Aduances in
ATOMIC, MOLECULAR, AND OPTICAL. PHYSICS VOLUME 36
Editors BENJAMIN BEDERSON New York University New York, New York HERBERT WALTHER Max-Plank-Institutf i r Quantenoptik Garching bei Miinchen Germany
Editorial Board P. R. BERMAN Uniuersity of Michigan Ann Arbor, Michigan M. GAVRILA F. 0. M. Institute uoor Atoom-en Molecuul’sica Amsterdam, The Netherlands M. INOKUTI Argonne National Laboratory Argonne, Illinois
W. D. PHILIPS National Institute for Standards and Technology Gaithersburg, Maryland
Founding Editor SIRDAVIDBATES
ADVANCES IN
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by
Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEWYORK,NEWYORK
Herbert Walther UNIVERSITY OF MUNICH AND MAX-PLANK INSTITUTFUR QUANTENOPTIK MUNICH, GERMANY
Volume 36
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This book is printed on acid-free paper. @ Copyright 0 1996 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.
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4
3 2 1
Contents vi i
CONTRIBUTORS
Complete Experiments in Electron-Atom Collisions
Nils Anderson and Klaus Bartschat I. I1 . 111. IV .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impact Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 8 24 80 83
Stimulated Rayleigh Resonances and Recoil-Induced Effects J.-Y. Courtois and G. Glynberg I. I1 . 111. IV . V.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stimulated Rayleigh Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recoil-Induced Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Recoil-Induced Effects in Atomic and Molecular Physics . . . . . . . . . . Conclusion References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
........................................
88 91 109 121 136 137
Precision Laser Spectroscopy Using Acousto-Optic Modulators W A . van Wijngaarden I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Optical Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Spectroscopy Using Frequency-Modulated Lasers . . . . . . . . . . . . . . . . . . IV . Hyperfine Structure and Isotope Shifts . . . . . . . . . . . . . . . . . . . . . . . . V . Starkshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V
141 142 148 152 166 179 180
vi
Contents
Highly Parallel Computational Techniques for Electron-Molecule Collisions Carl Winstead and Vincent McKoy
....................................... .......................................... 111. Computational Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Illustrative Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction
I1. Theory
V. Conclusion References
........................................ ........................................
183 186 191 209 217 218
Quantum Field Theory of Atoms and Photons Maciej Lewenstein and Li You I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I11. Hamiltonian of QFTAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Properties of BEC in Trapped Alkali Systems . . . . . . . . . . . . . . . . . . . . V . DiagnosticsofBEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Quantum Dynamics of Condensation . . . . . . . . . . . . . . . . . . . . . . . . . VII . Theory of Bosers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII . Nonlinear Atom Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
222 224 230 234 239 253 267 274 275 275
......................................... ............................
281 287
SUBJECX INDEX
CONTENTS OF VOLUMES IN THIS SERIAL
Contributors Numbers in parentheses indicate the pages o n which the authors’ contributions begin.
NILS ANDERSON (11, Niels Bohr Institute, Orsted Laboratory, Copenhagen, Denmark KLAUS BARTSCHAT (11, Department of Physics, Drake University, Des Moines, Iowa 50311
(871, Institut d’Optique Thkorique el AppliquCe, Orsay, J.-Y. COURTOIS Cedex, France G. GRYNBERG (871, DCpartement de Physique de I’Ecole Normale Supkrieure, Laboratoire Kastler-Brossel, Paris, Cedex 05, France MACIEJ LEWENSTEIN (221), Commissariat B 1’Energie Atomique, DSM/DRECAM/SPAM, Centre d’Etudes de Scalay, Gif-sur-Yvette, France VINCENT McKov (1831, A. A. Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91 125 WILLIAM ARIEVAN WIJNGAARDEN (140, Physics Department, York University, Toronto, Ontario, Canada CARLWINSTEAD(1831, A. A. Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 91125 LI You (2211, Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 20138
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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 36
COMPLETE EXPERIMENTS IN ELE CTRON-A TOM COLLISIONS NILS ANDERSEN Niels Bohr Institute 0rsted Laboratory Copenhagen, Denmark
KLAUS BARTSCHAT Department of Physics Drake Unicrersify Des Moines, Iowa
1. Introduction
. ..... ... .... . . . . . . . . .. . . ...... . .. . . . . . .
A.History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Scattering Amplitudes . . . .. . . . . ........ 11. Elastic Scattering. . . . . ... . . . . . . . ... .. A. Light Targets without Spin: He 1’s . . . . . . . . . . . . . . . . . . B. Light Targets with Spin: Na 3 2 S . . . . ....... . .... .. . C. Heavy Targets without Spin: Hg 6’S,. .. . .. . . ..... . . D. Heavy Targets with Spin: Cs 6 2 S , / 2 . ...... . ...... 111. Impact Excitation. . . . . . . . . . . . ..... ..... .. A. Light Targets without Spin in the Initial State: He 1’s + 2’P, 33P . B. Light Targets with Spin: Na 32S + 3 2 P . . . . . . . . . ........ C. Heavy Targets without Spin in the Initial State: Hg 6’S,, + 6’P, . . . . D. Heavy Targets with Spin: Cs 6*S,/, + 62P,/2,,,2 . . . . . . . . . .... E. Higher Angular Momenta: Excitation of He 1’s + 3’ D . . . .... . . IV. Conclusions . . . ... . . . . . . . . .. . . ... .... Acknowledgments . .. . . . . . . . . . ..... . . .. References . . . . . . . . . . . . . . . . .. . .. . . . .... . .
. . .... . . . . . . .. . . . ... . . . . . . . . . . . . . . . ... . .. ..... . . . .. . . .. . . .... . ... . . .... ...... . ...... . ... . . ... . .. .... . . . . .. . .. ...... ....... . ... .... . .. . . . .... .. .
1 1 4 8 8 11
15 20 24 28 34 48 70 71 80 83 83
I. Introduction A. HISTORY
“The most important experimental technique in quantum physics is the scattering experiment.” This statement is the opening sentence of a well-known textbook (Taylor, 1987) and highlights the central role that scattering experiments continued to play in the development of modern physics. A schematic diagram of a generic scattering experiment is shown 1
Copyright 0 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003836-6
2
N. Andersen and K. Bartschat
/
9 0 A
’0
B
FIG. 1 . Schematic diagram of a collision process.
in Fig. 1. The particles A and B are allowed to interact. When emerging from the interaction region, the collision partners generally have changed directions and may be in quantum states different from the initial ones. The aim of theoretical quantum physics is to model as accurately as possible the development of the system in the interaction region, for confrontation of the predictions with actual observables. The experimentalist aims at precisely defining the incident channel, as well as characterizing the collision products as closely as possible. For many important processes in nature, typical observables are averages over key parameters, such as incident directions, scattering angle, velocity, temperature, and so forth. However, the ultimate goal is to establish uniquely the relationship between the ket vectors and which determine the initial and final states of the system. A complete description in the quantum mechanical sense is succinctly expressed in terms of a corresponding set of complex-valued scattering amplitudes. What sets atomic physics apart in this context is that it may perhaps be the area of collision physics that today can present the largest systematic collection of fundamental processes for which this ideal has been achieved. This statement should not hide the fact that, in the words of Sir Harrie Massey, “The techniques required are very elaborate. Indeed, it is probably true that experiments in this field are among the most complicated in atomic physics today. They are very important for deepening our understanding of atomic collisions and it is essential that their complexity should not be allowed to obscure their importance” (Massey, 1983). In a series of influential papers, Bederson (1969a, b) coined the term perfkt scattering experiment to stimulate the development of experiments, or rather combinations of experimental approaches, that would result in a complete
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
3
determination of the quantum mechanical scattering amplitudes. His starting point was a systematic analysis of elastic and inelastic electron-atom collision experiments, using alkali atoms as an example and spin polarization as a key variable. At the time of formulation in 1969, many of these experiments could only be imagined as future goals, since few of the necessary key technologies had been developed to assist experimentalists and theoreticians in their common aim. In the intermediate period, this situation has dramatically improved, partly due to impressive advances in experimental and theoretical methods, but undoubtedly also due to the early, explicit formulation of a quest for a superior, systematic approach. Perfect scattering experiments have now been performed for several fundamental processes in electron-atom and atom-atom collision physics. At first sight, it may perhaps appear paradoxical that determination of scattering amplitudes, often considered somewhat “esoteric” (Callaway and McDowell, 1983) and abstract parameters mapping an event in Hilbert space, may provide us with a more concrete visualization of the collision dynamics. However, as we shall see, this is because the scattering amplitudes or density matrix parameters are ultimately related to directly observable properties, such as a change of spin direction of the scattered electron and, for impact excitation, characteristic properties of the photon radiation pattern that is emitted when the excited state decays. Thus, systematic exploitation of all available information enables a deeper insight into the collision dynamics than determination of a differential cross section-the “classical” observable in a scattering experiment-alone provides. An early example is the discussion by Kohmoto and Fano (1981) on the relationship of attractive and repulsive forces to the sign of the angular momentum transferred to the excited atomic electron in an electron-atom collision event (see Fig. 2). This chapter addresses the advances up to the present in complete electron-atom collision experiments. The aim is to present a series of key examples for fundamental scattering processes, together with the experimental techniques that have been used. The purpose is not a full presentation of all processes studied, nor of all data that have been accumulated; rather, it is to select examples of the most recent theoretical and experimental results that will enable the reader to assess the present level of achievement. We hope that the power of this approach will become evident along the way, in the sense that it provides an efficient framework for a systematic and complete test of the current theoretical understanding. In addition, it may produce specific recipes for ways to select experimental geometries that most efficiently test theoretical predictions, and it may reveal connections between apparently unrelated observables from
4
N. Andersen and K Bartschat
FIG.2. Propensity rule for positive orientation due to attractive forces between projectile and target in a collision process (from Kohmoto and Fano, 1981).
often very different and highly sophisticated experiments, thus providing valuable consistency checks. The presentation is structured in the following way. To begin with, a general analysis of scattering amplitude properties concludes in a recipe for determination of the number of independent parameters necessary to define a complete experiment for a given process. We then proceed to analyze in a systematic way a string of specific cases of elastic and inelastic collisions, with gradually increasing levels of sophistication. Finally, we comment on directions in which future studies could fruitfully be pursued.
B.
SCATTERING AMPLITUDES
As we have pointed the determination of important to derive number depends on
out, a “perfect” or “complete” experiment requires all independent scattering amplitudes. It is, therefore, the number of such independent amplitudes. This the assumptions about the dynamics of the collision
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
5
process, i.e., the individual projectile-target interactions that are taken into account. Throughout this chapter, we will assume that 1. All interactions conserve the total parity of the projectile-target system. 2. All interactions are invariant against time reversal. 3. Atomic hyperfine structure does not affect the collision process.
These assumptions are very well fulfilled for most atomic collisions and certainly for all cases discussed in this chapter. It is, however, important to realize that the resulting simplifications in the theory are not fufjiffed exact& in nature. Wi$ these assumptions in mind, transitions from an initial state IJ,M,,k l m , ) to a final state IJ,M,,z,m,) are described by scattering amplitudes f ( ~ , mM~p, , ; 6 )
=
( ~ , ~ , ; z IA ~ m~ , ~ , ; ~ , m , >(1)
where 9 is the transition operator. Furthermore, J, ( J , ) is the total electronic angular momentum in the initial (final) st_ate ,Of the target and M I ( M , ) its corresponding z component, whereas k, (kf) is the initial (final) momentum of the electron and m, (m,) its spin component. The For simplicity, we will scattering angle 8 is the angle between k, and if. often omit it in the notation, together with an overall normalization factor that is needed in explicit numerical calculations. In practical applications, one must define the scattering amplitudes with respect to a quantization axis for the angular momentum components. A standard choice for numerical calculations is the so-called “collision system,” where the incident beam axis is the quantization (2‘) axis and the y‘ axis is perpendicular to the scattering plane. On the other hand, the algebra often becomes simpler and many observables can be interpreted more easily in the “natural system” where the quantization axis z” coincides with the normal vector to the scattering plane and the X’ axis is defined by the incident beam direction. If not stated otherwise, we will use the natural coordinate frame throughout this chapter. The transformation of the scattering amplitudes from one system to another can be achieved in a straightforward way by transforming the initial and final states through standard rotation matrices and using the fact that the action of the Yoperator must be independent of the particular coordinate system. For details, we refer to Appendix A of Andersen et al. (1996). An important point for the discussion of scattering amplitudes is the fact that the abovementioned assumptions about the symmetry properties
6
N. Andersen and K Bartschat
of the projectile-target interaction lead to conservation laws through the Y operator. These, in turn, will cause interdependences between various scattering amplitudes or simply require some amplitudes to vanish. Consider, for instance, the conservation of the total parity. For our case of interest, electron-atom scattering in a plane, the process must be invariant against reflection in this plane. This reflection operation can be constructed as the parity operation, followed by a 180” rotation around the normal axis of the scattering plane. As shown in Appendix A of Andersen et al. (1996), the result for amplitudes in the natural coordinate frame is
where n, and n, are the parities of the initial and final states, respectively. Equation (2) shows that many amplitudes vanish in the natural frame (namely those where the exponent is an even or odd integer, depending on the product of the parities). This fact is one of the many advantages that can be used when formulating the general theory in this frame. Numerical calculations, on the other hand, are simpler in the collision frame, where one finds phase relationships ( f) between amplitudes with a given set of magnetic quantum numbers and those where the signs of all quantum numbers are reversed. We now discuss the number of independent scattering amplitudes. There are 4(2J, 1)(2Jf 1) possible combinations of magnetic quantum numbers. Due to parity conservation of all interactions determining the outcome of the collision process, the number of independent scattering amplitudes is cut in half, giving a total of 2(2J, + 1)(2Jf + 1) complex amplitudes for each transition between fine structure levels. Subtracting a common arbitrary phase, the total number of independent real parameters is thus 4(2J, + 1)(2Jf + 1) - 1. These are usually parameterized as one absolute differential cross section and 4(2JL+ 1)(2J1 + 1) - 2 dimensionless numbers, namely 2(2J, + 1X2Jf 1) - 1 relative magnitudes and 2(2J, + 1)(2Jf + 1) - 1 relative phases. Without going into details, we note that time reversal invariance of the projectile-target interaction leads, in general, to relationships between scattering amplitudes for inelastic excitation and “superelastic” deexcitation. In the special case of elastic scattering, such relationships may further reduce the number of independent amplitudes. An important example is elastic electron scattering from cesium atoms which will be discussed in Section 1I.D. In addition to the fundamental assumptions of parity conservation and time reversal invariance, it is sometimes also assumed that the total spin S
+
+
+
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
7
and the total orbital angular momentum L of the combined target + projectile system are conserved during the collision. This nonrelativistic approximation is generally good for electron collisions with light targets, such as helium or sodium. It is also the basis for the “fine-structure effect” (Hanne, 1983) in which observables for various fine structure transitions are related by algebraic factors. In the nonrelativistic approximation, the scattering amplitudes depend in a purely algebraic way on the spin quantum numbers, and transitions between fine structure levels are described by standard recoupling techniques. Specifically, the scattering amplitudes (1) can be expressed as
f ( M p p Mimi; 0 )
where S,, S,, M,. , MF,, L , , L , , M L , and M L , are the spins as well as the orbital angular momenta and the ’ corresponding z components of the initial and final target states, and
are nonrelativistic scattering amplitudes that describe transitions between orbital angular momentum states. Equation ( 3 ) expresses the conservation of the total spin S and its component
M,
=
M,,
through the Clebsch-Gordan Together with
f’( M L , , ML,; 0 )
=
+ m l = MLs,+ m,
(5)
coefficients ( j l ,m i ; j z , m 2 I j 3 , m3).
n,n,( - l)’‘,-M’~cfS(
MI>,,M [ , , ;0 )
(2a)
from parity conservation, this approximation reduces the number of independent scattering amplitudes for any transition between members of two
N. Andersen and K Bartschat
8
fine structure multiplets to NsN‘, where Ns is the number of total spin channels and NL
(2L; + 1)(2L/ =
+ 1) + ( - l ) L ’ + L ’ n ; n f 2
(6)
As in the more general case discussed previously, time reversal invariance may further reduce the number of independent amplitudes. This formalism yields, for example, the four independent amplitudes f : f! f; 1, and f l for the transitions 32S,/2 + 32P,/2,3/2in electron impact excitation of sodium, where the superscripts “ t ” and “s” refer to the “triplet” ( S = 1) and “singlet” (S = 0) total spin channels and the subscript ML, = i-1 to the orbital magnetic quantum number of the excited P state. Both ML,(= 0) and the scattering angle have been omitted in the notation. This compares with 24 (!) independent amplitudes for the transitions 62S1/2 + 62P1/2,3 / 2 in electron scattering from cesium atoms, namely 8 for 62S1/2-+ 6’PIl2 and 16 for 62S1,2 -+ 62P3/2.
11. Elastic Scattering We will start with complete experiments in elastic electron-atom scattering, going from the simplest case of a light target without spin and orbital angular momentum (such as helium) to more complicated situations such as light targets with spin (sodium), heavy targets without spin (mercury), and finally, heavy targets with spin (cesium). Except for the cesium target, these are characteristic examples of cases where complete experiments have, indeed, been performed or where only a small amount of information, such as the sign of a phase angle, is still missing. We will only consider initial atomic S states. While experiments for elastic electron scattering from targets with nonzero orbital angular momentum have been performed, these can only be described as the very first steps toward the complete determination of all independent scattering amplitudes. A. LIGHTTARGETS WITHOUT SPIN: He
1’s
From the point of view of a complete experiment, this is the simplest possible case in elastic electron-atom scattering. In the nonrelativistic approximation, there is only one independent scattering amplitude, namely f S = ’ / ’ ( M L ,= 0, M L , = 0; 0 ) = f that needs to be determined for each scattering angle. Figure 3 shows this amplitude as an arrow in the complex
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
9
f
FIG. 3. The scattering amplitude f for elastic e-He scattering.
plane, but given the arbitrary phase, the determination of the magnitude of this amplitude does, indeed, correspond to a complete experiment. This magnitude can be determined via the absolute differential cross section
a=lfl
2
(7)
it.,
f=G for each scattering angle 8. Figure 4 shows the apparatus of Brunger et al. (1991) that has been used for absolute differential cross section measurements for electron-helium scattering. It is a crossed electron-atom beam apparatus where the energy spread of the electron beam is reduced by a monochromator and the scattering angle can be varied by a rotatable electron spectrometer. For details of the experimental procedure we refer to the original publication, but it is important to point out that a major difficulty in this kind of experiment arises from the need of an absolute value of the differential cross section. As pointed out in Section I.B, one such value is required in all complete experiments-if additional parameters are needed, these can be determined on a relative scale. Figure 5 shows experimental data of Brunger et al. (1992) for elastic e-He scattering at energies of 1.5, 10.0, and 20.0 eV. Note how the differential cross section becomes peaked in the forward direction with increasing energy, a typical feature of such collisions. The data are compared with a recent “convergent close-coupling’’ (CCC) calculation of Fursa and Bray (1995). The agreement between experiment and theory is excellent.
10
N. Andersen and K Bartschat
,- Atomic
Electron ODtics
in
"
zoom
YONOCHROMATOR
1s-a
1
Target Region
CEM Detect
5
0
cm
FIG. 4. Schematic diagram of an apparatus for measurements of a differential cross section (from Brunger et al., 1991).
In summary, the complete experiment, in its simplest form, has been achieved for this case. This statement also applies to elastic scattering from other closed-shell targets, such as Be, Ne, or Ar, as long as the electron spin does not enter into the discussion. The situation will change when relativistic effects, such as the spin-orbit interaction between the projectile electron and the target nucleus, need to be taken into account. This will be further discussed in Section 1I.C.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
u OO
'
'
io.
60 90 '120' '150' Scottering Angle (deg) '
'
'
'
11
'180
FIG.5. Differential cross sections for elastic electron scattering from helium atoms at electron energies of 1.5, 10, and 20 eV. The experimental data of Brunger et al. (1992) are compared with the "convergent close-coupling" results of Fursa and Bray (1995).
B. LIGHTTARGETS WITH
SPIN:
Na 3 2 S
The next level of complexity is encountered for a target with spin one-half, as for example atomic hydrogen and the quasi-one-electron targets Li, Na, K, etc. The most detailed picture has been achieved for the Na 3's state, because it is experimentally more convenient than, in particular, H 1's. Due to the possibility of triplet and singlet scattering, two scattering amplitudes, f S = I'ML, = M L , = 0; 6 ) = f ' and f s = ' ( M Lf = M L , = 0; 6 ) = f ' , are required for the description. These are shown as two arrows in Fig. 6. We thus need three parameters, namely one cross section, one relative size, and one angle, to completely determine the amplitude pair. We
FIG. 6. The triplet and singlet amplitudes f' and f" for elastic e-Na scattering
12
N. Andersen and K. Bartschat
obtain the following expression for the cross section for unpolarized beams
where we have defined the relative sizes
If'12
w'
-
4%
and
with 3w'
+ w"1.
While the differential cross section may again be measured with a setup of the type shown in Fig. 4, the triplet and singlet probabilities have been determined in an experiment developed at NIST and sketched in Fig. 7 (McClelland et al., 1989). This is a crossed beam experiment in which both the electron and sodium beams are spin polarized before the interaction. The polarized electron beam is produced by shining circularly polarized Optical Pumping Detector
e
-
Linear Polarizer Laser Diode FIG. 7. Polarized-electron-polarized-atomscattering apparatus used in the NIST experiments for e-Na scattering, GaAs polarized electron source, scattering chambers, sodium oven, and Mott spin analyzer (from McClelland er al., 1989).
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
13
laser light on a GaAs crystal, while the sodium beam is illuminated by circularly polarized light tuned to the resonance transition before reaching the interaction zone. In the geometry shown, it is thus possible to spin polarize both beams perpendicular to the scattering plane, and flip their spin direction individually by inversion of the circular polarization of the photon beams. One may then measure the so-called spin asymmetry parameter
where P and PA are the spin polarization of the electron and atom beams, respectively. A relative phase angle, conveniently chosen as A (see Fig. 6), thus remains to be determined. In Munster, Hegemann et al. (1993) performed the experiment sketched in Fig. 8. Electrons with polarization P perpendicular to the scattering plane collide with unpolarized atoms, and the polarization P' of the scattered electron beam is determined by Mott scattering. The depolarization parameter T is defined as P' T ~ - P= 2 2 ( w'
+
cos A )
(13)
In this way cos A may be determined, and thereby A, except for the sign. This final ambiguity has not yet been resolved experimentally, but it could
180" spectrometer I arge
GaAsP cathode
0
Pockels cell
fi
U
HeNe laser
.__ ...
.6
"0 63
Mott detector FIG. 8. Experimental setup to measure the depolarization factor T = P ' / P for scattering of spin-polarized electrons from unpolarized sodium atoms (from Hegemann et al., 1993).
14
N. Andersen and K. Bartschat
be removed in a setup where the spin polarizations in the initial channel are orthogonal, say electron spin P in the forward direction and atom spin PA in the scattering plane perpendicular to the beam, and the final electron spin polarization component P ’ perpendicular to the scattering plane is determined. Then a measurement of
determines A uniquely (Andersen and Bartschat, 1993). A determination of the set
thus constitutes a complete experiment. Figure 9 shows theoretical “close-coupling plus optical potential” (CCO; Bray, 1992) and “convergent close-coupling’’ (CCC; Bray, 1994) predictions together with experimental results at 12 eV collision energy for the parameter set ( T , A , V ,cos A). The available experimental data for a; (Gehenn and Reichert, 1972) are unfortunately only relative, but the shape agrees well with the one found theoretically (see Fig. 10). A complete experiment has thus not yet been achieved at this energy, and the situation is similar at 4.1 eV. Nevertheless, we show in Fig. 10 the complete parameter set (uu;w‘;A). After conversion, the choice of sign for the experimental values of A has been guided by the theoretical curve. We notice satisfactory agreement between the theoretical predictions and the experimental results. In the literature, one may often find an alternative choice of amplitudes, f and g, related to “direct” and “exchange” scattering (Kessler, 1985). They are related to the triplet and singlet amplitudes through
and
f” = f + g Thus, if there is no exchange between the projectile and the target electron ( g = 0, or f ‘ = f’),then w‘ = w’= and A = 0, causing A = 0, T = 1, and V = 0. On the other hand, if exchange dominates (f = 0, or f ‘ = -fs), then again w‘ = w s = but A = - T , resulting in A = 0, T = 0, and V = 0.
a
a,
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
0
McClellond e l 01.
0
Scattering Angle (deg)
15
McClelland et 01.
Scattering Angle (deg)
FIG. 9. Experimental results for the parameter set ( T , A , cos A ) for elastic e-Na scattering at 12 eV. The experimental data of the Miinster (Hegemann et al., 1993) and the NIST (McClelland et al., 1992) groups are compared with CCO (Bray, 1992) and CCC (Bray, 1994) calculations. Also plotted are the theoretical predictions for the V parameter (see text).
C. HEAVYTARGETS WITHOUT SPIN:Hg 6’s”
We now move on to heavy targets where relativistic effects, in particular the spin-orbit interaction between the projectile electron and the target nucleus, can no longer be neglected. A classic example of such a target is mercury, with a nuclear charge of Z = 80 and the ground state configuration ( 6 s 2 ) ’ S , . Spin dependent effects in elastic electron scattering from this target have been studied extensively over the past three decades. Detailed discussions of this collision system were given by Kessler (1985, 1991) who used the collision coordinate frame with a “direct” amplitude f and a “spin-flip’’ amplitude g.
16
N. Andersen and K Bartschat
1000
100
10
1
0.1
O.OlL,’ 0.4
“ ‘
30
“
60
‘
”
90
’
”
120
’
“
150
’
I
I
180
. . wt 20
0
true
ghost
-1 .1
Scattering Angle (deg)
Scattering Angle (deg)
FIG. 10. Differential cross section u,, weight parameter w‘, and phase angle A for elastic e-Na scattering at 12 eV. The relative differential cross section measurements of Gehenn and Reichert (1972) have been normalized to the theoretical predictions. There are two possible experimental solutions for A, and we have chosen the “true” and the “ghost” set based on their agreement/disagreement with the theoretical results.
In the formalism outlined in Section I.B, we can use Eq. (2) to determine the number of independent amplitudes in the natural coordinate frame. Since the initial and final atomic states are identical and the target carries no electronic angular momentum, we find the two nonvanishing amplitudes
and
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
17
Note that there are no spin-flipsin the natural frame! However, using the semicl3sical form of the spin-orbit potential as being proportional to i.S: where 1 and are the orbital and spin angular momenta of the continuum electron, one sees immediately that the scattering potentials for spin “up” and spin “down” electrons on a given trajectory_are not identical if this potential has to be taken into account. (Note that 1 is perpendicular to the scattering plane for all classical trajectories.) Hence, this is the mathematical equivalent to the qualitative description of this effect given in Section 3.4.2 of Kessler (1985). The amplitudes f and f 1 can be related to the direct and spin-flip amplitudes by transforming from the collision to the natural flame, as outlined in Appendix A of Andersen et al. (1996). We find
and (Note the similarity of this transformation to the relationship between the triplet/singlet and direct/exchange amplitudes in the previous section.) A schematic picture of the amplitudes as two complex numbers is shown in Fig. 11. Apart from the magnitudes of f and f i, a complete experiment also requires the determination of the phase angle A. The latter replaces the angle y 1 - y 2 , i.e., the phase between the direct and spin-flip amplitudes in Kessler (1990, as the last independent parameter that needs to be determined. To measure the relative magnitude and phase of the two scattering amplitudes, the Munster group developed an experimental setup in which polarized electrons are scattered from unpolarized atoms and the electron
FIG. 11. The “spin up” and “spin down” amplitudes scattering.
fT
and f i for elastic e-Hg
18
N. Andersen and K Bartschat
polarization after the collision is determined. In contrast to e-Na scattering discussed in the previous section, all the information must now come from the preparation and analysis of the electron spin, since the target remains structureless during the whole collision process. This type of experiment allows for the determination of the so-called STU parameters. Because of the necessary extension to inelastic scattering (see Section 1111, Fig. 12 shows a schematic diagram of the “generalized” STU parameters (Bartschat, 1989) that fully describe the change of an arbitrary initial electron polarization through scattering from any ensemble of unpolarized target atoms for both elastic and inelastic scattering. There are seven relative generalized STU parameters with the following physical meaning: The polarization function S, gives the polarization of an initially unpolarized projectile beam after the scattering, while the asymmetry function S, determines the left-right asymmetry in the differential cross section for scattering of spin polarized projectiles. Furthermore, the contraction parameters T,, Ty, T, describe the change of an initial polarization component along the three Cartesian axes, whereas the parameters UKyand U,, determine the rotation of a polarization component in the scattering plane. Together with the absolute differential cross section a; for the scattering of unpolarized electrons, these eight parameters describe the maximum information that can be obtained from preparation and analysis of electron polarization alone.
i A
i
I
I
--
‘-1 I
‘+ SAP,
-7-
T X P X - uxypy
X”
1 +SAP, FIG. 12. Physical meaning of the generalized STU parameters for an initial spin potarization @,which is changed to a final spin polarization @‘through the scattering process (see text).
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
19
Since, in the present case, that is the maximum information about the total collision process, it is not surprising that the generalized STU parameters are not independent for electron scattering from targets without angular momentum. For elastic scattering from such targets, we find
where we have defined
and w1
If
LIZ
=-
2% with
wt
+ w1 = 1
Also, T,
=
1
( 22)
reflects the fact that spin components perpendicular to the scattering plane are conserved. Equations (19) can be inverted to yield the complete parameter set (q,; w ; A ) through wT
=
A
=
s+l ~
2
and ATAN2 ( - U , T )
where ATAN2 is the FORTRAN function for arg(T - iU).This shows the physical meaning of the phase angle A: It describes the rotation of an
20
N. Andersen and K Bnrtschnt
initial electron polarization component in the scattering plane through the collision. The most essential parts of the apparatus used by Berger and Kessler (1986) to measure the STU parameters for elastic electron scattering from mercury and xenon atoms are shown in Fig. 13. Experimental results from their work together with the absolute differential cross section a, of Holtkamp et a f . (1987) for elastic e-Hg scattering at 50 eV are shown in Fig. 14, followed by the set ( w T , A ) in Fig. 15. Note that all relative parameters are in very good agreement with the relativistic calculation of McEachran and Stauffer (1986), while the absolute differential cross section is generally overestimated by the theory. The most important reason for this disagreement is the neglect of “absorption,” i.e., loss of flux into inelastic channels, in the calculation. This problem can be remedied in a simple way by including at least a semiempirical complex absorption potential (Hasenburg et af., 1987).
D. HEAVYTARGETS WITH SPIN:Cs 6’S1/2
We finish this main section with a brief discussion of elastic electron scattering from a heavy target with spin, cesium ( Z = 5 5 ) with the ground state configuration 6’S1,,. This case requires extension of the formalism presented in Section 1I.B to account for relativistic effects. To begin with,
G A S P CATHODE
--
u
U
FIG. 13. Experimental setup for measurement of the change of the electron polarization vector in elastic scattering (from Berger and Kessler, 1986).
21
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
Holtkomp et 01.
- McEachron and Stauffer
0.1
”
’
0
30
’
60
’
’ . 90
“
120
’
150
.
180
0
30
60
90
120
150
180
30
60
90
120
150
180
-0.2 -
- McEachron ond Stauffer 0
30
60
90
120
-0.4 -
150
180
0
Scattering Angle (deg)
Scattering Angle (deg)
FIG. 14. Differential cross section a, and STU parameters for elastic e-Hg scattering at 50 eV. The experimental data of Holtkamp et al. (1987) for au and of Berger and Kessler (1986) for (S,T , U )are compared with results from a relativistic calculation by McEachran and Stauffer (1986).
inspection of Eq. (2) reveals that parity conservation allows for the following eight nonvanishing scattering amplitudes in the natural frame:
f ( + +, + +; 01,
f(+
- 9
+ -;
01,
f ( - +, - +; 01, f(-
- >
-
-; 0)
(24a)
f ( + -,
- + ; e l , f ( - + , + - ; e l , f ( + + , - - ; e > , f(-
fi.
-, + + ; e l (24b)
The amplitudes (24a) are with the abbreviations f for spin values “nonflip” amplitudes, whereas those listed in (24b) correspond to “double-flip” amplitudes, since the spins of both the projectile and the
22
N. Andersen and K Bartschat Hg 6 ' S 0
50eV
".n 7n I"
Berger and Kessler
- McEachran and Stauffer
0.60
0.50
0.40
0.30;'
" '
30
"
60
'
"
90
'
"
120
'
"
'
150
S c a t t e r i n g Angle ( d e g )
'
I
0
180
S c a t t e r i n g Angle ( d e g )
'
FIG. 15. Weight parameter w and phase angle A for elastic e-Hg scattering at 50 eV. The experimental data, obtained from the STU results of Berger and Kessler (1986), are compared with predictions from a relativistic calculation by McEachran and Stauffer (1986).
target electron are flipped, either through exchange or by explicitly spin dependent forces such as the spin-orbit interaction. In a pioneering paper, Burke and Mitchell (1974) analyzed this problem in detail and showed that time reversal invariance reduces the number of independent amplitudes from eight to six. They used a modified natural coordinate system with the quantization axis perpendicular to the scattering plane, the x (their q ) axis along the direction of the momentum This choice of coorditransfer 2, - gi,and the y ( p ) axis along if+ ii. nate system had been discussed earlier (Schumacher and Bethe, 1961, Wilmers, 1972) and yields a transparent system of equations for various spin dependent variables. Specifically, Burke and Mitchell gave equations for the following physical observables (their notation is given in parentheses): 1. The cross section a, (ZJ for scattering of unpolarized beams. 2. The asymmetry function S,(P,); S = S, = S, is a general result for elastic scattering that follows from time reversal invariance. 3. Another asymmetry function ( P , ) that determines a left-right asymmetry in the differential cross section for scattering of unpolarized electrons from polarized target atoms.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
23
4. Asymmetry functions (C,,, Cp!,,C,,, Cr,q7C,J that are generalizations of the exchange asymmetry A discussed in Section 1I.B. 5. Functions ( D i n ,D;!,, Diq> and (D:,f, D;!,, D&) that determine the atomic polarization after scattering of unpolarized electrons from polarized atoms (superscript “1”) or the electron polarization after scattering of polarized electrons from unpolarized atoms (superscript “2”); such parameters are generalizations of the T parameter. 6. Functions ( K ! , f ,K&, Kd,) and (K:,,, Ki,,, Ki,) that determine the atomic polarization after scattering of polarized electrons from unpolarized atoms (superscript “1”) or the electron polarization after scattering of unpolarized electrons from polarized atoms (superscript “2”); these are also generalizations of the T parameter. 7. Functions ( D j Y ,D i p , D;,, D i p ;K;y, K:*, Ki,, K i p ) that are the corresponding extensions of the U parameters for the cases listed in ( 5 ) and (6). Without going into details of the analysis, we point out that even a measurement of the entire set of parameters given by Burke and Mitchell does not yet correspond to a complete experiment. This can be seen immediately by simplifying their equations to the nonrelativistic case. Then all the S and U type parameters disappear, all the T type parameters become identical, C/,4= C , , = 0, and the remaining C type parameters reduce to the exchange asymmetry A of Eq. (12). As shown in Section II.B, however, V type measurements are necessary to obtain complete information about the relative phases of the scattering amplitudes. Given the difficulty of such experiments, we follow Burke and Mitchell and will not provide the generalized equations for such parameters-if such an experiment were to be performed, there should be plenty of time for the experimentalist to derive them while waiting for the signal in the polarization detector! Nevertheless, the e-Cs collision system is a prime example of a case where new information can be obtained by using spin polarized electron and atom beams. In a recent experiment, the Bielefeld group (Leuer et al., (199.5) determined the parameter set ( A “ ” ,A””, Aint),which corresponds to ( - C n n ,P,, P I ) of Burke and Mitchell. The experimental arrangement is shown in Fig. 16, followed by results for elastic e-Cs scattering at a collision energy of 7 eV in Fig. 17. The nonvanishing result for Aint is particularly interesting, since it is the first experimental verification of an interference effect between electron exchange and the spin-orbit interaction. A combination of both effects is required for such a result, as pointed out by Farago (1974) more than two decades ago.
24
N. Andersen and K. Bartschat Monochromator
Laser Diode
n. Polarizer
W
FIG. 16. Polarized electron-polarized atom scattering apparatus used in the Bielefeld experiment for e-Cs scattering (from Leuer et al., 1995).
111. Impact Excitation After elastic scattering, the next level of sophistication is reached when the atom is excited. This step, however, enables us to exploit the rich information contained in the pattern of photons that are emitted in optical decays of the excited state. (We will not deal with excitation of metastable states in the present chapter). For simplicity, the discussion will be focused on S ++ P and J = 0 J = 1 transitions, the main cases of interest for experimental studies to this date. Generalization of the formalism to higher angular momenta is straightforward, as will be illustrated by the example He 1's + 3'D, where interesting new aspects appear. Figure 1Na) shows schematically the basic problem. An atom A is excited from an S state to a P state by impact of an electron, which is deflected by an angle 8, as monitored by a detector in this direction. The excited P state of the atom may be characterized from a study of the radiation pattern Ce., photon direction and/or polarization). This can be achieved with the geometry shown in Fig. 18(b), where the scattered electron is detected in coincidence with the photons emitted in a specific direction, including a possible photon polarization analysis. Two
-
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
c
6
EE
25
0.1 -
.*
0.0-
*-
**
-0.1 -
0 h
*** -~*********=
Q -0.2-0.3 30 40 50 60 70 80 90 1001101201301401
Angle 0 [deg]
-0.31 '
" " " "
" " " " " " '
30 40 50 60 70 80 90 100110120130140150
Angle 0 [deg]
0.041
eV
-0.064 . , .
.
, . , . I . , . . . , . , . . 30 40 50 60 70 80 90 1001101201301401 '0 I
I
I
Angle 0 [deg] Fic. 17. Scattering asymmetries for elastic e-Cs scattering at 7 eV (from Leuer et al., 1995). The nonvanishing values of A'"' indicate the simultaneous importance of electron exchange and spin-orbit interaction.
N. Andersen and IC Bartschat
26
a
b
2"
C
FIG. 18. (a) Schematic diagram of a collisionally induced charge cloud of an atom A cxcited to a P state by an electron scattered at an angle 0. This event may be studied in two ways: (b) Photons emitted in the P + S decay are polarization analyzed (Stokes parameters) in a selected direction and detected in coincidence with the scattered particle. In the time-reversed scheme (c), the atom A is excited by photons coming in from a selected direction, and the number of particles B leading to deexcitation are detected as function of laser polarization. The hvo approaches yield essentially equivalent information.
approaches are here common:
1. Coherence analysis, i.e., a measurement of the Stokes vector ( P , , P,, P 3 ) in one or several suitably selected directions in space. 2. Correlation analysis, in which the angular distribution of the photons in one or two planes containing the collision center is mapped. This is equivalent to a measurement of the two linear light polarizations P , and P, . The correlation approach, therefore, gives less information than (1); nevertheless, it is often used in cases where photon polarization analysis is difficult, such as for resonance excitation of atomic hydrogen or the rare gases.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
27
Figure 19 shows a schematic diagram of the pioneering photon correlation experiment that was built by the Stirling group (Eminyan er al., (1975). The atomic beam (helium) emerges perpendicular to the plane of the diagram. Figure 20 illustrates a later refinement of the experiment, also by the Stirling group (Standage and Kleinpoppen, 19761, which for the first time allowed for a full Strokes parameter analysis of the photons emitted from a planar electron-atom collisional excitation process. An alternative method, which has also been used very successfully, exploits the time-reversed scheme. Figure 18(c) illustrates how this is done. A photon beam from a selected direction excites the atom A. One then maps the P + S deexcitation as function of photon polarization for electrons scattered at an angle 8. For incident energy E, it is equivalent to the information obtained from the reverse S + P excitation experiment at an impact energy of E + A E , where A E is the S-P energy difference. The exploration of the possibilities of this approach was pioneered by Hertel and coworkers (Hertel and Stoll, 1977). In what follows, we shall combine information obtained from all these approaches.
Electron pulses electron analyzer 127O electron monochromator
Q 4 4 Photon pulses
Photon detector
Constant frpction timing discnminator Constant frpqtion timing discnminator
FIG. 19. Experimental setup used by the Stirling group to study electron-photon correlations in electron impact excitation of He 2 ' P (from Eminyan et al., 1975).
28
N. Andersen and K Bartschaf
Analyzer
electrons e beam Scattering plane FIG. 20. Experimental setup used by the Stirling group for electron-photon analysis (from Standage and Kleinpoppen, 1976).
A. LIGHTTARGETSWITHOUT SPIN IN He 1's -+ 2lP, 33P
THE
INITIALSTATE:
We start by discussing the case He 1's + 2lP, a process described by the two scattering amplitudes f s = 1 ' 2 ( M L ,= 1, ML, = 0, 6 ) = f + l and f S = ' I 2 ( M L ,= - 1, ML, = 0; 6 ) = f - shown in Fig. 21. Transitions to the ML, = 0 state is forbidden due to conservation of reflection symmetry in the scattering plane. Hence, this process is determined by three parameters: the differential cross section, a parameter describing the relative size of the two amplitudes, and an angle that fixes their relative phase.
f-1
FIG. 21. The amplitudes f + , and accounting for electron spin.
f-l
for electron impact S
+
P excitation without
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
29
The two independent dimensionless parameters may be chosen as the angular momentum transfer = (O,O, L L ) ,with
(c)
L,
=
lf+1I2
- If-I12
If,
+ If-
It2
112
- lf+1I2 -
lf-1I2
-
=
a;
w+- w _
(25)
and the alignment angle y of the major symmetry axis of the charge cloud in the scattering plane, shown in Fig. 18(a). The alignment angle y , defined mod r , is related to the phase angle 6 (cf. Fig. 21) through
s = -2y*
(26)
7r
All this may be expressed compactly through the density matrix p, which in the IML) helicity basis is given by 1 p=uu2
1+ L, O 0 0 -P,eZiY o
-Ple-2iY
1 - L,
The parameters ( L , , y ) are related to the Stokes vector $ = (P,,P2,PJ for the light emitted in the + z direction in a subsequent P + S decay through P,
2
+ iP2 = P,e2iy = - -U If+llIf-lle-iS
(28a)
and P3 = -L,
(28b)
Determination of the parameter set
thus constitutes a perfect experiment, since the wavefunction of the He2’P state can be written as
1)
=
/=
/ 7
1 + 1) - e2;” -(1
-
L,)
I - 1)
(30)
This wavefunction is normalized as (I) I I)) = uu,consistent with the trace of the density matrix (27).
30
N. Andersen and K Bartschat
In this case, the degree of linear polarization, P! = JPf + P; is a shape parameter that describes the form of the electron charge cloud, with a length/width ratio in the scattering plane given by l / w = (1 +_PI)/ (1 - PI). The total degree of polarization of the emitted light, P = lPl, is unity, i.e., P 2 = L2, P: = 1 (31)
+
An early illustration of how these parameters can be used to visualize the outcome of the collision process is given in Figs. 22 and 23 (Andersen et
al., 1984). The upper part of Fig. 22 shows experimental results at 80 eV 1' ' '
x'*B
4
1
1
1
1
'
1
f' \
\ .
+,
\ \
-
\
8"
',
\
-
,,/'
0,.
-1.
I
1
50
1
I
1
'
I
'
I
8
I
I
I
1 . .
1 \
-
I
t
9
**-.
/
i ................... I
, , .o
I
1
1
1
'
80
1
........... I\*; .................... ....
180
....
I
1:::
]
\
I I I
? - \
,
\tl
0 '
0:
L-7-
, P A
,s*-{
-
-50.
10'
, 7--o-
I
/@
P)
0.5
,/' --
0
0
'1.0
1
A
:/I"
: 0
4
I
,
,
I
...... ................. ..... ,
,
I
,
,
- 280 I
,
,
I
,
l
FIG. 22. The collision-induced angular momentum transfer L , and alignment angle y (bottom) for He l l S + 2’P excitation at an incident electron energy of 80 eV (from Andersen ef al., 1984). The labels (a)-($ between the two figures refer to the situations displayed in Fig. 23. The data are taken from Eminyan et al. (1975) (0)and Hollywood ef al. (1979) (0).
31
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
I
8 It
I
f
h
i
i
Fro. 23. The shape and dynamics of the radiating He 2'P state for electron excitation at an incident electron energy of 80 eV (from Andersen et al., 1984). The scattering angles of interest are O"(af, 25"(b), 35"(c), 50"(d), 65"(e), 70"(f), 90°(g), 110"(h), 130"(i), and lSO"(j). The upper row (a-e) corresponds to counterclockwise rotation of the electron around the atomic core, whereas the lower row (f-j) corresponds to clockwise rotation.
for the orientation parameter L , , and the lower part results for the alignment angle y , here compared with the value predicted by the first Born approximation (FBA), which describes the excitation as creation of a p orbital aligned along the direction of the momentum transfer "k -+ The orientation parameter shows a characteristic dependence on the scattering angle 8: Starting at zero, positive values grow with increasing 8 until an almost circular state is observed, with the alignment angle following the FBA prediction until this point. From then on, the orientation decreases rapidly, changes sign, and approaches its other extremum value. This corresponds again to a circular state, but with opposite sense of rotation (i.e., clockwise) of the excited electron around the atomic core. During this interval, the alignment angle is almost perpendicular to the FBA prediction, with another rapid change near the minimum value of L , . At even larger scattering angles, the orientation and the alignment angle converge back to zero until, at backward scattering angle, the state is again in its initial shape, a p orbital aligned along the beam axis. Figure 23 shows the corresponding behavior of the charge cloud in the collision
zi.
N . Andersen and K Bartschat
32 1000
1
. .
I
.
,
I
0 (1 0-19
.
.
I
.
.
I
.
'
I
.
1 .o
.
cmz/sr) 0
Hall et 01.
0
CartWright e l 01.
1
0.5
1
0.0
?
-0.5
i
0.1
d '
.iO. '$0'
'sb'
1;O'
'140' 'lS6''o0
1 .o
1 .o
0.5
0.5
0.0
0.0
-0.5
-0.5
-1
.o
0
30
60
90
120
150
Scattering Angle (deg)
186'"O
30
60
0
Beijers et 01.
A
Eminyon e l 01.
90
120
V
30
60
90
150
180
Khokoo e l 01.
120
150
180
Scattering Angle (deg)
FIG.24. Differential cross section mu and Stokes vector ( P , , P , , P3)for He 1's -+ 2'P excitation at an incident electron energy of 50 eV. The experimental data of Hall et al. (19731, Cartwright et al. (1992), McAdams et al. (1980), Beijers et al. (19871, Eminyan et al. (1974), and Khakoo et al. (1986) are compared with CCC calculations of Fursa and Bray (1995). Except for the direct P3 measurement reported by Khakoo et al. (1986), the magnitude of P3 was obtained by assuming full coherence of the radiation, and the sign was guessed by using the Khakoo et al. data and the theoretical results as a guide.
plane, with the upper strip corresponding to L, > 0, and the lower one to L, < 0. The behavior may be conveniently summarized by plotting the corresponding movement of the Stokes vector on the PoincarC sphere (Andersen and Hertel, 19861, which, however, is outside the scope of the present chapter. The current state of the art is shown in Fig. 24, which summarizes experimental and theoretical values of the parameters (a,, P I ,P z , P 3 ) at an impact energy of 50 eV. These values are converted in Fig. 25 to the standard set (u,;L ,; y , Pl), of which the first three are independent,
33
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS 1000
100
10
0
Halletal.
0
Cartwriqht et 01.
1
0.1
0
30
30
60
90
120
150
186"'
60
90
120
150
186900
0
30
Scattering Angle (deg)
60
90
120
150
180
Scattering Angle (deg)
FIG.25. Differential cross section uu and coherence parameters (LL, f', y ) for He 1's + 2' P excitation at an incident electron energy of 30 eV. The data were transformed from the measurements shown in Fig. 24.
whereas the fourth determines the shape of the charge cloud. Due to experimental difficulties associated with the circular polarization measurement of the 2'P + l l S line at 58.4 A, the magnitude of P3 was, in most experiments, obtained via lP,l = i.e., assuming completely polarized light. Then the sign was guessed by using the theoretical results and the direct measurement of P3 by Khakoo et al. (1986) as a guideline. It is important to point out that the indirect approaches forfeit the opportunity for an experimental consistency check by using the independent measurement of all three components of the Stokes vector. The importance of such a consistency check will be further discussed in Section III.C, where it is seen to be much more critical than in the present case. Furthermore, a complete experiment requires the experimental determination of the sign
dm,
34
N. Andersen and K Bartschat
of P3. This has, indeed, been achieved for 50 eV and some other energies by the Windsor (Khakoo et al., 1986) and Perth groups (Williams, 1986). We note good agreement between the CCC theory of Fursa and Bray (1995) and experiment at all scattering angles, a situation that was far from being satisfactory during the early exploration stages of this process. In summary, He 1’s -+ 2lP excitation serves today as a standard example of a perfect experiment on inelastic scattering. Similar, but less extensive, results have been obtained at other energies, and for higher members of the He n’P series. We end this section by briefly summarizing the situation for excitation of the He 1’s + n3P series, using results for 33P at 30 eV as an example. This process is interesting since, for helium, excitation of a triplet state can only occur by electron exchange. T h t set of Stokes parameters measured for the 33P + 23S photons (3889 A) is not a unit vector due to the influence of the fine structure coupling, which will effectively depolarize the electron charge cloud between collision and decay. This effect, however, can be remedied by introducing so-called “reduced” Stokes parameters (Andersen et al., 1979), labeled by a bar. Appendix B of Andersen et al., (1988) gives the following equations:
E.,
and 41 - 27
F --P3 The reduced Stokes vector is again a unit vector. The results for the coherence parameters are displayed in Fig. 26. Again, we see a satisfactory agreement between theory and the set of complete experimental observables.
B. LIGHTTARGETSWITH SPIN:Na 3,s
+
32P
3
For Na32S -+ 32P excitation, the projectile and target spins of s = double the number of scattering amplitudes from two to four (recall that fo = 01, since we have the possibility of triplet ( t ) and singlet (s) scattering, bringing us to the situation of Fig. 27.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
35
1
0.1
0.01 1 .o
0.8 0.6 0.4
0.2 0.0
0
30
60
90
120
150
Scattering Angle (deg)
18
Scattering Angle (deg)
FIG. 26. Same as Fig. 25 for He 1's + 3 3 P excitation at an incident electron energy of 30 eV. The experimental data are obtained from measurements of Donnelly et al. (1988).
Neglecting an overall phase, we thus need determination of seven independent parameters for a complete experiment. In addition to the differ-
36
N. Andersen and K Bartschat
FIG.27. Schematic diagram of triplet (f)and singlet (s) scattering amplitudes in the natural frame for 'S j 2 P transitions by electron impact. Note that A + + 6' = A - + 6'.
ential cross section o;, six dimensionless parameters must be defined, three to characterize the relative lengths of the four vectors, and three to define their relative phase angles. As a start, we parameterize the density matrices in analogy to Eq. (27), as (see also Hertel et al., 1987)
and
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
37
where
++
- 4- and as = I,!+- I,!-. In the case where unpolarized with 6' = beams are used, the total density matrix becomes the weighted sum of the two matrices p s and p', i.e.,
where
N. Andemen and K Bartschat
38
The parameters w' and w s are related to the parameter r by Hertel et al. (1987) through W'
=
WS =
1 - ws 1- 3 w ~ 3ws W'
=-
=
(+'/us used
(38)
However, we prefer the use of w'.' to r for reasons of mathematical symmetry and simplicity. At this point we have thus introduced a total of six parameters, namely gU,w', L', , Ls, , y', and ys,leaving one parameter, a relative phase, still to be chosen. Inspection of Fig. 27 suggests, for example, the angle A + . Note that the fourth angle, A - is then fixed through the relation A + - A - = 8' - 8 s
= 2 ( y S- y ' )
(39)
The first equality sign follows from inspection of Fig. 27, the second one from Eq. (26), applied individually to the singlet and triplet components. We thus use the following complete set:
These parameters 1 . Allow for a complete description of the scattering process 2. Are a natural generalization of the parameters used for unpolarized beams 3. Can be interpreted in simple physical pictures 4. Are accessible in "partial" (i.e., noncomplete) experiments The reduced Stokes vector @ of an unpolarized beam experiment is given by the weighted sum of the singlet and triplet (unit) Stokes vectors @','as fi = 3w'@' + w s i j , s
(41)
from which the set of parameters ( L L ,y , P,) for an unpolarized beam experiment may be evaluated. In particular,
L,
=
3W'LL
+ w";
(42)
Since, in general, L\ # L: and y' # ys,this causes the (reduced) degree of polarization P to be smaller than unity, i.e., P I 1 for an unpolarized electron beam. We now want to express the STU and V parameters discussed in Section I1 in terms of the complete set of parameters (40). Provided that electron exchange is the only spin dependent effect of importance for the excitation, and that the fine structure energy splitting is negligible com-
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
39
pared with the initial and final energies of the projectile, the STU parameters get vastly simplified (see Appendix C of Andersen et al., 1996). The seven polarization, asymmetry, contraction, and rotation parameters for each fine structure level reduce to the following set of only four independent parameters (the superscripts denote the J value of the excited target state):
sP -= s 'P/ 2 =
- 2s;/2
sA -= s'/2 = A T
~ 1 / = 2
u
~ 1 1 2=
(43a)
-2sy T;/Z = ~ z 1
(43b) / = 2 ~ 3 1 2= ~ 3 1 = 2 ~ 3 1 2 X z Y
112 = - 2 ~ 3 1 2 = - 2 u Y ; / 2 XY
(434
UYX
XY
(43c)
The results are
s,
s,
=
1
=
+ WSL1
-W'L\
-[
a+p+ cos A + -
24 1
T = -[a+p+ 2 a; U V
1
=
-[
COSA-] - 2w'L:
COSA++ a - @ -COSA-]
+2
~ '
(44b) (44c)
a+p+ sin A + -
2% 1
=
(44a)
-[a+@+
2 a;
sin A + + a - p - sin A-1
(44e)
The amplitude sizes a+ - and p, may be eliminated from Eqs. (44) by using
P+
-a* =
2 flu
iw'wS(1 f LL)(l f L?)
(45)
We now investigate to what extent the perfect scattering experiment has been achieved to date. We begin with the key experiment performed by the NIST group (McClelland et al., 1989) and shown in Fig. 7, but with the beam overlap modified so that scattering may take place also from the excited state. Spin polarized electrons with polarization vector perpendicular to the scattering plane were scattered superelastically from spin polarized sodium atoms in the 32P state. This state was produced by pumping with circularly polarized laser light. By reversing the directions of the two polarizations individually, the experiment allows for the determination of L: , L; , and w' (for details, see Hertel et al., 1987).
40
N. Andersen and K Bartschat
This experiment does not determine the alignment angles y f and ys. However, the off-diagonal elements of Eq. (36) show that ple2iy = 3wrp;e2i~r+ wsp;e2i~’
(46)
As illustrated in cig. 28, $is complex equation corresponds to addition of the two vectors Pt and P/, multiplied by+weighting factors 3w‘ and ws, respectively, to form the resulting vector PI. Hence, elementary geometry (the true can be applied to obtain two pairs of solutions, ( y l , ySItrue solution) and (y ‘ , ys)ghost (the other possibility):
y”y*-
* 2
where the angles y, and )I are defined in the figure. Provided experimental data are available for the parameter set ( P I ,y , w‘, L: , L:) at a given collision energy and scattering angle, two sets of possible angles ( y f , y s ) can be determined. It was pointed out by Hertel et af. (1987) that the foregoing ambiguity is mathematically identical to the one found for S + D excitation processes (Andersen et al., 1983). This will be further discussed in Section 1II.E. The ambiguity could have been resolved with the following modified version of the NIST experiment. If the laser light propagates in the collision plane perpendicular to the electron beam direction, with the electron spin polarization still parallel or antiparallel to the atomic spin polarization,
FIG.28. Vector diagram corresponding to Eq. (46). For a discussion, see text.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
41
one can determine an asymmetry parameter B analogous to the A parameter of Eq. (12). The result is (Andersen and Bartschat, 1993) B = -
w‘(1 - P ; ) - w”1
1 - P,
-
P;)
(48)
Thus, a consistency check between the results of Eqs. (47) and (48) will eliminate the ghost solution. Such an experiment, unfortunately, has not been performed yet. The ambiguity for the alignment angles can, however, be removed with help from state of the art theory. While data for all the parameters necessary for determination of the two sets of angles ( y ‘ , y s ) have indeed been obtained for electron-sodium (de)-excitation, the energy and scattering angle combinations investigated by the Adelaide (Teubner and Scholten, 1992) and the NIST (McClelland et al., 1985; Scholten et al., 1991) groups, unfortunately, do not overlap at all. Consequently, we replace the missing experimental data for PI and y with theoretical results of Bray’s calculation (Bray, 1992) at a total energy of 4.1 eV to “invert” the NIST data at this energy. This approach seems justified in light of the excellent agreement between Bray’s theory and experiment for all available collision energies. The results of the inversion are shown in Fig. 29 for the angles y r and y s . The error bars on the experimental points were obtained by first changing the theoretical results for the set of input parameters
(w‘,L> ,L y ) (recall that = 41 - (,!,‘is)’ ) by a small amount and then looking at the effect on the inverted theoretical results for the two pairs of ( y ‘ , 7’). This gives partial derivatives of y f and y s with respect to the three input values and thus allows for the calculation of error bars. (This is a somewhat pessimistic estimate of the error bars, since we assume an independent determination of the parameters w‘,L: , L: .) Since the true and the ghost solutions evaluated from theory are in very good agreement with the experimental values, we select the “true” experimental data as those that follow the true theoretical solution. Note that the two sets of solutions can cross each other, and that it is impossible to stay on the “true” experimental curve by assuming, for example, a smooth angle and energy dependence of the phase angles. Next, we note from inspection of Eqs. (44) that information on the still missing phase difference between a singlet and triplet amplitude can only be obtained from STU parameters. The T parameter was measured by Hegemann et al. (1991, 1993) at 4.0 and 12.1 eV total collision energy, whereas the NIST experiments were performed at 4.1 and 10.0 eV, respectively. Again, due to the unfortunate lack of data from different experiments at the same energy, we demonstrate an inversion method for
42
N. Andersen and K Bartschat
"1 h
cn 0
z
ul
.?
theory
- ahost
30
o - 30 - 60
30
60
90
120
150
I
180
Scottering Angle (deg)
60 A
cn
- theory
____
I
ghost experiment
30-
FIG. 29. Alignment angles y' and y s calculated from the NET (McClelland et al., 1989) data for ( w f , L: , L:) and theoretical results for (P,, y ) from scattering amplitudes of Bray (1992) for electron impact excitation of the 3*P state of sodium at an incident electron energy of 4.1 eV; 0, two sets of inverted experimental data as well as true (-) and ghost (- - -) theoretical solutions (from Andersen and Bartschat, 1993).
the A angles by using theoretical data (Bray, 1992) for the parameter T at 4.1 eV. (If experimental results for the U or I/ parameter were available, evaluation of the missing phase angle is straightforward from Eqs. (44).) The idea of the second inversion procedure is illustrated in Fig. 30. Equation (444 for the T parameter corresponds to a nonlinear equation for A' and A-. In addition, the difference between these two angles is related to the difference between the alignment angles y' and y s through Eq.(39). Consequently, solutions for A + and A - can be found by searching for crossings between the lines determined by A cos A + + B cos A - = C
(49)
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS AcosA+
43
+ BCOSA- = C
180
120
60 h
[r
s
o
I
a - 60
-1 20
-1 80
-
FIG.30. Determination of the singlet-triplet phase angles A’ and A - from experimentally observable parameters (see text).
where the constants A , B, and C are evaluated from Eq. ( 4 4 ~ )and the in Fig. 30. Due to the lines defined by Eq. (39) labeled (7’ - yt)true,shost ambiguity in the sign of the arguments in the cosines, and if the ambiguity in the pair ( y t , y s ) is taken into account, one will usually find four solutions, only one of which is correct. This is illustrated in Fig. 30 for inversion of the theoretical data at a scattering angle of 40” and a total collision energy of 4.1 eV (Andersen and Bartschat, 1993). Note that the problem can be reduced to searches in the first quadrant, since the “ghosts” in the second and fourth quadrants may be found via intersections with the dashed lines in Fig. 30, which are mirror images of the difference lines in those quadrants seen in the first quadrant. Since the slopes of the dashed mirror lines are reversed compared with the original difference lines, the actual crossing points in the second and fourth quadrants can easily be reconstructed, while the only remaining crossing, in the third quadrant, is related to the one in the first quadrant through a simultaneous sign change in A\+ and A-. The results for A’ and A - as a function of the scattering angle for a collision energy of 4.1 eV are shown in Fig. 31. For simplicity, only one ghost solution (where A \ + > 0) is shown. Again, the theoretical results help
44
N. Andersen and K Bartschat
-::I -1 800
,
, l ,
30
,
60
7,
theory , , , ghost experiment
90
,
120
,
1
150
180
60 90 120 150 Scottering Angle (deg)
180
Scottering Angle (deg)
180 120 CTI
60
0)
s LI
o - 60 -1 20
-l8O0
experiment
30
FIG. 31. Singlet-triplet phase angles A' and A - (Andersen and Bartschat, 1993) calculated from data for (w',L: ,LS, ) of McClelland et al. (19891, the corresponding alignment angles ( y ' , y s ) presented in Fig. 29, and theoretical T parameter results from scattering amplitudes of Bray (1992) for electron impact excitation of the 3*P state of sodium at an incident electron energy of 4.1 eV; 0 , two sets of inverted experimental data as well as the true (-) and one ghost (- - -) solution (see text).
to identify, in most cases unambiguously, the true solution among the possibilities obtained from an inversion of experimental data alone. Figures 32 and 33 show the full set of results obtained in this way at collision energies of 4.1 and 10 eV, respectively. We notice a very satisfactory overall agreement between the theoretical predictions and the complete set of experimental parameters (40). Nevertheless, the importance of such detailed benchmark measurements is demonstrated very clearly by looking at the parameter L: at 10 eV. Only the most sophisticated CCC theory of Bray (1994) can reproduce the experimental results for this
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
45
Na(32P) 4.1 eV
O(deg)
FIG. 32. Survey of alignment and orientation parameters for excitation of the Na 3*P state by spin polarized electrons at an incident electron energy of 4.1 eV. The differential cross sections are given in units of &sr. The experimental data of the NIST and Miinster groups have been transformed to the parameter set (40); they are compared with CCC (-; Bray, 1994) and CCO (- -; Bray, 1992) results and a 10-state close-coupling calculation of the JILA group ( ......; Zhou et al., 1995).
46
N. Andersen and K Bartschat
FIG. 33. Same as Fig. 32 for an incident electron energy of 10 eV. The dotted line shows results from a second order distorted-wave calculation by Madison et al. (1992). The differential cross section data are from Srivastava and VuskoviE (1980).
observable; in contrast, the parameter L\ is much less sensitive to the quality of the theoretical model. Although a complete set of parameters at the present time could only be extracted from available experimental data after two guesses guided by theory, the discussion showed that a complete experiment is within reach
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
47
for this case, in particular if the experimental programs of the participating groups aim at common choices of angles and energies. Indeed, the somewhat complicated inversion procedure could be avoided and the perfect scattering experiment directly be achieved, for example, as follows: A measurement of the in-plane asymmetry parameter B could resolve the ambiguity in the two ( y ' , 7') pairs. Furthermore, since the equations for the pairs ( S , , U ) and ( T , V ) can be recast in a form similar to Eq. (41), determination of one of the pairs gives a geometrical ambiguity in the ( A + , A - ) pair, as discussed for ( y ' , y'). This ambiguity can be removed by measurement of one more of the remaining S T W parameters. If ( 7 , P I ) are known from unpolarized beam experiments and (L: , L: ,w ' ) from a NIST type experiment7 any three of the five parameters B,S,, T , U , and I/ will suffice to achieve a perfect scattering experiment. The scattering amplitude information contained in the atomic density matrix (i.e., the Stokes parameters) and the reduced density matrix of the scattered electrons (i.e.7 the STU parameters) is illustrated in Fig. 34. From a Stokes parameter analysis, one obtains information about the relative phase between the two fil and fi, amplitudes and the relative and f-,amplitudes, as well as the relative phase between the two sizes of all four amplitudes. However, the Stokes parameters do not depend on the relative phases between any triplet and singlet amplitude. The STU parameters, on the other hand, can be used to determine the relative phase A + between the two fil and fil amplitudes and the relative phase A - between the two fi, and f: amplitudes, provided that the relative sizes of all four amplitudes are known from a Stokes parameter measurement.
fsl
a
Stokes
b
STU
f?, FIG.34. (a) This diagram shows which relative amplitude sizes and phases can be evaluated from a Stokes parameter analysis. (b) This diagram shows which relative amplitude sizes and phases enter into the equations (44) for the STU parameters.
48
N. Andersen and K Bartschat
The preceding discussion also demonstrates how the inversion procedures may serve as consistency checks among separate experimental data sets. Consistent experimental data should always allow for inversion within experimental uncertainties. We end the discussion of Na32P excitation by pointing out one additional consistency test. Equation (44a) points to an interesting link between Stokes parameters and STU parameters. Nickich et al. (1990) performed a measurement in a similar geometry to the NIST setup. The essential difference was the choice of a target density high enough to ensure unpolarized target atoms due to the depolarization effect of radiation trapping. For each fine structure level, the asymmetry function
was measured for the superelastic transition 32P -+ 32S. Because of time reversal invariance of the interaction, however, the S, measurement for the deexcitation process is equivalent to determination of the polarization function S, for excitation. One can thus use Eq. (44a) to predict the results of this experiment from the data set (L:, L:,w') of the NIST experiment (Hertel et al., 1987; Balashov and Grum-Grzhimailo, 1991). The data of Nickich et al. (1990) and the results predicted from the NIST experiment (Scholten et al., 1991) are shown together with theoretical results calculated from Bray's amplitudes (Bray, 1992) in Fig. 35. The consistency between the two independent experimental data sets and their agreement with the theoretical prediction underlines the power of the present formalism. C. HEAVYTARGETS WITHOUT SPININ THE INITIALSTATE: Hg 6lS, + 63P1 For the Hg6lSO+ 63P, excitation process, the argument in Section 1.B yields six independent scattering amplitudes for a J = 0 + J = 1 transition, thereby requiring the determination of one absolute differential cross section, five relative magnitudes, and five relative phases. For heavy atoms, explicitly spin independent forces, such as the spin-orbit interaction or other relativistic effects, must be taken into account-in the T operator for the collision and/or in the description of the target states (e.g., by an intermediate coupling scheme). The large number of independent parameters reflects the additional degrees of freedom that the problem presents. Figure 36, for instance, shows that for electron spin polarization in the scattering plane the charge cloud symmetry axis is no longer restricted to
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
SD 0'4
49
r------
0.2
0.0
-0.2
-n
Y .
A I
0
20
40
60
80
100
120
140
160
180
Scattering Angle (deg) FIG. 35. Spin polarization function S, for electron impact excitation of the ( ~ P ) ~ P , / , , , / , states of sodium at an incident electron energy of 10 eV; data of Nickich et al. (1990) for and Sp(2P,/2) ( 0 )from the Sp(2PI/2(0)and S,(2P3,,) ( W k prediction of S,(2P,/2 (0) NIST data (Scholten et af., 1991) for (w',L i , L$ k the theoretical curves for S,(*P,/,) (-1 and S,(2P3,2X- - -1 were calculated from scattering amplitudes of Bray (1992).
Z
X
FIG. 36. Example of a tilted and twisted charge cloud due to an in-plane spin polarized incident electron beam (from Raeker er al., 1993). The figure is for Hg 63P, excitation with impact energy 8 eV, initial spin polarization Py = 1, and scattering angle 0 = 30".
50
N. Andersen and K Bartschat
*+
f+l f+l
+
f-1
J
FIG.37. Schematic diagram of scattering amplitudes in the natural frame for J =
1 transitions by electron impact. Note that A + + 6
=
=
0
+
A-+ 6 T.
this plane; instead, it may tilt away from the plane and even twist (Raeker et al., 1993). This will necessarily lead to considerable complications in the
algebra. Omitting J = 1, Ji = M i= 0, and 8, we parameterize the six scattering amplitudes f{Mf, mf,m i ) in Fig. 37 as (Andersen and Bartschat, 1994):
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
51
The quantities (51a-d) represent no-$@ amplitudes that leave the projectile spin unchanged, whereas (51e, f) describe the cases where the electron spin is j-l@ped. The up/down arrows correspond to the initial spin projection in the natural frame. As for e-Na excitation, we want to express the photon and electron polarization properties in terms of the amplitude parameters, as well as the density matrix for the excited state. The analysis is done most systematically in the reduced density matrix formalism, with an expansion of the density matrix elements in terms of state multipoles. In the following, we shall not repeat the derivations, which are given elsewhere (Andersen et al., 1996), but just state the results. Since most information is contained in the emitted light, we start by addressing the photon pattern. 1. Generalized Stokes Parameters The emitted radiation can be analyzed systematically in terms of the "generalized Stokes parameters" introduced by Andersen and Bartschat (1994). They are defined in such a way that all four possible combinations of photon polarization analyzer and initial electron polarizations enter on an equal footing. We use the following notation: The quantity 1: p x , y , ( p ) is the light intensity transmitted by a linear polarization analyzer oriented at an angle p for incident electron polarizations in the x , y , or z direction, with the light being observed in the direction denoted by A. Similar definitions are made for the intensities transmitted by circular polarization analyzers. This gives, for example, 1: pJ45")
=
t [ 1;
p,
+ cIp2);Pz]
(52a)
and the total intensity for unpolarized incident electrons can be constructed as 1,'
=
a[ 1Y,pJ450)+ I Y P J 4 5 O ) + IU+P,(1350)+ ZYPJ1350)]
(53)
We define the generalized Stokes parameters Q$m measured with a photon detector in the direction Ti and electron spin polarization along the direction rh by taking the three other independent linear combinations of the four intensities. The second subscript j = 1,2,3 refers to the sign combinations + - -, - - +, and -, whereas the first subscript i = 1,2,3 refers to the photon polarizer settings (O", 90"), (45",135"),
+
+
+ +
N. Andersen and K Bartschat
52
and a-,a+),as for the standard Stokes vector. See Fig. 38 for sign conventions. For example, we get Zy+pJOO) QYZ
=
l1 -
Z:pJOo)
+ ZY_p(Oo) - Zy+p,(900)- ZY_p19Oo) + ZY,(O") + Z'+p,(900)+ ZYPJ9O0)
(54a)
The standard three component Stokes vector is thus replaced by a 3 matrix. All denominators in Eqs. (541456) are equal to the intensity
X
3
I,' of
Eq. (53). The first three components (54a, 55a, 56a) form the usual Stokes vector (P1,P,,P,), as measured with an unpolarized beam. The other columns contain additional information, which we shall now explore. For this purpose, we first address the density matrix. It no longer describes the electronic charge could, but instead the excited state ( J = 1) distribution.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
53
FIG. 38. Coordinate frame for definition of generalized Stokes parameters (Andersen and Bartschat, 1994). The linear polarizer settings in the directions it = x , y , z are shown for polarizer angles p = o”, 45”, 9o”, and 135”, following the notation of Blum (1981). The incident electron beam is characterized by spin polarization components * P I , +P’, or * P z , as indicated.
Since this state radiates like a set of classical oscillators completely 1 analogous to a P, state, we shall maintain our previous notation by replacing the term “charge cloud density” by “oscillator density.” We also maintain the parameter name L , though, strictly speaking, .TI would be more appropriate. We now parameterize the density matrix for the classical oscillator density by further generalizing the method used previously for sodium. For excitation with an unpolarized electron beam, the density matrix for heavy atoms is decomposed into a pair of matrices, one having positive reflection symmetry with respect to the scattering plane and the other having negative reflection symmetry. The importance of excitation of states with negative reflection symmetry and the corresponding spin-flip processes is measured by the height parameter h (Andersen et al., 1988). We consider the two cases: 1. Electron beam polarization perpendicular to the scattering plane, i.e., along the z direction of the natural coordinate system. The natural extension of the earlier decomposition is a pair of density matrices, one for spin-up electron impact excitation and one for spin-down excitation, where
54
N. Andersen and K Bartschat
“up” and “down7’ correspond to the initial spin component orientation with respect to the scattering plane. Hence, 1+Ll
-P,?e2iY =
wtpT
0 0 O
1 - L,+
0
0
0
+ w”p“ 0
1+L;T
-p,+Te-2iyT\
0 0 -p; t e 2 i ~ t 0
1
0 l-L,+’
1 + L,+1
0
0
0 0
-p;
1 e 2 i ~I
,
!
-p,+~e-ziY’ 0 1 - L y
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
55
(58i)
2y'
=
2yJ
=
&- 4+*
*--
*,&
+
7T=
-6'
7T=
-6J +7T
- 7T
(59f)
(59g)
Consequently, the decomposition of the density matrix (57) is described in terms of the cross section uu and seven dimensionless independent parameters: (uu; W T , L y , L,f l , h' ,h' ; y
1,
y
1)
(60)
This set leaves three relative phases still unknown. Before searching for additional information about these phases using electron polarization in the scattering plane (to be discussed in (2) following), we first map the results of the radiation pattern analysis with electron polarization P, by showing the relationship between the parameter set (60) and the general-
56
N. Andersen and K Bartschat
ized Stokes parameters. For an electron beam polarized along the z axis, the generalized Stokes vector matrix for observation in the $2 direction is given in terms of the density matrix parameters by (Andersen et al., 1996):
ti
(IQ;;)=-
(1-h)P,
w'(l-h')P,'
(I-h)P,
wt(l-hr)PJ -w'(l-hi)f)
-wl(l-hl)ft
f[w'(I -3h')-wJ(l-3h')]
(1 - h)P,
w ' ( 1 - ~ ' ) P J-
+ [ . ' ( I- 3 / 1 9 - w J ( 1 - 3 h ' ) ]
w l ( ~-
$[w'(l -3hr)-wJ(1-3h1)]
hJ)fj
with the (normalized) light intensity
I =I,'
=
;(I
- h)
and (1 - h ) P i = ~ ' ( 1 h')Pi'
+~
i
' ( 1 hL)Pi"
=
1,2,3 ( 6 1 ~ )
We now assume that the height of the charge cloud for unpolarized electron impact, h, is known from a standard P4 measurement (Andersen et al., 19881, i.e., a P , measurement with photon detector along the y direction in the scattering plane. One may then use the sum and the difference of the elements in the+first two columns (i.e., six paLameters) of Eq. (60) to obtain w r ( l - h T ) P T / ( l - h ) and w L ( l - h l ) P i / ( l - h), where 9 = (P,' ,P 2 , P3f ) and @ = (P,' ,P$ ,P: ). Since the degrees of = I? ' I = 1 for the two initial spin projections, one polarization P + canfirstextract c T w T ( l - h T ) / ( l - h)and c i = w"1 - h c ) / ( l - h ) from the sum of the squares of the individual components, and subsequently L:', L T L , y ' , and y ' from the Stokes vectors S T and 3'. (Since c T c c = 1 theoretically, any measured deviation from this relationship should be remedied by renormalizing all elements of the generalized Stokes matrix by a common factor.) The last column determines [w (1 - 3h ) - w (1 - 3h )]/(1 - h), which, when combined with c and c allows for determination of w h ? , and h I.Thus, knowing h , the seven dimensionless parameters of (60) can be determined from QY. The independent determination of h cannot be replaced by Eq. (59~);this equation is not independent from the others and can only serve as a consistency check. Switching now to the matrices measured in the two remaining directions y and x , equations for the nonvanishing elements in the first row of the and (QY) are written here as a column for convenience matrices (Andersen et al., 1996): 9
''
+
',
',
(Qr)
- [ w f ( l - 3h')
+wJ(l - 3h1)]+ [w'(l
-[w'(l -3h')[w'(1
+h') -~
w'(1 - 3 h l ) ] ' ( + 1 h')]
-
- h')Pg + w l ( l - hL)P,']
[ w ' ( l -h')P,'
+ [w'(l
- w'(1 - hL)P,']
- h')P: - w J ( l
-hl)P,']
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
57
with the normalized light intensity
I =I,' and
(z]:[ -
-[(l
4
+ h ) + (1 - h ) P , ]
- 3h') +wl(l - " I ) ]
-[w'(l
=
3
=
-[w'(l - 3h')-
[w'(I
wl(l
~
3h1)]
+ h ' ) - w'(l + h i ) ]
+ [w'(l +
- h')P?
[w'(I -h')P,f
+w'(l
- h')P,']
- w'(l - h l ) P { ]
- [w'(l - h')PJ - w'(l - h')P,']
with the normalized light intensity
r=z,.=
3 -4[ ( I + h )
-
Except for the P4 measurement, we see from inspection that no additional information can be obtained from generalized Stokes parameters observed in other directions (such as x or y ) with electron beam polarization vector perpendicular to the scattering plane. On the other hand, such additional measurements with photon detectors in the scattering plane can provide valuable consistency checks. As found previously, three more relative independent phases are needed to determine the scattering amplitudes uniquely. In analogy to the sodium case, we define
A-E
++- $+ +-- $-
(64b)
A"
4" - I)~
(64c)
A+=
(64a)
Inspection of Fig. 37, however, shows that only two of these are independent, since A + - A - = 6 ' - S i = 2( Y ' - Y ' ) (65) in analogy to Eq. (39). Therefore, it remains to fix the phase of the spin-flip amplitudes fJ ' relative to the nonflip amplitudes. As will become clear, a convenient choice for the remaining phase angle is
'
611
4,-
(66)
$0
A complete set of independent parameters is then given by ( u Uw ;',
L I T L, , f L h, ' , h l ; y ' , y ' , A + , A", 6
')
(67)
i.e., one absolute cross section, five relative sizes, and five relative phases. Information about the remaining three phase angles may be sought for in
58
N. Andersen and K Bartschal
experiments with in-plane spin polarization, a possibility that we will now explore. 2. Electron beam polarization in the scattering plane. No additional information is obtained with such an electron polarization if the photons are observed in the z direction (Andersen and Bartschat, 1994). However, in the directions x and y one obtains eight nontrivial components, namely IQ,"i" and Z Q P with = A X , xy, yx, and yy. To clarify the algebraic structure of the general expressions for these components, we introduce the following abbreviations: A,
=
a+Po
(68a)
A,
=
P+ao
(68b)
A,
=
a-po
(68c)
A,
=
p-ao = 4+- *o
(684
w,
st l cpo = s T L =
w, =
*+-
w, =
cp-- $bo = S T J
( 69a) -Ao-A+
(69b)
- 8'
(69c)
w4=$b--40=ssJ -Ao-Ab'-SJ
(69d)
where the phases are defined in Fig. 37. With this notation, we find
(;g) :(
+
+ + + A3 cos w 3 - A, cos
-Al sin w , A, sin o2 A, sin w 3 - A , sin w, - A , cos 0,+ A, cos w , A , cos wg - A, cos w4
=
(;g)=?(
3 A , cos W , - A, cos W , - A , sin w1 + A , sin 0,- A, sin 3
(%)=5(
w,
0 ,
+ A, sin w,
)
+
- A l cos w 1 - A, cos w 2 + A 3 cos 0, A, cos W , A , sin w1 + A , sin w , - A , sin w , - A , sin w,
(;g)=-( ++ 3 A , sin w , 2 A , cos w ,
A, sin w , + A , sin w 3 + A , sin w4 A, cos w , + A3 cos w3 + A, cos w,
)
)
(70a)
(70b)
)
(70c)
(704
Inspection shows that a measurement of the generalized Stokes parameters in the y and x directions with in-plane electron beam polarizations P,, and P, will provide the four relative magnitudes and the four relative phase angles defined in Eqs. (69). If analysis in the z direction with polarization P, has been completed, all relative magnitudes are already
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
59
known, and the four A parameters may serve as consistency checks. For example, the parameters L I and L; can be derived as
Additional information may be obtained from the w angles. They determine the relative phase angles within the amplitude triplets (f,!,, f J l ,fd ) and
w3 =
6'
=
w2 - w4 = 8 1 =
-2y' -2yJ
+ 7-r + 7-r
( 72a) (72b)
Provided that y 'I and y are already known, these relations allow us to extract mo additional angles only, e.g., S and A' A-. This is possible by photon observation in the y direction only, as can be seen from Eqs. (70a, b) after eliminating w g and w4 with (72). For a perfect experiment, a determination of any of the three angles A + , A-, or A' will suffice. This can only be achieved by using the STU parameters, since, in the photon analysis, the incoherent summation over the unobserved final electron spins destroys the phase information between the amplitude triplets that belong to final electron spins up and down, respectively. The STU analysis will be presented in Section III.C.2. Before we turn to the experimental situation, we summarize the preceding discussion. Figure 39 shows the three generalized Stokes parameter matrices that can be measured in each of the three orthogonal coordinate directions. In the z direction, apart from the Stokes parameters P , ,P 2 , P3 (the first column) corresponding to an unpolarized beam, new information is obtained only with electron spin polarization in the z direction. From this matrix, provided P4 is also measured, the dimensionless parameter set (60) can be evaluated. For this spin direction, the three numbers in the third column should be identical and may thus serve as consistency checks for the three polarizer settings. Given this information, observation in the y direction with polarization P, adds nothing new, except for the parameter P4 that can be obtained without spin polarization. The two other nonzero components in the first row can be derived from the data obtained
'
+
60
N. Andersen and K Bartschat
5"
P, ,P,,P,,P,, Consistency checks{
*
: Unpolarized beam
: New information : Can Zero be derived from
FIG. 39. Summary of the information that can be obtained from generalized Stokes parameter measurements. For details, see text.
already, but their measurement may serve as another consistency check. For the spin directions P, and P,,, however, new information is derived from the two last elements of the second column, sufficient to evaluate two additional angles. At this point, we have thus extracted all information that is available in the photon polarization pattern. Observation in a third direction in the
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
61
collision plane, such as x , is equivalent to observation in the y direction. Thus, it can either replace this or serve as consistency check, since all elements may be predicted from the results obtained in the z and y directions. We now turn to the experimental situation as of today. Results from an electron-polarized-photon coincidence experiment with a spin polarized incident electron beam have been reported by Goeke et al. (1989) and by Sohn and Hanne (1992) for electron impact excitation of the (6s6pI3P, in mercury. We have used their data to calculate the generalized Stokes parameters for the case of initial electron spin polarization perpendicular to the scattering plane. Experimental results for an incident electron energy of 8 eV are shown in Fig. 40 together with theoretical results based on a five state Breit-Pauli R-matrix (close-coupling) calculation (Scott et af., 1983). Unfortunately, the full set of generalized Stokes parameters for the z direction could only be obtained for three scattering angles (loo, 20", and 30'1, due to some missing data for the circular polarization parameters. Although Q;: = Qi; = Q;; according to Eq. (61a), the experimental data for Q;; and Q;; do not agree within the specified error bars, thus indicating some internal inconsistency between them. We have therefore averaged these results and plotted them as x in the field for Q;;. Further consistency checks may be performed by using data from the y direction to evaluate, for example, the third column of (Q,") in yet another way. Inspection of Eqs. (61) and (62) shows that
We have solved for Qf; and plotted the results as v in the third column of Fig. 40(a). The trend is correct, but the data are not entirely consistent. Finally, using the available experimental data, we can predict experimental results for generalized Stokes parameters that have not been measured to date, such as those that would be observed in the x direction, i.e., along the incident beam axis. Inspection of Eqs. (611463) yields Q?
=
1 + P, 1 + P4 1 - p 1 p 4 QP; + 1 - P I P 4Qf;
1 + P,
1 + P4 QY'
- 1- ~ l3
1
Q;,. ~
=
4
p , - p4 1 - p1p4
( 74a)
62
N. Andersen and K Bartschat Hg 6'50 -+ 6 3 P ~
a
1.0
0.0
-1.0 1.0
0.0
-1.0 1.0
0.0
-1.0
0
90
18010
90 lJ(deg)
180' 0
90
180
FIG. 40. Generalized Stokes parameters for electron impact excitation of Hg(6s6pI3P, at an incident electron energy of 8 eV. 0, published results (Goeke el al., 1989; Sohn and Hanne, 1992); 0 , unpublished data of M. Sohn and G. F. Hanne; A , derived from measured polarization and asymmetry data of Goeke et al. (1989) and Sohn and Hanne (1992); X, average of measured ( A ) QG and Q;; data; V , prediction based on properties of the generalized Stokes parameters (see text). The experimental results are compared with predictions from a five-state Breit-Pauli R-matrix calculation based on Scott er al. (1983).
where we have used that Qff = P, and QP; = -P4. The predictions are plotted in Fig. 40(c). The agreement between the experimental results in Fig. 40(a)-(c) and the theoretical predictions is satisfactory, bearing in mind the complexity of the collision problem, the level of detail in the comparison, and the remaining inconsistencies within the experimental data set. The agreement between experiment and theory is good for the in-plane polarization
63
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS Hg 61S0 -+ @PI
b
.o
1.o
1
0.0
0.0
180' 0
90
-1.0 180
180' 0
90
180
-1.0
0
90
180' 0
90 @(dell)
1.o
0.0
-1.0 0
90
180' 0
90 Weg)
1.0
"
"
'
,
QB 0.0
-1.0 1.0
v ,
,
,
,
-1.0
-'
'
n
0.0
,
,
QZ
(2%
0.0
1.0
QE
'
-1.0 1.0
0.0
'
'
'
'
-1.0
64
N. Andersen and K Bartschat
parameters Q;; and Q42y shown in Fig. 4O(c), except for a theoretical underestimation of Q{$ at small scattering angles. Extraction of the set (60) of spin dependent density matrix parameters (except for the cross section a,) from the generalized Stokes parameters is complicated by depolarization effects from hyperfine structure interaction in mercury isotopes with nonvanishing nuclear spin (Andersen et al., 1996). The two sets of results plotted in Fig. 41 were calculated from published data of Goeke et al. (1989) and of Sohn and Hanne (1992) ( X 1, and also from unpublished data of Sohn and Hanne (+) for scattering angles of lo", 20", and 30". The important difference between these data sets is that the latter set corresponds to a full determination of the generalized Stokes parameters shown in Figs. 40(a) and (b), i.e., all parameters that can be determined from the photon radiation pattern with electron polarization perpendicular to the scattering plane. On the other hand, the first set lacks data for Q;; and Q;:. While data for Q;; are desirable for a consistency check between the three elements in the third column of the (QF) matrix, data for Q;; are mandatory for performing the evaluation procedure outlined following Eqs. (61). In particular, the normalization p + = P+ = 1, ensuring full coherence in the individual spin channels, was found to be crucial for the numerical stability of the inversion. Only a complete set of generalized Stokes parameters can be tested for this normalization and, consequently, be renormalized to fulfill this fundamental condition exactly. If any elematrix is missing, the normalizament in the first two columns of the tion condition can be used without verification to calculate this element (except for the sign). Obviously, this is a much less satisfactory procedure and leads to significantly larger systematical errors, as seen in Fig. 41. This illuminates the importance of complete data sets. In the preceding case, the additional data for Q;; not only provide more information about the spin dependent orientation, but also stabilize the inversion procedure for all other parameters through built-in internal consistency checks. Interesting features can be seen in Fig. 41 after transformation of the generalized Stokes parameters into the spin dependent coherence parameters defined in Eqs. (58). The differential cross sections for the two spin directions are fairly similar at all angles. The coherence parameters, however, are vastly different for the two incident spin projections. Note, for example, that L; > 0, whereas LI < 0 at small scattering angles! While the positive value of LIT is in agreement with well-established propensity rules, the negative value of L f L might seem surprising. Both L; and L; are nonzero for forward scattering, and Lf (0") = -Lf (0") by symmetry requirements. The alignment angles y and y show no similarities, with the directions of the major axes often being
(QY)
FIG. 41. Spin dependent coherence parameters for electron impact excitation of H g ( 6 ~ 6 p ) ~ Pat, an incident electron energy of 8 eV. The experimental data were calculated from the generalized Stokes parameters presented in Fig. 40 (see text), X , prediction based on data of Goeke et al. (1989) and Sohn and Hanne (1992); +, prediction based on unpublished data of M. Sohn and G. F. Hanne. The experimental data are compared with the results from a five-state Breit-Pauli R-matrix calculation based on Scott et al. (1983).
N . Andersen and K Bartschai
66
I I
t
t
c9
0
FIG. 42. Illustration of angular momentum transfer by spin-up (a) and spin-down (b) electrons, as predicted for electron impact excitation of Hg(6s6pI3P, for small scattering angles at an incident electron energy of 8 eV.
perpendicular to each other (y:' (0") = - yJ: (0") # 0). There is a large difference between the height parameters h and h (which measure the relative importance of spin-flips) in this angular range, with h assuming a maximum value of 75% near a scattering angle of 40". This means that spin-flips are very likely for incoming spin-down electrons, but those spin-down electrons, whose spin is not flipped, tend to transfer a negative angular momentum to the atom. This situation is illustrated in Fig. 42, which is the spin-resolved analog of the Kohmoto-Fano graph shown in Fig. 2.
'
'
2. Generalized STU Parameters We now focus on the generalized STU parameters for the spin polarization of the scattered electron beam. They can be written as (Andersen et al., 1996):
s,
= w' -
s,
=
w'(1
w' -
(75a)
2 h ' ) - 2'(1
T, = w ' ( l - 2 h ' ) TY
1
+~
-
' ( 1 2h')
= - ( a+P+ cos A + + a - P a,
2/29
(75b) =
1 - 2h
cos A - - aoPo cos A'}
(75c) (75d)
67
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
T,
1
=
-{ a + P + cos A + + a-p-
cos A - + a o P ocos A'}
(75e)
UU
u =--{
1
YX
a+P+ sin A + + a-p- sin A - - a o P osin A')
(75f)
UU
1
u' Y = - {
a;
a+@+sin A + + a - P - sin A - + croposin A'}
(75g)
For brevity, we kept the products of the amplitude magnitudes in the T and U parameters. They can be expressed in terms of density matrix parameters by using a*@* - JwTwl(l
-
h t ) ( l - h l ) ( l f L T t ) ) ( l f L , + J ) ) (76a)
UU
Note that the three parameters S,, S,, and T, may be predicted from coherence parameters extracted in electron-photon coincidence experiments and vice versa; the parameter T, = 1 - 2h can already be obtained from the coincidence setup with an unpolarized electron beam (Andersen et al., 19951, whereas prediction of S,, requires an electron beam polarized perpendicular to the scattering plane. On the other hand, measurements of the set ( S , , S , , T ) allows for the evaluation of the three relative sizes h T , h l , and w = 1 - w l. The two missing relative sizes L: and L T 1 can only be obtained by circular polarization measurements. The parameters Ty, T,, Uyx,and Uxy contain new information, namely the three phase angles ( A + , A', A-). We see that linear combinations of the form Ty T, and Uyx+f Uxy allow for determination of the two complex numbers a+P+e" + a _P-eiA- and PoageiAo,thereby providing the crucial phase difference A' not obtainable with the electronpolarized-photon coincidence technique without simultaneous electron spin analysis in the exit channel. Hence, the combination of techniques described in this subsection and Section III.C.l constitutes the ultimate goal of a peflect scattering experiment for the important J = 0 J = 1 excitation problem in its most general form. Figure 43 shows results for the generalized STU parameters at 8 eV. Unfortunately, experimental data only exist for the S, parameter. This reflects the fact that their determination is a major experimental effort. That a full determination is, however, within the capabilities of present day
,,
+
--j
68
N. Andersen and K Bartschat Hg 63P,
O(del3)
FIG. 43. Differential cross section and generalized STU parameters for electron impact excitation of Hg 63P, at an incident energy of 8 eV. The experimental data for S, are measurements by Borgmann et al. (1987) ( A ) and predictions from electron-photon coincidence data by Goeke et al. (1989) (0).The theoretical curves are from a five-state Breit-Pauli R-matrix calculation based on Scott et al. (1983) (-1 and a first-order semirelativistic distorted-wave calculation of Bartschat and Madison (1988) (- - - -).
technology is illustrated in Fig. 44, which presents a complete set of generalized STU parameters, measured in Munster at 40 eV for Hg6'S, + 63P, excitation. The foregoing considerations, and the fundamental difference between the information extracted from generalized Stokes and STU parameters,
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
69
Hg 63P2
FIG. 44. Differential cross section and generalized STU parameters for electron impact excitation of Hg 6'P2 at an incident energy of 40 eV. The experimental data are from Borgmann et al. (1987) ( A and Diimmler et al. (1990) ( 0 )for S,, Miiller and Kessler (1994) ( 0 ) for S, and T,, and Klose (1995) (0)for T,, T y , Uxy,and yYx. The theoretical curves represent first order and semirelativistic (- - - -; Bartschat and Madison, 1988) and full relativistic ( ... ; Srivastava et al., 1992) distorted-wave calculations.
may be succinctly summarized in the way shown in Fig. 45. Figure 45(a) illustrates that generalized Stokes parameter analysis in the z direction with electron spin polarization P, perpendicular to the scattering plane determines all relative amplitude magnitudes, and the phase relationship between the f+l and f-l amplitudes-provided P4 is also known. Addi-
N. Andersen and K Bartschat
70
f
d
FIG.45. (a) This diagram shows which relative amplitude sizes and phases can be evaluated from a generalized Stokes parameter analysis in the z direction with electron polarization P, and from in-plane measurements with polarizations P, and Py, respectively. (b) This diagram shows which relative amplitude sizes and phases enter into Eqs. (75) for the 7 only depend on the relative sizes of generalized STU parameters. Note that S, , S,, and ' the amplitudes and can, therefore, be predicted from a generalized Stokes parameter analysis.
tional analysis in the y (or x ) direction with in-plane spin polarization Py or P, yields the four relative phases within the two triples (fj1,fjl,f,,') and (f,!,,f?,, ), respectively. None of the relative phases (A', A', A - ) between up/down amplitudes for the same magnetic sublevel enter. For the generalized STU parameters, Fig. 45(b) illustrates that S,, S,, and T, depend only on the relative sizes of the amplitudes, while T,, Ty,Uy,, and Uxy also depend on the relative phases (A+,A',A-) within the three amplitude pairs (f:,, fj,),(fJl,fll), and (f,,', 1. No information on the relative phase angles between these pairs can be extracted from generalized Stokes parameter analysis.
fk
fk
D. HEAVYTARGETS WITH SPIN: Ca 62S1/2 -+ 62P1/2,3/2
As pointed out in Section I.B, the number of independent scattering amplitudes increases to 24 for these transitions, 8 for 62S1,2 + P2PlI2and 16 for 62S,i2 62P3/2.Given the complexity of the cases discussed so far, with a maxlmum of six independent amplitudes, a complete experiment for these transitions seems impossible in the foreseeable future. -+
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
71
Nevertheless, e-Cs excitation experiments may serve as a benchmark for testing scattering theories. This is illustrated in Fig. 46, which shows theoretical results for the differential cross section and the spin polarization and asymmetry functions for electron impact excitation of the two fine-structure states at an incident electron energy of 2.04 eV. The results were calculated within an R-matrix formalism, including the lowest five or eight target states in the close-coupling expansion and treating relativistic effects either by the Dirac or the Breit-Pauli Hamiltonian. While the Breit-Pauli (Bartschat, 1993) and Dirac (Thumm et af., 1993) results for the differential cross sections are more in line with each other than with previous Breit-Pauli results of Scott et af. 1984) (who used different target wavefunctions), this is not the case for the scattering asymmetry function S, and, in particular, for the spin polarization function S, (see the second row of Fig. 46). All calculations predict significant deviations from the statistical branching ratio of a(’P ) = 2 ~ r ( ~ P , / ~ ) that one would obtain in the approximation of the puk%ine-structure effect” (Hanne, 1983) by applying the nonrelativistic approximation for the scattering amplitudes. Note, however, that the Breit-Pauli results are in qualitative agreement with the prediction S,, A ( 2 P 1 / 2=) -2S,, A(2P3/2) that one obtains in the same approximation. This relationship is strongly violated in the S, data from the Dirac calculation and makes an experimental test highly desirable. As mentioned in Section III.B, the function S, for such transitions has been measured for electron scattering from sodium atoms (Nickich et af., 1990); i.e., an experimental determination seems possible with present technology. ANGULAR MOMENTA:EXCITATION OF He E. HIGHER
1’s + 3’D
We finish Section I11 with a discussion of ‘S + ‘ D transitions in light atoms, the simplest example of an L = 2 excitation. This case has been discussed in the literature over the years. Although the four Stokes parameters ( P I ,P2, P3, P 4 ) contain all information obtainable from the P decay, it is well known that radiation pattern of the subsequent D they are not sufficient to define a complete experiment; instead, two possible solutions are obtained (Andersen et al., 1983). The purpose here is not to review the early discussions, but to put this case into the context of the previous sections by demonstrating how the systematics developed there may be used as a tool for a transparent analysis of this case as well. Equation (6) reveals three nonzero amplitudes, as shown with our shorthand notation in Fig. 47. Then, in addition to a differential cross section, two relative sizes and two relative phases need to be determined to achieve a complete experiment.
-
72
N. Andersen and K Bartschat 2P1/2
20,. -L
,
NO -0
,
-
- -
< 15;:
-- .--. .
--
*
2p3/2 - *
- .
- - -
&state BP 5-state BP Scott et 01.
40
,
-.---.
: y
.. -.-. **
,
C
o h - - . -60- * -- -120 - . - --180 - I
. -
Angle 0 (deg)
- . - - = . - - - - ,
.---
L
-
.60 * -
- -
&state BP 5-state BP Scott et 01. 5-stote Dirac
- .120 * -
- -
Angle 0 (deg)
.:
'
.180
-.-. ---*
-0.5 I -1.06- - -
- *
-
- - - -
- - -
60 120 Angle 0 (deg)
-180'
60
120
Angle 0 (deg)
180
l - - - - lo* l I
120
Angle 0 (deg)
180
SA
-0 -1
.oo
60
120
180
Angle 0 (deg)
FIG. 46. Differential cross section, spin polarization function S,, and asymmetry function + ( ~ P ) ~ P , / , , , /transitions , in cesium at an incident energy of 2.04 eV. The Breit-Pauli results are from Scott et al. (1984) and Bartschat (1993), and the Dirac results are from Thumm et al. (1993) based on Thumm and Norcross (1992). S, for electron impact excitation of the (6s)'S,/,
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
FIG. 47. The amplitudes f + 2 , fo, and f-* for electron impact S accounting for electron spin.
+
73
D excitation without
For excitation from the 1’s ground state, only states with positive reflection symmetry with respect to the scattering plane enter. It is useful to keep in mind their shapes, as illustrated in Fig. 48 for the d orbital. Contrary to the cases studied before, the excited charge cloud may have a “height,” i.e., a nonzero component along the z axis, without change of reflection symmetry. Since the spherical harmonic Y2, is proportional to 3 cos20 - 1, the relative height cannot be unity even in the extreme case of all probability going to the excited d, orbital. In fact, the density components for the corresponding state along the directions x , y , z have the ratios 1:1:4, so that the maximum value of h is 4/6 = 2/3. In most cases studied until now, the four standard Stokes parameters ( P I ,P 2 , P 3 , P4) are measured for the dipole emission from the D + P decay, without observation of the second step, the P + S cascade.
FIG. 48. The angular part IY,,(O, +)I2 of the charge cloud for the magnetic M sublevels of a D state quantized along the vertical z axis (from Andersen et al., 1988). The arrows indicate the sense or rotation of the electron charge cloud around the z axis.
74
N. Andersen and K Bartschat
D
FIG. 49. Branching ratios for decay of an atomic D state for photon emission along the quantization axis. S -+ D electron impact excitation will only populate the magnetic sublevels with M = 0, + 2.
The algebra for this case becomes very transparent if we start by considering this stepwise process in more detail. Figure 49 (partly taken from Fig. C.2(c) of Andersen et al., 1988) shows the complete decay pattern as mapped by the photons emitted in the +z direction, including photon helicities and (for D -+ P) relative transition probabilities. Note that, in this direction, no photons originate from decays to or from the po orbital of negative reflection symmetry. The emission pattern is thus orbital composed by two different channels, one going through the and one going through the p-l orbital. These two channels may be studied individually if observed in coincidence with the corresponding cascade photon, most conveniently done in the +z direction through a circular polarization analyzer. If only the D -+ P photons are observed without regard of the cascade, the total signal is the weighted, incoherent sum of the two channels. We shall now state this line of thought in algebra, by first evaluating the signals observed for the two channels individually. The corresponding parameters are labeled " " and " - " according to the angular momentum projection of the two cascade photons. The problem resembles much the situation encountered in Section III.C, where the total signal for unpolarized beams was obtained as an incoherent sum over the two final spin projections. We first parameterize the three scattering amplitudes:
+
with analogous notation to that of Section 1II.C. We note from Fig. 49 that the d, orbital only contributes to the " + " channel, the dd, orbital only
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
75
contributes to the " - " channel, and the contribution from the do orbital is equally shared between the two. We thus get 1
+ -2
6 + =ff; 6-= f f z ,
( 78a)
ff;
1
+ -2
(78b)
ff;
We obtain the Stokes parameters by simple substitution into Eqs. (58a-d) (42,401for in the following way: ( a + ,a _ ) (n,, ao/ 61,( 4 + ,4-) the "+" component, and ( p + , p - ) (ff,/6, 4, (IcI+, IcI-) (40,4-2)for the " - "component, respectively. The divisor 6 accounts for the relative branching ratio; cf. Fig. 49. We find +
-+
-+
where we have defined the relative phase angles (cf. Fig. 47) 62
=42
6-2 =
and the alignment angles
? * by
4 0
-
40
-
4-2
+
76
N. Andersen and K Bartschat
For the angular momentum-type parameters, we obtain
i;= -2P,f(l - A + )
=
2
- a;/6 a;
+ a;/2
E [ - 2/3,2] and Note the dynamical ranges of these quantities, namely i ; whereas L, E [ - 2,21. For completeness, we give the height and weight parameters as functions of L: :
i 1E [ - 2,2/3]
W'=
4-i+,+i; With the differential cross section a, given by
a,
=
a;
2 + a; + a-*
(84)
a complete experiment will be defined by the set
(a;;i:, ii;Y + , 7 - )
(85)
Interestingly, inspection of Fig. 49 shows that the analysis can equally well be made in the opposite order, i.e., Stokes vector analysis of the P + S coincidence with helicity-selected D + P photons, with identical outcome. The optimal choice of order may thus be based on experimental convenience, such as photon wavelengths. We obtain the corresponding formulas for the noncoincident case by evaluation of the sum of the two channels with proper weights
and
with
++++-= 1
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
P;=
77
Jm
Equations (88a,b) reproduce the results stated in Eqs. ((2.44-46) of Andersen et af. (1988). The height parameter is given as the usual function of P I and P4, i.e., ( 1 + Pl)(1 - P4) 4 - ( 1 - P , ) ( 1 - P4)
h=
since (see Appendix C of Andersen et al., 1988) P -
+ P , ) - h(3 + P,) ( 1 + P I ) + h ( l - P,) (1
-
The traditional parameterization of this case is the set ( %; L,, Poo; Y , Pl)
(91)
with the latter four evaluated from the four Stokes parameters ( P , , P 2 , P 3 , P4). Since the two sets of relative sizes, (i:,L;) and ( L , , pool, are equivalent, one set may be converted into the other through the equations (recall that h = + p w )
78
N. Andersen and R Bartschat
and
h=
(2 - i 3 ( 2
+ i;)
2(4 - iT + i;)
As mentioned, the set (85) is complete, whereas the set (91) is consistent with two possible wavefunctions. This is demonstrated in Fig. 50 (Andersen et al., 19831, which shows a multipole expansion of a D state electron density. The mathematical structure of Eq. (88a) is identical to that of Eq. (46) shown in Fig. 28, so the ambiguity lies in the sign of 1
7) =
,(i.++ T - )
- y
(94)
with 7 defined mod ~ / 4 .The two possible shapes of the charge cloud are thus mirror images of each other in a plane through the z axis in the direction y. For the wavefunction itself, time reversal needs to be invoked in addition (Andersen et al., 1983). The proper sign is revealed by a coincidence analysis with the cascade photons. Generalization of the preceding formalism to cases L > 2 is straightforward, as well as the derivation of formulas for the corresponding triple, quadruple, etc. coincidence measurements, or partially incoherent steps on the way. Since such developments are still challenges for the future, we refrain from further discussion.
+
+
FIG.50. Decomposition of the D state electron density in the scattering plane (from Andersen et al., 1983). Negative and positive contributions are indicated as hatched and crosshatched areas, respectively.
EXPERIMENTS IN IN ELECTRON-ATOM ELECTRON-ATOM COLLISIONS COLLISIONS EXPERIMENTS
79 79
Turning now now to to the the experimental experimental situation, situation, results results from from photon photon cascade cascade Turning coincidence analysis are presently not available in the literature, but work coincidence analysis are presently not available in the literature, but work is in progress (Wang and Williams, 1996). In the meantime, we shall is in progress (Wang and Williams, 1996). In the meantime, we shall demonstrate inversion of experimental data sets for ( P , , P 2 , P 3 , P,), using demonstrate inversion of experimental data sets for (P,, P 2 , P 3 , P,),using 28 and and resolving resolving the the again the the geometrical geometrical technique technique introduced introduced in in Fig. Fig. 28 again sign ambiguity for the pair ($+, $-)-and thereby for T--by comparison sign ambiguity for the pair (T+,?-)-and thereby for 7-by comparison 1's+ +3'D 3'D with theoretical theoretical predictions. predictions. For For this this purpose purpose we we select select the the He He1's with excitation process at 40 eV, for which experimental results with good excitation process at 40 eV, for which experimental results with good quality are are available available (McLaughlin (McLaughlin etet al., al., 1994; 1994; Mikosza Mikosza etet al., al.,1994). 1994).The The quality theory selected selected for for comparison comparison isis again again the the CCC CCCcalculation calculation of of Fursa Fursa and and theory Bray (1995). (1995). Figure Figure 51 51shows shows results results for for the the four four Stokes Stokesparameters, parameters, which which Bray
FIG.51. Stokes parameters ( P , , P , , P 3 , P4) for electron impact excitation of He 1's + 3'D transition at an incident electron energy of 40 eV. The experimental data of Mikosza et al. (1994) and McLaughlin er al. (1994) are compared with CCC calculations of Fursa and Bray (1995).
80
N. Andersen and I2 Bartschat
are used to produce the set (91) displayed in Fig. 52. Before discussing the inversion procedure, we point out that the observed radiation is nearly incoherent P = 0) at scattering angles of 100" and 120", despite a completely coherent excitation in this case. Figure 53 displays the results of our inversion procedure, starting with the unique determination of i', and i; from the available L , and poo data. As in the case of Hg63P, excitation, the individual i: do not have to vanish for scattering in the forward and backward directions. Also note that the two individual channels correspond to nearly complete circular polarization for scattering angles between 80" and 120", whereas the average angular momentum transfer is almost zero. The nearly circular nature of the radiation in the individual channels also explains the difficulty in solving for the alignment angles j ~ *in this angular range. Nevertheless, Fig. 53 shows how a reliable theory can be used to distinguish between the true and the ghost solution. In fact, the remaining problems in the ( P I ,y ) * (T', T-) inversion can, in part, be traced back to the relatively large magnitude of PI (cf. Fig. 52). (Mathematically, this results in one side of the triangle in Fig. 28 being longer than the sum of the other two, in which case we set JI = x = 0 and thereby get j ~ + =?-= y.). Based on the good agreement between theory and experiment for the parameter set (i:,i; , y ) and the fact that the theoretical data are internally consistent (even if they do not describe nature perfectly), one might suspect that the experimental data for PI are slightly overestimated. If this were indeed true, the agreement between experiment and theory might further improve in Fig. 52.
IV. Conclusions The series of examples presented in this chapter shows that the field of quantum mechanically complete experiments has developed to considerable maturity within the area of electron-atom collisions since the time of formulation of a "perfect scattering experiment" more than 25 years ago. Today, a collection of simple elastic and inelastic scattering processes in fundamental systems serve as benchmarks for current state of the art scattering theory. Several more systems are close to completion, perhaps in many cases closer than was actually realized at the time of data accumulation. We found that the analysis of incomplete data sets can often be completed with the assistance of inversion procedures and .guidance from theoretical predictions. The examples presented in this chapter span the range from the simplest case, completely described by a single amplitude,
81
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS 10
u
(1 0 - ~ ~ ~ ~ 2 / ~ ~ )
He 3'D
40eV
1 .o
1
0.0 0.1 -1
.o - ccc
,2.O0'
0.01 1.0
,
,
1
,
.
I
,
,
1
,
,
1
'
'
I
'
30
' .
" "
60
90
'
"
120
'
"
'
150
'
180
90
'
:PI
"
.
60
0
.
1.0
,
I
,
,
1
,
,
I
.
.
I
. .
I
.
1 .o
'
: Po0 0.8 -
-
0.6 -
- 0.6
0.4
-
-
0.8
0.4
0
Scottering Angle (deg)
30
60
90
120
150
180
Scattering Angle (deg)
FIG. 52. Differential cross section a, and coherence parameters ( L , , P,, y , poo, P ) for electron impact excitation of He 1's + 3'D transition at an incident electron energy of 40 eV. The experimental data of Mikosza ef al. (1994) and McLaughlin et al. (1994) are compared with CCC calculations of Fursa and Bray (1995).
82
N. Andersen and K Bartschat
He 3 ’ 0
40eV
2.0
1 .o
0.0
-1
.o
-2.0
0
30
60
90
120
150
180
0
30
60
90
120
150
180
90 60 30
0 -30
- 60 -90
Scattering Angle (deg)
FIG.53. Sublevel resolved coherence parameters (i:,y’ ) and (i;, T-) for electron impact excitation of He 1’s + 3’D transition at an incident electron energy of 40 eV. The experimental points have been calculated from the data of Mikosza et al. (1994) (“down” triangles) and McLaughlin et al. (1994) (‘‘up” triangles). They are compared with results obtained from CCC scattering amplitudes of Fursa and Bray (1995). The “true” (?+, y - ) pair (full symbols) was guessed by using the theoretical results as a guide.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS
83
to a present maximum of six scattering amplitudes. All these cases were conveniently discussed within a common framework by parameterizing the change of the scattered electron polarization by means of generalized STU parameters and, for excitation, by a complete evaluation of generalized Stokes parameters for full characterization of the radiation pattern. Systematic mapping of the various dimensionless parameters provides us with a much closer insight into the detailed collision dynamics than the single differential cross section parameter would permit. On the other hand, one should not underestimate the need for accurate differential cross section measurements. It is the only absolute observable, and a truly complete experiment cannot be achieved with relative measurements alone. As is evident from our analysis, several of the existing cases deserve further experimental refinement in terms of a larger angular range or smaller error bars. It is also clear, however, that for cases of significantly greater complexity than those presented here, future progress will require further development of sophisticated coincidence setups and the ability to handle very long data accumulation times under stable conditions. For higher angular momenta and targets with spin, the ideal of completeness may quickly become an unrealistic goal within the foreseeable future. More progress is expected, in particular, from scattering and (de)excitation studies involving optically prepared states. We hope that the systematic framework presented here will serve as a helpful guide for the future exploration of this fascinating field.
Acknowledgments We thank Igor Bray and Al Stauffer for communicating data in electronic form, and John Broad for producing some of the figures. This work was supported, in part, by the Danish Natural Science Research Council (NA) and the United States National Science Foundation (KB).
References Andersen, N., and Bartschat, K. (1993). Comments At. Mol. Phys. 29, 157. Andersen, N., and Bartschat, J. (1994). J. Phys. B 27, 3189; Corrigeudurn (1996). ibid 29, 1149. Andersen, N., and Hertel, I. V. (1986). Comments At. Mol. Phys. 19, 1. Andersen, N., Andersen, T., Cocke, C. L., and Pedersen, E. H. (1979). J . Phys. B 12, 2541. Andersen, N., Andersen, T., Dahler, J. S. Nielsen, S. E., Nienhuis, G., and Refsgaard, K. (1983). J . Phys. B 16, 817. Andersen, N., Hertel, I. V., and Kleinpoppen, H. (1984). J. Phys. B 17, L901.
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Andersen, N., Gallagher, J. W., and Hertel, I. V. (1988). Phys. Rep. 165, 1. Andersen, N., Bartschat, K., and Hanne, G. F. (1995). J . Phys. B 28, L29. Andersen, N., Bartschat, K., Broad, J. T., and Hertel, I. V. (1996). Phys. Rep. (in press). Balashov, V. V., and Grum-Grzhimailo, A. N. (1991). Z. Phys. D: A4 Mol. Clusters 23, 127. Bartschat, K. (1989). Phys. Rep. 180, 1. Bartschat, K. (1993). J. Phys. B 26, 3595. Bartschat, K., and Madison, D. H. (1988). J. Phys. B 21, 2621. Bederson, B. (1969a). Comments At. Mol. Phys. 1, 41. Bederson, B. (1969b). Comments At. Mol. Phys. 1, 65. Beijers, J. P., Madison, D. H., van Eck, J., and Heideman, H. G. M (1987). J . Phys. B 20, 167. Berger, O., and Kessler, J. (1986). J . Phys. B 19, 3539. Blum, K. (1981). “Density Matrix Theory and Applications.” Plenum, New York. Borgmann, H., Goeke, J., Hanne, G. F., Kessler, J., and Wolcke, A. (1987). J . Phys. B 20, 1619. Bray, I. (1992). Phys. Reu. Left. 69, 1908. Bray, I. (1994). Phys. Reu. A 49, 1066. Brunger, M. J., and Buckman, S. J., Newman, D. S., and Alle, D. T. (1991). J. Phys. B 24, 1435. Brunger, M. J. Buckman, S. J., Allen, L. J., McCarthy, I. E., and Ratnavelu K. (1992). J. Phys. B 25, 1823. Burke, P. G., and Mitchell, J. F. B. (1974). J. Phys. B 7, 214. Callaway, J., and McDowell, M. R. C (1983). Comments At. Mol. Phys. 13, 19. Cartwright, D. C., Csanak, G., Trajmar, S., and Register, D. F. (1992). Phys. Reu. A 45, 1602. Donnelly, B. P., Neill, P. A., and Crowe, A. (1988). J . Phys. B 21, W21. Diimmler, M., Bartsch, M., GeeSmaM, H., Hanne, G. F., and Kessler, J. (19901, J . Phys. B 23, 3407. Eminyan, M., MacAdam, K. B., Slevin, J., Standage, M. C., and Kleinpoppen, H. (1974). J . Phys. B 7, 1519. Eminyan, M., MacAdam, K. B., Slevin, J., Standage, M. C., and Kleinpoppen, H. (1975). J. Phys. B 8, 2058. Farago, P. S. (1974). J . Phys. B 7, L28. Fursa, D. V., and Bray, I. (1995). Phys. Reu. A 52, 1279. Gehenn, W., and Reichert, E. (1972). Z. Phys. 254, 28. Goeke, J., Hanne, G. F., and Kessler, J. (1989). J. Phys. B 22, 1075. Hall, R. I., Joyez, G., Mazeau, J., Reinhardt, J., and Scherrnann, C. (1973). J. Phys. (Orsay, Fr.) 34, 827. Hanne, G. F. (1983). Phys. Rep. 95, 95. Hasenburg, K., Bartschat, K., McEachran, R. P., and Stauffer, A. D. (1987). J. Phys. B 20, 5165. Hegemann, T., Oberste-Vorth, M., Vogts, R., and Hanne, G. F. (1991). Phys. Reu. Lett. 66, 2968. Hegemann, T., Schroll, S., and Hanne, G. F. (1993). J. Phys. B 26, 4607. Hertel, I. V., and Stoll, W. (1977). Adu. At. Mol. Phys. 13, 113. Hertel, I. V., Kelley, M. H., and McClelland, J. J. (1987). Z. Phys. D: At.. Mol. Clusters 6, 163. Hollywood, M. T., Crowe, A., and Williams, J. F. (1979). J. Phys. B 12, 819. Holtkamp, G., Jost, K., Peitzmann, F. J., and Kessler, J. (1987). J . Phys. B 20, 4543. Kessler, J. (1985). “Polarized Electrons,” Springer-Verlag, Berlin and New York. Kessler, J. (1991). Adu. At. Mol. Phys. 27, 81. Khakoo, M. A., Becker, K., Forand, J. L., and McConkey, J. W. (1986). J. Phys. B 19, L209. Klose, M. (1995). Ph.D. Thesis, Universitat Munster, FRG.
EXPERIMENTS IN ELECTRON-ATOM COLLISIONS Kohmoto, M., and Fano, U. (1981). J. Phys. B 14, L447. Leuer, B., Baum, G., Grau, L., Niemeyer, R., Raith, W., and Tondera, M. (1995). Z. Phys. D : At. Mol. Clusters 33, 39. Madison, D. H., Bartschat, K., and McEachran, R. P. (1992). J . Phys. B 25, 5199. Massey, H. S. W. (1983). I n “Fundamental Processes in Energetic Atomic Collisions” (H. Lutz, J. S. Briggs, and H. Kleinpoppen, eds.) Plenum, New York. McAdams, R., Hollywood, M. T., Crowe, A,, and Williams, J. F., (1980). J . Phys. B 13, 3691. McClelland, J. J., Kelley, M. H., and Celotta, R. J. (1985). Phys. Reu. Lett. 55, 688. McClelland, J. J., Kelley, M. H., and Celotta, R. J. (1989). Phys. Rev. A 40, 2321. McClelland J. J. Lorentz, S. R., Scholten, R. E., Kelley, M. H., and Celotta, R. J. (1992). Phys. Reu. A 46, 6079. McEachran, R. P., and Stauffer, A. D. (1986). J. Phys. B 19, 3523. McLaughlin, D. T., Donnelly, B. P., and Crowe, A. (1994). Z. Phys. D : A t . Mol. Clusters 29, 259. Mikosza, A. G., Hippler, R., Wang, J. B., and Williams, J. F. (1994). Z . Phys. D: At. Mol. Clusters 30, 129. Miiller, H., and Kessler, J. (1994). J. Phys. B 27, 5933; corrigendum: ibid. B 28, 911 (1995). Nickich, V., Hegemann, T., Bartsch, M., and Hanne, G. F. (1990). Z. Phys. D: A t . Mol. Clusters 16, 261. Raeker, A,, Blum, K., and Bartschat, K. (1993). J . Phys. B 26, 1491. Scholten, R. E., Lorentz, S. R., McClelland, J. J., Kelley, M. H., and Celotta, R. J. (1991). J. Phys. B 24, Lf553. Schumacher, C. R., and Bethe, H. A. (1961). Phys. Reu. 121, 1534. Scott, N. S., Burke, P. G., and Bartschat, K. (1983). J. Phys. B 16, L361. Scott, N. S., Bartschat, K., Burke, P. G., Nagy, O., and Eissner, W. B. (1984). J . Phys. B 17, 3775. Sohn, M., and Hanne, G. F. (1992). J. Phys. B 25, 4627. Srivastava, R., Zuo, T., McEachran, R. P., and Stauffer, A. D. (1992). J . Phys. E 25, 2409. Srivastava, S. K., and VuskoviE, L. (1980). J. Phys. B 13, 2633. Standage, M. C., and Kleinpoppen, H. (1976). Phys. Rev. Lett. 36, 577. Taylor, J. R. (1987). “Scattering Theory,” Krieger Publishing, Malabar. Teubner, P. J. O., and Scholten, R. E. (1992). 1. Phys. E 25, L301. Thumm, U., and Norcross, D. W. (1992). Phys. Reu. A 45, 6349. Thumm, U., and Bartschat, K., and Norcross, D. W. (1993). J . Phys. E 26, 1587. Wang, J. B., and Williams, J. F. (1996). Aust. J . Phys. 49, 335. Williams, J. F. (1986). Aust. J . Phys. 39, 621. Wilmers, M. (1972). Ph.D. Thesis, Universitat Mainz, FRG. Zhou, H.-L., Whitten, B. L., Trail, W. K., Morrison, M. A., MacAdam, K., Bartschat, K., and Norcross, D. W. (1995). Phys. Reu. A 52, 1152.
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ADVANCES IN ATOMIC. MOLECULAR. AND OPTICAL PHYSICS. VOL. 36
STIMULATED RAYLEIGH RESONANCES A N D RECOILINDUCED EFFECTS J.-Y COURTOIS Institut d’Optique Thioorique el Appliqu6e Orsay. France
G. GRWBERG Laboratoire Kastler-Brossel Dkpartement de Physique de I’Ecole Normale Supkneure Pans. France
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Stimulated Rayleigh Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Stationary Two-Level Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Other Examples of Stimulated Rayleigh Resonances in Atomic Physics D . Examples in Molecular Physics .......................... E . Stimulated Rayleigh Resonances in Solid State Materials . . . . . . . . . . . . I11. Recoil-Induced Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . The Recoil-Induced Resonance as a Stimulated Rayleigh Resonance . . . . . B. Experimental Observation of Recoil-Induced Resonances C . The Recoil-Induced Resonance as Raman Processes between Different Energy-Momentum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Atomic Bunching in the Transient Regime E . Coherent Atomic Recoil Laser (CARL) IV . Other Recoil-Induced Effects in Atomic and Molecular Physics . . . . . . . . . . A . Recoil Effects in Saturated Absorption Spectroscopy B. Recoil Doublet of Optical Ramsey Fringes . . . . . . . . . . . . . . . . . . . . C . The Ramsey-Bordt Matter Wave Interferometer D . The Experiment of Kasevich and Chu ...................... E . Recent Advances in Atom Interferometry Based on the Photon Recoil . . . . F. Recoil-Induced Inversionless Lasing of Cold Atoms . . . . . . . . . . . . . . . G. Atomic Recoil and Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . H . Other Related Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Copyright Q 1996 by Academic Press. Inc. All rights of reproduction in any form resewed . ISBN 0-12-003836-6
88
.I.-Y. Courtois and G. Grynberg
I. Introduction Light scattering occurs as a consequence of fluctuations in the optical properties of a material medium. It is indeed well known that a completeZy homogeneous material can scatter light only in the forward direction (see, for example, Fabelinskii, 1968). A light scattering process is said to be spontaneous if the fluctuations that cause the light scattering are excited by thermal or by quantum mechanical zero-point effects. By contrast, a light scattering process is said to be stimulated if the fluctuations are induced by the presence of a light field. One of the simplest technique for investigating stimulated scattering mechanisms is pump-probe transmission spectroscopy, which is illustrated in Fig. l(a). This figure shows a material system interacting with two externally applied laser beams, namely an intense pump beam of frequency w and a weak probe beam of frequency wp = w 6. Under the most general circumstances, the probe transmission spectrum obtained by recording the intensity of the probe beam after transmission through the scattering medium as a function of the pump-probe frequency detuning 6 has the form shown in Fig l(b), in which Raman, Brillouin, and Rayleigh features are present. Raman scattering typically results from the interaction of light with the vibrational modes of the molecules constituting the material system. Brillouin scattering is the scattering of light from sound waves, that is, from propagating pressure (hence density) waves. Rayleigh scattering is the scattering of light from any other nonpropagating modulation of material observables. It is also known as quasi-elastic scattering because there is almost no frequency shift between the incident and scattered beams. The reason that these scattering processes can lead to stimulated amplification or absorption of the probe beam (see Fig. lb) is that the interference of the pump and probe fields contains a frequency component at the difference frequency S. The observables of the material system can be driven by this light interference, which therefore acts as a source for the generation of macroscopic observable modulations. Thus, the beating of the pump wave with the modulation of the material observable tends to reinforce the probe wave, whereas the beating of the pump wave with the probe field tends to reinforce the observable modulation. Under proper circumstances, the positive or negative feedback described by these two interactions can lead to exponential growth or decay, respectively, of the amplitude of the probe wave. More precisely, the interaction of the pump field with the observable modulation yields a macroscopic polarization of the material system having the same characteristics (frequency, polarization, wavevector) as the probe beam. However, the phase of this polariza-
+
STIMULATED RAYLEIGH RESONANCES
89
scattering medium
T
F 0P
=0+6
Brillouin
anti-Stokes I
0
6 FIG. 1. Investigation of stimulated scattering by pump-probe spectroscopy. (a) A pump beam of frequency w and a probe beam of frequency wp are sent into a nonlinear medium. Because of the excitation of an observable in the nonlinear medium, their propagations are coupled. The probe beam intensity is recorded versus S = up - w at the exit of the nonlinear medium. (b) Typical probe beam transmission spectrum. The stimulated Rayleigh resonance is centered at 6 = 0 and usually displays a dispersive shape. The lateral resonances correspond to stimulated Raman and Brillouin scattering.
tion generally differs from that of the probe field because the modulation of the material observable is phase-shifted with respect to the pump-probe excitation wave (typically, this phase shift results from the finite response time of the material system). In other words, the polarization of the medium exhibits a nonzero component being 7~/2 phase-shifted with respect to the probe beam. The work of the probe field onto this component leads to the modification of its intensity as displayed by the probe transmission spectrum. The fact that the energy transfer between the pump and probe beams via the material system exhibits resonances, or 2 component of the material polarequivalently that the ~ / phase-shifted
90
J.-Y Courtois and G. Grynberg
ization presents resonant enhancements, is therefore related to resonant variations of the phase shift in the medium response and/or of the modulation amplitude of the material observable. Stimulated Rayleigh scattering was discovered in dense molecular media at the end of the 1960s (Mash et al., 1965; Bloembergen and Lallemand, 1966; Chiao et al., 1966; Cho et al., 1967; Fabelinskii, 1968). It was then identified in stationary two-level systems (Mollow, 1972) and in optical crystals (Giinter and Huignard, 1988), where it is better known as the photorefractive effect. During the 1980s, stimulated Rayleigh processes induced by optical pumping were discovered in various dilute atomic systems (Grynberg et al., 1990). Experimental interest in this subject has been renewed because of the discovery of many original stimulated Rayleigh processes in laser-cooled atomic vapors (Courtois and Grynberg, 1992, 1993; Lounis et al., 1992; Hemmerich et al., 1994; Courtois et al., 1994). One of the most striking examples of these new mechanisms are the so-called “recoil-induced resonances,” which were first predicted and interpreted by Guo et al. (1992) in terms of stimulated Raman processes between energy-momentum states differing because of the momentum exchange between the pump and the probe fields during photon redistribution processes. Soon afterward, these resonances were observed experimentally and reinterpreted by Courtois et al. (1994) in terms of stimulated Rayleigh scattering involving atomic spatial bunching. Beside its intrinsic interest in the framework of nonlinear optics, the Rayleigh-Raman duality of recoil-induced resonances provides a clear illustration of the possible ambiguity in the identification of a photon recoil-induced effect. Such an ambiguity is often found in atomic and molecular physics because many effects that are easily explained in terms of momentum exchange between atoms and photons can be alternatively interpreted without involving explicitly the photon recoil. The organization of this chapter is as follows. We present in Section I1 the basic ideas about stimulated Rayleigh scattering by considering more particularly the situation where it arises from a relaxation process going on in the material system, and we describe a few experimental observations made in atomic and molecular physics. We then consider the case of nonstationary two-level atoms, and we derive the shape and characteristics of the recoil-induced resonances (Section 111). In particular, we show that these resonances can be interpreted either as originating from a stimulated Rayleigh effect or as a stimulated Raman phenomena between atomic energy-momentum states having different momenta. Finally, to make a clear distinction between the physical phenomena that pertain directly to recoil-induced processes (i.e., that actually permit the measurement of the photon recoil) and those for which the introduction of the
STIMULATED RAYLEIGH RESONANCES
91
recoil constitutes a mere physical convenience, we review in Section IV some indisputable manifestations of the photon recoil in atomic and molecular physics.
11. Stimulated Rayleigh Resonances This section is devoted to the presentation of the basic physical ideas about stimulated Rayleigh resonances (Section 1I.A) and of some experimental illustrations (Sections 1I.B-El. For the sake of clarity in the presentation, we restrict ourselves to those manifestations of stimulated Rayleigh scattering that are observable on pump-probe transmission spectra. However, the concepts presented hereafter can be readily extended for describing resonant variations of the nonlinear refractive index, or fourwave mixing (and phase conjugation) spectra. A. INTRODUCTION As previously mentioned, stimulated Rayleigh resonances displayed by pump-probe transmission spectra originate from the diffraction of the pump wave onto nonpropagating observable modulations that are phaseshifted with respect to the pump-probe interference pattern. It appears from this scheme that the interpretation of a stimulated Rayleigh resonance consists essentially in identifying (1) the modulated observable V , (2) its driving mechanism, (3) the physical origin of its phase shift with respect to the excitation, (4)the diffraction mechanism of the pump beam onto the observable modulation. In order to make a satisfactory compromise between generality in the theory and clarity in the presentation, we will only consider in this section the most commonly encountered situation, where the phase shift of the observable modulation originates from relaxation mechanisms taking place in the material system. This restriction will be relaxed in Section 111. We consider a set of atoms or molecules interacting with two incident beams E (pump) and Ep (probe) with respective frequencies w and wp =
0
+ 6:
E = R e [ E e x p { - i ( w t - kar)}] Ep = Re[ Ep exp{ --i( wpt For the sake of simplicity, the fields are presently considered as scalar quantities. Effects associated with the vectorial character (polarization) of the field will be discussed later on. We also assume that each beam
92
J.-Y Courtois and G. Grynberg
separately does not undergo linear absorption during propagation, although this assumption can be easily relaxed in the case of a weakly absorbing medium. The quantity 77 = Ep/Z,equal to the ratio of the probe to the pump fields, is assumed to be small (1771 << 1) and can therefore be used as a relevant parameter for perturbation expansions. The superposition of E and Ep yields an integerencepattem (Fig. 2a) that moves in the sample with the phase velocity u = S/lk - k,l. In a nonlinear material, this intensity modulation excites some observables of the medium and thus leads to the formation of a material grating that follows the motion of the interference pattern. A phase shift between the material grating and the interference pattern generally occurs because once excited, the observables of the medium do not decay instantaneously (Fig. 2b). The field at the exit of the medium in the probe direction is the sum of two contributions: (1) the transmitted probe beam and (2) the component of the pump beam diffracted onto the material grating. Note
Material grating
Interference pattern
FIG. 2. Excitation of a material grating in a pump-probe experiment. (a) Interference pattern created by E and Ep. This pattern moves with a velocity u proportional to 8 . (b) Because of the interference, the material is not uniformly excited. A modulation of the material observable is thus created that follows the light pattern with a delay because of the material response time.
93
STIMULATED RAYLEIGH RESONANCES
that because of the grating velocity, the frequency of this diffracted beam is Doppler-shifted and is identical to that ( u p of ) the transmitted probe. It is the interference between these two contributions that gives rise to the stimulated Rayleigh effect. In particular, when the interference is constructive, gain can be found. We aim now at presenting a more quantitative treatment of the gain mechanism. We assume that the polarization P of the medium is related to the total field at the same point by the relation P
= E,
Re(
x[8exp{ -i(
wt - k
*
r)} + gpeXP{ -i(
- kp’
.>}I}
(2)
In a nonlinear medium, the electrical susceptibility x is a function of one or several atomic observables V that are assumed to depend on the field intensity.’ For the present discussion, we assume that there is just one relevant observable, the time evolution of which is characterized by a singZe relaxation rate y. In other words, V evolves according to dV
- + y ( V - V,,) dt
=
0
(3)
where V,, is the equilibrium value of V for a time independent field intensity. Because the influence of the probe beam reduces to a small spatiotemporal modulation of the total field intensity, it is possible to deduce the probe-induced modification of the medium from the intensity variation of the steady state observable V,, in the vicinity of its value V,, obtained in the absence of the probe beam (the intensity then merely coincides with the pump intensity Z = l8I2/2). If the medium had an instantaneous response time to the pump-probe excitation, the effect of the probe beam would be readily obtained through the expansion of V,, up to first order in the small parameter Q = gP/g:
-i[ 6 t
-
(kp - k) * r]})
(4)
Equation (4) somehow characterizes the “instructions” given by the probe to the material system to adapt to its presence. How the medium actually complies with these instructions, though, depends on its response time.
’ In the general case, these observables may also depend on the field polarization.
94
J.-Y. Courtois and G. Grynberg
Using Eq. (3), we thus find that the actual steady state value for V in the presence of the probe reads: Y
y-iS
-
exp{ -i[ S t - (k, - k) r]))
(5)
The functional dependence of the macroscopic polarization on the observable V being given, it is possible to derive from (5) the expression of the susceptibility x of the medium up to first order in 7.This yields
Y
dV dl
y-iS
exp{-i[ S t
-
(k, - k)
- 1-11}
and hence the total polarization P is
P
= E,
Re Eo
( x ( V,) [ 8exp{ - i ( w t - k . r)} + 8, exp( -i(
+-Re 2
d x dV0 ( d V dl
[
w,t -
k,
*
r))]}
Y y-iS
- - 2YP8*-
xexp{-i[St-(k,-k).r]]
1
+ c . c . Pexp{-i(wf-k.r)}}
(7)
Because we are interested in the probe propagation through the material system, we only consider the component P, of the polarization that radiates a field in the probe direction. By setting
P,
=
Re[ 9, exp{ -i( wPt - k, * r)}]
with k, = ~ ~ < w , / and c )x ‘ the real part of x(V,), and by using the slowly varying envelope approximation, one obtains, as the propagation equation for the probe field,
For a nonabsorbing medium, i.e., a medium for which the imaginary part x(V,) is equal to 0, Eqs. (7) and (8) yield
x” of
STIMULATED RAYLEIGH RESONANCES
95
with g=
(
1 d-x 2dV
YS
OP
+i
dV d l Z c ~ ~ ) [ - y 2 + S 2 y 2 + S 2
The real part of g, which describes the modification (absorption or amplification) of the probe amplitude,* is thus a dispersive function of 6 for a nonabsorbing medium? In particular, the asymptotic value of g for S / y + + m is 0. This is because the translational motion of the light interference pattern becomes so fast that the medium is almost uniformly excited and the contrast of the material grating becomes vanishingly small. Equation (10) shows that the gain also vanishes as 6 goes toward 0. This is because in that case the pump-probe modulation is so slow that the medium can adapt quasi-instantaneously to the excitation wave; hence, the interference pattern and the material grating exactly coincide. The extra rr/2 phase shift undergone by the pump beam during diffraction on the grating leads to a transmitted probe being rr/2 phase-shifted with respect to the diffracted pump. Therefore, no constructive nor destructive interference takes place between these two waves. This shows that to observe gain for S = 0, it is necessary that another physical mechanism induces a spontaneous phase shift between the stationary light pattern and the material grating. An example of such a behavior can be found in photorefractive crystals such as BaTiO, (Giinter and Huignard, 1988). Finally, although we do not discuss the stimulated Rayleigh structures displayed by four-wave mixing spectra, we note that in such a situation, both the 0 and the rr/2 phase-shifted components of the material polarization contribute to the signal. Therefore, the physical contents of the Rayleigh resonances in probe transmission and in four-wave mixing generation are slightly different. B. STATIONARY TWO-LEVEL ATOMS The identification of a stimulated Rayleigh resonance on the probe transmission spectrum of an ensemble of stationary two-level atoms can be
* The gain coefficient for the intensity is 2 Re(g).
+
In the case of a weakly absorbing medium ( x'' << 1 x ' ) , the effective gain gCs includes the absorption of the probe originating from the imaginary part of x(V,,). For that situation, the gain coefficient is equal to
where g is given by Eq. (10).
96
.I.-Y. Courtois and G. Grynberg
traced back to a theoretical paper of Mollow (1972) and to the experimental work of Wu et al. (19771, who used a sodium atomic beam driven by a cw dye laser.4 In this system, which was widely studied in the literature because of its simplicity, stimulated Rayleigh scattering originates from the relaxation process associated with spontaneous emission. Later on, it was shown that the magnitude of the stimulated Rayleigh resonance can be significantly increased in the case where the relaxation mainly arises from dephasing collisions with a buffer gas (Grynberg et al., 1986; Gruneisen et al., 1988). In particular, the maximum value of the gain can be sufficiently large to achieve laser emission, as shown by GrandclCment et al. (19871, who used a cell filled with sodium atoms and a buffer gas interacting with a nearly resonant cw laser beam. Strictly speaking, in the case of two-level atoms (ground state a, excited state b ) the electrical susceptibility x does not depend on a single intensity dependent observable, so that the approach of Section 1I.A cannot be readily applied. In fact, it turns out that two time scales are present in the problem. The separation between these two time scales is particularly clear in the dressed-atom approach (Grynberg and Berman, 1989). One finds that the dressed-atom eigenstates follow adiabatically the time dependent field, whereas the population between the dressed levels has a response is the lifetime of the excited state b. time on the order of r-’,where rP1 The calculation of the gain coefficient is usually performed using the optical Bloch equation for the density matrix (Mollow, 1972; Agarwal and Nayak, 1984; Berman et al., 1985). For a nonresonant excitation (IAl 3 I‘, where A = w - wo is the detuning between the laser ( w ) and the atomic resonance ( w o ) frequencies) and in the limit where the resonance Rabi frequency R, is small compared with [A1 (R, = -D&Y/fi,where D is the matrix element of the electric dipole moment between the states a and 61, the stimulated Rayleigh gain for the case of collisional relaxation (Gxynberg et al., 1986) is found to be
In fact, none of these papers is devoted to the stimulated Rayleigh resonance. They both consider probe transmission through a set of atoms interacting with an intense pump beam. The probe spectrum consists of three resonances, the stimulated Rayleigh resonance being the weakest one. Because it is located at the center of the spectrum, the stimulated Rayleigh resonance in two-level atoms is often called the “central resonance of the Mollow absorption spectrum.”
STIMULATED RAYLEIGH RESONANCES
97
where .Nis the atomic density and ycoll is the relaxation rate of the optical coherence due to dephasing collisions. In the case of rudiutiue reluxution, the Rayleigh gain reads Re(g)
=
op D 2 fl; -N- --
6
2c eo?iA 4A4 S 2 +
r2 r
In both situations, the stimulated Rayleigh resonance has a dispersive lineshape (centered around S = O), with a peak-to-peak distance equal to 2 r . The only difference between the cases of radiative and collisional relaxations is the magnitude of the resonance, which is larger in the collisional case because the stimulated Rayleigh resonance appears to the third order in the small parameter fl,/IAI, whereas it appears to the fifth order in the case of radiative relaxation. To find the actual gain geA, one should add to Eq. (11) or (12) the linear absorption coefficient a”,which is equal to
where ycoll should be set to 0 in the case of radiative relaxation. The comparison between Eq. (13) and Eqs. (11) and (12) shows that the Rayleigh gain can overcome the linear losses provided that fl:/lAl r is sufficiently large. It is important to note that this parameter can be larger than 1 even in the perturbative limit5 where a , e IAI because a, can be much larger than r. One can also remark that the probe amplification always occurs for a frequency wp lying between o and w o . One unique and important interest of two-level atoms is that they are sufficiently simple to permit a derivation of the stimulated Rayleigh resonance in terms of elementary processes (GIynberg and CohenTannoudji, 1993), hence to shed light on the microscopic origin of stimulated Rayleigh scattering. It is important to emphasize that the quantum description of stimulated Rayleigh scattering is much more difficult than that of stimulated Raman or Brillouin scattering. In Raman scattering, for example, the relationship between spontaneous (Fig. 3a) and stimulated (Fig. 3b) Raman processes is relatively easy to understand because both processes are described by similar Feynman diagrams. This is because the initial ( a ) and final ( b ) atomic states of the scattering process are different. One thus finds that a stimulated process with absorption of a pump photon A nonpertubative formula for the gain can also be derived. The exact calculation confirms
the predictions found with the pertubative approach in the case of two-level atoms (Pinard and Grynberg, 1988).
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J.-Y. Courtois and G. Grynberg
la) FIG.3. Spontaneous and stimulated Raman processes. (a) Spontaneous Raman effect. The elementary process consists of the absorption of a pump photon followed by spontaneous emission of a photon with transition of the atomic system from level a to level b. (b) Quantum picture of the stimulated Raman process. The atomic system goes from level a toward level b, which is less populated, by absorbing a photon in the pump beam and emitting a photon in the probe beam.
w and stimulated emission of a probe photon wp takes place when the populations of the initial and final atomic states in the process are different, and that the stimulated Raman gain is proportional to this population difference.6 By contrast, in spontaneous Rayleigh scattering (Fig. 4) the initial and final atomic states of the process coincide, and there is no simple way of transforming the mechanism described by Fig. 4 into a stimulated process. The probability that an atom will absorb a pump photon and emit a stimulated photon in the probe mode is indeed exactly balanced by the probability of the reverse process where an atom absorbs a probe photon and emits a stimulated photon in the pump mode. In fact, the description A stimulated Raman gain between two levels having the same population can be triggered by a relaxation process (Bogdan et al., 1981). The physical origin of this uncommon Raman gain is very similar to that of the stimulated Rayleigh gain.
STIMULATED RAYLEIGH RESONANCES
99
\ \ \
%
FIG.4. Spontaneous Rayleigh scattering. The atom absorbs a photon in the incident beam and emits a photon having the same frequency.
of stimulated Rayleigh scattering in terms of Feynman diagrams is usually much more complicated than what appears on Fig. 4. It is actually so complicated that, to date, this approach has been developed only in the case of stationary two-level atoms. We summarize now the results obtained by Grynberg and CohenTannoudji (1993) in the limit of large frequency detunings from resonance (lAl >> I') and for radiative relaxation. The principle of the method is to identify all scattering processes that contribute to the absorption or the amplification of the probe and to calculate their contributions using scattering theory (Cohen-Tannoudji et al., 1992). The first contribution to probe absorption is the usual Rayleigh scattering process, which is described by the diagram shown in Fig. 4. This process is obviously independent of the presence of the pump and just constitutes a background term. The other contributions to probe absorption involve pump photons. In the limit where a , is small compared with JAl, the most obvious nonlinear absorption process is shown in Fig. 5(a). It corresponds to the nearly resonant excitation of the upper level through a hyper-Raman process with absorption of two photons, one of the pump beam and one of the probe beam, and spontaneous emission of two photons, one of frequency w 1 and one of frequency w 2 . These latter frequencies are not well defined, but their sum o ,+ w 2 must be equal to w + cop because of energy conservation. In order to characterize the probe absorption, one should include another diagram, which is as important as the one of Fig. %a). This other diagram, shown in Fig. 5(b), appears at a higher order in the pump field but exhibits a supplementary resonant intermediate step. In fact, the quantitative evaluation of the contribution of these two diagrams shows that their ratio is of the order of flf/IAI r. As mentioned, this quantity
100
J.-Y Courtois and G. GIynbeG
\ \
I
\
V
FIG. 5. Diagrams describing the main nonlinear contributions to probe absorption in the pertubative limit. The diagrams (a) and (b) have the same initial and final states and can thus interfere. The diagram (b) appears to a higher order in the pump field but has a supplementary resonant intermediate step.
can exceed unity even though R , -=K IAI, because R , can be much larger than r. Furthermore, note that these two diagrams have the same initiul andfinal states for both the atom and the field (in the two diagrams, there is just one photon that disappears in the pump and the probe field). As a result, the absorption cross section is not merely the sum of the cross sections associated with each diagram separately; it also involves an interference term. This interference term gives a positive or a negative contribution to the total cross section, depending on the sign of 6.
STIMULATED RAYLEIGH RESONANCES
101
FIG. 6. Main diagram describing probe amplification.
Probe amplification is much easier to describe because it involves a single diagram (Fig. 6). This diagram involves the absorption of three photons of the pump beam and spontaneous emission of two photons of frequencies a;and a;. This amplification diagram is actually quite similar to the second diagram (Fig. 5b) for probe absorption, and in effect the amplification cross section is just opposite to the absorption cross section associated with Fig. 5(b) alone. In summary, the total cross section is the sum of the absorption cross sections for the elastic Rayleigh scattering (Fig. 4) and for the hyper-Raman process of Fig. 5(a), plus the interference term arising from the interference between the diagrams of Figs. 5(a) and 5(b) (because the amplification cross section is canceled by the absorption cross section associated with Fig. 5b). For a given value of the frequency detuning A, the two first terms do not vary much when 6 is scanned over a range of several r. In contrast, the interference term gives a dispersive lineshape with a peak-to-peak distance equal to 2 r. Because the interference term can be larger than the two other terms, probe amplification is possible. It thus turns out that probe amplification does not arise from a population inversion, but from a destructive interference between two absolption processes.’ As expected, the final result for the probe amplification cross section is identical to the one found using the density matrix approach, but the diagrammatic method gives better insight into the physical mechanisms leading to the stimulated Rayleigh resonance. In particular, it shows that the dispersive shape arises from an interference among several scattering amplitudes and that the underlying process ’The principle of the gain process is analogous to the one usually considered in lasing without population inversion (Harris, 1989; Zibrov ef al., 1995).
102
J.-Y Courtois and G. Grynberg
involves a real excitation of the upper level, which gives its width to the Rayleigh resonance.
c. OTHER IN
EXAMPLES OF STIMULATED RAYLEIGH RESONANCES ATOMICPHYSICS
The Zeeman degeneracy of the atomic ground state leads to stimulated Rayleigh processes of a new kind. The relevant observables for the associated Rayleigh resonances are the magnetization and the alignment of the ground state, the relaxation times of which are usually much longer than that of the excited state due to spontaneous emission. Therefore, the corresponding stimulated Rayleigh resonances are much narrower. Such resonances were first observed by Grynberg et al. (1990) using a cw dye laser slightly detuned from the D, transition of sodium. More precisely, the experimental setup consisted of pump and probe beams having linear orthogonal polarizations, which were mixed in the sodium cell and split behind using two Glan prisms (Fig. 7). When the probe frequency was scanned, a Rayleigh resonance having the usual dispersive shape was observed on the probe transmission (Fig. 8a). The peak-to-peak distance was of the order of 0.6 MHz, which is typical for a ground state observable relaxation rate. The symmetry between the probe (Fig. 8a) and pump (Fig. 8b) absorptions was checked by recording simultaneously both transmitted intensities. The two curves of Fig. 8 display opposite shapes, which proves that the photons are always transferred from the high frequency field to the low frequency field in the conditions of the experiment.8 In fact, this redistributionof photons between the two fields has a close analogy with the forces involved in sub-Doppler radiatwe cooling (Dalibard and CohenTannoudji, 1989; Ungar et al., 1989). If instead of using two traveling waves propagating along the same direction, one considers atoms moving in the standing wave formed by two counterpropagating beams having the same frequency, the problem in the atomic rest frame is, because of the Doppler shift, identical to the problem of a stationary atom interacting with two copropagating beams having different frequencies. As a result of the redistribution of photons shown in Fig. 8, there is thus a transfer of momentum from the counterpropagating beams to the atom that permits the damping of the atomic velocity. The direction of the transfer of photons changes with the sign of the frequency detuning A. For the same value of A, opposite directions are found near the D, and D, resonances. The fact that the presence of one beam helps the transmission of the other is an example of electromagnetically induced transparency. In Fig. 8, the two fields are nearly degenerate. When a static magnetic field is applied, the same effect is observed when the frequency difference 6 between the two fields is of the order of the Zeeman shift between the Zeeman sublevels of the ground state (Vallet et al., 1992).
STIMULATED RAYLEIGH RESONANCES
103
scattering
medium /I
wP eY@ FIG. 7. Scheme of the experimental setup used to observe the stimulated Rayleigh resonance associated with optical pumping. The pump and the probe beams have crossed linear polarizations. Their sum produces a resultant field whose polarization is time modulated. The same type of setup can be used in stimulated Rayleigh wing experiments (see Section 1I.D).
The physical interpretation of the stimulated Rayleigh resonance of Fig. 8 is particularly simple in the case of atoms having an angular momentum J = 1/2 in the ground state, because the only relevant atomic observable is then the magnetization. The superposition of the probe and the pump fields of Fig. 7 yields a resultant field having a time dependent polarization.
FIG.8. Probe (a) and pump (b) transmission through a sodium cell versus 6 (frequency difference between probe and pump). The two beams have orthogonal polarizations and are tuned on the blue side of the D, transition ( A = 4.5 GHz). The two beams have an intensity of 100 mW, and their common transverse dimension corresponds to a waist of 0.4 mm. The dashed line of (a) corresponds to a fitted dispersive curve. The comparison between (a) and (b) demonstrates that amplification of the probe is associated with extra absorption of the pump (and vice versa). There is thus a redistribution of photons between the pump and the probe beams.
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J.-Y. Courtois and G. GIynberg
When the ' u circular component is predominant over the u- one, the atoms are preferentially optically pumped into the m = + magnetic sublevel. Symmetrically, at a later time when the u- component is predominant over the ' u one, the atoms are preferentially optically pumped into the m = - sublevel. The atomic magnetization thus exhibits a periodic oscillation. However, this magnetization is phase-shifted with respect to the optical pumping excitation because of the relaxation time of the ground state. For example, when the resultant field is linearly polarized, the atomic magnetization still differs from zero. Because a nonzero magnetization induces a Faraday rotation of a linearly polarized field inside the cell, one Cartesian component of the field increases at the expense of the other. This is the origin of the redistribution of energy between the probe and the pump fields (Grynberg et al., 1990). Stimulated Rayleigh spectroscopy is also extremely useful to gain insight into the dynamics of laser-cooled atoms. Apart from the recoil-induced resonances that will be studied in Section 111, they were used to measure the fiction coeflcient of cesium atoms in a one-dimensional u+-umolasses (Lounis et al., 1992; Courtois and Grynberg, 1993) and for characterizing the dynamics of the atomic center of mass in optical lattices (Courtois and Grynberg, 1992; Hemmerich et al., 1994). The former experiment (Fig. 9) was performed with a probe beam of frequency wp = w + 6 having the same polarization and direction of propagation as one molasses beam (of frequency 0). As a result from the interference between these two beams, the radiation pressure is time-modulated and the atoms are submitted to a time-modulated force.g Here again, a phase shift occurs between the applied force and the atomic motion because of the friction force experienced by the atoms in the molasses. As a result, a stimulated Rayleigh resonance is observed on the probe transmission spectrum, whose width gives a direct quantitative access to the friction coefficient." The value of the friction coefficient deduced from the experiment is in excellent agreement with the theoretical predictions (Lounis et al., 1992).
4
4
In fact, there are two contributions to this force. The first one is related to the modulation of the total intensity because of the interference between the probe and the pump beams. The second one results from an optical pumping modification by the probe, which in turn leads to a modification of the radiation pressure exerted by the pump beams on the atoms (Courtois and Grynberg, 1993). 10 The friction coefficient is not easily measured by any other methods. For example, temperature measurements are sensitive to the ratio between the momentum diffusion coefficient and the friction coefficient, but neither of these two coefficients can be deduced independently.
105
STIMULATED RAYLEIGH RESONANCES
-
I \
Ravleigh
1 Raman/
1
I
I
I
I
I
-400
-200
0
200
400
I
6 (Wz) FIG.9. Stimulated Rayleigh resonance in a one-dimensional u + - 6 molasses. (a> Scheme of the experiment. Cesium atoms are first trapped and cooled in a magneto-optical trap (Raab et al., 1987; Walker et al., 1990). The six molasses beams and the inhomogeneous magnetic field are then switched off, and counterpropagating beams of frequency w and respective polarizations u+ and u- are switched on. The dynamics of atoms in this one-dimensional u+-u- molasses is probed using a weak beam of frequency w + 6,which has the same circular polarization as the nearly copropagating molasses beam. The molasses beam and the probe beam intensities are respectively 10 and 0.1 mW/cmz. They are tuned on the red side of the 6S,,,(F = 4) -+ 6P3,,(F = 5 ) transition ( A = -10 MHz). The angle between the molasses beam and the probe beam is 5". (b) Probe transmission spectrum. The spectrum consists of three lines. The lateral resonances correspond to Raman transitions between differently light-shifted Zeeman sublevels. The central resonance is a stimulated Rayleigh resonance, which allows one to measure the damping of the atomic velocity in the molasses.
In the case of optical lattices (Castin and Dalibard, 1991; Verkerk et al., 1992; Jessen et al., 1992; Hemmerich and Hansch, 1993; Grynberg et al., 19931, atoms are trapped in micrometer-sized potential wells originating from the light shifts (Cohen-Tannoudji, 1962) of the atomic ground state sublevels in an optical field created by the interference of several light beams of frequency w (Fig. 10). One-, two-, and three-dimensional optical
106
J.-Y Courtois and G. Grynberg
-
0-
-50
-
STIMULATED RAYLEIGH RESONANCES
107
lattices were studied experimentally. Using a probe beam of frequency w,, = w 6, it is possible to excite some relaxation modes of an atom in the lattice. Certain relaxation modes involve the transfer of the atom from one well to another; other modes are associated with transitions between different quantized vibrational levels within the same well. These different relaxation modes have different relaxation rates, and they are excited with a weight that depends on the probe direction and polarization (Courtois and Grynberg, 1992). In the same optical lattice, the width and the shape of the stimulated Rayleigh resonance can therefore exhibit huge variations when the probe direction or polarization is changed.
+
D. EXAMPLES IN MOLECULAR PHYSICS It is traditional to split the stimulated Rayleigh processes occurring in molecular media into two groups. The first group is associated with observables having a very short relaxation time, in the range 10-13-10-10 sec, and corresponds to the so-called stimulated Rayleigh wing scattering (this term originates from the very broad wings of the associated resonances). The second group concerns media having relaxation times of the order of lo-' sec, which lead to much narrower Rayleigh resonances. Simulated Rayleigh wing scattering arises from the interaction of an intense pump beam with anisotropic molecules such as CS, or C,H,, which tend to align along the direction of the laser electric field (Mash el al., 1965; Bloembergen and Lallemand, 1966; Chiao et al., 1966; Cho et al., 1967). In the case of a cigar-like molecule such as CS,, this is because the polarizability a,, of the molecule along its cylindrical symmetry axis differs from the polarizability aI in the perpendicular directions. When the incident field E is linearly polarized, the molecules tend to align along the FIG. 10. Stimulated Rayleigh resonance in a one-dimensional optical lattice. (a) The interaction of atoms with counterpropagating beams having crossed linear polarizations leads to space-modulated light shifts that act as a periodic external potential for the atomic external degrees of freedom. The shape of the potential together with the position in energy of the quantized levels for the atomic motion are shown. (b) Probe transmission in a one-dimensional optical lattice. The probe beam has a linear polarization orthogonal to that of the copropagating lattice beam, and the angle between the beam directions is 3". (c) Probe transmission spectrum. The lattice and probe beam intensities are respectively 5 and 0.05 mW/cmz. They are tuned on the red side of the 6S,,,(F = 4) + 6P,,,(F = 5 ) transition (A = -35 MHz). The lateral resonances correspond to Raman transitions between adjacent vibrational levels in a potential well. The central stimulated Rayleigh resonance is the superposition of several dispersive curves having different widths.
108
J.-Y Courtois and G. Gtynberg
electric field polarization in order to minimize the dipole interaction energy. However, this tendency is partly balanced by thermal effects that spread the orientations of the molecular axis. As a result, the larger the energy q,(aI, - a,),!? compared with k,T, the larger the number of molecules whose axes coincide with the electric field. Thus, the medium exhibits a nonlinear susceptibility because the induced polarization depends on the average molecular orientation, which is a function of ~ ~- 6,)E2/k,T. ( a ~ A~stimulated Rayleigh process is therefore possible. It involves the relaxation time y-' of the molecular orientation, which depends on the temperature and on the viscosity of the medium. The stimulated Rayleigh wing effect can be observed with the experimental setup of Fig. 7. The combination of the probe beam linearly polarized along Oy and having frequency up= w S and the pump beam of frequency w linearly polarized along Ox leads to a total electric field whose direction exhibits a time dependent oscillation at the frequency 6. When 161 -ZK y , the molecules follow adiabatically the oscillation of the electric field. When IS1 x=-y the oscillation is too fast, and the molecules just experience an averaged field. The transition between these two regimes occurs for IS I = y , which is the width expected for this stimulated Rayleigh wing resonance. In molecular media the stimulated Rayleigh effect, strictly speaking, is generally associated with a periodic variation of the total field intensity such as the one considered in Section 1I.A. Depending on the experimental conditions, the molecules tend to accumulate at points of maximum or minimum intensity. In fact, the potential energy of a molecule having a scalar polarizability a in an electric field E is - $ a E 2 . If a is positive, the molecules tend to be attracted near points where the intensity is maximum (electrostrictiue eflect). In the presence of an interference pattern due to a probe and a pump beam, the molecular density is thus spatially modulated. Here again, the depth of the density modulation results from an equilibrium between the electrostrictive attraction and thermal effects. The relaxation process for the density fluctuations is associated with a spatial For the diffusion mechanism characterized by a diffusion coefficient 0,. field configuration of Fig. 2, the damping rate is of the order of D,(k - k ,)' and is therefore a decreasing function of the spatial period of the interference pattern. When light absorption in the medium is nonnegligible, local heating takes place near points of maximum intensity. In this case, which corresponds to what is known as themodifision, the density tends to decrease at points of maximum intensity. When a > 0, the shape of the stimulated Rayleigh line permits comparison of the relative magnitude of the electrostrictive and of the thermodiffusive contributions, because these two
+
STIMULATED RAYLEIGH RESONANCES
109
effects lead to dispersive lineshapes having opposite signs. Generally, however, the presence of impurities absorbing the laser radiation always favors the thermodiffusive effect."
E.
STIMULATED RAYLEIGH
RESONANCES IN SOLID
STATE MATERIALS
Stimulated Rayleigh resonances are also used in optically active solid state materials to get information on the relaxation times. The more detailed studies were performed in photorefractive crystals (Giinter and Huignard, 1988), where the stimulated Rayleigh resonance is often called two-wave mixing or two-beam coupling. These materials are characterized by a large Rayleigh gain that is used in photorefractive oscillators. A peculiarity of the stimulated Rayleigh effect in photorefractive materials is that a large gain can be found when the probe and the pump have the same frequency (this is, for example, the case in BaTiO,). The shape of the Rayleigh resonance then differs from a dispersion because a spontaneous phase shift takes place between the light interference pattern considered in Section 1I.A and the spatial modulation of the refractive index occurring through the electrooptic effect from the light-induced modulation of charge carriers in the medium. This provides an example of stimulated Rayleigh scattering that does not involve directly a relaxation process in the medium. Finally, it may be noticed that because of the slow response time of photorefractive materials, the Rayleigh resonance is generally narrow.
111. Recoil-Induced Resonances In this section, we consider more particularly the recently discovered recoil-induced resonances, which constitute one of the simplest examples of nonlinear optical response due to translational degrees of freedom in atomic physics. These resonances are of particular interest because they provide an elementary illustration of a stimulated Rayleigh mechanism involving no relaxation process. Moreover, their unusual characteristics will allow us to point out some possible ambiguities in the interpretation of stimulated scattering mechanisms. In particular, we will show that recoilinduced resonances can be identically interpreted in terms of stimulated Rayleigh (Courtois et al., 1994) or Raman (Guo et af., 1992) scattering, depending on whether the atomic motion is described in position or momentum space, respectively. This outstanding duality, which was previ" For
more details about this effect, see, for example, Boyd (1992, Chapters 8 and 9).
110
1-Y Courtois and G. Glynberg
ously encountered in free electron laser physics (Madey, 1971; Hopf et al., 1976; Kroll and MacMullin, 1978) and evoked by Bloembergen (1967) in the study of stimulated Rayleigh wing scattering, is typical of systems displaying a continuous energy spectrum. In the original theoretical proposal by Guo et al. (19921, recoil-induced resonances were interpreted in terms of stimulated Raman scattering between energy-momentum states differing because of the momentum exchange between the pump and probe fields during photon redistribution processes. In other words, they were attributed to a recoil-induced effect, whence their name. This theoretical prediction was soon followed by the experimental observation of recoil-induced resonances by Courtois et al. (1994), who proposed an alternative interpretation of the resonances in terms of stimulated Rayleigh scattering associated with an atomic spatial bunching process. As noticed by Courtois et a f . (19941, there is an obvious analogy between this physical description of the recoil-induced resonances and the physics of the free electron laser (Elias et al., 1976). In both cases, the particles (atoms or electrons) are indeed bunched in the potential created by the superposition of an intense and a weak field, and the diffraction of the intense field on the particle density modulation gives rise to an amplification of the weak field. This analogy can be extended from the so-called weak gain regime (which is considered in Sections 1II.A-D) to the situation of large gain, where a self-consistent description of the fields and the particles becomes necessary. The laser that can be achieved with atoms in this high regime is called the coherent atomic recoil laser (CARL) and has been investigated by Bonifacio et al. (1994) and by Bonifacio and De Salvo (1995a, b). A brief description of the CARL will be presented in Section 1II.E. A. THERECOIL-INDUCED RESONANCE AS RAYLEIGHRESONANCE
A
STIMULATED
We first consider the interpretation of recoil-induced resonances in terms of stimulated Rayleigh scattering because it follows straightforwardly from the considerations of Section 11, and we postpone the discussion of the experiment and of the Raman approach to Sections 1II.B and III.C, respectively. We consider again the situation of Section ILB, where an ensemble of two-level atoms interacts with a pump beam of frequency w and a probe beam of frequency wp = w 8.” Contrary to Section II.B,
+
12
Recoil-induced resonances can also be observed with more complex atomic transitions involving Zeeman sublevel degeneracy. A theoretical investigation of the recoil-induced resonances in the case of a u+-u- configuration for the pump field has been studied by Guo and Berman (1993).
STIMULATED RAYLEIGH RESONANCES
111
however, the atoms are no longer assumed to be stationary. As we shall see, atomic motion is actually of fundamental importance in the physics of recoil-induced resonances. We further assume that the fields are fardetuned from resonance (IAl >> r) and that their intensity is such that the atomic transition is unsaturated (a, << IAl, with the same notations as in Section 1I.B). The superposition of the pump and probe yields a time and spatial modulation of the light intensity, hence of the ground state light shift (Cohen-Tannoudji, 1962), that is analogous to the electrostrictive effect discussed in Section 1I.D. This light shift modulation acts as an external potential for the atomic translational degrees of freedom and results into a time and spatial variation of the atomic density. As will be shown in the following, this density modulation exhibits a nonzero rr/2 phase-shifted component with respect to the light shift modulation, which leads through diffraction of the pump beam to a stimulated Rayleigh resonance similar to the one considered in Section 1I.A. To describe more quantitatively the atomic dynamics, we introduce the phase space density distribution of the atoms f(r, p, t). Furthermore, because of the weakness of the probe field we perform an expansion of f up to first order in E,, (we use q = ZFp/iF as a perturbation parameter):
f ( r ,P,t ) = f o ( r ,PY t ) + f d r , PI t )
(14)
In Eq. (14), the zeroth order term f o stands for the phase space distribution in the absence of the probe field, and it will be assumed to correspond to a Maxwell-Boltzmann di~tribution'~ for an ensemble of atoms of mass M , spatial densityM, and temperature T : fo(r7
P)
=MfM
( Px)fM ( Py ) f M ( Pz)
Note that the phase space distribution f,, is solution of the unperturbed Liouville equation:
which involves no force since the pump field-induced light shift is spatially uniform (the pump beam is assumed to be a traveling plane wave). In other words, the system dynamics in the absence of the probe field, which 13
Because we have assumed that the frequency detuning A is much larger than legitimate to neglect the influence of radiation pressure.
r, it
is
112
.I.-Y. Courtois and G. Grynberg
is completely characterized by the Liouville operator Po, merely reduces to free propagation at the constant uelocity p/M. The influence of the probe field on the atomic medium is described by the first order term of Eq. (141, f, ,which can be derived from the Liouville equation using perturbation theory. By considering the terms of order 1 in Ep in the Liouville equation, one thus obtains
The physical significance of Eq. (17) is as follows: Except for the noncrucial presence of the phenomenological relaxation rate y, which was introduced for the sake of rigorousness in order to account for possible effects such as diffusion of atoms through the pump-probe interaction region,14 the left hand side term of Eq. (17) describes the dynamical evolution of the phase space density fi in the absence of the probe beam (it only involves the unperturbed Liouville operator Po). On the other hand, the term on the right hand side of Eq. (17)is a source term accounting for the interaction of the probe beam with the unperturbed atomic system (it depends only on the unperturbed phase space density distribution f,). More precisely, it describes the modification of the atomic momentum distribution under the influence of the dipole force associated with the modulation of the ground state light shift U,: Ul
D2
=
2fiA Re(8*ZP exp( - i [ ( k - k p ) * r + 611))
( 18)
which is of order 1 in Ep because it results from the interference between the probe and the pump beams. In summary, the effect of the probe beam is to modify the atomic momentum distribution with respect to (15) in a space dependent way, the atoms being accelerated or decelerated depending on the local slope of the optical potential associated with the light shifts. Once this modification has taken place, the dynamical evolution of the atoms reduces to free propagation at a constant velocity, although it has been modified with respect to steady state. In particular, the oscillation motion of the atoms in the optical potential is not accounted for by Eq. (171,and it would appear at higher perturbation orders. Before turning to the quantitative description of the system, let us first try and identify the basic physical mechanism involved in spatial bunching, and more particularly in the counterintuitive occurrence of a v/2 phasel4
The introduction of y is not absolutely necessary. Indeed, y disappears in the final result (see Eq. (22)).
STIMULATED RAYLEIGH RESONANCES
113
shifted component of the atomic density modulation with respect to the optical potential. To make clear that the relaxation mechanism associated with the phenomenological damping rate y is not the crucial mechanism for the phase shift, we will assume all along our qualitative discussion that y = 0. We consider the situation where the pump-probe frequency detuning 6 is fixed to a constant value. As previously discussed, the interference between the pump and probe waves results into a time and spatially modulated optical potential moving in the medium at the constant phase velocity u = S/lk - k,l in a direction that will be denoted by z for the sake of definiteness (we will not be interested in the other directions in the following discussion). It is therefore convenient to perform a change of Galilean frame so that, in the new referential, the spatially modulated optical potential appears as stationary. In the new frame, the unperturbed atomic momentum distribution fo appears as shifted with respect to the Maxwell-Boltzmann distribution by an amount of -Mu. In particular, in the case where u is positive, the atoms appear as undergoing an average translational motion characterized by a negative velocity. Furthermore, because of the properties of the Maxwell-Boltzmann distribution, the velocity distribution in the new frame is no longer symmetric with respect to the zero velocity. Therefore, there are more atoms having a small negative velocity than atoms having a small positive velocity. We now consider the influence of the optical potential on the atomic momentum distribution. More precisely, we restrict ourselves to the steady state modifications of the atomic density distribution. As a consequence, we will not be interested in the nonzero velocity groups (in the moving frame) because their contributions to the density modulation vanishes with time (this argument will be made clearer in Section III.D, devoted to the transient regime of recoil-induced resonances). Because atoms tend to be accelerated toward the bottom of the optical potential wells, the zero velocity group will be populated from the velocity groups having an infinitesimally small positive velocity at points where the potential slope is positive, whereas it will be populated from the velocity groups having an infinitesimally small negative velocity at locations where the potential slope is negati~e.'~ However, as previously mentioned, the unperturbed populations of the infinitesimally positive and negative velocity groups are not equal in the moving frame. As a consequence, in the case u > 0, where the infinitesimally negative velocity group is dominant, the zero velocity 15
The reason that only infinitesimally small velocities are involved is clear from the source term of Eq. (17), which shows that the modification of the zero velocity group is proportional to the deriuatiue of the unperturbed momentum distribution evaluated in p
=
0.
114
J.-Y. Courtois and G. Giynberg
group will be more populated at points of negative potential slope than at symmetrical points of positive potential slope (the conclusion being reversed in the opposite case, u < 0). In other words, a spatial modulation of the atomic density appears in the system, with a nonzero 7r/2 phase-shifted component with respect to the optical potential. Furthermore, it is possible 2 component to infer from this simple argument that the ~ / phase-shifted will cancel for u = 0, i.e., 6 = 0 (no spatial asymmetry in the modifications of the zero velocity group takes place in this situation), hence a zero probe gain, and that it will be maximum for pump-probe frequency detunings for which the imbalance associated with phase velocities u = & between the infinitesimally positive and negative velocity groups is maximum, hence a maximum probe gain or absorption. In summary, returning to the initial frame, the occurrence of spatial bunching of the atoms results from a probe-induced modification in the number of atoms having a velocity equal to the phase velocity of the light shift modification. The occurrence of a 7r/2 phase-shifted component of the spatial density with respect to the optical potential, i.e., of an asymmetry in the number of atoms accumulating on either side of the optical potential wells, is a purely dynamical effect that results from the imbalance between the number of atoms moving slightly faster or slower than the light shift modulation. We now return to the quantitative description of recoil-induced resonances, which will allow us to derive the precise shape of the resonance. The steady state solution of Eq. (17) is
When r, 0, << IAI, the macroscopic polarization of the ensemble of twolevel atoms is equal to
X
R e ( g e x p i [ k . r - o t ] +gPexpi[k;r
-
o p r ] ) (20)
115
STIMULATED RAYLEIGH RESONANCES
Using Eqs. (8), (9), (19), and (201, we find the stimulated Rayleigh gain g : o f2: g=2--
P M y - i S - i(k - k,) (k - k,)
D2
2c A2 4.5,kBT
*
-
p fo(r7P)
M
1
(21)
Under most circumstances, the width of the Maxwell-Boltzmann distribution is much broader than My/lk - k,l. Hence, in the calculation of the probe gain, Re(g), it is possible to replace y / { y 2 + [ S + (k - k,) p/M12} by a quantity proportional to the Dirac delta function
-
SD[
8 + (k - k,) *P/M]
As a result, Eq. (21) simplifies and yields
where we have introduced u2 = kBT/M and k, = 2k sin(O/2), 0 being the angle between k and k,. Thus, the lineshape of the recoil-induced resonance appears as the derivative of a Gaussian. In the case where the velocity distribution differs from a Maxwell-Boltzmann distribution, this result can be generalized, and one finds that the lineshape corresponds to the derivative of the velocity distribution. In other words, the recoil-induced resonance can be used to measure the atomic velocity distribution. It can also be noticed that the lineshape is the same for a positive and a negative frequency detuning A, which is in contrast with the behavior found with stationary two-level atoms (see Eqs. (11) and (12)). Another difference with this latter situation is that the gain is here proportional to a:, whereas it is proportional to f2; for stationary two-level atoms in the absence of collisions (see Eq. (12)). These differences are not surprising because the physical origin of the Rayleigh resonance if quite different in the two situations: for stationary atoms the resonance arises from the saturation of the atomic transition, whereas the relevant effect for moving atoms is a spatial bunching mechanism induced by the optical potential. OF RECOIL-INDUCED RESONANCES B. EXPERIMENTAL OBSERVATION
The first experimental indication for the existence of recoil-induced resonances was obtained incidentally with laser-cooled cesium atoms by Verkerk et al. (1992). However, their actual identification was only per-
116
J.-Y; Courtois and G. Grynberg
formed in 1994 (Courtois et al., 1994; Meacher et al., 1994). An example of the variation of the transmitted probe intensity observed when the probe frequency is scanned is shown in Fig. 11. This lineshape can be fitted very satisfactorily by the derivative of a Gaussian, and no other simple function yields a better fit. From such a recording, one can deduce the velocity disfribution, or equivalently, the kinetic temperature of the atomic sample. To evaluate the reliability of this measurement, the temperature deduced from the width of the recoil-induced resonance was compared with the temperature found from a ballistic measurement (time of flight method) for various cooling conditions. The result shown in Fig. 12 shows a very good agreement between the two kinds of measurements. During the course of the experiments several other characteristics of the recoilinduced resonances, such as the shape invariance as the sign of the frequency detuning from resonance is changed, were also checked. A very good quantitative agreement between experiment and theory was always obtained. One should, however, mention an intrinsic impediment of the temperature measurement using recoil-induced resonances. Because of their interaction with the pump and probe fields, atoms scatter photons. This results in heating of the atoms and a drift of the cloud due to radiation pressure. To get rid of the former effect, the experiments were carried out for
I 1.04 1 .oo
.96
-25 25 -6 (ICHZ) FIG. 11. Recoil-induced resonance: variation of the probe transmission versus 6. Cesium atoms are first cooled and trapped in a magneto-optical trap (Raab et al., 1987; Walker et al., 1990); then, the inhomogeneous magnetic field and the trap field are switched off and the beams necessary for the observation of the recoil-induced resonance are switched on. In the experiment of Meacher et al. (19941, the pump and probe beams were traveling waves having same linear polarization and making an angle 0 = 10" with each other. The frequency detuning A from the 6S,,,(F = 4) -+ 6P3,,(F = 5) resonance was on the order of -30 MHz (which corresponds to - 6 r) and the beam intensities were of the order of 0.5 mW/cm2. The fitted curve is the derivative of a Gaussian.
STIMULATED RAYLEIGH RESONANCES
117
FIG. 12. Plot of the temperature obtained from the peak-to-peak distance of the recoilinduced resonance as a function of that obtained by a time of flight technique. The fitted slope is 0.9 f 0.1.One possible reason for the uncertainty is the difficulty of accurately measuring the angle between the pump and probe beams.
different interaction times between the atoms and the fields, and the initial temperature was obtained by extrapolating the data to a zero interaction time. To avoid the drift of the atoms, Courtois et al. (1994) reflected the pump beam onto itself after rotating its polarization by 90". In this way, a one-dimensional optical lattice (see Section 11.0 was obtained along the pump beams axis, so that in principle the recoil-induced resonance involving the transverse translational degrees of freedom of the atoms was superimposed with a longitudinal Rayleigh resonance of the lattice (Courtois and Grynberg, 1992). However, it can be shown theoretically (Guo, 1994) that, for this field geometry, the longitudinal Rayleigh resonance is much weaker than the recoil-induced resonance. It actually turned out that all the experimental observations could be interpreted only in terms of recoil-induced resonances, and no indication of the occurrence of a longitudinal Rayleigh resonance was found in the spectra.
RESONANCE AS RAMAN PROCESSES C. THERECOIL-INDUCED BETWEEN DIFFERENT ENERGY-MOMENTUM STATES In the original approach by Guo et al. (1992), recoil-induced resonances were interpreted by describing the atomic motion in momentum space. From this viewpoint, recoil-induced resonances appear as resulting from stimulated Raman processes between atomic energy-momentum states (BordC, 1976) associated with different momenta. More precisely, stimulated Raman transitions with absorption of a pump photon and stimulated
118
J.-Y. Courtois and G. Glynberg
emission of a probe photon take place between energy momentum states ( E = p2/2M,p) and ( E = (p + hk,)2/2M,p hk,) with kL = k - k, because of momentum conservation during the atom-laser interaction. Because the populations of the initial and final states of these elementary Raman processes are different, a Raman gain is expected for the probe, with the following resonance condition deduced from energy conservation:
+
As shown by Eq. (231, there is a one-to-one correspondence between the pump-probe frequency detunings 6 and the abovementioned pairs of energy-momentum states. In other words, the entire velocity distribution is probed as 6 is scanned. It may be noticed that Eq. (23) is quite similar to the resonance condition of Compton scattering in atomic physics textbooks, the only difference being the substitution of the atomic mass for the electron mass. This is the reason that recoil-induced resonances are sometimes called stimulated optical Compton scattering (SOCS). More quantitatively, the transition rate from the energy-momentum hk,) can be destate ( E = p2/2M,p) toward ( E = (p hk,)2/2M,p rived from time dependent perturbation theory, which yields
+
+
(24) where 6,(E) = ( r / 2 . r r h ) [ ~ i n ~ ( E r / 2 h ) / ( E 7 / 2 is h )a~ ]function that tends toward the Dirac delta function S,(E) as r tends toward infinity. The actual transition rate from the momentum group p toward p + hk, results from the balance between the process described by Eq. (24) and the reverse transition process from the energy-momentum state ( E = (p + hkl)’/2M,p + hk,) toward ( E p2/2M,p). Because these processes are weighted by the steady state populations of the initial levels, the overall transition rate reads
+
p
M
h2k: 2M
where fo(p) is the steady state momentum distribution.
STIMULATED RAYLEIGH RESONANCES
119
In most situations of experimental interest, the momentum distribution is much broader than the momentum carried by a photon.16 As a result, it is possible to neglect the recoil energy (fik,)'/2M in the delta function and to introduce the derivative of the velocity distribution:
The amplification cross section for the probe beam is then obtained by 2fio, dividing Eq. (25) by the incident probe photon flux ~ ~ c l ~ ~ l ~ / and summing over all the velocity groups. One thus obtains
where the derivative of the velocity distribution should be taken along the direction of k,. Equation (27) is connected to the previously established result (22) through the relation 2Re(g) =Hi
(28)
In the case of a Maxwell-Boltzmann momentum distribution f J p , ), one recovers exactly the same shape and the same dependence as in Eq. (22). It may be noticed that the final result does not depend on the recoil energy. This is a supplementary argument that shows that the introduction of the recoil is a convenient method for deriving the resonance shape but is not physically essential. Recoil-induced resonances can thus be interpreted either in terms of stimulated Raman transitions between energy-momentum states of different momenta or as a stimulated Rayleigh process. This duality is not very surprising because of the close analogy between these resonances and the free electron laser gain mechanism, which is also known to support two equivalent interpretations: the first being associated with stimulated Compton scattering (Madey, 19711, and the second with a spatial bunching of the electrons (Hopf et al., 1976; Kroll and MacMullin, 1978). D. ATOMIC BUNCHING IN
THE
TRANSIENT REGIME
Finally, we consider the transient regime of recoil-induced resonances in order to clearly identify the role of the different velocity groups in the atomic bunching process. By solving the Liouville equation with the initial condition f(r, p, t = 0 ) = fo(r,p), one can characterize completely the 16
The case of a sub-recoil velocity distribution has been investigated by Berman et al. (1995).
120
J.-Y: Courtois and G. GIynberg
dynamics of atomic bunching in the optical potential. Assuming y corresponding solution of Eq. (17) can be readily obtained:
=
0, the
Using Eq. (21), it is possible to predict the variation of the probe transmission versus time. First, it is straightforward to check that in the limit of large values of t, the component of f l being 7r/2 phase-shifted with respect to the optical potential is identical to the value deduced from Eq. (19). One thus recovers the same expression for the recoil-induced gain in steady state regime. Second, one notices that because of the presence of the diffraction function (sin x ) / x in Eq. (29), the contribution to atomic bunching of the velocity groups having a velocity differing from the phase velocity of the optical potential becomes vanishing small as the time goes on, which confirms the statement made in Section 1II.A.
E. COHERENT ATOMICRECOILLASER(CARL) Because of the close analogy with the free electron laser, it is tempting to look into the recoil-induced gain in a regime where the amplification of the probe field is so strong that the pump and probe field amplitudes cannot be considered as a constant. As a result, the optical potential varies with space and time (or with time only, if the atoms are enclosed within a short cavity). This yields a complicated dynamics of atomic bunching, because a change in the optical potential modifies the atomic density modulation, which back-reacts on the probe propagation through the Maxwell equations. In this regime, which is the atomic analog of the so-called “super free electron lasers,” one should solve the coupled system of equations involving simultaneously the fields and the atoms. This strong gain regime was investigated by Bonifacio et al. (1994) and by Bonifacio and De Salvo (1995a, b). The results obtained in this regime are qualitatively different from those obtained in the weak gain regime discussed previously. In particular, the gain mechanism appears from an instability of the atom-field system. It is consequently no longer described by an antisymmetric function of 6, and it fluctuates in time.
STIMULATED RAYLEIGH RESONANCES
121
IV. Other Recoil-Induced Effects in Atomic and Molecular Physics As is well known, several physical phenomena that are often interpreted as recoil effects can be equivalently accounted for by using alternative approaches. This is, for example, the case of recoil-induced resonances that were presented in Section 111. We focus in this section on those processes, which actually permit the measurement of the atomic recoil and for which there is therefore no ambiguity in the involvement of the recoil. A definitive historical account of recoil-induced effects in atomic and molecular physics should critically evaluate all published papers on the subject. This task is clearly not compatible with the purpose of this chapter. The following survey singles out those contributions that were, in the opinion of the authors, of special significance in the field of recoil-induced phenomena. This necessarily introduces some measure of subjectivity, and the interested reader should supplement this review by careful analysis of all original references.” The general background of recoil-induced effects contains the following factors: 1. The development of ultrahigh resolution spectroscopy 2. A widespread interest in manipulating atomic and molecular motion with light 3. The development of laser cooling and trapping of neutral atoms 4. The emergence of matter optics, including atomic and molecular interferometry.
Saturated absorption spectroscopy, the subject of Section IV.A, provided the framework for the first theoretical proposal (Kol’chenko et d, 1968) and the first experimental observations (BordC and Hall, 1974; Hall and BordC, 1974; Hall et al., 1976) of the photon recoil. Soon afterward, the combination of sub-Doppler techniques with the Ramsey method of spatially separated fields led to the demonstration of a recoil doublet in optical Ramsey fringes (Barger et al., 1979) (Section 1V.B). BordC (1989) pointed out that the interaction geometries used for observing optical Ramsey fringes actually constituted atom interferometers based on the photon recoil, which allowed one to detect phase shifts induced by rotations or accelerations. As shown in Sections 1V.C and IV.D, important ”As previously, chronological dates given in this section refer to the date of publication. Note, however, that because of the differing publication delays of comprehensive articles, letters to the editor, and translations, this style may not entirely reflect the actual chronology of events.
122
1-Y. Courtois and G. Grynbeig
developments in atom interferometry (Riehle et al., 1991; Kasevich and Chu, 1991) originated from the contribution of BordC. Since these early experiments, promising proposals have been made for improving the sensitivity of atom interferometers based on the photon recoil, which should lead to future advances in atom interferometry (Section 1V.E). Lately, the theoretical prediction and the experimental observation of the so-called recoil-induced resonances (see Section 111) renewed the interest in photon recoil effects, especially in cold atomic samples. Among recent suggestions, it was proposed by Gheri et al. (1995) to realize a cw deep UV laser based on cooled metastable helium atoms, by taking advantage of the recoil shift’s exceeding the natural linewidth and the Doppler broadening of the lating transition (Section 1V.F). Finally, the importance of the photon recoil in laser cooling and some other related phenomena are briefly discussed in Sections 1V.G and IV.H, respectively. A. RECOILEFFECTSIN SATURATED AESORPTION SPECTROSCOPY Saturated absorption spectroscopy is a method for obtaining Doppler-free spectra of atoms or molecules that was developed by BordC (19701, Hansch (1977), and Letokhov and Chebotayev (1977). Schematically speaking, it involves recording the intensity of two counterpropagating beams as a function of their common frequency after transmission through the gaseous medium. When this frequency is such that the beams interact with different velocity groups, their respective absorptions do not depend on the presence of the other beam. By contrast, when the two beams interact with the same velocity group, their absorption is reduced. If the counterpropagating beams probe the same atomic or molecular transition, this phenomenon gives rise to a genuine saturated absorption resonance. If they probe different transitions, the corresponding resonance is called a crossover. The possibility of observing the momentum exchange between the radiation field and a quantum absorber using saturated absorption spectroscopy was first pointed out by Kol’chenko et al. (1968). Because of the molecular recoil during the interaction with light, every saturated absorption resonance is actually a doublet (with the exception of crossover resonances). The origin of this doubling for a two-level system is illustrated in Fig. 13. In a nonrelativistic approximation, the curve representing the total energy versus the momentum component along the optical z axis is a parabola in each level. Because of momentum conservation, molecules interacting with a monochromatic field of a given direction have to belong to two different velocity classes in the lower ( l a ) ) and in the upper (Ib)) levels (Fig. 13a). The momentum change is precisely equal to the light quantum momentum +hk (where the sign depends on the wave direction
123
STIMULATED RAYLEIGH RESONANCES
(a)
(b)
(c)
FIG. 13. Origin of the doublet structure of saturated absorption resonances due to photon recoil.
with respect to the z axis). There are two situations where one of the two waves can produce a change in the absorption of the other. The first is obtained when molecules in the upper level can interact with both waves (Fig. 13b). The corresponding energy exchanged with the light can be read directly on the vertical axis. From the equation of the parabola, E = p,2/2M, it is readily seen that this energy is smaller than the Bohr energy noA = Eb - E, by an amount equal to the one-photon recoil energy ER = h 2 k 2 / 2 M .Similarly, a second resonance is expected when molecules in the lower state can interact with both waves and for which the energy exchanged exceeds the Bohr energy by the same quantity (Fig. 13c). Each of these peaks has a width limited ultimately by the natural linewidth r of the excited state of the molecular transition, but practically by transit time broadening. The first experimental indication for the existence of this doublet structure was obtained with methane at a wavelength of 3.39 p m by BordC and Hall (1974) through a careful lineshape study. With suitable attention to the laser frequency stability and to the various spectral broadening mechanisms, it eventually become possible in 1976 to clearly resolve the 2.16-kHz splitting of methane (Hall and BordC, 1974; Hall et al., 19761, as shown in Fig. 14. The recoil effect was then observed by BordC et al. (1979) in the visible part of the spectrum with iodine. This experiment was challenging because the natural lifetime (-70 kHz) of the molecular transition was significantly larger than the recoil shift (5.9 kHz), hence the impossibility of resolving directly the splitting. However, BordC and his colleagues succeeded in observing the shift of the crossouer peaks through a differential
J.-Y Courtois and G. Grynberg
124 r
I
I
I
I
I
I
I
I
I
I
I
1
KILOHERTZ DETUNING FROM REFERENCE LASER FIG. 14. Derivative spectrum of the three main hyperfine lines of "CH, showing the recoil doubling (lower curve). A least squares fit (solid line) gives a width of 1.27 kHz HWHM and a recoil doublet splitting of 2.16 kHz.The upper curve, integrated from a sample of such data, shows that each hyperfine component is spectrally doubled by the recoil effect (Hall et al., 1976).
measurement technique, based on the fact that the recoil shifts have opposite signs for crossovers involving a common upper level and for those having a common lower level. More recently, saturated absorption spectroscopy provided the framework for the observation by Ishikawa et d.(1994) of the recoil shift of the linear absorption Doppler profile of the calcium intercombination line. The demonstration of this shift, which is reminiscent of the process depicted in Fig. 13(a), was achieved by checking the symmetry of the Doppler-broadened absorption profile with respect to the saturation dip, which was not recoil-shifted because the recoil doublet was unresolved (see Fig. 15). It is interesting to note that all the sub-Doppler nonlinear spectroscopy methods do not permit the observation of the recoil. For example, Doppler-free two-photon spectra are not sensitive to the recoil effect (Grynberg and Cagnac, 1977). This is because the momentum of an atom
125
STIMULATED RAYLEIGH RESONANCES 1300 1
I
I
I
I
I
I
I
1
1250 1200 ?
?
-
1150 -
e
5 1100 c
.-0
P
1050 -
<
1000
$ .a
-
950 -
'
900 -800
.-.....reversed profile I
I
I
I
-600
-400
-200
0
I
200 Detuning (kHz)
I
I
400
600
I 800
FIG. 15. The recoil shift of the Doppler profile with respect to the saturation dip can be seen if the measured absorption profile (full line) and the same profile reversed in frequency (dashed line) are fitted in the center of the saturation dip (Ishikawa et al., 1994).
does not change after absorption of two photons propagating in opposite directions. B.
RECOIL
DOUBLETOF OPTICAL
RAMSEY
FRINGES
The idea of associating sub-Doppler techniques with the method of spatially separated fields introduced by Ramsey (1950) in the microwave domain was suggested by BordC (1972) in order to circumvent the basic limitation to ultrahigh resolution in saturation spectroscopy due to transit time broadening." The first realistic experimental implementation, based on three equidistant standing waves (Fig. 16a), was proposed by Baklanov et al. (1976) and led to the first experimental demonstration of optical Ramsey fringes (Bergquist et al., 1977). However, because of imperfections in the standing wave character of the three field zones and of an intrinsic impediment of this interaction geometry, namely the occurrence of multiple photon redistribution processes between the oppositely traveling waves, the fringe contrast was not sufficient to demonstrate any recoil effect. Eventually, by successive improvements of their experiment setup, Barger 18
For a review of these techniques, see, for example, Ramsey (1995).
126
J.-Y. Courtois and G. Grynberg
Atom Bear
Standing wave
Standing wave
t
Standing wave
Beam
Laser beam
-T-
Laser beams
Laser beam
- T -
FIG. 16. Standard interaction geometries between laser waves and an atomic beam for observing optical Ramsey fringes. (a) Three equidistant standing waves method. (b) Four spatially separated traveling waves scheme.
et al. (1979) succeeded in observing for the first time the recoil doublet of
optical Ramsey fringes (Fig. 17). In an independent theoretical study, BordC (1977a, b) showed that Ramsey fringes could be also obtained with four spatially separated traveling waves (Fig. 16b). Two important advantages of the BordC scheme over the three standing waves method are the limited number of photon redistribution processes allowed during the atom-laser interactions, hence an improved fringe contrast, and its close connection with the original
STIMULATED RAYLEIGH RESONANCES
127
-
FREQUENCY
FIG. 17. Saturated absorption profile. Outer curve: Large scale profile showing a saturation dip. Inner curves: Expansion of the bottom of the saturation dip, showing Ramsey fringes for different zone separations. The fringe patterns (solid line) result from the superposition of two recoil-shifted components (dashed line) (Barger et al., 1979).
proposal by Ramsey. It follows that the physical origin of the optical Ramsey fringes and their recoil doublet is especially simple to understand in this situation. Let us recall that, in the usual Ramsey interaction scheme, the atomic polarization induced in a first field zone precesses freely until the atom crosses into a second one. Because the interaction between the field and the atomic polarization in the last zone depends on their relative phase shift, the final populations of the ground and excited states exhibit an oscillatory behavior with the frequency detuning and the time of flight between the two field zones. Unfortunately, in the optical domain, the distribution of first order Doppler shifts averages out the fringes. The four traveling wave interaction geometry circumvents this problem by repeating the usual Ramsey operation with a second pair of laser beams traveling in a direction opposite to that of the first pair. In this way, the Doppler dephasing experienced by the atoms during the first Ramsey sequence is compensated for during the second one. As a result, optical Ramsey fringes depending on the laser detuning alone survive the average over the atomic velocity distribution. The presence of a recoil doublet on optical Ramsey fringes can be readily understood by considering a two-level system {la),Ib)) of transition
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J.-Y: Courtois and G. Glynberg
frequency w, = ( E , - E J / h and by identifying the energy-momentum states (BordC, 1976) IE,,p, + m,hk) = la, m,) coupled by the laser fields in the different interaction zones (BordC et al., 1984) (a = a, b, and the integer m, indicates the net number of light momentum quanta fik that have been exchanged from the incident atomic momentum p,). In the first phase, the interaction with the pair of waves traveling in the +z direction (see Fig. 16b) couples the initial state la,O) to lb,l). The corresponding transition probability toward state Ib) exhibits the usual Ramsey oscillatory behavior with the time of flight T between the two field zones (see Fig. 16b) and with the detuning A , = w - w, - ku, - E R / h between the laser frequency ( w - ku,) in the atomic rest frame defined by v = p,/M and the transition frequency ( w A ER/h) between states la, 0) and Ib, 1). In the second phase, in which the interaction takes place with the pair of waves traveling along the -2 direction, the system evolves independently from la, 0) and from Ib, l),namely la, 0) becomes coupled to Ib, -1) whereas Ib, 1) is coupled to la, 2). The former interaction yields an oscillatory dependence of the probability transition toward Ib) (more precisely, toward Ib, -1)) with T and the detuning A(# = w - w, + ku, E R / h , whereas the latter leads to a transition probability toward the upper state (Ib, 1)) oscillating with T and A(;) = w - w, ku, 3 E R / h . Combining both phases, one finds that the probability of detecting an atom in the energy-momentum states Ib, -1) and Ib, 1) at the exit of the last field zone depends sinusoidally on the parameters ( A l A(#)T = 2(w - w, ER/fi)T and ( A I A(;,)@ = 2(w - w, ER/h)T, respectively, which are independent of the atomic velocity. Consequently, the overall probability of finding an atom in the upper state Ib) appears as the sum of two fringes patterns up and down shifted by the recoil shift ER/h, hence the recoil doublet. The first observation of optical Ramsey fringes using four spatially separated traveling waves was reported simultaneously with SF, at 10 p m by BordC et al. (1981) and with calcium at 656 nm by Helmcke et al. (1982). The advantage of this geometry over the three standing wave method in terms of fringes contrast was then demonstrated experimentally and theoretically by BordC et al. (1984). The first observation of a recoil-induced splitting of optical Ramsey fringes was reported by Helmcke and Morinaga (1986); (see also Helmcke et al., 1988).
+
+
+
c. THE RAMSEY-BORDBh h T I E R
+
+
+
WAVE INTERFEROMETER
In 1989, recoil-induced phenomena made a remarkable apparition in the developing field of matter-wave interferometry, after BordC (1989) demonstrated that the interaction geometry comprising four traveling
129
STIMULATED RAYLEIGH RESONANCES
waves, which was used to obtain optical Ramsey fringes in atomic spectroscopy, actually constituted an atom interferometer based on the photon recoil. In particular, it appeared that de Broglie phase shifts induced by rotation or accelerations were detectable through frequency shifts of the optical Ramsey fringes. The argument of BordC was as follows: Consider the interaction geometry of Fig. 16(b), where a beam of two-level systems {la), Ib)} interacts with two counterpropagating sets of pairs of copropagating traveling laser waves. Because of the energy and momentum exchanges during resonant absorption/emission processes in each interaction zone, the matter waves are coherently split into two components with an internal state la) and Ib) and with wavevectors differing by the laser wavevector k in the direction e, of light propagation. Any possible sequence of such processes can be represented by means of rays originating from the ground state ( l a ) ) incident beam as depicted in Fig. 18, where each segment is labeled by an energy-momentum state I a , m,) (see Section 1V.B). In this picture, the transition probability toward state Ib) in the last zone can be obtained from the amplitudes associated with the different quantum mechanical paths represented by the rays of Fig. 18, and this
b,+l
/ a,O
b,+'
/ a,O
LASER BEAM -d-
LASER
LASER
BEAMS -d-
BEAM
FIG. 18. Interaction geometry of the four traveling beams (u,*k e , ) with the atomic beam ( E a ,po). Each ray is labeled by its energy and momentum state la,rn,) (BordB, 1989).
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.I.-Y Courtois and G. Grynberg
leads to the optical Ramsey fringes discussed in Section 1V.B. In fact, one can readily show that among the different possible quantum mechanical paths, only the contributions associated with the closed circuits of Fig. 19 survive the average over the broad distribution of velocity components v,. These circuits actually correspond to the two recoil components of the Ramsey fringes. Moreover, they have a finite area and will therefore behave as matter wave interferometers with a phase difference sensitive to both internal and external degrees of freedom. Note that each of these interferometers has two output channels associated with two energymomentum states labeled by a diflerent internal state. If the system is now submitted to an external inertial field, the induced de Broglie phase corresponding to each arm of the interferometer will therefore be distinguishable from the other, no matter the photon recoil momentum being resolved. Thus, one expects a frequency shift of the Ramsey fringes to be observable. A considerable interest in atom interferometers based on the photon recoil followed the paper of BordC. In particular, the capabilities of the Ramsey-Bord6 interferometer for measuring a rotation-induced inertial field through the shift of the recoil doublet (Sagnac effect) was soon demonstrated by Riehle et al. (1991) in calcium. In a slightly different context, one can also mention the measurement by Zeiske et al. (1995) of the Aharonov-Casher phase using the same device.
D. THE EXPERIMENT OF KASEVICHAND CHU A few days after Riehle et al. (1991) reported the observation of the Sagnac effect, Kasevich and Chu (1991) demonstrated the sensitivity of atom interferometry to the gravitational field. Their interaction geometry, which was also based on the photon recoil, brought two important potential improvements over the optical Ramsey experiments. First, the two-level system employed by Kasevich and Chu comprised two ground state hyperfine sublevels 11) and 12) of laser-cooled sodium atoms, which were coupled via velocity-selective stimulated Raman transitions. The recoils from the two photons that excited these transitions were in the same direction, so that the effective wavevector was k,, = k , + k , = 2k, but the effective frequency of the transition was the hypefine splitting. In this way, they obtained the large momentum kicks they would have got with single violet photons, while they had the convenience and accuracy of working with microwave frequencies. Second, since both atomic levels were ground states, the linewidth of the stimulated Raman transition was limited only by the measurement time, which was quite long because the atoms originated from an atomic fountain.
131
STIMULATED RAYLEIGH RESONANCES
a
ATOMIC
BEAM
Y
a,O
LASER
LASER
LASER
BEAM
BEAMS
BEAM
LASER
LASER
LASER
BEAM
BEAMS
BEAM
FIG. 19. Closed circuits of matter wave rays, which correspond respectively to the higher frequency (a) and to the lower frequency (b) recoil components of the Ramsey fringes, and which constitute two distinct atom interferometers (BordC, 1989).
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The atom interferometer of Kasevich and Chu separates and recombines the atoms using a ?r/2-n-?r/2 sequence of Raman pulses. Starting from an atom with momentum component along the direction of the laser beams p and internal state Il), the first 7r/2 pulse puts the original state I1,p) into a superposition of states I1,p) and 12,p + 25k). After a time At, the wavepackets will have separated by an amount 25k A t / M . The ?r pulse then induces the transitions I1,p) + 12,p + 2 h k ) and 12,p + 2hk) + Il,p), and after another interval At, the two wavepackets merge again. By adjusting the phase of the final ?r/2 pulse, the atom can be put into either of the hyperfine states. Figure 20 illustrates the configuration employed by Kasevich and Chu, where the atomic separation is along the direction of motion. As shown by Kasevich and Chu (1990, if the laser frequencies are properly chosen to keep approximately in resonance with the atoms (the 7r/2 pulse must remain a 7r/2 pulse as the atom accelerates), then the phase difference between the two paths 1 + 2 + 4 and 1 + 3 + 4 of Fig. 20 depends on the acceleration of gravity g times the delay At squared, and on the relative phases of the lasers during the three pulses. It follows that the interference fringes displayed by the probability of finding the atoms in a given hyperfine sublevel at the output of the interferometer as the relative phases of the lasers are varied does not permit a measurement of g. However, by working with fixed relative laser phases, it is possible to detect relative uuriutions of g up to a precision of the order of lo-'.
FIG. 20. Diagram of the v/2-7i-v/2 pulse interferometer described in the text. The mechanical recoil from the fust w / 2 pulse coherently splits (position 1) the atomic wavepacket. A v pulse (position 2 and 3) redirects each wave packet's trajectory. By adjusting the phase of the second v/2 pulse (position 4), the atom can be put into either 11) or 12). In the experiment, the atoms are prepared in the 11) state (solid line) and detected in the !2) state (dashed line) (Kasevich and Chu, 1991).
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133
E. RECENTADVANCES IN ATOMINTERFEROMETRY BASEDON THE PHOTON RECOIL
As it is well known, the sensitivity of an interferometer is in many cases proportional to the surface delimited by its two arms (this is in particular the case for photon recoil measurements). In other words, the larger the area of the closed circuits of the kind depicted on Fig. 19, the better the sensitivity. Recently, two proposals (Weiss et al., 1993, 1994; BordC et al., 1994) have been made for improving significantly the performances of atom interferometers based on the photon recoil, one of which has already been experimentally implemented for performing an ultrahigh resolution measurement of h/M,-, (Weiss et al., 1993, 1994). The atom interferometer of Weiss et al. (1993, 1994) is the separated oscillatory field version of the experiment of Kasevich and Chu (see Section 1V.D). It consists basically of the same 7r/2 pulse sequence induced by four spatially separated traveling waves as the one employed in the Ramsey-BordC interferometer. The improvement in the sensitivity of the interferometer is achieved by sandwiching a series of up to 15 7r pulses between the middle two 7r/2 pulses, with alternating propagation directions (see Fig. 21). Each additional pulse adds one effective photon recoil to the center of mass velocity of each quantum mechanical path. In this way, for N 7r pulses the surface enclosed between the closed circuits of the interferometer is multiplied by N + 1. The potentialities of this technique for performing high precision measurements of h / M was demonstrated by Weiss et al. (1993, 1994) in the case of cesium, where an accuracy of lo-’ was obtained. The theoretical proposal of BordC et al. (1994) constitutes a further improvement over the interferometer of Weiss et al. (1993,1994). By using the same number of 7r pulses, it actually leads to an interferometer surface growing with the square of the number of 7r pulses, hence (potentially) much higher sensitivities. This is obtained with two series of 7r pulses sandwiched between the two first and the two last 7r/2 pulses (see Fig. 22). F. RECOIL-INDUCED INVERSIONLESS LASINGOF COLD ATOMS
Although it has already been shown by Einstein (1917) that a consistent thermodynamic description of atoms coupled to light requires the inclusion of photon momentum, the standard models for describing laser action in the infrared and visible frequency domains (Haken, 1984; Lax, 1968; Sargent et al., 1977) ignore the recoil imparted to an atom by emission or absorption of a laser photon. The reason that this simplifying assumption
134
J.-Y. Courtois and G. Grynbee Interferometer A
Interferometer 6
FIG. 21. Wavepacket trajectories for improved photon recoil measurements. The solid and dashed lines correspond to the two internal states of the atom. The photon arrows show the direction of the effective k vector for velocity-selective stimulated Raman transitions. A series of ?r pulses with alternating propagation directions between the middle two ?r/2 pulses helps to improve the sensitivity of the interferometer. Up to 15 ?r pulses were used in the experiment (Weiss et af., 1993, 1994).
normally constitutes a fairly good approximation is that the recoil-related changes in energy of the absorbed or emitted photons are usually much less than the natural linewidth of the lasing transition. When extending the theory of lasing into the UV and XUV domain, one has to think more carefully about recoil-related effects. For shorter wavelengths A, the recoil energy is indeed vastly increased according to the power law ER = h2/2MA2.However, because of higher mode densities in this frequency range, the spontaneous emission rates ( al / A 3 ) will be increased accordingly, and thus one should in principle be allowed to use the same models. In fact, in the UV domain, incoherent optical pumping is not efficient for achieving a steady state population inversion due to the presence of fast spontaneous decay. One therefore realizes that a viable gain scheme for UV radiation will need to provide a transition whose excited state doe not decay rapidly. Consequently, the recoil shift is likely to exceed the natural linewidth of the transition and will need to be accounted for. Hence, the atomic absorption and emission line for an atom initial& at rest will differ in frequency, and a spontaneously emitted photon will not be absorbed by a second atom also at rest.
135
STIMULATED RAYLEIGH RESONANCES
-I
\b-3
7712
- n - pulses
TI2
TI2
77 -
pulses
7712
FIG. 22. Wavepacket trajectories for an improved interferometer with three exchanged momentum quanta per interaction sequence. The two-level system interacts with effective multiphoton fields of opposite directions either perpendicular to or collinear with the atomic motion (BordC et al., 1994).
Following the ideas of Marcuse (19631, who first noted that inversionless maser action should be possible if the recoil shift exceeded the linewidth due to Doppler broadening (because the emitted photons would be redshifted and less likely to be absorbed by neighboring atoms), Gheri et al. (1995) recently proposed utilizing a Raman gain scheme involving cold atoms in a metastable state to create atomic inversion with sufficiently large stimulated emission probability, while benefiting from the recoil-mediated reduction of reabsorption. However, it was found that in the case of helium, which is one of the best candidates for implementing this proposal, the possibility of achieving laser threshold was presently beyond the state of the art experimental techniques. G. ATOMICRECOILAND
LASER COOLING
Momentum exchanges between photon and atoms are a key component in the understanding of the fundamental limits of laser cooling (CohenTannoudji, 1992). The lowest temperature T, that can be achieved with most atomic transitions is indeed limited by the one-photon recoil energy (k,T = f i 2 k 2 / 2 M ,where k , is the Boltzmann constant). This is because laser cooling results from sequences of laser photon absorptions followed by spontaneous emissions of photons, which permit transformation of the
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atomic kinetic energy into radiation energy. Because of the randomness in the direction of the spontaneously emitted photons, the lowest achievable atomic momentum is of the order of hk, and therefore the lowest atomic kinetic energy is of the order of h2k2/2 M. However, it turns out that this fundamental limit can be bypassed in the situation where the laser beams are tuned near a Jg = 1 + J, = 1 atomic transition. This is because such an atomic transition supports velocity-selective dark states (Ol’shanii and Minogin, 1992) that yield sub-recoil temperatures (Aspect et al., 1989).
H. OTHERRELATEDPHENOMENA This chapter has focused on those aspects of the historical developments that pertain directly to recoil-induced phenomena, i.e., on physical processes actually allowing the measurement of h / M . This restriction does not do justice to many parallel developments in allied fields, where the introduction of the photon recoil helps the understanding and improves the clarity of the physical interpretation, although it is not required for fundamental reasons. These phenomena, which are characterized by the absence of the parameter h / M in the final result, can be usually interpreted using alternative treatments involving no momentum transfer between photons and atoms. A glaring example of such effects is the physical mechanism of recoil-induced resonances. As shown in Section III.A, h / M is absent from the expression (22) of the probe gain, which can be interpreted in purely classical terms as a stimulated Rayleigh process involving atom bunching. We could also have mentioned the recent observation by Gould et al. (1986) of a well-resolved atomic diffraction spectrum of an atomic beam by a standing wave grating, radiation pressure, etc.
V. Conclusion In this chapter, we have shown that the key component of stimulated Rayleigh scattering is the excitation of a nonpropagating observable of a material system being phase-shifted with respect to the driving interference pattern between the probe and the pump beams. In the most common situation, where the phase shift in the medium response arises from relaxation processes, the stimulated Rayleigh resonance displayed by
STIMULATED RAYLEIGH RESONANCES
137
the probe transmission spectrum takes the form of a dispersion curve having a width equal to twice the relaxation rate, and therefore it provides useful information about the system dynamics. In other situations, such as in photorefractive materials or in recoil-induced resonances, the origin of the observable phase shift can be more subtle, and the shape of the stimulated Rayleigh resonance can differ significantly from a dispersion. The case of recoil-induced resonances has been more particularly investigated because it provides a particularly simple illustration of some possible ambiguities in the interpretation of stimulated Rayleigh processes. First, we have shown that recoil-induced resonances could be identically interpreted in terms of stimulated Rayleigh or Raman scattering, depending on the atomic motion’s being described in position or momentum space, respectively. This property, which has been known for years in the free electron laser community, is typical of systems displaying a continuous energy spectrum. Second, it appears from the comparison between both interpretations of recoil-induced resonances that the role of the photon recoil in a physical process can be somewhat elusive. It thus seems appropriate to distinguish between two groups of “recoil-induced” phenomena. The first group contains the processes that permit an actual measurement of the photon recoil or, equivalently, of h / M . The second group includes the effects whose interpretation in momentum space involves the photon recoil, even though they do not permit the determination of the recoil itself. Generally, an alternative explanation in position space that does not require the introduction of the recoil can be found for such effects. In our opinion, it is more legitimate to attribute to the photon recoil only those processes belonging to the first category. As an illustration of such effects, we have presented in this chapter a few examples of nonlinear spectroscopy experiments that could not be understood without taking into account the momentum exchanges between photons and atoms.
References Agarwal, G. S., and Nayak, N. (1984). J . Opt. SOC.Am. B 1, 164. Aspect, A., Arimondo, E., Kaiser, R., Vansteenkiste, N., and Cohen-Tannoudji, C. (1989). J . Opt. SOC. Am. B 6, 2112. Baklanov, Y. V., Dubetsky, B. Y., and Chebotayev, V. P. (1976). Appl. Phys. 9, 171. Barger, R. L., Bergquist, J. C., English, T. C., and Glaze, D. J. (1979). Appl. Phys. Lett. 34, 850. Bergquist, J. C., Lee, S. A., and Hall, J. L. (1977). Phys. Rev. Lett. 38, 159.
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Berman, P. R., Khitrova, G., and Lam, J. F. (1985). In “Spectral Line Shapes 3” (F. Rostas, ed.). de Gruyter, Berlin. Berman, P. R., Dubetsky, B., and Guo, J. (1995). Phys. Reu. A 51,3947. Bloembergen, N. (1967). Am. J. Phys. 35, 989. Bloembergen, N., and Lallemand, P. (1966). Phys. Rev. Lett. 16,81. Bogdan, A. R., Downer, M. W., and Bloembergen, N. (1981). Opt. Lett. 6,348. Bonifacio, R., and De Salvo, L. (1995a). Appl. Phys. B 60,S233. Bonifacio, R., and De Salvo, L. (1995b). Opt. Commun. 115,505. Bonifacio, R., De Salvo, L., DAngelo, E., and Narducci, L. (1994). Phys. Rev. A 50, 1716. BordC, C. J. (1970). C.R. Hebd. Seances Acad. Sci. 271B,371. Bord6, C. J. (1972). Ph.D. Thesis, UniversitC Paris VI, Paris. BordC, C.. J. (1976). C.R. Hebd. Seances Acad. Sci. 283B, 181. BordC, C. J. (1977a). C.R. Hebd. SeancesAcad. Sci. 284B, 101. BordC, C. J. (197%). In “Laser Spectroscopy 111” (J. L. Hall and X. X. Carlsten, eds.). Springer, Berlin. BordB, C. J. (1989). Phys. Lett. A 140, 10. BordB, C. J., and Hall, J. L. (1974). In “Laser Spectroscopy” (R. G. Brewer, and A. Mooradian, eds.). Plenum, New York. BordC, C. J., Carny, G., aria Decomps, B. (1979). Phys. Reu. A 20,254. Bord6, C. J., Avrillier, S., Van Lerberghe, A., Salomon, C., Bassi, D., and Scoles, G. (1981). J. Phys. Orsay (Fr..)C42, C8-Cl5. BordC, C. J., Salomon, C., Avrillier, S., Van Lerberghe, A., BrCant, C., Bassi, D., and Scoles, G. (1984). Phys. Rev. A 30, 1836. BordC, C. J., Weitz, M., and Hansch, T. W. (1994). In “Laser Spectroscopy XI” (L. Bloomfield, T. Gallagher, and D. Larson, eds.). AIP Press, New York. Boyd, R. W. (1992). “Nonlinear Optics.” Academic Press, San Diego, CA. Castin, Y., and Dalibard, J. (1991). Europhys. Lett. 14,761. Chiao, R. Y., Kelley, P. L., and Garmire, E. (1966). Phys. Reu. Lett. 17, 1158. Cho, C. W., Foltz, N. D., Rank, D. H., and Wiggins, T. A. (1967). Phys. Rev. Lett. 18, 107. Cohen-Tannoudji, C. (1962). Ann. Phys. (Paris) 7, 423. Cohen-Tannoudji, C. (1992). In “Fundamental Systems in Quantum Optics, les Houches, Session LIII” (J. Dalibard, J.-M. Raimond, and J. Zinn-Justin, eds.), p. 1. Elsevier, Amsterdam. Cohen-Tannoudji, C., Dupont-Roc, J., and Grynberg, G. (1992). “Atom-Photon Interactions.” Wiley, New York. Courtois, J.-Y., and Grynberg, G. (1992). Phys. Rev. A 46,7060. Courtois, J.-Y., and Grynberg, G. (1993). Phys. Reu. A 48, 1378. Courtois, J.-Y., Grynberg, G., Lounis, B., and Verkerk, P. (1994). Phys. Reu. Lett. 72, 3017. Dalibard, J., and Cohen-Tannoudji, C. (1989). J. Opt. SOC. Am. B 6,2023. Einstein, A. (1917). Phys. Z. 18, 121. Elias, L. R., Fairbank, W. M., Madey, J. M. J., Schwettman, H. A., and Smith, T. I. (1976). Phys. Reu. Lett. 36,717. Fabelinskii, I. L. (1968). “Molecular Scattering of Light.” Plenum, New York. Gheri, K. M., Ritsch, H.Walls, D. F., and Balykin, V. I. (1995). Phys. Reo. Left. 74, 678. Could, P. L., Ruff, G. A., and Pritchard, D. (1986). Phys. Rev. Lett. 56,827. GrandclCment, D., Grynberg, G., and Pinard, M. (1987). Phys. Rev. Lett. 59, 40. Gruneisen, M.T., MacDonald, K. R., and Boyd, R. W. (1988). J. Opt. SOC. Am. B 5, 123. Grynberg, G.,and Berman, P. R. (1989). Phys. Reu. A 39, 4016.
STIMULATED RAYLEIGH RESONANCES
139
Grynberg, G., and Cagnac, B. (1977). Rep. Prog. Phys. 40,791. Grynberg, G., and Cohen-Tannoudji, C. (1993). Opt. Commun. 96, 150. Grynberg, G., Le Bihan, E., and Pinard, M. (1986). J. Phys. Orsay, (Fr.) 47, 1321. Grynberg, G., Vallet, M., and Pinard, M. (1990). Phys. Reu. Lett. 65,701. Grynberg, G., Lounis, B., Verkerk, P., Courtois, J.-Y., and Salomon, C. (1993). Phys. Reu. Lett. 70,2249. Giinter, P., and Huignard, J.-P. (1988). Top. Appl. Phys. 61, 1. Guo, J. (1994). Phys. Reu. A 49,3934. Guo, J., and Berman, P. R. (1993). Phys. Reu. A 47, 1294. Guo, J., Berman, P. R., Dubetsky, B., and Grynberg, G. (1992). Phys. Reu. A 46, 1426. Haken, H. (1984). “Laser Theory.” Springer-Verlag, Berlin. Hall, J. L., and BordC, C. J. (1974). Bull. Am. Phys. Soc. [2] 19, 1196. Hall, J. L., BordC, C. J., and Uehara, K. (1976). Phys. Reu. Lett. 37, 1339. Hansch, T.W. (1977). Proc. Int. Sch. Phys. “Enrico Fermi” 44. Harris, S. E. (1989). Phys. Reu. Lett. 62, 1033. Helmcke, J., and Morinaga, A. (1986). CPEM Dig.86CH2267-3,123-124. Helmcke, J., Zevgolis, D., and Yen, B. U. (1982). Appl. Phys. B 28, 83. Helmcke, J., Glaser, M., Riehle, F., Morinaga, A., and Snyder, J. J. (1988) Sou. J. Quantum Electron. (Engl. Transl.) 18,758. Hemmerich, A., and Hansch, T. W. (1993). Phys. Rev. Lett. 70,410. Hemmerich, A.,Weidmiiller, M., and Hansch, T. W. (1994). Europhys. Lett. 27, 427. Hopf, F. A., Meystre, P., Scully, M. O., and Louisell, W. H. (1976). Opt. Commun. 18,413. Ishikawa, J., RieNe, F., Helmcke, J., and Bordt, C. J. (1994). Phys. Reu. A 49,4794. Jessen, P. S., Gerz, C., Lett, P. D., Phillips, W. D., Rolston, S. L., Spreeuw, R. J. C., and Westbrook, C. I. (1992). Phys. Rev. Lett. 69,49. Kasevich, M., and Chu, S. (1991). Phys. Reu. Lett. 67,181. Kol’chenko, A.P., Rautian, S. G., and Sokolovskii, R. I. (1968). Zh. E h p . Teor. Fiz.55, 1864; Sou. Phys-JETP (En@. Trunsl.) 28,986. Kroll, N. M., and MacMullin, W. A. (1978). Phys. Reu. A 17,300. Lax, M. (1968). In “Brandeis University Summer Institute Lectures (1966)” (M. ChrCtien, E. P. Gross, and S. Deser, eds.), Vol. 2. Gordon & Breach, New York. Letokhov, V. S., and Chebotayev, V. P. (1977). “Nonlinear Laser Spectroscopy.” SpringerVerlag, Berlin. Lounis, B., Courtois, J.-Y., Verkerk, P., Salomon, C., and Grynberg, G. (1992). Phys. Rev. Lett. 68,3029. Madey, J. M. J. (1971). J . Appl. Phys. 42, 1906. Marcuse, D. (1963). Proc. IEEE 51, 849. Mash, D. I., Morozov, V. V., Starunov, V. S., and Fabelinskii, I. L. (1965). JETP Lett. (Engl. Transl.) 2,25. Meacher, D. R., Boiron, D., Metcalf, H., Salomon, C., and Grynberg, G. (1994). Phys. Rev. A 50, R1992. Mollow, B. R. (1972). Phys. Reu. A 5, 2217. Ol’shanii, M. A., and Minogin, V. (1992). Opt. Commun. 89,393. Pinard, M., and Grynberg, G. (1988). J.. Phys. (Paris) 49, 2027. Raab, E.,Prentiss, M., Cable, A,, Chu, S., and Pritchard, D. (1987). Phys. Rev. Lett. 59, 2631. Ramsey, N. F. (1950). Phys. Reu. 78, 695. Ramsey, N. F. (1995). Appl. Phys. B 60,85. Riehle, F., Kisters, T., Witte, A., Helmcke, J., and BordC, C. J. (1991). Phys. Reu. Lett. 67, 177.
140
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Sargent, M., Scully, M. O., and Lamb, W. E. (1978). “Laser Physics.” Addison-Wesley, Reading, MA. Ungar, P. J., Weiss, D. S., Riis, E., and Chu, S . (1989). J . Opt. SOC.Am. B 6, 2058. Vallet, M., Pinard, M., and Grynberg, G. (1992). Opt. Commun. 87,340. Verkerk, P., Lounis, B., Salornon, C., Cohen-Tannoudji, C., Courtois, J.-Y., and Grynberg, G. (1992). Phys. Rev. Lett. 68, 3861. Walker, T., Sesko, D., and Wieman, C. (1990). Phys. Rev. Lett. 64, 408. Weiss, D. S., Young, B. C., and Chu, S. (1993). Phys. Rev. Lett. 70, 2706. Weiss, D. S., Young, B. C., and Chu, S . (1994). Appl. Phys. B 59, 217. Wu, F. Y.,Ezekiel, S., Ducloy, M., and Mollow, B. R. (1977). Phys. Rev. Lett. 38, 1077. Zeiske, K, Zinner, G., Riehle, F., and Helmcke, J. (1995). Appl. Phys. B 60, 205. Zibrov, A. S., Lukin, M. D., Nikonov, D. E., Hollberg, L. W., Scully, M. O., Velichansky, V. L., and Robinson, H. G. (1995). Phys. Reu. Lett. 75, 1499.
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 36
PRECISION LASER SPECTROSCOPY USING ACOUSTO-OPTIC MODULATORS W . A. VAN WZJNGAARDEN Physics Department, York Universiry Toronto, Ontario, Canada
....... . ... . . . .. .......... . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . .. . . .. . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . ..... . . . . . . . . . . . .. .. ... ..... .. . . . . .... .. . ... .. .. .... . . .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . , . . .... . ..... . ... . .. . . . ...... ..... .. . . . ... . ... .... . . . ..... .. . . ......... . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . ... .... . .. ........ . .. . . . . . . .... ..... .. . . . .. .. . . . .. . .. .........
I. Introduction. . ...... .. .. 11. Optical Spectroscopy . 111. Spectroscopy Using Frequency-ModulatedLasers . . . . A. Optical Modulators . . . . ..... . ..... . . . B. General Experimental Arrangement IV. Hyperfine Structure and Isotope Shifts . . . . . . . . . . . A. Background . . . ....... . . B. Ytterbium ( 6 ~ 6 p ) ~ P state , .. . . ... .... C. Sodium 3P,,, State . . . . .... ... . .. . D. Cesium 6P3/, State. . . .. . . V. StarkShifts.. . . . . . . , . . .. . A. Background . . . . . . . . . .. . . . . . B. Ytterbium ( 6 ~IS ),~+ (6s6p) 'P1 Transition . . . . . . . . . . C. Cesium 6P3,, + nS,/, ( n = 10-13) Transitions. . . . D. Cesium D Lines. . ... E. Precision Stark Shift Summary. . . .,. . . . VI. Concluding Remarks . . . . . .. . . . . . . . References. . . . . . . . . . . . . . . .
141 142 148 148 149 152 152 153 158 163 166 166 168 170 172 177 179 180
I. Introduction Precision optical spectroscopy is an invaluable tool to study atomic structure. Absolute measurements of optical transition frequencies can be readily made with an accuracy of parts in lo9 (Riis et al., 1994). However, in many experiments one is primarily interested in determining the energy interval separating two nearby states. Examples include fine and hyperfine structure splittings, isotope shifts, and the Lamb shift. Additional frequency shifts arise when electric and magnetic fields are applied to the atom. These energy splittings can be found by subtracting precisely determined frequencies of transitions occurring to the states in question. For 141
Copyright 0 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-003836-6
142
W. A. van Wqngaarden
example, consider an atom that has an optical transition at 5 X Hz to an excited state consisting of two hyperfine levels separated by 1 GHz. The hyperfine splitting can be determined with an uncertainty of 1 MHz if the transition frequencies to the two hyperfine levels are measured with accuracies of one part in 5 X lo8. A different approach is to measure the hyperfine splitting directly, which only requires a measurement accurate to one part in lo3. The latter task requires simpler and cheaper apparatus and is therefore in general easier. This chapter reports on a new spectroscopic method that uses a frequency-modulated laser to excite an atomic beam. It has an especially promising future given the rapid technological advances in developing new relatively inexpensive acousto-optic and electro-optic modulators. Most significantly, this new method is free of various systematic effects that have limited the accuracy of past experiments. This chapter is organized as follows. Section IT briefly reviews some of the advances made in optical spectroscopy during the last few decades. Principally, it discusses the use of Fabry-Perot etalons in conjunction with laser atomic beam spectroscopy. Interferometers have been extensively employed by numerous groups to determine many different kinds of frequency shifts. Section I11 describes three possible experimental arrangements using optically modulated laser beams to make frequency measurements. The advantages and limitations of these approaches are illustrated in Section IV by three specific examples of experiments that determined isotope shifts and hyperfine structure. Section V discusses some precision Stark shift measurements for optical transitions. It concludes with a summary of polarizability data having uncertainties of less than 0.5%. Sections IV and V also compare the results obtained using a variety of competing spectroscopic techniques. Finally, Section VI gives concluding remarks.
11. Optical Spectroscopy One of the first methods developed to study spectra was to use a spectrometer to examine the light emitted by atoms excited in a discharge lamp. The width of the observed spectral lines is limited to about a gigahertz by collisional and Doppler broadening. Hence, one typically observes several overlapping lines for a transition occurring in an atom having several stable isotopes. The signals produced by the less abundant isotopes can be enhanced by loading the lamp with isotopically enriched atomic samples. An example of such an experiment was done by Ross (1963) to study the hyperfine splittings and isotope shifts of the ytterbium ( 6 ~'So ) ~+ ( 6 ~ 63P1 ~ ) transition at 555.6 nm. Ytterbium has naturally occurring
PRECISION LASER SPECTROSCOPY
143
isotopes with atomic mass units 168 (0.13%), 170 (3.05%), 171 (14.3%), 172 (21.9%), 173 (16.1%), 174 (31.8%), and 176 (12.7%). The nuclear spin of the even isotopes is zero, whereas isotopes 171 and 173 have spins of and 5 I, respectively. The experimental resolution was further improved by cooling the lamp cathode with liquid nitrogen to reduce the Doppler width. The emitted light was filtered by a Fabry-Perot interferometer and subsequently detected with a spectrograph. The resulting hyperfine splittings and isotope shifts are listed in Table I and have uncertainties between 2 and 18 MHz. A similar experiment was performed by Chaiko (1970) using a somewhat more sophisticated spectrometer. His results have slightly smaller uncertainties and are in good agreement with the data obtained by Ross (19631, with the single exception of the "'Yb transition from the F = $ ground state hyperfine level to the F = $ hyperfine level of the excited state. The identification of this line is complicated by the very close proximity of a transition occurring in 171 Yb. Later, more accurate experiments have shown Chaiko's result to be in error. A significant improvement in resolution was obtained by using narrow linewidth lasers to excite atoms in an atomic beam. This greatly reduced the collisional broadening since the atoms travel in a vacuum. The laser and atomic beams intersect orthogonally to eliminate the first order Doppler shift. This method is ideally suited for studying transitions to short-lived excited states, since lasers generate high fluxes of photons and the peak cross section for resonance scattering given by
is large at optical wavelengths A. Hence, the fluorescence produced by the radiative decay of the excited state can be readily detected. A typical experiment is illustrated in Fig. 1. A n atomic beam consisting of two stable isotopes is considered. The laser frequency is scanned across the resonance while fluorescence is detected by a photomultiplier. The width of the observed spectral lines depends on various factors including the laser frequency jitter, the natural linewidth of the transition, and the laser power. Laser linewidths of 0.5 MHz are commonly achieved by commercial ring dye lasers that are electronically stabilized using an external cavity. The natural linewidth (FWHM) of a transition is given by Av,,,
1 = -
2TT
TABLE I ISOTOPE SHIFTS OF THE YTTERBIUM (6s)' IS,
Position relative to ~
+ ( 6 ~ 6 p ) ~ TRANSITION P, AT 555.6 NM 1
7
6 (MHZ) ~
~~
Spectral Line 173(5/2 + 7/2) 171(1/2 + 1/2) 176 174 172 170 173(5/2 + 5/21 168 171(1/2 -+ 3/21 173(5/2 + 3/2)
Ross (1963) -1460 f 6 - 1187 f 3 0 947 f 2 1940 f 9 3229 f 18 3247 f 9 4581 f 15 4761 f 9 4749 12
*
Chaiko (1970) -1451 f 6 -1184 f 6 0 947 + 2 1940 k 9 3232 f 6 3253 f 6 4761 f 9 4700 f 6
Broadhurst et al. (1974)
- 1434.2 f 0.6 - 1176.1 f 0.6 0 954.6 f 0.6 1955.9 f 0.6 3242.6 f 0.6 3267.2 f 0.6 4610.6 f 0.6 4758.7 f 0.6 4764.7 f 0.6
Clark et al. (1979)
- 1431.7 f 0.5 - 1177.2 f 0.5 0 954.8 k 0.5 1955.0 f 0.5 3241.5 f 0.5 3266.5 f 0.5 4610.1 f 0.5 4759.8 f 0.5 4760.4 f 0.5
Jin et al. (1991)
- 1432.5 f 4.3 - 1176.6 f 3.5 0 953.0 f 1.3 1954.0 f 2.0 3239.6 f 3.2 3264.3 f 9.8 4609.5 f 4.6 4759.5 f 14.3 4759.5 f 14.3
van Wijngaarden and Li (1994a) -1432.6 f 1.2 - 1177.3 f 1.1 0 954.2 f 0.9 1954.8 f 1.6 3241.5 f 2.8 3267.1 f 2.8 4611.9 f 4.4 4761.8 f 3.7 4761.8 f 3.7
3
a
C
a
3
25
5n
p
PRECISION LASER SPECTROSCOPY
145
FIG. 1. Laser atomic beam spectroscopy using a Fabry-Perot etalon. See text for a detailed discussion.
where T is the radiative lifetime of the excited state. For the ytterbium ( 6 ~ 6 p ) ~ Pstate, , the lifetime has been found to be 8.27 X lo-’ sec (Baumann and Wandel, 19661, corresponding to a natural linewidth of 190 kHz. This is much smaller than is typical for an electric dipole allowed decay to the ground state. For example, the sodium D , line has a linewidth of 9.7 MHz (Carlsson et al., 1992). Measurements should also be taken at low laser powers to avoid power broadening and to minimize optical pumping effects (Baird et al., 1979). The change in laser frequency is monitored by passing part of the laser beam through a Fabry-Perot etalon. The laser is transmitted through the interferometer, producing a so-called frequency marker whenever the laser frequency changes by an amount equal to the cavity’s free spectral range vFSR.For a confocal etalon, the latter is given by c VFSR =
(3)
W A. uan Wjngaarden
146
Here, c is the speed of light, n is the index of refraction of the material (usually air) occupying the interferometer, and L is the cavity length. Hence, the etalon length must be carefully measured and be stabilized against vibrations for accurate frequency calibration. This can be done using an HeNe laser, the wavelength of which is locked to an atomic transition in iodine as is illustrated in Fig. 2. A photodetector monitors the transmission of the HeNe laser through the interferometer and feeds back a servo signal to a piezoelectric crystal on which a cavity mirror is mounted (Clark et al., 1979). The mirror position is adjusted to keep the Fabry-Perot etalon locked to the laser wavelength.
Lt
HeNe Laser
--
1
Laser Frequency Servo
<
FIG. 2. Interferometer stabilization apparatus. The length of the Fabry-Perot etalon is stabilized using an HeNe laser, the wavelength of which is locked to an iodine transition.
147
PRECISION LASER SPECTROSCOPY
The accuracy of the frequency determination between the frequency markers is limited by nonlinearities of the laser wavelength scan. This may be further complicated by temporal fluctuations of the laser power that affect the amplitude of the frequency markers. The calibration can be improved by decreasing the free spectral range. However, cavities with lengths in excess of two meters are impractical. Baird et al. (1979) attempted to resolve this problem in their study of the hyperfine structure of the barium (6s)' 'So + ( 6 s 6 p ) ' P , transition by interpolating the frequency using a polynomial function to fit the marker positions. They report this permitted the determination of the laser frequency throughout the scan with an uncertainty of 0.25 MHz. An alternative approach has been developed by Broadhurst et al. (1974) in their study of the ytterbium (6s)* 'So + ( 6 s 6 p ) 3P1 transition. The laser frequency was monitored using a confocal Fabry-Perot etalon having a free spectral range of 300.31 f 0.02 MHz. The cavity length was stabilized to two parts in 10'" using an HeNe laser as is illustrated in Fig. 2. The interferometer length L therefore satisfied mhHeNe = 4nL
(4)
where m is the order of interference, A H e N e = 632.8 nm, and n is the index of refraction of the air occupying the cavity. The etalon length could be changed rapidly and reproducibly corresponding to a change of the interference order to m + 1. This was done immediately after a transmission maxima of the dye laser was observed during a scan of the dye laser wavelength across the ytterbium resonance. Hence, the Fabry-Perot etalon acted as two cavities and generated a double set of frequency markers separated by
For the ytterbium transition, hdye= 555.6 nm and A = 41.7 MHz. A complication was caused by air pressure fluctuations that perturbed the index of refraction differently at the dye laser and HeNe laser wavelengths. This affected the location of the frequency markers that were generated by the dye laser, whereas the cavity length was locked using the HeNe laser. The results listed in Table I (Broadhurst et al., 1974) were believed to have been corrected for these effects. However, a subsequent experiment by the same group (Clark et al., 1979) enclosed the interferometer in a vacuum chamber and obtained data that differed from the earlier results by as much as 4 MHz. This work also obtained improved values for the positions of the closely spaced 17'Yb ( F = + +F = and 173Yb
3)
148
W. A. van Wijngaarden
( F = $+ F = $1 lines using a magnetic field of 102 G. The zero field positions of these lines were found by computing the Zeeman shifts. Table I also lists measurements made by Jin et al. (1991). Their experiment is similar to that of Clark et al. (1979) except that their Fabry-Perot etalon only generated frequency markers every 297.7 MHz. They claim an accuracy of 0.1-0.2% for the even isotope shifts and 0.3-0.4% for the hyperfine splittings of the odd isotopes. The uncertainties listed in Table I were computed using their lower error estimates.
111. Spectroscopy Using Frequency-ModulatedLasers A. OPTICALMODULATORS Significant technological advances in materials during recent years have produced acousto-optic and electro-optic modulators that can frequency shift infrared, visible, and even some ultraviolet laser wavelengths. These modulators have found important uses in optical communications and in laser spectroscopy. Various articles describing the theory and applications of these devices have been written, including ones by Korpel (19881, Tran (19921, and Xu and Stroud (1992). This section focuses on making precision measurements using acousto-optic modulators. Many of the techniques presented can be readily adapted to use electro-optic modulators. Various groups have employed the latter devices to determine isotope shifts in mercury (Rayman et al., 1989), helium (Zhao et al., 1991), ytterbium (Deilamian et al., 1993), and hydrogen (Schmidt-Kaler et al., 1993). The operation of an acousto-optic modulator is briefly summarized in Fig. 3. A modulation frequency vAo is amplified and applied to a transducer. The latter generates a pressure or sound wave inside the acoustooptic material that alters the refractive index. This produces a moving grating that diffracts portions of an incident laser beam at frequency v. The frequencies of the outgoing laser beams are given by Vd =
v f nv,,
(6)
where n is an integer. The trajectory of the unshifted laser beam is unaffected, whereas the frequency-shifted laser beams are spatially deflected by up to a few milliradians. The amount of light diffracted into the various modes depends on the power of the modulation signal. In our experiments, a tellurium oxide crystal (Brimrose TEF 27-10) was used that had a diffraction efficiency of over 50% for shifting light at 590 nm by 220-320 MHz. The modulation signal was supplied by a computer control-
149
PRECISION LASER SPECTROSCOPY F'requrmcy Svntht?sizer
Amplifier . u - ' VAO
U
U + UAO
Pressure Wavefront in Acousto-optic Modulator FIG. 3. Operation of an acousto-optic modulator. Part of a laser beam of frequency u is shifted to produce diffracted beams at frequencies ud = u nvAo, where n is an integer and vAo is the modulatior. frequency.
*
lable frequency synthesizer (Hewlett Packard 8645A) with an accuracy of one part per million. This signal was then amplified to a power of 1 W. Acousto-optic modulators capable of frequency shifting light up to several gigahertz are commercially available. Their operation is straightfonvard and the technical requirements are relatively undemanding although some units require substantial power and may therefore need to be water cooled. B. GENERAL EXPERIMENTAL ARRANGEMENT
This section discusses various ways in which an acousto-optically modulated laser beam can be used to determine a frequency interval such as an isotope shift. An atomic beam is considered rather than a cell, since the fluorescent signals obtained using the latter are Doppler and collisionally broadened and therefore have inherently lower resolution. Figure 4(a) shows an atomic beam consisting of two isotopes that have transitions separated by frequency A vIso. An acousto-optic modulator (A01 frequency shifts part of a laser beam at frequency v by an amount vA0. The two laser beams at frequencies v and v - vAo are superimposed and intersect the atomic beam orthogonally. Fluorescence is monitored by a detector (Det) as the laser frequency is scanned across the resonance.
150
W A. van Wjngaarden 0 0
Isotope I Isotope 2
Atomic Beam 0
0
I#
-
0
0
0
0
.
0
0
0
0
0
0
0
22
72 0
0
0
TY 0
F
-
0
0
0
I !!
I v-vAO Laser
-
-Each isotope is excited once by each of the two laser beams, producing a double spectrum. The frequency axis can then be calibrated using the modulation frequency uAo, which separates each pair of peaks. The advantage of this method is its simplicity. Only a single laser and one detector are required. It is critical however, that the frequency-shifted and unshifted laser beams be collinear. A misalignment by an angle 8 produces a residual shift "res
=
"Dop
(7)
PRECISION LASER SPECTROSCOPY
151
For 8 equal to 1 mrad and a Doppler width of 0.5 GHz, Avres = 0.5 MHz. This affects the frequency calibration, since each pair of peaks in the fluorescent spectrum is then separated by vAo + Av,,,. The laser beams can be suitably collimated using alignment slits or by passing the laser beams through a single mode optical fiber (Tanner and Wieman, 1988a). Note that this problem does not occur when using an electro-optic modulator, which generates frequency sidebands that are not spatially separated. The accuracy of the experiment outlined in Fig. 4(a) is limited by nonlinearities of the laser frequency scan, which also affected experiments that use a Fabry-Perot etalon to calibrate the frequency as has been discussed in Section 11. However, the modulation frequency applied to the optical modulator can be changed quickly and conveniently, unlike the free spectral range of an interferometer. Hence, tests can be readily made to check for the existence of scanning nonlinearities. A second experimental setup is shown in Fig. 4(b). Two laser beams at frequencies v and v - vAo intersect the atomic beam at different locations. Fluorescence is monitored by two detectors (Det l and Det 2 ) as the laser frequency is scanned across the resonance. The isotope shift is determined by finding the modulation frequency vAo such that the two isotopes are simultaneously excited by the two laser beams. This method has the advantage of being relatively insensitive to any nonlinearity of the laser scan. However, two detectors are needed, and the acousto-optic modulation frequency must match the isotope shift. The latter requirement may not be possible in the case of a large isotope shift, since the synthesizer and amplifier needed to generate modulation frequencies in excess of 2 GHz are relatively expensive. A third option is to perform a so-called heterodyne experiment (Walther, 1974) as is shown in Fig. 4(c). The two isotopes are excited by the laser beams at frequencies v and v - vA0 at separate locations along the atomic beam. The signals obtained by the two detectors are used to lock the lasers to the transition frequencies. The isotope shift then equals the modulation frequency. This method has the advantage of obtaining data faster than either of the methods illustrated in Figs. 4(a) and 4(b) since the laser frequency does not need to be scanned across the resonance. It does, however, require fast electronic servo circuits that can accurately lock the laser frequency to the center of an atomic transition. A number of variations of this scheme exist. Some groups have locked two separate lasers to the different transitions (Hunter el al., 1991). The .frequency shift is then found by measuring the beat frequency of the two laser beams. This approach is in general only affordable when the transition can be excited using relatively inexpensive diode lasers.
152
W. A. uan Wijngaarden
The ultimate accuracy of an experiment that measures frequency shifts is limited by the natural linewidth of the transition. Very few groups have claimed the ability to determine the center of a resonance line to better than one part of a thousand of the linewidth, which for the sodium D , line represents only 10 kHz. A number of sometimes subtle effects including optical pumping (Holmes and Griffith, 1999, stray magnetic and electric fields, light shifts (Mathur et al., 1968), second order Doppler shifts, and the presence of nearby lines (van Wijngaarden and Li, 1995) can distort the natural Lorentzian lineshape and shift the observed line center away from the transition frequency.
IV. Hyperfine Structure and Isotope Shifts A. BACKGROUND Measurements of hyperfine structure and isotope shifts have yielded important information about nuclear structure (Arimondo et al., 1977). The hyperfine Hamiltonian consists of the magnetic dipole H M D and the electric quadrupole EEo terms: Hhyp
= HMD
+ HEXI
(8)
describes the interaction of the nuclear magnetic moment with the magnetic field generated by the electrons. It is given by
HMD
-1-
H M D = ahZ. J
(9)
where a is called the magnetic dipole constant, h is Planck’s constant, f i s the nuclear spin, and J”is the total electronic angular momentum. The electric quadrupole term originates from the Coulomb interaction between the electrons and a nonspherically symmetric nucleus. It is given by
HEQ=
-[
+
3 ( T - J ) ‘ (3/2)( T * f )- T’J”’ 21(21 - 1 ) J ( 2 J - 1)
where b is called the electric quadrupole moment coupling constant. This expression is valid only if the nuclear spin Z 2 1 and is zero otherwise. The eigenstates of the hyperfine Hamiltonian are designated by IFm, JZ)
153
PRECISION LASER SPECTROSCOPY
yhere m F is the azimuthal component of the angular momentum I. The corresponding eigenenergies are E,
where K
=
= F(F
F' = .?+
+
K 3 K ( K + 1) - 41( 1 1 ) J ( J + 1) ah - + bh 2 81(21 - 1 ) J ( 2 J - 1)
(11)
+ 1) - Z ( 1 + 1) - J ( J + 1).
B. YTTERBIUM ( 6 s 6 p ) 3P, STATE An experiment similar to that outlined in Fig. 4(a) determined the hyper-
fine structure and isotope shifts of the ytterbium (6s)*'So + ( 6 ~ 6 p ) ~ P , transition (van Wijngaarden and Li, 1994a). The apparatus is illustrated in Fig. 5. An atomic beam was generated by heating the ytterbium metal close to its melting point of 819 K. The atoms were collimated by a series of silts producing a beam having a divergence of about 1 mrad. The atomic beam traveled in a vacuum chamber that was pumped to a pressure of about 1 x lo-' Torr using a diffusion pump.
Amplifier
Chopper Ref. PM I
\
r
Ar+Laser
+
RingDye Laser
,
yy
3 '
A.O.Mod.
U +%
I
Frequency Synthesizer
FIG. 5. Apparatus for studying hyperfine and isotope shifts of the ytterbium (6s6pI3P, state. Details are given in the text (van Wijngaarden and Li, 1994a, by permission).
154
W.A. van Wijngaarden
The laser light was generated by a ring dye laser (Coherent 699) that was pumped by an argon ion laser. The dye laser at frequency v was frequency shifted by an acousto-optic modulator. The frequency-shifted and unshifted laser beams were superimposed using slits placed on either side of the vacuum chamber. The two laser beams were estimated to be aligned to better than 0.3 mrad. Fluorescence was detected by a photomultiplier and recorded by a digital lock-in amplifier (Stanford Research 850). The lock-in reference signal was provided by a chopper that modulated the laser light at a frequency of about 2 kHz. The demodulated signal was digitized by the lock-in at a rate of 128 Hz. The laser was scanned across the resonance at a speed of about 50 MHz/sec. Hence, a typical 7-GHz scan shown in Fig. 6 took about 2.5 min and consisted of nearly 20,000 data points. The lock-in fitted a Gaussian function to each peak to determine the position of its line center. The data shown in Fig. 6 were taken using an acousto-optic modulation frequency of 300.000 MHz. The laser power was attenuated to about 1 mW
N
2
W
s;
1 F
i
2
3
4
5
6
Laser Requency ( GHz )
FIG. 6. Excitation of the ytterbium (6s)* IS, + ( 6 ~ 63Pl ~ ) transition using frequencyshifted and unshifted laser beams (van Wijngaarden and Li, 1994a, by permission).
155
PRECISION LASER SPECTROSCOPY
with neutral density filters to reduce the power broadening of the line. The observed linewidth (FWHM) of the spectral lines was about 7 MHz. The frequency was calibrated as follows. First, the average number of points separating the eight pairs of peaks was determined for each wavelength scan. This number was then divided into 300.000 MHz to find the frequency interval between neighboring points. The frequency calibration was affected by the laser scan speed, which varied by up to 0.1% from scan to scan. Figure 7 displays a histogram showing the number of times a particular interval separating a pair of peaks generated by the frequencyshifted and unshifted laser beams was observed. The data are fit well by a Gaussian function, as would be expected for random fluctuations of the data about an average value. It was concluded that the laser frequency
Gaussian Center is at 811.9 f 0.7 pts.
I
8iO
860
880
Number of Data Points Corresponding to 300.000 MHz FIG. 7. Frequency calibration check of the laser scan of the ytterbium (6s)*'S0 + ( 6 ~ 63P, ~ )transition. The number of data points corresponding to the acousto-optic modulation frequency of 300.000 MHz was determined for 306 pairs of peaks obtained from 40 separate laser scans. The Gaussian function is centered about the average value of 811.9 0.7 data points. The frequency interval between two neighboring points is therefore 0.3695 0.0003 MHz (Li, 1995, by permission).
*
+
156
W A. van Wungaarden
jittered randomly and that no systematic scanning nonlinearity was observed at the part in a thousand level. The isotope shifts were obtained by averaging the data collected in 40 separate wavelength scans. The uncertainty of each result listed in Table I equals one standard deviation of the data about its average value. The results are consistent with the most accurate data obtained by Clark et al. (1979). The experimental accuracy is limited by the variation of the laser scan speed. The uncertainty of the isotope shifts exceed the transition's natural linewidth of 190 kHz. Hence, future experiments yielding significantly more accurate results are possible. The hyperfine coupling constants for 171,173 Yb can be extracted from the data given in Table I using Eq. (11). The resulting values of the magnetic dipole and electric quadrupole coupling constants obtained by Clark et al. (1979) and Li (1995) are listed in Table 11. The hyperfine structure has also been examined by several groups using the level crossing and optical double resonance techniques. Baumann et al. (1969) excited an atomic beam of ytterbium using light generated by a hollow cathode lamp. The light was directed in the x direction and was linearly polarized along the y axis. A magnet generated a field in the z direction. The field strength was measured using a proton resonance magnetometer. A photomultiplier detected fluorescence that was emitted in the y direction and linearly polarized parallel to the x axis. The magnetic field mixes the F = i, hyperfine levels of the 171 Yb ( 6 ~ 6 p ) ~ Pstate , and also the F = t, hyperfine levels of the 173Yb( 6 . ~ 63P, ~ ) state. The fluorescence polarization changes when two levels have the same energy, which is called a level crossing. The magnetic dipole hypefine constants were found by measur-
3 2,
TABLE I1 HYPERFINE CONSTANTS OF THE (6~6p)~P, STATE OF I7'YB AND '73YB
Hyperfine Constants (MHz) a('7'Yb)
3959.1 f 1.4 3957.72 f 0.11 3958.23 k 0.06 3957.97 0.47 3959.1 f 3.0
*
a(I7'Yb) - 1094.7 f
0.6
- 1094.35f 0.03 - 1094.32 f 0.04
-1094.20 f 0.60 - 1094.44f 0.84
b('73Yb)
Reference
- 826.9 + 0.9 -826.59 + 0.20 -825.90 0.09 -827.15 f 0.47 - 827.89 f 0.85
Baumann et al. (1969) Budick and Snir (1970) Wandel (1970) Clark etal. (1979) Li (1995)
+
PRECISION LASER SPECTROSCOPY
157
ing the magnetic fields H, at which the level crossings occurred and using the relations a(I7lYb)
H,(171Yb)
=
O.99987gJ
=
-O.33705gJ -Hc(173Yb) h
h
PB
Here, p B is the Bohr magneton and g, is the gyromagnetic factor. The latter quantity was determined in an optical double resonance experiment (Baumann and Wandel, 1968) that selectively populated the m, = 0 Zeeman sublevel of the (6s6p) 3P, state using light polarized parallel to the magnetic field. A radio frequency field was then applied that transferred some of the atoms to the m, = k 1 sublevels of the excited state. The gyromagnetic factor was determined by measuring the radiofrequency and the corresponding magnetic field strength. The result of 1.49285 k 0.00005 for g, is in excellent agreement with the value 1.49280 & 0.00004 that was obtained in a nearly identical experiment (Budick and Snir, 1967). Baumann et af. (1969) determined the electric quadrupole constant b('73Yb) using an optical double resonance experiment that measured the energy separating the F = and F = $ hyperfine levels of the ( 6 ~ 6 p ) ~ state P , at zero magnetic field. An improved level crossing experiment has been done by Budick and Snir (1970). The size of the fluorescent signal was increased by loading isotopically enriched ytterbium in the atomic beam oven. In addition, a more homogeneous magnetic field claimed to be uniform to one part in lo5 over a volume of 1 in.3 was used. This reduced the linewidth of the level crossing signals to 0.14 G, which is close to the limit set by the natural linewidth. The hyperfine coupling constants have also been determined by Wandel (1970) using an optical double resonance experiment that measured the splittings between the various hyperfine levels of the ( 6 ~ 6 p ) ~ state f ' ~ at zero magnetic field. His results shown in Table I1 claim a slightly improved accuracy compared with those obtained by Budick and Snir (1970). It is disturbing however, that the results obtained by these two experiments for ~ ( ' ~ ' y and b ) b(I7'Yb) disagree. The differences of about 0.5 MHz correspond to the uncertainty obtained in the experiment of Clark et al. (1979) that measured the frequency shifts using an optical spectroscopic method. Hence, an improved measurement such as outlined in Fig. 4(b) or 4(c)
158
W.A. van Wijngaarden
would be useful to help resolve the discrepancy between the data obtained by the level crossing and optical double resonance techniques. C. SODIUM 3P1/2 STATE
The hyperfine structure of the sodium 3P1,2 state has been intensively studied by many researchers using a variety of spectroscopic methods. A recent experiment (van Wijngaarden and Li, 1994b) scanned a laser across the D , line while fluorescence produced by the radiative decay of the 3P,,, state was detected, This line consists of four transitions between the F = 1 and F = 2 hyperfine levels of the 3S,,, and 3P,,, states as is illustrated in Fig. 8. A sample fluorescence signal is shown in Fig. 9. The frequency scan was calibrated using the ground state hyperfine splitting, which separates the first and third peaks as well as the second and fourth peaks appearing in the spectrum. The splitting between the hyperfine
F
2t A
- e
1772 MHz
1-
-
#
FIG. 8. Hyperfine levels of the sodium 3S,/,and 3 P , / , states.
159
PRECISION LASER SPECTROSCOPY
c 0.0
0.5
1.o
1.5
2.0
2.!
Laser Frequency ( GHz ) FIG. 9. Laser excitation of the sodium D , line. These data were taken using an apparatus similar to that shown in Fig. 5. Neutral density filters attenuated the laser power to reduce the power broadening of the line. The spectral linewidth was observed to be 12 MHz (FWHM), which compares with the natural linewidth of 9.7 MHz (van Wijngaarden and Li, 1994b, by permission).
levels of the 3S,,, state has been determined using Ramsey’s method of separated oscillatory fields (Ramsey, 1956) to be 1771.626129 f (1 X MHz (Beckmann et al., 1974). This result is significantly more accurate than any frequency generated by a commercial synthesizer, and therefore an acousto-optically modulated laser beam was not used to calibrate the laser scan. The experiment was, however, similar to that illustrated in Fig. 4(a) and therefore potentially suffered from similar systematic effects including scanning nonlinearities of the laser frequency. The resulting value of 94.44 + 0.13 MHz for the magnetic dipole hyperfine constant of the 3 P , , , state was found by averaging data produced by 200 separate wavelength scans. The uncertainty represents one standard deviation of the data about the average value. A comparison of this result with data found using other spectroscopic techniques therefore provides a strong test
160
W.A. uan Wijngaarden hhCNETIC
TABLE 111 DIPOLE HYPERFINE CONSTANT OF THE SODIUM 3P,,,
STATE
~
a (MHz)
94.45 f 0.50 94.3 f 0.2 94.25 f 0.15 94.465 f 0.010 94.42
* 0.19
94.05 f 0.20 94.44 f 0.13 94.4065 f 0.0015 91.7 91.40 93.02 88.56
Method Atomic Beam Magnetic Resonance Optical Double Resonance Saturation Spectroscopy Optical Heterodyne Quantum Beat Spectroscopy Fabry-Perot Spectroscopy Modulated Laser Spectroscopy Optical Heterodyne Many-Body Perturbation Theory Many-Body Perturbation Theory Many-Body Perturbation Theory Multiconfiguration Hartree-Fock Theory
Reference Per1 et al. (1955) Hartmann (1970) Pescht et al. (1977) Griffith et al. (1977) Carlsson el al. (1992) Windholtz et al. (1992) van Wijngaarden and Li (1994b) Young and Griffith (1994) Lindgren et al. (1976)
Johnson et al. (1987) Salomonson and Ynnerman (1991) Carlsson et al. (1992)
of the validity of using modulated laser beams to measure frequency intervals. Table I11 lists the most accurate data found in the literature for the magnetic dipole hyperfine constant of the 3P,,, state. Per1 et al. (1955) used the so-called atomic beam magnetic resonance method. A beam of sodium atoms was passed through three successive regions, A, B, and C. Atoms occupying the m, = - $ ground state Zeeman sublevel were deflected out of the beam in region A by an inhomogeneous magnetic field and subsequently blocked by a beam stop. In the intermediate region B, light generated by a lamp excited some of the atoms to the 3 P , / , state in the presence of a uniform magnetic field. A radiofrequency field was also applied, which induced transitions among the levels of the excited state. Some of the atoms decayed to the m, = - 3 ground state Zeeman sublevel and were focused onto a hot wire detector in region C by an inhomogeneous magnetic field. The beam intensity was then monitored as
PRECISION LASER SPECTROSCOPY
161
a function of the radiofrequency. This experiment is analogous to an optical double resonance experiment where a change of the beam intensity corresponds to a change of the fluorescent polarization. The accuracy was limited by various effects including a relatively poor signal to noise ratio as compared with later experiments that monitored fluorescence generated by using lasers to excite the 3P1,, state. Hartmann (1970) studied the 3P,,, state using an optical double resonance experiment. A lamp excited sodium atoms contained in a vapor cell. The atoms were subjected to magnetic fields of up to 5000 G that were measured using a proton resonance magnetometer. The field was uniform to one part in lo4. The hyperfine coupling constant was determined by measuring the radiofrequency needed to induce a transition between the levels of the excited 3P1,, state. The experiment also obtained a value for the gyromagnetic ratio g , ( 3 P l I 2 )= 0.66581 k 0.00012, which agrees well with the result of 0.66589 calculated using the Land6 formula (Corney, 1977). A saturation spectroscopy experiment has been done by Pescht et al. (1977). Two laser beams originating from the same dye laser were incident on a sodium cell. One laser beam saturated the 3S,,, + 3P,,, transition while the transmission of the second so-called probe laser through the cell was measured. The dye laser was tuned across the D , resonance, and a reduction in the absorption of the probe laser occurred whenever the first laser excited a hyperfine level of the 3P,,, state. The change in laser wavelength was initially monitored using a Fabry-Perot etalon having a free spectral range of 150 MHz. This permitted the hyperfine splitting of the 3P1,, state to be determined with an uncertainty of a few megahertz. The experimental accuracy was limited by temperature fluctuations that affected the interferometer’s optical length. An improved frequency calibration was performed using two HeNe lasers. The dye laser was locked to the Fabry-Perot cavity, which in turn was locked to a free running HeNe laser. This HeNe laser was frequency offset from a second HeNe laser, the frequency of which was lamp-dip stabilized to +50 kHz. The dye laser was then tuned by varying the frequency offset of the two HeNe lasers, which was measured using a frequency counter. Griffith et al. (1977) used an optical heterodyne experiment to measure the hyperfine structure. Two separate dye lasers were superimposed and excited an atomic beam at the same location. Fluorescence was detected by a single photomultiplier. The resulting signal was used to lock the two separate lasers to the 3S,,,(F = 2) + 3P,,,(F = 1) and the 3S,,,(F = 2) + 3P1,,(F = 2) transition frequencies, respectively. Portions of the two laser beams were then focused onto a fast photodiode, which measured the
162
W. A. van Wijngaarden
beat frequency. The magnetic dipole constant of the 3P,,z. state was found to be 94.46 f 0.01 MHz. This uncertainty is impressive since the authors mention that an angular separation of only 0.1 mrad between the two laser beams would produce an error of 0.1 MHz. This effect should reverse when the lasers are interchanged between the two transitions. Hence, the final value for the magnetic dipole constant is the average of equal numbers of beat frequency readings obtained with the first laser exciting initially the F = 1 the later the F = 2 hyperfine levels of the 3P,,, state. A recent conference proceeding (Young and Griffith, 1994) by the same group reports a value of 94.4065 f 0.0015 MHz for a(3P,,,). The ground state hyperfine splitting was also measured and found to be within a few kilohertz of the accepted value found using Ramsey’s method of oscillatory magnetic fields. This improved version of their earlier work used the saturated absorption signal observed in a sodium vapor cell to lock the frequency of one laser while the second laser excited an atomic beam. The 1.5 kHz claimed uncertainty is less than one part in 5000 of the natural linewidth of the sodium D , line. Unfortunately, no reason is given to explain why the results of their two experiments disagree by more than five times the standard deviation given in their earlier work. It is hoped that a more detailed account of their experiment will soon be published. Carlsson et al. (1992) studied the hyperfine structure using quantum beat spectroscopy. A mode-locked dye laser produced 6-psec pulses at a repetition rate of a few megahertz. The laser excited an atomic beam, producing an excited state that was a superposition of the hyperfine levels of the 3P1,z state. The resulting temporal oscillations of the fluorescent intensity decay curve were measured using photon counting. The laser pulse was detected by a photodiode, whereas the fluorescence was monitored by a Peltier-cooled microchannel plate photomultiplier tube. The time difference of these two signals was measured with a time to amplitude converter (TAC). The high signal to noise ratio of their data was demonstrated by a sample fluorescent decay curve that displays nearly 25 clearly visible temporal oscillations. Values of 94.363 f 0.054 and 94.485 f 0.031 MHz were found for the magnetic dipole hyperfine constant & P ~ / Z ) using TAC settings of 200 and 500 nsec, respectively. These two results were averaged and the 0.15% uncertainty of the time scale was included to give 94.42 k 0.19 MHz. The experiment was checked by determining the radiative lifetime of the 3P,,, state. A value of 16.35 f 0.06 nsec was obtained, which agrees very well with 16.38 f 0.08 nsec found previously by Carlsson (1988) using the same technique. Two other experiments using the so-called fast beam laser technique have obtained values of 16.40 f 0.03 (Gaupp et al., 1982) and 16.40 f 0.05 nsec (Schmoranzer et al., 1979).
PRECISION LASER SPECTROSCOPY
163
It should be noted that these data are among the most accurate lifetimes published in the literature. Umfer et al. (1992) studied the effects on the sodium D , line of magnetic fields of up to 10,000 G. A ring dye laser excited a sodium atomic beam. Spectra were obtained by scanning the laser frequency across the D,line and observing the fluorescence detected by a photomultiplier. The laser scan was calibrated by passing a portion of the laser beam through an etalon having a free spectral range of 197.5974 f 0.0003 MHz. The fluorescent peaks had a linewidth (FWHM) of 16 MHz. The magnetic field was determined using a nuclear magnetic resonance gaussmeter with an accuracy of better than 0.1 G. The magnetic field mixes the hyperfine levels of the excited state and thereby changes the various transition frequencies making up the D ,line. An analysis of the spectra determined a(3P,/,) to be 94.05 f 0.20 MHz. Table I11 lists the values found for the magnetic dipole hyperfine constant of the sodium 3P,,, state by seven independent research groups, which each used a different measurement technique. All the data are in agreement except the two conflicting results obtained by the group of Griffiths et al. This supports the validity of the experimental method used by van Wijngaarden and Li (1994b), which is closely analogous to that shown in Fig. 4(a). The theoretical estimates are in poor agreement with the measured values. Three independent many-body perturbation theory calculations have yielded values of 91.7 (Lindgren et al., 1976) 91.40 (Johnson et al., 19871, and 93.02 MHz (Salomonson and Ynnerman, 1991) for a(3P,/,). Carlsson et al. (1992) used wavefunctions found using a multiconfiguration Hartree-Fock computer code to obtain a value of 88.56 MHz. The discrepancy between theory and experiment indicates the need for improved theoretical models of many-body systems such as the sodium atom. D. CESIUM6P,,, STATE A measurement of the hyperfine structure of the cesium 6P3/2 state has been made by Tanner and Wieman (1988b) using the approach outlined in Fig. 4(b). Their apparatus is shown in Fig. 10. Light at 852 nm corresponding to the 6S,,! + 6P3/, transition in cesium was generated by a diode laser. The linewidth of the latter was reduced to about 20 kHz by passively locking the laser to a Fabry-Perot etalon external to the laser. Part of the laser beam was sent into a saturation spectrometer. The resulting signal was used to lock the external cavity so that the laser excited one of the hyperfine levels of the 6P3,., state. The remaining portion of the laser beam was frequency shifted by an acousto-optic modulator. The
W. A. van Wijngaarden
164
n
Saturation SDectrometer
”
*
Vacuum
Aox Optical
FIG. 10. Apparatus for studying the hyperfine structure of the cesium 6P3,2 state. Details are given in the text (adapted from Tanner and Wieman, 1988a, by permission).
frequency-shifted beam passes through an optical fiber and intersected the atomic beam orthogonally. Fluorescence was detected by a cooled silicon photodiode. The acousto-optic modulation frequency was adjusted to bring the light into resonance with the various hyperfine levels of the 6P-312 state. Data were taken using a laser intensity of 13 pW/cm2 to reduce optical pumping effects, which can shift the resonance in the presence of stray magnetic fields. The results obtained for the hyperfine level splittings are given in Fig. 11, and the magnetic dipole and electric quadrupole hyperfine constants are listed in Table IV. The measurement precision was limited by the drift of the locked laser frequency. Table IV also lists data obtained by several other groups. The hyperfine structure of the cesium 6P312 state was first studied by Buck et al. (1956). The method and apparatus were identical to that used to study the sodium 3P,,, hyperfine structure, which has been discussed in Section 1V.C (Per1 et al., 1955). A computational mistake in their paper was discovered by Violino (1969). Table IV lists the corrected values for the hyperfine constants. Three level crossing experiments have also examined the 6P312 state (Kallas et al., 1965; Violino, 1969; Svanberg and Rydberg, 1969). All of them used light generated by a lamp to excite cesium atoms in a cell. The results of Kallas et al. (1965) disagree with theory according to Arimondo
PRECISION LASER SPECTROSCOPY
F=5
165
1 5a
+ 5 b = 251.00 f 0.02 MHz
4~
- $ b = 201.24 f0.02
3 3~
- + b = 151.21 f0.02
t
FIG. 11. Measured hyperfine splittings of the cesium 6P3,, state (Tanner and Wieman, 1988b, by permission).
et al. (19771, leading them to conclude that the error was underestimated. Svanberg and Rydberg (1969) analyzed their data using a value for the gyromagnetic ratio of 1.345 that disagrees with the theoretical value of 1.3341 as well as a subsequent experimental measurement of 1.3340 & 0.0003 (Abele, 1975). Their experiment yields a value of a(6P3,,) = 50.31 f 0.04 MHz when the corrected value for g, is used. Two optical double resonance experiments have also been done. The experiments were similar to that carried out in sodium by Hartmann (19701, which has been described in Section 1V.C. Svanberg and Belin (1972) determined the frequency for transitions between hyperfine levels of the 6P3,* state in a zero magnetic field. The data was extrapolated to zero radiofrequency power. The experiment of Abele (1975) was done using a magnetic field of 1300 G. Data obtained by the various experiments are in reasonable agreement. However, the data of Tanner and Wieman (1988b) are 10 times more precise than that of the previous best measurement. The experimental
U? A. van Wijngaarden
166
TABLE IV HYPERFINE CONSTANTS OF THE CESIUM 6P3,Z ~
~
~
_
_
_
_
~
a (MHz)
50.67 f 0.11 50.9 f 0.5 50.45 f 0.08 50.72 k 0.03 50.31 k 0.05 50.02 f 0.25 50.275 f 0.003 49.785
_ ~
_
STATE
_
b (MHz)
Method
-0.46 f 0.53 Atomic Beam Magnetic Resonance -0.9 0.6 Level Crossing - 0.66 f 0.72 Level Crossing -0.38 f 0.18 Level Crossing - 0.30 k 0.33 Optical Double Resonance Optical Double Resonance - 0.53 k 0.02 Modulated Laser Spectroscopy Many-Body Perturbation Theory
+
Reference Buck et al. (1956) Kallas et al. (1965) Violino (1969) Svanberg and Rydberg (1969) Svanberg and Belin (1972) Abele (1975) Tanner and Wieman (1988b) Blundell et al. (1991)
results are substantially more accurate than a many-body perturbation theory computation (Blundell et al., 19911, as was also found to be the case for the sodium 3P,,, state.
V. Stark Shifts A. BACKGROUND Precise measurements of Stark shifts provide information about polarizabilities of atomic states that are important for describing a number of properties including charge-exchange cross sections, van der Waals constants, and dielectric constants (Bonin and Kadar-Kallen, 1993). Some of the first determinations of polarizabilities were made by measuring the deflection of an atomic beam as it passed through an inhomogeneous electric field (Chamberlain and Zorn, 1963). These experiments yielded results that had a typical accuracy of about 10%. Improved data was obtained by the Bederson group using both inhomogeneous magnetic and electric fields. An atomic beam is not deflected if the forces exerted by inhomogeneous magnetic H and electric fields E cancel, i.e., a E d E / d z = p dH/dz. The polarizability can then be found provided the field gradients and the magnetic dipole moment p are known. This technique has yielded results with uncertainties of a few percent (Molof et al., 1974). Recently, several new methods have been developed. The ground state polarizabilities of rubidium (Bonin and Kadar-Kallen, 1993) and uranium (Kadar-
167
PRECISION LASER SPECTROSCOPY
Kallen and Bonin, 1994) have been determined to 10% accuracy by measuring the deflection of an atomic beam as it passed through a standing wave generated using a Nd:YAG laser pulse. A significant improvement in accuracy has been obtained by Ekstrom et al. (1995) using atom interferometric methods. They determined the sodium ground state polarizability with an uncertainty of 0.3%. Various other methods to determine polarizabilities exist, and the reader is referred to the review articles written by Miller and Bederson (1977, 1988). The Stark shifts of a number of transitions have recently been measured with uncertainties as low as a few parts in lo4. This accuracy is substantially better than that attained in the best lifetime measurements (Rafac et af., 1994) and by experiments that determine transition oscillator strengths (van Wijngaarden et al., 1986). These data therefore permit stringent tests of atomic theory. Accurate theoretical models of many-electron atoms are especially important for improved tests of parity violation in atoms such as cesium (Hoffnagle et af., 1981; Noecker et af., 1988; Dzuba et al., 1989; Bouchiat, 1991; Blundell et al., 1991). A similar experiment has recently been proposed for ytterbium (DeMille, 1995). A thorough understanding of Stark shifts is also needed for experiments searching for an electron dipole moment (Hunter et af., 1988). Accurate Stark shift data are also desired for applications such as the measurement of electrode spacings (Neureiter et al., 1986) and the determination of electric fields in plasmas (Lawler and Doughty, 1995; Rebhan et af., 1981). An electric field can then be found by measuring the frequency shift of a spectral line. This has a significant advantage over methods that measure fields using invasive probe electrodes which can strongly perturb the plasma. The Hamiltonian describing the interaction of an atoms with an external electric field E is given by (Khadjavi et al., 1968) HStark =
-
[
35: a')
+
-J'"
a2 J ( 2 J -
1)
1z E2
where f i s the electronic angular momentum and the quantization axis z lies along the field direction. For J < 1, the second term vanishes. The terms a. and a2 are the scalar and tensor polarizabilities, respectively, and are defined by
ff2 =
rn
-
1
87r2 ( 2 5 + 3 ) ( J
+ 1)
A:j, f J J t [ 8 5J( -k 1) - 3 x ( J r
xf
I)] (16)
W.A. van Wijngaarden
168
where X = J'(J' + 1) - 2 - J ( J + 1). Here, ro is the classical electron radius, AJ,, is the wavelength for a transition between states J and J', and fJJ. is the transition oscillator strength. The eigenstates of the Stark Hamiltonian are IJm ), where m is the azimuthal quantum number. The corresponding eigenenergy is
1-
3m: - J ( J + 1) E 2 J ( 2 J - 1) 2 The eigenenergy corresponding to a hyperfine level IFm,) is given by Eq. (17) with J and m , replaced by F and m F ,respectively.
B. YTTERBIUM ( 6 . ~ )'So ~ + (6s6p) 3P, TRANSITION The Stark shift of the ytterbium ( 6 ~'So ) ~+ ( 6 . ~ 63P, ~ ) transition has been studied using the method outlined in Fig. 4(a) (Li and van Wijngaarden, 1995b). The apparatus shown in Fig. 12 was also used to study the Stark
1-F Lock-In 1
1!11_.1-
PMl Voltmeter
Chopper Ref. Voltage Divider L0Ck-h 2
A 0
Frequency Synthesizer
vAO
>
Amplifier
/by X
FIG. 12. Stark shift measurement apparatus. Details are given in the text (Li and van Wijngaarden, 1996, by permission).
PRECISION LASER SPECTROSCOPY
169
shifts of transitions in barium (Li and van Wijngaarden, 1995a) and calcium (Li and van Wijngaarden, 1996). The generation of an atomic beam and the fluorescent detection has been discussed in Section 1V.B. Ytterbium atoms were excited by a laser in a field-free region and in a uniform electric field. The electric field was generated using two highly polished stainless steel disks having a diameter of 7.62 cm. The spacing was determined to be 1.0163 0.0003 cm using precision machinist blocks, the size of which was specified to within 2.5 X cm. Plate voltages of up to 50 kV were continuously monitored using a precision voltage divider that reduced the voltage by a factor of 5000 with an accuracy of 0.01% (Julie Labs KV-50/01). The reduced voltage was measured by a voltmeter with an uncertainty of less than 0.002%. The electric field shifts the transition by an amount
Av
=
KE2
( 18)
where the Stark shift rate
Here, m, has been set to zero since the laser was linearly polarized along the quantization axis, which was specified by the electric field. Hence, only ~ ) state was populated. Equation (19) the m = 0 sublevel of the ( 6 . ~ 63P, holds for the even isotopes of ytterbium that do not have a nuclear spin. For 171,173Yb, the hyperfine interaction must also be considered. For simplicity, only the transition in I7'Yb to the F = level of the ( 6 ~ 6 p ) ~ P , state was studied, which has a Stark shift rate given by
,
The tensor polarizability of the 'PI state could then be determined by subtracting (20) from (19). The laser frequency was tuned across the transition while fluorescence produced by the radiative decay of the excited state to the ground state was detected by two photomultipliers (PM1 and PM2). The signals were processed by separate lock-in amplifiers. Spectra similar to that shown in Fig. 6 were obtained, and the frequency was calibrated as has been described in Section 1V.B. The results of nearly 500 wavelength scans taken at various electric fields are shown in Fig. 13. A least squares fit of a straight line y = kE2+ y o to the data yielded K = - 15.419 k 0.048 kHz/(kV/cm)2. The frequency shift at zero field y o was 5.33 MHz. This offset arises from a small difference of the intersection angle of the laser and atomic beams in the field and field-free regions. The tensor polarizability was found to be a2 = 5.81 f 0.13 kHz/(kV/cm)*. The result is in good agreement with
170
W.A. van WQngaarden
E2 ( kV
/ cm ) 2
FIG. 13. Frequency shift versus electric field squared for the ytterbium ( 6 ~ ) ’ ~ s ~ ( 6 ~ 63P1 ~ ) transition (Li and van Wijngaarden, 1995b, by permission). --f
5.99 f 0.34 kHz/(kV/cm)2 found by an optical double resonance experiment (Rinkleff, 1980) and 6.04 f 0.21 kHz/(kV/cm)2 obtained using quantum beat spectroscopy (Kulina and Rinkleff, 1982).
C. CESIUM6P3/2
+
nS,/, (n
=
10-13) TRANSITIONS
An example of a Stark shift measured as illustrated in Fig. 4(b) is an
experiment that studied the 6P3/2 (10-13)S,/2 transitions in cesium (van Wijngaarden et al., 1994). The apparatus was similar to that shown in Fig. 12. An oven generated two cesium atomic beams propagating in opposite directions. One atomic beam passed through a field-free region, whereas the other beam traveled through a uniform electric field. Atoms were excited from the 6S,/2 ground state to the 6P3/2 state by a diode laser that generated a few milliwatts of light at 852 nm. A ring dye laser then excited the 6P3/: + nS,/, ( n = 10-13) transition. Part of the dye laser was frequency shifted by an acousto-optic modulator. The unshifted laser beam at frequency Y excited the atoms in the field-free region, --j
PRECISION LASER SPECTROSCOPY
171
whereas the laser beam having frequency v - vAo excited the atoms passing through the electric field. Fluorescence from the field and field-free regions was recorded as the dye laser frequency v was scanned across the resonance. The frequency interval separating the fluorescent peak observed in the two regions was given by A = -hvAo - K E 2
(21)
where the shift rate is
K = -1{ 2
- a0(6P3/2)
- a2(6p3/2)}
(22)
A was plotted versus the electric field squared, as is shown in Fig. 14. A line was fit to the data and the field such that atoms in the field-free and field regions were simultaneously in resonance, i.e., A = 0 was found. The polarizabilities a0(nS,/,) were found using the small contributions of ao(6P312) = 407 and a2 = -65.1 kHz/(kV/cm)* calculated by Zhou and Norcross (1989). The results listed in Table V agree with those found
FIG. 14. Frequency separation A of fluorescent peaks observed in field and field-free regions versus electric field squared for excitation of the cesium 13S,,, state using an acousto-optic modulation frequency of 350 MHz (van Wijngaarden et a/., 1994, by permission).
172
W.A. uan Wjngaarden TABLE V SCALARPOLARIZABILITIES OF THE CESIUM (10-13)S,,2 STATES
n
Fredriksson and Svanberg (1977)
van Wijngaarden et al. (1994)
Theory
10 11 12 13
123 & 6 322 + 16 720 45 1650 + 170
119.06 f 0.28 309.70 f 0.26 713.48 f 0.58 1491.20 f 1.22
118 309 709 1490
*
by Fredriksson and Svanberg (1977) but are much more accurate. The latter group used a lamp that excited atoms in an atomic beam to the 6P3/2 state. The 6P3/2 + (10-13)S,/2 transitions were excited by a dye laser having a linewidth of about 75 MHz. Data was taken at fked dye laser frequency as follows. The electric field applied across the atomic beam was increased from 0 to a maximum of 7 kV/cm while fluorescence produced from the radiative decay of the excited nS,,, state was monitored by a photomultiplier. Flourescent peaks occurred whenever the dye laser excited one of the hyperfine levels of the 6P3/2 state to the Starkshifted n S , / , state. The Stark shift rate was then found using the hyperfine splittings of the 6P3/2 state along with the field strengths corresponding to the peak maxima. The experimental accuracy was limited by uncertainties in the electric field determination and by the accuracy of the hyperfine data of the 6P3/2 state then available. The data listed in Table V agree closely with results computed using the method developed by Bates and Damgaard (1949). These results were found using experimentally measured energies and assuming a Coulomb potential to describe the interaction of the valence electron and the nucleus plus the inner core electrons. This approximation has been used to compute polarizabilities in several alkali atoms (Gruzdev et al., 1991; van Wijngaarden and Li, 1994~1,and good agreement with the experimental data has been obtained for all but the lowest P states. This is not surprising since the Coulomb approximation best describes excited states that have minimal penetration of the inner electron core and have a small spin-orbit interaction.
D. CESIUMD LINES Several groups have studied the Stark shifts of the 6S,,, + 6P1/2,3/2 transitions in cesium. Hunter et al. (1988) used two glass cells loaded with
173
PRECISION LASER SPECTROSCOPY
cesium atoms. One cell was also filled with 6 Torr of nitrogen gas that pressure shifted the cesium resonance by 40 MHz. The second cell was made by gluing two metal plates onto a pyrex cylinder. These plates served as the electrodes that generated an electric field. The fluorescence signal observed by a photodiode in the pressurized cell was used to lock the laser to the transition frequency. A n acousto-optic modulator shifted part of the laser beam by 40 MHz. The frequency-shifted laser beam was incident on the second cell, across which an electric field was applied. The modulation frequency needed for the laser to excite the Stark-shifted resonance was measured at various electric field strengths. Data were taken at laser powers of about 1 pW to minimize ac Stark shifts, optical pumping, and saturation effects. The result for the D , line Stark shift is listed in Table VI. The 6S,,2 + 6P312 transition was excited to the various hyperfine levels of the The experiexcited state to determine the tensor polarizability ~t2(6P3/2). ment was done using a ring dye laser and repeated with a diode laser. The diode laser experienced slightly less frequency jitter and therefore produced data having a smaller statistical variation. The experimental accuracy was limited by the determination of the electric field. The cells were plagued with systematic uncertainties including leakage currents and needed to be coated with surfa-sil to minimize field inhomogeneities. Unfortunately, the coatings deteriorated noticeably after several months. 6P3l2 Tanner and Wieman (1988a) studied the Stark shift of the 6S,/2 transition using apparatus similar to that shown in Fig. 10. An atomic beam passes through two plates that were separated by 0.3950 k 0.0002 cm. One plate had a transparent conductive coating that permitted the laser to be transmitted, whereas the second plate was a gold-coated mirror. Electric fields were generated by applying voltages of up to 18 kV to the plates. Voltages were determined using a high voltage divider and a digital voltmeter. The divider drifted slightly with temperature, limiting the fracThe diode laser tional uncertainty of the voltage calibration to 6 X frequency was locked to the cesium resonance using a saturation signal observed in a cell. Part of the laser was frequency shifted by an acoustooptic modulator and excited the atoms passing between the field plates. The Stark shift was determined by measuring the modulation frequency needed to keep the atoms in resonance. The result listed in Table VI is somewhat lower than that found by Hunter et al. but has a five times smaller uncertainty. The accuracy was limited by the determination of the electric field. A very accurate measurement of the Stark shift of the D , line was done by Hunter et al. (1992) using the apparatus illustrated in Fig. 15. Two diode lasers excited the 6S,,, + 6P,,, transition at 894 nm. One laser was --f
CL
TABLE VI
4 P
PRECISION STARK SHIFTSUMMARY
Transition l-+u
Atom Ba
(6s)' 'So
-+
(6s6p)'PI
Polarizability a&)
-
2a2(u) - ao(l)
4 4 )
( 4 ~'SO) ~
cs
6S1/2 + 7s1/2 6P,/2 6S1/2
-+
( 4 ~ 4 p ) ~ ~ I a,(u)
+
6S,/2
+
6P3/2
-
2 a 2 ( ~)
a&) - ( Y O U ) a&) - a&) a,(u) - a,(l) a,(u) - (Y"(1) a,(u) - a,(l) a,(u) - ao(l) a,(u) - a,(l) a,(u) ff,(U)
6p3/2
12s1/2
a&) - ao(l) a,(u) - ao(l) a,(u) - an(/)
Wp
aJu)
-
ao(l)
~4P1/2
a,(u)
-
ao(1)
2p,/2
a&) - ao(l) a&) - ao(I) a,(u) - a,W a&) - a,(l)
+
6p3/2
+
6p3/2
6P3/2
K
4
Li
2SI/2
2s1/2
s
-+
-
'OSl/2
'lSl/2
+
* 2p3/2
Ci,(U)
Reference
57.06 k 0.12 k 0.10 - 10.79 f 0.29
Li and van Wijngaarden (1995a) Kreutztrager and von Oppen (1973) Hese et al. (1977)
24.628 k 0.082
Li and van Wijngaarden (1996)
- 10.72
4 u ) Ca
Value (kHz/(kV/cm)2 Y
+
1420.6 4.8 241.2 + 2.4 230.44 k 0.03 230.5 314.2 f 3.2 308.6 f 0.6 308.0 2.0 -64.7 -65.3 k 0.4 -65.1 118,720 280 309,360 f 260 713,140 f 580 1,490,900 f 1200
+
78.800
0.010
+
-9.243 0.004 -9.234 & 0.082 - 9.272 -9.281 If: 0.100 0.408 0.011
+
Watts et al. (1983) Hunter et al. (1988) Hunter et al. (1992) Zhou and Norcross (1989) Hunter et al. (1988) Tanner and Wieman (1988a) Zhou and Norcross (1989) Hunter et al. (1988) Tanner and Wieman (1988a) Zhou and Norcross (1989) van Wijngaarden et al. (1994) van Wijngaarden et al. (1994) van Wijngaarden et al. (1994) van Wijngaarden et al. (1994) Miller e r a / . (1994) Hunter ef al. (1991) Windholz et af. (1992) Pipin and Bishop (1993) Windholz et al. (1992) Windholz et al. (1992)
3 & C
3
i!a
TABLE VI (continued)
Atom
Transition I+u
Value (kHz/(kV/cd2Y
Polarizability
0.399
(YO(1)
a
Note that 1 kHz/(kV/cmI2 = 4.0189ai/h and at = 1.4818 X
Reference Pipin and Bishop (1993)
48.99 f 0.11 49.28 f 0.15 -21.97 k 0.10 40.56 f 0.14
Windholz and Neureiter (1985) Windholz and Musso (1989) Windholz and Musso (1989) Ekstrom et al. (1995)
122.306 f 0.016
Miller et al. (1994)
-91.8 k 0.4 13.29 f 0.06
Neureiter et al. (1986) Neureiter et al. (1986)
30.838 f 0.096 5.81 k 0.13 5.99 f 0.34 6.04 f 0.21
Li and van Wijngaarden (199%) Li and van Wijngaarden (1995b) Rinkleff (1980) Kulina and Rinkleff (1982)
cm3, where a, is the Bohr radius and h is Planck's constant.
176
W A. van Wijngaarden
Atomic Beam
- - - - _ _- - - 1
-------
FIG. 15. Stark shift measurement apparatus for studying the alkali D lines (Hunter el al., 1992, by permission).
locked to the D, line using the saturated absorption signal observed by a photodiode (PD) in a cell while the second laser excited an atomic beam as it passed through an electric field. Considerable care was spent designing the field electrodes to permit an accurate determination of the electric field. Two h/10 optical quartz flats were coated with indium tin oxide, which has a transmission coefficient of over 80% at 894 nm. The fluorescence was therefore transmitted through the electrodes and detected by a photomultiplier (PM). The plate separation distance of about 2.8 mm was precisely determined using four 80% reflecting aluminium pads 2 mm in diameter that were placed at the corners of a 1.3-cm square centered on each electrode. The aluminium pads on the two electrodes were aligned to form four separate Fabry-Perot etalons. The electrode spacing could then be monitored throughout the experiment by measuring the free spectral range of the four etalons using a ring titanium sapphire (Coherent 899-21) laser and a wavemeter. This permitted the electrode spacing to be determined with a fractional uncertainty of 40 ppm. Voltages of up to k7.5 kV were applied to the plates. The voltage was determined using a high voltage divider chain accurate to 60 ppm that was constructed using
PRECISION LASER SPECTROSCOPY
177
precision wire wound resistors. The reduced voltage was measured with a voltmeter to a precision of 30 ppm. The experimental procedure was as follows. Part of the second diode laser was frequency shifted 60 MHz by an acousto-optic (A01 modulator. The frequency-shifted laser beam was then used to lock the diode laser to the Stark-shifted transition found by passing the atomic beam through the electric field. Part of each of the two diode laser beams was focused onto a fast photodiode (FPD). The acousto-optic modulator shifts the beat note to a higher frequency that is relatively insensitive to noise effects, which in general have lower frequency components. The beat frequency was measured by a counter as a function of the electric field to determine the Stark shift. Data were taken at various laser powers and voltages to check for systematic effects. The results given in Table VI are substantially more accurate than data found in their earlier experiment (Hunter et al., 1988). Tanner and Wieman (1988a) also found results that were lower than those obtained by Hunter et al. (1988) in their study of the cesium D, line Stark shift. Hunter et al. (1992) attributed this discrepancy to an underestimate of the electric field uncertainty in their initial work. The improved experiment used an atomic beam instead of a cell for observing the Stark-shifted transition and therefore did not suffer from the various problems discussed earlier. Hunter et al. have also used diode lasers to study the Stark shifts of D ,lines in lithium (Hunter et al. 1990, potassium, and rubidium (Miller et al., 1994). Several theoretical estimates of the D line Stark shifts have been made. The most accurate is that of Zhou and Norcross (19891, which is listed in Table VI. They used a semiempirical potential composed of a Thomas-Fermi potential plus a term describing the polarization of the inner electron core. The various potential parameters were adjusted to obtain optimum agreement of computed and measured excited state energies (Weber and Sansonetti, 1987). This potential was then used to solve the Dirac equation for the single valence electron. The close agreement between the calculated values and the measured results obtained for the Stark shifts of the D lines and the tensor polarizability of the 6P3,2 state is impressive. E. PRECISION STARKSHIFTSUMMARY A summary of Stark shifts measured with uncertainties of less than 0.5% is given in Table VI. Many of these results have been found using experimental techniques that have already been presented and are therefore not further discussed. The group led by Windholz has studied a number of transitions in lithium (Windholz et al., 19921, sodium (Windholz and
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W. A. van Wjngaarden
Neureiter, 198.9, and samarium (Neureiter et al., 1986). They used a ring dye laser to excite an atomic beam in a field-free region and in a uniform electric field. Fields of up to 300 kV/cm were generated by applying high voltages to stainless steel plates. Silver-coated optical glass flats could not be used as electrodes since the coatings were destroyed by occasional sparks at these high voltages. The laser frequency was scanned across the resonance, and the fluorescence was detected by photomultipliers. The change in laser frequency was monitored by passing part of the laser beam through a Fabry-Perot interferometer as is shown in Fig. 1. The accuracy of their results was limited to a few tenths of a percent by uncertainty in the frequency marker positions generated using the etalon. For the case of the lithium D, line, their data are in good agreement with the more accurate results found by Hunter et al. (1991). Both of the experimental determinations are a few percent lower than a theoretical estimate (Pipin and Bishop, 1993). The latter used a combined configuration interaction Hylleraas method to calculate wavefunctions and matrix elements. An experiment by Ekstrom et al. (1995) used the recently developed technique of atom interferometry (Adams et al., 1994) to determine the ground state polarizability of sodium. Their apparatus consists of a threegrating Mach-Zender interferometer. The transmission gratings have a 200-nm period and generated two atomic beams separated by 55 pm. One beam passed through an electric field created by applying a voltage across two metal foils. This generated a relative phase shift between the two beams given by
where u is the velocity of the atoms, a. is the ground state polarizability of sodium, E is the electric field, and L is the length of the electric field region. The two atomic beams were then recombined, and the resulting interference pattern was studied using a hot wire detector that was mounted on a translation stage. Phase shifts of up to 60 rad were observed using fields of several kV/cm. The scalar polarizability a 0 ( 3 S , / , ) was determined to be 40.56 k 0.14 kHz/(kV/cm)’. The uncertainty is due to statistical and systematic effects. The latter was dominated by geometrical effects such as fringing electric fields that affect the interaction length L. These were studied using electrode foils of 7 and 10 cm in length and using guard electrodes to minimize the fringing fields. Another complication was modeling the velocity distribution of the atoms in the atomic beam to estimate the average velocity u. The final result for ffo(3S1/2) is substantially better
PRECISION LASER SPECTROSCOPY
179
than the value of 41.0 f 2.9 kHz/(kV/cm)2 (Hall and Zorn, 1974), which was determined by measuring the deflection of an atomic beam in an inhomogeneous electric field. The result obtained by the atom interferometric experiment can be combined with the measured Stark shifts of the sodium D lines (Windholz and Neureiter, 1985; Windholz and Musso, 1989) to obtain values of 89.55 f 0.18 and 89.84 k 0.19 kHz/(kV/cm)’ for the scalar polarizabilities of the 3 P , , , and 3P3/’ states, respectively.
VI. Concluding Remarks The technique of precisely measuring frequency shifts using acoustooptically modulated laser beams has been demonstrated in a number of experiments. The apparatus is relatively straightforward consisting of an atomic beam, a frequency-modulated laser beam, and a photomultiplier. Large fluorescence signals having little noise can be generated using either dye or diode lasers. The method has a number of advantages when compared with other techniques. It does not require large and very uniform magnetic fields, as are needed in level crossing and in optical double resonance experiments. The data analysis is also less complicated since only a Lorentzian or Gaussian function is fitted to the observed spectral line. Short temporal resolution using relatively expensive transient digitizers is also not required, as is the case in quantum beat spectroscopy. Most significantly, the frequency calibration is much simpler than using a Fabry-Perot etalon. Interferometers are plagued by numerous problems including vibrations and sensitivities to pressure and temperature fluctuations, necessitating the use of frequency-stabilized lasers locked to an atomic transition to stabilize the cavity. In contrast, computer-controlled frequency synthesizers can quickly and conveniently generate a much wider range of frequencies with an accuracy of one part per million. Over the past decade, experiments using acousto-optically modulated lasers have yielded isotope, hyperfine, and Stark shifts representing frequency intervals ranging from a few megahertz to several gigahertz. Data of unprecedented accuracy with uncertainties as low as several parts in lo5 have been obtained. These results pose a stringent test for theories of multielectron atoms. The measured Stark shifts can also be used in conjunction with data obtained using novel new methods such as atom interferometry to determine scalar polarizabilities of excited states that heretofore could not be determined. Hence, the method of using
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W. A. van Wjngaarden
frequency-modulated lasers is very versatile, and optical modulators are likely to play an increasingly important role in precision laser spectroscopy.
Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council of Canada and York University.
References Abele, J. (1975). 2. Phys. 274, 179. Adams, C. S., Carnal, O., and Mlynek, J. (1994). A h . At. Mol. Opt. Phys. 34, 1. Arimondo, E., Inguscio, M., and Violino, P. (1977). Reu. Mod. Phys. 49, 31. Baird, P. E. G., Brambley, R. J., Burnett, K., Stacey, D. N., Warrington, D. M., and Woodgate, G. K. (1979). Proc. R. SOC. London, Ser. A 365, 567. Bates, D. R., and Damgaard, A. (1949). Philos. Trans. R. SOC.London 242, 101. Baurnann, M., and Wandel, G. (1966). Phys. Lett. 22, 283. Baumann, M., and Wandel, G. (1968). Phys. Left. A 28A, 200. Baumann, M., Liening, H., and Wandel, G. (1969). 2. Phys. 221, 245. Beckrnann, A., Boklen, K. D., and Elke, D. (1974). 2. Phys. 270, 173. Blundell, S . A., Johnson, W. R., and Sapirstein, J. (1991). Phys. Reu. A 43, 3407. Bonin, K. D., and Kadar-Kallen, M. A. (1993). Phys. Reu. A 47, 944. Bouchiat, M. A. (1991). I n “Proceedings of the Twelfth International Conference on Atomic Physics, 1990” (J. C. Zorn and R. R. Lewis, eds.), p. 399. American Institute of Physics, New York. Broadhurst, J. H., Cage, M. E., Clark, D. L., Greenlees, G. W., Griffith, J. A. R., and Isaak, G. R. (1974). J . Phys. B 7, L513. Buck, P., Rabi, I. I., and Senitzky, B. (1956). Phys. Reu. 104, 553. Budick, B., and Snir, J. (1967). Phys. Lett. A 24A, 689. Budick, B., and Snir, J. (1970). Phys. Reu. A 1, 545. Carlsson, J. (1988). 2. Phys. D: At. Mol. Clusters 9, 147. Carlsson, J., Jonsson, P., Sturesson, L., and Froese Fischer, C. (1992). Phys. Scr. 46, 394. Chaiko, Y. (1970). Opt. Spectrosc. 20, 424. Chamberlain, G. E., and Zorn, J. C. (1963). Phys. Reu. 129, 677. Clark, D. L., Cage, M. E., Lewis, D. A., and Greenlees, G. W. (1979). Phys. Rev. A 20, 239. Corney, A. (1977). “Atomic and Laser Spectroscopy.” Clarendon Press, Oxford. Deilamian, K., Gillaspy, J. D., and Kelleher, D. E. (1993). J . Opt. SOC.Am. B 10, 789. DeMille, D. (1995). Phys. Reu. Lett. 74, 4165. Dzuba, V. A., Flambaum, V. V., and Sushkov, 0. P. (1989). Phys. Lett. A 140,493. Ekstrom, C. R., Schmiedmayer, J., Chapman, M. S., Hammond, T. D., and Pritchard, D. E. (1995). Phys. Rev. A 51, 3883. Fredriksson, K., and Svanberg, S. (1977). 2. Phys. A 281, 189. Gaupp, A. Kuske, P., and Andra, H. J. (1982). Phys. Reu. A 26, 3351. Griffith, J. A. R., Isaak, G. R., New, R., Ralls, M. P., and van Zyl, C. P. (1977). J. Phys. B 10, L91.
PRECISION LASER SPECTROSCOPY
181
Gruzdev, P. F., Soloveva, G. W., and Sherstyuk, A. I. (1991). Opt. Spectrosc. 71, 513. Hall, W. D., and Zom, J. C. (1974). Phys. Reu. A 10, 1141. Hartmann, W. (1970). Z . Phys. 240, 323. Hese, A,, Renn, A,, and Schweda, H. S. (1977). Opt. Commun. 20, 385. Hoffnagle, J., Telegdi, V. L., and Weis, A. (1981). Phys. Lett. 86A, 457. Holmes, B. W., and Griffith, J. A. R. (1995). J. Phys. B 28, 191. Hunter, L. R., Krause, D., Murthy, S., and Sung, T. W. (1988). Phys. Rev. A 37, 3283. Hunter, L. R., Krause, D., Berkeland, D. J., and Boshier, M. G. (1991). Phys. Reu. A 44, 6140. Hunter, L. R. Krause, D., Miller, K. E., Berkeland, D. J., and Boshier, M. G. (1992). Opt. Commun. 94, 210. Jin, W. G., Horiguchi, T., Wakusugi, M., Hasegawa, T., and Yang, W. (1991). J. Phys. SOC. Jpn. 60, 2896. Johnson, W. R., Idrees, M., and Sapirstein, J. (1987). Phys. Rev. A 35, 3218. Kadar-Kallen, M. A., and Bonin, K. D. (1994). Phys. Rev. Lett. 72, 828. Kallas, K., Markova, G., Khvostenko, G., and Chaika, M. (1965). Opt. Spectrosc. 19, 303. Khadjavi, A,, Lurio, A,, and Happer, W. (1968). Phys. Reu. 167, 128. Korpel, A. (1988). “Acousto-Optics.” Dekker, New York. Kreutztrager, A., and von Oppen, G. (1973). Z. Phys. 265, 421. Kulina, P., and Rinkleff, R. H. (1982). Z . Phys. A 304, 371. Lawler, J. E., and Doughty, D. A. (1995). Adv. At. Mol. Opt. Phys. 34, 171. Li, J. (1995). Ph.D. Dissertation, York University, Toronto, Ontario, Canada (unpublished). Li, J., and van Wijngaarden, W. A. (1995a). Phys. Rev. A 51, 3560. Li, J., and van Wijngaarden, W. A. (1995b). J. Phys. B 28, 2559. Li, J., and van Wijngaarden, W. A. (1996). Phys. Reu. A 53, 604. Lindgren, I., Lindgren, J., and Martensson, A. (1976). Z. Phys. A 279, 113. Mathur, B. S., Tang, H., and Happer, W. (1968). Phys. Reu. 171, 11. Miller, K. E., Krause, D., and Hunter, L. R. (1994). Phys. Reu. A 49, 5128. Miller, T. M., and Bederson, B. (1977). Adu. At. Mol. Opt. Phys. 13, 1. Miller, T. M., and Bederson, B. (1988). Adu. At. Mol. Opt. Phys. 25, 37. Molof, R. W., Schwartz, H. L., Miller, T. M., and Bederson, B. (1974). Phys. Reu. A 10, 1131. Neureiter, C., Rinkleff, R. H., and Windholz, L. (1986). J. Phys. B 19, 2227. Noecker, M. C., Masterson, B. P., and Wieman, C. E. (1988). Phys. Reu. Lett. 61, 310. Perl, M. L., Rabi, I. I., and Senitzky, B. (1955). Phys. Reu. 98, 611. Pescht, K., Gerhardt, H., and Matthias, E. (1977). Z . Phys. A 281, 199. Pipin, J., and Bishop, D. M. (1993). Phys. Re“. A 47, R4571. Rafac, R. J., Tanner, C. E., Livingston, A. E., Kukla, K. W., BerIy, H. G., and Kurtz, C. A. (1994). Phys. Reu. A 50, R1976. Ramsey, N. F. (1956). “Molecular Beams.” Oxford Univ. Press, London. Rayman, M. D., Aminoff, C. G., and Hall. J. L. (1989). J. Opt. SOC.A m . B 6, 539. Rebhan, U., Wiegart, N. J., and Kunze, H. J. (1981). Phys. Lett. 8 5 4 228. Riis, E., Sinclair, A. G., Poulsen, O., Drake, G. W. F., Rowley, W. R. C., and Levick, A. P. (1994). Phys. Rev. A 49, 207. Rinkleff, R. H. (1980). Z . Phys. A 296, 101. Ross. J. S. (1963). J . Opt. SOC.A m . 53, 299. Salomonson, S., and Ynnerman, A. (1991). Phys. Rev. A 43, 88. Schmidt-Kaler, F., Leibfried, D., Weitz, M., and Hansch, T. W. (1993). Phys. Reu. Lett. 70, 2261. Schmoranzer, H., Schulze-Hagenest, D., and Kandela, S. A. (1979). Symp. A t . Spectrosc. Sept. 1979, p. 195. Tucson, AZ. Svanberg, S., and Belin, G. (1972). Z. Phys. 251, 1.
182
W. A. uan Wijngaarden
Svanberg, S., and Rydberg, S. (1969). 2. Phys. 227, 216. Tanner, C. E., and Wieman, C. (1988a). Phys. Reu. A 38, 162. Tanner, C. E., and Wieman, C. (1988b). Phys. Reu. A 38, 1616. Tran, C. D. (1992). Anal. Chem. 64, 971. Umfer, C., Windholz, L., and Musso, M. (1992). Z. Phys. D: A t . Mol. Clusters 25, 23. van Wijngaarden, W. A., and Li, J. (1994a). J. Opt. SOC.A m . B 11, 2163. van Wijngaarden, W. A., and Li, J. (1994b). 2. Phys. D : At. Mol. Clusters 32, 67. van Wijngaarden, W. A., and Li, J. (1994~).J. Quant. Spectrosc. Radiat. Transfer 52, 555. van Wijngaarden, W. A., and Li, J. (1995). Can. J. Phys. 73,484. van Wijngaarden, W. A., Bonin, K., Happer, W., Miron, E., Schreiber, D., and Arisawa, T. (1986). Phys. Reu. Lett. 56, 2024. van Wijngaarden, W. A., Hessels, E. A., Li, J., and Rothery, N. E. (1994). Phys. Reu. A 49, R2220. Violino, P. (1969). Can. J. Phys. 47, 2095. Walther, H. (1974). Phys. Scr. 9, 297. Wandel, G. (1970). Z. Phys. 231, 434. Watts, R. N., Gilbert, S. L., and Wieman, C. E. (1983). Phys. Reu. A 27, 2769. Weber, K. H., and Sansonetti, G. (1987). Phys. Reu. A 35, 4650. Windholz, L., and Musso, M. (1989). Phys. Rev. A 39, 2472. Windholz, L., and Neureiter, C. (1985). Phys. Lett. 1094 155. Windholz, L., Musso, M., Zerza, G., and Jager, H. (1992). Phys. Reu. A 46, 5812. Xu, J., and Stroud, R. (1992). “Acousto-Optic Devices.” Wiley, New York. Young, T. P., and Griffith, J. A. R. (1994). Tech. Dig. European Quantum Electronics Conference, 1994, Amsterdam. Zhao, P., Lawall, J. R., and Pipkin, F. M. (1991). Phys. Reu. Lett. 66, 592. Zhou, H. L., and Norcross, D. W. (1989). Phys. Reu. A 40, 5048.
ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 36
HIGHLY PARALLEL COMPUTATIONAL TECHNIQUES FOR ELECTRON-MOLE CULE COLLISIONS CARL WNSTEAD and KINCENT McKOY A . A . Noyes Laboratory of Chemical Physics California Institute of Technology Pasadena, California
I. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Electron-Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Parallel Computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1.Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Computational Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Designing a Parallel Program . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Parallel SMC Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Illustrative Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Performance.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. CrossSections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Conclusion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
183 183 185 186 191 191 196 209 209 212 214 217 218
I. Introduction A. ELECTRON-MOLECULE COLLISIONS
Electron-molecule collisions at low impact energies have long been of fundamental interest because of the variety of phenomena exhibited in such collisions and the possibility for gaining insights therefrom into molecular spectra, particularly the electronic structure of the ground and low lying excited states (Schulz, 1973; Lane, 1980; Hall and Read, 1984; Allan, 1989). At the same time, the rates and outcomes of such collisions have been of great practical interest to those seeking to understand partially ionized gases, including the atmospheres of Earth and of other planets, gas lasers, and the edge regions of fusion plasmas. Recently, the increasing importance of low temperature plasmas in a variety of materials 183
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processing tasks has been a driving force behind the study of low energy electron-molecule collisions (National Research Council, 1991). These nonequilibrium plasmas, widely used in the manufacture of semiconductor microelectronics, rely on collisional energy transfer between hot electrons and cold molecules to generate reactive species, such as free radicals and ions, that can effect desired chemical and physical changes at an exposed surface. Though hot in human terms, the electrons in these plasmas, with effective temperatures of thousands and tens of thousands of Kelvin, fall in the low energy regime with respect to collisions because they have kinetic energies comparable with those of the molecular valence electrons (1 eV = 12,000 K). Though most of our current knowledge of electron-molecule collisions derives from experiment, accurate measurements of collision cross sections are in fact quite difficult, especially for inelastic processes, and few groups worldwide have undertaken this challenging work. Demand for cross section data already exceeds supply, and the list of “critical” but absent data grows longer daily. Moreover, the species of interest include not only stable molecules but those radicals and ions whose populations within the plasma may be significant, and experiments will be all the more difficult for such transient species. The lack of reliable cross section data threatens to impair the accuracy of numerical models of low temperature plasmas and thus ultimately to impede the development of computer-aided design and optimization tools for plasma reactors (National Research Council, 1991). What can theoretical studies contribute? The calculation of electron cross sections at high impact energies (several hundred electron volts or more) is relatively straightforward: Since the first Born approximation (Schiff, 1968) applies, the problem effectively reduces to that of calculating a matrix element between bound electronic states of the target molecule. The well-developed methods of modern quantum chemistry may be brought to bear in calculating accurate approximations to such bound states. At low energies (roughly speaking, below 100 eV), matters are far less simple. A proper accounting for the identity of electrons requires antisymmetrization of the projectile electron with the N electrons of the target; that is, a single wavefunction for the ( N 1)-electron system must be determined, subject to appropriate (scattering) boundary conditions. Even if, as is often the case, nuclear motion may be neglected, the resulting problem in continuum electronic structure is formidable. Moreover, it must typically be solved for many different collision energies and for all possible combinations of directions of incidence and departure. Yet much progress has nonetheless been made toward the development of widely applicable methods for carrying out such calculations.
+
HIGHLY PARALLEL COMPUTATIONAL TECHNIQUES
185
Most recent theoretical studies of electron-molecule collisions have relied on variational approximations to the scattering amplitude or to some closely related quantity, thereby avoiding direct numerical solution of Schrodinger’s equation. Using one such variational method, we have been able to carry out calculations of elastic and inelastic cross sections for a variety of molecules of interest in plasma processing, including such large molecules (in terms of electron count) as SiF,, AlC,H,, and C,F,. However, studies of many-electron systems remain numerically intensive despite the choice of an efficient theoretical approach, and our work has depended on exploiting the prodigious advances in computational power that have resulted from the development of massively parallel processors.
B. PARALLEL COMPUTATION The advent in recent years of commercial parallel computers has opened new possibilities in computational physics and chemistry. As machines based on large numbers of powerful microprocessors have begun to supplant conventional vector supercomputers, not only has the absolute performance of the largest machines increased at a sharply accelerated rate, but the ratio of performance to price has improved as well. As a result, many calculations that seemed impossibly vast 10 or even 5 years ago would now be more or less routine. However, applications of massively parallel computational methodology to problems in atomic and molecular physics have been rather limited, especially in comparison with other areas of physics (e.g., fluid dynamics). This relative lack of progress no doubt stems from a number of causes, including perhaps the perception that parallel machines are not suited to problems that do not have an obvious spatial decomposition, as well as the perception that any effort invested in the arduous hand parallelization of a complex program is soon to be vitiated by the advent of automatically parallelizing compilers. The chaotic competition between architectures and paradigms, accompanied by the frequent appearance and disappearance of vendors, certainly justifies a degree of hesitancy and skepticism also. Nevertheless, we would argue, the widespread application of massively parallel machines to atomic and molecular problems is not only feasible and appropriate, but overdue. While it is undeniably easier to conceive strategies for parallelizing physical problems that involve low dimensional spaces and local interactions, we will show by example that an obvious physical decomposition of the problem is not prerequisite to an effective computational decomposition. Moreover, automatic parallelization, while the topic of vigorous research, is enormously difficult, and the design and
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implementation of complex programs that scale efficiently to hundreds of processors seems likely to remain a task for humans into the foreseeable future. Granted, we must still choose from an array of parallel programming methods and target architectures, with little assurance that today’s favored archetype will be in favor-or even in existence-tomorrow. However, in the past couple of years there has been some narrowing (by bankruptcy) in the range of hardware choices, accompanied by the emergence of de fact0 standards for at least the message passing model of parallel software. The latter development affords some assurance that investments in program development can be recouped on a variety of architectures, ranging from networks of workstations to conventional shared memory vector supercomputers. In this chapter, we will describe our methodology for implementing the calculation of low energy electron-molecule scattering cross sections on massively parallel computers. Our emphasis here will be on the adaptation of a problem in molecular quantum mechanics to parallel computation and on the performance achievable, and our hope is that some of the techniques described here will be of use to others who are contemplating the parallelization of similar computations. The present work updates and extends a previous account (Winstead and McKoy, 1995) along similar lines; for a discussion of various computational methods for electron collisions and a survey of recent results obtained by those methods, see Winstead and McKoy (1996). In Section 11, we summarize the theory behind our numerical method, showing how we arrive at working equations for the scattering amplitude beginning from Schrodinger’s equation. Section 111 discusses the parallel implementation of our method and associated issues. An illustrative example is presented in Section IV, and concluding remarks are offered in Section V.
11. Theory The approach we use to solve the electron-molecule scattering problem is referred to as the Schwinger multichannel (SMC) method (Takatsuka and McKoy, 1981, 1984). The SMC method is a straightforward extension of the original variational method of Schwinger (1947) to multichannel problems in many-electron systems. A recapitulation here of its main points will aid in understanding the associated computational issues that are our main subject. Consider a molecule possessing N electrons and it4 nuclei. In the collision problems of interest, the nuclei can almost always be considered
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fixed, since the duration of the collision is short on the time scale of nuclear motion (Chase, 1956). The purely electronic Hamiltonian for the molecule is then
where atomic units (ti = e = me = 1) are employed, as they wjll be is the coordinate vector of electron i, and Z j and R j are throughout, the charge and coordinates of nucleus J . An electron colliding with this molecule interacts with it through a potential V,+
<
N
b + 1 =
c
i=l
The ( N
M
1 j
-t
Iri - r N + i l
-c J=I
ZJ +
Ir,+1
+
-RJI
(2)
+ lbelectron Hamiltonian for the total system is therefore H
=
Hmol - 3 V . + l + VN+,
= Ho
+
VN+l
(3)
where the zeroth order Hamiltonian Ho describes the system in the absence of the projectile-target interaction V,+ 1 . Continuum solutions of H subject to appropriate boundary conditions will provide all necessary information about the collision event, specifically the cross sections for various elastic and inelastic scattering processes. Direct determination of the continuum eigenstates of H is out of the question because of the high dimensionality of the problem. Instead, we use a variational principle that provides approximate solutions within a given trial space, or set of basis functions. Since we are interested in scattering, the variational principle that we use determines a stationary value f of the scattering amplitude f, whose square modulus yields the cross section. The form of this principle is
zin
In Eq. (4), and Loutare the projectile electron's wavevector at long and Yr(-) are eigenfunctions of H times before and after collision, *(+) subject, respectively, to outgoing- and incoming-wave boundary conditions
C. Winstead and I/. McKoy
188
(Newton, 19821, and c $ ~ and ~ 40,t are solutions to the interaction-free Hamiltonian Ho. Thus, for example,
4in
6 ,...,r N r N ) exp( iZi, -1-
=
+j(
(5)
j 1)+
* ~
where $i is the initial (usually ground) electronic state of the molecule; is constructed similarly from the final molecular electronic state which may or may not be the same as In Eq. (4) we have dropped the subscript N 1 on V, since no confusion is possible. The operator A ( + ) occurring in the last term on the right hand side may take more than one form. In the original Schwinger principle, which is suitable for potential scattering and single-channel problems, it is A ( + )= V - VG(+)V (6) 0
+
where G6+)is the outgoing-wave Green's operator associated with H,,
G ~ + ) ( E=) lim ( E - H , + i e ) - ' (7) €-o+ with E being the electronic energy of the system. Complications that arise in many-particle problems (Geltman, 1969) suggest that an alternative form (Takatsuka and McKoy, 1981, 1984) of A ( + )be used:
In Eq. (8), P is a projection operator that selects open channels, that is to say, energetically allowed scattering processes, and G$+) is the projected version of the Green's operator G6f). This alternative form allows us to obtain correctly antisymmetric solutions without including closed-channel states in the representation of the Green's operator, greatly facilitating a numerical implementation. The variational expression, Eq. (41, is reduced to working equations by introducing first many-particle and then one-particle basis sets. Thus, we assume that Yr(+) and Yr(-) can be approximated as linear combinations of a set of known ( N 1)-electron functions xi:
+
*(+'(Lout)
xi (6
=
Ci xi(
=
C yi( Z i n ) X i ( q
Zout)
+ * * * 3
rN+ 1
(9)
and similarly q(-)( Gh)
,*.*,
~
j 1)+
( 10)
1
As indicated in Eqs. (9) and (101, the+ expansion coefficients xi and yi depend on the wavevectors Gin and ,tout, though we will suppress this
HIGHLY PARALLEL COMPUTATIONAL TECHNIQUES
189
dependence henceforward. If the set of xi is complete, then the expansions of Eqs. (9) and (10) are exact; in practice, of course, the expansions are truncated at some finite size K . Introducing the truncated expansions into Eq. (4) and imposing the requirement of variational stability,
af
af
ax,
dy,
-=--
-0
for all i, we arrive at a system of linear equations; in matrix form, these are = b(i") ( 12) with Aij
=
( Xi1 A ( + )I X j )
a K x K square matrix, x the column vector of coefficients x i , i and b(in)the column vector of length K with elements
(13) =
1 , . . ., K ,
Solving Eq. (12) for x allows us to obtain the variational approximation to the scattering amplitude as - 2.rrf = b(o'Jt)fx
where
b(O"')
(15)
is the column vector
and the superscript t indicates Hermitian conjugate (i.e., complexconjugate transpose). Generally, we will need to:ompute the amplitude flZin, Lout)for many different directions finand kout of the initial and final wavevectors. The procedure for doing so is conveniently expressed by transforming the vectors b(in)and b(O"') into rectangular matrices, with each column corresponding to a different direction. The solution x then becomes a rectangular matrix as well, so that fe, as given by Eq. (15), is a matrix with rows labeled by Rout and columns labeled by fin. Before we can solve Eq. (12) for the coefficients x i , we must choose a form for the ( N + 1)-electron basis functions xi that will allow us to compute numerical values for the matrix elements A,,, bii"),and b i ( O u f ) . As in bound-state problems, the most convenient form is an antisymmetrized product of one-electron functions l(3,i.e., a Slater determinant, or better still, a spin eigenfunction formed from one or more Slater determinants, a so-called configuration state function (CSF). When such forms are used for
190
C. Winsteadand K McKoy
the xi,all matrix elements required in Eqs. 02-16), except those involving the Green’s function G V ) , reduce to combinations of simple integrals over the coordinates of either one or two electrons (since only one- and two-electron operators appear in the Hamiltonian H ) . We will have considerably more to say about these integrals in Section 111. The Green’s function we may treat by means of the spectral representation
where En means the energy of the molecular state t,bn and the sum is over all channels (allowed excitation processes). Introducing this representation of G6f) into our expression for A ( + )allows us to express matrix elements of the Green’s function in terms of matrix elements that are analogous to the bin) ( n = in,out) of Eqs.314) and (16). However, there remains an integration over the variable k’,for which we must resort to quadrature. The quadrature associated with the Green’s function term of A ( + )is in fact the major computational step in the SMC method (and in most other Schwinger-type methods as well). Having obtained an approximation to the scattering amplitude, we have nearly solved the collision problem. The differential cross section per unit solid angle, d u / d R , is readily obtained from Eq. (15) as 2
dR
kin
Equation (18) is in the body frame; that is, the directions 7fin and f,,, are defined with respect to an axis system fixed in the molecule. Most commonly, we are interested in scattering by a gas, which under the assumptions made amounts to a vast collection of randomly oriented molecules. In that case, the mzasured Gfferential cross section depends only on the angle 0 between kin and k,,, and corresponds to an+average_pver all possible orientations of the molecule yith respyt to kin and k,,,-or, equivalently, over all orientations of kin and k,,, with respect to the molecule, subject to the relative angle 0 being held fixed:
I
2
=
where
t$
is the unit vector in the direction of
7f.
cos 0)
(19)
HIGHLY PARALLEL COMPUTATIONAL TECHNIQUES
191
111. Computational Implementation A. DESIGNING A PARALLEL PROGRAM The parallel SMC program that is currently in use was not written “from scratch”; rather, it is the descendant of a preexisting sequential program that was used extensively on VAX and CRAY machines. As we will discuss, the coarse-grained parallel programming model that we have followed not only permits but encourages extensive reuse of existing code. However, achieving high scalability, meaning ready extensibility both to larger problems and to larger numbers of processors, required extensive top level reorganization of the program, as well as reformulation of some lower level steps. Subsequent program development has entailed further adaptations to running in a parallel environment. We introduce our discussion of the parallel SMC method by describing various considerations that enter into the design or redesign of a program for parallel execution. A guiding assumption will be that the object is to implement a computation whose potential demand for resources is virtually unlimited in such a way that the largest possible systems can be studied.
1. Architectures The choice of a target architecture, or type of machine on which the program is to run, is one of the most important decisions that must be made early in the design process. Since, as mentioned in Section I, a vigorous Darwinian competition among architectures is in progress, this choice must be made with some trepidation. Perhaps the best route is to choose a target that provides maximum flexibility. With this in mind, let us consider in rough outline the types of parallel machines most commonly encountered, as well as how well each type of machine can imitate other types. Many conventional vector supercomputers such as the CRAY C90 are of course also parallel computers, in that they contain more than one central processing unit (CPU). These PVPs (parallel vector processors) are characterized by a single, shared memory space and by a small number of fast processors. Because memory is shared, “communication” among the processors is very fast. Similar to PVPs are symmetric multiprocessor machines (SMPs), which typically have a small number of microprocessors sharing a common memory space. In contrast, MPPs, or massively parallel processors, are composed of large numbers of processors and almost always have distributed memory; that is, each processor (or pair, triple, . . . of
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processors) has associated with it a separate physical memory and a separate address space. Data sharing on an MPP requires communication between processors; however, on most modern MPPs, that communication takes place over a relatively fast network. Finally, we mention the NOW model, or network of workstations, typically consisting of a small number of processors, each with its own memory, communicating over a relatively slow network. In NOWs, especially, it may be important to assume heterogeneity, i.e., that processors differ in their speed, memory size, etc. (whether because of actual differences among the workstations or because some of them are busy with other jobs). As a general rule, architectures with more tightly coupled hardware can emulate well those of looser construction. Thus, PVPs and SMPs can be used as if they were MPPs or NOWs, whereas an MPP may run programs designed for a NOW better than any NOW. The greatest flexibility would therefore be achieved by assuming a NOW as the target architecture. However, we would argue that this assumption is overly restrictive. Low interprocessor communication speeds will make it impossible to implement efficiently many algorithms for operating on distributed data structures, while the assumption of heterogeneity may require excessive attention to elaborate load balancing procedures. Most seriously, the assumption of a small number of heterogeneous processors may discourage adequate attention to the issue of scalability, and thus ultimately limit the size of problem the resulting program can treat. Although a few methods (e.g., probabilistic simulations) that essentially involve running many independent instances of the same calculation are readily implemented for a NOW, in most cases the MPP is a better target architecture. Moreover, as network speeds improve in the future, some of the distinction between a NOW and a small MPP may be erased.
2. Programming Models There are many approaches that can be taken to parallel programming. At one pole is completely automatic parallelization of sequential programs by the compiler. For all but the simplest and smallest programs, this goal seems distant as it is lofty. A more modest approach is embodied in parallel programming languages, which permit the specification of data structures that are to be distributed over multiple processors and/or tasks that are to be carried out in parallel, leaving it up to the compiler and the run-time system to manage the actual distribution of data (possibly with some guidance from the programmer) and to supply the necessary communication among processors. Such languages aid the transition from sequen-
HIGHLY PARALLEL COMPUTATIONAL TECHNIQUES
193
tial programming by presenting the user with a single address space and by hiding the interprocessor communication, but they have significant drawbacks as well. Apart from the lack of widely supported standards, which may impair the portability (and ultimately the longevity) of programs written in parallel languages, there is the question of efficiency. Choosing the best way to implement an operation in parallel may require knowledge about the problem being solved (such as the range of typical input parameters) that is not available to the compiler; in the absence of such information, the compiler may choose a less than optimal algorithm or may be forced to sacrifice efficiency completely in order to assure that the resulting code is general (correct on any number of processors and for any dimensions). While there is continuous development in the field of parallel languages, performance of programs employing them is, to our knowledge, often disappointing at present. The opposite pole from automatic parallelization is explicit message passing. In this approach, the program is written from the point of view of a single processor with its own private memory. The programmer must provide a mechanism by which each processor can decide (based on its processor number) what tasks it is to do and what data items belong to it; when data are needed by other processors, or must be obtained from elsewhere, the programmer is responsible for providing the necessary communication calls. Such a program is trickier to write than a sequential program, but not inordinately so; new considerations such as maintaining a proper sequence of events across processors quickly grow familiar after the first few encounters. Within the message passing model there are further distinctions to be made, of which the most important is between task parallelism and data parallelism. One of the easiest approaches both to writing a new parallel program and to parallelizing an existing sequential program is the task parallel “master/slave” model of computation, in which one master processor advances through the list of work to be done, assigning tasks to the remaining slave processors. When finished with a task, the slaves report results to the master and/or advertise themselves as ready for future assignments. Provided the tasks are numerous and not too different in length, the master/slave model can provide good load balance, even on heterogeneous architectures. However, on further consideration, a number of disadvantages to the model become apparent. Since only the work, and not the data structures, is being partitioned, the per-processor memory (and not the aggregate memory) may limit the size of problem that can be addressed. On large numbers of processors, moreover, the master processor may be overwhelmed by the job of managing the many slaves, and slave processors may then fall idle waiting for instructions. Moreover, funneling
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C. Winsteadand I/: McKoy
all communication into and out of a single master processor does not take full advantage of bandwidth in the multiply connected communication networks found on MPPs. Finally, not all computations will have an obvious decomposition into discrete tasks that can be carried out independently. For example, multiplication for two very large matrices is best done in parallel with a very regular division of data and a tightly coordinated pattern of interprocessor communication. A master/slave implementation of matrix multiplication, though possible, could be expected to have vastly inferior performance. Data parallelism assumes that the processors will be carrying out (roughly) the same operations at (roughly) the same time, but operating on different data. Data parallelism is natural in many signal and image processing applications, and was central to the single-program, multipledata (SPMD) architecture of well-known early parallel computers such as the Connection Machine 1, in which instructions were broadcast to vast numbers of simple processors from a central source. However, less rigid forms of data parallelism are suitable to a wide variety of scientific computations. A key feature provided by data parallelism is the opportunity to decompose large data structures across the processors. Once the data are distributed, usually in some regular fashion, each processor follows the same steps, though operating only on its local data. While both the amount of data and the amount of work should be nearly equal among processors, it is neither necessary nor, in most cases, desirable that the processing steps be identical across processors. For example, we may wish each processor to call the same subroutine, but the branches that are taken within that subroutine may be different depending on the particular data assigned to each processor. Likewise, though it will generally prove necessary to synchronize the processors from time to time (e.g., where communication or 1 / 0 takes place), there is no advantage to insisting on synchronization at a fine scale, especially when the operations vary slightly among processors. In short, a course-grained, loosely synchronous approach, in which large tasks of nearly equal size are carried out between synchronization points, will often prove advantageous within the SPMD model.
3. Scalability
Planning for high scalability includes identifying those tasks within the program that grow most rapidly with a critical problem dimension. For example, in gravitational or molecular mechanics simulations, the impor-
HIGHLY PARALLEL COMPUTATIONAL TECHNIQUES
195
tant dimension is the number of interacting bodies n, and the force computation, scaling as n2 or n log n (depending on the algorithm), dominates for large n. In many-electron problems, both the number of one-particle basis functions and the number of many-particle basis functions may be important scaling parameters, and various tasks may scale as the fourth, fifth, or even higher powers of these parameters. Equally important is identifying data structures (e.g., arrays) that grow rapidly with problem size. Both the largest tasks and the largest data structures must be divided up among the processors in a truly scalable program; otherwise, the amount of memory available on a single processor, rather than the aggregate memory of all processors, will limit the maximum problem size. There is, alas, no universal formula that says how to divide up either the work or the data in an arbitrary calculation. However, important considerations are load balance, meaning an equal division of work among all processors (so that those with less to do are not left idle, waiting for the busiest processor to finish), and an equal division of the data. Most significant, also, is planning for communication. When data structures are divided up among processors, most operations on those data structures will require interprocessor communication. The pattern of division will determine not only the pattern but the amount of communication, and thus it will strongly affect the efficiency of the resulting program. To take a trivial example, matrices can be added without any communication at all if, and only if, corresponding elements are stored on the same processor. With an appropriate division of data, it is often the case that scaling laws operate to keep the overhead arising from communication low (Fox et al., 1988); for instance, the communication required to multiply two n X n matrices, which grows as n2,becomes negligible, for large enough n, compared with @n3) computational effort.
4. Further Considerations Current MPPs have certain common characteristics that also should be taken into account when designing a parallel program. As a rule, main memory is slow compared with that found on PVPs. A partial compensation is the presence of some amount of fast cache memory; however, only algorithms that can reuse the data in cache (e.g., Fourier transforms and matrix-vector or matrix-matrix multiplication) can sustain a large fraction of the microprocessor’s peak speed. Even then, it is important to take advantage of vendor-supplied libraries of optimized routines if available.
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C. Winstead and V. McKoy
On the other hand, main memory, when aggregated over all processors, tends to be large, and many procedures that would ordinarily require storing data on disk can be planned as “in core” operations on MPPs. This is an especially important point because disk storage may be very slow; moreover, disk space (at least that available to a single user) may not be significantly larger than memory space. A concern for portability would suggest avoiding any hardware or software features that are peculiar to a particular machine. This is made more difficult by the lack of true standards for communicating between processors, for writing files in parallel, etc. However, the Message Passing Interface (Walker and Dongarra, 1996) is emerging as a de facto standard for communication and is available on a number of MPPs and on NOWs. The most conservative approach is to assume only a minimal set of simple communication operations exists and to construct more elaborate operations from members of that set. Still, it is important to strike a balance between ease of implementation and portability. Most current message passing libraries include simple global operations such as finding the maximum value of a variable across all processors or summing corresponding elements of a vector across all processors; judicious use of such routines can be convenient and will not seriously impair portability.
B. PARALLEL SMC METHOD We now look more closely at the steps that carry us from the formulation of the scattering problem in terms of a variational principle, as described in Section 11, to actual numerical results, paying particular attention to the relative difficulty of the various steps and to the suitability of the calculations involved for execution on highly parallel machine. In line with the discussion in Section III.A, we have implemented the parallel SMC method using a coarse-grained, loosely synchronous SPMD model. This model, as we will see, is highly suitable to the matrix-oriented calculations that underlie the SMC method. We recall from Section I1 that the SMC method calculates cross sections by constructing and solving a system of linear equations, Eq. (12). As will be discussed, the solution of the linear system can be carried out in parallel (Hipes, 1989). However, the actual solution of the linear system is not, in fact, a significant component of the calculation; constructing the matrices A and b(“) is far more demanding. Accordingly, it is critical to implement the steps involved in constructing these matrices in a manner that is both efficient and scalable.
HIGHLY PAFULLEL COMPUTATIONAL TECHNIQUES
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1. One- and Two-Electron Integrals We recall that working equations are obtained by expanding the wavefunctions *(+Iand Yr(-) in terms of a basis set { x i } of many-electron functions, as in Eqs. (9) and (10). In variational calculations, it is important to employ basis functions that permit the approximate wavefunction to satisfy the boundary conditions. In the present case, the boundary conbehave, ) as any one electron’s coordinates are ditions require that * ( + taken to infinity, as
That is, W(+)asymptotically behaves as the superposition of the incident plane wave, multiplied by the initial target state, and expanding spherical waves, one for each open scattering channel, muciplkd by the appropriate target state and by the scattering amplitude, fon(kin,kout).The corresponding form for * ( -is)similar but, as implied by the - superscript, involves collapsing spherical waves exp( - ikr)/r. The important point to note about the boundary conditions is that *(* ) does not vanish asymptotically. Thus, we cannot, without further consideration, employ localized basis functions xi to represent the wavefunctions. This is a serious complication, since we wish to expand xi in terms of localized one-electron functions, specifically Cartesian Gaussian functions
where Nnlmo is a normalization constant and F= ( x , y , z ) . Fortunately, closer inspection reveals a very desirable property of the Schwinger variational method: the necessary matrix elements, Eqs. (13), (14), and (161, involve either the interaction potential V alone or VGV’V, as in Eq. (6). Since there can be no contribution to the required integrals from regions beyond the effective range of V , it is not necessary that the basis functions xi extend beyond the range of V in the Schwinger method. In the SMC method, in contrast to the original Schwinger method, A ( + ) contains additional terms that do not contain V. However, it can be shown (e.g., Winstead and McKoy, 1993) that these additional terms cancel each other in the asymptotic region, so that an expansion in localized basis functions remains permissible.
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C. Winstead and K McKoy
+
As already mentioned, we use for the ( N 1)-electron functions xi antisymmetrized products (Slater determinants) of one-electron functions, or rather spin-adapted combinations of Slater determinants, CSFs. The CSFs are not expressed immediately in terms of Cartesian Gaussians, but in terms of molecular orbitals a ( 3 , b(r3, c ( 3 , . .., obtained by solving a bound-state electronic structure problem. For example, the simplest approximate representation of the ground state of a closed shell molecule is
where the I... I notation indicates a Slater determinant, a and P are single-particle spin functions, and the molecular orbitals are obtained from a self-consistent field (SCF) calculation. When “virtual orbitals” spanning the orthogonal complement of the space of occupied orbitals are included, the molecular orbitals form an orthonormal basis convenient for the computation of many-electron matrix elements, for the representation of the N-electron target states qin and and for the representation of the ( N 1)-electron basis functions that are most effective in the scattering calculation, namely, those having the form of a target state times an additional one-electron function. The Cartesian Gaussians at last appear as a basis set for expansion of the molecular orbitals:
+
where Cia is a coefficient and the formal summation index i refers collectively to the parameters 1, m ,n , a,and R’ of Eq. (21). We defer detailed discussion of the procedure for forming manyelectron matrix elements from integrals involving one-electron functions, in order to focus on the integrals themselves. Here we only note that successive insertion of the expansions of Eqs. (9), (lo), and (23) into the expressions for the elements of A and b(“), Eqs. (131, (14), and (16), together with the use of expansions similar to Eq. (22) for the molecular states, gives rise to two major categories of integrals. The first category arises from the Hamiltonian and potential terms in matrix elements of A ( + )between CSFs xi and xi. It includes one-electron integrals (those involving the coordinates of a single electron), which we write as ( m I n), with ( mIn )
=
T,,
+ Urn,
(24)
HIGHLY PARALLEL COMPUTATIONAL TECHNIQUES
199
where T,, arises from the kinetic energy operator: Tmn =
1 -2 /d3rlm(r3v2ln(3
(25)
and Urn, from the electron-nuclear attraction:
Also in the first category are the two-electron integrals
that arise from the electron-electron repulsion term of the potential. The integrals in this category are familiar from bound-state molecular electronic structure calculations and can be evaluated analytically (Boys, 1950) when the 5, are Gaussians. The second category of integrals involves both Gaussians and a plane wave. These integrals arise from Eqs. (14) and (16) for the elements of b(i") and b(Out), and also, as described in Section 11, from the remaining term of A ( + ) ,the VG$+)Vterm, after the introduction of the spectral representation of Gb+), Eq. (17). Kinetic energy integrals do not occur, so that for the one-electron integrals we have -3
(mI k )
=
urn;
=
-/d3rlm(3E J
whereas the two-electron integrals are
These Gaussian-plus-plane-wave integrals can be evaluated analytically using techniques similar to those used for the Gaussian integrals of Eqs. (26) and (27) (Ostlund, 1975; Watson and McKoy, 1979). 2. Quadrature and Scaling
The number of integrals in the first category, Eqs. (24-271, is easily estimated. Given Ng Gaussians, there are Ng(Ng+ 1)/2 unique pairs of
200
C. Winstead and V. McKoy
Gaussians, and thus the same number of one-electron integrals (m I n). Far more numerous are the two-electron integrals (mn Ipq); there are approximately N:/8 of these, after making full use of the equivalences
To estimate the size of the second category of integrals, we must know not only the number of Gaussians, Ng,but also the number Ng of different wavevectors k' that will arise in the calculation. We can arrive at an approximate value of Ng as follows. Obviously N; depends in part on N E , the number of energies at which we wish to compute cross sections. Moreover, in a calculation involving N, open channels, we require at each energy N, different matrices b'"), Eq. (16), to evaluate a scattering amplitude for each channel. In general, the channels will have different excitation thresholds; thus, the residual kinetic energy of the electron, and hence the magnitude of the wavevector k', will be different in each channel, requiring an independent set of integrals ( m I k') and (mn I&). Meanwhile, the number of directions & in each channel, Nk, must be sufficient for the matrix representative f' of the scattering amplitude, obtained from (151, to provide an accurate representation of the continuous function Eq. + + f(ki,, kout).Because we are interested in scattering at low energies, where only a limited number of partial waves contribute to the scattering, this number is fairly modest; 100 to 200 values for 2 usually suffice. [An exception to the rule that only low angular momenta matter at low scattering energies occurs in the case of molecules possessing permanent dipole moments; however, that case can be handled by applying a correction after the SMC calculation is complete. For further discussion, see Winstead and McKoy (1996).] Thus, for the evaluation of the b'"), we require B(N,N,Ni) = @(102N,N,) values of k' in Eqs. (28) and (291, implying B(Io~N,N,N,~)integrals (mn I pk'). Since the integrals of Eqs. (28) and (29) also arise in the evaluation of VGj+)V,we must consider in more detail how that term is evaluated. Having introduced a spectral representation of G$+)in Eq. (171, we are left with a three-dimensional integral over a k' variable, which must be evaluated by numerical quadrature. There being a term in G$+)for each open channel, it might in fact appear necessary to perform N, separate quadratures. Indeed, G$+)also depends on the total energy E, and we might even suppose N, NE quadratures to be necessary. In either case the SMC method would be rendered impractical, for, as we are about to see, even one such quadrature implies a substantial computation. Fortunately, we can separate both the energy and the channel dependence from the
HIGHLY PARALLEL COMPUTATIONAL TECHNIQUES
201
quadrature. The key is careful treatment of the singularities of the integrand, which contain both the energy and the channel dependence (Lima et al., 1990). Let us write
where k:/2
=E
-
En and
The angular integral in Eq. (32) poses no special problem and may be evaluated either by a product of one-dimensional quadratures in the polar angles 6kt and 4k! or, more efficiently, by means of the quadratures developed by Lebedev (1975,1976,1977, 1994; Lebedev and Skorokhodov, 1992; Treutler and Ahlrichs, 1995) for functions defined on the surface :0 a sphere. Since we are no longer confined to small magnitudes of the k vector, however, we find that &(lo3)angular quadrature points is a more realistic estimate for this part of the calculation. The singular radial integrals of Eq. (31) we first separate into principal value and residue contributions:
- i,rrk,F,( k,)
(33)
where 9 indicates principal value. The residue term may be evaluated using the same integrals-used for the b(") matrices (in fact, using the matrices themselves). The principal value term may be written
jm k " dk' 0
9
',(''I ;/2 - kr2/2
=
dmdk'
kr2F,( k ' ) - kiFn( k,) k;/2 - kr2/2
( 34)
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C. Winstead and K McKoy
since m
g
p
1 n
-- 0
(35)
The form of the principal value integral given by the right hand side of Eq. (34) is much more tractable to quadrature. Since both the numerator and denominator vanish at k, , the integrand is smooth there, and we need not employ a set of quadrature points adapted to the value of k,; a single set of points can be used for all channels n and all energies E. In all our studies to date, we have found that Nk, = lo2 radial points in k‘, or fewer, are sufficient to converge the radial integrals. Thus, including the angular and Ng dependencies, evaluating VGbf’V requires @(lOsNi)of the integrals (mn I pL). To summarize, the scalings of the integral calculations are @(N:) f2r ] the (mn I p k ) the (mn I p q ) integrals, and @[(102N,N, l O S ) N ~for integrals, where in the latter case we have included both the integrals needed to evaluate b‘”) and the (additional) integrals necessary to evaluate VGY’V. A clearer idea of both the relative and absolute magnitudes of these tasks requires some estimate of Ng, N,, and N E . In calculations to date, N, has usually been less than 10, whereas a few dozen energies E are usually enough to characterize the energy dependence of the cross sections. The value of Ng depends, as mentioned earlier, on the size of the molecule; however, in most of our recent work it has ranged from somewhat less than 100 to somewhat more than 300. Thus, we may conclude (1) that the (mn Ip6) integrals are much more numerous than the (mn Ipq) integrals and (2) that @(lo”) integrals of the former type might not frequently be required. Needless to say, many floating point operations are required per integral, and a rather large computation is implied. Though, as we will see, other steps in the S,MC procedure can also be substantial, very often evaluating the (mn Ipk) integrals is the dominant task in a calculation.
+
3. Parallel Integral Evaluation Because the integrals ( m I n ) and (mn Ipq) are relatively less numerous and are independent of the energy E, we evaluate them “once and for all” within a setup program that reads a description of the molecule and the scattering problem and prepares input files for the main calculation. This setup program has not been parallelized, since it represents only a small fraction-of the total execution time. Parallelizing the evaluation of the (mn I p k ) integrals, however, is clearly essential.
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Since each integral (mn I p z ) can be evaluated independently, there is no difficulty in principle in parallelizing the work, although attention must be paid to load balance and to future communication. An additional consideration is the number of integrals it is practical to do at once: on current machines, it is not feasible to hold 10’l integrals in memory, and in any case, we may wish to break the calculation down into smaller pieces whose results can be saved periodically to disk. The scheme we e_mploy, which has proven generally satisfactory, is to consider the (mn Ipk) as a series of rectangular arrays Z;L:i, L with, rows labeled by an index to unique ( m ,n ) pairs and columns by the angle k , there being a different array I for each Gaussian p and each magnitude k. The processors are (conceptually) arranged into a corresponding square or rectangular array; successive rows of I$:~,J are then assigned to successive rows of processors, and successive columns of integrals to successive columns of processors. Since there are always more rows and columns o_f,integrals than of processors, each assignment is cyclic. Though the ( m I k ) integrals are much less numerous and need not be evaluated in parallel, it is convenient to “piggyback” their evaluation onto that of the (mn I p z ) integrals by adding to each integral in which the column label is again k. In this array I$:],J a row ( p I manner, every integral is allocated to a unique processor that is responsible for its calculation. We should mention that the allocation of work can itself be done in parallel; that is, rather than have a central authority broadcast assignments, each processor decides, based on its position in the processor array (which in turn is deduced from its processor number), which integrals to evaluate. Load balance in the scheme just described is achieved statistically. Depending on the particular Gaussians m and n involved, the amount of work required to evaluate (mn I p z ) can vary considerably; the dependence on k is much weaker. However, the number of ( m , n ) pairs is always much larger than the number of row of processors in our logical array, and the cyclic assignment of pairs to processor rows has a pseudo-randomizing effect. Thus, by the time each processor has finished its subset of I$:;, h , the fluctuations in the time required by individual integrals will have averaged out to a great extent. This is an example of the flexibility afforded by a coarse-grained, loosely synchronous approach. A second advantage of a course-grained approach is that it facilitates reuse of existing code. The integral evaluation routines can be carried over without modification into the parallel program, since no communication, synchronization, or further subdivision of work is required within them. Indeed, the only modifications we have made to these routines are optimizations that would also have been useful in the original sequential program.
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4. Integral Transformation Procedure When an array Z$k],f is complete, something must be done with the integrals before proceeding to subsequent values of p and k. Specifically, we must evaluate the contributio: of the current set of integrals to matrix such as arise in the evaluation of the elements of the type ( xjl V l~$,,(k>>, b(")and of the VGF)V term in A. The original, sequential SMC program followed a conventional path in first transforming from two-electron integrals over Gaussians to integrals over molecular orbitals, and then employing the latter integrals to construct many-electron matrix elements of V. However, it is difficult to conceive a parallel strategy in which the second step of this two-step procedure is efficient. Each integral over molecular orbitals will in general contribute to only a few many-electron matrix elements, and those are likely to be located on other processors than that where the integral resides. An intricate pattern of data movement, reflected in an intricate and error prone program, is thus likely to result. To avoid this, we fuse the transformation to molecular orbital integrals with the formation of many-electron matrix elements, so that a single, simple communication operation effects both transformations (Hipes et al., 1990). To provide a concrete example of this procedure, let us take 4n to be the product of a plane wave with the single-determinant, closed shell state of Eq. (22),while xj is that same closed shell state antisymmetrized with the jth virtual orbital. The usual rules (e.g., Szabo and Ostlund, 1982) for many-electron matrix elements then give
where the sum extends over the occupied molecular orbitals. To connect this expression to integrals over Gaussians, we introduce the expansion of Eq. (23), arriving at NR
c m j ( m12) + m=l
4
=c p= 1
c c
UEOCC
n,p=l
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where sums have been reordered and relabeled; the matrix
is, within a factor of N , the Gaussian basis representative of the oneelectron density matrix for the target state of Eq. (22). In Eq. (37), we have a direct connection between many-electron matrix elements and one- and two-electron integrals, one that is, moreover, suited to the parallel integral evaluation sch_eme we have described previously. Let the matrix elements ( x,l V I+,Jk)) be distributed over the logical array of processors as arrays, with j the row index, ff the column index, and k fixed %thin each array. Indeed, we recognize these arrays as the matrices b("'(k) defined in Eq. (14). Then we may write a matrix equation for b,$ k , : b("k) =
N.
C
K(p)I(Pk)
(39)
p= I
where
K'P),
in light of Eq. (371, is given by
except for the last column, which contains Cpi,and where the elements of I(pk)are the integrals Zc',",k!,~, with ( p I $) forming the bottom row. With these definitions, Eqs. (37) and (39) are equivalent. To summarizeiwe have developed a scheme for evaluating a subset of integrals (mn Ipk) and ( p I in parallel and for transforming them to final matrix elements b,!;k) by means of a series [represented by the summation on p in Eq. (39)] of dense matrix multiplications. Since we are free to distribute the matrix K(p) over the processors in any way we wish, we will arrange its distribution to conform with that of I(p"). The matrix multiplication in Eq. (39) then involves multiplication of local matrix subblocks on the processors, interleaved with communication of matrix subblocks among processors. Such a distributed matrix multiplication is easy to implement, efficient, and readily portable among machines. Although we have described the transformation procedure for a single, specific target state @ n , in general we are interested in several such states. Multiple channels are accommodated within the foregoing scheme by constructing additional matrices K(p) according to rules appropriate for the electronic structure of the target state and of the determinants xj. For open shell configurations, these rules may be more elaborate than Eq. (40), but the general outline is the same. Note that, unless the magnitude k of the wavevector changes, only the computation of the K(p) coefficient
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matrix and the matrix multiplication need be repeated to obtain matrix elements for all channels from a single set of integrals I(pk). In particular, employing a loop over k outside a loop on the channel index, we obtain all quadrature data for VGS;‘)V (except the residue, or “on-shell,” contributions) from a single evaluation of an integral set I(pk). The on-shell contributions, which enter into both VGS;‘)Vand b‘”),depend on both the channel label and the collision energy, and they are evaluated in a separate loop after the “off-shell” quadrature data are complete.
5. Scaling of Transfornation Step It is interesting to determine the scaling of the work involved in the integral transformation and compare it with that required by the actual integral evaluation. From Eq. (39) and the surrounding discussion, we see that for each value of k , we have N, matrices b ( n k )to construct, and that constructing each will involve Ng matrix multiplications. The matrices and Npair X being multiplied are, respectively, of dimension NCsFX Npair N i , where NCsF is the number of CSFs xi included in the calculation, and Npair = N,2/2. In most cases, the number of open channels is fairly small and CSFs corresponding to closed channels are excluded; we may then approximate NCsF as being on the order of N,Ng. We thus have b(N;N:N:) operations associated with the matrix multiplications. Taking nominal values of N; lo5, N, 1, and N6 100 (see the discussion in Section III.B.2) yields an estimate of operations, increasing rapidly as either N, or Ng is increased. Several observations may be made about this estimate and scaling. First, transforming the integrals is obviously a major operation in its own right; thus, performing the transformation in parallel (as we have done by implementing it as a series of distributed-matrix multiplications) is essential. Second, the relative expense of the computation of two-electron integrals and of the transformation of those integrals to b(nk)matrix elements is not fixed; rather, it depends on Ng and N,, the number of Gaussians and the number of channels. However, the time required for the two steps does not depend on operation counts alone. The transformation step, being a matrix multiplication, is very efficient on current MPPs because there are extensive opportunities for the reuse of data in cache. Using vendor-supplied multiplication routines in the subblockmultiplication kernel of this step can yield performance that, for large matrices, approaches the theoretical limit of the machine, even when the overhead of the communication among processors is included. On the other hand, the integral evaluation step involves fairly complicated logic, 51
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-
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numerous subroutine and math library calls, short loops, and other performance inhibiting features; thus, it tends to execute at a small fraction (10% is not unusual) of theoretical peak performance, even though it involves no communication overhead at all. Taken in combination, the high efficiency of the transformation step and the low efficiency of the computation step imply that computing the integrals tends to dominate in single- or few-channel calculations. For large numbers of channels, a more traditional transformation procedure, via two-electron integrals over molecular orbitals, is advantageous if we look at operation count alone, since its scaling with N, is less severe (Winstead and McKoy, 1995); yet, as we have noted, the effectiveness of an algorithm depends not on operation count alone, but also on operation speed and communication overhead. In practice, available memory and disk space, rather than the time required by the transformation procedure, have limited the number of open channels it is practical to consider. 6. Overall Program Outline
Having considered the two steps that form the computational heart of the parallel SMC calculation in some detail, we are ready to draw an outline of the parallel program as a whole. As mentioned earlier, the parallel program uses input files created by a sequential setup run that digests the problem description and evaluates those matrix elements involving Gaussians alone. At the beginning of the parallel run, all processors read these input files along with a control file containing such information as the quadrature to be used. Each processor decides, based on its processor number, where it sits in the logical array of processors, and therefore which data belong to it and what work it is to do. The processors then begin the computation of VG$+)Vusing the loop structure described in Section III.B.4. As each radial quadrature point k, is completed, the resulting b(nki)matrices are further reduced by performing the angular quadrature indicated in Eq. (32) to obtain a square, NCsFX NCsFmatrix Fn(ki)for each channel n. This quadrature may be written as F,
=
b(")tWb(n)
(41)
where W is a diagonal matrix of quadrature weights and, thus, has the form of distributed-matrix multiplication. The matrices F,(k,) are then saved to disk for later use, and the next k , taken up. When the ki loop is complete, the on-shell matrices b(")(zn)are computed for values of <, appropriate to the specified energy E. We are now in a position to read back the stored F,, matrices and to complete the
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construction of VGY'V, and hence of A(+), according to Eqs. (33) and (34). Since b(i")also provides the right hand side of our linear system, Eq. (12), the only remaining steps are the actual solution of the system and formation of scattering amplitudes and cross sections according to Eqs. (15) and (18). Because A is an NCsFX NCsFdense matrix, solving the linear system involves @(NZsF)= @(N:N;) operations (assuming once more that NCsF= NgN,, as is valid for small N,). Although the difficulty of the solution step grows rapidly with N,, in calculations to date it has been far outweighed by the integral evaluation and transformation. The solution may be carried out in parallel, using a parallel LU factorization routine (Hipes, 1989); however, when more control over possible illconditioning of A is desired, we simple save the A and b(")matrices to disk for later solution on a sequential machine, using singular value decomposition (Press et al., 1986). The b(") computation and solution are repeated for as many energies as are necessary to characterize the cross sections.
7. Input/Ouput ( I / O )
So far we have mentioned 1/0 in passing but have not discussed it as a discrete topic. Yet a consideration of the storage and retrieval of VGY)V quadrature data shows that 1/0 is potentially a major concern. As described in the preceding two subsections, for Nk radial quadrature points, we create and store N k N , square matrices of dimension NCsF= N'N,; thus, 1/0 volume scales as NkNiN:, and for modest numbers of channels, storage can reach into the gigabytes. Even assuming there is some place to put the quadrature data, the time needed to write and read back the data may become a bottleneck in the calculation if the 1 / 0 method is inefficient. Because the quadrature data set is written only once but is read back in its entirety at each energy where cross sections are computed, the efficiency of the read step is especially important. Working in our favor is a natural parallelism in the I/O: When a matrix Fn(ki) has been created by the SMC program, it exists in distributed form, with each processor owning a subblock. Thus, provided there is hardware and software support for parallel 1/0 on the MPP being used, the processors may simultaneously write their blocks of data to disk. Likewise, when reading these matrices back, each processor need retrieve only the blocks it has previously written, and again the operation may proceed in parallel on machines capable of parallel I/O. While such opportunities for 1/0 parallelism are a natural feature of the SPMD style of programming, and while current MPPs generally do support parallel 1/0 in some form, caveats are in order. First, the prospect
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of replacing an ordinary sequential write or read operation with dozens or hundreds of smaller, independent, and more or less simultaneous 1/0 transfers suggests that some rethinking of how the data should be structured may be appropriate. Moreover, there is currently little agreement about what a file system for a parallel machine should look like or what types of access it should support, and there are inherent difficulties in defining standards (e.g., what should be the outcome if two processors write to the same disk location at the same time?). The designer of an 1/0 strategy should be aware that completely different approaches may be called for on different machines, and that the approach taken can radically affect 1/0 performance. In our own work, for example, we have seen improvements in the read rate for the V G r ) V quadrature data of more than an order of magnitude simply by replacing a natural but naive parallel 1 / 0 strategy (one that matched the logical structure of the program) with a strategy tuned to the underlying hardware and to the strengths of the system software (Winstead et al., 1996).
IV. Illustrative Application A. GENERAL DESCRIPTION
As we have noted, the relative importance of different steps in the SMC method depends on the parameters that define the particular problem at hand, and the performance of those steps varies markedly. In discussing performance, it would be perfectly possible to take as an example a calculation with a large number of open channels and a small number of basis functions, in which the very large (and very efficient) matrix multiplications of the integral transformation phase would predominate. Nor would this be a completely unrealistic example, since many of our studies involve electronic excitation processes in which coupling among many channels is important. Such an example would present a highly favorable picture of the parallel SMC method's performance, since large matrix multiplications tend to execute at near-peak speeds. A less ideal but equally relevant example is a few-channel study on a larger molecule, for which a many-channel treatment may be unduly expensiye. In such a calculation the evaluation of two-electron integrals (mn Ipk) will predominate, owing to its @(N') scaling. We briefly describe in this section some performance data and cross section results from work of this kind currently in progress. Specifically, we look at singlechannel calculations of low energy elastic electron scattering by trimethylaluminum, AIC,H,, which is used in plasma deposition of aluminum and
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of aluminum oxide and nitride films. These calculations, which are part of an ongoing study of elastic and inelastic scattering by trimethylaluminum and its dimer, provide both a concrete example of the general approach that has been outlined and a context in which to touch on topics such as the use of symmetry and the use of contracted Gaussian basis sets that have heretofore been neglected in order to focus on matters most directly connected to parallelism. An electron scattering calculation using the SMC method begins with a conventional electronic structure calculation to determine an electronic wavefunction for the target molecule’s ground state and for any excited states of interest. In our study of trimethylaluminum, we employ a singleconfiguration, self-consistent field approximation to the ground state, of the form given by Eq. (22). The nuclear geometry we use is based on experimental values for the Al-C and C-H distances and the A-C-H angle (Almenningen et al., 1971); the AlC, backbone is planar, with the carbons forming an equilateral triangle. Because there is little if any barrier to rotation of the methyl groups, we choose, for convenience, a configuration of overall C,, point group symmetry, in which one H of each methyl is in the backbone plane. The Gaussian basis set used in the electronic structure calculation is chosen to give a good representation not only of the 20 occupied molecular orbitals but also of the “orbital” that describes the scattering electron. Since the character of the scattering wave function may be strongly energy dependent at low energies, and since we must describe that wavefunction over the range where the interaction V is significant, an extensive basis set that incorporates diffuse Gaussians is called for. Later, we will compare results from two basis sets. Each includes contracted Gaussians, which are linear combinations of so-called primitive Gaussians, Eq. (211, that give a compact description of occupied orbitals (particularly core orbitals), as well as uncontracted or primitive Gaussians that provide more flexibility in describing the valence and scattering orbitals. The first set is built around contracted atomic basis sets (Huzinaga, 1965; Dunning, 1970; Dunning and Hay, 1977) that may be written as (12s8p)/[6s4pI, (9s5p)/[4s3pI, and (4s)/[3s] for Al, C, and H, respectively, where numbers in parentheses refer to primitive and numbers in square brackets to contracted basis functions. To these are added uncontracted functions including one s, one p, and one d function on each C, one d on Al, and one p on each H. Note that each d represents six rather than five functions, since the linear combination ( x 2 + y 2 z2)exp(-ar2) is not excluded. Altogether, this basis includes 207 primitive and 147 contracted Gaussians. The second basis is characterized as (15s9p)/[6s4p] + 2d for Al, (lls6p)/[5s3p] + 2d for C, and (4s)/[3s] l p for H, and it employs atomic basis sets
+
+
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developed by Schafer et al. (1994). It comprises 240 primitive and 162 contracted Gaussians. The calculations described here are carried out in the static-exchange approximation, which in the present case implies that all of the ( N 1)electron configurations xi employed as a basis set for Yr(+) and Yr(-) are formed by adding a column containing one of the virtual orbitals to the Slater determinant describing the AlC,H, ground state. Since there are 20 occupied orbitals, we have 127 and 142 virtual orbitals, respectively, in the two basis sets, and we can form the same number of xi for use in the scattering calculation. However, because the interaction V is totally symmetric, we can separate the scattering calculation into independent calculations for each irreducible representation of the molecular point group. Computationally speaking, the matrix A$;) is block diagonal by symmetry. The largest single block is the 35 X 35 block for CSFs belonging to the A’ irreducible representation of C,, that is obtained in the larger Gaussian basis. As outlined in Section 111, the results of the electronic structure calculation, together with control information specifying the type of scattering calculation desired, are passed to a sequential program that prepares input files for the parallel program. For the present example, the sequential program requires only a few seconds of execution time on a CRAY Y-MP, since the matrix elements ( xilI/ I xj) that are its main responsibility can be determined from one-electron integrals and from orbital energies (eigenvalues of the Fock matrix) already computed by the electronic structure program. (Multichannel calculations may require an hour or more of time in the sequential program, which in its present form recomputes the two-electron integrals over Gaussians prior to forming the ( xil I/ I xj);this is an area for future optimization.) The sequential calculation is followed by the main, parallel calculation, whose largest computational component is the evaluation of VGY )V quadrature data. The parallel SMC program is able to make limited use of symmetry in this quadrature; specifically, it can take advantage of the existence of up to two orthogonal mirror planes to reduce by up to a factor of four the number of k angles for which integrals ( p 12) and ( m n I p z ) must be evaluated and transformed. The C,, group of AlC,H, has only one reflection plane, giving a factor of two savings. Regardless of molecular symmetry, quadrature points need extend over only half of k’ space, since ( p I -2) = ( p I i)*and (mn Ip - 2) = (mn I&*. Both calculations use the same quadrature for the VG$+)V matrix elements. There are 72 points in the radial dimension, including GaussLegendre points distributed over an inner region and Gauss-Laguerre
+
Zi
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points at larger values of k. The angular quadrature is a product of one-dimensional Gauss-Legendre quadratures in the polar angles 8, and &, with the number of quadrature points ranging from 324 (18 X 18) to 1024 (32 X 32) as the radius k increases. Altogether, the k' quadrature contains 26,384 symmetry-distinct points. Smaller quadratures will often suffice for smaller molecules; further, as noted in Section 111, an angular quadrature of the Lebedev type would have been more efficient. Nonetheless, the quadrature actually used is quite practical for the problem at hand.
B. PERFORMANCE Although, as has been emphasized, performance strongly depends on the nature of the calculation and on the machine used, it may nonetheless be useful to give some idea of the scale of the present calculations and the time they required. The performance figures given here refer to a CRAY T3D containing 256 processors, which was used for both calculations. This machine has a nominal peak speed of 38.4 GFLOP (38.4 X lo9 floating point operations per second) based on the per-processor peak speed of 150 MFLOP (150 X lo6 operations/sec), though in practice cache limitations seem to limit achievable speeds to slightly over 100 MFLOP per processor. From the basis set and quadrature dimensions given in the preceding subsection, we can determin5 that the VG$+)V calculation gives rise to 4.22 X 10" integrals (mn Ipk) over contracted Gaussians in the smaller basis set, and 5.64 X 10" integrals in the larger basis. In terms of primitive integrals, these numbers become 1.18 X 10l1 and 1.83 X lo1',respectively. Not all of these integrals need be evaluated, however, since some of them will be negligibly small. Based on prefactors that are easily computed, a simple estimate can be made of an integral's magnitude prior to its detailed evaluation; if the integral will be very small, it can be taken to be zero. This technique is particularly use@l in VG$+)V quadrature since, asymptotically in k, an integral (mn I p k ) behaves as exp(- ,$k2), with 6 depending on the nature of the Gaussians m,n, and p. As the quadrature proceeds to large values of k, more and more integrals are recognized as negligible and omitted, giving a significant speedup. For example, in the larger basis, evaluating integrals at the forty-first radial point in k requires 278 sec, whereas at the seventy-second and last point only 30 sec are needed. Operation counts for the integral evaluation are difficult to obtain since the algorithms used are intricate and since the operation count varies from integral to integral, but it is easy to compute the overall rate at which integrals are evaluated. This rate is 7.05 X lo6 integrals over contracted
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Gaussians per second in the smaller basis set and 5.64 X lo6 integrals per second in the larger basis; in terms of primitive Gaussians, the corresponding rates are 19.7 x lo6 and 18.3 x lo6 integrals per second. The drop in the rate per primitive integral in the larger basis is probably due to the greater number of d Gaussians, whose integrals are more expensive to compute, while the more substantial drop in the rate per contracted integral reflects both the increased number of d orbitals and the higher level of contraction (i.e., ratio of primitive to contracted functions in the basis). Operation counts are easier to compute for the integral transformation, since it involves matrix multiplication. Based on these counts, we obtain rates that range from 6.26 GFLOP for the smallest angular quadrature in the smallest basis to 7.86 GFLOP for the largest angular quadrature in the larger basis. Though quite respectable in absolute terms, these numbers represent only 25-30 MFLOP per processor, well below the 100 MFLOP achievable in the largest multiplications that will fit in memory. We expect (and have observed) better performance for calculations that involve larger matrices. Also, it must be borne in mind that our numbers include the communication and synchronization overhead of the distributed multiplication, and that the matrices being multiplied are far from square, which has a deleterious effect on the work to communication ratio. The aggregate time for evaluating the VG$+)Vquadrature data was 9870 sec in the smaller basis. Of this, 61% was spent in computing integrals and 31% in transforming them, with the rest accounted for by all other tasks (including 1/01 and by overhead such as load imbalance. For the larger basis, corresponding figures are 15,000 sec total time, with 67% spent evaluating integrals and 28% spent transforming them. These percentages are typical for a single-channel calculation, while, as we have noted, the transformation will rapidly grow in importance as the number of channels increases. When the calculation of the quadrature data is finished, energy dependent information is calculated and the A and b matrices assembled. Each energy of interest is currently done as a separate calculation, requiring, for the smaller basis, about 340 sec, and about 440 sec for the larger basis. Both numbers include considerable startup overhead (loading the program and reading input), and both include a large 1/0 component associated with reading the quadrature data set, which could probably be reduced by further optimization. As matters stand, calculating a cross section at any one energy is still a trivial exercise in comparison with generating the quadrature data, while calculating a set of several dozen results spanning the low energy range is a task comparable in size with the quadrature calculation.
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C. CROSSSECTIONS Integral elastic cross sections for trimethylaluminum from 0 to 50 eV impact energy are shown in Fig. 1. The results obtained in the two basis sets are seen to be substantially similar, with some differences in detail. This is evidence, though by no means conclusive proof, that the basis sets used are adequate for the elastic scattering calculation. Three prominent features of the cross section as a function of energy deserve attention. The rise in the cross section at very low energy is an artifact of the staticexchange approximation; though such a rise is experimentally observed in some molecules, we cannot say whether it occurs in AIC,H, without more elaborate calculations. The two maxima seen at about 2.5 and 8 eV are, however, expected to correspond to experimentally observable features. These maxima arise from shape resonances, or quasi-bound states resulting from the temporary trapping of the incident electron by the interaction potential I/. The static-exchange approximation, which neglects the re-
201
0
I
I
I
I
10
I
I
I
,
20
I
I
I
[
30
I
I
I
I
40
I
I
I
!
Energy (eV) FIG. 1. Integral elastic electron scattering cross sections for trimethylaluminum. Dashed curve is the cross section obtained using the smaller basis set; solid curve is obtained with the larger basis set.
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sponse of the molecular charge distribution to the presence of the projectile electron, yields a V that is too repulsive; thus, we anticipate that the shape resonances are located somewhat higher in energy by our calculation than they will be observed in experiment. Prior experience with other molecules suggests shifts of perhaps 1 eV for the narrow resonance at 2.5 eV, and 2-3 eV for the broad resonance. Symmetry decomposition of the cross section is very useful in the classification of resonances. As shown in Fig. 2, the narrow, low energy resonance is ascribable to the A” irreducible representation of C,, , while the broad maximum arises from a combination of features, one of which is an apparent resonance in E‘ symmetry. The A” resonance is easily classifiable as arising from temporary trapping in the empty 3p, orbital of aluminum (taking 2 as the normal to the AlC, plane). We have observed a similar resonance of the appropriate symmetry in another borane analog, boron trichloride, and expect it to be a common feature of these molecules. In BCl,, and perhaps also in AlC,H,, this resonance becomes a bound
Energy (eV) FIG. 2. Symmetry decomposition of the elastic electron cross section of trimethylaluminum obtained using the larger basis set. Curves are labeled by the irreducible representation of C,, to which the corresponding ( N + 1)-particle wavefunctions belong.
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anionic state when the nuclear framework is allowed to distort away from a planar structure (recall that the nuclei are held fixed in the course of the scattering calculation). The E’ resonance is also observed in BCl,, at nearly the same energy as in AIC,H,, and likely arises from the antibonding orbital conjugate to the AI-C or B-Cl e’ bonding orbital. If this is the correct explanation of the E‘ resonance, we could also expect to see an A’ resonance associated with the a‘ antibonding valence orbital. Indeed, Fig. 2 does show a weak broad feature at about 6 eV in A’ symmetry that is not visible in the summed cross section of Fig. 1, whereas a much stronger A; resonance is seen at 5 eV in static-exchange calculations on BCl,. Thus, we are able to associate each of the resonances observed in Figs. 1 and 2 with empty valence molecular orbitals of the target molecule. The differential elastic cross section, or cross section as a function of scattering angle, is shown at selected energies in Fig. 3. Without going into great detail, we can observe that resonances have a marked effect not only on the strength but also on the angular pattern of the scattering. This can be useful experimentally in identifying weaker resonances, such as the A’ 30
20
10
0 100
1
Angle (deg) FIG. 3. Differential elastic electron cross section for trimethylaluminum at (a) 2.5, (b) 5, (c) 8, and (d) 30 eV. Results shown are obtained in the larger basis.
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resonance, that are swamped by the nonresonant background in the integral cross section. Apart from resonance effects, scattering at the lowest energies tends to show weak angle dependence, being dominated by low partial waves, whereas cross sections at higher energies become progressively forward-peaked, as we might expect, although the semilogarithmic plot of Fig. 3(d) reveals that structure persists in the differential cross section at larger angles.
V. Conclusion Experimental cross sections for trimethylaluminum, the molecule discussed in the preceding section, are not available, despite its importance in plasma deposition of aluminum and of aluminum oxide and nitride. That such an absence of measurements is not unusual points out the importance of theoretical methods as sources of data having both scientific interest and practical value. We have shown that, using modern MPPs, such computations can be completed in times on the order of several hours, quickly enough to make convergence studies and surveys of groups of related molecules possible. We emphasize that parallel computers offer approximately two orders of magnitude greater performance than do sequential supercomputers for our application (which is only partly vectorizable), and they are therefore essential for extensive and detailed calculations on larger polyatomics. It is also important to recognize that the rapid scaling of computational effort with problem size implies that available computational resources will always sharply limit the size of problem that can be treated; no matter how far single-processor speeds advance, there will always be interesting problems that are only feasible on MPPs. Although massively parallel computers are not yet widely employed for the calculations arising in theoretical atomic and molecular physics, there is every reason to think that they can provide the same enormous performance boost to a variety of computations that they do to our electron-molecule scattering studies. Computation of matrix elements, linear-algebraic manipulations, and a rapid scaling of work with problem dimensions are, after all, features common to many numerically intensive problems, especially in quantum mechanics. The general approaches we have outlined, or similar approaches, should work well for any such problems. Certainly, a significant human investment in adapting or rewriting existing sequential programs is required. Yet the potential for welldesigned scalable applications to redefine the limits of the feasible is too valuable to be ignored.
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Acknowledgments This work was supported by the Air Force Office of Scientific Research, by the National Science Foundation (including support under the Grand Challenge project “Parallel 1/0 Methodologies for 1/0 Intensive Grand Challenge Applications”), and by SEMATECH, Inc. Use of the computational facilities of the JPL/Caltech Supercomputing Project and of the Concurrent Supercomputing Consortium is gratefully acknowledged.
References Allan, M. (1989). J. Electron Spectrosc. Relat. Phenom. 48, 219-351. Almenningen, A., Halvorsen, S., and Haaland, A. (1971). Acta Chem. S c a d . 25, 1937-1945. Boys, S. F. (1950). Proc. R . SOC. London, Ser. A 200,542-554. Chase, D. M. (1956). Phys. Reu. 104,838-842. Dunning, T. H. (1970). J. Chem. Phys. 53, 2823-2833. Dunning, T. H., and Hay, P. J. (1977). In “Methods of Electronic Structure Theory” (H. F. Schaefer, ed.), pp. 1-27. Plenum, New York and London. Fox, G. C., Johnson, M. A., Lyzenga, G. A., Otto, S. W., Salmon, J. K, and Walker, D. W. (1988). “Solving Problems on Concurrent Processors,” Vol. 1, pp. 50-62. Prentice-Hall, Englewood Cliffs, NJ. Geltman, S. (1969). “Topics in Atomic Collision Theory,” pp. 82-103. Academic Press, New York. Hall, R. I., and Read, F. H. (1984). In “Electron-Molecule Collisions” (I. Shimamura and K. Takayanagi, eds.), pp. 351-425. Plenum, New York. Hipes, P. G. (1989). “Caltech Concurrent Computation Program Report,” C3P-6526 (unpublished). Hipes, P. G., Winstead, C., Lima, M. A. P., and McKoy, V. (1990). In “Proceedings of the Fifth Distributed Memory Computing Conference” (D. W. Walker and Q. F. Stout, eds.), pp. 498-503. IEEE Computer Society Press, Los Alamitos, CA. Huzinaga, S. (1965). J. Chem. Phys. 42, 1293-1302. Lane, N. F. (1980). Reu. Mod. Phys. 52, 29-119. Lebedev, V. I. (1975). Zh. Vychisl. Mat. Mat. Fiz. 15(1), 48-54; J. Comput. Math. Math. Phys. 15(1), 44-51. Lebedev, V. I. (1976). Zh. Vychisl. Mat. Mat. Fiz. 16(2), 293-306; J. Comput. Math. Math. Phys. 16(2), 10-24. Lebedev, V. I. (1977). Sib. Mat. Zh. 18, 132-142; Sib. Math. J. 18, 99-107. Lebedev, V. I. (1994). Dokl. Akad. Nauk 338(4), 454-456; Russ. Acad. Sci. Dokl. Math. 50, 283-286. Lebedev, V. I., and Skorokhodov, A. L. (1992). Ross. Akad. Nauk Dokl. 324(3), 519-524; Russ. Acad. Sci. Dokl. Math. 45, 587-592. Lima, M. A. P., Brescansin, L. M., da Silva, A. J. R., Winstead, C., and McKoy, V. (1990). Phys. Rev. A 41, 327-332. National Research Council (1991). “Plasma Processing of Materials: Scientific Opportunities and Technological Challenges.” National Academy Press, Washington, DC.
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Newton, R. G. (1982). “Scattering Theory of Waves and Particles,” 2nd ed., pp. 178-187. Springer-Verlag, New York and Berlin. Ostlund, N. S. (1975). Chem. Phys. Lett. 34, 419-422. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1986). “Numerical Recipes,” pp. 52-58. Cambridge Univ. Press, Cambridge, UK. Schafer, A., Huber, C., and Ahlrichs, R. (1994). J. Chem. Phys. 100,5829-5835. Schiff, L. I. (1968). “Quantum Mechanics,” 3rd ed., pp. 324-326. McGraw-Hill, New York. Schulz, G. J. (1973). Reu. Mod. Phys. 45, 423-486. Schwinger, J. (1947). Phys. Reu. 72, 742. Szabo, A,, and Ostlund, N. S. (1982). “Modern Quantum Chemistry,” pp. 64-89. Macmillan, New York. Takatsuka, K., and McKoy, V. (1981). Phys. Rev. A 24, 2473-2480. Takatsuka, K., and McKoy, V. (1984). Phys. Reu. A 30, 1734-1740. Treutler, O., and Ahlrichs, R. (1995). J . Chem. Phys. 102, 346-354. Walker, D. W., and Dongarra, J. J. (1996). Supercomputer 12, 56-68. Watson, D. K., and McKoy, V. (1979). Phys. Rev. A 20, 1474-1483. Winstead, C., and McKoy, V. (1993). Phys. Rev. A 47, 1514-1516. Winstead, C., and McKoy, V. (1995). In “Modern Electronic Structure Theory” (D. R. Yarkony, ed.), Part 11, pp. 1375-1462. World Scientific, Singapore. Winstead, C., and McKoy, V. (1996). Adu. Chem. Phys. (to be published). Winstead, C., Pritchard, H. P., and McKoy, V. (1996). In “Proceedings of the Debugging and Performance Tuning for Parallel Computing Systems Workshop, 1994” (M. Simmons, A. Hayes, and D. Reed, eds.). IEEE Computer Society Press, Los Alamitos, CA.
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ADVANCES IN ATOMIC. MOLECULAR. AND OPTICAL PHYSICS. VOL . 36
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS M A CIEJ LE WENSTEIN Commissariat 6 l’Energie Atomique. DSM/DRECAM/SPAM Centre d’Etudes de Saclay Gif.sur.Yuette. France
LI YOU Institute for Theoretical Atomic and Molecular Physics Harvard-Smithsonian Center for Astrophysics Cambridge. Massachusetts
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . “Holy-Grail” of Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . . . B. Quantum Field Theory of Atoms and Photons I1. Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Basics of BEC B. The Quest for BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I11. Hamiltonian of Q F M P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Hamiltonian in Spatial Representation ..................... B. Fock Representation C . Ranges of Parameters for QITAP . . . . . . . . . . . . . . . . . . . . . . . . IV. Properties of BEC in Trapped Alkali Systems . . . . . . . . . . . . . . . . . . . A. The Ginsburg-Pitaevski-Gross Equation B. BEC for Atoms with a Negative Scattering Length . . . . . . . . . . . . . . C. Excitations in Trapped BEC V . Diagnostics of BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Coherent Weak Light Scattering ............ . . . . . . . . . . . . B. Scattering of Short Intense Pulses . . . . . . . . . . . . . . . . . . . . . . . . C . Incoherent Light Scattering VI . Quantum Dynamics of Condensation . . . . . . . . . . . . . . . . . . . . . . . . A . The Master Equation B. Sideband Cooling of an Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . C. Generalized Bose-Einstein Distribution D . Cooling of a Gas with Accidental Degeneracy . . . . . . . . . . . . . . . . . E. Sympathetic Cooling F. Evaporative Cooling
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VII. Theory of Bosers A. A Prototypical Boser B. BoserModels C. Boson Accumulation Regime VIII. Nonlinear Atom Optics. IX. Conclusions References.
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I. Introduction A. “HOLYGRAIL”OF ATOMICPHYSICS The observation of Bose-Einstein condensates (BEC) and other effects related to the quantum statistical properties of weakly interacting gases of atoms has recently become a “Holy Grail” of atomic physics (Burnett, 1995). A large part of theoretical and experimental research has been focused on cooling atoms confined in traps at relatively high phase-space densities (Doyle et al., 1991; Monroe et al., 1993; Ketterle et al., 1993; Setija et al., 1993; Spreeuw et al., 1994). These studies were started with the pioneering work on spin polarized hydrogen atoms (Silvera and Walraven, 1980, 1986; Silvera and Reynolds, 1992; Walraven and Hijmans, 1994; Greytak, 19951, where evaporative cooling techniques (Hess, 1986; Masuhara et al., 1988; Doyle et al., 1991) were developed. It is the application of the later method to the laser precooled alkali atoms (Petrich et al., 1995; Adams et al., 1995; Davis et al., 1995a) that has recently led to the remarkable observation of BEC in a rubidium vapor at J I M (Anderson et al., 1995). Evidence of BEC in a lithium gas with attractive interactions has been also reported (Bradley et al., 1999, whereas the MIT group has observed the condensation of a sodium vapor (Davis et al., 199%). These observations might lead in the future to fascinating applications, such as the development of a coherent source of atoms-i.e. a boser (Cirac et al., 1994a; Wiseman and Collet, 1995; Holland et al., 1995; Olshan’ii et al., 1995; Spreeuw et al., 1995; Guzmhn et al., 1995). Parallel to the experimental progress, a new theory has been formulated and developed to describe these emerging scenarios for quantum optics (cf. Lewenstein et al., 1994). Such a theory most conveniently fits within the framework of second quantization, since it has to account for the quantum statistical nature of atoms as bosons or fermions. In addition, since many of the diagnostic methods of cold atom systems are realized with available optical means (such as spectroscopy, light scattering, absorption), and since laser light is also very often used for cooling and trapping of atoms, the theory in question must also fully describe the interactions of
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atoms with photons, and thus it may be termed a quantum field theory of atoms and photons (QFTAP).
OF ATOMSAND PHOTONS B. QUANTUM FIELDTHEORY
One may argue that such a theory is not new and has already existed for a long time-it is nothing more than the application of the standard methods of many-body physics (second quantization) to standard quantum optical Hamiltonians. As we stressed (Lewenstein et al., 1994), however, there are several specific aspects of such a theory that have unique quantum optical characteristics and, to our knowledge, have not yet been discussed in standard textbooks on many-body field theory (Mahan, 1993; Fetter and Walecka, 1971; Negele and Orland, 1988; Nozisres and Pines, 1990; Abrikosov et al., 1975): 1. This theory deals with confined atomic systems in traps or beams. There exists some literature on BEC in an external potential (de Groot et al., 1950; Ginsburg and Pitaevski, 1958; Oliva, 1988; Lovelace and Tommila, 1987; Goldman et al., 1981; Huse and Siggia, 1982; Bagnato et al., 1987; Bagnato and Kleppner, 1991), but it deals mainly with the properties of atoms in their ground electronic states. The problem of the interaction of atoms with light in traps has not yet been discussed in detail. 2. In quantum optical systems, the atom-field interaction has a resonant character. Consequently, close to the resonance we can usually limit the description of the atomic energy level structure to a few relevant levels. The paradigm model of quantum optics is a two-level atom (Allen and Eberly, 1987). 3. The resonant character of the interaction allows us to simplify the theory with the so-called rotating wave approximation (RWA) of quantum optics. This approximation relies on the huge difference between characteristic time scales of optical oscillations, ( 1014-1015 Hz) and other typical scales of the system (kHi-GHz), such as those related to collisional dephasing, natural or cooperative line widths, Rabi frequencies of the external driving field, and so forth. 4. There are several other approximations characteristic of atomic physics and quantum optics that are not so frequently used in many-body theory, such as the dipole approximation (Power, 19641, or the strong field approximation (Lewenstein and You, 1993; You et al., 1995a). 5. Quantum optics has in recent years developed its own unique way of dealing with quantum noise and quantum fluctuations (Gardiner, 1991).
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The techniques of the quantum optical master equation (ME), quantum Langevin equations, and quasi-probability distributions are to a great extent alien to other areas of physics. Generalization of the ME techniques to many-body problems is already one of the unquestionable achievements of QFTAP. It is the purpose of this chapter to discuss some of the most recent developments of QFTAP. QFTAP is a rather young theory, So far, research in the QFTAP has been mainly concentrated on the following five topics: (a) properties of BEC in alkali systems; (b) diagnostics of BEC and cold atom systems; (c) dynamics of condensation and related phase transitions; (d) boser theory; (el nonlinear atom optics (NAO). The chapter is organized as follows. In Section I1 we remind the readers of the basics about BEC and the history of the quest for it. In Section I11 we introduce the Hamilonian of QFTAP and basic notation. In the next five sections we then review the five aforementioned topics of QFTAP in the following manner: we present physical processes of interest, discuss relevant interactions, discuss methods of description and the solution of the corresponding problems, and present some results. We conclude in Section IX.
11. Bose-Einstein Condensation A. BASICSOF BEC
The aim of this section is to review the basics of the theory of BEC. We base our discussion on the seminal paper by de Groot et al. (1950). Let us consider a system of noninteracting particles (bosons) of mass M in a box. Assume that the system is in thermal equilibrium with its surroundings. According to statistical mechanics, the state of the system can then be described by the Bose-Einstein distribution (BED),
where N ( E j ) = A$ denotes the mean number of particles occupying the state of energy E j , with the Ej’s designating the energy levels of a single noninteracting particle in a box; p = l/k,T in the standard thermodynamic notation, where k, is the Boltzmann constant and T is temperature. For simplicity, we assume that the (nondegenerate) ground state energy E , is zero. z denotes the so-called fugacity, and is related to the chemical potential p s 0 via the relation z = exp(Pp). There is an additional
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constraint in the system, because the total number of particles in the box, N, is fixed and conserved. Therefore,
Alternatively we may write Eq. (2) as
which may be viewed as an equation for determining the chemical potential p (or fugacity z) as a function of T and N . For example, in a box of linear size a, the energies are
E. = I
-
8M
(j,
+ l), + ( j , + 1)’+
(j,
+ l),
-
3
U2
where j l , j,, j , = 0,1,2,. .. . The corresponding plot of fugacity is shown in Fig. l(a). It was Bose (1924) and following him Einstein (1924, 1925) who observed in 1924 interesting properties of this system in the thermodynamic limit as N and a tend to infinity, such that the particle density n = N / a 3 remains constant. As we see from Fig. l(a), there is a critical temperature T, below which z = 1. In the low temperature phase a macroscopic number No of particles condense in the ground state [see Fig. l(b)l. For the case of particles in a box, N o / N behaves as 1 - (T/7’,)3/2. This phase transition has been termed Bose-Einstein condensation and has created a great deal of discussion and interest (Uhlenbeck, 1927; London, 1938a,b, 1939; Uzunov, 1993). It has rather unusual properties: It occurs in a system of noninteracting particles and is thus a direct consequence of quantum statistics; for an ideal gas in a box there is no spatial phase separation, no latent heat, and even no discontinuity in the specific heat. For potentials other than the box potential, these results are somewhat modified; additionally there are problems with defining the appropriate thermodynamic limit because the size of the system is not so precisely determined (de Groot et al., 1950). For macroscopic systems there is a heuristic approach to the problem of BEC, which, although not as precise, provides insight into the physics of the problem. Namely, for any potential in the “thermodynamic” limit, we may assume that the energy level spacings are small and substitute for
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z
4 TO
T
FIG. 1. (a) Temperature dependence of fugacity. Dotted line denotes the result for a finite system with N = 100, while the solid line represents the result in the thermodynamic limit with same fixed density. (b). Temperature dependence of the condensed fraction of atoms.
sums by integrals in Eq. (3). Introducing the density of states p ( E ) , we obtain N
= No
+ lrn dEN( E ) p ( E ) O+
(5)
where 0’ denotes the exclusion of the ground state. For instance, for any potential in 3D we have
where V * ( E ) denotes the classically accessible region in configuration space. For the case of a 3D box, we thus have
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where V is the volume of the box. Inserting this expression into Eq. ( 5 ) and letting p -+ 0, T + T,, we find the analytic expression (de Groot et al., 1950) N
2.612
where
A,
h =
(27rMk,T)’/’
is the thermal de Broglie wavelength, i.e., the de Broglie wavelength of a particle that has kinetic energy equal to k,T. Equation (8) has very important physical consequences, since it is qualitatively valid for any potential (Bagnato et al., 1987; Bagnato and Kleppner, 1991). Equation (8) states that BEC occurs when the density of atoms is such that their thermal de Broglie wavelengths overlap. For any potential, the semiclassical density of atoms in thermal equilibrium n(?) is given by (Bagnato et al., 1987; Bagnato and Kleppner, 1991)
Note that the density at the minimum of the potential V(Fmin)= 0, and at the critical temperature, p + 0, is quite generally given by Eq. (8). This expression defines a parameter region in which one should start to look for condensation. It should be stressed that the preceding discussion was based on a rather idealized case of a noninteracting gas. BEC occurs, of course, also in a weakly interacting gas, and as frequently pointed out, only then it exhibits a full richness of its amazing properties. In the noninteracting case there is, in principle, no reason for spatial phase correlations between atoms. In the weakly interacting case, not only do all the atoms occupy the same state, but they do exhibit phase correlations. Technically, to account for such correlations (just as in the laser theory) one frequently uses coherent states rather than Fock states to describe the condensate (see Section 111). The phase fluctuations have a phonon-like dispersion relation and are responsible for such phenomena as superfluidity or superconductivity (for a nice discussion of this point, see Burnett, 1995b).
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Since the theoretical prediction of BEC, experimentalists have been trying to realize it in the laboratory. The history of attempts to reach BEC in various systems of weakly interacting bosons is very well presented in Griffin et al. (1995). The first candidate was liquid helium He4, which undergoes a superfluid transition at low temperatures (Mahan, 1993). In 1938, in a fundamental paper in Nature, London (1938) argued that a superfluid phase transition was an example of a Bose-Einstein condensation (see also Nozieres and Pines, 1990). In 1947, Bogoliubov proposed that this system could be described as a dilute, weakly interacting Bose gas. Bogoliubov’s theory predicted some features of the superfluid transition quite well, but soon it was realized that its agreement with experimental data was rather accidental. Liquid helium He4 is in fact characterized by very strong interactions between the atoms. The phase transition is still characterized by off-diagonal long range order (Penrose, 1951; Yang, 19621, but it cannot be thought of as pure BEC (see Mahan, 1993; Uzunov, 1993). Another candidate for BEC is a gas of spin polarized hydrogen atoms, studied in particular by groups at the University of Amsterdam, MIT, and Harvard (Silvera and Walraven, 1980; Hess et al., 1987; Silvera and Reynolds, 1992; Walraven and Hijmans, 1994; Grettak, 1995; Silvera, 1995). Evaporative cooling (proposed by Lovelace et al., 1985, and Hess 1986) in magnetic traps, with the help of elastic collisions for thermalization (Lovelace et al., 1985; Masuhara et al., 1988; van Roijen et al., 1988; Doyle et al., 1991; Doyle, 1991; Luiten, 1993; Setija et al., 19931, has proven to work extremely well for spin polarized atomic hydrogen, and it is now being utilized for other atoms (see Davis et al., 1995b, for a nice theoretical model, and what follows for more details). In the relative high density regime, three-body collision-induced recombination of molecular hydrogen has caused major problems. Despite valiant attempts, however, BEC in spin polarized hydrogen has not yet been observed. Recently, it was proposed that BEC could be achieved more easily in metastable helium (Shlyapnikov et al., 1994). The evidence of BEC of biexcitons in CuCl has been reported in 1979 (Chase et al., 1979; Peyghambavian et al., 1983). BEC case into fashion again in 1980, when Hulin et al. (1980) demonstrated that excitons in Cu,O obey Bose-Einstein statistics of a (practically) ideal gas. There are two kinds of such excitons, ortho-excitons with total spin 1, and para-excitons with total spin 0. Both are characterized by high mobility and relatively long lifetimes. Since 1980 many experiments have been done, but only recently has an observation of BEC of para-
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excitons in stressed Cu,O been claimed (Lin and Wolfe, 1993; Fortin et al., 1993). The evidence for condensation of magneto-excitons in GaAs quantum wells (Butov et al., 1994) has also been reported. Finally, several groups (Petrich et al., 1995; Davis et al., 1995a,c; Adams et al., 1995; Anderson et al., 1995; Bradley et al., 1995) attempted during recent years to realize BEC in systems of cooled alkali atoms using combinations of the aforementioned techniques. Apart from Stanford group (Adams et al., 1999, which used all-optical traps, in experiments at JILA and MIT, atoms are both cooled and trapped initially in the so-called magneto-optical traps (MOT, cf. Raab et al., 1987; Monroe et al., 1993); in order to suppress losses (such as those due to inelastic collisions of excited atoms), the dark MOTS (Ketterle et al., 1993) are used. Laser cooling involves Doppler cooling to about few hundred micro-Kelvin, and further cooling in atomic molasses with the help of the polarization gradient method and the Sisyphus effect. This technique allows, in principle, velocities of the order of a few recoil velocities uR = hk,/M to be achieved, where k, is the photon wave number. All of these methods are thoroughly discussed in detail in Chu and Wieman (1989) and Gilbert and Wieman (1993). Employing a small magnetic bias field and a circularly polarized laser, the atoms are optically pumped into a single angular momentum state (for instance, F = 2, mF = 2 in the case of 87Rb). The lasers are then turned off, and a magnetic field is turned on to provide a static (or quasi-static) trap from which the method of evaporative cooling can be applied. Magnetic traps are based on simple principles. Energy levels of both hydrogen and alkali atoms in a magnetic field undergo Zeeman splitting. The potential energy in the magnetic field B’ is is the total magnetic moment. For atoms in these just -jiB*B’,where energy levels, there are “weak field seekers,” which can be confined to local minima of B’, and “strong field seekers,’’ which actually cannot be trapped in static fields. So far, in most experiments weak field seekers are trapped, but there is a lot of interest in trapping the strong field seekers as well, because these are lower energy states than the weak field seeker states and typically exhibit lower spin-flip rates, i.e., lower trap losses. Recently, there have been interesting attempts to trap strong field seekers using time dependent traps (Agosta et al., 1989; Spreeuw et al., 1994). To avoid undesired trap losses due to Majorana spin flips (which occur when atoms pass the regions of zero magnetic field), the JILA group used a time orbiting potential (TOP) trap (Petrich et al., 1995) (which at least traps weak field seekers). The MIT experiments were performed using the technique of “plugging the hole” (Davis et al., 1995a), that is, focusing a blue-detuned laser that generates a repulsive optical dipole force in the
zB
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zero magnetic field region. The evaporation is realized using a radiofrequency “scalpel,” i.e., tuning an rf signal to a resonant transition to another angular momentum state for the atoms from the outermost regions of the trap (i.e., the hot ones); those atoms flip their spins and thus escape from the trap. The evaporative cooling allows the achievement of temperatures in the range of hundreds of nano-Kelvin, and of phase-space densities exceeding by several orders of magnitude the critical value for BEC. One of the drawbacks of evaporative cooling is that it necessarily leads to the loss of atoms in each cooling cycle. A cooling method that does not lead to such a loss has been proposed recently. This method employs energy shifts of internal atomic levels in external (magnetic or electric) fields, transfer between the internal levels using stimulated Raman scattering, and thermalization via elastic collisions (Cirac and Lewenstein, 1995).
111. Hamiltonian of QFTAP In this section we shall introduce the Hamiltonian of QFTAP. Its derivation and its various approximate versions are thoroughly discussed in Lewenstein et al. (1994). The Hamiltonian fully accounts for the physics of the atom-atom and atom-photon interactions, and as such it is designed to describe the physical processes discussed in the following sections: final state of evaporative cooling consisting of thermalization due to atom-atom collisions, interactions of the cold atom system with a probing light, laser cooling, and so forth.
A.
HAMILTONIAN IN SPATIAL
REPRESENTATION
The Hamiltonian of the quantum field theory of atoms and photons in the length gauge reads (fi = 1) where &“g and & denote the free Hamiltonians of the ground and excited state atoms, respectively:
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qe(z,
where *&I?, a ) and s ) [q;(l?, a ) and qJ(Z?s,) ]denote atomic field annihilation (creationk operators. These operators annihilate (create) an atom at the position R, in the ground electronic state characterized by an internal index a , or in the excited electronic state characterized by an internal index s. These operators fulfill the standard canonical commutation relations (if atoms are+ bosons), ancj anticommutation relations, if atoms are fermions. V,,(R,a ) and V,,(R, s ) are the corresponding t',ap potentials, which are frequently assumed to be harmonic, i.e., l/tg, te( R , a ) = Mw& te R 2 / 2 , with wtg,te denoting the trap frequency [in fact, for comparison with experiments asymmetric harmonic potentials have to be used (Anderson et al., 1995; Baym and Pethick, 1996). Finally, M denotes the atomic mass, and w,, the electronic transition frequency. Similarly, the collisional part of the Hamiltonian is written in the so-called shape independent approximation, in which a zero-range effective atomic potential (based on the ladder approximation; see, for instance, Fetter and Walecka, 1971) replaced the bare atom-atom interactions (for details, see Lewenstein and You, 1996, and references therein):
The coefficients in front of those parts of the Hamiltonian are related to the corresponding scattering lengths of the atom-atom collisions in the corresponding channels, and bg,ee,ge - 47ragg,e e , J M . The groundexcited part, apart from the term due to short range atom-atom interactions, contains a contribution of the so-called contact term that constjtutes a part of the resonant dipole-dipole interactions: be, = bgt + 47r ldg,"'I2. The ground-excited part refers in fact to laser assisted collisions, because typically electronic excitation is achieved via laser-atom interactions. The Hamiltonian describing the dipole interaction with the light can be written as
+ h.c.
(17)
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Here "2, ( a+t y )denotes the annihilation (creation) operator of a photon of momentum k and polarization F,. Note that in the length gauge most of the resonant dipole-dipole interactions (due to photon exchange) between ground the excited atoms is contained in this part of the Hamiltonian. Finally, the electromagnetic field Hamiltonian is
B. FOCKREPRESENTATION In applications it is often convenient to use a Fock representation for the QFTAP Hamiltonian in an appropriately chosen basis. For instance, for very tight traps and not too high densities, collisional terms may be regarded as perturbations and the convenient choice of basis corresponds to creation and annihilation of atoms in eigenstates of the free atomic Hamiltonians, (12) and (13). In such a case we introduce the operators gas, eGr and their Hermitian conjugates, given by
where #(d, a), +;(l?, s ) are the corresponding single-atom wavefunctions. These operators (for bosonic atoms) fulfill standard bosonic commutation relations:
with all other commutators being zero. For fermionic atoms, these commutators have to be replaced by anticommutators. The Hamiltonian of the quantum field theory of atoms in this Fock representation takes a simple form: 2'=%+g +%,
+g + coll. terms
(23)
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233
where we have
with given by Eq. (18). Here w;, and m& are corresponding energies of the motional eigenstates in the trap potentials; e ( k ) is the mode density of the electromagnetic (EM) field, and it denotes a slowly varying function of k that is related to the natural linewidth (HWHM) of the corresponding transition via
8,rr2k,2
GS= I@212 I e( k)I2 3c with k ,
= wo/c.
The matrix elements +
-. -
v C a G s ( k )= ( g , n ’ , a I e p i k ‘ R I e , r % , ~ ) , (28) are the analogs of Franck-Condon factors. The last part “coll. terms” [see (14)-(16)1 denotes interactions responsible for various collisional processes. When collisions do play a role, it is more appropriate to use a basis corresponding to creation and annihilation of self-consistently constructed quasi-particle states (see Lewenstein et al. 1994 for details). C. RANGESOF PARAMETERS FOR QFTAP The Hamiltonian (1) is the starting point of all studies of QFTAP. Before we turn to the particular topics within QFTAP, it would be useful to specify ranges or parameters of the theory that are accessible experimentally. The trap frequencies (for ground electronic states) would typically be w, 2: (27r)lOO Hz. The alkali atoms can interact with resonant light of frequency - ( 2 ~ ) 4 . 0X 1014 Hz. The typical value of the natural linewidth (HWHM) for alkali atoms is y = ( 2 ~ 1 2 . 5MHz. The size of the ground state wavefunction for the condensate is of the order of a micrometer, compared with the resonant wavelength, A = 800 nm. Note that the size a of the ground state of the condensate for rubidium or sodium is typically a few times bigger than the size of the “bare” ground state of the trap
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M. Lewenstein and Li You
abare= 1/ ,/-. This effect is due to repulsive ground-ground state interactions, which are believed to increase the size of the condensate (see Section IV). The ground-ground scattering length is not known very precisely, but it is believed to be 85 < agg < 140 (in units of Bohr radius) for 87Rb(Gardner et af., 1999, ugg= -27 for 7Li (Moerdijk et af., 1995; Cot6 et al., 1994; Abraham et af.,1999, and ugg= 95 for sodium (Davis et uf., 1995a). Recently, there have been interesting suggestions that the scattering length can be manipulated somewhat in experiments by putting atoms in an external magnetic of electric field (Verhaar et af., 1993 or using repulsive states (Marcassa et af., 1994). In the experiments at JILA and Rice, only a few thousand atoms are in the condensate; the peak density is of the order of 2 X 10l2 atoms/cm3, and the temperature of the order of a few hundred nano-Kelvin. In the MIT experiment, the number of atoms N = 500,000, with a density = 4 X 1014 atoms/cm3 and the temperature just below a micro-Kelvin.
IV. Properties of BEC in Trapped Alkali Systems The first topic of QFTAF' that we want to discuss does not involve photons -we want to consider properties of BEC obtained in a final thermalization step of evaporative cooling. Neglecting for the moment the problem of the finite lifetime of the condensate (due to two- and three-body losses), the problem reduces thus to the study of equilibrium thermodynamics for the system of atoms in the specific ground electronic state described by the Hamiltonian
z = <+
(29)
with the sum over internal index in Eqs. (12) and (14) restricted to the one relevant state. The relevant interactions that are responsible for thermalization consist of elastic collisions described by Zgg. The Hamiltonian (29) determines properties of the stationary state. If the assumed temperature is sufficiently low, it determines in particular the properties of BEC, such as its size, shape, and so forth. One can then use this Hamiltonian to study dynamical properties of BEC, such as response to weak perturbations. In principle, the same Hamiltonian governs the nonequifibrium dynamics of approach toward the equilibrium, and of the process of condensation. The latter problem is, however, much more difficult, and will be discussed in Section IV. The problem of a homogeneous Bose gas is a textbook problem of many-body theory (Mahan, 1993; Fetter and Walecka, 1971; Negele and Orland, 1988; Abrikosov et af., 1975). Standard treatment involves a
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
235
so-called ladder approximation, which allows reduction of Z, to the local form (14) with prefactor proportional to the s-wave scattering length. That is the reason that the sign of scattering length plays a crucial role in equilibrium statistical mechanics of weakly interacting homogeneous Bose gases. The gases with positive agg undergo condensation at low temperature, whereas the ones with the negative scattering length undergo gas-liquid or gas-solid transitions (Stoof, 1994). Further analysis (for positive a,) is done within a framework of Bogoluibov-Hartree (BH) theory (sometimes termed a "pairing" theory). In the homogeneous case it is then elementary to find the spectrum and the creation and annihilation operators for the elementary quasi-particle excitations around the (uniform) condensate. The quasi-particle spectrum exhibits photon-like dispersion relation and, in consequence, leads to superfluidity (Burnett, 1995a). The problem of inhomogeneous weakly interacting Bose gases is much more difficult. Finding the shape of the condensate at zero temperature using a BH approach is already a difficult task, and it requires a nontrivial solution of a nonlinear Schrodinger equation. Originally, this equation was derived to describe vortices in the homogeneous system, and it is referred to in the literature as the Ginsburg-Pitaevski-Gross (GPG) equation (Ginsburg and Pitaevski, 1958; Pitaevski, 1961; Gross, 1963, 1966; Lifshitz and Pitaevski, 1980). A. THEGINSBURG-PITAEVSKI-GROSS EQUATION
In The BH approach, the wavefunction of the system of atoms at zero temperature is a produce of identical wavefunctions that minimize the energy functional Z = /dZ"(Z)(
-2M V 2 + v;g(q)'y(R7
where **(Z), W d denotes a complex-valued wavefunction, normalized as ~ & ~ * ( Z ) W=ZI.) The GPG equation has a form
IWz)I2
where p ( z ) = N is the density profile of the condensate. This equation has been discussed in literature in the context of spin polarized
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M. Lewenstein and Li You
hydrogen (Goldman et al., 1981; Huse and Siggia, 1982; Lovelace and Tommila, 1987; Oliva, 1988, 1989). In Goldman et al. (1981) and Huse and Siggia (1982), a useful quasi-classical method for solving the GPG equation has been proposed. The method consists of neglecting the kinetic energy term and setting P(R3 =
where O ( x ) = 1 for x > 0, and zero otherwise,Th? energy E can be determined from the normalization condition j d R p ( R ) = N. Such a solution becomes more and more precise as Nbgg a Nu,, grows. For spherically symmetric harmonic traps, the solution (32) predicts that the ratio of the effective size of the ground state aeffto the size of the bare ground state is aeff/a = (60Na,,/a)’/s. The recent experiments on alkali atoms have been analyzed in Baym and Pethick (1995) using (32) with asymmetric harmonic trap potentials; this analysis predicts enlargement of the size and the shape of the condensate and its shape in accordance with experimental data. The quasi-classical method breaks down close to the classical “turning points” (where the density vanishes but its derivative is discontinuous) and when NaE/abare is small. In such situations one has to use numerical methods. Two such methods, designed for parameter ranges corresponding to the experiments with cold alkali atoms, have been recently discussed in Edwards and Burnett (1995) and You et al. (1996). In Fig. 2 we show the results obtained with the method of You et al. (1996) for the case of spherically symmetric trap potential. B. BEC
FOR
ATOMSWITH
A
NEGATIVE SCATTERING LENGTH
In experiments at Rice University, evidence has been found for BEC of ’Li atoms with a negative scattering length, i.e., with effectively attractive interactions. This shown that Stoof s result [obtained for homogeneous case (Stoof, 1994)l must have a limited validity. The GPG equation (or, more generally, the variational principle that leads to the GPG equation) can be in principle used to study the case of a negative scattering length. Several authors have shown that, indeed, the energy functional (30) has a locally stable minimum for negative bE, provided N is not too large and nonlinearity is not too strong. This result is not very surprising when one realizes that the nonlinear Schrodinger equation (NISEI for atoms with a negative scattering length has in the homogeneous case the same form as the NLSE describing the propagation of light in optical fibers. The latter
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
-
0.3
h
Z
y
,
c
,
I
1
I
I
237
I
0.2
In
2
3
0.1
0
1
2
3 4 . 5 R (units of a)
6
7
8
FIG. 2. Comparison of various approximations to the condensate wavefunction. The solid line denotes the numerical solution of the nonlinear Schrodinger (GPG) equation, and the dotted line denotes the approximate solution given by quasi-classical approximation. The dot-dashed line is a Gaussian with an effective width 1.5 times larger than that of the noninteracting Gaussian ground state represented by the dashed line. The coupling strength in dimensionless scaled unit is N ( a , / a ) = 10.
NSLE is known to have marginally stable soliton solutions (Drummond et al., 1993). The numerical solutions of the GPG equation in a trap have also a soliton-like character, and they have been found in Ruprecht et al., 1995. For the parameters of the Rice experiment (Bradley et al., 19961, those solutions are stable provided N does not exceed a few thousand. An approximate way to describe such solutions employs the variational principle for the functional (30) and seeks solutions of the GPG equation in a Gaussian form (Baym and Pethick, 1996; Kagan et al., 1996). Such an approximate method gives results in good agreement with numerical results of Ruprecht et al. (1995). BEC for atoms with a negative scattering length was also predicted to occur in microtraps (Lewenstein and You, 1996) then the size of the trap is comparable with the coherence length of the condensate (i.e., the length In such a over which the condensate can heal, 6 = 1/ situation, the shape independent approximation for 3% breaks down, and one has to consider more realistic models of interatomic potential, which are nonlocal and contain both a repulsive core of a very short range and an extended attractive part. In small traps the atoms do not feel the tails of
4 s ) .
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M. Lewenstein and Li You
the attractive part of the potential, so that the effective atom-atom interactions turn out to be weakly repulsive, independent of the sign of the scattering length. The state of the condensate in such a case has, however, quite different scaling properties compared with the standard case described by (321, and the corresponding thermodynamical phase of the gas has been termed by us a super weakly interactinggas.
C. EXCITATIONS IN TRAPPED BEC To describe the weak excitations of trapped BEC and the quasi-particle spectrum, one should use the BH pairing theory. Detailed discussion of such a theory is presented for instance in de Gennes (1966; see also Fetter and Walecka, 1971; Lewenstein et al., 1994, and references therein). The quasi-particle annihilation and creation operators are constructed as linear combinations of both annihilation and creation operators of the bare atomic states. The coefficients in such linear combination fulfill a set of two coupled Schrodinger-like equations, with a “potential” determined self-consistently (assuming Bose-Einstein distribution for quasi-particles). Such an approach, however, is technically very difficult and, to our knowledge, has yet to be studied systematically in the context of cold alkali systems (at finite T, in particular). An alternative approach, developed by the Oxford group, consists of studying the time dependent GPG equation, in which the right hand side of Eq. (31) is replaced by i times the time derivative. In this manner, one can study the response of the condensate to changes of the trap potential, and the stability of the ground state solutions (for both positive and negative scattering lengths) (Ruprecht et al., 1995). To find elementary excitations (i.e., analogs of quasi-particles), one can first consider a driven GPG equation (Edwards et al., 19961, with two perturbation terms oscillating at some probe frequency o.Using linear response theory and setting the strength of the perturbation equal to zero, one can then identify the normal modes of the condensate and calculate the spectrum of quasiparticles. The linear response of the condensate to periodic perturbations (such as modulations of the trap frequency) can be used for diagnostics of the condensate. It is also interesting to go beyond the linear response theory and study nonlinear excitations in a trapped BEC (Edwards et al., 19961, which may lead to the generation of harmonics of the probe frequency and frequency mixing between the normal modes. These phenomena are the matter wave analogs of conventional nonlinear optics.
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
239
V. Diagnostics of BEC In actual experiments so far, two diagnostic methods are used: absorption imaging of the atomic cloud while it is trapped (Bradley et al., 1999, or after a sudden switching off the trap and a delay of several milliseconds (Andersen et al., 1995; Davis et al., 199%); the latter method provides a direct time-of-flight measurement of the velocity distribution. The theory of the ballistic expansion of a condensate (Holland and Cooper, 1996) and of a cloud of atoms at finite temperature (You and Holland, 1996) has been presented; it is in excellent agreement with experimental results of J I M . Theoretical studies have concentrated on the problems of light scattering and absorption in BEC, signatures of condensation in the absorption and scattering cross sections, coherent and incoherent spectra, and complex indices of refraction. Basically, two physical situations were considered: weak light scattering, and scattering of strong but short laser pulses. The light scattering theory has now been combined with realistic theory of optical imaging (taking into account the presence of lenses, apertures, etc.) (You and Lewenstein, 1996). The problems of light scattering are discussed in the following sections. Let us here only mention that another extremely interesting area of QFTAP concerns detection of BEC using atom counting theory. The theory of atom counting has not yet been developed to its final form [although important theoretical (Gardiner, 1994), and experimental (Kasevich, 1994) contributions have been discussed at the QFTAP Workshop at J I M in 1994, and at the workshop on “Collective Effects in Ultracold Gases,” in les Houches (Kasevich, 1996; Yasuda and Shimizu, 1996)l. Javanainen and Yo0 (19951, using the ideas borrowed from photon counting theory (Glauber 1963a,b; Grochmalicki and Lewenstein, 1990, have investigated the interference between two condensates, and they have devised an operational method to analyze the spontaneous symmetry breaking for an atomic system with an arbitrary number of atoms. A. COHERENT WEAKLIGHTSCATTERING We begin the analysis of the weak light scattering by considering calculation of the coherent scattering and absorption cross sections. This problem, although simpler than the calculation of the incoherent spectra, is already quite complex. We assume that the atomic system is in thermal equilibrium at a temperature T (below or above the condensation point), and that it interacts with an incident weak laser beam of the frequency wL.
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M.Lewenstein and Li You
The interaction Hamiltonian relevant for this problem is described by the dipole coupling term (17):
where for simplicity we consider a singlet s ground state to a vector p excited state and thus neglect internal state label for the ground state operators, while keeping the vector notation for the excited state o era tors. More precisely, the contact part of A? [proportional to IC&~ P, seege (1611 is also contributing to the atom-laser interactions, but it turns out that its role is negligible here (You et al., 1994). The Hamiltonian (33) and the contact term contain implicitly all the dipole-dipole interactions between atoms due to exchange of photons. These are the parts of Hamiltonian that lead to nonlinear ground-excited atom-atom interactions in the framework of nonlinear atom optics (see Section VIII). We can assume that the process of diagnostics by light is sufficiently fast so that the short range collisional terms in the Hamiltonian, Zg,&,, and the rest of Zg,can be safely neglected during the scattering process. This, however, does not mean that all of them can be totally disregarded. The reason is that the initial state of the system is determined by ground state atom-atom collisions, i.e., by Zg.
1. Scattering Equation To study the coherent scattering one employs the method of the Heisenberg equations derived from the Hamiltonian,
In the limit of weak light scattering, a linearization of the Heisenberg equations is performed, which physically expresses the fact that the equilibrium state is only weakly perturbed by the incident laser field. Technically, this can be achieved by using a perturbative Ansatz for the ground state operators g d t ) = gJ0) exp(-iEit). Such an approximation at low temperature is equivalent to the Bogoliubov Ansatz: replacing the atomic field by its c-number mean value describing the condensate. It neglects the spontaneous emission events that bring the system out of the equilibrium; such processes are approximately reintroduced into the theory by including the natural linewidth in the electronic transition frequency, oo-+ oo- iy.
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
241
As a result of the linearization, we arrive at the scattering equation for photon operators, which in the time domain takes the form
where the self-energy kernel is defined as
X(T ; 2; p , Z',
p') = (Zip*
Zi-,,,)e(k ) e ( k r ) 9 (p ;
z,I?')
(36)
while the reduced kernel P(T; 2, L') can be expressed in terms of initial distribution, Franck-Condon factors, etc. (for details, see You et al., 1994, 1996). The physical interpretation for the self-energy kernel is simple. It describes the amplitude for the process of the formation of a wavepacket in the excitej state trap potential due to the absorption of a photon of momentum k' and polarization p' at time t ' , followed by a free evolution of the wavepacket within the time interval 7 = t - t'. The free evolution consists primarily in quantum diffusion and drift caused by the momentum of the absorped photon, and it terminates at the time t when recombination to the gyund electronic state accompanied by emission of a photon of momentum k and polarization p takes place. As shown in You et al. (19961, in the range of parameters of interest, the reduced kernel can be represented approximately as a functional of the mean density of the gas:
where
is the form factor, i.e., the Fourier transform of the equilibrium density profile. r = y yd denotes an effective width of the line for a single atom, which apart from spontaneous emission effects also accounts for other dephasing processes such as excited wavepacket drift and quantum diffusion. We stress that the study of weak light scattering off the condensate in reality probes the mean density profile, and it does not depend on the density fluctuations, density-density correlations, etc.; as such, it gives only the indirect signatures of the quantum statistics.
+
242
M. Lewenstein and Li You
2. Optical Potential
Coherent weak field scattering off the condensate leads to various effects depending on the size and the density of the condensate. Different methods then have to be used to solve the scattering equation (35) in various regimes. This can be easily understood using the concept of an optical potential. In fact, the scattering equations can be transformed to a potential scattering problem (You et al., 1996), with the complex potential given by
with k , denoting the wavevector of the incident laser beam, and A = w,, - w,. The height of the potential (as compared with the energy of the incident photon, w,), and its extent (as compared with the wavelength), determine qualitative features of the scattering. Since the peak density is ii = N / a 3 , at the resonance A = 0 there are two relevant parameters, p ( A / 2 d 3 and k,a. Note that, in general, density parameter should be replaced by
3. Dense and Large Condensates The case k,a >> 1 and p(A/27rI3 >> 1 describes a large and dense condensate. Close to resonance (A = 01, the potential is then very high, and the photons cannot enter the condensate-depending on whether the density of the condensate varies slowly or not, one expects that the photon will be deflected or backscattered from the condensate. In this case, the proper treatment of the scattering equation involves solutions for a quasihomogeneous system, with a local constant index of refraction. This result was derived by Svistunov and Shlyapnikov (1990a, b) and Politzer (1991). These authors have analyzed the homogeneous case, for which the momentum is a conserved quantity (and therefore a good quantum number). They have demonstrated that the mixing of the photonic and atomic degrees of freedom leads to the formation of polaritons, i.e., collective excitations of photon and excited atom waves. The spectrum of the system necessarily displays an avoided crossing at the point where the photonic and atomic dispersion relations cross: w = kc = w o . This creates a gap in the spectrum (see Fig. 3) of the order of y p ( h / 2 d 3 for A + 03. At the
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
243
h
v)
c
c
3
W
k (Arb. Units) FIG. 3. In the case of a homogeneous system, the photonic degrees of freedom (slanted solid line) mix with the atomic excitation (solid line) to generate two branches of polaritons (represented in dotted lines). A frequency gap is formed that covers all k space. Within this gap, no propagating modes inside the condensate can exist, and they are thus totally reflected or deflected away.
resonance, the two branches of the dispersion relation differ by ( A / 2 7 7 ). In the polarization description, the resonant photons cannot enter the condensate because of the gap formation-the excitations at such resonant frequencies cannot exist iilside of the bulk of condensate. 4. Limit of Tight Traps
Another method of solving the scattering equation was proposed by Javanainen (1993), who assumed that the excited atomic field operators can be effectively replaced by a single bosonic operator that describes a collective excitation of an atomic wavepacket. Javanainen's approach becomes strictly valid for tight traps k L a < 1, and optically thin condensates p ( A / 2 d 3 = 1 (the approach neglects dipole-dipole shifts). Note that these two conditions imply that the validity regime of Javanainen's approach is N < 1, which is not particularly interesting. It is worth stressing, however, that when the approach is applied to larger traps k L a > 1, it describes the situation when the multiple scatterings occur, but the excited atoms wavepacket has the same form in each subsequent scattering event. That means that in practice the approach can be indeed
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M. Lewenstein and Li You
applied for k,a > 1 provided the form of the wavepacket in the emission-reabsorption cycles does not change much. Such situations take place in the optically thin media where the number of reabsorptions is small; similarly, such situations take place in the case of far-off resonant scattering. As a result scattering occurs mainly in the forward direction, and the spectrum has a Lorentzian shape with the width corresponding to a statistically enhanced spontaneous emission rate to the condensate:
Note this enhanced spontaneous emission must occur in a solid angle of a (linear) size l/k, a, because of approximate momentum conservation. This is the origin of the denominator in Eq. (41).
5. Moderate Size and Medium Density Condensates The most interesting case occurs when the density and the size of the condensate are moderate, so that C(A/2.rrl3 I 1 and k,a is of the order of 1-10. This case in fact corresponds to current experimental conditions. In this case, the photon energy exceeds the maximum of the optical potential, but the scattering process, nevertheless, cannot be treated perturbatively because of the relatively large extent of the potential. Obviously, in such a case the scattering will be mainly forward, and it will consist of multiple events of absorption and emission of a photon by condensed atoms. The scattering angle will, as in the previous case, be determined by the size of the condensate. It is interesting to realize, however, that in each of the subsequent absorption events the excited atom wavepacket changes its form, becoming broader and containing more components corresponding to momenta different from the incoming photon momentum. For this reason, as multiple scatterings occur the effect of the statistical enhancement of the spontaneous emission rate becomes less and less pronounced. In effect, the spectrum now does not contain one single Lorentzian; it consists, rather, of a sum of gradually narrowing Lorentzians. The spectrum thus becomes non-Lorentzian, and it exhibits a narrow cusp at the exact resonance, whose width r is determined by the single atom dephasing processes. This result was obtained by Lewenstein et a f . in 1993 (Lewenstein et al., 1994; You et af., 1994); for details of the derivation, see You et al. (1996). To derive this result from the scattering equation, a more sophisticated method must be used. From what we have said, it is clear that the considered regime of parameters corresponds precisely to the region of
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
245
validity of the so-called Glauber’s generalized diffraction theory (Glauber, 1959; for more recent applications of the Glauber’s theory, see, for instance, Bleszyhski et al., 1981; Abgrall et al., 1986; Bleszyiiski and Jaroszewicz, 1986). Alternatively, one can use an on-shell approximation to the scattering matrix, in which the photon energy remains unchanged in all of the multiple scatterings, and only the direction of its momentum varies (Taylor, 1987). Both theories give similar results (You et al., 1996). According to them, coherent light scattering exhibits the following clear signatures of condensation. First, the scattering occurs mainly in the forward direction with a divergence angle proportional to l/k,a determined by the size of the condensate, a; we stress that the enhancement of the scattering at small angles is a quantum statistical effect-it occurs because of the enhancement of the transitions to the condensate. It increases dramatically at the critical temperature of the Bose-Einstein transition, as shown in Politzer (1995) for the case of a homogeneous system. Second, both scattering and absorption cross sections exhibit broad non-lorentzian N y / ( k , a)2 in spectra with an overall (statistically enhanced) width of the spectral wings. Those spectra gradually narrow closer to the resonance, r at the resonance. and they contain a cusp of a characteristic width (See Fig. 4.)
-
N
6. Low Density Limit
Finally, in the low density limit ~ ( A / L ? T )-e ~ 1 [or, for off-resonant scattering, see (5.8)], and for not too large condensates, Born approximation may be used to solve the scattering equation. Such solutions are (up to the scaling of the coupling constant) equivalent to the solutions obtained in the problem of a single photon scattering from a single trapped atom. A systematic and elegant analysis of this problem is discussed in Gajda et 01. (1996). These authors present a “phase diagram” of the characters of the scattering in the parameter space of trap frequency and photon recoil energy (as compared with the natural linewidth), identifying various regimes such as standard dipole scattering, time-resolved scattering, instantaneous scattering, and so forth. B. SCA~TERING OF SHORTINTENSE
PULSES
Another method of diagnostics concerns the scattering of short laser pulses off the system of cold (bosonic) atoms (Lewenstein and You, 1993). The Hamiltonian that describes the relevant interactions has the same form as in the case of weak light scattering, Eq. (34). If the system is driven by a short coherent laser pulse, and if such a pulse is strong enough and
246
M. Lewenstein and Li You 1 0 0 ~ " ' " "
"
'
h
N
.c
0
.-+K v)
3
v
f
b K
.-c0 V
m
L n v) v)
e
u
Ol C
.-
m +
Na,,=l
0 0
i
ln
-0.5
-0.25
0.0 (GHz)
0.25
0.5
wL-w~
FIG.4. Typical - cross sections obtained from the on-shell . _ results for the scattering approximation for three different values of the interaction strength Nu, = 10,000a, 100a, l a , respectively, and r = y = (2~ 12.5MHz. The solid lines represent the results obtained from the density profile of the numerical solutions of the GPG equation. The dotted line represents the results obtained for the quasi-classical density profiles. The dashed lines serve as a reference and are obtained for noninteracting gases with the Gaussian density profile.
short enough, we may neglect spontaneous emission (and dipole-dipole interaction) effects during the pulse and substitute the electric field operator entering the interaction Hamiltonian in Eq. (17) by a c-number. The pulses we intend to use should have duration T~ I300 psec or shorter, i.e., width yL = 1 / = ~ 3 ~X 109-10" Hz.The first estimate shows that indeed y L B y = 2.5 MHz, i.e., the spontaneous emission may be legitimately neglected during the time of interaction of the pulse with the atoms. Note, however, that this estimate is misleading, since the atoms will
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
247
respond collectively and the effective spontaneous emission (due to bosonic final state enhancement) rate yerfis greatly enhanced, as discussed in the previous section. In practice the assumption has to be checked selfconsistently to assure that the total number of emitted photons N,,, is much smaller than the total number of atoms N . 1. Linearization in Strong Light Scattering
In You et al. (1995a), we have analyzed the foregoing assumption in great detail; moreover, we have shown that during the interaction with such short pulses the effects of both atomic collisions and (for loose atomic traps) the trap potential can be neglected. Assuming that the neglect of spontaneous emission and dipole-dipole and collisional interactions is valid, one substitutes for the product of the electric field operator and the absolute value of the electronic transition dipole moment by
where R is the peak Rabi frequency of the laser pulse. Here, F ( y , t ) is the temporal envelope of the pulse; it is chosen to be real and assumed to have a bell shape with a maximum at t = 0 equal to 1. We+assume that the pulse has the form of a plane wavepacked moving in the k direction with a central frequency wL and a linear polarization Z'. With the preceding substitution, the Heisenberg equations that follow from the Hamiltonian (34) become linear. Thus, at resonance, wL = oo+ + k i / 2 M , and in the rotating frame in which gz + e - i E f g,, r fz ,-1(E;+w,)I fz, they take a particularly simple form: -+
where we have introduced a new notation for the annihilation and creation of wavepackets of excited states that originate from the Zth state of the ground state potential,
248
M. Lewenstein and Li You
Note that these annihilation and creation operators describe independent wavepackets, i.e., they obey the standard bosonic commutation relations. Equations (43) and (44) can be easily solved, and the electromagnetic field can be constructed during the interaction with the pulse. Note, however, the initial conditions for these equations correspond to a thermal equilibrium distribution and are highly nontrivial; that is, they have to account for atomic collisions that are responsible for the thermalization process. The physical picture is now the following: Each of the n' levels of the ground state oscillator (when populated) creates an independent wavepacket which is a superposition of the excited state wavefunctions. The population oscillates coherently between the n'th ground state and the corresponding excited state wavepacket. The system behaves as if it consists of a set of independent two-level atoms coherently driven by the laser pulse. If the area of the pulse is a multiple of 2m, the system will be left in the same initial state after the pulse is gone.
6,
2. Scattering of 2mK Pulses In Lewenstein and You (1993) and You et al. (1995a), we have shown that scattering of 2mK pulses allows for the detection of the condensation. We have calculated the spectrum of scattered photons defined as
which contains both coherent and incoherent parts. Note that as defined, such a spectrum also carries information about angular distribution. Similarly, we considered the total number of emitted photons that can be obtained by integrating the spectrum:
Obviously, N,,,can also be divided into coherent and incoherent parts. Both of these quantities display critical behavior as temperature lowers. In particular, the number of scattered photons increases dramatically below T, (or, in another words, as the number of condensed particles grows; see Fig. 5). The dominant coherent spectrum at high temperatures exhibits forward scattering restricted to extremely narrow angles. The atoms behave as a set
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
249
108
I 0’ C
Z 7J
106
c
0
8 1 o5
r
Z
(b)
lo8 1 o7
C
U r
106 105
Z
1 o4
1 o3 loo
lo2
lo4
lo6
lo8
No
FIG. 5. The dependence of the number of coherently (triangles) and incoherently scattered photons (diamonds) on the condensate occupation No for (a) T~ = l / y L = 100 (psec) and (b) T~ = l/yL = 10 (psec) with N = lo8.
of random scattering objects in this case, and constructive phase matching can only occur close the forward direction, so that
For T = 10 p K , Po, = 5 X lo-’, the scattering will cover only a tiny solid angle with half-angle s 1.0 x In contrast, at low temperatures the coherent part of the spectrum gains a contribution from the condensate,
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M.Lewenstein and Li You
where No denotes the number of the condensed atoms. As we would expect, the coherent scattering now covers a much larger solid angle (determined by the condensate size) with half-angle 1.0 x lo-’ for condensates of the few micrometers (see Fig. 6). The idea of using short, intense laser pulses for studying BEC has been developed further in Zeng et al. (1995). These authors studied the case of atoms with degenerate ground states and proposed to use pulse excitation to create two (interacting) condensates in two degenerate levels.
-
C . INCOHERENTLIGHTSCATTERING Calculation of incoherent light scattering spectra and of a complex index of refraction is a much more difficult task than the analysis of the coherent scattering. The studies of these problems have just begun and are far from complete. They are, however, very challenging, since the incoherent scattering carries the information about density fluctuations, and thus it exhibits direct signatures of quantum statistics. Morice et al. (1995; for detail, see Morice, 1995) have formulated a ) ~1 for the systematic approach valid in the low density limit p ( h / 2 7 ~ < calculation of the complex index of refraction up to the second order terms of the density expansion for a homogeneous gas. The beautiful thing about this calculation is that it takes systematically into account multiple scattering of photons within the pairs of close atoms, giving rise to the resonant van der Waals interaction. Even though the effects of those interactions in the considered regime of parameters are not very significant, the developed method is of great value. It relies on deriving the hierarchy of equations for correlation functions, and breaking such a hierarchy at the appropriate level, in analogy to the methods used in the plasma theory (Ichimaru, 1993) and the theory of liquids (Yvon, 1937). The temperature dependence of the refractive index gives a clear signature of quantum statistical effects, even above the critical temperature (see Fig. 7). The reason for that is that the refractive index depends functionally on the density-density correlation function. The result of Morice et al. (upon neglecting of the van der Waals interactions) has been rederived by us using Glauber’s diffraction theory (You et al., 1995b). The incoherent spectrum has been also calculated for the case of scattering of short but intense laser pulses (You et al., 1995a). As in the previous case, the incoherent spectrum probes the density-density correlation, and as such it gives a direct signature of quantum statistics. Finally, in Javanainen (1995; for details, see Javanainen and Ruostekoski, 1995) off-resonant light scattering from ideal Bose and Fermi gases at low temperatures was considered. To this aim, quantum mechani-
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
251
FIG. 6. The angular and spectral dependence of the scattering amplitude squared for (a) coherent scattering, and (b) incoherent scattering, for No = 5 X lo7 with N = lo8.
M. Lewenstein and Li You
252
a -7.00~10"
7 -7.25~10-~ "c
-7.50~10-~
-7.75~10-~ I
.om1
i I1111111 I I1111111
.001
I I lilllll
I 11111111
.I
.01
1
I I1
10
MkBT/h2k2
i I
I I 111111'
.0001
.001
I I 1111111
.01
I I 1111111
.1
I I 1111111
1
I I 1111111
10
MkBT/h2k2 FIG. 7. (a) Real part, (b) imaginary part of the refraction index as a function of temperature for bosons (solid curve) and distinguishable atoms (dotted curve). The detuning 6 = r, and the density is p y 3 = 0.5. The vertical line indicates the condensation temperature. By courtesy of Olivier Morice.
cal Heisenberg equations were solved in the Born approximation. The resulting scattering spectrum (i.e., the temporal Fourier transform of the electric field correlation function) is proportional to the Fourier transform of the spatiotemporal density-density correlation function. As a result, the spectrum contains two peaks below the critical temperature: one corresponds to the situation in which a condensed atom is taken out of the condensate during the scattering, the other to the situation when a
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
253
noncondensed atom is transferred into the condensate. The appearance of these two peaks (separated by an energy difference of the order of two photon recoils) in the spectrum indicates condensation.
W . Quantum Dynamics of Condensation The quantum dynamics of condensation in a system of cold atoms is a very complex many-body problem. In certain situations, the onset of the condensation processes may be analyzed using quantum Boltzmann equations (Reichl, 1980). However, as pointed out in Kagan et al. (1992), this method cannot be used when a coherent condensate has already built up; in such a case it is better to use a time dependent nonlinear Schodinger equation approach. In quantum optics, there exists a very powerful tool to describe dynamics-the quantum master equation (ME) (Gardiner, 1991). The formulation and analysis of the ME in the second quantized framework is one of the recent achievements of QITAF’. The ME allows one to study atom number fluctuations in each of the trap levels as well as quantum coherences. Thus, it provides a more complete description of the cooling process. In particular, it may be used, in principle, to describe uniformly the dynamics of condensate formation (from the kinetic to the coherent phase of the process). Apart from the area of quantum optics, master equations for quantum Bose or Fermi gases have also been used in statistical physics (Spitzer, 1970; Denton et al., 1973; Spohn, 1991; Przenioslo et af., 1991; Barszczak and Kutner, 1991; Kutner and Barszczak, 1991; Kutner et af., 1995). There, however, the ME is usually postulated starting from general statistical requirements. It describes the approach toward the thermal equilibrium described by the Bose-Einstein or, correspondingly, Fermi-Dirac distributions, it fulfills detailed balance conditions, and sometimes it conserves certain order parameters. The dynamics it generates might have some universal properties, but it does not usually have a direct physical interpretation in terms of interactions with specific energy reservoirs. In most of the quantum optical examples, however, the ME is derived via the elimination of the “bath” degrees of freedom starting from a general theory that describes a very specific physical situation. The eliminated “bath” has a direct physical interpretation-it consists of photons, colliding atoms, etc. Each of the jumps between the states of the system described by the ME usually corresponds to a well-defined physical process of photon emission, absorption, atom-atom collision, etc. For instance, the statistics of photons emitted in the laser cooling process might be directly
254
M. Lewenstein and Li You
obtained from the statistics of the jumps between the states of the system (see Cirac et al., 1994b). Finally, the ME is a starting point for stochastic formulation of the quantum dynamics, and for Monte Carlo simulation of the wave function (Carmichael, 1993; Dalibard et al., 1992; Gardiner et al., 1992; Gisin and Percival, 1992); the many-body stochastic wavefunction approach has also been recently developed (Imamoglu and You, 1995; Imamoglu and Yamamoto, 1994). So far, the ME technique has been applied to the studies of quantum dynamics of laser-cooled gases in microtraps (Cirac et al., 1994b,c, 1995; Lewenstein and Cirac, 1996), and to the study of sympathetic cooling (Lewenstein et al., 1995). This is also the principal method used to describe a boser (see Section VII) and is frequently employed in the area of nonlinear atom optics (see Section VIII). Although these studies have predicted several interesting effects, none of the model problems corresponds to situations already realizable in experiments. The applications of the ME to the problem of evaporative cooling (i.e., a situation corresponding to the experiments on BEC) have only begun (C. W. Gardiner and P. Zoller, 1996; Quadt et al., 1996).
A. THE MASTER EQUATION As we stressed, the form of the ME is specific for the problem considered. In the problems that involve laser cooling or light scattering, the ME results from the elimination of the vacuum photon modes. Here we shall follow Cirac et al. (1994b) and consider a prototypical problem of laser cooling of a one-dimensional bosonic or fermionic gas (i.e., we assume that the gas is confined in a trap that is very tight in two dimensions and allows for atomic motion in one dimension only, say along the z axis). Using standard methods of quantum optics (Gardiner, 1990, one derives the ME governing the evolution of the system in the Markov-Born approximation. a, vg, v,, I wL - wol -=K w L , w o ,c / a This approximation is valid for Ny, (Cirac et al., 1994b1, where R is the Rabi frequency induced by an external laser field, vg, v, are trap frequencies for the ground and excited electronic states, and a is the typical size of the sample of atoms. In the present section, we will deal with moderate numbers of atoms and small traps (we neglect effects of atom-atom collisions!), and therefore this approximation is justified. The ME for the density matrix p written in the frame rotating with the laser frequency reads
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
255
where the Hamiltonian becomes
z=x+qas +SdiP
(51)
and the harmonic trap potentials
q
=
c Ivggk, + I
-~)ekrl
(52)
m
with 6 = w, - w,, the laser detuning. The interactions with an external laser are described by
where the coupling parameters factors given by
v;,
are the analogs of Franck-Condon
qkr = (rn,eIsin(k,z - 4)u+ll,g)
(54)
where u+= le)(gl. The laser standing wave is directed here along the z axis, and C#I is its phase. The part describing dipole-dipole interactions between ground and excited atoms is %ip
=
0
C
Am/i’rn,etmg/,g,em’
(55)
I, I ’ , m , m ’
where 0 = + 1 for bosons and fermions, respectively, whereas (9stands for the principal part of the integral over k )
and
Here, for transitions with Am,
whereas for Am,
= +_
=
0,
W ( u )= $(l - u’)
(5 8 )
+ u’)
(59)
1, W ( u )= i ( 1
In writing Eqs. (55) and (56), we have extracted the single atoms Lamb shift that simply renormalizes w o , but included the contribution from (otherwise neglected) counterrotating terms. The latter contribution is
256
M. Lewenstein and Li You
essential for the calculation of the shifts that are due to dipole-dipole interactions of atoms (Cohen-Tannoudji et al., 1989). The spontaneous emission part of the ME is
The foregoing second quantized ME constitutes the starting point for the series of papers on dynamics of laser cooling (Cirac et al., 1994b,c, 1995; Lewenstein and Cirac, 1996).
B. SIDEBAND COOLINGOF
AN
IDEAL GAS
Standard methods of cooling do not allow one to reach temperatures low enough for the condensation-this is one of the reasons for using the evaporative cooling technique. There are, however, several cooling schemes that are not limited by a so-called photon recoil limit. One of them is the method of sideband cooling (Diedrich et al., 1989; Cirac et al., 19921, which works in the so-called Lamb-Dicke limit when the size of the trap is small in comparison with the wavelength, so that the Lamb-Dicke parameter q = k,a < 1. In such a limit, one can locate the trap in a node of the standing wave of the laser (4 = O), so that the dominant processes of atom-light interaction will correspond to those represented in a diagram in Fig. 8. Because the trap is stiff, the recoil energy is much smaller than the trap frequency, and the spontaneous emission cannot change the motional state of an atom. The external laser field does change the motional state, but only by one quantum. Evidently, if the trap frequencies for the ground and excited states are equal and the laser is red-detuned, the laser will cool atoms. Conversely, for positive detunings the laser heats the atoms. Since we can assume that laser is not too strong, we may safely assume that the atoms are hardly excited, and we can then eliminate adibatically from the ME the excited electronic state contributions. The resulting ME is still very complex: Due to the enormous degeneracy of the harmonic oscillator levels in a manyatom system (see Section VI.D), the off-diagonal density matrix elements between such degenerate states do not decay to zero. In practice, however, assuming small anharmonicites of the trap levels (which actually may be
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
257
-7T FIG. 8. Energy level scheme including the center-of-mass degrees of freedom. Ig) and le) denote internal ground and excited levels, whereas Ik) stands for the kth level of the harmonic trap. Note that, in general, the trap frequencies for the ground and excited levels can be different. In the Lamb-Dicke limit, only the transitions indicated take place.
self-induced by the atom-atom interaction), we arrive at a simple ME describing the dynamics of ground state atoms: i,
=
r-
30
c (rn + 1)(2A, pAt, - A t , A ,
2
p - pAt,A,)
m=O
r+ +- C 2
m=O
(rn
+ 1)(2AkpA,
-A,AI,p-pA,Ak)
(61)
,,
with A, = gAg,+ r, = N,, denoting the statistically enhanced spontaneous emission rate, and
The preceding ME, valid for arbitrary N , can be interpreted as a jump process with transition rates between the neighboring trap levels. This enables us to simulate the time dependent dynamics of the system. In contrast to the familiar Monte Carlo simulations in statistical mechanics for lattice models in thermodynamic equilibrium, our simulation has an immediate physical meaning since each of the jumps is associated with an optical pumping transition and, thus, the emission of a fluorescence photon. The simulation gives us not only the trap level distribution but also the photon statistics of the emitted light.
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M. Lewenstein and Li You
The stationary state of the gas (which exists provided the detuning is negative) turns out to have a canonical form with an effective temperature defined via the relation
Stationary distributions of atoms are described by Bose-Einstein or Fermi-Dirac distributions. The Monte Carlo simulations here allow direct monitoring of the dynamics and observation of, for example, the evolution of populations of motional states. This is illustrated in Fig. 9 where we consider the dynamics of bosonic gas starting from the situation in which all atoms were in the fifth state. As cooling proceeds, bosons undergo collective and practically simultaneous jumps to the lower states of the trap. Figure 10 shows a comparison of dynamics for bosons and fermions. Again, the basic qualitative difference consists of the bunching tendency of bosons. As we mentioned each of the jumps in the Monte Carlo simulations has an interpretation here as a photon emission action. Jump statistics thus provide direct insight into the statistics of emitted fluorescence photons. Interestingly, although the mean fluorescence intensity is the same for bosons and fermions, photon statistics (for example, dispersion of a mean delay time between the two successive emissions) exhibits significant dependence on the quantum statistical nature of atoms (Fig. 11).
C. GENERALIZED BOSE-EINSTEIN DISTRIBUTION If the same method of sideband cooling is applied to the case in which the excited state potential is stiffer than the ground state (v, > v,), a new effect is possible. A laser that is blue-detuned for low motional levels will necessarily become red-detuned for high motional levels. As a result, the atom will be heated from below and cooled from above. For a single atom, such an interplay of heating and cooling leads typically to a final distribution centered close to the state for which the actual detuning is close to zero. In Cirac et al. (1995) it was shown that such a mechanism leads, when N is sufficiently large, to a perfect condensation of bosons in a single trap level, not necessarily the lowest one. Moreover, for a given value of laser detuning, there are in general several neighboring states into which the condensation may occur (see Fig. 12). The system thus exhibits multistability and hysteresis effects when the detuning is adiabatically and cyclically
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
a
259
1
4
Z
0.5
0
5
10
15
20
25
30
z b 1
zs 0.5
0
A
A
h
X
0.1
z
0.2
0.3
FIG. 9. (a) Relative populations of the states IS), 14), D), 12), 11) and 10) (n, = N,,,/N), as a function of the scaled time for N = 100 bosom, u = lOOOy, 6 = -5OOy. Initially, all atoms occupy the state IS). Each of the populations exhibits a well-separated peak of height = 1 as time increases. The dimensionless time parameter is T = (qn/r,)’yt. (b) Same as (a), but for N = 500.
changed (see Fig. 13). The stationary states of the system display some algebraic similarities to Bose-Einstein distributions (BEDS), and they have been termed generalized BEDS.
D.
COOLING OF A
GAS WITH ACCIDENTAL DEGENERACY
All of the results discussed so far rely on the fact that off-diagonal elements of the density matrix vanish. As mentioned, this is true provided
260
1
-275
I .yI " . .
-650
~=362
2 0 5
I 0
o
r
i
40
60
c"
d 0
0
20 m
0
20
40
60
m
FIG. 10. Comparison of the quantum Monte Carlo dynamics for N = 50 atoms, v = lOOOy, 6 = -5007. The plots on the left hand side correspond to bosons, whereas those in right hand side correspond to fermions. Initially, in both cases, atoms occupied the levels 4-53. The values of the dimensionless time T = (T$I/T,)'~~ are indicated.
the trap potentials are sufficiently anharmonic. In the opposite limit, when the potential is close to perfectly harmonic, the effects of accidental degeneracy dominate the dynamics of the system. Let us enumerate by rSi the eigenstates of a single-atom Hamiltonian in the rotationally symmetric harmonic trap of frequency o,where rSi is a natural number in one dimension, is a pair of natural numbers in 2D, a triple in 3D, etc. When we consider an ensemble of N atoms, the states of such an ideal gas can be written in the Fock representation as In<,n i , . . . >, where the n,- denote the occupation numbers of the corresponding S t h
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
261
c
4
FIG. 11. Mean dispersion of the delay time between emission of two successive photons and mean fluorescence intensity (inserts) as functions of the detuning for N = 10 and u = 1Oy (top), for N = 50 and v = 1OOy (bottom), for bosons (‘‘0”) and fermions (“x”). The results were obtained from numerical simulations of the master equation.
eigenstate. For noninteracting atoms, there are two kinds of degeneracies in such a system. First, there is a degeneracy of energy levels due to rotational invariance; that is, for the states for which the sum of rid's with a fixed sum of the components of 6 ,which are themselves fixed. Obviously, such degeneracies are not present in 1D. We shall not discuss them here, since we shall focus on the case of one-dimensional gas. Second, there exists an accidental degeneracy, due to the particular symmetry of the harmonic potential. This degeneracy occurs even in the case of 1D: for instance, for the states 10,2,0,. . . ) and ll,O, 1,0,. .. >.Here, the state with two atoms in the first energy level has an energy 2 X w , which is equal to
262
M. Lewenstein and Li You
* z a
6/r FIG. 12. Phase diagram for the bosonic system as a function of laser detuning and the dimensionless parameter d, = N,/(2ug). In each closed area, we indicate the possible phases by the level that can be occupied ( y = r).
the energy of the state with one atom in the ground level and another atom in the second excited level (1 X Ow + 1 X 20). As we said, both kinds of degeneracies are lifted up if one considers anisotropic trap with anharmonic energy levels. If one then assumes that the resulting energy level shifts are larger than cooling rates, one can evoke standard secular arguments to reduce the ME to a diagonal form in the basis of the bare ideal gas states (Cirac et al., 1994b,c). In the opposite case, i.e., when the effects of accidental degeneracy dominate the dynamics of the system, it turns out that 1. There exist nN(l) so-called vacuum states loL,,) that are annihilated m gig,, + by the “jump” operator A = C:= d
,,
A
lo/,,>
=
0
(64)
where the index 1 = 0, or 1 = 2,3,. .., N energy of the corresponding states,
- 1
indicates the bare
m
c wnat,an lo,,,)
=
0 1 IO/,J,
(65)
n=O
whereas s = 1, ..., n N ( l ) . Each of the vacuum states is a linear combination of the accidentally degenerate energy eigenstates. The number of the accidentally degenerate states of the energy wl in the N atom system p N ( l )is given by a solution of the partition problem
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
IcX0.5 a
i l . ! R
'
--'-.-. I
'\
$0.5
Ilj:] Jj--;
0
263
I'
0
\\
---
100
50
-
u
150
0
50
100
150
0
50
100
150
IcX0.5
0 C
0
-'
rcx0.5 J=pl
J -7q 1
I
50
100
u
150
0
I
u
I
0
1
v
I
0
-
6/r
50
0
-
' f
0
6rr
50
FIG. 13. Occupation probability 7ik of the ground ( k = 0, solid line) and excited ( k = 1, dashed line) harmonic levels as a function of the laser detuning 6/r for N = (a) 100, (c) 1000, and (e) 2000. Fluorescence intensity (in arbitrary units) as a function of the laser detuning 6/r for N= (b) 100, (d) 1000, and (f) 2000. The results were obtained by solving the master equation with Monte Carlo simulations. Only (e) and (f) show hysteresis behavior. The parameters are ve = 1050r, v8 = 1OOOr.
j
of the number theory (Hardy and Ramanu'an, 19181, and is extravagantly large (c.f. p,(l) = O(exp(.rr 21/3 )) for 1 I N ) . The number of the vacua is given by n,(l) = p,(l) - p N ( l - 1).The very existence of multiple vacuum states is thus a direct consequence and, at the same time, a signature of the accidental degeneracy. 2. The vacuum states are orthonormal, (O,, I O,,, $,) = S,,,Sssr. 3. The Fock-Hilbert space of the system splits into an infinite number of Fock subspaces corresponding to each of the vacuum states. The Fock states in the (1, s)th subspace are constructed as
with k = O,1,. . . . They are also mutually orthonormal, and they are eigenstates of the energy operator with the corresponding eigenvalue w ( f + k ) . They are also highly degenerate (for k + 1 = k' + 1').
264
M. Lewenstein and Li You
The dynamics exhibits in the Lamb-Dicke limit two time scales. On a faster time scale of the order of T - ~ it, is nonergodic, i.e., it does not mix the different 1 subspaces. After a short time, all coherences between the Ikl,,) and lki,,s,) for k # k’ vanish. Within each 1 subspace, the system approaches the thermal equilibrium characterized by the density matrix diagonal in k, with undamped off-diagonal elements for s # s’, and some temperature (related to the temperature of the “heat bath,” i.e., the system that provides energy dissipation). The dynamics, however, cannot be reduced to a Poisson jump process (i.e., a sequence of random jumps between the various I kc s ) states with the transition probabilities governed by the detailed balance conditions characteristic for the thermal equilibrium). The reason is that coherences corresponding to 1 # 1‘, k = k ’ , as well as to 1 = l ’ , but s # S ’ do not vanish. The ergodicity is restored on a much longer time scale, of the order 77-4, when the dynamics begins to mix various 1 subspaces. The coherences for 1 # 1’ die out, and eventually the system approaches an equilibrium state described by the “canonical” distribution with respect to the energy, with arbitrary coherences between the states with the same k, 1, but s # s’. The latter coherences are then damped on an even longer time scale.
E. SYMPATHETIC COOLING Another route to cold samples of particles is sympathetic cooling (Wineland et al., 1978, 1985; Phillips et al., 1985; Drullinger et al., 1980; Larson et al., 1986; Gabrielse et al., 1989, 1990; Lewenstein et al., 1995, and references therein). With this technique, a gas of particles (,‘A”) is cooled via its interactions with another gas (,‘B”) which is already at a low temperature. Typically, one can assume that the number of particles in B is very large and/or that they are kept cold by another mechanism (such as laser cooling or evaporate cooling). Then B can be regarded as a thermal bath, and therefore the final temperature of A will be very close to that of B. Here, as in the case of evaporative cooling, the required thermalization occurs due to particle-particle collisions. To our knowledge, the idea of sympathetic cooling of neutral particles, and in particular atoms, has not been exploited in the literature. In Lewenstein et al. (19951, such a possibility was discussed concentrating on the following physical situation: the gas of alkali atoms B is confined in a large and rather loose trap, such as magneto-optical trap (MOT). qpically, for alkali atoms such traps have frequencies of the order of 10-100 Hz and sizes of a few micrometers. The gas B is cooled by some mechanism (laser cooling, evaporative cooling, etc.) to a temperature T,. The temperature T, might still be relatively high for the B atoms, which are additionally
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
265
assumed to be relatively heavy. The gas A is composed of other alkali atoms, assumed to be stored in a tight trap such as a far-off resonance dipole trap (FORT) (Miller et al., 1993) located inside the MOT. Tight traps may have frequencies in the range of few kilohertz and sizes of = 0.1 p m . A atoms have smaller mass, but not necessarily much smaller than that of B atoms. In Lewenstein et al. (19951, the ME method has been applied to a model describing the quantum dynamics of sympathetic cooling. We considered a gas of particles A trapped in an harmonic potential and interacting with other particles B that can be regarded as a bath at a given temperature. The interactions between the particles are due to atom-atom collisions, which we have modeled using the standard shape independent potential approximation,
where a:., a,.,, bt(Z>, b ( z ’ ) denote creation and annihilation of A and B atoms in the harmonic oscillator, or plane wave states, respectively, whereas
whereas +&?’) is the wavefunction corresponding to the Zth level of the harmonic oscillator, and C is a constant depending on the &dimensional scattering length a%.For example, in three dimensions,
with p the reduced mass. Again, the methods borrowed from quantum optics were used to derive an ME for the reduced density operator of the system A. The ME describes cooling through transitions between different trap levels. The rates at which these transitions occur depend on the specific properties of the atomic collisions, as well as on the characteristics of the trap and the temperature of the atoms of the bath. For processes involving transitions from n” to Z and 6’to 6 that fulfill
c
I=x,y..
(It,
.
-4)= a ,
( m , - m ’I ) I=x,y
.. .
=
-a
(70)
266
M. Lewenstein and Li You
the rates are
x n ( Z ) [ n ( Z ’ )+ l ] S [ ~ ( k )- ~ ( k ’+) ahv]
(71)
where a is defined through (701, so that a > 0 ( a < 0) corresponds to processes decreasing (increasing) the energy. In principle, the rates (71) contain all the information concerning the cooling process. The main result (Lewenstein et al., 1995) consists of analyzing these rates and deriving accurate analytic formulae for them. The results and the techniques developed (stationary phase for other asymptotic methods) can be generalized to study other problems, such as an analogous problem of evaporative cooling of atoms in a loose MOT trap with a tight FORT in the center.
F. EVAPORATIVE COOLING As we already have mentioned, perhaps one of the most difficult problems in quantum dynamics concerns the formation of a condensate in the final (thermalization) stage of evaporative cooling. Here again, the relevant atom-atom interactions are elastic collisions. Several authors have attempted to estimate the time scale of the formation of the condensate in a homogeneous system and came to contradictory conclusions (Levich and Yakhot, 1977; 1978; Snoke and Wolfe, 1989; Eckern, 1984; see also Kagan et al., 1992). As carefully discussed in Kagan et af. (1992), most of the authors used quantum kinetic equations and applied them in the so-called coherent region in which a coherent condensate has been already formed, and in which kinetic equations are no longer valid, strictly speaking. Stoof pointed out that the evolution is characterized by the two time scales: the time scale of the slow growth of the condensate, preceeded by the much faster nucleation of the coherent population of the zero momentum state (Stoof, 1991, 1992). Kagan et af. performed more appropriate analysis, dividing the evolution into three regimes: coherent regime (in which the dynamics is essentially governed by the time dependent GPG equation), kinetic region in a linear regime (which concerns the hottest atoms, which do not exhibit degeneracy effects and can be well described by a quasiequilibrium distribution), and kinetic region in a nonlinear regime (which concerns colder atoms, which exhibit degeneracy). The paper by Kagan et al. does describe the basic physics of the condensate formation, but it has
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
267
two drawbacks: (i) it does not address the condensation in trapped systems; (ii) it lacks uniformity in the description of various stages of dynamics. For these reasons, several authors have attempted to study the dynamics of evaporatively cooled gas with the help of the ME technique (C. W. Gardiner and P. Zoller, 1996; M. Wilkens and M. Lewenstein, unpublished, 1995). These can be done either by modeling interactions of atoms with some external heat bath of a given temperature, or by treating the system as closed. The latter approach corresponds more closely to the situation realized in experiments. Here, the ME can be derived by eliminating some of the degrees of freedom: either by using some kind of coarse graining procedure, or by eliminating the hot atoms (i.e., those that, as pointed out in Kagan et af. (1992), evolve according to linearized Boltzmann equations, and whose state can be conveniently described as a quasi-equilibrium distribution with time dependent parameters). This problem, however, has not yet been solved, and it remains one of the most challenging problems of the quantum critical dynamics. In Quadt et af. (1995), the authors derived an ME for the lowest mode in the trap, eliminating all the others. The equilibrium state in such a case is a grand canonical ensemble with a Hamiltonian accounting for atom-atom collisions. The collisions cause nonclassical properties of the equilibrium state (sub-Poissonian statistics of the fluctuating number of condensed particles). From what we have said, one should expect that such a phenomenological treatment is definitely physically sound in the limit of tight traps when the behavior of the lowest state can indeed be separated from other states of the trap. It cannot, however, be valid if condensation occurs via the nonlinear kinetic region discussed in Kagan et af. (1992) for a homogeneous system (i.e., an “infinite” trap). The quantum dynamics of gases of cold atoms can be very well treated with the help of the ME technique, and it exhibits, in our opinion, an enormous richness of interesting physical and mathematical phenomena, such as multistable, exotic stationary states, multistage dynamics, and so forth. There are still, however, basic questions to be answered, and further studies are required to gain more understanding of the new physics, including the development of other statistical physics tools, such as diffusion equations, hydrodynamic limits, and so forth.
VII. Theory of Bosers The idea of a coherent source of atomic matter waves has been developed by several groups independently: Holland and Burnett at Oxford, BordC at UniversitC Paris-Nord, Cirac, Lewenstein and Zoller at J I M , Gardiner in
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New Zealand, Spreeuw and Wilkens in Konstanz, and others. For the first time, it was thoroughly discussed and several models were proposed at the Workshop on QFTAP at JILA in 1994 by Burnett, Cirac, Gardiner, Holland, Lewenstein, and Zoller. Since then several proposals have been formulated (Wiseman and Collet, 1995; Holland et al., 1995; Olshan’ii et al., 1995; Spreeuw, 1995; G u z m h et al., 1995). One should stress that most of the proposals are quite speculative, and for the moment they seem to be quite far from experimental realization. In this section, we shall discuss in some detail a prototypical model of a boser formulated by Cirac, Lewenstein, and Zoller, and presented at the JILA Workshop. Before doing that, however, it is worth mentioning that the same ideas have been developed in the context of condensation of excitons. In particular, a model of a boser that generates a coherent population of nonequilibrium excitons has been proposed (Imamoglu and Ram, 1996). Also, the first experimental evidence of spontaneous buildup of the coherent exciton polariton population in a microcavity has been reported (Pau et al., 1996). A. A PROTOTYPICAL BOSER Boser theory is formulated in analogy to laser theory. A boser is an open system into which atoms can be pumped and from which they can be taken away. The pumping is assumed to be incoherent and consists of putting atoms (in some quantum states) into a black box, called a boser. Atoms are lost also in an incoherent way-in a continuously working boser, atomic losses occur analogously to photon losses from a cavity; in a pulsed boser, the output is realized by letting the atoms leave the boser periodically in analogy to Q-switched lasers. Inside a black box, there is an atomic cavity -usually it is some kind of an atomic trap with well-defined mode structures. Bosing consists in mode selection, and it is caused essentially by quantum statistics. As soon as the atoms start to occupy one trap level (i.e., one atomic cavity mode), other atoms that enter the system will tend to do the same. Of course, this effect must be mediated by some kind of interactions. Such interactions in fact favor transitions to some state-the boser models are based on cooling schemes and contain dynamical mechanisms that help to populate the ground state of the trap. In contrast to the standard laser theory (in which atom-photon coupling has a coherent character and may, for instance, cause Rabi oscillations), the mode selection mechanisms in bosers are frequently incoherent. Until now, two kinds of mechanisms were considered: (i) those based on elastic atom-atom collisions, analogous to the ones used in evaporative cooling (Holland et al., 1996); (ii) those based on laser cooling and spontaneous transitions
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(Wiseman and Collet, 1995; Olshan’ii et al., 1995; Spreeuw, 1995; GuzmQn et al., 1996). The prototype model of a boser consists of two atomic modes (two trap levels): the bosing level (ground state of the trap), described by the annililation and creation operators b,, bi; and the pumping mode, described by b,, b!. The boser dynamics is most conveniently described using the ME technique. The ME for a density matrix p of the two-level cw boser model is
i,
=
-iv[ b:b,’ p ]
The first term here describes free evolution (with zero frequency for the ground state, and some trap frequency v for the pumping mode). The second and third terms describe the statistically enhanced transitions from level 1 to 0 at the rate r+, and from 0 to 1 at the rate r- , respectively. Of course, we expect that bosing will only be possible if the mode selection occurs, i.e., r+>r- , so that the system prefers transitions to the ground state. The fourth term in Eq. (72) describes incoherent pumping of atoms into the level 1 at the rate K , . Finally, the last term describes incoherent atom losses from the trap at the rate K,. Further analysis of this model follows the standard lines of quantum optics. If the rate I?+ is also larger than the pumping rate K , , the pumping level can be eliminated adiabatically. Introducing Glauber’s Prepresentation for the ground state operators (Glauber, 1963a, b) (i.e., a diagonal representation in coherent states 1 a )o)
it is then possible to derive a semiclassical equation for the mean value of the ground state field amplitude,
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which has exactly the same form as the semiclassical equation describing a single-mode laser. The cooperation number C is here defined by
(74) whereas atom saturation number is defined by
r+(75) r+-rAs we see, bosing is possible, provided r+> K ] > r _ and C > 1. In such a no =
K1
case, just as in a laser, the ground state amplitude attains a nonzero stationary value, la12 = n,,,(C - 1). Using standard methods of the quantum noise theory (Gardiner 19911, it is easy to show the foregoing model describes indeed an analog of a laser. To this aim, first we demonstrate that the distribution of the number of atoms in the ground state undergoes a transition from a thermal-like to a Poissonian-like form at the boser threshold. Second, we show that above the threshold the boser spectrum is very narrow, since it essentially arises from the phase difSusion process. In fact, the boser bandwidth far above the threshold is ybos= K O ( r - / K I ) and is much less than the trap loss rate (i.e., the analog of the atomic cavity width) K". We stress that proving those two coherence properties is essential for any model of the boser.
B. BOSERMODELS Various boser models employ different mode selection mechanisms. The model of Holland et al. (1996) assumes that atoms (in the ground electronic state) are pumped into a magnetic trap, where they undergo elastic collisions, just as in the case of evaporative cooling. Due to the complexity of the problem, the authors in fact limit the number of accessible levels to three: a ground level, an intermediate pumping level (to which the atoms are incoherently pumped), and a high energy level from which evaporation-like losses might occur. First, the high energy level is eliminated adiabatically, resulting in a two-level model very much analogous to the one just discussed. The model works in the limit when the collisional redistribution rate (i.e., the rate at which two atoms from the pumping level collide, sending one to the high energy level and the other to the ground state) is fast. Unfortunately, in the same limit collisions between the atoms in the ground state of the trap become faster, which might seriously reduce the coherence properties of the boser.
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In Wiseman and Collet (19951, a boser model employing dark state cooling has bee proposed (Aspect et al., 1988). Ideally, dark states are atomic states (of atoms at rest) that are not coupled to the laser field. In the dark state, cooling atoms may undergo laser excitation from nondark states and spontaneously emit into the dark state. If they enter the dark state, however, they remain there forever. In Wiseman and Collet (19951, a many-body ME for dark state cooling is analyzed. Again all atomic field operators other than for the ground state are eliminated, and the ME for the ground state operators is considered. Two other models in which pumping into the bosing mode is provided, as in the previous case, by spontaneous emission concern two opposite limits. Olshan’ii et al. (1995) considered a general case of pumping atoms into a ground state level of a large trap from a single (electronically) excited level. In an experimental realization, this could correspond to pumping atoms from other ground electronic states in the hyperfine manifold to the final ground state via Raman transitions. If in such a process the effects of photon reabsorption and optical thickness are disregarded, the system does exhibit a boser transition, and it may accumulate atoms in the ground state of the trap (or, strictly speaking, the state from which losses are minimal). Interestingly, in this model spontaneous emission does not select any level-condensation occurs due to quantum statistical enhancement of the emission to the level from which the loss is the smallest. The same model has been analyzed taking into account reabsorption effects. It turns out that due to the large absorption cross section of atoms ( = A’), reabsorption effects may totally prevent the boser action in large traps. Using a noisy laser might help to reduce absorption cross sections, and to restore the possibility of bosing. Spreeuw (1995) considered a similar model but in the Lamb-Dicke limit, Their system consists of metastable Ar atoms trapped in a single minimum of a so-called dark optical lattice created by crossed laser beams. Such a trap is in fact a blue-detuned FORT. The trap is loaded with laser-cooled atoms, which undergo the transfer into the trapped states through spontaneous emissions. The mode selection takes place here because the ground state of the trap turns out to have the lowest loss rate (due to the scattering of trap-laser photons) and the highest pumping efficiency. Finally, in G u z m h et al. (19961, a one-dimensional atomic cavity is considered in which atoms are ‘‘localized’’in a quantum ground state with respect to the transverse motion (with the help of a FORT) and can occupy a discrete set of motional (standing wave) states aligned with the cavity axis (Zhang et al., 1995). In contrast to other schemes, the nonlinear effects of quantum statistics are provided here by atom-atom interactions
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due to near-resonant dipole-dipole forces. Even though the lasers used for cooling and trapping the atoms are far from resonance, and atoms are thus hardly excited, they are subjected to effective two-body dipole-dipole forces. For practical calculation, the authors reduce the model to a three-level case: with the highest pumping level and two active cavity levels.
C. BOSONACCUMULATION REGIME
Obviously, the quantum statistical effects are necessary to achieve the boser action. Three of the models we have discussed involve spontaneous emission. It is therefore challenging to study the role of quantum statistical effects in spontaneous emission more carefully, taking this account reabsorption effects and dipole-dipole interactions. In Cirac and Lewenstein (1996), we have considered the behavior of an atomic Bose-Einstein condensate in the presence of an atom in an excited electronic level. We analyzed the boson accumulation regime (BAR), defined by the relation No >> qev,N - No, where N is the total number of atoms in the ground electronic state, No is the number of atoms in the condensate, and ale"is the number of levels to which the excited atom can effectively decay. In this regime, quantum statistical effects related to the boson nature of the atoms predominate in the process of spontaneous emission. Simple arguments (based on rate equations) suggest that the proportion of atoms in the condensate decreases after the spontaneous decay. Using a more appropriate approach based on an ME description, we have demonstrated that, in general, these simple arguments may lead to erroneous conclusions. Under certain conditions, the ME approach predicts that the proportion of atoms in the condensate increases. We have given an interpretation of this phenomenon in terms of quantum interferences between processes that include reabsorption of the emitted photons and the dipole-dipole interactions between the atoms. Namely, in the BAR limit, the processes that give the leading contributions are presented in Figs. 14 and 15. The process in Fig. 14 is dominant and describes spontaneous emission accompanied by a direct transition to the condensate; processes (a) and (b) in Fig. 15 contribute in the first order of the ratio alev/N,or ( N - N o ) / N , both decrease the condensate fraction, but both start and end in the same state. Their amplitudes necessarily interfere, and amazingly their interference is destructive! Finally, process (c) in Fig. 15 contributes also in the first order of the ratio alev/N, or ( N - N o ) / N , and increases the condensate fraction.
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a
273
10’
FIG. 14. The model consists of one (internal) excited level and a + 1 ground levels. Initially, there is one atom excited and N atoms in the ground state, distributed among the a + 1 levels.
The surprising physical effect considered in Cirac and Lewenstein (1996) may help to pump atoms into a condensate and to compensate for atomic losses in the atomic trap. Mathematically, the main achievement of this paper is the demonstration that in the BAR it is possible to construct a systematic expansion of the solutions of the ME in a series of parameters alev/N, or ( N - N , ) / N . To estimate these parameters, we may assume a
b
C
FIG. 15. (a) Process in which the excited atom decays directly into the kth ground level. (b) Process in which the excited atom decays into the condensed level; the emitted photon is absorbed by an atom in the same level, and it subsequently decays into the kth level. (c) Process in which the excited atom decays into the condensed level; the emitted photon is absorbed by an atom in the kth ground level, and it decays back into the condensed level.
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that the excited atoms have a mean energy of the order of one recoil, so that one can estimate ale" = ( ~ , / h v )where ~ , eR is the recoil energy and v is the trap frequency. For the current experiments dealing with BEC, ~ , / ( h v )2 30, which would require a number of condensate atoms N s 25,000. According to Anderson et al. (1995) and Bradley et al. (19951, this seems to be within the reach of experiments in the near future. On the other hand, it seems that the BAR has been achieved, or is close to being achieved, in the MIT experiment (Davis et al., 1995~).Another interesting situation to test predictions of Cirac and Lewenstein (1996), however, will ) 3; in this case, the BAR condition will be be the one in which ~ , / ( h v = fulfilled with a small number of particles. To realize such a situation, the trap frequency must be of the order of 1 kHz (or larger). This is the typical case for a dipole (FORT) trap. We expect that studies of the ME equation in the BAR will soon bring further fascinating results.
VIII. Nonlinear Atom Optics As we have already mentioned in Section IV, nonlinear excitations of BEC may lead to various matter wave analogs of nonlinear optics (Edwards et al., 1996). Such phenomena, however, deal with atoms in their ground electronic state and are thus mediated by elastic collisions described by Zg [see (1411. It was, however, suggested by Meystre and his collaborators, and Zhang and Walls, that similar phenomena could also occur in the area of atom optics, and these were termed nonlinear atom optics (NAO). Atom optics is developing vary rapidly. It mainly concerns manipulations of the matter waves in analogy with light waves (Mlynek et al., 1992), and often it involves manipulations of the matter waves using laser light. In such situations (even if the lasers used are far from resonance, and atoms are hardly excited), the resonance dipole-dipole interactions between the excited-ground state atoms play an essential role, and they involve quantum statistical effects, if the temperatures of the atomic beam are sufficiently low and densities sufficiently high. The relevant interactions for kinds of phenomena involve 3&(161, but in the first place the interactions are due to exchange of photons, described by%, (17). Eliminating photon degrees of freedom from the theory, one arrives at an effective (nonlinear) potential for atoms, and one treats the atom by a Hartree-type approximation. This approach is completely analogous to elimination of atomic degrees of freedom in order to derive equations of nonlinear optics. In this sense, Maxwell-Bloch equations provide a unified view of nonlinear optics
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and nonlinear atom optics (Castin and Molmer, 1995). Zhang (1993) has studied, in this manner, atom correlations induced by dipole-dipole forces. Effective atom-atom potential has been frequently approximated by a local potential, in analogy to Eq. (16). In view of the long range character of the dipole-dipole forces such approximations might be questionable (Castin and Mdmer, 1995). Nevertheless, using such an approximation, Lenz et al. (1993,1994; Schernthanner et al., 1994) and Zhang et al. (1994; Zhang and Walls, 1993, 1994) described nonlinear quantum statistical effects that lead to creation and propagation of atomic Thirring solitons. The basic theoretical tool of their analysis is a set of two coupled nonlinear Schodinger equations (for ground and excited state atoms). The theory of NAO was further developed in the works devoted to the study of spontaneous emission effects on atomic solitons. In particular, a self-consistent Born-Markov-Hartree-Fock ME for nonlinear atom optics has been derived (Lenz and Meystre, 1994; Schernthanner et al., 1995). The two-body dipole-dipole interaction potential between atoms in a nonlinear optical cavity (Zhang et al., 1995) is an essential ingredient of the boser model (Guzmh et al., 19961, discussed in the previous section.
IX. Conclusions We hope that the readers of this chapter will appreciate its main message: In the advent of the recent experiments on Bose-Einstein condensation, quantum optics and atomic and molecular optics have entered a new phase! Theoretical quantum optics is merging with many-body theory, condensed matter physics, and statistical physics, and it is becoming a theory of equal, if not higher, complexity. Quantum optics does not only use the methods of quantum field theory and many-body physics-it also starts to contribute its own methods to open new paths in quantum field theory.
Acknowledgments This chapter would not have been written if we did not have the honor and pleasure of collaborating and exchanging ideas with Ignacio Cirac, Jinx Cooper, and Peter Zoller over the last few years. We acknowledge also fruitful and enlightening discussions with, and the help in the preparation of this chapter of, Vanderlei Bagnato, Keith Burnett, Yvan Castin, Claude Cohen-Tannoudji, Eric Cornell, Alex Dalgarno, Jean Dalibard, John Doyle, Ralph Dum, Mark Edwards,
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Mariusz Gajda, Crispin Gardiner, Roy Glauber, Rick Heller, Murray Holland, Atac Imamoglu, Juha Javanainen, Wolfgang Ketterle, Pierre Meystre, David Politzer, Gora Shlyapnikov, Christoph Salomon, Boudewijn Verhaar, Jook Walraven, Carl Wieman, and Martin Wilkens.
References Abgrall, Y., Belaidi, R., and Labarsouque, J. (1986). I n “Coherence, Cooperation and Fluctuations,” (F. Haake, L. M. Narducci, and D. F. Walls, ed.) p. 46. Cambridge Univ. Press, Cambridge, UK. Abraham, E. R. I., Alexander, W. I., Sackett, C. A., and Hulet, R. G. (1995). Phys. Reu. Lett. 74, 1315. Abrikosov, A. A., Gorkov, L. P., and Dzyaloshinski, I. E. (1975). “Methods of Quantum Field Theory in Statistical Physics.” Dover, New York. Adams, C. S., Lee, H. J., Davidson, N., Kasevich, M., and Chu, S. (1995). Phys. Reu. Lett. 74, 3577. Agosta, C. C., Silvera, I. F., Stoof, H. T. C., and Verhaar, B. J. (1989). Phys. Reu. Lett. 62, 2361. Allen, L., and Eberly, J. H. (1987). “Optical Resonance and Two-Level Atoms.” Dover, New York. Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E., and Cornell, E. A. (1995). Science 269 198. Aspect, A., Arimondo, E., Kaiser, R., Vansteenkiste, N., and Cohen-Tannoudji, C. (1988). Phys. Reu. Lett. 61, 826. Bagnato, V., and Kleppner, D. (1991). Phys. Reu. A 44, 7439. Bagnato, V., Pritchard, D. E., and Kleppner, D. (1987). Phys. Reu. A 35, 4354. Barsznak, T., and Kutner, R. (1991). J. Stat. Phys. 62, 389. Baym, G., and Pethick, C. (1996). Phys. Reu. Lett. 76, 6. Bleszyiiski, M., and Jaroszewicz, T. (1986). In “Coherence, Cooperation and Fluctuations” (F. Haake, L. M. Narducci, and D. F. Walls, eds.), p. 52. Cambridge Univ. Press, Cambridge, UK. Bleszyiiski, M., Glauber, R. J., and Ostlund, P. (1981). Phys. Lett. 104, 389. Bogoliubov, N. N. (1947). J. Phys. (Moscow) 11, 231. Bose, S. N. (1924). Z. Phys. 26, 178. Bradley, C. C., Sackett, C. A., Tollet, J. J., and Hulet, R. G. (1995). Phys. Reu. Len. 75, 1687. Burnett, K. (1995). Science 269, 182. Burnett, K. (1996). Contemp. Phys. 37, 1. Butov, L. V., Zrenner, A., Abstreiter, G., Bohm, G., and Weimann, G. (1994). Phys. Reu. Lett. 73, 304. Carmichael H. J. (1993). “An Open Systems Approach to Quantum Optics,” Lect. Notes Phys. Monogr. Springer, Berlin. Castin, Y., and M~lmer,K. (1995). Phys. Rev. A 51, R3426. Chase, L. L., Peyghabarian, N., Grynberg, G., and Mysyrowicz, A. (1979). Phys. Rev. Lett, 42, 1231. Chu, S., and Wieman, C. E., eds. (1989). Special issue of J. Opt. Soc. Am. B 6(11). Cirac, J. I., and Lewenstein, M. (1995). Phys. Reu. A 52, 4737. Cirac, J. I., and Lewenstein, M. (1996). Phys. Rev. A 53, 2466.
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
277
Cirac, J. I., Blatt, R., Zoller, P., and Phillips, W. D. (1992). Phys. Reu. A 46, 2668. Cirac, J. I., Lewenstein, M., and Zoller, P. (1994b). Phys. Reu. Lett. 72, 2977; erratum: ibid. 73, 2010 (1994). Cirac, J. I., Lewenstein, M., and Zoller, P. (1994~1,Phys. Reu. A 50, 3409. Cirac, J. I., Lewenstein, M., and Zoller, P. (1995). Phys. Reu. A 51, 2899. Cohen-Tannoudji, C., Dupont-Roc, J., and Grynberg, G. (1989). “Photos and Atoms, Introduction to Quantum Electrodynamics.” Wiley, New York. C6t6, R., Dalgarno, A., and Jamieson, M. J. (1994). Phys. Reu. A 50, 339. Dalihard, J., Castin, Y., and Molmer, K. (1992). Phys. Reu. Lett. 68, 580. Davis, K. B., Mewes, M.-O., Joffe, M. A., Andrews, M. R., and Ketterle, W. (1995a). Phys. Reu. Lett. 74, 5202. Davis, K. B., Mewes, M. O., and Ketterle, W. (1995b). Appl. Phys. /Purf B] 60(2), 155. Davis, K. B., Mewes, M.-O., Andrews, M. R., van Druten, N. J., Durfee, D. S., Kurn, D. M., and Ketterle, W. (199%). Phys. Reu. Lett. 75, 3969. de Gennes, P. G. (1966). “Superconductivity of Metals and Alloys.” Benjamin, New York. de Groot, S. R., Hooyman, G. J., and ten Seldam, C. A. (1950). Proc. R . Soc. London, Ser. A 203, 266. Denton, R., Miihlschlegel, B., and Scalapino, D. J. (1973). Phys. Reu. B 7, 3589. Diedrich, F., Bergquist, J. C., Itano, W. M., and Wineland, D. J. (1989). Phys. Reu. Lett. 62, 403. Doyle, J. M. (1991). Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA (unpublished). Doyle, J. M., Sandberg, J. C., Yu, A. I., Cesar, C. L., Kleppner, D., and Greytak, T. J. (1991). Phys. Reu. Lett. 67, 603. Drullinger, R. E., Wineland, D. J., and Berquist, J. C. (1980). Appl. Phys. 22, 365. Drummond, P. D., Shelby, R. M., Friberg, S. R., and Yamamoto, Y. (1993). Nature (London) 365, 307. Eckern, U. (1984). J . Low. Temp. Phys. 54, 333. Edwards, M., and Bumett, K. (1995). Phys. Reu. A 51, 1382. Edwards, M., and Reprecht, P. A., Burnett, K., and Clark, C. W. (1995). Phys. Reu. A (in press). Einstein, A. (1924). Sitzungsber. K. Preuss. Akud. Wiss., 261. Einstein, A. (1925). Sitzungsber. K. Preuss. Akud. Wiss., 3. Fetter, A. L., and Walecka, J. D. (1971). “Quantum Theory of Many-Particle Systems.” McGraw-Hill, New York. Fortin, E., Fafard, S., and Mysyrowicz, A. (1993). Phys. Reu. Lett. 70, 3951. Gabrielse, G., Fei, X., Orozco, L. A., Tjoelker, R. L., Haas, J., Kalinowsky, H., Trainor, T. A., and Kells, W. (1989). Phys. Reu. Lett. 63, 1360. Gabrielse, G., Fei, X., Orozco, L. A., Tjoelker, R. L., Haas, J., Kalinowsky, H., Trainor, T. A., and Kells, W. (1990). Phys. Reu. Lett. 65, 1317. Gajda, M.,Krekora, P., and Mostowski, J. (1996). Phys. Reu. A . (in press). Gardiner, C. W. (1991). “Quantum Noise.” Springer, Berlin. Gardiner, C. W., Parkins, A. S., and Zoller, P. (1992). Phys. Reu. A 68, 4363. Gardiner, C. W., and Zoller, P. (1996). (preprint). Gardner, J. R., Cline, R. A., Miller, J. D., Heinzen, D. J., Boesten, H. M. J. M., and Verhaar, B. J. (1995). Phys. Reu. Lett. 74, 3764. Gilbert, S. L., and Wieman, C. E. (1993). Opt. Photon. News 4, 8. Ginzburg, V. L., and Pitaevskii, L. P. (1958). Sou. Phys.-JETP (Engl. Trunsl.) 7, 858. Gisin, N., and Percival, I. C. (1992). J. Phys. A : Math. Gen. 25, 5677. Glauber, R. J. (1959). Lect. Theor. Phys. 1, 315.
278
M. Lewenstein and Li You
Glauber, R. J. (1963a). Phys. Rev. 130, 2529. Glauber, R. J. (1963b). Phys. Rev. 131, 2766. Goldman, V. V. Silvera, I. F., and Leggett, A. J. (1981). Phys. Rev. B 24, 2870. Greytak, T. J. (1995). In ‘‘Box-Einstein Condensation,” (A. Griffin, D. W. Snoke, and S. Stringari, eds.) BEC-93, p. 131, Cambridge Univ. Press, Cambridge, UK. Griffin, A., Snoke, D. W., and Stringari, S., eds. (1995). “Bose-Einstein Condensation.” Cambridge Univ. Press, Cambridge, UK. Grochmalicki, J., and Lewenstein, M. (1991). Phys. Rep. 208(3). Gross, E. P. (1963). J . Math. Phys. 4, 195. Gross, E. P. (1966). In “Physics of Many Particle Systems: Methods and Problems,” (E. Meeron, ed.), vol. 1, p. 231. Gordon and Breach, New York. GuzmBn, A. M., Moore, M., and Meystre, P. (1996). Phys. Rev. A 53, 977. Hardy, G. H., and Ramanujan, S. (1918). Proc. London Math. SOC.17, 75. Hess, H. (1986). Phys. Reu. B 34, 3476. Hess, H. F., Kochanski, G. P.,Doyle, J. M., Masuhara, N., Kleppner, D., and Greytak, T. J. (1987). Phys. Rev. Lett. 59, 672. Holland, M., and Cooper, J. (1996). Phys. Rev. A 53, R1954. Holland, M., Burnett, K., Gardiner, C., Cirac, J. I., and Zoller, P. (1996). Phys. Reu. A (in press). H u h , D., Mysyrowicz, A., and B e d i la Guillaume, C., Phys. Reu. Lett. 45, 1970. Huse, D. A., and Siggia, E. D. (1982). J . Low Temp. Phys. 46, 137. Ichimaru, S. (1993). “Basic Principles of Plasma Physics: A Statistical Approach.” AddisonWesley, New York. Imamoglu, A., and Ram. R. J. (1996). Phys. Lett. A 214, 193. Imamoglu, A., and Yamamoto, Y. (1994). Phys. Lett. A 191, 425. Imamoglu, A., and You, L. (1995). Phys. Reu. A 50, 2642. Javanainen, J. (1993). Phys. Reu. Lett. 72, 2375. Javanainen, J. (1995). Phys. Reu. Lett. 75, 1927. Javanainen, J., and Ruostekoski, J. (1995). Phys. Rev. A 52, 3033. Javanainen, J., and Yoo, S. M. (1996). Phys. Rev. Lett. 76, 161. Kagan, Yu. M., Svistunov, B. V., and Shlyapnikov, G. V. (1992). Sou. Phys.-JETP. (Engl. Tmnsl.) 74, 279. Kagan, Yu. M., Shlyapnikov, G. V., and Walraven, J. T. M. (1996). Phys. Reu. Lett. 76, 2669. Ketterle, W., Davis, K. B., Joffe, M. A., Martin, A,, and Pritchard, D. E. (1993). Phys. Rev. Lett. 70, 2253. Kutner, R., and Barszczak, T. (1991). J. Star. Phys. 65, 813. Kutner, R., Kehr, K. W., Renz, W., and Przenioslo, R. (1995). J. Phys. A: Math. Gen. 28,923. Larson, D. J., Berquist, J. C., Bollinger, J. J., Itano, W. M., and Wineland, D. J. (1986). Phys. Rev. Lett. 57, 70. Lenz, G., and Meystre, P. (1994). I n “Quantum Optics IV” (J. D. Harvey and D. F. Walls, eds.). Springer, Berlin. Lenz, G., Meystre, P., and Wright, E. M. (1993). Phys. Rev. Lett. 71, 3271. Lenz,G., Meystre, P., and Wright, E. M. (1994). Phys. Rev. A 50, 1681. Levich, E., and Yakhot, V. (1977). Phys. Rev. B 15,243. Levich, E., and Yakhot, V. (1978). J. Phys. A: Math. Gen. 11, 2237. Lewenstein, M., and Cirac, J. I. (1996). Phys. Rev. A. (in press). Lewenstein, M., and You, L. (1993). Phys. Rev. Lett. 71, 1339. Lewenstein, M., and You, L. (1996). Phys. Rev. A. 53, 909. Lewenstein, M., You, L., Cooper, J., and Burnett, K. (1994). Phys. Rev. A 50, 2207. Lewenstein, M., Cirac, J. I., and Zoller, P. (1995). Phys. Rev. A 51, 4617.
QUANTUM FIELD THEORY OF ATOMS AND PHOTONS
279
Lifshitz, E. M., and Pitaevskii, L. P. (1980). “Statistical Physics.” Pergamon, London. Lin, J. L., and Wolfe, J. P. (1993). Phys. Reu. Lett. 71, 1222. London, F. (1938a). Nature (London) 141, 643. London, F. (1938b). Phys. Reu. 54, 947. London, F. (1939). J . Chem. Phys. 43, 49. Lovelace, R. V. E., and Tommila, T. J. (1987). Phys. Rev. A 35, 3597. Lovelace, R. V. E., Mahanian, C., Tomilla, T. J., and Lee, D. M. (1985). Nature (London) 318, 30. Luiten, J. (1993). Ph.D. Thesis, Universiteit van Amsterdam (unpublished). Mahan, G. D. (1993). “Many-Particle Physics.” Plenum, New York. Marcassa, L., Muniz, S., de Queiroz, E., Zilio, S., Bagnato, V., Weiner, J., Julienne, P. S., and Suominen, K.-A. (1994). Phys. Reu. Lett. 73, 1911. Masuhara, N., Doyle, J. M., Sandberg, J. C., Kleppner, D., Greytak, T. J., Hess, H. F., and Kochanski, G. P. (1988). Phys. Reu. Lett. 61, 935. Miller, J. D., Cline, R. A., and Heinzen, D. J. (1993). Phys. Reu. A 40, 4567. Mlynek, J., Balikin, V., and Meystre, P. (1992). Appl. Phys. B 54, Spec. Issue on optics and interferometry with atoms. Moerdijk, A. J., Stwalley, W. C., Hulet, R. G., and Verhaar, B. J. (1995). Phys. Reu. Lett. 72, 40. Monroe, C. R., Cornell, E. A., Sackett, C. A., Myatt, C. J., and Wieman, C. E. (1993). Phys. Reu. Lett. 70, 414. Morice, 0. (1995). Ph.D. Thesis, Ecole Normale Supkrieure (unpublished). Morice, O., Castin, Y., and Dalibard, J. (1995). Phys. Reu. A 51, 3896. Negele, J. W., and Orland, H. (1988). “Quantum Many-Particle Systems.” Addison-Wesley, New York. Oliva, J. (1988). Phys. Reu. B 38, 8811. Oliva, J. (1989). Phys. Reu. B 39, 4197. Olshan’ii, M., Castin, Y., and Dalibard, J. (1995). In “Laser Spectroscopy X I ” (M. Ignuscio, M. Allegrini, and A. Lasso, eds.). World Scientific, Singapore. Pau, S., Cao, H., Jacobson, J., Bjork, G., Yamamoto, Y., and Imamoglu, A. (1996). Phys. Rev. A (in press). Penrose, 0.(1951). Philos. Mag. [7] 42, 1373. Petrich, W., Anderson, M. H., Ensher, J. R., and Cornell, E. A. (1995). Phys. Reu. Lett. 74, 3352. Peyghambarian, N., Chase, L. L., and Mysyrowicz, A. (1983). Phys. Reu. B . 27, 2325. Phillips, W. D., Prodan, J. V., and Metcalf, H. J. (1985). At. Phys. 9, 338. Pitaevskii, L. P. (1961). Sou. Phys.-JETP (Engl. Trunsl.) 13, 451. Politzer, H. D. (1991). Phys. Rev. A 43, 6444. Politzer, H. D. (1995). Preprint. Power, E. A. (1964). “Introductory Quantum Electrodynamics,” Vol. 24. Longman, London. Przenioslo, R., Barszczak, T., Kutner, R., Guzicki, W., and Renz, W. (1991). Int. J . Mod. Phys. C 2, 450. Quadt, R., Wiseman, H. M., and Walls, D. F. (1996). Quunrum Opt. (in press). Raab, E., Prentiss, M., Cable, A,, Chu, S., and Pritchard, D. E. (1987). Phys. Reu. Lett. 59, 2631. Reichl, L. E. (1980). “A Modem Course on Statistical Physics.” Univ. of Texas Press, Austin. Ruprecht, P. A., Holland, M. J., Burnett, K., and Edwards, M. (1995). Phys. Reu. A 51,4704. Schernthanner, K. J., Lenz, G., and Meystre, P. (1994). Phys. Reu. A 50, 4170. Schernthanner, K. J., Lenz, G., and Meystre, P. (1995). Phys. Reu. A 51, 3121.
280
M. Lewenstein and Li You
Setija, I. D., Werij, H. G. C., Luiten, 0. J., Reynolds, M. W., Hijmans, T. W., and Walraven, J. T. M. (1993). Phys. Reu. Leff.70, 2257. Shlyapnikov, G. V., Walraven, J. T. M., Rahmanov, U. M., and Reynolds, M. W. (1994). Phys. Rev. Lett. 73, 3247. Silvera, I. F. (1995). In “Bose-Einstein Condensation” (A. Griffin, D. W. Snoke, and S. Stringari, eds.), BEC-93, p. 160, Cambridge, Univ. Press, Cambridge, UK. Silvera, I. F., and Reynolds, M. (1992). J. Low Temp. Phys. 87, 343. Silvera, I. F., and Walraven, J. T. M. (1980). Phys. Reu. Letf.44, 164. Silvera, I. F., and Walraven, J. T. M. (1986). h o g . Low Temp. Phys. 10, 139. Snoke, D. W., and Wolfe, J. P. (1989). Phys. Reu. B 39, 4030. Spitzer, F. (1970). Adu. Math. 5, 246. Spohn, H. (1991). “Large Scale Dynamics of Interacting Particles.” Springer, Berlin. Spreeuw, R. J. C., Gerz, C., Goldner, L. S., Phillips, W. D., Rolston, S. L., Westbrook, C. I., Reynolds, M. W., and Silvera, I. F. (1994). Phys. Reu. Lett. 72, 3162. Spreeuw, R. J. C., Pfau, T., Janicke, U., and Wilkens, M. (1995). Europhys. Lett. 32, 469. Stoof, H. T. C. (1991). Phys. Rev. Left. 66, 3148. Stoof, H. T. C. (1992). Phys. Reu. A 45, 8398. Stoof, H. T. C. (1994). Phys. Reu. A 49, 3824. Svistunov, B. V., and Shlyapnikov, G. V. (1990a). Sou. Phys.-JETP (Engl. Trunsl.) 70, 460. Svistunov, B. V., and Shlyapnikov, G. V. (1990b). Sou. Phys.-JETP (Engl. Trunsl.) 71, 71. Taylor, J. R. (1987). “Scattering Theory: The Quantum Theory of Nonrelativistic Collisions.” Wiley, New York. Uhlenbeck, G. E. (1927). “Over Statistische Methoden in de Theorie der Quanta.” Nijhoff, The Hague. Uzunov, I. (1993). “Theory of Critical Phenemena.” World Scientific, Singapore. van Roijen, R., Berkhout, J. J., Jaakkola, S., and Walraven, J. T. M. (1988). Phys. Reu. Lett. 61, 931. Verhaar, B. J. Gibble, K., and Chu, S. (1993). Phys. Reu. A 48, R3429. Walraven, J. T. M., and Hijmans, T. W. (1994). Physica B (Amsferdum]197,417. Wineland, D. J., Drullinger, R. E., and Walls, F. L. Phys. Reu. Letf. 40,1639. Wineland, D. J., Itano, W. M., Berquist, J. C., Bollinger, J. J., and Prestage, J. D. (1985). At. Phys. 9, 3. Wiseman, H. M., and Collet, M. J. (1995). Phys. Lett. A 202, 246. Yang, C. N. (1962). Rev. Mod. Phys. 34, 694. Yasuda, M., and Shimizu, F. (1996). (preprint) You, L., and Holland, M. (1996). Phys. Rev. A 53, R1. You, L., and Lewenstein, M. (1996). J. Res. NIST (in press). You, L., Lewenstein, M., and Cooper, J. (1994). Phys. Reu. A 50, R3565. You, L., Lewenstein, M., and Cooper, J. (1995a). Phys. Rev. A 51, 4712. You, L., Lewenstein, M., and Glauber, R. J. (1995b). Bull. Am. Phys. Soc. [2] 40(4). You, L., Lewenstein, M., Glauber, R. J., and Cooper, J. (1996). Phys. Rev. A . 53, 329. Yvon, J. (1937). “Recherches sur la ThCorie CinCtique des Liquides.” Hermann, Paris. Zeng, H., Zhang, W., and Lin, F. (1995). Phys. Rev. A 52, 2155. Zhang, W. (1993). Phys. Leff.A 176, 225. Zhang, W., and Walls, D. F. (1993). Quantum Opt. 5, 9. Zhang, W., and Walls, D. F. (1994). Phys. Rev. A 49, 3799. Zhang, W., Walls, D. F., and Sanders, B. C. (1994). Phys. Rev. Left. 72, 60. Zhang, W., Meystre, P., and Wright, E. (1995). Phys. Rev. A 52, 498.
Index A
Absolute differential cross section, 16, 20 Acousto-optic modulator, 148-149 Alkali systems, Bose-Einstein condensates, 234-238 Atom counting, 239 Atomic bunching, in the transient regime, 119-120 Atom interferometry, 121, 130-133 Atom optics, 274 Automatic parallelization, 192-193
B Barium, Stark shift, 174 Bogoluibov-Hartree (BH) theory, 235,238 Bose-Einstein condensates (BEC), 222, 225-230 light scattering coherent weak light, 239-245 incoherent light, 250-253 short intense pulses, 245-250 in trapped alkali systems, 234-238 excitations, 238 Ginsburg- Pitaevski-Gross equation, 235-236 negative scattering length, 236-238 Bose-Einstein distribution (BED), 224, 258-259 Boser, 254, 267-273 Bosing, 268, 269 Boson accumulation regime, 271-273 Brillouin scattering, 88
impact excitation, 70-71 Stark shifts, 170-175 Close-coupling plus optical potential (CCO), 14 Coherent atomic recoil laser (CARL), 110, 120 Coherent light scattering, 239-250 Computers parallel SMC method, 191-217 architecture, 191-192, 212-213 load balance, 195 performance, 212-213 programming models, 192-193,209-212 scalability, 194-195 Convergent close-coupling (CCC) calculations, 9, 14 Cooling with accidental degeneracy, 259-264 evaporative cooling, 229 Bose-Einstein condensates, 234-238 quantum dynamics, 266-267 laser cooling recoil-induced effects, 135-136 stimulated Rayleigh scattering, 104 sideband cooling, ideal gas, 256-258 sympathetic cooling, 264-266 Crossover, 122 Cross sections, trimethylaluminum, 214-21 7
D
Data parallelism, 194 Degeneracy, accidental, cooling a gas with, 259-264 Differential cross section, 9, 10, 19, 216-217
C
Calcium, Stark shift, 174 Cesium electron-atom scattering, 8, 20-24, 25 hyperfine structure and isotope shifts, 163-166
E Elastic scattering, 8 heavy targets, 15-24, 25 light targets, 8-15 281
282
INDEX
Electron-atom collisions, 1-3, 80, 83 elastic scattering, 8 cesium, 8, 20-24, 25 helium, 8-11 mercury, 15-20 sodium, 8, 11-15 generalized Stokes parameters, 51-66 generalized STU parameters, 66-70 higher angular momenta, helium, 71-80 impact excitation, 24, 26-28 cesium, 70-71 helium, 28-34 sodium, 34-48 scattering amplitudes, 3-8 Electron-molecule collisions, 183-186 application, 209-217 highly parallel computation, 191-209 theory, 186-190 Evaporative cooling, 229 Bose-Einstein condensates, 234-238 quantum master equation, 266-267
F Fabry-Perot etalon, laser atomic beam spectroscopy, 142, 145, 151 Far-off resonance dipole trap (FORT), 264, 266 First Born approximation, 31 Free electron laser, 110 Frequency-modulated lasers, 148-152
G Generalized diffraction theory, 245 Generalized Stokes parameters, 51-66 Generalized STU parameters, 66-70 Ginsburg-Pitaevski-Gross (GPG) equation, 235-236
H Helium Bose-Einstein condensate, 228 electron-atom scattering, 8-11 impact excitation, 28-34, 71-80 Hydrogen, Bose-Einstein condensate, 228
Hyperfine structure, 152-153 cesium, 163-166 sodium, 158-163 ytterbium, 153-158
I
Impact excitation, 24, 26-28 heavy targets, 48-71 light targets, 28-48 Incoherent light scattering, 250-253 Integral elastic electron cross section, 214-217 Interferometry atomic interferometry, 121, 133 Fabry-Perot etalon, 145-146, 151 Ramsey-BordE matter wave interferometer, 128-130 Inversionless lasing, of cold atoms, 133-135 Isotope shifts, 152-153 cesium, 163-166 sodium, 158-163 ytterbium, 142-143,144,147-148,153-158
L Laser atomic beam spectroscopy, 142-152 Laser cooling recoil-induced effects, 135-136 stimulated Rayleigh scattering, 104 Lasers coherent atomic recoil laser (CARL), 110, 120 free electron laser, 110 frequency-modulated lasers, 148-152 inversionless lasing of cold atoms, 133-135 light scattering, 245-250 magneto-optical traps, 229, 264 recoil-induced inversionless lasing of cold atoms, 133-135 Laser spectroscopy, 142-148 hyperfine structure and isotope shifts, 152-166 optical modulators, 148-149 Stark shifts, 166-179 Light scattering, 88, 236-237, 239 Brillouin scattering, 88 coherent weak light scattering, 239-245
283 incoherent light, 250-253 Raman scattering, 88 Rayleigh scattering, 88 recoil-induced resonance, 109-137 short intense pulses, 245-250 stimulated Rayleigh scattering, 90-109 stimulated scattering, 88 Lithium, Stark shift, 174
M Magneto-optical traps, Bose-Einstein condensate, 229, 264 Many-body theory, 223, 224, 253, 275 Massively parallel processors (MPPs), 191-192,217 electron-molecule collisions, 185, 191-195 Master equation (ME), 253-259 many-body theory, 224 nonlinear optics, 274 sideband cooling of ideal gas, 256-258 sympathetic cooling, 265 Mercury electron-atom scattering, 15-20 impact excitation, 48-70 Message passing, 193
N Network of workstations (NOWs), 191 Nonlinear atom optics (NAO), 273-274 Nonlinear optics, 236-238, 273-274 Nonlinear Schrodinger equation (NLSE), atoms with negative scattering length, 236-237
0
Optical lattices, stimulated Rayleigh scattering, 104-107 Optical modulators, 148-149 Optical potential, weak light scattering, 242 Optical Ramsey fringes, recoil doublet, 125-128 Optical spectroscopy, 142-148
P Pairing theory, 235, 238 Parallel SMC method architecture, 191-192, 212-213 electron-molecule collisions, 185, 191-217 electron scattering computations, 196 input/output, 208-209 integral transformation, 204-206 one- and two-electron integrals, 197-199 parallel integral evaluation, 202-203 program outline, 207-208 quadrature and scaling, 199-202 scaling of transformation step, 206-207 load balance, 195 programming models, 192-194,209-212 scalability, 194-195 Perfect scattering experiment, 2-3, 80 Photorefractive crystals, stimulated Rayleigh resonance, 169 Photorefractive effect, 90 Polaritons, 242 Potassium, Stark shift, 174 Precision laser spectroscopy, 141-142, 152-180 Pump-probe spectroscopy, 88, 89
Q Quantum dynamics boser, 254, 267-274 condensation in cold atoms, 253 generalized Bose-Einstein distribution, 258-259 master equation, 253-259 nonlinear atom optics, 273-274 sideband cooling of ideal gas, 256-258 Quantum field theory of atoms and photons, 223-224 Bose-Einstein condensates, 234-275 Hamiltonian of, 230-234 master equation, 253-259 many-body theory, 224 nonlinear optics, 274 sideband cooling, 256-258 sympathetic cooling, 265 Quantum master equation, 253
284
INDEX
Quantum optics, 223 light scattering, 88, 236-237, 239 coherent weak light, 239-245 incoherent light, 250-253 recoil-induced resonance, 109-137 short intense pulses, 245-250 stimulated Rayleigh scattering, 90-109 many-body theory, 223, 224,253,275 nonlinear atomic optics, 273-274 Quasi-elastic scattering, 88
R Raman scattering, 88 Ramsey-Bordt matter wave interferometer, recoil-induced effects, 128-130 Ramsey fringes, recoil doublet, 125-128 Rayleigh scattering, 88 Recoil doublet, optical Ramsey fringes, 125-128 Recoil-induced effects, 121-122 atom interferometry based on atom recoil, 133 inversionless lasing of cold atoms, 133-135 Kasevich-Chu experiment, 130-132 laser cooling, 135-136 Ramsey-BordC matter wave interferometer, 128-130 recoil doublet of optical Ramsey fringes, 125-128 in saturated absorption spectroscopy, 122-125 Recoil-induced resonance, 90,109-1 10, 137 atomic bunching in the transient regime, 119-120 coherent atomic recoil laser, 120 experimental observation, 115-117 as Raman process between different energy-momentum states, 117-119 as stimulated Rayleigh resonance, 110-1 15 Rotating wave approximation, 223-224 Rubidium, Stark shift, 175
S
Samarium, Stark shift, 175 Saturated absorption spectroscopy, recoil effects, 122-125
Scalability, highly parallel computational techniques, 194-195 Scattering, 88 Scattering amplitudes, 3-8 Scattering equation, weak light scattering, 240-241,243 Scattering experiments, 1-3, 80, 83 elastic scattering, 8 heavy targets, 15-24, 25 light targets, 8-15 highly parallel computation, 196-209 impact excitation, 24, 26-28 heavy targets, 48-71 higher angular moments, 71-80, 81, 82 light targets, 28-48 parallel SMC method, 196-209 input/output, 208-209 integral transformation, 204-206 one- and two-electron integrals, 197-199 parallel integral evaluation, 202-203 program outline, 207-208 quadrature and scaling, 199-202 scaling of transformation step, 206-207 pump-probe spectroscopy, 88, 89 scattering amplitudes, 3-8 Schwinger multichannel (SMC) method, 186, 190, 191 parallel SMC, 196-209 Sideband cooling, ideal gas, 256-258 Single-program multiple-data architecture, 194 Sodium electron-atom scattering, 8, 11-15, 71 hyperfine structure and isotope shifts, 158-163 impact excitation, 34-48 Stark shift, 175 Spectroscopy frequency-modulated lasers, 148-152 laser atomic bean spectroscopy, 142-148 optical spectroscopy, 142-148 precision laser spectroscopy, 141-142, 152-180 pump-probe spectroscopy, 88, 89 saturated absorption spectroscopy, 122- 125 Stark shift, 166-168 barium, 174 calcium, 174 cesium, 170-175
INDEX lithium, 174 potassium, 174 precision data, 173, 177-179 rubidium, 175 samarium, 175 sodium, 175 ytterbium, 168-170 Stimulated Rayleigh scattering, 88,90,91-95, 136-137 laser-cooled atoms, 104 in molecular physics, 107-109 optical lattices, 104-107 recoil-induced resonance as, 110-1 15 in solid-state materials, 109-1 15 stationary two-level atoms, 95-102 sub-Doppler radiative cooling, 102-104 Stimulated Rayleigh wing scattering, 107 Strong light scattering, 245-250 Sub-Doppler nonlinear spectroscopy, 124 Sub-Doppler radiative cooling, 102-104 Symmetric multiprocessor machines (SMPs), 191 Sympathetic cooling, 264-266
285 T
Time orbiting potential, BEC, 229 Trapped alkali systems, Bose-Einstein condensates, 234-238 Trimethylaluminum cross sections, 214-217 electron scattering, 185, 209-212 Two-beam coupling, 109 Two-wave mixing, 109
W Weak light scattering, 239-245
Y Ytterbium isotope shifts, 142-143,144,147-148, 153-158 Stark shift, 168-170
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Contents of Volumes in This Serial Volume 1
Volume 3
Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall andA. T. Amos Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, B. H. Bransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K . Taknyanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J . P. Toennies High-Intensity and High-Energy Molecular Beams, J . B. Anderson, R. P. Andres, and J . B. Fen
The Quanta1 Calculation of Photoionization Cross Sections, A. L. Stewart Radiofrequency Spectroscopy of Stored Ions I: Storage, H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Budick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C. Wolf Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J . Wood
Volume 2
Volume 4
The Calculation of van der Waals Interactions, A . Dalgamo and W. D. Dauison Thermal Diffusion in Gases, E. A. Mason, R. J . Munn, and Francis J . Smith Spectroscopy in the Vacuum Ultraviolet, W. R. S. Garton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A . R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J. de Heer Mass Spectrometry of Free Radicals, S. N. Foner
H. S. W. Massey-A Sixtieth Birthday Tribute, E. H. S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Bates and R. H. G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, P. A . Fraser Classical Theory of Atomic Scattering, A . Burgess and I. C . Perciual Born Expansions, A. R. Holt and B. L. Moiselwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke
287
288
CONTENTS OF VOLUMES IN THIS SERIAL
Relativistic Inner Shell Ionizations, C. B. 0. Mohr Recent Measurements on Charge Transfer, J. B. Hasted Measurements of Electron Excitation Functions, D. W. 0. Heddle and R. G. W. Keesing Some New Experimental Methods in Collision Physics, R. F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M. J . Seaton Collisions in the Ionosphere, A. Dalgamo The Direct Study of Ionization in Space, R. L. F. Boyd
Volume 5 Flowing Afterglow Measurements of IonNeutral Reactions, E. E. Ferguson, F . C. Fehsenfeld, and A . L. Schmeltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. G. Dehmelt The Spectra of Molecular Solids, 0. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A . Ben-Reuuen The Calculation of Atomic Transition Probabilities, R. J . S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations sAs’”p~, C. D. H. Chisholm, A . Dalgamo, and F. R. Innes Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle
Volume 6 Dissociative Recombination, J. N. Barhley and M. A . Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A . S. Kaufman
The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa The Diffusion of Atoms and Molecules, E. A . Mason and T. R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A . E. Kingston
Volume 7 Physics of the Hydrogen Master, C. Audoin, J . P. Schermann, and P. Griuet Molecular Wave Functions: Calculations and Use in Atomic and Molecular Processes, J. C. Browne Localized Molecular Orbitals, Hare1 Weinstein, Ruben Pauncz, and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J . Gerratt Diabatic States of Molecules-Quasi-Stationary Electronic States, Thomas F. O’Malley Selection Rules within Atomic Shells, B. R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H. S. Taylor, and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A . J . Greenfield
Volume 8 Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C . Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C . Y. Chen and AugustineC. Chen
CONTENTSOF VOLUMES IN THIS SERIAL Photoionization with Molecular Beams, R. B. Cairns, Halrtead Harrison, and R. I. Schoen The Auger Effect, E. H. S. Burhop and W. N. Asaad Volume 9
Correlation in Excited States of Atoms, A. W . Weiss The Calculation of Electron-Atom Excitation Cross Sections, M . R. H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi O h The Differential Cross Section of LowEnergy Electron-Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy Volume 10
Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong, Jr. and Serge Feneuille The First Born Approximation, K. L . Bell and A. E. Kingston Photoelectron Spectroscopy, W. C . Price Dye Lasers in Atomic Spectroscopy, W . Lunge, J . Luther, and A. Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett A Review of Jovian Ionospheric Chemistry, WesZey T . Huntress, Jr. Volume 11
The Theory of Collisions between Charged Particles and Highly Excited Atoms, I. C. Perciual and D. Richards Electron Impact Excitation of Positive Ions, M. J. Seaton
289
The R-Matrix Theory of Atomic Process, P. G. Burke and W. D. Robb Role of Energy in Reactive Molecular Scattering: An Information-Theoretic Approach, R. B. Bernstein and R. D. Lmine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M . F. Golde and B. A . Thrush
Volume 12
Nonadiabatic Transitions between Ionic and Covalent States, R. K. Janev Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R. -J. Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M. Broyer, G. Goudedard, J . C. Lehmann, and J . vigue' Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, WIhelm Raith Ion Chemistry in the D Region, George C. Reid
Volume 13
Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M. Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, PaulR. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I. V. Hertel and w. Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J . Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K. Nesbet
290
CONTENTS OF VOLUMES IN THIS SERIAL
Microwave Transitions of Interstellar Atoms and Molecules, W . B. Somewille
Volume 14 Resonances in Electron Atom and Molecule Scattering, D. E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J. Jamieson, and Ronald F. Stewart (e, 2e) Collisions, Erich Weisold and Ian E . McCarthy Forbidden Transitions in One- and TwoElectron Atoms, Richard Marrus and Peter J . Mohr Semiclassical Effects in Heavy-Particle Collisions, M. S. Chikf Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in Ion-Atom Collisions, S. V. Bobasheu Rydberg Atoms, S. A . Edelstein and T. F. Gallagher U V and X-Ray Spectroscopyin Astrophysics, A. K. Dupree
Volume 15 Negative Ions, H. S. W. Massey Atomic Physics from Atmospheric and Astrophysical Studies, A. Dalgarno Collisions of Highly Excited Atoms, R. F. Stebbings Theoretical Aspects of Positron Collisions in Gases, J. W . Humberston Experimental Aspects of Positron Collisions in Gases, T. C. Griflth Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion-Atom Charge Transfer Collisionsat Low Energies, J. B. Hasted Aspects of Recombination, D. R. Bates The Theory of Fast Heavy Particle Collisions, B. H . Bransden
Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H . B. Giibody Inner-Shell Ionization, E. H. S. Burhop Excitation of Atoms by Electron Impact, D. W . 0. Heddle Coherence and Correlation in Atomic Collisions, H . Kleinpoppen Theory of Low Energy Electron-Molecule Collisions, P. G . Burke
Volume 16 Atomic Hartree-Fock Theory, M. Cohen and R. P. McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R . Diiren Sources of Polarized Electrons, R. J. Celotta and D. T. Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopyof Laser-Produced Plasmas, M. H . Key and R. J . Hutcheon Relativistic Effects in Atomic Collisions Theory, B. L. Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experiment, E . N. Fortson and L. W e t s
Volume 17 Collective Effects in Photoionization of Atoms, M. Ya. Amusia Nonadiabatic Charge Transfer, D. S. F. Crothers Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Superfluorescence, M. F. H . Schuurmans, Q. H. F. Vrehen, D. Polder, and H. M. Gibbs Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M . G . Payne, C. H . Chen, G. S. Hurst, and G . W . Foltz
CONTENTS OF VOLUMES IN THIS SERIAL Inner-Shell Vacancy Production in Ion-Atom Collisions, C. D. Lin and Patrick Richard Atomic Processes in the Sun, P. L . Dufion and A . E. Kingston
Volume 18
Theory of Electron-Atom Scattering in a Radiation Field, Leonard Rosenberg Positron-Gas Scattering Experiments, Talber? S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, J . Morellec, D. Normand, and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A. S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B. R. Junker Direct Excitation in Atomic Collisions: Studies of Ouasi-One-Electron Systems, N. Anderson and S. E. Nielsen Model Potentials in Atomic Structure, A . Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D. W. Norcross and L. A. Collins Quantum Electrodynamic Effects in FewElectron Atomic Systems, G. W. F. Drake
Volume 19
Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B . H. Bransden and R. K. Janeu Interactions of Simple Ion-Atom Systems, J . T. Park High-Resolution Spectroscopy of Stored Ions, D. J. Wineland, Wayne M. Itano, and R. S. Van Dyck, Jr. Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K. BIum and H. Meinpoppen
291
The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, F. JenZ The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Fuubel Spin Polarization of Atomic and Molecular Photoelectrons, N . A. Cherepkou
Volume 20
Ion-Ion Recombination in an Ambient Gas? D. R. Bates Atomic Charges within Molecules, G. G . Hall Experimental Studies on Cluster Ions, T . D. Mark and A. W. Castleman, Jr. Nuclear Reaction Effects on Atomic InnerShell Ionization, W. E . Meyerhof and J.-F. Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and Dauid A . Armstrong On the Problem of Extreme UV and X-Ray Lasers, I. I. Sobel’man and A. V. Vinogradou Radiative Properties of Rydberg States in Resonant Cavities, S. Haroche and J . M. Ralmond Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction-Rydberg Molecules, J . A . C. Gallas, G. Leuchs, H. Walther, and H . Figger
Volume 21
Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O’Brien, Pierre Meystre, and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen Theory of Dielectronic Recombination, Yukap Hahn
292
CONTENTS OF VOLUMES IN THIS SERIAL
Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu Scattering in Strong Magnetic Fields, M. R . C. McDowell and M. Zarcone Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M. More
Volume 22
Positronium-Its Formation and Interaction with Simple Systems, J. W. Humberston Experimental Aspects of Positron and Positronium Physics, T . C. Grifith Doubly Excited States, Including New Classification Schemes, C. D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H . B. Gilbody Electron-Ion and Ion-Ion Collisions with Intersecting Beams, K. Dolder and B. Pearl Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion-Atom Collisions, R . Anholt and Harvey Gould Continued-Fraction Methods in Atomic Physics, S. Swain
Volume 23
Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R . Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Hany M. Quiney Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D. E . Williams and Ji-Min Yan Transition Arrays in the Spectra of Ionized Atoms, J . Bauche, C. Bauche-Amoult, and M . Klapisch Photoionization and Collisional Ionization of Excited Atoms Using Synchroton and Laser Radiation, F. J . Wuilleumier, D. L. Ederer, and J.L. Picque'
Volume 24
The Selected Ion Flow Tube (SIDT): Studies of Ion-Neutral Reactions, D. Smith and N. G. A d a m Near-Threshold Electron-Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S. J . Smith and G . Leuchs Optical Pumping and Spin Exchange in Gas Cells, R. J . Knize, Z. Wu, and W . Happer Correlations in Electron-Atom Scattering, A. Crowe
Volume 25
Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgarno: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements. Thomas M. Miller and Benjamin Bederson Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson Differential Scattering in He-He and He+-He Collisions at KeV Energies, R. F. Stebbings Atomic Excitation in Dense Plasmas, Jon C . Weisheit Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M. Sando and Shih-I Chu Model-Potential Methods, G . Laughlin and G. A. Victor Z-Expansion Methods, M. Cohen Schwinger Variational Methods, Deborah Kay Watson Fine-Structure Transitions in Proton-Ion Collisions, R. H . G . Reid Electron Impact Excitation, R. J. W. Henry and A. E. Kingston
CONTENTS OF VOLUMES IN THIS SERIAL Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher The Numerical Solution of the Equations of Molecular Scattering, A. C . Allison High Energy Charge Transfer, B . H . Bransden and D. P. Dewangan Relativistic Random-Phase Approximation, W . R. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G. W. F. Drake and S. P. Goldman Dissociation Dynamics of Molecules, T. Uzer
Polyatomic
Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F. van Dishoeck The Abundances and Excitation of Interstelllar Molecules, John H . Black
Volume 26
Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S. Stein Electron Capture at Relativistic Energies, B. L . Moiseiwitsch The Low-Energy, Heavy Particle Collisions -A Close-Coupling Treatment, Mine0 Kimura and Neal F. Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V. Sidis Associative Ionization: Experiments, Potentials, and Dynamics, John Weiner, FranGoise Masnou-Sweeuws, and Annick Giusti-Suzor On the p Decay of I8’Re: An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen, Leonard Rosenberg, and Lany Spruch
Progress in Low Pressure Mercury-Rare Gas Discharge Research, J . Maya and R. Lagushenko
293
Volume 27
Negative Ions: Structure and Spectra, Dauid R . Bates Electron Polarization Phenomena in Electron-Atom Collisions, Joachim Kessler Electron-Atom Scattering, I. E. McCarthy and E. Weigold Electron-Atom Ionization, I. E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, V. I. Lengvel and M. I. Haysak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule Volume 28
The Theory of Fast Ion-Atom Collisions, J . S. Briggs and J . H. Macek Some Recent Developments in the Fundamental Theory of Light, Peter W . Milonni and Surendra Singh Squeezed States of the Radiation Field, Khalid Zaheer and M. Suhail Zubairy Cavity Quantum Electrodynamics, E . A. Hinds Volume 29
Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L . W. Anderson Cross Sections for Direct Multiphoton Ionization of Atoms, M. V. Ammosou, N. B. Delone, M . Yu. Iuanou, I. I. Bondar, and A. V. Masalov Collision-Induced Coherences in Optical Physics, G. S. Aganval Muon-Catalyzed Fusion, Johann Rafelski and Helga E . Rafelski Cooperative Effects in Atomic Physics, J. P. Connerade Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J . H. McGuire
294
CONTENTS OF VOLUMES IN THIS SERIAL
Volume 30 Differential Cross Sections for Excitation of Helium Atoms and Helium-Like Ions by Electron Impact, Shinobu Nakazaki Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and 1. C. Nickel The Dissociative Ionization of Simple, Molecules by Fast Ions, Colin J . Latimer Theory of Collisions between Laser Cooled Atoms, P. S. Julienne, A. M. Smith, and K . Burnett Light-Induced Drift, E. R. Eliel Continuum Distorted Wave Methods in Ion-Atom Collisions, Derrick S. F. Crothers and Louis J . Dub6
Volume 31 Energies and Asymptotic Analysis for Helium Rydberg States, G. w. F. Drake Spectroscopyof Trapped Ions, R. C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H . Walther Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, Rudolf Duren, and Jacques Robert Atomic Physics and Non-Maxwellian Plasmas. Michkle Lamoureux
Volume 32 Photoionization of Atomic Oxygen and Atomic Nitrogen, K. L. Bell and A. E. Kingston Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B. H . Bransden and C. J . Noble Electron-Atom Scattering Theory and Calculations, P. G. Burke Terrestrial and Extraterrestrial H; , Alexander Dalgamo Indirect Ionization of Positive Atomic Ions, K. Dolder
Quantum Defect Theory and Analysis of High-Precision Helium Term Energies, G. W. F. Drake Electron-Ion and Ion-Ion Recombination Processes, M. R. Flannery Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H . B. Gilbody Relativistic Electronic Structure of Atoms and Molecules, I. P. Grant The Chemistry of Stellar Environments, D. A. Howe, J . M. C. Rawlings, and D. A. Williams Positron and Positronium Scattering at Low Energies, J. W . Humberston How Perfect are Complete Atomic Collision Experiments?, H . Kleinpoppen and H . Hamdy Adiabatic Expansions and Nonadiabatic Effects, R. McCarroll and D.S. F. Crothers Electron Capture to the Continuum, B. L . Moiseiwitsch How Opaque Is a Star? M. J . Seaton Studies of Electron Attachment at Thermal Energies Using the Flowing AfterglowLanemuir Technique, David Smith and Pairik Spang1 Exact and Approximate Rate Equations in Atom-Field Interactions, S. Swain Atoms in Cavities and Traps, H. Walther Some Recent Advances in Electron-Impact Excitation of n = 3 States of Atomic Hydrogen and Helium, J . F. Williams and J . B. Wang
Volume 33 Principles and Methods for Measurement of Electron Impact Excitation Cross Sections for Atoms and Molecules by Optical Techniques, A. R. Filippelli, Chun C. Lin, L. W. Andersen, and J . W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Analysis of Scattered Electrons, S. Trajmr and J . W . McConkey
CONTENTS OF VOLUMES IN THIS SERIAL Benchmark Measurements of Cross Sections for Electron Collisions: Electron Swarm Methods, R . W . Crompton Some Benchmark Measurements of Cross Sections for Collisions of Simple Heavy Particles, H. B. Gilbody The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Bany I. Schneider Analytic Representation of Cross-Section Data, Mitio Inokuti, Mineo Kimura, M . A. Dillon, Isao Shimamura Electron Collisions with N,, 0, and 0: What We Do and Do Not Know, Yukikazu Itikawa Need for Cross Sections in Fusion Plasma Research, Hugh P. Summers Need for Cross Sections in Plasma Chemistry, M. Capitelli, R . Celiberto, and M . Cacciafore Guide for Users of Data Resources, Jean W. Gallagher Guide to Bibliographies, Books, Reviews, and Compendia of Data on Atomic Collisions, E . W . McDaniel and E . J. Mansky Volume 34
Atom Interferometry, C. S. Adams, 0. Carnal, and J . Mlynek Optical Tests of Quantum Mechanics, R. Y. Chiao, P. G. Kwiat, and A. M . Steinberg Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andreas Buchleifner Measurements of Collisions between LaserCooled Atoms, Thad Walker and Paul Feng The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas, J . E . Lawler and D. A. Doughty Polarization and Orientation Phenomena in Photoionization of Molecules, N . A. Cherepkov Role of Two-Center Electron-Electron Interaction in Projectile Electron Excitation and Loss, E . C. Montenegro, W . E. Meyerhof, and J . H . McGuire
295
Indirect Processes in Electron Impact Ionization of Positive Ions, D . L . Moores and K . J . Reed Dissociative Recombination: Crossing and Tunneling Modes, David R. Bates
Volume 35
Laser Manipulation of Atoms, K . Sengstock and W . Ertmer Advances in Ultracold Collisions: Experiment and Theory, J . Weiner Ionization Dynamics in Strong Laser Fields, L . F. DiMauro and P. Agostini Infrared Spectroscopy of Size Selected Molecular Clusters, U.Buck Femtosecond Spectroscopy of Molecules and Clusters, T . Baumer and G. Gerber Calculation of Electron Scattering on Hydrogenic Targets, I. Bray and A. T . Stelbovics Relativistic Calculations of Transition Amplitudes in the Helium Isoelectronic Sequence, w. R. Johnson, D. R. Planfe, and J . Sapirstein Rotational Energy Transfer in Small Polyatomic Molecules, H. 0. Everitt and F . C. De Lucia
Volume 36
Complete Experiments in Electron-Atom Collisions, Nils Overgaard Andersen, and Klaus Bartschat Stimulated Rayleigh Resonances and Recoil-Induced Effects, J.-Y. Courtois and G. Gtynberg Precision Laser Spectroscopy Using Acousto-Optic Modulators, W . A. van Wijngaarden Highly Parallel Computational Techniques for Electron-Molecule Collisions, Carl Winstead and Vincenf McKoy Quantum Field Theory of Atoms and Photons, Maciej Lewensfein and Li You
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