Laser Spectroscopy: Proceedings of the XVIII International Conference: ICOLS 2007: Telluride, Colorado, USA, 24-29 June 2007

The remaining task is to detect the variances of the final state. Again the second pulse is used as the read out of the atomic state, but this time the input is not varied but held constant. The variance of the atomic state can be retrieved from the light measurement with help of eqs. 2 and 3 . In Fig.2(a) one can see the outcomes of a light measurement. From this the atomic state can be reconstructed as shown in 2 (b). In the case of the displayed measurement, where the displacement corresponds to a photon number of fi = 5, the fidelity is F = 0.58 f .02, which clearly lies above the classical limit of 0.5.5 For a limited range of input states the fidelity is maximized with a gain different from one. For input distributions with widths of < n >= 2,5,10,20,200 fidelities of F 2 = 0.64 f 0.02, F 5 = 0.60 f 0.02, F ~= o 0.59 f 0.02, F ~ = o 0.58 f 0.02 and I7200 = 0.56 f 0.03 can be extrapolated from our measurement^,^ which should be compared with the achievable classical fidelities5 F t l a s s i c a l - 0.60, ~ g c l a s s i c a l= 0.545, F $ a s s i c a l - 0.52, F ; m s i c a l 0.51 and F;&jssical= 0 .50 . There are different possibilities to improve the performance of this experiment. The protocol as it is suffers from residual noise introduced by the initial entangling interaction. By including higher order temporal modes and utilizing squeezed light in the entangling arm those extra noise contributions can be lessened and the fidelity increased.

117 (a)

(b) probability density . •!.

I

"'... ..' 'I ' ' J. - i.'-'t- i- '•- .

verifying pulse: ^

Fig. 2. From the light measurement (a) the atomic state can be reconstructed (b). There are two probability distributions indicated. The one without coloring is the best possible classical teleportation. It has been shown,5 that the best possible achievable classical fidelity F is 0.5. Compared to the best classical teleportation (uncolored graph) the teleported state (colored graph) is narrower.

2.3. Single atom squeezing In section 2.1 the two-cell setup was discussed briefly. From eqs. 4 and 5 the possibility of a mapping protocol of light onto atoms arises, where after the interaction one of the light quadratures is automatically written on the atoms. Then y of the light is measured and fed back to the atoms, so that: Pout = Pout + g • yout = -yin, if the gain is adjusted properly. This experiment has been conducted successfully.6 However, the fidelity of the mapping protocol is limited by the residual input noise from the other atomic quadrature (see eq. 4). This can be partly overcome by squeezing Xin. Here the multilevel structure of the Cesium atoms is utilized to reach this goal. By creating a suitable superposition of the even magnetic sublevels m = 4, 2,0,... a spin squeezed state7 can be achieved, at the expense of a decrease in the macroscopic spin Jx. As a result, one of the normalized transversal spin components x or p has a variance smaller than ^. Experimentally such a superposition state can be obtained by inducing Raman transitions with two light beams. The important features of the experiment are sketched in Fig.3(a). Figure 3(b) shows preliminary results for the experiment. The crosses are measured variances of the squeezed quadrature, obtained for 10000 measurements with equal pulse duration, power and detuning. For first tests a single cell setup was used and the Faraday interaction utilized for the read out of the two atomic quadratures. A noise reduction of (30 ± 10)% compared to the standard quantum limit is achieved. The dotted line shows the maximum achievable squeezing with

118

Fig. 3. In (a) the squeezing experiment is shown, where two beams, which are off resonantly coupled to the Dl line (A = —550 MHz), are sent through the atoms along the macroscopic atomic orientation to create a superposition of m = 4, 2,0,.... The two beams are detuned by twice the magnetic sublevels splitting, given by the Larmor frequency Ci = 322 kHz. The Faraday interaction is used to determine the noise reduction of the atomic quadratures. Figure (b) shows measurement results. The solid line indicates the noise level of a coherent spin state (CSS) with the equivalent macroscopic spin. The crosses and the circles show the measured squeezed and anti-squeezed variance, whereas the dotted line gives the calculated optimum squeezing.

the used interaction in the absence of decoherence predicting approximately 70% noise reduction. Different effects limit the performance of the experiment. The decay of the created state during the preparation and the read-out introduces constraints. Furthermore, the second order Zeeman and Stark shifts lead to the fact that the atomic spin can not be described by a Larmor precession with a single frequency. Experimentally, one needs to strike a balance between the limiting effects by choosing an optimal power and timing, as well as detuning, of the Raman beams to achieve optimal noise reduction.

3. Dispersive measurements on dipole trapped cold Cs atoms In a different experimental setup we investigate non-destructive probing of laser cooled Cs atoms confined by a focussed laser beam and restricted to populate the two ground level clock states (F — 3,mp = 0) and (F = 4, mp = 0) and superpositions thereof. As is well known such a two level system is formally equivalent to spin 1/2 system and we can form a collective pseudo-spin for the atomic ensemble in much the same way as introduced in section 2.1. The system is conveniently illustrated in the Bloch sphere picture Fig. 4.

119

fz ]F=4,mF=0>

Spin Squeezed State

1

!F=3,m,=0> Uncertainty disk prior to (2ND measurent

Fig. 4. Bloch sphere representation of collective pseudo- spin. Beginning from situation with all the atoms in one of the clock states (a pole on the sphere) and applying a Tf/2pulse produces an equal superposition of both clock states (the equatorial plane - here assumed to point in the a;-direction) with an uncertainty disk associated to it. Such a so-called coherent spin state will give rise to projection noise when measuring the z-axis spin projection. Performing a QND measurement produces a spin squeezed state.

3.1. Atom-light interaction A probe laser beam propagating through the trapped atoms will experience an index of refraction according the population of atoms in either clock state. This leads to a state dependent phase shift of light which we can measure by comparing to a reference beam in free space using a Mach Zehnder interferometer.8 Again, light in the two-path interferometer can be described using an angular momentum algebra in a similar way as for the polarization states in section 2.1. It turns out that the interaction between the collective angular momenta for light and atom leads to a quantum nondemolition (QND) measurement of an atomic pseudo spin projection when the phase shift of light is detected. The outcome of this measurement can in principle be used to infer spin squeezing, i.e. a reduction of uncertainty in one spin component (Sz) on the expense of an increased uncertainty in another component (Sy)9'w (See Fig. 4). 3.2. Rabi Oscillations As a first step towards the observation of projection noise and production of spin squeezed states on the clock transition, we have studied the coherent evolution of a spin polarized sample.11 The atoms are prepared in the (F = 3, mp = 0) state and Rabi oscillations on the clock transition are driven with a resonant 9.1 GHz microwave field. During Rabi flopping

120

the atoms are probed with short (200 ns) off resonant (150 MHz blue detuned from 6Sri/2(-F = 4) —> 6P3/2(F' = 5) transition.) laser light pulses. The probe laser beam will be shifted in phase proportionally to the number of atoms in the (F = 4, mF = 0) state. Figure 5 shows an example of non-destructive probing of Rabi oscillation using our Mach Zehnder interferometer. Using this method makes it possible to follow the coherent evolution of the ensemble quantum state in "real time". Having established that the atomic ensemble can be controlled coherently, we have the tools at hand to produce the coherent spin state Fig. 4. The challenge is now to detect the atomic quantum fluctuations in the recorded phase shift for independently prepared ensembles and that this variance can be reduced by using information gained in a previous QND measurement.

(b)

A A A A A A A A * A

=yvy\/vvv^vv '"%

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~4C8

' CM ~

flea

ioaa

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Fig. 5. Rabi Oscillations measured as a state dependent phase shift using our interferometer, (a) An average over 50 experimental runs, (b) Single experimental run.

4. Non-classical states of light 4.1. Gaussian states With the purpose of being a resource for the experiments presented in sections 2 and 3, we have a setup for generating various non-classical states of light. The heart of the experiment is an optical parametric oscillator (OPO) pumped below threshold. Employing a nonlinear PPKTP crystal (periodically poled potassium titanyl phosphate), the blue pump beam at frequency 2o>o is down-converted into several longitudinal cavity modes centered around the frequencies ujk = w0 +

k = . . . , -2, -1, 0, +1, +2, . . . ,

121

Fig. 6. Results of homodyne measurement and reconstruction of the two different nonGaussian states; Wigner function and density matrix in number state representation. Left: Photon subtracted squeezed vacuum (kitten state). Note the predominance of odd photon numbers in the density matrix diagonal. Right: Single photon state.

where WA is the free spectral range of the cavity. The field emitted from the OPO in the degenerate mode U>Q is in a squeezed vacuum state with one of the field quadratures being less noisy than the vacuum level. We are currently able to produce a very pure state with -6.5 dB squeezing versus 10 dB anti-squeezing. As mentioned in section 2.2, the squeezed vacuum can be used to improve the fidelity of the memory protocol. Apart from the u>o mode, the non-degenerate longitudinal modes are pairwise correlated such that e.g. cj_i and w+i are in a two-mode squeezed state. In the OPO the two modes are produced in the same spatial and polarization modes, but since they have different frequency they can be separated via a cavity resonant on w_ 12 4.2. Non-Gaussian states The single mode and two-mode squeezed states are indeed non-classical states, but they are still Gaussian. As demonstrated by Wenger et al,13 it is possible to de-gaussify the states by conditioning on detection of a photon in a part of the field. In the case of the squeezed vacuum, we reflect on a beam splitter a small fraction towards a single photon counter. When this photon counter clicks, we have effectively subtracted a photon from the remaining transmitted field, turning it into a superposition of odd-photon number states (the initial squeezed vacuum is ideally an even-photon number superposition), sometimes referred to as a 'Schrodinger kitten'.14 If we instead focus on the two-mode squeezed state, detection of a photon in the w-\ mode will de-gaussify the correlated u}+\ mode. If the pump intensity is sufficiently low, the generated state in w+i will be a single pho-

122

detector Fig. 7. Experimental configuration for the detection of backscattered light from a trapped BEC. The inset illustrates momentum detection after time of flight.

ton state.15 This scheme for a single photon source is similar to the standard down-conversion scheme of triggering on one photon from a down-converted pair, except that they are usually performed by single pass pulsed pumping. In that case the bandwidth of the state will be several GHz or nm wide. Due to the cavity enhancement by the OPO, we can reduce the bandwidth to the order of 10 Mhz while still keeping a high production rate of ~10,000 s"1. This property - which is shared by the kitten states - is very important for future perspectives, where storage of such non-Gaussian states in atomic memories is a possibility. The wavelength of the source is frequency tunable around the 852 nm Cs line. In order to characterize the non-Gaussian states, we measure them in a broadband homodyne detection setup. The conditional states appear in a temporal mode centered around the trigger time and with a shape roughly determined by a double-sided exponential decay with the decay constant of the OPO. Hence it is necessary to temporally filter the continuous homodyne detection signal to properly measure the non-Gaussian state and not the background squeezed state. After acquiring several thousand quadrature points at different phase angles we can reconstruct the density matrix and Wigner function of the state. The results, presented in Fig. 6, shows clearly non-Gaussian Wigner functions which even have deep negative regions around the origin. The purity of the states are between 60% and 70%. 5. Atom-Light interface with quantum degenerate atoms

The dispersive coupling between Cs atoms and light discussed in the previous sections can, of course, also be applied to different atomic species. The coupling strength K introduced in Sec.2 between the collective variables of atoms and light can be conveniently expressed as K2 = ao?7, the product of on-resonance optical depth ao and time integrated spontaneous emission rate r\. For a cold atomic sample optical depth is monotonous in phase

123

Fig. 8. Left: atomic momentum distribution measured after 45 ms of free expansion with the depleted original condensate to the right and the recoiling atoms to the left surrounded by an isotropic halo, populated by light scattering and by s-wave collisions during expansion; Right: Background corrected photodetector signal showing the time dependent reflectivity of the atomic sample;

space density, so that quantum degenerate bosonic samples (BEG) offer the ultimate coupling strength for a given amount of dissipation by spontaneous emission. Furthermore, the absence of Doppler broadening allows to resolve the excited state hyperfine structure in the optical excitation, which gives rise to a different effective interaction between atomic internal states and light polarization states with more complex input/output relations.16 For atomic samples far below the recoil temperature the momentum degree of freedom becomes accessible as an additional quantum system with a well denned initial state, where the coupling between light and atomic momentum states can be formulated in a language analogous to the polarization/angular momentum case. In first experiments we try to assess the achievable coupling strength by studying super-radiant Rayleigh scattering off EEC's.17 Experimentally we prepare EEC's of 87Rb atoms by standard rf evaporation techniques inside a magnetic trap of the loffe-Pritchard type.18 The cigar shaped clouds, containing ~ 5 • 105 atoms with no discernible thermal fraction, are probed along the long axis with IfjW pulses of circularly polarized light, focused to a waist of 20fj,m and detuned by —2.6GHz from the 5si/2(F = 1) —> 5pi/2(F' = 2) transition (see Fig.7). The elongated geometry of the sample favors repeated scattering and super-radiant gain in the direction of the long axis. We detect light scattered in the backward direction and measure the atomic momentum distribution by shadow imaging after time-of-flight expansion (see Fig.8) clearly demonstrating that the

124

super-radiant regime is reached. Ongoing experiments aim to verify the strict correlations between scattered photons a n d recoiling atoms implied by momentum conservation.

6. Acknowledgements

This research is funded by DG and through the EU projects COVAQUIAL,

QAP, a n d EMALI. References 1. J. F. Sherson, B. Julsgaard and E. S. Polzik, Deterministic atom-light quantum interface, in Adv. At. Mol. Opt. Phys. 54, eds. P. Berman, C. Lin and E. Arimondo (Elsevier, 2006) pp. 82-131. 2. K. Hammerer, E. S. Polzik and J . I. Cirac, Phys. Rev. A 72, 064301 (2005). 3. L. Vaidman, Phys. Rev. A 49, 1473 (1994). 4. J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac and E. S. Polzik, Nature 443, 557 (2006). 5. K. Hammerer, M. M. Wolf, E. S. Polzik and J. I. Cirac, Phys. Rev. Lett. 94, 150503 (2005). 6. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek and E. S. Polzik, Nature 432, 482 (2004). 7. M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993). 8. P. G. Petrov, D. Oblak, C. L. Garrido Alzar, N. Kjaergaard and E. S. Polzik, Phys. Rev. A 75,033803 (2007). 9. A. Kuzmich, N. P. Bigelow and L. Mandel, Europhys. Lett. 42, 481 (1998). 10. D. Oblak, P. G. Petrov, C. L. Garrido Alzar, W. Tittel, A. K. Vershovski, J. K. Mikkelsen, J. L. Sorensen and E. S. Polzik, Phys. Rev. A 71,043807 (2005). 11. P. Windpassinger et al., in prep. (2007). 12. C. Schori, J. L. S~rensenand E. S. Polzik, Phys. Rev. A 66, 033802 (2002). 13. J. Wenger, R. Tualle-Brouri and P. Grangier, Phys. Rev. Lett. 92, 153601 (2004). 14. J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mdmer and E. S. Polzik, Phys. Rev. Lett. 97, 083604 (2006). 15. J. S. Neergaard-Nielsen, B. M. Nielsen, H. Takahashi, A. I. Vistnes and E. S. Polzik, Opt. Express 15,7940 (2007). 16. 0. S. Mishina, D. V. Kupriyanov, J. H. Miiller and E. S. Polzik, Phys. Rev. A 75,042326 (2007). 17. S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard and W. Ketterle, Science 285,571 (1999). 18. T. Esslinger, I. Bloch and T. W. Hansch, Phys. Rev. A 58,R2664 (1998).

DEGENERATE FERMIGASES

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AN ATOMIC FERMI GAS NEAR A P-WAVE FESHBACH RESONANCE D . S. JIN, J. P. GAEBLER, AND J. T. STEWART JILA, Quantum Physics Division, NIST and Department of Physics, University of Colorado, Boulder, 440 UCB Boulder, CO 80309-0440, USA

Abstract: Atomic scattering resonances, called Feshbach resonances, have been used to create molecular Bose-Einstein condensates and Fermi superfluids. Past work has focused on s-wave, or non-rotating, pairs created from two fermionic atoms. Here we report on investigations of pair creation in an ultracold Fermi gas of 40Katoms near a p-wave Feshbach resonance.

1. A p-wave Feshbach Resonance 1.1. Introduction and Motivation Ultracold gases of atoms are powerful model systems for exploring many-body quantum phenomena. A unique feature of these systems is that the experimenter can actually control the interactions between the particles through the magic of a Feshbach resonance. By going to the strongly interacting regime in a Fermi gas of atoms, it is now possible to create a Fermi superfluid state [l]. This state results from the pairing of atoms; a pair of correlated fermions is itself a composite Bose particle, which can form a Bose condensate and thus give rise to superfluidity. This basic phenomenon can be seen in many other Fermi systems including superconductors, superfluid liquid 3He, and nuclear matter. The simplest type of pairing is s-wave pairing. In this case, the pairing is isotropic in space and does not involve orbital angular momentum. The Fermi superfluid state realized in ultracold gases, using either 40K atoms or 6Li atoms, is an s-wave superfluid. For ultracold gases, unlike for dense Fermi systems, we understand extremely well the microscopic origin of the interactions. Indeed, we control those interactions in order to form the Fermi superfluid state. If, as in the case of current experiments, the interactions are controlled using an swave Feshbach resonance: then the resulting pairs are obviously s-wave. Now, let us consider the possibility of using a non-s-wave Feshbach resonance to create non-s-wave pairs. We know that non-s-wave pairing occurs in high T, superconductors (d-wave) and in superfluid liquid 3He (p-wave). These condensed matter systems have some unique properties because of the non-s-wave pairing. Non-s-wave pairing is anisotropic and can give rise to an a

An s-wave Feshbach resonance, as discussed here, couples an s-wave scattering state of two atoms to an s-wave (non-rotating) molecule. The p-wave Feshbach resonance discussed in this paper couples a p-wave scattering state of two atoms to a p-wave (rotating) molecule.

127

128

anisotropic pairing gap. In addition, there are now different possible quantum numbers describing the pairs; these correspond to the different projections of the orbital angular momentum of the pair. For example, in the case of p-wave pairing, with one quanta of orbital angular momentum (L=l), the projection, mL, can be - 1 , 0, or 1. This allows for multiple superfluid states, and the opens the possibility of quantum phase transitions between distinct superfluid states. In addition to providing access to these intriguing features in a uniquely controllable model system, p-wave superfluidity in an ultracold Fermi gas has also been discussed as an interesting system for topological quantum computing.

1.2. Not One, But Two Resonances In ultracold Fermi gases, magnetic-field tunable p-wave Feshbach resonances have been observed for both 40Katoms and 6Li atoms. In 2003 our group reported the observation of a p-wave resonance between spin-polarized 40K atoms [3]. As a first step toward pursuing the possibility of using this resonance to create correlated fermion pairs and p-wave superfluidity, we discuss here experiments studying weakly bound p-wave molecules created using this Feshbach resonance. More details about these experiments can be found in Ref. [3].

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B (Gauss) Figure 1. Loss in atom number observed near a p-wave Feshbach resonance between atoms. spin-polarized 40K

A Feshbach resonance arises from the coupling of two-particle scattering states to a bound state. And, therefore, it turns out that one can use a variety of

experimental techniques near a magnetic-field tunable Feshbach resonance to very efficiently convert atoms painvise into weakly bound molecules. Perhaps the simplest technique is simply to set the magnetic-field strength to a value near the resonance position and wait. Figure 1 shows the loss in the number of atoms that we observe for such an experiment near the 40Kp-wave Feshbach resonance. A striking feature in this data is the splitting of the p-wave resonance into two distinct resonances. This was first observed in Ref. [2] and explained in Ref. [4]. The two resonances correspond to p-wave scattering with different projections (onto the direction of the magnetic field) of the pair orbital angular momentum mL. The lower field resonance corresponds to mL = +1 or mL= -1, while the higher field resonance corresponds to mL=O. These two resonances are separated by about 0.5 Gauss; this splitting is due to a small energy difference that comes from the magnetic dipole interaction between atoms. The loss in the observed number of atoms seen in Figure 1 is consistent with the conversion of atoms into weakly bound Feshbach molecules. Since we measure the atom number using resonant absorption imaging of the gas, and since molecules, in general, do not absorb the same color light as free atoms, we expect that the creation Feshbach molecules would appear as a loss of atoms in our measurment. More direct evidence of p-wave Feshbach molecule creation is presented later in section 2.2. 2. p-wave Molecules

2.1. Molecule Energy

Another technique that has been used to create s-wave Feshbach molecules is to set the magnetic-field strength to a value near the resonance position and then add a small-amplitude sinusoidal modulation to the magnetic field. This oscillating magnetic field can resonantly couple free atoms pairs to the Feshbach molecule state[5][6]. By varying the frequency of the magnetic-field modulation and looking for the resonant loss of atoms, we can determine the resonant frequency for a particular magnetic-field detuning from the Feshbach resonance. Then, repeating the measurement for a variety of magnetic-field values, we mapped out the energy difference between free atom pairs and Feshbach molecules. This is shown in Figure 2. Compared with similar plots of molecule energies for an s-wave resonance, Figure 2 illustrates several new features of the p-wave resonance.

130 The p-wave resonance is split into two distinct resonances, as discussed in the previous section. Here, we directly measure the energy splitting of the two p-wave Feshbach molecule states. 2. For magnetic fields above the Feshbach resonance, there exists a metastable state with a well-defined energy. Such a state is not seen for the s-wave resonances that have been used for Fermi superfluidity. 3. The Feshbach molecule has a binding energy that depends linearly on the magnetic-field. In contrast, for the s-wave resonances that have been used for Fermi superfluidity, the s-wave Feshbach molecule energy depends quadratically on the magnetic-field detuning from resonance.

1.

-1.0

-0.5

0

0.5

1.0

1.5

Figure 2. Energy of the Feshbach molecule relative to free atoms vs. magnetic-field detuning from the (lower) Feshbach resonance. This plot is taken from Ref. [3]. A negative energy corresponds to a true bound-state while a positive energy corresponds to a metastable “quasi-hound” resonance state.

2.2. “Seeing”p-wave Molecules The existence of the quasi-bound state for fields above the Feshbach resonance provides a way to more directly probe the p-wave Feshbach molecules. After creating molecules (by holding the magnetic-field for a few ms near the resonance) we can increase the magnetic-field to a value above the resonance. The molecule state then adiabatically becomes a quasi-bound state with an energy greater than that of free atoms. This quasi-bound state is metastable

131

because of the p-wave centrifugal barrier. The height of this centrifugal barrier, in units of frequency, is about 5.8 MHz. This is much larger than typical kinetic energies in the gas (the Fermi energy is typically about 10 kHz) and also much larger than the quasi-bound state energies explored here (see Figure 2). However, the quasi-bound state will eventually dissociate into free atoms by quantum mechanical tunneling through the centrifugal barrier. The resulting free atoms will have a relative kinetic energy that is defined by the quasi-bound state energy.

Figure 3. Images of dissociated p-wave Feshbach molecules. A linear grayscale indicates the optical depth, with white corresponding to more absorption of the resonant probe light. The images are taken after ballistic expansion from the trap and therefore show the velocity distribution of the atoms resulting from dissociation of the Feshbach molecules. The left image corresponds to the mL = ±1 resonance and the right image corresponds to the mL = 0 resonance. The image plane is transverse to the quantization axis, which is defined by the external magnetic field. The images show the expected angular distributions for p-wave pairs.

Figure 3 shows images of p-wave molecules taken using this technique. After the Feshbach molecules are created, we remove all remaining free atoms using resonant laser light. Since the molecules do not absorb this light, they remain unperturbed. We then increase the magnetic-field strength to a value above the resonance and allow the now-quasi-bound molecules to dissociate. The resulting free atom clouds are shown in Figure 3. The left and right images are taken using the mL = ±1 resonance and the mL = 0 resonance, respectively. Both images are taken after ramping the magnetic-field to the same positive detuning from the relevant resonance and for the same expansion time after

132 turning the trap off. The difference in the clouds’ sizes and shapes then reflects the different angular distribution of atoms in the p-wave pairs. 2.3. Creating p-wave Molecules

As mentioned in section 1.2. it is possible to produce p-wave molecules simply by setting the magnetic field to a value near the resonance and waiting. With our method to see the molecules we could dynamically probe this process. It should be noted that we do not currently understand this molecule creation process. Moreover, the data presented in this section represent our first investigations, for the p-wave resonance, of the region where many-body effects can be expected to play an important role in determining the behavior of the gas.

t

Time (ms) Figure 4. (a) Measured atom number as a function of time that the magnetic field is held at the mL = i 1 resonance. (b) Measured molecule number for the same hold time on resonance. The plot is taken from Ref. [3]. The inset shows the timing sequence for this experiment. The number of molecules is measured using the dissociation technique described in the text (solid line.) The number of atoms not in molecules is measured by ramping the field below the resonance (dashed line).

133 Figure 4 shows the atom and molecule populations as a function of time. In this experiment, we quickly changed the magnetic field from a value far from the resonance to a value near the resonance and held for a variable amount of time. The data was taken for the magnetic-field value where we observe the highest conversion efficiency to molecules. It can be seen that the molecule population quickly reaches its maximum value in about 1 ms, and then slowly decays on a timescale of order 10 ms. The atomic population monotonically decays, indicating the presence of an inelastic decay process.

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Figure 5. Number of molecules created for a 1 ms hold vs. magnetic-field detuning from the resonance. The data suggest that molecule creation occurs only when the p-wave resonant state has a positive energy that is less than the maximum collision energy between atoms in the Fermi gas.

If we set the hold time at the near resonant magnetic field to be 1 ms and then vary the value of the field, we observe that molecule creation occurs only over a small range of magnetic-field values near the resonance; this can be seen in Figure 5 . The molecule creation feature is observed to be asymmetric with a width that scales with the Fermi energy of the gas. This suggests that the measured width of the resonance feature reflects the atomic kinetic energy distribution rather than an intrinsic energy width of the resonance. In other words, the energy width of the resonance is narrower than the distribution of kinetic energies in the gas. The s-wave resonances that have been used to

134

realize super fluidity in Fermi gases have been in the opposite, broad resonance limit where the resonance simultaneously affects all collision energies in the gas.

2.4. Molecule Lifetimes Using the technique to see the p-wave molecules we could measure their lifetimes. Note that our imaging technique is state selective and we only detect the p-wave Feshbach molecules (and not other molecule states or free atoms). We could also measure the quasi-bound molecule lifetimes. The result of these measurements is shown in Figure 6.

-200

-100

0

I0

Energy (kHz) Figure 6 Lifetimes of the p-wave molecules as a function of their energy This plot is taken from Ref [3] Negative energies (left) correspond to bound molecules, while positive energies (nght) correspond to quasi-bound pairs Data for the mL = 0 resonance are shown in open symbols, while mL= *1 are closed symbols The two dotted lines on the left indicate the averages of the measured bound state lifetimes The solid line on the nght is a theory curve for the quasi-bound lifetimes

For the bound molecules (negative energy relative to the free atoms), we find that the lifetime does not depend strongly on the binding energy (and therefore does not depend strongly on the magnetic-field detuning from

135 resonance). This is very different from the case of the s-wave resonances that have been used for Fermi superfluidity. For those s-wave molecules, the lifetime has been seen to change by orders of magnitude over a similar range of binding energies. The dotted lines in Figure 6 indicate the average measured lifetimes for bound molecules created at each resonance. On the right side of Figure 6 we show the measured lifetimes for the quasibound p-wave molecules. Here, we see a very strong dependence on the pair energy. The solid line in Figure 6 shows a zero-free-parameter theory prediction by John Bohn [ 3 ] for the lifetime due to quantum mechanical tunneling through the p-wave centrifugal barrier. The expected (and measured) dependence of the lifetime on energy is a power law with a power of -312. The good agreement between the experiment and the theory show that the dominant decay mechanism for the quasi-bound p-wave pairs is dissociation into free atoms by tunneling through the centrifugal barrier. It should be noted that this is not an inelastic process and so does not cause any heating of the gas. 2.5. Future Prospects

One motivation for exploring a p-wave Feshbach resonance in an ultracold Fermi gas is the possibility of creating a p-wave superfluid state and exploring the many-body behavior of this new state. The measured lifetimes of the pwave Feshbach molecules tell us something about how difficult this will be to achieve. For bound p-wave molecules, our measured lifetimes of 1 or 2 ms are short compared to thermalization times for the trapped gas. This short lifetime is a serious problem for future prospects for creating Bose-Einstein condensates of these p-wave molecules. There are two different decay mechanisms for our pwave molecules. One is collisional vibrational quenching, where a molecule collides with a free atom or with another molecule. This inelastic collision produces a more tightly bound, and therefore lower energy, molecule state and releases a relatively large amount of energy. The second decay mechanism is not due to collisions, but rather can occur for single molecules in isolation. This decay is due to dipolar spin relaxation, where the molecule decays into a pair of free atoms whose hyperfine spin states have lower energy than the original internal states of the two atoms paired in the molecule. This process only exists when the original atom states are not the lowest energy states in a magnetic field. This is the case for our 40K p-wave resonance. It is possible to consider creating p-wave molecules using a different pwave resonance where dipolar spin relaxation of the molecules would not be

136

time (ms) Figure 7. P-wave Feshbach molecule lifetime after removal of free atoms. The molecules were created at the mL=Oresonance. The solid line is a fit to an exponential decay, which gives a lifetime of 7 1 ms.

*

possible. While such a resonance does not exist for 40K atoms, it does for fermionic 6Li atoms. However, collisional vibrational quenching of the molecules would remain as a possible inelastic decay channel. To see if collisional decay of the p-wave molecules was important we measured the molecule lifetime after removing all remaining free atoms. (The conversion efficiency from atoms to molecules was about 20%, so the remaining free atoms would be the most likely collision partner for the molecules.) Free atoms were removed using a pulse of resonant light to heat them out of the trap. The molecule lifetime was then measured as described previously. The results of this measurement are shown in Figure 7. The measured molecule lifetime without atoms present was 7 + 1 ms. This is significantly longer than the previously measured molecule lifetime, but still relatively short. This measured lifetime without atoms present is consistent with a prediction by John Bohn for the molecule lifetime due to dipolar relaxation [3]. This result suggests that atom-molecule collisions do play an important role in limiting the lifetime of the p-wave Feshbach molecules. For future work it will be important to understand if it is possible to create non-s-wave molecules with better stability against collisional decay. Arguably, the most interesting many-body physics will occur on the quasibound molecule side of the resonance. Here, there lifetimes shown on the right side of Figure 6 are even shorter, but this is not necessarily a bad thing. The

137 measured lifetime of the quasi-bound molecules comes from dissociation into free atoms by tunneling through the centrifugal barrier. This is an elastic process and, in fact, this tunneling is the process that can establish the pairing required for Fermi superfluidity. Very close to the resonance, where one would like to explore the many-body physics of the Fermi gas, inelastic collisional decay may become important. Exploring this behavior very close to resonance will be important in assessing the possibility for achieving p-wave Fermi superfluidity.

2.6. Conclusion Using a Fermi gas of 40K atoms, we have been able to create and detect pwave Feshbach molecules. We have explored several novel aspects of this resonance, such as the existence of a quasi-bound state that dissociates by tunneling through the p-wave centrifugal barrier. Measurements of the molecules lifetime suggest that inelastic collisional decay presents a serious challenge for future experiments attempting to investigate equilibrium manybody physics with this system.

Acknowledgments

This work was supported by the National Science Foundation and by NASA. References

1. C. A. Regal and D. S. Jin, Adv. Atom. Mol. Opt. Phys. 54, 1 (2007). 2. C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys. Rev. Lett. 90, 053201 (2003). 3. J. P. Gaebler, J. T. Stewart, J. L. Bohn, and D. S. Jin, Phys. Rev. Lett. 98, 200403 (2007). 4. C. Ticknor, C. A. Regal, D. S. Jin, and J. L. Bohn, Phys. Rev. A 69,042712 (2004). 5. M. Greiner, C. A. Regal, and D. S. Jin, Phys. Rev. Lett. 94,070403 (2005). 6. S. T. Thompson, E. Hodby, and C. E. Wieman, Phys. Rev. Lett. 95, 190404 (2005).

BRAGG SCATTERING OF CORRELATED ATOMS FROM A DEGENERATE FERMI GAS R. J. BALLAGH, K. J. CHALLIS, and C. W. GARDINER Jack Dodd centre for Photonics and Ultra-Cold atoms Physics Department, University of Otago, Dunedin, New Zealand *E-mail: [email protected] www.physics. otago. ac. nz/research/jackdodd We formulate a treatment of Bragg scattering of a Fermi gas in a BCS state, based on the time-dependent Bogoliubov de Gennes equations. We solve these equations in three dimensions to present a quantitative analysis of the scattering. We find that, in addition to the expected scattering of atoms by the Bragg momentum transfer tzq, a comparable fraction of atoms is scattered into a spherical shell in momentum space, centered at &/2. The atoms in the scattered shell are pair correlated, and we present an analytic model that provides an interpretation of the correlated scattering mechanism, and explains the key parameter dependencies.

1. Introduction Superfluidity in fermion systems arises due to momentum correlations between pairs of particles (the phenomenon of Cooper pairing). Techniques for probing these pair correlations in ultra-cold atomic gases have concentrated primarily on observations of the energy gap associated with the pairing (e.g., Ref. 1).Bragg scattering has been suggested in a number of theoretical studies (e.g see Refs. 2-8) as a means for characterising Fermi gases. In this paper we show that Bragg scattering can be used t o coherently probe and manipulate a degenerate Fermi gas, and that it reveals a unique signature of the pairing correlation. Bragg scattering, with a light potential of the form Acos(q . r - w t ) , has proven to be a versatile tool for manipulating and characterising BoseEinstein condensates. A long Bragg pulse is used in the so-called momentum spectroscopyg , where a narrow momentum range of the condensate is selected and incremented in momentum by an amount hq. Short Bragg pulses 138

139 on the other hand can coherently divide the condensate spatial wavefunction, giving one of the packets a momentum kick of fiq. This has enabled the implementation of atomic beam-splitters and mirrors, and the exploitation of the large scale spatial coherence of condensates in atom interferometry" . The correlations in a Fermi gas are of a different nature: they are due to attractive collisional interactions, and reside primarily on the Fermi surface. Observations of the associated energy gap using single photon processes destroy the correlation. However, Bragg scattering, which is a coherent two-photon process, has the potential to manipulate the correlation into a different form which, as we show in this paper, may be directly observed. The regime we will consider is for weak collisional interactions, for which the initial equilibrium state and its pair correlations have been successfully described by the BCS model" . This provides an appropriate starting point for our treatment, but we find the Bragg scattering is sensitive to the value of the collisional interaction at the Fermi surface, and a better treatment of the collisional interaction is required than for the conventional BCS theory, as we outline in section 2. The dynamical behaviour of the atom field under the influence the Bragg potential is calculated using the formalism of time-dependent Bogoliubov de Gennes equations. We sketch the derivation of these equations in section 2.2, and present numerical solutions for the case of a homogeneous three dimensional gas. As expected, the Bragg potential induces scattering of some of the atoms by momentum fiq. However, the key result of our calculations is that a new phenomenon of Bragg scattering atoms occurs, giving rise to a scattered spherical shell of atoms centered at momentum fiq/2. There is a threshold Bragg frequency for this shell t o form, and its radius increases with w . The atoms in the shell are correlated spin-up spin-down pairs, scattered from Cooper pairs on the Fermi surface. In the final section of the paper, we develop an analytic model that describes the underlying mechanisms for this new signature for Cooper pairing in a degenerate Fermi gas. We show in detail how it results from laser mediated transfer of the initial correlation to a different region of momentum space.

2. Time-dependent Bogoliubov de Gennes equations 2.1. Treatment of the BCS state

Our treatment is based on a BCS-like approach for the description of a degenerate Fermi gas, but with an improved treatment of the interatomic

140

collisions. In the conventional BCS approach collisions are described by a contact interaction potential, which has an infinite momentum-space range, and leads to the divergence of the pairing mean-field potential (a central object of the theory). The pair potential is typically renormalised to a finite value by introducing a momentum-space cutoff. The BCS ground state is cutoff independent, but in the limit that the cutoff tends to infinity, the renormalisation process requires that the collisional interaction strength becomes arbitrarily small. The renormalisation process described above is not sufficient for the case of Bragg scattering. We find that the correlated-pair scattering depends quantitatively on the initial pair correlations at the Fermi surface (see section 4.2), and therefore on the value of the collisional interaction potential at that momentum. We use a more sophisticated treatment of the interatomic collisions, in which the momentum-space range of the collisional interaction potential appears explicitly and is determined from experimental measurements. We assume a homogeneous Fermi gas with equal numbers of spin ‘up’ and spin ‘down’ atoms (denoted here as and -, respectively). In the absence of the Bragg field, the many-body Hamiltonian has the form

+

4cy

where is the field operator for spin state a , ‘Ho = ( - f i 2 / 2 M ) V2, M is the atomic mass, and p is the chemical potential. The interaction Hamiltonian f i c o l l n is the most general translationally invariant two-body collisional Hamiltonian for s-wave interactions between fermions of opposite spins12 ;

x

4-(Y (R

-

Here R is the centre-of-mass coordinate, r and r‘ are the relative coordinates of the two particles, and V ( r , r’) is a non-local collisional interaction potential. Our approach is to approximate V ( r , r’) by the separable potentia1l3 ,

V(r, r’) = g F ( r ) F * ( r ’ ) , (3) where F(r) has a finite range 0,even parity, and is normalised to 1. For a particular form of F(r),the interaction V(r,r’) is characterised by the interaction strength g and the range 0.

141

For a given range CT,the interaction strength g can be determined by solving the Lippman Schwinger equation for the separable potential. At low energy this gives

where T ~ B = 47rh2a/M is the two-body T-matrix, with a the s-wave scattering length. The parameter y is given by -

if(k)i2d3k

M ( 2 4 3 ~

k2

'

(5)

and depends on the range c7 through the function f ( k )= F(r)e-"k"d3r. Szymariska et al. assume a Gaussian form for the function F(r),and then determine g and c7 for a particular pair of colliding atoms, far from a Feshbach resonance. For computational convenience, we approximate the collisional interaction potential using the step function f ( k ) = Q ( k c - lkl), and then choose the wave-vector cutoff k , to match the parameters g and y with those used by Szymariska et al. By this method, we find that for 40K atoms prepared in the ( F = 9 / 2 , m ~= -9/2) and ( F = 9 / 2 , m ~= -7/2) Zeeman states, the wave-vector cutoff is k , = 0.0154/a~,h,. Therefore, for a typical Fermi momentum of k~ 0.5 x 10-3/ag0h,14 , we find that k, 30k~. In the spirit of BCS, we introduce mean-field potentials and approximate kcolln by the sum of the dominant single-particle interaction terms, which are of the form (GiGa)G',&a and ( ~ ~ ~ ~ a Assuming ) ~ a ~ that p a the BCS correlation length greatly exceeds the range of the collisional interaction potential, we obtain the Heisenberg equations of motion

-

-

The quantities W and X contain the usual Hartree and pair potentials, but with 'smearing' functions that sample the field operators over the collsional range, i.e.,

X ( r , r ' , t ) = A ( r , t ) F * ( r- r'),

(7)

.

142

with the Hartree and pair potentials given by

The prefactors in these mean-field definitions (i.e., T ~ B and g, respectively) are chosen here in order to agree with the standard renormalised BCS theory. 2.2. The Bragg formalism

We assume that the Bragg field does not change the particle spin, and describe the effect of the Bragg field on the Fermi gas by adding V B = ~ ~ ~ Acos(q.r-wt) to 7-10. The new single particle Hamiltonian 7-1 = 7 - 1 o + V ~ ~ ~ ~ ~ is used in the dynamic field equations (6). To solve those equations we introduce a time-dependent form of the well-known Bogoliubov transformation,l1>l5i.e.,

In equation (8), the 9 k e are t i m e - i n d e p e n d e n t quasi-particle annihilation operators defined to obey the standard fermion commutation relations. The dynamic evolution of the gas is described by the evolution of the amplitudes U k ( r , t ) and Z)k(r,t ) . The quasi-particle modes are populated according to the equilibrium mean-value rules (?ka?k,p) t = bkk’bapfik and (?ka?k,p) = 0. The Fermi function is f i k = l/[exp(Ek/kBT) 11, with T the temperature and the quasi-particle energies Ek of the gas in the ground state being measured relative to the chemical potential p. The equations of motion for the amplitudes U k ( r , t ) and ?&(I-, t ) are derived from equation (6) giving

+

+

1

W(r, r’, t ) U k ( r ’ ,t)d3r’

+

J

X(r,r’, t ) U k ( r ’ , t)d3r’,

and

-

J W(r, r’,

t)’Uk(r’,t)d3r’

+

J

X*(r,r’, t ) U k ( r ‘ ,t)d3r’,

in which W and X act as projectors in momentum space, with ranges 2k, and k , respectively. We solve the coupled sets of equations (9) and (10) in three dimensions, for a large set of modes k,within the regime k ~ l u l< 1, for which the BCS theory is valid.

143 The initial state is a standard BCS ground state, found from the time-independent form of equations (9) and (10) with the Bragg field off. Figure l(a) shows the momentum space column density /n(k, t)dkz, where the number density of the gas at momentum Kk is n(k, t) = C(^(k, £) a (k, t ) } . The momentum space field operator is defined as (/>a(k, t) = /^ Q (r,i)e~ l k ' r d 3 r/L 3 / 2 , where L3 is the computational volume, and the normalisation constant C is chosen such that J n(k, t)d?k = 1. The unit of energy is chosen to be the Fermi energy EF = hujp, and the unit of wavevector is the Fermi wavevector defined in f f k p / Z M = Ep. In this paper we consider only zero temperature, in which case the Fermi energy is related to the chemical potential via EF = /z + (1 — gjT^B^U . The momentum radius of the cloud is approximately hk'F, where the effective Fermi wave vector k'F is defined in terms of the effective chemical potential by fj,' = EF - U = H2k'F2/2M (e.g., Ref. 16). Column Density

|
(a)

(b)

O -4

-4

kx

kx

Fig. 1. Column density, and pair correlation function |<7o(k)| in the kz = 0 plane, of a three-dimensional homogeneous Fermi gas at zero temperature. Parameters are fee = 30.0/CF and fcFo = -0.427 [i.e., [/(O) = -0.256.EF and A(0) = 0.0491EF]. In general, we define the correlation function between a pair of atoms with momenta fi(k' + k) and ?i(k' — k) as

k)},

(11)

Figure l(b) shows the pair correlation function <7o(k) for the initial state i.e., the initial pair correlation between atoms with momenta fik and — ftk, in the kz = 0 plane. It takes the maximum possible value of 1/2 at |k| « k'F. 3.

Results of Bragg scattering

The result of applying a Bragg pulse of length tp to an initial BCS state is shown in Fig. 2, where q is in the x direction, and for clarity we have

144

chosen the case of scattering well outside the Fermi sea, i.e., q > 2kp. Column Density u> — 9.2wF

(b)

(a)

^jjjjp'

^SP

uj = 15.6u>F

J

(c)

0* 8

-

4

0

MM

4

-

4

(d)

f 0

4

MM

Fig. 2. Column density, and pair correlation function |G q / 2 (k)| in the kz = 0 plane, of a Bragg scattered three-dimensional homogeneous Fermi gas at zero temperature. The plotted quantities are saturated on the chosen scales in order to clearly observe the features. Parameters are A = l.SOEf, q = 4.80fcF, t = 8.22/ojF, kc = 30.0fcF, kFa = -0.427, and (a), (b) u; = 9.2wF, and (c), (d) ui = 15.6wF.

In the momentum space column density plots (left-hand column), the initial state is a spherical atom cloud centered at k = 0. A fraction of the initial population has been scattered by momentum /iq, by the usual singleparticle Bragg scattering (e.g., Ref 17). This is understood in terms of the well known resonance condition

(12) which equates the net (photon) energy HUJ supplied by the Bragg field to the change in kinetic energy of a particle scattered from momentum fik' to H(k' + q). For a given u> and q, the particular momentum components in the initial distribution that are resonant with the Bragg field have a momentum ftk', where k' satisfies equation (12). The momentum-space width of the transition is S w ^/(
145

Of more interest is the correlated-pair scattering," which gives rise to a spherical shell centered at momentum hq/2 in Figs. 2(a) and (c). The correlated-pair scattering has a marked dependence on the Bragg frequency w,appearing as a tight cloud a t a threshold value Wthres [see Fig. 2(a) and equation (4.1) below] but with a radius that increases as w increases, as seen in Fig. 2(c). In the right-hand column of Fig. 2 we plot the correlation function Gq12(k),which shows that the atoms in the spherical shell are correlated as spin-up spin-down pairs, with momenta h(q/2+k) and h(q/2k). The maximum value of lGq,2(k)l in Fig. 2(d) is 0.394. We have investigated the dependence of the correlated-pair scattering on a range of system parameters. For the example shown in Fig. 2 , approximately 0.2% of the atoms are scattered into the pair-correlated spherical shell. The number of scattered pairs grows linearly with the length of the Bragg pulse (until t 7 0 / w ~ )The . number of pairs scattered increases quadratically with the initial pair function A (0) , as shown in Fig. 3. Approximately 6% of the initial population is scattered into the pair-correlated shell for A(0) = 0.293 ( k ~ = a -0.689).

-

AIEFl

Fig. 3. Number of scattered pairs vs the initial pairing field A(0) (on a log-log scale). Parameters are A = 1 . 8 0 E ~q, = 4 . 8 0 k ~ ,t = 8 . 2 2 / w ~ ,Ic, = 3 0 . 0 k ~ and , w =1 0 . 8 ~ ~ .

The number of scattered pairs also increases as the Bragg amplitude A increases, or as the Bragg wavevector q decreases. However, if lql becomes too small the correlated-pair scattering is supressed because the scattered pairs fall within the Fermi sea. At finite temperatures, the correlated scattering decreases roughly in proportion to the decrease in the initial pair function, i.e., as A2 ( 0 ) .

146

4. Mechanism for Correlated-Pair Scattering

The fundamental mechanism for correlated-pair scattering is the formation by the Bragg fields of a moving grating in the pair function, which we can describe as A(r, t ) M A, ( t ) A, ( t )exp [i (q . r - w t ) ] . The correlated pairs then scatter from this grating to produce the spherical shell illustrated above.

+

4.1. Geometry of correlated scattering An explanation can be given for the geometric behaviour of the correlatedpair scattering, noting that in a correlated Bragg scattering event the photons provide net energy hu and net momentum hq to the initial pair of correlated atoms. Before the event, the individual momenta are (hk’, -hk’) and total momentum 0. After scattering, the pair has total momentum hq, distributed amongst the individual atoms as (ri (q/2 krel) h ( q / 2 - krel)). Energy conservation requires the photon energy to equal the change in energy of the scattered atoms, i.e.,

+

Equation (13) describes a sphere of momentum radius krel centred on q/2 and we have used the fact that initially, the most strongly pair correlated atoms reside on the Fermi surface (i.e.] lk’l = k ~ )At . the correlated scattering threshold frequency, the radius of the scattered sphere is krel = 0, giving

4.2. Analytic Model A more detailed understanding of the correlated scattering can be obtained by noting that due t o the periodic nature of the Bragg potential, the solutions to equation (9) and (10) must have Bloch form, and can be expressed

147

as

*hZ

dt

=t..;(k)a;(t)

1 + ;A

[&-l(t)

+ C Urn(t)If(k/2 + nq/2

+~k+,(t)] -

mq/2)12alc-m(t)

m

+ C~m(t)f*(k+nq-mq)blc-m(t),

(18)

m

where b z ( k ) = Ek+nq-p-nb, and E k = h 2 k 2 / 2 M .A similar equation is found for dbk/dt by substituting Eqs. (14) - (17) into Eq.(lO). By analysing resonance conditions and coupling terms, we can show that for each mode k, the dominant coefficients are uk, u f , bk,, bk, as we have also confirmed from numerical simulations of the full time-dependent Bogoliubov de Gennes equations. In terms of the dominante coefficients, the pair potential grating amplitude is

In the initial BCS state, the amplitude A, is zero because the only nonzero coefficients are uk (for Ikl 2 kf.) and bk (for Ikl 5 kf.). The process of correlated scattering begins by first creating A, by single-particle scattering of atoms on the Fermi surface, with Ikl M kf.. This process can be approximately described by a truncated version of equation (B), namely

which converts u i to uy , creating a non-zero pair-potential grating amplitude, namely A, M -9 CFermi Surface uy(t)b:* ( t ) .We emphasize that this is a coherent process which transforms the initial correlated pair (hk, -hk) to

148

a new correlated pair (hk+ q, -tik). This process forms part of the correlation Gq,2(k),i.e. that which is represented by the outer circles in Fig. 2(c) and (d), whose radii are independent of the Bragg frequency w . Once A, is formed, it provides a coupling from uk to b k l , which is resonant for the region of k in the correlated shell (i.e., at q / 2 f krel). The population in the correlated shell is given by lbk112. We can follow the evolution of b k l ( t )approximately by truncating equation (18), and the corresponding equation for d b k l d t , to yield the set of equations

ih& dt

[

] [ AT@) =

Ek

-

p’

bkl(t)

I):;$[

+ + h]

-Ek--q A1 ( I-1’ t)

3

(21)

which can be understood in terms of the well known Rabi problem. The coupling frequency is A, [which is seeded as discussed above, but must be calculated self consistently according to equation (19)],but b k , is generated only if (Ek

-

P’)

-

(-Ek-q

+ p’ + h) 0. M

(22)

Equation (22) can be rewritten as

which is equivalent to equation (13). Alternatively it can be written w Wthres = li2k:el/M, which shows that above the threshold frequency, any additional energy provided by the Bragg field (i.e., h) is distributed equally between the two atoms of the scattered pair as kinetic energy. From equations (19) and ( 2 l ) , we find that the total amount of correlated scattering g 2 . Therefore, the parameter g must be accurately known in order to quantify the correlated scattering, which was a motivation for the detailed collisional formalism of section 2.1.

-

References 1. C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Hecker Denschlag and R. Grimm, Science 305,p. 1128 (2004). 2. J. Ruostekoski, Phys. Rev.A 61, p. 033605 (2000). 3. A. Minguzzi, G. Ferrari and Y. Castin, Eur. Phys. J . D 17, p. 49 (2001). 4. M. Rodriguez and P. Torma, Phys. Rev. A 66, p. 033601 (2002). 5 . H. P. Buchler, P. Zoller and W. Zwerger, Phys. Rev. Lett. 93, p. 080401 (2004). 6. Bogdan Mihaila, Sergio Gaudio, Krastan B. Blagoev, Alexander V. Balatsky, Peter B. Littlewood and Darryl L. Smith, Phys. Rev. Lett. 95, p. 090402 (2005).

149 7. R. Combescot, S. Giorgini and S. Stringari, Europhys. Lett. 75 (2006). 8. Bimalendu Deb, J . Phys. B 39 (2006). 9. J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, D. E. Pritchard and W. Ketterle, Phys. Rev. Lett. 82, p. 4569 (1999). 10. M. Kozuma, L. Deng, E. W. Hagley, J. Wen, R. Lutwak, K. Helmerson, S. L. Rolston and W D Phillips, Phys. Rev. Lett. 82, p. 871 (1999). 11. P. G. de Gennes, Superconductivity of metals and alloys (W. A. Benjamin, Inc., New York, 1966). 12. G. Bruun, Y. Castin, R. Dum and K. Burnett, Eur. Phys. J . D 7, p. 433 (1999). 13. M. H. Szymanska, K. G6ra1, T Kohler and K Burnett, Phys. Rev. A 72, p. 013610 (2005). 14. C. A. Regal, M. Greiner, S. Giorgini, M. Holland and D. S. Jin, Phys. Rev. Lett. 95, p. 250404 (2005). 15. J. B. Ketterson and S. N . Song, Superconductivity (Cambridge University Press, Cambridge, 1999). 16. N. Nygaard, G. M. Bruun, C. W. Clark and D. L. Feder, Phys. Rev. Lett. 90, p. 210402 (2003). 17. P. B. Blakie and R. J. Ballagh, J . Phys. B 33, p. 3961 (2000). 18. K. J. Challis, R. J . Ballagh and C. W. Gardiner, Phys. Rev. Lett. 98, p. 0930002 (2007).

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SPECTROSCOPY AND CONTROL OF ATOMS AND MOLECULES

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STARK AND ZEEMAN DECELERATION OF NEUTFCAL ATOMS AND MOLECULES

s. D. HOGAN, E. VLIEGEN, D. SPRECHER, N. VANHAECKE*, M. ANDRIST, H. SCHMUTZ, U. MEIER, B. H. MEIER AND F. MERKT Laboratorium fur Physikalische Chemie, ETH Zurich CH 8006 Zurich, Switzerland Argon and hydrogen atoms excited to Rydberg Stark states in supersonic expansions have been decelerated using inhomogeneous electric fields. In the case of hydrogen, the atoms have been decelerated from an initial velocity of 700 m/s to zero velocity in the lab frame using time-dependent inhomogeneous electric fields and subsequently stored in two- and three-dimensional traps. The dynamics of the Rydberg atoms in the traps and the phasespace characteristics of the decelerated atoms have been characterized by pulsed field ionization and imaging techniques. Multi-stage Zeeman deceleration of ground state H and D atoms has been demonstrated. Using this technique H atoms, traveling at 420 m/s, have been decelerated to half of their initial kinetic energy.

1. Introduction The development of general methods with which to control the translational motion of atoms and molecules in the gas phase is of interest in high-resolution spectroscopy and in studies of reactive collisions at very low collision energies. Laser cooling which has been very successful in stimulating research on (u1tra)cold atoms is not suited to generate cold samples of molecules in the gas phase. Over the past years, many promising techniques have been introduced that enable the production of such samples: photoassociation of cold atoms [ 11, Stark deceleration of polar molecules [ 2 ] , buffer gas cooling [3], low velocity filtering from effusive beams of polar molecules [4], billiard collisions [5]. Several methods rely on the use of cold samples formed in supersonic expansions and aim at stopping them in the laboratory frame. The methods employed so far include multi-stage Stark deceleration of polar molecules [6], optical Stark deceleration [7], and Rydberg Stark deceleration [8]. Recently, the

*

Permanent address: Laboratoire Aime Cotton du CNRS, UniversitC de Paris-Sud, 91405 Orsay, France

153

154 Zeeman equivalent of the multi-stage Stark deceleration method has also been realized experimentally [9]. We summarize here recent progress in Rydberg Stark deceleration and in multi-stage Zeeman deceleration of free radicals with emphasis on the work carried out in Zurich on atomic samples of argon and hydrogen. The methods of Rydberg Stark deceleration and of Zeeman deceleration can also be applied to molecular samples: Because all atoms and molecules possess Rydberg states, Rydberg Stark deceleration can in principle be applied to any particle in the gas phase. Zeeman deceleration is restricted to atoms and molecules subject to an electron Zeeman effect at low magnetic fields. Fig. 1 illustrates schematically the Stark effect in Rydberg states of atomic hydrogen (panel a), and of a nonhydrogenic atom or a molecule (panel b), and the Zeeman effect in the ground state of atomic hydrogen (panel c). Quantum systems undergoing a linear Stark or Zeeman effect can be thought of as having an electric belec) or magnetic (,urn,,) dipole moment which directly couples to the applied external fields according to equation (1).

A quantum state the energy of which is lowered (raised) in the presence of a field is often referred to as a "red-shifted" (blue-shifted) state, or a "high-field seeking" (low-field seeking) state. Particles undergoing a Stark or a Zeeman effect are subject to a force f in inhomogeneous fields which is proportional to the field gradient

In inhomogeneous fields, particles in low-(high-)field seeking states are accelerated in the direction of decreasing (increasing) field strength. If the force is directed perpendicularly to the beam propagation axis, it can be exploited to deflect an atomic beam or split it into different components, as in the wellknown experiments of Gerlach and Stem. If the acceleration vector points in the direction parallel or antiparallel to the beam propagation axis, the particles in the beam are accelerated or decelerated in the longitudinal dimension.

155

Figure 1. The Stark effect in Rydberg states of the H atom (a) and of nonhydrogenic atoms (Na) (b). (c) represents the Zeeman effect in the ground state of atomic hydrogen (adapted from Refs. [9,10 and 11]).

The energy levels of the H atom in an electric field of strength F are given to first approximation, in atomic units, by [10] E = E{- H2n2 + (3/2)nkF,

(3)

where n is the principal quantum number, E\~ 1/2 Hartree the ionization energy and A: is a quantum number which runs from - (n - I - \mf\) to + (n - 1 - \mt\) in steps of two. At zero electric field, the potential is Coulombic so that the energy eigenvalues do not depend on -6 or mt and have a degeneracy factor of n2. When an electric field is applied, the energy level structure fans out in linear manifolds of Stark states. States with k < 0 are red-shifted or high-field seeking, states with k > 0 are blue-shifted or low-field seeking, and states with k = 0 have, to a good approximation, a field-independent energy. Atoms and molecules excited to k = 0 Rydberg Stark states thus do not decelerate nor accelerate in an inhomogeneous electric field. If a H atom excited to a high-field seeking state in an electric field F flies out of this field, it decelerates, whereas an atom in a lowfield seeking state accelerates. The loss/gain in kinetic energy is equal to the Stark energy AE = (3/2)nkF.

(4)

The Stark effect in other atoms and molecules differs from that of the H atom in that (i) the penetrating low •£ states undergo a quadratic Stark shift at low fields and (ii) the crossings between Stark states at high field, which are exact in H, are avoided as a result of the non-Coulombic nature of the potential at short range (see Fig. 1). These qualitative differences between hydrogenic and nonhydrogenic systems must be considered when planning Rydberg Stark deceleration experiments and analyzing their results [12]. Nevertheless, the

linear manifolds of the nonpenetrating Stark states are sufficiently similar to those of H that Rydberg Stark deceleration is applicable to any particle in the gas phase. A property of Rydberg atoms and molecules particularly attractive for atom and molecule optics experiments is the fact that the dipole moment of the extreme components of the Stark manifolds scales as n2 and becomes very large at high n values (at n = 20 the dipole moment of a Ikl n state amounts to more than 1000 Debye). A qualitative understanding of the Zeeman deceleration of radicals can be gained from similar considerations taking into account the fact that the Zeeman shifts are proportional to the magnetic quantum number ms (or mJ) and the magnetic field strength. However, the Zeeman shifts of a radical are very small compared to the Stark shifts of a Rydberg atom or molecule, even if magnetic fields of more than 1 Tesla are used. For the Zeeman shift of a radical to be on the order of 1 ern-', a magnetic field in the range 1-2 Tesla is required. To achieve a similar Stark shift in the ground electronic state of a polar molecule, electric fields of 100 kV/cm or more must be applied. In contrast, a field of only 25 V/cm can induce Stark shifts of more than 1 cm-' in Rydberg states with n > 25. Given that the kinetic energy of a particle of mass number A4 moving at a velocity of 400 m/s in a supersonic beam of xenon amounts to

-

Ekin /

(hc cm-' )= 6.7 M

(5)

and that pulsed electric fields beyond 150 kV/cm and pulsed magnetic fields beyond 1-2 Tesla are difficult to generate experimentally, one can immediately conclude that while a single deceleration stage may be sufficient to completely decelerate a beam of Rydberg particles, the complete deceleration of a radical beam in a Zeeman decelerator will necessitate many (of the order of 10 to 100) successive deceleration stages, i.e., approximately as many as required for the complete Stark deceleration of a polar molecule in its ground electronic state. The Zeeman deceleration of radicals thus possesses many common features with the method of multi-stage Stark deceleration of polar molecules developed and successfully used to slow down and even trap polar molecules [2,6].

2. Stark deceleration and trapping of Rydberg atoms Breeden and Metcalf [13] and Wing [14] were the first to propose experiments in which inhomogeneous electric fields would be used to control

157 the translational motion of Rydberg atoms. Selected sentences from their abstracts "A method is proposed to decelerate neutral atoms in a thermal beam for the purposes of precision spectroscopy, atomic clocks, or loading into shallow traps. Rydberg atoms exhibit large dipole moments..." [13] and "Excited neutral atoms having positive Stark energies may be trapped near magnitude minima of electrostatic fields. At moderate field strengths Rydberg atoms have trap depths comparable to ambient kT [...]. Sustained trapping and cooling should be possible, allowing Doppler-free few-atom spectroscopy and novel collision studies" [14] make it clear that they had envisaged a wide range of applications. Their suggestions were first implemented in the laboratory of Tim Softley at Oxford where time-independent inhomogeneous fields were used to deflect beams of krypton Rydberg atoms [15] and both deflect and decelerate beams of hydrogen Rydberg molecules [8]. In these early proof-of-principle experiments, the change of kinetic energy did not exceed a few percent.

(a)

Photolysis La«*r (193nm}

JIL

Etectrottes

'"/'

Phosphor Screen

ceo Camera

NHjArjlrtO) 4

vuvauv /

mcp Quarts Capfflary

(c)

Figure 2. (a) Schematic representation of a Rydberg Stark decelerator. The capillary and photolysis laser are specific to experiments on radicals. The electrode configurations with equifield lines used to decelerate and trap Rydberg atoms are shown in panels (b) and (c), respectively (adapted from Refs. [16] and [20]).

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The work in Zurich, which is summarized below, has focused so far on the deceleration of argon [ 12,16,17] and on the deceleration, mirroring and trapping of H atoms [18-201. A schematic illustration of the experimental set up is depicted in Fig. 2a, and Figs. 2b and 2c show electrode configurations used to decelerate (2b), and trap the Rydberg atoms in two dimensions (2c). Fig. 2c also displays the electric field distribution along the y (symmetric distribution) and z (asymmetric distribution) directions. The atomic beams are produced in a supersonic expansion at the exit of a pulsed nozzle. To generate the H atoms, ammonia in a lO:l, Ar:NH3 gas mixture is photolyzed in a quartz capillary mounted at the exit of the nozzle [ 181. The atoms move at a velocity in the range 650-700 m l s and are photoexcited to Rydberg Stark states in the electric field generated by a set of electrodes depicted in Figs. 2b-c. The atoms then propagate through the electrode setup in the direction of a microchannel plate detector the anode of which is connected to a phosphor screen. The Rydberg atoms ionize in the field of the detector and the ionization signal is converted into both a time-of-flight profile (extracted on the back plate of the MCP) and an image recorded by a charged-coupled-device (CCD) camera located behind the phosphor screen. When the Rydberg atoms are reflected back or trapped in the electrode set up, they do not reach the MCP detector. However, they can be field ionized after an adjustable delay by applying large pulsed voltages on the electrodes. The polarity of the pulsed voltages is chosen so as to extract the ions toward the MCP detector. The TOF of the ions can be used to determine the position of the Rydberg atoms at the time of field ionization. Indeed, ions produced close to the exit electrodes (3 and 4 in Figs. 2b and 2c) arrive later than ions produced close to the entrance electrodes (1 and 2 in Figs. 2b and 2c). In the first deceleration experiments on argon Rydberg atoms [ 12,161, the main limitation associated with the use of time-independent fields was identified as being related to the adiabatic traversal of the avoided crossings in the Stark map that occur at fields beyond the Inglis-Teller field, i.e., the field at which the Stark states of neighboring n values first cross (See Fig. 1). The adiabatic traversals of these crossings make it impossible to gain (or lose) more kinetic energy than the Stark energy at the Inglis-Teller field [12]. Much more kinetic energy can be removed using time-dependent fields [12] and an optimal deceleration experiment is one in which the voltages applied to the electrodes in the decelerator are constantly adjusted to the instantaneous positions of the Rydberg atoms so that the deceleration takes place at constant electric field and maximal field gradient [ 16,171. The results of an experiment using timedependent electric fields are summarized in Fig. 3 and demonstrate that the time

159 dependence of the applied electric fields (here an exponentially decaying function, see Ref. [16] for more details) can be tailored to enhance the deceleration efficiency and that a kinetic energy more than three time larger than the Stark shift at the operating field can be removed in a single deceleration stage. Under these conditions, a small acceleration of the particles in the transverse dimensions is unavoidable. However, the translation temperature of the decelerated sample can be kept well below 1 K as was demonstrated in Ref. [ 171 by imaging the cloud of decelerated atoms and comparing the experimental images with images calculated with a trajectory simulation program. The Rydberg Stark deceleration experiments on atomic hydrogen have led to the development of a mirror [ 191 in which a cloud of H Rydberg atoms initially moving at a velocity of 700 m / s is stopped over a distance of 1.8 mm in less than 5 ps, and reflected back to the initial spatial distribution. The mirroring process is illustrated in Fig. 4, in which panel (c) shows the results of measurements of the central position of the atom cloud during the mirroring process, panel (a) the simulated distributions of atoms in the yz plane at selected times during the mirroring process and panel (b) images of the actual distributions at times of 4 and 6 ps.

Figure 3 Comparison of the time-of-flight (TOF) profiles measured after decelerationiacceleration of n = 15 argon Rydberg atoms with time- independent and time-dependent inhomogeneous electric fields (a) and their simulation (b). The dotted line corresponds to a reference measurement with k = 0 Rydberg states which do not accelerate or decelerate. The inner dashed and full profiles represent measurements on blue- and red-shifted states with time-independent electric fields for which the change of kinetic energy corresponds closely to the Stark shift. The outer dashed and full TOF profiles represent the corresponding measurements with time-dependent fields in which the changes of kinetic energy are approximately three times the Stark shifts. The fact that the red-shifted states are decelerated when using time-independent fields and accelerated using time-dependent fields is a consequence of the opposite signs of the field gradients that result from the different voltage configurations (adapted from Ref. [ 161).

160

Recently these experiments have culminated in the development of two[20] and three- [21] dimensional traps for Rydberg atoms in selected Stark states. The electrode configuration used for the three-dimensional trapping experiments is depicted in Fig. 5a. In the experiments, the deceleration to zero velocity in the laboratory frame is achieved using time-dependent inhomogeneous electric fields generated by applying exponentially decaying voltages to the electrodes as described above for the deceleration of argon. The voltages decay to small nonzero final values corresponding to the electric field distributions displayed in Figs. 5b and 5c. The trap is narrower in the y and z dimension than in the x dimension so that a large breathing motion of the trapped Rydberg atom cloud can be observed by imaging techniques, as depicted in Fig. 5d. Typically around 104-105 atoms can be trapped at each measurement cycle in a volume of about 1x1x6 mm3, as determined from the amplitude of the field ionization signals and the size of the images. The trap losses result primarily from radiative processes although collisions with atoms in the trailing part of the gas pulses and transitions induced by black-body radiation also play a role (see Refs. [20,21] for a more complete discussion). These results represent the first demonstration of a three-dimensional trap for state-selected Rydberg atoms.

Figure 4. Operation principle of a Rydberg atom mirror. The electrode configuration used for deceleration and mirroring corresponds to that displayed in Fig. 2b or 2c. (a) Simulated evolution of the Rydberg atom cloud during the mirroring process which shows the focusing of the cloud taking place at around 4-5 us. (b) Experimental ion images at times of 4 and 6 us illustrating the focusing effect apparent in the simulations presented in panel (a), (c) Measured positions (full dots) of the center of the Rydberg atom cloud during the mirroring process. The open circles correspond to a reference measurement for which the electrode voltages were kept at 0 V (adapted from Ref. [19]).

Because all atoms and molecules have Rydberg states, the Rydberg Stark deceleration and trapping techniques outlined above have, at least in principle, the potential of being universal methods to generate translationally cold samples

161

of atoms and molecules. The atoms and molecules in these samples are in electronically excited states and exhibit very large dipole moments, which may turn out to be an attractive feature for quantum information processing. The radiative lifetimes of Rydberg Stark states at n values beyond 20 exceeds 100 (is and can be made much longer (i.e. > 1 ms) by optically preparing Stark states with mt > 3. The possibility of decelerating and trapping H atoms may offer opportunities for precision measurements of the Rydberg constant and may lead to new experimental schemes [22] for research on anti-hydrogen which is currently generated in Rydberg states [23]. Our future plans include the deceleration and trapping of hydrogen and NO molecules.

Figure 5. Electrode configuration (a) with equifield lines in the yz (b) and xz planes (c) used to trap hydrogen atoms in n — 30, k = 25 Rydberg states in three dimensions, (d) Breathing motion of the Rydberg atom cloud in the x dimension observed experimentally by ion imaging (adapted from Ref. [21]).

3. Zeeman deceleration of hydrogen and deuterium Atoms and molecules with unpaired electrons (so-called free radicals) tend to be particularly reactive. Several chemical reactions involving radicals are barrierless and exothermic processes and are expected to still take place at very low temperature [24]. By virtue of their unpaired electrons, free-radicals are subject to an electron Zeeman effect (see Fig. Ic). The translational motion of free radicals can thus be controlled by inhomogeneous magnetic fields and the

162

use of such fields in atom optics experiments has a long and rich history in physics starting with the experiments of Gerlach and Stern which enabled the measurements of magnetic moments and led to the experimental observation of space quantization [25-27]. A natural way to generate cold samples of free radicals at rest in the laboratory frame consists of using a supersonic expansion in combination with a multi-stage Zeeman decelerator of the type depicted in Fig. 6 which was recently constructed and tested at ETH [9]. The decelerator is placed along the axis of a supersonic beam of radicals generated using a versatile radical source developed in our laboratory [28]. The decelerator consists of several copper solenoids (length 7.8mm, inner diameter 5 mm, 42 windings, diameter of the copper wire 300 |j,m, operating current up to 250 A, maximal field on axis 1.4 Tesla) through which current pulses with rise and fall times, 5/r and 8tf, of less than 5 j^s can be applied (see Fig. 7b). Figure 7 [panel (a)] shows the magnetic field distribution in and around a solenoid, and a magnetic field pulse measured using a small pick-up coil placed inside the solenoid [panel (b)] [29]. s»I»a-ilf3ai9i

Lc

Quartz ««piilay

To

Pulsed

Figure 6. Schematic representation of the Zeeman decelerator developed at ETH. The pulsed valve delivering the supersonic beam is separated from the solenoids by a skimmer. At the end of the array of solenoids the supersonic beam is intersected at right angles by vacuum ultraviolet and/or ultraviolet laser beams between two parallel metallic plates used to extract the ions produced by photoionization toward a microchannel plate detector (adapted from Ref. [29]).

As the free radicals in the beam approach each solenoid in the decelerator, they are subject to an increasing magnetic field and the atoms in magnetic sublevels with positive Zeeman shifts are decelerated. Before the atoms reach the position of maximal field strength, the magnetic field is rapidly turned off to prevent the atoms from being accelerated when leaving the solenoid. The deceleration cycle is repeated at each successive solenoid in the decelerator. In optimizing the sequences of pulses to be applied to the solenoids

163

care has to be taken to (i) preserve phase stability during the whole deceleration process both in the transverse and longitudinal dimensions, and (ii) prevent the magnetic field from returning to zero in order to avoid transitions to other magnetic sublevels (see Ref. [29] for more details). At the end of the decelerator, the atoms are photoionized between two metallic plates and the ions are extracted by an electric field pulse to an MCP detector. Time-of-fiight profiles are recorded by monitoring the ion signal as a function of the time delay between the production of the radicals and the laser pulse used to photoionize them. (a)

(b)

Figure 7. (a) Magnetic field distribution in and around one of the solenoids calculated for an operating current of 250 A. (b) Temporal profile of a typical magnetic field pulse measured using a pick-up coil placed in the middle of the solenoid (adapted from Refs. [9,29]).

To demonstrate the operational principle of the Zeeman decelerator, Fig. 8a shows a set of deceleration measurements carried out on hydrogen atoms in their ground state. The lowest trace shows a time-of-flight profile corresponding to a reference measurement with the decelerator turned off and provides information on the velocity distribution of the H atoms in the beam. Comparing this reference measurement with the time-of-flight profiles obtained with pulse sequences designed to decelerate atoms of selected initial velocities, one observes two main differences: First, a late peak, corresponding to the decelerated atoms, is observed and its position shifts to longer times of flight as the selected initial velocity of the atoms to be decelerated decreases, as expected. Second, a clear increase of the signal at the edge of the main time-offlight peak is observed. This signal increase arises from the lateral guiding effect of the decelerator that results from the fact that the magnetic field in the solenoids increases as one moves away from their symmetry axis (see Fig. la). The profiles presented in Fig. 8b were obtained in full three-dimensional

164

simulations of the trajectories of the H atoms in the beam. The simulations involved no adjustable parameters. The analysis of these simulations and the good agreement between experimental and simulated data enable one to determine the final velocities of the decelerated atoms which are given above each trace in Fig. 8b. From the comparison between experimental and simulated time-of-flight profiles, one can conclude that close to half of the initial kinetic energy can be removed with only seven stages. The last panel (c) of Fig. 8 compares two deceleration measurements carried out on H and D atoms under otherwise identical experimental conditions (a pulse sequence optimized for the same initial velocity, the same carrier gas (krypton), etc.). As expected from the larger mass, the decelerated peak in the case of D appears at earlier times than in the case of H. We attribute the better signal-to-noise ratio of the D measurement to the fact that the H atoms are located closer to the edge of the gas expansion cone than the heavier D atoms and that, consequently, fewer H atoms pass through the skimmer placed at the entrance of the decelerator (see Fig. 6) and eventually reach the detector.

Figure 8. Zeeman deceleration of H atoms using a seven-stage Zeeman decelerator. (a) Measured time-of-flight profiles after deceleration with magnetic field pulse sequences optimized for atoms moving at selected initial velocities. (b) Simulations of the experimental time-of-flight profiles for comparison with the experiments. The velocities indicated in (a) represent the velocities of the atoms for which the magnetic field pulse sequence was calculated. Those indicated in (b) represent the final velocities of the decelerated bunches of atoms. (c) Comparison of time-of-flight profiles of H and D atoms obtained after Zeeman deceleration (adapted from Ref. [29])

These measurements represent the first demonstration of multi-stage Zeeman deceleration. This method may become useful to generate cold samples of free radicals. A 100-stage decelerator is currently under construction with which we plan to decelerate, and ultimately trap, molecules such as NH and oxygen.

165

4. Conclusions Two new methods have been described with which translationally cold samples of atoms and molecules can be generated. Both methods rely on the cooling of internal and translational degrees of freedom that takes place in supersonic expansions and involve decelerating the particles in the supersonic beam from a high velocity (between 400 and 2000 m/s depending on the carrier gas) to zero velocity in the laboratory frame. Rydberg Stark deceleration exploits the very large electric dipole moments of atoms and molecules in Rydberg states. With inhomogeneous electric fields of moderate strength, Rydberg atoms and molecules can be decelerated to zero velocity over only a few millimeter and in a few p, and loaded in two- or three-dimensional electrostatic traps. Zeeman deceleration relies on the linear electron Zeeman effect exhibited by most atoms and molecules with unpaired electrons. We hope that both methods will turn out to be useful tools in atomic and molecular physics.

Acknowledgments We thank Prof. T. P. Softley, Dr. S. Willitsch and Dr. M. Tomaselli for many discussions and their precious input to the work presented here. This work is supported financially by ETH Zurich and the Swiss National Science Foundation under Project Number 200021-1 13886.

References 1. A. Fioretti, D. Comparat, A. Crubellier, 0. Dulieu, F. Masnou-Seeuws, and P. Pillet, Phys. Rev. Lett. 80,4402 (1998). 2. H. L. Bethlem, G. Berden, and G. Meijer, Phys. Rev. Lett. 83, 1558 (1999). 3. J. M. Doyle, B. Friedrich, J. Kim, and D. Patterson, Phys. Rev. A 52, 2515(R) (1995). 4. S. A. Rangwala, T. Junglen, T. Rieger, P. W. H. Pinkse, and G. Rempe, Phys. Rev. A . 67,043406 (2003). 5. M. S. Elioff, F. F. Valentini, and D. W. Chandler, Science 302, 1940 (2004). 6. H.L. Bethlem and G. Meijer, Znt. Rev. Phys. Chem. 22,73 (2003). 7. R. Fulton, A. I. Bishop, and P. F. Barker, Phys. Rev. Lett. 93, 243004 (2004).

166 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23. 24. 25. 26. 27. 28. 29.

S. R. Procter, Y. Yamakita, F. Merkt, and T. P. Softley, Chem. Phys. Lett. 374,667 (2003). N. Vanhaecke, U. Meier, M. Andrist, B. H. Meier, and F. Merkt, Phys. Rev. A. 75,03 1402(R) (2007). T. F. Gallagher, Rydberg Atoms, (Cambridge University Press, Cambridge, 1994). M. L. Zimmerman, M. G. Littman, M. M. Kash and D. Kleppner, Phys. Rev. A. 20,225 1 (1979). E. Vliegen, H. J. Worner, T. P. Softley, and F. Merkt, Phys. Rev. Lett. 92, 033005 (2004). T. Breeden and H. Metcalf, Phys. Rev. Lett. 47, 1726 (1981). W. H. Wing, Phys. Rev. Lett. 45,631 (1980). D. Townsend, A. L. Gooodgame, S. R. Procter, S. R. Mackenzie, and T. P. Softley, J. Phys. B. 34,439 (2001). E. Vliegen and F. Merkt, J. Phys. B. 38, 1623 (2005). E. Vliegen, P. Limacher, and F. Merkt, EPJD. 40,73 (2006). E. Vliegen and F. Merkt, J. Phys. B. 39, L241 (2006). E. Vliegen and F. Merkt, Phys. Rev. Lett. 97,033002 (2006). E. Vliegen, S. D. Hogan, H. Schmutz, and F. Merkt, Phys. Rev. A 76, 023405 (2007). S . D. Hogan and F. Merkt, submitted for publication. A. Kellerbauer et al., submitted to NIM B for publication as contribution to the proceedings of the Low-Energy Positron and Positronium Workshop, Reading, 2007 M. Amoretti et al., Nature 419,456 (2002). I. R. Sims and I. W. M. Smith, Ann. Rev. Phys. Chem. 46, 109 (1995). W. Gerlach and 0. Stem, 2. Phys. 9,349 (1922). W. Gerlach and 0. Stern, 2. Phys. 9,353 (1922). W. Gerlach and 0. Stern, Ann. Phys. 16,673 (1924). S. Willitsch, J. M. Dyke and F. Merkt, Helv. Chim. Acta 86, 1152 (2003). S. D. Hogan, D. Sprecher, M. Andrist, N. Vanhaecke, and F. Merkt, Phys. Rev. A. 76,023412 (2007).

GENERATION OF COHERENT, BROADBAND AND TUNABLE SOFT X-RAY CONTINUUM AT THE LEADING EDGE OF THE DRIVER LASER PULSE AURELIE JULLIEN, THOMAS PFEIFER, MARK J. ABEL, PHILLIP M. NAGEL, STEPHEN R. LEONE AND DANIEL M. NEUMARK Departments of Chemistry and Physics, University of California, Berkeley, CA 94720, and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Coherent soft x-ray continuous spectra are experimentally generated at the leading edge of the driving laser pulse by temporal confinement of the high-order harmonics emission. The analysis relies on the measurement of individual half-cycle cutoffs [ l ] and the proposed interaction regime opens the route to the production of spectrally tunable isolated attosecond pulses.

1

Basics of attosecond pulses production

Recent developments in attosecond sources [2-41 are motivated by the desire to study electronic dynamics and electron-electron correlations on the atomic time scale. In particular, pioneering studies demonstrate the direct probing of ultrafast electronic motions following the creation of an inner-shell hole [5, 61. In another experiment, a subfemtosecond XUV pulse probes the motion of an electronic wave packet under the influence of the electric field of an infrared pulse [7]. Very recently, the real-time observation of electron tunnelling in an atom exposed to a strong laser field with attosecond resolution is reported [8]. The ability to produce coherent attosecond laser pulses is opening the way to a wide field of investigation devoted to the measurement of various electronic processes in atoms, clusters and molecules. The continuous trend towards shorter laser pulses has made major progress because of Kerr-lens modelocking, self-phase modulation broadening, and dispersion control, but is limited by the intrinsic barrier of the visible optical period to the femtosecond timescale. To go beyond this femtosecond barrier, laser sources with higher carrier frequencies are needed. The generation of high-order harmonics (HHG) of intense laser pulses focused into a gas provides a tabletop coherent soft x-ray source. The HHG process can be described by a three step semiclassical model [9]. In the first step 167

of the interaction, an electron is released by tunnel ionization of an atom or molecule and, in the second step, accelerated by the laser field. The electron is driven back by the reversing electric field and can recombine with its parent ion leading to the emission of a photon whose energy depends on the recombining electron kinetic energy. The recombination occurs on an ultrashort time scale and leads to a broad spectral emission. The process is repeated every intense half-cycle for a multicycle laser pulse. Therefore, high-order harmonic generation is considered theoretically as a potential solution for attosecond pulse production [lo]. An experimental breakthrough was achieved when it was confirmed that the HHG process can emit a train of attosecond pulses [ l 11, the duration of which can reach 130 as [12]. So far, to extract a single attosecond pulse from this train, few-cycle carrierenvelope phase (CEP) stabilized laser pulses have been required [7, 131. When the CEP value is close to 0 (cosine-shaped pulse), the maximum of the electric field coincides with the maximum of the pulse envelope; the most energetic harmonic burst of XUV light is emitted for the highest field strength and consequently only once per pulse for this particular phase value. Then, adequate spectral filtering selects this highest energetic part of the harmonic spectrum and an attosecond pulse can be isolated. This method has led to the production of isolated pulses with a duration of 250 as (10 eV bandwith around 100 eV), about 1/10 of the optical period of a Ti:sapphire laser [7]. Another approach has combined few-cycle phase-stabilized pulses with a polarization gating technique [ 141 to confine emission of harmonic radiation to one optical cycle, enabling the generation of 130 as single pulses [ 151. Here, we present experimental evidence for the confinement of the emission of harmonic radiation to the leading edge of the driving laser pulse due to an ionization gating mechanism, without the need for polarization control [ 161. We demonstrate that this interaction regime allows the generation of quasicontinuum harmonic spectra, with a broad bandwidth and a tunable central frequency. These spectra are strongly indicative of an isolated attosecond pulse.

2

Confinement of HHG production at the leading edge of the driver pulse: towards tunable isolated attosecond pulse generation

The proposed analysis method relies on the recent work of Haworth et al. [l]: when the electric field of the driver pulse is composed of several half-cycles whose extrema vary significantly from one to the other, which corresponds to a rapid change for the pulse envelope on the timescale of an optical cycle (particularly true for few-cycle pulses), the harmonic cutoff emission of each individual half-cycle is visible on the harmonic spectrum as a local maximum.

169

Measurement of the positions of these individual half-cycle cutoffs (HCO) allows the precise knowledge of the absolute phase value of the laser pulse to be obtained. Extending this analysis, we show that it is also possible to determine conditions when the harmonic radiation is confined to the leading edge of a few cycle driver pulse. Our experiments have been carried out with a commercial Ti:sapphire femtosecond laser delivering 800 pJ, 25 fs pulses at 3 kHz repetition rate. A hollow-core fiber filled with neon (1.7 bar) and a chirped mirror compressor provide external pulse compression to the few-cycle regime (about 6 fs) [17, 181. A pair of glass wedges is used to finely adjust the pulse dispersion. Laser pulses are CEP stabilized (root mean square fluctuations rms 250 mrad). Highorder harmonics are produced by focusing the driver pulses into a 2 mm long gas cell filled with 180 mbar of neon. Two Zr filters (200 nm thickness each) remove the near infrared light and low order harmonics. The remaining radiation is sent to a soft-x-ray spectrometer composed of a Si3N4transmission grating and a soft-x-ray-sensitive CCD.

60

70

80

90

100

130

120

130

140

Photon energy (eV) Figure 1. High-harmonic spectra obtained for different CEP values (the initial offset w value is unknown). On the first graph, the transmission of the Si3N4 (grating material) is shown by the dashed line.

Figure 1 shows spectra registered when varying the CEP value around an unknown initial offset. The overall minimum transmission occurring beyond 100

170

eV corresponds to the absorption edge of Si in the spectrometer grating. The measured spectra extend up to 130 eV. In the whole spectral range explored (60 eV - 130 eV), a slow modulation carrying the harmonic peaks and local spectral maxima is visible. This modulation is CEP-dependent and the energy of the maxima (HCO) shifts towards higher photon energy when the relative CEP value is increasing. As the energy of each HCO is dependent on the amplitude of the corresponding electric-field extremum, it can provide interesting information about the absolute phase and the intensity envelope of the laser pulse. In order to extract this information from the HCO behavior, we measured harmonic spectra for a relative CEP varying from 0 to 2co in a/8 steps. To determine more easily the HCO energy, the harmonic modulation (at 2ojlaser) is removed by Fourier analysis and the spectra are normalized by the CEP-averaged spectrum. Fig. 2 shows the resulting spectra plotted as a function of CEP. Positions of the HCOs are indicated. 130 0 Norm, spsctrat

1

Intensity

1

2 3 4 5 Relative CEP value (rad)

6

Figure 2. Harmonic spectra as a function of the CEP. The harmonic modulation is removed and all the spectra are normalized by a CEP-average spectrum. The HCOs positions (local maxima) for each spectrum are marked with white points. One can observe the quasi linear dependence of the HCOs positions with the CEP.

Experimental data are fitted by a simple calculation of the HCOs positions as a function of the carrier-envelope phase using the following approximation: EHCO = Ip +3.17(E/2o)iaser)2 (in atomic units, E is the extremum field value of the half cycle, Ip is the ionization potential of the target gas) [1]. If we assume that the last cutoff is emitted at the maximum pulse intensity, the best result

171 corresponds to a 4.5 fs driver pulse and the HCO prediction clearly does not match the experimental behavior (fig. 3a). Indeed, for a fixed value of the phase, the gap between two successive HCOs is not decreasing when the harmonic photon energy is increasing, as would be expected for half-cycles reaching the maximum of a Gaussian intensity envelope. The HCO energy evolves linearly with the phase and does not exhibit any saturation. 130

(b)

x i •' ^ it .

- 1 0 1 3 3 4 Absolute CEP value (rad)

5

-

1

t

.

i

.+J 1^1 <

<

0 1 2 3 4 5 AbsclutB CEP value
Figure 3, Ex peri men tally recorded HCO energy as a dinclion of CEP (black squares). Considering a driver pulse of 4.5 & and the last cutoff corresponds to the maximum pulse intensity (a, dotted line), the fit does not describe the experimental HCO behavior. A fit to the leading edge of a 9 fs pulse assuming an ionization gate {b, dotted line) results in a good agreement with the experimental data.

The observations are understood if we include ionization of the medium in the HCO analysis. Cao et al theoretically demonstrated that a very intense pulse (in the range of 1015 Wcm"3) can create a Fast depletion of the ground state population, occurring on the leading edge of the laser pulse [19]. In our measurements, the experimental intensity in the harmonic cell reaches this high value, since the cell is positioned very close to the laser focus. In that case, the efficiency of harmonic generation decreases when the ionization yield increases and stops when the ground state is depleted. Ionization acts as a gate, as the harmonic emission is shut down not only on the trailing edge of the pulse, but also before the maximum pulse intensity can interact with the harmonic generation gas. We note that the gate could also be achieved by phase mismatching due to ionization, even without a complete depletion of the ground state. To include this gating phenomenon in the HCO calculation, we consider that the efficiency of the HHG process depends on the ground state population and drops rapidly to 0 on the leading edge. Fig. 3b shows a fit of the experimental HCO energy when the harmonic emission is restricted to the leading edge of a 9 fs pulse whose peak intensity would allow, without

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ionization gating, emission of harmonic radiation up to 144 eV. Good agreement with experimental measurements is obtained. The duration of the driver pulse during the interaction is longer than expected. But, ultrashort laser pulses are extremely sensitive to excessive or insufficient dispersion compensation, which can be induced by the wedge positions, compressor setting or ionization-induced pulse reshaping due to the gas in the harmonic cell. As a consequence, the HCO analysis can be a useful tool to optimize in situ the pulse duration (see below). Notice also that a fit to the leading edge of an 8 fs or 8.5 fs pulse (not shown here) can also give convincing results. This uncertainty of the laser pulse duration prevents an extraction from our data of the absolute CEP value, as was done in [l]. This can be understood by the fact that the CEP is defined as the offset between the electric field extremum and the maximum of the pulse intensity envelope. In our case, the ionization gating mechanism prevents HCO investigations after the closure of the gate, and assumptions about the pulse duration and the pulse shape would be necessary to estimate the absolute CEP value. However, as we show below, in the ionization-gating regime, the CEP stabilization is required but a knowledge of the absolute phase value is no longer essential to produce isolated attosecond pulses. The direct consequence of this particular regime is the ability to spectrally isolate the contribution of adjacent half-cycles if the leading edge is sufficiently steep to provide significant intensity discrepancy from one half-cycle to another. In our experiment this condition is achieved by making the pulse shorter (about 6 fs by optimizing the compression parameters) and consequently more intense. Two resulting representative spectra are shown in figure 4(a). 250

.zc 150

.. 0

fe loo L

3P: 50 0

60

70

80

90

Photon energy (eV)

100

110

Time (fs)

Figure 4. (a) Tunable and broadband quasi continuum harmonic spectra for two CEP values. These spectra are obtained when the pulse is shorter and consequently the rising edge is steeper (neon pressure : 240 mbar). The remaining modulation is weak and would lead to satellite attosecond pulses with a relative intensity of less than 10% (b). The Fourier transform is calculated assuming a flat spectral phase for the solid line spectrum of (a).

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As only one maximum is visible, we estimate that a single half-cycle is mainly responsible for the harmonic emission in the considered spectral range. As a consequence, the spectra exhibit a quasi-continuous shape, strongly indicative of an isolated attosecond pulse. The remaining weak spectral modulation results from a weaker attosecond pulse generated by the previous or the next half-cycle. The relative intensity of this (these) parasitic pulse(s) is estimated to be less than 10% by Fourier transformation of the experimental spectrum (fig. 4b). Another noteworthy feature is the width of these quasi-continuous spectra (about 15 eV), which is broader than the usual continuum generated at the peak of the pulse with the same duration. This is the result of the large difference in intensity between two consecutive half-cycles on the leading edge, which becomes smaller near the peak of the pulse. It also becomes clear from figure 4(a) that in this regime the CEP is a parameter to tune the central wavelength of the continuum. The CEP adjusts the intensity of the most intense half-cycle before the closure of the ionization gate and therefore the photon energy of the last HCO. One is no longer restricted to work with a CEP stabilized to 0, but the absolute phase can be used to spectrally tune the attosecond pulse to the preferred energy for possible experiments. 3

Conclusion

To conclude, we have experimentally demonstrated that broadband continuum soft x-ray continuum spectra can be generated at the leading edge of the driver pulse, by means of an ionization gating mechanism. This regime enables the isolation of part of the XUV emission of an individual half-cycle on the leading edge, opening the way to produce isolated attosecond pulses with a large bandwidth and a tunable central frequency controlled by the carrier-envelope phase. Acknowledgments

This work is supported by a MUM program from the Air Force Office of Scientific research, contract No. FA9550-04-1-0242. Portions of the laboratory were supported by the Director, Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under contract DE-AC0205CH11231. T. P. acknowledges support of a Feodor Fellowship of the Alexander von Humboldt Foundation. References

1. 2.

C. A. Haworth, L. E. Chipperfield, J. S. Robinson, P. L. Knight, J. P. Marangos and J. W. G. Tisch, Nature Physics 3,52 (2007). P. Agostini and L. F. DiMauro, Rep. Prog. Phys. 67,813 (2004).

174

3. 4. 5.

6.

7.

8.

9. 10.

11. 12.

13.

14.

15.

16.

17. 18. 19.

A Scrinzi, M Yu Ivanov, R Kienberger and D M Villeneuve, J. Phys. B: At. Mol. Opt. Phys. 39, R1 (2006). P.B. Corkum and F. Krausz, Nature Phys. 3,381 (2007). M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann and F. Krausz, Nature 419,803 (2002). M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, Th. Westenvalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, J. Electron. Spectrosc. 137,259 (2004). R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, T. Westerwalbesloh,U. Kleineberg, U. Heinzmann, M. Drescher and F. Krausz, Nature 427,817 (2004). M. Uiberacker, T. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schroder, M. Lezius, K. L. Kompa, H. G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher and F. Krausz, Nature 446,627 (2007). P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993). M. Y. Ivanov, P. B. Corkum, T. Zuo and A. Bandrauk, Phys. Rev. Lett. 74, 2933 (1995). P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, P. Balcou, H. G. Muller and P. Agostini, Science 292, 1689 (2001). Y. Mairesse, A. de Bohan, L.J. Frasinski, H. Merdji, L.C. Dinu, P. Monchicourt, P. Brtger, M. Kovacev, R. Taieb, B. Carre, H.G. Muller, P. Agostini, P. Salieres, Science 302, 1540 (2003). A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hansch and F. Krausz, Nature 421,611 (2003). 0. Tcherbakoff, E. Mevel, D. Descamps, J. Plumridge and E. Constant, Phys. Rev. A 68,043804 (2003). G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Sagira, S. De Silvestri and M. Nisoli, Science 314,443 (2006). T. Pfeifer, A. Jullien, M. J. Abel, P. M. Nagel, L. Gallmann, D. M. Neumark and S. R. Leone “Generating coherent broadband continuum soft-x-ray radiation by attosecond ionization gating”, submitted to Opt. Exp. (2007). M. Nisoli, S. De Silvestri and 0. Svelto, App. Phys. Lett. 68,2793 (1996). L. Gallmann, T. Pfeifer, P. M. Nagel, M. Abel, D. M. Neumark and S. R. Leone, App. Phys. B 86,56 1566 (2007). W. Cao, P. Lu, P. Lan, X. Wang and G. Yang, Phys. Rev. A 74, 063821 (2006).

CONTROLLING NEUTRAL ATOMS AND PHOTONS WITH OPTICAL CONVEYOR BELTS AND ULTRATHIN OPTICAL FIBERS Dieter Meschedet , Wolfgang Alt , and Arno Rauschenbeutel* Insitut fur Angewandte Physik, Universitat Bonn, 591 15 Bonn, Germany t E-mail: meschede0uni-bon71.de www.iap. uni- bonn. de We store Caesium atoms in a 1D standing wave optical dipole trap with no more than one atom per site and sufficient spacing t o individually control the properties of each atom. The Caesium ground state hyperfine levels are used as qubit states, and all single-atom qubit operations have been realized in the past already, thereby realizing a neutral atom quantum register for a bottom-up approach towards quantum information processing with neutral atoms. With a second optical dipole trap we extract atoms from their trapping sites and insert them into to other micropotentials. This allows us to regularize the spacing of stored atoms, to insert two atoms into one and the same micropotential in order to induce interactions, and to insert atoms in a controlled way into high finesse optical resonators. In a new line of research we are exploring the potential of light confined by micro-structured thin optical fibers for controlled light-matter interaction in quantum optics. We have constructed fully tunable bottle micro-resonators and we have found that we can very efficiently detect and manipulate atoms or molecules in the immediate vicinity of fibres with sub-wavelength diameters. Keywords: Quantum information with neutral atoms, ultrathin fibres.

Introduction One great promise of controlling microscopic particles at the ultimate quantum level draws on the perspective of studying complex many particle systems with fully controlled interactions, for instance for applications in quantum information processing or quantum simulations.' For neutral atoms, there are two complementary approaches currently pursuing this attempt: The bottom-up and the top-down approach. The top-down approach relies 'Nvw address: lnsritiir fiir Physik, Illliversitat ?rlainz, hlaiiiz, C;criiiany

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176

on 2D- and 3D- optical lattices filled with exactly one atom per lattice site. Such systems have become available with the realization of the superfluid Mott insulator transition. It has been shown already that highly correlated many-particle states can be generated with this system.2 In this report we describe in the first part advances with the bottom-up approach which for neutral atoms may be symbolized with Fig. 1 showing small strings of neutral Caesium atoms stored in a standing wave optical dipole trap (DT, ID optical lattice) which can be operated as an optical conveyor belt.3 In the second part we give preliminary results about a new concept of controlling light matter interaction with ultrathin optical fibres. 1. Controlling atoms in optical conveyor belts

Fig. 1. Fluorescence image of sorted strings of 1 to 7 neutral Caesium atoms stored in an optical conveyor belt. The conveyor belt consists of two counterpropagating laser beams at 1.03 /^m wavelength forming a ID optical lattice with ~ 0.5 /urn periodicity. Here, equidistant strings of 1-7 atoms were created by rearranging an inital string of atoms with random distributions in the ID lattice.4 The elongated shape of the spots is due to the pancake shaped micropotentials which are observed edge-on.

At the level of single atoms we have shown earlier (for an overview see ref. 5) that we can tightly control the internal dynamics of every atom stored in the conveyor belt. We have found that the quantum states of the well known atomic clock microwave transition connecting the hyperfine ground states F=3 —> F=4 form an excellent qubit system offering long coherence

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times of tens of ms in the optical dipole trap. Individual addressability with a resolution of about 2.5 pm has been achieved, too, with a magnetic field gradient method. With strong experimental command realized for all single particle (or qubit) operations, the next important step is to prepare small many particle systems (typically 2-10 atoms) and to deterministically induce interactions between individual particles. We have taken first steps towards this goal with an accelerated experimental routine for the generation of a desired small number of atoms (Sect. l.l),by showing that we can deterministcally induce atom-atom interactions (Sect. 1.2), and by inserting atoms into and removing them from an optical high finesse cavity (Sect. 1.3). 1.1. Number-triggered loading of atoms 100-

9080; 70c 60-

-

P

2

2

50-

a 40-

302010-

. . . . . . . . . . . . . . . . . . . . . .

Our 1D lattice is always loaded from a magneto-optical trap (MOT) providing a small number of atoms with Poissonian distribution. In Fig. 2 we show the strongly enhanced preparation efficiency for generating a desired number of neutral atoms in our present system. Since the loading process is fast compared with all other experimental procedures it is simply interrupted and rapidly repeated until the desired number of atoms is found ("number-triggered loading"). One cycle takes typically 0.2 s. The success probability for larger atom numbers drops as a consequence of 1and 2-atom losses from the trap during preparation. We have found that during the loading routine atoms undergo light induced collisions favouring single occupation of the lattice sites. Details are found in ref. 6.

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1.2. Induced atom-atom interaction The sorting procedure used to regularize the atom strings shown in Fig. 1 is realized by extracting atoms from the ID lattice with an optical tweezers and re-inserting the atoms at evenly spaced locations which are separated by more than 10 /im in the example. At such large separations an atom already residing in the DT is not affected by the second optical tweezers. The spatial resolution of the placement procedure is indicated by the distribution of atoms over the micropotentials shown in Fig. 3. At zero separation, i.e. insertion into exactly the same micropotential one expects from the data in Fig. 3 a probability of about 20% that the second atom is indeed inserted into exactly the same micropotential as the first one. To experimentally detect the operation we have used light induced collisional loss of the atom pair which occurs if and only if two atoms are present in exactly the same site. We have found a success rate of 16% in agreement with theoretical expectations. While this interaction - which actually resembles a chemical process - is still highly dissipative, it demonstrates that interactions can be deterministically induced at the level of exactly two atoms.

04-

23

25

27

28

31

33

35

Final distance between two atoms I)vml2]

Fig. 3. Spatial distribution of the placement routine with two atoms.7 The modulation of the distribution reflects the periodicity of the ID optical lattice. The resolution of the procedure can be improved by tightening the focus of the optical tweezers and by lowering atom temperature.8

1.3. Conveyor belts and optical high finesse cavities With cold atoms stored in dipole traps and interacting with the field of a high finesse optical cavity in the strong couling regime of cavity QED many impressive experimental results have been achieved already.9 The

179

strong coupling regime is realized with large atom-field coupling constants, <7/7 = \/14tom/Knode & and low field relaxation rates re, i.e. with large g2/(2K^) and g/K values. Controlled transport of atoms within10 and into11 a high finesse cavity has been reported previously. We have realized an apparatus that allows to not only simultaneously insert 1-10 atoms prepared by the methods described above into a cavity but also to take them out again and detect their quantum state (Fig. 4).

conveyor belt, fast transport

cavity parameters: finesse 1.1-106 Qvalue4-10 8 mode volume 1.1 -10s A,3

Fig. 4. Shuttling of atoms across an optical high finesse cavity in the strong coupling regime. At constant velocity the temporal dependence translates into a spatial profile of the atom-field coupling strength. The average resonant photon number was 0.5 here. The signal on the right has been averaged over 16 individual transits.

Atoms are prepared and detected about 4.6 mm outside the cavity with parameters indicated in Fig. 4. They are then transported into the cavity where interactions in the strong coupling regime of cavity QED take place. After the interaction atoms can be taken back to the original position where their presence as well as their quantum state can be detected. In a high finesse cavity driven by resonant laser radiation the transmission is expected to strongly drop when an atom is inserted with transmission due to vacuum Rabi splitting.9 In a preliminary experiment we have observed strong reduction of transmission (Fig. 4) when an atom is shuttled forward and backward across the cavity mode, mapping the spatial profile of the atom-field coupling strength. We observe long residence times of our atoms in the cavity of several s indicating that cavity mediated laser cooling is active.12 a

7 is the spontaneous emission rate, Vmo^e is the cavity mode volume, and V^tom = c
180

A characteristic of our cavity is a very high finesse (F ~ 1.1 • 106) corresponding to a photon life time of T = (2/t)"1 = 0.19 yus and for the Cs transition at A = 852 nm with natural line width 27/(2?r) = 5.2 • 106 s^1 to a ratio 7/K = 6. Thus damping in our system is dominated by atomic relaxation due to spontaneous emission. For comparison we have displayed in Fig. 5 the characteristic parameters of recent cavity-QED experiments using different set-ups. relaxation photon dominated

atom dominated

flj -o o

I 1
D5

<5

1 0.01

0.1

1

10

7/K Fig. 5. Overview: Relevant strong coupling parameters of cavity QED in recent experiments. Data points are explicited in Tab. 1.3.

#

Laboratory

(all in 106 1 2 3 4 5 6 7

8

Optical cavities with dipole traps 2. 6 Caltech15 16* 3.8 13 16* MPI Quantum Optics16 1.4 3 30 1.25 3 MPI Quantum Optics17 3.1* 1.3 3 Georgiatech11 17* 3.5 7 2. 6 18* 0.43 University of Bonn 145 Optical cavities on atom chips 215* 53 3 137 ENS Paris18 3 Imperial College19 97* 2180 .75 Microwave cavities, with superconductors and Rydberg atoms 1 105 .125 4.4-103 Yale University14 ENS Paris20 .05 6-10-7 0..003t 7-105

9 *. calculated for the strongest transition; t : from atom transit time.

4.2

11 2.5 4.8 42 3.9

0.05 840

9-10 4

The dividing line here is chosen to be 7/K = 1 where atomic relaxation

181

equals cavity relaxation. In a simplified manner of speaking one can say that physical information about the dynamics of the coupled atom-cavity system is gained to the left of this line by observing photons emitted from the system. On the right side it is more profitable to derive information about the status of the system by monitoring the dynamic evolution of long lived and coherent atomic quantum states. This is well known from the microwave domain which has been deeply explored by experiments at the ENS Paris.13 More recently it has been shown that this regime can be realized experimentally also with superconducting circuit systems.’ With our system it should be possible to venture into this regime also with an optical cavity-QED set-up which has the specialty of controlled insertion of several atoms simultaneously into the system. 2. Thin and ultrathin fibres

Having conquered communication technologies already optical fibres have begun to revolutionize also other areas of controlled propagation of light waves. For instance micro structured fibres, so called photonic crystal fibres, are now widely applied for efficient applications of nonlinear processes.” On of the simplest micro structuring methods is to reduce the thickness of standard 125 pm diameter optical fibres. We have begun to explore the properties of such thinned fibres with modulated cross section for applications in quantum optics. Here we report on an advantageous concept for ”bottle micro resonators” (Sect. 2.1), and on ultrathin fibre applications as an ”infinite focus” (Sect. 2.2). 2.1. Bottle resonators - tunable micro cavities

As outlined in Sect. 1.3 the realization of cavity QED processes for e.g. quantum information processing requires large atom-field coupling strengths g/y = & & o m / V m o d e . Built from discrete mirror substrates the performance (Vmode, K ) of the cavities used in experiments 1-5 in Tab. 1.3 seems to be limited by technical imperfections, and no clear route towards substantially improved values is in sight. An alternative is offered by micro resonators reyling on total internal reflection such as glass spheres or toroids.” Very high Q values in the lo8 to lo1’ range with small damping rates K rivaling the performance of conventional cavites have been demonstrated. More importantly, micro resonators promise further enhancement of the g-factor, since their mode volume is only of order 103X3 instead of 105X3,resulting in an order of magnitude enhancement of g.

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We have found a novel geometry which can be fabricated with a fibre pulling apparatus where the optical fibres are thinned while they are heated with a flame or with a CO2-laser.23 The resonators look like little

photodiode

I °o

bottle resonator

•

excitation! laser Fig. 6. "Bottle resonators". The light ray travels along the circumference of 16 /urn diameter optical fibres. Due to angular momentum conservation turning points occur for identical diameters where the caustics generate strong field enhancement and provide efficient coupling ports. Light is coupled in and out of the bottle resonator by coupling to the evansecent field of an ultrathin fibre, see Sect. 2.2. On the right side we show a microscope image of a test fibre drawn by our appartus. For practical bottle resonators the diameter modulation is only a few % and not visible.24

bottles (Fig. 6) which may be considered an extremely prolate deformation of spherical resonators. The conceptual advantages of such resonators24 include a small free spectral range (100 GHz compared to I THz for spherical resonators), straightforward strain tuning over more than one free spectral range, and two input/output ports for convenient coupling by ultrathin fibres. Preliminary results in our laboratory indicate that such "bottle resonators" behave precisely as theoretically expected.

2.2. An "infinite" focus For coupling of external light fields to the bottle resonator described above the evanscent light field of ultrathin fibres provides a very convenient coupling mechanism (Fig. 6). The thin fibres are drawn on our pulling appartus, and Fig. 7 shows both the taper section transferring the light field of a conventional, weakly guiding fibre into the strongly guided mode of an ultrathin fibre. The electron microscope image on the lower left shows the

183 0,6

0,4

0,2

Fig. 7. The infinite focus. Upper left: Fibre taper section; lower left: ultrathin fibre, electron microscope image. Right: Calculation of the effective mode area of the guided wave (Aeff) and the quantity Aeff/R which determines the sensitivity for surface spectroscopy.

ultrathin section with diameter ~ A/2. For an intuitive understanding of the properties of the evanescent light field in this limit it is useful to define an effective area of the guided mode by Aes(X) = P(A)// s u r f(A) where P(X) is the total power travelling in the guide, /surf is the surface field strength.25 The area AeS is a universal function26 of R/X (right side of Fig. 7, R the ultrathin fibre radius). It shows a pronounced maximum of order A2 near _R/A=0.25. Thus all material particles in the immediate vicinity of the surface including adsorbed molecules or atoms bound in surface trapping states are subject to a light field resembling a very tightly focused laser beam. In contrast to free space focusing, however, this condition is extended from the short depth of focus (~ A) along the entire stretch of the ultrathin fibreL, resulting in an enhancement factor of order L/X ~ 103 — 104. We have carried out an initial experiment with organic molecules adsorbed on the surface of an ultrathin fibre and found that we obtain very high sensitivity due this "infinite focus" situation.25 Acknowledgments

We wish to thank the students that have worked on this project: Y. Miroshnychenko, I. Dotsenko, M. Khudaverdyan, S. Reick, T. Kampschulte, A. Stiebeiner, E. Vetsch, F. Warken, and our colleague M. Sokolowski. Support by the Deutsche Forschungsgemeinschaft and the European Commission is gratefully acknowledged. 3. References References

1. ARDA roadmap: http://qist.lanl.gov; http://qist.ect.it/Reports/reports.htm.

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Report:

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2. I. Bloch, Nature Physics 1,23 (2005). 3. D. Schrader, S. Kuhr, W. Alt, M. Mueller, V. Gomer, D. Meschede, Appl. Phys. B 73 819 (2001). 4. Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, D. Schrader, A. Rauschenbeutel, Nature (London), 442,151(2006). 5. D. Meschede and A. Rauschenbeutel, in: Adv. A t. Mol. Phys. 53, 76 (2006), G. Rempe and M. 0. Scully, eds. 6. L. Forster, W. Alt, I. Dotsenko, M. Khudaverdyan, D. Meschede, Y. Miroshnychenko, S. Reick, A. Rauschenbeutel, New. J . Phys. 8,259 (2006). 7. Y.Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, D. Meschede, S. Reick, A. Rauschenbeutel, Phys. Rev. Lett. 97,243003 (2006). 8. Y.Miroshnychenko, W. Alt, I. Dotsenko, L. Forster, M. Khudaverdyan, A. Rauschenbeutel, D. Meschede, New J . Phys. 8,191 (2006). 9. R. Miller, T.E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer and H. J. Kimble, J. Phys. B: A t. Mol. Opt. Phys. 38,551, (2005). 10. S. Nuamann, M. Hijlkema, B. Weber, F. Rohde, G. Rempe, and A. Kuhn, Phys. Rev. Lett. 95, 173602 (2005). 11. K. M. Fortier, S. Y. Kim, M. J. Gibbons, P. Ahmadi, M. S. Chapman, arXiv:quant-ph/0703156vl;Phys. Rev. A , in print (2007). 12. Stefan Nuamann, Karim Murr, Markus Hijlkema, Bernhard Weber, Axel Kuhn, and Gerhard Rempe, Nature Physics 1, 122 (2005). 13. S. Haroche and J. M. Raimond, Exploring the Quantum (Oxford Univ. Press 2006). 14. D. I. Schuster et al., Nature 445,515 (2007). 15. A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, Phys. Rev. Lett. 98,193601 (2007). 16. T.Puppe, I. Schuster, A. Grothe, A. Kubanek, K. Murr, P. W. H. Pinkse, and G. Rempe, Phys. Rev. Lett. 99,013002 (2007). 17. T. Wilk, S. C. Webster, H. P. Specht, G. Rempe, and A. Kuhn, Phys. Rev. Lett. 98,063601 (2007). 18. Y . Colombe, T.Steinmetz, G. Dubois, F. Linke, D. Hunger and J. Reichel, arXiv:0706.1390~1 [quant-ph] 19. M. Trupke, J. Goldwin, B. Darqui, G. Dutier, S. Eriksson, J. Ashmore, and E.A. Hinds, Phys. Rev. Lett. 99,063601 (2007). 20. S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Delglise, U. Hoff, M. Brune, J. M. Raimond, S. Haroche, Nature 449,297 (2007). 21. P. Russell, Science, 299,358 (2003). 22. K. Vahala, Nature, 424,839, (2003). 23. T.A. Birks and Y. W. Li, J. Lightw. Tech., 10,432 (1992). 24. Y. Louyer, D. Meschede, A. Rauschenbeutel, Phys. Rev. A , Rapid Comm, 72,031801(R) (2005). 25. F. Warken, E.Vetsch, D. Meschede, M. Sokolowski, and A. Rauschenbeutel, Opt. Exp., in print, (2007). 26. F. Le Kien, J. Q. Liang, K. Hakuta, V. I. Balykin, Opt. Commun., 242,445 (2004).

SPECTROSCOPY ON THE

SMALL SCALE

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WIDE-FIELD CARS-MICROSCOPY CHRISTOPH HEINRICH, ALEXANDER HOFER, STEFAN BERNET, MONIKA RITSCH-MARTE Division f o r Biomedical Physics, Innsbruck Medical University, Innsbruck, Austria * E-mad: monikarztsch-marteoi-med.ac.at We have developed a non-scanning version of Coherent anti-Stokes Raman Scattering (CARS) microscopy. In contrast to the more common confocal implementation, where one scans tightly focussed femto- or picosecond pulses over the sample, we use nanosecond pulses and a special excitation geometry that is designed to satisfy the phase matching condition over the whole field of view. This allows fast image acquisition, even snapshots with a single shot of nanosecond laser pulses are possible.

Keywords: CARS microscopy, vibrational imaging, Raman spectroscopy

1. Introduction Making use of nonlinear effects in optical microscopy requires intense electromagnetic waves, i.e. pulsed laser beams. Continuous technical advances in the development of laser sources (and highly sensitive detectors) have led to laser systems that are ideally suited for the task. Depending on the specific multiphoton setup, femtosecond (fs), picosecond (ps), or even nanosecond (ns) laser sources are used. Coherent anti-Stokes Raman scattering (CARS) is a third order nonlinear process that is suitable for nonlinear microscopy. CARS mixes two laser pulses at frequencies wp and ws which generate a blue shifted signal at frequency W A S = 2 wp - ws.This signal is resonantly enhanced, if the frequency difference wp - ws approaches the resonant frequency of molecules inside the medium. This effect was first demonstrated by Maker and Terhune in 1964 with discrete frequencies and became a practical tool in the 1970s when tunable dye laser for continuous scanning were developed. At that time CARS was mainly used as a spectroscopic tool, especially for the analysis of combustion processes. 187

188

The first ” proof-of-principle” CARS-microscope (CARSM) was set-up by Duncan et al. in 1982 The authors used dye lasers delivering ps-pulses in the visible region. This resulted in a rather strong nonresonant CARS background that almost overwhelmed the spectrally selective signal, Thus today nearly all CARSM setups utilize laser pulses in the near-infrared (NIR) region.

’.

2. Features of a CARS-microscope CARSM is a spectrally (chemically) selective method that allows functional mapping of the targeted substances in the unstained sample. Recently it has attracted much interest in the biomedical sciences, since the method enables imaging of cells components within living cells with three dimensional sectioning capability. This interest was triggered by the pioneering work of Xie and coworkers in 1999. Since then it has conquered a ”niche” in vibrational microscopy, as on the one hand CARS is more sensitive than spontaneous Raman scattering due to stimulated emission, and on the one hand it offers better resolution than IR microscopy, because the emitted signal is blue-shifted with respect to the excitation laser pulses. CARS imaging of living cells can be done at video frame rate and even faster ’. On the ”negative side” CARS microscopy has to deal with an inherent nonresonant, i.e. spectrally independent, background, which negatively influences the signal-to-noise ratio. Thus it is crucial to adapt the microscopic CARS setup in a way to suppress this nonresonant background. Many approaches have been developed to this end. For instance, detecting the CARS signal in the backward direction (epi-CARS) delivers high contrast for scatterers smaller than the wavelength of light An ensemble of dipoles lined up in a plane orthogonal to the propagation of the excitation fields emit a symmetrical radiation pattern in axial direction. Coherent addition of the radiation fields from an ensemble of dipoles lined up in axial direction, however, generates a large signal in forward and a weak signal in backward direction, due to constructive and destructive interference. Thus detecting the CARS signal in the backward direction avoids the large forward-going signal from bulk water, but at the cost of a lower signal. Another possibility is to utilize the polarization difference between the resonant and nonresonant CARS fields Adjusting the analyzer in front of the detector orthogonally with respect to the nonresonant polarization, the background can be suppressed efficiently. Unfortunately, the absolute strength of the Raman resonant CARS signal is likewise reduced, which limits the sensitivity for the detection of weak Ranian resonances. This

‘.

’.

189

dilemma can be circumvented by the interference of a weak CARS signal with an intense nonresonant field acting as a local oscillator, an approach that has been named heterodyne-CARS '. Other approaches utilizing interferometry ', modulation techniques with lock-in amplifiers or timedomain techniques by means of pulsed-sequenced detection have been demonstrated. It is also possible to combine epi-CARS with polarization sensitive detection and correlation spectroscopy to monitor dynamic processes within biological samples

'',

''

'.

3. WIDE-FIELD CARS MICROSCOPY

A standard CARS microscope utilizes a confocal setup working with tightly focussed beams of NIR laser pulses and with oil-immersion objectives of high numerical aperture (NA = 1.4). The confocal setup has the advantage of high spatial resolution. The tight focussing in space leads to an increased spread in momentum space and thus "relaxes" the phase matching condition. Image formation is carried out by scanning the sample (or the laser) which, however, limits the acquisition time. Wide-field CARS microscopy avoids scanning taking an image the whole sample "all at once". The schematic setup of our wide-field CARS setup13 is sketched in Fig.1. The idea is to use a non-collinear beam geometry, distributing the intensity of ns pulse trains homogeneously over the whole sample region of interest. We actually use a markedly non-collinear beam geometry ("extremely folded box-CARS" ), which utilizes an ultra-dark field condenser to deliver a cone of light from above, while the Stokes beam is coupled through the objective of the inverted microscope from below. This gives rise to an anti-Stokes beam that counter-propagates with respect to the Stokes beam, which can conveniently be read out through the microscope objective. Note that the high numerical aperture of the dark-field condenser provides a narrow sheet of light illumination which very effectively suppresses the nonresonant CARS background. Based on the axial confinement of the nonlinear interaction zone and the intrinsic nonlinear intensity dependence, this system offers good optical sectioning capability. We have also investigated using a structured illumination technique based on laser speckles, Dynamic Speckle Illumination (DSI), which was recently invented by J. Mertz l4 and coworkers, to improve the sectioning capability even further. Our laser system consists of a frequency doubled and tripled diodepumped Nd:YAG laser and and an optical parametric oscillator (GWU Lasertechnik). The 1064 nm line of the laser system is currently used as the

190

Fig. 1. Schematic setup of the wide-field CARS-microscope: The laser pulses enter the microscope from different directions. The pump pulse is coupled in from above through a dark-field condenser whereas the Stokes pulse comes from the backside of the inverted microscope and travels through the objective. As a consequence of the phase matching condition the generated anti-Stokes signal counter-propagates with respect to the Stokes beam and is imaged by the same objective on an intensified CCD-camera. Multimodefibers are used for beam delivery in order to destroy the spatial coherence.

Stokes beam, whereas the 355 nm line drives the optical parametric oscillator, which emits ns-pulses that are continuously tunable between 410 nm and 2600 nm and are used as the pump beam. The sample is typically immersed in a liquid (e.g. water) and sandwiched between two glass coverslips.

(B)

(C)

(D)

Fig. 2. Spectrally selective imaging of a mixture of polymer particles. (A) presents a reference dark-field image of the test sample. (B) and (C) show the acquired CARS images obtained at 3052cm"1 (yellow) and 2953 cm^ 1 (red), respectively. (D) displays an overlays of the images from (B) and (C).

The performance of the wide-field CARS-microscope is demonstrated on test samples consisting of two kinds of polymer beads of 1.5/mi size. One

191

species is polystyrene (PS) and the other polymethylmethacrylat (PMMA). Both substances exhibit strong Raman resonances that are close to each other, namely PMMA at 2953cm-1 and PS at 3052cm-1. Fig.2 demonstrate the high spatial resolution as well as the spectral selectivity of our CARS setup. Fig.2 (A) shows a reference dark-field image of the test sample clearly indicating the position of the various polymer beads. Besides in Fig.2 (B) one can see the detected CARS signal when the difference of the laser pulses is tuned to the Raman resonance of PS at 3052 cm"1. Fig.2 (C) present the CARS signal obtained by changing the beating frequency of the laser pulses to 2953cm-1. Fig.2 (D) overlays both CARS signals encoded with different colors. A comparison of the overlay with the reference darkfield image shows that the brighter beads in the dark-field image perfectly match with the beads that light up when tuning the excitation laser frequencies to 3052cm-1 indicating PS beads, as it should be, since the higher refractive index of PS (nps = 1.55) compared to PMMA (UPMMA = 1.49) results in a stronger signal under dark-field illumination.

Oark-ffeW iHuminalon 1 CARS fi> 5850cm"1

Fig. 3. (color) CARS image of an example of a living alveolar type II lung cell. Left: Dark-field illumination. Middle: Detected CARS signal at 2850 cm^ 1 exciting aliphatic CH2 stretching vibration. Right: Overlay of both images.

To demonstrate the non-invasiveness of the method we applied it to visualize phospholipid-rich vesicles inside living alveolar type II cells from rats. These cells are specialized to generate specialized vesicles, the so-called "lamellar bodies", that contain pulmonary surfactant. These vesicles are released from the cells by exocytosis to provide the monomolecular lipid film of dipalmitoyl phosphatidylcholine (DPPC) on the surface of lung alveoli that is required to lower the surface tension to the level necessary for optimal gas exchange. Lamellar bodies contain various lipid and protein components of varying composition, according to the special requirements of the organ. The lamellar bodies of the lung epithelium typically have a diameter range of 1 - 2,5 //m. Fig.3 is an example for imaging these cell

192

components by means of wide-field CARS microscopy. The left image in Fig.3 presents an example of a lung cell under dark-field illumination. Next one can see the CARS signal when tuned to the aliphatic CH2 stretching vibration at 2850cm-l generating a strong signal for all lipids and phospholipids. The five white circles in the dark-field image now light up bright and show that these structures contain DPPC. An overlay of both images is shown on the right. Recently Toytman et al. l5 have presented another type of wide-field microscopy where the excitation geometry of the laser beams does not satisfy the phase-matching condition. Thus, in a homogenous medium such as the solvent alone, no CARS signal is generated effectively. In this approach, somewhat reminiscent of the idea of dark-field microscopy, only light that is scattered from structures in the sample to satisfy the phase matching can contribute to the CARS signal. This avoids the nonlinear background from the bulk, but a disadvantage seems to be the fact that one has to rely on "accidentally" satisfying phase matching in regions of the sample, which might give rise to only weak CARS signals, or might make the interpretation of images of unknown structure and orientation difficult. Nevertheless the authors have successfully implemented their scheme and given proofof-principle. 4. APPLICATIONS A N D OUTLINE

Since lipids give particularly strong CARS signals, it is very likely that CARSM will become an established tool in lipid metabolism research, which is a very "hot" topic at present. Many groups are currently setting up CARS experiments with living cells, e.g. trying to visualize changes in the location, distribution, or concentration of fatty acids. It has to be stressed, however, that CARSM is not a single-molecule method. Currently the sensitivity typically is in the order of lo6 vibrating modes per focal volume, which allows the imaging of single lipid bilayers l6 and single cellular membranes, but not the monitoring of single biomolecules and their dynamics. The fact that CARSM is a labelling-free method could be a strong advantage in an area of biomedical research that is presently growing very rapidly: There is growing concern about possible unknown risks of nanoparticles, which are increasingly used in the emerging "nanomedicine", e.g. as contrast agents or for drug delivery. To date there often exists only limited or unsatisfactory information on the "fate" of the nanoparticles in the human body. Being an optical method, diagnostics deep within the body is impossible - but CARSM might be able to assist cell or tissue studies on

193

the up-take or clearance the nanoparticles and their degradation products, for instance, whether they aggregate or where they accumulate. The fact that there is no need for staining or any other treatment of the particles which could possibly change their behavior in the tissue - can give CARSM a special advantage here. This work was supported by the Austrian Science Fund FWF (Project P16658-N02). References 1. P. Maker, R. Terhune. ”Study of optical effects due to an induced polarization third order in the electric field strength.” Phys. Rev. A 137, 801 (1965). 2. M. Duncan, J . Reintjes, and T. Manuccia. ”Scanning coherent anti-Stokes Raman microscope.” Opt. Lett. 7,350 (1982). 3. A. Zumbusch, G.R. Holtom, X.S. Xie. ”Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering.” Phys. Rev. Lett. 82, 4142 (1999). 4. C.L. Evans, E. 0. Potma, M. Puoris’haag, D. Cbte, C.P. Lin, and X.S. Xie.”Chemical imaging of tissue in vivo with video-rate coherent anti-Stokes Raman scattering microscopy. Proceedings of the National Academy of Sciences 102, 16807 (2005). 5. C. Heinrich, S. Bernet, and M. Ritsch-Marte. ”Nanosecond microscopy with spectroscopic resolution.” New J. Phys. 8, 36 (2006). 6. A. Volkmer, J.X. Cheng, and X.S. Xie. ”Vibrational imaging with high sensitivity via epidetected coherent anti-Stokes Raman scattering microscopy.” Phys. Rev. Lett. 87, 023901 (2001). 7. J.X. Cheng, L.D. Book, and X.S. Xie. ”Polarization coherent anti-Stokes Raman scattering microscopy.” Opt. Lett. 26, 1341 (2001). 8. E.O. Potma, C.L. Evans, X.S. Xie. ”Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging.” Opt. Lett. 31, 241 (2006). 9. C.L. Evans, E.O. Potma, and X.S. Xie. ”Coherent anti-Stokes Raman scattering spectral interferometry: determination of the real and imaginary components of nonlinear susceptibility chi(3) for vibrational microscopy.” Opt. Lett. 29, 2923 (2004). 10. F. Ganikhanov , C.L. Evans, B.G. Saar, X.S. Xie. ”High-sensitivity vibrational imaging with frequency modulation coherent anti-Stokes Raman scattering (FM CARS) microscopy. Opt. Lett. 31, 1872 (2006). 11. A. Volkmer, L.D. Book, and X.S. Xie. ”Time-resolved coherent anti-Stokes Raman scattering microscopy: Imaging based on Raman free indiction decay.” Appl. Phys. Lett. 80, 1505 (2002). 12. T. Hellerer, A. Schiller, G. Jung, and A. Zumbusch. ”Coherent anti-Stokes Raman scattering (CARS) correlation spectroscopy.” 3, 630 (2002). 13. C. Heinrich, S. Bernet, and M. Ritsch-Marte. ”Wide-field Ccherent antiStokes Raman scattering microscopy.” Appl. Phys. Lett. 84, 816 (2004). 14. C. Ventalon, J. Mertz. ”Quasi-confocal fluorescence sectioning with dynamic speckle illumination”, Opt. Lett. 30, 3350 (2005).

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15. I. Toytman, K. Cohn, T. Smith, D. Simanovskii, and D. Palanker. ” Widefield coherent anti-Stokes Raman scattering microscopy with non-phasematching illumination.” Opt. Lett. 32, 1941 (2007). 16. E.O. Potma and X.S. Xie. ”Detection of single lipid bilayers with coherent anti-Stokes Raman scattering (CARS) microscopy.” Journal of Raman Spectroscopy 34,642 (2003).

ATOM NANO-OPTICS A N D NANO-LITHOGRAPHY V.I. Balykin, P.N. Melentiev, A.E. Afanasiev, S.N. Rudnev, A.P. Cherkun, and V.S. Letokhov Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow region, 142190 Russia P.Yu. Apel, and V.A. Skuratov Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980 Russia V.V. Klimov Lebedev Physical Institute, Russian Academy of Sciences, Leninski pr. 53, Moscow, 11 9991 Russia

1. Introduction Semiconductor devices are currently constructed on a microscopic scale, predominately using optical lithography. The fabrication of structures at scales smaller than the current limits is a technological goal of great practical, but also fundamental interest: material structures with dimensions in the 10 nm range represent a bridge between the classical and the quantum mechanical world. In practice, it is desirable to have the ability to build any nanostructures with atomic precision using any atomic species. To date, no single approach meets this demand. Rather, there are a number of techniques, each of them possessing some advantages and having some drawbacks. In particular, there are known difficulties for further development of well-known techniques: Conventional optical lithography is diffraction limited down to 100 nm; charged particle beam lithography suffers from the serial nature of patterning and Coulomb repulsion; scanning probes, as they manipulate single atoms, are generally too slow; and self-assembled fabrication still requires a better understanding of the physical processes. Micro and nanofabrication of material structures is denoted generically as atom lithography. 195

196

We report here on the new approaches in atom lithography that is based on the use of (1) the spatially on nanometer scale localized laser fields; (2) the atom nanopencil; (3) the atom pinhole camera, and (4) the laser induced quantum atom adsorption on a surface.

2. Nanometer scale localized laser fields The ultimate goal in the field of atom nano-optics is the formation of nanometer-sized ensembles and beams of neutral atoms. It is known that laser fields capable of forming such atomic ensembles must be well localized. Standing light waves - the best known laser fields - are finding widespread use in atom optics. The first application of standing light waves in atom nano-optics was the localization of atoms in a nanometer-sized space and their placement in channels. The idea of channeling atoms in a standing light wave has led to the development of techniques of atom lithography that allow periodic one- and two-dimensional structures to be produced on a surface.' On the basis of general physical considerations, it is evident that the use of spatially localized fields (and, accordingly, spatially localized atomic potentials) can offer new possibilities for the construction of atomoptical elements. To date, we have knowledge of only two types of laser fields that are well enough localized in space to make this possible: surface (evanescent) light waves and light that arises in the vicinity of structures with a characteristic size of less than the wavelength. Evanescent waves find widespread use in atom optics for the reflection, localization and cooling of atoms. A drawback of evanescent waves is that they are localized exclusively on a 2D surface. The three dimensional localized light field is that which results after diffraction by an aperture the size of which is small in comparison to the wavelength of light. In this case, a local 3D maximum of field intensity is formed near the aperture. The magnitude of the maximum is governed mainly by the size of the aperture. A substantial drawback of the field structures considered above is the accompanying standing or traveling light wave: when atoms move in a standing light wave they can undergo spontaneous decay processes that, in many cases, undesirable. The light nanofield configurations that are free from the abovementioned shortcomings are the following.' Two plane conductive plates spaced a distance d of the order of or smaller than the wavelength of light apart form a plane waveguide for the laser radiation coupled into it from one side. If the electric field strength vector of the laser radiation is normal to the plane of the waveguide, the radiation can propagate through the

197

waveguide, no matter how thin it is. Now let two small coaxial apertures be made in the conductive screens that form the waveguide. If the diameters a of these apertures are smaller than the wavelength of the radiation coupled into the waveguide, little radiation escapes from the waveguide through the apertures, but the light field near them is strongly modified. Figure 1 shows the light field intensity distribution near the apertures inside and outside the waveguide when the thickness of the waveguide is equal to the radius of the apertures. As can be seen in Fig. 1, there is a light field intensity minimum in the direction normal to the plane of the waveguide. Such a light field configuration can be called as a photon hole. Its characteristic size is determined by the size of the apertures, the thickness of the waveguide and its volume V ~ cfd
Fig. 1. Electromagnetic-field intensity for a photon hole.

198

If two conductive screens with coaxial apertures are spaced a distance of d = A/2 apart and the electric field strength vector is parallel to the waveguide plane than the intensity distribution in such a light field is different than in the previous case, Fig. 2. As can be seen, the field drops off rapidly outside the waveguide in the direction normal to the waveguide and has its maximum at the center of the waveguide. This maximum is caused by the constructive interference of the fields scattered by the apertures. Such a light field configuration can be called as a photon dot. The maximum intensity at the photon dot is twice as high as the field intensity in the absence of apertures.

x/a

a

Fig. 2. Electromagnetic-field intensity for a photon hole.

The characteristic size of a photon dot or a photon hole is in the nanometer region, which allows for nanometer-sized atomic ensembles to be formed. Let us consider as an example two possible uses for photon dots and photon holes: the focusing and localization of atoms. When an atom is exposed to laser light, electric field of the laser light induces in the atom an oscillating dipole moment. If the light field amplitude at the atom position is spatially nonuniform, a gradient force on an atom develops. It is precisely the gradient force that is used to modify the trajectory of the atoms. When the detuning of the laser radiation frequency relative to the atomic transition

199

frequency is positive, an atom in the laser light configuration is drawn into the weak field region. In the case of a photon hole, the nanometer- sized weak field region is surrounded by the strong field inside the waveguide; if the light field frequency detuning is positive, the atoms that fly through the apertures in the waveguide walls will be attracted to the axis of the system-in other words, they will be focused. A photon dot draws atoms in negative frequency detuning and also provides focusing for the atomic beam that passes through the apertures in the waveguide walls. It has been ~ h o w nthat ~ > ~an atomic beam can be focused to a spot of the order of the de Broglie wavelength, which for a thermal beam amounts to a few Angstroms. The photon dot and photon hole light-field configurations have extreme points at which the gradient force is zero. Such light-field configurations are naturally considered as the possible atom trap c~nfigurations.~ The photon dot light-field configuration is stable and is truly three dimensional, while in a photon hole light-field configurations either the axial or the radial motion of the atom will be infinite since no matter what the sign of the frequency detuning. The extreme photon hole point is a saddle point. A number of schemes can be used to make the atomic motion in the photon hole region finite. One is based on the use of frequency detuning that varies in sign and over time, making it possible to localize atoms dynamically in a way similar to the localization of ions in high-frequency electromagnetic traps. 3. Atom nanopencil "Atom nanopencil" is an aperture in a thin material foil with a diameter ranging from 1 nm to 1 pm. As atomic beam passes through the aperture arbitrary patterns can be written by moving the aperture relative to a suitable substrate behind the foil. The aperture size defines the feature size of the written atom structures. A quantitative theoretical analysis shows that the van der Waals forces acting on the motion of atoms in the individual tapered channels of the aperture and the diffraction of the atomic wave to determine the final size of the deposited nano~tructures.~ To produce a set of appropriate microholes for use as atom nanopencils, nanosieves have been built and characterized by using the atomic force microscope. The nanosieve has multiple openings of equal diameter, randomly situated in a dielectric foil. We have nanosieves with openings ranging from 40 nm to 1000 nm. In our experiment, a mica or glass substrate was positioned at a distance around 10 pm behind the nanosieves. By moving the nanosieves across an atomic beam, we are able to create atom nanostruc-

200

tures on the various substrates. Fig. 3 shows an example of such nanostructures: we create chromium nanolines with heights in the range of 1-10 nm, widths of 180 nm and lengths of 1200 nm. The total amount of nanolines produced in one process exceeds 107 and occupy an area of 2 x 2 mm on the substrate surface.

Atomic Nanostructure (Cr) 2.5}itri

3 atomic layers

Fig. 3. Nanostructure of Cr atoms built with the atom nanopencil.

4. Atom pinhole camera

The most difficult problem in atom optics is the problem of high-resolution focusing of neutral atoms, which is promising for the nondestructive method for probing the surface at the atomic level, as well as for the creation of nanostructures on the surface. The main difficulty is the creation of the interaction potential of the atom with the electromagnetic field that is close to an "ideal" lens for atoms. We experimentally implement another approach to the problem of focusing and construction of an image in atom optics, which is based on a well known idea of "optical pinhole camera". The pinhole camera in optics is a camera without lens. Light forming an image passes through a small hole. In our experiment with the atom pinhole camera the atomic beam passes through a set of holes in a metal mask and thereby forms, by analogy with optics, a "glowing" atomic object of a given geometry. The atoms pass through the mask, propagate in vacuum along

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rectilinear trajectories, similar to light rays, and are incident on a pinhole. The great many pinholes were created in a thin plastic film.6 As a thin film, we used a track membrane of an asymmetric structure.6 The initial material for the track membrane was a Hostaphan RES (Hoechst AG) polyethylene terephthalate film 5-10 /um thick. The film was irradiated by a 253-MeV accelerated krypton ion beam at the U-400 accelerator (Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research). Then, the film was irradiated by ultraviolet radiation from one side; after that, chemical etching was performed. The thin film with a large number (n =107 — 10s cm~ 2 ) of nanoholes (50 nm diameter) is placed at a distance of 90 mm from the mask. Each hole of the film is a pinhole camera for atoms, which forms its individual image of the object on the substrate surface placed at a distance of I = 5 jum behind the film. Figgure 4(a) shows 2x2 p,m surface section with the image of the "cross" object. In addition to the almost completely formed cross images, the figure also exhibits structures with images of only its part. This occurs owing to the partial blocking of atoms forming the image of the cross of atoms and because the axes of various holes of the track membrane are nonparallel to each other.

Fig. 4. Nanostructures of Cr atoms on the glass surface that are obtained using the pinhole camera and the "cross" atomic object. Surface sections of sizes (a) 2x2 /nm and (b) 800 x 800 nm are shown. The nanostructures are measured by means of an atomic force microscope.

Figure 4(b) shows the detailed image of one of the crosses. As seen in the figure, the cross consists of partially resolved subnanostructures that are images of separate holes of the object mask. The width of the nanostructures is approximately equal to 70 nm, which well corresponds to the passage of

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the atomic beam through the holes of pinhole cameras and is determined by the sum of the hole diameter d = 50 nm and the mask image diameter do = 0.5 mm.

5. Laser-Induced Quantum Adsorption of Atoms on a Surface The adsorption of an atom (molecule) on a surface is a natural process of the trapping of the atom (molecule) in a surface potential well. The probability of trapping of the particle on the surface is determined both by the electronic structures of the particle and surface and by the thermodynamics of the collision of particles with the surface. In addition to the general physical interest in the process of the adsorption of particles on the surface, the adsorption of particles underlies the modern industry of microand nanoelectronics based on the methods of molecular beam epitaxy or gas phase epitaxy. For this reason, the control of processes of the adsorption of particles on the surface are of both fundamental and large applied importance. We propose a new mechanism of loading atoms into the surface potential well (i.e., their adsorption on a surface) and demonstrate the implementation of this scheme for Rb atoms adsorbed on the surface of a YAG We also show the possibility of producing micro- and nanostructures of arbitrary shape that consist of atoms localized on the dielectric surface. The proposed mechanism of the loading of atoms into the surface trap is based on the energy-pooling effect, i.e., inelastic collision of two excited atoms followed by the transition of one of them to the ground state and the other one to the highly excited state.g The defect of the internal energy is compensated by the kinetic energy of the atoms. When the atomic collision occurs inside the surface potential well, an atom can be trapped in this potential well. Figure 5a shows the scheme of the low-lying levels of the Rb atom. The 5 2 P 3 p level of the Rb atom is populated due to the absorption of 780-nm laser light. The energy of the 5 0 level is close to the 5 P 5 P asymptotic energy of the Rb2 molecule and, therefore, can be populated owing to the energy-pooling collisions of excited atoms:

+

Rb(5P)

+ Rb(5P) + A E

4

Rb(50)

+ Rb(5S),

(1)

where A E = 93 K is the difference between the total kinetic energies before and after collision (energy defect). As a result of collision process (1) of two atoms, one atom passes to the ground state and the other atom passes to

203

(a)

.

7-

5

.sp+sp

--

m

0

-

i

- . ~ . ~2.0 . - 3.0. ~ . - . 0.5

1.0

1.5

2.5

Atom to surrace distance, nm

Fig. 5. (a) The energy level diagram for Rb atom. (b) The mechanism of loading of Rb atom in the surface trap.

the 5D excited state. The energy defect is compensated by the energy of the translational atomic motion; i.e., the kinetic energy of two atoms after collision event (1) decreases by 93 K. If the collision of two excited atoms occurs near the bottom of the surface potential well, the loss of the kinetic energy in the atomic collision can lead to the localization of an atom in the surface potential, i.e., to its laser induced adsorption, as schematically shown in Fig. 5b. The quantum adsorption of Rb atoms is experimentally implemented on the surface a YAG crystal. The temperature of the surface can be varied from room temperature to 240 'C. Laser radiation is tuned t o resonance with the Dz line of Rb atom and a laser beam passing through and perpendicular to the surface. The laser beam diameter is varied between 0.5 and 2 mm and the maximum power of laser radiation is equal to 70 mW. The energy-pooling process is identified by detecting the blue fluorescence of Rb atoms (A= 420.2 and 421.6 nm) from the laser beam.g Laser induced quantum adsorption opens the possibility of creating atomic micro- and nanostructures with a given geometry on the dielectric surface. Such a possibility is illustrated in Fig. 6 on the example of creation of the three letters P, R, and L on the surface. Atomic structure

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a

Fig. 6. (a) The mask consists of 35 holes in a thin metal foil, (b) The microstructure of Rb atoms in the form of three letters P R L that are obtained by the laser induced quantum adsorption.

of the shape PRL was produced in a following way: (1) 35 openings in a metal foil were done to produce a mask for these letters; (2) the mask was illuminated by the laser light; (3) the image of the mask was projected onto the surface and during the time interval t = 5 min were produced 35 microtraps on the surface reproducing the letters P, R, and L of size about 50 j,m. 6. Acknowledgments This work was supported by the Russian Foundation for Basic Research (project nos. 06-02-16301-a, 06-08-01299-a, and 05-02-16370-a). References 1. J.J. McClelland et al., Science 262, 877 (1993). 2. V.I. Balykin, V.V. Klimov, V.S. Letokhov, Atom Nanooptics, In "Handbook of Theoretical and Computational Nanotechnology", eds., M. Rieth and W. Schommers, American Scientific Publishers (2006). 3. V.I. Balykin, V.V. Klimov and V.S. Letokhov, Atom Nano-Optics, Optics & Photonics News, 16, 33 (2005). 4. V. Balykin, V. Klimov, Letokhov, JETP Lett., 78, 8 (2003). 5. A.E. Afanasiev, P.N. Melentiev, and V.I. Balykin, JETP, (2007) (to be published). 6. V. I. Balykin, et all., JETP Lett., 84, 466 (2006). 7. A.E. Afanasiev, P.N. Melentiev, and V.I. Balykin, JETP Lett., 86, 198 (2007). 8. A.E. Afanasiev, P.N. Melentiev, and V.I. Balykin, Phys.Rev.Lett., 2007 (to submitted). 9. Z.J. Jabbour, et.al., Phys. Rev. A 54, 1372 (1996).

PINHEAD TOWNTALK, PUBLIC LECTUREAND MOUNTAINFILM

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THE QUANTUM REVOLUTION TOWARDS A NEW GENERATION OF SUPERCOMPUTERS

R. BLATT Institut f u r Experimentalphysik, Universitat Innsbruck, Technakerstr. 25, A-6020 Innsbruck, Austria, and Institut fur Quantenoptik und Quanteninformation (IQOQI), Osterreichzsche Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria Computers that operate with quantum processes promise unprecedented computational power for some algorithms, much more than could be obtained by classical machines. The implementation of such a quantum computer requires the precise control and manipulation of individual quantum systems, a task that can be achieved by quantum optical means and the use of precision laser spectroscopy.

Keywords: Quantum computing, entanglement, trapped ions

1. Introduction

Computing machines have always been a dream of mankind and many mechanical tools have been developed to allow for faster and more complicated calculations. While the tools of the 19th century were mostly of mechanical nature, the technical evolution of the 20th century allowed the construction of fast electronic switches and hence the development of powerful computing machines. This led eventually to the revolution of technology by the ubiquitous availability of personal computers and its corresponding use in all areas of our daily life. 2. Computers and technology

The technical advance in computer technology is well illustrated by Moore's law' that states that nowadays computing power, as indicated by the number of transistors in a processor or the available memory on a chip, doubles approximately every 24 months. Surprisingly enough, this empirical law 207

208 holds since more than 40 years and provides a guideline for the manufacturing industries. Also, the number of atoms that are required, for example, to store the information of one bit, decreases in a similar way.2 While the storage of a single bit required about 10’’ atoms in 1962, only lolo atoms were necessary in 1988 and this trends still holds. Assuming an ongoing development, a simple extrapolation reveals that around 2020 only a single atom would carry the information of one bit. Clearly, a t this stage the laws of quantum mechanics will govern the storing and retrieving of information; in practice, however, quantum mechanical effects will become important a long time before that. Therefore, it seems natural to investigate whether using quantum physics for information processing can actually be used to our advantage. Already in the 1980s Deutsch and Feynman have discussed information processing using quantum physic^,^,^ however, at that time this subject was more an academic exercise since it was neither known how quantum information processing could be particularly useful nor how it could be implemented. This situation changed in 1994 when Peter Shor came up with an algorithm5 to factor large numbers that required only polynomial efforts (in terms of the digits of the number in question) while a classical computer needs an exponential overhead to solve this problem. This application, which is of enormous impact for cryptology, and the later found fast data base search by Lov Grover‘ led to an intense search for physical systems that allow one to really implement quantum computing. Thus, the mid 1990s mark the start of the quest for a quantum computer that is still ongoing and inspiring wide fields in physics today. Meanwhile, quantum information processing has matured as a multidisciplinary area in physics and computer science that ranges nowadays from most fundamental theoretical concepts to implementations using technologies from laser physics and laser spectroscopy to solid state and condensed matter physics and the field is still growing and widening. 3. Quantum bits, registers and gate operations

The smallest unit of classical information is the bit, usually represented by a switch either in ”up”/”down” or ”0”/”1” position. More generally, the quantum system comprised of the two levels 10) and 11) can be written also as the superposition

I+)

= COlO)

+Clll)

(1)

209

and this two-level system is commonly known as a quantum bit, or short, qubit. While a classical bit is best visualized as a switch, the qubit must be described as an arrow pointing somewhere on the surface of a sphere as is indicated in Fig. 1. For a quantum register, a row of several qubits is

T

Fig. 1. Classical information (a,b) and quantum information (c-f). (c,d) show the system in state |0), |1) respectively, (e) describes a superposition of equal weights |co|2 = |ci|2 = l/-\/2 and (f) shows a general, usually unknown superposition.

formed and the corresponding quantum state of the entire register must be considered. Clearly, this state can be, and generally is, in a superposition of all the pertaining states of the individual qubits. This generalizes the superposition concept that we commonly encounter in individual quantum systems to the entire system that seemingly consists of separate two-level systems. However, that point of view is no longer valid since even a manipulation of any one of the qubits in a row will change the quantum state of the entire system. Hence, the quantum register must usually be considered as entangled. Superposition and entanglement are the two features that generally distinguish a classical computer from a quantum computer and their control and manipulation will eventually enable one to speed up computational processes. Analogous to a classical computer, a set of gate operations is required that allow one to formulate a calculation step by step. It has been shown in 1995 that two operations are sufficient to provide a universal set of quantum operations, i.e. that allow one to implement arbitrary computations.7 Aside from the single qubit operation, which implements an arbitrary positioning and rotation of the arrow (see Fig. 1) given on the sphere, it is necessary to allow for conditional operations equivalent to the classical XOR operation in computer science. With such operations, the state of a target qubit can

210

be changed depending on the state of a control qubit. The corresponding truth table is shown in Fig. 2 and looks exactly like the classical XOR operation, however, it needs to hold for all coherent superpositions. Thus,

|0) 1}

11)11} |1)|0)

Fig. 2. Truth table of a CNOTgate operation. The state of a target bit is flipped if and only if the control bit is in state " 1". The notation indicates that this must hold for all superpositions.

I*

the concept of a quantum computer can be visualized as shown in Fig. 3. Starting from an arbitrary input state

E

ne{0,l}"

cn\ni,.. .,

El

(2)

the computation works as a series of one and two-qubit operations according to the specific algorithm under consideration and the outcome is just another superposition f ( \ x ) } with an operation f that can be described by a unitary operator. To this point a quantum computation is completely reversible. The outcome of the calculation is then obtained by a measurement that projects the system on its eigenstates and yields classical information in terms of zeros and ones for the individual qubits. The realization concepts of a specific quantum computer and its implementation vary widely depending on the quantum system considered, the quantum operations that are performed and the measurements that are taken to obtain classical information. On the other hand, there are a few building blocks and requirements that can be generally defined and described and that are common to all quantum computers. The requirements for a system to be considered for the implementation of a quantum computer are currently known as the so-called DiVincenzo criteria.8 Irrespective of the specific system, a quantum computer clearly needs (1) storage sites for the quantum information, i.e. qubits that can be arranged to form a (scalable) quantum register, (2) the possibility to initialize the qubits to

211

Quantum Processor

Fig. 3. Scheme of a circuit model quantum computer illustrated with 3 qubits. An arbitrary input superposition state x) is processed with a series of one and two-qubit operations described by a unitary operation F(\x)). (0,if) denote single qubit rotations, where (6, if) may vary from pulse to pulse depending on the algorithm.

arbitrary states, (3) long unperturbed computation (coherence) times, (4) a universal set of gate operations to allow for universal computations and (5) a highly efficient measurement of the qubit states to read out the result of the computation. In order to achieve large scale quantum computation, (6) the system under consideration should allow for a conversion between the stored, i.e. static and flying qubits that (7) can be faithfully transmitted between two quantum computer nodes. During the last decade a large number of systems has been and still is investigated for its suitability to implement a quantum computer.9 In particular, quantum optical systems of atoms and ions in traps, using the tools of laser spectroscopy, are among the most promising candidates for such a device. 4. Quantum computer with trapped ions

One of the first and seminal schemes to implement a scalable quantum computer was proposed by I. Cirac and P. Zoller in 1995.10 They considered a string of laser cooled trapped ions in a Paul trap as quantum register and formulated how a CNOT-gate operation can be realized with laser pulses that individually address the ions. The crucial idea was that the harmonic motion of ions in the trap can be used as a quantum bus. That allows one

212 to map the excited state of the controlling qubit to the motion and thus enables state manipulation of the target qubit conditioned on the motion and thus the controlling qubit. However, this requires that the ion string is optically cooled to the ground state of the harmonic oscillator which can be achieved in ion traps by sideband cooling.11 Moreover, the state of laser cooled ions in traps can be detected with nearly 100% efficiency using the "shelved electron technique", where the absorption of a single photon results in a lack of a huge number of fluorescence photons on a monitoring transition.11 As is well known from precision spectroscopy, trapped ions also offer extremely long coherence times12 and therefore provide an ideal system for the implementation of a quantum computer. Fig. 4 shows schematically

70 pm Fig. 4. Sketch of a linear ion trap holding a string of laser cooled ions that can be individually addressed for quantum state manipulation. With a CCD camera the fluorescence light is detected and allows for efficient state detection. how a string of laser cooled ions is manipulated with focussed laser beams. State detection is achieved by observing resonance fluorescence with a CCD camera. With such setups, single qubit operations were performed, and CNOT-gate operations have been implemented with trapped ions13"16 using the Cirac-Zoller idea as well as other proposals based on geometric phase changes.17 5. Simple quantum computations With a small trapped ion quantum computer, both the Boulder13 and the Innsbruck14 groups have demonstrated simple quantum algorithms. As a

213

basic building block, the CNOT gate operation (or an equivalent phase gate operation) was used to generate Bell states at the push of a These subsequently could be used as a resource for a demonstration of the teleportation of an atomic ~ t a t e . Non-classical ’ ~ ~ ~ ~ entangled states of three particles, in particular the GHZ and W states were deterministically and analyzed using quantum state t~mography.’~ A quantum Fourier transform was implementedz4 and error correction was demonstrated by the Boulder Multi-partite entanglement was achieved and analyzed for a 6-ion GHZ statez3 and an 8-ion W state,z6 the latter demonstrating for the first time a quantum byte. This experiment for the first time also manifested the power of quantum computation: While the creation of an eight-ion entangled state takes just about 1 ms to achieve, its analysis via quantum state tomography required a lot more efforts. In order to obtain all density matrix elements (i.e. 256x256 entries) more than 6500 different qubit rotations had to be applied that took more than 10 hours of uninterrupted running time of the quantum computer. Moreover, the data analysis on a classical computer required a raw computing time of several days on a computer cluster. This clearly demonstrates that even with relatively few qubits highly complex states can be created. 6. Future developments

With the ion trap quantum computer quantum information processing clearly has become reality, albeit on a small scale yet. On the other hand, there exist already architectures to scale such a system up.28 Of course, this still requires enormous efforts in physics and technology, especially in order to meet the requirements for error correction. While this is technically involved, there is however not a real roadblock in sight at this time. Therefore, current efforts are dedicated especially towards the development of yet smaller, so-called segmented ion traps that allow one to move the ions and thus the information around.2g With such chip traps available, there is good hope for an even broader application of quantum information processing. In particular, the use of entanglement for precision ~ p e c t r o s c o p y seems ~~>~~ a very promising avenue for further enhanced measurements and sensor technology.

7. Conclusion While the ion trap quantum computer seems farthest advanced at this time, it is quite foreseeable that other quantum optics and laser spectroscopy

214

technologies, e.g. based on atoms in lattices, quantum dots, or superconducting qubits will become very strong alternatives soon. The steadily growing field of quantum information has enormously profitted from the fields of laser spectroscopy, laser cooling and precision measurements. On the other hand, quantum information processing provides us with new tools that allow us to further enhance our understanding of fundamental physics as well as to develop future devices. References 1. G. Moore, Electronics 38, Nr. 8, April 19, (1965); see also http://www.intel.com/technology/mooreslaw/. 2. R. W. Keyes, IBM J. R&D. V32, N1, Jan 1988, p.26 3. D. Deutsch, Proc. R. Soc. London A 400, 97 (1985). 4. R. Feynman, Int. J. Theor. Phys. 21, 467 (1982). 5. P. W. Shor, Proceedings of the 35th IEEE Symposium on Foundations of Computer Science, pages 124-134. IEEE, 1994. 6. L. Grover, in Proceedings of the 28th Annual ACM Symposium on the Theory of Computation (ACM Press, New York, 1996), pp. 212219. 7. A. Barenco, C.H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, T. Sleator, J . A. Smolin, and H. Weinfurter, Phys. Rev. 52, 3457 (1995). 8. D. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001). 9. for the US roadmap see http://qist.lanl.gov/ and for the European roadmap see http://qist.ect.it/. 10. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). 11. D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003) and references therein. 12. D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, J. Res. Natl. Inst. Stand. Technol. 103, 259 (1998) and references therein. 13. D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenkovic, C. Langer, T. Rosenband, and D. J. Wineland, Nature 422, 408-411 (2003). 14. F. Schmidt-Kaler, H. Haffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner and R. Blatt, Nature 422, 412415 (2003). 15. J. P. Home, M. J. McDonnell, D. M. Lucas, G. Imreh, B. C. Keitch, D. J . Szwer, N. R. Thomas, S. C. Webster, D. N. Stacey and A. M. Steane, New Journal of Physics 8, 188 (2006). 16. P. C. Haljan, P. J . Lee, K-A. Brickman, M. Acton, L. Deslauriers, and C. Monroe, Phys. Rev. A72, 062316 (2005). 17. K. M ~ l m e rand A. S~rensen,Phys. Rev. Lett. 82, 1835 (1999); G. J. Milburn, S. Schneider, and D. F. V. James, Fortschr. Phys. 48, 801 (2000). 18. Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland; Phys. Rev. Lett. 81, 3631 (1998).

215 19. M. Riebe, H. Haffner, C. F. Roos, W. Hansel, J. Benhelm, G. P. T. Lancaster, T. W. Korber, C. Becher, F. Schmidt-Kaler, D. F. V. James, R. Blatt, Nature 429, 734 (2004). 20. M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri and D. J. Wineland, Nature 429, 737 (2004). 21. C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, R. Blatt, Science 304, 1478 (2004). 22. D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J . Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J . Wineland, Science 304, 1476 (2004). 23. D. Leibfried, E. Knill, S. Seidelin, J . Britton, R. B. Blakestad, J . Chiaverini, D. B. Hume, W. M. Itano, J . D. Jost, C. Langer, R. Ozeri, R. Reichle and D. J . Wineland, Nature 438, 639-642 (2005). 24. J. Chiaverini, J. Britton, D. Leibfried, E. Knill, M. D. Barrett, R. B. Blakestad, W. M. Itano, J . D. Jost, C. Langer, R. Ozeri, T. Schaetz,ddagger D. J. Wineland, Science 308, 997-1000 (2005). 25. J . Chiaverini, D. Leibfried, T. Schaetz, M. D. Barrett, R. B. Blakestad, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, R. Ozeri, D. J. Wineland, Nature 432, 602-605 (2004). 26. H. Haffner, W. Hansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Korber, U. D. Rapol, M. Riebe, P. 0. Schmidt, C. Becher, 0. Guhne, W. Dur and R. Blatt, Nature 438, 643-646 (2005). 27. C. F. Roos, G. P. T. Lancaster, M. Riebe, H. Haffner, W. Hansel, S. Gulde, C. Becher, J. Eschner, F. Schmidt-Kaler, R. Blatt, Phys. Rev. Lett. 92,220402 (2004). 28. T. S. Metodi, D. D. Thaker, A. W. Cross, F. T. Chong, I. L. Chuang, arXiv:quant-ph/0509051vl. 29. D. Kielpinski, C. Monroe, D. J . Wineland, Nature 417, 709-711 (2002). 30. P. 0. Schmidt, T. Rosenband, C. Langer, W. M. Itano, J. C. Bergquist, and D. J . Wineland, Science 309, 749-752 (2005). 31. C. F. Roos, M. Chwalla, K. Kim, M. Riebe, R. Blatt, Nature 443, 316-319 (2006).

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COLD ATOMSAND MOLECULESI

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ULTRACOLD & ULTRAFAST: MAKING AND MANIPULATING ULTRACOLD MOLECULES WITH TIME-DEPENDENT LASER FIELDS C. P. KOCH

Institut fur Theoretische Physik, Freie Universitat Berlin, Arnimallee 14, 14195 Berlin, Germany E-mail: christiane.koch@physik. fu- berlin. de

R. KOSLOFF The Fritz Haber Research Center, T h e Hebrew University of Jerusalem, Jerusalem 91904, Israel

E. LUC-KOENIG and F. MASNOU-SEEUWS Laboratoire Aim6 Cotton, C N R S and Universite' Paris Sud, B i t i m e n t 505, 91405 Orsay, France

R. MOSZYNSKI Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland While ultracold matter brought quantum effects onto the macroscopic scale, ultrafast lasers made quantum dynamical phenomena observable in real-time. Bringing the two together seems natural and holds the promise of employing quantum interferences in an unprecedented way. Photoassociation provides an optimal framework for the merger since in principle it relies only on the presence of optical transitions. Combining it with coherent control where the potential energy surfaces governing the dynamics can be 'shaped', a general route toward stable ultracold molecules is obtained.

1. Introduction

The formation of ultracold (T 5 100 pK) molecules along with the creation of quantum degeneracy in molecular gases offers exciting perspectives for investigating ultracold chemistry, implementing new schemes for quantum computing, building a molecule laser, and performing high precision measurements.' Moreover, since at ultralow temperatures only a single or very few quantum states contribute to the dynamics of a process, the ultracold 219

220 regime holds the promise of achieving coherent control in an unprecedented way. Coherent control relies on the superposition principle of quantum mechanics, and in particular on the ability to manipulate the relative phases of different quantum pathways. In a theoretical approach one can formulate control as an inversion problem, asking for the 'potential' (containing the potential energy surfaces as well as the interaction of the molecule with an external field) which generates the desired dynamics. For state-tostate transfer, i.e. when starting from a given initial state with the goal of reaching a certain target state, efficiencies close to 100% can be obtained. This was demonstrated for the example of stabilizing molecules in a highly excited vibrational state just below the dissociation limit to the vibrational ground state.2 The experimental approach is based on feedback loops where a genetic algorithm steers a pulse shaping device and the outcome of each measurement is used to create the next generation of pulses. At high temperatures prevailing in experiments to date, averaging over many thermally populated states is implied such that quantum interference effects are washed out or hidden. Thermal averaging can be avoided by going to very low temperatures, and it thus seems natural to combine the techniques of coherent control with ultracold gases. However, first experiments aimed a t photoassociating ultracold atoms to form molecules with femtosecond laser pulse^^,^ had to face a number of difficulties due to the high peak intensities of short lasers, interference with the trapping light, and the slow timescales of ultracold collisions. Nevertheless, coherent photoassociation could provide a means to finally realize control of a binary reaction. In the following, a brief overview over the theory of coherent photoassociation will be given. The concept of the pump-dump scheme is introduced in section 2, followed by examining in section 3 which potentials are required to make this scheme work. Section 4 will present a new proposal to engineer favorable potentials by an external field in order to ensure a successful route toward ground state molecules, and section 5 concludes. 2. The concept of pump-dump photoassociation

In photoassociation, two colliding ground state atoms are transferred to an electronically excited state due to the interaction with a laser which is red-detuned with respect to the atomic resonance f r e q ~ e n c y .Depending ~)~ on the shape of the excited state potential, molecules in the electronic ground state can be obtained by spontaneous emission or by applying a

221

11900 11800 11700 3

i3 h

f W

100 0

-100 -200

20

40 60 80 100 internuclear distance [ Bohr radii ]

120

Fig. 1. Pump-dump photoassociation, shown here for Cs2: A first pulse (pump) excites part of the initial ground state population, creating a wavepacket in the excited state. A second pulse (dump), delayed in time and blue detuned with respect to the first pulse, transfers part of the excited state wavepacket into bound levels of the ground state.

second laser field, blue-detuned with respect to the first one. In order to render this molecule formation scheme coherent, two requirements need to be met: (i) the timescale of the process should be shorter than that of spontaneous emission, and (ii) the time-reversal symmetry needs to be broken in order to avoid back-stimulation. Both conditions can be fulfilled by applying picosecond pulses, possibly chirped, in a pump-dump sequence, cf. figure This concept was first applied to the formation of Cs2 and R b ~ , ~ 1two ' systems that had extensively been studied in experiments using continuous wave lasers. The effect of the pump pulse shall be examined first.g Due to the finite bandwidth of the pump pulse, the resonance condition is fulfilled for a range of internuclear distances around the Condon radius, giving rise to the concept of the photoassociation The pulse intensity can be chosen such as to deplete the initial population within the photoassociation window, creating a hole in the ground state wavefunction. This is depicted in the lower panel of figure 2. The corresponding pulse energies are fairly low, amounting to a few nano-Joule.' The pump pulse creates a wavepacket in the excited state whose vibrational distribution is shown in the upper panel of figure 2. The width of this distribution is given by the spectral bandwidth of the pulse (inversely

222 8x10

•a

a! 4xlOH 'o OH

10 15 20 binding energy [ cm ]

25

20 40 60 80 100 internuclear distance R [ Bohr radii 1 Fig. 2. Applying a photoassociation pump results in a hole in the ground state distribution (bottom) and an excited state wavepacket, represented by its vibrational distribution (top), shown here for Rb2.

proportional to the duration of a transform-limited pulse rp as used in figure 2). The detuning Ap of the pulse determines the peak position of the distribution.8 The small feature close to zero detuning should be noted: It appears despite the spectral amplitude of the laser field being almost zero, and it is due to the huge free-bound Franck-Condon factors which increase by several orders of magnitude as the detuning, or, respectively, binding energy, becomes smaller. Thus it is imperative to employ red-detuned narrowbandwidth pulses8 or, alternatively, to remove spectral components exciting the atomic resonance from a broad-bandwidth pulse.3'4 Once transferred to the excited state, the wavepacket starts to evolve under the influence of the excited state potential, cf. figure 1. The oscillation period for binding energies of one to a few wavenumbers is on the order of 200 ps for the heavy alkali dimers Rb2 and Cs2- The time delay between the pump and dump pulses should be chosen such that the overlap between the time-dependent wavepacket and the bound levels of the ground electronic state becomes maximal. This is the case when the wavepacket reaches the inner turning point, cf. figure 1. Two different regimes for the dump pulse are then conceivable:7 a narrow-bandwidth pulse populates a single ground state level, while a broad-bandwidth pulse achieves maximum electronic ground state population which is distributed over several vibrational levels. In both cases, the dump pulses are short compared to the vibrational period of the excited state potential, and the dynamics can be understood in terms of an instan-

223 taneous two-level s y ~ t e m . ~ Since, > ~ > lin~ analogy to the pump pulse, weak fields and no 'shaping' of the dump pulse beyond a linear chirp is assumed, the transition probabilities are determined solely in terms of spectral amplitudes and Franck-Condon factors. Therefore, how much of the excited state wavepacket can actually be transferred to molecular levels of the ground state, depends crucially on the shape of the excited state potential.

3. The role of the excited state potentials The pump-dump scheme involves an excitation and a stabilization step, and the overall efficiency is determined by those of both steps combined. Figure 3 illustrates possible underlying physical mechanisms: Efficient excitation is afforded by the long-range attractive -1/R3 behavior of the excited state potential (upper panel, shown here is the Cs2 O;(p3/2) state), yielding large free-bound Franck-Condon factors. The -1/R3 behavior is caused by the resonant dipole interaction between two identical atoms where one is in its electronic ground state, while the other one is in the first excited state. Hence this behavior is observed in almost all excited state potentials of homonuclear dimers correlating to the n s n p asymptote (with the exception of cases where the 1/R3-term is accidentally cancelled such as in

+

O,(Pl/2)).

--

11900

5

y

11800

Y

8 11700 20

40 60 80 I00 internuclear distance Ia,u -1

120

~

--

2mo 1 1900

5

' & 16000 I

-,llS00 >

p 12000

D 8 11700

B

so00

1 100

20

40 60 80 100 internuclear distance [ a. ]

120

20

40 60 80 internuclear distance [ a, ]

100

Fig. 3. Excited state potentials with -l/R3 long-range behavior provide an efficient mechanism for the pump step (upper panel), while the dump step (lower panels) can be facilitated by a softly repulsive +l/R3 potential wall at intermediate distances (left) or by resonantly coupled excited state potentials (right).

224

x10

Timefps] 600_^g Energy f cm ]

[cm l]

Fig. 4. Overlap of the excited state wavepacket with ground state levels \{
In general, stabilization is difficult. Two mechanisms which lead to efficient stabilization are shown in the lower panel of figure 3. A repulsive + l/ J R 3 -term at intermediate range, shown for the double-well 0~(j»3/2) potential of Cs2 slows the wavepacket down, leading to a piling up of probability amplitude at the inner turning point of the long-range well.7 Due to the softly repulsive nature of the potential wall, the wavepacket stays close to this turning point for a few picoseconds such that it can easily be caught by the dump pulse. The wavepacket dispersion can be minimized, i.e. the wavepacket can be focussed at the inner turning point, by applying an appropriately chosen negative chirp to the pump pulse.7'9'10 About 20% of the overall excited state population can be transferred to bound levels of the ground state potential, the remaining part, if addressed by the dump pulse, is dissociated.7 The dissociation can be attributed to the high-momentum components which the wavepacket acquired while propagating on the 1/-R3 potential and which cannot be projected onto bound levels of the ground state with its l/R6 long-range behavior. Two excited state potentials which are coupled resonantly by e.g. spinorbit interaction provide a second mechanism for stabilization to bound levels by the dump pulse (lower right panel of figure 3).8'11 Some of the vibrational wavefunctions of the coupled potentials show then a doublepotential structure with four turning points. While the outermost turning point implies good free-bound Franck-Condon factors for the pump pulse, the next turning point at intermediate distances leads to greatly improved bound-bound Franck-Condon factors for the dump pulse.12 If none of the two stabilization mechanisms is present which is true for most excited state

225

potentials, it is not possible to form ground state molecules by the dump pulse except in the very last vibrational levels. This is demonstrated in figure 4 which compares the time-dependent overlap of the excited state wavepacket with bound ground state levels for the resonantly coupled 0+ states and the generic lg states of Rb2 (the binding energy of the excited state wavepacket is identical in both cases). 4. Engineering favourable potentials

The potentials which govern the dynamics can be 'shaped' by applying laser fields to a molecular system, thus steering the dynamics in the desired direction. This is the fundamental message to be learnt from coherent control. However, if this concept is applied straightforwardly to two generic potentials coupled by a laser field, all work needs to be done by the field. In an example where molecules in levels close to the dissociation limit of the electronic ground state potential shall be transferred to their vibronic ground state, very high pulse energies and many Raman steps are then required.2 The high intensity of a short pulse can be avoided by employing a train of pulses over a longer time leading to coherent population accumulation.13 The number of necessary Raman steps can be minimized by joining the concepts of resonant coupling and of shaping the potentials, i.e. by inducing resonant coupling by an external field. This concept is visualized in figure 5 with potentials corresponding wavepacket motion

25000 20000

coupling

I"115000

pump

'fi 1000 _!i ° So 5000 O) 0) 500 0 -500 -1000

dump

5

10

15

20

25

30

50 100 150

internuclear distance R [ Bohr radii ] Fig. 5. Engineering resonant coupling by a strong off-resonant laser field.

226 to the calcium dimer. A strong off-resonant infrared laser field couples two excited state potentials, while the pump and dump pulses couple the ground state to one of the excited states. The coupling laser modifies the wavepacket dynamics in the excited state such that dumping to deeply bound levels of the ground state, including u = 0, becomes fea~ib1e.l~ Depending on the target level, the frequency of the coupling laser field should be in the infrared to near infrared with intensities on the order of lo8 - lo9 W/cm2. The coupling laser field can be realized using nanosecond pulses, while the pump and dump pulses should be picosecond pulses as discussed before in section 2 for the alkali dimers.

5 . Conclusions

The formation of stable molecules from ultracold atoms with short laser pulses in a pump-dump scheme was presented. Picosecond pump pulses are expected to be the most efficient as they can best make use of large free-bound F'ranck-Condon factors occuring close t o atomic resonance while still being much shorter than the excited state lifetimes. In order to obtain ground state molecules with binding energies larger than one wavenumber, special stabilization mechanisms are required. Softly repulsive potential walls a t intermeditate distance or resonantly coupled excited states lead to appreciable overlap of the time-dependent excited state wavepacket with ground state levels a t the time when the wavepacket is close to its inner turning point. However, most excited potentials do not show such a favorable topology. Pump-dump photoassociation creates then only molecules in the very last vibrational levels which are extremely weakly bound. If these molecules shall be transferred to their vibronic ground state, a large number of Raman transitions is required. However, building on the coherent control idea of shaping the potentials, an additional strong off-resonant laser field can be employed which modifies the excited state dynamics such that transitions to target levels including 'u = 0 become feasible. The 'inverse' concept of de-coupling (instead of coupling) two excited state potentials with a strong off-resonant infrared laser pulse has already been realized successfully.l5 Our work demonstrates the potential of combining ultracold molecular processes and coherent control concepts.

Acknowledgements

This work has been supported by the Deutsche Forschungsgemeinschaft (C.P.K.) and by the European Commission in the frame of the Cold Molecule Research Training Network under contract HPRN-CT-200200290.

References 1. J. Doyle, B. Friedrich, R. V. Krems and F. Masnou-Seeuws, Eur. Phys. J . D 31, 149 (2004). 2. C. P. Koch, J. P. Palao, R. Kosloff and F. Masnou-Seeuws, Phys. Rev. A 70, p. 013402 (2004). 3. W. Salzmann, U.Poschinger, R. Wester, M. Weidemiiller, A. Merli, S. M. Weber, F. Sauer, M. Plewicki, F. Weise, A. Mirabal Esparza, L. Waste and A. Lindinger, Phys. Rev. A 73, p. 023414 (2006). 4. B. L. Brown, A. J. Dicks and I. A. Walmsley, Phys. Rev. Lett. 96, p. 173002 (2006). 5. F. Masnou-Seeuws and P. Pillet, Adv. in At., Mol. and Opt. Phys. 47, 53 (2001). 6. K. M. Jones, E. Tiesinga, P. D. Lett and P. S. Julienne, Rev. Mod. Phys. 78, p. 483 (2006). 7. C. P. Koch, E. Luc-Koenig and F. Masnou-Seeuws, Phys. Rev. A 73, p. 033408 (2006). 8. C. P. Koch, R. Kosloff and F. Masnou-Seeuws, Phys. Rev. A 73, p. 043409 (2006). 9. E. Luc-Koenig, R. Kosloff, F. Masnou-Seeuws and M. Vatasescu, Phys. Rev. A 70, p. 033414 (2004). 10. E. Luc-Koenig, F. Masnou-Seeuws and M. Vatasescu, Eur. Phys. J . D 31, 239 (2004). 11. C. M. Dion, C. Drag, 0. Dulieu, B. Laburthe Tolra, F. Masnou-Seeuws and P. Pillet, Phys. Rev. Lett. 86, 2253 (2001). 12. H. K. Pechkis, D. Wang, Y. Huang, E. E. Eyler, P. L. Gould, W. C. Stwalley and C. P. Koch, Phys. Rev. A 76, p. 023425 (2007). 13. A. Pe’er, E. A. Shapiro, M. C. Stowe, M. Shapiro and J. Ye, Phys. Rev. Lett. 98, p. 113004 (2007). 14. C. P. Koch and R. Moszyriski, in preparation. 15. B. J. Sussman, D. Townsend, M. Y. Ivanov and A. Stolow, Science 314, 278 (2006).

BOSE-EINSTEIN CONDENSATES ON MAGNETIC FILM MICROSTRUCTURES M. SINGH, S. WHITLOCK, R. ANDERSON, S. GHANBARI, B. V. HALL, M. VOLK, A. AKULSHIN, R. McLEAN, A. SIDOROV AND P. HANNAFORD ARC Centre of Excellence for Quantum-Atom Optics, and Centrefor Atom Optics and Ultrafast Spectroscopy, Swinburne University of Technology, Melbourne, Australia 3122 Ernail: phannaford@swin. edu.au We report on recent experiments with BECs and ultracold atoms in magnetic microtraps created by magnetic film microstructures on an atom chip. We describe the use of RF spectroscopy plus absorption imaging of ultracold atoms as a sensitive high resolution technique to map the magnetic field topology of the magnetic film. We find that the distribution of condensate atoms between the wells of an asymmetric double-well potential on the magnetic film can provide a highly sensitive technique for determining potential gradients, such as gravitational gradients, using an atom chip. Finally, we report on the use of periodic magnetic film microstructures to generate a magnetic lattice for manipulating BECs and ultracold atoms.

1. Introduction

At the 2001 Laser Spectroscopy conference in Snowbird the Tubingen and Munich groups reported the realization of a Bose Einstein condensate (BEC) in a magnetic microtrap created by current-carrying micro-wires on a substrate [ 1, 21. Such ‘atom chips’ can produce tightly confining magnetic potentials using modest electric currents, thus simplifying and speeding up the production of a BEC. In addition, atom chips allow precise control of the motion and position of the condensate, and they are compact and relatively robust, making them attractive for BEC-based devices and applications. Since Snowbird more than 20 groups around the world have now produced BECs om an atom chip. The performance of atom chips based on current-carrying conducting wires has, however, certain limitations. Current instabilities and current noise can limit the lifetime and coherence properties of a condensate, and high current densities can lead to excessive heating and breakdown of the conductors [3]. In addition, small imperfections in the wires can lead to tiny spatial deviations in the current flow which can fragment the atom cloud [4], while thermal fluctuations associated with Johnson noise in the conducting wires can lead to spin flips and a loss of atoms close to the surface of the chip [5,61. 228

229 At the last Laser Spectroscopy conference, in Aviemore, we reported the observation of a BEC on a permanent magnetic film atom chip [7, 81 based on perpendicularly magnetised TbGdFeCo magneto-optical films previously developed for our atom optics experiments [9-111. Magnetic film microstructures can produce highly stable, tightly confining magnetic potentials without ohmic heating, and they can produce very fine-structured magnetic potentials, with periodicities down to about 1 pm [l 11, and complex magnetic potentials such as ring-shaped structures. Although the potentials from magnetic film microstructures are necessarily static, they can be combined with magnetic potentials from current-carrying conductors when time-dependent magnetic fields are required, e.g., when loading the atom chip. BECs on permanentmagnet atom chips have also recently been produced using videotape [12], a CoCrPt hard disk [ 131 and FePt thin films [14]. In this paper we report on recent experiments with BECs and ultracold atoms on perpendicularly magnetised magnetic film microstructures on an atom chip. We describe a sensitive high resolution technique based on RF spectroscopy plus absorption imaging of ultracold atoms to map the magnetic topology of the magnetic film. 1151. We find that the distribution of condensate atoms between the wells of an asymmetric double-well potential on the magnetic film can provide a highly sensitive technique for determining potential gradients, such as gravitational gradients, using an atom chip 1161. Finally, we report on the use of periodic magnetic film microstructures to generate a magnetic lattice for manipulating BECs and ultracold atoms [ 17-191 2.

The Magnetic Film Atom Chip

Our atom chip consists of a perpendicularly magnetised 1 pm-thick TbGdFeCo magneto-optical film deposited on a 300 pm-thick glass substrate mounted on a silver foil ‘circuit’, which provides time-dependent magnetic fields for initial trapping and loading of ultracold atoms in the magnetic film trap. The silver circuit has U- and Z-shaped wires for a surface magneto-optical trap (MOT) and a Ioffe-Pritchard magnetic trap. The magnetic film microtrap is produced by the field above an edge of the film plus a bias field Bblasto provide radial confinement and two end-wires to provide weak axial confinement [8] (Fig. 1). The magneto-optical films are prepared in-house by magnetron sputtering at substrate temperatures of 100-200°C 1111. We typically use films comprising six 150 nm layers of Tb10Gd6Fe8&04separated by 100 nm layers of Cr. The films have high perpendicular anisotropy with excellent magnetic homogeneity, high remanent magnetization (- 3 kG), high coercivity (- 2 kOe) and a

230

Figure 1. Schematic of the magnetic film atom chip. Long range inhomogeneity in the magnetic film leads to fragmentation of the trapped atom cloud when close to the surface. After [15].

nominally high Curie temperature (~ 300°C). The magneto-optical films are coated with a reflecting gold film for use in a mirror MOT. The atom chip is mounted in the vacuum chamber with the magnetic film facing down. Initially, about 2 x 108 87Rb atoms are collected in the mirror MOT located 5 mm below the surface, optically pumped into the |F=2, mF=+2> tapping state and subsequently transferred to the Z-wire microtrap. After a preliminary RF evaporative cooling stage, the atoms are transferred to the magnetic film microtrap by adiabatically reducing the current in the Z-wire to zero. Forced RF evaporative cooling of the atom cloud in the magnetic film microtrap leads to a BEC with about 105 atoms [7, 8]. Temperature measurements on a dilute cloud of ultracold atoms indicate a remarkably low heating rate of 3 nK s"1 in the magnetic film trap, compared with 270 nK s"1 in the Z-wire trap, and a trap lifetime of about 5 s, which is limited by time-varying stray magnetic fields. 3. Spatially Resolved RF Spectroscopy to Probe Magnetic Field Topology The small kinetic energies and small spatial extent of the cloud of ultracold atoms can provide a sensitive high resolution (~ 5 urn) probe of corrugations in the magnetic potential [20]. Here, we employ precision RF spectroscopy plus high resolution imaging of tapped ultacold atoms to probe the magnetic field topology along the edge of the magnetic film [15]. After allowing the atom cloud to expand along the 5 mm edge of the magnetic film by reducing the current in the end-wires, a ramped RF field is applied perpendicular to the trap axis to resonantly out-couple atoms to untapped magnetic states at positions where the RF frequency matches the Zeeman splitting of the atoms. At the end of the ramp the resonant frequency approaches a final cut-off frequency vf corresponding to the bottom of the tap, and the atom distribution is described by a truncated Boltzmann distribution which is characterised by a spatially dependent truncation parameter [15].

231

j~ 0.8 P ' >, 0.4 L

I

I.

-S.S

-2

-1.5

I

- 1

I

-0.5

i

0

I

O.B

U_

1

J

J

1.6

2

2.8

longitudinal position z (mm) Figure 2. Absorption images for a 87Rb cold atom cloud in the magnetic microtrap located 67 urn below the edge of the magnetic film, for a RF cut-off frequency vf of (a) 1238 kHz, (b) 890 kHz, (c) 766 kHz and (d) 695 kHz. Trap parameters: Sbias = 5.7 G, field offset B0 = 0.82 G, cor = 27i x 1070 Hz. Initial cloud temperature T= 10 uK. After [15]. (3 (z, Vf) = [mFh vf - mpgfUB \ Bz(z) + B0 \ }lkBT,

where B0 is an offset field and T is a fit parameter that characterises the nonequilibrium distribution during truncation. Figure 2 shows absorption images for a cold atom cloud in the magnetic microtrap located 67 um below the magnetic film edge for four values of the RF cut-off frequency Vf. For Vf < 1.3 MHz significant fragmentation of the atom cloud is observed, and for Vf < 0.9 MHz well separated regions appear corresponding to atoms in the lowest potential wells. Using an iterative procedure we extract values of I B^y(z) \ and obtain the reconstructed magnetic field profiles shown in Fig. 3 for a number of trap heights. We find that the amplitude of the observed magnetic corrugations falls off with distance y as ^,-1.85 ±0.3 yj^ characteristic period of the corrugations is about 390 um for j > 1 0 0 u m with additional higher frequency components appearing as the atom cloud is brought closer to the film. The amplitude of the corrugated potential is about three orders of magnitude larger than that estimated from the roughness (~ 50 nm) of the polished film edge, but is consistent with the y'2 behaviour predicted for long range inhomogeneities in the magnetisation of the film [15]. Investigating this further, measurements taken with the atom cloud positioned below the magnetic film at x= 100 um from the edge show significant fragmentation while those recorded below the non magnetic film at x = -100 um from the edge show almost no corrugation, thus confirming that the fragmentation results from spatial variations of the magnetisation in the body of the magnetic film rather than from fluctuations along the edge of the film.

232

I -2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

longitudinal position z (mm) Figure 3. Magnetic field profiles B,(z) measured using spatially resolved RF spectroscopy (solid lines) for various distances y below the edge of the magnetic film. The dotted lines correspond to measurements of the magnetic film edge using a scanning magneto-resistance probe, which measures the By component of the corrugated magnetic field. The relative longitudinal offset has been adjusted for optimal agreement. After [15].

After removing the magnetic film atom chip from the vacuum chamber, the magnetic corrugation was further characterised using a home-built magnetoresistance microscope, which allows measurements over a wide range of distances with a spatial resolution of about 50 pm. The magneto-resistance measurements, indicated by the dotted lines in Fig. 3 , show a remarkable correlation with the FW spectroscopy cold atom data, thus verifying the validity of the RF spectroscopy cold atom technique. The observed magnetic inhomogeneity in the magnetic films is attributed to deterioration during vacuum bake out of the magnetic film atom chip (140°C for 4 days). Assuming the inhomogeneity originates from reversal of magnetic domains, we conclude that the mean magnetisation of the film is about 90% of the saturation value. After remagnetising the film the level of inhomogeneity was reduced by about a factor of 10. 4. Dynamic Splitting of a BEC in an Asymmetric Double Well

In the reconstructed magnetic field profiles in Fig. 3 , we identify a double-well potential in the region near z = 0 when the magnetic trap is brought close to the surface of the film by increasing Bbiasin the x direction [16]. The double well originates from higher spatial frequency components of the magnetisation inhomogeneity at distances close to the magnetic film. Figure4 (left) shows a series of high resolution absorption images as a BEC is slowly brought (in 0.5 s) from about 170 pm below the film, where the

233

Figure 4. Left: dynamic splitting of a BEC in a double-well potential. High resolution absorption images are shown as the BEC is slowly moved from about 170 (xm below the film to about 57 ^m below the film. Right: characterisation of the double-well potential as a function of trap-surface separation is performed using two-component clouds. The well separation A,, barrier height (3 and trap asymmetry A are shown schematically in (c). After [16].

potential behaves as a single well, to 57 um below the film, where it behaves as a double well. Under these conditions the BEC dynamically splits into two, with about equal numbers of atoms in each well. The double well is characterised by taking absorption images for various trap heights y and determining the well separation A, and barrier height (3 as a function of y (Fig. 4, right) [16]. The trap frequencies of the two wells, obtained from dipolar oscillations, are identical within a few percent. The asymmetry A in the double well can result in a marked difference in the number of atoms in the two wells when the barrier height P < the chemical potential u. Slight tilts of the atom chip with respect to gravity, e.g., from tiny movements of the optical table, were found to have a marked effect on the distribution of the condensate between the two wells. The gravitational gradient is cancelled by applying a small magnetic field gradient provided by a current imbalance ± 57 between the two end-wires on the chip. The trap asymmetry A can be calibrated in situ against 87 by adiabatically splitting a condensate in a symmetric double well with barrier height P > ^ and using spatially resolved RF spectroscopy, as described in Sect. 3. A condensate with a relatively small number of atoms is initially prepared in a single well at y = 170 um below the film and then Bbias is slowly increased to produce a tailored double well at y = 155 um, where the well separation A, ~ 70 um and the barrier height p = V* u. The splitting time ts is chosen to be larger than A'1 to ensure the condensate remains in the ground state of the potential well.

234

'-2S3

-2CQ

-ISO

-100

-50

0

50

100

150

2S&

25S

Asymmetry A (Hz) Figure 5. Fractional number difference versus asymmetry A of a double-well potential. The asymmetry is varied by changing the end-wire current imbalance 87. The experimental points and line of best fit (dashed line) compare well with the simple analytic result (solid line). After [21].

Absorption images of the condensate are then recorded as 87 is ramped over ±117 mA, and the fractional atom number difference (NR + NL)/(NR -NL) = AN/N is recorded against 87, and hence A. From the distribution of the data (Fig. 5) we estimate a single-shot sensitivity ASNR = 16 Hz, which is limited by the shot-to-shot variation in the number of atoms in the condensate. From this result we infer a single-shot sensitivity to gravity gradients of 8g/g ~ 2 x 10"4. Using a simple model in which the double-well potential is represented by two uncoupled 3D harmonic oscillators we obtain [16] AAMVw 1.65 (47r)(a0/15aA02/5 A/TO, where a is the scattering length, ao is the ground state harmonic oscillator length, and TO is the geometric mean trap frequency. Inserting experimental parameters into this expression yields a straight line in good agreement with the experimental data (Fig. 5). The sensitivity could be further enhanced by decreasing the trap frequency TO, thereby lowering the chemical potential n, or by using multiple double-well potentials on a single chip. For 100 double wells and reasonable values for the parameters we estimate a possible single-shot sensitivity of ASNR ~ 0.04 Hz, or 8g/g ~ 5 x 10~7. 5. Periodic Magnetic Lattices Optical lattices produced by the interference of intersecting laser beams are widely used to manipulate BECs and clouds of ultracold atoms, e.g., in quantum tunnelling experiments. Here, we report on the use of periodic magnetic lattices based on permanent magnetic films as an alternative to optical lattices [17-19].

235

(a)

(b)

Figure 6. (a)-(c) ID periodic array of parallel magnets with perpendicular magnetisation, (d)—(f) Calculated contour plots of the magnetic field in the central region in the yOz plane with (d) no bias fields, (e) bias field Bly = -15 G, and (f) bias fields Blx=-20G, B,y = -l5 G. 1001 magnets, a = 1 urn, t = 0.05 urn, /, = 1000.5 urn, and 4nM, = 3.8 kG. 7 G contour spacing. After [17].

Magnetic lattices based on magnetic films have a number of distinctive characteristics: (i) They are highly stable with low technical noise; (ii) There is no spontaneous emission; (iii) The atoms need to be prepared in low magnetic field-seeking states, allowing RF evaporative cooling in situ and the use of spatially resolved RF spectroscopy; (iv) They can have large and controllable barrier heights and large trap curvature leading to high trap frequencies; (v) They can be constructed with a wide range of periods, down to about 1 jam, and they can have complex potential shapes; (vi) They are necessarily static, precluding use in time-dependent lattice experiments; and (vii) 2D and ID magnetic lattices, but not 3D, can be constructed. Thus magnetic lattices can be considered as complementary to optical lattices, in much the same way as magnetic traps are complementary to optical dipole traps. We first consider a simple ID magnetic lattice produced by a single periodic array of parallel magnets of thickness t with perpendicular magnetisation, period a, and bias fields B\x and B\y parallel and perpendicular to the grooves, respectively (Fig. 6) [17]. For B\x = B\y = 0 (Fig. 6(d)), the magnetic field falls off exponentially with distance z above the surface, representing the case of a magnetic mirror [9, 10]. For B\x = 0, B\y = -15 G (Fig. 6(e)), the magnetic field develops 2D magnetic traps with zero potential minima; this configuration can give rise to spin flips and is not suitable as a

236

Figure 7. (a) 2D periodic array consisting of two crossed layers of parallel magnets with perpendicular magnetisation, (b) 3D plot of the magnetic field. 1001 magnets, a = 1 urn, ?, = 0.322 urn, <2 = 0.083 urn, /, = /,= 1000.5 urn, 47tMz = 3.8kG, B,,= -4.08G, BU, = -6.05G, S,2 = -0.69 G. S min = 2.7 G, zmin = 1.22 urn, AU* = ALP' = 485 uK, AU Z = 307 nK, to, = CD, = 2n x232 kHz,
lattice. For Blx = -20 G, B\y = -15 G (Fig. 6(f)), the magnetic field has 2D magnetic traps with non zero potential minima. For an infinite ID magnetic lattice, the potential minimum, trap height, and barrier heights are given by [17] Bmm = Blx\; zmm = (a/2n) In [B0yl\Bly\] B,

where B0y = B0 (1 - e kt) ekl and k = 2n/a. The potential minimum, trap height and barrier heights can be controlled by means of the bias fields 5 u and B\y, We now consider a 2D magnetic lattice produced by two crossed periodic arrays of parallel magnets, thicknesses t\ and ti, separated by distance s (< a/2n), and with bias fields B]x, B}y (Fig. 7(a)). Figure 7 (b) shows the calculated magnetic field for the parameters given in the caption. For an infinite symmetrical 2D magnetic lattice, the trap minimum, trap height, and barrier heights are given by [17]

Bl

with the constraint B\y = cQB\x for a symmetrical lattice. The c\s are dimensionless constants that involve geometrical constants a, s, t\ and t2 of the magnetic arrays and B0x = B0 (1 - e~*'2) e^+'1+'2). The potential minimum, trap position and barrier heights can be controlled by varying BIX and Biy. Other configurations of 2D magnetic lattices have also been proposed [17, 19].

237

Figure 8. Absorption images, after 8 ms time of flight, of a 87Rb condensate after it has been brought to various distances below the 10 urn-period magnetic microstructure, ranging from z ~ 6 um (first image) to z ~ 150 \an (final image). Bv = 0 (perpendicular to grooves).

6. Realisation of a Permanent-Magnet Lattice for Ultracold Atoms We have constructed a ID magnetic lattice with period a = 10 (am and dimensions 10 mm x 10 mm using a six-layer structure of perpendicularly magnetised TbGdFeCo magneto-optical films deposited on a grooved silicon wafer. The magnetic microstructure is coated with a reflecting gold film for use in the mirror MOT. The magnetic microstructure is mounted on an atom chip, similar to that described in Sect. 2, but with a 30 mm long Z-wire, in addition to the 5 mm long Z-wire, perpendicular to the grooves for effective loading of ultracold atoms into the magnetic lattice. To ensure minimal demagnetisation during vacuum bakeout care was taken not exceed a temperature of 100°C. First, a BEC with ~ 2 x 105 87Rb atoms is prepared in the 5 mm Z-wire trap 250 um below the chip surface where the magnetic microstructure has a negligible effect. The current in the Z-wire is then slowly ramped down to bring the condensate closer to the surface so that it interacts with the magnetic microstructure for less than 5 ms before being released. In this initial experiment the bias field perpendicular to the grooves was switched off, so the microstructure is not yet a trapping magnetic lattice. Absorption images are taken after a time of flight (TOP) of 8 ms. As the condensate is brought closer (from ~ 150 um to ~ 6 (am) to the magnetic microstructure, the TOP images change from a tiny cloud to sharply defined crescent-shaped structures, indicating that the condensate has interacted with the magnetic microstructure. The crescent-shaped structures are attributed to reflection of the atoms from a periodic corrugation in the magnetic potential caused largely by a magnetic field produced by residual current in the Z-wire. When the temperature of the atom cloud is raised above the BEC transition the crescent-shaped structures broaden markedly due to the larger trap volume. Bias fields of B\x = —15 G, B\y = -30 G are then turned on to create a magnetic lattice of 2D traps with a barrier height of about 1.2 mK. Absorption images taken after loading about

238

Holding Time (ms) Figure 9. Left: Absorption image, after 14 ms time of flight, of a thermal cloud of ~ 5 x 10s 87Rb atoms at 15 uK trapped in a 10 urn-period ID magnetic lattice at 6 - 8 urn from the surface of the atom chip. Bx = —15 G, By — -30 G. The image is taken in a direction parallel to the grooves. Right: Decay of the thermal cloud after release from the magnetic lattice. The lifetime is 0.34 s.

5 x 105 87Rb atoms in a 15 uK thermal cloud from the 5 mm Z-wire trap into the 10 mm x 10 mm magnetic lattice and after 14 ms TOP indicate that atoms are trapped in about 100 of the 1000 lattice sites at 6 - 8 urn from the surface (Fig. 9, left). By releasing the atoms from the magnetic lattice, the lifetime of the atoms trapped in the magnetic lattice is found to be 0.34 s (Fig. 9, right). With our current pixel image resolution of 9 urn it was not possible to resolve atoms in individual lattice sites of the 10 um-period magnetic lattice. We also note that at this resolution there is little evidence of fragmentation even though the atoms are trapped very close (6-8 urn) to the magnetic microstructure. In future it should be possible to populate a large fraction of the 1000 lattice sites, by loading from the 30 mm long Z-wire trap, and to resolve atoms in the individual lattice sites using spatially resolved RF spectroscopy. We also plan to implement RF evaporative cooling on the trapped atom clouds to create condensates on individual sites of the magnetic lattice and to study the diffraction of the condensate atoms when released from the magnetic lattice. We also plan to implement a 2D magnetic lattice on an atom chip and to construct magnetic lattices with smaller periods (1-5 um) in order to study quantum tunnelling in ID and 2D magnetic lattices. One of the major challenges will be to fabricate periodic magnetic microstructures having sufficiently smooth magnetic potentials very close to the surface to produce high-quality small-period magnetic lattices that preserve quantum coherence. During the preparation of this manuscript a preprint appeared in which ultracold rubidium atoms were trapped in > 30 lattice sites of a 2D FePt magnetic lattice with period 20 urn [22].

239 7. Summary We have implemented perpendicularly magnetised magnetic film microstructures to trap and manipulate BECs and ultracold atoms on an atom chip. We have developed a sensitive high resolution technique, based on RF spectroscopy plus absorption imaging of ultracold atoms, to map the magnetic field topology of the magnetic film. We have used a double-well potential close to the magnetic film to dynamically split a BEC. We find that the distribution of condensate atoms between the two wells can provide a highly sensitive technique for determining potential gradients, such as gravitational gradients, on an atom chip. Finally, we have loaded and trapped small clouds of ultracold atoms in a 1D permanent-magnet lattice on an atom chip. Acknowledgments

This project is supported by the ARC Centre of Excellence for Quantum-Atom Optics and a Swinburne University Strategic Initiative grant. References

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

H. Ott et al., Phys. Rev. Lett. 87,230401 (2001). W. Hansel et al., Nature 413,498 (2001). D. C. Lau et al., Eur. Phys. J. D 5, 193 (1999). D. W. Wang et al., Phys. Rev. Lett. 92,076802 (2004). M. P. A. Jones et al., Phys.Rev. Lett. 91,080401 (2003). J. Esteve et al., Phys. Rev. A 70, 043629 (2004). B. V. Hall et al., Laser Spectroscopy XVII, (World Scientific, 2005), 275. B. V. Hall et al., J. Phys. B: At. Mol. Opt. Phys. 39,27 (2006). A. Sidorov et al., Compt. Rend. 2, Series IV, 565 (2001). A. Sidorov et al., Acta Phys. Pol. B 33,2137 (2002). J. Y. Wang et al., J. Phys. D: Appl. Phys. 38,4015 (2005). C. D. J. Sinclair et al., Phys. Rev. A 72,031603(R) (2005). M. Boyd et al., preprint cond-mat/0608370 (2006). T. Fernholz et al., preprint cond-mat/07052569 (2007). S. Whitlock et al., Phys. Rev. A 75, 043602 (2007). B. V. Hall et al., Phys. Rev. Lett. 98, 030402 (2007). S. Ghanbari et al., J. Phys. B: At. Mol. Opt. Phys. 39, 847 (2006). R. Gerritsma and R. J. C. Spreeuw, Phys. Rev. A 74,043405 (2006). S. Ghanbari et al., J. Phys. B: At. Mol. Opt. Phys. 40, 1283 (2007). S. Wildermuth et al., Nature 435,440 (2005). S. Whitlock, PhD Thesis, Swinburne University of Technology, 2007. R. Gerritsma et al., preprint physics/07061170 (2007).

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COLD ATOMS AND MOLECULES11

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ULTRACOLD METASTABLE HELIUM-4 AND HELIUM-3 GASES W. VASSEN, T. JELTES, J.M. MCNAMARA, A.S. TYCHKOV, W. HOGERVORST Laser Centre Vrije Universiteit Amsterdam, The Netherlands K.A.H. VAN LEEUWEN Dept. of Applied Physics, Eindhoven Univ. of Technologv, The Netherlands V. KRACHMALNICOFF, M. SCHELLEKENS, A. PERRIN, H. CHANG, D. BOIRON, A. ASPECT, C.I. WESTBROOK Lab. Charles Fabry de 1’Institut d’Optique, Univ. Paris-Sud, Palaiseau, France We discuss our work to obtain a condensate containing more than lo7 atoms and the first degenerate Fermi gas in a metastable state. Sympathetic cooling with Helium-4 is used to cool lo6 Helium-3 atoms to a temperature T/TF< 0.5. The ultracold bosonic and fermionic gases have been used to observe the Hanbury Brown and Twiss effect for both isotopes, showing bunching for the bosons and antibunching for the fermions. A proposal for high resolution spectroscopy at 1.557 pm, connecting both metastable states directly, is discussed at the end.

1. Introduction Helium in the metastable 2 3SI state has been Bose condensed in 2001 by two French groups [ 1,2]. Since then research on ultracold metastable helium gases has been concentrated on determining the scattering length, measuring loss processes, studies of the hydrodynamic regime and producing longrange molecules. In 2005 two other groups realized BEC in metastable helium [3,4]. Studies of atom-atom correlations in ultracold clouds of 4He bosons were started around that time as well, exploiting the unique detection properties of metastable atoms [5]. In this contribution we will discuss experiments performed in Amsterdam. We will discuss the setup in which a BEC containing over 10’ atoms and a degenerate Fermi gas (DFG) of metastable 3He containing over lo6 atoms was realized. Next we discuss experiments where the Amsterdam and OrsayRalaiseau groups joined forces measuring the Hanbury Brown and Twiss effect both for a gas of ultracold 243

244

4

bosons and a cloud of ultracold fermions, demonstrating atom bunching for He and atom antibunching for 3He.

2. Helium level structure and relevant parameters Relevant energy levels are shown in Fig. 1. The helium atom is strongly LScoupled effectively leading to two spectra, that of parahelium, where the electron spins are antiparallel and that of orthohelium where the spins are parallel. The ground state of orthohelium (ls2s 3S1 or He*) can be laser cooled with laser light of 1083 nm exciting the ls2s 3SI - ls2p 3P (2 3S - 2 3P) transition. The 2 'S1 state, populated in a DC discharge, has a lifetime of -8000 s which is infinite for all practical purposes. Its 20 eV internal energy provides the unique detection possibilities: (almost) everything a He* atom hits gets ionized and the released electron can be counted with high efficiency using electron multipliers or microchannel plate (MCP) detectors. IONIZATION LIMIT

24

-

23 22 -

metastable state: is2s 3S, (4He*) F= i/2,3/2(3He*)

4.8 eV

Figure 1: Level scheme showing the relevant energy levels for laser cooling and trapping of 4He in the 2 3SIstate (M,=+l in a magnetic trap). The 2 'S state is split in a F=3/2 and a F=1/2 state (AE=6740 MHz) in the case of 'He. The F=3/2 state has the lowest energy and is used for cooling and trapping (M~=+3/2in the magnetic trap).

245

Losses in ultracold He* gases, either in a magneto optical trap (MOT) or a magnetic trap, are mainly due to two-body inelastic Penning ionization where one ion and one ground state atom are produced in a collision. The loss rate constant for this process in an unpolarized gas is quite large: -1x10"'° cnrVs in the dark and ~4xlO" 9 cm3/s in the light of a MOT [6,7]. When the atoms are in a fully stretched state (as in a magnetic trap) these losses are suppressed in first order and the rate constant drops to ~lxlO~ 1 4 crnVs. In the fully stretched state of 4He* (|J,Mj>= !,+!>) Penning losses are therefore acceptable and a EEC can be realized. The 4He*-4He* scattering length a44 that determines many properties of a EEC has recently been measured with high accuracy: a44(exp)= +7.5105 (25) nm [8]. Recent ab initio calculations of the molecular potential have allowed an astonishing accuracy in the calculation of a44 as well and approach the experimental accuracy: a44(theory)= +7.562 (28) nm [9]. Mass scaling of the potential then provides a theoretical value of a34(meory)= +27.1 (5) nm for the scattering length determining the collision rate between spin-polarized 3He* and 4He* atoms [9].

3. The experimental apparatus Laser cooling and trapping of He* atoms is performed at a wavelength of 1083 nm. As the isotope shift is 33 GHz we use two separate laser systems, each laser locked to the cycling transition of the relevant isotope in a DC discharge cell. The experimental setup is shown in Figs. 2 and 3. We use a liquid nitrogen cooled DC discharge source, which uses recycling of helium atoms from the turbo pump exhaust via molecular sieves.

jA*.

**».

Jtom».

MOT

dc discharge source laser collimator

Zeeman slower

+1° deflection Figure 2: Schematic or the experimental setup used to trap 3He* and 4He* atoms. The collimation section (two-dimensional) significantly improves the MOT loading. A deflection zone (in the horizontal plane) prevents ground state atoms entering the 2 m long Zeeman slower.

246

Figure 3: Schematic of one-dimensional Doppler cooling in the cloverleaf magnetic trap (left) and UHV chamber (right; the arrow indicates the absorption imaging beam). The coils are located in plastic boxes positioned inside the re-entrant window flanges.

We typically load 2 x 1Q9 4He* atoms or 1 x 1093He* atoms in 1 s in a MOT at a temperature of 1 mK. After a short spin polarization pulse the atoms are loaded into a spherical cloverleaf trap matching the MOT cloud. The high (24 G) bias magnetic field in this geometry allows efficient 1-dimensional Doppler cooling to a temperature of 0.15 mK (in 2 s), shrinking the cloud considerably and increasing the phase space density about a factor 600 to a value around 10"4 - 10"5 [3]. In another 2 s the cloud is compressed at small bias field (~3 G). The trap lifetime is 2 - 3 minutes (either 3He or 4He). To produce a BEC we perform a 15 s rf evaporative cooling ramp realizing a BEC containing (1.5-4) x 1Q7 4He* atoms [3]. We monitor the BEC applying absorption imaging at 1083 nm (not very efficient due to the low efficiency of CCD cameras at 1083 nm) or on an MCP detector mounted directly under the trap (see Fig. 4). The latter technique is very sensitive allowing us to monitor a BEC after up to 75 s trapping inside the magnetic trap. To monitor the growth of a condensate we use a second MCP detector that attracts all ions produced by Penning ionization. As soon as the condensate starts to grow a dense cloud forms and the ion signal suddenly increases [3]. The lifetime of our condensate is about 1 s. We studied this as a function of the thermal fraction and found that a large thermal fraction reduces the lifetime of the condensate considerably. We attribute this to transfer of condensate atoms to the thermal cloud during condensate decay. A simple model that assumes thermal equilibrium during the decay confirms this and allows extraction of the two- and three-body loss rate constants [3,10].

247

210

170

180

190

200

210

200

210

250

180 190 Tims of flight (ms)

Figure 4: Time-of-flight pictures of a EEC (upper figure), a DFG (middle figure) and a mixture (lower figure). The upper figure shows three plots, each with a slightly different end rf frequency, showing a thermal cloud above the EEC temperature, a mixture of BEC and thermal cloud, and a pure BEC with the typical inverted parabola shape. In the middle figure a fit to a Fermi-Dirac distribution is shown from which we extract a temperature T=0.45 TV. In the lower figure the dashed-dotted line shows the BEC contribution to the signal and the dashed line the DFG contribution.

248 70 (XI

2 50 e

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44,

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i"

XI 10 120

110

160

IHI

im

I-,,

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llmE 1mj

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Figure 5: He* atom laser, realized by repeatedly output coupling of a fraction of the BEC in 250 MHds ramps. The time-of-flight signal shows the effects of mean-field repulsion.

We can couple out a small fraction of the atoms from a condensate applying an rf ramp as shown in Fig. 5. We applied repeatedly a 250 MHz/s rf ramp to couple atoms out of the condensate showing a typical atom laser pulse shape, revealing mean-field repulsion. To produce a degenerate Fermi gas a we load a MOT with both isotopes [3,6,7,11,12]. With an equal number of both isotopes in our gas reservoir we trap 1 x lo94He* atoms and 7 x 10' 3He*atoms in a MOT. However, we can not cool so many fermions and therefore reduce the number of fermions to 110% of the number of bosons. The large heteronuclear scattering length provides almost ideal conditions for sympathetic cooling and we obtain a DFG containing 2 x lo6 3He* atoms at T/TF=0.45 (Fig. 4) as well as a degenerate mixture (also Fig. 4). Three-body losses probably limit the lifetime of this mixture to -1 0 ms [ 121.

4. Hanbury Brown and Twiss experiments In 2005 the atomic analogon of the Hanbury Brown and Twiss (HBT) effect was demonstrated for metastable 4He* atoms released from a cloverleaf magnetic trap very similar to the one in Amsterdam [5]. The HBT effect, first demonstrated for light in the fifties of the 20th century [ 131, represents the measurement of the two-body second-order correlation function

displaying the joint probability of detecting two atoms (or photons) at locations r and r'. For incoherent sources this function equals 1 for large separations and will tend to 2 for bosons and 0 for fermions, for detector

249 separations smaller than the correlation length /: bosons bunch and fermions antibunch. In the Amsterdam experiments we demonstrated bunching for 4 He* and antibunching for 3He* trapped in identical traps at the same temperature [14]. For this purpose the position sensitive detector (PSD) of the Orsay (now Palaiseau) group was transported to Amsterdam and mounted under the Amsterdam UHV chamber (which is shown in Fig. 3). The detection part of the setup is shown in Fig. 6, showing on the left the PSD and on the right a schematic of the measurement. The correlation function for a measurement of atoms released from a harmonic trap is given by [14] 2 T( )

(Ax, Ay, Az) = 1 ± r/exp

with correlation lengths in each of the three spatial directions /; = ms where sf — ^ kBT /mcof , m the mass of the atom, T the temperature, t the drop time and co, the trap frequency. The contrast 77 (0 - 1) depends on the detector resolution in all three spatial dimensions. This two-particle detector resolution is 6 nm in the vertical (z-) direction and 500 |am in the horizontal plane of the detector. As the correlation length in the x-direction is much smaller than the detector resolution in that direction (we have a cigar-shaped trap with long axis in the x-direction) 77 is on the order of 1/15.

Figure 6: Position-sensitive MCP detector and schematic of the experiment. The inset of the figure on the right conceptually shows the two 2-particle amplitudes that interfere to give bunching or antibunching.

250 As the detector resolution is very good in the vertical plane, we plot the measured correlation function along the z-direction in Fig. 7. This plot was obtained by averaging over about 1000 clouds per isotope, with -lo4 detector points per shot. The temperature was 0.5 pK. We also measured the correlation function along the y-direction, however with substantial broadening along the horizontal axis due the 500 pm resolution in the x-y plane [14]. Fig. 7 clearly demonstrates antibunching for fermions and bunching for bosons, over a macroscopic distance of -1 mm. The measured ratio of the correlation lengths is 1.3 0.2, which is as expected, realizing that the cloud sizes of both isotopes in the trap are equal, the drop times are equal and the mass ratio is 4/3. Also the individual correlations lengths agree very well with the formula 1, =ht/ms, We observe a small discrepancy in the amount of contrast for both isotopes. There are two possible experimental imperfections that affect the bosons and fermions differently and that may explain this, i.e. the non-sudden trap switch-off and the determination of the detector resolution function.

-1 0

2

3

Separation Az (mm)

Figure 7: Normalized correlation functions for 4He* in the upper plot and 'He* in the lower plot, measured in the same trap at the same temperature of 0.5 pK.Error bars correspond to the square root of the number of pairs in each time bin. The line is a fit to a Gaussian function. Correlation lengths of 0.75 0.07 mm and 0.56 f 0.08 mm for fermions and bosons, respectively, are extracted.

*

251

T

0.5

1

0,5

Temperature

1.5

Temperature (^K)

Figure 8: Measured temperature dependence of correlation length and contrast. The curves are calculations and describe the measurements reasonably well.

We took data for fermions at 0.5 uK, 1 .0 |^K and 1 .4 uK. The fit results for /z and 77 are shown in Fig. 8 together with the calculated results. The limited resolution in the horizontal plane can be circumvented by using a negative atom lens, reducing the apparent size of the cloud in the horizontal plane. This does not affect the correlation length in the vertical plane but does affect the contrast via the correlation length ly, which increases. To implement this we focused a 300 mW laser beam, with elliptical waist and 300 GHz blue detuned from the 1083 nm transition, for 0.5 ms through the expanding 3He* cloud after switching off the magnetic trap. The results are shown in Fig. 9 and qualitatively agree with expectations. 1JS-,

«^i**-M^M* ,&f~r*V

1 %, § emO

0

1

1

3

Separation Az (mm) Figure 9: Normalized correlation functions along the z (vertical) axis for 3He*, demonstrating the effect of a diverging atomic lens in the x-y plane. The light circles are without lens and the dark squares with lens. The dip is deeper with the lens because the correlation length in the ydirection is increased which affects the contrast (and not the correlation length in the zdirection).

252

a unbundling

Pair separation (ms) Figure 10: Raw data of an atomic lens experiment with fermions. The data show the numbers of pairs measured for a 25 us time bin (1 ms corresponds to a vertical separation of 3.5 mm). They are averaged over 1000 shots.

We wish to emphasize that the antibunching effect in Fig. 7 is big enough that significant processing of the data is not necessary. Fig. 10 shows the raw data of one of our runs before normalizing the data. Normalization (to get the data of Figs. 7 and 9) is done by dividing the raw data by the autoconvolution of the sum of the 1000 single-particle distributions. Therefore the raw data show the antibunching effect as well as the Gaussian spatial distribution of the cloud.

5. Proposed metrology experiment As one of the possible applications of samples of ultracold 4He* and 3He* atoms we envision to excite the "forbidden" 2 3S| - 2 'S0 transition at 1.557 urn [15]. This transition is only magnetic-dipole (Ml) allowed with an Einstein A-coefficient of 6.1 x 10"8 s"1 [16]. The high-resolution potential is very large as the natural linewidth of the transition is 8 Hz, fully determined by the 20 ms lifetime of the 2 'S0 metastable state due to two-photon (2E1) decay to the ground state. Measuring this transition will directly link the orthohelium and parahelium system (see also Fig. 1). The absolute frequency (for both isotopes) will provide sensitive tests of atomic theory for twoelectron systems, measuring QED contributions which are large for S-states. Measuring the isotope shift, the main theoretical inaccuracy is in the difference in mean nuclear charge radius of the 4He and 3He nucleus.

253

To measure the transition we theoretically investigated two setups: (1) spectroscopy with a laser-cooled and collimated atomic beam as provided from the Zeeman-slower and (2) spectroscopy in a one-dimensional optical lattice, loaded from a magnetically trapped and evaporatively cooled cloud. For both situations we solved the optical Bloch equations simplifying the model neglecting the decay from 2 'S to 2 3S as well as the decay from 2 3S to 1 'S (see Fig. 11). For the beam experiment we assume a 100 m/s Zeemanslowed atomic beam, transversely cooled to twice the Doppler limit. With an atomic beam intensity of 1011 atoms per second, a 2 W 100 kHz laser beam (Icmxlcm excitation region) will provide an on-resonance Rabi frequency Q= 61 s"1, which for an excitation time of 100 us will lead to an excited state population P22=4.6 x 10"7. A considerable flux of 5 x 103 atoms per second in the 2 'S state then may be expected.

Figure 1 1 : Relevant energy levels for 2 Si — 2 S0 spectroscopy and decay rates.

The expected spectroscopic linewidth is 400 kHz in this case, limited by Doppler broadening. Using 1083 nm light resonant with the 2 3Si - 2 3P2 transition the nonexcited atoms can simply be deflected realizing a (hopefully) zero-background signal on a metastable atoms detector. A schematic view of the proposed setup, with three laser-atomic-beam crossings depicted, is shown in Fig. 12.

2 ifycdSmuSani deffeoion IWBnm

2-'S,rallimat dsfieaion tSBrni!

Fig. 12: Schematic of the proposed setup to measure the 2 3Si — 2 'S0 transition in a beam.

254

The one-dimensional lattice has more potential to achieve high accuracy. In the experiments described in Section 4 an ultracold cloud at temperatures around 1 pK of 3He*, 4He*, or a mixture was presented. Such a cloud may be transferred to a far off-resonance dipole trap (FORT). When a single retroreflected beam is used a one-dimensional lattice results which, when a wavelength of 1.557 pm is used, can easily trap a cloud from our cloverleaf trap at a temperature of a few pK. The experiment may be performed using the same 2 W laser as discussed in the beam experiment as the trapping potential is insensitive to the wavelength on a scale of an experiment on the forbidden transition. A narrower linewidth laser, though, is useful here as the width of the spectroscopy signal will now be limited by the laser linewidth as Doppler broadening is suppressed by Lamb-Dicke narrowing. Detection is now more challenging as the 2 'S atoms are anti-trapped and slow. Photo ionization, or detection of ions due to Penning ionization with trapped 2 3S atoms, are then options to consider. The accuracy of the spectroscopy in the lattice will be limited by how well the AC Stark shift due to the lattice potential can be controlled and measured. Calculations predict a 2 MHz shift of the transition at the maximum of the potential (the atoms are trapped in the light as the 1.557 pm wavelength is red detuned from the 1083 nm transition). The best experiment therefore will be to trap at a magic wavelength (where the AC Stark shifts of the metastable states are equal), which will not be easy as the highest magic wavelength available is 410 nm at which wavelength very high laser power is required to trap He* atoms. A second magic wavelength occurs at 35 1 nm and may be more suitable for spectroscopy on this forbidden transition.

References 1. A. Robert, 0. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. Westbrook, A. Aspect, Science 292,461 (2001). 2. F. Pereira Dos Santos, J. LConard, J. Wang, C.J. Barrelet, F. Perales, E. Rasel, C.S. Unnikrishnan, M. Leduc, C. Cohen-Tannoudji, Phys. Rev. Lett. 86, 004359 (2001). 3. A.S. Tychkov, T. Jeltes, J.M. McNamara, P.J.J. Tol, N. Herschbach, W. Hogervorst, W. Vassen, Phys. Rev. A 73,031603(R) (2006). 4. R.G. Dall, A.G. Truscott, Opt. Comm. 270,255 (2007). 5. M. Schellekens, R. Hoppeler, A. Perrin, J.V. Gomes, D. Boiron, A. Aspect, C.I. Westbrook, Science 310, 648 (2005).

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6. R.J.W. Stas, J.M. McNamara, W. Hogervorst, W. Vassen, Phys. Rev. A 73,032713 (2006). 7. J.M. McNamara, R.J.W. Stas, W. Hogervorst, W. Vassen Phys. Rev. A 75,06271 5 (2007). 8. S. Moal, M. Portier, J. Kim, J. DuguC, U.D. Rapol, M. Leduc, C. CohenTannoudji, Phys. Rev. Lett. 97, 023203 (2006); for a small correction to a44see S. Moal, thesis ENS Paris (2006). 9. B. Jeziorski, Private communication (2007). 10. P. Zin, A. Dragan, S. Charzynski, N. Herschbach, P. Tol, W. Hogervorst, W. Vassen, J. Phys. B 36, L149 (2003). 11. R.J.W. Stas, J.M. McNamara, W. Hogervorst, W. Vassen Phys. Rev. Lett. 93,053001 (2004). 12. J.M. McNamara, T. Jeltes, A.S. Tychkov, W. Hogervorst, W. Vassen Phys. Rev. Lett. 97,080404 (2006). 13. R. Hanbury Brown, R.Q. Twiss, Nature 177,27 (1956). 14. T. Jeltes, J.M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, C.I. Westbrook, Nature 445,402 (2007). 15. K.A.H. van Leeuwen, W. Vassen, Europhys. Lett. 76,409 (2006). 16. K. Pachucki, Private communication (2006).

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SINGLE ATOMS AND QUANTUM OPTICS I

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RECENT PROGRESS ON THE MANIPULATION OF SINGLE ATOMS IN OPTICAL TWEEZERS FOR QUANTUM COMPUTING A. Browaeys*, J. Beugnon, C. Tuchendler, H. Marion, A. Gaetan, Y.Miroshnychenko, B. DarquiB, J. Dingjan, Y.R.P. Sortais, A.M. Lance, M.P.A. Jones, G. Messin, P. Grangier

Laboratoire Charles Fabry de l’lnstitut d’optique, CNRS, Univ. Paris-sud, Campus Polytechnique, R D 128, 91 I27 Palaiseau cedex, France *E-mail: antoine. browaeysQinstitutoptique.f r This paper summarizes our recent progress towards using single rubidium atoms trapped in an optical tweezer t o encode quantum information. We demonstrate single qubit rotations on this system and measure the coherence of the qubit. We move the quantum bit over distances of tens of microns and show that the coherence is preserved. We also transfer a qubit atom between two tweezers and show no loss of coherence. Finally, we describe our progress towards conditional entanglement of two atoms by photon emission and twophoton interferences.

Quantum computing has been proposed to solve certain classes of computational problems, such as factoring and searching, faster than using a classical computer’ . In addition, one could engineer these quantum computers in such a way that they could perform simulations of quantum systems. From a fundamental point of view, a quantum computer can be thought as a collection of two-level systems, well isolated from the environment, which can interact with each other in a controlled way. Building such a quantum computer may therefore help to understand decoherence of a macroscopic quantum system towards a classical system. The practical implementation of a quantum computer relies on a physical system that constitutes a good approximation of a two-level system. Among all the systems proposed so far, neutral atoms present the advantage of well controlled manipulations of the internal and external degrees of freedom. Furthermore, neutral atoms offer built-in scalability when the atoms are trapped in periodic potentials. Following this route, we have chosen to encode the quantum information

259

260 on two hyperfine states of a single rubidium atom trapped in an optical tweezer. Using this system, it has been demonstrated that several tweezers can be arranged in an array, with each sites well localized and adressable by optical methods2i3 . This paper describes how we trap and observe a single atom in an optical tweezer created by focusing a far-off resonant laser down to a micrometer size waist. We then show the coherent manipulation of the two-level system and characterize the coherence of this quantum bit. As a first step towards the controlled interaction between two atoms trapped in an array of tweezers, we demonstrate a scheme where the qubit is transfered between two tweezers, with no observed loss of coherence and no change in the external degrees of freedom of the atom. Additionnally, we move the atom over distances that are typical of the separation between atoms in an array of optical traps, and show that this transport does not affect the coherence of the qubit. Finally, we are working towards the conditional entanglement of two atoms trapped in tweezers separated by 10 microns. We have shown two key ingredients of this protocol: the controlled emission of a single photon by a single atom and the two-photon interference of photons emitted by two atoms. 1. Diffraction-limited optics for single-atom manipulation

Our optical tweezer is a far off-resonance dipole trap, with a size of about one micrometer. We produce this tweezer by focusing a laser beam down to the diffraction limit of a large numerical aperture aspherical lens. We use a lens manufactured by LightPath Technologies with a numerical aperture of 0.5. The working distance between the lens and the focal point is 6 mm, large enough to allow a good optical access around the lens. We have separately tested the lens using a wavefront analyzer. We have found a RMS deviation of the resulting wavefront of less than X/30 over the whole numerical aperture, thus demonstrating that the aspherical lens is diffraction limited. The optical layout, consisting of the aspherical lens and standard optical elements, is shown in figure 1. The dipole trap laser beam, produced by a 850 nm laser diode, is sent throught a single mode optical fiber and is shaped a t the output of the fiber by a triplet lens. The beam goes through the viewport of the vacuum chamber inside which the aspherical lens is placed. Before putting this system together, we have checked on a separated bench that the whole optical system is diffraction limited (see reference4 for more details).

261

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tes

i FocaS

1p«|

Pinhole

Singlet

f

<

Vacuum chamber

>

• optical tweezer (850 nm) imaging system (780 nm}

Fig. 1. Optical setup of the single-atom trapping (solid line) and imaging systems (dotted lines).

The same lens is used to collect the fluorescence light (780 nm) emitted by the atom trapped in the tweezer. This fluorescence light is collected outside the vacuum chamber using a confocal imaging system, as represented in figure 1. The light is sent onto an avalanche photodiode, used in counting mode, and a CCD camera. This imaging system is also diffraction limited and has a spatial resolution of 1 micrometer4 . We use this optical system to trap and observe single atoms in the tweezer. The tweezer is loaded from the cloud of rubidium atoms cooled in an optical molasses. Figure 2 shows an example of the signal obtained on the avalanche photodiode versus time. The steps in the signal correspond to the fluorescence of a trapped atom at 780 nm, induced by the molasses laser. The absence of double steps is an indication that only individual atoms are trapped. This single-atom trapping is made possible by a "blockade mechanism" studied in detail in reference5 . 2. Single-atom quantum bit

Our quantum bit is encoded on the |0) = \F — 1,M = 0), |1) = \F = 2, M = 0} hyperfine sublevels of a rubidium 87 atom. This choice of levels for the qubit provides the advantage of zero first-order sensitivity to magnetic fields. We initialize the qubit in state |0) by optical pumping, with an efficiency of 85 %. We read the state of the qubit using a state selective measurement limited by the quantum projection noise. For this purpose, we send a laser on resonance with state |1) and the state F' = 3 connected by a transition

262 50000-

41000

41500

time (ms)

42000

histogram

Fig. 2. (a) Fluorescence of a single atom measured by the avalanche photodiode. (b) Histogram of the measured fluorescence recorded over 100 sec. The two well separated peaks in this histogram correspond to the absence of atom and the presence of exactly one atom.

at 780 nm. Radiation pressure expels the atom out of the trap, if the atom is initially in state |1). Otherwise, if the atom is in state |0), the laser leaves the atom unaffected. We then check for the presence of the atom. This method therefore maps the internal state of the atom on the presence or the absence of the atom at the end of the sequence. We drive the qubit transition with two Raman lasers, one of them being the dipole trap. The two beams are colinear and sent through the same optical fiber and the large numerical aperture lens. Due to the tight focusing of the two lasers, we observe a Rabi frequency of the two-photon transition as high as 2?r x 6.7 MHz. Figure 3 shows the population of state |0) for two durations of the Raman pulse6 . We study the coherence of the quantum bit using Ramsey spectroscopy. We apply a first Tr/2 Raman pulse to prepare the atomic state (|0} + |l})/-\/2We let the system evolve and we apply a second ?r/2 pulse. The signal exhibits oscillations at a frequency given by the detuning of the Raman lasers with respect to the qubit transition. The decay of the contrast of the oscillations as a function of the time interval between the two pulses is the signature of the loss of coherence. We attribute this decay to the residual motion of the atom in the trap, together with the fact that the two states of the qubit experience a slightly different trapping potential, leading to a dephasing of the quantum bit. Our best 1/e dephasing time is 630 //sec.

263

a"

oi...

50 '

100

150

200

250

300

Pulse length (ps)

Pulse length (ps)

Fig. 3. Single-atom Rabi oscillations. Fraction of atoms in F = 1, as a function of the Raman pulse length, at low (a) and high (b) intensity. Details can be found in reference6 .

For details, see reference6 . We rephase this dephasing by inserting a 7r pulse between the two 7r/2 pulses. Using this spin-echo technique, we observe a revival of the oscillations after a time as long as 40 ms, corresponding to a 70 fold improvement with respect to the coherence time of the qubit, as shown in figure 4.

3. Transport and transfer of atomic qubits Neutral atoms are promising candidates for the realization of a large-scale quantum register. To perform a quantum computation, a key feature is the ability to perform a gate between two arbitrary qubits of the register. As a first step, we have demonstrated a scheme where the atomic qubit is transfered between two tweezers, and then transported over several tens of micrometers. We show that these manipulations of the external degrees of freedom preserve the coherence of the qubit, and do not induce any heating. The distance travelled is typical of the separation between atoms in an array of dipole traps. These techniques can also be useful to position an atom at the node of the electromagnetic field in an optical cavity for QED experiments7 . To show that the transfer preserves the coherence of the quantum bit, we prepare a superposition ( 1 0) Il))/JZin a first tweezer using a first 7r/2 pulse. We then decrease the depth of the first tweezer while increasing the

+

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Time between xi2 pulses (ms) Fig. 4. Example of the spin-echo signal. Figure (a) shows the revival of the oscillations after the 7r pulse has been applied. Figure (b) shows the amplitude of the echo signal for different durations of the spin-echo sequence.

depth of a second tweezer superimposed with the first one. After a dwelling time of 200 psec, we transfer the atom back to the initial tweezer and apply a second 7r/2 pulse. We vary the time between the two pulses around 200 psec and observe the corresponding Ramsey oscillations. We have shown that the amplitude of the Ramsey signal remains unchanged whatever the depth of the second trap is. We have also checked that when the depths of the two traps are identical, the “temperature” of the atom is unchanged. We have also moved the tweezer when the qubit is prepared in a superposition (10) Il))/fi. We have transported the atom up to f9 pm

+

265

away from the axis of the focusing lens, and brought it back to its initial position8 . The entire round trip takes 6 ms. As this time is longer than the dephasing time of the quantum bit, we apply a TT pulse when the tweezer is at its turning point, as shown in figure 5. We have measured that the ampli-

(a)

Initial position

Target position

-§
(b) 1.0oj

0.8 -

"5. 0.6 10

I °-4 ai

I °-2 ^ 0.0-1 - 8 - 6 - 4 - 2 0 2 4 6 8 Trap displacement {pm)

Fig. 5. (a) Principle of the moving qubit experiment, with the position of the tweezer when the various pulses of the sequence are applied, (b) Amplitude of spin-echo signal versus the amplitude of the displacement.

tude of the spin-echo signal remains constant when we varied the amplitude of the displacement. This indicates that the coherence of the quantum bit

266 is preserved during the transport. Finally, we have measured the phase of the Ramsey sequence during the transfer experiment and the phase of the spin-echo signal during the transport. We have modelled this phase evolution and found a good agreement with the result of the experiment. This understanding of the phase is crucial for a possible implementation in a quantum computer where qubit phases need to be controlled.

4. Towards conditional entanglement of two atoms The ability to generate entanglement is a key feature that any physical system should exhibit to be useful for any quantum information processing task. For example, entanglement is essential in teleportation protocols between two parties. Entanglement is also a necessary ingredient in the two approaches to quantum computing. In the quantum circuit approach' , entanglement is generated during the course of the implementation of the algorithms. In the cluster state approachg , it even constitutes the starting point of the calculation. It is therefore important to be able to generate and control the entanglement in this quantum system. Among all the methods proposed to entangle two neutral atoms, conditional entanglement based on photon emission is promising, as it does not require any direct interaction between the atoms. A simplified scheme is presented in figure 6 (see for example referenceslo>l1).We isolate three levels in the atom. The upper level is connected t o the two logic levels by an optical transition, with frequencies v1 and 7 4 . Each atom can decay to level 10) and 11) by emitting a photon with equal probability. The scheme entangles the internal state of the atom with the frequency of the emitted photon, generating for atom A the state (IOA, v1) 11A, v 2 ) ) / a .The state of the two photons and two atoms A and B is therefore

+

1

1

16)= -(IOA,

Jz

v1)

+ 11A, v2))8 -Jz (\oB,

v1)

+ IlB, v2)) .

Re-arranging the terms, this state can be re-written as

267

When the two photons are recombined on a beam-splitter, a two-photon interference prevents coincidences on the two detectors when the two-photon state before the beam-splitter is z'l, z'l) and |z/2, z^}- The absence of simultaneous coincidence is also true when the two-photon state is v\, v^} +11/2, v\), as the two-photon amplitudes of each component exactly cancel out (see for example reference11). Therefore a simultaneous detection event on the two detectors heralds the preparation of the entangled atomic state (\QA-, IB) — |lA>O.B))/V / 2- In this scheme, two-photon interferences acts as a "filter", and the preparation is heralded by the double detection. AtomB

Atom A

-

Rg

Detector 1 ,>

Detector 2

Fig. 6. Principle of a conditional entanglement of two atoms A and B based on the entanglement between each atom and an emitted photon followed by the recombination of the two photons on a 50/50 beam-splitter (BS).

Three ingredients are required to implement this protocol. The first one is the triggered emission of a single photon by a single atom. The second one is the observation of two photon interferences, and the third one is the ability to entangle an atom with an emitted photon. In the following sections we describe our implementation of the two first steps. We note that the third step has been realized by two groups in the recent years12'13 . 5. Single atom as a single-photon source

We control the emission of single photons by a single atom placed at the focal point of a large numerical aperture lens by sending -K pulse on the optical transition connecting (F = 2,M = 2) and (F' = 3,M' = 3), see figure 7. The duration of the pulses is 4 ns, and the separation between the pulses is 200 ns. The emitted photons all have the same
268

We collect 0.6 % of the emitted photons on an avalanche photodiode. We have characterized the single-photon nature of the source by measuring the coincidences on two photodetectors placed in the imaging system14 . The resulting curve is shown in figure 7(c). The absence of coincidence at zero delay is the signature that single photons are emitted by the atom. A careful analysis of this curve shows that the probability that the source emits two photons following the same excitation pulse is 1.8%. This number is in good agreement with a calculation taking into account the 4 ns duration of the excitation pulse, which is not negligible with respect to the 26 ns lifetime of the upper state.

(b) Atom

Counter K- pulse // 780 nm //

lens Excitation pulse (C)

500

g 400 •g 300 "3 .£ 200 o ° 100

0

-1000 -500 0 500 1000 1500 Delay (ns)

Fig. 7. (a) Principle of the single-photon source based on an atom trapped at the focal point of a large numerical aperture lens, (b) Relevant hyperfine levels of rubidium 87 used in the experiment, (c) Histogram of the coincidences measured on the two single-photon counters.

6. Interference of two photons emitted by two atoms

When two indistinguishable single photons are fed into the two input ports of a beam-splitter, the photons will leave together from the same output

269

port. This is a quantum interference effect, which occurs because the two possible paths, where the photons leave in different output ports, interfere destructively. This effect was first observed in parametric downconversion by Hong, Ou and Mandel15 . We have shown the interference of two photons emitted by two atoms trapped in two tweezers separated by 6 microns (see details in reference16). The two atoms are excited by the same 2 ns-laser pulse following the procedure described in the previous section. We recombine the two photons of same polarization on an optical setup equivalent to a 50/50 beam-splitter and we measure the coincidences at zero delay on two photodiodes placed in the outpout ports of the beam-splitter, as represented in figure 8(a). If the two-photon interference were perfect, one should not measure any coincidence as the two photons must leave in the same output port.

(a)

•BS

Pulsed excitation -100

r

T

0

i

100

Translation (\im)

Fig. 8. (a) Principle of the two-photon interference experiment. The photons emitted by two atoms A and B are recombined on a 50-50 beam-splitter (BS), and the coincidences are measured using two single-photon detectors (b) Influence of the wavefront matching. We plot the amplitude of the residual coincidence signal at zero delay for various relative distance between the two photon modes, translated parallel to each other. For perfect interference, the signal should go all the way down to zero when the two modes are not translated.

To analyze the visibility of the interferences, we varied the spatial overlap between the photons propagating in free space. In order to do so, we translated the spatial mode of one photon with respect to the other one. The result is represented in figure 8(b). For our best overlap, we find a visibility of the interferences of 60%, coming from the difficulty to mode-match the two photons. This number is a measure of the indistinguishability of the two interfering photons. This mode-matching can be improved by coupling the photons to single mode fibers.

270

7. Conclusion We have demonstrated some basic manipulations of a single quantum bit encoded on a n atom trapped in an optical tweezer. We have shown internal state rotation at the single atom level and we have measured the internal coherence of the qubit state. We have also demonstrated two necessary ingredients of a conditional entanglement protocol. Our current estimate gives an efficiency t o produce one entangled pair on the order of 10-6-10-5. We anticipate a rate of entangled pair production of one every 100 seconds.

Acknowledgments: We acknowledge financial support from IFRAF, ARDA/DTO and the European Integrated project SCALA. LCFIO is CNRS UMR8501. M.P.A. Jones and A.M. Lance are supported by Marie Curie Fellowships. A. Gaetan is supported by a DGA Fellowship.

References 1. M.A. Nielsen, & I.L. Chuang Quantum Computation and Quantum Information. Cambridge University Press (2000). 2. S. Bergamini, et al., J . Opt. SOC.A m . B 21, 1889 (2004). 3. R. Dumke, et al., Phys. Rev. Lett. 89, 097903 (2002). 4. Y.R.P. Sortais, et al., Phys. Rev. A 75, 013406 (2007). 5. N. Schlosser, G. Reymond and P. Grangier, Phys. Rev. Lett. 89, 023005 (2002). 6. M.P.A. Jones, J. Beugnon, A. Gaetan, J. Zhang, G. Messin, A. Browaeys, P. Grangier, Phys. Rev. A 75, 013406(R) (2007). 7. S. Nussmann, M. Hijlkema, B. Weber, F. Rohde, G. Rempe, and A. Kuhn, Phys. Rev. Lett. 95, 173602 (2005). 8. J. Beugnon, et al., in press Nature Physics (2007); arXiv:0705.0312 [quantPhl . 9. R. Raussendorf and H. Briegel, Phys. Rev. Lett. 86, 5188 (2001). 10. C. Simon, W.T.M. Irvine, Phys. Rev. Lett. 91, 110405 (2003). 11. D. L. Moehring, M. 3. Madsen, K. C. Younge, R. N. Kohn, Jr., P. Maunz, L.-M. Duan, C. Monroe and B.B. Blinov, J . Opt. SOC.A m . B 24, 300 (2007). 12. B.B. Blinov, D.L. Moehring, L.-M. Duan, C. Monroe, Nature 428,153 (2004). 13. J. Volz, M. Weber, D. Schlenk, W. Rosenfeld, J. Vrana, K. Saucke, C. Kurtsiefer and H. Weinfurter, Phys. Rev. Lett. 96, 030404 (2006). 14. B. Darqui6, et al., Science 309, 454-456 (2005). 15. C.K. Hong, Z.Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044-2046 (1987). 16. J. Beugnon, et al., Nature 440, 779 - 782 (2006).

PROGRESS IN ATOM CHIPS AND THE INTEGRATION OF OPTICAL MICROCAVITIES E. A. HINDS,* M. TRUPKE, B. DARQUIE, J. GOLDWIN and G. DUTIER Center for Cold Matter, Imperial College London, Prince Consort Road, London, SW7 ZAZ, UK *E-mail: ab-ed. hindsQimperial.ac.uk http://www3. imperial. ac. uk/ccm/ We review recent progress at the Centre for Cold Matter in developing atom chips. An important advantage of miniaturizing atom traps on a chip is the possibility of obtaining very tight trapping structures with the capability of manipulating atoms on the micron length scale. We recall some of the pros and cons of bringing atoms close to the chip surface, as is required in order to make small static structures, and we discuss the relative merits of metallic, dielectric and superconducting chip surfaces. We point out that the addition of integrated optical devices on the chip can enhance its capability through single atom detection and controlled photon production. Finally, we review the status of integrated microcavities that have recently been demonstrated at our Centre and discuss their prospects for future development. Keywords: Atom chip; Integrated optics; Cavity QED; Quantum information.

1. Introduction

Atom chips offer a powerful way to miniaturize experiments in atomic physics.' Microstructures on the surface of the chip produce magnetic and electric fields which can be used to confine and manipulate cold atoms above the substrate surface. Experiments have shown that it is possible to transport and split cold neutral atom clouds or BoseEinstein condensates in tight traps near the surface of a chip using either microscopic patterns of permanent magnetization in a film or microfabricated wire structures carrying currents or charges. Large field gradients can be generated close to a microstructured surface, creating tight traps with oscillation frequencies up to 1MHz at the micron scale where tunneling and coupling between traps become important. This makes atom chips suitable for applications in matter-wave interferometry,' quantum ~ e n s i n g atomic ,~ quantum N

271

272 information p r o ~ e s s i n gand , ~ the study of low-dimensional quantum gases.6 However many of these potential applications require the ability to detect a small number of atoms, to create versatile microscopic trapping configurations, and to manipulate the atomic quantum state, both internal and motional. Optical methods exist to accomplish these tasks, but are only just now starting to be miniaturized and integrated into atom chips. One microscopic optical device we are studying is an atom detector based on a pair of optical fibres facing each other. These single mode fibres have been machined at their ends, providing small beam waists a t the foci. Light transmitted through one fibre and collected by the other should allow detection of 5 to 10 atoms placed in the gap,7 either through their absorption (on-resonance imaging) or by the optical phase shift (dispersive imaging). The fibres can also be used to generate a one-dimensional standing-wave pattern, making an optical lattice in which the atoms can be manipulated by the optical dipole force. However, in order to bring the optical sensitivity down to the single atom level, we require an optical cavity to increase the atom-light interaction strength. The integration of microcavities with the trapping and guiding capabilities of atom chips opens new possibilities for experiments in atomic physics. In an optical microresonator, the interaction strength of an atom with a photon in the cavity mode increases with decreasing mode volume, as 1/fi.8 This is evident in the expression for the vacuum Rabi frequency in the cavity,

where p is the dipole moment of the atomic transition and wc is the resonance frequency of the cavity. This interaction is damped by the decay rate of the excited atomic population, which is 27 in free space, and of power in the cavity, which is 2 6 = m / ( L F ) .Here L is the cavity length and F is the finesse of the resonator. The driving and damping rates are succinctly compared in the single atom cooperativity, C = g2/2&,y. The total spontaneous emission rate of an atom in a cavity can be made to differ greatly from the free-space value, depending on the cooperativity. With the cavity tuned to the atomic transition frequency and with 6 < g, the natural decay rate becomes 2y(l 2C). The second term represents emission of photons into the cavity mode at a rate 4Cy = 2g2/r;. The resonators used successfully in other experiments have reached single-atom cooperativity values exceeding 50,’ but are not ideal for atom chip experiments as they consist of comparatively large components, mak-

+

273 ing it impossible to bring the intense part of their mode close t o the trapping structures on the chip surface. For this reason, research is being carried out on a variety of integrable microcavity designs. These include microtoroids, fibre-fibre microcavities and our fibre-chip cavities. Microtoroids offer the highest quality factors among these, and are well-suited for scalable atomchip integration as they can be produced by standard silicon microfabrication techniques. Strong coupling between single atoms and the field of a microtoroid resonator has been demonstrated" with atoms passing through the evanescent field in free-fall. To couple light into and out of these resonators, a tapered fibre is used. The fibre position needs to be adjusted so that the evanescent field at the taper overlaps with the evanescent component of the microtoroid mode. This is the only manual procedure required to make the devices operational. While the strong-coupling condition has been experimentally fulfilled for the first time for such a device, the challenge of reliably positioning and trapping atoms in the evanescent field of these devices with the required accuracy has yet to be surmounted. By contrast, atoms can be placed directly and accurately into the region of highest field strength of Fabry-Perot-type resonators. For this reason, the efforts of several research groups are currently focussed on this type of resonator. Several experimental groups have already succeeded in positioning atoms accurately within the mode of an optical resonator using optical, electrostatic and magnetic transport techniques. The positioning of a Bose-Einstein condensate in a microcavity on a chip by means of current-carrying wire guides has also been recently demonstrated using a fibre-coupled microcavity.ll The resonators used in that work are however not ideally suited for scalable integration, as they are constructed in a sequence of mostly manual steps. Here we report on recent experiments at our Centre on the detection of atoms with on-chip micocavities, whose design aims to combine the advantages of microfabrication and fibre-coupling with the open access of a Fabry-Perot resonator. In the next section we describe two typical atom chips that are in use in our laboratory to magnetically trap atoms either with current carrying wires or permanent magnet structures. Section 3 deals with detection of atoms and enhanced emission of photons using on-chip microcavities. We conclude with an outlook in Sec. 4.

274

2. Magnetic atom chips

To give a concrete example of an atom chip, we show in Fig. 1 an interferometer chip7 currently being used in our laboratory. The reflective gold surface is used to form a mirror magneto-optical trap (MOT) as a reservoir of pre-cooled atoms. The coils used for the MOT are seen in the figure. The chip consists of a 3 /zm layer of gold, thermally evaporated onto a silicon substrate, in which wires have been lithographically defined by ion-beam milling. Currents in the wires allow cold 87Rb atoms to be magnetically trapped and split, in a matter-wave analogue of an optical beam splitter.2'3 The inset in Fig. 1 is a microscope image showing the four parallel wires at the heart of the interferometer, which have a center-to-center distance of 300 /zm between the thick outer pair.

Fig. 1. Current carrying wire atom chip and chip mount used at Imperial College by Eriksson et al.7 The inset shows a high resolution microscope image of the central region of the chip where atoms are trapped and manipulated.

With metallic chips such as this one, the homogeneity and stability of cold atom clouds can be compromised close to the room temperature surface. Two main phenomena have been identified: (i) spatial imperfections of the wire cause the current to flow non-uniformly and make the atom trap rough, leading to fragmentation of the atom clouds,12'13 and (ii) thermal fluctuations of the magnetic field near the surface, caused by current noise in the conductor, drive spin flips of the atoms, thereby inducing loss and destroying quantum coherences.14 The fragmentation can be controlled by using fabrication methods that give very homogeneous wires with smooth

275 edges and uniform thickness. The magnetic noise field can be altered by adjusting the thickness of the surface and by changing the material in order to alter the complex c ~ n d u c t i v i t y . ~ ~ ~ ~ ~ We have also been using a chip based on patterns of permanent magnetization, written on commercial videotape. This exhibits very much longer spin flip times because the thermal noise currents are suppressed by the high resistivity of the videotape. A pattern of sinusoidal magnetisation (in plane), with a period of about 100pm, allows the confinement of atoms at distances less than 100 pm from the surface of the chip in an array of long, thin traps. These have a very high aspect ratio (> lo3), with tight transverse confinement that can bring ultra-cold atoms into the one-dimensional regime.l7>l8As well as offering a long spin-flip lifetime for BECs trapped near its surface,17 the permanent magnet atom chip has the benefit of low power dissipation. Some fragmentation of these videotape traps has been observed and is due primarily to inhomogeneity of the magnetic layer. Preliminary experiments on Pt/Co multilayer magnetic thin filmslg indicate that these are a promising alternative t o videotape. Other permanent magnet materials are being studied in the group of P. Hannaford, also reporting at this confererence. Superconductors offer another way to reduce the magnetic noise level and increase the spin-flip lifetimes by many orders of magnitude. Indeed, atoms have already been trapped near superconducting surfaces in two laboratories.20,21A recent calculation22 shows that rubidium atoms trapped 1 pm away from a superconducting niobium surface at liquid helium temperature should have a long spin flip lifetime, in excess of 1000s. Under these conditions, the cold atoms offer a new way to probe the superconducting surface because they are very sensitive to the local magnetic field. For example, cold atoms would be able to image the vortices of a thin superconductor close to the Kosterlitz-Thouless transition or more generally to study vortex dynamics in a type II s u p e r c o n d ~ c t o r . ~ ~ 3. Chips with optical micro-cavities

We have recently tested a new type of optical microcavity, which combines the advantages of microfabrication and fibre-coupling with the open access of a Fabry-Perot resonator. The plano-concave resonator is formed between an isotropically etched hemisphere on the surface of a silicon chip and the plane end of single-mode fibre. Both surfaces are coated with a high-reflectivity multilayer dielectric film. Our collaborators at the University of Southampton have fabricated large arrays of concave mirrors on

276

silicon chips and we have used them to build cavities24 with F > 5000 and Q > 106. More recently, we have demonstrated both the detection of atoms and the enhanced emission of photons into the mode of the cavity.25 The apparatus is shown in Fig. 2. Two fibres with plane dielectric mirrors on their ends are held in grooves on a glass-ceramic substrate, facing two of the mirrors in the silicon array. The silicon mirror chip is mounted on a piezo-electric translator (PZT), which is adjusted to tune the upper cavity to the free-space atomic resonance. In this work the lower cavity was not used. A mirror is used to form a reflection MOT directly above the cavities.

APD

Probe Fig. 2.

Fibre

Mirror chip

Schematic diagram of apparatus used in the experiment.

For these experiments, we used cavities with F = 280 and L = 133 //m, giving K = 27r x 2 GHz. The beam waist of the mode is we = 4.6 /mi, giving a vacuum Rabi frequency of 2g = 1-n x 200 MHz for 85Rb atoms at an antinode of the cavity mode, driven with circularly polarised light on the closed transition \F — 3,mp = ±3) —> \F' = 4,mp/ — ±4). This transition has a natural lifetime 27 = 2?r x 6 MHz, yielding a single-atom cooperativity of C = 0.8. The cooperativity increases linearly with the number A^A of atoms in the cavity. Since these are not generally at an antinode, the total cooperativity is Ctot = jC^,n=i A r «) where /(r n ) is the fraction of peak intensity at the position of the n th atom. The factor | accounts for an average of Clebsch-Gordan coefficients over all the F = 3 Zeeman sublevels. The effective number of atoms interacting with the cavity field is then defined as Nf = C to t/(f C).

277 3.1. Atom detection

z , n

320

'v)

m

0

--

v

300

a, 280

c

3

0

0 260 10

20

30

40

50

60

Time (ms) Fig. 3. Change in reflected probe light as atoms traverse the cavity. Laser, cavity, and atom frequencies are equal: W L = wc = W A . Average time-of-flight signal of 34 MOT 0.7 atoms in the cavity on releases, with an integration time of 2 5 0 p . There are average at the peak.

-

-

To put atoms into the microcavity, we release a cloud of 2 x lo7 85Rb atoms from the MOT, 7mm above. Weak resonant probe light (1 p W ) is incident on the cavity through a beam splitter and the reflected photons are counted by an avalanche photodiode (APD, see Fig. 2). When the cavity is far from resonance with the light, the reflected intensity is I,,, = 419 x lo3 s-l, which drops to I m i n = 272 x lo3 s-l at resonance. As displayed in Fig. 3, this rises to a peak of Iatoms = 315 x lo3 s-l when the atoms pass through the cavity. From these three count rates we derive the peak cooperativity of the atom-cavity system using the relation

+

where Ptot= 2CtOt 1 is an effective Purcell factor, and v = 1- ,/is the empty cavity fringe visibility. From our experiments, we determine a total cooperativity of Ctot 0.23 at the peak of the atom signal, corresponding to an effective atom number of only Nzff = 0.7. We have measured the change in the peak reflected intensity when we scan the detuning of the probe light ALA= (WL - w ~ ) / ykeeping , the cavity frequency equal to the atomic resonance frequency w c = W A . Figure 4

-

278

0.701, -10

,

, 0

,

,I

10

E -10

0

10

Detuning (MHz)

Fig. 4. Peak fraction of power reflected versus detuning ALA when w c = W A for two different initial MOT numbers. The fits (solid lines) yield average atom numbers of (a) (Niff)= 1.1 (b) (Niff)= 0.6.25

displays, for two different values of the initial atom number in the MOT, Iatoms/Imax versus ALA,for which we expect

assuming g/y >> ( ~ , A L A The ) . solid lines in Fig. 4 are fits to the data, taking into account fluctuations of Ctot and the probe laser l i n e ~ i d t h . ' ~ These lineshapes confirm that we have better than single atom sensitivity.

3.2. Noise suppression The arrival of atoms in the cavity is signaled not only by an increase in the reflected light level, but also by a decrease in the intensity noise. To measure this, we dropped the cloud 48 times, recording the reflected intensity in 10 ps bins. We then determined for each bin the ratio of variance to mean, taken over the 48 drops. After correcting our count rate for the measured APD dead-time of 4 4 n ~ we , ~found ~ the correct variance-to-mean ratio f ( n ) V a r ( n ) / ( n at ) each time bin ( n being the counts already corrected for the dead-time) by including the 10% loss at the beamsplitter of Fig. 2 and the 60% loss due to the quantum-efficiency of the APD, according to fcorr(n)1 = [ f ( n )- 1]/0.54. The resulting values, plotted in Fig. 5, show a strong

279

20

30

40

50

Time (ms) Fig. 5. Ratio of variance to mean for photon count rate reflected from the cavity during passage of atom cloud (corrected for APD dead-time and losses). Vertical dashed line: arrival time of the peak number of atoms. Horizontal line at 1.0: shot noise level. The sample size is 48 atom cloud drops.

reduction of the noise during the passage of the atom cloud. In addition to the photon shot noise, intensity fluctuations are due to small shifts of the cavity length away from its resonance. The effect of these mechanical fluctuations is expected to be less pronounced when atoms are present, and this partially explains the decrease. However many of our experiments seem to show additional noise reduction, the origin of which requires further investigation.

3.3. Photon generation

A further effect observed with our microcavity is the enhancement of spontaneous emission. For this experiment, we once again release a cloud of atoms into the resonator. When the atom number in the cavity reaches its highest value, we switch on a resonant excitation beam, aimed at the atoms from below the cavity. This beam pushes the atoms out of the cavity and also pumps them into the dark F = 2 state. The loss of interacting atoms produces a sharp drop in the cavity reflection signal, as shown in Fig. 6 (a). We then perform the same experiment but without any probe light. This time, a photon peak is recorded at the moment the excitation laser is turned on, as shown in Fig. 6 (b). These are fluorescence photons

280 I

I I

,

,

,

,

’

1.o 0 10

20

30

40

50

60

Time (ms) Fig. 6. (a) Sharp drop in the cavity reflection signal due to atom loss and optical pumping when the excitation laser is turned on. (b) Cavity-enhanced spontaneous emission collected via the fiber when the atom cloud is excited. The dotted line shows coincidence of the photon pulse with the turn-on of the excitation laser.

emitted into the cavity mode at the Purcell-enhanced rate 4CtOty,as mentioned in the introduction, and transported to the APD through the fibre. This is confirmed by the fact that no photons are collected when the cavity is detuned from the atomic transition or when there are no atoms in the cavity. 4. Outlook

These experiments have shown that this type of microresonator is suitable for the detection of numbers of atoms smaller than one, and can be used t o generate photons by enhanced spontaneous emission. The latter is of particular interest for quantum-cryptography and quantum-computation schemes. These cavities have by no means yet reached their full potential. In the laboratory, with materials of the same type, we have created cavities with F > 500 and I m i n / l m a x < 15%. Both values are improvements by approximately a factor 2 compared to the published results, which will considerably increase the detection signal, as can be deduced from Eq. (2). We had previously shown that we can even achieve finesse values of F > 5000,

281

albeit with a low fringe-extinction of I m i n / l m a x 5 At present, these values are limited by scattering losses caused by the surface roughness of the silicon mirror-substrate. Working with the group of M. Kraft at the university of Southampton, we expect in the near future to have smoother cavity mirrors that can operate well within the strong-coupling regime of cavity quantum-electrodynamics, in which the coherent exchange of quantum information between atoms and photons in the cavity becomes possible. These microcavities have the virtues that they are open, giving atoms direct access to the peak of the mode pattern, they couple the light in and out in a simple way through an integrated fiber, and they can be fabricated in large numbers. Being integrated into a silicon substrate, they can be readily combined with other micromachined components on atom chips. Some examples include the magnetic trapping and guiding structures described in Sec. 2, which allow the deterministic transport of atoms in and out of the cavity, integrated actuators for tuning the resonator,26 or pyramidal micro-mirrors to realize an array a of single-atom sources from an array of microscopic M O T S . Work ~ ~ is in progress to produce a fully integrated device above which neutral atoms can be trapped, guided, detected and manipulated in situ, using magnetic and optical fields.

Acknowledgments This work was supported by EU networks Atom Chips, Conquest, and SCALA, by the Royal Society, and by EPSRC grants for QIPIRC, CCM programme and Basic Technology. The atom chip devices in use at Imperial College were fabricated by the nanosystems group of Professor M. Kraft a t the University of Southampton.

References 1. J. Fortagh and C. Zimmermann, Rev. Mod. Phys. 79,235 (2007). 2. T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, S. J. and P. Kriiger, Nut. Phys. 1, 57 (2005). 3. E. A. Hinds, C. J . Vale and M. G. Boshier, Phys. Rev. Lett. 86, 1462 (2001). 4. P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. Hansch and J. Reichel, Phys. Rev. Lett. 92, 203005 (2004). 5. T. Calarco, E. A. Hinds, D. Jaksch, J. Schmiedmayer,J. I. Cirac and P. Zoller, Phys. Rev. A 61, 022304 (2000). 6. L. Pricoupenko, H. Perrin and M. Olshanii (eds.), Quantum Gases in Low Dimensions, J. Phys. IV France Vol. 116, (EDP Sciences, 2004). 7 . S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A.

282

8.

9. 10.

11. 12. 13. 14.

15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. 0. Gollasch and M. Kraft, Eur. Phys. J D 35,135 (2005). P. R. Berman (ed.), Cavity Quantum Electrodynamics Advances in Atomic, Molecular and Optical Physics, Supplement 2, (Academic Press, New York, 1994). K. J. Vahala, Nature 424,839 (2003). T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J . Vahala and H. J . Kimble, Nature 443,671 (2006). Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger and J. Reichel, arXiv:0706.1390 (2007). M. P. A. Jones, C. J. Vale, D. Sahagun, B. Hall, C. C. Eberlein, B. E. Sauer, K. Furusawa, D. Richardson and E. A. Hinds, J . Phys. B 37,L15 (2004). Z. Moktadir, B. Darqui6, M. Kraft and E. A. Hinds, J . Mod. Opt. 54,2149 (2007). M. P. A. Jones, C. J. Vale, D. Sahagun, B. V. Hall and E. A. Hinds, Phys. Rev. Lett. 91,080401 (2003). P. K. Rekdal, S. Scheel, P. L. Knight and E. A. Hinds, Phys. Rev. A 70, 013811 (2004). S. Scheel, P. K. Rekdal, P. L. Knight and E. A. Hinds, Phys. Rev. A 72, 042901 (2005). C. D. J . Sinclair, E. A. Curtis, I. L. Garcia, J. A. Retter, B. V. Hall, S. Eriksson, B. E. Sauer and E. A. Hinds, Phys. Rev. A 72,031603 (2005). C. D. J. Sinclair, J . A. Retter, E. A. Curtis, B. V. Hall, I. L. Garcia, S. Eriksson, B. E. Sauer and E. A. Hinds, Eur. Phys. J . D. 35,104 (2005). S. Eriksson, F. Ramirez-Martinez, E. A. Curtis, B. E. Sauer, P. W. Nutter, E. W. Hill and E. A. Hinds, Appl. Phys. B 79,811 (2004). T. Nirrengarten, A. Qarry, C. ROUX,A. Emmert, G. Nogues, M. Brune, J.-M. Raimond and S. Haroche, Phys. Rev. Lett. 97,200405 (2006). T. Mukai, C. Hufnagel, A. Kasper, T. Meno, A. Tsukada, K. Semba and F. Shimizu, Phys. Rev. Lett. 98,260407 (2007). U. Hohenester, A. Eiguren, S. Scheel and E. A. Hinds, Phys. Rev. A 76, 033618 (2007). S. Scheel, R. Fermani and E. A. Hinds, Phys. Rev. A 75,064901 (2007). M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka and M. Kraft, Appl. Phys. Lett. 87,211106 (2005). M. Trupke, J. Goldwin, B. Darqui6, G. Dutier, S. Eriksson, J. Ashmore and E. A. Hinds, Phys. Rev. Lett. 99,063601 (2007). C. Gollasch, Z. Moktadir, M. Kraft, M. Trupke, S. Eriksson and E. A. Hinds, J . Micromech. Microeng. 15,S39 (2005). M. Trupke, F. Fernandez-Ramirez, E. A. Curtis, J. P. Ashmore, S. Eriksson, E. A. Hinds, Z. Moktadir, C. Gollasch, M. Kraft, G. V. Prakash and J . J . Baumberg, Appl. Phys. Lett. 88, 071116 (2006).

SINGLE ATOMSAND QUANTUM OPTICS I1

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QUANTUM OPTICS WITH SINGLE ATOMS AND PHOTONS H. J. Kimble Norman Bridge Laboratory of Physics MC 12-33, California Institute of Technology, Pasadena, CA 91125, USA www.its.caltech. edu/Nqoptics/ An overview of research in the Caltech Quantum Optics Group is presented. We are attempting t o harness atomic radiative processes “one-by-one” and thereby to achieve full quantum control of the optical field. Applications include the realization of a quantum interface between matter and light, and the implementation of quantum networks for quantum information science and quantum many-body physics.

1. Introduction

The general theme of research in the Caltech Quantum Optics Group is an investigation of the dynamics of open quantum systems in manifestly quantum (i.e., nonclassical) domains. The research seeks to develop the means for the deterministic control of the evolution of open quantum systems in a prescribed fashion at the level of the individual constituents. Within this setting, the investigations relate to strong coupling in optical physics whereby nonlinear interactions require only single atomic and field excitations. In qualitative terms, the goal is to move beyond traditional optical physics into a new regime with dynamical processes involving single quanta taken one by one. Three experiments are discussed, namely cavity quantum electrodynamics (cavity QED) with a conventional Fabry-Perot resonator, cavity QED with lithographically fabricated microresonators, and quantum information processing with atomic ensembles. This contribution is focussed specifically on research in the Caltech Quantum Optics Group and, with apology, does not provide an overview of the many exciting advances being made by other groups around the world. At the 2007 ICOLS, there were several presentations related to the themes discussed here, t o which the reader is referred. Included are amazing re-

285

286

sults in cavity QED presented by S. Haroche from ENS, as well as new experiments in this area described by E. Hinds from Imperial College and D. Meschede from Hannover. Likewise, landmark experiments involving entanglement and teleportation with atomic ensembles were reported by E. Polzik from the Niels Bohr Institute and future prospects by E. Giacobino from the Laboratoire Kastler Brossel. 2. Cavity QED with single trapped atoms

A long-standing ambition in the field of cavity quantum electrodynamics has been to trap single atoms inside high-Q cavities in a regime of strong coupling. A critical aspect of this research is the development of techniques for atom localization that are compatible with strong coupling and that do not interfere with cavity QED interactions.’>’ Our group first achieved this goal in 2003 for single Cs atoms stored in a state-insensitive, intracavity far-off resonance trap (FORT).3 The observed lifetime of 3 s represented an advance of lo3 - lo4 beyond the first realizations of trapping in cavity QED.4,5Unlike typical FORTS, where the AC-Stark shifts for excited and ground states are of opposite sign, our trap3 causes only small shifts to the relevant transition frequencies, thereby providing advantages for coherent state manipulation of the atom-cavity system. The capabilities reported in Ref.3 have enabled a series of advances our group, which are reviewed in Ref.‘ An essential component of this work has been the development of cooling schemes, which lead to atomic localization Ax:,,i,~ = 33 nm, Aptransverse Y 5.5 pm,’ and more recently, to AxazialN 8.3 nm.8 2.1. Photon blockade

Sufficiently small metallic and semiconductor devices at low temperatures exhibit “Coulomb blockade,” whereby charge transport through the device occurs on an electron-by-electron basis. In 1997, Imamoglu et al. proposed that an analogous effect might be possible for photon transport through an optical system by employing photon-photon interactions in a nonlinear optical ~ a v i t y . ~ In Ref.1° we reported the first observation of photon blockade, in our case, for the light transmitted by an optical cavity containing one atom strongly coupled to the cavity field.ll For coherent excitation at the cavity input, we investigated the photon statistics for the cavity output and demonstrated the nonclassical character of the transmitted field, which exhibited both photon antibunching and sub-Poissonian photon statistics. We

287

observed modulation of the cavity transmission arising from the oscillatory motion of the atom trapped within the cavity mode, which allowed us to infer a maximum kinetic energy E/ks ~ 250 ^K.

2.2. Control of the center-of-mass motion in cavity QED In recent years, great strides have made by many groups to localize atoms for strong coupling to the fields of optical cavities. However, until our work in Ref.,8 it had not previously been possible to access the quantum regime for the atomic center-of-mass motion in cavity QED. In Ref.8 we achieved localization to the ground state of motion for one atom trapped in an optical cavity in a regime of strong coupling. Resolved sideband cooling to the ground state was accomplished with a coherent pair of intracavity Raman fields. To deduce the resulting state of atomic motion, we introduced a new scheme for recording Raman spectra by way of the strong interaction of the atom with a resonant cavity probe. Our scheme is the cavity QED equivalent of state detection in free space by quantum-jump spectroscopy and achieved a confidence level for state discrimination > 98% in 100 /is. From these Raman spectra, we inferred that the lowest vibrational level n = 0 of the axial potential with zero-point energy huja/2kB = 13 fj,K was occupied with probability PQ — 0-95 for one trapped atom. We are working now to utilize this advance to investigate quantum kinematics in a quantized light field. The setting is cavity QED in a domain of strong coupling for both the internal (i.e., atomic dipole and cavity field) and external center-of-mass degrees of freedom. Examples include the transfer of quantized states of atomic motion to quantum states of light, and conversely,12 as well as measurements that surpass the standard quantum limit for sensing atomic position.13

y\r...y\r Fig. 1. Illustration of the protocol of Ref.14 for quantum state transfer and entanglement distribution from system A to system B. By expanding to a larger set of interconnected cavities, complex quantum networks can be realized.

288

a) to detectors

F=3

F=4

Fig. 2. (a) Schematic of the experiment for reversible state transfer.16 The probe A(4) resonantly drives the cavity through input mirror Min; the classical field Q(t) excites the atom transverse to the cavity axis. Photons emitted from the output mirror M0ut are directed to a pair of avalanche photodiodes. (b) Atomic level diagram. Double arrow g indicates the coherent atom-cavity coupling, and Q(t) is the classical field. The cavity and O field are blue-detuned from atomic resonance by A.

2.3. Reversible state transfer between light and a single trapped atom In the initial proposal for the implementation of quantum networks,14 atomic internal states with long coherence times serve as 'stationary' qubits, stored and locally manipulated at the nodes of the network. Quantum channels between different nodes are provided by optical fibers, which transport photons ('flying' qubits) over long distances. A crucial requirement for such network protocols is the reversible mapping of quantum states between light and matter. Cavity quantum electrodynamics (QED) provides a promising avenue for achieving this capability by using strong coupling for the interaction of single atoms and photons.6 Within this setting, reversible emission and absorption of one photon can be achieved by way of a dark-state process involving an atom and the field of a high-finesse optical cavity. The basic scheme, illustrated in Fig. 1, involves a three level atom with ground states |a) and |6) and excited state e). An optical cavity is coherently coupled to the atom on the 6 <-> e transition with rate g, and a classical field fi(t) drives the atom on the a <-> e

289

transition. If the R field is ramped adiabatically off t on, then state la, n) evolves into ( b , n I), and state ( b , n ) remains unchanged, where lu,n), Ib,n) denotes a state in which the atom is in ground state a, b and there are n photons in the cavity. Ramping R on 4 off implements the reverse transformation. This process can be used to generate single photons by preparing the atom in la)and ramping R off 4 on, thereby effecting the transfer la,0) + Ib, 1) with the coherent emission of a single photon pulse from the cavity.15 A distinguishing aspect of the protocol is that it should be re~ersible,'~ so that a photon emitted from one system A can be efficiently transferred to another system B. Furthermore, it should be possible to map coherent superpositions reversibly between the atom and the field:

+

(cop)

+ c1la)) 8 10)

Ib) c 3(

d o )+ 4 1 ) ) .

(1)

Although single-photons have been generated in diverse physical systems, most such sources are not in principle reversible, and for those that are, no experiment has verified the reversibility of either the emission or the absorption process. In Ref.16 we achieved an important advance related to the interface of light and matter by explicitly demonstrating the reversible mapping of a coherent optical field to and from the hyperfine ground states of a single, trapped Cesium atom. As illustrated in Fig. 2, we map an incident coherent state A(t) with fi = 1.1photons into a coherent superposition of F = 3 and F = 4 ground states with transfer efficiency = 0.057. We then map the stored atomic state back to a field state. The coherence of the overall process is confirmed by observations of interference between the final field state and a reference field that is phase coherent with the original coherent state, resulting in a fringe visibility w, = 0.46 f 0.03 for the adiabatic absorption and emission processes. We thereby provide the first verification of the fundamental primitive upon which the protocol in Ref.14 is based. Our experiment represents an important step towards the realization of cavity QED-based quantum networks, wherein coherent transfer of quantum states enables the distribution of quantum information across the network.

<

3. Cavity QED with microtoroidal resonators

In Ref.,17 we achieved strong coupling between single Cesium atoms and the fields of a microtoroidal cavity fabricated from SiOZ." As illustrated in Fig. 3 , laser cooled atoms are dropped from above the microtoroid, with some small number of atoms transiting through the external evanescent field

290

Fig. 3. (a) Simple schematic of our experiment17 showing a cloud of cold Cs and associated trapping lasers above an array of microtoroidal resonators. Light from the probe beam Pin is coupled into a resonator by way of the fiber taper, with the forward propagating output Pp coupled into the taper, (b) Illustration of an SiC>2 microtoroidal resonator, fiber taper, and atom cloud above. The calculated field distribution for the lowest order resonator mode is shown by the color contour plot on the right. Cold Cesium atoms fall through the external evanescent field of this mode and are thereby strongly coupled to the resonator's field.

of the resonator. By recording the dependence of the forward propagating power Pp on the frequency detuning A.AC between the atom and cavity resonances for individual atom transits, we inferred the maximum accessible coupling rate (7, K, Ai^ 1 ), where the rate of atomic spontaneous decay 7/2?r = 2.6 MHz, the cavity field decay rate K/2ir = 18 ± 3 MHz, and At PS 2/zs is the average transit duration through the evanescent field of the toroidal resonator. Our measurements are the first to demonstrate strong coupling for single atoms interacting with an optical resonator other than a conventional Fabry-Perot cavity. Our goal with this research is to realize complex quantum networks consisting of multiple atom-resonator systems. Such capabilities could have broad impact in Quantum Optics and Quantum Information Science, including the implementation of quantum networks for quantum computation

291

and communication,14'19'20 as well as quantum metrology21 23 and quantum information processing on atom chips.24 A principal motivation is also the investigation of quantum many-body systems by way of quantum networks. 4. Quantum information with atomic ensembles

In the new science of quantum information, distributed quantum networks play an important role, including for quantum computation, communication, and metrology. An approach of particular importance has been the seminal work of Duan, Lukin, Cirac, and Zoller (DLCZ) for the realization of quantum networks based upon entanglement between single photons and collective excitations in atomic ensembles.25 In recent years, critical experimental capabilities related to the DLCZ protocol have been achieved in several laboratories around the world. However, it has not previously been possible to integrate these individual capabilities into a system with the minimal functionality required for scalable quantum networks. In Ref.,26 Fig. 4. Laboratory setup26 for entanglewe achieved an important mile- ment of two pairs of atomic ensembles for the implementation of quantum communistone in this quest. As illustrated cation by way of the protocol of Ref.28 The in Fig. 4, we created, addressed, elongated ovals represent cylinders of Ceatoms, each containing approximately and controlled pairs of atomic en- sium 105 atoms, with 1 excitation shared remotely sembles at each of two quantum at a distance of 3 meters between each pair nodes, thereby leading to the first in an entangled state. realization of entanglement distribution in a form suitable both for quantum network architectures and for entanglement-based quantum communication schemes involving independent operations on parallel chains of quantum systems.25

292

In our experiment, single collective excitations in a pair of atomic ensembles are employed to encode a coherent qubit at each of two quantum nodes ( L , R ) separated by 3 meters.27 These effective qubits are remotely entangled by suitable measurement^^^ and conditional quantum evolution.28 Entanglement is verified by mapping the stored quantum states at the ( L ,R ) nodes from the respective atomic ensembles to propagating light fields. The states of these light fields, after filtering through local operations and classical communication, lead to photoelectric counting statistics that significantly violate a Bell inequality, demonstrating entanglement between the quantum nodes. Our system is thereby capable of implementing entanglement-based quantum communication protocols, such as quantum cryptography and quantum t e l e p ~ r t a t i o n . ~ ~ Importantly, entanglement is achieved between pairs of ensembles at the L , R nodes in an asynchronous fashion, with the quantum memory providing the capability to wait independently for entanglement of the two pairs.28 This capability is critical for enabling the quantum repeater architecture of Ref.25 with parallel “chains” of ensembles, as for our L , R nodes, thereby avoiding exponential resource scaling. Combined with extensions of the protocol of Ref.25 and improvements in the coherence time of the quantum memory,2g the capabilities achieved in Ref.26 can enable the robust distribution of entanglement and thereby scalable quantum networks.

5. Acknowledgements First and foremost, I gratefully acknowledge the contributions of the Caltech Quantum Optics Group, who carried out the experiments that I have described. More information about these people and their projects can be found in the references that follow and at our group website [www.its.caltech.edu/~qoptics/]. This work is supported by the National Science Foundation and by the Disruptive Technology Office.

References 1. H. J . Kimble, C. J. Hood, T. W. Lynn, H. Mabuchi, D. W. Vernooy, and J. Ye in Laser Spectroscopy: X I V International Conference, eds. R. Blatt, et al. (World Scientific, Singapore, 1999), 80. 2. H. Katori, T. Ido, and M. Kuwata-Gonokami, J . Phys. SOC.Jpn. 68,2479 (1999). 3. J. McKeever, J.R. Buck, A.D. Boozer, A. Kuzmich, H.-C.Nager1, D.M. Stamper-Kurn, H.J. Kimble, Phys. Rev. Lett. 90, 133602 (2003). 4. J. Y e , C. J. Hood, T. Lynn, H. Mabuchi, D. W. Vernooy and H. J . Kimble, IEEE Transactions on Instrumentation and Measurement 48, 608 (1999).

5. J. Ye, D. W. Vernooy, and H. J. Kimble, Phys. Rev. Lett. 83, 4987 (1999). 6. R. Miller, T. E. Northup, K. M. Birnbaum, A. Boca, A. D. Boozer, and H. J. Kimble, J. Phys. B: At. Mol. Opt. Phys. 38, S551 (2005). 7. A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, Phys. Rev. Lett. 93, 233603 (2004). 8. A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, Phys. Rev. Lett. 97, 083602 (2006). 9. A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, Phys. Rev. Lett. 79, 1467 (1997). 10. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Nature 436, 87 (2005). 11. L. Tian and H. J. Carmichael Phys. Rev. A46, R6801 (1992). 12. A. S. Parkins and H. J . Kimble, Journal Opt. B: Quantum Semiclass. Opt. 1,496 (1999); Frontiers of Laser Physics and Quantum Optics, eds. Z. Xu et al. (Springer, Berlin, 2000), 132; Phys. Rev. A 61, 052104 (2000). 13. P. Storey, M. Collett, and D. F. Walls, Phys. Rev. Lett. 68, 472 (1992) and Phys. Rev. A47, 405 (1993); R. Quadt, M. Collett, and D. F. Walls, ibid , 74, 351 (1995). 14. J . I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Phys. Rev. Lett. 78, 3221 (1997). 15. J . McKeever, A. Boca, A. D. Boozer, R. Miller, J . R. Buck, A. Kuzmich, and H. J. Kimble, Science 303, 1992 (2004). 16. A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, Phys. Rev. Lett. 98, 193601 (2007). 17. T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, H. J. Kimble, T. J. Kippenberg, and K. J . Vahala, Nature 443, 671 (2006). 18. S. M. Spillane, T. J . Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, Phys. Rev. A 71, 013817 (2005). 19. L.-M. Duan and H. J. Kimble, Phys. Rev. Lett. 92, 127902 (2004). 20. H.-J. Briegel, S. van Enk, J.I. Cirac, P. Zoller, in The Physics of Quantum Information, D. Bouwmeester, A. Ekert, A. Zeilinger, Eds. (Springer, Berlin, 2000), pp. 192-197. 21. V. Giovannetti, S. Lloyd, L. Maccone, Science 306, 1330 (2004). 22. V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006). 23. S. M. Roy and S. L. Braunstein, quant-ph/0607152. 24. P. Treutlein, T. Steinmetz, Y. Colombe, P. Hommelhoff, J. Reichel, M. Greiner, 0. Mandel, A. Widera, T. Rom, I. Bloch, T. W. Hansch, Fortschritte der Physik 54, 702 (2006). 25. L.-M. Duan, M. Lukin, J. I. Cirac, and P. Zoller, Nature 414, 413 (2001), here referred to as DLCZ. 26. Chin-Wen Chou, Julien Laurat, Hui Deng, Kyung So0 Choi, Hugues de Riedmatten, Daniel Felinto, and H. J. Kimble, Science 316, 1319 (2007); published online in Science Express, 5 April, 2007. 27. C. W. Chou, H. de Riedmatten, D. Felinto, S. V. Polyakov, S. J. van Enk, and H. J . Kimble, Nature 438, 828 (2005).

294 28. D. Felinto, C. W. Chou, J. Laurat, E. W. Schomburg, H. de Riedmatten, and H. 3. Kimble, Nature Physics 2, 844 (2006) (advanced online publication 29 October, 2006). 29. D. Felinto, C.W. Chou, H. de Riedmatten, S.V. Polyakov, and H.J. Kimble, Phys. Rev. A 72, 053809 (2005).

OPTICALATOMICCLOCKS

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FREQUENCY COMPARISON OF Al+ AND Hg+ OPTICAL STANDARDS T. ROSENBAND*, D. B. HUME, A. BRUSCH, L. LORINIt, P. 0. SCHMIDT$, T. M. FORTIER, J. E. STALNAKERS, S. A. DIDDAMS, N. R. NEWBURY, W. C. SWANN, W. H. OSKAYT, W. M. ITANO, D. J. WINELAND, AND J. C. BERGQUIST National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305 *E-mail: [email protected] We compare the frequencies of two single ion frequency standards: 27Alf and Ig9Hgf . Systematic fractional frequency uncertainties of both standards are and the statistical measurement uncertainty is below 5 x below Recent ratio measurements show a reproducibility that is better than

Although single-ion optical frequency standards promise a potential accuracy of lo-'' or better,'>2 this long-standing goal has not yet been realized due to various technical difficulties. Here we report progress for the NIST lg9Hg+ and 27Al+ single-ion standard^,^)^ as their systematic fractional frequency uncertainty approaches In these measurements, the fourth harmonics of two clock lasers are locked t o the mercury and aluminum clock transitions at 282 and 267 nm respectively. An octave-spanning self-referenced Ti:Sapphire femtosecond laser frequency comb (FLFC)5 is phase-locked to one clock laser, and the heterodyne beat-note of the other clock laser with the nearest comb-tooth is measured. The various beat-note and offset frequencies can be combined to yield a frequency ratio, which is independent of the Cs-based definition of the second, allowing this ratio to be measured even more accurately than the fundamental unit of time can be realized. In recent comparisons of the frequencies of the two clock lasers included here, an octave-spanning selfreferenced fiber comb laser6 has provided a second independent measure of t Present address: Istituto Nazionale di Ricerca Metrologica (INRIM), Torino, Italy f Present address: Institut fur Experimentalphysik, Universitat Innsbruck, Austria §Present address: Department of Physics and Astronomy, Oberlin College, Oberlin, OH TPresent address: Stanford Research Systems, Sunnyvale, CA

297

298 the frequency ratio. The 27Al+ 'So t 3Po standard, which uses Quantum Logic Spectroscopy7 has been described previously.* Briefly, one 27Al+ ion is trapped together with a 'Be+ ion, which provides sympathetic cooling. The 27Alf clock state is mapped to 'Be+ repetitively through their coupled motion, allowing for up to 99.94% clock state detection fidelity.8 With the ability to detect the clock state comes the ability to detect state transitions, whose probability depends on the clock laser frequency. The clock laser is locked to the atomic transition by alternating between upper and lower slopes of the atomic resonance curve, and keeping the transition rates equal. At the operating field of 1 gauss, the Zeeman structure is split by several kilohertz, and the individual Zeeman components are well resolved. We alternate between the extreme states of opposite angular momentum (mF = f 5 / 2 ) , which allows compensation of magnetic field shifts up to second The accuracy of the aluminum standard is limited to 2.2 x primarily due to uncertainties in the second-order Doppler shifts. For ions confined in Paul traps, there are two types of motion: secular motion, which is the harmonic motion of the trapped particle, and micromotion, which occurs when the ion is displaced from the rf-null of the confining field. Both types of motion have rms-velocities of about 1-2 m/s, with similar uncertainties. Excess micromotion results when slowly fluctuating electro-static fields in the ion trap displace the ion from the null position. These quasi-static fields are monitored and corrected by interleaving micromotion-test experiments with the clock interrogations. Tests are performed by monitoring the strength of radial-to-axial coupling of certain normal modes via 'Be+,'' and applying compensation voltages at the ion trap to minimize this coupling. With real-time corrections, the stray electric fields are nulled to (0 f 10) V/m, allowing an estimate of the time dilation uncertainty.l' The total fractional frequency clock shift due to the extreme case of a 10 V/m field along both radial directions is 3.2 x and we estimate the fractional frequency shift caused by residual (uncompensated) stray electric fields to be (2 f 2) x when the clock is operating. Secular mode causes deviations in the secular kinetic energy from the Doppler-cooling limit. Among the 6 normal modes" of the 27Alf/gBe+ionpair, two radial modes with large amplitude on 27Al+ , and "The two axial normal-mode frequencies are 5.80 MHz and 2.62 MHz, and the frequencies of the poorly damped radial modes are u, = 3.50 MHz and vy = 4.64 MHz. The other pair of radial modes, which is well damped, has frequencies of 14.06 MHz, and 13.02

MHz.

299 small amplitude on 9Be+ are damped only weakly by the laser-cooled 9Be+ ion. Prior to clock interrogation, we enhance this damping by twisting the ion-pair with a static electric field of about 300 V/m in order to begin each experiment a t the Doppler-cooling limit. We also apply 'Be+ -Dopplercooling light continuously during each 100-ms clock interrogation, keeping the four well-damped modes at the Doppler-cooling limit. However, the two poorly damped radial modes have damping rates of y M 60 s-l, and heating rates of (&) = 400 f 300 quanta/s, and (hY)= 100 f 100 quanta/s, leading to 6.6 5 and 1.5f1.5 excess motional quanta. The fractional clock shift per radial motional quantum is 8 x for both modes, and at the Doppler-cooling limit both modes would contain 3 f 1 quanta. Conservatively we estimate that the average total motional quantum number for the two radial modes during each clock interrogation is 14 f 10 quanta, leading to a second-order Doppler shift of (-1.1 f 0.8) x Other important shifts are the blackbody radiation shift,14 which is very small in 27Al+115and the quadratic Zeeman shift, which has been accurately calibrated by varying the magnetic field, and measuring the shift in the 27Al+ /l''Hg+ ratio, together with the linear Zeeman splitting v1 between the 7r-polarization-excited m = f % lines. The resulting shift is v2 = v; x 1.0479(7) x 10W8/Hz.

*

1 i 1 -; i 1

Table 1. 27Al+ 'So + 3Po and lg9Hg+ 2 S ; + 2 D s shifts in units of lo-'' 2 of fractional frequency. Evaluations of first-order Doppler shifts, background gas collisions, and the gravitational red shift are in progress. shift Micromotion Secular motion Blackbody rad. 313 nm Stark DC Quad. Zeeman AC Quad. Zeeman Electric quadrupole TOTAL

I I

Av~i -20

I

U A ~ AvHg

20

I

UHg

-4

4

-1203 0 0 -1210

10 10 16

4 :

-685 0 0 -735

0.5 0 0.5 22

I I

limitation static electric fields Doppler cooling DC polarizability polarizability, intensity calibration trap magnetic fields B-field orientation

The lg9Hg+ ion standard is based on the 2 S + + 2D+ electric-quadrupole tran~itionA . ~ 194 nm laser cools the ion to the Doppler-cooling limit via the allowed 2S4 + 2P; transition] and a fiber laser frequency-quadrupled to 282 nm excites the clock transition. The clock state of the "'Hg+ ion is measured directly via quantum jumps in the scattering fluorescence rate of the 194 nm laser.16 Systematic uncertainties in the lg9Hg+ standard are listed in Table 1.Here the largest uncertainty is a possible quadratic Zeeman

300 shift due to unbalanced rf-currents in the ion trap. The magnitude of this shift is conservatively estimated by assuming the worst case scenario; that all of the rf-currents flowing in the ring electrode (in the middle of which the ion is held) are fully unbalanced and travel in a half loop along one side of the ring only. The maximum rms-current (IRF < 1 mA) at the site of the trap is set by the trap capacitance (C < 70 ,)?!f the rf-frequency (12 MHz) and the maximum voltage applied (1100 V). This produces an rms-field of approximately 4 x lop7 T a t the center of the ring electrode for a ring diameter of 1 mm. A field of this magnitude causes a fractional frequency shift of the order 1 x The return path for the rf-current through the endcap electrodes ideally produces no net field at the center of the trap, but if they are unbalanced, then the field produced can only help compensate the field generated by the asymmetrical flow of rf-currents through the ring electrode. Otherwise, the rms-field produced at the site of the ion is less than 4 x T. The electric-quadrupole shift, which has previously limited the accuracy of the lg9Hg+standard is constrained below by averaging over three orthogonal magnetic field direction^.'^)^^ Second-order Doppler shifts are easier to control for lg9Hgf than for 27Al+ , because the heavy mercury ion moves less in response ambient electric fields than the lighter aluminum ion does. Near the Doppler-cooling limit, the total time-dilation shift due to secular motion is ( - 3 f 3 ) x 10-lS. Micromotion is carefully compensated," leading to a similar shift of (-4 4) x The quadratic Zeeman coefficient in lg9Hg+ was calibrated in an analogous way to that of the 27Alf standard. Here the shift is u:! = -u; x 1.23564(37) x 10-ll/Hz, where u1 is the linear Zeeman splitting between the 7r-polarization-excited m F = f 2 lines. Both atomic clocks were operated simultaneously, while the FLFCs recorded their frequency ratio every second. Figure l ( b ) shows Allan deviation (an estimate of the statistical measurement uncertainty vs. measurement duration) of a typical ratio measurement. For measurement durations r greater than 100 s, the uncertainty is given by 7 x 1 0 - 1 5 / f i . A deviation from this slope at long measurement times, which would indicate fluctuating systematic shifts, has not been observed. Figure l ( a ) shows the history of measurements of the u ~ ~ + / v Hratio. ~ + - While the systematic uncertainties in Table 1 are only valid for the last data point, the consistency of the earlier measurements provides confidence in the reproducibility of this result. Currently, both standards are being evaluated for first-order Doppler shifts, which would appear if the ion motion was correlated with the probe

*

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(b) Fig. 1. Fractional stability of the ratio vAl+/vHg+ (a). History of measurements of V AI+ lvHg+ O")1 Error bars are statistical, and the shaded area represents the statistical uncertainty of the weighted mean. These data points do not contain the complete set of systematic corrections listed in Table 1, and will be revised in the future.

beam. By probing alternately from opposite directions, such motion can be detected, and averaged away. Possible causes for such a shift would be stray charge buildup in the ion trap, which is correlated with the interrogation pulses, or correlated mechanical motion caused by shutters. We have not observed a direction-dependent shift for either standard. Besides their obvious application to frequency metrology and precision time-keeping, more accurate atomic clocks might also help answer a fundamental question in physics. Are the constants of nature really constant, or are they changing in time, or dependent on the gravitational potential in which they are measured? Frequency ratio measurements of dissimilar atomic clocks can help answer these questions. In particular, the ratio of atomic resonance frequencies depends on the fine-structure constant a.19'20 Repeated accurate measurements of the i^Ai+/l/Hg+ frequency ratio will provide constraints on present-day changes of the fine-structure constant.

Acknowledgements

This work is supported by ONR, DTO, and NIST. P.O.S. acknowledges support from the Alexander von Humboldt Foundation. This work is a contributions of NIST, and is not subject to U.S. copyright.

302 References 1. H. G. Dehmelt, IEEE Trans. Inst. Meas. 31,p. 83 (1982). 2. D. J. Wineland, W. M. Itano, J. C. Bergquist and R. G. Hulet, Phys. Rev. A 36,2220(Sep 1987). 3. W. H. Oskay, S. A. Diddams, E. A. Donley, T. M. Fortier, T. P. Heavner, L. Hollberg, W. M. Itano, S. R. Jefferts, M. J. Delaney, K. Kim, F. Levi, T. E. Parker and J. C. Bergquist, Phys. Rev. Lett. 97, p. 020801 (2006). 4. T. Rosenband, P. 0. Schmidt, D. B. Hume, W. M. Itano, T. M. Fortier, J. E. Stalnaker, K. Kim, S. A. Diddams, J. C. J. Koelemeij, J . C. Bergquist and D. J . Wineland, Phys. Rev. Lett. 98, p. 220801 (2007). 5. T. M. Fortier, A. Bartels and S. A. Diddams, Opt. Lett. 31,p. 1011 (2006). 6. W. C. Swann, J. J. McFerran, I. Coddington, N. R. Newbury, I. Hartl, M. E. Fermann, P. S. Westbrook, J. W. Nicholson, K. S. Feder, C. Langrock and M. M. Fejer, Opt. Lett. 31,3046 (2006). 7. P. 0. Schmidt, T. Rosenband, C. Langer, W. M. Itano, J. C. Bergquist and D. J . Wineland, Science 309,p. 749 (2005). 8. D. B. Hume, T. Rosenband and D. J. Wineland, Phys. Rev. Lett. 99, p. 120502 (2007). 9. J. E. Bernard, L. Marmet and A. A. Madej, Opt. Comm. 150,p. 170 (1998). 10. M. D. Barrett, B. DeMarco, T. Schaetz, V. Meyer, D. Leibfried, J. Britton, J. Chiaverini, W. M. Itano, B. JelenkoviC, J. D. Jost, C. Langer, T. Rosenband and D. J . Wineland, Phys. Rev. A 68,p. 042302 (2003). 11. D. J . Berkeland, J . D. Miller, J. C. Bergquist, W. M. Itano and D. J . Wineland, J. Appl. Phys. 83,p. 5025 (1998). 12. Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J . Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe and D. J. Wineland, Phys. Rev. A 61, p. 063418(May 2000). 13. L. Deslanriers, S. Olmschenk, D. Stick, W. K. Hensinger, J. Sterk and C. Monroe, Phys. Rev. Lett. 97,p. 103007 (2006). 14. W. M. Itano, L. L. Lewis and D. J . Wineland, Phys. Rev. A 25, p. 1233 (1982). 15. T. Rosenband, W. M. Itano, P. 0. Schmidt, D. B. Hume, J . C. J. Koelemeij, J. C. Bergquist and D. J. Wineland, Blackbody radiation shift of the 27alS Is0 - 3pO transition arXiv:physics/0611125. 16. J . C. Bergquist, R. G. Hulet, W. M. Itano and D. J . Wineland, Phys. Rev. Lett. 57,1699(0ct 1986). 17. W. M. Itano, J . Res. NIST 105,p. 829 (2000). 18. W. H. Oskay, W. M. Itano and J. C. Bergquist, Phys. Rev. Lett. 94,p. 163001 (2005). 19. J . K. Webb, V. V. Flambaum, C. W. Churchill, M. J. Drinkwater and J . D. Barrow, Phys. Rev. Lett. 82,884(Feb 1999). 20. S. G. Karshenboim, V. Flambaum and E. Peik, Atomic clocks and constraints on variations of fundamental constants arXiv:physics/0410074.

SR OPTICAL CLOCK WITH HIGH STABILITY AND ACCURACY* A. LUDLOW, S. BLATT, M. BOYD, G. CAMPBELL, S. FOREMAN, M. MARTIN, M. H. G. DE MIRANDA, T. ZELEVINSKY, AND J. YE JILA, National Institute of Standards and Technology and University of Colorado Boulder, Colorado 80309-0440, USA E-mail: [email protected]

T. M. FORTIER, .I. E. STALNAKER, S. A. DIDDAMS, C. W. OATES, z. w . BARBER, AND N. P O L I ~ National Institute of Standards and Technology 325 Broadway, Boulder, Colorado 80305, USA We report on our recent evaluations of stability and accuracy of the JILA Sr optical lattice clock. We discuss precision tools for the lattice clock, including a stabilized clock laser with sub-Hz linewidth, fs-comb based technology allowing accurate clock comparison in both the microwave and optical domains, and clock transfer over optical fiber in an urban environment. High resolution spectroscopy (Q > 2 x 1OI4) of lattice-confined, spin-polarized strontium atoms is used for both a high-performance optical clock and atomic structure measurement. Using a Ca optical standard for comparison, the overall systematic uncertainty of the Sr clock is reduced to < 2 x

1. Introduction Optical atomic clocks derive their advantages in stability and accuracy from the enhanced measurement precision arising from large line quality factors (Q) that now exceed lOI4 [ l , 21. The lattice clock technique [ l , 3-71 presents an intriguing situation in which one benefits from a large signal-to-noise ratio ( S N ) provided by a neutral atom ensemble, while enjoying many of the spectroscopic features of single trapped ions. The lattice potential provides Lamb-Dicke confinement where the resonance feature of interest is not influenced by atomic motion, enabling high precision, high accuracy measurements. The weakly allowed 'So - 3P0 transition in alkaline-earth(-like) atoms is particularly well suited for such a scheme due to the existence of a magic wavelength where the ac Stark shift from the trapping laser is identical for the two clock states, and the insensitivity of the states to lattice polarization. The accuracy of a lattice clock based on "Sr is expected to reach below while *This work is supported by NIST, ONR, and NSF. talso with Dipartimento di Fisica and LENS, Universita di Firenze, INFN-Sezione di Firenze, Via N. Carrara 1 1-50019 Sesto Fiorentino (FI), Italy

303

304 the stability could reach below at 1 s when probed by a sufficiently stable optical local oscillator. Such accuracy and precision goals will require focused efforts in both laser development and precision spectroscopy. In this paper we report the present status of our clock system, including local oscillator performance, clock comparison technology using fs-combs, high resolution spectroscopy, and the current level of accuracy and stability of the lattice clock.

2.

OPTICAL CLOCKWORK

2.1. Stable Optical Local Oscillator For the purposes of operating an optical clock with the best possible stability, it is necessary for the probe laser to have an exceedingly long phase-coherence time. Development of such an oscillator to probe the Sr clock transition at 698 nm has therefore been one of the central focuses of our work. The stabilization scheme for our laser is discussed in detail elsewhere [8]. Briefly, a gratingstabilized diode laser is locked to a high-finesse ultra-low-expansion (ULE) cavity mounted in a vertical orientation to reduce fluctuations of the cavity length due to vibrations [9]. The cavity is under vacuum and mounted on a compact, passive vibration-isolation table.

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Figure 1. Allan deviation (with a linear drift removed) for a heterodyne beat between two independent clock laser systems at 698 nm. The -1 x lo-'' stability at 0.1 - 1000 s is consistent with the estimated thermal noise limit. The inset shows the beat between the lasers is well below 1 Hz for an integration time of a few seconds.

The clock laser performance has been characterized in a number of ways, including direct comparison of two similar systems at 698 nm, comparison to highly stabilized lasers at other colors using the fs-comb, and by precision atomic spectroscopy. Direct comparison between the two 698 nm systems via heterodyne beat reveals narrow linewidths often below 300 mHz for integration times of a few seconds (see the inset of Fig. 1). The Allan deviation shown in

305 Fig. 1 reveals that the fractional frequency noise of the beat i s -1 x at 0.1 1000 s, consistent with the expected limit set by thermal-mechanical noise in the cavity mirrors and mirror coatings [lo].

2.2. Optical Frequency Comb Clockwork and Precision Fiber Transfer The femtosecond optical comb plays an essential role for clock development. It provides a coherent link between high accuracy clocks operating in either the optical or microwave domains. For absolute frequency measurements, a directoctave-spanning, self-referenced frequency comb similar to that reported in [ 111 is stabilized by the clock laser at 698 nm. The comb provides a measurement of the laser frequency relative to a microwave signal derived from a hydrogen maser calibrated to the NIST primary Cs fountain clock [12]. To enable the microwave comparison, a diode laser operating near 1320 nm i s amplitudemodulated and transmitted over a 3.5-km fiber optic link from NIST to JILA [13, 141. The instability of the frequency-counting signal i s -2.5 x for a 1s integration time. The periodic stretching and compressing of the fiber, associated with daily temperature cycles, has been found to cause frequency offsets as large as Stabilization of the fiber link is thus implemented for high accuracy clock comparisons [14].

-+ 1 Hz

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Averaging Time (s) Figure 2. (a) Optical beat between the fiber-transferred laser and the NIST comb at 1064 nm. The fs-comb in JILA is stabilized to the 698 nm clock laser, distributing the stability across the optical spectrum. A JILA 1064 nm Nd:YAG laser is stabilized to the comb and used as a transfer laser to NIST over a 3.5 km fiber which is optical-phase stabilized. At NIST, a second fs-comb stabilized to the 1123 nm Hg+ clock laser is used to compare the two stable oscillators. The beat linewidth between the NIST comb and the transfer laser is 1 Hz. (b) Instability of the transfer systems. Open diamonds are typical passive instabilities of the 7- and 32-km fiber links. Closed diamonds (circles) are for the 32-km (7-km) fiber transfer system. Black triangles represent in-loop performance. The solid line represents 1 radian of accumulated phase noise during the averaging time.

The comb and fiber link also support transmission of stable optical frequencies between JILA and NIST. First, the comb i s used to distribute the Sr clock stability across the optical spectrum, allowing use of a transfer laser at

306 1064 nm. We have examined the precision of the distribution across the comb spectrum by comparing our 698 nm clock laser with a similar independent system operating at 1064 nm [9]. The resulting beat between the stabilized comb and the 1064 nm laser has a linewidth below 0.5 Hz, verifying the ability of the comb to hlly preserve the optical phase coherence. For transfer of the clock signal, a 1064 nm laser is phase-locked to the comb and sent through the fiber to the NIST laboratory where it is measured relative to a number of high accuracy optical clocks [15, 16, 71 using a second fs-comb [17]. To eliminate noise induced by the transfer, the 1064 nm light is partially reflected back to our lab allowing interferometric stabilization of the link [ 181. Figure 2 (a) shows a beat between the transfer laser and a NIST fs-comb that is stabilized to a local oscillator used for the Hg+ clock [19]. The beat has a resolution-bandwidthlimited linewidth of 1 Hz, demonstrating that the phase coherence of the two clock lasers is not degraded by the two fs-combs and the fiber transfer process. In addition, Fig. 2(b) summarizes results of passive and phase-stabilized optical transfer for 7 km (the round trip length of our fiber link) and 32 km fiber lengths. The transfer noise of the stabilized fiber links is -1 x at 1 s [ 181.

3.

High resolution Spectroscopy

Lattice-confined fermionic 87Sratoms are used for an accurate atomic reference. 87 Sr atoms are laser cooled in two stages [20, 41, first via a broad (32 MHz) and then a narrow (7.4 kHz) cycling transition. A vertical one-dimensional optical lattice is overlapped with the atomic cloud as the atoms are cooled in the second stage magneto-optic trap. After the cooling cycle is finished, roughly lo4 atoms remain trapped in the lattice, at a temperature of -2 pK. 3.1. Spectroscopy in the Magic Wavelength Lattice The lattice potential is sufficiently deep such that the 'So - 3P0clock transition is probed in the Lamb-Dicke regime along the lattice axis where the spatial dimension of the confinement is small compared to the interrogation wavelength. The resulting absorption spectrum is free of Doppler or recoil effects, allowing high line Q's to be achieved. When the transition is strongly saturated, spectra such as that in Fig. 3 (a) are observed. Here the narrow transition is accompanied by motional sidebands. The sideband spectrum is useful for direct measurement of the thermal-mechanical properties of the trapped atoms, including temperatures and trap motional frequencies [4]. The narrow central feature serves as the atomic reference as it excludes any motional or Stark broadenings when the magic wavelength lattice is used. In the unsaturated case, this carrier transition has a width of -5 Hz, shown in Fig. 3(b), representing a line Q of loi4.

307 3.2. Nuclear Spin Effects The limit to the achievable line Q in Fig. 3(b) is mainly due to the magnetic sensitivity of the clock transition. The hyperfine state mixing, which leads to the -1 mHz linewidth of the otherwise forbidden clock transition, increases the magnetic moment of the 3P0 state, resulting in a small differential g-factor between 3P0and 'So. The resultant sensitivity of the clock transition to magnetic fields requires careful clock design to ensure frequency stability and accuracy. I

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To determine the magnitude of the differential g-factor, we have performed NMR-like measurements on various mF-sublevels of the clock transition in the presence of a bias magnetic field. Figure 4(a) shows the absorption spectrum in the presence of a bias field using linearly polarized light to drive ten possible Ktransitions (AmF= O). In Fig. 4(b), the probe light polarization is rotated by 90" from the quantization axis such that nine 0- (AmF= -1) and nine o+ (AmF= +1) transitions are observed simultaneously. While the n-transitions would be sufficient for measurement of the differential g-factor magnitude, the o+'transitions provide more information as the spectrum depends on both the magnitude and the sign of the effect (i.e. the state mixing increases the 3P0gfactor). Furthermore, calibration of the field magnitude is automatic if the ground state magnetic moment is known. Spectroscopy of the resolved sublevels gives us precise measurement of the differential g-factor, yielding 108.4(4) x mF HdG [21, 11. The resolved spectrum has also allowed experimental investigation of the tensor light shift sensitivity of the clock states,

308 demonstrating that they have a negligible impact on the current 1-D lattice clock P11. 3.3. Hz-Resolution Optical Spectroscopy With the degeneracy of the clock states lifted in the presence of a magnetic field, the resolved transitions are explored free-of-broadening from the statedependent

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Laser Detuning (Hz) Figure 4. Spectroscopy of resolved mF-sublevels in the presence of a magnetic bias field. (a) Ten x-transitions are observed when the probe is linearly polarized along the quantization axis set by a 0.5843 magnetic field. (b) When the probe polarization is orthogonal to the quantization axis, 18 o transitions are seen, under a bias field of 0.69 G. In both (a) and (b), peaks are labeled according to the ground state mF origin of the transition. In (b) the spectrum consists of two overlapped sets of nine evenly spaced peaks, the excitation polarization is labeled for clarity.

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Figure 5. High-resolution spectroscopy of the clock transition for mF = 512. (a) Ultraspectrum achieved with a 500-ms Rabi pulse. (b) Ramsey narrow (Q > 2 x spectrum for 25-ms Ramsey pulses separated by a 80-ms evolution time, resulting in a 10 Hz FWHM for the central fringe. The inset shows the spectrum for longer pulse and freeevolution times of 80 ms and 200 ms, respectively. In both (a) and the inset of (b), the spectral resolution is limited by laser frequency noise during acquisition times of -30 s.

Zeeman and Stark shifts, revealing the true spectroscopic limit of our system. An example spectrum is shown in Fig. 5(a), where the 3Po(mF= 5/2) population is measured as the laser is tuned across the resonance. The resulting resonance [I], near the -1.8 Hz Fourier limit set by the width is 2.1(2) Hz (Q > 2 x 500-ms interrogation time. Laser frequency noise prevents longer probe times as the beat between our two 698 nm lasers broadens to nearly 2 Hz when integrated over the relevant timescale for scanning the line (about 30 s). Ramsey spectroscopy can also be performed, as shown in Fig. 5(b). The Doppler-free absorption spectrum and long interaction time provided by the lattice confinement allows use of long, low-intensity pulses for the Ramsey sequence, resulting in simple spectra with a small number of fringes. 4.

Clock Performance

The high-resolution spectroscopy results establish confidence in the lattice clock technique as a tool for precision measurement. The high stability oscillator, large line Q, and relatively large S/N can result in clock stability at the 1 x 10.'' level at 1 s. Another important issue is the level of accuracy that can be achieved with the lattice system. 4.1. Accuracy Evaluation (2006): Degenerate Sublevels We first present an accuracy evaluation using degenerate sublevels without a bias magnetic field. To evaluate the accuracy of the "Sr lattice clock an interleaved scheme is used where the parameters of interest are quickly varied

31 0 during a scan of the transition, allowing extraction of the shift coefficient using the precision of the laser cavity as a reference. In this way, we explore a variety of systematic effects, with dominant contributions to the uncertainty budget being the lattice shift, density shift, and Zeeman shift. Operating at a lattice wavelength of 813.4280(5) nm, we find that the ac Stark shift for the typical For the typical density of -5 x 10" ~ m -the ~, trap depth is -2.5(6.0) x collision shift is constrained to be below 3.3 x Each of these effects is consistent with zero and limited by the statistical uncertainty associated with hundreds of measurements. The Zeeman sensitivity is found to be non-zero but < 5.3 x when the residual magnetic field is controlled to 5 mG. Due to the differential g-factor, we zero and monitor the field to this level using the transition linewidth. The total uncertainty for lattice spectroscopy under these operating conditions is 9 x [5]. As the spectroscopy systematic uncertainties are evaluated, an absolute frequency measurement is performed using the fiber link and a Cs-calibrated Hmaser. After a 24 hour run, the frequency of the 87Sr 'S0-3Po transition is determined to be 429,228,004,229,874.0 + 1.1 Hz. The total uncertainty of 2.5 x is dominated by the statistical uncertainty in the frequency comparison (1.4 x and the maser calibration (1.7 x lo-") [5]. Figure 6 summarizes the recent absolute frequency measurements made by the groups in JILA [4, 51, Paris [6, 221, and Tokyo [23]. The excellent agreement between independent groups speaks strongly for the 87Srlattice clock as a frequency standard. 8C n

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311

4.2. Accuracy Evaluation: Optical Clock Comparison with SpinPolarized Samples Under its current operating parameters, the Sr lattice clock has a potential stability limit from quantum projection noise at 7 x 10"16 at 1 s. In practice, the stability is estimated to be 2 x 10"'5 at 1 s, limited by the Dick effect of the optical local oscillator [24]. Such measurement stability cannot be fully exploited by comparison to current state-of-the-art microwave standards. Rather, accuracy evaluation of optical atomic clocks greatly benefits from direct comparison against other optical clocks with similar stability. Using the optical carrier transfer system described earlier, we now evaluate systematic uncertainties of the JILA Sr lattice clock by measurement against the cold Ca optical clock at NIST [15]. Figure 7 shows the measurement potential of a clock comparison between the 87Sr lattice clock and the Ca clock. The stability at 1 s is ~ 3 x 10~15 and a precision of 3 x 10"16 is achieved in only 200 s of averaging. This frequency discrimination at relatively short averaging times facilitates high precision measurement of systematic shifts of the Sr clock. To reduce measurement sensitivity to long time-scale (> 1000s) frequency drifts of the Ca standard, we measure frequency differences between sets of 100-s data under various Sr parameters to be evaluated. By collecting many sets of these data, we characterize systematic shifts below the 1 x 10"16 level. Srvs.NlSTMaser| Srvs.MSTCa I

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Figure 7. Microwave and optical clock comparisons using the fiber link discussed in the text. The Allan deviation for the Sr - Maser comparison is shown as squares, yielding a stability of-2.5 x 10'13 T '"2. The advantage of an optical clock is apparent from the Sr Ca comparison (circles), which typically yields stability of 3 x 10'15 at 1 s.

31 2

Figure 8. Spectroscopy of the x transitions under a bias magnetic field. Blue (dark gray) is with atomic polarization to mF = + 912, and red (light gray) is without. The polarization is achieved by optical pumping using the 'SOF=9/2 - 3PIF=7/2 transition. To reduce clock sensitivity to some key experimental parameters, we spinpolarize the Sr sample to the stretched nuclear spin states, mF = & 9/2, and then perform spectroscopy with a small bias magnetic field to lift spin state degeneracy (see Fig. 8). By stabilizing the clock laser to the average of the n transitions from both spin states, we remove shifts originating from the lStorder Zeeman effect and the vector light shift due to the lattice confinement. The clock remains sensitive to 2"d order Zeeman shifts. As we vary the bias magnetic field and measure the clock frequency versus Ca, we confirm that the 1'' order Zeeman effect is eliminated, resulting in a clock shift of 1.6(2.3) x 1017 G-2, . Furthermore, we measure the 2"d order Zeeman shift at 5.8(0.8) x resulting in a 2"d order frequency shift of 2.3(0.2) x In addition to Zeeman shifts, we characterize systematic shifts resulting from: AC Stark effect from the lattice and probe lasers, atom servo error, blackbody AC Stark effect, atomic collisions, and higher order motional effects. A preliminary summary of these effects is given in Table 1. One of the largest observed shifts originates from density-dependent Sr-Sr interactions. Measurement of the clock shifts indicate that these interactions depend on both the nuclear spin state as well as the electronic excitation. We are currently investigating these effects in more detail and it is anticipated that a zero-densityshift condition will be found and implemented for the clock operation. The stated density shift and uncertainty in Table 1 are given for our current operating conditions. The total systematic shift uncertainty is < 2 x indicating the strong performance of the Sr lattice clock as a potential primary frequency standard. We expect this uncertainty to continue to decrease.

313

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Table 1. Systematic frequency corrections for the Sr lattice clock, and their respective uncertainties. All quantities are expressed in fractional frequency of Conclusion We have presented our recent results for an optical atomic clock based on strontium atoms confined in an optical lattice. The local oscillator is now performing at the 1 x lo-’’ level at 1 - 1000 s, and the fiberkomb transfer system has been shown to support precision frequency transfer at the level of 1 x at 1 s. The lattice clock technique, combined with the high precision laser, has yielded the highest line Q (> 2 x achieved in coherent spectroscopy, with a good S i N for further enhanced stability. Nuclear-spin effects in the lattice clock have been explored, including a measurement of the differential g-factor of the clock transition. An accuracy evaluation is performed, reducing the lattice clock systematics below lo-’’, and an absolute frequency measurement with a 1 Hz uncertainty is made, which is in excellent agreement with that of other groups. Comparison of the Sr clock with the NIST at 1 s. Using the high Ca clock revealed the clock instability is < 3 x precision optical comparison and spin-polarized samples, the lattice clock systematic uncertainty has been reduced to 2 x and it will continue to decrease in the near future. Acknowledgments We gratefully acknowledge contributions from J. Bergquist, T. Parker, S. Jefferts, and T. Heavner for help with the fiber-link-based laser comparison and the absolute frequency measurements made via the Sr - Maser comparison. References 1. M. M. Boyd et al., “Optical atomic coherence at the 1-second time scale,”

Science, vol. 314, pp. 1430-1433,2006.

314

2. R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Sub-dekahertz ultraviolet spectroscopy of 199Hg+,” Phys. Rev. Lett., vol. 85, p. 2462,2000. 3. M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature, vol. 435, pp. 321-324,2005. 4. A. D. Ludlow et al., “Systematic study of the 87Sr clock transition in an optical lattice,” Phys. Rev. Lett., vol. 96, p. 033003, 2006. 5. M. M. Boyd et al., “87Sr lattice clock with inaccuracy below 10-15,” Phys. Rev. Lett., vol. 98, p. 083002,2007. 6. R. Le Targat et al., “Accurate optical lattice clock with 87Sr atoms,” Phys. Rev. Lett., vol. 97, p. 130801, 2006. 7. Z. W. Barber, C. W. Hoyt, C. W. Oates, L. Hollberg, A. V. Taichenachev, and V. I. Yudin, “Direct excitation of the forbidden clock transition in neutral 174Yb atoms confined to an optical lattice,” Phys. Rev. Lett., vol. 96, p. 083002,2006. 8. A. D. Ludlow et al., “Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1x10-15,” Opt. Lett., vol. 32, pp. 641-643, 2007. 9. M. Notcutt, L.-S. Ma, J. Ye, and J. L. Hall, “Simple and compact 1-Hz laser system via an improved mounting configuration of a reference cavity,” Opt. Lett., vol. 30, pp. 1815-1817, 2005. 10. K. Numata, A. Kemery, and J. Camp, “Thermal-noise limit in the frequency stabilization of lasers with rigid cavities,” Phys. Rev. Lett., vol. 93, p. 250602,2004. 11. T. M. Fortier, D. J. Jones, and S. T. Cundiff, “Phase stabilization of an octave-spanning Tixapphire laser,” Opt. Lett., vol. 28, pp. 2198-2200, 2003. 12. T. P. Heavner, S. R. Jefferts, E. A. Donley, J. H. Shirley, and T. E. Parker, “NIST-F 1: recent improvements and accuracy evaluations,” Metrologia, vol. 42, p. 41 1,2005. 13. J. Ye et al., “Delivery of high-stability optical and microwave frequency standards over an optical fiber network,” JOSA B, vol. 20, pp. 1459-1467, 2003. 14. S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum., vol. 78, p. 021 101,2007. 15. C. W. Oates, E. A. Curtis, and L. Hollberg, “Improved short-term stability of optical frequency standards: approaching 1 Hz in 1 s with the Ca standard at 657 nm,” Opt. Lett., vol. 25, pp. 1603-1605,2000. 16. W. H. Oskay et al., “Single atom optical clock with high accuracy,” Phys. Rev. Lett., vol. 97, p. 020801,2006. 17. T. M. Fortier, A. Bartels, and S. A. Diddams, “Octave-spanning Ti:Sapphire laser with a repetition rate > 1 GHz for optical frequency measurements and comparisons,” Opt. Lett., vol. 31, pp. 1011-1013,2006.

31 5 18. S. M. Foreman et al., “Coherent optical phase transfer over a 32-km fiber with long-term instability
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AUTHOR INDEX A Abel, M.J. 167 Afanasiev, A.E. 195 Akulshin, A. 228 Alberti,A. 89 Alt, W. 175 Anderson, R. 228 Andrist, M. 153 Apel, P. Yu. 195 Aspect, A. 11,243

Campbell, G. 303 Cancio, P. 75 Challis, K.J. 138 Chang,H. 243 Cherkun, A.P. 195 Chwalla, M. 53 Clement, D. 11 Cornell, E.A. 3 D Darquie, B. 259, 271 De Angelis, M. 89 De Miranda, M.H.G. 303 de Natale, P. 75 Deleglise, S. 63 Diddams, S.A. 297 Dingjan, J. 259 Dutier, G. 271

B Bailly, D. 103 Ballagh, R.J. 138 Balykin, V.I. 195 Bartalini, S. 75 Bergquist, J.C. 297 Bernet, S. 187 Bernu, J. 63 Bertoldi, A. 89 Bethlem, H.L. 103 Beugnon, J. 259 Blatt, R. 53,207 Blatt, S. 303 Bloch, I. 23 Boiron, D. 243 Borri,S. 75 Bouyer,P. 11 Boyd,M. 303 Browaeys, A. 259 Brune,M. 63 Brusch,A. 297 Buning, R. 103 C Cacciapuoti, L.

F Fernholz, T. 113 Ferrari, G. 89 Foreman, S. 303 Fortier, T.M. 297 Fortson, E.N. 39

G Gaebler, J.P. 127 Gaetan, A. 259 Gagliardi, G. 75 Galli, I. 75 Gardiner, C.W. 138 Ghanbari, S. 228 Giorgini, A. 89 Giusfiedi, G. 75 Gleyzes, S. 63

89 31 7

31 8

Goldwin, J. Grangier, P. Guerlin, C.

271 259 63

H Hall, B.V. 228 Hannaford, P. 228 Hannemann, S. 103 Haroche, S. 63 Heinrich, C. 187 Hilliard, A.J. 113 Hinds, E.A. 27 1 Hofer, A. 187 Hogan, S.D. 153 Hogervorst, W. 243 Hume, D.B. 297 I Itano, W.M. 297 Ivanov,V. 89

J Jeltes, T. 243 Jensen, K. 113 Jin, D.S. 127 Jones, M.P.A. 259 Jullien, A. 167

Leone, S.R. 167 Letokhov, V.S. 195 Lorini, L. 297 Luc-Koenig, E. 2 19 Ludlow, A. 303 Lugan, P. 11 M Maddoloni, P. 75 Malara, P. 75 Marion, H. 259 Martin,M. 303 Masnou-Seeuws, F. 219 Mazzotti, D. 75 McLean, R. 228 McNamara, J.M. 243 Meier, B.H. 153 Meier, U. 153 Melentiev, P.N. 195 Merkt, F. 153 Meschede, D. 175 Messin, G. 259 Miroshnychenko, Y. 259 Monz,T. 53 Moszynski, R. 2 19 Muller, J.H. 113 Murphy, M.T. 103

K Kaper, L. 103 Kim,K. 53 Kimble, H.J. 285 Kjaergaard, N. 113 Klimov, V.V. 195 Koch, C.P. 219 Kosloff, R. 219 Krachmalnicoff, V. 243 Krauter, H. 113 Kuhr, S. 63

N Nagel, P.M. 167 Neergaard-Nielsen, J.S. Neumark, D.M. 167 Newbury, N.R. 297 Nielsen, B.M. 113

L Lamporesi, G. 3, 89 Lance, A.M. 259

P Perrin,A. Pfeifer, T.

0 Oblak, D. 113 Olausson, C. 113 Oskay, W.H. 297

243 167

113

319

Poli,N. 89 Polzik, E.S. 113 Prevedelli, M. 89

R Raimond, J.-M. 63 Rauschenbeutel, A. 175 Riebe,M. 53 Ritsch-Marte, M. 187 Roos,C.F. 53 Rosenband, T. 297 Rudnev, S.N. 195 S Salumbides, E.J. 103 Sanchez-Palencia, L. 11 Sayrin, C. 63 Schellekens, M. 243 Schindler, K. 53 Schmidt, P.O. 297 Schmutz, H. 153 Schweikhard, V. 3 Sherson, J.F. 113 Sidorov, A. 228 Singh, M. 228 Skuratov, V.A. 195 Sorrentino, F. 89 Sortais, Y.R.P. 259 Sprecher, D. 153 Stalnaker, J.E. 297 Stewart, J.T. 127 Swann, W.C. 297

T Tino, G.M. 89 Trupke,M. 271 Tuchendler, C. 259 Tung, S. 3 Tychkov, A.S. 243

U Ubachs, W.

103

V Van Leeuwen, K.A.H. Vanhaecke, N. 153 Vassen, W. 243 Vervloet, M. 103 Vliegen, E. 153 Volk, M. 228 W Westbrook, C.I. 243 Whitlock, S. 228 Windpassinger, P. 113 Wineland, D.J. 297

Y Ye, J.

303

”

L

Zelevinsky, T.

303

243