) is an Alfred P. Sloan fellow. References 1. G. C. Bjorklund, “Frequency-Modulation Spectroscopy: A
New Method for Measuring Weak Absorptions and Dispersions,” Opt. Lett. 5,15 (1980). 2. N. C. Wong and J. L. Hall, “Servo Control of Amplitude Modulation in Frequency-Modulation Spectroscopy: Demonstration of Shot-Noise-Limited Detection,” J. Opt. SOC.Am. B 2, 1527 (1985). 3. J. F. Ward, C. C. Wang, and C. E. Wieman, “A Proposal for
4.
5. 6.
7.
8. 9. 10. 11.
Improved OH Detection Using Frequency Modulation Spectroscopy,” Bull. Am. Phys. SOC.30,643 (1985). For a comprehensive overview and collection of reprints, see I. P. Kaminow, An Introduction t o Electrooptic Deuices (Academic, New York, 1974). J. T. Ruscio, “A Coherent Light Modulator,” IEEE J. Quantum Electron. QE-1,182 (1965). I. P. Kaminow and W. M. Sharpless, “Performance of LiTaO3 and LiNbO3 Modulators at 4 GHz,” Appl. Opt. 6,351 (1967). N. H. Tran, T. F. Gallagher, J. P. Watjen, G. R. Janik, and C. B. Carlisle, “High Efficiency Resonant Cavity Microwave Optical Modulator,” Appl. Opt. 24,4282 (1985),and references therein. T. F. Gallagher, N. H. Tran, and J. P. Watjen, “Principles of a Resonant Cavity Optical Modulator,” Appl. Opt. 25,510 (1986). M. Born and E. Wolf, Principles of Optics, (Pergamon, New York, 1980), pp. 327-328. E. Hecht and A. Zajac, Optics (Addison-Wesley, Reading, MA, 1976), p. 308. T. W. Hiinsch and B. Couillaud, “Laser Frequency Stabilization by Polarization Spectroscopy of a Reflecting Reference Cavity,” Opt. Commun. 35,441 (1980). 1 May 1987 / Vol. 26, No. 9 / APPLIED OPTICS
1695
432-iim source based on efficient second-harmonic generation of
GaAlAs diode-laser radiation in a self-locking external resonant cavity
Carol E. Tanner* and Carl E. Wieman Join1 Inslitule for 1.uborulory Astrophysics trnd I I i t : I~e~?urlrni:nl of Physics. Uriiwrsily of Colortrtln, I h l l t l n r . Colorotlo ti:
.iUJ
Recoivctl Doccrnbcr 5. .I9RII: iicccl~tulApril 10. 1989 Using a polassiutn niol)ute cryslitl i n ii n i t d i l ' i c d scll'-lt~ckiiig~ i ~ ) ~ ~ , ~ ~ ~ -~l iiv~i tt yi w* ,l ( lIi;ive ~ ~ ~ fiw q w i w y dotililrtl I lie 865-nm output from a CaAlAs laser diode. !Villi 12.4 inn' ol'input piiwer wc have chtninetl H iinidireclioiinl output of 0.215 m W a t 432 am. In contrast l o prcvious diode tlou1)liiig exl)erinients, the oufput was both single frequency ahd circular Gaussian. With better optics, substant.ially higher ctmversioii efficiencies should be possible using.this technique.
A compact, all-solid-state laser operating in the blue or violet regions of the spectrum is of interest for optical storage, reprographics, and olher applications. Various techniques for producing such a device have been described. Those based on diode-laser-pumped, solid-state laser technology include intracavity harmonic generation of a 946-nm Nd:YAG laser1,' and sum-frequency u p c o n v e r ~ i o n . ~Direct -~ second-harmonic generation (SHG) using the output of a diode laser has also been investigated by a number of researchers."-7 Unfortunntely, single-pass SHC is Iimited in efficiency by the low power of available diode lasers."." Cerenkov waveguide doublers, while more efficient, produce crescent-shaped outputs that are undesirable for many application^.^ In this Letter we report the demonstration of a novel, self-locking technique for SHG using the output of a diode laser in an external resonant cavity. This device provides for both high conversioi~efficiency and good spatial beam q uali t y . SHG in external resonant cavities was originally proposed by Ashkin et al.8 as a method for improving the conversion cfficiency of low-power laser sources. Iteccnt research by ICozlovsky et uL9 on the upconvers i 0 II o f sing 1c - f r e q 11en c y , d i o d e - 1a s e r - I,u ni p e cl Ntl: 1'AG lasers has sh(Jwn that conversion efficiencies in excess of 60% are possible a t input powers of50 m W. Ilall and ICimble~O havc achieved even higher et'liciencies using an elaborate syst.etn to obtain a cavity that is r e w i a i i t for both lhe frtndamental and the second harmonic. Howwer, these techniques require complicated feedback circuitry to lock the laser emission frequency to a resonance oT the external cavily. Such techniques are even more difficult to apply to semiconcluctor lasers owing to the broad spectral width of their emission. Resonant power multiplication of diode-laser radiation in a high-finesse external cavity can lie simplc, 0146-9592/89/14073 1 -03$2.00/0
769
however, if a portion of the resonant intracavity field is fed t m l c to the diode laser. This is known as the selflocking power-buildup cavity.11 By eliminating direct backreflection from the input mirror af the external cavity and by feeding back a small portion of the intracavity resonant field to the diode laser, one can force the locking of the diode-laser frequency to a resonant frequency of the external cavity over a considerable range of exlernal cavity lengths and/or diode-laser currents. li1,l2 This passive frequency-locking technique eliminates the need for electronic feedback circuitry. Additionally, the linewidth reduction that has been tlernonstrated to occur when the diode laser is locked to the external cavity makes it possible to use a resonator having a finesse t h a t is higher then would otherwise he practical. By applying the self-locking power-buildup cavity principle t o SHG of diode-laser radiation in potassium niobate (1
770 732
OPTICS LET'I'ERS / Vol. '14. No.I4 / 111ly15. iotig through the isolator to the diode laser, locking it to the
where P, is the power circulating within t.he cavity and YSH is t h e nonlinear conversion efficiency. 'I'he circulating power is related t o the incidenl po\ver 113'
where P I is the incident IxJWer, TI a i d 111 ' arc the power transmission and reflectivity of the i n p i i t mirror, respectively, and R,,, is the cavity rcl leclancc parameter.8 R,,, is defined in terms of the unitlircclional nonlinear output coupling, Lsll, the unitlirectionnl transmission loss of the crystal, Lc,,and the reflcclivity of the output mirror, R,, by (3)
In Eq. (2), M is a mode-match parameter, no1 included in the previous analysis, t h a t quantifies the match of the diode-laser input beam with the fundamental mode of the resonator. I t is equal to the percentage of power from the input beam t h a t is matched into the fundamental mode of the external resonator. Analytically, if the input beam were expanded in terms of the eigenmodes of the resonator, M would be the coefficient of the TEMoo term. A representation of the self-locking, external, resonantly doubled diode is shown in Fig. 1. T h e output of a single-stripe, index-guided diode laser operating near 865 nm is collimated by lens L1 and is passed through an optical isolator before being imaged by lens L2 onto the input mirror of the external resonant cavit y formed by mirrors M I and M2. The focal lengths of L1 and L2 are chosen t o optimize the match between the input beam and the TEMoo mode of the external resonator. A KNbO:! crystal (Virgo Optics, Port Richey, Florida) located inside the external resonator is oriented so t h a t the input polarization is optimal for SHG using d : ~ . . T h e KNbO:] crystal is heated t o the correct phase-matching temperature (slightly above room temperature) with a simple resistance heater. Both second-harmonic and transmitted fundamental radiation are incident upon the dichroic mirror, M3, which transmits the harmonic but reflects a portion of the fundamental. Infrared radiation transmitted by the external cavity is imaged back HALF-WAVE PLATE
&
FEEDBACK 865-nm
865-nm DIODE LASER
f
/
MAGNETO-OPTIC ISOLATION
\
OUTPUT
\
OUTPUT
POTASSiUM NIOBATE
POLARIZING B E A M SPLITTER
Fig. 1. Schematic of the self-locking, external, resonantly doubled diode.
cavity resonance. Self-locking of the diode laser to the external cavity requires t h a t direct backreflections be eliminated eit h e r by isolalion or by appropriate cavity design and that a porlion of the resonant intracavity field be returned to the diode laser." A linear, standing-wave cavity was selected for initial experiments because of its basic simplicity. T h e necessary isolation and feedback between the cavity and the diode laser is accomplished as follows. T h e second polarizer of the magneto-optic isolator is actually a polarizing beam splitter t h a l is oriented t o transmit light polarized parallel to the b axis of the KNbO:j crystal4 (parallel to the plane of Fig. 1). T h e input polarization and Faraday rotator are adjusted to provide maximum isolation against direct feedback and maximum transmission through the output polarizer. Light of orthogonal polarization (perpendicular t o the plane of Fig. 1) brought into the polarizer in a direction opposite to t h a t of the rejected input beam is returned, with negligible attenuation, t o the diode laser. Light transmitted by the external cavity is imaged back through the output polarizer along the rejected beam path. T h e intensity of this feedback is adjusted by rotating its polarization with a half-wave plate. Locking behavior is obtained over a wide range of feedback levels, estimated to be between 0.1%and 1%. This method has two advantages over other possible feedback schemeslOJ1that could be used with a linear cavily. First, it allows one to maintain linear polarization wiihin the external cavity, which is necessary for efficient type I SHG. Second, the amount of feedback can be easily optimized without affecting the coupling into the cavity. We expect that unidirectional ring resonators, in conjunction with other, simpler, feedback schemes, will give improved device performance. T h e diode laser used in our experiments operated in a single longitudinal mode and, over a limited range of external cavity lengths and/or diode-laser currents, would stably lock t o a single transverse mode of the external cavity. T h e cavity used for our experiments consisted of a flat input mirror coated t o give 98% reflectivity a t the fundamental and a 9.375-mm-radius, concave, output mirror with a reflectivity a t the fundamental of 99.7%. T h e cavity spacing was adjusted during the experiment t o optimize the mode match with the input beam and was estimated t o be 6 t o 8 mm. T h e KNbO:] crystal was antireflection coated a t 865 nm on both faces and positioned less than 0.25 mm from the input mirror. T h e peak unidirectional harmonic output of 0.215 mW was obtained with a 12.4-mW input beam a t an intracavity, circulating, infrared power of 248 mW. T h e observed infrared buildup factor of approximately 20 was significantly less than t h a t observed for the same cavity without KNb03 in which a power multiplication of 52 was measured. A theoretical buildup factor of 109 was calculated based on a total absorption and scatter loss of 0.2% per mirror, giving a modematch parameter of 48%. T h e significant reduction in intracavity power when the KNb03 crystal was inserted corresponds to an added, single-pass, linear loss of
771 july 15, 1989 / Vol. 14, No. 14 / OPTICS LETTERS
733
understood now b u t m a y be d u e t o temperature inhoapproxinia tcly 0.8% : t i i d a nonlinear conversion loss mogcneity in t h e crystal oven or improper alignment 0.087%. of the incoming beam with respect to t h e crystalloIf w e use. these values a n d t h e equations above, it is graphic axes of t h e KNb03. possible to prcdicl qtrantitatively t h e effec1.s of‘potenI n summary, we have achieved highly efficient, extial improveineiit,s i l l this device. First, with the proper opt.ics it should I)e possible to achieve a ~ n o d ~ - ~ ~ ~ n ternal, ~ c l i resonant SHG at 432 n m using radiation from an 865-nm GaAlAs diode laser in a linear standingparameter of nea1.1y 1,O, which would c~u;~clruple t.he wave resonator containing KNb03. Passive locking to liarnionic output. Prirthermore, a retlriction in t.he t h e external buildup resonator has been demonstrated loss d u e to the lLNI)O;\ crystal w o d t l allow greater rising the self-locking power-buildup cavity technique. i n t rii ca vi t y build 11p an ti, 11e nce , i m p roved n c ) n 1i n car IVith lower-loss optics a n d crystals, this technique con\wsiun. KNbO:, hdlc losses a t 1064 n n i are reshould allow one t o frequency double low-power diode ported to be O.%~%/CJII.’:~ We espect thc iiii5-11m lasers with efficiencies greater t h a n 10%. losses to be coinparahle arid, for this reason, to be insignificant relative to the coating a n d m i r r o r losses. * P r e s e n t address, National Institute for Standards ‘I’he surface finish on the sainple used for this esperia n d Technology, Gaithersburg, Maryland 20899. m e n t was clearly suhstnndard a n d is t)elievecl t,o be responsible for the excess loss. Improved polishing Iteferciices t.echniques are expected to reduce t h e t,ol.d crystal losses t o less than 0.5%, allowing unitlircct.ioiia1 con1. G . J , I)ixon, %. M. Zhang, R. S. I“. Chang, and N. Djen, Opt. Lett. 13, 137 (1988). version efficiencies great.er t h a n 10% to be ol)t,ained 2. W.1’. Risk and W. Lenth, Opt. Lett. 12, 993 (1987). with 12.4-mW input. 3. W. 1’. Risk, J.-C. Baumert, G. C. Bjorklund, F. M. SchellT h e observed conversion efficiency per milliwat,t of‘ e n l ~ c r gand , W. Lenth, Appl. Phys. Lett. 52, 85 (1988). circulaling power was also less t h a n would be pretlict4. G. J . Dixon, D. W. Anthon, F. Hilgers, M.Ressl, and T. ed from t h e published value of d:,2.5 In cn1cul:Ltiiig a I’icr, i n Digest of Conference on Lasers a n d Electrolower limit on t h e theoretical conversion, we iiss\iiiie a Optics (Optical Society of America, Washington, D.C., %‘-pni waist located a t t h e input face of t h e nonlinear 1988),paper FE5. crystal a n d a circulating power of 215 mW. ‘ r h e con5. J . C. 13auinert, J. Hoffnagle, and P. Gunter, Proc. SOC. focal parameter corresponding t o this waist is 1..0cm, I’hoto-Opt. Instrum. Eng. 492, 374 (1984). i.e., exactly twice the length of t h e crystal used in our 6. M. I<. Chun, L. Goldberg, and J. F. Weller, Appl. Phys. Lett. 53, 1170 (1988). experiment. Since the waist is located a t one end of 7. ‘1’. ’I’aniuchi, I(. Yamamoto, and G. Tohmon, in Proceedt h e crystal, t h e harmonic power for the 5-mni crystal ings o/ 1988 LEOS Anttud Meeting (Institute of Elecwould be 0.25 times t h a t for a 1-cm crystal in which t h e trical and Electronics Engineers, New York, 1988), pawaist was located a t the center. T h e expression for per VL2.4. harmonic power is therefore1“ 8. A . Ashltin, G. D. Boyd, and J. M. Dziedzic, IEEE J. C~uaiitumElecton. QE-2, 109 (19GG). 9. W. ,J. I<wlovsky, C. D. Nabors, and R. L. Byer, IEEE J. Quantum Electron. QE-24, 913 (1988). where w is t h e angular frequency of t h e ftincta~iiental, 10. J. Hall, H. J. Kimble, S. F. Pereira, and M. Xiao, Phys. d:lz is t h e nonlincar susceptibility, which is equal t.o 20 Itev. A. 38,4931 (1988). pin/V, kl,, is lhe fundainental wave numi)cr, cI, is t.he 1 1 . C. I+;. Tanner, 13. P. Masterson, and C. E. Wieman, Opt. permitlivity of free space, n,,,is t h e index of rcl’ract.ion Lett. 13, 357 (1988). a t t h e fundamental, c is t h e velocity of light, PI,,and 12. B. Dahmaiii, L. Hollberg, and R. Drullinger, Opt. Lett. P?,,.are the fundamental a n d harmonic powers, respec12,876 (1987). t.ively, a n d I, is t h e length of the crystal. From this 13. Y. lJeJnzIk!lIand T. Fukuda, Jpn. J. Appl. Phys. 12, 841 espressiori t.he theoretical harmonic o u t p u t at. a circu( 1973 ) . lating power ot’ 215 m W is 0.82 mW. ‘l.’he observed 14. It. I,. 13yer, i n Nonlinear Optics, P. G. Harper and B. S. harmonic conversion is less than this value h y a factor Wherrett, eds. (Academic, San Francisco, Calif., 1977)’ I,. 43. of ali~iost4. Tlik reduction in efficieiicy is n o t well
~~
~
Using diode lasers for atomic physics Carl E. Wieman Joint Institute for Laboratory Astrophysics University of Colorado and National Institute of Standards and Technology and Physics Department, Uniwrsity of Colorado, Boulder. Colorado 80309-0440
Leo Hollberg National Institute of Standards and Technology, Boulder, Colorado 80303
(Received 20 April 1990; accepted for publication 2 August 1990) We present a review of the use of diode lasers in atomic physics with an extensive list of references. We discuss the relevant characteristics of diode lasers and explain how to purchase and use them. We also review the various techniques that have been used to control and narrow the spectral outputs of diode lasers. Finally we present a number of examples illustrating the use of diode lasers in atomic physics experiments. 1. INTRODUCTION
Much of current atomic physics research involves the interaction of atoms and light in some way. Laser sources that can be tuned to particular atomic transitions are now a standard tool in most atomic physics laboratories. Traditionally this source has been a tunable dye laser. However, semiconductor diode lasers have been steadily improving in reliability, power, and wavelength coverage, while steadily decreasing in cost. It is now possible to have a diode laser system that will produce more than 10 mW of tunable light with a bandwidth of 100 kHz for a cost of less than $1000. Furthermore, the diode laser and power supply will fit in a 1-/ box, require very little power or cooling water, and, under the right conditions, will be within a few MHz of the desired wavelength as soon as it is turned on. A further virtue is that the amplitude is very stable compared to most other laser sources so that it is relatively simple to make sensitive absorption or fluorescence measurements. These features make diode lasers increasingly attractive alternatives for a variety of uses. In this review we present something of a users’ guide aimed at atomic physicists who might want to start using diode lasers in their work. In this spirit we present the advantages of diode lasers as well as their negative features (there are severalserious ones) and the best ways to deal with them. We begin in Sec. I1 by discussing relevant basic laser characteristics. Section I11 addresses some of the most important issues involved in actually setting up and using diode lasers in atomic physics experiments, such as purchasing the correct laser, tuning it to the desired frequency, and how to avoid destroying it. Section IV discusses optical and electronic feedback to control the laser output frequency. In the past, one of the main drawbacks to using diode lasers in atomic physics was the difficulty in obtaining narrowband, easily tunable output. However, the developments covered in Sec. IV have improved this situation considerably. Section V then discusses a selected set of applications of diode lasers in atomic physics, focusing primarily on work that has been made possible by these new wavelength control techniques. TOkeep this article to a reasonable length there are necessarily many things we do not discuss. For example, we do 1
Rev. Scl. Instrum. 62 (1). January 1991
not cover progress made on producing special diode lasers that work in marvelous ways, because these developmental lasers are not available to most atomic physicists. We restrict ourselves to lasers that are standard commercial devices. Another notable omission is the extensive work in molecular spectroscopy that has been done using lead salt diode lasers in the middle and far infrared regions of the spectrum. Camparo’ completed an extensive review on the use of diode lasers in atomic physics in 1985. That review covered much of the basic physics and characteristics of diode lasers, and the various uses that had been made of them. At that time most devices were simple free-running lasers without feedback, and they had been used principally for optical pumping and low resolution spectroscopy. In the present review, although there will necessarily be some overlap of that discussion, we will emphasize the recent developments including “stabilized lasers” (ones with feedback for frequency control), and new types of applications. There is also an excellent review entitled “Coherence in Semiconductor Lasers” by Ohtsu and Tako2 that discusses applications in a variety of areas. II. BASIC LASER CHARACTERISTICS
The basic physics of diode lasers is presented in many references3 and will not be repeated here. However, we do wish to single out a few features that are relevant to much of the later discussion. The construction of a typical semiconductor diode laser is shown in Fig. 1. The devices are extremely small and yet are capable of reasonable cw output powers with high electrical to optical efficiency. The laser light is generated by sending a current (the “injection current’’) through the active region of the diode between the nand p-type cladding layers. This produces electrons and holes, which in turn recombine and emit photons. The laser’s emission wavelength is determined by the band gap of the semiconductor material and is very broadband relative to atomic transitions. The spatial mode of the laser is defined by a narrow channel in the active region that confines the light. This confinement of the transverse laser mode is achieved either through the spatial variation of injection current density (gain guided) or by spatial variations in the index of
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FIG. 1. Illustration of the different layers of semiconductor and the typical dimensions of a diode laser. The rectangular shapc of the gain region leads to the oval radiation pattern. (Figure adapted from Ref. 4, with permission. )
refraction due to changes in the materials used in the laser's construction (index guided). A wide variety of laser designs have been used. Most of the atomic physics done with diode lasers has been carried out using index-guided GaAlAs lasers in the near infrared region of the spectrum. Unless otherwise noted, this type will be assumed in the subsequent discussion. These lasers typically produce single-mode (spatial and longitudinal) output powers of 5-1 5 mW with just the cleaved facets of the semiconductorserving as thelaser's mirrors. Output powers up to z 50 mW are commercially available from what are essentially the same devices with the addition of a high re10
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Foward current I F (mA) FIG.2. Output power vs injection current for a typical laser. The sudden change in the slope of each curve marks the onset of laser action, and the current at that point is the threshold current for the laser. The different curves show how the temperature affects the laser output. (Figure adapted from Ref. 4. with permission.) 2
Rev. Sci. Instrum., Vol. 62, No. 1, January 1991
FIG.3. cw diode laser output powers displayed as a function of wavelength. The various crosshatched boxesare taken from manufacrurerscatalogs and represent the distributions in wavelength and power that are advertised. In some cases it is possible to obtain lasers outside of these normal distributions. The range shown here is the distribution of lasers from the manufacturers and should not be confused with the tuning range of any specific laser which is much smaller. (The manufacturers shown in this table include; Hitachi, Mitsubishi, Nec, Oki, Sharp, and Spectra Diodeand are listed only as being representative, not as an endorsement; other sources should be equally suitable. We have experience with some of these devices but we have not verified all devices nor that the manufactures will deliver all of the devices as advertised. )
flectance coating on the back surface and a reduced reflectance coating on the output facet. Higher power lasers that have special designs such as multistripes or very wide single stripes are also available. These will likely be useful in the future, but at the present time the difficulty and expense in obtaining these special designs at a desired wavelength is a formidable problem. The output power of a typical semiconductor diode laser as a function of the injection current and temperature is shown in Fig. 2. This shows the abrupt onset of laser action at the threshold current and the increase in the threshold with temperature. Alternatively, for fixed injection current, the laser's output increases rapidly as the temperature is lowered. Figure 3 shows the power and wavelength characteristic of cw diode lasers that are readily available from commercial sources. These are divided into five main bands as driven by commercial applications. The two bands in the infrared, at 1300 and 1500 nm, have been developed primarily for fiber optic systems and are based on quartenary InGaAsP semiconductors. The lasers in the 750-890 nm region are based on AlGaAs and have applicationsin consumer and commercial electronics as well as some fiber-opticapplications. T h e relatively new visible diode lasers near 670 nm are based on InGaAlP semiconductorsand have many projected applications in commercial optoelectronic systems In applying these various lasers to spectroscopy we find that their spectral characteristics vary considerably.Those in the 750-870 nm region have the best overall characteristics. A. Beam spatial characteristics Because the light is emitted from a small rectangular region (on the order of 0.1 p n by 0.3 p m ) the output of a Diode lasers
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diode laser has a large divergence (see Fig. 1) . A typical output beam will have a divergenceangle (full width at 50% intensity) of 30"in the direction perpendicular to the junction, and 10" in the parallel direction. Normally the beam is collimated using a lens with a smallfnumber. If the laser is operating in a single transverse mode, the collimated beam will be elliptical, but it can be made nearly radially symmetric using anamorphic prisms. In addition to the different divergence angles, the output beam of most diode lasers is astigmatic. This astigmatism can also be compensated when necessary.' A typical diode laser wavefront collimated in this fashion will often have a significant amount of structure. This is primarily a function of the quality of the first collimating lens and the amount it apertures the beam. However, with relatively inexpensive lenses and spatial filtering, one can obtain wavefronts with good Gaussian profiles. This can be done with relatively little loss in power ( 10%), but sometimes the optical feedback from the filtering elements can be a problem. If one is forced to work with a laser that is emitting light in a number of transverse modes, a highly structured beam is unavoidable unless one sacrifices a substantial amount of power.
B. Amplitude spectrum For most applications, the amplitude and spectral characteristics of the light are more important than the spatial characteristics. When compared to other tunable laser sources, the amplitude noise on diode lasers is relatively small, but it can vary considerably depending on the actual laser and its operating conditions. Figure 4 shows the amplitude noise spectrum of a typical single-modediode laser taken with a fast photodiode and an rfspectrum analyzer. The natural scale for the measurement of laser amplitude noise is the quantum-fimited shot-noise level that is indicated on the figure for this detected power. The general structure of the amplitude noise shows peaking at the lowest Fourier frequencies (f<500 kHz). At intermediate frequencies (500
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There are two natural scales to the oscillation spectrum of diode lasers. The first scale is set by the spacing of the cavity modes of the laser which is on the order of 160 GHz ( -0.35 nm). The second is the linewidth of a single cavity mode. At the present time, it is quite routine to obtain "single-mode" lasers in the near infrared region (750 to 850 nm, AlGaAs lasers) although the visible (670 nm, InGaAlP laDiode lasers
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f (H=) FIG. 5. Spectral density of laser frequency noise for an AlGaAs diode laser. The dots are experimental data which are compared to theory as represented by the lines. The frequency noise peaks at low Fourier frequencies and again at the relaxation oscillation frequency. The theory shows that current induced temperature fluctuations are responsible for the low frequency peak, carrier induced index changes cause the high frequency resonance, and spontaneous emission contributes a flat background noise. (Figure adapted from Ref. 1 1 , with permission.)
sers) and infrared lasers ( 1300 and 1500 nm, InGaAsP lasers) ordinarily will have their power distributed in many modes. The distributed feedback (DFB) lasers in the infrared will run in a single mode but have a limited tuning range. We might mention a few caveats about nominally single-mode lasers. First, they are not absolutely single mode in that there are small but readily observable amounts of power in numerous other modes that can occasionally be a problem. Second, a laser that has been sold as a single-mode laser will often not run in a single mode for all injection currents. The lasers will atways be multimode for very low injection currents, but even at much higher currents there are often ranges of current and temperature where the laser will continue to operate in several modes. For the remainder of this section we will consider only the linewidth of a single mode. For some applications, in-
cluding laser spectroscopy, coherent communications, and precision measurements, the diode laser’s linewidth can be a serious problem. Many factors contribute to the linewidth of diode lasers; the most fundamental of these is that the laser cavities are so short that the Schawlow-Townes linewidth is significant (typically a few M H z ) . ~The linewidths of common single mode diode lasers are larger than the SchawlowTownes value because of fluctuations in the carriers, the temperature, and in the complex susceptibility of the laser” (typically linewidths vary from about 10 to 500 MHz). Linewidths of 20 to 50 MHz are typical for low power indexguided lasers. The spectrum of the frequency noise is similar (and coupled) to the spectrum of the amplitude noise. An example of a frequency noise spectrum of a diode laser” is shown in Fig. 5 . Unfortunately, we see that the frequency noise is large and extends to very high frequencies. The high frequency FM noise is a problem for two reasons: (1) it is difficult to make electronics that are fast enough to correct the frequency fluctuations and ( 2 ) even when the linewidth is narrowed by some optical or electronic means there is usually residual high frequency phase noise that contarninates some types of measurements.
D.Tuning characteristics A diode laser’s wavelength is determined primarily by the band gap of the semiconductor material and then by the junction’s temperature and current density.’* The band gap that determines the general range of the laser’s wavelength is unfortunately not under the control of the laser user. The range of wavelengths that are readily available is shown in Fig. 3, but a given commercial laser will only (and incompletely) tune over a wavelength interval of about 20 nm. Thus the user must buy a laser that is doped to operate within the tuning range of the wavelength of interest. The laser frequency tunes with temperature because both the optical path length of the cavity and the wavelength dependence of the gain curve depend on temperature. Unfortunately these temperature dependencies are quite different. For example, in the AlGaAs devices the optical length of the cavity changes about + 0.06 nm/K, while the gain curve shifts about + 0.25 nm/K. This results in a temperature tuning curve which, in an ideal device, is a staircase with sloping steps (see Fig. 6 ) . The slope of each step is just the tuning of that cavity mode, while the jump between steps corresponds to hopping from one longitudinal mode to the next ( ~ 0 . 3 5nm) due to the shifting of the gain curve. The spectral “gaps” encountered in the laser tuning as it jumps from one step to the next are the biggest drawback to using diode lasers in atomic physics. We will discuss below some empirical techniques that have been developed for dealing with this behavior in an isolated laser, as well as techniques for using external optical feedback to reduce or eliminate the problem. In practice, although the overall tuning follows the staircase pattern described, often a laser may jump several cavity modes at once, and then perhaps jump back at a slightly higher temperature as shown in Fig. 6. The choice of which mode the laser will hop to next is often extremely sensitive to optical Feedback. ‘‘Room temperature” commercial diode lasers are typi-
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30 K about room cally rated to operate in a range of temperature. This provides a tuning range ofabout 21 nm for AlGaAs lasers. The elevated temperatures generally cause noticeable degradation in the laser’s lifetime. For example, the data sheets indicate that a typical lifetime of about lo5h is reduced by a factor of 5 when the temperature is increased by 10 K. Other performance characteristics may also be degraded at higher temperatures. The temperature dependence of the laser characteristics vanes substantially for different models and types of lasers; however, detailed information is often provided by the manufacturer. Lasers can actually be cooled far more than 30 K (experience shows that at least some of the commercial lasers can be operated down to liquid He temperatures); the lower temperatures bring some additional complications such as the need to protect the laser from water condensation and mechanical stress due to large temperature changes. For very large temperature changes one must use the Varshni equationI3 to predict the wavelength, rather than the linear approximation given above. In addition to depending on the temperature, the laser wavelength also depends on the injection current. Changes in the injection current affect both the diode temperature and change the carrier density which also changes the index of refraction, and these in turn affect the wavelength. For time scales longer than about 1 ps the current tuning can simply be thought of as a way to change the temperature rapidly, because the carrier density contribution to the index-of-refraction tuning is relatively small. The current affects the junction temperature because of Joule heating, and the resulting tuning curve for wavelength versus current looks much the same as that versus temperature. Only for times shorter than -1 ps is the temperature rise small enough that the carrier density effect is not ovenvhelmed.14 For typical AlGaAs diode lasers the variation in the lasers’ frequency is z - 3 GHz/mA for frequencies below about 1 MHz, then it drops to =: - 300 MHz/mA for frequencies from 1 to 3000 MHz. It rises again to =: - 1 GHz/mA at the frequency of the relaxation oscillation (typically 3 GHz) and then drops off rapidly above this frequency. The crossover behavior between the different regions of F M response can cause considerable complication in high-speed modulation and frequency control systems for diode lasers. One of the important advantages of diode lasers over other optical sources is that their amplitude and frequency can be modulated very easily and rapidly by changing the injection current. Unfortunately, when the injection current is modulated, one obtains both AM and FM, and these are not independent. The simplest useful picture of the modulation response ofdiode lasers is that the AM and F M are both present but with different sensitivities.” Also modulation can be complicated by the fact that the relative phase between the AM and the FM changes as a function of modulation frequency. To a good approximation, the AM and the FM are linear with the injection current but the F M modulation index can be more than ten times the AM index. This means that the amplitude change can be ignored in many atomic physics applications. Thus the laser can be scanned over spectroscopic features and/orjumped back and forth to
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specific frequencies just by applying the appropriate modulation to the injection current. Applications of various modulation capabilities are discussed in Sec. V B. E. Visible lasers
The new visible diode lasersI6 which operate near 670 nm are exciting for spectroscopic and other applications. Because of the newness of these devices we have had only limited experience with them. The primary problem with these lasers is their broad spectral width. Most are gain-guided rather than index-guided, which typically means that they will have very poor spectral characteristics. We find that the visible lasers usually run multimode, with limited regions of single-mode operation. In addition, even in the regions of single-mode operation (and with very limited data) the linewidths of unstabilized devices are 300 MHz o r larger. They also have a transverse mode structure that has a very large and asymmetric divergence (7” and 40‘). Without compensation, this produces a laser beam with a linear shape rather than a round spot. This structure could be corrected, but it is still difficult to obtain optics designed specifically for these lasers. Finally, the output power of visible diode lasers has a very steep dependence on temperature and injection current which can lead one easily to a light intensity induced laser failure! 111. PRACTICAL GUIDE TO USING DIODE LASERS
Having discussed the basic behavior of diode lasers, we will now present some of the practical issues involved in setting up and using diode lasers for atomic physics. Although we do not discuss lasers with feedback in this section most of the discussion applies to that case as well. A. Purchasing diode lasers
Probably the single biggest frustration in using diode lasers in atomic physics experiments has nothing to do with scientific or technical issues. Instead it is the difficulty in purchasing diodes that have been doped to emit at the desired wavelength (or close enough to reach it with ternperature tuning). Because of this unfortunate fact, it seems worthwhile to provide some discussion to prepare the potential user for some of the problems that must be handled. These arise because the atomic physics market is an utterly insignificant fraction of the total diode laser market, and diode laser production is overwhelmingly Japanese. Therefore, unless you are fortunate enough to have some direct contact, you will probably be dealing with a distributor who is unfamiliar with your special requirements and rather distant from the supplier. With this background we will now explain what is involved in getting the laser you desire. In the manufacturing process there will normally be a target wavelength which the company is trying to produce, but there will be some scatter in what is actually produced, and they will advertise this as the range over which lasers are available. Thus the first issue is to get the company to screen the lasers and select the desired wavelength. Some manufacturers of lasers will do this and some will not. However, it is quite common for the local distributor for a company that does select to say that it is Diode lasers
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impossible to get wavelength selected lasers, or simply refuse to handle them. Sometimes extended phone discussions with the national headquarters and/or the distributor will solve this problem. Once you have found a distributor willing to order selected lasers for you, there are still some additional problems. First, if you want a wavelength that is well out on the tail of the distribution of wavelengths produced, you could have a substantial delay. Several years ago there were many anecdotes about 1.5-2 year waits for delivery, but our impression is that in recent years this has happened less frequently, and the quoted delivery times (typically 2-3 months) have been fairly reliable. However, there is still the problem that after waiting several months for lasers, one may receive lasers that have been selected for a wavelength that is not the requested wavelength. Based on our considerable direct and anecdotal experience concerning this phenomenon, the problem usually seems to be with the distributors and their lack of familiarity with handling wavelength selected lasers. There are two alternatives to going through the time consuming and frustrating purchasing process just described. The first is to buy a wafer doped to your specifications that will then produce a very large number of lasers. The second, and more common, approach is to find a secondary supplier of diode lasers who will provide wavelength selected lasers. There are a number of suppliers who buy large numbers of lasers from various manufacturers and then resell them, often with optional packages including power supplies and/or optics. Frequently such companies will wavelength select lasers from their stock. Of course this is usually more expensive than obtaining them directly from the manufacturer. Electronics distributors are also often willing to select lasers from their stock. In concluding this discussion we strongly recommend that if you are starting an experiment with diode lasers, buy several lasers at once. Because of the difficulties and delays in purchasing lasers, combined with the possibility that some lasers may never reach the desired transition and others may die abruptly, it is highly advisable to keep a number of spares on hand. Since the typical cost of AlGaAs lasers is $100$300, this is a worthwhile investment. However, with the special high power and long wavelength lasers that are more expensive, it may not be feasible to afford this luxury. B. Mounting of diodes and related optical elements
Once the laser has arrived, it must be set up for use. Two important initial considerations about mounting the laser, regardless of application, are optical feedback and temperature control. One needs to consider the best arrangement to minimize unwanted optical feedback and, if desired, provide controlled optical feedback for frequency control. This usually determines the overall physical layout. The details of the mounting are then determined by considerations of how to keep the temperature of the laser as stable as possible. This is discussed below. Normally, the optimum design in terms of temperature and mechanical stability, as with any laser system, i s one that is as compact as possible and reasonably rigid. However, because diode lasers and associated optics can be much smaller and lighter than other types of lasers, 6
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the scale of the assembly is often quite different. In Fig. 7 we show a simple laser mount. The laser itself comes attached to a small heat sink. This is mounted with good thermal contact to a small metal plate to which a heater (or thermoelectric cooler) and sensing thermistor are bonded. This plate is then attached to a solid block through a connection that is mechanically rigid, but a poor thermal conductor. The collimating lens is then mounted directly onto this block. A vast variety of types (and prices) of lens are available. Making a rather sweeping generalization, larger lenses that will be mounted farther away from the laser tend to cause less feedback, but it is more difficult to keep the feedback constant than with a small lens close to the laser. The transverse lens position is relatively unimportant; one can simply translate it to the correct position by hand and then tighten down the mounting screw. However, the focusing (longitudinal) adjustment is much more critical, so it is desirable to have some sort of fine screw adjustment on it. A specific warning to the beginning diode laser user about microscope objectives for collimating lenses: Standard objectives may seem to be a desireable choice because they are so readily available, but they are usually quite lossy for infrared wavelengths, and the optical feedback problems are often quite severe. A superior, low cost alternative is one of the many plastic spheric lenses made specifically for diode lasers. Another choice whenbeam quality is not important is the use of inexpensive gradient index ( G R I N ) lenses. C. Temperature and current control
As a general rule, the day-to-day repeatability of the laser system performance is determined primarily by the temperature stability. The most direct factor is the temperature dependence of the laser output frequency. However, we have found that the benefits of good temperature stability extend beyond the diode itself. If there is any external feedback, the laser frequency is always somewhat sensitive to the positions of the optical elements and these change with temperature. One almost unavoidable example is back scattering Diode lasers
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from the collimating lens that feeds back to the laser. If this is not too large and is fairly constant in phase, often it is not even noticeable. However, if variations in the temperature of the apparatus start causing the distance and hence the phase of the feedback light to vary, it can cause quite unwelcome and, at the time, highly mysterious variations in the laser wavelength or linewidth. Thus, poor temperature control can affect the laser performance in a variety of ways. Although it is adequate for some purposes to use only a single stage of temperature control on the laser, we have found that in the long run it is usually worth the small extra effort to add a second stage of temperature control on the base of the mount or on the box enclosing the laser assembly. A detailed discussion of temperature control servoloops has been given by Williams" and we will not dwell on them here. The basic idea is to use a thermistor as one leg in a balanced bridge circuit, and any voltage across the bridge is amplified and used to drive a heater or cooler. As discussed in Ref. 17, one must carefully consider the thermal time delays and the time constants in the electronics to achieve optimum performance. However, we have found that if effort is made to reduce the thermal mass and the thermal delays by mounting the sensing thermistor and the laser in close proximity to the heater, a relatively simple circuit will achieve stabilities on the order of 1 mK. A circuit diagram for such a simple controller is shown in Fig. 8. To achieve this kind of performance, the reference voltage and the bridge resistors must have low temperature coefficients. In addition because of thermal radiation and air currents, it is advisable to enclose the laser mount in some sort of metal container. This
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has the added benefits of keeping dust out of the system and isolating the laser system from acoustic vibrations. The last important element in setting up a diode laser is the current control circuit. It is difficult to recommend a generally useful circuit because the requirements of various experiments can be so different. In every case one needs to start with a low-noise current source and some protection against unwanted transients that can destroy the laser. Such a setup can range from a battery and a potentiometer plus a few diodes and capacitors, to quite elaborate circuits. Often the primary consideration in selecting a circuit is its current modulation capabilities. In Fig. 9 we show two sample circuits. See also Bradley and co-workers." As discussed in Sec. I11 G, it is important to mount the protective diode right on the laser mount. D. Tuning to an atomic transition
The main difficulty in using diode lasers in most atomic physics experiments is the problem of tuning them to the desired wavelength, usually that of some atomic transition. We have developed some empirical procedures for addressing this problem. To find a transition we set the laser current at the value we would like to operate, and then tune the temperature of the laser to near the correct value by observing the laser's wavelength on a spectrometer. ( A note of warning-if one follows the usual procedure of sending a collimated beam into the spectrometer, which no doubt has metal slits, the backscattered light from the slits will often cause the laser wavelength to shift and make the measure-
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ment meaningless. ) Once the laser is near the correct temperature, the current is then rapidly ramped back and forth over a large range, while the mount temperature is adjusted by small increments. The fluorescenceor absorption from an atomic absorption cell is used to determine when the desired transition wavelength is reached. Because of the tuning gaps, there may be no combination of current and temperature that produces the desired wavelength. However, there may
also be several and one must choose which is most desirable. If maximum laser power is needed, a temperature should be chosen where the laser is on the transition a t a high current. On the other hand, if only a small amount ofpower is needed it is better to pick a low current point so that the diode lifetime will be maximized (although the linewidth and amplitude noise will be somewhat larger). We repeat our previous warning: Never buy a single laser and expect to do an experi-
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ment with it. There is always a significant probability that a given laser will have a tuning gap just at the output wavelength of interest. We have found that, if no external feedback is used, this probability varies between about 30% and 60% for different batches of lasers operating near 852 nm. As discussed below, additional optical feedback to tune the laser can almost always force it-to oscillate at the desired wavelengths, but of course this results in a more complicated device and procedure.
E. Avoiding unwanted optical feedback
We have mentioned several times that the laser performance is affected by optical feedback. One of the unique features of diode lasers relative to most other lasers is their astonishing sensitivity to such feedback. This can be both a blessing and a curse. In Sec. IV we will discuss the blessing aspect; here we will consider the curse. As discussed in several references, the sensitivity of diode lasers to optical feedback arises from a combination of factors. First, the gain curve is very flat as a function of wavelength; second, the cavity finesse is quite low; and third, the cavity is very short. As a result, the overall gain of the system has an extremely weak dependence on wavelength and there are relatively few photons in the cavity, so that the lasing frequency is very easily perturbed. In addition, when light is returned to the laser it acts as a photodetector, generating more carriers in the junction and affecting the net laser gain. Detailed quantitative discussions are given in the references, but it is probably more useful here to give some general rules for laboratory work using a free-running laser with uncoated output facets. We have already mentioned the problem with scattering from spectrometer slits. In general, we can say that if the laser beam is collimated and hits a surface which scatters strongly, such as burnished metal or a white lab wall, there will be a significant effect on the laser wavelength (frequency shifts on the order of 10 MHz or greater, and easily observable variations in the laser amplitude as the distance changes), but if the surface has a dull black appearance there will probably be little effect. If the beam is going through a focus a larger fraction of the scattered light will be focused back into the laser thus increasing feedback sensitivity. We have found that, as a general rule, one cannot put anything, even good optical glass, at a focus without causing a large perturbation on the laser. These rules must be considered only as very crude guides since there are significant differences in sensitivity among the various types of lasers. How much feedback can be tolerated generally depends on how the laser is being used and the stability of the feedback. If the amplitude and phase of the feedback are constant, it can often be ignored unless one needs to scan the laser’s wavelength. Lasers with an output facet coated for reduced reflectance are more sensitive to optical feedback than uncoated lasers. On the other hand, lasers set up to have strong optical feedback to control the wavelength are correspondingly less sensitive to stray feedback. When using a laser in an experiment where simple positioning of the optical elements does not avoid feedback, one 9
Rev. Sci. Instrum., Vol. 62, No. 1, January 1991
must use some form of optical isolator. The simplest and least expensive isolator is an attenuator in the beam, but this of course is useful only when one can afford to waste most of the laser power. An alternative is a circular polarizer that does not waste light, but only works if the incident and feedback light have the same polarization. This is still fairly inexpensive. When the polarization of the return light is different from the incident light, which happens if there are any birefringent elements in the beam, it is necessary to use a Faraday i ~ o l a t o rUnfortunately .~ these usually cost several times as much as the laser itself.
F. Aging behavior A key consideration for any experimenter is the reliability of the laser. With diode lasers this has two different aspects. The first is euphemistically referred to as catastrophic failure, which usually means the experimenter did something to destroy the laser. The second is a gradual change in the behavior of the laser, referred to as “aging.” Normally one must gain considerable experience with diode lasers before aging becomes relevant compared to catastrophic failure. We will discuss aging first. Although we have not done careful statistical studies of aging behavior we can provide some observations based on our experiences with on the order of 100 lasers. First, while operation at high temperatures and/or high currents can contribute to aging, the original manufacturing process seems to be the dominant factor. We have had some lasers that we used extensively for two or three years before they showed any aging while others changed markedly in just a few weeks. Usually all the lasers in a given manufacturing run will age in a similar fashion. There are a variety of ways that the characteristics will change with age. One that is hardly noticeable because it is usually slow is that the tuning steps will gradually move. This means that one will occasionally have to adjust the current and/or temperature to keep the laser on the atomic resonance. This becomes serious if eventually the shift causes one to lose the transition entirely, but the opposite can also happen: namely, a laser which formerly could not be tuned to the resonance may age until it can. Aging drift rates can be as large as 30 M H z / h and may be due in part to increases in the thermal conductivity of the laser to the heat and changes in the laser due to nonradiative recombination. There are two other symptoms of aging that are more immediately detrimental. The first of these is an increasing tendency for the laser to emit in more than one longitudinal mode, which often means that higher and higher currents must be used to obtain single mode output. The second characteristic is that the spectral width of the single mode output can gradually increase from its initial 20-30 MHz up to several hundred megahertz.
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G. Catastrophic failure modes, or 1001 ways to kill a laser
In a lab where diode lasers are applied to a variety of problems and there is a constant flux of students and guest workers, it is a highly unusual laser that can age gracefully and die of the “natural causes” listed above. It is far more common for the lasers to be abruptly destroyed. We have Diode lasers
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discovered many ways this can be accomplished, and we list these here in the hope it may save others from repeating our experiences. A major weakness of diode lasers is that a very brief transient, which causes too much current to flow or produces too large a back voltage across the junction, can be fatal. A common way this can happen is switching transients when turning the laser on or off. Although one obviously must put in protective circuitry to deal with transients when the laser power supply is switched on and off intentionally, it is also necessary to be concerned with the potential effects of accidental disconnection of the power supply, or intennittent connections in power cables or the ac power line. Other sources of transients are discharges of static electricity and high voltage arcs in other parts of the lab. These last two are minimized by goad grounding and shielding procedures. Also the protection diode normally used to prevent excessive back voltage should be as close to the laser as possible. If there is even a fraction of a meter of moderately shielded cable between the laser and the protective diode, a high voltage spark nearby will often destroy the laser. We should point out that a laser which has been electrocuted will often continue to emit laser light, but the wavelength, threshold, and output spectrum will be quite different. Although electrocution is by far the most common form of death for diode lasers, the lasers that are not mounted in a protective can are also quite delicate and one must be careful not to touch the laser itself, or the tiny current leads, while mounting the laser or associated optics. Considering all of the possible failure modes for the diode lasers, we find that, on the average, the lasers need to be replaced approximately every six months. This average includes lasers dying for whatever reason or becoming impossible to tune to the appropriate wavelength. Presumably the replacement rate would be much less if the lasers were left alone and not always being tampered with in evolving experiments. This average rate is for moderately experienced users and failure is much more frequent with inexperienced users.
IV. TECHNIQUES FOR CONTROLLING AND NARROWING LASER OUTPUT SPECTRA
Several techniques, both optical and electronic, have been devised to narrow the linewidth and control the center frequency ofdiode lasers. As mentioned earlier, theeffects of optical feedback on semiconductor diode lasers are both profound and complicated, but they have been studied in detail.18 In particular, the effects of optical feedback on the spectral properties of these lasers can be found in Ref. 20. However, the essence of the optical feedback methods is the simple idea that by increasing the quality factor (Q)of the laser’s resonator, the linewidth will be reduced. The simplest implementation of the optical method for spectral narrowing is just to reflect back to the laser a small fraction of its output power. The basic electronic method uses feedback to the laser’s injection current to control the laser’s frequency. Both the simple optical and simple electronic methods have severe limitations in terms of general applicability. More elaborate frequency control systems are usually necessary to deal with the variety of diode lasers that are 10
Rev. Sci. Instrum., Vol. 62, No. 1, January 1991
commercially available. No one method has been found to narrow the linewidth and control the frequency of all types of diode lasers. Here we will briefly review some of the more successful methods. We should mention that there are many errors that one will find in the literature relative to measurements of diode laser linewidths and frequency stability. The most common mistake is that the experimenters will measure the residual noise on an error signal inside a servo loop and use this information to infer a frequency stability. Such measurements do not guarantee that the error signal actually represents laser frequency fluctuations, and they are fraught with systematic errors. Picque and co-workers*’ and Pevtschin and Ezekiel’’ have made more realistic measurements and pointed out some of the possible pitfalls. A. Simple feedback
Some limited success in narrowing the linewidth of a diode laser has been achieved by using simple optical elements to feed back to the laser some of its output power. Among the optical elements that have been used for feedback are simple mirrors,” e t a l ~ n s , ’gratings,” ~ fiber cavities,26 and phase conjugate mirror^.^' The resulting systems must be described as lasers with complex resonators where the diodes’ facets and the external optical elements all play a role in creating the net resonator structure as seen by the semiconductor gain medium. By providing optical feedback from a small mirror or glass plate placed close to one of the laser facets, sometimes one can force a multimode laser to oscillate on a single mode, or induce a single-mode laser to tune across the forbidden “gaps.”’* The limitations to this method are that it does not work with all lasers, and still requires some sort of frequency reference to stabilize the laser’s frequency. Also, as with any complex resonator, it can be difficult to achieve long-term stable performance with such systems. B. External cavity lasers
The method of the “external cavity laser” uses antireflection (AR) coatings on the diode laser chip, and some external optics to provide the laser resonator. The external optics may contain frequency selective elements such as a gratingz9 and/or etaions,30 with the grating being the least expensive and most common. A variety of geometries are possible and one example of an external cavity laser is shown in Fig. 10. For the nonspecialist these systems can be a challenge because they require having the laser diode facets AR coated, which is usually an expensive and nontrivial procedure. In addition one may need access to both output facets, which is often difficult given the packaging of commercial devices. In principle, this method should work with all types of diode lasers, although only a limited number of systems have been built, and they are not commercially available. A number of variations on the basic external cavity idea have been demonstrated. These include the use of prisms, birefringent filters, and various other clever optical schemes ’ An (of particular interest is the paper by Belenov et d 3 ). example is putting AR coatings on one facet of the laser chip and building pseudo-external cavity lasers as discussed in Diode lasers
I0
782 ANTIREFLECTION COATED
DIODE LASER
\ /
OUTPUT MIRROR
ANTIREFLECTION COATED LENSES
FIG. 10. External-cavity diode laser. This schematic shows an AR coated diode laser chip used in an external cavity with a grating for wavelength selectivity. (Figure from Fleming and Mooradian,Ref. 29, reproducedwith permission.)
the next section. This technique has been useful in controlling the spectral properties of 670-nm diode lasers.32 C. Pseudo-externalcavity lasers
FIG. 1 I. A pseudo-externalcavity laser using a standard commercial laser and a diffraction grating for feedback. The flex pivot is used to obtain the precisefocusingof the collimating lens which is necessary. The peizoelectnc transducer under the grating is used for fine frequency tuning by changing the length of the cavity. Other tuning options are to have the piezo-electric transducer on the other side of the mount so that it rotates as well as translates the grating, or to do the tuning using an uncoated piece of glass which is mounted on a galvanometernear Brewster’s angle and is precisely rotated. Not shown is the temperature control elements (sensors and heaters or coolers) that must be used on the diode and may be used on the baseplate.
Various laboratories have shown that if only one of the facets of a diode laser is AR coated, the laser’s linewidth can be narrowed and its oscillation frequency controlled by providing external frequency selective optical feedback. Fortunately, many of the high power ( > 15 mW) commercial lasers already have reduced reflectance coatings on the output facet and high reflectors on the back facet. Gibble and Swann and collaborators have demonstrated that without modification these commercial lasers with coatings can be spectrally narrowed and tuned effectively with grating feedback.” An example of such a system is shown in Fig. 11. This method is a combination of the two listed above in that it is not really an external cavity laser, since the laser oscillates without the extra feedback, but the system can operate so that external feedback dominates that from the low reflectance facet. These devices have fundamental linewidths on the order of 100kHz or less, but the true linewidth is usually dominated by the mechanical and thermal instabilities in the length of the grating-laser cavity. This linewidth can be several hundred kilohertz or larger if the cavity length is not electronically stabilized. We and others have found such lasers to be quite well suited for a variety of applications. Their virtues are that they use standard commercial lasers and inexpensive components, and the grating allows tuning over a range larger than could be easily covered by temperature tuning. Also the tuning can be continuous over much larger spectral ranges, and it is quite easy to set and keep the laser frequency on particular atomic transitions. Minor difficulties in thedesign are that the grating alignment is moderately sensitive, and to achieve the necessary degree of collimation of the laser light, thedistance between the laser and the collimating lens must be adjusted to within a few micrometers or less. In practice it is adequate to have the lens position adjusted with a fine-pitch screw, with the position set so that the beam appears the same diameter to the eye over a path of several meters. There are tradeoffs one can make in the detailed design between greater feedback, and hence more frequency con-
trol, versus coupling out more power. The exact performance will depend strongly on the reflectance of the output facet of the laser. For illustrative purposes, wi will describe the behavior of such a laser system using a Sharp LTOl5 laser (mention of this product does not represent a recommendation and we expect other lasers to behave similarly), with ~ 7 0 %feedback to the laser. As the grating is rotated, the laser will tune continuously over a region of about 80 GHz spanning the peak of the semiconductor laser cavity resonance. To obtain continuous tuning, the grating is mounted so that, as it rotates, the change in the cavity resonant frequency due to the change in the cavity length matches the angle tuning of the grating.” A small (2 cm) piezoelectric speaker disk placed between the grating tilt screw and the block on which the grating is mounted is a convenient and inexpensive way to rotate the grating smoothly. There will then be a gap in the tuning of 70-80 GHz until one reaches the next mode of the small semiconductor cavity. This pattern repeats itself as the laser is tuned farther from its gain peak, with the range of continuous tuning gradually decreasing. The maximum range the laser wavelength can be pulled from its gain peak is about +_ 10 nm, at which point the tuning range per diode mode is about 10 GHz. With the feedback reduced to 25% to increase the output coupling, the maximum range ofcontinuous tuning is about f 6 GHz, and the farthest the frequency can be pulled is about 6 nm. Gaps in the tuning are easily filled by adjusting the laser current or temperature to shift the frequency of the laser diode modes. The output power is about 10 mW with this output coupling.
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Diode lasers
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11
783 Mod.--.(-
tf:zral
Locking
Cavity Monitor Cavity
FIG.12. Schematic diagram of an optical feedback locking system for laser diodes. In the geometry shown optical feedback (type 11) occurs when the lasers frequency matches the resonance frequency of the optical locking cavity. This type of feedback locks the laser’s frequency to the cavity resonance frequency and narrows the laser’s linewidth. The monitor cavity is used as a diagnostic and does not affect the laser‘s frequency because of the optical isolator. (Figure from Ref. 59, reproduced with permission.)
D. Feedback from h i g h 4 optical cavities
Another optical feedback locking system that we have used quite successfully is diagrammed in Fig. 12. In this system we use weak optical coupling of the laser’s output to a high+ optical res~nator.~’ The geometry is such that the laser sees optical feedback from the Fabry-Perot cavity only when the laser’s frequency matches a resonance of this cavity. The result is that the coupled (laser plus cavity) system haslowerlossesat a cavityresonanceand thelaser’sfrequency automatically locks up to the cavity resonance. In this way the laser’s linewidth can be reduced to a few kilohertz and the laser’s center frequency is stabilized to the cavity. One of the limitations of this system is that it requires some additional slow electronics to keep the laser locked to the same cavity mode for long times and to keep the laser synchronized with the cavity for long scans (greater than about 1 GHz). Without the additional electronics, the laser will stay locked to the same cavity mode for times that vary from less than 1 to as much as 30 min, depending on the stability of the cavity and the temperature and current controls for the laser. By reducing the distance between laser and cavity to a few centimeters and mounting the cavity and laser in a pressure-tight temperature-controlled enclosure, we have been able to extend this to several hours without electronic control. We and others use this system with good results with unmodified commercial lasers operating in the diode laser wavelength bands at 780,850, and 1300 nm. With rather limited experience, we have been disappointed in the results of this method for unaltered gain-guided lasers operating at 670nm. However, the method appears to work with the new index-guided versions that have some AR coating.36 The major advantages of this system of frequency control are that the linewidth is very narrow and the stability is determined by the external cavity, which can be made very rigid and insensitive to changes in temperature and atmospheric pressure. Also, the laser retains some high speed modulation ca12
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pabilities. The disadvantages are the sensitivity to cavity laser separation mentioned above, and the fact that laser tuning range (and gaps) is still essentially the same as that of the basic unstabilized laser. There are a number of optical configurations that can produce resonant optical feedback from a high Q resonator, but in practice the V configuration shown in Fig. 12 is a very convenient one. The alignment procedure is straightforward and after a little practice can be done fairly quickly when one recognizes the proper signal shapes. During the alignment and afterwards it is best to sweep the laser’s frequency over several orders of the Fabry-Perot and to monitor the power transmitted through the cavity. The effects of feedback from the cavity to the laser will be obvious in the shape of the transmitted fringe. By sweeping many cavity modes the effects of the feedback can be observed. The system requires only a small amount of the laser’s output power ( 1%) for locking but for alignment it is easier to start with higher powers ( - 10%) so that it iseasy tosee the beams.Afterone becomes familiar with aligning the optical feedback lock, the feedback power can be reduced. This 10% of the output is split off and sent through an attenuator and a lens that approximately mode-matches the beam into the high-Q FabryPerot cavity. An easy way to determine that the mode matching is correct is to ensure that the return beam reflected from the Fabry-Perot cavity is about the same size as the beam that is incident on the cavity. Next, one looks to see that the input beam is centered on the Fabry-Perot’s input mirror and that the reflected beam is at a small angle relative to the input beam. The divergence angle can be quite small but it is necessary that the return beam is a few spot diameters away from the input beam by the time it returns to the laser. One should also observe that there are two output beams transmitted through the cavity on resonance (see Fig. 12). By setting the attenuator for very low powers one will see the usual Fabry-Perot peak structure in the transmitted light; then as the feedback is increased the laser will begh t o lock to the cavity resonance and the Fabry-Perot shape will broaden and become squarish. The fringe will typically broaden to a few hundred megahertz, corresponding to the range over which the cavity controls the laser’s frequency. The feedback power can be increased to enhance the locking range and to reduce further the laser’s linewidth. When the feedback becomes too strong the system will become unstabie.37 The phase of optical feedback (distance from the laser to the cavity) is optimum when the power transmitted through the cavity is maximum. One can force the lock to the optimum condition by modulating the path length from the laser to the cavity (with a PZT) and detecting the transmitted power. This signal is then demodulated with a lock-in amplifier and fed back electronically to control the position of the PZT. Variations in the optical feedback phase can be minimized by good mechanical stability and by making the distance from the laser to the cavity small. In fact we often operate these optically locked systems without active electronic control. Although commercial cavities are often adequate it is useful to be able to make simple inexpensive tunable Fabry-
-
Diode lasers
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Perot cavities. The simplest method is to glue laser quality mirrors onto a cylindrical Invar or quartz tube, with a PZT disk under one of the mirrors so that the cavity can be scanned. If one of the mirrors can slide inside a tubular mount, the cavity can be aligned and set to the confocal condition by aligning it with a laser before the last mirror is glued in place. For the best stability it is useful to have a hermetically sealed cavity. In these simple cavities this can be accomplished with a glue that does not unnecessarily stress the mirrors (such as Tracon 2 143D epoxy, mention of this product does not represent a recommendation). An alternative design uses a fine thread on the Invar spacer to set the cavity to the proper length. For most of our diode laser spectroscopy we use reference cavities with finesses of about 100 and free-spectral ranges of 0.25-8 GHz. E. Other optical feedback techniques
Other monolithic or semimonolithic extended cavity diode lasers with good spectral qualities have been reported in the literature,'* but they are usually laboratory test devices that are not commercially available and usually not tunable. Thereare indications that some of these systems wiII become commercially available in the near future, but it is not likely there will be broad spectral coverage anytime soon. F. Electronic feedback
The other competing method for laser frequency stabilization is electronic feedback. It is straightforward to use electronic feedback to lock the center frequency of diode lasers to cavities or atomic absorption lines using standard laser stabilization methods.39 On the other hand, narrowing the laser's linewidth with electronic servos is a much more difficult task. The problem is that the frequency noise spectrum extends to such high frequencies that most servosystems are not able to act fast enough to correct the laser's frequency fluctuations. A few groups-* have been successful in developing very fast (few ns delay time) electronic servosystems that can be used to narrow the linewidth of diode lasers if the laser's intrinsic linewidth is not too broad to start with. Linewidths on the order of 100 kHz have been achieved with these techniques. Electronic systems have the advantage over most of the optical methods in that they do not degrade the modulation characteristics of the lasers. However, a disadvantage is that they do not extend the laser's tuning range. Unfortunately, considerable expertise in electronics is required to design and build the necessary circuits. G. Hybrid systems
Hybrid designs are of two basic types. In the first case, electronic feedback is simply used to tune a cavity that controls the laser frequency by optical feedback. In the designs described in Secs. IV B, IV C , and IV D, for example, this would entail moving the end mirror or grating using a piezoelectric transducer. Typically this is done to achieve longterm frequency stability by locking the cavity frequency (and thereby the laser) to an atomic or molecular transition. Promising results have also been obtained with a second 13
Rev. Scl. Instrum., Vol. 62, No. I , January 1991
type of hybrid system which uses electronic feedback in addition to optical feedback to reduce the laser's l i n e ~ i d t h . ~ ~ In this hybrid optical-electronic system the optical feedback is used to initially narrow the laser to the point that less sophisticated electronics (than that mentioned in Sec. IV F) can further narrow the linewidth and stabilize the laser's frequency. With such systems it should be possible to achieve frequency stabilities that are limited only by one's ability to provide an adequate frequency discriminator. Using this type of servosystem, phase locks between diode lasers have been d e m ~ n s t r a t e d . ~ ~ H. injection locking
It is also possible to stabilize the frequency of diode lasers by optical injection locking." The frequency of an unstabilized diode laser can be locked to a spectrally narrow, master laser oscillator by coupling a small amount of power from the master laser into the slave diode laser. This technique may be useful for some applications (e.g., generating higher power) but obviously does not meet many spectroscopic needs, because if a good master laser were available the diode laser would not be needed. The notable exception is that injection locking allows one to use the high powers available from arrays or wide channel lasers and still maintain the precise frequency control which can be obtained with a feedback-stabilized di0de.4~
I. Summary
For some spectroscopic applications the spectral characteristics of commercial diode lasers are good enough as they stand, but for many others that is not the case. Also, unstabilized lasers can be quite difficult to tune in a controlled manner. Unfortunately, none of the existing frequency stabilization methods (perhaps with the exception of the external cavity system) are able to narrow the linewidth and control the frequency of all of the types of semiconductor lasers that are potentially useful for atomic physics. We do not mean to discourage the potential user of high resolution lasers, because at many wavelengths the existing solutions work very well and provide tunable narrow linewidth lasers from commercially available diodes. Such devices have the advantages of very narrow linewidths and excellent amplitude stability, and they are much simpler and more reliable to tune to a particular frequency than are unstabilized lasers. V. APPLICATIONS
Having discussed how to build and use diode lasers, we will now discuss some of their applications in atomic physics. Of the numerous possibilities, we have chosen to describe a few novel applications that demonstrate some of the new capabilities of diode lasers that have not previously been appreciated. Many of the examples are related to our own work. We do not wish to imply that this is the only work in this field; rather these are simply a number of representative samples that are most familiar to us. A. Optical pumping
One of the earliest applications of a diode laser to atomic physics was as the light source for optical pumping and probDiode lasers
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ing of a polarized medium. Many applications of this type are reviewed by Camparo.’ Here we want to mention briefly some interesting new optical-pumping applications using traditional laser approaches. Diode lasers have been used extensively in frequency standards and atomic clocks. In these applications they optically pump the atoms of interest, and then detect the atoms that have undergone the microwave frequency clock transitions. Fortuitously, the atoms used in some of our best frequency standards (rubidium and cesium) have resonanceline wavelengths that are easily produced by semiconductor lasers. The work in this area started in earnest in the mid7 0 ’ and ~ ~ continues ~ with promising results Significant research and development efforts to incorporate diode lasers into rubidium, cesium, and other atomic frequency standards are under way around the world. These efforts are driven by vast improvements in signal-to-noise ratios, ultimate accuracy, and the relative simplicity obtained by using diode lasers. Figure 13 is a schematic diagram of a diode-laser-pumped cesium atomic clock. The lasers are first used to prepare the atomic beam by putting all the atoms into the appropriate hyperfine level using optical pumping. Then, after the atoms have passed through the microwave region, a laser excites the initially depleted level. Thus the microwave transitions are detected by the increase in the downstream fluorescence. In an optical-pumping experiment of a different sort, Streater, Mooibroek, and W ~ e r d m a nused ~ ~ the light from a diode laser to drive very successfully a rubidium “optical piston.” The optical piston is an atomically selective drift that is driven by laser light. We list this under optical pumping because a key element in the experiment was the novel scheme of using the relaxation oscillation sidebands of the diode laser to assist in optical pumping of Rb atoms. In this particular case the 3-GHz ground state hyperfine splitting of 85Rbmatches the relaxation oscillation frequency of the 780nm lasers. Thus a single laser allowed them to excite atoms in both hyperfine ground states, which greatly enhanced the operation of the “piston.”
B. Applications using fast frequency modulation
One of the unique aspects of semiconductor diode lasers is the ease with which their amplitude and frequency can be modulated. The amount of modulation can be both large and very fast. For common diode lasers the modulation response extends out to a few gigahertz, with a few special lasers capable of modulation rates of more than 10 GHz. This modulation capability has applications in spectroscopy as well as other fields. As mentioned earlier, modulation of the diode laser’s injection current produces both AM and FM, but in many cases of atomic transitions the frequency modulation is the dominant effect and we will focus attention on it here. The frequency modulation is dominant simply because the atomic linewidth corresponds to such a small fractional change in frequency. We can. picture the frequency modulation of the laser either in the time domain or in the frequency domain, depending on the particular application. If we first think in the frequency domain, modulation of the injection current at high frequency produces a laser spectrum with modulation sidebauds. The application of modulation sidebands to all types of lasers has proven very useful for spectroscopy and other application^.^^ The principal advantage of these optical heterodyne methods is that they increase the signal-tonoise ratio in detecting absorption and dispersion signals. These techniques have been extended to diode lasers by B j ~ r k l u n d , ~Lenth,52 ’ and others.” The diode laser case is complicated somewhat by the additional A M that accornpanies the desired FM. Lenth in particular has sorted out the effects of having both AM and FM modulation simultaneousIy.52 We have used optical heterodyne methods with frequency modulated diode lasers to detect alkali atoms and optical cavity resonances with high sensitivity. External frequency modulators have been also used with InGaAsP lasers to detect NH, using EM s p e c t r o ~ c o p yO . ~ h~ t s and ~ ~ collabora~ tors have achieved excellent results with optically pumped R b clocks by using frequency-modulated laser diodes for
/\Photodetector
FIG. 13. Diagram of an optically
Light
pumped cesium atomic frequency standard. The cesium atomic beam travels in a vacuum through the first laser interaction region where theatornsareoptically pumped, it then traverses the microwave cavity region and then on to the second laser interaction region where the atoms are detected. The population in the ground state hyperfine levels is indicated along the path by the Fand m, values. The microwave clock transition is between the F = 4 . m, = O . and the F = 3, m, = 0 states. As shown here the system uses two lasers with different frequencies for optical pumping and a third laser for detection. (Figure from Ref. 2, reproduced with permission.)
U Laser I 14
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Diode lasers
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Optical Spectrum, 367 MHz Modulation FIG. 14. Modulation response of an optically locked diode laser. The spectrum shown here is taken with the detector (Det.) and monitor cavity system shown in Fig. 12. The laser's injection current is modulated at 367 MHz through the modulation port (Mod.). The modulation frequency is rationally related ( $ times) to the free spectral range of the locking cavity. In this case the optical lock is stable and provides an array of narrow linewidth laser side bands each spaced by the modulation frequency. Lasers without optical locking have similar modulation response but with broad linewidths. (Figure from Ref. 57, reproduced with permission.)
optical pumping and high sensitivity detection. Other applications of frequency-modulated diode lasers include communication systems, length measurements, range finding, and laser radar. A great deal ofspectroscopic work with diode lasers and frequency modulation has been done with the lead salt diode lasers operating in the infrared. Methods have been developed using modulation at two frequencies ("two-tone" modulation) to detect molecular species with high sensitivity.'(' The techniques employed with these lasers are also generally applicable to the GaAs lasers. If we increase the strength of the modulation of the injection current, it is possible to spread out the diode laser's spectrum into many discrete sidebands. These sidebands can cover spectral windows of many gigahertz. Figure 14 shows a spectrum of a diode laser whose injection current (97 mA dc) is modulated at 367 MHz with an amplitude of 4 mA. This modulation produces eight discrete sidebands spaced by the modulation frequency, in addition to the original carrier, and does not otherwise broaden the spectrally narrow laser ( z10-kHz linewidth). Potential applications of this high modulation index case include laser frequency control, fixed local oscillators for coherent communication systems, and atomic and molecular physics experiments where numerous frequencies are required. Some applications of diode laser frequency modulation are best viewed in the time domain. For example, rapid chirping or jumping of the output frequency has been used successfully in several experiments. Watts and Wieman5' demonstrated that the output frequency of a laser can be changed by 10 G H z and become stable to within a few megahertz in a time of 2 ps. This was accomplished by simply changing the injection current through the laser using an appropriately tailored pulse. With this high speed switching it was possible to optically pump all the atoms in a cesium 15
Rev. Scl. Instrum., Vol. 62, No. 1, January 1991
beam into a specific magnetic sublevel (F,M,) using a single laser. Normally this would be impossible because cesium has two ground state hyperfine levels that are separated by 9.2 GHz. However, in a fraction of the time it took the atomic beam to pass through the laser beam, the laser frequency was switched to excite both levels. Robinsons9 has also used this frequency jump technique to polarize a rubidium cell with a single laser. This capability to rapidly sweep the laser frequency over a large range in a controlled manner is unique. In these experiments it was used to reduce the number of lasers needed for optical pumping, but it may find greater utility in studies of various types of transient phenomena.' A recent application of the frequency chirping capability of diode lasers is the stopping of atomic beams. Ertmer et aZ.* showed that by frequency chirping the light from a dye laser it was possible to slow and stop a beam of sodium atoms. T o achieve the necessary frequency scan a sophisticated broadband electro-optic modulator was developed. A corresponding frequency chirp is very easy to produce with a diode laser by merely adding a small sawtooth ramp to the injection current. Watts and Wieman used this technique to stop a beam of cesium atoms6' with a surprisingly simple and inexpensive apparatus. Salomon er have carried out a similar experiment to study the optimization of the cooling. They showed that by adding 110-MHz sidebands to the laser, they could enhance the number of stopped atoms by 50%. Recently Sheehy eta[." have used frequency chirped diode lasers to stop rubidium atoms. These various experiments demonstrated that, in addition to slowing atoms, the frequency modulation capability of diode lasers is also well suited to probing atomic velocity distributions rapidly enough to avoid distortion. The resulting 15-m/s velocity spread of the nearly stopped atoms reported in Ref. 61 was limited by the 30MHz linewidth of the diode laser. To improve on this, Sesko, Fan, and Wieman64 stopped atoms in a similar manner using acavity-stabilized laser of the typediscussed in Sec. IV D. In this case, the frequency was ramped by changing both the laser current and the length of the stabilization cavity (using a piezoelectric transducer). The speed of the cavity response was less than that of the laser, but was still adequate for atom stopping. With a cavity-stabilized stopping laser, the number of stopped atoms was found to improve dramatically and the final velocity spread was substantially smaller. We have obtained similar results by stopping atoms with the light from grating cavity lasers of the type discussed in Sec. IV C. C. High resolution spectroscopy
We mentioned briefly the improvements that cavity locked lasers have made for atom stopping. This is only one of a large number of experiments made possible by optical feedback stabilization. This technique provides a very narrow bandwidth source which is valuable for high resolution spectroscopy and other experiments where narrow linewidth is important. Althoughoften neglected in the English literature, a great deal of forefront work on diode laser stabilization and applications to high resolution spectroscopy has been done by Velichanski and co-workers.6s Other examples of high resolution and high sensitivity spectroscoDiode lasers
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F‘ = 3
4
5
are the work of Martin and co-workers.68They have demonstrated time-resolved spectroscopy of BaCl and velocitymodulation spectroscopy of N;’ . Avila et aLb9 have used diode lasers to make a high precision measurement of the absolute energy of the 6P3,, state in cesium. The laser frequency was locked to the cesium resonance line by observing fluorescence from a collimated cesium beam. A wavemeter was then used to measure the absolute frequency to 12 MHz ( 3 parts in 10’). In the work of Gibble and Gallagher,67a pseudo-external cavity laser with grating feedback was used to study velocity changing collisions in rubidium. Using a saturated absorption technique and this continuously tunable narrowband laser, they were able to measure collisional distortions of thesaturated absorption lineshapes which were a small fraction of the 10-MHz natural linewidth. Feedback-stabilized diode lasers are uniquely simple and inexpensive sources of very narrowband tunable light. For this reason, they will undoubtedly be widely used for high resolution spectroscopy in the future. D. High sensitivity spectroscopy
py are described in Refs. 66-69. Figure 15 illustrates the capabilities of cavity stabilized lasers for high resolution spectroscopy. This figure shows a fluorescence spectrum of the 6S-6P3,, transition in cesium observed in a single 100-ms oscilloscope trace. The spectral linewidths were limited entirely by the 6-MHz natural linewidth of the transition, and had no contribution from the laser linewidth. Figure 15 also illustrates how the amplitude stability of the laser leads to very good signal-to-noise ratios. This laser was used to make precise measurements of the Stark shifts and the hyperfine structure of the W,,, state of cesium. These measurements had an uncertainty of only 20 kHz. Good examples of applications that use the unique modulation capabilitiesof diode lasers for precision spectroscopy
Laser Frequency-
FIG. 16. Fluorescence measurements of the 6s F = 4 to 6P,,> F = 5 resonance line in a cesium atomic beam. The laser’s frequency noise causes excess amplitude noise on the cesium fluorescence signal. This noise is surprisingly irregular as can be seen by comparing it to the noise observed on the Fabry-Perot cavity transmission signal that was taken simultaneously.We see the frequency noise shows up on the sides of the cavity transmission fringe whereas it is enhanced on the peah and one side of the fluorescence signal. This noise disappears when the la%erslinewidth is narrowed. 16
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Another developing area of application of diode lasers is to high sensitivity spectroscopy. This includes the application to optical pumping, optical heterodyne or FM spectroscop^,^'-^^ ultrasensitive detection,” optogalvanic spectroscopy,” and other areas:’ Also, significant work in sensitive spectroscopy has been done using the lead salt diode lasers72 which we have not addressed in detail in this article. Often these applications are just extensions (to diode lasers) of well-established techniques of spectroscopy. On the other hand, there are a few applications that utilize the unique capabilities of the diode lasers. Diode lasers stabilized by optical feedback have been found to be much better than unstabilized lasers for high precision measurements of fluorescenceor absorption where the signal-to-noise ratio is important. In some cases the feedback not only reduces amplitude noise on the laser, but also reduces some previously unexpected noise on atomic spectra associated with the laser FM spectrum. In using diode lasers for precision spectroscopy some workers have reported that the measured signal-to-noise ratios are much lower than expe~ted.’~ For example, we observe large amounts of excess noise in diode laser-induced fluorescence and direct absorption measurements of the F = 4 to F = 5 resonance line in cesium (see Fig. 16). This noise results from the frequency fluctuations in the diode laser and is eliminated when the diode laser’s Iinewidth is reduced by using the high+ cavity feedback lock. Two surprising features are observed in this excess noise relative to what one would expect from simple rate equation analysis of the laser induced transition: first, it is much too large; and second, the noise can also be asymmetric with respect to laser detuning from the resonance. By properly treating coherence and atomic saturation with a noisy laser source, Zoller and c o - ~ o r k e r s ?have ~ provided an explanation of these observations. Two significant instances in which the elimination of this noise has been a major concern are the new optically pumped cesium clocks being constructed at NIST and other Diode lasers
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national standards labs4’ and the precise measurement of parity nonconservation in cesium using an optically pumped beam.’> In these experiments the use of feedback-stabilized lasers has been essential because of the reduced noise in the atomic fluorescence.
E. Self-locking power buildup cavities Another application of feedback from high-Q cavities is the self-locking power-buildup cavity, which can provide large enhancements in laser power. One obvious shortcoming of diode lasers is the small output power, but buildup cavities allow one to overcome this limitation in some experiments where there is little optical absorption. The basic idea ofsuch a cavity is the sameas discussed in Sec. IV D, but the optics are arranged so that a large fraction of the laser power is coupled into the locking cavity. This results in the intracavity power becoming much higher than the output power of the laser. Buildup cavities have been used with lasers for many years; however these always required some electronic feedback system to keep the cavity and laser frequencies the same. With a diode laser, the need for electronic feedback can be eliminated because the optical feedback locks the laser to the cavity resonance. Enhancements as high as 1500 in the one-way power buildup76have been obtained with such “self-locking buildup cavities.” These have provided a circulating power of 45 W. Tanner, Masterson, and Wieman’” used a self-locking buildup cavity to study how a strong blue-detuned standing wave can collimate a cesium atomic beam. Using this technique they were able to reduce the divergence of the beam from 15 to 4 mrad using only a 10-mW laser source. In another application, over 1 mW of tunable light at 432 nm was produced by placing a potassium niobate crystal inside such a self-locking buildup cavity to produce the second harmonic of the diode laser light.” With the proper lasers and crystals, it may be possible to use this approach to achieve several milliwatts of narrowband tunable light over much of the blue region of the spectrum. When this technology is combined with injection-lockeddiode arrays or arrays that are themselves optically locked to buildup cavities,” many tens of milliwatts will be available. On the other hand, for applications not requiring a large amount of second-harmonk light, Ohtsu7’ has shown that it is possible to do spectroscopy with the small amount of second-harmonic light that is emitted from ordinary diode lasers. F. Trapping and cooling atoms using diode lasers
Another area in which stabilized diode lasers have been used very successfully is in cooling and trapping neutral atoms. Such experiments often require several different laser frequencies with linewidths of less than 1 MHz, and thus they are quite expensive to set up using dye lasers. This expense precludes the use of dye laser-cooled and/or trapped atoms in a wide range of atomic physics experiments for which they would otherwise be well suited. The use of cavityand grating-stabilized diode lasers overcomes this problem. Sesko and co-workersWhave recently cooled cesium atoms using a three-dimensional standing wave tuned to the red side of the cesium resonance line (so called “optical molas17
Rev. Sci. instrum., Vol.62, No. 1, January 1991
ses”). This has produced atomic samples containing 5X lo7 atoms with a temperature of 100pK. In a related experiment he and co-workers then observed the cesium “clock” transition between the two ground hyperfine states in this cold sample.” The low temperature allowed an interaction time of 20 ms and hence a linewidth of only 44 Hz, in an interaction volume of less than 1 cm3. In other experiments Sesko ef 0 1 . ~ ’have succeeded in trapping atoms using a &man shift spontaneous force trap created with diode lasers. Using four stabilized lasers, samples of up to 4 X 10’ atoms were trapped, and trapping times of greater than 100 s were obtained.’’ A variety of interactions between the trapped atoms were studied. In both the optical molasses experiments and these trapped atom experiments, features of the diode lasers other than their cost have been important. There were a number of probe studies where very low amplitude noise and ease of rapid frequency changes were invaluable. Inexpensive diode lasers are proving to be valuable tools for atomic spectroscopy. They offer continuously tunable sources of radiation with linewidths under 100 kHz and powers of 30 mW or more. They can cover much of the near infrared region of the spectrum and are moving into the visible. These devices can (though they may not) be simple and highly reliable. Their low cost and ease of use make multilaser experiments quite feasible, and more importantly, diode lasers allow laboratories with limited resources to still be involved in “state-of-the-art” laser-atomic physics experiments. These include the areas of laser cooling and trapping, nonlinear optics, and high precision spectroscopy. ACKNOWLEDGMENTS
The authors acknowledge valuable discussions with, and contributions made by J. Camparo, J. L. Hall, G. Tino, V. L. Velichanski, M. Ohtsu, H. Robinson, R. Drullinger, R. Hulet, K.Gibble, W. Swann, C. Tanner, B. Masterson, D. Sesko, and R. Watts. Readers can obtain a copy of the references including the titles of the articles by contacting the JILA Publications Office. C. Wieman is pleased to acknowledge support in this work from the Office of Naval Research and the National Science Foundation through the University of Colorado.
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tron. QE-20, 1045 (1984). I'M. Hashimoto and M. Ohtsu. J. Opt. SOC.Am. B 6, 1777 (1989); L A . Wang, D. A.Tate, H. Riris, andT. F.Gallagher, J. Opt. SOC.Am. B6.871 (1989). s4 T.Yanagawa, S. Saito, S.Machida, and Y. Yamamoto, Appl. Phys. Lett. 47, 1036 (1985). "M. Hashimoto and M. Ohtsu, IEEE J. Quantum Electron. 23, 446 ( 1987); M. Hashimoto and M. Ohtsu (unpublished); H. Furuta and M. Ohtsu, Appl. Opt. 28, 3737 (1989). I6 J. Reid, M. El-Sherbiny, B. K. Garside, and E. A. Ballik, Appl. Opt. 19, 3349 ( 1980); D. T. Cassidy and J. Reid, Appl. Phys. B 29,279 ( 1982); D. E. Cooper and J. P. Watjen, Opt. Lett. 11.606 (1986); D. E. Cooper and R. E.Warren, J. Opt. Soc.Am. B 4,470 (1987); C. Clinton and D. Cooper, Opt. Lett. 14, 1306 (1989). "L. Hollberg and M.Ohtsu, Appl. Phys. Lett. 53,944 (1988). "R. Watts and C. Wieman, Opt. Commun. 57,45 (1986). 59 H. Robinson (Duke Univ.) (private communication). -W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985). '' R. N. Watts and C. E. Wieman, Opt. Lett. 11, 291 (1986). "C. Salomon, J. Dalibard, A. Aspect, H. Metcalf, and C. Cohen-Tannoudji, Phys. Rev. Lett. 59, 1659 (1987); C. Salomon. H. Metcalf, A. Aspect, and J. Dalibard. in LaserSpectroscopy VII, edited by W. Persson and S.Svanberg (Springer, Berlin, 1987), p. 404;C. Salomon and J. Dalibard, C. R. Acad. Sci. Paris 306, 1319 (1988); C. Salomon and J. Dalibard, in Frequency Standards and Merrology, edited by A. De Marchi (Springer, Berlin, 1989); 3. Dalibard, C. Salomon, A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, in Proceedings of rhe 11th International Conferenceon Atomic Physics,edited by S. Haroche. G. Grynberg, and J. C. Gay (World Scientific, Singapore, 1989). "B. Sheehy, S-Q. Shang, R. Watts, S.Hatamian, and H. Metcalf, J. Opt. Soc.Am. B6.2165 (1989). MD.Sesko, C. G. Fan, and C. E. Wieman, J. Opt. SOC.Am. B 5. 1225 (1988). "'Yu. A. Bykovskii, V. L. Velichanskii, I. G. Goncharov,andV. A. Maslov. Sov. Phys. Semicond. 4.580 (1970); V. L. Velichanskii, A. S.Zibrov, V. S.Kargopol'tsev, 0.R. Kachurin, V. V. Nikitin. V.A. Sautenkov, G. G. Kharisov, and D. A. Tyurikov, Sov. J. Quantum Electron. 10, 1244 (1980); V. A. Sautenkov, V. L. Velichanskii, A. S. Zibrov, V. N. Luk'yanov, V. V. Nikitin, and D. A. Tyurikov, Sov. J. Quantum Electron. 11, 1131 (1981); Y. A. Bykovskii, V. L. Velichanskii, I. G. Goncharov. A. P.Grachev, A. S.Zibrov, S. I. Koval, and G. T. Pak, Sov. J. Quantum Electron. 19,730 (1989). * C . E. Tanner and C. E. Wieman, Phys. Rev. A 38, 162 (1988); C. E. Tanner and C. E. Wieman, Phys. Rev. A38.1616 ( 1988); G. M. Tino, L. Hollberg, A. Sasso, M.Inguscio, and M. Barsanti (to be published). 07K. E. Gibble and A. C. Gallagher, in XVI ICPEAC (in press). "T. Gustavsson and H. Martin, Phys. Scripta34,207 ( 1986); B. Lindgren, H. Martin, and U. Sassenberg, in LaserSpecrroscopy VIII, edited by W. Persson and S.Svanberg (Springer, Berlin, 1987). p. 402; G. Gustafsson. H. Martin, and P. Weijnitz, Opt. Commun. 67, 112 (1988). 09G.Avila, P. Gain, E. de Clercq, and P. Cerez, Metrologia 22, 11 1 ( 1986). "J. Reid. M. El-Sherbiny. B. K. Garside, and E. A. Ballik, Appl. Opt. 19, 3349 (1980); W. Lenth and M. Gehrtz, Appl. Phys. Lett. 47, 1263 ( 1985); L. G. Wang, H. Riris, C. B. Carlisle, and T. F. Gallagher, Appl. Opt. 27, 207d (1988); C. Clinton and D. Cooper, Opt. Lett. 14, 1306 (1989); L.-G. Wang, D. A. Tate, H. Riris, and T. F. Gallagher, J. Opt. SOC.Am. B 6,871 (1989). 7 ' S.Yamaguchi and M. Suzuki, Appl. Phys. Lett. 41,597 ( 1982);S. Yamaguchi and M. Suzuki, IEEE J. Quantum Electron. QE-19, 1514 (1983). '* D. T. Cassidy and J. Reid, Appl. Phys. B 29,279 ( 1982); M. Gehrtz, G. C. Bjorklund, and E. A. Whittaker, J. Opt. SOC.Am. B 2, 1510 (f985); J. S. Wells, A. Hinz,andA. G. Maki, J. Mol. Spectrosc. 114,351 (1985);D. E. Cooper and J. P. Watjen, Opt. Lett. 11,606 (1986); D. E. Cooper and R. E. Warren, J. Opt. Soc. Am. B 4,470 (1987); D. E. Cooper and R. E. Warren, Appl. Opt. 26, 3726 (1987); A. Hinz, J. S. Wells, and A. G. Maki, 2. Phys. D 5, 351 (1987); D. E. Cooper and C. B. Carlisle, Opt. Lett. 13,719 (1988);C.B.CarlisleandD. E.Cooper,Appl.Opt. 28,2567 (1989); C. Clinton and D. Cooper, Opt. Lett. 14, 1306 (1989). 73 V. Giordano, A. Hamel, G. Theobald, P. Cerez, and C. Audoin, Metrologia 25, 17 ( 1988); L. Hollberg, Applied Laser Spectroscopy, NATO AS1 series, edited by M. Inguscio and W. Demtroder (Plenum, New York) (in press). 74 Th. Haslwanter, H. Tirsch, J. Cooper, and P. Zoller, Phys. Rev. A 38, Diode lasers
19
791 5652 (1988); H.Ritsch, P.Zuller, and I. Cooper. Phys. Rev. A 41.2653 (1990). "S. L. Gilbert and C. E. Wieman, Phys. Rev. A 34. 792 ( 1986); M.C. Noecker, 8.P. Masterson, and C. E. Wieman, Phys. Rev. Lett. 61. 310 (1988). 7bC.E. Tanner, B. P. Masterson, and C. E. Wieman, Opt. Lett. 13, 357 (1988). Insubsequentwork wehaveachieved higher buildup (1500) and higher circulating power (45 W).
" G . J. Dixon, C. E. Tanner, and C. E.Wieman, Opt. Lett. 14,731 (1989). "M. K.Chun, L. Goldberg, and J. F. Weller. Appl. Phys. Lett. 53, 1170 (1988). 7qM.Ohtsu, Electron. Lett. 25, 22 ( 1989). wD.W. Sesko and C. E. Wieman, Opt. Lett. 14,269 (1989).
'' D. Sesko, T.Walker, C. Monroe, A. Gallagher. and C. Wieman, Phys. Rev. Lett. 63,961 (1989); T. Walker, D. Sesko, and C. Wieman, Phys. Rev. Lett. 64,408 (1990).
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Rev. Sci. Instrum., Vol. 62, No. 1, January 1991
Diode lasers
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Frequency stabilization of a diode laser using simultaneous optical feedback from a diffraction grating and a narrowband Fabry-Perot cavity H. Patrick and C. E. Wieman Joint Institute for Laboratov Astrophysics, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309-0440 Received 6 June 1991; accepted for publication 12 July 1991 We present a system which uses optical feedback from a diffraction grating to provide primary control over the frequency tuning of a diode laser, and employs simultaneous optical feedback from a narrowband Fabry-Perot cavity to reduce the linewidth. The frequencies of the optical feedback from the diffraction grating and narrowband cavity are electronically locked using a novel technique. The reduction in the linewidth is measured as a reduction in the frequency noise seen on the side of a Cs-saturated absorption line. The presence of optical feedback from the narrowband cavity reduces the frequency noise by a factor of 3 compared to the frequency noise obtained using the grating feedback alone.
I. INTRODUCTION In recent years, diode lasers have been increasingly used as tunable light sources in optical communications and spectroscopy, due to their low cost and relative simplicity compared to other tunable sources. A free-running diode laser may be tuned over a total range of typically 15 nm by changing its temperature and injection current. However, because the temperature and current tuning curves do not match, the tuning exhibits “gaps” in frequency that are not accessible to that particular diode laser. In addition, the linewidth of free-running diode laseis is generally on the order of 20 MHz. This wide linewidth is unacceptable for many applications. Because diode lasers are highly susceptible to optical feedback, various schemes exploiting this susceptibility have been developed to address the issues of tunability, linewidth narrowing, and sensitivity to temperature change. Impressive linewidth narrowing (by a factor of 1000 or greater) and frequency stability equal to that of the reference cavity has been achieved by optical locking of diode lasers to the resonant optical feedback from a narrowband confocal Fabry-Perot cavity,’-3 but the tuning remains limited to that of the free-running laser: A second technique, using a grating in a Littrow configuration to form a cavity with the back facet of a standard commercial diode laser, achieves greater tunability than that of the free-running diode laser, eliminates the tuning gaps, and narrows the linewidth to several hundred kHz.’ Our system combines the tunability of the grating cavity laser with the superior linewidth narrowing associated with optical feedback from a narrowband cavity. To achieve the necessary frequency match between the narrowband cavity and the grating cavity we use a novel locking scheme. This scheme gives a very robust lock using rather simple optics and electronics. The setup is based upon a grating cavity laser, in which optical feedback from a grating controls the diode laser frequency. A small amount of additional optical feedback comes from the type I1 resonant reflection (maximum on resonance and minimum off resonance) from a tunable 2593
confocal Fabry-Perot cavity (CFP). When the resonant frequency of the CFP is within 20 MHz of the frequency of the grating cavity, the grating cavity laser optically locks to the CFP. This reduces the linewidth of the grating cavity laser output. In principle, if one tunes the frequency of the CFP so that it always nearly matches that of the grating cavity (or vice versa), the optical lock will be maintained. In practice, however, the thermal drifts in the lengths of the two cavities cause the frequencies to separate by too large an amount to maintain the optical lock. When this happens the laser output frequency spectrum behaves as if there were no CFP. To prevent loss of the optical lock it is necessary to have an electronic feedback system which counteracts any drifts and locks the grating cavity frequency to that of the CFP. The primary challenge in making a useful laser system of this type is to find an error signal for the electronic feedback system which will allow a robust lock. We produce an error signal for this purpose by sending an extra beam through the CFP at a large angle to the CFP axis. The transmission of the error signal beam exhibits broad peaks, which can be offset in frequency from the type I1 transmission peaks, and used to side-lock the grating cavity laser to the CFP using simple electronics. II. APPARATUS
Figure 1 is a schematic of the experimental apparatus. The grating cavity laser consists of a standard 838 nm Sharp LTOl5 diode laser, mounted in an aluminum block which is temperature controlled to the level of 10 mK. The output of the diode is collimated onto a blazed grating in the Littrow configuration. About 25% of the light is diffracted back into the diode, so that the grating forms a low finesse cavity with the diode’s rear facet. The frequency of the laser output is primarily determined by the frequency of the feedback from the grating cavity, because of the low reflectivity of the output facet of the diode. This frequency may be changed by simultaneously changing the angle of the grating and the length of the grating cavity, either by means of adjusting screws on the grating mount, or by
Rev. Sci. Instrum. 62 (1I ) , November I991 0034-6748/91/112593-03$02.00
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FIG. 1. Schematic of the grating cavity Iaser and CFP, showing the beam paths for the type I1 remnant optical feedback, and the error signal beam. The output bcam goes to a Cs-saturated absorption s p trometer, which is not shown.
GRATING CAVITY LASER
applying a voltage to PZT-G. There is a total tuning range of f and - 10 nm from the free.-running diode laser wavelength. The output beam is the M=O reflection from the grating. A beamsplitter picks off about 10% of this output and sends it to the CFP. The CFP has a length of 5 cm, a free-spectral range of 1.5 GHz, and a finesse of approximately 125. The length of the CFP may be tuned by applying a voltage to PZT-C. The beam which provides optical feedback to the laser is aligned in a V mode, such that the type I1 resonant optical feedback (as shown in Fig. 1) goes back to the laser and the direct reflection off the input mirror does not. This reduces undesirable off-resonance optical feedback. DET-T monitors the type I1 transmission. To obtain the error signal for electronically locking the grating cavity frequency to the frequency of the CFT,a beamsplitter picks off 40% of the laser output (though a much smaller amount would work just as well). This error signal beam is then sent through the CFP at a large angle to the axis, producing transmission peaks with the same free spectral range as the type I1 transmission, but with a broad line shape suitable as a side-lock error signal. The DET-EStrace in Fig. 2 shows the shape of this transmission. The frequency of the DET-ES peaks can be offset with respect to the DET-T peaks by changing the alignment of the error signal beam. The side-lock error signal is fed back to PZT-G, through a slow electronic servo with a unity gain frequency of about I00 Hz.This servo keeps the grating cavity frequency and the CFP frequency commensurate for many hours at a time. The CFP and grating cavity frequencies can be tuned over a range of 3 GHz (limited by the range of the high voltage amplifier to PZTC ) and remain electronically locked. 111. RESULTS
Rev. -1.
-PZT-G
VOLTAGE d
(42 MHzlDIV)
Figure 2 shows the effect of the CFP feedback on the output of the laser, as shown by the saturated absorption spectrum (top trace), DET-ES (middle trace), and DETT (lower trace). The frequency of the laser i s scanned by applying a ramp to PZT-G while the CFP resonant frequency is held constant. When the grating cavity and CFP 2694
frequencies do not match, the ramp tunes the laser smoothly over the Cs 6SlI2(F= 4) to 6P3,*(F = 5 ) transition and cross over resonance as seen in the top trace. When the grating cavity frequency nearly matches the CFT frequency (the region marked with two arrows), the laser optically locks to the CFP. This is seen on the top trace as an abrupt change in slope on the side of the 4-5 transition, indicating that the laser frequency scan due to the motion of the grating is being counteracted by the stationary CFP resonance. In the lower trace, the effect on DET-T is a sudden increase in transmission over the locked range, since the laser frequency is held near the CFP resonance by the optical lock. The exact shape of the transmission in this range is determined by the phase of the laser fieId at the input mirror, and thus changes if the distance from the laser to the CFP changes. Finally, the
Instrum., Val. 62, No. 11, November 1991
FIG. 2. Upper trace: A portion of the Cs 155'~~ to 6P3n saturatedabsotp tioa spectrum. Middle trace: DET-EStransmission. Lower trace DET-T transmission. The grating cavity frcqucncy is scanned by applying a m p to PZT-G.Note the flattened areas in all three traces. which is marked by the arrows, indicating optical locking of the grating cavity laser to the CFP.
Laser frequency rtablllrrtlon
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FIG. 3. Frequency noise spectra seen on the saturated absorption photodetector. In both cases, the laser is tuned to halfway up the side of the Cs 4-5 line. The Cs 4-5 line has a height of 290 mV, and a width of 18.1 MHz. Upper trace: Grating cavity laser alone. Lower trace: Grating cavity laser optically and electronically locked to the CFP. 1.o
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I
I
I
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I
I
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Hz
locking appears as a flattened area in the DET-ES transmission (middle trace). The output of the laser is sent to a Cs-saturated absorption spectrometer.6This allows us to monitor the effect of the laser linewidth on the frequency noise seen on an atomic transition. To measure the frequency noise produced by the grating cavity laser alone, we first tune the laser to halfway up the side of the 4-5 transition by adjusting the injection current and the offset to PZT-G. The frequency noise seen on the satufated absorption spectrometer is then sent to a 0-100 kHz HP3562A spectrum analyzer, and averaged over 20 scans. To measure the frequency noise produced by the grating cavity laser when it is optically locked to the CFP, we first tune the grating cavity laser to the frequency of the 4-5 transition as above. We then tune the CFP to the same frequency, electronically lock them together, and observe the frequency noise on the side of the 4-5 line. Figure 3 shows typical frequency noise spectra with and without optical and electronic locking of the grating cavity laser to the CFP. The optical lock to the CFP reduces the frequency noise on the atomic transition by a factor of 3 compared to that of the grating cavity laser alone. The electronic lock has no effect over the frequency scale shown, except to keep the frequencies of the grating cavity and CFP in agreement over the time of the measurement. The slope on the side of the 4-5 line was 62.4 MHz/ V, giving a noise of 190 Hz/(Hz)”* at a frequency of 60
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kHz, compared with 580 Hz/(Hz)’” at 60 kHz for the grating cavity laser alone. We have checked that the amplitude noise gives a negligible contribution to these spectra. Determination of the laser linewidth requires knowledge of the functional form of the spectra, and was not undertaken. However, for applications in atomic spectroscopy the reduction of frequency noise seen on an atomic transition is the quantity of interest. Our hybrid scheme reduces this frequency noise, while retaining the full tuning range of the grating cavity laser. ACKNOWLEDGMENTS
The authors wish to thank B. Pat Masterson for many helpful discussions, and W. Swann for design and construction of the grating feedback lasers. We are pleased to acknowledge support from the ONR and NSF.
’
B. Dahmani, L. Hollberg, and R. Drullinger, Opt. Lett. 12, 876 (1987). ’Ph. Laurent, A. Clairon, and Ch. Breant, IEEE J. Quantum Electron. QE-25, 1131 (1989). ’J. Harrison and A. Mooradian, IEEE J. Quantum Electron. QE-25, 1152 (1989). “Ph. Laurent, D. Bicout, Ch. Breant, and A. Clairon, in Laser Specrroscopy IX (Academic. New York, 1989), p. 243. ’C. E. Wieman and L. Hollberg, Rev. Sci. Instrum. 62, 1 (1991). Provides a review of diode lasers in atomic physics and extensive list of references. 6W.Demtroder, Loser Spectroscopy (Springer, Berlin, 1982), p. 485.
Laser frequencystabilization
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A magnetic suspension system for atoms and bar magnets C. Sackett, E. Cornell, C. Monroe, and C. Wieman Joint Institute for Laboratory Astrophysics and the Department of Physics, University of Colorado, Boulder, Colorado 80309-0440
(Received 21 January 1992; accepted 20 September 1992) A three dimensional magnetic confinement system is presented which will trap both macroscopic and atomic magnetic dipoles. The dipole is confined by dc and oscillating magnetic fields, and its motion is described by the Mathieu equation. Most aspect3 of the dynamics of the trapped objects depend only on the ratio of the magnetic moment to the mass of the dipole. Similar motion was observed for masses varying over 21 orders of magnitude (from 1 atom to 0.2 g). The trap is constructed from inexpensive permanent magnets and small coils which are driven by 60 Hz line current. The design of the trap as well as the behavior of the trapped particle are discussed herein.
I. INTRODUCI'ION People have always been fascinated by the sight of an object suspended in space, free from contact with any material substance. Demonstrations of such an event are often the centerpiece of magic shows, NASA public relations, and presentations on high temperature superconductivity. Recently, atomic physicists have developed a particular interest in these phenomena. Their work would probably make an excellent magic trick for a stage act, but is of 304
course well grounded in physical theory. Electric or magnetic fields, produced by some relatively distant source, are used to suspend and confine isolated ions, electrons, and atoms. When confined in this manner, the particles are free from the usual perturbations produced by nearby atoms, and thus can be studied with unprecedented precision. The Nobel prize in physics was awarded to Dehmelt and Paul for their pioneering work in the field of ion trapping. While trapping of charged particles is a mature field, the trapping of neutral particles using magnetic fields is more difficult 0 1993 American Association of Physics Teachers
Am. J. Phys. 61 (4), April 1993
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and very new. A significant development in this field was the recent demonstration of a new type of trap which uses an oscillating magnetic field gradient to trap cesium atoms by interacting with their magnetic dipole moments. lV2 We realized that the behavior of a particle in such a trap depends only on the ratio of the magnetic moment to mass, and not on either individually. Since macroscopic ferromagnetic dipoles can have moment to mass ratios which are comparable to or larger than that of an isolated cesium atom, this suggests that an ac trap could be used to trap a macroscopic magnet. We are interested in this macroscopic magnet trap for three rather unrelated reasons. First, a large scale model provides an excellent demonstration of the atom trapping technique and is a helpful pedagogical aid. Second, the technique is a new method of magnetic suspension and confinement. It has several advantages over previous techniques, and SO might prove useful in other applications. Finally, the physics of the system is interesting, and merits study for its own sake. In this paper, we first discuss the problem of magnetic confinement in general, and the theory for our particular trap. We then describe our apparatus, and the constraints upon its design. In the final section, we discuss the behavior of the system, and compare it with theory. 11. MAGNETIC TRAPS
Consider the problem of confining an uncharged magnetic dipole in three dimensions. In a magnetic field B, a dipole with moment p has a potential energy - p B. The principle of virtual work may be used to find the force on the dipole, say -Fflz=dU= -d(pzBz). In general then, the result3 is F = V ( p . B ) . For an ordinary bar magnet placed in a slowly changing field, p will align itself with B, so that p . B becomes Ip 1 I B I, and the magnet will be accelerated in the direction of increasing field magnitude by a force 1 p I V I B I. Hence, the dipole would be stably confined at any local maximum in IBI. Unfortunately, Maxwell’s equations prohibit such a maximum in free space.4 Although a bar magnet cannot be trapped in free space using a purely static magnetic field, there are several alternative approaches to confinement. Perhaps the simplest is to magnetically confine the dipole in one or two dimensions, using some other force in the remainder. A common example of this is the toy in which a ring magnet is placed around a hollow tube. When a smaller magnet is placed inside the tube, the field confines it to the plane of the ring, and the tube prevents it from sticking to the side of the magnetic ring. Although the small magnet is confined, the presence of contact forces is undesirable in many cases. Another way to trap a magnetic dipole in a static field is to have it spinning. If the angular momentum of the object is non-negligible, and aligned with the dipole moment, the dipole will tend to precess about the magnetic field, at some fixed angle. If we set the system up with p and L antiparallel to the field, the dipole will be attracted to a minimum in I B I. Field minima of this sort can occur, so this is a physically realizable method of confinement. Although it is difficult in practice to work with a rapidly spinning macroscopic object, the technique may be readily applied to an atom, where the projection of spin onto B is quantized. Spin polarized atomic gases have, in fact, been confined in such “weak field seeking state” traps.’
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Finally, a dipole may be confined in a dynamic field. An example of this is the (rather more expensive) toy which magnetically suspends an object, but uses optical sensors and a feedback mechanism to continually adjust the field so as to correct for the instability of the static equilibrium. Systems of this type require complicated electronics, and though they may provide a nice example of feedback and control systems for the engineer, they are not particularly interesting to the physicist.6 This article presents a quite different dynamic method, which uses sinusoidally oscillating magnetic fields. The technique is similar to the Paul trap for ions, in which an oscillating electric potential can be arranged in such a way that the motion of an ion in the field is stable.’ The trapping mechanism of such dynamic traps is analogous to the confinement of a marble placed at the center of a rotating saddle. Along any given direction, the marble is alternately attracted to and repelled from the center. If the saddle is rotating fast enough, the marble will not have time to roll off before being pushed back, and will be stably confined. A dipole placed in a suitable alternating magnetic field can be similarly confined, as shown in the next section. A trap of this nature was first developed using pure electric fields, to trap ions. The magnetic version was first proposed by Lovelace and Tommila for use with atomic hydrogen2 Electrodynamic traps far macroscopic particles have also been built, most recently by Winters and Ortjo-
111. THEORY OF ac MAGNETIC CONFINEMENT
Our trap, like the Paul (or “rf”) ion trap, holds a particle in an oscillating quadrupole potential. We achieve this using an axially symmetric magnetic field B,(r,z,t)=B,+1/2
(kdC+k,,cos R t ) ( 2 - ? / 2 ) ,
B,( r,z,t) = - ( kdc+ k,, cos Rt)zr,
(la) (Ib)
where kdc = d2@/d2 and k,, = d 2 c / d 2are the curvatures of the constant and oscillating components of the field, respectively. If the constant B, is much larger than all other terms, the magnitude of the field can be approximated by B, and the field direction will be predominantly axial and constant in time. A dipole placed in the field will align its moment with B and experience a force I p I (VB,). The resulting classical equations of motion can be cast in the form of Mathieu equations d2~i/dT2+(nj+2qicos2T)xi=.0, xI=rx2=z,
(2)
where m is the mass of the particle, a,=4pk,,/mR2, q, =2pk,,/mR 2 , a,= - 4 2 , q,=q,/2, and T=Rt/2. Solutions to the Mathieu equation are described in various refe r e n c e ~ .The ~ important fact is that the solutions are bounded for certain values of the coefficients a and q. Physically, this means that if the frequency R and curvatures k,, and k, are adjusted properly, the dipole will be confined. We use the relationships ar= -2a, qz=2q, to determine regions of the a,-q, plane where the stability condition is met for both the axial and the radial coordinates simultaneously. If a, and qr lie in one of these “stability regions,” the dipole can be confined in three dimensions. A graph of the experimentally most convenient stability region is shown in Fig. 1. Sackett et al.
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constructed of a material with permeability not equal to one, which will again change the field within the object. IV. APPARATUS
-0.5
-1 .o
1
0
\
0.5
1.0
1.5
Fig. 1. The stability region of the aZ-qz plane. Solutions of the Mathieu equation are governed by the parameters a and q, which are proportional to the curvature of the dc and ac fields, respectively. For certain values, solutions exist which will he hounded for all time, while for other values all solutions diverge to infinity. The relationship between the parameters for the various components allows the motion to be completely characterized by one pair of values, a, and qr Plotted is one of the regions of the o,-qz plane for which particle motion will be confined in all three dimensions.
Even when stable, the motion of the dipole is complicated. For small q, we can approximate the solution of Eq. (2) by
x(T)=A(1+q/2cos2T)cosBT,
(3)
where, to fourth order in q,
a’= l3/2 -a.
(4)
In each dimension, then, the motion consists of a large slow oscillation at frequency 00=PR/2, on which is superposed a small rapid oscillation at the driving frequency SZ. It is conventional in the ion trapping literature to refer to these separate components as the secular motion and micromotion, respectively. The secular motion is that of a particle placed in a harmonic potential well with spring constant mu:, while the micromotion is a smaller oscillation at the frequency of the ac field and with amplitude that varies with time, proportional to the secular displacement. We originally contructed this sort of trap to contain cesium atoms. After successfully doing so, we noted that the essential parameters a and q depended only on the ratio of magnetic moment to mass. Since macroscopic ferromagnets can have a moment to mass ratios comparable to that of an individual atom, this trap can work for a bar magnet in essentially the same way as for an isolated atom. Differences between the microscopic and macroscopic system arise in several possible ways. First, if the object to be trapped is reasonably large compared to the trap itself, the magnetic field will vary somewhat across the object. Thus the cohesive forces holding the object together will effect its motion. Second, if the object is metallic, screening currents will be induced which will change the field inside the material. Skin depths for good conductors at 60 Hz are a few millimeters, so this effect will be important for an appreciably sized object. Finally, the object to be trapped may be 306
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To stably confine a particle with a given ratio of magnetic moment to mass p / m , the magnetic field must meet several requirements. It must have dc and ac components with (1) a dc magnitude large compared to the ac magnitude; (2) a dc gradient to balance gravity, d I B I /dz= mg/ p, at a position of small or zero ac gradient; and (3) an ac frequency R, axial dc curvature kdo and axial ac curvature k,, such that the parameters a, and q, lie in the region of stability shown in Fig. 1. We were faced with several additional practical constraints. As the device was intended to be a demonstration model, it had to be reasonably portable and to not require bulky power supplies. Therefore we chose R2/2a=60 Hz, and used permanent magnets to produce the dc field. It was also desirable that the confinement region be clearly visible. Finally, for simplicity we did not want to use water or forced air cooling of the ac coils; thus the rate of heat dissipation limited the amount of current we could use. The values of a, and qr both depend on the moment to mass ratio, p / m . The design of the apparatus therefore depends on the specific material to be trapped. A larger ratio is desirable because it requires smaller fields for a given trap depth. We therefore use fragments of a commercially available Nd/Co/Fe alloy magnet, with moment to mass ratio of roughly 100 erg/G/g, about 2.5 times that of a spin-polarized cesium atom. The fragments are irregularly shaped since they are obtained by chipping pieces off of a larger magnet. Less expensive materials, with lower mass to moment ratios, could certainly be used, but would require proportionally more driving current. This would be feasible in a water-cooled system. The precise determination of p / m for these fragments is difficult, but for purposes of designing the trapping fields, various simple measurements suffice. A technique we find convenient is to run dc current through a large coil of known dimensions, and measure how much current is needed to lift the fragment against gravity. We then know the field gradient required to cancel gravity, from which we determine p / m . The final design used to meet these constraints is sketched in Fig. 2, and described in detail in the Appendix. It consists of three ring-shaped permanent magnets and a coil assembly. The two larger magnets are iron, and are mounted 1.50 in. above and 1.69 in. below the trap center, so that their fields add but their gradients nearly cancel. The third magnet is a thin ring of plastic magnet, mounted 0.2 in. above the plane of the trapping region. It is used to finely adjust the field curvature and gradient. The combined dc field of the assembly at trap center was predominantly axial, with B z 150 G, d B / d z z 10 G/cm, and d’B/ d2=: 8 G/cm’. A dc curvature of 8 G/cm2 and a p / m of 100 (erg/G)/g makes a, equal to 0.02. We see in Fig. 1 that q, must be at least 0.2, which requires an ac curvature of 160 G/cm’. In order to provide this curvature while insuring that the ac field be small relative to the dc field, we use a set of four coils symmetrically mounted in pairs and connected in series. The current sense of the outer pair is opposite the sense of the inner pair, so that their respective ac fields nearly cancel at the trap center, while their positioning was Sackett et al.
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the forces felt in this process are significant, and tend to pull the fragment out of the forceps. Finally, the nonidealities mentioned at the end of Sec. I11 become significant for fragments larger than a few millimeters. V. BEHAVIOR OF THE TRAPPED MAGNET
Fig. 2. Arrangement of magnets and coils in the trapping apparatus. The two large permanent magnets on the top and bottom provide a large field at trap center, while the magnet in the center provides a small correction to the gradient and curvature. Coils are driven with 60 Hz ac current, in the sense indicated.
such that the ac curvatures added. We calculate that the coils provide an rms magnitude per ap lied current of 2.6 G/A, and a curvature of 41.4 G/cm P/A. Powered by a Variac, the coils can maintain an ac current of up to 9 A rms without burning the insulation on the wires. Although constructing the trap was not difficult, loading it presented a challenge. The magnetic fragment must be inserted through the large fringing fields of the coils and magnets, and released at the proper position with the correct orientation. We do this by holding the fragment in nonmagnetic forceps. The orientation is set by placing the fragment on top of the upper permanent magnet, and then grabbing it with the forceps held vertically. A simple jig is then used to position the forceps so that the fragment is held in the center of the trap, where it is released. We found that the fragments tended to stick to plastic forceps, but that copper (and presumably any other nonmagnetic metal) works adequately. Because it is difficult to release a fragment exactly at rest in the center of the trap, it is easiest to turn the current up to 6 A for loading. Once the initial motion caused by loading has been damped out by air resistance, the trap current may be gradually reduced to as low as 3.5 A. All of the authors have been able to repeatably load fragments up to 5 mm in diameter, and one (C.A.S.) has loaded fragments of nearly 1 cm. Tinkering with the position of the shim magnet (part B in Fig. 5 ) is sometimes required. Larger fragments do not require more current than smaller ones, but are more difficult to load. One source of difficulty is that large fragments must be positioned more precisely than smaller ones, since they occupy a large fraction of the trapping region. However, this is somewhat compensated for by the fact that the micromotion forces in large fragments can be felt through the forceps. The fragment may then be positioned by feeling the decrease in the vibration at trap center. Loading is also difficult because as the fragment is inserted into the trap, it must closely pass one of the large permanent magnets. For larger fragments, 307
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As shown in Sec. 111, a magnet in an ac magnetic trap is expected to execute simple harmonic secular motion, on which is superposed a small vibration at the driving frequency. The motion of a relatively large fragment, for which approximation as a point dipole is not accurate, is considerably altered. The motion is quite complicated and fascinating, but is difficult to describe quantitatively. However, for small perturbations from equilibrium, it did prove possible to analyze the frequency spectrum of the motion. This analysis, as well as a qualitative description of the motion, is presented below. The motion of a trapped fragment is not simple, even for very small amplitude motion. It depends strongly on the details of the excitation, but in general is characterized by strong coupling between the two horizontal translational modes and the various rotational modes. This coupling is presumably due to variation in trapping forces across the fragment, as well as irregular torques produced by air resistance. As an additional complication, the fragment is essentially free to rotate about its magnetic moment vector, which is not necessarily along a principle axis and thus couples to other motions. Damping by air resistance occurs on a time scale of approximately a minute, but varies with the size and shape of the fragment. The micromotion is barely perceptible by eye, showing up as a fuzziness in the profile of the fragment when it is displaced from the center. The volume in which the particle was stably confined was disk shaped, roughly 1 mm high in the axial direction and up to 2 cm radial diameter. The size of this disk was observed to decrease as the ac curvature was increased. To study the frequency components of the motion, a HeNe laser was used to illuminate the fragment from the side, so that the shadow of the fragment moved across a photodiode. We then took a Fourier transform of the output of the photodiode. A sample spectrum for a secular amplitude of approximately 0.5 mm is shown in Fig. 3. Rewriting Eq. ( 3 ) as separate frequency components, we obtain x , ( t ) =A(cosP,flt/2+q,/4
cos[ l-PJ2)at (5)
+qj/4cos(1+PJ2)atl.
So, if we examine the spectrum near a=60 Hz, we expect to see two pairs of sidebaEds; one at the axial secular frequency P, and one at the radial frequency P,. The two outer pairs of sidebands in Fig. 3 correspond to these frequencies, the axial being the outermost. The innermost pair of sidebands is due to axial rotation, while the peak at 60 Hz itself is due to misalignment of the trap components so that the point at which the dc gradient exactly cancels gravity does not lie at the center of the ac coils. The equilibrium position therefore has a slight secular displacement and corresponding 60 Hz micromotion. For larger secular oscillations the Fourier spectrum rapidly becomes more complex, and individual components are impossible to identify. By varying the ac current, the various secular frequencies can be changed. The axial and radial frequencies may then be used to determine u, and q, by inverting Eq. (4). Sackett et al.
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799 A
50
60
70
Hz Fig. 3. Frequency spectrum of fragment motion for small perturbation. The two outer pairs of sidebands are the axial and radial secular oscillation. The inner sidebands are due to axial rotation, and 60 Hz peak is an artifact of trap misalignment.
Figure 4 shows an expanded graph of the stability region in the a,<, plane, with our data plotted for several ac currents. Since a,=4kdc/mR2 does not depend on the ac field, we expect a variation in the ac current to change only qn so that the points in Fig. 4 should lie on a horizontal line. That they do not is probably due to the misalignment of the permanent magnets. If they are imperfectly aligned, so that the dc field gradient does not exactly cancel gravity at the center of the ac coils, the equilibrium position of the fragment is displaced. The amount of this displacement depends on the ac fields, so varying the ac current alters the equilibrium position of the particle. The dc curvature kdc also varies with position, so that a, will vary slightly with ac current. One additional aspect of the trap which we found interesting was the possible use of eddy currents to damp out
0.02
-0.02
0
0.10
0.20
0.30
0.40
0.50
qz
Fig. 4. Expanded section of Fig. 1, showing points at which trap has heen operated. Contours show lines of constant 8. Ditferent data points correspond to different ac currents, and hence different values of qr We expect, then, that the points should lie on a horizontal lie. The ar.omalous variation in az is due to trap misalignment. 308
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Fig. 5. Drawing of the apparatus. Dimensions are in inches. Parts A are torroidal iron magnets, with outer diameter 2.9, inner diameter 1.2, and thickness 0.5. Part B is a trim magnet with OD 2.6, ID 2.1, and thickness 0.25. The position of part B relative to trap center is adjustable by set screws. Parts C are the coil assembly, shown in detail in Fig. 6. Parts D are three aluminum support posts, bolted to parts C.
secular motion. This effect is quite dramatic, and can be observed by placing a copper plate near the fragment. When the plate was placed about a millimeter from the center of a small fragment, the damping time decreased from over 1 min to 0.25 s. VI. CONCLUSION We have constructed a magnetodynamic trap, capable of confining a 0.2 g object in three dimensions. The trap is constructed of simple, low cost materials, and requires only 50 W of power at 60 Hz. The behavior of a trapped particle is described reasonably well by the Mathieu equation. The capacity of our trap is limited by the rate of power dissipation, with lower ratios of magnetic moment to mass requiring proportionally greater ac current and stronger dc magnets. The apparent limit on the size object our trap can contain is the size of the trap itself, although the behavior of larger objects will likely be less precisely modeled by the Mathieu equation. Since larger coils require more current to produce the same curvature, more efficient power dissipation would enable the trap to hold both larger objects and less magnetic materials. We perceive the primary value of the trap to be pedagogical, as it provides a fascinating example of physics accessible to the undergraduate student. It is one of relatively few experiments based on purely classical mechanics suitable for an advanced physics lab. At the same time it demonstrates a technique which is being used currently in the trapping and cooling of neutral atoms. If designed as a fixed lab apparatus instead of a portable demonstration model, several drawbacks of the trap could be remedied. A water cooling system for the ac coils would allow exploration of the high curvature regions of the stability diagram. The use of a dc coil rather than permanent magnets to supply the required dc field would provide a second adjustable parameter, while also allowing fine corrections of the field alignment. We feel that after improvements such as Sackett et aL
308
800 Tapped holes. 6 Places
I
I
I
-
0.79 00
0.65
11800
0 . 2 7 ~24
m Trap center
I -
’
I
I
I I
I
I I I
I
1
I
1
I I I
,-~ I /
I
Fig. 6. Mechanical drawing of coil assembly. Dimensions in inches, with only critical dimensionsshown. Wire coils are wound within the channels. Tapped screw holes are located equilaterally about the top and the bottom coil form.
these are made, the trap will make an excellent, and inexpensive, addition to the undergraduate laboratory repertory.
ACKNOWLEDGMENTS This work was supported by the Office of Naval Research and the National Science Foundation. C.S. acknowledges a fellowship from the National Science Foundation.
APPENDIX: DESIGN SPECIFICATIONS For readers wishing to duplicate our apparatus, we describe our final design. A scale drawing of the apparatus is shown in Fig. 5. The main permanent magnets we used (parts A in Fig. 5 ) were obtained from Edmund Scientific, part number A37,621 at $7.00 each. The trim magnet (part B) was cut to shape from a sheet of flexible magnetic material. The material is soft and can easily be cut by a variety of means. A drawing of the coil forms used (part C) is shown in
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Fig. 6. The forms were constructed of aluminum, and wound with 22 gauge copper magnet wire, with 44 turns in the larger grooves and 24 turns in the smaller. The four coils are connected in series. The two large grooves should be wound in the same direction, and the two s ~ a l l e r grooves in the opposite direction. The coils can be powered by running line voltage through a low-current variable transformer, providing say 2 A at 0 to 120 V. This current is then fed to a step down transformer, which yields 20 A at 0 to 12 V. The appropriate transformers can be obtained from any electronics supplier, at a cost of around $80. Adjustibility of the current is not strictly necessary for the operation of the trap, so if cost is a significant constraint, a single transformer, or even a lamp bank could be used to power the coils. The coils and magnets are held in place by three aluminum supports, placed axially around the trap. One of the supports is shown in Fig. 5 (part D). A 1/2 in. diam aluminum post was attached to one of the supports, and used as a jig to position the forceps when loading the trap. The jig was constructed by attaching a screw to the forceps, and mounting them on an (aluminum) optical post. The forceps assembly was then mounted on the large post using post clamps, obtainable from an optics supplier. ‘E. A. Cornell, C. Monroe, and C. E. Wieman, “Multiply loaded, ac magnetic trap for neutral atoms,” Phys. Rev. Lett. 67, 2439-2442 (1991). ’R. V. E. Lovelace, C. Mehanian, T. J. Tommila, and D. M. Lee, “Magnetic confinement of a neutral gas,” Nature 318, 30-35 (1985). ’R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, (Addison-Wesley, Reading, MA, 1964), Vol. 11, Sect. 15-1. 4W. Ketterle and D. Pritchard, “Trapping and focusing ground-state atoms with static fields,” Appl. Phys. B 54, 403406 (1992). 5V. S. Bagnato et a l , “Continuous stopping and trapping of neutral atoms,” Phys. Rev. Lett. 58, 2194-2197 (1987); A. Migdall et aL, “First observation of magnetically trapped neutral atoms,” Phys. Rev. Lett. 54, 25962599 (1985). %ee, for example, the 1992 Edmund Scientific Company catalogue, p. 177. ’F. G. Major and H. G. Dehmelt, “Exchange-collisiontechnique for the d spectroscopy of stored ions,” Phys. Rev. 170, 91-107 ( 1968). ‘H. Winter and H. W. Ortjohann, “Simple demonstration of storing macroscopic particles in a Paul trap,” Am. J. Phys. 59, 807-813 (1991). ’Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, Applied Math Series Vol. 5 5 (National Bureau of Standards U.S.,Washington DC, 1966), pp. 721-751; (reprinted by Dover, New York, 1965.)
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Laser vibrometer based on opticalfeedback-induced frequency modulation of a single-mode laser diode P. A. Roos, M. Stephens, and C. E. Wieman
We describe a sensitive and inexpensive vibrometer based on optical feedback by diffuse scattering to a single-mode diode laser. Fluctuations in the diode laser’s operating frequency that are due to scattered light from a vibrating surface are used to detect the amplitude and frequency of surface vibrations. An additional physical vibration of the laser provides an absolute amplitude calibration. The fundamental bandwidth is determined by the laser response time of roughly lo-’ s. A noise floor of 0.23 nm/Hz’’’ at 30 kHz with 5 X of the incident light returning is demonstrated. This instrument provides an inexpensive and sensitive method of noncontact measurement in solid materials with low or uneven reflectivity. It can be used as a vibration or velocity sensor. Key words: Vibrometer, optical sensor, noncontact measurement, surface displacement.
1. Introduction
The laser vibrometer described in this paper is made from a $20 laser diode; less than 0.005% of the incident light is required to return to the laser. A sensitivity of 0.23 nm/Hzl” at 30 kHz has been measured with this small amount of light. Current nondestructive optical methods, such as those applied to quality assurance, predictive maintenance, and acoustic research,l-6 are limited by the need for moderately to highly reflective surfaces,7,8 constraints on the distance to the measurement surface,g and expensive components such as Bragg modulators or frequency stabilized 1asers.lO This vibrometer provides an inexpensive, sensitive alternative to displacement detection and requires extremely low feedback power. In this device the signal arises from the phase relationship between the light leaving the laser and the light returning from the surface of interest, which is similar to other laser vibrometer measurement techniques.8JO-13 By using induced frequency changes as a signal, this sensor takes advantage of
P. A. Roos and C. E. Wieman are with the Department of Physics and Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colorado 80309. M. Stephens is with the National Institute of Standards and Technology, Boulder, Colorado 80303. Received 5 February 1996; revised manuscript received 12 April 1996 6754
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the extreme sensitivity of the lasing frequency of single-mode laser diodes to optical feedback.14 Near-infrared light from a laser diode is directed onto a vibrating object. The object scatters a small fraction of light back into the laser diode cavity. The optical feedback alters the frequency of the emitted laser light. A small portion of the light is diverted to a Fabry-Perot (FP) cavity, where changes in the laser operating frequency (LOF) are analyzed. The LOF changes provide information about the amplitude and frequency of the surface vibrations. Also, they can be used to determine the velocity of the test object. The effects of optical feedback on the LOF depend not only on the motion of the object but also on the backscattered power and the distance to the object. Therefore a calibration mechanism was developed to isolate the effects of surface motion on the LOF. This was done by mounting the laser diode onto a bimorph piezoelectric transducer (PZT) and translating it back and forth with a calibrated amplitude and frequency. This calibration mechanism is uniquely applicable to a laser diode because the laser is so small. The vibrometer described here provides an inexpensive, noncontact method for measuring vibrations in solid materials with low reflectance. Its sensitivity is higher than that of sensors that use laser-diodefeedback interferometry,lz and, while not as good as some commercial laser Doppler vibrometers, it is less
802
expensive and operates reliably with one tenth the scattered light. Owing to their high-gain active medium, 1ow-Q optical cavity, and extraordinarily small size, semiconductor lasers are extremely sensitive to optical feedback. When an object is placed in the beam path, causing external feedback, an external cavity is formed between the object and the back facet of the laser, altering the gain of the system. This results in a slight frequency shift of the emitted light. In the case of weak optical feedback (the operating condition of this sensor) the external feedback can be considered a small perturbation t o the laser gain. The effects of weak optical feedback on the LOF are described by the Lang-Kobayashi equations,l4 which result in the relation
-C sin[+ + w0TeXt+ A~T,,],
tt)
Aw=Asin - + y ,
2. Theory of Operation
AwTBxt=
ut, then
(1)
where w, is the LOF in the absence of optical feedis the back, Am is the resulting change in LOF, rext round-trip time for photons returning from the external surface, and = tan-' a;here a is the linewidth enhancement factor,15 a constant with typical value@ ranging from 6 to 10. C is given by
+
where y = 4.rrLo/Xis a constant phase. The velocity can be extracted by measurement of the modulation frequency of the LOF. For example, with a laser wavelength of 830 nm, an object moving at 1 m/s would produce a 2.4-MHz modulation in the laser frequency. This modulation, and hence the velocity, can be accurately determined with a simple counter. This technique measures only the magnitude of the velocity normal t o the laser and not the sign of the motion. Detecting small vibrations is somewhat more complicated. For a surface vibrating a t a frequency w f and amplitude b << L,
L = L o + b sin(w+),
(6)
where Lo is the initial distance to the surface. If 4nb/X << 1, Eq. (4) can be written as
4rrb A w = Asin(+) A
where R is the laser facet reflectance (typically 3% for a commercial, antireflection-coated laser or 30% for an uncoated laser), rL is the round-trip time for photons in the diode laser cavity (typically 10-l' s), and fext is the fraction of power reflected back into the laser cavity; feA will depend on the reflectance of and the distance to the test surface. Expanding Eq. (1)for small Aw yields
between the laser and The round-trip photon time rext the test surface and the LOF wo can be written as T,,~ = 2L/c and w, = 27rc/X, where c is the speed of light, L is the distance between the laser and the test surface, and A is the unperturbed wavelength of the laser diode. Since is a constant-phase factor it can be set to - ~ / 2for simplicity, and Eq. 3 can be written
+
(4) where A = -(c/2L) C. If L is large relative to its variation, then the amplitude A is approximately constant, and the laser frequency is modulated sinusoidally with changes in L. This modulation can be used to measure velocity or detect small vibrations. For a simple velocity measurement, when the external surface moves with velocity u , the frequency of the change in LOF determines the velocity. For example, if L = Lo +
-
Here J, is the nth-order Bessel function and the approximation J , (x) x / 2 has been used. Since Lo is constant, Aw varies sinusoidally in time with angular frequency wf and amplitude
47rb
A' = A- A cos(
4rrL0
.
Note that the approximation leading to Eqs. (7) and ( 8 ) requires that b << h / 4 1 ~ .For small vibration amplitudes the signal is linear with b, as predicted by Eq. (7); for larger amplitudes the signal appears sinusoidal. Inspection of Eq. (4)with Lo set t o nX/8 for simplicity (n a positive integer) shows that the signal is maximized for b = A/8, or for the peak-topeak vibration amplitude equal to h/4. Largeramplitude vibrations result in a sinusoidally varying s.+gnal. While in principle the vibration of the external surface could be measured using only Eq. (8), in practice there are two difficulties with this simplified view. 1 December 1996 / Vol. 35, No, 34
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First, the amplitude of the LOF moddation given by Eq. (8) depends on the distance Lo. If ~ T L , / A~2-1~12, where TZ = 1 , 3 , 5 . . . . the LOF modulation will be small. On the other hand, if 4rrLO/A nTr (n = 1, 2 , 3 . , . ), the signal will be maximized. Therefore if the distance Lo drifts by more than A/2, the signal can vary from its maximum value to zero. Second, the LOF modulation amplitude A' depends on both the amplitude of vibration and the amount of light returning t o the diode laser. The signal from an object with a small vibration amplitude but with large feedback power is indistinguishable from that of an object with a large vibration amplitude but small feedback power. A calibration mechanism was devised to determine the vibration amplitude independently of the feedback power and drifts in Lo. Physical vibration of the laser, with a known amplitude and frequency of motion, was used to produce a calibration LOF modulation. The response of the LOF to a vibrating external object relative to the stationary laser is identical to the response when the laser vibrates relative to a stationary external object. Hence the laser vibration causes an LOF modulation with an amplitude determined by the feedback power, the region of modulation on the LOF curve, and a known amplitude of laser vibration. If the object vibrates simultaneously with but a t a different frequency than the calibration motion of the laser, then a LOF modulation corresponding to the same feedback power and distance Lo but a different and unknown vibration amplitude will be produced. The relative amplitudes of the two sine waves therefore depend only on their corresponding vibration amplitudes. Because the signal a t the calibration frequency (the frequency a t which the laser is being vibrated) corresponds to a known vibration amplitude, the ratio of the amplitudes can be used to determine the vibration amplitude of the object. Both amplitudes will vary according to the amount of feedback power and with the drift of Lo, but their ratio will remain constant. If the laser is vibrated with frequency w, and amplitude n so that
..........
..............
-
Lo = L, + n sin(w,t),
(9)
then by inserting Eq. (9) into Eq. (7) and letting 4nn/A << 1, we find
The calibration signal is necessary only for object vibration amplitudes smaller than h/2. Larger am6756
APPLIED OPTICS / Vol. 35, No. 34
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.......
TEMPERATURE CONTROL BOX
O,ODE
METER
U
U
Fig. 1. Diagram of vibrometer. The laser diode and collimation lens were mounted in a closed box for temperature stability. The beam exited the box and was directed onto a calibrated PZT test surface. A small portion of the beam was sent to a FP cavity that was used as a frequency discriminator. The intensity of light reaching the test surface was controlled by a half-wave plate and polarizing beam splitter. A small portion of the light was sent to a p-i-n photodiode to monitor the intensity of light reaching the test surface.
plitudes correspond to more than one period of the sine wave; in that case the amplitude of the motion can be extracted simply by counting fringes (and fractions of a fringe). The following section describes the realization of the sensor. The research focused on measurement of small vibrations. 3. Apparatus
The vibration sensor is shown in Fig. 1. The light from a 5-mW laser diode was directed onto a bimorph PZT that was used as the test object. The PZT was 0.8 m from the laser diode. The PZT surface was a diffusive silver color that scattered light but gave no specular reflection. The amplitude of the PZT motion relative to the applied drive voltage was independently calibrated as a function of frequency in a Michelson interferometer. Approximately 4% of the initial laser beam was diverted by a glass slide and directed into a confocal FP cavity that was used as the LOF discriminator. The FP cavity had a 7.5GHz free spectral range and a finesse of 100. An adjustable A/2 plate and polarizing beam splitter were placed in the beam path to regulate the power reaching the test object and therefore the power returning t o the laser. A second glass slide placed after the beam splitter diverted a small portion of laser light onto a p-i-n photodiode. This signal was used to monitor the light reaching the test surface and was used to estimate the backscattered power returning to the laser. The laser beam was collimated but was not focused onto the measurement surface. (The beam could have been focused onto the surface to increase the sensitivity, but it was not necessary.) Figure 2 is a diagram of the mechanism that vibrated the laser. A 6-mm hole was drilled in the center of a PZT disk with a carbide bit. Lightweight wires were soldered t o each side of the PZT. A nonconductive washer and a transistor jack were epoxied
804
INJECTION CURRENT WlRES
UPRIGHT PORTION OFMOVNTlNGBLOCK
PZr WITH HOLE
NONCONOUCTlNG TRANS STOR WASHER ACK
USER
VLt
OiODt
CURRENT
J+ I
I+"
___
PHOTO-
COMPENSATION VOLTAGE
8
V CI
COMPARISON
Fig. 3. Slow laser-frequency servo. Frequency fluctuations were detected with a photodiode located behind the FP cavity. Corrections were made with small changes to the laser diode injection current.
Fig. 2. Calibration mechanism. The whole laser was mounted on a PZT to allow physical vibration of the laser.
to the outer surface of the PZT with the leads of the transistor jack protruding through the hole and out the back of the PZT. This setup was epoxied to a laser mounting block a t four isolated points to allow the PZT to flex. The injection current wires soldered t o the transistor jack were insulated to ensure no electrical conduction with the PZT. The laser diode was then fitted into the compatible transistor jack. This system was placed in a n aluminum box to reduce temperature-induced changes in the LOF. A hole, sealed with a microscope slide, was drilled at a 20" angle t o the normal of the side of the box to allow light to exit the box while minimizing the effects of optical feedback from the slide. The spectral properties of the laser light were verified with an optical spectrum analyzer. The laser was operating with the desired single-longitudinal mode a t -775 nm with 85 m A of injection current. Frequency discrimination was achieved with the FP interferometer. The central value of the laser frequency was held constant by locking it to the side of the FP resonance and monitoring the unservoed, high-frequency modulations. The error signal was derived from the light transmitted through the FP cavity. The photodiode signal from the transmitted light was filtered and fed back to the laser diode injection current as shown in Fig. 3. The reference voltage of the servo was set so that at low frequencies the transmitted light power was held constant at half of the FP cavity's peak transmittance. LOF modulations resulting from the test object's vibration were then easily detected as fast modulations (above the servo bandwidth) in the light transmitted through the FP cavity. The subsequent voltage modulations were observed on an oscilloscope and a spectrum analyzer. The amplitudes of the peaks produced on
the spectrum analyzer were proportional to the LOF modulation amplitude. To calibrate the motion of the bimorph PZT for an applied drive voltage, a very small mirror was glued to the front surface of the PZT and was used as one mirror in a Michelson interferometer. The PZT response as a function of driving frequency was measured. PZT limitations restricted measurements to frequencies below 50 kHz. The PZT that vibrated the laser was calibrated similarly. To ensure a proper calibration signal from the vibration of the laser, it was important to minimize the effects of optical feedback from surfaces other than the test PZT. The collimation lens was epoxied to the front of the laser so that the two objects moved together. All optical components were thoroughly cleaned. Attenuators and slight misalignment of the FP cavity were used t o reduce unwanted optical feedback from the FP cavity to a negligible level. An optical isolator between the FP cavity and the laser diode would be a better, but more expensive, solution to this problem. Alternatively, the light could be coupled into a n off-axis mode of the cavity. 4.
Results
The three factors that influence the signal amplitude are distance t o the object, feedback power, and surface-vibration amplitude. The effects of each were compared with the expected behavior outlined in Section 2. The measured relationship between the vibration amplitude of the test surface (PZT) and the resultant sensor signal, as measured on the spectrum analyzer, is shown in Fig. 4. The test PZT was vibrated a t 35 kHz. The vibration amplitude was varied while the feedback power and Lo were held constant. As expected from Eq. (4), the sensor signal was maximized for peak-to-peak vibration amplitudes of 114 wavelength (195nm). The Ao measured at the vibration frequency with our test surface at a distance of 0.8 m and a vibration amplitude of 195 nm was approximately 10 MHz. Motions with amplitudes greater than h/4 were observed but were not analyzed quantitatively (see Fig. 5). A fringe-counting technique would be necessary for such measurements. 1 December 1996 / Vol. 35, No. 34 / APPLIED OPTICS
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805
. . , 1 lo, ,
0
0
20
,
40
,
,
,
,
,
,
60 SO 100 120 140 160 Object Vibration Amplitude (nm)
180
200
Fig. 4. Sensor signal as a function of peak-to-peak object vibration amplitude for constant Lo and feedback power. The filled circles represent data points; their size represents the measurement uncertainty. The roll-off at larger amplitudes is expected because of the sinusoidal dependence of the signal on vibration amplitude. The dashed curve is a normalized plot of Eq. (4) with A = 49.6, h = 775 nm; L is one half of the peak-to-peak object vibration amplitude. Note that the approximation that led to Eq. (7) is not valid for vibration amplitudes for which b >> h/47~.
The dependence of the sensor signal on feedback power is shown in Fig. 6. The light returning to the laser diode was estimated to be 0.005% of the light incident upon the PZT by measurement of the fractional power of scattered light a t a p-i-n photodiode 0.2 m from the object and multiplication of that value by the estimated ratio of the solid angle of the diode laser to the solid angle of the p-i-n photodiode. The adjustable X/2 plate and a polarizing beam splitter were then used to change the feedback power below this level. A glass slide diverted a small portion of the incident beam onto a p-i-n diode to monitor the attenuation (see Fig. 1). The PZT vibration amplitude was fixed a t 97 nm, and the distance between the laser and the PZT was held constant by manual
Fig. 5. Oscilloscope signal obtained while the test object is vibrated with an amplitude greater than h / 2 . The top trace is the ramp signal sent to the test PZT. The bottom trace is the sensor signal. The signal is sinusoidal, as expected. 6758
APPLIED OPTICS / Vol. 35, No. 34 / 1 December 1996
0
0
1
2 3 Fractional Feedback Power (*105)
4
5
Fig. 6. Sensor signal as a function of feedback power for constant Lo and object-vibration amplitude. The dashed curve is a normalized plot of Eq. (2) with C equal to 4870 X fex;”. The filled circles represent the data points; their size represents the measurement uncertainty. The data fit the predicted feX:” dependence well. The maximum feedback power was 0.005% of the incident light.
adjustment of the dc voltage to the PZT (to compensate for slow, temperature-induced drifts in Lo). In the next set of measurements the diode laser was vibrated as described in Section 2 to eliminate sensitivity to such drifts as well as any change in feedback power. Figure 7 shows a typical spectrumanalyzer signal with a 3.3-M-fzcalibration-vibration frequency. The peaks at of - o, and a t 2w, predicted by Eq. (10) are barely visible above the noise. The previous measurements were repeated, and they agreed with predictions without any need to compensate for thermally induced drift in the value of Lo. The calibration frequency was chosen arbitrarily to be 6.4 kHz and the object-vibration frequency used was 19.4 kNz. The test surface’s vibration amplitude was measured by comparison of the magnitude of the vibration signal a t 19.4 k€€z with the magnitude of the calibration signal a t 6.4 kHz (which corresponded to a known vibration am-
Fig. 7. Photograph of the calibration and object peaks on the spectrum analyzer. The laser was vibrated at 6.4 kHz with an amplitude of 97 nm, and the object was vibrated at 19.4 kHz with the same amplitude.
806 Table 1. Values of Calibration Peak, Test Surface Peak, and Ratio of the Peaks as a Function of Intensity Returning to the Laser Diode for Constant Test-Surface and Calibration-Vibration Amplitudesa
Normalized Intensitv
Calibration Peak Amplitude for 6.4 kHz (mV)
Test Surface Peak Amplitude for 19.4 kHz (mV)
Test Surface/ Calibration
4.52 3.69 2.79 1.70
4.88 3.75 2.80 1.76
1.08 1.02 1.00 1.04
~~
1.0 0.78 0.57 0.29
"Magnitudes of the peaks are measured on a spectrum analyzer at the appropriate frequency. Thermal drifts of Lo were permitted so the magnitudes of the peaks will not necessarily follow the feXt1lz dependence predicted by Eq. (2). Note that, although the magnitudes of the signals vary by a factor of 3 over the intensity range, the ratio of the signals varies by less than 5%.
plitude). Table 1 shows a set of data taken over different values off,,. An attenuator was used to change the fraction of light returning to the laser. Note that, although the amplitudes of the individual peaks changed by more than a factor of three, the ratio of the peaks changed by less than 5%. Repeatability of the calibration ratios was examined while vibration amplitude, feedback power, and distance to the laser were separately varied. When the vibration amplitude or feedback power was varied, the standard deviation of the ratios decreased as the inverse square root of the number of averages, as expected from random noise fluctuations. However, variations beyond those expected from random noise were obtained for different values of Lo. A closer analysis revealed that the noise of the sensor changed when drifts in Lo were larger than X/2. This systematic effect appears to be connected with spurious optical feedback from the FP cavity. While a further analysis of this systematic effect is warranted, we expect that an optical isolator, better attenuation of the light reflected from the cavity, or coupling into the cavity off-axis will solve the problem. The noise floor of the instrument is shown in Fig. 8. It includes noise due to changes in FP cavity mirror
Fig. 8. Noise floor of the sensor as a function of frequency. 1 kV/Hz'/' = 0.015 nm/Hz'''.
spacing caused by thermal variations and seismic vibrations (dominant a t low frequencies) as well as intrinsic noise from the laser and electronics noise. The displacement noise a t 35 H z is 0.23 nm/Hz1I2. 5. Discussion
The results show that this instrument can be used to measure small vibrations in solid materials with extremely low feedback while maintaining high sensitivity. This section discusses further modifications and fundamental limits for frequency range and dynamic range. Flexibility with respect to the returning light power needed for measurement makes this method of vibrometry unique. Movements as small as a few tenths of a nanometer were detected with less than 0.005% of the incident power returning to the laser. This means that it is possible to measure small vibrations of objects that have extremely low reflectance. Alternatively, the diode sensor can be positioned far from the object being measured. The minimum feedback power needed will depend on the vibration amplitude to be measured. The fundamental limit to any vibration measurement will be the intrinsic laser-frequency noise. The power spectrum o f the intrinsic noise increases with decreasing frequency17 to -10 MHz. Therefore, the ultimate sensitivity of a sensor limited by the laserfrequency noise will be higher a t high frequencies away from the relaxation oscillations (near 2 GHz). The vibrometer's inherent bandwidth is very large. The range will be limited a t low frequencies by the servo bandwidth. While the bandwidth of the servo implemented here was about 3 kHz, this limit was not optimized and should be lowered to less than 100 Hz (the trade-off here is low-frequency sensitivity of the sensor versus robustness of the lock to the FP cavity resonance; a more robust locking method such as that described by H a s c h and Coullaud could be used's). The upper limit of vibration frequency in this demonstration was -50 k H z and was completely limited by the response of the bimorph PZT used as the test object and the electronics used to drive it. A diode laser's response to optical feedback is much faster (of the order of 1ns)17J8 and is the fundamental limiting factor for high-frequency measurements. The analysis of the signal at frequencies greater than the FP cavity linewidth will be complicated by the change in the response of the cavity, since the cavity will act as a phase, rather than a frequency, discriminator. An increase in signal can be obtained by increasing the feedback power. At high feedback power the LOF fluctuations can no longer be approximated by the Lang-Kobayashi equations. Within this limit, however, there are several ways to increase the feedback power and therefore increase the signal. The beam can be focused onto the measurement surface to increase feedback power dramatically. Surfaces with higher reflectance and surfaces that are closer to the laser will also increase the signal. The properties (finesse and free-spectral range) of 1 December 1996 / Vol. 35, No. 34
/ APPLIED OPTICS
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the FP cavity used will also affect the sensitivity. No attempt was made to optimize these properties. Using a dielectric coating to adjust the reflectance of the front facet of the diode laser could also improve the sensitivity14 [this can be seen from Eqs. (1)and
(211.
brometer. The vibrometer’s unique measurement technique allows for measurement of physically inaccessible vibrating surfaces that have small mass and very low reflectance. The vibration measurement is not the only application of optical feedback-induced frequency modulation in laser diodes. The sensor can also be used as a velocimeter. The method described in this work will require substantially lower returning light power than other forms of vibrometry, maintain excellent sensitivity characteristics, and provide flexibility in terms of vibration amplitude and frequency as well as the distance to the measurement surface, all a t a very low cost.
No careful measurements of the working range of the vibrometer were made. In all the measurements described here, the test surface was 0.8 m from the diode. The range will depend on the reflectivity of the test object. (Of course, this sensor will be subject to the same problems as normal interferometers; i.e., air currents will become an important source of noise.) Vibration amplitudes as high as 550 nm were observed, but the upper limit of detectable vibration We thank Eric Cornell for suggesting the calibraamplitudes was not investigated. tion mechanism. This work was funded by the OfThis vibrometer takes advantage of a different asfice of Naval Research and the National Institute of pect of the same physical phenomenon as sensors employing laser-diode-feedback interfer~rnetry.~JlJ~Standards and Technology. However, it measures vibrations by looking a t freReferences quency rather than intensity changes in the laser 1. S. G. Anderson, “Complextests exploit laser technology,”Laser diode’s emitted light. The frequency is a more senFocus World 71-78 (July 1994). sitive discriminator. The vibrometer therefore re2. J . P. Nokes and G. L. Cloud, “The Application of Three Interquires far less (approximately a factor of 1000) ferometric Techniques to the NDE of Composite Materials,” in backscattered light to operate. The displacement Interferometry VI: Applications, R. J. Pryputniewicz, G. M. sensors described in Ref. 9 measure changes in freBrown, and W. P. Jueptner, eds., Proc. SPIE 2004, 18-26 quency caused by optical feedback, but because the (1993). light sources used are not single-mode diode lasers, 3. P. Sriram, J. I. Craig, and S. Hanagud, “Scanninglaser Doppthe sensitivity to optical feedback is greatly reduced ler techniques for vibration testing,” Exp. Tech. Phys., 21-26 and the sensors required that 25% of the incident (Nov./Dec. 1992). 4. M. Samuels, S. Patterson, J. Eppstein, and R. Fowler, “Low light be returned to the laser t o achieve a comparable cost, handheld lidar system for automotive speed detection and sensitivity. law, in Laser Radar VII: Advanced Technology for ApplicaThis sensor requires far less scattered light than tions, R. J. Becherer, ed., Proc. SPIE 1633, 147-159 (1992). other optical noncontacting methods of vibrometry, 5. A. R. Duggal, C. P. Yakymyshyn, D. F. Fobare, and D. C. such as Michelson interferometry10J9 and laserHurley, “Optical detection of ultrasound with a microchip laDoppler vibrometry.2.3 It is also less costly, because ser,” Opt. Lett. 19, 755-757 (1994). it uses inexpensive diode lasers and does not require 6. D. J. Anderson, J . D. Valera, and J. D. C. Jones, “Electronic equipment such as Bragg modulators. Its sensitivspeckle pattern interferometry using diode laser stroboscopic ity is comparable t o Michelson interferometry but is illumination,” Meas. Sci. Technol. 4, 982-987 (1993). not as good as laser-Doppler vibrometry. 7. P. J . de Groot, G. M. Gallatin, and S. H. Macomber, “Ranging and velocimetry signal generation in a backscatter-modulated The sensitivity of relatively inexpensive nonoptical laser diode,”Appl. Opt. 27, 4475-4480 (1988). systems such as capacitive or inductive sensors i s 8. D. A. Oursler and J. W. Wagner, “Full-fieldvibrometry using a comparable with or (at low frequencies) better than Fabry-Perot Btalon interferometer,” Appl. Opt. 31, 7301-7308 the sensor described here. The major advantage of (1992). this (and most optical sensors) over nonoptical sys9. K.-L. Deng and J . Wang, “Nanometer-resolution distance meatems is the increased range of the sensor. The diode surement with a noninterferometric method,” Appl. Opt. 33, can be placed more than 1 m away from the test 113-116 (1994). object, while capacitive and inductive sensors typi10. Sun Yusheng, Zhang ZiDong, and Cai Kanze, “Improvements cally have bulky packages that need t o be placed in a laser heterodyne vibrometer,” Rev. Sci. Instrum. 63, within a few millimeters of the test object. 2974-2976 (1992). 6. Conclusion
The effects of very small amounts of optical feedback on a laser diode’s operating frequency can be used to measure small amplitude vibrations. The effects of the feedback on the LD operating frequency match the predictions. A calibration mechanism to differentiate between large vibrations with small amounts of optical feedback and small vibrations with large amounts of optical feedback was demonstrated. There are many possible applications for this vi6760
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11. J. Kato, N. Kikuchi, I. Yamaguchi, and S. Ozono, “Optical feedback displacement sensor using a laser diode and its performance improvement,” Meas. Sci. Technol. 6, 45-52 (1995). 12. S. Donati, G. Giuliani, and S. Merlo, “Laser diode feedback interferometer for measurement of displacements without ambiguity,” IEEE J. Quantum Electron. 31, 113-119 (1995). 13. E. T. Shimizu, “Directional discrimination in the self-mixing type laser Doppler velocimeter,” Appl. Opt. 26, 4541-4544 (1987). 14. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. QE-16, 347-355 (1980).
808 15. D. Lenstra and J. S. Cohen, “Feedback noise in single-mode semiconductor lasers,” in Laser Noise, R. Roy, ed., Proc. SPIE 1376,245-258 (1990). 16. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers-An Overview,” IEEE J. Quantum Electron. QE-23, 9-29 (1987). 17. C. E. Wieman and L. Hollberg, “Using diode lasers for atomic
physics,” Rev. Sci. Instrum. 62, 1-20 (1991). 18. T. W. Hansch and B. Coullaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35, 441-444 (1980). 19. M. Imai and K. Kawakita, “Optical-heterodyne displacement measurement using a frequency-ramped laser diode,” Opt. Commun. 78, 113-117 (1990).
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Frequency-stabilized diode laser with the Zeeman shift in an atomic vapor Kristan L. Corwin, Zheng-Tian Lu, Carter F. Hand, Ryan J. Epstein, and Carl E. Wieman
We demonstrate a robust method of stabilizing a diode laser frequency t o an atomic transition. This technique employs the Zeeman shift to generate an antisymmetric signal about a Doppler-broadened atomic resonance, and therefore offers a large recapture range as well as high stability. The frequency of a 780-nm diode laser, stabilized to such a signal in Rb, drifted less than 0.5 MHz peak-peak (1part in lo9)in 38 h. This tunable frequency lock can be constructed inexpensively, requires little laser power, rarely loses lock, and can be extended to other wavelengths by use of different atomic species. 0 1998 Optical Society of America OCIS codes: 140.0140,140.2020,300.1030.
1. Introduction
2.
Lasers with stable frequencies are essential in many fields of research. In addition, they are used commercially in precision machining tools, gravimeters, and laser vibrometers. He-Ne lasers have been the industry standard for many years,l but they are bulky, energy inefficient, and have limited tube lifetime. Diode lasers offer an improvement in all these areas and moreover can be stabilized to atomic transitions. Typical methods of stabilization,* although practical in some laboratory settings, are not reliable enough for use in commercial equipment. Using a technique originally demonstrated with a LNA (Lal-JVd,MgAl,,O,g) laser in helium,3 we developed a robust diode laser stabilization scheme that will be useful in both commercial instruments and research laboratories.
The frequency of a diode laser with grating feedback depends on the current, temperature, and external diffraction grating position.2 With the laser cavity in a Littrow configuration (see Fig. l), the output beam reflects off the grating, while the first-order beam diffracts back into the laser diode. The optical feedback from the grating is spectrally narrowed and peaked at a frequency that can differ from the bare diode central frequency. Thus this feedback narrows the laser linewidth to <1 MHz and forces the central frequency to nearly that of the feedback signal. To tune the laser central frequency, the grating is tilted by applying a voltage to a piezoelectric transducer (PZT). Over time, the central frequency will drift because of temperature, current, and mechanical fluctuations. This drift can be reduced by stabilizing the laser to an external reference. In addition, small, rapid fluctuations in laser frequency, which contribute to the laser linewidth, can be reduced by rapidly controlling the diode laser current. In one popular method of stabilizing the diode laser frequency, some of the output light is sent into a saturated absorption spectrometer. The diode laser frequency is then locked to either the side or the peak of the narrow saturated absorption features,2.4shown in Fig. 2. These narrow lines offer the advantage of a steep slope, where the slope is the change in the fractional absorption signal with laser frequency. Side-locking to this slope is accomplished by electrically controlling the PZT voltage so that the saturated absorption signal is maintained at a particular level. However, a disadvantage of side-locking is
When this research was performed all the authors were with JILA, University of Colorado and National Institute of Standards and Technology; and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440. 2.-T. Lu is now with the Physics Division, Building 203,Argonne National Laboratory 9700 South Cass Avenue, Argonne, Illinois 60439. C. F. Hand is now with New Focus, Incorporated, 2630 Walsh Avenue, Santa Clara, California 95051. R. J. Epstein is with the Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556. Received 9 October 1997;revised manuscript received 2 February 1998. 0003-6935/98/153295-04$15.00/0 0 1998 Optical Society of America
Diode Laser Frequency Stabilization
20 May 1998 / Vol. 37,No. 15 / APPLIED OPTICS
809
3295
810
ui
Po'arLZer
Beam spii
Vapor Cell in B field
:Laser Diode: .____.______
Fig. 1. Schematicof a DAVLL system. Here we show the entire beam passing through the lock, but in actuality, only a small amount of power is picked off from the main beam and enters the locking apparatus.
Frequency (MHz)
that fluctuations in beam alignment and intensity will alter the lock point and cause drift in the laser frequency. Peak-locking is less sensitive to these fluctuations, but has its own disadvantages related to phase-sensitive detection: Either the output of the laser is modulated directly or expensive electro-optic components are used t o modulate only the light entering the spectrometer. A further disadvantage of both peak and side locks is their small capture range, which prevents them from recovering from perturbations that shift the laser frequency by more than -30 MHZ. 3. The Dichroic-Atomic-Vapor Laser Lock Signal
To overcome the aforementioned disadvantages with the conventional locks, we developed a dichroicatomic-vapor laser lock (DAVLL). This technique employs a weak magnetic field to split the Zeeman components of an atomic Doppler-broadened absorption signal and then generates an error signal that depends on the difference in absorption rates of the two components. The subtraction technique minimizes the frequency drifts that are due to changes in line shape and absorption that typically limit the utility of Doppler-broadened absorption features for frequency stabilization. The DAVLL lock offers the
"Rb
-0.4
85Rb
I
-1500
-1000
-500
I
I
0
500
Relative Laser Frequency (MHz) Fig. 2. Oscilloscope trace o f (a) the signal from a saturated absorption spectrometerand (b)the DAVLL signal, as the diode laser is scanned across Rb resonances with the PZT. A laser can be locked to either of the two circled zero crossings of the DAVLL signal. These features are due to the 87Rb F = 2 F = 1,2, 3 and the "Rb F = 3 F = 2 , 3 , 4 transitions. The frequency of the lock point can be tuned optically by rotating the quarter-wave plate, or electronically by adding an offset voltage to the signal. ---f
---f
3296
APPLIED OPTICS 1 Vol. 37,No. 15 1 20 May 1998
Fig. 3. Origin of the DAVLL signal shape. (a) A Dopplerbroadened transition in Rb in the presence of no magnetic field. (b)The same transition, Zeeman shifted in a 100-G magnetic field, when circularly polarized light is incident on the vapor. (c) The same as (b), but with the opposite circular polarization. (d) The difference between (c) and (b) giving the DAVLL signal. In this idealized case, the arrow indicates that the off-resonant signal is zero.
advantages over saturated absorption: large recapture range, simplicity, low cost, and no need for frequency modulation. As shown in Fig. 3(a), a Doppler-broadened absorption feature is detected when a laser beam (with wave vector k = K2) passes through a Rb vapor and the laser's frequency is scanned across a transition. In the absence of a magnetic field, we obtain the same signal regardless of the laser polarization (E). However, if a uniform magnetic field (B = B2)is present and the laser is circularly polarized (e = &+), the central frequency of the absorption feature increases Fig. 3(b)]. If the laser has the opposite polarization (E = 6-) [Fig. 3(c)], the central frequency decreases. By subtracting the two absorption profiles Fig. 3(d)], we obtain an antisymmetric signal that passes through zero and is suitable for locking. A DAVLL signal with a steep slope causes the lock to be less sensitive to noise sources that mimic laser frequency changes, such as laser intensity noise. A rough comparison with a typical saturated absorption setup in our laboratory shows that the DAVLL slope is comparable with that of the saturated absorption lines. This may seem surprising a t first, because the linewidths of the saturated absorption lines (FWHM -20 MHz) are much smaller than those of the DAVLL lines (-500 MHz peak-peak) Fig. 2(b)]. However, the heights of the saturated absorption features range from -1/3 to 1/30 of the onresonant Doppler-broadened absorption fraction, whereas the DAVLL signal height is twice that absorption fraction. By approximating the slope as the linewidth divided by the signal height, we estimate that the slope of the largest saturated absorption peak is only four times bigger than the DAVLL slope. The slope of the DAVLL signal is also affected by the magnetic field. The separation of the two Zeeman-shifted absorption peaks must be large
81 1
enough to give a sizable capture range, but small enough to give a large slope through the unshifted resonance. In addition, the Zeeman-shifted absorption peaks broaden with increasing field, because the various transitions contained within one Dopplerbroadened feature shift different amounts. We found that 100 G maximizes the slope, and therefore represents the best compromise between increased separation and increased broadening. However, the dependence of slope on the magnetic field is not strong, so if B is varied by a factor of 2 it should not significantly alter the lock performance. 4. Apparatus
A schematic of the diode laser and optics used to generate the DAVLL signal is shown in Fig. 1. The SDL 780-nm diode laser is tuned by use of a diffraction grating, as described above. The output beam from this laser passes through a beam splitter, and a small amount of power is split off to be used for locking. After passing through a small aperture, the resulting beam passes through a linear polarizer. Pure linear polarization is equivalent to a linear combination of equal amounts of two circular polarizations. This beam (2.5-mm diameter, 0.5 mW) next passes through a cell-magnet combination, consisting of a glass cell filled with Rb vapor and a 100-G magnetic field. The magnet is made of rings of rubber-embedded permanently magnetic material, spaced appropriately and glued together concentrically around the glass cell.5 To generate the DAVLL signal, the absorption profiles of the 6 , light must be subtracted from that of the &. To accomplish this, after exiting the cell, the two circular polarizations are converted into two orthogonal linear polarizations by passing through a quarter-wave plate. Then the two linear polarizations are separated by a polarizing beam splitter, and the resulting two beams are incident on two photodetectors whose photocurrents are subtracted. As the frequency of the laser is scanned across an atomic transition, an antisymmetric curve is generated, as shown in Figs. 2 and 3. The diode laser is then locked by feeding back a voltage to the PZT so that the DAVLL signal is maintained at the central zero crossing. We align the optics by orienting the fast axis of the quarter-wave plate at 45" to the axis of the output polarizing beam splitter, so that equal intensities are incident on the two photodetectors when the laser is far detuned (>1GHz) from the Rb resonances [see Fig. 3(d)]. The DAVLL system is least susceptible to drifts when the off-resonant signal gives no net photocurrent, and the lock is therefore very near the center of the unshifted resonance, as shown in Figs. 2 and 3. We tuned the locked laser frequency either by adding an electronic offset or by rotating the quarter-wave plate. The latter optical method changes the frequency by weighting one circular polarization more heavily than the other. This type of offset is more stable than the electronic offset because the lock point is always a t a zero in net photocurrent,
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Fig. 4. Measured beat frequency between two DAVLL systems over a 38-h period. Variations in the beat frequency indicate the limits of the laser stability to be approximately 500 kHz peakpeak. These data show a stability of 27 kHz rms during an 11-h period at night when environmental factors such as room temperature and air currents are more stable. The discontinuities at the end of the run are due to incomplete shielding of the detection photodiodes from room lights. The run was stopped when a laser mode hopped, but aRer we adjusted the current to return the laser to the proper mode, it returned to the same frequency.
which occurs when the powers incident on the two photodetectors are equal. Thus with optical offsets, the lock point maintains its insensitivity to laser intensity fluctuations. 5. Characterization of Frequency Stability
To monitor the frequency stability of the laser lock, we stabilized two separate lasers each to their own DAVLL system. We locked them to the same Doppler-broadened feature (85Rb F = 3 -+ F ' ) with different optical offsets, typically approximately 25 MHz apart. A portion of the light from each laser was combined at a beam splitter and copropagated onto a fast photodetector (125 MHz). The resulting beat note, corresponding to the difference between the two laser frequencies, was fed into a high-speed counter. By reading the counter every 5 s, a computer monitored the laser stability over periods ranging from 12 to 38 h. In this way, the difference between the two laser frequencies was monitored over many days, under different conditions. The beat frequency was stable to 2.0 MHz peak-peak while the temperature of the laboratory, and therefore of the optical components, varied a couple of degrees throughout the day. When the cells (with attached magnets) were enclosed in a copper pipe and crudely temperature stabilized, the stability improved to 500 kHz peak-peak over 38 h, as shown in Fig. 4. The cell-magnet combinations have measured dependences of 1.0 MHz/"C and 1.7 MHz/"C. We attribute this drift to a temperature-dependent birefringence of the cell windows, because the lock point is more sensitive to birefringence than to any other parameters. This is expected and observed, as discussed below. To confirm that optical offsets are more stable than electronic, we used an optical offset to tune one laser 120 MHz away and found that the drift rate was still 20 May 1998 / Vol. 37, No. 15 / APPLIED OPTICS
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comparable. When similar frequency offsets were applied electronically, the drift increased to 3 MHz peak-peak. If the two lasers drift in a correlated manner, then the difference frequency remains constant so the above measurement is insensitive to it. To confirm that this was not occurring, we measured the stability of one DAVLL system by beating it against a second diode laser that was locked to a peak of a saturated absorption feature. Because the physics of the two locks is quite different, we expect drifts in the two systems to have different dependencies. In this case, we observe a stability of 200 kHz peakpeak over 12 h, which is consistent with the result previously described. From this we conclude that the two DAVLL systems were not driRing in a correlated manner, and the stability of the beat frequency can be interpreted as the stability of the absolute frequency. The frequency stability of the lock can also be predicted without comparing two separate systems. We can convert the stability of the off-resonant signal level (Fig. 3) to an equivalent frequency stability by multiplying the fluctuations in photocurrent by the slope of the central resonant DAVLL signal. This calculation reliably predicts the frequency stability of the locked system and is therefore a simple, useful diagnostic. The agreement between the predicted and measured stability also indicates that the primary source of drift is changing birefringence of the optical components, because birefringence equally affects the signal levels both on and off resonance. As a final testament to the lock‘s stability, we used these lasers to maintain a Rb magneto-optic trap for many days without adjusting the lasers that were locked to DAVLL systems. The above results were obtained by use of zeroorder glass/polymer retarders, calcite GlanThompson input polarizers, and calcite Wollaston prism beam splitters. Comparable stability was also found when we used less expensive optics, including a plastic film polarizer, a plastic film retarder (X/4 at 540 nm), and a single calcite crystal (used as a polarizing beam splitter). In contrast, we found that some dielectric polarizing beam-splitting cubes give a large temperature dependence. The DAVLL lock was found to be robust because of the very broad locking signal. In fact, we applied mechanical perturbations to the optical table as high as the table’s damage threshold (including banging on the table with a hammer) and were unable to
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knock the lasers out of lock. The lasers jumped once every couple days, apparently because one of the lasers jumped to a different mode of the laser chip. These jumps were usually attributable to temperature drifts in the laser chip, but could occasionally be caused by a fast electromagnetic pulse such as that produced by our turning on a large nearby argon-ion laser. These types of mode hops are not observed in diodes with good antireflection coatings because the chip resonances are greatly suppressed. Therefore a DAVLL system constructed with such diodes would likely never lose lock. 6. Conclusion
We have shown that the DAVLL lock provides an effective method for stabilizing a diode laser to a very broad, stable atomic reference. In comparison with saturated absorption locks, this system stays locked for much longer periods of time and requires fewer optics, less electronics, and less laser power. It can also be quite compact and inexpensive. This simple, robust stabilization scheme should work for a number of atomic and molecular species at a variety of wavelengths and is an appealing option whenever a continuous stable laser frequency is desired. We thank Brian DeMarco and Neil Claussen for helpful discussions. This research was funded by the National Science Foundation (NSF), and R. J. Epstein was supported by NSF through the Research Experience for Undergraduates program at the University of Colorado. References and Notes 1. L. S.Cutler, “Frequency stabilized laser system,”U.S. patent 3,534,292 (13 October 1970). 2. C. E.Wieman and L. Hollberg, “using diode lasers for atomic physics,”Fkv. Sci. Instrum. 62,l-20 (1991);K. B.MacAdam,A. Steinbach,and C. Wieman, “Anarrow-bandtunable diode laser system with grating feedback and a saturated absorption spectrometer for Cs and Rb,”Am. J. Phys. 60,1098-1111 (1992). 3. B.Cheron, H. Gilles, J. Havel, 0. Moreau, and H. Sorel, “Laser frequency stabilization using Zeeman effect,” J. Phys. I11 4, 401-406 (1994). 4. For a discussion of saturated absorption spectroscopy, see W. Demtroder, h e r Spectroscopy (Springer-Verlag,Niw York, 1996). 5. We used the materialwith part number PSM1-250-3X36Cfrom the Magnet Source, 607-T S. Gilbert St., Castle Rock, Colo. 80104, 1-800-525-3536. Although uniformity is not critical to stability, we minimized variations to 5% along the field axis of symmetry by spacing the inside rings closer together than the outer ones.
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