Advances in Atomic, Molecular, and Optical Physics, Volume 42

Advances in

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 42

Editors BENJAMIN BEDERSON New York University New York, New York HERBERT WALTHER Max-Planck-lnstitut fur Quantenoptik Garching bei Munchen Germany

Editorial Board P. R. BERMAN University of Michigan Ann Arbor, Michigan M. GAVRILA E 0.M. Instituut voor Atoom-en Molecuulfysica Amsterdam, The Netherlands M. INOKUTI Argonne National Laboratory Argonne, Illinois

W. D. PHILLIPS National Institute for Standards and Technology Gaithersburg, Maryland

Founding Editor SIRDAVIDR. BATES

Supplements 1. Atoms in Intense Laser Fields, Mihai Gavnla, Ed. 2. Cavity Quantum Electrodynamics, Paul R. Berman, Ed. 3. Cross Section Data, Mitio Inokuti, Ed.

ADVANCES IN

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by

Benjamin Bederson DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK

Herbert Walther UNIVERSITY OF MUNICH AND MAX-PLANK-INSTITUT QUANTENOPTIK MUNICH, GERMANY

Volume 42

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Contents CONTRIBUTORS

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

ix

Fundamental Tests of Quantum Mechanics Edward S . Fry and Thomas Walther I . Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Bell Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111. Loopholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IV. Experiments Based on Photons . . . . . . . . . . . . . . . . . . . . . . . . . V. Experiments Based on Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Contrasts between Photon- and Atom-Based Experiments . . . . . . VII. Greenberger-Home-Zeilinger. . . . . . . . . . . . . . . . . . . . . . . . . . . VIII . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX . Outlook and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

4 6 8 9 21 23 24 24 25 25

Wave-Particle Duality in an Atom Interferometer Stephan Durr and Gerhard Rempe 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Bragg Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. The Atom Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Delayed Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Storing Which-Way Information . . . . . . . . . . . . . . . . . . . . . . . . VII . Interferometer with Which-Way Information . . . . . . . . . . . . . . . . VIII . Quantum Erasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX . Incomplete Which-Way Information . . . . . . . . . . . . . . . . . . . . . X . Wigner Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI1. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI11. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 33 36 42 48 49 54 57 60 65 69 69 69

Atom Holography Fuji0 Shimizu I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Atomic Beam Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Design of Thin-Film Hologram . . . . . . . . . . . . . . . . . . . . . . . . . IV. Quality Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Other Possible Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 74 77 89 90 92

vi

Contents

VII . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92 92

Optical Dipole Traps for Neutral Atoms Rudolf Grimm. Matthias Weidemuller, and Yurii B . Ovchinnikov I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Optical Dipole Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I11. Experimental Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Red-Detuned Dipole Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Blue-Detuned Dipole Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 97 108 123 145 162 165 165

Formation of Cold (T 5 1 K) Molecules J . T Bahns. P. L. Gould. and W C. Stwalley

...................................... ......................... 111. Optical Cooling Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Formation of Cold Molecules Via Laser-Induced Photo-association . . V. Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . VI . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction

11. Nonoptical Cooling Techniques

172 191 195 206 219 219 220

High-Intensity Laser-Atom Physics C. J . Joachain. M . Dorr, and N . J . Kylstra I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Multiphoton Processes in Atoms and Ions . . . . . . . . . . . . . . . . . . 111. Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Conclusions and Future Developments . . . . . . . . . . . . . . . . . . . . V. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

226 227 237 278 279 279

Coherent Control of Atomic. Molecular. and Electronic Processes Moshe Shapiro and Paul Brumer I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11. Preparation and Dynamics of a Continuum State

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

I11. Bichromatic Control of a Superposition State . . . . . . . . . . . . . . . IV. The Coherent Control Principle . . . . . . . . . . . . . . . . . . . . . . . . . V. Weak-Field Coherent Control: Unimolecular Processes . . . . . . . . . VI . Strong-Field Incoherent Interference Control . . . . . . . . . . . . . . . . VII. Coherent Control of Bimolecular Processes . . . . . . . . . . . . . . . . . VIII. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287 289 296 304 304 325 332 342

Contents

IX. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii 343 343

Resonant Nonlinear Optics in Phase Coherent Media M. D. Lukin, l? Hemmer, and M . 0. Scully I. Introduction.

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

11. Review of Atomic Coherence Studies. . . . . . . . . . . . . . . . . . . . . 111. Resonant Enhancement of Nonlinear Optical Processes:

IV. V. VI. VII. VIII. IX. X. XI. XII.

Theconcept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Nonlinear Optical Enhancement . . . . . . . . . . . . . . . . Resonant Enhancement of Four-wave Mixing Processes . . . . . . . . Physical Origin of Nonlinear Enhancement . . . . . . . . . . . . . . . . . Optical Phase Conjugation in Double-A Systems . . . . . . . . . . . . . For Optical Aberration Correction in Double-A Medium. . . . . . . . Nonlinear Spectroscopy of Dense Coherent Media . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347 350 353 357 363 367 37 1 375 376 3 82 384 384

The Characterization of Liquid and Solid Surfaces with Metastable Helium Atoms H. Morgner I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Reactions of Metastable Helium Atoms with Matter:

General Survey.

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

III. Quantitative Evaluation of MIES Data . . . . . . . . . . . . . . . . . . . .

IV. Discussion of Selected Systems . . . . . . . . . . . . . . . . . . . . . . . . . v. summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

387 393 438 449 482 483 483

Quantum Communication with Entangled Photons Harald Weinfirter I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11. Entanglement: Basic Features . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Quantum Communication with Entangled States . . . . . . . . . . . . .

IV. The Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. The Quantum Communication Experiments . . . . . . . . . . . . . . . . VI. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

490 492 494 503 519 528 530 530

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

535

CONTENTS OF VOLUMESINTHISSERIES . . . . . . . . . . . . . . . . . . . . . . .

545

SUBJECT INDEX

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Contributors Numbers in parentheses indicate pages on which the author’s contributions begin.

J. T. BAHNS(172), Department of Physics, University of Connecticut, Storrs, CT 06269 PAULBRUMER (287), Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Canada M5S 1Al

M. DORR(226), Department de Physique, Universitk du Louvain, B-1348 Louvain-la-Neuve, Belgium STEPHAN DURR(29), Fakultat fur Physik, Universitat Konstanz, 78457 Konstanz, Germany EDWARD FRY (l), Department of Physics, Texas A & M University, College Station, TX 77843-4242 P. L. GOULD(172), Department of Physics, University of Connecticut, Storrs, CT 06269

RUDOLFGRIMM(95), MPI dur Kernphysik P.O. Box 103980, 69029 Heidelberg, Germany P. HEMMER (347), Max-Planck-Institut fur Quantenoptik, 85748 Garching, Germany C. J. JOACHAIN (226), Physique Theorique, Universitk Libre de Bruxelles, Campus Plaine CP 227, Boulevard du Triomphe, B-1050 Bruxelles, Belgium N. J. KYLSTRA(226), Optics Section, Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom

M. D. LUKIN(347), Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138 H. MORGNER (387), Institut fur Experimentalphysik, Universitat WittenHerdecke, Stockumer Strafe 10, D-58448 Witten, F.R. Germany ix

X

Contributors

YURII B. OVCHINNIKOV (99, National Institute of Standards and Technology, PHY B167, Gaithersburg, M D 20899 GERHARD REMPE(29), Fakultat fur Physik, Universitat Konstanz, 78457 Konstanz, Germany M. 0. SCULLY(347), Sensors Directorate, Air Force Research Laboratory, Hanscom AFB, MA 01731 MOSHESHAPIRO (287), Chemical Physics Department, The Weizmann Insitute of Science, Rehovot, Israel 76100 FUJIOSHIMIZU (73), Institute for Laser Science, University of ElectroCommunications, Chofu-shi 182-8585, Japan (172), Department of Physics, University of Connecticut, W. C. STWALLEY Storrs. CT 06269 (l), Department of Physics, Texas A & M University, THOMAS WALTHER College Station, TX 77843-4242

MATTHIASWEIDEMULLER (93, Max-Planck-Institut fur Kernphysik, 69029 Heidelberg, Germany HARALD WEINFURTER (490), Sektion Physik der Universitaet Muenchen, 80799 Muenchen, Germany Max-Planck-Institut fur Quantenoptik, 85748 Garching, Germany

Advances in

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 42

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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 42

FUNDAMENTAL TESTS OF QUANTUM MECHANICS EDWARD S. FRY and THOMAS WALTHER Physics Department, Texas A&M University, College Station, Texas

I. Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bell Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loopholes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiments Based on Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiments Based on Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Entanglement in the Micromaser. .................... B. no-Particle Entanglement Based on the Photo-Dissociation of a Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . ...... I . Generation of the Entangled State . . . . . . . . . . . . . . . . . . . . . 2. Conditional Detection Probability g . . . . . . . . . . . . . . . . . . . . 3. Spin Analysis and Detection of the Hg Atoms . . . . . . . . . . . . 4. Quantum-Mechanical Prediction . . . . . . . . . . . . . . . . . . . . . . 5. Enforcement of the Locality Condition. . . . . . . . . . . . . . . . . . VT. Contrasts Between Photon- and Atom-Based Experiments. . . . . VII. Greenberger-Home-Zeilinger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Outlook and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. 111. IV. V.

1 4 6 8 9 10

12 13 15 15 21 21 21 23 24 24 25 25

Abstract: We review recent advances in fundamental tests of quantum mechanics, particularly those involving entangled Einstein-Podolsky-Rosen (EPR) pairs. We will only briefly mention the latest down-conversion experiments and will instead concentrate on the atom-based tests. Specifically, we will discuss the first atomic entanglement in the micromaser; we will then provide an extensive review of an experiment based on photodissociation of the dimer lWHg2.The isotope 199Hghas nuclear spin I = 1/2, and those molecules in a state with total nuclear spin I = 0 are selected. It is an exact experimental realization of Bohm's well-known version of the EPR gedankenexperiment. This experiment will test the Bell inequalities in a regime very different from those using photons.

I. Historical Overview Within only 30 years at the radically changed the way mechanics had in predicting supported the new theory.

beginning of this century, quantum mechanics we view physics. The success that quantum experimental results was startling and strongly However, many were concerned about the 1

Copyright 1 0 2000 by Academic Press All rights of reproduction in any furm reserved. ISBN 0-12-003842-O/ISSN Io.19-2.50XIO $30.00

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Edward S. Fry and Thomas Walther

implications of quantum mechanics, specifically regarding action at a distance, and its general interpretation. These concerns were voiced by Albert Einstein, Boris Podolsky, and Nathan Rosen (who are generally referred to in this context as EPR) in their famous paper of 1935 (Einstein et al., 1935), in which they argued that quantum mechanics was not a “complete” theory. Presumably, additional parameters, for which the term hidden variables ( H v ) was later coined, would be required in order to make the theory complete. Einstein, Podolsky, and Rosen illustrated their concern with a gedankenexperiment using a two-particle system entangled in position and momentum. Bohm’s version of this EPR gedankenexperiment provides a conceptuallynice presentation of the argument; it is based on the entanglement of two spin-I /2 particles (Bohm, 1951), whose wave function can be written

Quantum mechanics predicts that the total spin of this two-particle system is zero, and for measurements of both components in one specific direction, it predicts that the results will be opposite with unit probability. Thus, for a given two-particle system, if particle 1 is measured and found to have spin up (t) in some direction, then particle 2 must necessarily be in a definite state with spin down (I) in that same direction. Furthermore, this knowledge of the spin of particle 2 is obtained without in any way disturbing it; only particle 1 is disturbed by the measurement. Consequently, EPR would argue that the spin of particle 2 must have a “real”, fixed component of spin pointing down in that particular direction. Of course, since the spin of particle 1 could have been measured in any direction, particle 2 should therefore have a corresponding “real” component of spin for every direction. However, quantum mechanics does not permit knowledge of the absolute orientation of an individual spin (all spin components); it teaches us that this is impossible. Quantum mechanics only provides probabilities for the outcomes of measurements of the components of spins in various directions. This situation was untenable in the view of Einstein and many others, and it led to their belief that quantum mechanics was an “incomplete” theory. Specifically, quantum mechanics does not permit knowledge of more than one spin component of a particle, but the EPR gedankenexperiment leads us to conclude that all components are “real” and that it should therefore be possible to determine them. Moreover, to circumvent the EPR argument, we would have to invoke a seemingly instantaneous communication of relative spin orientation between the two particles. Such arguments violated Einstein’s strong belief in causality, and he rejected this action-at-a-distance behavior of quantum mechanics. For an introductory article on action-at-a-distance, see

FUNDAMENTAL TESTS OF QUANTUM MECHANICS

3

Hardy, 1998. For a good tutorial of a different view of the EPR question, see Cantrell and Scully (1978) as well as more recent discussions (Mohrhoff, 1999). The discussions remained purely philosophical in nature until 1964, when John Bell showed (based on Bohm’s classic version of the spin-1/2 particle EPR gedankenexperirnent) that any hidden-variable theory satisfying a physically reasonable condition of locality (LHV theory, e.g., a classical theory) will yield statistical predictions that must satisfy restrictions for certain correlated phenomena (Bell, 1964) such as occur with entangled states [cf. Eq. (l)]. These restrictions are now called Bell inequalities, and they show that the strength of classical statistical correlations is limited. In contrast, quantum mechanics predicts much stronger statistical correlations than such classical theories, and Bell explicitly demonstrated that the quantum-mechanical prediction for the statistical result of correlation measurements on an ensemble of two-particle entangled states can violate the inequalities. In summary, he showed that the classical counterparts of quantum mechanics, the local hidden-variable theories (LHV), always restrict the statistical correlations so that the Bell inequalities are satisfied; whereas the quantum-mechanical predictions for the statistical correlations can violate the Bell inequalities. Thus, for the first time it was possible, at least in principle, to distinguish experimentally between the LHV and quantummechanical pictures. A crucial point is that no particular LHV theory is specified. The test of a Bell inequality is general and leads to discrimination between any LHV and quantum mechanics. The caveat is that the strongest form of the Bell inequalities - the Bell-Clauser-Home (BCH) inequality (Clauser and Horne, 1974) - must be tested. The BCH inequality contains no additional assumptions and therefore provides the strongest possible test. It is remarkable that this equation has actually not yet been tested (cf. Section 111.). The principal problem of experimental tests of these Bell inequalities lies in the fact that experimental imperfections generally preclude a definitive test; that is, they shift the result such that the Bell inequalities are no longer violated. In practice, additional auxiliary assumptions had to be introduced in order to make physically realizable experiments possible with existing technology (Clauser et al., 1969). Specifically, the strong BCH inequality is a ratio of the rate of coincidence events between detectors to the rate of single detection events at each detector. The auxiliary assumptions make it possible to write the inequality as a ratio of coincidence events to coincidence events. Several such experiments, involving correlations between two photons in an atomic cascade or from parametric down-conversion, have been completed (for a review of those experiments, see Pipkin, 1978; Clauser and Shimony, 1978; Duncan and Kleinpoppen, 1988; Chiao et al., 1994; Shih, 1999). They

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Edward S. Fry and Thomas Walther

generally gave results in agreement with quantum mechanics and violated a weaker form of Bell inequalities derived using the auxiliary assumption required for the case of low detection efficiency. In this article we will briefly discuss the strong Bell inequality and what is involved in testing it. We will then summarize the existing loopholes and briefly describe the optical tests performed since the last review of the field by Y. R. Shih (Shih, 1999). We will then discuss experimental tests based on atoms, with particular emphasis on the photo-dissociation of mercury dimers. This is followed by a discussion about principal differences between optical and atomic tests and then by a brief mention of the advances in the field due to the Greenberger-Home-Zeilingerstates. We conclude with a short discussion of some applications in quantum communication and a summary.

11. Bell Inequalities The test of the strong BCH inequality requires the measurement of coincidence rates Rtt ( e l , O * ) for the simultaneous detection of one particle of the entangled pair at detector 1 with spin up (“t”)in the direction 81 and of the other atom at detector 2 with spin up (‘‘T”) in the direction €I2; the singles rates Rir (ei),which are defined as the rate of detection of particles with spin up in the direction €Ii at detector i, where i = 1 or 2, must also be measured. The parameter for which the pair is entangled (i.e., polarization, timeposition, or spin) or the type of particle (i.e., photon or atom) that is entangled is not significant to the argument at this point. However, for simplicity we restrict our discussion to atomic systems with an entanglement of the spin according to Eq. (1). The BCH inequality is formulated in terms of the ratio of coincidence rates to singles rates for four different combinations of angles for the spin measurement (Clauser and Horne, 1974; Clauser and Shimony, 1978). In the present context, the strong BCH inequality can be expressed as

where 81,e; are two values of angles for the first spin analyzer and 02, 0; are two values of angles for the second spin analyzer. The quantum-mechanical predictions, including experimental imperfections for the above-mentioned coincidence and singles rates, can be written (Fry et al., 1995; Walther and

FUNDAMENTAL TESTS OF QUANTUM MECHANICS

5

Fry, 1997b) as follows:

where q is the detector efficiency for the atoms and is assumed to be the same for both detectors. E + and E - are defined as E + = EM E m and E- = EM - E m , respectively, where EM is the transmission of the analyzers for one spin component, and E , is the leakage through the analyzer for the other spin component. E , and cM are positive definite and are assumed to be identical for both analyzers. Specifically, E + and E - are measures of the capability to discriminatebetween the two spin components in a measurement for optimum discrimination, j if - E + = E - = 1. The two detectors are assumed to be symmetric; they have identical geometries, and f is the detector acceptance solid angle; g is the conditional probability that if one of the atoms of the entangled pair enters the aperture of one detector, then the other atom of that pair enters the aperture of the other detector; N is the total number of entangled pairs per unit time. In an actual experiment the goal will be to choose the angles 81,02, 0{, and 0; such that when the quantum mechanical predictions Eqs. (3) and (4)are used in Eq. (2), the resulting SQ, provides a maximum violation of the BCH inequality. We find that for 01,02, el,,and 0; equal to 135", O", 225", and 90", respectively, the prediction is

+

S ~ ~ ( 1 3 5 ~ , 0 ~ , 2 2 5=-qg&+(l+h(:)'). ~1, 9 0 ~ ) 2

(5)

In order to obtain the largest possible violation of the BCH inequality by the quantum-mechanical prediction, the right-hand side of Eq. ( 5 ) must exceed 1, the parameters q, g, and EM should be as large as possible, and E, should be as small as possible. Finally, it should be noted that any test of the Bell inequalities is statistical in nature because the measured values are ratios of coincidence count rates to singles count rates. Moreover, the measured values must be determined with a sufficiently small error that the error in the measured value of S is much less than the magnitude of the violation (the amount by which the predicted SQM exceeds 1). Specifically, it is not the magnitude of the violation, but rather its comparison to the error limits of an experimentallydetermined value of S, that is a measure of the significance of the violation. As an aside, note that some

6

Edward S. Fry and Thomas Walther

experimental parameters must be measured in order to evaluate SQM,and therefore SQMwill generally have errors associated with it.

111. Loopholes As discussed earlier, the strongest form of the Bell inequalities has not yet been tested; no experiment to date has been entirely free of loopholes (Pearle, 1970; Clauser and Home, 1974; Clauser and Shimony, 1978; Zeilinger, 1986; Santos, 1996). Specifically, one or more of the following loopholes were present: (1) the spatial correlation loophole, (2) the detection efficiency loophole, and (3) the enforcement of locality (the communication loophole). The first loophole - the spatial correlation loophole - relates to the ideal case in which the two-particle entangled state is generated by a two-body decay that involves only the two particles that are entangled. In this case, momentum conservation ensures a strong spatial correlation that is necessary because the experimenter must make sure that both particles of an entangled pair end up in the detectors. In the experiments based on the two-photon cascade in an atom, the atom acts as a third body that can take some of the momentum and thus significantly reduce the spatial correlation between the two photons. Santos and coworkers published a proposal to close this loophole by repeating the atom cascade experiments and closing the spatial correlation loophole by measuring the atomic recoil (Huelga et al., 1994). But this loophole has been effectively closed by the experiments performed several years ago in which correlated photons were produced by parametric down-conversion. The second loophole - the detection efficiency loophole - originates from experimental imperfections. Low detection efficiencies reduce the observable correlation, because the lower the detection efficiency, the lower the probability that both partners of an entangled pair will be detected. In particular, the BCH inequality, Eq. (2), involves ratios of coincidence rates to singles rates. If the detection efficiency is loo%, the ratio of coincidence rates to singles rates has a maximum of unity, but as the detection efficiency decreases below loo%, the maximum value of this ratio decreases rapidly. In fact, for low detection efficiencies, the ratio is so low that the quantum mechanical predictions for SQ, can never violate the inequality. Hence an additional assumption known as the fair sampling assumption was introduced. Physically, the assumption is that the subensemble of those pairs of particles for which both partners are detected behaves in exactly the same way as the remaining subensemble of those pairs for which at least one partner is missed in the detection process as a result of insufficient detection efficiencies. This leads to weaker forms of the Bell inequalities that involve ratios of coincidence rates to coincidence rates; that is. the coincidence rates are normalized to

FUNDAMENTAL TESTS OF QUANTUM MECHANICS

7

some maximum coincidence rate. All tests of the Bell inequalities to date have tested these weaker forms. Unfortunately, it is possible to construct LHV theories that show the strong correlations of quantum mechanics in the limit of low detection efficiencies (Pearle, 1970; Clauser and Horne, 1974). Hence a rigorous test of the Bell inequalities requires high detection efficiencies. From Eq. (5) it can be shown a minimum efficiency q = 0.82 is necessary for a rigorous test, assuming that all other experimental parameters are optimized (i.e., g = E + = E- = 1). However, it has also been shown that for asymmetric entangled states (states for which the two components have weights other than 1/&) an efficiency of 0.67 would be sufficient (Eberhard, 1993). From this discussion, it is clear that atoms could provide an ideal basis for these experiments, because photo-ionization schemes provide high detection efficiencies for nearly all elements of the periodic table (Hurst et al., 1979). As pointed out by Santos (1996), and as is evident in the quantum-mechanical predictions [Eq. ( 5 ) ] , high detection efficiency alone is not enough to test the BCH inequality. High detection efficiencies must be accompanied by a high probability of actually detecting both particles of an entangled pair in their respective detectors - that is, a high conditional probability g of finding particle 2 in detector 2 provided that particle 1 entered detector 1. The third loophole - enforcement of the locality condition (also known as the communication loophole) - requires that the correlation measurement on the two particles be completely independent at each analyzer. Specifically, it is necessary to guarantee that there is no communication between the two analyzers the two spin analyses: they must be outside each other's space-time light cone. This means that the time it takes for selection of a spin analysis direction and the analysis must be short compared to the time required for a light signal to propagate from one analyzer/detector to the other. This requires a relatively large separation of the analyzer/detectors. The first attempt to enforce the locality condition was pursued by Aspect and coworkers (Aspect et al., 1982a). Their experimental scheme did not allow a strict enforcementof the locality condition, because they used periodic switching of the polarizer orientation rather than random orientations; also the choice of the switching period was unfortunate (Zeilinger, 1986). Very recently, however, Zeilinger et al. succeeded in closing this loophole (Weihs et al., 1998). They utilized a down-conversion source and coupled the entangled photon pairs into two fibers that led to two detector arrangements that were separated by 400 m. At each detector, fast electro-optic modulators rotated the plane of polarization between two randomly selected values, 0" and 45".This is equivalent to randomly rotating the subsequent analyzing polarizer. The principle of the setup is very similar to Aspect's experiment, with the important difference that the switching was random. The type of process used to determine the random number is of significance if the locality

8

Edward S. Fry and Thomas Walther

loophole is to be closed. A pseudo-random, computer-generated number is certainly not enough. Consequently,Zeilinger et af. used a light-emitting diode and a beamsplitter as their random generator. Zeilinger's success represents a very important step in the clarification of the EPR argument. However, because the experiment was performed with a detection efficiency of approximately 5%, the detection loophole is not yet closed. The ultimate test must be one that enforces locality and simultaneously closes all other loopholes associated with previous tests. The flight time of the particles to their respective detectors does not influence the arguments in the discussion of the locality condition. It is solely the communication between the two analyzer/detectors that is important. However, the time of flight for the particles is important to a discussion of the lifetime of the entangled state. It is conceivable that the strong correlations predicted by quantum mechanics exist only for a limited time. This is the reason why the experiment by Gisin and coworkers (Tittel et al., 1998) represents a major step forward. They performed a Bell inequality test of the Franson type (Franson, 1989) with time-position entanglement, where the two detectors were separated by more than 10 km.They measured correlations consistent with quantum mechanics after taking into account the losses in the fiber due to imperfect transmission. The large separation requires a minimum lifetime of around 30 ps for the entangled state; this is very long compared to the 40 ns in Zeilinger's experiment. Herein lies one of the benefits of using atoms. Atom speeds are generally much less than the velocity of light, and even with relatively small separations of the detectors, a very long lifetime (milliseconds) of the entanglement can be tested via the Bell inequalities.

IV. Experiments Based on Photons As discussed earlier, most experiments performed so far have been based on photons in polarization or time-position entanglement. The first experiments to give agreement with quantum mechanics were those of Freedman and Clauser (1972), Clauser (1976), and Fry and Thompson (1976). These were followed by the well-known experiments by Aspect et af.(1 982a, 1982b). The first experiments employing two-photon parametric down-conversion were those of Alley and Shih (1987), followed by those of Ou and Mandel(l988). Some of the more recent tests (cf. Fig. 1) have been performed by Kwiat, Zeilinger, and Shih (Kwiat et al., 1995)and by Kwiat et al. (1998). In the most recent experiment, a more refined version of the generation of entanglement via parametric down-conversion was utilized. By combining two thin nonlinear type-I crystals whose nonlinear axes are at 90" each other and using

FUNDAMENTAL TESTS OF QUANTUM MECHANICS

9

FIG. I . Most recent experiments testing the foundations of quantum mechanics were performed with type41 nonlinear down-conversion processes. The uv photons are down-converted into two IR photons of the same wavelength (degenerate case). The figure shows the cones on which the down-converted photons can exit the nonlinear crystal. The top half is polarized horizontally, the lower half vertically. At the intersection of the cones, both polarizations can exist, and hence the photons are entangled.

45" polarized pump light, Kwiat et al. produced the entangled state

1 19)= - ( I H H )

Jz

+expiiSIW))

where the phase iS can be adjusted by tilting the crystals. They achieved the highest production rates of entangled states to date, even surpassing their previous type-II-based source by a factor of 10.

V. Experiments Based on Atoms Experiments based on atoms differ considerably from the photon-based experiments (see also Section VI). For example, entanglement can be produced only by direct manipulation of the atomic degrees of freedom. Several proposals to generate EPR pairs of atoms for experimental tests exist in the literature (Oliver and Stroud, 1987; Freyberger et al., 1996; Lo and Shimony, 1981; Fry et af., 1995). The proposals by Oliver and Stroud and by Freyberger et al. are based on entanglement of atoms in the micromaser. The latter is an extension of the former in which experimentally more realistic conditions, such as nonperfect detection efficiencies and velocity distributions of atoms, are considered. The proposal by Fry et al. (1995) is based on the photo-dissociation of Hg dimers, and the proposal by Lo and Shimony (1981) is based on photo-dissociation of Na dimers.

10

Edward S. Fry and Thomas Walther

FIG. 2. Generic micromasersetup for the generation and detection of entanglement between atoms.

A. ENTANGLEMENT IN THE MICROMASER In a recent article, Englert ef al. (1998) reviewed the entanglement of atoms in a micromaser. We will therefore emphasize only a few key ideas relevant to entanglement in the micromaser and to the experiment of Haroche’s group (Hagley er al., 1997). Entanglement in the micromaser is produced through the interaction of two completely independent Rydberg atoms with a common radiation field in a micromaser cavity. Rydberg atoms are used because their lifetimes are very long and the atom can be treated as a two-level system (two Rydberg levels) interacting with a single mode of the radiation field. In analogy with a spin- 1/2 particle, we designate the higher-energy Rydberg level as It) and the lowerenergy level as 11).A generic experimental arrangement is shown in Fig. 2. Atoms in It) first pass through a classical microwave (Ramsey field 1) field, which prepares them in the superposition state

where 4 I is the phase of the classical field and 81 is the Rabi angle. The state IS,) corresponds to a pure It) or 11) state in another quantization direction. The atoms then pass through a high-Q micromaser cavity, which is resonant with the microwave transition between the two Rydberg levels. Finally, the atoms pass through a second Ramsey field and into a detection system that provides a measurement result or J. for each atom. Consider the entanglement of two atoms initially in the state It) passing successively through the apparatus of Fig. 2. Assume that Ramsey field 1 is off so that atom 1 enters the micromaser cavity in the state It). Suppose that the cavity is in the vacuum state and the interaction strength and time are

FUNDAMENTAL TESTS OF QUANTUM MECHANICS

11

chosen such that the cavity acts as a x/2 pulse; that is, we will find the atom with probability 1/2 in the excited It) or the ground state 11).Correspondingly, the cavity is in the vacuum state (if the atom remains in the excited state) or has one photon stored in it (if the atom underwent a transition to the ground state). We can therefore write the combined atom/cavity state as

Now, suppose Ramsey field 1 is turned on and adjusted so that the second atom is driven to the ground state 11)before entering the cavity. We also choose the interaction with the cavity ( x pulse) such that, if a photon is present in the cavity, the atom undergoes a transition to the excited state It).Now, after atom 2 leaves the cavity, we find the following two-atom/cavity wavefunction,

We see that the cavity part is separable and we are left with an entangled state of the two atoms that is just like that of the two spin-1/2 particles. Practical limitations such as the finite lifetime of the photons in the cavity will result in much smaller correlations than indicated by Eq. (9). Other practical limitations arise from the arrival statistics of the atoms at the micromaser cavity and from imprecise control of the interaction between the atoms and photons that results from variations in the time of flight of atoms through the micromaser. The first experimental demonstration of entanglement between atoms in a micromaser was made by the group of Haroche (Hagley et al., 1997), using atoms in circular Rydberg states. They began with a thermal beam of rubidium atoms and used a laser to optically pump all atoms out of the F = 3 hyperfine level and into other ground state hyperfine levels. By means of a second laser at a 55" angle to the atom beam, they were able to use the Doppler effect to optically pump ground state atoms in a narrow velocity distribution back into the F = 3 level. In the next step, atoms within a thin velocity-segment in the already velocity-selected F = 3 atoms were excited to the required circular Rydberg state using a diode laser to drive the first step of the excitation, followed by radio frequency transitions to the circular Rydberg states (Nussenzveig et al., 1993). The resulting velocity resolution of the atoms in the Rydberg states was f0.4 m/s; also, because the lasers were pulsed, the position of an atom was known to within f l mm, and the interaction time with the fields could be controlled. The entangled state of two atoms was produced as discussed in the previous paragraph, the first atom being excited to Rydberg state IT) and the second to Rydberg state 11) before entering the micromaser cavity.

12

Edward S. Fry and Thomas Walther

The entanglement was verified by measuring P t l , P ~ TPtt, , P l l , where the first (second) subscript corresponds to the result of a measurement on the first (second) particle, e.g., Prl is the probability of finding atom 2 in the ground state 11)when atom 1 can be found in the excited state 1 T). For a pure EPR pair, the probabilities Ptl and P i t should both be 1/2, whereas the Ptt and Pl1 should both be zero. The observed values of Ptl and Plt were both less than 1/2 as a result of three main factors. First, the entanglement is reduced because of the finite lifetime of the photons in the cavity. Second, it was found that the x pulse required to prepare atom 2 in the micromaser cavity was not perfect (it transferred only 80% of the atoms). Third, the detectors showed a relatively high false rate; that is, the 11)detector clicks with a 10% probability even though the atom is in the upper 11) state, and similarly (13%) for the It) detector. The relatively low detection efficiency does not reduce the correlations because the mean number of atoms was low, and the measurements were postselective for coincidences (only those events are registered where both detectors are triggered in the expected time interval between the atoms). In a subsequent elegant experiment, they applied a 7r/2 analyzing pulse (e.g., Ramsey field 2 of Fig. 2) and observed the probabilities PLt and Ptt as a function of the frequency of Ramsey field 2. Their results are consistent with the interpretation that measurement of the first atom of the pair determines “at-a-distance” the state of the spatially separated second atom. This state of the second atom then precesses in Ramsey field 2, and if the frequency of Ramsey field 2 is varied, then the final angle of the second atom varies so that PLTis modulated sinusoidally as Hagley et al. observed. Problems for tests of the Bell inequalities based on the entanglement in the micromaser are the spatial separation of the atoms when they become entangled, and the fact that the entanglement is not generated in one step but requires several steps. Moreover, the detection of the atoms occurs sequentially and would be a problem for the enforcement of Einstein locality. These problems do not arise for experiments in which the two-particle entanglement is produced via the photo-dissociation of a molecule.

B. TWO-PARTICLE ENTANGLEMENT BASEDON DISSOCIATION OF A MOLECULE

THE

PHOTO-

The wavefunction for a diatomic homonuclear molecule consisting of two atoms, each with nuclear spin 1/2, and with total electron and nuclear spin angular momentum zero, can be expressed in terms of separated atom basis states (Fry et al., 1995) as

FUNDAMENTALTESTS OF QUANTUM MECHANICS

13

FIG.3. The Hgz-dissociation-based EPR experiment at Texas A&M University.

where subscripts 1 and 2 label the two atoms, and It) and 11)indicate spin up and spin down, respectively. This state is identical to that of the two spin- 1/2 particles in Bohm’s classic version of the EPR gedunkenexperiment (Bohm, 1951; Clauser and Shimony, 1978). Thus a spatially “separated” entangled state suitable for testing Bell inequalities can be prepared by molecular dissociation. An experimental implementation based on the dissociation of the mercury dimer 199Hg2will now be outlined (see Fig. 3).

1. Generation of the Entangled State Choice of the isotope 199Hgis based on its nuclear spin I = 1/2. A molecule consisting of two 199Hgatoms (a 199Hg2dimer) will be produced in a supersonic beam and will be photo-dissociated via stimulated Raman excitation. Specifically, a laser at M 266 nm drives a transition in the v 58-0 band of the ’EC,. system. A second laser at 355 nm completes the stimulated Raman transition to a continuum level of the ground state and leaves the dissociated atoms with a center-of-mass (CM) kinetic energy of 1.17 eV. Because detection of the two spatially separated Hg atoms will be achieved with pulsed lasers, precise knowledge of their arrival time at the detector is crucial. Not only does the stimulated emission process determine this time very accurately, but the narrow laser linewidth also ensures a narrow distribution of energies in the dimer continuum state and thus leaves the two Hg atoms with a very narrow distribution of kinetic energy. Indeed, the latter contribution to the uncertainty in the velocity toward the detectors is negligible compared to the uncertainties due to the laboratory velocity distribution of the dimers.

14

Edward S. Fry and Thomas Walther

The excitation laser radiation at 266 nm is produced by a frequencytripled Alexandrite laser running at 798 nm. This laser has an intracavity phase modulator that reduces the linewidth by a factor of 4.It has a pulselength of 120 ns and a typical linewidth of 11 MHz (Nicolaescu et al., 1998). We chose Alexandrite for its long pulselength and hence narrow Fourier-transformlimited linewidth. The stimulated transition down to the continuum state is driven by an excimer pumped dye laser operating at 355 nm. The technique to generate the entangled state given by Eq. (lo), involves selecting two 199Hgatoms with a total nuclear spin Z = 0 and is based on the specific symmetry rules of the total wavefunction for a homonuclear diatomic molecule. Because 199Hgis a fermion with nuclear spin 1/2, the 199Hg2dimers must have an antisymmetric total wavefunction with respect to an exchange of the two nuclei in order to conform to the Pauli principle. Based on the symmetry properties of the nuclear, rotational, vibrational, and electronic wavefunctions of 199Hg2dimers in their ground state, it can be shown that the antisymmetric nuclear spin singlet states are associated with symmetric evenN rotational levels, and the symmetric nuclear spin triplet states are associated with antisymmetric odd-N rotational levels (Fry et al., 1995; Herzberg, 1950; Walther and Fry, 1997b), where N is the rotational quantum number. Now, the selection rule for the rotational quantum number in a homonuclear molecular electronic transition is that N can only change by f 1 in a ( ~ u ) ~ C-, 'C; + transition. Because the stimulated Raman process involves two transitions the excitation beam and Raman beam in (Fig. 4), the overall change must be 0, f 2 . Consequently, if the initial state of the dimer has even N, the final continuum state must also have even N. In summary, nuclear singlet states (Z = 0) can be selectively excited using transitions starting with even N. Because of the angular momentum selection rules for the excitation (266 nm)

FIG. 4. Schematic of the preparation of the entanglement between the two Hg atoms. The entanglement is produced by a two-photon Raman process. One laser beam excites the dimer in an excited state, and the second beam pumps the atoms on to a dissociative part of the molecular ground state energy surface. The atoms separate, and the spin analysis can be performed.

FUNDAMENTALTESTS OF QUANTUM MECHANICS

15

and stimulated emission (355 nm) transitions, the final dissociating level of the 'C; ground state must also have even N and hence zero total nuclear spin. The entangled singlet state is thus produced by initially exciting transitions with the 266-nm laser, which start from even-N rotational states. This selection works even in the case when the ( ~ U ) ~excited C ; state is mixed with a 'II state (Walther and Fry, 1997b). The purity of the entangled singlet state will be very high. Since both electronic states involved in the dissociation process are states with angular momentum A = 0, and the total nuclear spin state is I = 0. Any interaction between the electron spins with the nuclear spins of the atoms should be negligible.

2. Conditional Detection Probability g The probability that one atom of a dissociating dimer enters a detector, given that the other atom entered the other detector, is called the conditional probability g; we saw it in Eqs. (3) and (5). It is a function of the size of the dissociation volume (source volume), the angular distribution of the dissociating dimer fragments, the size and position of the detector apertures, and the spread in the velocities of the dissociating fragments. The source volume is the common intersection of the excitation and dissociation (Raman) laser beams with the supersonicdimer beam (cf. Fig. 4). The optimum detectorposition is determined by the vector addition of the initial dimer velocity with the corresponding CM velocity of the dissociating atoms. In principle, g can reach values close to unity. For a fixed detector size and position, the bmaller the spread in the initial velocities of the Hg2 dimers, the larger the conditional probability g. The spread in dimer velocities can be reduced considerably by using the Doppler effect to select spectroscopically only those molecules whose speed lies within a very narrow velocity window. This can be achieved by aligning the 266-nm excitation laser beam (which has a very narrow frequency spread, Av = 33 MHz) so that it nearly copropagates or counterpropagates with the molecular beam. Monte Car10 simulations show that for this laser linewidth, together with the other parameters used in our experimental setup, a value of g > 0.94 can be achieved (cf. Fig. 5 and Fry et al., 1995). Other pairs of atoms originating from the dissociation of different isotopomers will have different velocities toward the detectors and will be suppressed in the detection step; this is the case regardless of whether the other isotopomers have the same or different total mass. 3. Spin Analysis and Detection of the Hg Atoms

The requirements of the spin analysis and detection scheme are high detection efficiency and high discrimination between spin states. The high discrimination and detection efficiency will be achieved using a two-step

Edward S. Fry and Thomas Walther

16

0

0.2

0.4 0.6 0.8 radlus of dissociation region / mm

' 0.92

1

FIG.5. Monte Car10 simulation of the conditional probability g. We repeated the simulation in Fry et al. (1995) with the relevant data of our laser system. Because of restrictions of time of flight, some of the atoms are not detected for large radii of the source volume.

excitation-ionizationprocess. Detection of the Hg+ ion and its photo-electron yields the required high detection efficiency. Immediately following the entrance aperture to each detector, two independent laser beams (253.7 nm and 197.3 nm) illuminate the Hg atoms (cf. Fig. 6 ) .As shown in Fig. 7, the first laser drives a transition from the (6s2)6'So ( F = 1/2) ground state (level 1) to the (6s6p) 63Py ( F = 1/2) state (level 2). ~ The second laser drives a transition from level 2 to the (6p2)6 3 P autoionizing state (level 3). Spin Analysis. The analyzing beams for the two dectors at 253.7 nm lie in parallel planes and are at angles 81, (32 to the +z-axis (cf. Fig. 6 ) . The angles 81, (32 of these 253.7-nm laser beams define the directions in which each atom's nuclear spin component is observed. For a ground state lg9Hg ( F = 1/2) atom, the quantum numbers for the two components of angular momentum in a given direction are mF = &1/2; because J = 0, these are just the components of nuclear spin in that direction. If the 253.7-nm laser beam mF must decrease by 1 in the transition. has left-circular polarization (0-), Thus only ground state atoms for which the projection of the angular momentum (nuclear spin) in the direction of propagation of the left-circular polarized laser beam is mF = + 1/2 can be excited to the 63Py ( F = 1/2) state and subsequently ionized; see Fig. 7. Conversely right-circular polarization could be used to excite the mF = -1/2 state, which could then be ionized.

FLTNDAMENTAL TESTS OF QUANTUM MECHANICS

17

FIG.6. The detection planes are parallel to the dimer beam and the Raman laser used for preparation of the entangled state. The laser at 253.7 nm is the analysis laser, which performs the spin analysis. The quantization axis is the propagation direction of this laser. Ionization is achieved via the 197.3-nm beam.

/

,

, ’ ,/’

;

//,’

,

,,

/ ’

/

’

,’ I

,/ ,

,,

,/

,/

,’/, / /

Ionization limit 197.3 nm

22 GHz

1-

__

-

-

F=3/2 6s6p

Level 2 F=1/2

FIG.7. Photo-ionization scheme used for spin analysis of the Hg atoms. If the Hg atom is in “spin up” (mF = 1/2) with respect to the quantization axis given by the propagation direction of left-circular (0-) polarized 253.7-nm beam, it will be ionized in a two-photon step. Transitions of the “spin down” (mF = - 1/2) are suppressed because of the large detuning from the F = 3/2 state.

Because the transition from the mF = -1/2 to the 63PT (F = 3/2) mF = -3/2 is only detuned by 22GHz, a small fraction of Hg atoms in the spin mF = -1/2 state might still be ionized. This represents the dominant leakage in our analyzer scheme. Imperfect polarization of the 253.7-nm laser beam also leads to a leakage, but this should be negligible.

18

Edward S. Fry and Thomas Walther

Hg Atom Zonizafion. The degree of ionization was calculated by numerically solving the corresponding master equations, assuming Fourier-transformlimited Gaussian pulses with a pulselength of 8 ns (Fry et al., 1995). A maximum value EM = 99.5% can be achieved for the ionization efficiency. The high efficiency of the analyzer/detector is due to the large oscillator strengths of the two transitions and to the favorable ratio of non-radiative to radiative decay rate out of level 3. The 0.5% loss in the ionization process arises mainly from radiative decay out of the intermediate level to the other mF sublevel of the ground state. Because of the circular polarization of the excitation, this other mF sublevel cannot be excited back to the intermediate level. Transitions through the mF = f 3 / 2 hyperfine states are highly suppressed (E,,, < 0.075%), so a very high degree of discrimination between the two spin states is achieved. Crucial to these high efficiencies is the exact timing of the two laser beams. Assuming 8-ns laser pulses, the two laser pulses must be delivered to the detectors within a relative time window of M 4 ns. Otherwise, the achievable ionization rates drop dramatically. In order to minimize the time uncertainty between the two laser pulses, we developed a single flashlamp pumped Ti:Sapphire laser system that produces the two wavelengths simultaneously (Walther et al., 1998).The laser is actually operated as a nanosecond regenerative amplifier that is injection-seeded with two external cavity cw diode lasers at 761.1 nm and 789.9 nm. (The latter operate in a single longitudinal mode and are frequency-stabilized to an external reference.) Because the laser is a regenerative amplifier, it is unnecessary to stabilize the cavity length to match the seed lasers (which would be very difficult for two wavelengths simultaneously). Pulses at both wavelengths are output simultaneously when the cavity is dumped, and we have shown that the relative pulse amplitudes can be adjusted by adjusting the relative intensities of the seed lasers. The relative timing between pulses can be easily adjusted over a few nanoseconds by adjusting the optical path lengths to the detectors. The required 253.7 nm laser source is produced by frequency tripling (double, the sum) the 761.1 nm output. The 197.3 nm laser source is provided by frequency doubling and then two steps of frequency summing.

Hg Atom Defection. After the spin analysis and photo-ionization, the Hg+ and/or the associated photo-electron must be detected with high efficiency and low background. The latter is particularly important because the signal must be reduced to a level at which there is negligible probability of more than one Hg atom in the detection region at a time (otherwise, coincidences between atoms from different dissociating molecules will be observed). The detectors must also have large acceptance angles in order to achieve high values of the conditional probability g (see Section 2) (Fry e f al., 1995).

FUNDAMENTAL TESTS OF QUANTUM MECHANICS

19

Atom detection is via both the resulting ion and the photo-electron. This dual detection has important consequences, because the measurements of singles rates between electron and ion at each detector, and of coincidence rates between detectors, provide independent determinations of the absolute efficiency for electron and ion detection at each detector, as well as of the conditional probability g (Fry et al., 1995; Fry, 1973). These parameters can be internally evaluated from the same data used to test the strong Bell inequality - an important advantage of this scheme. Moreover, other factors like E + and E - can be measured via simple intensity- and polarization-related sets of measurements without relying on quantum-mechanical arguments (Walther and Fry, 1997a). Hg ions will be detected by using four electrostatic lenses first to extract them from the photo-ionization region and then to focus and collimate them to a small-diameterbeam. They are then accelerated (kinetic energy M 25 keV) into an aluminum-coated surface at an angle of incidence of 74" to produce a burst of secondary electrons. The high kinetic energy increases the yield of secondary electrons. For our design, an average of 13 secondary electrons should be produced per ion. Finally, the secondary electrons will be detected with a large input-cone channeltron. The photo-electrons will be extracted from the photo-ionizationregion and guided with the help of electric fields onto a Cu:Be electrode, where secondary electrons will be produced. These will then also be detected using a large input-cone channeltron. The Cu:Be surface should produce, on average, five to eight secondary electrons. The electrodes and input lenses were designed such that photo-electrons originating from the laser beams hitting parts of the detector and other surfaces are suppressed. The present design leads to very high detection efficiencies. The overall probability for detecting an electron or ion is given by

where p ( 0 ) is the probability of detecting no secondary electron; ph(n) is the probability of producing n secondary electrons per ion or electron, when h is the average number of secondary electrons produced; and P c h is the probability for detecting an electron in a channeltron ( p c h M 0.9). pn(n) is given by a Poisson distribution p h ( n ) = $e-' (Benetti et al., 1991). For the high number of secondary electrons produced in our scheme, the sum in Eq. (1 1) is negligible, and the detection probability is reduced from unity only by the probability pn(0) that no secondary electron is generated. As depicted in Fig. 8, overall detection efficiencies close to unity for both

20

Edward S. Fry and Thomas Walther

FIG. 8. Overall efficiency of the Hg detection as a function of secondary electrons produced per electron and Hg ion. We assume an OR decision; that is, the Hg is detected when either the electron or the ion is detected. Our experimental setup should produce more than 5 secondary electrons per electron and 12 secondary electrons per ion.

the electron and the ion should be possible. This has been confirmed by preliminary measurements on the performance of our detector (Fry et al., 1998). An important consequence of the double detection scheme with an OR decision, however, is that background or noise counts must be negligible. Therefore, it is especially important to minimize the Hg partial pressure, but the partial pressure of all other residual gases should also be minimized (an ultrahigh vacuum of < torr). The detector and all surfaces on the line of sight to it must be cooled to liquid nitrogen temperatures to freeze out background Hg atoms (Fry et af., 1995). Finally, it is essential to suppress photo-electrons produced by scattered photons. Nickel plating all metal surfaces eliminates the production of photo-electrons by the 253.7-nm laser beam. Spurious scattering of the 197-nm laser beam must simply be avoided. Of course, for detection efficiencies close to unity, it will be possible to employ an AND detection scheme, which makes background signals, especially from photo-electrons produced by stray light photons, much less important.

FUNDAMENTAL TESTS OF QUANTUM MECHANICS

21

4. Quantum-Mechanical Prediction

The quantum-mechanical prediction SQMof Eq. ( 5 ) can now be estimated by using the quantities g = 0.94, q = 0.99, E + = 0.996, and E- = 0.994, which were derived in the preceding sections. 5 ' ~ ~ ( 1 3 5 ~ , 0 ~ , 2 2 5=~1.112 , 9 0 ~>) 1.

(12)

Thus the BCH inequality is clearly violated by the quantum-mechanical predictions. It should be emphasized that the experiment not only is able to test LHV theories against quantum mechanics but also provides completely independent mechanisms to simultaneously determine the experimental parameters required for the quantum-mechanical prediction (see the preceding section and Fry and Walther, 1996). 5. Enforcement of the Locality Condition

The remaining loophole, the enforcement of Einstein locality (also known as the communication loophole), can be closed by employing electrooptic modulators (EOM). Specifically, the EOM, together with a polarizing beamsplitter, can change the propagation direction of the excitation laser beam and hence the component of spin angular momentum being observed. Rough estimates show that a separation between our detectors of approximately 5 m will be necessary to close the locality loophole in our setup. These estimates include allowances for selecting a random number, switching the EOM, and firing the analysis detection lasers. The distance could be reduced by using picosecond laser systems for the dissociation/ionization process.

VI. Contrasts Between Photon- and Atom-Based Experiments Whenever one is testing a fundamental concept, it is vital to do the study over as wide a range of the parameters as possible. (As an extreme example, one frequently refers to separate laws of conservation of energy and conservation of mass, but we know neither is conserved by itself. In fact, we really have conservation of mass-energy, the conversion between the two being E = mc2; this becomes apparent in studies at high energies.) The experiment with 199Hgdimers described here dramatically extends the parameter range over which Bell inequalities can be tested. With respect to

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Edward S. Fry and Thomas Walther

previous Bell inequality experiments that have been done with photons, the experiment with 199Hgdimers has four different, but related, important aspects. (1) Massive particles versus massless photons. Nonrelativistic massive particles obey the nonrelativistic Schrodinger equation. Photons are very different; see Chapter 1 of Scully and Zubairy (1997) for an excellent discussion. They point out, “The interference and diffraction of matter waves is the essence of quantum mechanics. However the corresponding behavior in light is described by the classical Maxwell equations.” In his classic text, Kramers (1958) says, “one can not speak of particles in a radiation field in the same sense as in the (nonrelativistic) quantum mechanics of systems of point particles.” Clearly, a Bell inequality test with massive particles is in a regime very different from those with photons. ( 2 ) Lifetime of the entangled state. In photon experiments the entangled state generally exists for only a few nanoseconds before being annihilated at the detectors. Even in the recent experiment of Gisin et al. (Tittel et al., 1998), it existed for only 30 microseconds. Because the 199Hgatoms travel relatively slowly compared to the speed of light, the two atom entangled state must continue to exist at large spatial separations for approximately a millisecond in our initial experiment (and for tens of milliseconds in later experiments). This is an increase of several orders of magnitude in the time scale for an entangled state lifetime. ( 3 ) Fermion versus boson. The photon is a boson; a 199Hgatom is a fermion. Thus the new Bell inequality experiment will be done with particles that obey completely different quantum statistics. (4)Inside versus on rhe light cone. Any massive particle must have a velocity less than the velocity of light and must therefore trace out a world line inside the light cone. By contrast, photons travel with the speed of light and must therefore always be on the light cone. Einstein locality plays a crucial role in the Bell inequalities, so experimental tests done well inside the light cone are especially important compared to all the photon tests done on the light cone. In particular, because photons travel with the velocity of light in any reference frame they cannot be strictly localized, and the concept of causality becomes somewhat muddled. There is nothing in quantum mechanics to suggest that any of these four aspects might lead to a classical-like interpretation and to statistical results that satisfy a Bell inequality; for example, there is no suggestion of a time scale or distance scale for the validity of quantum mechanics. On the other hand, there is nothing in non-relativistic classical mechanics to suggest that conservation of energy is not a stand-alone, fundamental law. Thus it behooves us to test the Bell inequalities in very different regimes from those

FUNDAMENTAL TESTS OF QUANTUM MECHANICS

23

of previous studies. The entangled state produced by dissociation of 199Hg2 dimers provides exactly this opportunity.

VII. Greenberger-Horne-Zeilinger A quite different approach to testing the issues brought forward by Einstein, Podolsky, and Rosen was proposed several years ago by Greenberger, Horne, and Zeilinger (GHZ) (Greenberger et al., 1989, 1990). The concept is based on the entanglement of more than two particles. They showed that classical theories cannot mimic the strong correlations predicted by quantum mechanics for many-particle entangled systems. Hence it is unnecessary to test an inequality in these systems; rather, a single measurement of the strong correlation is, in principle, sufficient to disprove LHV theories. A GHZ experiment would be definitive and nonstatistical in the sense that no statistical quantities such as rates must be compared; simple observation of the strong correlations is sufficient. The existence of the strong correlations must, of course, still be indisputably established outside any systematic experimental errors. The main problem with the GHZ approach has been the difficulty of producing an entanglement of three or more particles. One possibility is the above-mentioned micromaser technique. Earlier this year, another approach by a group at Los Alamos (Laflamme et al., 1998) succeeded in producing a GHZ state using NMR. They produced an entanglement between the proton and carbon spins in trichloroethylene. However, the requirement for a test is that the entangled particles be spatially separated; the microscopic separation in a molecule is not sufficient. There exists a proposal by Zeilinger's group (Zeilinger et al., 1997) to generate a three-particle entangled state from two entangled photon pairs produced by nonlinear down-conversion. Using an arrangement of polarizing beamsplitters and h/2 plates as well as a beamsplitter, they combine the photon pairs of each source so that there will be some events in which they measure a coincidence among all four outputs. For such events a GHZ state has been generated. Recently, they succeeded in implementing this scheme (Bouwmeester et al., 1999). But instead of two distinct sources, Zeilinger and coworkers used a single 200-fs uv pulse that occasionally produces two entangled pairs of photons by down-conversion in a BBO crystal. On average, they created and detected two entangled pairs every 150 seconds. This is, however, sufficient to demonstrate the generation of a separated multiparticle entanglement. Unfortunately, they have not yet measured a strong correlation, because proving the existence of the GHZ state results in its annihilation.

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Edward S. Fry and Thomas Walrher

VIII. Applications The properties of entanglement and the physics involved have attracted interest in recent years far beyond the foundations of quantum mechanics. Quantum information, particularly quantum teleportation, quantum cryptography, and quantum computation, rely heavily on the concept of entanglement. In quantum communication schemes, two people, generally referred to as Alice (transmitter) and Bob (receiver), exchange information by the use of quantum entanglement. Recent review articles on quantum information are available; see the review article by Steane (1998) or the collection of articles in Physics World (Rodgers, 1998). Last year, the groups of Zeilinger and DeMartini independently achieved the quantum teleportation of photons (Bouwmeesteret al., 1997;Boschi et af., 1998). Though not comparable to the (unrealistic) teleportation of macroscopic objects in the manner of “Star Trek,” the present experiments demonstrate some startling possibilities. In both experiments, however, the capability of teleportation was subject to restrictions. In the Zeilinger experiment, the successful teleportation was shown by a coincidence measurement of two photo detectors at the emitter (Alice) and receiver ends (Bob). This approach both limited the efficiency of teleportation to 25% and led to an immediate annihilation of the teleported photon, such that no further measurements could be performed on it. In the case of DeMartini’s experiment,the state of the input photon was not arbitrary. A major advance in teleportation was recently made by Kimble and coworkers. They succeeded in teleporting coherent photon states of the radiation field by means of squeezed state entanglement - that is, teleporting states with continuous variables (Furusawa et af., 1998). They achieved a teleportation efficiency of 100% and were able to measure the fidelity of the teleportation by comparing the parameters of the teleported state with the original. Buttler and coworkers have for the first time brought quantum cryptography to the outside world (Buttler et al., 1998). They delivered a quantum cryptography key over the distance of 1 km in a free-space outdoor environment rather than in a fiber/laboratory setup. This is a first tentative step toward using quantum cryptography in satellite communications and the like.

IX. Outlook and Perspectives We hope to have shown, with this review article, that quantum mechanics still poses open questions and suggests the possibility of future applications. Quantum mechanics, even 50 years after its formulation, is still full of surprises.

FUNDAMENTAL TESTS OF QUANTUM MECHANICS

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To summarize our molecular dissociation EPR experiment, we have shown that the Hg isotope 199Hgis ideally suited for a loophole-free test of the Bell inequalities. It is possible to generate an entangled singlet state of spin-1/2 nuclei by dissociating a 199Hg2dimer using a stimulated Raman transition. Especially important features are the extremely long lifetimes (milliseconds) for the entangled states that can be achieved with atoms as compared to those achieved with photons (nanoseconds to microseconds); the production of an entanglement of fermions rather than bosons; the entanglement of massive rather than massless particles; and the fact that the test is done inside the light cone rather than outside. In addition, because nuclear spins are being observed, the system is relatively robust with respect to external influences, and photo-ionization provides the high detection efficiencies necessary to close the detection loophole. In conclusion, Bohm’s original gedankenexperiment is now becoming an experimental reality.

X. Acknowledgments This research was supported by the Robert A. Welch Foundation grant No. A-1218 and by the National Science Foundation grant PHY-9732459. The authors thank John Clauser and Marlan Scully for many helpful discussions on the theory and for encouragement, as well as Robert Kenefick, Remus Nicolaescu, and Cechan Tian for their experimental contributions and active participation.

XI. References Alley, C. O., and Shih, Y. H. (1987). “A new type of EPR experiment.” In M. Namiki (Ed.). Proc. Inr. Symp. Foundations of Quantum Mechanics, p. 47. Aspect, A,, Dalibard, I., and Roger, G. (1982). Phys. Rev. Lett. 49, 1804. Aspect, A,, Grangier, Ph., and Roger, G. (1982). Phys. Rev. Let?. 49,91. Bell, J. S. (1964). Physics I , 195. Reprinted in J. S. Bell. (1987). Speakable and unspeakable in quantiim mechanics. Cambridge University Press (New York). Benetti, P., Fossati, G., Rosella, M., Tornaselli, A., and Sigon, F. (1991). “Design and test of a Daly-type detector for RIMS.” In S. Hurst (ed.) Conf on Res. Ionization Spectroscopy, p. 373. Institute of Physics. Bohrn, D. (1951). Quantum physics. Prentice-Hall (New York). Boschi, D., Branca, S., DeMartini, F., Hardy, L., and Popescu, S . (1998). Phys. Rev. Lett. 80, 1121-1 125. Bouwmeester, D., Pan, J.- W., Mattle, K., Eible, M., Weinfurter, H., and Zeilinger, A. (1997). Nature 390, 575. Bouwrneester, D., Pan, J.- W., Daniel], M., Weinfurter, H., and Zeilinger, A. (1998). Phys. Rev. Lett. 82. 1345-1349.

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Buttler, W. T., Hughes, R. J., Kwiat, P. G., Lamoreaux, S. K.,Luher, G. G., Morgan, G. L., Nordjolt, J. E., Peterson, C. G., andsimmons, C. M. (1998). Phys. Rev. Lett. 81,3283-3286. Cantrell, C. D., Scully, M. 0. (1978). Phys. Reports 3,499-508. Chiao, R. Y., Kwiat, P. G., and Steinberg, A. M. (1994). In B. Bederson and H. Walther (Eds.). Adv. At. Mol. Opt. Physics, vol. 34, p. 35. Academic Press (New York). Clauser, J. F., and Home, M. A. (1974). Phys. Rev. D 10, 526. Clauser, J. F., and Shimony, A. (1978). Rep. Pmg. Phys. 41, 1881. Clauser, J. F., Home, M. A., Shimony, A., and Holt, R. A. (1969). Phys. Rev. Lett. 23,880. Clauser, J. F. (1976). Phys. Rev. Lett. 36, 1223. Duncan, A. J., and Kleinpoppen, H. (1988). In F. Selleri (Ed.). Quantum mechanics versus local realism - The Einstein-Podolsky-Rosen experiment, chapter 7, pp. 175-218. Plenum Press (New York). Eberhard, P. H. (1993). Phys. Rev. A 47,747-750. Einstein, A., Podolsky, B., and Rosen, N. (1935). Phys. Rev. 47,777. Englert, B.- G., Loffler, M., Benson, 0.. Varcoe, B., Weidinger, M., and Walther, H. (1998). Fort. Phys. 46, 897-926. Franson, J. D. (1989). Phys. Rev. Lett. 62, 2205-2208. Freedman, S. J., and Clauser, J. F. (1972). Phys. Rev. Lett. 28,938. Freyberger, M., Aravind, P. K., Home, M. A., and Shimony, A. (1996). Phys. Rev. A 53, 1232. Fry, E. S. (1973). Phys. Rev. A 8, 1219. Fry, E. S., and Thompson, R. C. (1976). Phys. Rev. Lett. 37, 465. Fry, E. S., and Walther, Th. (1996). In R. S. Cohen and J. Stachel (Eds.). Experimental metaphysics - Quantum mechanical studies for Abner Shimony, vol. I. Kluwer Academic (Dordrecht ). Fry, E. S., Walther, Th., and Li, S. (1995). Phys. Rev. A 52,4381. Fry, E. S., Walther, Th., and Kenefick, R. (1998). Physica Scripta "6.47-51. Furusawa, A., Ssrensen, J. L., Braunstein, S. L., Fuchs, C. A., Kimble, H. J., and Polziak, E. S. (1998). Science 282, 706-709. Greenberger, D. M., Home, M. A,, and Zeilinger, A. (1989). In M. Kafatos (Ed.).Bell's theorem, quantum theory, and conceptions of the universe, pp. 73-76. Kluwer Academic (Dordrecht ). Greenberger, D. M., Home, M. A., Shimony, A., and Zeilinger, A. (1990). Am. J. Phys. 58.1 13 1. Hagley, E., Maitre, X., Nogues, G., Wunderlich, C., Brune, M., Raimond, J. M., and Haroche, S. (1997). Phys. Rev. 79, 1-5. Hardy, L. (1998). Contemporary Phys. 39, 419429. Herzberg, G. (1950). Molecular spectra and molecular structure. D. Van Nostrand (Princeton, NJ). Huelga, S. F., Ferrero, M., and Santos, E. (1994). Eumphys. Lett. 27, 181. Hurst, G. S., Payne, M. G., Kramer, S. D., and Young, J. P. (1 979). Rev. of Mod. Phys. 5 1,767. Kramers, H. A. (1958). Quantum mechanics. North-Holland (Amsterdam). Kwiat, P. G., Mattle, K., Weinfurter, H., Zeilinger, A., Sergienko, A. V., and Shih, Y.(1995). Phys. Rev. Lett. 75, 4337. Kwiat, P. G., Waks, E., White, A. G., Appelbaum, I., and Eberhard, Ph. H. (1998). quant-ph/ 9810003. Laflamme, R., Knill, E., Zurek, W. H., Catasti, P., and Mariappan, S. V. S. (1998). Phil. Trans. R. SOC. Land. A 356, 1941-1947. Lo, T. K., and Shimony, A. (1981). Phys. Rev. A 23,3003-3012. Mohrhoff, U. (1999). Am. J. of Phys. 67, 330-335. Nicolaescu, R., Walther, Th., and Fry, E. S. (1998). Number CTuM69 in CLEO '98, p. 179. OSA.

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Nussenzveig, P., Bemardot, F., Brune, M., Hare, J., Raimond, J. M., Haroche, S., and Gawlik, W. (1993). Phys. Rev. A 48, 3991-3994. Oliver, B. J., and Stroud, C. R. (1987). J. Opt. Soc. Am. B 4, 14261428. Ou, Z. Y., and Mandel, L. (1988). Phys. Rev. Lett. 61,50. Pearle, P. M. (1970). Phys. Rev. D 2, 1418. Pipkin, F. M. (1978). In D. Bates and B. Bederson (Eds.). Adv. At. Mol. Opt. Physics, vol. 14, p. 281. Academic Press (New York). Rodgers, P. (1998). Physics World (special issue), vol. 11, pp. 33-57. Santos, E. (1996). Phys. Lett. A 212, 10-14. Scully, M. O., and Zubairy, M. S. (1997). Quantum optics. Cambridge Univ. Press (New York). Shih, Y. H. (1999). In B. Bederson and H. Walther (Eds.). Adv. At. Mol. Opt. Physics, vol. 41, p. 1. Academic Press (New York). Steane, A. (1998). Rep. Prog. Phys. 61, 117-173. Tittel, W., Brendel, J., Zbinden, H., and Gisin, N. (1998). Phys. Rev. Lett. 81, 3563-3566. Walther, Th., and Fry, E. S. (1997a). In M. Ferrero, E. Santos, and S. Huelga (Eds.). New developments on fundamental problems in quantum physics, vol. 81 of Fundamental theories ofphysics. Kluwer Academic (Dordrecht). Walther, Th., and Fry, E. S. (1997b). Zeitschcf: Nutu8orschung 52a, 20-24. (Proceedings of the workshop in honor of E.C.G. Sudarshan.) Walther, Th., Liao, Y., Nicolaescu, R., Pan, X. J., and Fry, E. S. (1998). Number CTuC6 in CLEO '98, p. 69. OSA. Weihs, G., Jennewein, Th., Simon, C., Weinfurter, H., and Zeilinger, A. (1998). Phys. Rev. 81(23), 5039-5043. Zeilinger, A. (1986). Phys. Lett. A 118(1), 1. Zeilinger, A., Home, M. A., Weinfurter, H., and Zukowski, M. (1997). Phys. Rev. Lett. 78, 3031-3034.

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ADVANCES IN ATOMIC, MOLECULAR, AND OPnCAL PHYSICS, VOL. 42

WAVE-PARTICLEDUALITY IN AN ATOM INTERFEROMETER STEPHAN DURR and GERHARD REMPE Fakultat fur Physik, Universitat Konstanz, Konstanz, Germany

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

11. Experimental Setup

A. Atomic Beam B. Microwave Field . . . C. Internal State Prepara

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

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

111. Bragg Reflection

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

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

A. Generalized Storing Scheme. . . . . . . . . . . . . . . .

X. Wigner Function .......................... XI. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MI. Acknowledgments .

29 33 33 34 35 35 36 36 42 42 45 48 48 49 50 52 54 51 60 60 62 64 65 69 69 69

I. Introduction Wave-particle duality means that a quantum object can exhibit either wave or particle properties, depending on the experimental situation. The wave nature gives rise to interference phenomena, whereas knowledge about the path taken by the object testifies to its particle nature. The crucial point is that it is impossible to observe wave and particle properties simultaneously. When wave-particle duality was introduced in the early days of quantum mechanics, many physicists felt uncomfortable about this concept and 29

Copyright ic, ?OW hy Academic Press All rights of reproduction in any form reserved. ISBN 0-12-003842-011SSN 104')-250X/00 $3O.(K)

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Stephan Durr and Gerhard Rempe

searched for ways around it. Einstein, for example, suggested (Bohr, 1949) obtaining information about the way of a photon through a double-slit apparatus by measuring the recoil given to a collimation slit mounted in front of the double slit. He claimed that in this arrangement, an interference pattern and the photon’s way could be observed simultaneously. Bohr’s refutation (Bohr, 1949) of this gedunkenexperiment elucidates the principle of complementarity: He pointed out that the observation of the recoil requires a movable slit with a very well-defined initial momentum. But in such an arrangement, Heisenberg’s position-momentum uncertainty relation predicts such a large position uncertainty of the slit that the interference pattern is smeared out. Hence one can either use a fixed slit to obtain interference fringes or use a movable slit to perform a which-way measurement. Historically, this BohrEinstein dialogue was one of the landmarks of the development and understanding of quantum mechanics. Quantum mechanics enters Bohr’s argument only at one point: where Heisenberg’s uncertainty relation is employed. The importance of the uncertainty relation in which-way experiments is also illustrated in Feynman’s light microscope (Feynman et al., 1965). In this gedunkenexperiment, electrons are illuminated immediately after they pass through a double slit with slit separation d . The position of the electron can be determined from the scattered light with an accuracy of the order of the light wavelength, Az M h. Hence the uncertainty relation implies that after the scattering process, the electron’s momentum uncertainty is of the order of Ap, M h/h. The microscopic origin of the momentum disturbance is the photon recoil, as already pointed out by Heisenberg (1927). For h < d, a which-way measurement is performed, but the photon recoil destroys the interference fringes. On the other hand, for h > d, the photon recoil is too small to wash out the fringes, but the electron’s way cannot be determined because of diffraction. Feynman concluded that “if an apparatus is capable of determining which hole the electron goes through, it cannot be so delicate that it does not disturb the pattern in an essential way. No one has ever found (or even thought of) a way around the uncertainty principle” (Feynman et al., 1965). An experimental realization of Feynman’s light microscope is very difficult, because precise knowledge about the way of the particle through the interferometer must be obtained with high efficiency - a challenging task. Nevertheless, several steps in that direction have been made in the field of atom interferometry (Berman, 1997). Atoms are ideal test objects, because their internal structure gives rise to a large cross section for the interaction with near-resonant light. This makes it possible to investigate the influence of the spontaneous emission of a photon onto atomic interference patterns. With this aim, several experiments have been performed recently (Pfau et al., 1994; Clauser and Li, 1994; Chapman et ul., 1995). They clearly demonstrate the

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change of an interference pattern as a result of photon recoil, but which-way information was not acquired. The gedankenexperiments discussed above emphasize the importance of Heisenberg’s uncertainty relation in which-way experiments. However, in 1991, Scully, Englert, and Walther raised the question whether wave-particle duality is always enforced by the uncertainty relation. They discussed a new gedankenexperiment, in which the way of each atom through a double slit is determined using quantized microwave fields as which-way detectors. In this scheme, no net momentum is transferred to the atom during the interaction with the which-way detector, so the loss of interference fringes is not related to the uncertainty relation. Instead, entanglement- or more general-correlations between the which-way detector and the atomic motion destroy the interference. This new explanation for the loss of interference represents a substantial development, because it no longer refers to classical concepts, such as momentum transfer. Instead, an exclusive quantum feature entanglement - is employed. Of course, entanglement or correlations are present in any which-way experiment. But in the foregoing examples, the mechanical effects of the which-way detector on the particle’s motion can explain the loss of interference as well so that the effect of the correlations is hidden. Hence these schemes cannot be regarded as a proof that correlations alone are sufficient to destroy the interference. Experimental indications for the absence of interference due to correlations can be found in different fields of physics. Such correlations are, for example, responsible for the lack of ground state quantum beats in time-resolved fluorescence spectroscopy (Haroche, 1976). Another example is correlated photon pairs created in parametric down-conversion crystals. If a “signal” photon can be generated along two different paths, then interference cannot be observed because of the entanglement with the “idler” photon (Zou et al., 1991; Herzog et al., 1995). In a further experiment (Eichmann et al., 1993), the interference of light scattered from two trapped ions was investigated. Here, which-way information could be stored in the ions’ internal states, leading to a loss of interference. Correlations also play an important role in neutron interferometers: Flipping the neutron spin selectively in one arm of the interferometer destroys the interference pattern (Rauch et al., 1975; Badurek et al., 1986). The correlation between the which-way detector and the particle’s motion need not always be perfect. Instead, the degree of correlation can be varied so that a continuous transition between a wave and a particle picture is possible. In this intermediate regime, one obtains only incomplete which-way information and retains interference fringes with a reduced visibility. Stimulated by the first theoretical work by Wooters and Zurek (1979), investigators (Jaeger et al., 1995; Englert, 1996) recently found a fundamental limit, connecting

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Stephan Diirr and Gerhard Rernpe

the fringe visibility with the maximum obtainable amount of which-way information. The reduction of the visibility due to the storage of incomplete which-way information has been observed in several experiments; for example, changing the polarization of the light field in only one arm of a Mach-Zehnder interferometer reduces the fringe visibility (Schwindt et al., submitted). But this effect can be described in terms of purely classical electrodynamics that is, without entanglement. This is different in the down-conversion experiments we have mentioned, which are based on entangled quantum states. In one of these experiments (Zou et al., 1991), the degree of correlation between the “signal” and the “idler” photon could even be tuned, so that interference fringes with a reduced visibility were created. In another experiment (Brune et al., 1996), transitions between atomic Rydberg states were induced with two Ramsey fields (Ramsey, 1950), and interference fringes were observed in the excitation probability. Between the two Ramsey fields, the atoms passed through a high-quality microwave cavity, thereby shifting the phase of the intracavity field depending on the atom’s internal state. A measurement of this phase shift could (at least in principle) reveal in which state the atom was. By tuning the magnitude of the phase shift, one could adjust the degree of correlation, leading to reduced visibility of the Ramsey fringes. Finally, an Aharanov-Bohm experiment with electrons in a semiconductor was performed recently (Buks et al., 1998). The fringe contrast was gradually reduced by performing a weak measurement of the particle’s way through the interferometer. Technically, this was achieved by coupling a quantum dot in one arm of the interferometer to a quantum point contact. In this article, we will describe several experiments with a novel atom interferometer. Bragg reflection from a standing light wave is used as a beamsplitter for atoms. The atomic beams are recombined with a second standing light wave, and a spatial interference pattern is observed in the far field. Which-way information can be stored in internal atomic states by adding a microwave field. This allows us to study the origin of quantummechanical complementarity (Diirr et al., 1998a). In addition, a quantum eraser can be realized. It is also possible to store incomplete which-way information and to perform a quantitative test of wave-particle duality (Diirr et af., 1998b). The article is organized as follows: After describing the experimental setup (Section 11), we discuss Bragg reflection (Section III), present the atom interferometer (Section IV), and discuss delayed-choice aspects (Section V). We then explain the mechanism used to store which-way information (Section VI) and investigate how its presence changes the interference pattern (Section VII). After discussing quantum-eraser measurements (Section VIII),

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33

we report experiments where incomplete which-way information is stored (Section IX). Next, we illustrate the evolution of the wavefunction in the atom interferometer using the Wigner function (Section X) and finally give a conclusion (Section XI).

11. Experimental Setup A. ATOMICBEAM Figure 1 shows a scheme of the experimental setup, which has been described in more detail by Kunze et al. (1997a). Light forces (Adams and Riis, 1997) are employed to load a magneto-optical trap (MOT) with 85Rb atoms from background vapor. After trapping, the atoms are cooled further in optical molasses to a temperature of about 10 pK.Then the cloud of atoms is released by turning off the laser light. Freely falling through the apparatus,

20cm

photomultiplier

galvo

detection beam

Fici. 1. Scheme of the experimental setup. *'Rb are atoms dropped from a magneto-optical trap. They form a pulsed atomic beam, which is collimated by two mechanical slits. Then the atoms pass the interaction region containing a standing light wave and a microwave resonator. In the far field, the spatial distribution of atoms is monitored via laser-induced fluorescence.

34

Stephan Diirr and Gerhard Rempe TABLE I ATOMIC BEAM.

PARAMETERS OF THE

Longitudinal velocity in the interaction region de Broglie wavelength in the interaction region Time of flight from the (effective) source to the interaction region Time of flight from the interaction region to the detection region Width of the upper collimation slit Width of the lower collimation slit Transverse momentum width after collimation

vx = 2.0mI.s = 2.3nm

hde

t,,,,e rdef

= 176ms

= 92111s

O.lOmm 0.45 mm

Anmm

= 0.25

Fez4

Fe=3

5p2P3/2

Fe=2 F e z1

cycling

1

g: : ; : : ;

I repumping

I

FIG.2. Level scheme of the D2 line of 85Rb. The ground state 5s 2S1/2 is split into two hyperfine components with total angular momentum Fg = 2 and Fg = 3. The excited state 5p 2P3/2is split into four hyperfine components with total angular momentum F, = I , . . . ,4. The magnetic sublevels are not shown.

the cloud forms a pulsed atomic beam, which is collimated by two mechanical slits. The first slit, 1 cm below the MOT, is 0.1 mm wide, and the second slit, M 20cm below the MOT, is 0.45 mm wide. Table I lists the parameters of the atomic beam, and Fig. 2 shows a scheme of the relevant atomic levels. B. MICROWAVE FIELD The interaction region, right below the second slit, contains a standing light wave inside a microwave resonator. The field inside the resonator can be excited with an antenna powered by an external microwave source. The interaction time with the microwave field is controlled by switching the input power on and off. The microwave oscillates at a frequency of OM^ M 2x x 3.035 GHz and induces transitions between the Fg = 2 and the Fg = 3 hyperfine components of the atomic ground state (see Fig. 2). Only Am, = 0

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

35

transitions are induced, because the oscillating magnetic field is parallel to an externally applied 100 pT magnetic bias field. In addition, this bias field Zeeman-shifts out of resonance all transitions except Fg = 2, mF = 0 tf Fg = 3, mF = 0. C. INTERNALSTATEPREPARATION After release from the optical molasses, most of the atoms are optically pumped into the Fg = 3, mF = 0 state using linearly polarized light, as described by Kunze et al. (1997a). In the interaction region, these atoms are transferred to the Fg = 2,mF = 0 state with a microwave 7c pulse. Some atoms remain in the Fg = 3, mF # 0 states even after optical pumping. They are subsequently pushed away using a laser beam resonant with the Fg = 3 tf F, = 4 “cycling” transition, a method similar to that described by Gibble and Chu (1993). After scattering M 50 photons, these atoms no longer reach the detection region, whereas atoms “hidden” in the Fg = 2 state continue their way downward. D. STANDING LIGHTWAVE The standing light wave is created by retroreflecting a laser beam from a flat mirror mounted inside the microwave resonator. The interaction time of the atoms with the standing light wave is controlled by switching the light on and off. The atoms fall through the interaction region with a velocity of only v, = 2.0 m/s. This allows us to perform the whole interferometer experiment with only one standing light wave, which is switched on and off twice. In order to illuminate all atoms with light of approximately the same intensity, we choose a large laser beam waist of ox= 10 mm (l/e2 radius of intensity) in the vertical direction and of my = 8.4 mm in the horizontal direction. However, the finite temperature of the atomic cloud after release from the optical molasses leads to an expansion of the cloud. During the flight to the interaction region, the cloud size increases to about 15 mm (full width at half maximum, FWHM) in the vertical direction. It is therefore impossible to illuminate all atoms with a pulsed light field of the same intensity. Therefore, we select a small fraction of the atoms with a vertical size of 2 mm by gating the photomultiplier signal (see below) only 1 ms. The intensity variations of the standing light wave due to the finite laser beam waist, as seen by these atoms, are less than 6% root mean square (rms). Additionally, the transverse profile of the standing wave exhibits intensity wiggles caused by imperfections in the optical elements. We measured the total transverse intensity variations to be 12%nns.

36

Stephan Diirr and Gerhard Rernpe

E. FLUORESCENCE DETECTION

The atomic position distribution is observed 45 cm below the MOT, in the far field of the interaction region. For that purpose, the atoms are excited with a resonant laser beam, and the fluorescence photons are detected with a photomultiplier. Before entering the vacuum chamber, the laser beam is reflected from a mirror mounted on a galvo drive. Tilting the galvo allows us to scan the laser beam along the (horizontal) z-direction and to monitor the atomic position distribution. The detection laser beam propagates nearly parallel to the (horizontal)y-axis and has an elliptic Gaussian profile. Its horizontal waist of oz= 50 pm determines the position resolution of the detection system. Depending on the frequency of the detection laser, atoms in different internal states can be detected. To detect only atoms in the Fg = 3 state, we tune the frequency of the detection laser into resonance with the Fg = 3 * F, = 4 “cycling” transition. To detect atoms in the Fg = 2 and Fg = 3 state simultaneously, we add a “repumping” laser beam resonant with the Fg = 2 * F, = 3 transition. In principle, only atoms in the Fg = 2 state could be detected by applying a Fg = 2 H F, = 1 laser, but in practice, offresonant optical pumping into the Fg = 3 state leads to a very low detection efficiency in this scheme. To circumvent this problem, an additional microwave n pulse is applied before the atoms leave the interaction region. A subsequent detection of atoms in the Fg = 3 state then corresponds to the direct detection of atoms in the Fg = 2 state. The state-selectivedetection of atoms constitutes an internal state measurement. This measurement, as discussed so far, projects onto the Fg = 2 or the Fg = 3 state. In order to project onto an arbitrary state of the internal degree of freedom, a microwave pulse with suitable pulse area is applied before the atoms leave the interaction region, and finally, only atoms in the Fg = 3 state are detected. To detect, for example, atoms in the superposition state (IFg = 2) IFg = 3))/&, a n/2 pulse is used.

+

111. Bragg Reflection In the field of atom optics (Adams et al., 1994), a variety of methods to split and recombine atomic beams have been proposed and demonstrated. One of these methods is based on Bragg reflection of atoms from a standing light wave (Martin et al., 1988; Kunze et al., 1996; Giltner et al., 1995a; Bernet et al., 1996).This method provides a beamsplitter, which is particularly useful for applications in atom interferometry, because it creates only two output beams without reducing the total atomic flux (in contrast to, for example, a mechanical double slit). This beamsplitter relies on the fact that nonresonant

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

37

FIG.3. Scheme of the beamsplitter for atoms. Atoms enter a standing light wave, which creates a periodic potential. From this periodic potential, a part of the atoms is Bragg-reflected, while the rest of the atoms is transmitted.

light creates a conservative potential for the atoms, which is called light shift or ac-Stark shift. This potential is proportional to the light intensity, which in a standing wave is a periodic function of position. It follows that the light-shift potential is periodic, too. A beam of atoms entering a standing light wave, as shown in Fig. 3, can be Bragg reflected from this periodic potential. This process is similar to Bragg reflection of X-rays from the periodic structure of a solid-state crystal, but with the role of matter and light exchanged. Here the light creates the periodic structure, from which the matter wave is reflected. In order to achieve Bragg reflection, the angle 8 under which the atomic beam enters the standing light wave must fulfill the Bragg condition

Here, the atomic de Broglie wavelength hm = h / p is determined by the atomic momentum, p; the order of reflection, n, is an integer; and the spatial period of the light intensity equals half a light wavelength, 1 = h/2. The Bragg condition can be rewritten as a condition for the transverse atomic momentum p z = p sin 9 (horizontal in Fig. 3), yielding p z = nhk, where k = 2n/h is the wave vector of the light. After Bragg reflection, the atomic beam is split into two components: one transmitted beam with unchanged momentum and one Bragg-reflected beam with the sign of the transverse momentum, p z , reversed. The reflectivity of this beamsplitter - that is, the fraction of reflected atoms - can be tuned by varying the parameters of the atom-light interaction. For applications in an atom interferometer,one usually prefers a reflectivity of 50%. Here we will calculate the probability amplitudes for reflection and transmission, Eqs. (1 1) and (12), in order to determine the parameters required to create a 5050 beamsplitter and in order to explore the phase shifts occurring

38

Stephan Durr and Gerhard Rempe

in the beamsplitter. Readers who are not interested in this calculation may continue with Section IV. The calculation is based on the model introduced in Bernhardt and Shore (1981), Pritchard and Gould (1985), Kazantsev et al. (1990), Wilkins et al. (1991), Schumacher et af. (1992), Marte and Stenholm (1992), and Durr et al. (1996). We consider a two-level atom inside a standing light wave. The detuning of the light frequency from the atomic resonance is assumed to be so large that excitation and spontaneous emission can be neglected. In this regime, the atom experiences only the light-shift potential V ( z ) = 4hxcos2(kz). The potential depth is determined by the light-shift parameter x = R2/(4A), which depends on the peak traveling-wave Rabi frequency R and on the atom-light detuning A = Wlight - oat,. Note that x is proportional to the light intensity. Restricting the model to one dimension, we obtain the Hamiltonian

h 2 a2 H ( z ) = - - - 4ti~cos’(k~) 2m az2

+

where m is the mass of the atom. We assume that the initial atomic state is a plane wave with transverse momentum p z , which we normalize with respect to the photon momentum, tik, by writing

p z = nhk

(3)

Here, we do not restrict n to be an integer. This makes it possible to investigate the influence of deviations from the exact Bragg condition. The corresponding atomic momentum state is denoted by In). Using cos2(kz) = (2 exp exp(2ikz)}/4, we find (-2ikz)

+

+

where orec = hk2/(2m) is the recoil frequency. Equation (4) shows that the light couples momentum states with a momentum difference of 2hk. This can easily be explained, because a standing wave consists of two counterpropagating traveling waves. An atom can absorb a photon out of one of these traveling waves, followed by an induced emission into the other traveling wave. This Raman-like two-photon process transfers two photon momenta to the atom. Alternatively, an induced emission into the same traveling wave can occur so that the atomic momentum remains unchanged. This process is already possible in a single traveling light wave and gives rise to an energy

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

39

shift, namely the light shift hx. In the standing wave, the light shifts of the two traveling waves add up and create the term 2hxln) in Eq. (4). In the following, we focus on first-order Bragg reflection-that is, n M 1 -and include small deviations from the exact Bragg resonance:

n = 1+An

with

/ A n [ < <1

(5)

+

According to Eq. (4), the initially populated momentum state 11 An) is coupled to the states {. . . , I - 3 An), I - 1 An), 11 An), 13 An), . . .}. Whereas the states I - 1 An) and 11 An) are nearly degenerate, the kinetic energies of the states 13 An) and 11 An) differ by M 8h0,,. We << 80, - where this consider the case of weak coupling - that is mismatch between the kinetic energies prevents the transfer of significant population to the state 13 +An). A similar argument applies to the state I - 3 An). This allows us to perform a two-beam approximation; that is, we neglect the coupling to all momentum states except I - 1 An) and I 1 An). In a matrix representation with respect to the states

+ + +

+

+

+

+

+

+

1x1

+

+

the Hamiltonian is

The diagonal elements of this matrix contain the kinetic energies of the . off-diagonal elements represent momentum states and the light shift 2 h ~The the light-induced coupling between the states. The difference between the kinetic energies and their mean value are

We assume that the light intensity is switched on instantaneously at time t = 0, stays constant during the interaction, and is switched off instantaneously at time r = tpuise.The time evolution operator for such a single light pulse is UBS= exp(-iHrpulse/h).A short calculation using Eq. (7) yields

40

Stephan Diirr and Gerhard Rempe

with the amplitude coefficients for transmission and reflection t l = cos(Rtp,l,/2)

+- i AEkin h a sin(Qtpulse/2)

rl = -i -~ 2Xi n (f2 t ~ , , ~ ~ ~ / 2 )

0

(11) (14

the Pendellosung frequency

and an overall phase shift due to the light-shift terms in the diagonal elements of H in Eq. (7),

This yields the probabilities for reflection and transmission

and, of course,

The first factor in Eq. (15) is a Lorentzian as a function of An and describes the finite acceptance angle for Bragg reflection. This has been investigated experimentally in Durr and Rempe (1999). For atoms entering the standing light wave exactly at the Bragg resonance - that is, An = 0 - the second factor describes a sinusoidal oscillation as a function of tpulse(or x), which is called Pendellosung (Martin et al., 1988; Kunze et al., 1996; Durr et al., 1996; Durr and Rempe (1999); Giltner et al., 1995b; Ewald, 1917). Both effects, the finite acceptance angle and the Pendellosung, are visible in Fig. 4, which displays the atomic momentum distribution after Bragg reflection for different values of the laser power, P, of the standing light wave. The initial momentum distribution, visible for P = 0, is centered around p z M - hk. At P M 50 pW, atoms with a momentum sufficiently close to the Bragg condition are Bragg-reflected to p z M +hk. The fraction of Braggreflected atoms exhibits Pendellosung oscillations as a function of the laser

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

41

FIG.4. Bragg reffection of atoms from a standing light wave. The atomic momentum distribution after the interaction is shown for different values of the laser power, P , in the standing light wave. The initial momentum distribution (at P = 0) is centered around -hk. Atoms with a momentum sufficiently close to the Bragg condition can be first-order Braggreflected to f f i k . The fraction of Bragg-reflected atoms exhibits Pendellosung oscillations as a function of the laser power. The data were recorded with atoms falling through a continuous standing light wave with a vertical waist of ox= 35 pm, corresponding to an interaction time of x 17 ps (from Kunze et al., 1996).

power. In addition, the acceptance angle increases for larger values of the laser power. We conclude that a Bragg beamsplitter allows full control over both the reflection probability and the momentum width of the reflected beam. In particular, the acceptance angle can be adjusted by varying the light-shift parameter, x. This makes it possible to match the momentum width of the Bragg resonance to the momentum width of the incoming atomic beam. The latter amounts to An = k 0.25 (see Table I), so that according to Eq. ( 1 3 , light fields with 1x1 >> 0.5 o, are required. Taking into account that the two-beam M 0.7 approximation is valid for small values of 1x1 only, light pulses with orec are applied in the experiment. The pulse duration, tpulse,is chosen to fulfill = n/4, as is necessary to create a 5050 beamsplitter for the condition (x(tpulse atoms with An = 0. For these parameters, the reflection probability for atoms with An = 50.25 is 45%, sufficiently close to the desired value of 50%. Table I1 summarizes the parameters of the atom-light interaction.

1x1

42

Stephan Durr and Gerhard Rempe TABLE 2 PARAMETERS OF THE ATOM-LIGHT INTERACTION.

Light wavelength Bragg angle Recoil velocity Recoil frequency Light shift parameter [see Eq. (2)] Duration of one Bragg pulse Separation between Bragg pulses (except Fig. 6b,c)

h = 780nm 9 = hde/h = 3 m a d ,v = 6.0mm/s 0 , = 2n: x 3.8kHz 1x1 PZ 0.7 om tpulsc = ~ / ( 4 l ~=l 45 ) w t X p = 105 ps

IV. The Atom Interferometer As shown in Fig. 5 , an atom interferometer can be realized with two Bragg beamsplitters. The first standing light wave splits the incoming atomic beam, A, into two beams, B and C. After free propagation during a time interval tsep, the two beams are shifted by a transverse distance d with respect to each other. Then a second standing light wave splits each atomic beam again into two components. Now two beams, D and E, are traveling to the left, while beams F and G are traveling to the right. In the far field, each pair of overlapping beams produces a spatial interference pattern. The experiment employs only one standing light wave, which is switched on and off twice, so that the vertical axis in Fig. 5 represents time rather than distance. For first-order Bragg reflection, the transverse distance

is determined by the separation time tsep and the recoil velocity v,, = tik/m. This scheme is similar to a double-slit experiment with slit separation d. This analogy suggests a fringe period of Ap = h/d in momentum space. In the following two subsections, we will discuss the validity of this naive expectation. Again, readers may wish to skip these calculations and directly continue with Section 1V.C. A. PLANE WAVETHEORY With the results of the preceding section, the shape of the fringe pattern can be calculated. We begin by writing down the time evolution operator describing the free flight between the two light pulses during the time interval tsep:

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

43

FIG.5. Scheme of the atom interferometer. Bragg reflection from a standing light wave splits the incoming atomic beam, A, into two beams, B and C. A second standing light wave splits the beams again. In the far field, a spatial interference pattern is observed (Reprinted by permission from Durr et al., 1998a, Nature, Macmillan Magazine Ltd.)

where

is the phase difference that the two momentum states accumulate during the free flight because of the difference between their kinetic energies. In order to describe the complete interferometer, the time evolution operators corresponding to the two light pulses and to the free flight must be multiplied, which yields

Here, the amplitude coefficients for transmission and reflection are

44

Stephan Durr and Gerhard Rempe

with tl and rl defined in Eqs. (11) and (12). The reflection amplitude in Eq. (22) is the sum of two terms. The fist term describes an atom that is transmitted through the first beam-splitter and reflected from the second beam-splitter. This term corresponds to beam E in Fig. 5. The second term corresponds to beam D. Defining y = 2argt1

(23)

that is, t l = It1 leiY/2,the relative phase between the two amplitudes in Eq. (22) is a y. The probabilities for reflection and transmission can easily be calculated, yielding

+

and, of course, 2 Ittotall

+

2 lrtotall

=1

(25)

For the parameters of our experiment, the factor 41rl1*1t11*in Eq. (24) varies between 1 and 0.99. This small variation can be neglected, yielding

+

The total reflection probability oscillates as a function of a y. In order to calculate the shape of the fringe pattern, we must evaluate the dependence of a and y on An. Whereas a is linear in An according to Eq. (19), the dependence of y on An is more complicated and must be determined from Eqs. (1 1) and (23). However, ]An1is small in the experiment, so y can be linearized in An. ~ l we ~ obtain ~ Using l ~ l t =~n/4,

For the parameters of our experiment, Eq. (27) differs less than 3% from the exact result. Whereas a is the relative phase shift, the momentum states accumulate because of their different kinetic energies during tsep, y takes into account the finite interaction time with the standing light waves, 2tpulse. Inserting Eqs. (19) and (27) into Eq. (26) yields

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

45

This is the central result of our calculation. The reflection probability oscillates sinusoidally as a function of An. This oscillation is due to interference between beams D and E. Depending on their relative phase, ct y, the interference is constructive or destructive. A similar argument applies to the interference between beams F and G, creating the corresponding antifringes in JttotalJ2according to Eq. (25). In the limit of long separation times (tsep >> tpulse), the period of the interference fringes as determined by Eq. (28) is An,,,, M 2x/(4~,,t,ep).This result is identical to Ap,,, = h / d as expected from the analogy to a double-slit experiment. We conclude by noting that from a mathematical point of view, the Hamiltonian Eq. (7) is analogous to that of a two-level atom driven by a nearresonant light field in the absence of spontaneous emission. In particular, the Pendellosung corresponds to Rabi oscillations, and the spatial fringes of the interferometer correspond to Ramsey fringes.

+

B. WAVEPACKET THEORY In every realistic situation, the initial state of the atom is a wave packet, not a plane wave. The state vector can be described, for example, in a rescaled momentum representation

I$)

(29)

= jdnJl(n)ln)

where $(n) is the wave function, and n = p,/hk is the rescaled transverse momentum, as in Eq. (3). In our experiment, the initial wave function, $in(n), is nonvanishing only in a small region centered around n = 1. From Eq. (20) the final wave function after the interaction with the standing light waves can be calculated, yielding

The corresponding momentum distribution is

Note that the cross term

+ 2)$:n(n)$in(n + 2)

(32)

ttota~(n)rtota~(n

and its complex conjugate vanish, because $in(n) is nonzero only in a small region with half-width An = 0.25, so the product (n)$in(n 2) is zero for all values of n.

$rn

+

46

Stephan Diirr and Gerhard Rempe

The first term in Eq. (3 1) describes the right half of the interference pattern shown in Fig. 5 , and the second term describes the left half. The factor \ttotd(n)l2produces sinusoidal fringes according to Eqs. (25) and (28). The envelope of these fringes, l+in(n)[2,is determined by the collimation properties of the incoming beam. In the experiment, the spatial far-field distribution 1 + ( . ~ ) 1 ~ is observed. Within a Fraunhofer approximation, this would be determined by

where tdet is the time of flight from the interaction region to the detection region. The Fraunhofer approximation reflects the simple fact that an atom with transverse velocity nvrec travels the transverse distance z = n'llrectdet during tdet. In particular, the center of the reflected beam and that of the transmitted beam have a velocity difference of 2vreC,so the distance between them in the detector plane is 2Vrectdet = 1.1 mm. The Fraunhofer approximationis not strictly valid for the parameters of our experiment, because the time of flight from the source to the interaction region, t source, has to be taken into account, yielding

with

Each term in Eq. (34) describes one-half of the interference pattern, as in Eq. (31). The argument, n, in the transmitted part is similar to that in Eq. (33), but with fdet replaced by t,,,,, tdet. This reflects the fact that the atom already travels a transverse distance during its flight from the source to the interaction region. The reflected part is shifted with respect to the transmitted part by 2'Urectdet. Here tsource does not appear, because the change of the atomic transverse velocity by 2vrec takes place in the interaction region. It is easy to see that for tsource= 0, Eq. (34) reduces to Eqs. (31) and (33). In the experiment, the effective source of atoms is the cloud after optical

+

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

47

54-

2 3-

7

.-

C

a

w

2-

C

;1 0-1

1

0

position / mrn ............

a

E

3-

7

.c 2a

4-

5 8

1-

0.

1

I

I

-1

0

1

d

position / rnrn .............

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

-1

0

1

position / mm FIG.6. Spatial fringe pattern in the far field of the interferometer. The solid line is a fit to the experimental data (circles). The dashed line shows the independently measured envelope. Parts (a), (b), and (c) were recorded for a splitting d = 1.3 pm, 3.1 pm, and 5.0 pm, respectively.

48

Stephan Diirr and Gerhard Rempe

pumping. This is because the atomic momentum changes as a result of photon recoils. This has been taken into account in the value of t,,,,, in Table I. Note that in Eq. (34) we chose z = 0 at the source, whereas in the following, we will always choose z = 0 midway between the transmitted and reflected beams.

C. EXPERIMENTAL RESULTS Figure 6 shows spatial interference patterns observed in the far field. The data were obtained with tpulse = 45 ps. We chose tsep = 105 ps with d = 1.3 pm in Fig. 6(a), tsep = 255 ps with d = 3.1 pm in Fig. 6(b), and tSep= 415 ps with d = 5.0 pm in Fig. 6(c). The solid lines in Fig. 6 represent fits to the experimental data. The dashed lines represent the measured beam envelope, which consists of two broad peaks. The right peak is due to beams F and G, with a shape determined by the momentum distribution of the initial beam A. The left peak is a combination of beams D and E. It is a Bragg-reflected picture of the right peak. The fringe patterns under these two broad peaks are complementary;that is, the interference maxima in the left peak correspond to interference minima in the right peak, and vice versa, as expected from Eq. (25). The best-fit values for the fringe period are 620 pm, 320 pm, and 210 pm, respectively, in good agreement with the theoretical expectation from Eqs. (28) and (34), with tsource= 176 ms and fdet = 92 ms. The best-fit values for the visibility are (75 f l)%, (44 f l)%,and (18 f l)%,respectively. To the best of our knowledge, the pattern displayed in Fig. 6(a) has the highest visibility of spatial fringes ever observed in an atom interferometer. The reduction of the fringe visibility evident in Figs. 6(b) and 6(c) is due to the finite width of the detection laser beam and of the upper collimation slit.

V. Delayed Choice The experiments described so far have already revealed a lot about waveparticle duality. Let us first recall the simple setup considered in Section 111: Atoms pass through a single beamsplitter and are detected in the far field. Each detector click reveals whether the atom is in beam B or beam C. Of course, each individual atom gives a click for either beam B or C. We never find half an atom in beam B and the other half in beam C; that is, each atom has a well-defined way, corresponding to a particle picture. Let us now add the second beamsplitter. The creation of interference fringes can be explained only in a wave picture: The de Broglie wave is split into two components that are recombined later. Depending on the relative

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

49

phase shift accumulated in the meantime, the two components interfere constructively or destructively. To create the interference pattern, each atom must move along both ways simultaneously. The atom therefore exhibits a “split personality”: Sent through the interferometer, it exhibits wave properties, whereas detection behind a single beamsplitter reveals its particle character. In this context, von Weizsacker (1931 and 1941) and Wheeler (1978) raised the question of at what time the atom decides between behaving like a particle or like a wave. Does it happen when the atom passes through the first beamsplitter, or does the atom wait until the detection? Quantum mechanics gives a clear answer: The decision is made at the detection. In particular, the system is in a coherent superposition of the two possibilities before the wavefunction collapses in the measurement process. Whether this is correct can be tested in so-called delayed-choice experiments, where the setup allows one to insert or remove the second beamsplitter after the atom has already passed through the first beamsplitter. The first delayedchoice experiments were performed in light interferometers (Hellmuth et al., 1987). Our experiment offers the possibility of delayed choice without further modifications. Because a pulsed standing light wave is used, there is enough time to decide whether a second light pulse shall be applied after the first pulse is over. The simple fact that interference fringes are observed in our experiment therefore demonstrates that during the first light pulse, the atom cannot decide between particle-like and wave-like behavior, because it cannot “know,” at this moment, whether a second light pulse will be applied.

VI. Storing Which-Way Information In the foregoing discussion about delayed choice, the second beamsplitter is removed in order to determine whether the atom is in beam B or beam C. This is a very rude method, because it completely destroys the interferometer. It is, of course, much more interesting to look for the atom’s way inside the interferometer. For that purpose, a second physical system, called the which-way detector, must be added. With a suitable interaction, the state of the which-way detector must be modified depending on the way the atom takes. A later measurement on the which-way detector then reveals the atom’s way. We use two internal states of the atom as a which-way detector: the Fg = 2, mF = 0 state and the Fg = 3, mF = 0 state (see Fig. 2), which in the following discussion are labeled 12) and 13),respectively (see Fig. 7a). In this section, we will explain how which-way information can be stored in the

50

Stephan Diirr and Gerhard Rempe

a)

c)

b)

le>-

T

Yight

13>OmW

12>-

w

FIG. 7. (a) Simplified level scheme. (b) Light-shift potentials as a function of position. (c) Sandwiching the first standing light wave between two microwave pulses stores which-way information in internal atomic states (see text). Shown here is the simplified case discussed in Eqs. (43) and (44).

population of these two states. We start by investigating how the properties of a single Bragg beamsplitter depend on the internal atomic state. A. BEAMSPLITTER FOR Two INTERNALSTATES

Only two atomic levels were considered in the preceding calculations. However, the level scheme of real atoms used in the experiment is somewhat more complex (see Fig. 2). We explicitly take into account the hyperfine splitting of the ground state, but we replace all the hyperfine states of the excited level by only one (virtual) state le) for the following reason. Consider an atom initially prepared in state 12). In a detuned standing light wave, each allowed transition from state 12) to an excited state creates its own light-shift potential for the atom. The total potential is obtained by simply adding the individual potentials. For light with a fixed frequency, the situation can therefore be modeled by assuming only one (virtual) excited state le). A similar argument applies to an atom in state 13). This justifies the use of the simplified level scheme shown in Fig. 7(a). Because the standing light wave does not induce any coupling between the states 12) and 13), Eq. (10) holds for each of the two internal states separately. But the light-shift parameter x is different for atoms in states 12) and (3).We denote the two values of the light shift by x2 and x3 and the corresponding values of t l , rl, and p by t2, r2, and p2, and t 3 , r3, p3, respectively. As indicated in Fig. 7(b), the light frequency is tuned between the 12) t+ Ie) resonance and le) resonance. For atoms in state 12),the light is red-detuned and the the 13) light shift is negative, whereas for atoms in state 13), the light is blue-detuned and the light shift is positive. In order to obtain the same reflectivity of the

-

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

51

beamsplitter for atoms in both internal states, the light frequency is carefully adjusted to give x 2 = -x3

Using Eqs. (1 1) through (14), this yields

(37)

p2 = - p3 and

t2 = t3

r2 = -r3

(38) (39)

It is useful to introduce the (not normalized) state vectors I+*), and so on, which denote the external degree of freedom for atoms in each beam in Fig. 5. The phases are chosen with respect to atoms in state 13). It follows that atoms in state 13) that passed through the first beamsplitter are described by the state vector

whereas for atoms in state 12), the state vector is

According to Eq. (14), p = 2xtpulser so a 5050 beamsplitter with x3tpulse = n/4 has p3 = -p2 = 7c/2 and e1(p3-pz) = -1. Equations (40) and (41 ) reveal that the reflection probability is independent of the internal atomic state. However, the amplitude of the wave function experiences two phase shifts depending on the internal state: 0

0

an overall phase factor ei(p3-p2) a n phase shift for atoms reflected in state 12)

The physical origin of these two phase factors is easy to understand. The first phase, p3 - p2, arises simply because the atoms travel in the internal-statedependent light-shift potential during tpulse. The accumulated phase shift increases linearly with tpulse. In contrast to this, the second phase shift is always n - independent of all parameters. An analogy for this n phase shift can be found in classical optics, where a light wave reflected from an optically thicker medium experiences a n phase shift, whereas reflection from an optically thinner medium or transmission into an arbitrary medium does not cause any phase shift. This is analogous to the situation in our experiment, because an atom in 12) sees a negative light-shift potential, corresponding to an optically thicker medium, whereas an atom in 13) sees a positive potential,

52

Stephan Durr and Gerhard Rempe

corresponding to an optically thinner medium. Hence an atom experiences a n: phase shift only if it is reflected and in (2). In the following discussion, we show how this n phase shift can be used to store which-way information by applying a microwave field.

B. COMBINATION WITH THE MICROWAVE FIELD The microwave field is resonant with the 12) tf 13) transition and induces Rabi oscillations between these states. Using the basis { 12), 13)) for a matrix representation of the internal degree of freedom, the Rabi oscillations are described by the time evolution operator

=

(cos(cp/2) sin(cp/2)

- sin(cp/2)

cos(cp/2)

)

where we have chosen an interaction picture in which the states 12) and 13) have the same energy; a rotating-wave approximation has been performed, and cp denotes the pulse area. In order to convert the 7c phase shift in Eq. (41) into a population difference between the hyperfine states, two microwave n:/2 pulses are applied. They form a Ramsey scheme as shown in Fig. 7(c). The atom is initially prepared in state 12). Then a microwave n:/2 pulse is applied, converting the internal state to the superposition state ((3) 12))/& Next, the atom interacts with the standing light wave. Because of the superposition principle, the interaction can be described for the 12)-component and the 13)-component of the state vector separately. Using Eqs. (40) and (41), the state of the system after the interaction with the standing light wave becomes

+

where the phase factor ei(b3-P2) has been ignored for the moment. Equation (43) reveals that the n: phase shift effectively creates an entanglement between the internal and the external degrees of freedom of the atom. This entanglement is the key point for the storage of which-way information. Finally, the second microwave pulse acting on both beams (the transmitted and the reflected) converts the internal state of beam C to state 13), whereas beam B is converted to state -12). Thus the state vector after the pulse sequence shown in Fig. 7(c) is

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

-1 000

-500

0

500

53

1000

microwave detuning / Hz

50 40

E

v-

30

-1000

-500

0

500

1000

microwave detuning / Hz

-1000

500 microwave detuning / Hz

-500

0

1000

FIG. 8. Determination of the differential light shift via Ramsey spectroscopy. The number of atoms that are in beam C and simultaneously in state 13) is measured as a function of the detuning of the microwave frequency from the atomic resonance. Applying the pulse sequence shown in Fig. 7(c), parts (a), (b), and (c) were recorded at constant x with tpulse= 0 p,22 p,and 44 p,respectively. The fringe amplitude varies because of the Pendellosung. The phase shift of the Ramsey fringes reveals the differential light shift x3 - x2. From fits (solid lines) to the data, we obtained (x3 - x2)/(20,,) = 0.73 f0.02.

Stephan Diirr and Gerhard Rempe

54

Equation (44) shows that the internal state is correlated with the way taken by the atom. The which-way information can be read out later by performing a measurement of the internal atomic state. The result of this measurement reveals which way the atom took. If the internal state is found to be 12), the atom moved along beam B; otherwise, it moved along beam C. So far, we have discussed only the effect of the K phase shift. For a 5050 beamsplitter the additional phase factor ei(p3-p2)equals - 1. Taking this into account, we find that Eq. (44) is modified to

so the states 12) and 13) are exchanged, but this is not essential. The storing process discussed here employs a standing light wave sandwiched between two microwave pulses. It is similar to the technique used by Kunze et al. (1994, 1997b). In passing, we note that the differential light shift x3 - x2 can be measured by scanning the microwave frequency o m around the atomic resonance, similar to the technique of Muller et al. (1995). Counting the number of atoms that are in beam C and simultaneously in state 13), we obtain Ramsey fringes as a function of o m ,as shown in Fig. 8. Without the standing light wave, the population of state 13) exhibits a maximum at the atomic resonance. With the standing light wave, the value of the microwave frequency at this maximum is shifted as a result of the phase difference p3 - pz. Because tpulse is known, the frequency shift of the Ramsey fringes reveals the differential light shift x3 - x2.

VII. Interferometer with Which-Way Information After considering a single beamsplitter, we now return to the complete interferometer. Sandwiching the first Bragg beamsplitter between two microwave 7t/2 pulses stores the which-way information in the internal atomic state, as described above. The state vector after the interaction with the first beamsplitter sandwiched between the two microwave pulses is given in Eq. (45). The second beamsplitter transforms this state vector into

Note that the second Bragg beamsplitter causes a K phase shift for but the population of the internal states remains unchanged. Alternatively, the second (instead of the first) standing light wave can be sandwiched between the two microwave 71/2 pulses. Then the internal state

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

55

contains the information whether the atom was reflected or transmitted in the second standing light wave. The corresponding state vector is

instead of Eq. (46). There is no fundamental difference between these two schemes. In both cases, which-way information is stored and can be read out by measuring the internal state. Clearly, once the atom’s way through the interferometer is known, the interference fringes must vanish. The experimental result is shown in Fig. 9(a): There are no fringes! In order to read out the which-way information, only atoms in state 13) were detected. All other parameters are the same as in Fig. 6(a). Note that the total number of atoms detected here is only half as large as in Fig. 6(a), because half of the atoms are in state 12) where they are not detected. We determined experimentally that the same result is obtained if only atoms in state 12) are detected. Quantum mechanics explains this loss of interference fringes in the following way: If the internal state of an atom is found to be 13), the state vector Eq. (46) collapses to

Because the beams D and F do not overlap in the far field, they do not interfere. The situation is analogous for atoms detected in state 12). But what happens if we do not read out the which-way detector? In this case, we do not have any knowledge about the atom’s way. Can we observe fringes now? No! The mere fact that which-way information is stored in the detector and could be read out destroys the interference fringes. Otherwise, one could first observe the fringes and later read out the detector, which would clearly be in conflict with wave-particle duality. The experimental result is shown in Fig. 9(b). The data were recorded with the same parameters as in Fig. 9(a), but atoms in both internal states were detected, so the which-way detector was not read out. Again, there are no fringes. To calculate the spatial distribution, P ( z ) , of atoms in the far field, the internal degree of freedom must be “traced out.” Using Eq.(46) this yields under the left peak of the envelope

because here the spatial wavefunctions +F(~) and $c(z) vanish. The first two terms describe the mean intensity under the envelope. Interference could

56

Stephan Diirr and Gerhard Rempe

5

2

4

7 3 .-c 0 2

i -

c

;1

0 -1

0

1

position / m m

-1

0

1

position / m m FIG.9. Atomic far-field pattern obtained in the interferometer with which-way information stored in the internal atomic state. In part (a), only atoms in state 13) are detected so that the which-way information is read out, in contrast to part (b), where all atoms are detected. In both cases, the interference fringes are lost as a result of the storage of which-way information.

be created only by the last two terms, but they vanish because (2 13) = 0. Precisely the same entanglement that was required to store the which-way information is now responsible for the loss of interference. In other words, the correlations between the which-way detector and the atomic motion destroy the interference. This explanation for the loss of interference, as proposed by Scully, Englert, and Walther (1991), was criticized by the Auckland group (Storey et al., 1994). They argue that every realistic which-way detection scheme involves such a strong localization of the particle that the uncertainty relation

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

57

implies a momentum transfer sufficiently large to wash out the fringes. This point of view has been discussed controversially during the last five years (Englert et al., 1995; Storey et al., 1995; Wiseman and Harrison, 1995; Wiseman et al., 1997). An important issue in this discussion is “What constitutes a momentum transfer?” In addition to the classical notion of a momentum transfer probability distribution, the Auckland group also considers quantum amplitudes to describe momentum transfer. They find that this “quantum” momentum transfer cannot be less than that required by the uncertainty relation but that the “classical” momentum transfer can vanish. This is the case, for example, in the Scully-Englert-Walther scheme, in contrast to Einstein’s recoiling slit and Feynman’s light microscope, where “classical” momentum transfer occurs in line with the uncertainty principle. However, the results obtained by the Auckland group are valid only for schemes involving a mechanical double slit. It is a peculiarity of double-slit experiments that any which-way measurement yields position information simply because the two beams are spatially separated. The arguments advanced by the Auckland group do not apply to schemes with spatially overlapping beams, such as our experiment, where beams B and C (see Fig. 5) are never separated in position space; the width of these beams is 450 pm and their transverse shift, d, is only a few micrometers. Hence storing which-way information does not imply any storage of transverse position information. Therefore, Heisenberg’s uncertainty relation cannot imply any momentum transfer. Nevertheless, the interference fringes are lost. This clearly demonstrates that the entanglement alone is sufficient to destroy the interference pattern, as has been discussed in more detail by Diirr et al. (1998a).

VIII. Quantum Erasure One might argue that the loss of interference fringes that we have described is not surprising at all, because atoms in state 12) and atoms in state 13) are different objects, so they cannot interfere -just like apples and oranges. However, this analogy is too naive, as the following experiment will show. In order to test this argument, an observable of the internal state with eigenvectors

is measured. Clearly, the result of such a measurement does not reveal any which-way information. Even worse, the which-way information is lost in an irrecoverable way because of the collapse of the wavefunction. Hence the information is not only unknown but also no longer available. In

58

Stephan Diirr and Gerhard Rempe

other words, this particular measurement erases the which-way information. This process is called quantum erasure, a concept proposed by Scully and Driihl(l982) and first demonstrated in neutron interferometry (Summhammer et a/., 1983). This complete loss of which-way information allows us to regain interference fringes. This can easily be shown by projecting the state vector Eq. (47) onto, for example, ((3) 12))/&, with the result

+

0

-1

i

position / mm

5

E

.____________. ______________

4

7-3 .-c f 2 3

8 1

0 -1

0

1

position / mm FIG. 10. Quantum erasure. The which-way information is stored in the internal state as in Fig. 9. In part (a) only atoms in state (13) 12))/fi were detected, in part (b) only atoms in state (13) - 12))/&. This internal-state measurement erases the which-way information. Both subensembles exhibit interference fringes.

+

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

59

Obviously, the collapse of the wavefunction destroyed the entanglement between the internal and external degrees of freedom. The far-field pattern of this subensemble of atoms is displayed in Fig. 10(a), where interference fringes are clearly visible. Figure 10(b) displays the position distribution of atoms detected in state (13) - 12))/&. The corresponding state vector is

Again, the far-field pattern exhibits interference fringes, but now the fringe contrast is reversed; that is, the positions of interference maxima and minima are exchanged with respect to Fig. 10(a). This is due to the R phase shifts between beams D and E and between beams F and G. We emphasize that both subensembles exhibit interference fringes. With apples and oranges, this would be impossible. Note, however, that interference fringes are regained only in the subensembles. The pattern of the full ensemble can be obtained as the (incoherent) sum of the patterns in Figs. 10(a) and 10(b). This sum is clearly identical to the pattern shown in Fig. 9(b), where no fringes are visible. Another interesting aspect of quantum erasure is that it relies on the coherent superposition of the possible ways. If the system was in a statistical mixture of the two possibilities, a quantum erasure experiment could not produce any interference fringes. From this point of view, the quantum eraser tests whether the interaction with the light and the microwave is a unitary evolution. Because the patterns shown in Fig. 10 have the same fringe visibility as the initial pattern, we conclude that no significant dissipation occurred. The concept of quantum erasure can be generalized to partial erasure of the which-way information. Consider, for example, the measurement of an observable with eigenstates

1 +cos-12), sin-13) 2 2 1

1 rl cos-13) - sin-12) 2 2

(53)

This measurement yields full which-way information for q = 0, whereas it completely erases the information for q = ~ / 2 For . intermediate values of 1, only partial erasure is achieved; that is, the internal-state measurement reveals incomplete which-way information. The two subensembles in such a partialerasure experiment exhibit interference fringes with a reduced visibility. Hence, partial wave and partial particle character of one and the same atom can be observed simultaneously.

60

Stephan Durr and Gerhard Rempe

IX. Incomplete Which-Way Information Partial quantum erasure thus offers the possibility of a continuous transition between the wave character and the particle character of atomic subensembles. In all these schemes, full which-way information is stored, but the read-out process is more or less efficient. Alternatively, it is interesting to study the case where only incomplete which-way information is stored, so that full which-way information can never be obtained - no matter how the internal state is read out. In these intermediate situations, interference fringes with a reduced visibility are observed in the total ensemble, in contrast to partial quantum erasure, where only subensembles show interference.

A. GENERALIZED STORING SCHEME In the experiments described so far, which-way information was stored using two microwave pulses with area n/2. This scheme is now generalized to arbitrary pulse areas cp. It is sufficient to consider the case where the areas of both microwave pulses are identical. In this case, the state vector of an atom after passing through the first beamsplitter sandwiched between the two microwave pulses becomes

instead of Eq. (45).It follows that no which-way information is stored for cp = 0, whereas full which-way information is stored for cp = n/2. Obviously, incomplete which-way information is stored for intermediate values of cp. As a consequence, the atomic far-field distribution, Eq. (49), is modified to

Hence the visibility of the interference pattern is reduced by a factor lcos cp I with respect to the maximum value V,,, obtained for cp = 0. Figure 11 shows the spatial interference pattern obtained for different values of cp. All other parameters are the same as in Fig. 6(a). Obviously, the fringes are not shifted; only their visibility is reduced. For cp > n/2, the fringe contrast is reversed because cos cp is negative. Figure 12 displays the measured fringe visibility V as a function of cp, when all atoms are detected and the second standing light wave is sandwiched between the two microwave pulses. The solid line shows the theoretical

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

61

4

cn3 E

F

.r= 2 cn +

c = 1 0 0

0 -1

0

1

position / mm 4v)

E

3-

7

.-C 2cn * C

=

0

1-

0

O T

I

-1

I

0

I

1

position / mm FIG. 1 1 . Interference patterns recorded with a variable amount of which-way information stored. In part (a), with cp = n/3. the fringe visibility is reduced. In part (b), with cp = n, the visibility is restored and the fringe contrast is reversed.

prediction

with V,,, = 0.72 determined from the data point at cp = 0. The reduction of V,,, from unity is due to the finite sizes of the detection laser beam and the upper collimation slit and due to background counts in the fluorescence detection, as discussed in more detail in Durr et al. (1998b). The fact that an increase of the pulse area to cp = 7c fully restores the interference pattern provides further evidence for the conclusion that the

62

Stephan Durr and Gerhard Rempe

60% 40%

20%

0%

0

n/4

n;/2

3d4

n

microwave pulse area cp FIG. 12. Visibility V as a function of the microwave pulse area cp. The solid line is the theoretical expectation Eq. (56).

reduction of the visibility in our experiment is due purely to entanglement, as already discussed in Section VII. B. DISTINGUISHABILITY OF THE WAYS Application of the generalized storing scheme stores incomplete which-way information. This means that from a later measurement of the internal atomic state, we cannot determine the atom’s way with certainty. Obviously, there is still a lack of information after this internal-state measurement. This lack of information can be quantified in various ways. One possibility is to employ the standard measure used in information theory, which is related to the concept of entropy in thermodynamics (see, for example, Shannon, 1948). But any monotonic function of this measure of information would also be an acceptable candidate. In the context of which-way experiments, the most convenient of these candidates is the “distinguishability of the ways,” which can be defined as follows. Assume that an observable W in the Hilbert space of the internal atomic state is measured, in order to read out the which-way information. Let p ( Wi, B) and p ( Wi, C) denote the joint probabilities that the eigenvalue Wi of the observable W is found and that the atom moved along beam B or C, respectively. If Wi is found, the best guess that one can make about the way is to opt for beam B if p(Wi,B) > p(Wi, C) and for beam C otherwise. This yields the “likelihood for guessing the right way”

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

63

LWdepends on the choice of the observable W , because this determines what fraction of the stored which-way information is read out. Consider, for example, the case where full which-way information is stored: Choosing W carefully, one can reach LW = 1, whereas an unfortunate choice of W (as in a full quantum erasure measurement) could result in LW = 1/2, so that one might just as well toss a coin. To measure quantitatively how much whichway information is stored, the arbitrariness of the read-out process must be eliminated. This motivates the definition of the “distinguishability of the ways” (Jaeger et al., 1995; Englert, 1996)

D = -1

+ 2max{Lw) W

using the maximum of LW that is reached for the best choice of W. The ways cannot be distinguished at all if D = 0, and they can be held apart completely i f D = 1. In order to measure the likelihood, Lw, we perform the experiment with only one standing light wave sandwiched between the two microwave pulses, while the other standing light wave is removed. This allows us to measure the atom’s external and internal states simultaneously, because the far-field position of an atom now reveals whether the atom is in beam B or beam C. These measurements yield the joint probabilitiesp ( Wi,B) and p ( Wi, C), from which Lw can be inferred. In order to obtain the distinguishability,D, a suitable observable Wept. has to be found that maximizes Lw. This observable can be measured as explamed in Section 1I.E: An additional (third) microwave pulse is applied before the atoms leave the interaction region, and finally, only atoms in state 13) are detected. It can be shown that with the state vector Eq. (54), the third microwave pulse must have an area of 7c/2 - cp (or 3x/2 -cp etc.) and that this measurement should yield

Figure 13 displays the measured values of D as a function of the area cp of the first two microwave pulses. Simultaneously, the area of the third microwave pulse was tuned, too, in order to measure Wopt for all values of cp. According to Eq. (59), D should reach unity at cp = 4 2 , but in the experiment, we find Dm, = 0.81 f0.02. As discussed in Diirr et al. (1998b), this reduction is due to background counts in the fluorescence detection and to intensity variations of the standing light wave. Both effects reduce the measured value of D by a constant factor, independent of cp, so that the quantity actually measured in the experiment is

64

Stephan Diirr and Gerhard Rempe

100% 1

.c 80% .a ca

60%

0% 0

n14

n12

3n14

n

microwave pulse area ‘p FIG. 13. Distinguishability D as a function of the microwave pulse area cp. The solid line is the theoretical expectation Eq. (60).

instead of Eq. (59). The solid line in Fig. 13 displays Eq. (60) with = 0.81. Note that the distinguishability (as well as the visibility) is an ensemble property. There is no measurement that can be performed on a single atom to yield the value of D (or V). Only an ensemble of atoms can build up an interference pattern or reveal the probabilities from which D can be inferred. In addition, a simultaneous measurement of D and V is not possible. This is because the second beamsplitter must be removed in order to measure the degree of correlation between the internal state and beams B and C, from which D is inferred; but this beamsplitter is needed to produce interference fringes in order to measure V.

D,,,

C . DUALITY RELATION The visibility and the distinguishability quantify how much wave character and how much particle character, respectively, can be attributed to the atom. Obviously, D = 1 enforces V = 0, and V = 1 requires D = 0. Hence in the intermediate regime, the following question arises: Is there a limit on the amount of which-way information that can be stored for a given value of the visibility? The answer is “yes”; such a limit exists in form of the duality relation (Jaeger et al., 1995; Englert, 1996) D2

+ V 2 I1

(61)

This relation is a fundamental limit in quantum mechanics, which applies to every two-beam interferometer and to every type of which-way detector.

a

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

65

~

100%

60% 40% 20% -

p

& A -

=*€*€€€$*$

80%

(VNm,)*

+ (D/Dm=)2

microwave pulse area cp FIG.14. Experimental test of the duality relation based on the data from Figs. 12 and 13. (D(cp)/Dmax)2 ( V ( ( P ) / V , , , is ~ )plotted ~ as a function of cp. According to the duality relation, the data points may not exceed unity. The open circles are close to unity by definition (from Diirr, 1998b).

+

Connecting D and V, the duality relation generalizes the concept of waveparticle duality to the regime of incomplete which-way information and makes a quantitative test of wave-particle duality possible. In our experiment, the reduction of the measured values of the visibility V and the distinguishability D is well understood. In order to test the duality relation, it is therefore justified to divide the measured data from Figs. 12 and 13 by Vm, and D, respectively. The result is shown in Fig. 14, where

is plotted as a function of cp. The data at cp = 0 and cp = x / 2 (open circles) are close to unity by definition. All other data (full circles) are below unity, which means that we find good agreement with the duality relation.

X. Wigner Function All information about a quantum system can be extracted, for example, from its state vector, its density matrix, or - equivalently - its Wigner function:

66

Stephan Durr and Gerhard Rernpe

Here $ ( p ) denotes the wave function in the (not rescaled) momentum representation. Equation (63) is equivalent to the standard definition (Wigner, 1932). The Wigner function is a quasi-probability function on phase space, and it allows an elegant illustration of a quantum state. The position (or momentum) probability distribution of the state can be obtained by integration of the Wigner function along the momentum (or position) axis, respectively. This section discusses the evolution of the Wigner function of an atom moving through our experimental setup. The calculation is based on the theoretical model discussed in the preceding sections. Although realistic parameters of the atom-light interaction are chosen ( x = Ore,, Xtpulse = n/4, and tsep= 4/0,), the parameters of the incoming atomic beam have been modified to minimize the computational effort. In particular, we made two simplifying assumptions. First, we assume that the atomic source creates a Gaussian atomic beam that is initially in a minimum uncertainty state. For a reasonable value of the momentum width of the emitted beam, An,, = 0.25, this corresponds to an extremely small size of the source, Azms = h/n. Second, we assume a much shorter time of flight from the source to the = 4n/w,,. These assumptions greatly interaction region, namely f simplify calculation of the Wigner function. Nevertheless, the results display most of the relevant features that would be obtained for realistic parameters. A realistic treatment would have to take into account that the atomic beam is not diffraction-limited; rather, it is emitted from a thermal source and then collimated with two relatively broad mechanical slits. In addition, the source is much larger, and the beam inside the interferometer is much broader, than assumed in the calculation. Results of our computation are shown in Fig. 15. Figure 15(a) shows the initial Wigner function at the source. Because of our simplifying assumption, the Wigner function is described by a Gaussian. In order to fulfill the Bragg condition, the distribution is centered around n = 1. In Fig. 15(b), the atomic cloud has reached the interaction region. The free flight during tsource leads to a shearing of the Wigner function. The size of the cloud (in position space) has increased. The tilt of the principal axes of the ellipsoid with respect to the coordinate system is due to correlations between the atomic momentum and the position that build up during the free flight. Figure 15(c) shows the Wigner function immediately after the first beamsplitter (without any microwave pulses). The incoming beam is split into a transmitted beam, which is less populated but is hardly affected otherwise, and a Bragg-reflected beam, which is shifted by two photon momenta. This beam creates the lower peak centered around n = -1. Because the reflected and transmitted beams are in a coherent superposition, the Wigner function oscillates in the region between the two peaks. Note

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

67

FIG. 15. Wigner function. Light (dark) regions represent positive (negative) values of the Wigner function. (a) At the idealized source, (b) after free flight, (c) after the first beamsplitter, (d) before the second beamsplitter, (e) after the interferometer, (f) after the interferometer with which-way information stored.

68

Stephan Durr and Gerhard Rempe

that these oscillations average to zero when the momentum distribution is calculated but that they give rise to sinusoidal interference fringes when the (near-field) position distribution is plotted. These fringes describe the atomic standing wave formed by the two counterpropagating atomic beams, which are produced in the Bragg beamsplitter. The following free flight during tsep simply shears the Wigner function again, as displayed in Fig. 15(d). Figure 15(e) shows the Wigner function after the interaction with the second standing light wave. Each of the four peaks corresponds to one of the beams in Fig. 5: the lower left to D, the lower right to E, the upper left to F, and the upper right to G. Oscillations are visible halfway between each pair of peaks, indicating that the four beams are in a coherent superposition. A later observation of the far-field position distribution corresponds to a measurement of the momentum distribution. The oscillations between the two upper peaks (beams F and G) create the right half of the interference pattern observed in the experiment. The oscillations between the two lower peaks (beams D and E) create the left half of the interference pattern. The oscillations between the upper half and the lower half of the Wigner function describe near-field interference fringes, which are not observed in the experiment. Let us now add microwave pulses to store which-way information, so that an entanglement between the internal and external atomic state is created. Because the Wigner function displays only the external degree of freedom, it cannot visualize the complete quantum state of the system any more. Information about the quantum state can be displayed in the form of a Wigner function only if the entanglement is removed. This can be achieved either by performing an internal-state measurement, associated with a collapse of the wavefunction, or by “tracing out” the internal state, if no internal-state measurement is performed. The result for the latter, more interesting case is shown in Fig. 15(Q for atoms that passed through the interferometer with the first beamsplitter sandwiched between two microwave n/2 pulses: Storing which-way information, but ignoring the internal state, obviously destroys the oscillations between the left and the right peaks of the Wigner function. Hence interference in the far field (between beams D and E, or between beams F and G) is lost now. The remaining oscillations give rise to near-field interference fringes only between beams D and F and between beams E and G, which are not observed in the experiment. Note that the tiny recoil of the microwave photons has not been taken into account in this computation. Again, this section shows that the loss of interference, as described by the disappearance of the relevant oscillations in Fig. 15(f), is due only to the entanglement with the internal degree of freedom.

WAVE-PARTICLE DUALITY IN AN ATOM INTERFEROMETER

69

XI. Conclusion Atom interferometry enables us to address fundamental aspects of the quantum theory of measurement in real experiments. We have shown that the loss of interference fringes in a which-way experiment need not be due to Heisenberg’s uncertainty relation. We have also demonstrated how interference can be restored by erasing the which-way information. In addition to this, we investigated a new regime, where only incomplete which-way information is available. This allowed us to study the continuous transition between wave and particle pictures and to test wave-particle duality in a quantitative manner. Future experimental investigations in this field might focus on the investigation of decoherence effects in order to study the transition between quantum and classical physics, leading to a more detailed understanding of the processes underlying the collapse of the wavefunction (see, for example, Wheeler and Zurek, 1983; Zurek,1991; Haroche, 1998).

XII. Acknowledgments We thank T. Nonn for help in the experiment and acknowledge fruitful discussions with S. Kunze. This work was supported by the Deutsche Forschungsgemeinschaft.

References Adams, C. S., and Riis, E. (1997). Progr: Quantum Electron. 21, 1. Adams, C. S., Sigel, M., and Mlynek, J. (1994). Phys. Rep. 240, 145. Badurek, G., Rauch, H., and Tuppinger D.(1986). Phys. Rev. A 34, 2600. Berman, P. R., Ed. (1997). Atom interferometry. Academic Press (New York). Bemet, S., Oberthaler, M. K., Abfalterer, R., Schmiedmayer, J., and Zeilinger, A. (1996). Phys. Rev. Lett. 77, 5160. Bernhardt, A. F., and Shore, B. W. (1981).Phys. Rev. A 23, 1290. Bohr, N. (1949).In P. A. Schilpp (Ed.), Albert Einstein: Philosopher-scienrist, Library of Living Philosophers (Evanston, IL). Reprinted in Wheeler and Zurek, 1983. Brune, M., Hagley, E., Dreyer, J., Maali, X., Wunderlich, C., Raimond J. M.,and Haroche, S. (1996). Phys. Rev. Lett. 77,4887. Buks, E., Schuster, R., Heiblum, M., Mahalu, D.,and Umansky, V. (1998). Nature 391, 871. Chapman, M. S., Hammond, T.D., Lenef, A., Schmiedmayer, J., Rubenstein, R. A., Smith, E., and Pritchard, D. (1995). Phys. Rev. Lett. 75,3783. Clauser, J. F., and Li, S. (1994). Phys. Rev. A 50, 2430. Diirr, S., Kunze, S., and Rempe, G. (1996). Quantum Semiclass. Opt. 8, 531. Dun; S., Nonn, T., and Rempe, G. (1998a). Nature 395, 33. Dun; S., and Rempe, G. (1999). Phys. Rev. A 59, 1495. Diirr, S., Nonn, T., and Rempe, G. (1998b). Phys. Rev. Len. 81,5705.

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Eichmann, U., Bergquist, J. C., Bollinger, J. J., Gilliban, J. M., Itano, W. M., Wineland, D. J., and Raizen, M. G. (1993). Phys. Rev. Lett. 70,2359. Englert, B.-G. (1996). Phys. Rev. Lett. 77, 2154. Englert, B.-G., Scully, M. 0.. and Walther, H. (1995). Nature 375, 367. Ewald, P. P. (1917). Ann. Physik 54, 519. Feynman, R. P., Leighton, R. B., and Sands, M. (1965). In The Feynman lectures onphysics, vol. III, chap. 1. Addison-Wesley (Reading, MA). Gibble, K., and Chu, S. (1993). Phys. Rev. Lett. 70, 1771. Giltner, D. M., McGrown, R. W., and Lee, S. A. (1995a). Phys. Rev. Lett. 75, 2638. Giltner, D. M., McGrown, R. W., and Lee, S. A. (1995b). Phys. Rev. A 52, 3966. Haroche, S. (1976). In Shimoda K. (Ed.), High-resolution laser spectroscopy (Topics in Applied Physics, vol. 13) Springer (New York). Haroche, S. (1998) Phys. Today 5 1 (7). 36. Heisenberg, W. (1927). Z. Phys. 43, 172. Hellmuth, T., Walther, H., Zajonc, A., and Schleich, W. (1987). Phys. Rev. A 35, 2532. Herzog, T. J., Kwiat, P. G., Weinfurter, H., and Zeilinger, A. (1995). Phys. Rev. Lett. 75, 3034. Jaeger, G., Shimony, A., and Vaidman L. (1995). Phys.Rev. A 51, 54. Kazantsev, A. P., Surdutovick, G. I., and Yakovlev, V. P. (1990). Mechanical action of light on atoms. World Scientific (London). Kunze, S., Rempe. G., and Wilkens, M. (1994). Europhys. Lett. 27, 115. Kunze, S., Diirr, S., and Rempe, G. (1996). Europhys. Lett. 34, 343. Kunze, S., Diirr,S., Dieckmann, K., Elbs, M., Ernst, U., Hardell, A,, Wolf, S., and Rempe, G. (1997a). J. Mod. Opt. 44, 1863. Kunze, S., Diekmann, K., and Rempe, G. (1997b). Phys. Rev. Lett. 78,2038. Marte, M., and Stenholm, S. (1992). Appl. Phys. B 54,443. Martin, P. J., Oldaker, B. G., Miklich, A. H., and Pritchard, D. E. (1988). Phys. Rev. Lett. 60,515. Miiller, J. H., Bettermann, D., Rieger, V.,Sengstock, K., Stem, U., and Ertmer, W. (1995). Appl. Phys. B 60, 199. Pfau, T., Spalter, S., Kurtsiefer, C., Ekstrom, C. R., and Mlynek, J. (1994). Phys. Rev. Lett. 73, 1223. F’ritchard, D. E., and Gould, P. (1985). J. Opt. Soc. Am. B 2, 1799. Ramsey, N. F. (1950). Phys. Rev. 78,695. Rauch, H., Zeilinger, A., Badurek, G., Wilfing, A., Bauspiess, W., and Bonse, U. (1975). Phys. Lett. A 54, 425. Shannon, C. E. (1948). Bell Syst. Techn. J. 27, 379. Schumacher, E., Wilkens, M., Meystre, P., and Glasgow, S. (1992). Appl. Phys. B 54, 451. Schwindt, P. D. D., Kwiat, P. G., and Englert, B.- G. (submitted). Scully, M. 0.. and Driihl, K. (1982). Phys. Rev. A 25, 2208. Scully, M. O., Englert, B.- G., and Walther, H. (1991). Nature 351, 11 1. Storey, E. P., Tan, S. M., Collett, M. J., and Walls, D. F. (1994). Nature 367, 626. Storey, E. P., Tan, S. M., Collett, M. J., and Walls, D. F. (1995). Nature 375, 368. Summharnmer, J., Badurek, G., Rauch, H., Kischko, U., and Zeilinger, A. (1983). Phys. Rev. A 27, 2523. Weizsacker, C. F. v. (1931). Z Phys. 70, 114. Weizsacker, C. F. v. (1941). Z. Phys. 1 18,489. Wheeler, J. A. (1978). In Marlow, A. R., (Ed.) Mathematical foundations of quantum theory Academic Press (New York). Wheeler J. A., and Zurek, W. H. (1983). Quantum theory and measurement. Princeton University Press (Princeton, NJ). Wigner E. (1932). Phys. Rev. 40, 749.

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Wilkens, M., Schumacher, E., and Meystre, P. (1991). Phys. Rev. A 44, 3130. Wiseman, H. M., and Harrison, F. E. (1995). Nature 377, 584. Wiseman, H. M., Harrison, F.E., Collett, M. J., Tan, S. M., Walls, D. F., and Killip, R. B. (1997). Phys. Rev. A 56, 55. Wooters, W. K., and Zurek, W. H. (1979). Phys. Rev. D 19, 473. Reprinted in Wheeler and Zurek, 1983. Zou, X. Y., Wang, L. J., and Mandel, L. (1991). Phys. Rev. Lett. 67, 318. Zurek, W. H. (1991). Phys. Today 44 (lo), 36.

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ADVANCES IN ATOMIC, MOLECULAR, AND OFTICAL PHYSICS,VOL.42

ATOM HOLOGRAPHY FUJI0 SHIMIZU Institute for Laser Science, University of Electro-Communications, Chofu-shi, Japan

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

13

11. Atomic Beam Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Coherent Flux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 74

I. Introduction. . . . . . .

111.

IV. V.

VI. VII. VIII.

B. Source and Detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. An Example: Metastable Neon Beam Source . . . . . . . . . . . . . . . Design of Thin-Film Hologram . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Generation of a Simple Pattern . . . . . . . . . . . . . . . . . . . . . . . . . B. Fraunhofer Hologram by Subdivision of a Cell. . . . . . . . . . . . . . C. Hologram with Imaging Function . . . . . . . . . . . . . . . . . . . . . . . D. Gray-Scale Hologram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quality Evaluation . . . . . . . . ....................... Other Possible Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Phase Hologram . . . . . . ......................... B. Bragg Hologram by a St ght Wave . . . . . . . . . . C. Optical Reconstruction of Atomic Interference . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . ....................... References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

16 I1 81 81 84 86 89 90 90 91 91 92 92 92

I. Introduction Holography is a technique to manipulate the wavefront of a wave. An arbitrary wave pattern is generated by passing a wave of simpler form through a film called a hologram that has an appropriate complex transmission function. The technique was demonstrated with an optical wave by Gabor (1948) with the expectation to improve the imaging quality of an electron microscope. For the optical wave a wide variety of holographic techniques were developed since the 1960s after the invention of the laser. Holography has been used as a tool to store and reconstruct three-dimensional images of an object. Holography is applicable in principle to any matter wave with monochromatic energy that satisfies the Helmholz equation. The process of holography may be divided into two steps. In the first step, the interferometric pattern between the object and reference waves is recorded on the hologram. In the second step, the wave for the reconstruction is sent through the hologram and is diffracted to produce the object wave. The waves in the first step may be optical waves or any matter waves, or the interference pattern of the two waves may be constructed by computer calculation. In the latter case, the 13

Copyright 02000 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-003842-011SSN 1049-25OXlOO$30.00

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Fuji0 Shimizu

technique is called computer holography. The wave in the second step must be the one through which we would like to observe the object. Owing to the short de Broglie wavelength of particles with mass, however, there were few examples of matter-wave holography. In 1979 Tonomura et al. read optically the interference pattern of electron waves that was recorded on the detector of an electron microscope. The technique has been used to observe phase objects in electron microscopes (see, for example, Tonomura, 1995). Another example of matter-wave holography is limited to atoms, in which an arbitrary image of atoms was produced by a computer-generated hologram (Fujita et al., 1996, Morinaga et al., 1996). Although the wave nature of atoms and molecules was experimentally verified nearly 70 years ago (Estermann and Stem, 1930), there was little development until Gould et al. (1986) studied diffraction of a Na atomic beam by a transmission-type grating made of SiN film. Interferometry of atoms has made large progress in the last 10 years because of the development of laser cooling. In the early 1990s, various types of atom interferometers were demonstrated. They include Young's double-slit interferometer (Carnal and Mlynek, 1991) and the Mach Zender interferometer that uses gratings as beamsplitters and deflectors (Keith et al., 1991). They used thermal beams that have de Broglie wavelength in the range of 10-"m, which was too short to observe the interference pattern as a two-dimensional image by a detector with spatial resolution. Such a mode of detection is possible with a sufficiently cooled atomic beam. Shimizu et al. (1992) recorded Young's double-slit interference pattern of laser-cooled neon atoms on a microchannel plate detector. The first interferometric manipulation of atoms that aimed at atomic image formation was the focusing of an atomic beam by Fresnel lens (Carnal et al., 1991). Production of an atomic pattern by a computer-generated hologram, demonstrated by our group, is the generalization of those techniques. We describe in this article the present status of holographic manipulation of atoms and comment on the possible future of this technique.

11. Atomic Beam Source A. COHERENT FLUX The flux of an atomic beam is limited by collisions of atoms in the beam. It increases with the velocity of atoms. However, for the holographic application, the hologram has to be illuminated coherently. Consider that atoms are released through a hole with diameter a from a reservoir containing atoms with average velocity v and density n,. The number of atoms passing through

75

ATOM HOLOGRAPHY

the hole per unit time F is approximately x n,va 2 F =16

However, only those atoms that are emitted within the diffraction angle h/a can be used for interferometry. Therefore, the coherent flux Fcoh is

h2

Fcoh = -n,va

2

16a2

h2ns 16m2v

=-

Atoms should not collide with each other in the beam. The atomic density in the beam is largest at the exit and keeps the same density as in the source approximately by the distance of a. To avoid collisions within this distance, the atomic density must be smaller than

n,

1 cJa

<-

where c~ is the collision cross section. Inserting this relation into the above equation we obtain the maximum coherent flux h2

Fcoh

< 16m2ova

This equation tells us that to obtain larger flux, one should use colder atoms and a smaller source. The absolute value of Fcoh in the above expression is many orders of magnitude smaller than the value of an optical beam. At very low temperature, the collisional rate (JV,rather than the collisional cross section 0,is constant. For laser-cooled alkali atoms, (JV lo-'* m3 s-l. Inserting a magneto-optical trap of typical size (a lop3m and m 3x kg), one obtains Fcoh < 3 x lo4s-I, which is 11 orders of magnitude smaller than the photon flux of a 1-mW He-Ne laser. Considering that approximately lo6 atoms are necessary to draw a black-and-white image with the resolving power of lo3 x lo3, this number is by no means sufficient. The atomic beam that is released from a Bose-Einstein condensate (BEC) of atoms can improve the situation (Anderson et al., 1995; Davis et al., 1995; Bradley et al., 1997;Mewes ef al., 1997).However, the present production rate of BEC atoms is not higher than the limiting value of a thermal source [Eq. (l)]. N

N

N

AND DETECTOR B. SOURCE

A wide range of atoms can be shaped into a beam at room temperature or slightly elevated temperature. The de Broglie wavelength of atoms at room

76

Fuji0 Shimizu

temperature is on the order of lo-" m. Because of the technical limitation of interferometric components, it is difficult to obtain a diffraction angle larger than rad. This limits the size of the reconstructed image. Laser-cooled atoms are free from this restriction. However, it is not practical to use an atomic beam slower than 1m/s on earth, becasue such beams are constantly accelerated by gravity. It is, nevertheless, useful to make the atomic beam from a colder source, because the acceleration by gravity compresses the velocity spread. Up to this time, alkali, alkali earth, metastable rare gas, Yb, and Cr atoms have been laser-cooled and trapped. Chromium is stable on a solid surface and has been used to demonstrate nanometer-scale deposition of atoms by a standing wave of light (Timp et al., 1992; McClelland et al., 1993). Patterns of alkali and metastable rare gas atoms can be recorded on solid surfaces by using proper photoresistant materials (Kreis et al., 1996; Nowak et al., 1996; Bard et al., 1997; Younkin et al., 1997). The pattern of the laser-cooled atoms can be detected with high sensitivity by resonance fluorescence of the cooling transition, because a single atom emits many spontaneous photons during the illumination. The resolution of optical detection is limited by the photon recoil during the absorption and spontaneous emission processes. Metastable rare gases are sensitive to charged particle detectors that have charge-amplifying capability, such as a micro-channel plate (MCP) detector, and thus they can be detected with very high sensitivity. The spatial resolution of an MCP is on the order of 10 pm. A method to obtain higher resolution was developed by Kurtsiefer and Mlynek (1997). They let metastable atoms hit a metal surface and imaged the pattern of ejected Penning electrons on an MCP by using magnifying electron optics. C. AN EXAMPLE: METASTABLE NEONBEAMSOURCE

We describe here a neon atomic beam in the metastable ls~('P0)state that was used in the work in the following sections. Figure 1 shows the apparatus. 2 ) are generated by a weak discharge in Metastable atoms in the l s ~ ( ~ Pstate the channel of the nozzle. The nozzle was made of sapphire and cooled by liquid nitrogen by contact. To increase the flux, the metastable atoms are deflected and collimated by resonant laser beams at 640 nm that travel zigzag between two sets of curved mirrors (Shimizu et al., 1990). The atoms are decelerated in the Zeeman slower and trapped in a magneto-optical trap with a four-laser-beam configuration (Shimizu et al., 1991). A laser beam at 598 nm is focused into the cloud of trapped atoms, and the Is5 atoms are transferred to the ls3 state by optical pumping. Because the ls3 atoms are free from the trapping force, they fall vertically in response to gravity.

77

ATOM HOLOGRAPHY Transfer Deflection

source

18, neon beam

FIG.1. The schematics of a Is3 metastable neon beam source.

111. Design of Thin-Film Hologram Although the basic principle of holography is the same for any kind of wave, actual implementation of atom holography is considerably different from that of optical holography. First, atoms do not transmit through solid material coherently. As a result, holograms for atoms that have been demonstrated to date used thin films with real holes, and their transmission function is binary. Holograms with continuous transmission functions may be constructed if one utilizes coherent reflection from a solid surface. It is also possible, in principle, to make a hologram with near resonant light wave that has a continuous function. The second difference is that the flux of an atomic beam is many orders of magnitude smaller than that of an optical wave. The number of atoms determines the maximum complexity of the pattern that can be drawn. Third, atoms are constantly accelerated by gravity and deflected downward. For cold atoms, the resulting de Broglie wavelength can be much shorter than that at the initial state. Let us consider that a monochromatic atomic wave is emitted from a point source at r = (0,0,O)and passes through the hologram placed on the plane z = -II with the transmission function f ( r )(see Fig. 2). When the diffraction angle is small, the complex amplitude of the wave F(R) on the screen at z = -I1 - 12 is expressed to a good approximation by F(R) = A

s

f(r)ei@(r9R)dr

where the accumulated phase

@(r,R ) = -

pdl=

"I

-

h

v2dt

78

Fuji0 Shimizu

Z

.

Source

-r

HoI ogram

Screen FIG.2. Trajectory of an atom from the source to the screen through the hologram.

ATOM HOLOGRAPHY

79

is calculated along the classical pass of the atom that passes through r and R. In this expression, p and v are the classical momentum and the velocity of the atom, respectively. The transmission function of the hologramf(r) is obtained by solving F ( R ) from Eq. (2). Assuming that the gravity force is along the z-axis and that the horizontal velocity is much smaller than the vertical velocity at z = 11, one can expand @ in a power series of r and R as follows:

where m is the mass of the atom; v,, vh, and V d are the velocity at the source, hologram, and screen, respectively; and tl and f 2 are the transit times of the atom from the source to the hologram and from the hologram to the screen, respectively. The first line of the equation is the phase between the source and the hologram, and the second line is the phase between the hologram and the screen. Because the first of the two lines are independent of r and R, the pattern on the screen may be obtained by taking the second terms of two lines only:

This expression is a Fourier integral of the transmission function f ( r ) multiplied by quadratic phase factors. Therefore, the transmission function of the hologram is obtained by the inverse Fourier transform of the atomic image F(R) on the screen as follows:

The range of validity of this expression is estimated from the magnitude of the fourth-power terms o f t and R in Eq. (3). Considering that h / ( m v ) is the de Broglie wavelength of the atom and that vf2 is approximately the length between the hologram and the screen, the condition is given by

80

Fuji0 Shimizu

where 1 is the distance between the hologram and the screen, and R is the size of the image or hologram. When this condition is not satisfied, one has to solve Eq. (2) directly to obtain the transmission functionf(r). Although Kirchhoffs diffraction theory guarantees arbitrary three-dimensional wave generation by specifying the boundary condition on a surface, real implementation is limited by the physical restriction of holograms. The hologram is a passive component. Therefore, the amplitude of f ( r ) cannot exceed unity. When the reconstruction beam illuminates the hologram uniformly, the amplitude off(r) must be more or less uniform to make the average transmission of atoms lie in a practical range. When 1 >> R, the gross structure of f ( r )is the Fourier transform of the image F ( R ) .If the image has a uniform phase, or is “specular,” its Fourier transform shows an extremely large peak around the origin r = (0,O).Even if we set If(0,O)l = 1, the rest of the hologram becomes almost opaque. Uniform amplitude distribution on the hologram plane is obtained only when the phase or the amplitude of the object rapidly changes. The situation is the same for any wave, including an optical wave. The object may have a curved surface to obtain a three-dimensional view. This does not mean the generation of an arbitrary wave field. The wave that is scattered from the diffuse surface of the object has random characteristics and cannot be used for practical purposes except on the imaging surface. For the purpose of pattern generation of atoms, the random phase does not affect the outcome. In special cases, three-dimensional waves with definite phases can be constructed.One such case is when the hologram is placed close to the image plane. Another example occurs when the image is very small so that the image is reconstructed from a large area of the hologram. The Fresnel lens of atoms (Camel el al., 1991) may be interpreted as this kind of hologram. To construct the hologram, we divide the imaging area and the hologram surface by 2N square meshes, where N is an integer. We assign a single value to each cell of the mesh. To make the image diffuse, a random phase factor exp{i+rand(m,,m y ) } is first multiplied on each value F(m,,m y ) of the image plane, where m, and my show the position of the cell along the x- and y-axes. Then the transmission function of the hologramf(j,, j y ) is calculated by the digitized from of Eq. (4),

using the fast-Fourier-transform algorithm. In Eq. (5), d and D are the cell length of the hologram and image, respectively. There are many ways to

81

ATOM HOLOGRAPHY

express the complex valuef(j,, j y ) of the hologram. We describe below two methods that we used to make atomic images.

A. GENERATION OF A SIMPLE PATTERN Generation of a simple pattern is often possible without randomization of phase. In this case, both phase and amplitude have a predictable function in the entire dimensional space. However, as a tool to generate a two-dimensional pattern of atoms, the hologram without phase randomization generally either is inefficient or produces only a localized simple pattern. Nevertheless, such a hologram is useful as a component for atom optics, such as beamsplitters, and as a tool to produce a simple pattern, particularly in the near Fraunhofer diffraction region. Figure 3 shows the diffraction pattern of atoms generated by a two-dimensional array of rectangular holes. The hole has the dimensions 250 by 500 nm and is separated from the adjacent holes by 250 nm [Fig. 3(a)]. The hologram was a SiN4 film 100 nm thick and 0.5 pm square. It was illuminated by a point atomic source described in Section II.C from 41 cm above the hologram. The diffractionimage was taken by a micro-channel plate detector placed 46 cm below the hologram. The diffraction pattern [Fig. 3(b)] has, as expected, the same shape as an optical diffraction pattern produced by the holes of similar shape. The figure reflects the characteristics of the source and the metastable Ne atomic wave. The efficiency of the optical pumping from the Is5 state to the 1s3 state is approximately 50%, and the remaining atoms decay to the ground state by emitting a 74-nm the VUV photon. The zeroth-order diffraction pattern of the VUV photons is clearly seen overlapping on the atomic pattern. Its size is approximately 1.5 times larger, because photons move in a straight path, whereas atoms take parabolic paths. In principle, for the same reason, the photon image can be removed by tilting the atomic path from the vertical direction. The metastable atoms may be quenched while passing through the hologram. The intensity of the first-order diffraction pattern relative to the zeroth pattern in the figure is slightly larger than that expected from the calculation. This shows that the effective size of the hole is smaller than the real size, a difference that is probably caused by the quenching of metastable atoms that pass near the 100-nm wall of the hole of the hologram. The quenching distance obtained from the intensity ratio is 12 nm. This distance must be taken into account if one tries to control the transmission function by the shape of the hole. B. FRAUNHOFER HOLOGRAM BY SUBDIVISION OF A

CELL

The method that follows is basically the same as that developed by Lohmann and Paris (1967). (See also Brown and Lohmann, 1966, for the computer

82

Fuji0 Shimizu b

b

b

b

b

b

00

no no 0 . b . b b

b

FIG.3. Diffraction pattern from a two-dimensional array of rectangular holes. (a) The pattern of holes. (b) The atomic pattern.

hologram in optics). The complex number is expressed by controlling the shape and position of the hole in a cell. Let us consider an imaginary plane that is tilted by the angle 8, = sin(h/d) from the hologram plane along the xaxis. For the wave that is diffracted perpendicularly to the reference plane, the distance between the hologram and the reference plane changes by h as the distance along the x-axis moves by the cell length d (see Fig.4). Therefore, the phase of the atomic wave relative to the reference plane changes as 2 m / d .

83

ATOM HOLOGRAPHY

Z h

d

< I

I

I I

I

Hologram surface

> x

-2 + i

FIG.4. A method to express a complex number on the hologram. Each point (cell) of the hologram is divided into 4 x 4 subcells. The complex transmission coefficient of the cell is determined by the number and position of open subcells (white area in the figure). The example of the figure shows the number that is proportional to -2 i.

+

The partial wave coming out from the slit on the hologram that is open at the distance Ax from the edge of the cell and has a length Ay has the complex amplitude proportional to Ayexp(27ciAxld) on the reference palne. If the angular spread Bi of the atomic image is small, 2Nd8i << h, this technique expresses a complex number to good accuracy. However, a narrow slit results in a wider diffraction of the atomic wave, and atoms are dispersed to images of many diffraction orders. In Fig. 4, a cell is divided into 4 x 4 square subcells. The phase is expressed in units of n/2, and the amplitude is digitized to five linear steps. The example in the figure shows the transmission function proportional to -1 2i. Figure 5 shows the result of the holography that was designed to draw a letter F by this method (Fujita et al., 1996). The hologram was designed to produce the image at infinite distance by the source placed at infinite distance. The hologram had 256 x 256 cells. Each cell was divided into 4 x 4 subcells as shown in Fig. 4. The hologram was a SiN4 film lOOnm thick and with an area of 350 x 350 pm. The size of the unit cell was 1400 x 1400nm, and the

+

84

Fuji0 Shimizu

FIG.5. The atomic pattern reconstructed by the hologram that is coded by the method described in Fig. 4.

size of a hole was 350 x 350 nm. The atomic beam source, the hologram, and the MCP detector were palced vertically. Figure 5 shows the image that was reconstructed on the MCP. Because the hole size is one-fourth of the cell size, atoms are diffracted approximately over the f4th-order diffraction patterns. Patterns on all orders along the y-axis are seen in the figure. Along the x-axis, some patterns disappear because of the way the phase is expressed. The phase between the adjacent subcells jumps by nN,/2 for the N,th diffraction pattern along the x-axis. At the angle of the N, = 4N lth diffraction, the phase is accurately expressed to produce the designed image. Therefore, the real image appears at this angle. At the angle (4N 2 ) h / d , the phase changes by 5c, and the interference between the odd and even subcells cancels the wave. At the angle 4N h / d , all holes have a positive real value producing a nondiffracted pattern. Similarly, the conjugate patterns are produced at the angle (4N 3 ) h / d .

+ +

+

C. HOLOGRAM WITH IMAGINGFUNCTION In the preceding example, the size of the hole is much smaller than the size of the cell, and images of many orders have nearly equal intensities. There is a more direct way of expressing the transmission function of the hologram,

ATOM HOLOGRAPHY

85

which can concentrate a large fraction of atoms in a single set of images (Morinaga et al., 1996). To approximate the complex transmission function f ( j x ,j y ) by a binary function, we add the conjugate functionft(jx, j y ) to make& j y ) real:

Then we set a threshold f, to approximatef,(j,, j y ) with the binary function

where a hole is open for the cell with f b ( j x ,j y ) = 1. This procedure is straightforward, and the characteristics of the atomic pattern on the MCP are easily understood from Eqs. (6) and (7). The diffraction is a linear process, and each term of the transmission function, Eq.(6), produces an independent pattern. The conjugate image is created from the conjugate termft(jx, j y ) . The second procedure is similar to adding a constant to the transmission function and produces the nondiffracted image whose shape is close to the projection of the hologram area on the detector. The appearance of three images is a general characteristic of an intensity thin-film hologram. The image is constructed by summing amplitudes of the partial waves that are diffracted from each cell. Therefore, the area on which an image can be drawn is the area covered by the partial waves coming from a single hole. When the hole has a square shape with the side of the cell d, this is a sin function with the first zero at the angle of the first-order diffracted wave fsin(h/d). The higher-order diffraction patterns cancel out, and the majority of atoms fall within this angle. The hologram must be designed to form the image within this angle. When the hole is smaller than d, atoms are diffracted at a larger angle, and the higher-order diffraction patterns do not disappear generally. Because the normal and conjugate images appear on the side of the nondiffracted pattern, they may overlap with the zeroth-order images, unless the image is written within the angle fd/2h. Figure 6(b) shows the holographic reconstruction of the image by this method (Morinagae et al., 1996). The hologram is designed to focus a point atomic source on the MCP with tl = 0.286 s and f2 = 0.123 s. The hologram had 1024 x 1024 cells with the size of 500nm. The thresholdf, was determined so that approximately 8% of the cells were open. Figure 6(a) is the scanning microscope image of a part of the hologram pattern, where black squares are the holes. Figure 6(b) shows the atomic pattern obtained after 2 hours of accumulation and contains approximately 2 x lo5 atoms. The letters

86

Fujio Shimizu

NEC in the lower part of the figure constitute the reconstructed image. The square in the middle is the nondiffracted beam that shows the projection of the pattern of the hologram. The conjugate image covers the entire picture area and is not distinguishable from the background, because it is completely out of focus on the detector.

D. GRAY-SCALE HOLOGRAM The number of atoms required to draw a black-and-white pattern is approximately the number of cells 2 2 N .If one tries to write a photographic pattern with the gray scale of Ns, then the number scales by the same factor N g , and a long exposure time is necessary with currently available coherent atomic sources. The situation can be eased by increasing the area of the hologram. Because the size of the cell is determined by other factors, such as the size of the image and the distance between the hologram and the imaging plane, a larger hologram means a larger array size of the FFT transformation. When high resolution is not necessary, one can avoid massive FFT by dividing the hologram area into smaller sections and by operating FFT independently.

FIG. 6. The hologram with imaging function at a finite distance. (a) Scanning microscope image of a part of the hologram. The black squares are holes. (b) The reconstructed atomic image.

ATOM HOLOGRAPHY

87

-1IIlIU(b) FIG.6. (Continued)

The translational shift of the hologram position by r, causes the phase shift of (rnlhq) (r, . R) in the integrand of Eq. (4). If one choose

m

-(r,

. 0)= 2x

fit2

the phase correction becomes a multiple of 2x. Therefore, the integral Eq. (4) is the same for all sections if the same random function & a n d ( ~ x , my)is used. This condition is automatically satisfied when the length of the image is set equal to the distance between the zeroth and the first diffraction spots. Figure 7 shows the image that was generated by the multiply constructed hologram (Kishimoto et al., 1998; Shimizu et al., 1998). The hologram was made by packing 4 x 4 smaller holograms that had 1024 x 1024 cells. In this example, independent random phase factors were multiplied to

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FIG. 7. A gray-scale hologram. (a) The original photograph. (b) A part of the hologram pattern. (c) Computer reconstructionof the image. (d) The atomic pattern. The actual size of the figure is l O m m by 5 mm.

ATOM HOLOGRAPHY

89

each section, and the FFT were repeated 16 times. The cell size of the hologram was 200nm, and tl = 0.289 s and t 2 = 0.156s. Figure 7(a) is the original pattern, and Fig.7(b) was a part of the hologram. Because the image is placed to the side of the image area, the hologram has holes stretched along the vertical direction. To avoid breaking the hologram film by unidirectional stress, long holes were cut into several pieces so that the maximum length did not exceed three unit cells lengths, 3d. Figure 7(d) is the reconstructed atomic image. The conjugate image is on the right-hand side of the zeroth-order pattern and is not shown in the figure. The accumulation time was 2 hours. The ratio of the open cell was approximately 25%. To avoid forming unsupported islands, the hole size was slightly smaller (160 nm) than the cell size. This generated images of higher-order diffraction. The conjugate image of the (- 1,O)th-orderdiffraction is clearly seen on the left of the zeroth-order image.

IV. Quality Evaluation The ultimate spatial resolution of a wave is the wavelength of the atomic wave,

which can be obtained using a focusing lens with a large aperture. A highquality lens for atoms that can image relatively complex patterns is yet to be developed. When the hologram is the sole optical component to produce an atomic image, the situation is different. The hologram described in the previous section is composed of holes of finite size, where the phase variation of the atomic de Broglie wave within the hole cannot be controlled. Suppose we make a binary image using a hologram with the number of cells 2N x 2N.The maximum information that the hologram can carry is the same as the number of cells 22N.Therefore, the resolving power along an axis cannot exceed 2N. The maximum resolution can be obtained only when the partial waves from all cells contribute to each point of the image. To satisfy this condition over the entire image area, the image size must be larger than the size of the hologram. This means that the resolution of the image cannot exceed the size of a hole of the hologram. The size of the atomic source, the velocity spread of the atoms at the hologram plane and the flatness of the hologram surface are other factors that affect the resolution. Geometrical resolution from the source is the diameter of the source a multiplied by the magnifying factor of the hologram. To obtain the resolving power of 2N intrinsic to the hologram, the atomic

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wave has to illuminate the hologram coherently. This gives an additional condition

where dN = 2Nd is the size of the hologram. The resolving power is also limited by the relative velocity spread Aw/v of atoms at the hologram plane. The phase has to be accurate to within 7c to reconstruct the designed atomic pattern. When the vertical position of the hologram is changed by Az, the pass length of the atom from the source to the screen changes by approximately AzdN/(wt). The hologram surface has to be kept flat so that this value does not exceed h/2. The presence of background atoms is an another factor that affects the quality of the reconstructed image. Mathematically, the transformation from the hologram transmission amplitudef ( x ,y) to the image amplitude F ( X , Y) is unique. The difference between the real transmission and background functions A f ( x ,y) = f ( x , y) -fb(jx, j y ) prodcues spurious images and background. Because f b is a binary function, the amplitude [Aft is of the same magnitude as In examples in the previous section, a random phase factor was multiplied on each cell area. Therefore, the correlation length of Af is the cell length, and Af(x,y) produces random background on the screen, whose total intensity is of the same order as the total intensity of the transmitted wave. Good contrast is not generally expected if the pattern covers a large portion of the image area.

If[.

V. Other Possible Techniques A. PHASEHOLOGRAM The hologram described in the previous sections is a thin intensity hologram, where at best 22%of transmitted atoms contribute to make the image. A phase hologram theoretically has a maximum efficiency of 100%.Although there is no transparent solid for atomic waves, one can use the Zeeman effect or the second-order Stark effect to produce phase shift on an atomic wave. Both effects have been used to focus and trap slow atoms. For the holographic application, the required magnetic or electric field is much weaker, because the phase shift of 2n is sufficient to achieve arbitrary wavefront manipulation. This value is easily realizable with moderate electric potential or current for the micron-size structure of the hologram (Shimizu et al., 1992; Ekstrom et af., 1995).

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BY A STANDING LIGHTWAVE B. BRAGGHOLOGRAM

An alternative method to produce phase shift is optical dipole interaction by an off-resonant light wave. The diffraction of atoms by a standing wave of light has been used as beamsplitters and reflectors of an atom interferometer. Although no complex atomic pattern has been generated, the diffraction by light may be considered a primitive form of atom holography. Bernet et al. (1997) discussed various modes of interferometric operations in terms of holography. Soroko (1997) discussed an arbitrary atomic wave generation by light. When the thickness of the standing wave is much larger than the wavelengths of the laser and atoms, three-dimensional momentum conservation must be satisfied. The process is equivalent to four-wave mixing, in which two waves are optical waves and the remaining waves are atomic de Broglie waves. If a plane wave is used in one of the optical waves, a planar atomic wave that has the same wave vector generates a scattered atomic wave identical to the second optical wave. This is aberration-free transfer of an optical image into an atomic image. The technique is, unfortunately, not really practical, because the gravity acceleration of atoms has a more serious effect on the phase matching. The accumulated phase shift due to the acceleration becomes larger than n after a vertical drop of

This value is only 50 pm for Ne atoms at v = lm/s, and the atomic beam interacts with the optical wave coherently only for approximately 10 optical wavelengths. If the optical waves are thin, the atomic beam is diffracted to the direction that conserves the momentum only in the plane of optical waves. This is the case for a thin-phase hologram. However, it will not be possible to form an arbitrary pattern of the hologram with all optical waves propagating in a thin sheet. It is more practical to pulse the atomic beam. Unlike photons, atoms move slowly, and it is not difficult to define the interaction region within a thin layer, which functions as a thin-phase hologram.

c. OPTICAL RECONSTRUCTION OF ATOMIC INTERFERENCE So far, we have discussed only the manipulation of atomic de Broglie waves by a hologram that was generated by other means. One may consider the reverse process, where the interferencepattern of atomic waves that is created by spatial variation of potential on atoms is read by optical wave, and the pattern of the phase shift caused by the potential is reconstructed as an optical

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image. This was the case in the electron holography demonstrated by Tonomura et al. (1979). The recording techniques of the atomic pattern were described in the previous section. For alkalies and metastable rare-gas atoms, recording on photoresists followed by etching will produce a hologram that is adequate for optical reconstruction. Detection by resonant fluorescence or by an MCP detector possesses higher sensitivities. However, relatively poor spatial resolution will limit the complexity of the reconstructed image.

VI. Conclusions The present experimental status of atom holography is rather primitive. However, it is a promising technique for atom manipulation, because it handles atoms in mass, the patterning is completely general, and it controls the pattern from a distance. The technique can also be used to investigate the spatial phase and amplitude structure of an atomic de Broglie wave. However, to make atom holography practical, the flux of the coherent atomic beam has to be improved considerably.

VII. Acknowledgments All holograms presented in this article are fabricated by J. Fujita of NEC Fundamental Research Laboratories, who, together with S. Matsui, was the first to suggest atom holography by using a Lohmann-type binary hologram (Lohmann and Paris, 1967; Onoe and Kaneko, 1979).

VIII. References Anderson, M. H., Ensher, J. R., Mathews, M. R., Wieman, C. E., and Cornell, E. A. (1995). Science 269, 198. Bard, A., Berggren, K. K., Wilbur, J. L., Gillaspy, J. D., Rolston, S. L., McClelland, J. J., Phillips, W. D., Fkntiss, M., and Whitesides, G. M. (1997). J. Vuc. Sci. Technol. B, 15, 1805. Bernet, S., Abfalterer, R., Keller, C., Oberthaler, M. K., Schmiedmayer, J., and Zeilinger, A. (1997). J. Imaging Sci. Technol. 41, 324. Bradley, C. C., Sackett, C. A,, Tollett, J. J., and Hulet, R. G. (1997). Phys. Rev. Lett. 78, 985. Brown, B. R., and Lohmann, A. W. (1996). Appl. Opt. 5,967. Carnal, O., and Mlynek, J. (1991). Phys. Rev. Lett. 66, 2689. Carnal, O., Sigel, M., Sleator, T., Takuma, H., and Mlynek, I. (1991). Phys. Rev. Lett. 67, 3231. Davis, M. 0.. Mewes, M.-O., Andrews, M. R., van Druten, N. J., Durfee, D. D., Kum, D. M., and Ketterle, W. (1995). Phys. Rev. Lett. 75, 3969. Ekstrom, C. R., Schmiedmayer, J., Chapman, M. S., Hammond, T. D., and Pritchard, D. E. (1995). Phys. Rev. Lett. 51, 3883.

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Estermann, I., and Stem, 0. (1930). Z. Phys. 61,95. Fujita, J., Morinagae, M., Kishimoto, T., Yasuda. M., Matsui, S., and Shimizu, F. (1996). Nature 380, 691. Gabor, D. (1948). Nature 161, 777. Gabor, D. (1949). Proc. Roy. SOC.(London) A 197,454. Gould, P. L., Ruff, G. A,, and Pritchard, D. E. (1986). Phys. Rev. Lett. 56, 827. Keith, D. W., Ekstrom, C. R., Turchette, Q.A., and Pritchard, D. E. (1991). Phys. Rev. Len. 66, 2693. Kishimoto, T., Fujita, J., Mitake, S., and Shimizu, F. (1998). Private communications. Kreis, M., Lison, F., Haubrich, D., Meschede, D., Nowak, S., Pfau, T., and Mlynek, J. (1996). Appl. Phys. B, B63, 649. Kurtsiefer, C., and Mlynek, J. (1997). Appl. Phys. B B64. 85. Lohmann, A. W., and Paris, D. P.(1967). Appl. Opt. 6, 1739. McClelland, J. J., Scholton, R. E., Palm, E. C., and Celotta, R. J. (1993). Science 262, 877. Mewes, M.-0.. Andrews, M. R., Kurn, D. M., Durfee, D. S., Townsend, C. G., and Ketterle, W. (1997). Phys. Rev. Lett. 78, 582. Morinaga, M., Yasuda, M., Kishimoto, T., Shimizu, F., Fujita, J., and Matsui, S. (1996). Phys. Rev. Lett. 77, 802. Nowak, S., Pfau, T., and Mlynek, J. (1996). Appl. Phys. B B63, 203. Onoe, M., and Kaneko, M. (1979). Denshi Tsushin Gukkai Zasshi 62, 118 (in Japanese). Shimizu, F., Shimizu, K., and Takuma, H. (1990). Chem. Phys. 145, 327. Shimizu, F., Shimizu, K., and Takuma, H. (1991). Opt. Lett. 16, 339. Shimizu, F., Shimizu, K., and Takuma, H. (1992). Phys. Rev. A 46, R17. Shimizu, F., Fujita, J., Morinaga, M., Kishimoto, T., and Mitake, S. (1998). Proc. Int. Con$ Atomic Physics. Soroko, A. V. (1997). J. Phys. B 30, 5621. Timp, G., Behringer, R. E., Tennant, D. M., Cunningham, J. E., Prentiss,M., and Berggren, K. K. (1992). Phys. Rev. Lett. 69, 1636. Tonomura, A., Matsuda, T., and Endo, J. (1979). Jpn. J. Appl. Phys. 18, 9. Tonomura, A. (1995). “Electron holography,” Pmc. Int. Workshop on Electron Holograpahy. Younkin, R., Berggren, K. K., Johnson, K. S., Prentiss, M., Ralph, D. C., and Whitesides, G. M. (1997). Appl. Phys. Lett. 71, 1261.

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ADVANCES IN ATOMIC, MOLECULAR, AM) OFTICAL PHYSICS, VOL. 42

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS RUDOLF GRIMM and M A P H I A S WEIDEMULLER Max-Planck-lnstitut f i r Kernphysik, Heidelberg, Germany

YURII B. OVCHINNIKOV National Institute of Standards and Technology, Gaithersburg, MD

I. Introduction .................................... 11. Optical Dipole Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Oscillator Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Multilevel Atoms .............................. 111. Experimental Issues .............................. A. Cooling and Heat .............................. B. Experimental Techniques . ......................... C. Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Red-Detuned Dipole Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Focused-Beam Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Standing-Wave Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

V. Blue-Detune

eTraps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

B. Hollow-Beam Traps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Evanescent-Wave Traps . . . . . . . . . . . . . . . . . . ........ VI. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 97 98 102 108 109 117 121 123 125 133 138

141 145 146 150 155 162 165 165

I. Introduction Methods for storage and trapping of charged and neutral particles have very often served as the experimental key to great scientific advances, covering physics in the vast energy range from elementary particles to ultracold atomic quantum matter. The ultralow-energy region became experimentally accessible as a result of the dramatic developments in the field of laser cooling and trapping that have taken place over the last two decades (Stenholm, 1986; Minogin and Letokhov, 1988; Arimondo et al., 1992; Metcalf and van der Straten, 1994; Chu, 1998; Cohen-Tannoudji, 1998; Phillips, 1998). 95

Copyright 0 2000 by Academic F’ress AU rights of reproduction in any form reserved. ISBN 0-12-003842-01ISSN 1049-25OXl00$30.00

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For charged particles, the strong Coulomb interaction can be used for trapping in electric or electromagnetic fields (Bergstrom et al., 1995; Ghosh, 1995). At the very low temperatures reached by laser cooling, single ions show a variety of interesting quantum effects (Wineland et al., 1995), and ensembles form a crystalline ordered state (Walther, 1993). Laser cooling in ion traps has opened up new possibilities for ultrahigh-precision spectroscopy and related fundamental applications (Thompson, 1993). It is a very important feature of ion traps that the confining mechanism does not rely on the internal structure of the ion, which is therefore accessible for all kinds of experiments. For neutral atoms, it has become routine to produce ensembles in the microkelvin region, and many experiments are being performed with such laser-cooled ultracold gases. It is thus possible to trap the atoms by much weaker mechanisms as compared to the Coulomb interaction. Traps for neutral atoms can be realized on the basis of three different interactions, each class having specific properties and offering particular advantages. 0

0

Radiation-pressure traps operating with near-resonant light (Pritchard et al., 1986; Raab et al., 1987) have a typical depth of a few kelvins, and because of very strong dissipation, they make it possible to capture and accumulate atoms even from a thermal gas. In these traps, the atomic ensemble can be cooled down to temperatures in the microkelvin range. The trap performance, however, is limited in several ways by the strong optical excitation: The attainable temperature is limited by the photon recoil, the achievable density is limited by radiation trapping and light-assisted inelastic collisions, and the internal dynamics is strongly perturbed by resonant processes on a time scale on the order of a microsecond. Magnetic traps (Migdall et al., 1986; Bergeman et al., 1987) are based on the state-dependent force on the magnetic dipole moment in an inhomogeneous field. They represent ideal conservative traps with typical depths on the order of l00mK and are excellent tools for evaporative cooling and Bose-Einstein condensation. For further applications, a fundamental restriction is imposed by the fact that the trapping mechanism relies on the internal atomic state. This means that experiments involving the internal dynamics are limited to a few special cases. Furthermore, possible trapping geometries are restricted by the necessity to use arrangements of coils or permanent magnets. Optical dipole traps rely on the electric dipole interaction with fardetuned light, which is much weaker than all mechanisms discussed above. Typical trap depths are below 1 millikelvin. The optical excitation can be kept extremely low, so that such a trap is not limited by the light-induced mechanisms present in radiation-pressure traps. Under

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appropriate conditions, the trapping mechanism is independent of the particular sublevel of the electronic ground state. The internal groundstate dynamics can thus be fully exploited for experiments, which is possible on a time scale of many seconds. Moreover, a great variety of different trapping geometries can be realized as, for example, highly anisotropic or multiwell potentials. The subject of this review is atom traps of the last class described, along with their unique features as storage devices at ultralow energies. Historically, the optical dipole force, acting as a confining mechanism in a dipole trap, was first considered by Askar’yan (1962) in connection with plasmas as well as neutral atoms. The possibility of trapping atoms with this force was considered by Letokhov (1968), who suggested that atoms might be one-dimensionally confined at the nodes or antinodes of a standing wave tuned far below or above the atomic transition frequency. Ashkin (1970) demonstrated the trapping of micron-sized particles in laser light based on the combined action of radiation pressure and the dipole force. Later he suggested three-dimensional traps for neutral atoms (1978). Bjorkholm et ul. (1978) demonstrated the dipole force on neutral atoms by focusing an atomic beam by means of a focused laser beam. In a great breakthrough, Chu et ul. (1986) exploited this force to realize the fist optical trap for neutral atoms. After this demonstration, enormous progress in laser cooling and trapping was made in many different areas, and much colder and denser atomic samples became available for the efficient loading of shallow dipole traps. In the early 1990s, optical dipole forces rapidly began to attract interest not only for atom trapping, but also in the emerging field of atom optics (Adams er al., 1994). In this review, we focus on dipole traps realized with fur-detuned light. In these traps, an ultracold ensemble of atoms is confined in a nearly conservative potential well with very weak influence from spontaneous photon scattering. The basic physics of the dipole interaction is discussed in Section 11. The experimental background of dipole trapping experiments is then explained in Section 111. Specific trapping schemes and experiments are presented in Sections IV and V, where we explore the wide range of applications of dipole traps and consider particular examples.

11. Optical Dipole Potential Here we introduce the basic concepts of atom trapping in optical dipole potentials that result from the interaction withfur-detuned light. In this case of particular interest, the optical excitation is very low, and the radiation force due to photon scattering is negligible compared to the dipole force. In Section II.A,

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we first consider the atom as a simple classical or quantum-mechanical oscillator to derive the main equations for the optical dipole interaction. Then, in Section I1 B, we discuss the case of real multilevel atoms - in particular, the alkali atoms used in the great majority of experiments.

A. OSCILLATOR MODEL The optical dipole force arises from the dispersive interaction of the induced atomic dipole moment with the intensity gradient of the light field (Askar'yan, 1962; Kazantsev, 1973; Cook, 1979; Gordon and Ashkin, 1980). Because of its conservative character, the force can be derived from a potential, the minima of which can be used for atom trapping. The absorptive part of the dipole interaction in far-detuned light leads to residual photon scattering of the trapping light, which sets limits on the performance of dipole traps. In the following discussion, we derive the basic equations for the dipole potential and the scattering rate by considering the atom as a simple oscillator subject to the classical radiation field. 1. Interaction of Induced Dipole with Driving Field

When an atom is placed into laser light, the electric field E induces an atomic dipole moment p that oscillates at the driving frequency w. In the C.C. and p(r, t) = usual complex notation E(r, t ) = e,!?(r)exp(-iwt) ej(r) exp( -iot) c.c., where e is the unit polarization vector, the amplitude 3 of the dipole moment is simply related to the field amplitude E by

+

+

Here ct is the complex polarizability, which depends on the driving frequency 0.

The interaction potential of the induced dipole moment p in the driving field E is given by 1 2

udip = - - (pE) =

1 2EOC

- -Re(ct) I

where the angular brackets denote the time average over the rapid oscillating and the factor reflects the fact that terms, the field intensity is I = 2~0c1,!?1~, the dipole moment is an induced, not a permanent, one. The potential energy of the atom in the field is thus proportional to the intensity I and to the real part of the polarizability, which describes the in-phase component of the dipole

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oscillation as responsible for the dispersive properties of the interaction. The dipole force results from the gradient of the interaction potential

It is thus a conservative force, proportional to the intensity gradient of the driving field. The power absorbed by the oscillator from the driving field (and re-emitted as dipole radiation) is given by Pabs = (pE) = 2wIm($*)

0

= - Im(a)l EOC

(4)

The absorption results from the imaginary part of the polarizability, which describes the out-of-phase component of the dipole oscillation. Considering the light as a stream of photons hw, the absorption can be interpreted in terms of photon scattering in cycles of absorption and subsequent spontaneous reemission processes. The corresponding scattering rate is

We have now expressed the two main quantities of interest for dipole traps, the interaction potential and the scattered radiation power, in terms of the position-dependent field intensity Z(r) and the polarizability a(w). We point out that these expressions are valid for any polarizable neutral particle in an oscillating electric field. This can be an atom in a near-resonant or far offresonant laser field or even a molecule in an optical or microwave field. 2. Atomic Polarizability

In order to calculate its polarizability cr, we first consider the atom in Lorentz’s model of a classical oscillator. In this simple and very useful picture, an electron (mass m,,elementary charge e) is considered to be bound elastically to the core with an oscillation eigenfrequency 00 corresponding to the optical transition frequency. Damping results from the dipole radiation of the oscillating electron according to Larmor’s well-known formula (see, e.g., Jackson, 1962) for the power radiated by an accelerated charge. It is straightforward to calculate the polarizability by integration of the equation of motion 2 rmi w i x = - e E ( t ) / m , for the driven oscillation of

+

+

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R. Grimm, M. Weidemiiller; and Yu. B. Ovchinnikov

the electron, with the result

Here

202 ro= 671e&Ome c3

(7)

is the classical damping rate due to the radiative energy loss. Substituting e 2 / m e=~ K E O C ~ ~ , /and O ~ introducing the on-resonance damping rate I' = roo= (oo/o)21',,we put Eq. (6) into the form

In a semiclassical approach, described in many textbooks, the atomic polarizability can be calculated by considering the atom as a two-level quantum system interacting with the classical radiation field. One finds that when saturation effects can be neglected, the semiclassical calculation yields exactly the same result as the classical calculation with only one modification: In general, the damping rate (correspondingto the spontaneous decay rate of the excited level) can no longer be calculated from Larmor's formula but is determined by the dipole matrix element between the ground and excited states,

For many atoms with a strong dipole-allowed transition starting from the ground state, the classical formula Eq. (7) nevertheless provides a good approximation to the spontaneous decay rate. For the D lines of the alkali atoms Na, K, Rb, and Cs, the classical result agrees with the true decay rate to within a few percent. An important difference between the quantum-mechanical and the classical oscillator is the possible occurrence of saturation. At too high intensities of the driving field, the excited state gets strongly populated and Eq. (8) is no longer valid. For dipole trapping, however, we are essentially interested in the far-detuned case with very low saturation and thus very low scattering rates (rSc<< I?). We can therefore use expression Eq.(8) as an excellent approximation for the quantum-mechanical oscillator.

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3. Dipole Potential and Scattering Rate

With the above expression for the polarizability of the atomic oscillator, the following explicit expressions are derived for the dipole potential and the scattering rate in the relevant case of large detunings and negligible saturation:

These general expressions are valid for any driving frequency o and show two resonant contributions: Besides the usually considered resonance at 0 = 0 0 , there is also the so-called counter-rotating term resonant at 0

= -00.

In most experiments, the laser is tuned relatively close to the resonance at such that the detuning A _= o - 00 fulfills IAl << WO. In this case, the counter-rotating term can be neglected in the well-known rotating-wave approximation (see, e.g., Allen and Eberly, 1972), and one can set o/oo M 1 . With a few exceptions discussed in Section IV, this approximation will be made throughout this article. In this case of main practical interest, the general expressions for the dipole potential and the scattering rate simplify to 00

The basic physics of dipole trapping in far-detuned laser fields can be understood on the basis of these two equations. Obviously, a simple relation exists between the scattering rate and the dipole potential, ll

which is a direct consequence of the fundamental relation between the absorptive and dispersive response of the oscillator. Moreover, these equations show two very essential points for dipole trapping:

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R. Grimm, M. Weidemiillel; and Yu. B. Ovchinnikov

Sign of detuning. Below an atomic resonance (“red” detuning, A < 0), the dipole potential is negative, and the interaction thus attracts atoms into the light field. Potential minima are therefore found at positions with maximum intensity. Above resonance (“blue” detuning, A > 0), the dipole interaction repels atoms out of the field, and potential minima correspond to minima of the intensity. According to this distinction, dipole traps can be divided into two main classes, red-detuned traps (Section IV) and blue-detuned traps (Section V). Scaling with intensig and detuning. The dipole potential scales as ]/A, whereas the scattering rate scales as ] / A 2 .Therefore, optical dipole traps generally use large detunings and high intensities to keep the scattering rate as low as possible at a certain potential depth.

B. MULTILEVEL ATOMS In real atoms used for dipole trapping experiments, the electronic transition has a complex substructure.The main consequence is that the dipole potential in general depends on the particular substate of the atom. This can lead to some quantitative modifications and to interesting new effects. In terms of the oscillator model we have discussed, multilevel atoms can be described by state-dependent atomic polarizabilities. Here we use an alternative picture that provides very intuitive insight into the motion of multilevel atoms in fardetuned laser fields: the concept of state-dependent ground-state potentials (Dalibard and Cohen-Tannoudji, 1985, 1989). Alkali atoms are discussed as of great practical importance, in order to clarify the role of fine structure, hyperfine structure, and magnetic substructure.

1. Ground-State Light Shifts and Optical Potentials The effect of far-detuned laser light on the atomic levels can be treated as a perturbation in second order of the electric field - that is, it is linear in terms of the field intensity. As a general result of second-order timeindependent perturbation theory for nondegenerate states, an interaction (Hamiltonian 3-11) leads to an energy shift of the ith state (unperturbed energy Ei)that is given by

For an atom interacting with laser light, the interaction Hamiltonian is = -bE, where fi = -er represents the electric dipole operator. For the

3-11

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relevant energies &i, one has to apply a “dressed state” view (CohenTannoudji et al., 1992), considering the combined system “atom plus field.” In its ground state, the atom has zero internal energy, and the field energy is nho according to the number n of photons. This yields a total energy &i = nho for the unperturbed state. When the atom is put into an excited state by absorbing a photon, the sum of its internal energy hoo and the field energy (n - 1)ho becomes &, = hoo (n - 1)ho = -hag nho. Thus the denominator in Eq. (15) becomes &i - &j = hAy. For a two-level atom, the interaction Hamiltoman is ‘FI1 = - p E , and Eq. (15) simplifies to

+

+

for the ground and excited states (upper and lower signs, respectively); we have used the relation I = ~ E O C and I B Eq. ~ ~ (9) to substitute the dipole matrix element with the decay rate r. This perturbative result obtained for the energy shifts reveals a very interesting and important fact: The optically induced shift (known as the light shift or ac Stark shift) of the ground state exactly corresponds to the dipole potential for the two-level atom [Eq.(12)]; the excited state shows the opposite shift. In the interesting case of low saturation, the atom spends most of its time in the ground state, and we can interpret the light-shifted ground state as the relevant potential for the motion of atoms. This situation is illustrated in Fig. 1.

ho

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For applying Eq. (15) to a multilevel atom with transition substructure,' one has to know the dipole matrix elements p, = (eilplgi) between specific electronic ground states Igi) and excited states le,). It is well known in atomic physics (see, e.g., Sobelman, 1979) that a specific transition matrix element

can be written as a product of a reduced matrix element 1IpJIand a real transition coefficient c,. The fully reduced matrix element depends on the electronic orbital wavefunctions only and is directly related to the spontaneous decay rate I'according to Eq. (9). The coefficients cd, which take into account the coupling strength between specific sublevels i and j of the electronic ground and excited states, depend on the laser polarization and the electronic and nuclear angular momenta involved. They can be calculated in the formalism of irreducible tensor operators or can be found in corresponding tables. With this reduction of the matrix elements, we can now write the energy shift of an electronic ground state Igi) in the form

where the summation is carried out over all electronically excited states lei). This means that for a calculation of the state-dependent ground-state dipole potential Udip,i= A&i,one has to sum up the contributions of all coupled excited states, taking into account the relevant line strengths c$ and detunings A,.

2. Alkali Atoms Most experiments in laser cooling and trapping are performed with alkali atoms, because their closed optical transitions lie in a convenient spectral range. The main properties of alkali atoms that are relevant for dipole trapping are summarized in Table I. As an example, the full-level scheme of the relevant ns + n p transition is shown in Fig. 2(a) for a nuclear spin I = 3/2, as in the case of 7Li, 23Na,39741K, and 87Rb.Spin-orbit coupling in the excited I Perturbation theory for nondegenerate states can be applied in the absence of any coupling between degenerate ground states. This is the case for pure linear II or circular of polarization, but not for mixed polarizations where Raman couplings between different magnetic substates become important; see, for example, Deutsch and Jessen (1997).

105

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

TABLE I ALKALI ATOMSTHAT ARE RELEVANT TO OPTICAL DIPOLE TRAPPING: TRANSITION WAVELENGTHS h ~AND , Lo,, FINE-STRUCTURE SPLITTING Abs, NUCLEAR SPINI, GROUND-STATE HYPERFTNE SPLITTING Aws, EXCITED-STATE HYPERFINE SPLITTING Ahs, NATURALLINEWIDTH r, RECOILTEMPERATURE TmC(FOR THE D2 LINE), AND THE CORRESPONDING HEIGHTh,, = kaTRk,,/(mg)IN THE FIELD OF GRAVITY. PROPERTIESOF THE

~

670.9, 670.9 ‘Li 7 ~ i 23Na 589.0, 589.6 39K 766.7, 769.9

10 510 1500

WK 4’ K

85Rb 780.0, 794.8 7200 87Rb ‘ 3 3 c s 852.1, 894.3 16600

1 312 312 312 3 312 512 312 712

228 804 1772 462 1286 254 3036 6835 9192

5.9

4.6 18 I12 34 100 17 213 496 604

9.9 6.2

5.9

5.3

7.1 6.1 2.4 0.84 0.81 0.79 0.37 0.36 0.20

1000

740 88 18 17 16 3.7 3.5 1.3

F’=2 F’=l

F=2

(b)

- - - L’=l

1

b0

(4 - - - J’= +’

11

J’= Vz

J = V2

FIG.2. Level scheme of an alkali atom. (a) Full substructure for a nuclear spin I = j. (b) Reduced scheme for very large detunings that exceed the fine-structure splitting ([A1>> Ah). (c) Reduced scheme for large detunings in the range A h 2 IAI >> A m , A h s .

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R. Grimm, M. Weidemuller; and Yu. B. Ovchinnikov

state (energy-splittinghALS)leads to the well-known D-line doublet 2S1/2-+ 2 P 1 / 22,P 3 / 2 . The coupling to the nuclear spin then produces the hyperfine structure of both ground and excited states with energies h A ~ and s hALs, respectively. The splitting energies, obeying A & >> AWS>> A’Ws,represent the three relevant atomic energy scales. On the basis of Eq. (18), one can derive a general result for the potential of a ground state with total angular momentum F and magnetic quantum number mF, which is valid for both linear and circular polarization as long as all optical detunings stay large compared with the excited-state hyperfine splitting AfiFs:

Here g F is the well-known Land6 factor, the transition frequency 00 refers to the line center of the D doublet, and P characterizes the laser polarization (P = 0, f l for linearly and circularly of polarized light). The detunings A ~ , and F A ~ ,refer F to the energy splitting between the particular ground state 2S1/2,F and the center of the hyperfine-split 2P3/2 and 2P1/2excited states, respectively. The two terms in parentheses in Eq. (19) thus represent the contributions of the 0 2 and the D1 line to the total dipole potential. In order to discuss this result, let us first consider the case of very large ~ ~A;,), detunings greatly exceeding the fine-structuresplitting ( [ A ,1,, I A F , >> in which we can completely neglect the even much smaller hyperfine splitting. Introducing a detuning A with respect to the center of the D-line doublet, we can linearly expand Eq. (19) in terms of the small parameter AgS/A:

Although the first-order term describes a small residual dependence on the polarization P and the magnetic substate mF, the dominating zero-order term is just the result obtained for a two-level atom [Eq. (12)]. The latter fact can be understood in terms of a simple argument: If the fine structure is not resolved, then the detuning represents the leading term in the total Hamiltonian, and the atomic substructure can be ignored in a first perturbative step by reducing the atom to a very simple s --+ p transition; see Fig. 2(b).

* The assumption of unresolved excited-state hyperfine structure greatly simplifies the calculation according to Eq. (18) because of existing sum rules for the line strength coefficients c;; see also Deutsch and Jessen (1997).

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

107

Such a transition behaves like a two-level atom with the full line strength for any laser polarization, and the ground-state light shift is thus equal to that of a two-level atom. This single ground state couples to the electronic and nuclear spin in exactly the same way as it would without the light. In this simple case, all resulting hyperfine and magnetic substates directly acquire the light shift of the initial atomic s state. In the more general case of a resolved fine structure but an unresolved 1 1 , ,I A F ,>> ~ ~A w s ) , one may first consider hyperfine structure (Aks2 ~ A F the atom in spin-orbit coupling, neglecting the coupling to the nuclear spin. The interaction with the laser field can thus be considered in the electronic angular momentum configuration of the two D lines, J = t J ’ = $, In this situation, which is illustrated in Fig. 2(c), one can first calculate the light shifts of the two electronic ground states mJ = ki and, in a later step, consider their coupling to the nuclear spin. Here it is important to distinguish between linearly and circularly polarized light: For linear polarization, both electronic ground states (mJ= f1)are shifted by the same amount because of simple symmetry. After coupling to the nuclear spin, the resulting F, mF states have to remain degenerate like the two original mJ states. Consequently, all magnetic sublevels show the same light shifts according to the line strength factors of 2/3 for the DZ line and 1/3 for the D1 line. For circular polarization (a*),the light lifts the degeneracy of the two magnetic sublevels of the electronic 2S; ground state, and the situation gets more complicated. The relevant line strength factors are then given by $(1 f mJ) for the 0 2 line and +(1 2 m ~ for ) the D1 line. The lifted degeneracy of the two ground states can be interpreted in terms of a “fictitious magnetic field” (Cohen-Tannoudji and Dupont-Roc, 1972; Cho, 1997; Zielonkowski et al., 1998a), which is very useful in explaining how the lifted mJ degeneracy affects the F , m F levels after coupling to the nuclear spin. According to the usual theory of the linear Zeeman effect in weak magnetic fields, coupling to the nuclear spin affects the magnetic substructure such that one has to replace gJmJby gFmF, where gJ and g F denote the ground state Land6 factors. Applying this analogy, and using gJ = 2 for alkali atoms, we can replace mJ by igFmF to calculate the relevant line strength factors f(2 fg F m F ) and f ( 1 7 gFmF) for the 0 2 and ~1 lines, respectively. These factors lead to the mF-dependent shifts for circularly polarized light in Eq. (19). One finds that this result stays valid as long as the excited-state hyperfine splitting remains unresolved. For the photon-scattering rate rscof a multilevel atom, the same line strength factors are relevant as for the dipole potential, because absorption and light shifts are determined by the same transititon matrix elements. For linear polarization, in the most general case A >> A h s considered here, one

i

i.

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R. Grimm, M. Weidemiiller, and Yu. B. Ovchinnikov

thus explicitely obtains

This result is independent of mF but in general depends on the hyperfine state F via the detunings. For linearly polarized light, optical pumping tends to distribute the population equally among the different mF states and thus leads to complete depolarization. If the detuning is large compared to the groundstate hyperfine splitting, then all substates F,mF are equally populated by redistribution via photon scattering. For circular polarization, Zeeman pumping effects become very important; they depend on the particular detuning regime, and we do not want to discuss them in more detail here. It is interesting to note that the general relation between dipole potential and scattering rate takes the simple form of Eq. (14) either if the contribution of one of the two D lines dominates for rather small detunings or if the detuning is large compared to the fine-structure splitting. Our discussion on multilevel alkali atoms shows that linearly polarized light is usually the right choice for a dipole trap,3 because the magnetic sublevels mF of a certain hyperfine ground state F are shifted by the same amounts. This requires only a detuning large compared to the excited-state hyperfine splitting, a condition that is usually very well fulfilled in dipole trapping experiments. If the detuning also exceeds the ground-state hyperfine splitting, then all substates of the electronic ground state are equally shifted, and the dipole potential becomes completely independent of mF and F. For circularly polarized light, there is a mF-dependent contribution, which leads to a splitting analogous to a magnetic field. This term vanishes only if the optical detuning greatly exceeds the fine-structure splitting.

111. Experimental Issues Here we discuss several issues of practical importance for experiments on dipole trapping. Cooling and heating in the trap are considered in Section III.A, followed by a summary of the typical experimental procedures in Section 1II.B. Finally, in Section III.C, the particular role of collisions is discussed. 3An interesting exception is the work by Corwin er al. (1997) on trapping in circularly polarized light; see also Section IV.A.2.

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

109

A. COOLING AND HEATING Atom trapping in dipole potentials requires cooling to load the trap and eventually also to counteract heating in the dipole trap. We briefly review the various available cooling methods and their specific features with respect to dipole trapping. Then we discuss sources of heating, and we derive explicit expressions for the heating rate in the case of thermal equilibrium in a dipole trap. This allows for a direct comparison between dipole traps operating with red and blue detuning.

1. Cooling Methods Efficient cooling techniques are an essential requirement to load atoms into a dipole trap because the attainable trap depths are generally below 1 mK. Once the atoms are trapped, further cooling can be applied to achieve high phasespace densities and to compensate for the effects of possible heating mechanisms (see Section III.A.2) that would otherwise boil the atoms out of the trap. The development of cooling methods for neutral atoms has proceeded at breathtaking speed during the last decade, and numerous excellent reviews have been written illuminating these developments (Foot, 1991; Arimondo et al., 1992; Metcalf and van der Straten, 1994; Sengstock and Ertmer, 1995; Ketterle and van Druten, 1996; Adams and Riis, 1997; Chu, 1998; Cohen-Tannoudji, 1998; Phillips, 1998). In this section, we briefly discuss methods for cooling atoms in dipole traps. Sections IVand V describe the experimental implementation of the cooling schemes in particular trap configurations. Doppler Cooling. Doppler cooling is based on cycles of near-resonant absorption of a photon and subsequent spontaneous emission resulting in a net atomic momentum change per cycle of one photon momentum hk, with k = 2n/h denoting the wavenumber of the absorbed photon. Cooling is counteracted by heating due to the momentum fluctuations by the recoil of the spontaneously emitted photons (Minogin and Letokhov, 1987). Equilibrium between cooling and heating determines the lowest achievable temperature. For Doppler cooling of two-level atoms in standing waves (“optical molasses”), the minimum temperature is given by the Doppler temperature ksTD = hr/2. Typical values of the Doppler temperature are around 100pK, which is just low enough to load atoms into a deep dipole trap. The first demonstration of dipole trapping (Chu et al., 1986) used Doppler cooling to load the trap and keep the atoms from being boiled out of the trap. With the discovery of methods reaching much lower temperatures, Doppler cooling has lost its importance for the direct application to dipole traps.

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R. Grimm, M. Weidemiillec and Yu. B. Ovchinnikov

Polarization-Gradient Cooling. The Doppler temperature is a somewhat artificial limit because it is based on the simplifying assumption of a two-level atom. It was soon discovered that atoms with a more complex level structure can be cooled below TO in standing waves with spatially varying polarizations (Lett et af., 1988). The cooling mechanisms are based on optical pumping between ground-state Zeeman sublevels. The friction force that provides cooling results either from unbalanced radiation pressures through motioninduced atomic orientation or from a redistribution among the photons in the standing wave (Dalibard and Cohen-Tannoudji, 1989). In the latter case, the cooling force can be explained in terms of the so-called Sisyphus effect for a moving atom. The atom loses kinetic energy by climbing up the dipole potential induced by the standing wave of the trapping light. Upon reaching the top of this potential “hill,” the atom is optically pumped back into the bottom of the next potential “valley,” from where it starts to climb again (Dalibard and CohenTannoudji, 1985). Polarization-gradient cooling can be achieved in standing waves at frequencies below an atomic resonance (red-detuned molasses) (Lett et al., 1988; Salomon et al., 1990) as well as above an atomic resonance (blue-detunedmolasses)(Boiron etal., 1995; Hemmerich etal., 1995). In bluedetuned molasses, atoms primarily populate states that are decoupled from the light field, resulting in a reduction of photon scattering but also in a smaller cooling rate as compared to red-detuned molasses (Boiron et al., 1996). With polarization-gradient molasses, one can prepare free-space atomic samples at temperatures on the order of lOT,,, with the recoil temperature N

being defined as the temperature associated with the gain in kinetic energy by emission of one photon. For the alkali atoms, recoil temperatures are given in Table I. The achievable temperatures are much below the typical depth of a dipole trap. Polarization-gradientcooling therefore allows efficient loading of dipole traps (see Section III.B.1) either by cooling inside a magneto-optical trap (Steane and Foot, 1991) or by cooling in a short molasses phase before transfer into the dipole trap. Besides enhancing the loading efficiency, polarization-gradient cooling was directly applied to atoms trapped in a dipole potential by subjecting them to near-resonant standing waves with polarization gradients (Boiron et al., 1998; Winoto et al., 1999). Because the cooling mechanism relies on modification of the ground-state sublevels by the cooling light, a necessary condition for efficient cooling is the independence of the trapping potential from the Zeeman substate, which, in contrast to magnetic traps, can easily be fulfilled in dipole traps as explained in Section II.B.2.

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

111

Raman Cooling. The recoil temperature marks the limit for cooling methods based on the repeated absorption and emission of photons, such as Doppler cooling and polarization-gradient cooling. To overcome this limit, different routes were explored to decouple the cold atoms from the resonant laser excitation once they have reached small velocities. All approaches are based on transitions with a narrow line width, which are thus extremely velocityselective, such as dark-state resonances (Aspect et al., 1988) or Raman transition between ground-state sublevels (Kasevich and Chu, 1992). With free atomic samples, the interaction time, and thus the spectral resolution determining the final temperature, was limited by the time the thermally expanding atomic cloud spent in the laser field (Davidson et al., 1994).Taking advantage of the long storage times in dipole traps, Raman cooling was shown to work efficiently for trapped atomic samples (Lee et al., 1994; Lee et al., 1996; Kuhn et al., 1996). The basic principle of Raman cooling is as follows: Raman pulses from two counterpropagating laser beams transfer atoms from one ground state 11) to another ground state 12),transferring 2hk m~mentum.~ Using sequences of Raman pulses with varying frequency width, detuning, and propagation direction, pulses can be tailored that excite all atoms except those with a velocity near v = 0 (Kasevich et al., 1992; Reichel et al., 1995; Kuhn et al., 1996). The cooling cycle is completed by a light pulse resonantly exciting atoms in state 12) in order to pump the atoms optically back to the 11) state through spontaneous emission. Each spontaneous emission randomizes the velocity distribution so that a fraction of atoms acquire a velocity M 0. By the repetition of many sequences of Raman pulses followed by optical pumping pulses, atoms are accumulated in a small-velocity interval around v = 0. The final width of the velocity distribution - that is, the temperature - is determined by the spectral resolution of the Raman pulses. In dipole traps, high resolution can be achieved by the long storage times. In addition, motional coupling of the degrees of freedom through the trap potential made it possible to cool the atomic motion in all three dimensions with Raman pulses applied along only one spatial direction (Lee ef al., 1996; Kuhn et al., 1996). Resolved-Sideband Raman Cooling. Cooling with well-resolved motional sidebands, a technique well known from laser cooling in ion traps (Ghosh, 1995), has also been applied to optical dipole traps (Hamann et al., 1998; Pemn et al., 1998; Vuletic et ul., 1998; Bouchoule et al., 1999). For resolvedsideband cooling, atoms must be tightly confined along (at least) one spatial dimension with oscillation frequencies oosc large enough to be resolved by Raman transitions between two ground-state levels. In contrast to the Raman The two states are, for example, the two hyperfine ground states of an alkali atom.

112

R. Grimm, M. Weidemiiller; and Yu. B. Ovchinnikov

cooling discussed in the preceding paragraph, the confining potential of the trap is therefore a necessary prerequisite to the application of sideband cooling. Atomic motion in the tightly confining potential is described by a wavepacket formed by the superposition of vibrational states ~n.,,).In the LambDicke regime, where the rms size of the wavepacket is small compared to the wavelength of the cooling transition, an absorption-spontaneous emission cycle almost exclusively returns to the same vibrational state it started from (An.,, = 0). The Lamb-Dicke regime is reached by trapping atoms in dipole potentials formed in the interference pattern of far-detuned laser beams (see Sections 1V.B and 1V.C). Sideband cooling consists of repeated cycles of Raman pulses tuned to excite transitions with An.,, = -1, followed by an optical pumping pulse involving spontaneous emission back to the initial state with An.,, = 0. In this way, the motional ground state nos, = 0 is selectively populated because it is the only state that does not interact with the Raman pulses. Achievable temperatures are limited only by the separation and the width of the Raman sidebands determining the suppression of off-resonant excitation of the no,, = 0 state. A particularly elegant realization of sideband cooling was accomplished by using the trapping light itself to drive the Raman transition instead of applying additional laser fields (Hamann et al., 1998; Vuletic et al., 1998). For this purpose, a small magnetic field was applied, shifting the energy of two adjacent ground-state Zeeman sublevels relative to each other by exactly one vibrational quantum fro.,,. In this way, the bound states ImF;nosc)and I mF - 1; nosc - 1) became degenerate, and Raman transitions between the two states could be excited with single-frequency light provided by the trapping field (degenerate sideband cooling) (Deutsch and Jessen, 1998). The great advantage of degenerate sideband cooling is that it works with only the two lowest-energy atomic ground states being involved, resulting in the suppression of heating and trap losses caused by inelastic binary collisions (see Section 1II.C). Evaporative Cooling. Evaporative cooling, orginally demonstated with magnetically trapped hydrogen (Hess et al., 1987), has been the key technique for achieving Bose-Einstein condensation in magnetic traps (Ketterle and van Druten, 1996). It relies on the selective removal of high-energetic particles from a trap and subsequent thermalization of the remaining particles through elastic collisions. Evaporative cooling requires high densities to ensure fast thermalization rates, and it requires large initial numbers of particles because so many trapped atoms are removed from the trap by evaporation. For evaporative cooling to be effective, the ratio between inelastic collisions (causing losses and heating) and elastic collisions (providing thermalization and evaporation) has to be large.

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

113

In dipole traps, inelastic processes can be greatly suppressed when the particles are prepared in their energetically lowest state. However, the requirement of large particle numbers and high density poses a dilemma for the application of evaporative cooling to dipole traps. In tightly confining dipole traps such as a crossed dipole trap, high peak densities can be reached, but the trapping volume, and thus the number of particles transferred into the trap, is small. On the other hand, large trapping volumes that yield large numbers of trapped particles provide only weak confinement, yielding small elastic-collision rates. This is why only one experiment has been reported on evaporative cooling in dipole traps starting with a small sample of atoms (Adams et al., 1995). By precooling large ensembles in dipole traps with optical methods explained in the preceding paragraphs, much better starting conditions for evaporative cooling are achievable (Engler et al., 1998; Vuletic et al., 1998; Winoto et al., 1999), and this means that evaporative cooling is still an interesting option for future applications. Adiabatic Expansion. When we adiabatically expand a potential without changing its shape, the temperature of the confined atoms is decreased without increasing the phase-space den~ity.~ In dipole potentials, cooling by adiabatic expansion was realized by slowly ramping down the intensity of the trapping light (Chen et al., 1992; Kastberg et al., 1995). In far-detuned traps consisting of micropotentials induced by interference, adiabatic cooling is particularly interesting when the modulation on the scale of the optical wavelength is slowly reduced without modifying the large-scale trapping potential - for example, by changing the polarization of the interfering laser beams. Atoms are initially strongly localized in the micropotentials, resulting in high peak densities. After adiabatic expansion, the temperature of the sample is reduced, as is the peak density. However, the density averaged over one period of the interference structure is not modified.

2. Heating Mechanisms Heating by the trap light is an issue of particular importance for optical dipole trapping. A fundamental source of heating is the spontaneous scattering of trap photons, which, because of its random nature, causes fluctuations of the radiation force.6 In the far-detuned case considered here, the scattering is completely elastic (or quasi-elastic if a Raman process changes the atomic 'Adiabatic changes of the potential shape leading to an increase of the phase-space density are demonstrated in Pinkse et al. (1997) and Stamper-Kurn et al. (1998b). Under conditions typical of a dipole trap, the scattering force that results in traveling-wave configurations stays very weak compared to the dipole force and can thus be neglected.

114

R. Grimm, M. Weidemullel; and Yu. B. Ovchinnikov

ground state). This means that the energy of the scattered photon is determined by the frequency of the laser, not of the optical transition. Both absorption and spontaneous re-emission processes show fluctuations, and thus both contribute to the total heating (Minogin and Letokhov, 1987). At large detunings, where scattering processes follow Poisson statistics, the heating due to fluctuations in absorption corresponds to an increase of the thermal energy by exactly the recoil energy Er, = k~Trec/2per scattering event. This first contribution occurs in the propagation direction of the light field and is thus anisotropic (so-called directional diffusion). The second contribution is due to the random direction of the photon recoil in spontaneous emission. This heating also increases the thermal energy by one recoil energy Ere, per scattering event, but distributed over all three dimensions. Neglecting the general dependence on the polarization of the spontaneously emitted photons, one can assume an isotropic distribution for this heating mechanism. Taking into account both contributions, the longitudinal motion is heated on an average by 4E,/3 per scattering process, whereas each of the two transverse dimensions is heated by EreC/3.The overall heating thus corresponds to an increase of the total thermal energy by 2Erec in a time I';. For simplicity, we do not consider the generally anisotropic character of heating here; in most cases of interest, the trap mixes the motional degrees on a time scale faster than or comparable to the heating. .We can thus use a simple global three-dimensional heating power &at =E corresponding to the increase of the mean thermal energy E of the atomic motion with time. This heating power is directly related to the average photon-scattering rate by

rSc

It is well known that for intense light fields close to resonance, in particular in standing-waveconfigurations,the induced redistribution of photons between different traveling-wave components can lead to dramatic heating (Gordon and Ashkin, 1980; Dalibard and Cohen-Tannoudji, 1985). In the far off-resonant case, however, this induced heating falls off very rapidly with the detuning and is usually completely negligible as compared to spontaneous heating. In addition to the fundamental heating in dipole traps, Savard et al. (1997) have pointed out that technical heating can occur because of intensity fluctuations and pointing instabilities in the trapping fields. In the first case, fluctuations occurring at twice the characteristic trap frequencies are relevant; they can parametrically drive the oscillatory atomic motion. In the second case, a shaking of the potential at the trap frequencies increases the motional amplitude. These issues have not been studied in detail yet, but they will strongly depend on the particular laser source and its technical noise spectrum. Several experiments have indeed shown indications of heating in dipole

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

115

traps by unidentified mechanisms (Adams et al., 1995; Lee et al., 1996; Zielonkowski et al., 1998b; Vuletic et al., 1998), which may be related to fluctuations of the trapping light. 3. Heating Rate For an ultracold atomic gas in a dipole trap, it is often a good assumption to consider a thermal equilibrium situation, in which the energy distribution is related to a temperature T. The further assumption of a power-law potential then allows one to derive very useful expressions for the mean photonscattering rate and the corresponding heating rate of the ensemble, which also illustrate important differences between traps operating with red- and bluedetuned light. In thermal equilibrium, the mean kinetic energy per atom in a threedimensional trap is ,!?kin = 3 k B T/2. Introducing the parameter K = Epot/Ehn as the ratio of potential to kinetic energy, we can express the mean total energy E as

For many real dipole traps as described in Sections IV and V, it is a good approximation to assume a separable power-law potential with a constant offset Lro of the form U(X,y , z ) = uo

+ a,xn' + a2yn2 + a3zn3

(25)

In such a case, the virial theorem can be used to calculate the ratio between potential and kinetic energy:

For a three-dimensional harmonic trap this gives K = 1, for an ideal threedimensional box potential K = 0. The relation between mean energy and temperature, Eq. (24), allows us to re-express the heating power that results from photon scattering, Eq. (23), as a heating rate

describing the corresponding increase of temperature with time.

116

R. Grimm, M. Weidemuller; and Yu. B. Ovchinnikov

1

--

U

I

FIG.3. Illustration of dipole traps with red and blue detuning. In the first case, a simple Gaussian laser beam is assumed. In the second case, a Laguerre-Gaussian LGol “doughnut” mode is chosen that provides the same potential depth and the same curvature in the trap center (note that the latter case requires e 2 times more laser power or smaller detuning).

r,,

The mean scattering rate can, in turn, be calculated from the temperature of the sample, according to the following arguments: Equation (14) relates the average scattering rate to the mean dipole potential u d i p experienced by the atoms. In a pure dipole trap7 described by Eq. (25), the mean optical potential is related to the mean potential energy EPt, the mean kinetic energy ,!?kin, and the temperature T by udjp

+

= U O E p t = Uo

+ K E b n = UO+-3K2 T

(28)

This relation allows us to express the mean scattering rate as

On the basis of this result, let us now discuss two specific situations that are typical for real experiments as described in Sections IV and V; see illustrations in Fig. 3. In a red-detuned dipole trap (A < 0), the atoms are trapped in an intensity maximum with Uo < 0, and the trap depth U = [Uol is usually large compared to the thermal energy ~ B TIn. a blue-detuned trap (A > 0), a potential minimum corresponds to an intensity minimum, which in an ideal case means zero intensity. In this case U O= 0, and the potential depth U is determined by the height of the respulsive walls surrounding the center of the trap. For red- and blue-detuned traps with U >> kBT, Eqs. (27) and (29) yield the following heating rates:

’

This excludes hybrid potentials, in which other fields (gravity, magnetic, or electric fields) are important for the trapping.

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

117

Obviously, a red-detuned trap shows linear heating (which decreases when ks T approaches whereas heating behaves exponentially in a blue-detuned trap. Note that in blue-detuned traps, a fundamental lower limit to heating is set by the zero-point energy of the atomic motion, which we have neglected in our classical consideration. Equations 30a and 30b allow for a very illustrative direct comparison between a blue- and red-detuned trap: The ratio of heating at the same magnitude of detuning [A[is given by

u),

This comparison shows that blue detuning offers substantial advantages in two experimental situations: 0

U

0

K

>> k B T , very deep potentials for tight confinement << 1, box-like potentials with hard repulsive walls

In other words, when a harmonic potential of moderate depth is to be realized for a certain experiment, the advantage of blue detuning is not substantial. The choice of red detuning may be even more appropriate, because the better concentration of the available laser power in such a trap allows one to use larger detunings to create the required potential depth.

B. EXPERIMENTAL TECHNIQUES I . Trap Loading The standard way to load a dipole trap is to start from a magneto-optical trap (MOT). This well-known radiation-pressure trap operating with near-resonant light was first demonstrated by Raab et af., in 1987 and has become the standard source of ultracold atoms in many laboratories all over the world. A MOT can provide temperatures down to a few lOT, when its operation is optimized for sub-Doppler cooling (see Section IH.A.1). This sets a natural scale for the minimum depth of a dipole trap as required for efficient loading. Because of their lower recoil temperatures (Table I), heavier alkali atoms require less trap depth than the lighter ones and thus allow for larger detunings. For the heavy Cs atoms, for example, dipole traps with depths as low as 10 pK can be directly loaded from a MOT (Zielonkowskiet al., 1998b).Trap N

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R. Grimm, M . Weidemiiller; and Yu. B. Ovchinnikov

loading at much lower depths can be achieved with Bose-Einstein condensates (Stamper-Kum et al., 1998a). For dipole trap loading, a MOT is typically operated in two stages. First, its frequency detuning is set quite close to resonance (detuning of a few natural linewidths) to optimize capture by the resonant scattering force. Then, after the loading phase, the MOT parameters are changed to optimize sub-Doppler cooling (Drewsen et al., 1994; Townsend et al., 1995). Most important, the detuning is switched to much higher values (typically 10- 20 linewidths), and eventually the laser intensity is lowered. For the heavier alkali atoms,8 this procedure provides maximum phase-space densities for trap loading. Another option is to ramp up the magnetic fields of the MOT to compress the sample spatially (Petrich et al., 1994). The dipole trap is filled by simply overlapping it with the atomic cloud in the MOT before the latter is turned off. In this procedure, it is advantageous to switch off the magnetic field of the MOT a few milliseconds before the laser fields are extinguished,because the short optical molasses cooling phase that results establishes the lowest possible temperatures and a quasi-thermal distribution in the trap. For practical reasons, the latter is important because a MOT does not necessarily load the atoms into the very center of the dipole trap. When MOT position and dipole trap center do not coincide exactly, loading results in excess potential energy in the dipole trap. When the MOT light is extinguished, it is very important to shield the dipole trap from any resonant stray light, particularly if very low scattering rates ( 51 s-') are to be reached. The MOT itself can be loaded in a simple vapor cell (Monroe et al., 1990). In such a setup, however, the lifetime of the dipole trap is typically limited to less than 1 s by collisions with atoms in the background gas. If longer lifetimes are required for a certain application, the loading of the MOT under much better vacuum conditions becomes an important issue, as in experiments on Bose-Einstein condensation. Loading can then the accomplished from a very dilute vapor (Anderson et al., 1994), but more powerful concepts can be realized with a Zeeman-slowed atomic beam (Phillips and Metcalf, 1982), with a double-MOT setup (Myatt et al., 1997), or with slow-atom sources based on modified MOTS (Lu et al., 1996, Dieckmann et al., 1998). Regarding trap loading, a dipole trap with red detuning can offer an important advantage over a blue-detuned trap. When MOT and dipole trap are turned on simultaneously, the attractive dipole potential leads to a local density increase in the MOT, which can substantially enhance the loading process. In very deep red-detuned traps, however, the level shifts become too *The lightest alkali atom, Li, behaves in a completely different way. Here optimum loading is accomplished at larger detunings, and optimum cooling is obtained relatively close to resonance (Schiinemann et al., 1998).

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large for the cooling light, and efficient loading requires rapid alternation between cooling and trapping light (Miller et al., 1993).

2. Diagnostics The atomic sample in a dipole trap is characterized by the number of stored atoms, the motional temperature of the ensemble (under the assumption of thermal distributon), and the distribution of population among the different ground-state sublevels. Measurements of these important quantities can be made in the following ways. Number of Atoms. A very simple and efficient method that is often used to determine the number of atoms in a dipole trap is to recapture them into the MOT and to measure the power of the emitted fluorescence light with a calibrated photo-diode or CCD camera. In this way, it is quite easy to detect down to about 100 atoms, but even single atoms may be observed in a more elaborate setup (Haubrich et al., 1996). This recapture method works particularly well if it is ensured that the MOT does not capture any other atoms than those released from the dipole trap. This is hardly possible in a simple vapor cell setup, but it is quite easy if an atomic beam equipped with a mechanical shutter is used for loading the MOT. In contrast to the completely destructive recapture, several other methods may be applied. The trapped atoms can be illuminated with a short resonant laser pulse of moderate intensity to measure the emitted fluorescent light. This can be done also with spatial resolution by using a CCD camera; see Fig. 18(b)for an example. If the total number of atoms is not too low, the detection pulse can be kept weak enough to avoid trap loss by heating. Furthermore, absorption imaging can be used (see Fig. 7), or even more sensitive and less destructive dark-field or phase-contrast imaging methods can be applied to monitor Bose-Einstein condensates sensitively (Andrews et al., 1996; Bradley et al., 1997; Andrews et al., 1997). Temperature. In a given trapping potential U(r), the thermal density distribution n ( r ) directly follows from the Boltzmann factor,

n(r) = noexp

(

2)

--

The temperature can thus be derived from the measured spatial-density distribution in the trap, which itself can be observed by various imaging methods (fluorescence, absorptive, and dispersive imaging). For a three-dimensional

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R. Grimm, M. Weidemiillel; and Yu. B. Ovchinnikov

harmonic potential U(r) = im (02 x 2 + o ; y 2 tion is Gaussian in all directions,

+ 01z2),the resulting distribu-

with oi = o ; ' J w . The temperature is thus related to the spatial extensions of the trapped atom cloud by

T = -m( T i m2 i

2

(34)

kB

Obviously, it is very important to know the exact trap frequencies to determine the temperature precisely; a practical example of such a measurement is discussed in connection with Fig. 7(b). This way of measuring the temperature is limited by the resolution of the imaging system and therefore becomes difficult for very tightly confining potentials. A widely used and quite accurate, but completely destructive, way to measure temperatures is the time-of-flight method. The trap is turned off to release the atoms into a free, ballistic flight. This has to be done in a rapid, completely nonadiabatic way; otherwise, an adiabatic cooling effect (see Section 1II.A.1) would influence the measurement. After a sufficiently long ballistic expansion phase, the resulting spatial distribution, which can again be observed by the various imaging methods, directly reflects the velocity distribution at the time of release. Another method, which is not limited by the natural linewidth of the optical transition, is to detect the Doppler broadening of Raman transitions between ground states, using a pair of counterpropagating laser beams (Kasevich and Chu, 1991). Internal Distribution. The relative population of the two hyperfine ground states of an alkali atom (see the level scheme in Fig. 2) can be measured by application of a probe pulse resonant to the closed subtransition F = I + 1/2 + F' = I + 3/2 in the hyperfine structure, which is well resolved for the heavier alkali atoms (see Table I). The fluorescence light is then proportional to the number of atoms in the upper hyperfine state, F = I 1/2. If, in contrast, a repumping field is present in the probe light (as it is always used in a MOT),the fluorescence is proportional to the total number of atoms, because all atoms are immediately pumped into the closed excitation cycle. The normalized fluorescence signal thus gives the relative population of the upper hyperfine ground state (F = I + 1/2); such a measurement is discussed in Section IV.A.2. A very sensitive alternative, which works very well in

+

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shallow dipole traps, is to blow the total upper-state population out of the trap by the radiation pressure of an appropriate resonant light pulse. Subsequent recapture into the MOT then shows how many atoms have remained trapped in the shelved lower hyperfine ground state. The distribution of population over different magnetic substates can be analyzed by Stern-Gerlach methods. When the atomic ensemble is released from the dipole trap and ballistically expands in an inhomogeneous magnetic field, atoms in different magnetic sublevels can be well separated in space. Such an analysis has been used for optically trapped Bose-Einstein condensates (Stamper-Kurn et al., 1998a; Stenger, 1998); an example is shown in Fig. 8(b). Another possibility, which can be easily applied to shallow dipole traps, is to pull atoms out of the trap by the state-dependent magnetic force, a technique that Zielonkowski et al. (1998b) have used for measuring the depolarizing effect of the trap photon scattering; see the discussion in Section IV.B.3.

C. COLLISIONS It is a well-known experimental fact in the field of laser cooling and trapping that collisional processes can lead to substantial trap loss. Detailed measurements of trap loss under various conditions provide insight into ultracoldcollision phenomena, which have been the subject of extensive research (Walker and Feng, 1994; Weiner, 1995). Here we discuss the particular features of dipole traps with respect to ultracold collisions. The decay of the number N of atoms in a trap can be described by the general loss equation

Here the single-particle loss coefficient a takes into account collisions with the background gas in the vacuum apparatus. As a rule of thumb, the l/e lifetime T = 1/aof a dipole trap is 1s at a pressure of 3 x lO-’mbar. This is about three times lower than the corresponding lifetime in a MOT because of the larger cross sections for collisional loss at lower trap depth. The two-body loss coefficient p describes trap loss due to ultracold binary collisions and reveals a wide range of interesting physics. In general, such trap loss becomes important if the colliding atoms are not in their absolute ground state. In an inelastic process, the internal energy can be released into the atomic motion, causing escape from the trap. Because of the shallowness of optical dipole traps, even the collisional release of the relatively small amount of energy in the ground-state hyperfine structure of an alkali atom always N

R. Grimm, M. Weidemiiller; and Yu. B. Ovchinnikov

122

E

0.1

0

0.5

1

1.5 time (s)

2

2.5

3

FIG.4. Decay of Cs atoms measured in a crossed-beam dipole trap (see Section 1V.C) realized with the 1064-nm light of a Nd:YAG laser. The initial peak density is 2.5 x 10l2~ m - ~ . When all atoms are in the lower hyperfine ground state (F = 3), the purely exponential decay (l/e-lifetime 1.1 s) is due to collisions with the background gas. When the atoms are in the upper hyperfme level (F = 4). a dramatic loss is observed as a result of hyperfine-changing collisions (loss coefficient p M 5 x lo-" cm3/s). Unpublished data, courtesy of C. Salomon.

leads to trap l0s.5.~Hyperjne-changingcollisions,which occur with large rate coefficients p of typically 5 x 10-"cm3/s (Sesko er al., 1989; Wallace er al., 1992), are thus of particular importance for dipole trapping. Alkali atoms in the upper hyperfine state (F = I + 1/2) can show very rapid, nonexponential collisional decay, in contrast to a sample in the lower ground state (F = I - 1/2). This is impressively demonstrated by the measurement in Fig. 4,which shows the decay of a sample of Cs atoms prepared either in the upper or the lower hyperfine ground state. For the implementation of laser cooling schemes in dipole traps, it is thus very important to keep the atoms predominantly in the lower hyperfine state; several schemes for meeting this requirement are discussed in Sections IV and V. Trap loss can also occur as a result of light-assisted binary collisions involving atoms in the excited state. The radiative escape mechanism (Gallagher and Pritchard, 1989) and excited-state fine-structure-changing collisions strongly affect a MOT, but their influence is negligibly small in a dipole trap because of the extremely low optical excitation. It can become important, however, if near-resonant cooling light is present. Another important mechanism for trap loss is photoassociation (Lett er al., 1995), a process In contrast, a MOT operated under optimum capture conditions is deep enough to hold atoms after hyperfine-changing collisions.

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in which colliding atoms are excited to bound molecular states, which then decay via bound-bound or bound-free transitions. Dipole traps indeed represent a powerful tool for photoassociative spectroscopy (Miller et al., 1993b). Three-body losses, as described by the coefficient y in Q. ( 3 3 , become relevant only at extremely high densities (Burt et al., 1997; Stamper-Kurn et al., 1998a) far exceeding the conditions of a MOT.In a collision of three atoms, a bound dimer can be formed and the third atom takes up the released energy, so all three atoms are lost from the trap. Because a far-detuned dipole trap allows one to suppress binary collision losses completely by putting the atoms into the absolute internal ground state, it represents an interesting tool for measurements on three-body collisions; an example is discussed in Section IV.A.4. In contrast to inelastic collisions releasing energy, elastic collisions lead to a thermalization of the trapped atomic ensemble. This also produces a few atoms with energies substantially exceeding kBT. A loss of these energetic atoms in a shallow trap leads to evaporative cooling (see Section III.A.l) and is thus of great interest for the attainment of Bose-Einstein condensation. In terms of the basic physics of elastic collisions, dipole traps are not different from magnetic traps, but they offer additional experimental possibilities. By application of a homogeneous magnetic field, atomic scattering properties can be tuned without affecting the trapping itself. Using this advantage of dipole trapping, Feshbach resonances have been observed with Bose-condensed Na atoms (Inouye et al., 1998) and thermal Rb atoms (Courteille et al., 1998). Moreover, an intriguing possibility is to study collisions in an arbitrary mixture of atoms in different magnetic substates (Stenger et al., 1998).

IV. Red-Detuned Dipole Waps The dipole force points toward increasing intensity if the light field is tuned below the atomic transition frequency (red detuning). Therefore, already the focus of a laser beam constitutes a stable dipole trap for atoms as first proposed by Ashkin (1978). The trapping forces generated by intense focused lasers are rather feeble, which was the main obstacle for trapping neutral atoms in dipole traps. Attainable trap depths in a tightly focused beam are typically in the millikelvin range, orders of magnitude smaller than the thermal energy of room-temperature atoms. One therefore had to first develop efficient laser cooling methods for the preparation of cold atom sources (see Section 1II.A. 1) to transfer significant numbers of atoms into a dipole trap. In their decisive experiment, S. Chu and coworkers (1986) succeeded in holding about 500 sodium atoms for several seconds in the tight focus of a red-detuned laser beam. Doppler molasses cooling was used to load atoms

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R. Grimm, M. Weidemiiller; and Yu. B. Ovchinnikov

L

7r FIG.5. Beam configurations used for red-detuned far off-resonance traps. Shown below are the corresponding calculated intensity distributions. (a) Horizontal focused-beam trap. (b) Vertical standing-wavetrap. (c) Crossed-beamtrap. The waist wo and the Rayleigh length ZR are indicated.

into the trap, which was operated at high intensities and considerable atomic excitation in order to provide a sufficiently deep trapping potential. Under these circumstances, the radiation pressure force still significantly influences the trapping potential due to a considerable rate of spontaneous emission. With the development of sub-Doppler cooling (see Section III.A.l) and the invention of the magneto-optical trap as a source for dense, cold atomic samples (see Section 1II.B.l), dipole trapping of atoms regained attention with the demonstration of a far-off resonant trap by Miller et al. (1993). In such a trap, spontaneous emission of photons is negligible, and the trapping potential is given by the equations derived in Section 11. Since then, three major types of traps with red-detuned laser beams have been established, all based on combinations of focused Gaussian beams: focused-beam traps consisting of a single beam, standing-wave traps wherein atoms are axially confined in the antinodes of a standing wave, and crossedbeam traps created by two or more beams intersecting at their foci. These different types of traps are schematically depicted in Fig. 5. We shall discuss these trap configurations and review applications of the different types for the investigation of interesting physical questions. Section 1V.A deals with focused-beam traps, Section 1V.B presents standing-wave traps, and Section V1.C discusses crossed-beam traps. Far-detuned optical lattices

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trapping atoms in micropotentials formed by multiple-beam interference represent a trap class of their own. Atoms might get trapped in the antinodes of the interference pattern at red detuning from resonance, but also in the nodes when the light field is blue-detuned. Section 1V.D is devoted to recent developments in far-detuned optical lattices. A. FOCUSED-BEAM TRAPS A focused Gaussian laser beam tuned far below the atomic resonance frequency represents the simplest way to create a dipole trap providing threedimensional confinement [see Fig. 5(a)]. The spatial intensity distribution of a focused Gaussian beam (FB)with power P propagating along the z-axis is described by

where r denotes the radial coordinate. The l/e2 radius w(z)depends on the axial coordinate z via

where the minimum radius wo is called the beam waist and Z R = mv$i denotes the Rayleigh length. From the intensity distribution, one can derive the optical potential U ( r , z )cx Zm(r,z) using Eq. (lo), (12), or (19). The trap depth U is given by U = I U (r = 0, z = 0)1. The Rayleigh length ZR is larger than the beam waist by a factor of xwo/h. Therefore, the potential is much steeper in the radial direction than in the axial direction. To provide stable trapping, one has to ensure that the gravitational force does not exceed the confining dipole force. Focused-beam traps are therefore mostly aligned along the horizontal axis. In this case, the strong radial force U/WOminimizes the perturbing effects of gravity." If the thermal energy k B T of an atomic ensemble is much smaller than the potential depth U , the extension of the atomic sample is radially small compared to the beam waist and is axially small compared to the Rayleigh range. In this case, the optical potential can be well approximated by a simple N

l o A new type of trap for the compensation of gravity was presented by Lemonde et al. (1995). It combines a focused-beam dipole trap providing radial confinement with an inhomogeneous static electric field along the vertical z-axis, inducing tight axial confinement through the dc Stark effect.

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R. Grimm, M. Weidemuller; and Yu. B. Ovchinnikov

cylindrically symmetric harmonic oscillator,

2 112 The oscillation frequencies of a trapped atom are given by o r= (4fi/rnwo) in the radial direction and by w, = (2.0/mz;)''* in the axial direction. According to Eq. (25), the harmonic potential represents an important special case of a power-law potential for which thermal equilibrium properties are discussed in Section 111.

I . Collisional Studies In their pioneering work on far off-resonance traps (FORT), Miller et al. (1993a), at the University of Texas in Austin, have observed trapping of *'Rb atoms in the focus of a single, linearly polarized Gaussian beam with detunings from the D1 resonance of up to 65 nm. The laser beam was focused to a waist of 10 pm, creating trap depths in the millikelvin-range. Between lo3 and lo4 atoms were accumulated in the trap from lo6 atoms provided by a vapor cell MOT. Small transfer efficiencies are a general property of traps with tightly focused beams resulting from the small spatial overlap between the cloud of atoms in the MOT. Typical temperatures in the trap are below 1 mK, which results in densities close to 10l2atoms/cm3. Peak photon-scattering rates were a few hundred per second, leading to negligible rates of loss by photon heating as compared to losses by background gas collisions. High densities achieved in a tightly-focused beam, in combination with long storage times, offer ideal conditions for the investigation of collisions between trapped atoms. The trap lifetime without cooling illustrated in Fig. 6(a) showed an increase of the lifetime by about an order of magnitude for increasing detunings at rather small detunings. At larger detunings, the lifetime was found to be determined by the Rb background pressure of the vapor-cell to a value of about 200 ms. The shorter lifetimes at smaller detunings were explained in a later publication (Miller et al., 1993b) by losses through photoassociation of excited Rb2 dimers that were induced by the trapping light. This important discovery has inspired a whole series of experiments on ultracold collisions investigated by the Austin group with photoassociation spectroscopy in a dipole trap. Instead of using the trapping light, photoassociation was induced by additional lasers in later experiments. The investigations comprise collisional properties and long-range interaction potentials of ground state atoms, (Cline et al., 1994b; Gardner et al., 1995), shape resonances in cold collisions (Boesten et al., 1997; Tsai et al., 1997), and the observation of Feshbach resonances (Courteille et al., 1998).

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40

790 8M) 810 820 830 840 850 860 870 Dipde Trap Wavelength (nm)

0.1

0.01 f .I 790

810 830 850 trap laser wavelength (nm)

1 870

FIG. 6. Measurement of trap lifetimes and hyperfine relaxation times for Rb atoms in a fardetuned focused-beam trap. (a) Trap decay time as a function of the trapping-beam wavelength. From Miller et al. (1993a). (b) Time constant trel for hyperfine population relaxation versus trapping-beam wavelength, in comparison to the mean time ts between two spontaneous scattering events. From Cline et al. (1994), 0Optical Society of America.

2. Spin Relaxation If the detuning of the trapping light field is larger than fine-structure splitting of the excited state, photon scattering occurs almost exclusively into the elastic Rayleigh component. Inelastic Raman scattering changing the hyperfine ground state is reduced by a factor of about 1/A2 as compared to Rayleigh scattering. The Austin group demonstrated this effect by preparing all trapped 85Rbatoms in the lower hyperfine ground state and studying the temporal evolution of the higher hyperfine state (Cline et al., 1994a). The relaxation-time constant as a function of detuning is plotted in Fig. 6(b) in comparison to the calculated For large detunings, average time between two photon-scatteringevents T~=rsc. the relaxation-time constant ~ , 1 is found to exceed T~ by two orders of magnitude. This shows the great potential of far-detuned optical dipole traps for manipulation of internal atomic degrees of freedom over long time intervals.

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R. Grimm, M. Weidemiiller;and Yu. B. Ovchinnikov

Using the spin state dependence of the dipole potential in a circularly polarized light beam (see Section II.B.2), Corwin et al. (1997) from the University of Colorado at Boulder have investigated far-off resonance dipole traps that selectively hold only one spin state. The small spin relaxation rates in dipole traps may have useful applications in the search for beta-decay asymmetries and atomic parity violation. 3. Polarization-Gradient Cooling Polarization-gradient cooling in a focused-beam trap has been investigated by a group at the ENS in Paris (Boiron et al., 1998). The trapped atoms were subjected to blue-detuned molasses cooling in a near-resonant standing wave (see Section 1II.A.1). Previously, the same group had performed experiments on polarization-gradient cooling of free atomic samples that were isotropically distributed. A density-dependent heating mechanism was found to be limiting the achievable final temperatures for a given density (Boiron et al., 1996). This heating was attributed to reabsorption of the scattered cooling light within the dense cloud of cold atoms. Of particular interest was the question of whether the anisotropic geometry of a focused beam influences heating processes through multiple photon scattering. Cesium atoms were trapped in the focus (wg = 45 pm) of a 700-mW horizontally propagating Nd:YAG laser beam at 1064 nm. The trap had a depth of 50 pK, and the radial trapping force exceeded gravity by roughly one order of magnitude. From a MOT containing 3 x lo7 atoms, 2 x lo5 atoms were loaded into the trap.' Polarization-gradient cooling was applied for some tens of milliseconds, yielding temperatures between 1 and 3pK, depending on the cooling parameters. To avoid trap losses by inelastic binary collisions involving the upper hyperfine ground state (see Section III.C), the cooling scheme was chosen in such a way that the population of the upper hyperfine ground state was kept at a low level. An absorption image of the trapped cesium atoms at T = 2 pK is depicted in Fig. 7(a), which shows the rod-shaped cloud of atoms. The distribution of atoms had a radial extension (T, = 6 pm and an axial size crZ = 300 pm corresponding to a peak density of 1 x lo'* atoms/cm3. The picture was taken 30ms after the cooling had been turned off. The time interval is large compared to the radial oscillation period (w,/2x = 330Hz) but short compared to the axial oscillation period (oZ/2x = 1.8 Hz). In this transient regime, the axial distribution had not yet reached its thermal equilibrium extension of (T, = 950 pm, which leads to a decrease in density. For the radial extension, the measured value coincided with the expectation for thermal

'

-

I'

The Nd:YAG beam was continuously on during the MOT loading.

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

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FIG.7. Properties of a horizontal focused-beam trap for Cs atoms. (a) Absorption image of the atom distribution. The transverse rms radius in the focal plane is 6 pm. (b) Measurement of the vertical oscillation period (see text). Adapted from Boiron et al. (1998).

equilibrium as determined by the measured temperature and oscillation frequency [see Eq. (34)l. To measure the transverse oscillation frequency, the Nd:YAG beam was interrupted for about 1ms, during which time the cold atoms were accelerated by gravity to a mean velocity of about 1cm/s. After the trapping laser had been turned on again, the atoms vertically oscillated in the trap. The oscillation in vertial velocity was detected by measuring the mean arrival time of the atoms at a probe laser beam 12 cm below the trap [the ordinate in Fig. 7(b)] as a function of the trapping time intervals [the abscissa in Fig. 7(b)]. The measured vertical oscillation frequency o r= 330Hz is consistent with the value o r= 390 Hz derived from the trap depth and the beam waist. The temperatures measured in the dipole trap were about 30 times lower than one would expect on the basis of the heating rates found for an isotropic free-space sample at the corresponding densities (Boiron et al., 1995). No evidence was found for heating of the trapped sample. The reduction of density-dependent heating is a benefit derived from the strongly anisotropic trapping geometry of a focused-beam trap. Because of the much smaller volume-to-surface ratio of an atomic cloud in the focused-beam trap as compared to a spherical distribution, photons emitted during cooling have a higher chance to escape without reabsorption from the trap sample, which causes less heating through reabsorption. 4. Bose-Einstein Condensates

Optical confinement of Bose-Einstein condensates was demonstrated for the first time by a group at M.I.T. (Stamper-Kumet al., 1998a). Bose condensates represent the ultimately cold state of an atomic sample and are therefore captured by extremely shallow optical dipole traps. High transfer efficiencies can be reached in very far-detuned traps, and the photon-scattering rate

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R. Grimm, M. Weidemiiller;and Yu. B. Ovchinnikov

acquires negligibly small values [see Eq. (14)]. Various specific features of dipole traps can fruitfully be applied to the investigation of many aspects of Bose-Einstein condensation that were not accessible formerly in magnetic traps. Sodium atoms were first evaporatively cooled in a magnetic trap to create Bose condensates containing 5 - 10 x lo6 atoms in the 3S112(F= 1, mF = -1) state (Mewes et al., 1996). Subsequently, the atoms were adiabatically transferred into the dipole trap by slowly ramping up the trapping laser power and then suddenly switching off the magnetic trap. The optical trap was formed by a laser beam at 985 nm (396 nm detuning from resonance) focused to a waist of about 6 pm. A laser power of 4 mW created a trap depth of about 4pK, which was sufficient to transfer 85% of the Bose-condensed atoms into the dipole trap and to provide tight confinement. Peak densities up to 3 x lOI5 atoms/cm3 were reported, representing unprecedented high values for optically trapped atomic samples. Condensates were observed in the dipole trap even without there initially having been a condensate in the magnetic trap. This strange effect could be explained by an adiabatic increase of the local phase-space density through changes in the potential shape (Pinkse et al., 1997). The slow increase of the trapping laser intensity during loading leads to a deformation of the trapping potential created by the combination of magnetic and laser fields. The trapping volume of the magnetic trap was much larger than the volume of the dipole trap. Therefore, phase-space density was increased during deformation, whereas entropy remained constant through collisional equilibration. Using the adiabatic deformation of the trapping potential, a 50-fold increase of the phase-space density was observed in a later experiment (Stamper-Kurn et al., 1998b). The lifetime of atoms in the dipole trap is shown in Fig. 8(a) in comparison to the results obtained in a magnetic trap. In the case of tight confinement and high densities (triangles in Fig. 8), atoms quickly escape through inelastic intratrap collisions. Long lifetimes (circles in Fig. 8) were achieved in the dipole trap and the magnetic trap, respectively, when the trap depth was low enough for collisionally heated atoms to escape the trap. Trap loss was found to be dominated by three-body decay with no identifiable contributions from two-body dipolar relaxation. The lifetime measurements delivered the threebody loss rate constant y = 1.1(3) x cm6/s [see Eq. (35)] for collisions among condensed sodium atoms. Figure 8(b) demonstrates simultaneous confinement of a Bose condensate in different Zeeman substates mF = 0, f l of the F = 1 ground state. To populate the substates, the atoms were exposed to an rf field sweep (Mewes et al., 1997). The distribution over the Zeeman states was analyzed through Stern-Gerlach separation by pulsing on a magnetic field gradient of a few

OPTICAL DLPOLE TRAPS FOR NEUTRAL ATOMS

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FIG. 8. Bose-Einstein condensates of Na atoms in focused-beam traps. (a) Trapping lifetime in optical traps and magnetic traps. The number of condensed atoms versus trapping time is shown. Closed symbols represent the data for optical traps with best transfer efficiency (triangles) and slowest decay (circles). Open circles represent data for a magnetic trap with optimized lifetime. The lines are fits based on single-particle losses and three-body decay. (b) Optical trapping of condensates in all hyperfine spin states of the F = 1 ground state. An absorption image after 340 ms of optical confinement and subsequent release from the trap is shown. Hyperfine states were separated by a pulsed magnetic field gradient during time of flight. The field view of the image is 1.6 by 1.8 mm. Adapted from Stamper-Kurn et al. (1998a).

G/cm after turning off the dipole trap. It was verified that all F = 1 substates were stored stably for several seconds. After the first demonstration of optical trapping, a spectacular series of experiments with Bose-Einstein condensates in a dipole trap was performed by the M.I.T. group. By adiabatically changing the phase-space density in the combined magnetic and optical dipole trap (see above), Stamper-Kurn et al. (1998b) were able to cross the transition to BEC reversibly. Using this technique, the temporal formation of Bose-Einstein condensates could be studied extensively (Miesner et al., 1998). The possibility of freely manipulating the spin of trapped atoms without affecting the trapping potential has led to the observation of Feshbach resonances in ultracold elastic collisions (Inouye et al., 1998) and to the investigation of spin domains and metastable states in spinor condensates (Stenger et al., 1998; Miesner et al., 1999).

5. Quasi-Electrostatic Traps The very interesting case of quasi-electrostaticdipole trapping has not so far been considered in this review. When the frequency of the trapping light is much smaller than the resonance frequency of the first excited state w << w, the light field can be regarded as a quasi-static electric field polarizing the atom. Quasi-electrostatictraps (QUEST) were first proposed (Takekoshi et al., 1995) and realized (Takekoshi and Knize, 1996) by a group at the University

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R. Grimm, M. Weidemiiller; and Yu. B. Ovchinnikov

of Southern California in Los Angeles. In the quasi-electrostatic approximation w << 00, one can write the dipole potential as

where uSutdenotes the static polarizability (w = 0). The light-shift potential of the excited states is also attractive, unlike the far-off-resonant interaction discussed before. Atoms can therefore be trapped in all internal states by the same light field. Because the trap depth in Eq. (39) does not depend on the detuning from a specific resonance line, as in the case of a FORT, different atomic species or even different molecules may be trapped in the same trapping volume. For the ground state of alkali atoms, Eq. (39) is well approximated by applying the quasi-static approximation to Eq. (lo), which gives

Compared to a FORT at a detuning A, the potential depth for ground-state atoms in a QUEST is smaller by the factor 2A/w0. Therefore, high-power lasers in the far-infrared spectral range have to be employed to create sufficiently deep traps. The CO2 laser at 10.6 pm, which is commercially available with cw powers up to some kilowatts, is particulary well suited for the realization of a QUEST (Takekoshi et al., 1995). An important feature of the QUEST is the practical absence of photon scattering. The relation between the photon-scattering rate and the trap potential can be derived from Eqs. (10) and (1 1) in the quasi-electrostaticapproximation

When compared to the corresponding relation for a FORT given by Eq. (14), the dramatic decrease of the photon-scattering rate in a QUEST becomes s-', showing that the QUEST obvious. Typical scattering rates are below represents an ideal realization of a purely conservative trap. Takekoshi and Knize (1996) have realized trapping of cesium atoms in a QUEST by focusing a 20-W C02 laser to a waist of 100pm, resulting in a trap depth of 115 pK. Around lo6 atoms prepared in the F = 3 state were loaded into the trap from a standard MOT. The atom loss rates of ~ 1 s - lwere consistent with pure losses through background-gas collisions. Hyperline relaxation times were found to exceed 10 s.

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133

B. STANDING-WAVE TRAPS A standing wave (SW) trap provides extremely tight confinement in axial dimension, as can be seen from Fig. 5(b). The trap can be realized by simply retroreflecting the beam while conserving the curvature of the wavefronts and the polarization. Assuming small extensions of the atomic cloud, one can write the potential in the form

with the standing wave oriented along the z-axis. The potential depth is four times as large as the corresponding trap depth for a single focused beam discussed in Section 1V.A. As for a single focused beam, radial confinement is provided by a restoring force c/wo.The axial trapping potential is spatially modulated with a period of h / 2 . Atoms are strongly confined in the antinodes of the standing wave (restoring force - f i k ) , resulting in a regular onedimensional lattice of pancake-like atomic subensembles. When aligned vertically, the axial confinement greatly exceeds the gravitational force mg. One can therefore use a rather shallow trap, just deep enough to trap a precooled ensemble,which results in low photon-scattering rates [see Eq. (14)] and large loading efficiencies. The tight confinement along the axial direction leads to large oscillation frequencies w, = k(2i?/n~)’’~at the center of the trap. The oscillation frequency decreases when moving along the z-axis because of the decreasing light intensity. At low temperatures, the energy of the axial zero-point motion $ h a z in the center of the trap can become of the same order of magnitude as the thermal energy iks T even for moderate trap depths.’* In this regime, the axial atomic motion can no longer be described classically but has to be quantized, and the vibrational ground state of the axial motion is substantially populated. The axial spread of the wavepacket is much smaller than the wavelength of an optical transition (Lamb-Dicke regime), giving rise to spectral line-narrowing phenomena. One might even enter a regime where the wavepacket extension comes close to the s-wave scattering length, leading to dramatic changes in the collisional properties of the trapped gas.

-

1. Optical Cooling to High Phase-Space Densities

Well-resolved vibrational levels and Lamb-Dicke narrowing, as realized along the axis of a standing-wave dipole trap, are requirements for the Using the recoil temperature Tree for the cooling transition at the wavelength introduced in Section III.A.I, one can write the zero-point energy as ( h o / k ) ( k B ~ , i r / 2 ) ” * .

134

R. Grimm, M . Weidemiillel; and Yu. B. Ovchinnikov

application of optical sideband cooling as explained in Section III.A.l. By employing degenerate-sideband Raman cooling, high phase-space densities of an ensemble containing large particle numbers have been achieved by Vuletic et al. (1998) at Stanford. In a vertical far-detuned standing wave, peak phase-space densities of about 1/180 have been obtained with lo7 cesium atoms, corresponding to a mean temperature of 2.8pK and a peak spatial density of 1.4 x lOI3 atoms/cm3. The trap was generated by a Nd:YAG laser with 17-W single-mode power at h = 1064 nm. The large beam waist of 260 pm created a trap depth of from a blue160 pK, which resulted in high loading efficiencies ( ~ 3 0 % detuned molasses released from a MOT) and small photon-scattering rates ( ~ s-'). 2 Atoms oscillated at 0,/2n: = 120 Hz in the radial (horizontal) direction and at 0,/2n: = 130 kHz in the axial (vertical) direction. A dramatic dependence of the trap lifetime on the hyperfine state of the cesium atoms was observed, similar to that shown in Fig. 4 in Section 1II.C. About 1 x lo7 atoms were contained in the trap populating 4700 vertical potential wells. Degenerate Raman sideband cooling was applied between vibrational states of a pair of Zeeman-shifted magnetic sublevels in the lowest hyperfine ground state, as explained in Section 1II.A.1. Suppressed collisional losses through inelastic binary collisions at high densities are greatly suppressed because population in the upper hyperfine state is kept extremely low in this cooling scheme. The Raman coupling was provided by the lattice field itself (Deutsch and Jessen, 1998). Axial and radial temperatures evolved differently during cooling, as indicated in Fig. 9. The axial direction was directly cooled, causing the axial temperature to drop quickly to T, = 2.5 pK. The radial temperature followed by collisional thermalization with a time constant of 150 ms. After sideband cooling was turned off, the temperature increased at a rate of 4 pWs, limiting the achievable final temperatures. This heating rate was much larger than the rate estimated on the basis of photon scattering; this might indicate additional heating sources such as laser noise. The achieved high thermalization rates, in combination with large particle numbers, would provide excellent starting conditions for subsequent evaporative cooling. A different approach to optical cooling of large particle numbers to high phase-space densities was followed by a group at Berkeley (Winoto et al., 1999),who applied polarization-gradientcooling to cesium atoms in a vertical standing-wave trap. The standing wave was linearly polarized, resulting in equal light shifts for all magnetic sublevels of the atomic ground state (see Section II.B.2). Therefore, polarization-gradient cooling, which relies on optical pumping between the ground-state sublevels (see Section 1II.A. 1), could be applied to the trapped sample in a very efficient way. A phase-space density of about with the large number of lo8 atoms was reached.

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

0

0

500

135

I000

cooling time [ms] FIG.9. Resolved-sideband cooling of Cs atoms in a vertical standing-wave trap. Evolution of vertical (solid squares) and horizontal (open squares) temperatures. Cooling is applied only along the tightly confining vertical axis; the horizontal degrees of freedom are indirectly cooled through collisional thermalization.From Vuletic et al. (1998).

2. Quantum Interference The regular arrangement of atoms in a standing-wave dipole trap has amazing consequences when a Bose condensate is loaded into such a trap. Macroscopic interference of Bose-condensed 87Rb atoms tunneling from an extremely shallow one-dimensional lattice under the influence of gravity has been observed by Anderson and Kasevich (1998) at Yale University. The dipole potential was created by a vertical standing wave at 850nm (detuning of 65 nm from the D 1line) with a waist of 80 pm, an order of magnitude larger than the transverse radius of the condensate. The condensate of lo4 atoms was coherently distributed among M 30 very shallow potential wells. The wells supported only one bound energy band below the potential edge, which is equivalent to U N ke Tree, where Tree is the recoil temperature [see Eq. (22)] at the wavelength of the trapping field. The gravitational field induces an offset between adjacent wells. For weak potential gradients, the external field can be treated as a perturbation of the band structure associated with the lattice. In this limit, wavepackets remain confined in a single band. The external field drives coherent oscillations at the Bloch frequency, as has been demonstrated with ultracold atoms confined in an accelerated, far-detuned, one-dimensional standing wave by Ben Dahan et al. (1996) at ENS in Paris and by Wilkinson et al. (1996) at University of Texas in Austin. The shallow potential wells allow for tunneling of particles into unbound continuum states. In the Yale experiment,the lifetime of the atoms confined in

136

I<. Grimm, M. Weidemiillel; and Yu. R. Ovrhinnikov

the lattice was purely determined by the tunneling losses. For a lattice of depth U = 1 . I kgTrec, the observed lifetime was ~ 5 0 m sEach . lattice site can be seen as a point emitter of a deBroglie wave. Interference between the different emitter outputs led to the formation of atom pulses falling out of the standing wave, quite similar to the output of a mode-locked pulsed laser. The pulse repetition frequency wreP= mg h / 2 h was determined by the gravitational increment between two adjacent walls. The repetition frequency can be interpreted as the difference in chemical potential divided by ti. This indicates the close relation of the observed effect of coherent atomic deBroglie waves to the ac Josephson effect resulting from quantum interference of two superconducting reservoirs.

3. Spin Manipulation Zielonkowski et al. (1998b) from the MPI fur Kernphysik in Heidelberg have used a vertical standing-wave trap for the manipulation of spin-polarized atoms. Using a large-volume, shallow, red-detuned standing-wave trap kept the depolarizing effects of photon scattering and atomic interactions at a low level while allowing for large numbers of stored atoms. Cesium atoms were trapped in a retroreflected 220-mW beam at a wavelength of h = 859 nm (detuning of 6.1 nm above the D2 line). The waist of the beam in the interaction region was 0.50mm. The maximum potential depth amounted to only 17 pK, which was deep enough to load atoms directly from a MOT at sub-Doppler temperatures. Despite the shallow potential depth, the confining force exceeded the gravitational force by about three orders of magnitude. The transfer efficiency from the MOT into the shallow trap was about 14%,resulting in about lo5 trapped atoms at a lifetime of T = 1.9 s, as shown by the stars in Fig. 10(a). In a first experiment, the spin state m F = 0 was selected by a Stern-Gerlach (SG) force. The force was created by horizontally shifting the zero of the MOT quadrupole field with respect to the position of the dipole trap: see Fig. 10(b). In this way, only atoms with m F = 0 are trapped by the dipole trap; all other magnetic substates are pulled out of the trap by the magnetic dipole force. The depolarizing effect of the trap light was determined by measuring the lifetime TSG of the mF = 0 atoms [see the circles in Fig. 10(a)] and comparing this value with the trap lifetime without Stern-Gerlach selection. ~ ' it possible to The depolarization rate r d e p o , = ~ / T S G - 1 / ~= 0 . 9 ~ made determine the photon-scattering rate from calculated values of the m F-state branching ratios for spontaneous scattering. The resulting scattering rate agrees well with the expectation based on Eq. (21). In a second experiment, spin precession in a fictitious magnetic field (Cohen-Tannoudji and Dupont-Roc, 1972; Zielonkowski et al., 1998a) was

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

137

+4 +3 +2

(a)

(b)

.,-I

+1

h

::D 2

25

1 c,

0

u

2

a

0.0

0.6

1.0

storage t i m e

1.6

(8)

2.0

m,,= 0

-25

-60 -1.0

-1

-0.5

2

0.0

0.5

(mm)

FIG. 10. Spin manipulation of Cs in a vertical standing-wave trap. (a) Storage time with all ground-state sublevels populated (stars) and for the Stern-Gerlach selected mF = 0 state (circles). Shown is the trapped particle number N relative to the number of atoms trapped in the MOT before transfer No. The line represents an exponential fit to the data yielding a time constant of 1.9 s for the unpolarized sample and 0.7 s for the mF = 0 polarized atoms. (b) Trap potentials for the combined magnetic quadrupole and optical trap used for Stern-Gerlach selection of the t n F = 0 state from the F = 4 hyperfine ground state. The zero point of the magnetic quadrupole has been horizontally shifted with respect to the center of the standingwave trap. Atoms with r n F # 0 are expelled from the dipole trap by the magnetic field gradient. Adapted from Zielonkowski et al. (1998b).

demonstrated. The field was induced by an additional off-resonant circularly polarized laser beam, which induced a light-shift scaling linear with mF as discussed in Section II.B.2. The resulting splitting corresponds to a fictitious magnetic field of 50mG. The mF = 0 state was selected by a short SternGerlach pulse resulting in a macroscopic magnetization of the sample. The population of the same state was analyzed after a 150-ms delay. Between preparation and analysis, the atoms interacted with a pulse of the fictitious field laser in combination with a holding magnetic field. The mF = 0 population oscillates with the duration of the laser pulses, which can be directly interpreted as the Larmor precession of the spin in the superposition of fictitious and holding magnetic field. A 27t and a 471rotation of the magnetization were observed.

4. Quasi-Electrostatic Lattices A standing wave created by the light at 10.6pm from a C02 laser creates a QUEST (see Section IV.A.5) with a spacing between the axial potential wells that is large compared to the transition wavelength of the trapped atoms. At the MPI fur Quantenoptik in Garching, such a lattice of mesoscopic potential wells was realized with rubidium atoms (Friebel et al., 1998a, 1998b). A 5-W CO;! laser beam was focused to a waist of 50 pm, creating a trap depth of about 360 pK. Up to 3 x 105atoms could be loaded into the horizontal standing-wave trap which had a lifetime of 1.8 s limited by backgroundgas collisions. Temperatures around 10 pK were achieved by polarizationgradient cooling in the trap.

138

R. Grimm, M . Weidemiillel; and Yu. B. Ovchinnikov I

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FIG.11. Excitation spectra of Rb atoms trapped with a CO?laser at 10.6l m , creating quasielectrostatic traps. The spectrum of a standing-wave trap (left-hand graph) is compared to the spectrum from a single-beam trap (right-hand graph). The horizontal axis is the fluorescence of atoms illuminated after 0.6 s of trapping time. The vertical axis gives the modulation frequency of the trap light intensity. Parametric resonances at the oscillation frequencies and at twice their value are observed. The jump at 100 Hz in both graphs is an artifact resulting from a change of the modulation depth and modulation time. Adapted from Friebel et al. (1998a).

The vibrational frequencies of atoms inside the potential wells were measured by modulating the laser intensity in order to drive parametric excitation of the atomic oscillations. When the modulation frequency equals twice the vibrational frequency 20,, (or subharmonics 20,,/n), atoms are heated out of the trap, leading to a reduced lifetime. This effect was demonstrated by varying the modulation frequency and measuring the number of atoms that were left in the trap after a fixed trapping time of 600 ms. The remaining atoms were detected by switching on a resonant light field and recording the fluorescence. The left-hand graph in Fig. 11 shows the excitation spectrum for the standing-wave trap. Parametric resonances at 2 lcHz and at 32 kHz are attributed to excitations of the radial and axial vibrations, respectively. A subharmonic resonance at 16 kHz is also observed. In the right-hand graph of Fig. 11, the excitation spectrum of a single-focused-beam dipole trap is presented. The trap was realized by interrupting the retroreflected laser beam that had formed the standing wave. The radial resonance shifts to 1.6 kHz (subharmonic at 0.8 kHz) because of the four-times-lower trap depth. The reduction of the axial vibrational frequency is more dramatic because it scales as 1 / ( 2 k z ~relative ) to the standing-wave trap (including the factor 4 in trap depth). The axial resonance is now found at 80 Hz (subharmonic at 40 Hz). C. CROSSED-BEAM TRAPS A single focused beam creates a highly anisotropic trap with relatively weak confinement along the propagation axis and tight confinement in the

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

139

perpendicular direction [see Figs. 5(a) and 7(a)]. In a standing-wave trap, the anisotropic atomic distribution of the single-beam trap is split into anisotropic subensembles with extremely tight confinement along the axial direction. Crossing two beams with orthogonal polarization and equal waist under an angle of about 90°, as indicated in Fig. 5(c), represents an obvious way to create nearly isotropic atomic ensembles with tight confinement in all dimensions. In this case, the dipole potential for small extensions of the atomic cloud can be approximated as

It should be noted that the effective potential depth is only U / 2 , because atoms with larger energy can leave the steep trap along one of the beams. 1. Evaporative Cooling

Crossed-beam dipole traps provide a good compromise between decent trapping volumes and tight confinement, and they are therefore suited for the application of evaporative cooling as explained in Section III.A.l. Adams et al. (1995) at Stanford used a crossed-beam configuration oriented in the horizontal plane for evaporative cooling of sodium. A single-mode Nd:YAG laser generated two beams of 4 W each, focused to a waist of 15pm and crossing under 90". The trap depth was close to 1mK. The polarizations of the two beams were chosen orthogonal, which results in a spin-independent trapping potential for the ground states because of the large detuning of the 1064-nm light from the two fine-structure lines of sodium (see Table I). After transfer from a MOT, evaporative cooling was started with -5000 atoms at a temperature of 140 pK and a peak density of 4 x 10l2atoms/cm3, as indicated by the dotted lines in Fig. 12. To force evaporation of highenergetic particles, the Nd:YAG power was exponentially ramped down from 8 W (trap depth 900 pK) to 0.4 W (trap depth 45 pK) within 2 s. After evaporation, 500 atoms were left in the trap. Temperature was reduced by a factor of 35 to 4 pK. Even so, density decreased by about an order of magnitude because the restoring force in the trap is reduced when ramping down the intensity. To keep the density at a high value for efficient evaporation, one would have to compress the cloud further. However, compared to the initial conditions, phase-space density was increased by a factor of 28 in the experiment, indicating the potential of evaporative cooling for the enhancement of phase-space density in dipole traps.

--

N

R. Grimm, M. Weidemiiller;and Yu. B. Ovchinnikov

140

.-3

1.0

C

3

$ $

l a

v

0.5

!? 3

G

0.0 I

I

I

I

I

1

I

1

-60 -40-20 0 20 40 60 Position (microns)

0

-100

I 100

1

Position (microns)

FIG. 12. Evaporative cooling of Na atoms in a crossed-dipole trap. (a) Atom density distribution before (dotted line) and after (solid line) evaporative cooling of Na in a crosseddipole trap. The density decreased by a factor of about 7. (b) Time-of-flight measurement of the temperature before (dotted line) and after (solid line) evaporative cooling. The temperature decreased from 140 pK to 4 pK.From Adams ef al. (1995).

2. Interference Effects Let us now consider the more general case of two beams of equal waist w o propagating in the xy-plane and intersecting at the focal points under an arbitrary angle. The angle between the beams and the x-axis is &$. This configuration includes as special cases the single-focused-beamtrap ($ = 0), discussed in Section IV.A, and the standing-wavetrap (4 = 90"),discussed in Section 1V.B. In the harmonic approximation, the trapping potential can be written as UCB(X,Y,Z) N -U(y) .

(

1 - 2 7 - 2- - 2wx x2 w yy2 2 wo z2)

+

(44)

The potential radii wx,yare given by w: = (cos2$ / w i sin2 $/2zi)-' and w: = (sin2+/wi + cos2 $/2zi)-'. The dependence on y of the trap depth U reflects interference effects between the two waves that lead to a modulation of the trap depth on the scale of an optical wavelength. If both beams are linearly polarized along the y-direction (linlllin), interference results in a pure intensity modulation with period D = h/(2 sin $), yielding h(y) = h,, cos (ny/D). Trapped atoms are therefore bound in the antinodes of the interference pattern, forming a one-dimensional lattice along the y-direction with lattice constant D.In the case of orthogonal polarization of the two beams (linllin), the intensity exhibits no interference effects, but the polarization is spatially modulated between linear and circular with the

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

141

same period D. As discussed in Section II.B.2, the light shift depends on the spin state of the atoms giving rise to a spatial modulation of the potential. For detunings A large compared to the fine-structure splitting A;,, the potential depth is modulated with a relative amplitude {g.rnFA;,/A; see Eq. (20). Malung use of these interference effects, the ENS group investigated Raman cooling (Kuhn et al., 1996) and sideband cooling (Perrin er al., 1998; Bouchoule et al., 1999) in a crossed-beam trap for cesium atoms. The trap consisted of two Nd:YAG laser beams (1= 1064nm) propagating along a vertical xy-plane with a power of about 5 W in each beam. The beams crossed at their common waists (wg M 100pm) under an angle = f53" with the horizontal x-axis. The hyperfine splittings of the ground and excited states were small compared to the detuning of the laser from the D1 and Dz lines of cesium at 894 nm and 852 nm, respectively. For the linllin case, the comparatively large fine-structure splitting of cesium (Aks/A M 5 ) led to spatially modulated potentials with a small yet significant modulation amplitude of M U/l5 for the stretched Zeeman states l r n ~ = l I 1/2. The trap was loaded from lo7 cesium atoms in a MOT. The loading efficiency and the shape of the atomic cloud strongly depended on the laser polarizations of the dipole trap (Kuhn er al., 1996).The potential wells formed by interference of the laser beams were used for resolved-sideband cooling with Raman transitions between the F = 3 and F = 4 hyperfine ground states. Great differences in the performance of sideband cooling were found for the linlllin case and the linllin case because of the different character of the potential wells being weakly or strongly modulated, respectively (Perrin et al., 1998). By optimizing the sideband cooling in the linlllin configuration, single vibrational states (motional Fock states) could be prepared in the onedimensional standing wave. Atoms were first sideband-cooled into the lowest vibrational state Inosc = 0), from which they could be transferred into other pure Inosc)states by Raman transitions at multiples of the resolved vibrational sidebands (Bouchoule et al., 1999).

+

+

N

D. LATTICES When adding more laser beams, one can design a whole variety of interference patterns to create two- and three-dimensional lattices confining the atoms in micropotentials of submicron extension (Deutsch and Jessen, 1998). When this approach is combined with efficient cooling methods, a significant population of the vibrational ground state can be achieved. Many important aspects of these optical lattices have been studied extensively for nearresonant trapping fields that simultaneously provide tight confinement and dissipation (Jessen and Deutsch, 1996; Hemmerich et al., 1996; Grynberg and Triche, 1996).

142

R. Grimm, M . Weidemidler; and Yu. B. Ovchinnikov

In this chapter, we have so far concentrated on red-detuned traps because of the conceptual differences in the practical realization of dipole traps compared to the blue-detuned traps discussed in the next chapter. In the case of threedimensional far-detuned lattices, this distinction becomes faint because both lattice types are realized through appropriate interference patterns of multiple beams. The atoms are trapped either in the antinodes (red detuning) or in the nodes (blue detuning) of the interference pattern. The main difference between red and blue detuning lies in the photon-scattering rates, as discussed in Section III.A.3. Here, we briefly present novel developments for both types of far-off-resonance lattices reported by several groups (Anderson et al., 1996; Muller-Seydlitz et al., 1997; Hamann et al., 1998; Boiron et al., 1998; DePue et al., 1998). Localized wavepackets oscillating in conservative microtraps are promising systems in which to study fundamental questions related to quantum-state preparation, coherent control, and decoherence of macroscopic superposition states. Furthermore, optical lattices can serve as prototype systems for the study of condensed-matter models based on periodic arrangements of weakly interacting particles. Anderson et al. (1996) have confined lithium atoms in three-dimensional optical lattices formed in the intersection of four laser beams. For lithium, the fine-structure splitting is only 10 GHz, leading to ground-state optical potentials that are independent of the light polarization and the atomic spin state. A face-centered cubic lattice with a nearest-neighbor spacing of 1.13 h was realized by a four-beam configuration. Three-dimensional lattices with periodicity much larger than h could be created by reducing the angles subtended by each possible pair of the four lattice beams.13 Up to lo5 atoms were trapped in the lattice. By reducing the intensity of the trapping light so that only the coldest atoms from a MOT were confined in the lattice, a trapped ensemble with an rms velocity spread corresponding to 1.8 pK was prepared. By adiabatically expanding deep potential wells (see Section 1II.A.l), cooling at the expense of smaller spatial density could be achieved. The temporal evolution of metastable argon atoms stored in the intensity nodes of a blue-detuned optical lattice was investigated by Muller-Seydlitz et al. (1997) at Konstanz University. The lattice of simple-cubic symmetry was formed by three mutually orthogonal standing waves with othogonal linear polarizations resulting in isotropic potential wells with a lattice constant of h/2 = 397 nm. About lo4 atoms are initially captured in the lattice. Figure 13 shows time-of-flight spectra for variable trapping times. After a certain storage time, the light intensity is ramped down, releasing one bound state after the other. For increasing storage times, the population of excited bands in the l 3 In a different approach to creating structures with large periodicities, Boiron et al. (1998) used interference from multiple beams emerging from a holographic phase grating.

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

143

250

200 150 100

50

I

-:

storage time T*,

FIG. 13. Time-of-flight spectra of metastable argon atoms trapped in the nodes of a threedimensional far-off-resonance optical lattice. The spectra were taken after various storage times T. Atoms were released from the trap by slowly ramping down the trapping light intensity. Higher-energetic bound states arrive first, deeper-confined states later. After long storage times, only atoms in the lowest bound state are found. From Miiller-Seydlitz et al. ( 1998).

potential wells decreases faster than the population of the vibrational groundstate band, leaving about 50 atoms populating the motional ground state after storage times of about 450ms. Two processes could be identified for this state-selective loss of particles: First, atoms in higher excited bands have a higher probability of leaving the finite extension of the trapping field (wg = 0.55 mm) by tunneling. Second, atoms interacting with the far-offresonance trapping light (detuning 2 nm from resonance for the metastables) are optically pumped into the electronic ground state of argon and are therefore lost from the trap. The probability for an optical pumping process depends on the spatial overlap between the optical lattice field and the atomic wavepacket confined to the potential well. The smaller the spatial extension of the wavepackets, the smaller the excitation rate, and it is smallest for the motional ground state. The reduced photon-scattering probability of deeper-bound states is an important specific property of a blue-detuned optical lattice. The first demonstrationof resolved-sidebandRaman cooling in a dipole trap, explained in Section 1II.A.1 was performed with cesium in a two-dimensional

R. Grimm, M. Weidemiiller; and Yu. B. Ovchinnikov

144

llq, 40

20

0

FIG. 14. Resolved-sideband Raman cooling of Cs atoms in a two-dimensional far-offresonance lattice. Shown is the inverse Boltzmann factor I/qs as a function of the applied magnetic field. When the magnetic field tunes the lattice-field-induced Raman coupling to the first-order (at B, x 0.12 G) or the second-order (at B, x 0.24 G) motional sidebands, Raman cooling is most efficient, resulting in the peaks of I/qs. Solid circles are data points; the solid line is a fit to the sum of two Lorentzians. Inset: Corresponding kinetic temperatures measured to determine the Boltzmann factor. The dashed line indicates the kinetic temperature of the vibrational ground state. From Hamann et al. (1998).

lattice by a group at the University of Arizona in Tucson (Hamann et al., 1998). The two-dimensional lattice consisted of three coplanar laser beams with polarizations in the lattice plane. A magnetic field perpendicular to the lattice plane was applied to Zeeman-shift the motional states I r n ~= 4;nose) and ImF = 3;nosc- 1) of the upper hyperfine ground state F = 4 into degeneracy (see Section 1II.A. 1). By adding a small polarization component orthogonal to the lattice plane, Raman coupling between magnetic sublevels with Am, = f l was introduced. In Fig. 13, the inverse Boltzmann factor l / q ~= exp(ho,,,/kBT) is plotted versus the magnetic field. The inverse Boltzmann factor reaches a maximum when the two motional states are shifted into degeneracy by the magnetic field so that sideband cooling by the lattice field becomes effective. The second, smaller peak shows well-resolved cooling on the second-order Raman sideband (An.,, = -2). As shown by the inset in Fig. 14, the thermal energy of the atoms closely approaches the zeropoint energy in the potential wells, indicating a population >95% of the vibrational ground state. Unity occupation of sites in a three-dimensional far-off-resonance optical lattice has been realized by the Berkeley group (DePue et al., 1999). In the

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

145

regime of unity occupation, interactions between highly localized atoms have dramatic effects, and studies of collisional properties of tightly bound wavepackets become possible. The necessary high densities were achieved by applying polarization-gradient cooling to the three-dimensional lattice filled with lo8 cesium atoms and by subsequent adiabatic toggling between the three-dimensional lattice and a one-dimensional standing wave. The threedimensional lattice was formed by three mutual orthogonal standing waves with controlled time-phase differences. After the atoms were cooled in the lattice, the horizontal beams of the lattice were adiabatically turned off, leaving the atoms in a vertical standing-wave trap. Because of their low temperature (700nK), the atoms were essentially at rest at their respective positions. Under the action of the transverse trapping potential, the atoms radially collapsed toward the trap center. All atoms arrived simultaneously at the center after about a quarter of a radial oscillation period. Thereby, the density in the trap center was transiently enhanced by a factor of 10, reaching 6 x 10’2atoms/cm3. At the moment of peak density, the horizontal lattice beams were adiabatically turned on again. A substantial fraction of lattice sites were then multiply occupied and underwent fast inelastic collisions. After multiply occupied sites had decayed, single atoms cooled to near their vibrational ground state occupied 44% of the lattice sites.

V. Blue-Detuned Dipole Traps Laser light acts repulsively on the atoms when its frequency is higher than the transition frequency (“blue” detuning). The basic idea of a blue-detuned dipole trap is thus to surround a spatial region with repulsive laser light. Such a trap offers the great advantage of atom storage in a “dark” place with low influence of the trapping light, which minimizes such unwanted effects as photon scattering, light shifts of the atomic levels, and light-assisted collisional losses. According to the discussion following Eq. (3 I), this advantage becomes substantial in the case of hard repulsive optical walls [K << 1 in Eq. (31)J or large potential depth for tight confinement (fi >> k ~ 7 ‘ ) . Experimentally, it is not quite so simple and straightforward to realize a blue-detuned trap as it is in the red-detuned case, where a single tightly focused laser beam already constitutes an interesting dipole trap. Therefore, the development of appropriate methods to produce the required repulsive “optical walls” has played a central role in experiments with blue-detuned traps. Three main methods have been applied for this purpose: Light sheets, produced by strong elliptical focusing of a laser beam, can be used as nearly flat optical walls (Davidson etal., 1995; Lee etal., 1996).Hollow laser beams can provide spatial confinement in at least two dimensions (Yang et af.,1986).

146

R. Grimm, M. Weidemiiller, and Yu. B. Ovchinnikov

Evanescent waves, formed by total internal reflection on the surface of a dielectric medium, represent nearly ideal mirrors to reflect atoms (Cook and Hill, 1982; Dowling and Gea-Banacloche, 1996). In most blue-detuned traps, gravity is used to close the confining potential from above. Such traps are referred to as gravito-optical traps. As a further experimental possibility, which we have already discussed in Section IV.D, atoms can be trapped in the micropotentials of far blue-detuned optical lattices (Miiller-Seydlitz er al., 1997). In this section, we discuss various blue-detuned traps that have been realized experimentally and their particular features; an overview is given in Table 11. In the following, these traps are classified in terms of the main method applied for producing the optical walls: light-sheet traps (Section V.A), hollow-beam traps (Section V.B), and evanescent-wave traps (Section V.C).

A. LIGHT-SHEET TRAPS In experiments performed at Stanford University, Davidson et al. (1995) and Lee et al. (1996, 1998) have realized light-sheet traps of various configurations and applied them for rf spectroscopy on trapped atoms and for optical cooling to high phase-space densities. The light sheets were derived from the two strongest lines of an all-line argon-ion laser. A laser power of up to 10 W at 5 14nm and up to 6 W at 488 nm was focused with cylindrical lenses to cross sections of typically 15 pm x 1 mm. For the Na atoms used in the experiments (resonance line at 589 nm), this leads to maximum light-sheet potentials in the order of 100 pK. When two light sheets are combined and overlap in space, there are two ways to avoid perturbing interference effects, which could open escape channels in the optical potential. First, if the two light sheets have different frequencies, then the relevant potential is determined by the time average over the rapid-beat note, in which the interference averages out. Second, if the light sheets have the same frequencies but orthogonal polarizations, then interference leads to a spatial modulation of the polarization. In the case of large detunings greatly exceeding the fine-structure splitting, as in the Stanford experiments, the polarization modulation has a negligible effect on the dipole potential [see Eq. (20)]. In the first experiment (Davidson et al., 1995), two horizontally propagating light sheets were combined to form a vertical “V” cross section. This configuration already provides three-dimensional confinement, because the trapping potential is closed along the propagation direction as a result of the divergence of the tightly focused light; see Fig. 15(a). Vertically, the atoms are kept in the trap by gravity. The gravito-optical trap thus has the form of a boat with a length of about 2 mm (for trap parameters, see Table 11).

TABLE 11 PARAMETERS OF VARIOUS EXPERIMENTALLY REALIZED BLUE-DETUNED DIPOLETRAPS. Trap

Confining fieldsa

Atomic species

Light-sheet trap Davidson et a[. (1995)

LS gravity

Na

Inverted pyramid Lee et al. (1996)

LS

Na

gravity

Detuning. depth

# Atoms, transfer

Cooling. temperature

Spccialty

3000

no

-

very long coherence times

1OOTHz 20 pK

4 x 105

Raman 1 .O pK

high phase-space densities

-

IOOTHz

10 pK

Plugged doughnut beam Kuga et al. (1997)

HB

Rb

60 GHz 40 PK

108 30%

molasses 13pK

large number of atoms

Single-beam trap Ozeri et al. (1998)

HE5

Rb

250 GHzC

105

no

single laser beam

Conical atom trap Ovchinnikov et al. (1988)

HE3 gravity

cs

3 GHz -1 mK

106 80%

molassesd -1OpK

high loading efficiency

Gravity-optical cavity Aminoff et al. (1993)

EW gravity

cs

1-10GHz

107

no

ten resolved bounces

Grav.-opt. surface trap Ovchinnikov et al. (1997)

EWkJB gravity

cs

16 30%

reflection 3 PK

atoms very close to surface

1m K e

1GHz' 100pK

LS: light sheets, HB: hollow beam, E W evanescent wave. See Torii er al. (1998). 'Trapping studied in a wide detuning range between 50 GHz and 15THz. Pure reflection cooling demonstrated at a trap detuning of 30 GHz. Number refers to vertical motion only. The horizontal depth was much lower (5 pK). Number refers to evanescent wave. The hollow-beam detuning was much larger (100GHz).

E

a

'

c

P

4

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R. Grimm, M. Weidemuller; and Yu. B. Ovchinnikov

FIG.15. Light-sheet trap used for rf spectroscopy. (a) Laser intensity produced in a horizontal plane 30 pm above the intersection of the two focused sheets of light. The x- and vdimensions are in microns, and the intensity is normalized to the peak laser intensity. (b) The central Ramsey fringes observed for rf-induced hyperfine transitions for a measurement time of 4 s. From Davidson et al. (1995).

Using this light-sheet trap, Davidson ef al. have stored about 3000Na atoms and have impressively demonstrated the advantages of blue-detuned dipole traps for spectroscopic applications. By using the method of separated oscillatory fields, they have measured Ramsey fringes of the F = 1, m F = 0 -+ F = 2, mF = 0 hyperfine transition of the Na ground state. For the excitation of this transition, an rf traveling wave with a frequency of 1.77 GHz was used. The Ramsey fringes were measured by applying two n/2 rf pulses separated by a time delay of up to 4 s. Initially, all trapped atoms were optically pumped into the lower hyperfine state (F = 1 ). After application of the two rf pulses, the number of atoms transferred into the upper state (F = 2) was measured by applying a short pulse of light resonant with the cycling F = 2 -+ F' = 3 transition and detecting the induced fluorescence. The two central Ramsey fringes observed by varying the rf frequency for a pulse delay of 4 s are shown in Fig. 15(b). By analyzing the dependence of the fringe contrast on the delay between the two rf pulses, a 1 / e coherence decay time of 4.4 s was obtained. In the same experiments, the Stanford group measured the mean residual light shift (ac Stark shift) of the hyperfine transition frequency as caused by the trapping light. From the frequency of the central Ramsey fringe, a corresponding shift of 270 mHz was observed. The absolute light shifts of the two hyperfine sublevels are larger by the ratio of the optical detuning (- 90 THz) to the hyperfine splitting (1.77 GHz) and thus amounted to 14 kHz.This number directly gives the average dipole potential u d i p M h x 14kHz M k B x 0.7 pK experienced by the atom and also allows one to determine an average photon-scatteringrate of 0.01 s-' according to Eq. (14).

-

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

149

106

106 r 0

b

104

2

103

s

102

0

10

30 40 Time (rec) 20

50

FIG. 16. Inverted-pyramid trap used for Raman cooling. (a) Schematic of the trap geometry. (b) Lifetime measurement performed after Raman cooling at a background pressure of - 3 x lo-" mbar. From Lee et al. (1996).

The authors also mention similar experiments performed in a red-detuned dipole trap realized with a Nd:YAG laser. In this case, the longest observed coherence times were -300 times lower, which highlights the advantage of the blue-detuned geometry for in-trap spectroscopy. In a later experiment (Lee et al., 1995), the light-sheet trapping was improved by a new trap geometry in the form of an inverted pyramid. As illustrated in Fig. 16(a), this trap was produced by four sheets of light. Because of the much larger trapping volume provided by the pyramidal geometry, this trap could be loaded with 4.5 x lo5 atoms, which constitutes an improvement over the previous configuration by more than a factor of 100 (see also Table 11). Lee et al. have also tested a similar, tetrahedral box trap, the performance of which was inferior to the inverted pyramid. The particular objective of the experiment by Lee et al. was to obtain high phase-space densities of the trapped atomic ensemble by optical cooling. The authors therefore applied one-dimensional Raman cooling (Kasevich and Chu, 1992) as a subrecoil cooling method; see also Section III.A.l. The geometry of the inverted-pyramid trap is very advantageous for getting dense atomic samples because of a strong spatial compression with decreasing temperature. In an ideal inverted pyramid, the density n would scale as TP3, in contrast to a three-dimensional harmonic oscillator, where n o( T - 3 / 2 . Moreover, this trap configuration provides very fast motional coupling between the different degrees of freedom, which is particularly interesting in a one-dimensional cooling scheme. About 4.5 x lo5 atoms were loaded at a temperature of 7.7 pK and a peak density of 2 x 10" cmP3.Application of a sequence of Raman cooling pulses

150

R. Grimm, M. Weidemiiller, and Yu. B. Ovchinnikol,

during the following 180ms reduced the temperature of the atoms to 1 pK, and, keeping practically all atoms in the trap, a peak density of 4 x 10" cmP3 was reached. This density increase by a factor of 20 resulting from a temperature reduction by a factor of about 7 indicates that the trap was already in a regime where the confining potential behaves harmonically rather than as in an ideal pyramid. This may be explained by the finite transverse decay length of the light-sheet potential walls. As a result of the Raman cooling, the atomic phase-space density was increased by a factor of 320 and reached a value that was about a factor of 400 from Bose-Einstein condensation. After an initial loss of atoms observed in the first second after the Raman cooling process, an exponential decay of the trapped atom number was measured with a time constant of 7s; see Fig. 16(b). This loss did not significantly depend on the background pressure below lo-'' mbar. which points to the presence of an additional single-particle loss mechanism. This loss was consistent with an observed heating process in the trap that exceeded by 30 times the calculated heating by photon scattering. This heating of unknown origin was identified as the main obstacle to implementing evaporative cooling in the inverted-pyramid trap. The experiments moreover showed evidence that ground-state hyperfine-changing collisions, ejecting atoms out of the trap, were limiting the maximum density achievable with Kaman cooling to 1 0 ' ~cmp3. In a later experiment, Lee and Chu (1998) used the inverted-pyramid trap to prepare a Raman-cooled sample with spin polarization in any of the three magnetic sublevels of the lower hyperfine ground state (F = I). A small bias magnetic field was used to lift the degeneracy of the ground state, and appropriate polarization was chosen for the Raman cooling light. The attained temperature and phase-space density were similar to the unpolarized case. This experiment can be seen as a nice illustration of the general advantage that dipole trapping leaves the full ground-state manifold available for experiments.

B. HOLLOW-BEAM TRAPS Hollow blue-detuned laser beams, which provide radial confinement, are particularly interesting and versatile tools for the construction of dipole traps. Hollow laser beams may be divided into two classes: beams in pure higherorder Laguerre-Gaussian (LG) modes and other hollow beams that cannot be represented as single eigenmodes of an optical resonator. The main difference is that LG beams preserve their transverse profile with propagation (only converging or diverging in width), which is of particular interest for atom guiding (Dholalua, 1998; Schiffer et al., 1998). Other, non-LG hollow beams can change their profile substantially, thus offering additional features of interest for atom trapping.

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

151

A Laguerre-Gaussian mode LGpl is characterized by a radial index p and an azimuthal index 1. A hollow beam with a “doughnut” profile is obtained for p = 0 and 1 # 0 with an intensity distribution given by

where P is the power and wo is the waist of the beam. For 1 = 0, this equation gives the transverse profile of a usual Gaussian beam (TEMm mode) as described by Eq. (36). The higher the mode index 1, the larger the ratio between the beam radius and the width of the ring - that is, the harder the repulsive optical wall radially confining the atoms, the weaker the heating by photon scattering.

1. Plugged Doughnut-Beam Trap The dougnut-beam trap shown in Fig. 17(a) was realized by Kuga et al. (1997) at the University of Tokyo; see also Table 11. The LGo3 doughnut beam (WO = 0.6mm) was derived from a laser that was forced to oscillate in a Hermite-Gaussian HGo3 mode by insertion of a thin wire into the cavity. An astigmatic mode converter (Beijersbergen et al., 1993) then transformed the beam into the LGo3 mode. Because a LG beam does not provide axial confinement, the hollow beam was plugged by two additional laser beams (diameter 0.7 mm), which were separated by 2 mm and perpendicularly intersected the hollow beam. The plugging beams were derived from the recycled doughnut beam. An exponential decay of the stored atom number was observed with a time constant of about 150ms [see Fig. 17(b)],which was explained by heating out of the trap by photon scattering. The authors also observed that trapping in a

FIG. 17. (a) Plugged doughnut-beam trap. (b) Measurement of the storage time (filled squares). The filled circles refer to a two-dimensional trap formed by the doughnut beam alone without plugging beams (no axial confinement). Adapted from Kuga et al. (1997).

R. Grimm, M. Weidemiiller,and Yu. B. Ovchinnikov

152

LGol mode showed inferior performance with very short lifetimes of a few milliseconds only, which highlights the benefit of “hard” optical walls for reducing heating by photon scattering. In subsequent experiments (Torii et al., 1998), the storage time of the trap was improved by application of a pulsed optical molasses cooling scheme to a value of 1.5 s, which then was dominated by collisions with the background gas. This noncontinuous molasses cooling scheme allows one to cool the atoms down to nearly the same temperatures as in a continuous cooling scheme, but it suppresses trap losses by light-assisted collisions and the interference of light shifts from the molasses light with the trapping potential. Moreover, potentially severe losses by hyperfine-changing collisions (see Section 1II.C) can be strongly reduced by keeping the atoms in the lower hyperfine ground states in the off-times of the molasses. In such a pulsed cooling scheme, the off-time can last as long as heating by photon scattering (see Section III.A.2)does not significantly degrade the trap performance. The on-time can be as short as the typical cooling time in the molasses.

2. Single-Beam Trap A blue-detuned trap based on a single laser beam has been demonstrated by Ozeri et al. (1999) at the Weizman Institute in Israel; see also Table 11. The trapping beam was produced in an experimentally very simple way by passing a single Gaussian beam through a phase plate of appropriate size, which slufted the center of the beam by a phase angle of IT. In the focus of such a beam, destructive interference leads to a reduced or even vanishing light intensity. By choosing the proper ratio between the diameter of the phase plate and that of the laser beam, Ozeri et al. obtained a darkness ratio (central intensity normalized to maximum intensity) of 1/750. Close to the center of this singlebeam trap, the intensity increases radially with the fourth power of the distance (as in the case of a LGo2 beam), and axially the dependence is quadratic. In order to measure the average light intensity experienced by the atoms in the trap, Ozeri et al. studied the relaxation of hyperfine population caused by photon scattering from the trapping light in a manner similar way to the approach of Cline et al. (1994) in a red-detuned far-off-resonance trap; see also Section IV.A.2.From measurements performed at a detuning of 0.5 nm, it was concluded that the average intensity experienced by a trapped atom was as low as 1/700 of the maximum intensity. In another series of measurements, they observed that at constant temperature, the photon-scattering rate scaled linearly with the inverse detuning. This observed behavior represents a nice confirmation of Eq. (29), which for a power-law potential directly relates the average scattering rate to the temperature with an inverse proportionality to the detuning. N

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

153

In a recent proposal, Zemanek and Foot (1998) considered a blue-detuned dipole trap formed by two counterpropagating laser beams of equal central intensities but different diameters. Along the axis of such a standing-wave configuration, completely destructive interference would lead to minima of the dipole potential with zero intensity. These traps would be radially closed because of the incomplete destructive interference at off-axis positions. This resulting linear array of three-dimensional dipole traps could combine the interesting features of standing-wave trapping schemes (see Section 1V.B) with the advantages of blue detuning. Experimentally, it seems straightforward to realize such a trap by retroreflection and appropriate attenuation of a single slightly converging laser beam.

3. Conical Atom Trap A single-beam gravito-optical trap was recently demonstrated by Ovchinnikov et al. (1998) at the MPI fur Kernphysik in Heidelberg. The conical atom trap (CAT), illustrated in Fig. 18(a), is based on a conical hollow beam and combines experimental simplicity with several features of interest for the trapping of a large number of atoms at high densities: high loading efficiency, tight confinement, low collisional losses, and efficient cooling.

FIG. 18. (a) Illustration of the conical atom trap (CAT). (b) Fluorescence image of atoms in the CAT (lower, elongated blob) combined with an image of atoms in the MOT (upper spot). The dashed lines indicate the conical trapping field. Adapted from Ovchinnikov et al. (1998).

R. Grimm, M. Weidemiiller; and Yu. B. Ovchinnikov

154

In the experiment (for parameters, see Table 11) the upward-directed conical trapping beam was generated by using an arrangement of two axicons and one spherical lens. In the focal plane, the beam profile was roughly Gaussian with a diameter of about 100 pm. Within a few millimeters from the focus, the beam evolved into a ring-shaped profile with a dark central region resembling a higher-order Laguerre-Gaussianmode. The opening angle of the conical beam was about 150mad. The trap was operated relatively close to resonance as compared to the other blue-detuned traps we have discussed. With an optical detuning of 3 GHz (12 GHz) with respect to the lower (upper) hyperfine ground state of Cs, a large potential depth was realized, which, together with the funnel-like geometry, facilitated the transfer of as much as 80% of all atoms from the MOT into the CAT. The relatively small detuning requires efficient cooling for removing the heat that results from photon scattering from the trapping light. For this purpose, an optical molasses was applied continuously to cool atoms in the upper hyperfine ground state. This cooling, however, takes place with an inherently reduced duty cycle and is thus similar to the pulsed molasses cooling scheme of Torii er al. (1998), which was discussed before. In the molasses scheme applied in the CAT, the short phases of cooling in the upper hyperfine ground state (typically 50 ps) are self-terminating by optical pumping into the lower hyperfine level. There the atoms stay for a much longer time (typically 2 1 ms) until they are repumped by the light of the intense conical trapping beam. This inherent duty cycle for the molasses provides cooling phases long enough to remove heat efficiently, while the average population of the upper hyperfine state is kept very low. This efficiently suppresses trap loss due to hyperfine-changing collisions (see Section 1II.C). At an atomic number density of 10” cm-3 lifetime measurements of the trapped atoms (l/e lifetime of 7.8 s because of background-gas collisions) showed no significant loss due to ultracold collisions, which indirectly confirmed the predominant population of the lower hyperfine ground state. The CAT was also operated in a pure “reflection cooling” mode without any molasses cooling -conditions similar to those theoretially considered by Morsch and Meacher (1998). Reflection cooling (Ovchinnikov et af., 1995a and b) is based on the inelastic reflection of an atom from a blue-detuned light field, as discussed in more detail in the following section in connection with evanescent waves. In a Sisyphus-like process, the atom is pumped from the strongly repulsive lower to the weakly repulsive upper hyperfine state. A closed cooling cycle requires a weak repumping beam to bring the atom back to the lower state. Such a beam was applied in the CAT experiment from above. The detuning of the conical beam was increased to a few ten GHz to optimize reflection cooling. Because of the lower potential depth, the loading N

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

155

was less efficient ( w 10% transfer from the MOT instead of the 80% reached before), but stable background-gas-limited trapping was achieved. Without the repumping beam, atoms were rapidly heated out of the trap. These observations clearly demonstrated reflection cooling in a blue-detuned trap made of free-propagating light fields. However, optimum reflection cooling requires very steep optical walls so that only evanescent waves are suited to reach temperatures close to the recoil limit by this mechanism, as will be discussed in Section V.C.3. C. EVANESCENT-WAVE TRAPS A hard repulsive optical wall with nearly ideal properties can be realized by a blue-detuned evanescent wave (EW) produced by total internal reflection of a laser beam from a dielectric-vacuum interface. In the vacuum, the EW intensity falls off exponentially within a typical distance of h/2n from the surface and thus provides a very large gradient. The use of an EW as a mirror for neutral atoms was suggested by Cook and Hill (1982), who also proposed to trap atoms in a box realized with EW mirrors. The first experimental demonstration of an atom mirror was made by Balykin et al. (1987) by grazing-incidence reflection of a thermal atomic beam. A few years later, Kasevich et al. (1990)observed reflection of cold atoms at normal incidence. Since then, many experiments have been conducted with EW atom mirrors. An extensive review on EW atom mirrors and related trapping schemes has been given by Dowling and Gea-Banacloche (1996).We thus concentrate on the essential issues and on some interesting new developments. 1. Evanescent-Wave Atom Mirror

The exponential shape of the repulsive optical potential of a far-detuned EW leads to simple expressions for the basic properties of such an atom mirror. The EW intensity as a function of the distance z from the surface is given by I(z) = 10exp(-2z/A)

(46)

where the l / e 2 decay length A = 3L(2ndn2sin28- 1) depends on the angle of incidence 8 and the refractive index n of the dielectric. The maximum EW intensity ZO is related to the incident light intensity ZI by Io/Zl = 4ncos2 e / ( n z - I). The repulsive dipole potential of an atom mirror is independent of the magnetic substate if the detuning is large compared to the excited-state hyperfine splitting and the EW is linearly polarized (see Section 11); the latter is obtained for an incoming linear polarization perpendicular to the plane of

156

R. Grimm, M. Weidemiiller; and Yu. B. Ovchinnikov

incidence (TE polarization). In the interesting case of low saturation, the EW dipole potential can then be calculated according to Eq. (19). A very important quantity used to characterize the EW reflection process is its probability of taking place coherently - that is, without spontaneous photon scattering. The probability of an (in)coherent reflection can be calculated by integrating the intensity-dependent scattering rate rsc[Eq. (21)] over the classical trajectory of the atom in the repulsive potential. For large enough laser detuning (still close to one of the D lines), the resulting small probability psp<< 1 for an incoherent reflection process is given by

mA

PSQ

r

= --

ti A"'

(47)

where W I is the velocity component perpendicular to the surface. The light shift of the ground-state sublevel integrated over time in a single reflection process corresponds to a phase shift

mA ti

= -wl

experienced by the atom. The simple relation psp= is a result of the fundamental connection between the absorptive and dispersive effects of the interaction with the light field; see also Eq. (14). As an interesting consequence of the exponential shape of the EW potential, both Eqs. (47) and (48) do not depend on the intensity of the light field, as long as the potential barrier is high enough. For higher (lower) intensities the reflection just takes place at larger (smaller) distances from the surface. Very close to the surface (z ,< 100nm), the van der Waals attraction becomes significant. Its main effect is to reduce the maximum potential barrier provided by the EW mirror (Landragin et al., 1996b). For small kinetic energies, the reflected atoms do not penetrate deeply into the EW and thus do not feel the surface attraction. We have thus neglected the effect of the van der Waals force in Eqs. (47) and (48),and we will continue to do so.

2. Gravito-Optical Cavities A great deal of interest in EW atom mirrors has been stimulated by the intriguing prospect of building resonators and cavities for atomic de Broglie waves (Balykin and Letokhov, 1989;Wallis etal., 1992). The simplest way to realize such a scheme is a single-atom mirror on which the atoms classically bounce as on a trampoline. Such a trapping scheme is referred to as a gravitooptical cavity.

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

157

The time tb between two bounces in a gravito-optical cavity is related to the maximum height h and the maximum velocity u1 of the atoms by ?b = 2 m = 2vl/g, where g is the gravitational acceleration. Using Eq. (47) for the probability of photon scattering per bounce, the average photonscattering rate can be expressed as

Analogously, using Eq. (48),the mean light shift 6W,, = UQ/A experienced by the bouncing atom is obtained as

It is a remarkable consequence of the exponential shape of the EW potential that and 601, do not depend on the energy of the atom. For an atom with less energy in a gravito-optical cavity, the higher bounce rate exactly compensates for the decrease in scattering probability and light shift. The average scattering rate can be interpreted as the decoherence rate of the gravito-optical resonator due to photon scattering. It also determines the heating power according to Eq. (23). The mean light shift is of interest for possible spectroscopic applications of gravito-optical cavities. The eigenenergies of the vertical modes in a gravito-optical cavity can be approximately calculated by idealizing the EW potential as a hard wall (Wallis e? al., 1992). In this case, the vertical potential has the shape of a wedge, and the energy of the nth vertical mode is given by

r,,

r,,

(51) where ov= ( 9 7 ~ ~ m g ~ / 8 his) "a ~characteristic frequency. For example, for cesium atoms, &/2n = 2080 Hz, which corresponds to a temperature of A o , / k ~ N 95 nK. Consequently, the population of a single vertical mode with many atoms, a challenging issue for future experiments, requires cooling below this temperature. As an important step for EW trapping, the ENS group in Paris observed the bouncing of atoms in a stable gravito-optical cavity (Aminoff et al., 1993). The atom mirror used in the experiment (diameter 1.5 mm) was produced on a concave spherical substrate (radius of curvature 2cm) to obtain additional transverse confinement of the atomic motion (Wallis et al., 1992); the resulting transverse trap depth was about 5 pK. N

158

(4

R. Grimm, M. Weidemiiller, and Yu. B. Ovchinnikov Photodiode

Trapping Beams

@ beam

us

Ti: Saphire Laser Beam

104

0

100

200

300

400

500

time (ms)

FIG. 19. Observation of atoms bouncing in a gravito-optical cavity. (a) Experimental setup. (b) Number of atoms detected in the probe beam for different times after their release (points). The solid curve is the result of a correspondingMonte Car10 simulation. Adapted from Aminoff et al. (1993).

In the setup sketched in Fig. 19(a), about lo7 cold atoms were dropped onto the mirror from a MOT located 3 nun above the prism (corresponding bounce period t b = 50 ms). The bouncing atoms were detected by measuring the fluorescence in a resonant probe beam, which was applied to the atoms after a variable time delay. The experimental results displayed in Fig. 19(b) show up to 10 resolved bounces, after which the number of atoms dropped below the detection limit. The measured loss per bounce of -40% was attributed to photon scattering during the EW reflection process “5% according to Eq. (47)], photon scattering from stray light from the mirror ( w lo%), collisions with the background gas (- lo%), and another, unidentified source of loss (- 20%). An explanation for the latter may be the diffusive reflection from an EW mirror, as observed later by Landragin et al. (1996a). The ENS gravitooptical cavity represents the first atom trap realized with evanescent waves. 3. Gravito-Optical Su$ace Trap A new step for gravito-optical EW traps was the introduction of a dissipative mechanism, following suggestions by Ovchinnikov et al. (1995a) and Soding er al. (1995) to cool atoms by inelastic reflections. In a corresponding experiment at the MPI fiir Kernphysik in Heidelberg, Ovchinnikov et al. (1997) realized the gravito-optical surface trap (GOST) schematically shown in Fig. 20(a); see also Table 11. This trap facilitates storage and efficient cooling of a dense atomic gas closely above an EW atom mirror. Aflat atom mirror was used in the GOST, and horizontal confinement was achieved by a hollow, cylindrical laser beam, far blue-detuned from the

OPTICAL DIPOLE TRAPS FOR NEUTRAL ATOMS

159

FIG.20. (a) Schematic of the gravito-optical surface trap (GOST). (b) Illustration of the evanescent-wave cooling cycle. The dot indicates an atom that approaches the dielectric surface in the lower hyperfine state; scatters an EW photon; leaves the EW in the upper; less repulsive ground state; and is finally pumped back into the lower state.

atomic resonance. This beam, with a ring-shaped transverse intensity profile providing very steep optical walls, was generated using an axicon (Manek er al., 1998).A Laguerre-Gaussian beam of similar performance would require extremly high order (about LGo, 100). The optical potentials of the GOST thus come very close to ideal hard walls, which leads to a strong reduction of photon scattering from the trapping light even relatively close to resonance. Cooling by inelastic EW reflections (evanescent-wave Sisyphus cooling) is the key to stable trapping in the GOST. The basic mechanism, which was experimentally studied before in grazing-incidence reflection of an atomic beam (Ovchinnikov er al., 1995b; Laryushin et al., 1996) and normalincidence reflection of cold atoms (Desbiolles er al., 1996), is based on the splitting of the 2S1/2ground state of an alkali atom into two hyperfine sublevels. In the case of linear EW polarization, the atom can be modeled as a three-level scheme (Soding er al., 1995; see also Section II.B.2) with two ground states separated by Ams and one excited state, for which the hyperfine splitting can be neglected. An inelastic reflection takes place when the atom enters the EW in the lower ground state and, by scattering an EW photon during the reflection process, is pumped into the less repulsive upper state; see Fig. 20(b). The cooling cycle is then closed by pumping the atom back into the lower hyperfine state with a weak resonant repumping laser [coming from above in Fig. 20(a)]. The mean energy loss A E l from the motion perpendicular to the surface per inelastic reflectionisgivenbyAEl/El = -:A,s/(A +A,), whereEl = mv:/2 is the kinetic energy of the incoming atom and A is the relevant detuning with respect to the lower hyperfine state. With the probability pspfor an incoherent reflection according to Eq. (47), the branching ratio q for spontaneous

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R. Grimm, M. Weidemiiller; and Yu. B. Ovchinnikov

scattering into the upper ground state, and the bounce rate t;' = g/2wl, EW Sisyphus cooling can be characterized by a simple cooling rate:

The vertical motion is damped exponentially because the average photonscattering rate is independent of the kinetic energy; see Eq. (49). The final attainable temperature is recoil-limited to a value of 10Trec,similar to a polarization-gradient optical molasses (see Section 1II.A.1). The GOST was experimentally realized for Cs atoms, the high mass of which is very favorable for gravito-optical trapping. In the gravitational potential, the recoil temperature Tree corresponds to a height hrec = ks Tree/ (mg), which for the heavy Cs atoms is particularly low (hrec= 1.3 pm; see also Table I). As a consequence, Cs atoms cooled close to the recoil limit can be accumulated very close to the dielectric surface. In the experiment, trapped atoms were observed for storage times up to 25 s, corresponding to more than 10.000 (unresolved) bounces. The observed exponential loss with a 1/ e lifetime of 6 s was completely consistent with collisions with the background gas. By time-of-flight diagnostics, a temperature of 3.0 pK was measured, which is quite close to the theoretical cooling limit. In thermal equilibrium, the number density as function of the distance from the surface follows from Eq. (32) (in this case, equivalent to the barometric equation), which for the measured temperature of 3.0 pK corresponds to an exponential decay with a 1/e height as low as 19 pm. The cooling dynamics observed in the experiments is shown in Fig. 21. The initial horizontal temperature is determined by the temperature of the MOT, whereas the much higher initial vertical temperature results from the release of the atoms at a height of -800 pm. With increasing storage time, the vertical temperature follows a nearly exponential decay with a time constant of 400 ms, in good agreement with the cooling rate according to Eq. (52) (with A/2n = 1 GHz, Aws/27c = 9.2 GHz, and q = 0.25). The temperature of the horizontal motion, which is not cooled directly, follows the vertical one with a clearly visible time lag and approaches the same final value of about 3 pK. This apparent motional coupling can be fully explained by a small diffusive component of the reflection from the EW atom mirror, as observed by Landragin et al. (1996a). A very important feature of the GOST and the evanescent-wave cooling mechanism is the predominant population of the absolute internal atomic ground state: The lower hyperline level is populated by more than 99.99% of the atoms. A unique feature of EW cooling is that the trapping and cooling light does not penetrate the atomic sample. Moreover, a possible reabsorption

-

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25 b4 3. v 20 n

Q

s 15

4 (d

k

Q

10

Q

5

z

4

U

0

1

2

time (s)

3

4

FIG.21. Cooling dynamics in the COST. The vertical (open circles) and horizontal (closed circles) temperatures measured for about Id trapped Cs atoms are plotted as a function of the storage time. The solid lines are theoretical fits based on Eq.(52) and the assumption of an EW reflection with a small diffusive, nonspecular component. From Ovchinnikov et al. (1997).

of scattered trap photons is strongly reduced because of the large surface area for photons to escape. Hence trap loss by ultracold collisions (see Section 1II.C) and other density-limiting mechanisms is strongly suppressed as compared to a MOT. One can therefore expect EW Sisyphus cooling to work very well up to number densities of 1013cm-3 or even higher (Siiding ef al., 1995). In the first COST experiments, this interesting regime was out of reach because the trap could be loaded with only lo5 atoms, leading to peak densities of 2 x 10" cmV3.In present experiments with the GOST,performed with a substantially improved loading scheme, the high-density regime of evanescent-wave cooling is being explored. The COST offers several interesting options for future experiments on dense atomic gases (Engler et al., 1998). By detuning the EW and the hollow beam very far from resonance, one can achieve a situation in which the photon-scattering rate is far below 1 s-'. For a sufficiently dense gas, one can then expect very good starting conditions for evaporative cooling, which may be implemented by ramping down the EW potential. This appears to be a promising route to quantum degeneracy of Cs, which, because of anomalously fast dipolar relaxation, seems not to be attainable in a magnetic trap (Siiding et al., 1998; GuCry-Odelin et al., 1998). Other interesting applications of the GOST, arising from its particular geometry, are related to the possible formation of a two-dimensional quantum gas. The vertical motion is much more strongly confined than the horizontal one, so that the corresponding level spacings of the quantized atomic motion differ by nearly six orders of magnitude. In such a highly anisotropic situation,

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Bose-Einstein condensation is predicted to occur in two distinct steps (van Druten and Ketterle, 1997). First, the vertical motion condenses into its ground state (Wallis, 1996). Then, in a second step occurring at much lower temperatures, the system condenses to its absolute ground state. The highly anisotropic nature of the trapping potential may be further enhanced by addition of a second, attractive evanescent wave (Ovchinnikov er al., 1991). This can create a wavelength-sized potential well close to the surface, which could be efficiently loaded by elastic collisions. The situation would then resemble that of atomic hydrogen trapped on liquid helium, for which evidence of a Kosterlitz-Thouless phase transition has been reported (Safonov er al., 1998). The prospect of realizing such a system with alkali atomsI4 is of particular interest for studying the effects of quantum degeneracy in a twodimensional system.

VI. Concluding Remarks In this review, we have discussed the basic physics of dipole trapping in fardetuned light, the typical experimental techniques and procedures, and the different trap types presently available, along with their specific features. In the experiments discussed, optical dipole traps have already shown great promise for a variety of different applications. The particular advantages of dipole trapping can be summarized as follows: 0

0

0

The ground-state trapping potential can be designed to be either independent of the particular sublevel or dependent in a well-defined way. In the lirst case, the internal ground-state dynamics under the influence of additional fields behaves in the same way as in the case of a free atom. Photon scattering from the trap light can take place on an extremely long time scale exceeding many seconds. The trap then comes close to the ideal case of a conservative, nondissipative trapping potential, allowing for long coherence times of the internal and external dynamics of the stored atoms. Light fields allow one to realize a great variety of different trap geometries, such as highly anisotropic traps, multiwell potentials, mesoscopic and microscopic traps, and potentials for low-dimensional systems.

l 4 Other experimental approaches to two-dimensional systems based on dipole trapping make use of the particular properties of the optical transition structure in metastable noble gas atoms (Schneble et al., 1998; Gawk ef al., 1998).

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Regarding these features, the main research lines for future experiments in optical dipole trapping may be seen in the following fields: Ultracold Collisions und Quantum Gases. The behavior of ultracold, potentially quantum-degenerate atomic gases is governed by their specific collisional properties, which strongly depend on the particular species, the internal states of the colliding atoms, and possible external fields. In this respect, dipole traps offer unique possibilities, because they make it possible to store atoms in any substate or combination of substates for the study of collisional properties and collective behavior. Experiments along these lines have already been performed (Gardner et al., 1995; Tsai et al., 1997; Stenger et al., 1998; Miesrier et ul., 1998b), but these efforts have opened the door to many other studies involving other atomic species, or even mixtures of different species (Engler el al., 1998). Of particular importance is the trapping of atoms in the absolute internal ground state, which cannot be trapped magnetically. In this state, inelastic binary collisions are completely suppressed for energetic reasons. In this respect, an ultracold cesium gas represents a particularly interesting situation, because Bose-Einstein condensation seems attainable only for the absolute ground state (Soding et al., 1998; Gukry-Odelin et al., 1998). As a consequence, an optical trap may be the only way to realize a quantum-degenerate gas of Cs atoms. Tuning of scattering properties by external fields is another very interesting subject. In this respect, Feshbach resonances at particular values of magnetic fields play a very important role. In optical traps, one is completely free to choose any magnetic field without changing the trap itself. Using this advantage, investigators have observed Feshbach resonances for sodium and rubidium (Inouye ef al., 1998; Courteille et al., 1998), and future experimental work will certainly explore corresponding collisional properties of other atomic species. Highly anisotropic dipole traps offer a unique environment for the realization of low-dimensional quantum gases. In this respect, interesting trapping configurations include standing-wave traps (Section IV.B), optical lattices (Section IV.D), evanescent-wave surface traps (Section V.C), and combinations of such trapping fields (Gauck et al., 1998). New phenomena could be related to a stepwise Bose-Einstein condensation (van Druten and Ketterle, 1997) and to modifications of scattering properties in cases where the atomic motion is restricted to a spatial scale on the order of the s-wave scattering length. Another fascinating approach would be to study collisional properties of ultracold fermions (such as the alkali atoms 6Li and 4oK) with the challenginggoal of producing a quantum-degenerateFermi gas. Direct optical

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cooling in a standing-wave dipole trap by degenerate sideband cooling, similar to the method of Vuletic et al. (1998), seems to be a particularly promising route. Spin Physics and Magnetic Resonance. Because dipole traps allow for confinement with negligible effect on the atomic ground-state spin, experiments related to the coherent ground-state dynamics in external fields can be performed in essentially the same way as in the case of free atoms. In such experiments, the trap would provide much longer observation times than are attainable in atomic beams or vapor cells. As a further advantage, trapped atoms stay at the same place, which keeps inhomogeneties of external fields very low. First demonstrationsalong this line are the experiments by Davidson et al. (1995) and Zielonkowski et al. (1998b), as discussed in Sections V.A and IV.B.3, respectively. In principle, a dipole trap constitutes an appropriate environment to perform any kind of magnetic resonance experiment with optically and magnetically manipulated ground-state spins (Suter, 1997).This could be of interest, for example, for storing and processing quantum information in the spin degrees of freedom. A possible, very fundamental application would be the measurement of a permanent electric dipole moment of a heavy paramagnetic-atom-like cesium (Khriplovich and Lamoreaux, 1997). Such an experiment, testing timereversal symmetry, could greatly benefit from extremely long spin lifetimes and coherence times of spin polarization, which seem to be attainable in dipole traps. Trapping of Other Atomic Species and Molecules. Optical dipole traps rely neither on a resonant interaction with laser light nor on spontaneous photon scattering. Therefore, any polarizable particle can be trapped in powerful, sufficiently far-detuned light. For this purpose, the quasi-electrostatic trapping with far-infrared laser sources, such as CO2 lasers (see Sections IV.A.5 and IV.B.4), appears to be very attractive. The trapping mechanism could be applied to many other atomic species or molecules that are not accessible to direct laser cooling. The problem is not the trapping itself, but the cooling to the very low temperatures required for trap loading. There may be several ways to overcome this problem. One possible way to load a dipole trap could be based on buffer-gas loading of atoms or molecules into magnetic traps (Kim et al., 1997; Weinstein et al., 1998) and subsequent evaporative cooling. The effectiveness of cryogenic loading and subsequent evaporative cooling is demonstrated by the recent attainment of Bose-Einstein condensation of atomic hydrogen confined in a magnetic trap (Fried et al., 1998), without any optical cooling involved.

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Similar strategies based on buffer-gas loading could open ways for filling optical dipole traps with many more atomic species than presently available or even with molecules. Another possible way could be the production of ultracold molecules by photoassociation of laser-cooled atoms (Fioretti ef al., 1998). The translational energies of these molecules can be low enough for loading into farinfrared dipole traps. An experiment by Takekoshi et af. (1998) indeed reports evidence of dipole trapping of a few CSZmolecules in a C02-laser beam, which may be just the beginning of a new class of experiments in dipole trapping. Optical dipole traps can be seen as storage devices at the low end of the presently explorable energy scale. We are convinced that future experiments exploiting the particular advantages of these traps will reveal interesting new phenomena and hold many surprises.

VII. Acknowledgments We would like to thank Hans Engler, Markus Hammes, Moritz Nill, Ulf Moslener, Martin Zielonkowski, and in particular Inka Manek from the Heidelberg cooling and trapping group for many useful discussions and for assistance in preparing the manuscript. Further useful discussions are acknowledged with Steven Chu, Lev Khaykovich, Takahiro Kuga, Heun Jin Lee, Roee Ozeri, Christophe Salomon, and Vladan Vuletic. We are grateful to Dirk Schwalm for generously supporting the work on laser cooling and trapping at the MPI fur Kernphysik. One of us (Yu.B.0.) acknowledges a fellowship by the Alexander von Humboldt-Stiftung. Our work on dipole trapping is supported by the Deutsche Forschungsgemeinschaft in the frame of the Gerhard-Hess-Programm.

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ADVANCES IN ATOMIC, MOLECULAR, AND OpllCAL PHYSICS, VOL. 42

FORMATION OF COLD (T 5 1K) MOLECULES J. T. BAHNS, P. L. GOULD, and W C. STWALLEY Department of Physics, University of Connecticut, Storrs, Connecticut

I. Introduction. ........... .................... A. Traditional Techniques for Formation of Low-Temperature ................. Molecules. . . . . . . . . . . . . . . . 1. Supersonic Molecular Beams . . . . . . . . . . . . . . . . . . . . . . . . 2. Matrix Isolation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . B. Other Approaches to Molecular Cooling. . . . . . . . . . . . . . . . . . . C. Atom Cooling and Trapping. . . . . . . . . . . . . . . . . . . . . . . . . . . D. Applications of Cold Molecules . . . . . . . . . . . . . . . . . . . . . . . . 1. Spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................ 2. Manipulation of Molecules. . 3. Cold Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . .. . 4. Nucleation and Metastability. . 5. Bose-Einstein Condensation of es. . . . . . . . . . . . . . . . 6. Degenerate Fermi Gases of Molecules . . . . . . . . . . . . . . . . . . 7. The Molecule Laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Nonoptical Cooling Techniques . . . . . . . . . . . . . . . . . . . . . A. Helium Cluster Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Helium Buffer-Gas Cooling . C. Trap Compression and Evapor D. Three-Body Processes Including Resonances. . . . . . . . . . . . . . . . 111. Optical Cooling Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Incoherent Optical Cooling of Molecules . . . . . . . . . . . . . . . . . . B. Coherent Optical Manipulation of Molecules . . . . . . . . . . . . . . . IV. Formation of Cold Molecules Via Laser-Induced Photoassociation . . . A. Spontaneous Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2. Observations ..................... B. Proposed Stimul ......................... V. Conclusions and Fu ................... VI. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. References ....................................

172 173 174 175 175 176 177 178 180 185 186 187 190 191 191 191 192 192 194 195 196 205 206 206 206 21 1 215 219 219 220

Abstract: The formation of low-temperature molecules with translational energy distributions characterized by T 5 1 K is reviewed. Such molecules can in principle be produced optically or nonoptically or by photoassociation of ultracold ( T 5 1 mK) atoms. Recent results producing cold - and especially ultracold - alkali metal dmers and producing cold paramagnetic molecules are highlighted, along with their potential applications and scientific significance. 171

Copyright 02000 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-003842-01ISSN 1049-250x100 $30.00

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I. Introduction The purpose of this review is to clarify and summarize the many proposals for formation of gas-phase molecules with subKelvin translational and internal temperatures (or average energies (divided by Boltzmann’s constant k) if not in a thermal distribution) and to discuss the exciting and very recent results achieved in implementing these proposals. Indeed, in many cases discussed below, we are discussing population transfer (e.g., into the lowest ground electronic state rovibrational level of a molecule), and not standard cooling by lowering the temperature of a thermal distribution of molecules. In fact, a thermal distribution of ultracold (T 5 1 mK) molecules in the true thermodynamic sense cannot exist, because the true equilibrium state consists of a solid of the same stoichiometry with an absolutely negligible vapor pressure. However, for low-density, low-temperature atomic and molecular gases, the time required to reach true equilibrium (a solid) can be much longer than the corresponding experimental studies. Only the translational degrees of freedom clearly equilibrate thermally on a short time scale. The possibilities for nonthermal distributions among other (internal) degrees of freedom are great, as discussed especially in Section I.D.3. Cold is a relative term. The person on the street might accept 273 K as the hot-cold boundary, for example, whereas 2.7 K, the current black-body temperature of the universe, might be more acceptable cosmologically. We choose 1 K as our boundary to exclude two very powerful and pervasive techniques for producing molecules in the 1-100 K range: supersonic molecular beam expansions (Section I.A. 1) and matrix isolation spectroscopy (Section I.A.2). We reserve the term ultracold for T 5 1 mK. We then survey, in Section I.B, the broad field of molecular cooling (and heating), most of which will not yield or have not yet yielded cold molecules, emphasizing optical techniques. The powerful techniques of atom cooling and trapping relevant to this review are briefly summarized in Section I.C. The major applications/motivations for production of cold molecules are summarized in Section I.D, which includes the following subsections: (1) Spectroscopy, (2) Manipulation of Molecules, (3) Cold Collisions, (4) Nucleation and Metastability, (5) Bose-Einstein Condensation of Molecules, (6) Degenerate Fermi Gases of Molecules, and (7) The Molecule Laser. Finally, in the remainder of this section, we will describe in greater detail the organization of the rest of this review into sections on Nonoptical Cooling Techniques (Section 11), Optical Cooling Techniques (Section 111),Formation of Molecules Via Photoassociation of Cold Atoms (Section IV), and Conclusions and Future Directions (Section V).

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A. TRADITIONAL TECHNIQUES FOR FORMATION OF LOW-TEMPERATURE MOLECULES

There are two powerful and widely used techniques for formation of lowtemperature molecules in the 1-100 K region: the supersonic molecular beam technique and the matrix isolation technique. The primary motivation for introducing these techniques was to simplify complex spectra by greatly reducing the internal partition function (the number of significantly populated internal states) and, in the supersonic molecular beam, the Doppler width. In a sense, many proposals for the formation of subKelvin molecules have evolved from these approaches. Thus we wish here to discuss, but in no sense comprehensively review, these extensive areas of study. It is also useful to examine the plot of energy/temperatureversus deBroglie wavelength [Fig. 1, a reformulation of Fig. 1 in Weiner ef al. (1999), where many of the proposals and results discussed in this review are indicated]. Not only does the reduced partition function at low T simplify spectra, but it also

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enhances quantum effects in collisions and many body interactions [such as Bose-Einstein condensation (BEC)]. 1. Supersonic Molecular Beams

Molecular (and atomic) beams have been widely used for most of this century to study fundamental properties of atoms and molecules (see Anderson et al., 1966; Ramsey, 1985; Scoles 1988). In the 1950s, Kantrowitz and Grey (1951) and Kistiakowsky and Slichter (195 1) proposed that the pressure behind the aperture emitting the beam be significantly increased, converting the beam from “effusive” to “supersonic” and producing an increased intensity and quite significantly - decreased internal (rotational and vibrational) and translational temperatures. Actually, the mean longitudinal translational energy of the atoms or molecules increased somewhat, while the spread in longitudinal translational energy decreased substantially (and was comparable to the spread in transverse translational energy). For example, early work (e.g., Sinha et al., 1973 and Wu et al., 1978 and references therein) on supersonic expansions of alkali metal vapor (predominantly atoms with a few percent molecules in the gas behind the nozzle) produced an enhanced concentration of molecules and significantly reduced internal (especially rotational) and translational temperatures. However, the temperatures were still quite high compared to those reviewed here (e.g., a 1370K expansion produced 7Li2 with a 50 K rotational temperature and a 190K vibrational temperature (Wu et al., 1978). A major advance came with the coexpansion of rare gases with other species, pioneered by Becker and Henkes (1956). Spectroscopic applications grew rapidly after the work of Smalley et al. (1974). A few percent or less of a species was coexpanded with rare gases such as He and Ar. The Ar was particularly effective in relaxing vibration, whereas He produced very low rotational temperatures in the < 10K regime. Toennies and Winkelmann (1977) achieved a temperature of 15 mK in a 4He expansion. However, the rare gases (and H2) ultimately cluster as the temperature produced by the expansion is reduced. Even 4He2 (and therefore 4He,) has a nonzero binding energy of 1.31 mK (Uang and Stwalley, 1982; Schollkopf and Toennies, 1994 and 1996; Tang et al., 1995; Luo et al., 1996), and thus clustering will occur. One of the techniques that can lead to subKelvin temperatures is the helium cluster “pickup” technique employed especially by Scoles et al. and by Toennies et al., discussed in Section 1I.A. In a sense, it is a version of matrix isolation spectroscopy as well, where the atom or molecule can either sit on the surface of the cluster or be embedded within the cluster. The temperature here is limited by the evaporative cooling of 4He clusters ( ~ 3 7 0 m K or ) (much more expensive) 3He clusters ( N l00mK).

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In principle, spin-polarizedatomic H (Stwalley and Nosanow, 1976; Silvera and Walraven, I980), which has recently been Bose-Einstein-condensed (Fried et al., 1998),is the ultimate “rare gas,” because there is no bound state 00 (Uang and Stwalley, 1980A). for its many-body system in the limit n Even 3He, for which no 3He2 species exists (Uang and Stwalley, 1982), has cluster bound states probably starting with n = 29 (Barranco et al., 1997) and a superfluid phase (Lee, 1997) as n -+ 00. Thus, in some cases, it might be possible to use spin-polarized H in supersonic molecular beam expansions (assuming no low-temperature exoergic reaction with H can occur). However, to the best of our knowledge, no such experiments are under way. --f

2. Matrix Isolation Spectroscopy The use of a rare gas (or other usually inert species, such as H2 or N2) solid matrix to trap and cool a molecule to low temperature, such as 4 K for liquid 4He cooling, has been used since the 1950s (Lewis and Kasha, 1944; McClure, 1954; Becker and Pimentel, 1956; Robinson et al., 1957; Bass and Broida, 1960)to obtain assignable spectra of high-temperature, highly reactive species. Recently, the technique has been extended beyond solids to include liquid 4He (e.g., Takahasi et al., 1993) (liquid 3He is also possible but unexplored). Toennies and Vileson (1998) have recently reviewed the spectroscopy of doped liquid 4He. Such spectra suffer from “matrix shifts” as well as inhomogeneous (matrix) broadening, but they provide a great deal of useful information about highly reactive species not readily obtained by any other technique. In recent years, Jacox (1988, 1990, 1994, 1998) has compiled and reviewed a great deal of matrix isolation spectroscopic data. As noted above, helium cluster spectroscopy (Section II.A) is an extension of matrix isolation spectroscopy. Likewise, helium buffer gas cooling (to ~ 3 0 0 m Kfor 3He and to 1.4K for 4He) technique of Section 1I.B is an extension from traditional solid-state matrix isolation and liquid He “matrix isolation” to a gaseous “matrix isolation.” Again, gaseous spin-polarized H might be an improvement in nonreactive situations over even 3He, with (uniquely) an extremely high vapor pressure (no stable solid or liquid form) at very low temperatures. Note also that a gaseous “matrix isolation” provides a gaseous molecule with a near zero “matrix shift” from the unperturbed gas phase spectrum. N

B. OTHERAPPROACHES TO MOLECULAR COOLING The first discussion of cooling atoms and molecules with light (“luminorefrigeration”) of which we are aware was presented by Kastler (1950), although the example elaborated was an atom, Na. Since then, a number

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of other concepts (and some experimental results) have emerged, including use of near-resonance radiation (Hansch and Schawlow, 1975; Wineland and Dehmelt, 1975; Letokhov et al., 1976; Berman and Stenholm, 1978; see also Section 1.C); spontaneous anti-Stokes scattering (Djeu, 1978; Djeu and Whitney, 1981); selection of slow particles (Vasilenkoet al., 1995); sequential laser cooling of rotation, translation, and vibration (Bahns er al., 1996); and laser cooling of multilevel solids (Kushida and Guecic, 1968; Mungan et al., 1997, 1997A; Edwards et al., 1998). Only the near-resonance approach has provided (for atoms) subKelvin temperatures of interest here (and also trapping), as discussed in Section I.C. The sequential laser cooling scheme (Bahns et al., 1996) has the potential of achieving subKelvin temperatures, but this has not yet been demonstrated. It is discussed in detail in Section 111. Alternatively, one can use already translationally subKelvin atoms, cooled by the techniques of Section I.C, and photoassociate them (Lett et al., 1995; Weiner et al., 1999; Stwalley and Wang, 1999) to form translationally subKelvin molecules. Such molecules can then be trapped and internally cooled, as discussed in Section IV. Finally, there are the nonoptical methods of cold molecule formation, two of which [helium cluster cooling (Section 1I.A) and helium buffer gas cooling (Section II.B)] were mentioned above. Two other nonoptical approaches, trap compression and evaporative cooling (Section II.C) and three-body processes including resonances (Section II.D), are also covered in Section 11. c . ATOMCOOLING AND TRAPPING One of the most impressive advances in science in the past two decades has been the development of techniques to cool atoms (to < 1 mK) and then trap them in ultrahigh vacuum, using lasers but no cryogenics. These techniques include slowing and even stopping an atomic beam, damping the motion of atoms away from an intersection of laser beams (“optical molasses”), trapping of atoms at temperatures of pK and densities of 10” atoms/cm3 in magneto-optical traps (MOTS)and far-off resonance traps (FORTS),and BoseEinstein condensation (BEC) of atoms at nK temperatures and densities of lOI4 atoms/cm3 in magnetic and optical traps. Recent reviews of this field include Metcalf and van der Straten (1994), Dowling and Gea-Banaloche (1996), Adams and Riis (1997), and Weiner et al. (1999). See also the lectures of the 1997 Physics Nobel Laureates (Chu, 1998; Cohen-Tannoudji, 1998; and Phillips, 1998). Because molecules exist in a far greater variety of internal states than atoms, cooling and trapping techniques are not as straightforward as for atoms. In particular, simple atoms often have a two-level cycling transition, where millions of photons can be absorbed and emitted between two levels

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without loss to other atomic levels. Similar cycling transitions are extremely rare (or even nonexistent) for simple molecules. One general approach (Bahns et al., 1996) is to generate a set of frequencies (e.g., 20 for the Cs2B’III,X ‘ E l cooling transition example of Bahns et al., 1996) such that a given molecule can absorb and emit photons at least l@times, with every individual absorption and emission event corresponding to one of these possible frequencies. Although many of the techniques for cooling and trapping atoms are not applicable to molecules, the use of laser light that is very far from resonance minimizes spontaneousemission and therefore the problem of optical pumping. Tightly focused laser light tuned well below a resonance line has been successfully used to trap atoms (Chu et al., 1986; Miller et al., 1993). Such a far-off resonance trap (FORT) operates via the interaction of the oscillating dipole moment induced in the atom with the laser field. For light tuned below an atomic resonance, the atom is attracted to a region of strong field. A focused laser beam, having an absolute maximum of intensity at the focus, thus acts as a trap for cold atoms. The depth of the trap depends on the energy of interaction of the dipole with the field and is proportional to the laser intensity Z and to the inverse of the laser detuning A. Spontaneous decays can occur, even for far-off resonant light, causing heating of the atoms in this otherwise conservative potential. This photon scattering rate is proportional to Z/A2. If the figure of merit is the trap depth divided by the scattering rate, then it is preferable to operate at large A, as long as the trap is deep enough to confine the atoms. For light tuned well below the D1 and D2 lines of an atom, an interference effect in the spontaneous Raman scattering suppresses the spin relaxation (optical pumping) that would otherwise accompany this heating (Cline et al., 1994). The extreme case of large detuning is to use an infrared laser (e.g., a COz laser). In this case, the frequency is so far below the atomic resonance that the field is best considered quasistatic. Such a quasi-electrostatic trap (QUEST) of Takekoshi et al. (1995) has been successfully used to trap Cs atoms (Takekoshi and Knize, 1996) and CS2 molecules (Takekoshi et al., 1998). D. APPLICATIONS OF COLD MOLECULES Although a great deal of useful information on atoms and molecules can be obtained at superKelvin temperatures, it is clear that the most precise spectra will be obtained at the lowest observation temperature. Likewise, “hot” molecules will be relatively hard to trap (typically in subKelvin traps) or otherwise manipulate (deflect, focus, reflect, interfere, etc.). Collisions of thermal molecules typically involve over lOOA in collisional angular momentum, as well as thousands or millions or more internal states in each molecule. Thus it is not surprising that quantum effects in general, or quantum

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resonances (involving a single internal state in each molecule and a single state of collisional angular momentum) in particular, tend to be “washed out” - that is, they make a negligible contribution to the observed average cross section. Moreover, in the absence of walls (e.g., in electromagnetic traps) and at relatively low densities where two-body (not three-body) processes dominate, such molecules can be very highly metastable with respect to the thermodynamically stable molecular (or other) solid. Such low temperatures at modest (metastable) density offer the interesting possibility of highly quantum-mechanical behavior in the gas phase (Stwalley and Nosanow, 1976), e.g., BEC. A further extension is to the ideal atomic or molecular beam, i.e., the “atom laser” (Mewes et al., 1997) and the “molecule laser” (Julienne et al., 1998), where all species in a single quantum state move coherently in the same direction with the same velocity. Here we briefly discuss the following applications of subKelvin molecules: spectroscopy, manipulation of molecules, cold collisions, nucleation and metastability, Bose-Einstein condensation of molecules, degenerate Fermi gases of molecules, and the molecule laser.

1. Spectroscopy A major motivation for matrix isolation and seeded supersonic molecular beam techniques was higher resolution molecular spectroscopy.The need to go even colder is not as strong because these two traditional techniques already provide a great deal of information. Nevertheless, subKelvin spectroscopy offers the opportunity to virtually eliminate the Doppler effect. Optical and magnetic traps (see Section I.D.2) eliminate “transit time broadening,” although they do introduce new (but often well-understood) terms in the molecular Hamiltonian. Clearly, resolution of closely spaced levels is an important motivation in many molecular systems and may justify the additional complexities of cooling to below 1 K. One example of high-resolution spectroscopy that has blossomed at subKelvin temperatures is the photoassociation of ultracold atoms, first proposed in 1987 (Thorsheim et al., 1987) and recently reviewed (Lett et al., 1995; Weiner et al., 1999; Stwalley and Wang, 1999). Colliding pairs of ultracold atoms have very low thermal kinetic energies [e.g., 7MHz in our 39K magneto-optical trap at 300pK (Wang et al., 1996)l. Thus free-bound absorption lines are as sharp as ordinary bound-bound lines. Moreover, colliding pairs of atoms reach relatively short distances (say, < 50 A) only if the collision is nearly head-on and the corresponding centrifugal barriers are surmounted (e.g., collisional angular momentum 1 = 0, 1, and 2 for 300 pK 39K). Because the probability of free + bound photoassociation decreases with decreasing internuclear distance R, this spectroscopy is ideally suited for

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observing very closely spaced molecu!ar levels with their vibrational wavefunctions concentrated at large (> 10A) internuclear distances (“long-range molecules” (Le Roy, 1973; Stwalley, 1978). Such levels have been studied since 1970 by conventional molecular spectroscopy (Le Roy and Bernstein, 1970; Stwalley, 1970) but with considerable difficulty because the long-range molecule spends very little time near its inner classical turning point. It can be shown that the vibrational spacings and rotational spacings go to zero in the limit that the vibrational quantum number approaches its value at dissociation V D (Le Roy, 1973; Stwalley, 1978). An example of a vibrationally resolved ultracold photoassociation spectrum of 39K is given in Fig. 2; a rotationally resolved spectrum is given in Fig. 3. “Pure long-range molecules” are weakly bound, long-range molecular states where both the inner and outer classical turning points are at large internucleardistance (Stwalley, 1978; Wang et al., 1996A;Stwalley and Wang, 1999). Such species cannot be populated directly from thermally populated levels of molecular ground states. We have succeeded in populating both the 0;(Wang et al., 1997) and the 1, (Wang et al., 1998) pure long-range states

v-9

0g

10

~=124

125

11

12

126

127

13

128 129

’El I

13039.9

I

I

13040.5

14

15

16

17 18 I9 20 21 22 23

130

135

I

I

13041.1

I 13041.7

I

13042.3

-’

Laser frequency (m

FIG.2. The vibrationally resolved photoassociation spectrum of ultracold (300 1K) 39K atoms recorded by “trap loss” (decrease in atomic fluorescence due to excited molecule formation)just below the 4s 4p3,, asymptote (at 13042.876cm-’) (Wanget al., 1997). Three vibrational progressions are shown: the 0; “pure long-range molecule” state (where levels v = 9-23 are shown at the top), the 0: state (where the vibrational numbering is uncertain, but 17 levels are shown at the bottom), and the 1, state (where levels = 124-139 are shown at the bottom). These states correlate with the repulsive 1 311g(0= 0-component), the attractive b311, (R = O+ component) and the attractive 1 ‘II, states, respectively, at short internuclear distance. Each level shows rotational structure (as in Fig. 3) at higher resolution.

+

J. Z Bahns, I? L. Gould, and W C. Snyalley

180

0

200

400

600

800

1000

Relative Laser frequency (MHz)

FIG.3. The rotationally resolved photoassociation spectrum of ultracold (300 pK) 39K atoms recorded by resonance-enhancedmultiphotonionization of the rotational levels J' = 0-4 of the o' = 0 vibrational level of the 0; "pure long-range molecule" state of 39K2 (Wang ef al., 1997).

of 39K2 using photoassociation. Others have populated these states in Na2, Rb2, and Cs2 (see Stwalley and Wang, 1999, and references therein). For 39K2, these fascinating states are at 28 A and 39 A, with binding energies of 6 cm-' and 0.5 cm-' , respectively. Recently, stimulated Raman photoassociation via one of these states has been demonstrated in ultracold 87Rb(Wynar et al., to be submitted). It is likely that the availability of samples of ultracold atoms and molecules will provide additional examples where new and interesting spectroscopic information can be obtained. For example, ultracold polar molecules may find use in precision spectroscopy searches for the dipole moment of the electron.

2. Manipulation of Molecules The ability to manipulate the translational motion of atoms with microstructures and various types of electromagnetic fields has spawned the very active field of atom optics (Adams et al., 1994). There are strong analogies between atoms and light in the area of geometric optics, where classical atomic trajectories are the equivalent of optical rays. For example, atomic beams can be focused by atomic lenses and reflected by atomic mirrors. Furthermore, in analogy with physical light optics, atomic deBroglie waves diffract and interfere. Because the forces exerted on neutral atoms by

FORMATION OF COLD (T 5 1 K) MOLECULES

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electromagnetic fields are generally small, significant geometrical optics effects are expected only when the atoms are moving rather slowly - i.e., laser-cooled. Because the deBroglie wavelength of room-temperature atoms is very small (e.g., 0.02 nm for a typical Na atomic beam), physical optics effects are also enhanced with slower atoms. One can thus anticipate that advances in the production of cold molecules will generate interest in extending atom optics techniques to molecules, especially in light of potential applications such as interferometry and lithography. In addition, the manipulation of molecules represents an important step in the large jump from simple atomic systems to more complex mesoscopic and macroscopic systems. Although there are many similarities between atom (or molecule) optics and traditional light optics, there are important distinctions. The fact that atoms have mass, whereas photons do not, implies that deBroglie waves, in contrast to light waves, have dispersion in vacuum. In fact, chromatic aberration due to a distribution of atomic momenta can be particularly severe in many atom optics elements. Obviously, these same considerations apply to molecules. Another important difference between atom optics and traditional light optics is the possibility of using dissipation (e.g., laser cooling) to increase brightness. In light optics, optical elements generally preserve brightness at best. If direct laser cooling of molecules becomes practical, then this ability to increase brightness will apply to molecules as well. At present, however, atom optical elements involving spontaneous emission are not applicable to molecules because of the severe problem of optical pumping. Manipulation of molecular beams with applied electric and magnetic fields has been used fruitfully for many decades to measure important properties of molecules (Ramsey, 1985 and Scoles, 1988). These techniques generally take advantage of the fact that the deflection of a molecule by an inhomogeneous field is state-selective. Resonant transitions between states are driven and detected by a change in the deflection. An important variation (Hill and Gallagher, 1975) was to use the resonant field itself to cause the deflection. In this experiment, polar CsF molecules were sent through a microwave cavity, experiencing a standing-waveelectric field resonant with a rotational transition. Molecules in the appropriate states were deflected by the interaction of the electric dipole moment with the strong electric field gradient of the standing wave. As discussed in the introduction, the lack of demonstrated laser-cooling techniques for molecules and, more generally, the inability to have sustained interactions with light (due to optical pumping), have prevented the extension of many of the tools of atom optics to molecules. However, a notable exception is the use of microfabricated diffraction gratings to diffract and interfere molecules (Chapman et al., 1995). The fact that the atom-optical elements were physical structures (as opposed to near-resonant light fields) allowed

182

J. 7: Bahns, P. L. Gould, and W C.Stwalley

them to work equally well for molecules as for atoms. In this work, a pure (>98%) beam of Na2 molecules was obtained from a seeded supersonic expansion by deflecting the Na atoms out of the beam with resonant radiation pressure. This selected beam was then incident on a series of three equally spaced 200-nm period gratings, which formed a Mach-Zehnder interferometer. This particular geometry yields a white-light interferometer whose interference pattern is independent of the deBroglie wavelength. The first grating split the molecular beam, and the second grating reflected the various orders, causing them to recombine at the location of the third grating. This third grating was scanned to read out the resulting interference pattern. Because the interfering beams were physically separated in the middle of the interferometer, it was possible to induce a phase shift in one of the beams by passing it through a gas cell. This allowed the complex index of refraction of Na2 passing through Ne to be measured. More specifically, the ratio of real to imaginary parts of the index was determined. These are related to the phase shift and attenuation of the interference signal, respectively, and yield useful information on the long-range interaction between Na2 and Ne. This initial experiment on molecular interferometry clearly indicates the power and potential of the technique. Superficially, it is somewhat surprising that the interferometer works as well for molecules as it does for atoms. Because the molecules occupy a large number of rovibrational states with different internal energies, one might think that this would wash out the interference pattern. However, as pointed out by the authors, the interferometer effectively makes each particle interfere with itself, rendering the internal state distribution irrelevant. It is also interesting to note that the diffraction of molecules (e.g., by a single grating) is by itself a useful diagnostic for molecular beams. A prime example is the use of such a diffraction pattern to observe directly and verify the existence of the He;! molecule (Schollkopf and Toennies, 1994, 1996). A related experiment determined the size of this dimer (- 6.2 nm) by its inhibited passage through nanoscale (- 100-nm) openings (Luo et al., 1996). The large electric fields available from pulsed lasers have been used to exert significant forces on room-temperature molecules (Stapelfeldt et af., 1997; Sakai et al., 1998). The lasers employed (Nd:YAG at 1.06pm and C02 at 10.6 pm) were very far from any resonances of the molecules (CS2 and I2), so that the interaction was via the static polarizability. The strong electric field induces an electric dipole moment in the molecule, and the interaction of this dipole with the field results in a potential energy. If the laser intensity varies spatially, this position-dependent potential leads to a “nonresonant dipole force” capable of focusing or trapping molecules (Seideman, 1996; see also Friedrich and Herschbach, 1995a). For a positive polarizability, the molecule is attracted toward a region of high field, e.g., toward the waist of a focused

FORMATION OF COLD (T 5 1 K) MOLECULES

183

Gaussian laser beam. This force is essentially the same as that utilized in trapping atoms and microscopic objects (e.g., dielectric spheres) in focused laser beams. In the pulsed-laser experiments (Stapelfeldt et al., 1997; Sakai et al., 1998),a molecular beam was passed transversely through a focused laser beam. The nonresonant dipole force deflected the molecules toward the axis of the laser beam (i.e., toward the region of highest intensity), imparting to them a velocity that depended on their initial location in the intensity gradient of the focused beam. This velocity was measured by subsequently ionizing the molecules with a tightly focused femtosecond pulse and measuring their time of flight to a detector. Molecules with an imparted velocity toward the detector would arrive earlier than those deflected away from the detector. Notably, the initial transverse velocities of the molecules were negligible compared to those imparted by the deflection process. The relative positions of the deflecting and ionizing lasers were then scanned to map out the transverse velocity as a function of position in the deflecting beam. The expected dispersion-shaped curve was observed, indicating that the laser beam did indeed act as a onedimensional lens to focus the molecules. The focal length of this lens was calculated to be typically several hundred micrometers. In these experiments, the depth of the dipole potential was on the order of 5 meV, several orders of magnitude deeper than typical traps for neutral atoms. It should be possible to increase this by an order of magnitude before significant ionization of the molecules occurs. This type of strong-field manipulation appears to be quite general, because it does not rely on internal resonances. Of course, the potential here is transient, lasting only as long as the laser pulse (typically, tens of nanoseconds). The use of intense cw laser light has also been proposed for trapping molecules (Friedrich and Herschbach, 1995). Pendular states, as opposed to freely rotating states, would be produced by the alignment of the induced dipole moment with the field. The energy of these pendular states depends on the intensity, resulting in a potential well for the molecules. However, the cw intensities required for reasonable trap depths [e.g., for 3Hebuffer-gas loading at 300 mK (see Section II.B)] appear to be technically challenging. Based on their success with atoms (see Section I.C), the extension of far-off resonance traps (FORTS) to molecules appears promising. Because multiple transitions are possible in molecules, it is important that the trapping light be detuned below all of these transitions. Also, the detunings should be very large in order to minimize spontaneous Raman transitions, thereby maintaining the state selectivity. The cw Nd:YAG laser at 1.064pm appears to be a good candidate for trapping K2 molecules because it is below resonance with all transitions from the v = 0 level of the X state. With 5 W of power focused to a 50-pm waist, we estimate a trap depth of - 4 0 0 p K and a photon scattering rate of -3 s-' for K2 (X, v = 0). The dominant contributions are from the

184

-

J. Z Bahns, P. L. Gould, and W C. Stwalley

-

A state (w 10) and the B state (v 2). Although this trap is not very deep, it is sufficient to hold molecules produced by photoassociation of ultracold atoms (Fioretti et al., 1998; Takekoshi et al., 1999; Nikolov et al., 1999; Comparat et al., 1999). The utility of the Nd:YAG laser for trapping heavier alkali dimers is questionable because it may not lie below all the relevant transitions. The CO;! laser at 10.6pm, on the other hand, should work for any of the alkali dimers. It also has the advantage of a very small scattering rate. This type of low-frequency laser trap is a quasi-electrostatic trap (QUEST) (Takekoshi et al., 1995); that is, the potential energy of a molecule in the laser field is due to the static polarizability (see Tarnovsky et al., 1993 for alkali dimer values). For 20 W focused to a 50-pm waist, the trap depth for K2 will be 600 pK. In a very exciting recent development (Takekoshi et al., 1998), this type of trap has been successfully used to trap Cs2 molecules. This experiment produced cold (150 pK) Cs2 from cold Cs atoms confined in a MOT. The molecules were produced at the focus of a COz laser that confined them for -0.5 s. The trap depth was estimated to be -350pK. Detection of the molecules was by ionization with a frequency-doubled (532-nm) pulsed Nd:YAG laser, followed by time-of-flight mass spectrometry. This sensitive detection scheme allowed the small number of molecules trapped to be observed (- 3 detected). However, continuous formation of molecules by three-body recombination of trapped atoms could mimic a long molecule trapping time. This possibility was dismissed by pushing the atoms out of the trap (with a resonant 852 nm laser beam) shortly after the initial loading of molecules. Trapping with light tuned above resonance has been achieved with atoms (Davidson et al., 1995; Kuga et al., 1997). In contrast with the red-detuned FORTSdiscussed previously (Section I.C), atoms are attracted to the region of minimum intensity of blue-detuned light, e.g., the axis of a “doughnut” beam. This has the advantage of confining the atoms in near darkness (as opposed to a region of high intensity), thereby reducing the photon-scattering rate. However, at a finite temperature, the molecules will sample a finite range of intensities, so the scattering is not completely eliminated. As a practical matter, if light is tuned above a given molecular transition, there are likely to be other transitions nearby, so the wavelength must be carefully chosen to fall in a gap, significantly detuned from other allowed transitions. Magnetic fields can also be used to trap paramagnetic atoms and molecules. Indeed, all atomic Bose-Einstein condensates (BECs) produced thus far have employed magnetic traps (Anderson et al., 1995; Davis et al., 1995; Bradley etal., 1997), although transfer of a BEC from a magnetic to an optical trap has also been achieved (Stamper-Kurnet al., 1998; Inouye et al., 1998). Details of magnetic trapping of a given paramagnetic molecule are complex (Friedrich er al., 1998), although some results are available for 0 2 and NO (Friedrich

-

FORMATION OF COLD (T 5 1 K) MOLECULES

185

et al., 1998), VO (Weinstein et al., 1998A), and CaH (Friedrich et al., 1999). In another very exciting recent report, magnetic trapping of a molecule, CaH, has been observed (Weinstein et al., 1998), with 10' molecules being trapped at -0.4 K for 0.5 s. Clearly, the era of trapped subKelvin molecules is just beginning, including large numbers of CaH at 0.4K and barely detectable numbers of Cs2 at 0.15 mK, orders of magnitude colder. We expect dramatic improvement in these parameters in the near future. Ultimately, such manipulated subKelvin molecules (with specified stoichiometry, e.g. GaAs) may be useful in fields such as lithography, where, for example, nanometer scale structures of Cr have been prepared by deposition of cold Cr atoms (McClelland et al., 1993).

3. Cold Collisions

The collisions of subKelvin atoms have already provided a great deal of interesting information reviewed in Walker and Feng (1994), Weiner (1995), and Weiner et al. (1999). In particular, such collisions (as in the photoassociative spectroscopy discussed in Section I.D. 1) involve only a few partial waves (e.g., collisional angular momenta 1 = 0, 1, and 2 for 39K at 300 pK). Such atomic collisions are highly quantum-mechanical and are frequently dominated by shape resonances (Schutte et al., 1972 and Boesten et al., 1996) or Feshbach resonances (Stwalley, 1976; Uang and Stwalley, 1980; Uang et al., 1981; Tiesinga et al., 1993; Inouye et al., 1998; Courteille et al., 1998; Roberts et al., 1998). The availability of ultracold molecules offers the opportunity to study a variety of atom-molecule and ultimately molecule-molecule collisions. In Table I, we list specific processes for the example of 39K2 that we feel can be studied using existing (Nikolov et al., 1999) and proposed improved (Band and Julienne, 1995; Vardi et al., 1997) methods of molecule formation (see also Section IV). Each process can potentially be studied theoretically by close-coupling methods. Analogous calculations of several of such processes involving H + H2 or He H2 collisions have recently been carried out by Dalgarno et al. (Balakrishnan et al., 1997, 1997A, 1998; Forrey et al., 1998, 1999) and are planned for K K2 by R. C8tt. As indicated in the comments in Table I, such processes provide important information for the possible achievement of molecular BEC discussed in Section I.D.5. The overall stability of a 39K2 (0, 0) BEC depends on the scattering length a for 39K2(0,O) 39K2 (0,O)collisions (process 7) (a positive scattering length indicating stability), and the efficiency of the evaporative cooling depends on the elastic cross section, which is proportional to a2.If high densities of 39K2(0,O) cannot be achieved, sympathetic cooling with 39K

+

+

+

186

. I . I: Bahns, P: L . Gould, and W C. Stwalley TABLE I

POSSIBLE

TYPES OF PROCESSES

TO BE STUDIED USING ULTRACOLD SOURCES, AND 3 9 ~ AVAILABILITY. 2

Process 1.

2. 3.

8.

9. 10.

ASSUMING 39'41K

Comments

39K+ 39K2 (v" = 0, J" = 0)- 39K+ 39K2(0,0)

sympathetic cooling of 39K2(0,0) by 39K for BEC same 39K+ 39K2 ( 0 , l )+ 39K+ 39K2 ((41) ortho --t para conversion, possibly 39K+ "K2 ( 0 , l ) 39K+ 39K2 (0,O) preventing sympathetic cooling of 3 9 ~ (0.1) 2 vibrational quenching reaction (no other 3 9 K 4 1 K ( 0 , J ) level can be produced) near-resonant reaction (exoergic) evaporativecooling of 3 9 ~ (0, 2 0)/ scattering length and stability of 39K2 (0,O)BEC 39K2 (0, I ) + "K2 (0, I ) + 39K2 (0, I ) + 39K2 ( @ I ) evaporativecooling of 3 9 ~ (0, 2 I)/ scattering length and stability of 39K2 (0,l) BEC relaxation that destroys BEC 39K2 (0, I ) + 3 9 ~ (2o , i ) 3 9 ~ (0, 2 o)+ 3 9 ~ (0,1) 2 same 39K2(o,1)+39~2(o,1)+ 3 9 ~ 2 ( o , o ) + 3 9 (0~0) ~2

-

-

or 41K would probably be used instead, in which case processes 1 and 5, respectively, would be critical to know. A further possibility is that metastable ortho 39K2 (0, 1) can undergo BEC [using the evaporative cooling process for 39K2 (0, l)]. In principle, relaxation such as processes 9 and 10 could occur, but they are likely to have very small cross sections. Other processes (e.g., 4 and 6) are less significant for BEC but are of considerable fundamental interest. It is also worth noting that none of the collisions shown can produce K3. Thus it is not clear how K3 can be formed in ultracold collisions; if K3 does not form, this will prevent nucleation to clusters and ultimately to the solid. It is conceivable that the initial collisions corresponding to processes 4 and 6 could also access Feshbach resonances in the lowest electronically excited state of K3, which would then rapidly radiate (in tens of nanoseconds) to the ground state of K3. 4. Nucleation and Metastability

The collisions of ultracold atoms and molecules have very few product channels compared to thermal collisions of the same species. For alkali metals, the dissociation energy of the ground state trimer to the ground state of the dimer and an atom is only about half that of the precisely known (Stwalley

FORMATION OF COLD (T 5 1 K) MOLECULES

187

and Wang, 1999) dissociation energy of the ground state of the dimer to two atoms. Thus at ultracold temperatures, there appears to be no obvious way to produce trimers (and higher clusters) from atoms and ground state dimers. An interesting question is whether a single "seed crystal" (a cluster with N 2 3 atoms) can rapidly nucleate (and destroy) the metastable gas. Similar considerations have been invoked in the discussion of the metastability of high-density spin-polarized atomic hydrogen (Stwalley et al., 1980). Such a seed crystal might be injected or formed by atom-molecule or moleculemolecule photoassociation or by tuning to a Feshbach resonance using a magnetic field, for example. 5. Bose-Einstein Condensation of Molecules

The proposed realization of Bose-Einstein condensation (BEC) in a dilute gas of atomic hydrogen (Stwalley and Nosanow, 1976) was finally achieved some 22 years later (Fried et al., 1998). Meanwhile, the interest in cold H spread to laser-cooled alkali atoms (Section 1.C). BEC was achieved in 1995 for 87Rb (Anderson et al., 1995) and subsequently for 23Na(Davis et al., 1995) and 7Li (Bradley et al., 1997), the latter despite its instability due to a negative triplet scattering length (Moerdijk et al., 1994; CSti et al., 1994).Table I1 lists some TABLE I1 SELECTED COMPOSITE BOSON SPECIES. THESIGN OF THE CORRESPONDING SCATTERING LENGTHa IS G I V E N IF KNOWN.

AN ASTERISK (*) INDICATES THAT A DILUTE-GAS BEC HAS BEEN ACHIEVED.

Species

a/lal

BEC

References

'H

+

*

Fried et al. (1998). Uang and Stwalley (1980A) Uang and Stwalley (1980A) Uang and Stwalley (1982) Bradley et al. (1997) Davis et al. (1995) Bohn et al. (1999) Bohn et al. (1999) Boesten et al. (1996), Tsai et al. (1997) Anderson et al. (1995) Kokkelmans et al. (1998), Legere and Gibble (1998) Stwalley (to be submitted) Stwalley (to be submitted) Stwalley (to be submitted)

'H E T 4He ' ~ i 23Na 39K 4' K *'Rb

+ ++

"Rb

+

'33cs

-

'H2 2H2 (-D2) 3H2 (-T2)

+ +

a Triplet

-

*

*

-

*

(a3C:) scattering length for H and alkali metals; singlet scattering length for 4He and H2 species.

J. T. Bahns, l? L . Gould, and W C. Stwalley

188

TABLE 111 NUCLEAR SPIN STATES (AND THEIR DEGENERACIES) FOR HOMONUCLEAR HYDROGENAND ALKALI DIATOMIC MOLECULES IN THEIR GROUND ELECTRONIC STATES FOR J = 0 AND FOR J = 1. ~~

C(21m 1.W

1,

Atom

I,

'H *H 'H 'Li 7 ~ i 23Na 39 K 40 K 41 K

112 1 112 1 312 312 312 4 312 512 312 712

85Rb "Rb 133Cs

Molecule

J=O

J=l

J=O

+ 1) J=l 3 3 3 3 10 10 10 36 10 21 10 36

composite Boson species for which the sign of the scattering length is relatively well known to be positive or negative (or small with uncertain sign); Fermion species are discussed in the next subsection. Of particular interest has been the case of 133Cs,where the triplet scattering length has been known to be large in magnitude with an uncertain sign because of a resonance near zero energy (Arndt et al., 1997; Leo et al., 1998). Very recently, it has been established to be negative in sign (Kokkelmans et al., 1998; Legere and Gibble, 1998). There are several complications in going from atomic to molecular BEC. First, ground state homonuclear diatomics are usually singlets, so the only (weak) magnetic moment such a molecule possesses is due to its nuclear spin. For some species, such as H2 (0,O) (para Hz), I = 0, so there is absolutely no magnetic moment for magnetic trapping. An optical trap must therefore be used. More generally, the nuclear spin state and its degeneracy complicate BEC, as summarized in Table 111. For example, for '33Cs2(0, 0), there are 28 nuclear spin states. Ignoring any splitting among these states in the absence of a magnetic field, one requires 28 times the total density at a given temperature to achieve 28 simultaneous and degenerate BECs. More generally, there should be some very small terms to break this huge degeneracy. The question will be whether the splittings are larger or smaller than the BEC transition temperature T,, which is typically 5 1 pK.

FORMATION OF COLD (T 5 1 K)MOLECULES

189

FIG.4. The transition temperature Tc (K)of a BEC decreases as the square root of the mean interparticle spacing r. Interparticle interactions of the identical particles in a BEC decrease as the cube of r for dipole-dipole interactions and as the fifth power of r for quadrupole-quadrupole interactions. Thus for BEC on the 1K temperature scale or lower, the molecular dipole-dipole interactions will cause significant deviations from the Bose ideal-gas approximation.

In addition, the intermolecular interactions will often be stronger than the interatomic. Figure 4 shows a plot of the log of the BEC T, versus the log of the average interparticle spacing. It also shows typical interparticle interactions for a magnetic dipole moment of 1 nuclear magneton on each identical particle (NN), for a magnetic dipole moment of 1 Bohr magneton (BB), for an electric dipole moment of 1 atomic unit (DD), and for an electric quadrupole moment of 1 atomic unit (QQ). BEC in dilute vapors such as 4He or "Mg, each with I = 0, would involve no multipole moment interactions between the identical particles. Other Bosonic atoms, such as 26Mg(I = l), would involve very weak interactions, with the NN curve well below the BEC curve in Fig. 4. The Bosonic atoms that have now undergone BEC ('H, 7Li, 23Na,and 87Rb)have far stronger interparticle interactions, but the BB curve is still well below the BEC curve for T, in the nanokelvin or even the microkelvin range. Nevertheless, interesting

190

J. T Bahns, l? L . Gould, and W C. Stwalley

“spinor” effects are expected and observed when a mixture of atomic hyperfine BECs interact (Stenger et al., 1998, and Miesner et al., 1999), as in liquid 3He (Lee, 1997). For molecules without dipole moments, simple BEC seems quite possible as well, for example, H2 (0,O) (para H2), for which a positive scattering length of 11.3 a0 and correspondingly stable BEC can be estimated (Stwalley, to be submitted). Sympatheticcooling of H2 (0,O)with an alkali BEC, for example, should be possible because no inelastic or reactive collisions can occur (except those that change the alkali atom hyperfine state). These inelastic collisions can be totally suppressed by using the lowest hyperfine state [e.g., 23Na(F = 1, M F = 1) in Inouye et al., 19981for the sympathetic cooling. The situation is more problematical for Bosonic molecules like ’Li ‘Hor ‘H 19F with large dipole moments (although a heteronuclear, homopolar molecule such as 39K41Kmight still be straightforward, with an extremely small electric dipole moment). In particular, as shown in Fig. 4,the dipole-dipole interaction (curve DD) between dipole moments of order 1 atomic unit is comparable to the BEC transition temperature T, even at nanokelvin temperatures. Thus, as in the case of liquid 4He, liquid 3He (where “Cooper pairing” occurs), Cooper pairs in superconductivity, exitonic BEC, etc., the BEC of molecules with large (- 1 a.u.) electric dipole moments will not be well described as a near ideal BEC.

6. Degenerate Fermi Gases of Molecules Like BEC for Bosonic species, degenerate Fermi gas effects should occur when the mean interparticle spacing in a Fermi gas is comparable to its thermal deBroglie wavelength. Experiments to probe such effects in atoms are under way at Rice (6Li) and JILA (40K) and Harvard ( 53Cr).HD is probably the simplest molecular species for which such effects could be observed. It is worth noting that all identical fermion-identical fermion cross sections go to zero in the limit T + OK. This is because Fermi-Dirac symmetry requires antisymmetry of identical particles. Thus s-wave ( 1 = 0) scattering is forbidden, whereas p-wave (1 = 1) scattering vanishes in the limit T + 0 K because there is a centrifugal barrier (e.g., 300 pK for two 39Katoms). Thus at nK temperatures, Fermi particles are “transparent” to each other, and sympathetic cooling with a nonidentical ultracold particle (e.g., Bosons in a BEC) must be used. The best theoretically studied degenerate Fermi gas is 6Li, for which it has been proposed (Stoof et al., 1996; Houbiers et al., 1997) that a mixture of two atomic hyperfine states (to avoid the “transparency” problem) might produce a superfluid transition at a transition temperature as high as 11 nK (Houbiers et al., 1997).The authors, however, quite reasonably admit that the uncertainty

FORMATION OF COLD (T 5 1 K) MOLECULES

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in their estimates is large, and we anxiously await further experimental and theoretical results in this fascinating area.

7. The Molecule Laser Recently, the concept of an “atom laser” has been introduced and demonstrated (Mewes et al., 1997; Ketterle and Miesner, 1997)to describe a coherent state-selected atomic beam produced by an atomic BEC, essentially a “matter wave” laser of atoms, not photons. The analogous “molecule laser” has now also been proposed by Julienne et al. (1998), where a stimulated Raman rather than a radiofrequency outcoupler is used to produce a coherent, state-selected molecular beam in the direction of net two-photon recoil. Such beams represent the ideal limit of atomic and molecular beams and should ultimately produce important scientific and technological advances.

11. Nonoptjcal Cooling Techniques Although the emphasis in this review is on highly selective optical techniques for formation of cold molecules, there are important nonoptical techniques as well. These techniques include the helium cluster cooling technique (Section ILA), the helium buffer-gas cooling technique (Section KB), trap compression and evaporative cooling techniques (KC), and three-body processes including resonances (1I.D). A. HELIUMCLUSTER COOLING The advantages of helium free jet expansions for formation of very cold ( w 0.4 K) liquid droplets (large clusters) has long been recognized (Buchenau et al., 1990 and references therein). The cold temperature is maintained by evaporation of He atoms. Doping such clusters provides a unique liquid matrix for extension of matrix isolation technology to lower (subKelvin) temperatures. The liquid nature of the matrix also provides an energetically subtle option for the dopant to sit on the cluster surface or to bury itself in the depths of the cluster. The spectroscopic studies of dopants on or in large helium clusters goes back to 1992, when infrared spectra of SF6 and (sF6)2 buried inside the clusters were obtained bolometrically (Goyal et al., 1992). Further experiments employing mass spectrometry for detection obtained rotational resolution (Hartmann el al., 1995) and established an internal temperature of 0.37-0.38 K (Hartmann et al., 1996), consistent with evaporativecooling. Other work on IR spectroscopy includes interior H20 and its clusters (Froechtenicht et al.,

192

J. T. Bahns, l? L. Gould, and W C. Stwalley

1996). Electronic spectroscopic studies have included excited states of He:! (Jiang et af., 1993; Northby et al., 1995), the surface-bound alkali metal dimers (Stienkemeieret al., 1995, 1995A, 1995B, 1996; Callegari et af.,1998; Higgins et al., 1998) and trimers (Higgins et al., 1996, 1996A) and other species (Hartmann et af., 1996, 1996a, and Reho et al., 1997; Stienkemeier et af.,1997).This field has recently been revised (Toennies and Vilesov, 1997; and Whaley, 1998). It is clear that a great deal of interesting spectroscopy will be done in and on such clusters. Of particular significance, in our opinion, is the spectroscopic preference for high spin multiplicities (less energy is released when these species are formed from atoms and consequently less desorption from the cluster). High spin multiplicity species, such as ' C i N, (Ferrante and Stwalley, 1983, 1986), are virtually unobserved experimentally. B. HELIUM BUFFER-GAS COOLING The use of 3He and 4He to cool atomic and molecular species, proposed by Stwalley (1984) (see also Dolye et al., 1995), has recently been demonstrated by Doyle, Friedrich, and their coworkers. In particular, this group has cooled and magnetically trapped Eu (Kim et af.,1997), Cr (Weinstein et al., 1998B), and CaH (Weinstein et al., 1998). When 3He is used, subKelvin temperatures of 300 mK are achieved, whereas use of 4He yields temperatures of 0.81.4K. The species are produced by laser ablation of appropriate solids into the helium buffer gas. For the molecule CaH, approximately lo8 molecules are - ~ a temperature of 400 f 50 mK for a trapped at a density of 8 x lo7 ~ r n and decay time of 0.5 s (Weinstein et af.,1998). Virtually all these molecules are in the electronic, vibrational and rotational ground state. About half of the molecules deposited into the trapping region were trapped. Thus improvement in production methods should lead to even larger numbers of trapped molecules. It should also be extendable to other paramagnetic molecules. Magnetic trapping is the key first step toward evaporative cooling (or sympathic cooling) to pK or even nK temperatures.

-

c.

TRAP

COMPRESSION AND EVAWRATIVE COOLING

The nonoptical formation of diatomic molecules from atoms requires threebody collisions, the rate of which increases as the atomic density cubed. The special case of resonances is discussed in Section 1I.D. Two straightforward ways of increasing atomic density in atom traps are trap compression [e.g., by increasing the magnetic field gradient in a magneto-optical trap (MOT)] and evaporative cooling (e.g., to BEC in a magnetic trap as in Anderson et af., 1995; see also the review of Ketterle and Van Druten, 1996).

FORMATION OF COLD (7'5 I K) MOLECULES

-4

12

eps 0

193

e3L I

I

2

I

I

I

I

6

4

I

I

8

I

I

10

I

I 12

N

FIG.5 . The calculated formation rates (units of cm6/s) for high rovibrational levels of energy E (v,N)(with respect to dissociation) of the lowest triplet (a 'C:) state of Z3Na2in the zero temperature limit (Moerdijk et al., 1996). It is assumed that the recombining atoms are doubly spin-polarized and interact only via s-wave collisions on the adiabatic quartet potential surfaces. No quasibound states above dissociation are formed because they are much higher in energy (at E l k > 1 mK above the dissociation limit).

Although there are apparently no experiments that directly measure the molecular states formed by subKelvin three-body recombination, there are a number of studies of three-body loss (e.g., Stamper-Kurn et al., 1998), where such loss was measured for BECs of 23Naboth in a magnetic trap and in an optical trap. Reassuringly, both rates were found to be the same and in rough agreement with two calculations (Moerdijk et al., 1996; Fedichev et al., 1996), which were an order of magnitude larger and an order of magnitude smaller, respectively. The former calculation is of particular interest because it predicts the branching of the recombination into different rovibrational levels of the lowest triplet molecular state, exclusively formed from doubly spin-polarized atoms. Figure 5 shows the results of Moerdijks et al. (1996) for 23Na2formation, where 86% of the recombination goes into the single most weakly bound level [E(15,O)lk = 0.002 K]. Three-body recombination in "Rb has been used to probe higher-order coherences in a BEC (Burt et al., 1997). The lower rate observed in a BEC compared to a thermal sample demonstrated the reduced density fluctuations in a BEC. The noncondensed

194

J. T. Bahns, P. L . Gould, and W C. Stwalley

rate agreed well with corresponding theory (Fedichev et al., 1996). Threebody recombination may also contribute to formation of translationally ultracold Cs2 molecules (Takekoshi et al., 1999). In the three-body loss work, it has been assumed that the loss was exclusively due to molecule formation, although in principle, three-body inelastic collisions could also contribute. However, because two-body inelastic collisions are often unimportant, as in Stamper-Kurn et al., (1998), such threebody inelastic collisions should not be important. It might also be noted that for the lowest hypefine state, which can be optically but not magnetically trapped (e.g., IF = 1, MF = 1) in 23Na),no such inelastic three-body processes are energetically possible except as the magnetic field goes to zero.

D. THREE-BODY PROCESSES INCLUDING RESONANCES The character of three-body recombination can be altered if there is a twobody long-lived resonance, either single-channel (shape) or multichannel (Feshbach). Such considerations were invoked in 1976 (Stwalley, 1976) in connection with the stability of spin-polarized hydrogen and are further discussed elsewhere (Stwalley et al., 1980; Uang and Stwalley, 1980; Uang et al., 1981). Essentially, the “resonance” is a long-lived intermediate in a two-body collision; thus the three-body process becomes a sequence of two two-body collisions. At the low energies of ultracold collisions, very long resonance lifetimes (microseconds)can still be within the thermal distribution of collision energy. For a given atom A in a specific hyperfine state, this corresponds to A

+ A +A $ ( v , J )

(1)

where # indicates a shape (quasibound) resonance and ( v , J ) represent the vibrational and rotational quantum numbers of the resonance, followed by A$(v,J)

+A

+

A2(v,J- 1)

+A

(2)

[and similar inelastic processes that yield other (v,J ) levels of lower energy], where the lower levels produced are generally either truly bound levels or orders of magnitude more stable quasibound (predissociative) levels. The resonance theory of three-body recombination was formulated in Roberts et al. (1969) for the case of H atom recombination, where the theory predicts a maximum rate at 65 K attributable to a single broad resonance, v = 14, J = 5 of X ‘CLH,, with an energy of 65.8 K with respect to dissociation and a width of 25 K (predissociation lifetime of 0.31 ps) in Le Roy and Bernstein (1971). This single resonance increases the rate by over an N

FORMATION OF COLD (T 5 1 K) MOLECULES

195

order of magnitude from the nonresonant background rate (Roberts et al., 1969). The Dz X 'Cp' (v = 21, J = 1) resonance implies a similar, more dramatic maximum near the energy of the quasibound level (- 124mK above the D D asymptote) (Stwalley, 1976). Essentially, such resonances are formed with a very long time delay and are then relaxed rotationally with a large cross section to more stable levels. To be somewhat more quantitative, a 70K width corresponds to a predissociation lifetime of 1 ps, a 70 pK width to 1 ps (Le Roy and Bernstein, 1971). Thus shape resonances with microKelvin widths at microKelvin temperatures will dramatically enhance three-body recombination rates. Such considerations apply not only to single-channel shape resonances but also to multichannel Feshbach resonances, for example, magnetic-fielddependent hyperfine-induced X Cp' - a 3 C i mixed-state resonances in H2 (Stwalley, 1976), Li2 (Uang et al., 1981), Rb2 (Tsai et al., 1997), and CSZ (Weickenmeieret al., 1985). Such resonances have recently been observed as a function of magnetic field in trapped 85Rb(Courteille et al., 1998; Roberts et al., 1998) and in a Na BEC (Inouye et af., 1998). Not surprisingly, tuning a magnetic field can bring Feshbach resonances into the thermal collision energy range of trapped atoms and BECs (pK and nK, respectively), allowing step 1 (where # now indicates a Feshbach resonance) to proceed, thereby enhancing the three-body recombination and three-body loss. This enhanced three-body loss has been observed in a Na BEC (Inouye et af., 1998 and Stenger et al., 1999). Shape resonances cannot, of course, be as readily tuned with a magnetic field.

+

'

111. Optical Cooling Techniques Current proposals for direct optical manipulation (here we mean cooling, heating, deflection, trapping, diffraction, etc.) of molecules can be broadly classified as either incoherent (resonant optical pumping with spontaneous decay) or coherent (Raman-like). Current incoherent methods appear to be technically more complex (requiring multiple frequencies) but are clearly capable of manipulating (cooling, heating, and trapping) macroscopic quantities of molecules. Starting with a thermal distribution of molecules, the ultimate goal is to cool and trap a large translationally ultracold (< 1mK) molecular sample in the lowest rovibrational level of the ground electronic state (with TROT= TVIB= OK). Although great progress has been made in recent years in manipulating atoms (Section LC), molecules have proven to be resilient to atomic optical manipulation methodologies. This is true mainly because of the increased spectral complexity and multiple spontaneous decay channels (lack of closed

196

J. I: Bahns, I! L. Gould, and W C. Stwalley

energy level schemes for recycling population). By changing vibrational and rotational state, molecules can readily shift themselves out of resonance in any known atomic cooling scenario. The analogous problem occurs in atomic laser cooling when there are multiple decay channels to nonresonant hyperfine levels. In this case, the addition of one (generally nearby) “repumping” laser is sufficient to counter this optical hyperfine pumping. But even for the simplest diatomic molecules, optical rovibrational pumping is far more complex than for atoms, typically requiring 19 or more “repump” lasers (Bahns et al., 1996). By limiting the number of states and frequencies involved, there have been a limited number of experimental demonstrations of optical manipulation of gaseous molecules. The only successful incoherent cooling experiment was reported by Djeu and Whitney (1981), who succeeded in cooling C02 molecules in a cell (AT 1 K) by pumping the P(20) transition of the 100-001 band with a 300 W cw C02 laser. However, it was necessary to use collisional equilibration with a hot buffer gas (Xe at 300°C 0.1 Torr) for population of the lower (100) pump level. Successful radiative (single-photon-momentum) deflection of molecules in a highly collimated beam has also been demonstrated by Herrmann et al. (1979). Coherent transfer between two molecular levels formulated by Gaubatz et al. (1990) and Bergmann et al. (1998) by the STIRAP (stimulated Raman adiabatic passage) process has proven to be exceedingly efficient. Presently, adaptations of STIW-like variants for sequential population transfers through multiple intermediate states [such as ladder schemes and RCAP (Raman chirped adiabatic passage), reviewed in Bergmann et al. (1998)l are formulated but remain to be demonstrated for molecules. N

A. INCOHERENTOPTICALCOOLING OF MOLECULES Sequential incoherent optical cooling of thermally populated molecular samples is assumed to require the simultaneous generation and precise frequency control of many laser lines. If a molecule is Raman active, it can in principle be cooled by the sequential technique. For the discussion,a general nomenclature designates the degree of freedom cooled followed by the type of optical transition employed (e.g., RE is cooling rotation using an electronic transition). Table IV lists the other relevant combinations, of which only those utilizing electronic transitions appear practical (RE, TE, and VE), on the basis of having the shortest radiative lifetimes (typically tens of nanoseconds). Sequential optical cooling of molecules, as formulated by Bahns et al. (1996), is restricted to electronic transitions. Only the proposed order rotation-translation-vibration (RE-TE-VE) appears to be practical. The

FORMATION OF COLD (T 5 1 K)MOLECULES

197

TABLE IV MOLECULAR CCOLING TYPES.

OVERVIEW OF OPTICAL

Degree of Freedom

Optical Transition

Cooling 5 p e

Electronic/Vibration/Rotation Vibration/Rotation Rotation

TE* TV

Electronic/Vibration/Rotation Vibration/Rotation

VE*

Rotation

Electronic/Vibration/Rotation Vibration/Rotation Rotation

RE* RV** RR

Electronic

Electronic/Vibration/Rotation

EE

EXTERNAL: Translation

INTERNAL: Vibration

TR

vv**

* Proposed in Bahns et ol. (1996). **Demonstrated in Djeu and Whitney (1981).

most challenging step, TE, can be avoided if one begins with a sample of ultracold atoms and performs PAS (photoassociativespectroscopy,Section IV) (Thorsheim et al., 1987; Band and Julienne, 1995) followed by VE cooling, assuming cold rotational energies. After selection of an appropriate set of rovibrational states of sufficient closure, an ensemble of laser frequencies can be generated in a MSF (multiple single-frequency)laser, described later in this section. For vibration, the degree of closure depends on the number of ground vibrational levels in the set. A set of ground vibrational levels, however, is sufficiently closed when spontaneous decays to all levels outside the set have a negligible effect on the population during the time required for cooling. As a general rule, the product of the number of photons needed to cool a degree of freedom times the probability of leaving the cooling cycle (per scattering event) should not exceed unity. For TE cooling Csz (B-X), the minimum set of ground vibrational levels turns out to be 10 (i.e., 21’’ = 0-9 coupled to 21’ = 0) for an initial speed of 2 2 O d s . Spontaneous decays that involve rotational selection rules (e.g., A J = 0, +/ - 1) do not cause a closure problem. Generally, shorter cooling times imply proportionately smaller sets of vibrational levels for achieving closure. RE cooling, illustrated in Fig. 6, is accomplished with the MSF laser (see below). This is accomplished by stepping sequentially (a “P-step” procedure) from a P branch frequency P(J&) to P(J& - 2), then to P(J& - 4), . . . , and finally to P(JN= 4). The first step transfers all population of para molecules from J& to J& - 2 because while stimulated absorption and emission are occurring between the levels (v‘’ = 0, J ’ = J& - 1) and

J. ‘I: Bahns, P. L. Gould, and W C. Stwalley

198 J‘=l -

J’=3

FIG.6. Rotational cooling to J” = 0 and 2 levels on an electronic transition (RE cooling) exemplified for the A’C: - X’C; or B In, - X’C,f transitions of the para form of an alkali dimer with odd nuclear spin (e.g., ‘ 3 3 C ~ 2 39K2). r The P branch frequencies would be stepped in order from P(JMAx)to P(JMA~ - 2) . . . to P(4). leaving all para population in J” = 0 or 2 levels.

(v”,JLm), spontaneous emission (“optical pumping”) occurs to the levels (v”,J” - 2). The second step similarly transfers population from the YAX - 2) levels to the (v”,J L A x - 4) levels. The final step transfers (v”,JMAX population from the (d’, 4)levels to the (v”,2) levels. There is no reason to rotationally cool the para molecules below J ” = 2 because of the TE and VE steps discussed below. Assuming saturation of the various P(J”) transitions, the time T required for transferring the population of the set of (w”,J”) levels to the set of (d’, J” - 2) levels can readily be estimated from the effective radiative lifetime of the (v’,J ’ = J ” - 1) level from the Einstein A coefficients:

The effective radiative lifetime can also be readily estimated from the true total radiative lifetime in the absence of resonant P branch [P(J”)] radiation (ignoring the rotational dependence of A v ’ + v t ~ ) : TRADe

’ 9’

(-)TR4D 2 J N-

1

(4)

For the B ‘II, - X C bands of Cs2 given in Table V, T ~ =D25 ns for the B ‘II,,w’ = O rotational levels. Thus 5 TRADe is approximately 275 ns and 292 ns for J” = 6 and J” = 4,respectively. The analogous “R-step” procedure beginning at R(0) and ending at R(JL,) provides the mechanism for rotational population transfer from J ” = 0 to J ” = JLAX.It is important to realize that although R and P

FORMATION OF COLD (T 5 1 K) MOLECULES

199

TABLE V EXAMPLES OF SEQUENTIAL RE - TE - VE COOLING OF PARA133Cs2USINGA MSF LASER OPERATING ON THE 20B In, (0’= 0,J’ = 1 ) - x ‘El (W” = 0 - 9, J ” = 0 AND 2) TRANSITIONS, FOR WHICH EACHFREQUENCY IS PRECISELY KNOWN. ~

~~

~

~~

~

~

~~

Cooling 5Pe

Pump Transition

Pump Plus Selected Raman Sidebands

Irradiation Time (1s)

Total Irradiation Time (ps)

RE RE TE

Po (6) Po (4) Po (2)

0.215 0.292 1 lo00

0.215 0.567 1 lo00

VE

Po (2)

P,,01(6) (v” = 0-9) Pwv(4)(v” = 0-9) P,fl(2) (v” = 0-9) R,,t(O) (w” = 0-9) P,fl(2) (w” = 0-9) R”o~(0)(w” = 1-9)

1.95

1lo00

branches overlap in a molecular band, rotation can be “cooled” or “heated” independently with narrow-band laser excitations. This is possible because R(J) and P(J) lines do not normally coincide in frequency, making it impossible for a given rotational population to interact simultaneously with P(J) and R(J) excitation. Rotationally hot “super rotor” molecules (which could be vibrationally and translationally cold) might be prepared by the R-step technique (Li et al., to be submitted). Such “super rotor” states, potentially with significantly greater energy than bond dissociation energies (e.g., J 2 106 for 6Li2), should exhibit novel collisional dynamics. Short infrared pulses have also been proposed as a means to spin up molecules to high angular momentum states (Karczmarex et af., 1999). Immediately following the RE step, TE cooling, illustrated for even J” molecules assuming the lowest three (rather than the lowest 10 as in Bahns ef af.,1996) vibrational levels form a “closed cycle,” as shown is Fig. 7. For the homonuclear alkali diatomic molecules, selection of the set of the two lowest even J” lines [R(O) and P(2)] can be used to TE cool para molecules. Conversely, selection of the set of the two lowest odd J ” [R( 1) and P(3)] can be used to TE-cool ortho molecules (for 6Li2 and 40K2, these para and ortho designations are reversed). RE-cooled ortho and para molecules can be TE-cooled simultaneously, using an additional pump laser. TE cooling of molecular translation is analogous to the more familiar atomic chirp cooling (Prodan and Phillips, 1984; Ertmer et al., 1985). Molecular TE cooling rates are also comparable to the atomic case and are limited by the 25 ns radiative lifetime of the upper state. For example, in Table V, TE slowing of a Cs2 beam on the B-X bands from o 220m/s, is estimated by Bahns et af. (1996) to require 1 1 ms (equivalent to 1.1 x lo5 spontaneous photon-scattering events). This time will depend on the details of the saturation of various R and P lines, laser polarization, etc. N

N

200

J. T. Bahns, P. L. Gould, and W C. Stwalley V

v " a , J"=O

Y

FIG.7. Translational chirp cooling of J" = 0 and 2 levels on an electronic transition (TE cooling) exemplified for the A I C - X I C or B I II, - X I C transitions of the para form of an alkali dimer with odd nuclear spin (e.g., 133C~2, 39K2). The diagram assumes that only three (rather than 10 as in Bahns er al., 1996) vibrational levels d'are needed for a "closed cycle." The six laser frequencies shown would be simultaneously chirped over the Doppler profile as in atomic c h r p cooling of translation using the MSF laser described below and in Bahns et af. (1996).

Immediately following the TE cooling step, VE cooling is implemented, as illustrated in Fig. 8 (for para molecules, assuming again only the lowest three (rather than the lowest 10 as in Bahns et al., 1996) vibrational levels are Franck-Condon-coupled to v' = 0). VE cooling to v" = 0, J" = 0 is initiated by removing (blocking) the single R(O), v" = 0, J" = 0 tf v' = 0, J' = 1, MSF laser side band. This causes the entire para population to be transferred (optically pumped) unidirectionally to v" = 0, J" = 0. If all but the blocked transition are saturated, the time required for this step depends on the effective radiative lifetime of the blocked transition, given by Trade = I [A,I=o,J L ~ - + , G O , ~ ~ 0 = 1 390ns. For the Cs2 example in Table V, > 99% transfer in this step can be considered complete in roughly 5 effective radiative lifetimes, or 1.95ps. This completes the (unique) RE-TE-VE sequence. The total time required, roughly 11ms, is due to the lack of speed and efficiency of TE cooling compared to RE and VE cooling (these are compared in Bahns er af., 1996). Both the total frequency change of the pump laser, 3.6 GHz, and the rates for both RE steps and TE chirps are experimentally achievable. Generally, by making the appropriate choices of states in the RE, TE, and VE cooling steps, nearly complete population transfer to a single rovibrational level is possible. The Cs2 (B-X)example constitutes a 40 1 level (280 -+ 1 state) transfer (v" = 0-9 and J" = 0, 2, 4, 6 -+ v" = 0, J f f = 0). N

N

--f

FORMATION OF COLD (T 5 1 K) MOLECULES

20 1

v"=2J"=2 v"=2J"=o v%lJ"=2 V'klJ"=o

v"=OJ"=2 v'k0, J"=O FIG.8. Vibrational cooling to the V" = 0, J" = 0 lowest level on an electronic transition (VE cooling) exemplified for the A 'C: - X ' C l or B In, - X'C; transitions of the para form of an alkali dimer with odd nuclear spin (e.g., 133Cs2.39K2).The d' = 0,R (0) MSF laser transition would be blocked, so spontaneous emission (optical pumping) would transfer all population from the coupled manifold of levels to V" = 0,J" = 0.

The MSF laser for performing the above cooling sequence is illustrated in Fig. 9. Again using Cs2 as the example, one begins in (a) with a single narrowband pump (master) laser (tunable and chirpable). The appropriately closed set of sidebands is generated when (b) the pump laser multiply traverses a sample of molecules contained in the Raman gain medium (here, a separate supersonic beam of C S ~ )Inelastic . resonant-stimulated Raman (RSR) scattering generates the ensemble of weaker narrow-band sidebands, which copropagate with the pump beam. The ensemble is then "filtered" by a selector (subtractive spectrometer) and amplified by a high-gain (> lo6) broad-band (- 25 nm) amplifier (Titanium Sapphire). RE, TE, and VE cooling outputs are illustrated in parts (c), (d), and (e), respectively, in Fig. 9 for the Cs2 example. For the RE cooling sequence, only P(J") lines are transmitted by the selector, giving the 10 frequencies required for the "P-step" (for a 280 --t 60 state population transfer). For TE cooling, the 20 frequencies are transmitted by the selector during the sub-Doppler chirp of the master laser. Critical to TE cooling, the Doppler tuning of each RSR sideband (S,i) is related to the pump detuning (6,) by 6si = ( k s i / k p ) & p

(5)

where ksi and kp are the wave vectors of the ith sideband and pump, respectively. This is vital to TE cooling because sub-Doppler chirping of the master pump laser results in synchronously chirped sidebands, all resonant

J. T. Bahns, P. L. Gould, and W C. Stwalley

202

C

output for cooling

M2

I/ I(( , ,

v"=O,P(J)

+V'd

I

v'no-a

\:

Pump

Pump + Stimulated Raman

b

I P chirp: P(J)

P(2) and R(0)

R(O),v"=O blocked C

V

FIG.9. The multiple single-frequency laser scheme for generating the frequencies needed for sequential RE, TE, and VE cooling. A single tunable, narrow-band pump laser (a) pumps the Raman gain medium (here a separate supersonic beam of Csz). The appropriate set of Raman sidebands are generated and then selected and amplified for use in cooling. If no selection is employed, all sidebands (P,,.( J ) , v" = 1-9 and Ruj# (J), v" = 0-9) are produced (b). If population transfer (RE cooling) from J to J - 2 is desired, the pump laser is tuned to the PO( J ) line and the Put<( J ) (v" = 1-9) as well as the PO(J)pump line are selected and amplified (c) (the R,. (J),(21'' = 0-9) lines are blocked). If TE cooling of the Doppler profiles of a previously REcooled sample is desired, the pump laser is tuned to the Po (2) line and all sidebands are selected and amplified (d); then the PO(2) pump laser is chirped over the Doppler profile, causing all sidebands to be simultaneously chirped bringing about TE cooling of all 20 lower rovibrational levels. If population transfer (VE cooling) of a previously RE- and TE-cooled sample to the v" = 0, J" = 0 level is desired, then the PO (2) pump line is again used to generate all sidebands, all of which except for the & (0)line are then selected and amplified (e); only the & (0) line is blocked.

FORMATION OF COLD (T 5 1 K) MOLECULES

203

with the same-velocity group of molecules. As molecules decelerate, Doppler compensation occurs simultaneouslyon all sidebands (ksi)when just the pump laser detuning is chirped (d6,ldt). This can be seen in the expression for the Doppler-compensateddetuning rate of the ith sideband (d&/dt), given by d6si/dt = (ksi/kp)d6p/dt = ksi(dv/dt)

(6)

Note that the deceleration of resonant molecules (dvldt) is independent of sideband (k,J (i.e., ksi appears on both sides of the last equality in equation 6). Therefore, the entire ensemble of decelerating molecules are always resonant with each sideband throughout the TE chirp, insuring that all molecules cool (decelerate) at the same rate, irrespective of sideband. In the final step (VE cooling), the selector simply blocks the 0 4 band R(0) sideband (this is easy to perform because it has the highest frequency). This causes the populations of all optically coupled states to be transferred to the “dark” state, v” = 0, J” = 0 (60 + 1 state transfer). It might be noted that the pump- and MSF-stimulated Raman sidebands will be narrowband sidebands. Thus the RE cooling should be carried out with these laser beams perpendicular to the molecular beam, with the small transverse Doppler width of the beam matched to the laser line widths, so that all molecules are RE-cooled. The TE chirp cooling must be carried out with the laser beams and the RE-cooled molecular beam counter-propagating along the beam axis. The RE-, TE-cooled molecules can then be VE-cooled from any direction. As a Raman gain medium for the MSF laser cooling of alkali dimers, we have developed a novel new pulsed high-temperature supersonicbeam (to be described in a future submission to Rev. Sci. Instrum.). This beam, now operating with potassium, involves injection of 0.3-Hz pulses of high-pressure (> 10 atmospheres) Ar from a standard pulse valve into a cell containing highpressure potassium vapor (currently 20 torr; potentially 150 torr). This pulse opens a poppet valve for 50 ms, producing a high-density slit-shaped pulse of K and Kz,with transverse translational and rotational temperatures of 60 K (based on single-shot, rotationally resolved B-X spectra). Besides serving as a Raman gain medium, the pulsed beam can also be used to produce large densities of state-selected cold molecular ions. Studies of ultracold molecular ions are currently possible by conversion of ultracold molecules via state-selected multiple resonance photo-ionization. Tsai et al. (1995) and Chang et al. (submitted) have demonstrated production of cold Na2f in a sodium heat pipe by all optical triple resonance (AOTR) of Na2, using the scheme illustrated in Fig. 10: N

N

J. I: Bahns, F? L. Gould, and W C. Shvalley

204

-

Na(3s)+Na'+c'

40-

30-

-,

"0 e

K

FIG. 10. Scheme for production of state-selected (v+ = 0, N+ = 0) N a i molecular ions (Tsai et al., 1995, Chang et al. (submitted)). If a translationally cold or ultracold source of Na2 was used, one would have an ideal source of state-selected ultracold Nai ions.

A'CT(1,J") + h V 2

+

31Cgf(0,J')

+ hv3 + nl'AF)(O,J)(E > 39,558.06 f0.06cm-') nl'AP'(0,J) X2C;(vf = 0, N + = 0) + e-

3'C;(O,J')

+

(8) (9) (10)

Excitation of Rydberg states lying slightly above the ionization threshold gives rise to state-selective autoionization, restricted to the lowest level of the dimer ion. Similar experiments have been carried out previously by

FORMATION OF COLD (T 5 1 K) MOLECULES

205

all optical double resonance in K2 (Leutwyler ef al., 1981; Broyer ef al., 1983). Accurate energetics permit precise multiple resonance schemes for state-selective production and spectroscopy of other cold alkali dimer and trimer ions (Stwalley and Bahns, 1993). Cooling gas-phase molecules at higher densities where collision effects become important can result in heating via quenching of excited states, reabsorption of anti-Stokes radiation, diffusion (or convection at sufficiently high densities), thermal conduction, multiphoton ionization, etc. We do not attempt to consider the array of such processes here (see, however, Section I.D.3). Without ways of overcoming losses due to these competing processes, there is currently little hope for direct incoherent optical cooling in the presence of frequent collisions. Coherent techniques that rely on the preservation of phase are also expected to be sensitive to collisions. In summary, direct sequential incoherent optical cooling of diatomic molecules using the MSF laser approach, currently under development, appears promising, particularly for faster and more efficient RE and VE cooling. These can also be used to cool molecules formed by incoherent photoassociation of ultracold atoms (Section PI).TE cooling of molecules presents the greater technical challenge but appears feasible. Ultracold stateselected molecular ions can potentially be synthesized from translationally ultracold molecules. B. COHERENT OPTICAL MANIPULATION OF MOLECULES Proposals for direct cooling of molecules by coherent processes have been recently formulated theoretically. One method, developed by Bartana ef al. (1993), uses a series of shaped pulses to cool molecular internal degrees of freedom. The procedure, analogous to a heat pump, maintains a cooled vibrational population on the molecular ground state surface and, simultaneously, a vibrationally hot population on an excited electronic surface. Experimental implementations and the possibility of extension to the ultracold regime await testing. In another method demonstrated theoretically by Garraway and Suominen (1998), non-Franck-Condon state-selective transfers of molecular wavepackets from the ground state potential through an intermediateto a stationary, displaced (in R) excited state have been proposed. The method, adiabatic passage by light-induced potentials (APLIP), uses a counterintuitive sequence of short high-intensity (TW/cm2) red-detuned laser pulses to couple displaced electronic states via a third intermediatedisplaced state. Preliminary modeling for adiabatic transfers via the sequence X I E i - A'C; - 2lrII,inNa2 allows complete transfers despite negligible Franck-Condon overlaps between states. Practical application (particularly for a A configuration) provides additional

J. Z Bahns, t? L. Gould, and W C. Stwalley

206

hope for efficient transfers of molecular wavepackets accompanied by large simultaneous changes in R and E. Unfortunately, the predicted threshold intensities for APLIP suggest significant losses due to nonlinear optical processes, such as multiphoton ionization. In summary, incoherent and coherent techniques have been proposed for cooling and manipulating thermally populated diatomic molecules. Experimental tests of these methodologies have recently begun.

IV. Formation of Cold Molecules Via Laser-Induced Photoassociation A. SPONTANEOUS DECAY 1. Proposals

The formation of translationally ultracold molecules, as discussed above, can be accomplished by starting with trapped ultracold atoms and photoassociating these to produce translationally ultracold molecules. This idea was first introduced by Thorsheim et al. (1987), who presented the theory of laserinduced photoassociation applied to sodium atoms via singlet and triplet manifolds (see also Section I.D.1). The scheme involves excitation of cold sodium atoms with a photoassociation laser (hvpA), which pumps ultracold atoms to a long-range high vibrational level of singlet or triplet excited electronic states Na + Na

+ hVpA

+ Nal.

(11)

Because the kinetic energies of ultracold atoms are exceedingly small (smaller than or comparable to the widths of bound-bound transitions), these free-bound transitions are state-selective. Spontaneous radiative decays of Na; via bound-free emission, (14

Na; - + N a + N a + h v

tend to dominate the decay. However, bound-bound emission also occurs,

'

where the final Na2 is either in the electronic ground state (X EL) or the lowest triplet state (a3CT). These bound-bound emissions tend to favor highly vibrationally excited lower-state vibration levels but are generally not

FORMATION OF COLD (T 5 1 K) MOLECULES

207

20

15

h

r

€ sX

10

FI

w

5

0

-5 4

2

6

8

10

12

14

16

18

20

R(A) FIG.11. Single photon photoassociationof ultracold 39Katoms to the v = 191 level of the A C: state 39K2 yields formation of translationally ultracold 39K2molecules in vibrationally excited levels (e.g., v = 36) of the X ' E l ground electronic state (Nikolov et al., 1999).

state-selective. The bound-free emission usually leads to the production of translationally hot sodium atoms that quickly leave the trap. With minor modifications, the analogous scheme for cold potassium (X ' X i ) molecules is illustrated in Fig. 11 based on Nikolov et al. (1999), which is further discussed in Section IV.A.2. For illustration, magnetooptically trapped (MOT) potassium atoms are photoassociated to w = 191 of the A'C; state K

+ K + hVpA

-+

K; (A'Cf, v = 191)

(14)

Spontaneous radiative decay here also is partitioned between bound and continuum states, K ; ( A ' C ~ , V =1 9 1 ) + ( K + K ) o r ( K z X ' C i ) + h v

(15)

Again, most decays are bound-free, but a small fraction (0.15%) is boundbound, leading to a distribution of high-lying vibrational levels, with the Franck-Condon factors suggesting that the most populous is v N = 36.

J. I: Bahns, F! L. Gould, and W C. Stwalley

208

It appears that trying to make cold 21’’ = 0, J” = 0 molecules by onephoton photoassociation will not work very well (here with the aim of ultimately trapping them for times long compared to reciprocal collision frequencies or sufficiently long for equilibration between internal and external degrees of freedom). The result will be mostly hot atoms (leading to significant trap loss), and the few molecules produced will be vibrationally hot, with virtually no chance of reaching w” = 0. Rotational heating is less problematic because photoassociation selects only partial waves leading to low rotational quanta (s, p, and d waves), and rotational selection rules allow a greater level of optical control. A better method for producing w” = 0, J” = 0 molecules is needed because optical trapping potentials are too weak to contain molecules heated by V-T or R-T energy transfer collisions with atoms or other molecules. In response to this dilemma, Band and Julienne (1995) proposed a twocolor photoassociative scheme for sodium that provides not only a large improvement in molecular fractions but also a spontaneous decay gateway to ultracold v” = 0 molecules. Their three-step scheme, here adapted to twophoton potassium atom photoassociation (Wang et al., 1997A). is shown in Fig. 12. The first step involves photoassociation of free atoms to high-lying, 35

30

25

0 0

20

0 + . I

15

5

3 10

0

2

4

6

0

10

12

14

16

R(A) FIG. 12. The Band and Julienne (1995) scheme for translationally ultracold molecule formation by two-color excitation, adapted to known states of 39K2.

FORMATION OF COLD (T 5 1 K) MOLECULES

209

long-range levels of the l'n, state (lg at long range in Hund's case c). K+K+hv1 +Kl(llIIg)

(16)

This state has a significantly longer radiative lifetime than the A C: state, which allows molecules to survive more vibrational periods before decaying spontaneously. Therefore, absorption at the l'n, inner turning point by a second laser (hv2) via K;(llIIg) +hv2 + Kl*(nllAu)(e.g.,5'nu,v= 11)

(17)

will have a higher probability. Optimization requires small detuning adjustments which allow for power shifts. Spontaneous decay of the Rydberg state to the ground state,

, V33) +hv K ; * ( ~ ' ~ , ,= L 11) J + K ~ ( X ' C , ~<

(18)

is entirely bound-bound, with roughly 10% of the emission probability to as shown in the Fig. 13 Franck-Condon factors. Estimates for potassium using typical experimentalparameters (Wang et al., 1996) predict a formation rate of lo6 K2 v = 0 molecules per second. Branching of Rydberg emissions to lower-lying excited 'Ci states is expected to be small (<5%), although the relevant potentials and transition dipole moments are somewhat uncertain. Because little selectivity of terminal vibrational levels is possible in this scheme, vibrationally hot molecules would need to be removed (i.e., by photo-ionization or photodissociation) or cooled to V" = 0, J" = 0 by VE MSF LJ" = 0,

N

c 0

U

c

600

Y2

400

0

e 200 0 0

2

4

6

8

10

12

14

16

18 20 22

24

28 28 30 32 34

v" in X state

FIG. 13. Franck-Condon factors for the 5 'IT, (v = 11) in Fig. 12.

+X

' E l (v") transition shown

J. T. Bahns, I? L. Gould, and W C. Stwalley

210

L

5

Distance r FIG. 14. Schemes of C6t6 and Dalgamo (1997, 1999) for production of various levels (e.g.. u4 = 10, u;' = 0) of the a 'C: state of translationally ultracold 'Liz.

laser cooling prior to trapping. Note that rotational cooling is not necessary because the photoassociation step could selectively produce J' = 2 of the llIIgstate, the Rydberg excitation step could produce J = 1 of the v = 11 level of the 51111,state, and the spontaneous emission could produce J" = 0 and 2 (para molecules) exclusively. More recently, a somewhat different two-color scheme for producing cold (triplet) molecules (e.g., 7Li2 a3C;) has been formulated by C6t6 and Dalgarno (1997). Their method provides large improvements in terminal vibrational state selectivity. The scheme, illustrated in Fig. 14 for producing 7Li2a3C,f (v" = 0) molecules begins with photoassociation to a high vibrational level of the l3C; state: 7Li +'Li

+ hv,

-+ 'Li;(l3C;,4

= 58)

(19)

Spontaneous decay of LiI(w', = 58) is primarily bound-free, but a significant fraction ( ~ 1 1 % )is bound-bound, of which -98% is to a single level, vll 1 - 10: 7 L i ; ( 1 3 C p , = 58) -+ 7Li2(a3C:,wy = 10)

+ hv

(20)

FORMATION OF COLD (T 5 1 K) MOLECULES

21 1

A second laser frequency is chosen that excites w',' = 10 molecules to a lower vibrational level (wk = 10) of the same 13Cg' state: 7Li2(a3Cr,wy = 10) + hvl

+ 7Li;(13Ci, wi =

10)

This level spontaneously decays back to the ground state with an increased bound-bound emission fraction (-22%): 7Li; ( 13C,f,wi = 10) + 7Li2 (a3C:, w; = 0) + hv2 of which -53% is to vg = 0. Recent refinements of the C8t6 and Dalgarno scheme (1999) now include stimulated as well as spontaneous sequences for producing ground singlet (X Cg') and lowest triplet (a 'C,') 7Li2 and 6Li2 molecules. The scheme works best for making triplet molecules. The most favorable spontaneous pathway so far for singlet molecules uses the A'C; state and produces primarily X'Cg' (w; 30) molecules (16.7%), with a smaller relative percentage of decays (-3 x going to w; = 0 [again, with possible postselection of N; = 0 (or 1) molecules].

-

2. Observations Recently, ultracold (- 100 pK) molecules from photoassociated trapped atoms have been directly observed. The first observations of ultracold molecules (cesium dimers in the lowest triplet state and later in the ground singlet state), were made by Fioretti et al. (1998, 1999) using the scheme depicted in Fig. 15(a). To form translationally ultracold molecules, they used an 852-nm laser (tunable from 0 to 300 GHz to the red of the 6 2P3/2 atomic asymptote) to form excited 0; and 1, molecules at long range. C s + C s + h v l +Cs;[O;(l'C:)

-

and l,(l'rIu)]

(23)

For cesium, these states have a unique double minimum structure resulting from avoided crossings at 15 Bohr and 18Bob, respectively. FranckCondon overlaps of the outer wells at these turning points with the lowest triplet and ground singlet states is large, providing efficient spontaneous decay channels for cold molecules: N

J. I: Bahns, t? L. Could, and W C. Stwalley

212

FIG.15. The Csz long-range potentials. (a) The long-range photoassociation (1I ) to the 0; “pure long-range molecule” state, followed by spontaneous emission to the a ’C; lowest triplet state, producing translationally ultracold triplet molecules. (b) The ionization detection of these translationally ultracold triplet Csz molecules by two-photon one-color resonance-enhanced ionization (Fioretti et al., 1998).

In addition, the triplet photoassociation spectrum exhibited two “giant” resonances at -64 and -193GHz. These were interpreted to be due to spontaneous decays originating from the inner well of the 1’ X i state, coupled via tunneling with the outer well where photoassociation occurs. For detection, molecules were two-photon-ionized using (Fig. 15b) Cs2(a3C:,v” > 8 and X’C:,v’’

N

140)

+ 2hv(716nm) + Cs2(5d311,) -, Cs,f + e-

(25)

followed by ion counting with a microchannel plate. Dimer ion counts, typically < 200 s-’, were discriminated against background Cs+ using time of flight. To discriminate against counts from ionized MOT laser-excited states, MOT lasers were turned off for a period long compared to radiative lifetimes prior to the ionizing laser pulses. Spatial and temporal mapping of molecules falling from the trap region by gravity gave a translational temperature of 300 pK.Note that for these high levels (a3Cr,v” > 8 and X I C , + , v N> 140), there is significant hyperfine mixing of singlet and N

FORMATION OF COLD (T 5 1K) MOLECULES

213

triplet character (Weickenmeier et al., 1985). In addition, the single-photon resonances in the two-photon-ionization spectrum near 716 nm are not assigned, so there may be singlet ungerade states in this region as well. Moreover, the lack of assignments means the internal levels populated in the a3C; and X'C; states are not yet known. Photoassociation signals characteristically tend to weaken with decreasing R due to repulsion of low-angular-momentum collisions by the centrifugal barrier [h2J(J 1) - f12)/(8112 pR2)] (R 150ao for cesium). In studies of cooperative effects of photoassociation and trapping lasers, Fioretti et al. (1999a) found that excitations at large R (- 1500ao) by the trapping laser to an attractive excited state lead to an enhancement (Sanchez-Villicanaet al., 1996) in small-R (< 30 ao) collisions of higher angular momentum (J 6). Small-R collision enhancements were monitored with a photoassociation laser tuned to the pure long-range 0; (v' = 4) state. In studies of trap loss versus ion (Csl) signals, Comparat et al. (1999) found that traploss signals were sensitive to excited l,, O;, and 0s states, but not to the 1, state. However, ion signals were found to be sensitive only to excited 0; and 1, states. This established a definitive link between ion signals and translationally ultracold molecule formation in the lowest triplet and in the singlet ground state. Similar observations of Cs2 molecules, but with much less spectroscopic detail, have been reported by Takekoshi et al. (1998, 1999). Recent observations of cold singlet ground state K2 molecules by Nikolov et al. (1999) used the original MOT photoassociation scheme of Wang et al. (1997), shown in Fig. 11. Here, spontaneous decays following photoassociation to the v' = 191 long-range level of the A 'C,+(O,+) state populated high vibrational levels (e.g., v" = 36) of the ground X'C; state. To detect cold K2, near-resonant two-color photoionization was used. The first step (resonant excitation) was from the X to the B state at 712nm; the second step was ionization of B-state molecules by a doubled YAG pulse, as shown in Fig. 16. Time-of-flight mass spectroscopy was used to discriminate between Ki and K + background signals detected with a channeltron. Again, gating of MOT and photoionization lasers eliminated dimer ion signals due to photoionization out of excited states directly populated by MOT lasers. The rotationally resolved photoassociation spectrum shown in Fig. 17 established that molecule production (as expected) was through the Of state rather than nearby 0; or 1, states. The estimated molecule production rate was < 2000 s-' , consistent with known photoassociation rates for this MOT (< lo7 s-') and with fractional Franck-Condon factors for bound-bound A-X decays (-0.15%). Measurements of cold molecules leaving the detection region gave an estimated molecular speed of -30cm/s (-300pK). Very recently, high production rates to low and intermediate vibrational levels of

+

-

-

J. 1: Bahns, P. L. Gould, and U! C. Stwalley

214

A

532 nm (ionizes) 20

B 'n,

\

15

el2

2700 nm

10

X'cs'

5 0 -

K ( 4 9 + K(4P)

0

30

20

10

R

40

6,

FIG. 16. Diagram of the resonance-enhancedtwo-photon two-color ionization scheme used to state-selectively detect translationally ultracold 39K2X Cp' (v" = 36) molecules.

'

the ground electronic state of K2 have realized (Nikolov et al., submitted) using the two-color method discussed in Section 1V.A.1. One potential means for conversion of w" = 36 state molecules to w" = 0 is the STIRAP process (Gaubatz et al., 1990; Bergmann et al., 1998), illustrated in Fig. 18. Using a counterintuitive sequence of pulses, w'' = 36 molecules could be coherently converted with 100% efficiency to w" = 0 molecules by adiabatic passage through the intermediate A'C: state (e.g., w' = 18). Deeply bound (w'' = 0, J" = 0) ultracold Kz molecules produced this way, although small in number, would offer a multitude of possibilities for high-resolution spectroscopy and collisional studies (Weiner et al., 1999). Ultracold molecular ions can be formed directly from colliding ultracold atoms via two-color photoassociation as illustrated in Fig. 19 for sodium. The scheme, similar to the neutral molecule scheme proposed by Band and Julienne (1995, Fig. 12), utilizes photoassociation to a long-lived 1, (l'rIg) level with the first photon. The second photon is tuned to excite Rydberg state levels from the inner turning wall of the 1'IIgstate. Rydberg state levels, lying slightly above Na; (w+ = 0, N+ = 0), autoionize to produce translationally, N

FORMATION OF COLD (T 5 1 K)MOLECULES

215

Trap loss spectrum

ion spectrum I

12970.71

I

I

I

I

I

I

12971.21

I

I

12971.71

Laser frequency (cm-' )

FIG.17. Comparison of the trap loss photoassociation spectrum of ultracold 39K atoms (top) and the 39K: molecularion signal [from ionization of X ' C l (w" = 36) 39K2 as in Fig. 161 as a function of photoassociation laser frequency (LpA in Fig. 11).

rotationally, and vibrationally ultracold molecular ions. Ultracold ions can be trapped in a hybrid MOT-Paul trap, presently under development by W. Smith and coworkers at the University of Connecticut. B. PROFWSED STIMULATED PROCESSES The major limitation of photoassociative methods employing spontaneous decay is the lack of state selectivity. Single or multiply resonant photoassociation can convert free atom pairs into single rovibrational electronically excited-state molecules. Unfortunately,these excited states invariably decay to a myriad of lower bound and continuum states. Furthermore, single-photon photoassociation followed by spontaneous emission populates mainly highly excited bound states in accordance with Franck-Condon overlap. In response to these shortcomings, several approaches have been proposed, all relying on some form of coherent photonprocesses interacting with an initially trapped atomic gas or Bose-Einstein condensate (BEC). Vardi et al. (1997) have developed a theory for stimulated Raman radiative recombination from initial continuum states that is a variant of the STIRAP process (Bergmann et al., 1998). In the STIRAP process, bound state populations can be transferred from one level of a three-level system, through

216

J. T. Bahns, P. L . Gould, and W C. Stwalley

f

A

’c;

v=l8

STIRAP X’X;

G 0

K(4s) + K(4s)

s 10

20

30

40

R FIG. 18. The combination of our demonstrated method of forming X’C; (71” = 36) translationally ultracold 39Kz molecules (Nikolov er al., 1999) with the well-documented stimulated Raman adiabatic passage (STIRAP) technique (see Bergmann et al., 1998 and references therein) could produce small numbers of translationally ultracold X’C: (d’ = 0) 3 9 ~ molecules. 2

an intermediate state, to a third bound level with 100% efficiency using a counterintuitive sequence of optical pulses. Spontaneous decays from the intermediate state can be completely eliminated (the intermediate state becomes a “dark” state). Using two strong ( 105-108W/cm2) counterintuitive 100 Hz optical pulses ( ~ 2 0 n s )applied to ultracold sodium vapor, Vardi et al. (1997) predicted conversion fractions of -6 x 10-6/pulse leading to 50% of the sample being converted in a time on the order of minutes. Such a conversion would be completely state-selective. Javanainen and Mackie (1998, 1999) have proposed a phenomenological Hamiltonian for the analysis of photoassociation in a BEC. Their analysis focuses on two examples of coherent photoassociation: nonresonant oscillations of atomic and molecular populations in condensates and conversions of condensate populations from atomic to molecular, driven by an adiabatic frequency-sweptlaser. Computed temporal molecular fractions for both cases predict fractional conversions approaching 100% on short (Rabi-like) time scales.

FORMATION OF COLD (T 5 1 K) MOLECULES I

I

Naz

I

I

I

-

K L A

30-

v

I

Na(3s) + Na(3d)

-

Na(3s) + Na(4s)

L2

‘E

I

X*,i

-

h

I

nl1z;+

40-

-

I

217

3 1,’s

-

Na(3s) + Na(3p)-

mo201 LLI F X

-

A’,

&

LPA

--

-

10-

Na(3s) + Na(3s)

FIG. 19. A photoassociative ionization scheme for production of state-selected (e.g., N+ = 0) translationally ultracold N a i ions from ultracold Na atoms. This is the photoassociative analog of the molecular scheme shown in Fig. 10.

v+

= 0,

Julienne er al. (1998) have performed coupled-channels calculations for a quasi-cw variant of the Raman photoassociative process of Vardi et al. (1997) (see also Mackie and Javanainen, submitted). The model for this process, depicted in Fig. 20, begins with two colliding atoms of energy E on the ground state potential V, (R). Next, atoms are photo-associated with optical coupling matrix element (01) to an excited intermediate state level (q) on potential V1 with laser frequency o I . Level 2rl can decay either by spontaneous decay (yl) or by stimulated emission ( 0 2 , 0 2 ) to a specific arbitrary level of the ground state (212). Once in level w2, molecules are no longer bound by the atomic trap. Cold molecules are coherently “emitted” from the condensate on the artificial molecular channel Vm0,(R),due to atomic center-of-mass motion (minus the

218

J. T. Bahns, P. L. Gould, and W C. Stwalley

Ql

4 LOSS

-E

G r o u n d State VdR)

8 2

/

Artificial Molecular C h a d

Vm, (R)

FIG. 20. Diagram of the scheme of Julienne er al. (1998) for production of state-selected diatomic molecules from an atomic BEC.

difference in photon momenta). Calculations of molecule production rates for 1 ps pulses as a function of collisional energy (Elk < 400nK) and detuning predicted large photoassociation rate coefficients of k,, > lo-’’ cm3s-l, capable of exceeding spontaneous losses by 1 to 2 orders of magnitude with large fractional transfers to molecules on a time scale short compared to the oscillation period of the atomic trap, particularly for low (- 1 nK) temperature collisions. If such coherent transfer processes are realized, there are exciting prospects for the future. Notably, coherent photoassociation could form the basis for the “molecule laser” proposed by Julienne et al. (1998) (see also Javanainen and Mackie, 1998, 1999),analogous to the recently demonstrated “atom laser” of Mewes et al. (1997). This would offer an additional armada of possibilities for high-resolution scattering, spectroscopy, and other studies using coherent matter waves. Very recently, Wynar et al. (submitted) have observed a stimulated Raman resonance between colliding pairs of 87Rb atoms and a molecular state of N

FORMATION OF COLD (T 5 1 K) MOLECULES

219

”Rb2 near dissociation (bound by -0.02 cm-I) in a Bose-Einsteincondensed condensate). A sharp resonance (< 1 MHz in width) is observed cloud ( ~ 2 0 % via the = 1, J = 0 level of the 0; pure long-range electronic state below the Rb (52S1/2)+ Rb (5’P3p) asymptote. The DUMP (bound-bound) laser frequency is generated from the PUMP (free-bound) laser using an acoustooptic modulator. Although the translationally ultracold molecules produced have not yet been directly detected, these experiments represent the first demonstration of the stimulated Raman photoassociation of ultracold atoms discussed above.

V. Conclusions and Future Directions The formation of molecules with subKelvin translational temperatures is now possible by a variety of nonoptical and optical techniques, both in cold (1 mK-1 K) and in ultracold (< 1 mK) regimes. It is likely that translationally ultracold molecules will soon be detected at submicroKelvin temperatures, for example, from formation in atom BECs. Greatly improved production rates are also expected. Prospects are also good for producing these translationally cold molecules state-selectively by a variety of schemes such as optical pumping, stimulated Raman photoassociation, and multiphoton ionization. Evaporative cooling of molecules could become an important method for producing ultracold molecules. Application of such techniques to forms of high-resolution spectroscopy in addition to photoassociative spectroscopy are expected. The trapping of such molecules has been achieved, and other cold molecule manipulations (“molecule optics”) will undoubtedly be demonstrated soon. Cold collisions and their applications in turn to low-temperature gas metastability (and its destruction by nucleation), to molecular BEC (and “molecule lasers”), and to degenerate molecular Fermi gases will surely be topics of high interest and excitement as well.

VI. Acknowledgments The authors particularly acknowledge their experimental and theoretical collaborators on formation of ultracold molecules: Yehuda Band, Keith Burnett, Ed Eyler, Paul Julienne, Jing Li, Anguel Nikolov, Marin Pichler, He Wang, Xiaotian Wang, Carl Williams, and Guoxing Zhao. The authors also acknowledge helpful discussions with a great many individuals, including Claude Amiot, Mike Andrews, Vanderlei Bagnato, Klaas Bergmann, Robin CBtC, Alex Dalgarno, John Doyle, Oliver Dulieu, Wolfgang Ernst, Bretislav

220

J. Z Bahns, P. L. Gould, and W C. Stwalley

Friedrich, Bob Gordon, Chris Greene, Dan Heinzen, Randy Hulet, Juha Javanainen, Gwang-Hi Jeung, Paul Lett, Li Li, Marjatta Lyyra, Matt Mackie, Mircea Marinescu, Francoise Masnou-Seeuws, Fred Mies, Jan Northby, Bill Phillips, Goran Pichler, Pierre Pillet, Amanda Ross, Giacinto Scoles, T m a r Seideman, Moshe Shapiro, Win Smith, Peter Toennies, Boudywyn Verhaar, John Weiner, Tsutomu Yabuzaki, Alexandra Yiannopoulou,and Warren Zemke. This work was supported in part by NSF CHE96-12207 and CHE 97-32467.

VII. References Adams, C. S., Sigel, M., and Mlynek, J. (1994). Phys. Rep. 240, 143. Adams, C. S., and Riis, E. (1997). Prog. Quant. Electc 21, 1. Anderson, J. B., Andres. R. P., and Fenn, J. B. (1966). Advan. Chem. Phys. 10, 275. Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E., and Cornell, E. A. (1995). Science 269, 198. Amdt, M., Ben Dahan, M., Guery-Odelin, D., Reynolds, M. W., and Dalibard, J. (1997). Phys. Rev. Lett. 79, 625. Bahns, J. T., Stwalley, W. C., and Gould, P. L. (1996). J. Chem. Phys. 104, 9689. Balakrishnan, N., Kharachenko, V., Forrey, R. C., and Dalgarno, A. (1997). Chem. Phys. Lett. 280, 5 . Balakrishnan, N., Forrey, R. C., and Dalgarno, A. (1997A). Chem. Phys. Let?. 280, 1. Balakrishnan, N., Kharachenko, V., Forrey, R. C., and Dalgarno, A. (1998). Phys. Rev. Lett. 80, 3224. Band, Y. B., and Julienne, P. S. (1995). Phys. Rev. A 51, R4317. Barranco, M., Navarro, J., and Poves, A. (1997). Phys. Rev. Lett. 78,4729. Bartana, A., Kosloff, R., and Tannor, D. J. (1993). J. Chem. Phys. 99, 196. Bass, A. M., and Broida, H. P. (Eds.). (1960). Formation and rrapping offree radicals. Academic Press (New York). Becker, E. D., and Pimentel, G. C. (1956). J. Chem. Phys. 25,224. Becker, E. W., and Henkes, W. (1956). Z. Physik 146, 320. Berman, P. R., and Stenholm, S. (1978). Opt. Commun. 24, 155. Bergmann, K., Theuer, H., and Shore, B. W. (1998). Rev. Mod. Phys. 70, 1003. Boesten, H. M. J. M., Tsai, C. C., Verhaar, B. J., and Heinzen, D. J. (1 996). P hys. Rev. Lett. 77,5 194. Bohn, J. L., Burke, J. P., Greene, C. H., Wang, H., Gould, P. L., and Stwalley, W. C. (1999). Phys. Rev. A 59,3660. Bradley, C. C., Sackett, C. A., and Hulet, R. G. (1997). Phys. Rev. Lett. 78, 985. Broyer, M., Chevaleyre, J., Delacretaz, G., Martin, S., and Woste, L. (1983). Chem. Phys. Lett. 99,206. Buchenau, H., Knoth, E. L., Northby, J., Toennies, J. P., and Winkler, C. (1990). J. Chem. Phys. 92, 6876. Burt, E. A., Ghrist, R. W., Myatt, C. J., Holland, M. J., Cornell, E. A., and Wieman, C. E. (1997). Phys. Rev. Lett. 79, 337. Callegari, C., Higgins, J., Stienkemeier, F., and Scoles, G. (1998). J. Phys. Chem. 102, 95. Chang, E. S., Li, J., Zhang, J., Tsai, C. C., Bahns, J., and Stwalley, W. C. Submitted to J. Chem. Phys. Chapman, M. S . , Ekstrom, C. R., Hammond, T. D., Rubenstein, R. A., Schmiedmayer, J., Wehinger, S . , and Pritchard, D. E. (1995). Phys. Rev. Left. 74,4783.

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ADVANCES JN ATOMIC, MOLECULAR. AND OPTICAL PHYSICS. VOL. 42

HIGH-INTENSITY LASER-ATOM PHYSICS C. J. JOACHAIN Physique The'orique, Universite' Libre de Bruxelles, CP 227, Boulevard du Triomphe, B-I 050 Bruxelles, Belgium Dipartement de Physique, Universite' de Louvain, 8-1348 Louvain-la-Neuve, Belgium Email: [email protected].

M. DORR Physique Thiorique. Universite' Libre de Bruxelles, CP 227, Boulevard du Triomphe, B-1050 Bruxelles, Belgium Max-Born-Institut, D-I 2489, Berlin, Germany Email: [email protected].

N . KYLSTRA Physique Thiorique, Universite' Libre de Bruxelles, CP 227, Boulevard du Triomphe, B- I050 Bncxelles, Belgium Optics Section, Blackett hboratory, Imperial College, London SW7 2BZ United Kingdom Email: [email protected] I. Introduction. . .

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toms and Ions. . . . . . . . . . . . . . . . . . . A. Multiphoton Ionization and Above-Threshold Ionization. . . . . . . . B. Harmonic Generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Laser-Assisted Electron-Atom Collisions . . . . . . . . . . . . . . . . . 111. Theoretical Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Basic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Perturbation Theory and Its Breakdown . . . . . . . . . . . . . . . . . . C. Semiperturbative Methods .......................... 1 . Essential States. . . . . . . . . . . . . . . , . . . . . , . . , . , . . . . . . 2. Laser-Assisted Electron-Atom Collisions. . . . . . . . . . . . . . . . . D. Floquet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Basic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Sturmian-Floquet Method. . . . . . . . . . . . . . . . . . . . . . . . 3. High-Frequency Floquet Theory . . . . . . . . . . . . . . . . . . . . . . 4. Floquet Theory for Laser-Assisted Electron-Atom Collisions . . E. R-Matrix-Floquet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . F. Low-Frequency Methods . . . . . , . . . . . . . . . . . . . . . . . . . . . . . G. Numerical Solution of the Time-Dependent Schrodinger Equation. 1 . Single Active Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Several Electrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Relativistic Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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IV.Conclusions and Future Developments . . . . . . . . . . . . . . . . . . . . . .

V. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract: The development of lasers capable of delivering short pulses of very intense radiation, over a wide frequency range, has led to the discovery of new, nonperturbative multiphoton processes in laser interactions with atomic systems. In this article, we first give a survey of the main properties of multiphoton processes such as the multiphoton ionization of atoms, the emission by atoms of high-order harmonics of the exciting laser light, and laser-assisted electron-atom collisions. We then review the theory of these processes, giving particular attention to ab-initio nonperturbative methods such as the Sturmian-Floquet approach, the R-matrix-Floquet theory, and the numerical integration of the time-dependent Schr6dinger equation. We also discuss relativistic effects that occur at ultra-high intensities. We conclude by consideringpossible future developments of high-intensity laser-atom physics.

I. Introduction In recent years, intense laser fields have become available over a frequency range extending from the infrared to the ultraviolet, in the form of short pulses yielding intensities of the order of or exceeding the value I, = 3.5 x 10l6W cm-2, corresponding to the atomic unit of electric field strength &, = 5.1 x 109Vcm-’. Such laser fields have been obtained using the “Chirped Pulse Amplification” (CPA) scheme, in which laser pulses are stretched, amplified, and then compressed (Strickland and Mourou, 1985). They are strong enough to compete with the Coulomb forces in controlling the electron dynamics in atomic systems. As a result, atoms and molecules in intense laser fields exhibit new properties that have been discovered via the study of multiphoton processes. These modified properties generate new behavior of bulk matter in intense laser fields, with wide-ranging potential applications such as the study of ultra-fast phenomena, the development of powerful high-frequency ( X W and X-ray) lasers, the investigation of the properties of plasmas and condensed matter under extreme conditions of temperature and pressure, and intense field control of atomic and molecular reactions. Over the last ten years, laser intensities have increased by more than four orders of magnitude (Mourou et al., 1998), up to lo2’ W cmP2, where relativistic effects in laser-atom interactions become important. In this article we shall review the field of high-intensity laser interactions with atoms and ions. Section I1 is devoted to a survey of the new phenomena discovered by studying atomic multiphoton processes in strong laser fields. The theory of these processes is discussed in Section III, where the main nonperturbativemethods currently used are examined. Possible future developments of this rapidly growing area of physics are considered in Section IV.

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We refer the reader to the book edited by Gavrila (1992a) and the articles by Burnett et al. (1993), Joachain (1994), DiMauro and Agostini (1995), and Protopapas et al. (1997a) for previous reviews of the subject.

11. Multiphoton Processes in Atoms and Ions In this section, we shall discuss three important multiphoton processes occurring in atoms and ions: multiphoton ionization,harmonic generation, and laser-assisted electron-atom collisions. A. MULTIPHOTON IONIZATION AND ABOVE-THRESHOLD IONIZATION We begin by considering the multiphoton (single) ionization (MPI) reaction:

where q is the charge of the target atomic system A, expressed in atomic units (a.u.), hm is the photon energy, and n is an integer. This process was first observed by Voronov and Delone (1969, who used a ruby laser to induce seven-photon ionization of xenon, and by Hall et al. (1965), who recorded two-photon ionization from the negative ion I-. In the following years, important results were obtained by several experimental groups, in particular at Saclay where the dependence of the ionization yields on the intensity and the resonance-enhancement of MPI were studied. A crucial step was made in our understanding of MPI when experiments detecting the energy-resolved photo-electrons were performed. In this way Agostini ef al. (1979) discovered that the ejected electron in the reaction (1) could absorb photons in excess of the minimum required for ionization to occur. The study of this excess-photon ionization, known as “above-threshold ionization” (ATI), has been one of the central themes of multiphoton physics in recent years. A typical example of AT1 photo-electron energy spectra, obtained by Petite et al. (1988), is shown in Fig. 1. The spectra are seen to consist of several peaks, separated by the photon energy hm. As the intensity I increases [see Fig. 1(b)], peaks at higher energies appear, whose intensity dependence does not follow the lowest-order perturbation theory (LOPT) prediction according to which the ionization rate for an n-photon process is proportional to I”. Another remarkable feature of the AT1 spectra, also apparent in Fig. 1, is that as the intensity increases, the low-energy peaks are reduced in magnitude. The reason for this peak suppression is that the energies of the atomic states are Stark-shifted in the presence of a laser field. For low laser frequencies

228

C. J. Joachain, M. Dorr, and N. Kylstra

-I

2

L

6

8

I0

12

I4

Electron energy (eV) FIG.1. Electron energy spectra showing AT1 of xenon at a wavelength h = 1064 nm. (a) I = 2 x 10” W cm-’. (b) I = lOI3 W cm-’. From Petite et al. (1988).

(e.g., a Nd-YAG laser with hw = 1.17eV), the AC Stark shifts of the lowest bound states are small in magnitude. On the other hand, the induced Stark shifts of the Rydberg and continuum states are essentially given by the electron ponderomotive energy Up,which is the cycle-averaged kinetic energy of a quivering electron in a laser field and is given (for non-relativistic velocities) by

where me is the absolute value of its charge, and &O is the electric field strength. It is worth stressing that the ponderomotive energy Up is proportional to I / 0 2 and may become quite large. For example, in the case of the Nd-YAG laser, U p = ti0 = 1.17 eV at the intensity I N 1013W cm-2. Because the energies of the Rydberg and continuum states are shifted upwards

229

HIGH-INTENSITY LASER-ATOM PHYSICS 0.5

x

P

Q)

c -0.5

I, 0.0

13

12 1

1

0.2

1

1

1

1

0.4

1

1

1

1

0.6

1

1

1

1

0.8

1

1

l

1.0

Intensity (arb. u.) FIG.2. Illustration of the mechanism responsible for the suppression of low-energy peaks in AT1 spectra. For low laser frequencies, the intensity-dependent ionization potential of the atom, l,,(l),is such that I,(I) N I, + Up and hence increases linearly with the intensity I (thick curve). Ionization by n photons, which is possible at the intensity I1 for which nfiw 2 I, Up,is prohibited at the higher intensities 12 and 13, where n + 1 photons are needed to ionize the atom. Also illustrated is the mechanism responsible for the resonantly induced structures appearing in AT1 spectra for short laser pulses. At the intensity 12, a Rydberg state (thin line) has shifted into multiphoton resonance with the ground state.

+

relative to the lower bound states by about Up, there is a corresponding increase in the intensity-dependent ionization potential Z p ( Z ) of the atom, so that I , ( ] ) 1: Zp + Up, where Zp = -Ei denotes the ionization potential of the field-free initial state of energy Ei. If this increase is such that nho < Zp Up, then ionization by n photons is energetically forbidden (see Fig. 2). However, atoms interacting with smoothly varying pulses experience a range of intensities, so that the corresponding peak in the photo-electron spectrum will not completely disappear, as seen in Fig. l(b). For short (sub-picosecond) pulses, the AT1 peaks exhibit a substructure (Freeman et al., 1987) because the intensity-dependent Stark shifts bring different states of the atom into multiphoton resonance during the laser pulse (see Fig. 2). This fine structure is not seen in long-pulse experiments because each electron regains its ponderomotive energy deficit from the field as it escapes the laser pulse adiabatically. Highly resolved spectra have been obtained recently by Hansch et al. (1997, 1998). A direct, simultaneous measurement of both the energy and the angular distribution of the photo-electrons has been performed by the group of Helm (Helm and Dyer, 1994; Schyja et al., 1998). In Fig. 3, we show the result

+

230

C. J. Joachain, M. Dorc and N. Kylstra

FIG.3. Density plot of the spatial distribution of the electrons ejected from a Xe atom irradiated by a laser pulse with h = 800nm and I = I .6 x lOI3 W crn-’. The laser polarization axis is along the vertical, and the radial distance from the center is proportional to the photoelectrons’ velocity. The innermost ring corresponds to the first AT1 peak, for which the dominant orbital angular momentum is 1 = 4. Courtesy of Schyja and Helm.

of the deconvoluted camera image of the spatial distribution of the photoelectrons. The rings correspond to the AT1 peaks, and the angular structure in the rings gives directly the dominant orbital angular momentum of the electrons emitted within a particular AT1 peak. For the present case, the dominant orbital angular momentum quantum number of the continuum electron in the lowest AT1 peak is 1 = 4,which indicates that the dominant resonance is an f-state. For increasing laser field strengths approaching the Coulomb field binding the electron ( I > loL4W cm-2), and for low laser frequencies, the sharp AT1 peaks of the photo-electron spectrum gradually blur into a continuous distribution (Augst et al., 1989; Mevel et al., 1993). In this regime, ionization can be interpreted by using a quasi-static model in which the bound electrons experience an effective potential formed by adding to the atomic potential the contribution due to the instantaneous laser electric field (see Fig. 4). This quasi-static approach was used by Keldysh (1965) to study tunneling ionization in the low-frequency limit and was pursued by several authors (Faisal,

231

HIGH-INTENSITY LASER-ATOM PHYSICS

\

\

X \

A, \

FIG.4. Schematic diagram showing (a) tunneling ionization and (b) over-the-barrier ionization. The dashed line corresponds to the contribution to the potential energy due to the instantaneouslaser electric field. The solid line correspondsto the full effective potential energy.

1973; Reiss, 1980; Ammosov et al., 1986). An important quantity in these studies is the Keldysh adiabaticity parameter y, defined as the ratio of the laser and tunneling frequencies, which is given by y=

JX

(3)

where ,Z is the field-free atomic ionization potential. For small y, tunneling dynamics will dominate, the transition from multiphoton to tunneling ionization taking place in the region around y = 1. Above a critical intensity Z, (which is equal to 1.4 x 1014W cm-2 for atomic hydrogen in the ground state), the electron can classically “flow over the top” of the barrier [over-thebarrier (OTB) ionization], so that field ionization occurs and the atom ionizes in about one orbital period (Shakeshaft et al., 1990). The semiclassical, “recollision picture” developed recently (Corkum, 1993; Kulander et al., 1993; Lewenstein et al., 1994) is based on the idea that strong field ionization dynamics at low frequency proceeds via several steps. In the first (bound-free) step, an electron is liberated from its parent atom by tunneling or OTB ionization. In the second (free-free) step, the interaction with the laser field dominates, a fact that was noted earlier by using a simple classical picture of a quivering electron (Kuchiev, 1987; van Linden van den Heuvell and Muller, 1988). As the phase of the field reverses, the electron is accelerated back towards the atomic core. If the electron returns to the core, a third step takes place in which scattering of the electron by the core then leads to single or multiple ionization, while radiative recombination leads to harmonic generation. This semiclassical three-step model has been very useful for explaining, in terms of classical trajectories and return energies, a number of features found

232

C. J. Joachain, M . D6rc and N. Kylstra 106 105

c

m

10‘ 103

v

v)

4

102

c

101

g

100

3

10-1

10-2 10-3

0

10

20

30

40

50

60

70

80

90

Electron Energy (eV) FIG.5. Photo-electron counts as a function of photo-electron energy, for various noble gases, at a laser wavelength of k = 630nm and an intensity I N 2 x lOI4 W cm-* (3 x lOI4 for He). From Paulus et al. (1994b).

in recent experiments. In these experiments, the use of kilohertz-repetitionrate, high-intensity lasers has allowed a precise measurement of photoelectron total yields and energy- and angle-differential spectra over many orders of magnitude in yield. These experimental results have revealed the existence of a “plateau” in the AT1 photo-electron energy spectra (Paulus et al., 1994b; Hansch et al., 1997). An example of such spectra is shown in Fig. 5. For the photo-electrons emitted at certain energies, sharp peaks can appear in the angular distributions (Yang et al., 1993; Paulus et al., 1994a). These angular peaks are sometimes called “rescattering rings,’’ because the distributions are symmetric around the axis of polarization of a linearly polarized laser field. The prominent groups of AT1 peaks that stand out within the plateau have also been studied in atomic hydrogen by Paulus et al. (1996) and in argon by Hertlein et al. (1997) and Muller and Kooiman (1998). Measurements of AT1 electron spectra in an elliptically polarized field have been carried out by Paulus et al. (1 998). The observed ellipticity dependence of emission rates in individual AT1 peaks was interpreted in terms of interference of electron tunneling at different times during an optical cycle. B. HARMONIC GENERATION Atoms interacting with a strong laser field can emit radiation at higher-order multiples, or harmonics, of the angular frequency o of the “pump” laser. For

233

HIGH-INTENSITY LASER-ATOM PHYSICS 1 O'O

10'

Xe 10 Torr Ar 15 Torr 0 No 40 Torr 0 Ha 70 Torr

t I\

1

L

2

f

2

10'

10' '01

I

25

50

75 100 Hormonic Order

125

FIG.6. Harmonic emission spectra of various noble gases at a wavelength = 1053 nm and an intensity I N 1.5 x lOI5 Wcm-'. From L'Huillier and Balcou (1993).

an initial state of a given parity, the harmonic frequencies appear at odd multiples of the laser frequency,i.e., oq= qo with q = 3,5,. . . .The observation of the third harmonic in noble gases was made by New and Ward (1967). The availability of intense lasers has made it possible to observe high-order harmonics (McPherson et al., 1987; Wildenauer, 1987; Ferray et al., 1988; Macklin et al., 1993; L'Huillier and Balcou, 1993; Miyazaki and Takada, 1995; Nagata et al., 1996; Preston et al., 1996). As an example, we show in Fig. 6 the emission spectra of various noble gases, obtained by L'Huillier and Balcou (1993), who observed the harmonic q = 133 in neon at an intensity I N 1.5 x 1015Wcm-*. In the recent experiments with ultrashort laser pulses at very high intensities (Zhou et al., 1996; Christov et al., 1996; Spielmann et al., 1997; Chang et al., 1997; Schniirer et al., 1998; Rundquist et al., 1998), the atoms experience only a few laser cycles. The highest harmonic frequencies and harmonic orders (q N 297) have been observed under these condition!, reaching into the water window, corresponding to wavelengths around 2.7 A. Other experimental developments have been the spatially resolved measurements of the time dependence (Tisch et al., 1998) and the direct measurement of the temporal coherence (Bellini et al., 1998) of high-order harmonics. The theoretical treatment of harmonic generation by an intense laser pulse focused into a gaseous medium has two main aspects. First, the microscopic, single-atom response to the laser field must be analyzed. The single-atom spectra must then be combined to obtain the macroscopic harmonic fields

234

C. J. Joachain, M. Dorc and N. Kylstra

generated from the coherent emission of all the atoms in the laser focus; this is done by using the single-atom polarization fields as source terms in the Maxwell equations. We shall only discuss here the microscopic aspect of the problem. The power spectrum of the emitted radiation is proportional to the modulus squared of the Fourier transform of the dipole acceleration, a(t), (Burnett et al., 1992; Krause et al., 1992a) the latter being given by

d2 a(t) = -d(t) dt2

(4)

where d ( t ) is the laser-induced atomic dipole moment

d(t) =

(W- eRl*'(t))

(5)

Here 1 9 (t)) denotes the atomic state vector in the presence of the laser field and N

R=Eri i= 1

is the sum of the coordinates ri of the N atomic electrons. When the atom is driven by a monochromatic field, the emitted harmonic radiation can be simply calculated from the induced dipole moment, which can now be expanded as

d(t) =

E [d(qo)e-'qO' + c.c.]

(7)

4

where C.C. denotes the complex conjugate, and d ( - q o ) = [d(qo)]*because d(t) is real. The rate of emission of photons of frequency qo is then proportional to Id(qo)12. At high laser intensities, the harmonic intensity distribution exhibits a rapid decrease over the first few harmonics, followed by a plateau of approximately constant intensity and then a cut-off, corresponding to an abrupt decrease of harmonic intensity. It is important to note that the existence of a plateau is a nonperturbative feature. Perturbation theory is applicable only in the weak-field regime, where the harmonic intensity decreases significantly from one order to the next (Potvliege and Shakeshaft, 1989). It was discovered in the framework of time-dependent Schrodinger equation (TDSE) calculations (Krause et al., 1992b) that the cut-off angular frequency a, of the harmonic spectrum is given approximately by the relation

ho, N I,

+ 3u,

(8)

HIGH-INTENSITY LASER-ATOM PHYSICS

235

In the three-step “recollision model,” the maximum returning kinetic energy of a classical electron recolliding with the atomic core is given by 3.2 Up, so that the highest energy that can be radiated is I, 3.2 U p ,in good agreement with the TDSE calculations and with experiment. An exciting new development is the possibility of using high-order harmonics to generate pulses of extremely short duration, in the range of hundreds of attoseconds (1 as = lo-’’ s). There currently exist several proposed methods of attosecond pulse generation. The first one (Schafer and Kulander, 1997) 0 which involves the use of a very short fundamental laser pulse ( ~ 2 fs) should make it possible to generate single harmonic pulses of subfemtosecond duration. The second one (Corkum et al., 1994) is based on the high sensitivity of harmonic generation to the degree of ellipticity of the fundamental laser pulse. Indeed, invoking the three-step model, it is easy to show that harmonics are essentially produced when the polarization of the laser field is linear, because otherwise the electrons would not return near the parent ion. By creating a laser pulse whose polarization is linear only during a short time (close to a laser period), the harmonic emission can be limited to this interval, so that single sub-femtosecond pulses could be produced. Recent theoretical work (Antoine et al., 1996a, 1996b; Salier2s et al., 1999) also predicts that high-order harmonics generated by an atom in a linearly polarized intense laser pulse form a train of ultrashort pulses, corresponding to different trajectories of electrons that tunnel out of the atom and recombine with the parent ion (see Fig. 7). Under appropriate geometrical conditions, the

+

I

I

0.2

0

0

0.2

0.4 0.6 Time (optical prriod)

0.8

1 .o

FIG.7. Time profile of the filtered harmonic signal, including the 41st to 61st harmonics (solid line), emitted by a macroscopic medium, generated by a laser pulse of intensity 6.6 x lot4W cm-’. The dashed line is the single-atom response. From Antoine et al. (1996a).

236

C. J. Joachain, M. Dorr; and N. Kylstra

macroscopic propagation in an atomic jet allows the selection of one of these trajectories, leading to a train of attosecond pulses, with one pulse per half-cycle. C. LASER-ASSISTED ELECTRON-ATOM COLLISIONS

An electron scattered by an atom (ion) in the presence of a laser field can absorb or emit radiation. Because these radiative collisions involve continuum states of the electron-atom (ion) system, they are often called “free-free transitions” (FFT).In weak fields, only one-photon processes have a large enough probability to be observed. However, as the field strength is increased, multiphoton processes become important. Examples of laser-assisted electron-atom collisions are “elastic” collisions: e- + A ( i ) + n f i o + e- + A ( i )

(9)

inelastic collisions: e-

+ A ( i ) + nhw

-+

e-

+A ( f )

(10)

and single ionization (e, 2e) collisions: e- + A ( i )

+ nfiw + A + ( f ) + 2e-

(11)

where A(i) andA(f) denote an atom A in the initial state i and the final statef, respectively, and A+(f) means the ion A+ in the final statef. Positive values of n correspond to photon absorption (inverse bremsstrahlung), negative values of n to photon emission (stimulated bremsstrahlung), and n = 0 to a collision process without net absorption or emission of photons, but in the presence of the laser field. A review of laser-assisted electron-atom collisions has been given by Ehlotzky er al. (1998). Direct information on laser-assisted electron-atom collisions is obtained by performing three-beam experiments, in which an atomic beam is crossed in coincidence by a laser beam and an electron beam, and the scattered electrons, having undergone m,are recorded. Several experiments of this kind have been done, in which the exchange of photons between the electron-atom system and the laser field has been observed in laser-assisted elastic (Weingartshofer et al., 1977; Weingartshofer er al., 1983; Wallbank and Holmes, 1994) and inelastic (Mason and Newell, 1987; Wallbank er al., 1988; Wallbank eral., 1990;Luan er al., 1991)processes. As an illustration, we show in Fig. 8 the results of Weingartshofer er al. (1983) for laser-assisted “elastic”

HIGH-INTENSITY LASER-ATOM PHYSICS

'1

237

€,= 15.800 eV

e=

ISSO

Energy hw 1 FIG.8. Energy spectrum of electrons scattered by argon atoms in the presence of a C02 laser of photon energy fio = 0.117 eV. The open circles correspond to the experimentaldata; the full line is drawn to guide the eye. The abscissa gives the final electron energy in units of the photon energy with the origin fixed at the initial electron energy Ei = 15.8eV. The scattering angle 8 = 155". The laser intensity I = lo*Wcm-2. From Weingartshofer et al. (1983).

electron-argon scattering. We remark that even at the modest intensity of lo8W cm-*, as many as eleven photon emission and absorption transitions were observed. As seen in Fig. 8, the relative intensities of two successive FFT peaks are of the same order of magnitude, which indicates that perturbation theory cannot be used to analyze these results. Thus, as in the case of the other multiphoton processes discussed above, nonperturbative methods must be developed.

111. Theoretical Methods In this section, following a brief summary of the basic equations of the theory, we shall give a survey of the main methods which have been used to study laser-atom interactions at high intensities. The most appropriate theoretical framework for studying the interaction of an atom with a laser field depends on the characteristics of the laser pulse and of course on the atom or ion being studied. For ow purposes, the relevant laser

238

C. J. Joachain, M. Dorr; and N. Uylstra

parameters are the frequency, maximum intensity, and duration of the laser pulse. With respect to the laser pulse duration, we can, in a nonrigorous way, distinguish between three regimes: the long-pulse, short-pulse, and ultrashortpulse regime. In the long-pulse regime, which is typically characterized by s; we note that the atomic unit of time is 2.4 x s) or nanosecond longer laser pulses, multiphoton processes in the atom can be characterized by rates that depend only on the angular frequency and maximum intensity of the laser. The exact details concerning the turn-on and turn-off of the pulse are not relevant, except for the requirement that the atom evolves adiabatically in the laser field. In this regime, the rates can be calculated using time-dependent perturbation theory when the field is weak, whereas for strong or resonant fields the Floquet approach, which will be discussed below, can be utilized. In the short-pulse regime (pulse durations of the order of picoseconds, or s), the evolution of the atom in the laser field can still be assumed to be adiabatic, apart from isolated resonances between quasi-bound states. In this regime, the shape of the laser pulse is important, so if a comparison is to be made with experiment, the relevant quantities to be calculated are the transition probabilities determined from the instantaneous intensity-dependent rates of the atom in the laser field (Potvliege and Shakeshaft, 1992). When comparing with experiment, the spatial profile of the laser in the interaction region will also have to be considered (Potvliege and Shakeshaft, 1992; Rottke et al., 1994). Once the rates have been calculated for the range of intensities of the pulse, the final probabilities can be determined for a given laser pulse shape. Finally, in the ultra-short-pulse regime (of the order of femtoseconds, or s), the evolution of the atom cannot be considered to be adiabatic, so the only recourse is to obtain information about the multiphoton processes from a direct integration of the time-dependent Schrodinger equation, as discussed in Section 1II.G. Obviously, the boundaries which delimit the three regimes cannot be precisely defined. With respect to the frequency of the laser, we can again distinguish between three regimes. In the low-frequency regime the ionization processes can be viewed in terms of a quasistatic effective potential given by the instantaneous electric field of the laser and the Coulomb potential. In the high-frequency regime the electron orbital frequency is less than the driving laser field frequency and thus an adiabatic approach eliminating the fast laser oscillation proves fruitful. Finally, in the intermediate regime, the photon energy is comparable to relevant atomic transition energies and resonant multiphoton processes will typically play an important role. Regarding the laser intensity, we can subdivide the low-frequency regime into a multiphoton ionization regime at the low-intensity end, an intermediate tunneling ionization regime, and the high-intensity over-the-barrier ionization regime. For intermediate and high frequencies, the laser-atom interaction can

HIGH-INTENSITYLASER-ATOM PHYSICS

239

be described perturbatively when the laser intensity is weak, whereas semiperturbative methods are useful at moderate intensities and close to resonances, when a relatively small number of atomic states are deemed important in describing the laser-atom interaction. Finally, completely nonperturbative methods are required at high intensities. The frequency and intensity regimes described above are to a degree interdependent. Moreover, although such a classification can usually be made for any atom, the intensities and frequencies that characterize the different regimes will depend strongly on the particular atom and on its initial state; e.g., a frequency that is low for the ground state can be high for an excited state. A. BASICEQUATIONS

In order to study the interaction of an atomic system with a laser field, we shall use a semiclassical approach in which the laser field is treated classically, while the atomic system is studied by using quantum mechanics. This approach is entirely justified for the intense fields considered here (Faisal, 1986; Mittleman, 1993; Joachain, 1994). We neglect for the moment relativistic effects and treat the laser field in the dipole approximation as a spatially homogeneous electric field &(t),the corresponding vector potential being A(t), with &(t)= -dA(t)/dt. For example, if the field is linearly polarized, we have

where i is the unit polarization vector, &O is the electric field strength, F ( t ) is the pulse shape function, and 4 is a phase. We note that for a chirped pulse, either o or is time-dependent. Our starting point is the time-dependent Schrodinger equation

+

a

i A - q x ,t ) = H ( t ) \ k ( X ,t ) at

where @(X,t) is the wave function and X denotes the ensemble of the atomic electron coordinates (i.e., their position coordinates ri and spin variables). The Hamiltonian H ( t ) of the system is given by

+

where Hat = T V is the time-independent field-free atomic Hamiltonian. Here T is the sum of the electron kinetic energy operators and V is the sum of

240

C. J. Joachain, M.Dorr; and N. Kylstra

the two-body Coulomb interactions. The laser-atom interaction term is e Hint(t)= - A(?) P m a

e2N +A2(r) 2m

where N is the number of electrons and N

P=):Pi i= I

is the total momentum operator. The term in A’ can be eliminated from the Schrodinger equation ( 13) by performing the gauge transformation

(17)

which gives for Qv(X,t) the Schrodinger equation in the velocity gauge

On the other hand, if we return to the Schrodinger equation (13) and perform the gauge transformation

Q(X,t)

[;

] Qt(X,

= exp - - A(?) R

r)

where R is the sum of the coordinates ri of the electrons [see Eq. (6)], we obtain the Schrodinger equation in the length gauge

As we shall see below, it is sometimes convenient to study the interaction of an atomic system with a laser field in an accelerated frame called the Kramers frame (Kramers, 1956; Henneberger, 1968). Starting from the Schrodinger equation (18) in the velocity gauge, we perform the unitary transformation

(21)

HIGH-INTENSITY LASER-ATOM PHYSICS

24 1

where m

is a vector corresponding to the displacement of a “classical” electron from its oscillation center in the electric field & ( I ) . The Kramers transformation, Eq. (2l), therefore corresponds to a spatial translation, characterized by the vector a(?),to a new frame moving with respect to the laboratory frame in the same way as a “classical” electron in the field &(t).In this accelerated Kramers frame, the new Schrodinger equation for the wave function $A(X,t) is

so that the interaction with the laser field is now incorporated via a(t)into the potential V, which becomes time-dependent. We note that in the case of a linearly polarized monochromatic field E ( t ) = i&o cos (at)

(24)

a(t) = i o cos (at)

(25)

we have

where o=-

f?&O

ma2

is called the “excursion” amplitude of the electron in the field. B.

PERTURBATION

THEORY AND ITS BREAKDOWN

At low intensities (such that the electric field strength &O is much smaller than the atomic fields relevant to the process considered) time-dependent perturbation theory (see Faisal, 1986) can in general be used to study multiphoton processes. The simplest form of this approach is called lowest (nonvanishing) order perturbation theory (LOPT). For example, in the case of an n-photon ionization process from an initial (unperturbed) bound state I$i ) , LOPT

242

C. J. Joachain, M.Don; and N. Kylstra

predicts that the ionization rate r, is given by

r, f

where T!) is the LOFT transition matrix element for the absorption of n photons and the sum is over allowed final states l\Ilf). Thus, if HO = Ha, is the “unperturbed” (field-free) Hamiltonian and G o ( E ) = (E - H o ) - ’ is the corresponding Green’s operator, one has, in the length gauge

where Ei is the energy of the unperturbed initial state. Similar LOFT expressions can be written down for other multiphoton processes such as harmonic generation and laser-assisted electron-atom collisions. The calculation of the LOFT transition matrix element Tjr) is in general a difficult task, particularly for high-order multiphoton processes and (or) for complex atoms. The simplest case is that of nonresonant MPI in oneelectron atoms for which LOFT has been applied successfully for intensities I < 1013W cmP2 and angular frequencies such that fiw >> Up (see Gontier and Trahin, 1980; Crance, 1987). Discrepancies from the perturbative I” power law, which are found at higher intensities, signal the breakdown of perturbation theory, as do other strong-field phenomena such as the “peak suppression” in AT1 spectra, the existence of a plateau in high-order harmonic generation, or successive FFT peaks of comparable height in laser-assisted electron-atom scattering. Let us now return to the transition matrix element, Eq. (28). Using the spectral representation of the Green’s operator G o ( E ) ,namely

with ~~1+~, = Ek I\I~~), we can write

~i,”’in the more explicit form

HIGH-INTENSITYLASER-ATOM PHYSICS

243

which shows that LOFT always fails for resonant multiphoton processes such that Ei + r h o = Ek,, for a particular r E { 1 , 2 , .. . ,n - 1). In this case, modifications of the theory are required, in which the resonantly coupled states are treated in a nonperturbative way, whereas the other states are treated by using perturbation theory. This approach belongs to the category of semiperturbative methods, to which we now turn our attention. C. SEMIPERTURBATIVE METHODS

We shall now describe two semiperturbative approaches, in which some of the interactions are treated in a nonperturbative way, and the remaining ones are treated by using perturbation theory. The first is the method of essential states, which we shall apply to resonant multiphoton ionization (REMPI). The second is a semiperturbative approach to laser-assisted electron-atom collisions.

1. Essential States When resonances are present, and for not too intense laser fields, the multiphoton processes will usually be dominated by the resonant contributions. A convenient way of accounting for this is by formulating an approach based on the Feshbach projection operator formalism (Feshbach, 1958,1962). Two projection operators are defined, P and Q, that project onto the space of, respectively, the field-free states that are resonantly coupled by the field and its complement space. Writing the wave function as * ( t ) = P*(t)

+ Q*(t)

(31)

and inserting this expression into the time-dependent Schrodinger equation, the following two coupled equations are obtained:

Now, by defining the Green's operator in the Q space,

C. J. Joachain, M.Don; and N. Kylstra

244

the second of Eqs. (32) can be formally solved, so that an effective timedependent Schrodinger equation in the P space is found:

(ih

)

- PH(t)P - PH(t)QGQ(t)QH(t)P( P @ ( t ) )= 0

(34)

The Green’s operator GQ can be approximated by accounting for the nonresonant part of the interaction using perturbation theory. Finally, by expanding the wave function P @ ( t ) in the basis of the resonantly coupled, unperturbed states

one obtains a system of coupled first-order differential equations for the resonantly coupled, or “essential,” states of the system d dt

ih - C(t) = H e f f ( t ) C ( t ) where the matrix Heff ( t ) represents the effective Hamiltonian which treats the resonantly coupled states exactly and takes into account the rest of the spectrum in a perturbative way. Various formulationsof essential-statesmethods have been widely employed to study a range of resonant, multiphoton phenomena. Examples include resonantly enhanced multiphoton ionization, Rydberg wavepacket formation and evolution (see Zobay and Alber, 1996; Mecking and Lambropoulos, 1998) and ionization suppression by quantum interference (see Fedorov and Movsesian, 1988; Fedorov et al., 1990). Further applications of the method of essential states are discussed in the review articles of Lambropoulos and Tang (1992) and Burnett et al. (1993).

2. Laser-Assisted Electron-Atom Collisions The theoretical study of electron-atom collisions in a laser field is in general very complex, because in addition to the difficulties associated with the treatment of field-free electron-atom scattering, the presence of the laser field introduces new parameters (such as the laser frequency, intensity, and polarization) which can influence the collisions. It is therefore of interest to start with a simpler problem, such that the target atom is modeled by a center of force, i.e., a potential V(r), and hence does not exhibit any internal

HIGH-INTENSITY LASER-ATOM PHYSICS

245

structure. For a laser field treated as a spatially homogeneous, linearly polarized, monochromatic electric field &(t) = iEo cos (cot), the Schrodinger equation in the velocity gauge is

a

ih- Qv(r,t ) = at

e + V(r) + -A(t) m

1

p Qv(r, t )

(37)

with A(t) = i A o sin(cot) and A 0 = -€o/co. Let us first study the motion of the electron in the presence of the laser field, but without scattering potential ( V = 0). The corresponding Schrodinger equation

is readily solved to give the (nonrelativistic) Gordon-Volkov wave function (Gordon, 1926; Volkov, 1935)

Xk(r,t) = ( 2 ~ ) - exp ~ ’ {i[k. ~ r - k a(t) - E k t / h ] }

(39)

where k is the electron wave vector, Ek = h2k2/2m is its kinetic energy, and a(?)is the displacement vector given by Eq. (25).

It is worth stressing that the Gordon-Volkov wave function, Eq. (39), is the exact solution of Eq. (38) and hence contains the effect of the laser field on the “free” electron “to all orders.” Defining a0 = 010i, this effect is characterized by the dimensionlessquantity k 010 = p i . k,where p = kclo = ke€o/(mo2).Thus a low-order perturbative treatment of the laser interaction with the “free electron” will be valid only if Ik .sol<< 1. In particular, for fast electrons, the parameter p is larger than unity even for laser fields of moderate intensity, so that the exact (nonperturbative) Gordon-Volkov wave functions, Eq. (39), must be used to treat the laser coupling with the “free” electron. Let us now return to the full Schrodinger equation, Eq. (37). Using the Gordon-Volkov wave functions, Eq. (39), and treating the interaction potential V(r) to first order (i.e., in first Born approximation,B1)Bunkin and Fedorov (1966) have shown that the differential cross section for scattering with the exchange of n photons is given (in a.u.) by +

246

C. J. Joachain. M. Dorr; and N. Kylstra

where d o B ’ / d Ris the field-free first Born differential cross section, J, is an ordinary Bessel function of order n,A = k i - kf is the momentum transfer of the collision (k and kf being the initial and final momenta of the electron, respectively), and k f (n)= ( k ? 2no)”*. The Bunkin and Fedorov formula, Eq. (40), in which the electron interaction with the laser field is treated in a nonperturbative way (using the Gordon-Volkov wave functions) whereas the interaction potential V(r) is treated by first-order perturbation theory, provides the simplest example of the use of semiperturbative methods in laser-assisted collisions. Let us now consider collisions of electrons with “real” atoms, having an internal structure, in the presence of a laser. Three types of interactions must then be taken into account. First, the interaction between the unbound electron and the target atom takes place, as in the field-free case. Second, the laser field interacts with the unbound electron. Third, the laser field interacts with the target atom and therefore “dresses” the atomic target states. In order to deal with this problem, which is much more complex than laserassisted potential scattering, Byron and Joachain ( 1984) have proposed a semiperturbativetheory which we now briefly outline. We consider the case for which the incident electrons are fast ( E k i 2 100eV), so that the electron-atom interaction can be treated perturbatively by using the Born series (see Joachain, 1983). Assuming again that the laser field can be described as a monochromatic, linearly polarized, and homogeneous electric field, the interaction between the laser and the projectile electron can be treated exactly, as in the case of potential scattering, by using a Gordon-Volkov wave function. On the other hand, the laser-target atom interaction can be treated by using first-order time-dependent perturbation theory, provided that the electric field strength lo remains small with respect to the atomic unit of electric field strength. (This is a required condition on laser-assisted collisions, because otherwise the target atom would be ionized by the laser.) Furthermore, the laser frequency must not be too close to resonance with an atomic transition between the initial or the final state of the atom and any intermediate atomic state. This semiperturbative theory has been applied to the three kinds of laserassisted scattering processes Eqs. (9)-( 11): “elastic” collisions, inelastic collisions, and (e, 2e) reactions (see the review articles of Francken and Joachain, 1990a; Joachain, 1994; Ehlotzky ef al., 1998). The results show the importance of the dressing of the atomic states by the laser field, even at relatively modest intensities. As an example, we show in Fig. 9 the differential cross section for “elastic” electron-atomic hydrogen scattering with the absorption of one laser photon. The full results, including the target dressing, exhibit important departures from the results in which target dressing has been neglected, particularly at small scattering angles.

+

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247

Scattering Angle (deg.) FIG.9. Differential cross section for “elastic” electron-hydrogen scattering in a laser field of fio = 2 eVand a peak laser electric field of 0.02 a.u., with the absorption of one net photon by the scattered electron from the laser field. The direction of the momentum transfer A is parallel to the field polarization i. The incident electron energy is 100eV. Solid curve: full results; dashed curve: neglecting target dressing. From Byron et al. (1987).

D. FLOQUET THEORY Having introduced perturbative and semiperturbativemethods, in this Section we consider a fully nonperturbative approach for studying laser-atom interactions, namely the Hoquet theory.

1. Basic Equations Let us now restrict our attention to monochromatic laser fields of angular frequency o and of arbitrary polarization. The HamiltonianH ( t ) of the system

248

C. J. Joachain, M. Don; and N. Kylstra

+

is then periodic, H(t T) = H ( t ) , where T = 27~/0.The Floquet method (Floquet, 1883; Shirley, 1965; Chu 1985) can therefore be used to write the wave function @(X,t) in the form

where the time-independent quantity E is called the quasi-energy, and the function F ( X , t) is periodic in time, with period T, so that it can be expressed as the Fourier series

n=-cc

The functions F n ( X ) are called the harmonic components of @(X,r ) . Using Eqs. (41) and (42), we obtain for *(X,t) the Floquet-Fourier expansion

n=-cc

If we also make a Fourier analysis of the interaction Hamiltonian, +cc n=-cc

and substitute both Eqs. (43) and (44) into the Schrodinger equation (13), we obtain for the harmonic components F n ( X ) a system of time-independent coupled differential equations:

with n = 0, f 1, f2, . . . . These equations, together with appropriate boundary conditions, form an eigenvalue problem for the quasi-energies, which we can write as

(HF- E ) F = O

(46)

where the Floquet Hamiltonian HF is an infinite matrix of operators. In the case of multiphoton ionization, the quasi-energies are complex and can be

HIGH-INTENSITY LASER-ATOM PHYSICS

249

expressed as E = Ei+ A - i -

r 2

(47)

where Ei is the energy of the initial unperturbed (field-free) state and A is the AC Stark shift of the state. The physical meaning of r can be deduced by noting that the integral over a finite volume of the electron density, averaged over one cycle, decreases in time like exp(-rt/h). Hence the characteristic lifetime of an atom described by the Floquet state, Eq. (41), is h / r , which means that r / h is the total ionization rate of that state. We also note that, in the velocity or the length gauges, the interaction Hamiltonian can be written in the form

where H+ and H- are time-independent operators. The coupled equations (45) then take the simpler form

and the Floquet Hamiltonian HF is a tridiagonal matrix of operators. 2. The Sturmian-Floguet Method

The Floquet theory has been used extensively to study multiphoton processes in atomic systems. In particular, detailed calculations have been performed for one-electron atoms. Denoting by r the electron position vector, and following Maquet et al. (1983), the system of coupled equations (49) can be solved by expanding each harmonic component Fn(r)on a discrete basis set:

where the YLM are spherical harmonics and the radial functions S i L are complex Stunnian functions (Rotenberg, 1970). The parameter IC is chosen as complex, allowing the computation of quasi-bound (Siegert) states, in complete analogy to the complex-rotation (or dilatation) transformation method (Chu, 1985). The complex, discrete Sturmian basis is particularly appropriate for obtaining highly accurate solutions to the coupled Floquet equations for one-electron atoms moving in Coulomb or modified Coulomb effective potentials. The Sturmian-Floquet method has been applied extensively to

C. J. Joachain, M. Dorr, and

250

0

1

2

3

N. Kylstra

4

5

6

Electron Energy (eV) FIG. 10. Yield of photo-electrons into the lowest three AT1 channels, versus photo-electron energy, for ionization of H(1s) by a 608-nm pulse whose peak intensity is 6.5 x I O l 3 Wcm-* and whose duration is 0.5 ps. The solid curve is the result of Sturmian-Floquet calculations, and the dashed curve represents the experimental data (Rottke et al., 1994). Some of the subpeaks are labelled by the dominant configuration of the resonant Floquet state.

study multiphoton ionization and harmonic generation in atomic hydrogen (see the review by Potvliege and Shakeshaft, 1992) and other systems modeled by effective single-active-electronpotentials. As an example, we show in Fig. 10 the photo-electron spectrum for the lowest three AT1 peaks (Dorr et al., 1990a; Rottke et al., 1994) for the multiphoton ionization of H(1s) compared with the experimental data of Rottke et al. (1990). The subpeaks are due to Rydberg states moving in and out of resonance when the intensity of the pulse rises and falls (Freeman et al., 1987), as explained in Section 1I.A. Angular distributions for some of the more prominent subpeaks within the different AT1 peaks have also been obtained and compared with the Sturmian-Floquet calculations (Rottke et al., 1994). Another interesting prediction of Floquet calculations is the occurrence of light-induced states (LIS). The emergence of LIS has been seen in onedimensional model-atom calculations (Bhatt et al., 1988;Bardsley et al., 1988) and has been analyzed in terms of shadow poles (Dorr and Potvliege, 1990; Dorr et al., 1991). Shadow poles have been discussed within the context of Sturmian-Floquet calculations (Potvliege and Shakeshaft, 1988, 1992; Fearnside et al., 1995). These poles arise from solutions of the Floquet equations corresponding to unphysical boundary conditions. They can, however, lead to physically observable consequences following the crossing

HIGH-INTENSITYLASER-ATOM PHYSICS

25 1

of a multiphoton threshold, when a shadow pole becomes a physical pole. This can give rise to the appearance of “new discrete states” in certain laser intensity regions. Although LIS arising from such shadow pole resonances have been seen in the calculations mentioned above, as well as in recent ones (Wells e f al., 1998), experimental verification has yet to come forth. 3. High-Frequency Floquet Theory

Floquet theory has been used to study the behavior of atoms interacting with high-intensity monochromatic laser fields whose frequency is much larger than the threshold frequency for one-photon ionization (Gavrila, 1992b, 1997). This approach is formulated in the Kramers frame introduced in Section III.A. For the case of an atom with one active electron, the Schrodinger equation (23) satisfied by the wave function @ A is

a

ih - @A(r, t ) = at

[C+ -

+

1

V ( r a ( t ) ) @A(r,t )

where a(t)is the displacement vector, given by Eq. (25) for the case of linear polarization. Because Eq. (5 1) has periodic coefficients, one can seek solutions having the Floquet-Fourier form, Eq. (43). Making also a Fourier analysis of the potential

one obtains for the harmonic components F n ( r ) of !PA(r,t), as defined in Eq. (42), the system of coupled equations

Gavrila and coworkers have shown that in the high-intensity, highfrequency limit, the atomic structure in the laser field is essentially governed by the potential V O ( Q ,r ) which is the static (time-averaged) “dressed” potential associated with the interaction potential V in the Kramers frame. Detailed calculations of the energies and eigenstates, performed for the hydrogen atom, show that the atom undergoes “dichotomy”; i.e., the electronic cloud in the static, “dressed” potential splits into two disjoint parts. This is illustrated in Fig. 11 for the ground state of atomic hydrogen in a

252

C. J. Joachain, M. Dorc and N. Kylstra

FIG. 1 1 . (a) High-frequency-limit energy eigenfunction of the ground state of atomic hydrogen in the static “dressed“ potential VO(a0,r) in the Kramers frame for a0 = 20a.u., for the case of linear polarization. @(x, 0, z) is the wave function in the xz plane, where the z-axis is chosen along the direction of the polarization vector t . (b) 1QI2 is the corresponding timeaveraged probability density in the laboratory frame obtained by averaging the probability density over one oscillation of the laser field. From Pont ef al. (1988).

linearly polarized laser field. With increasing ao,the wave function undergoes radiative stretching along the polarization axis. As a0 approaches 20 a.u., a saddle appears, as shown in Fig. ll(a), and two pronounced maxima are formed around the endpoints f a0 2 of the “classical” electron excursion. In Fig. ll(b), the time-averaged density is shown in the laboratory frame. Moreover, the ionization potential of the ground state of Vo(a0,r) decreases with increasing ao. An important prediction of the high-frequency Floquet theory (HFFT) is that at sufficiently high intensity, and when the frequency of the laser field is substantially larger than the threshold frequency for one-photon ionization, the ionization rate decreases when the intensity increases. This phenomenon, called adiabatic stabilization, is essentially due to the fact that the quiver motion of the bound electron(s) in the field becomes large, thus inhibiting the ionization process. It was initially investigated by Gersten and Mittleman (1976) and was studied in detail for atomic hydrogen, using not only the approximate high-frequency Floquet theory (Pont et al., 1988; Pont and Gavrila 1990; Gavrila, 1992b; Vos and Gavrila, 1992; Gavrila, 1997) but also the Sturmian-Floquet theory (Dorr et al., 1991; Potvliege and Smith, 1993b) and the R-matrix-Floquet theory (Dorr et al., 1993), which we shall consider in Section 1II.E. It should be noted that in contrast to the high-frequency Floquet calculations, both the Sturmian-Floquet and the R-matrix-Floquet methods provide exact Floquet results for atomic hydrogen. As an example, we show in Fig. 12 the total rate I’ for ionization of H(ls), as obtained from the Sturmian-Floquet method (Dorr et al., 1991)

HIGH-INTENSITY LASER-ATOM PHYSICS

Intensity (I o

253

’ ~w/cm’)

FIG. 12. Total ionization rate versus intensity for H(1s) in a linearly polarized laser field of angular frequency o = 0.65 a.u. The solid line corresponds to the Sturmian-Floquet (DOH et al., 1991) and the R-matrix-Floquet (DOIT et al, 1993) calculations, which are in excellent agreement. The LOPT results are given by the broken line.

and the R-matrix-Floquet theory (Dorr et al., 1993) for an angular frequency o = 0.65 a.u., which is larger than the threshold value (a= 0.5 a.u.) for onephoton ionization. The Floquet results are seen to increase linearly at low intensities, as predicted by lowest-order (in the present case, firstorder) perturbation theory (LOFT). They have a peak near 10l6W cm-’ and then decrease with increasing intensity, thus exhibiting the stabilization behavior. In the Floquet analysis discussed above, a monochromatic laser field of constant intensity is assumed to be present at all times. In reality, an atom cannot be instantaneously released into a super-intense field, but must be subjected to a laser pulse of finite duration. This implies that during the laser pulse turn-on, where the intensity is lower, substantial ionization will occur. In other words, before the atoms experience a super-intense field where they can stabilize, they must pass through a “death valley” where their lifetime is extremely short (Lambropoulos, 1985). Therefore, if adiabatic stabilization is to be observed experimentally, the laser pulse risetime must not be so long that saturation (complete ionization) occurs during the turn-on. However, it must be long enough so that the atom adiabatically remains in the ground state in the Kramers frame. By comparing moquet calculations with timedependent calculations, it has been shown that these criteria can both be fulfilled (Pont and Shakeshaft, 1991; Zakrzewski and Delande, 1995; Piraux and Potvliege, 1998). As the laser pulses become shorter, excited states of the

C. J. Joachain, M . Dorr; and N. Kylstra

254

I

-alCT .-Cfn

n ”

0

5

10

fluence ( J / c ~ ’ > FIG. 13. Photo-ionization yield of the 5 g circular state of neon as a function of the laser pulse fluence (in Jcm-’). The open circles were measured with low peak intensity (up to 1.2 x I O l 3 W cm-2, 1 ps). The solid curve represents a fit using the theoretical perturbative rate (including depletion). The solid circles were measured using shorter pulses, with the same fluence but more intense (up to 1.2 x W c d , 0.1 ps). The yield due to these pulses hardly increases with fluence, which indicates stabilization. From de Boer et al. (1993).

Kramers static potential V O ( C Ir) O ,become populated. As a result, dichotomy becomes less apparent (Kulander et al., 1991; Reed et al., 1991), but stabilization still persists, as will be discussed in Section 1II.G.1. Due to the adiabaticity condition and the high-frequency, high-intensity requirements, the experimental verification of adiabatic stabilization is a very difficult task. Fortunately, the stabilization conditions can be met at lower intensities and frequencies if the atom is prepared in an initial state that is not the ground state but a “circular” Rydberg state with large n, 1, and Irnl quantum numbers, such that its lifetime in the “death valley” regime is sufficiently long (Vos and Gavrila, 1992; Potvliege and Smith, 1993). An experiment of this kind, using two laser pulses, was carried out by de Boer et a1. (1993, 1994) to study the stabilization behavior of the hydrogenlike, “circular” 5g state of neon. A first pulse prepares the initial state. Comparison of the single-photon ionization yield, due to a second laser pulse (see Fig. 13), for both short (0.1-ps) intense and long (1-ps) less intense pulses, shows a suppression of ionization as the intensity increases, in accordance with the theoretical prediction of stabilization. The experimental

HIGH-INTENSITY LASER-ATOM PHYSICS

255

results are in fair agreement with theoretical calculations (Piraux and Potvliege, 1998). In addition to adiabatic stabilization, which occurs at high intensities and high frequencies, several mechanisms of ionization suppression have been suggested that rely on interference between ionization probability amplitudes from a coherent wavepacket of states (Fedorov and Movsesian, 1988; Parker and Stroud, 1989; Fedorov et al., 1990; Piraux et al., 1991; Burnett et al., 1991; Tikhonova and Fedorov, 1997). These mechanisms, referred to as dynamic stabilization, require intensities of the order of 1014W cm-2 but are destroyed at higher intensities. The stabilization in this case arises from an interaction of resonances with comparable (overlapping) widths (see also the discussion on laser-induced degeneracies in Section II1.E). 4. Floquet Theory for Luser-Assisted Electron-Atom Collisions

Floquet methods can also be applied to the study of laser-assisted electronatom scattering. We assume now that, on average, the laser intensity does not vary much on time scales which are of the order of typical scattering times. The scattering process can therefore be described in terms of time-independent transition rates. Starting with the Floquet-Fourier form of the wave function and expanding the Floquet-Fourier components in terms of partial waves, Floquet-closecoupling (FCC) equations can be derived that are satisfied by the corresponding radial functions. These equations and their method of solution are very similar to the close-coupling equations appearing within the context of electron-atom scattering. This will be discussed in more detail in Section 1II.E on the R-matrix-Floquet theory. Dimou and Faisal(l987) have studied laserinduced resonances in laser-assisted electron-proton scattering using the FCC equations. Within the context of the HFFT,Gavrila and Kaminski (1984) have considered scattering in a high-frequency laser. A momentum space formulation of the problem can be obtained by deriving an integral equation for the scattering T-matrix elements, the FloquetLippmann-Schwinger (FLS) equation, in complete analogy with the field-free case. In this approach, the time-dependent Schriidinger equation is first recast into the form of a time-dependent Lippmann-Schwhger equation. Using the Floquet-Fourier form of the wave function and the Gordon-Volkov propagator, an integral equation is found for the Floquet-Fourier coefficients, from which the Floquet-Lippmann-Schwingerequation is obtained for the required on-shell T-matrix elements (Kylstra and Joachain, 1998). By solving numerically the FLS equation for laser-assisted low-energy electron scattering by various potentials, Kylstra and Joachain found that for a low frequency o = 0.0043 a.u. (corresponding to a C02 laser), the differential cross section

C. J. Joachain, M.Dorr; and N. Kylstra

256

do”/dR corresponding to the scattering process ki--+ kf accompanied by the transfer of n photons, is given to good approximation by the Kroll and Watson (1973) formula

where d o ( k f,k;)/d R is the field-free differential cross section corresponding to the transition k f -+ kj,and ky(j = i, f)are shifted momenta such that ky = kj nwao/(A.ao). Finally, we mention that the Floquet theory has also been used to construct “dressed” atomic target states in a nonperturbative way in order to analyze laser-assistedelectron-atom collisions (Francken and Joachain, 1990a, 1990b; Dorr et al., 1994) within the semiperturbative theory discussed in Section III.C.2. This approach is particularly useful to study resonant cases, where the laser frequency is close to a transition frequency in the atom.

+

E. R-MATRIX-FLOQUET THEORY The R-matrix-Floquet (RMF) theory is a nonperturbative approach that has been proposed by Burke, Francken, and Joachain (1990, 1991) to analyze atomic multiphoton processes in intense laser fields. The RMF theory treats multiphoton ionization, harmonic generation and laser-assisted electronatom collisions in a unified way. It is completely ab-initioand is applicable to an arbitrary atom or ion, allowing an accurate description of electron correlation effects. Let us consider an atomic system, composed of a nucleus of atomic number 2 and N electrons, in a laser field that is treated classically as a spatially homogeneous electric field E ( t ) . Although more general cases (such as twocolor fields) have been studied, we shall assume for the moment that the laser field is monochromatic and linearly polarized, so that the electric field & ( t ) is given by Eq. (24). Neglecting relativistic effects, the atomic system in the presence of this laser field is then described by the time-dependent Schrodinger equation [see Eqs. (13)-( 15)]

at

1

. P + -A2 ( t ) 9(X,t ) e2m 2N

(55)

where the vector potential is A ( t ) = i A o sin (or) with A 0 = -&o/o.We shall be interested in the following three processes involving at most one unbound electron: multiphoton single ionization of atoms and ions, harmonic generation and laser-assisted electron-atom (ion) elastic and inelastic collisions.

HIGH-INTENSITY LASER-ATOM PHYSICS

257

According to the R-matrix method (Wigner, 1946; Wigner and Eisenbud, 1947),configuration space is subdivided into two regions. The internal region is defined by the condition that the radial coordinates ri of all N electrons are such that ri 5 a (i = 1,2,. . . , N ) , where the sphere of radius a envelops the charge distribution of the target atom states retained in the calculation. In this region, exchange effects involving all N electrons are important. The external region is defined so that one of the electrons (say electron N) has a radial coordinate rN 2 a, and the remaining N - 1 electrons are confined within the sphere of radius a. Hence, in this region, exchange effects between the “external” electron and the remaining N - 1 electrons can be neglected. Having divided configuration space into an internal and an external region, we must solve the time-dependent Schrodinger equation (55) in these two regions separately. This is done by using the Hoquet method, which, as we have seen in Section III.D, reduces the problem to solving an infinite set of coupled time-independent equations for the harmonic components F,, (X)of the wave function @(X,t). The solutions in the internal and external regions are then matched on the boundary at r = a. In the internal region it is convenient to use the length gauge, because in this gauge the laser-atom coupling tends to zero at the origin. We remark that in this region the Hoquet Hamiltonian HF is not Hermitian, due to surface terms at r = a arising from the kinetic energy operator in Ha,. These surface terms can be eliminated by introducing a Bloch operator (Bloch, 1957) Lg, so that H F + LB is Hermitian in the internal region. Following the R-matrix procedure (see Burke and Berrington, 1993), an elaborate basis set is then constructed, in which the operator H F + L g is diagonalized. Using the spectral representation of this operator, one obtains on the boundary the relation

‘

~ ( a=) R(E) [r

$1

r=a

where ~ ( rdenotes ) the set of reduced radial wave functions (i.e., radial wavefunctions multiplied by r ) and ‘R(E) is the R-matrix in the length gauge. The logarithmic derivatives of the reduced radial wavefunctions on the boundary r = a, which provide the boundary conditions for solving the problem in the external region, are then given by Eq. (56). In the external region, we have only one electron ( r 2~a), whose dynamics is studied by using the velocity gauge. Here a simple close-coupling expansion can be used for the harmonic components, because exchange effects between this “external” electron and the remaining N - 1 electrons are negligible. The resulting set of coupled differential equations is then solved, subject to boundary conditions at r = a and r -+ 00. At r = a, the matching of

258

C. J. Joachain, M. Dorr, and N. Kylstra

the internal- and external-region solutions provides "R(E), the R-matrix in the velocity gauge. The coupled equations are solved from r = a to a large value r = a' of the radial coordinate by propagating the R-matrix "R(E).The solutions obtained in this way are matched at r = a' with the solutions satisfying given boundary conditions for r + 00, calculated by using asymptotic expansions (Dorr et al., 1992). The boundary conditions for r + 00 are formulated in the Kramers frame, because in this frame the channels decouple asymptotically. These boundary conditions differ according to the process considered. For the case of multiphoton ionization and harmonic generation, there are only outgoing waves corresponding to Siegert boundary conditions. It is then found that solutions will occur only for certain complex values of the energy [see Eq. (47)]. From the knowledge of the eigenvectors, one may obtain all the other physical quantities, such as the branching ratios into the channels, the angular distribution of the ejected electrons, etc. In the case of laser-assisted electron atom (ion) collisions, one must impose S-matrix (or T- or K-matrix) asymptotic boundary conditions (see Joachain, 1983). The scattering amplitudes and cross sections are then given in terms of the elements of the S-, T-, or K-matrix. The RMF theory has been outlined above for the case of a monochromatic laser field. Recently, the Rh4F method has been generalized to bichromatic laser fields such that &(I) = i [El cos (0, t)

+ E2 cos ( W , t + 44

(57)

where i is the (common) unit polarization vector, &I and Ez are the amplitudes of the electric fields oscillating with the angular frequencies wl and w2, respectively, and 4 is a phase. This allows, for example, the study of multiply resonant processes (van der Hart, 1996a; Kylstra et al., 1998b) and of coherent interactions between the fundamental laser frequency and one of its harmonics. We shall now discuss some applications of the R-matrix-Floquet theory. Except for two-color processes, all the results presented below have been obtained for monochromatic, linearly polarized laser fields as described by Eq. (24). We have already shown in Fig. 12 the total RMF ionization rate as a function of the intensity for H( 1s) in a laser field of high angular frequency (w = 0.65 a.u.), in connection with our discussion of adiabatic stabilization. In Fig. 14, we consider the multiphoton ionization of H(1s) at an angular frequency w = 0.184 a.u. corresponding to a KrF laser. At low intensities, three-photon absorption is required for ionization. We show in Fig. 14(a) the RMF results for the total ionization rate. In Fig. 14(b), we display the branching ratios into the dominant ionization channels for the absorption of three photons. In both Figs. 14(a) and 14(b) we note the striking differences between the lowest-order perturbative and the RMF values.

259

HIGH-INTENSITY LASER-ATOM PHYSICS 0.00201.

s

. I . .

.

I

I

,

.

I . .

.

I . .

.,. ..

I/.

i

0.7 0.6 0.5

\c=l

4

.- 0.60 0

Y

0.1 0.0

Intensity (I 014 W/crn2) FIG.14. (a) Total ionization rate and (b) branching ratios into the lowest ionization channel where n = 3 photons have been absorbed, resolved into the angular momentum components, and plotted against intensity, for H( 1s) in a laser field of angular frequency o = 0.184 a.u. The solid lines correspond to the RMF calculations. The results of lowest (third)-order perturbation theory are given by the broken lines. The triangles in part (b) indicate the values obtained by using the Sturmian-Floquetmethod (Potvliege and Shakeshaft, 1992). From Dorr et al. (1993).

We now turn to two-electron atomic systems, for which the RMF theory has provided nonperturbative multiphoton ionization rates including electron correlation effects (hrvis et al., 1993). As a first example, we display in Fig. 15 the RMF results for the multiphoton detachment of H-,obtained at an angular frequency o = 0.0149 a.u., such that at least two photons are necessary to detach an electron (Dorr etal., 1995a).The total detachment rate is shown, as well as the partial rates into the two- and three-photon detachment channels. At low intensities the total detachment rate increases perturbatively, with the second power of the intensity, because the dominant detachment channel is the two-photon channel. At these low intensities the three-photon partial rate is very small, being proportional to the third power of the intensity. When the intensity reaches the value 6 x 10'' W cm-*, the twophoton channel closes, due to the dynamic Stark shift, and only three- and

260

C. J. Joachain,

M. Dorc and N. Kylstra

Energy (a.u.) -0.0280

1.5

-0.0290

-0.0300

-

Intensity (w/crn2>

XlO'O

FIG. 15. Total (solid curve) and partial (into the n = 2, upper broken curve, and n = 3, lower broken curve, photon channels) RMF detachment rates of H- in a laser field of angular frequency o = 0.0149 a.u., versus intensity. The circles correspond to the calculated results. From Mrr et al. ( I 995a).

higher-photon detachment processes are possible. Above this intensity the difference between the total detachment rate and the partial rate into the threephoton detachment channel is due to higher-order processes. As a second example of two-electron systems in a laser field, we show in Fig. 16 the ionization rate into the two-photon channel, versus the photoelectron energy, for He in a laser field of intensity 10l2W cm-*. The Rydberg series of peaks visible below the n = 1 threshold corresponds to one-photon resonances due to intermediate 'P bound states, whereas the series of resonances below the n = 2 threshold consists of two-photon resonances due to ' S and 'D autoionizing states. The RMF theory has allowed to study ab-initio a wide variety of resonance effects in multiphoton ionization (Purvis et al., 1993; Dorr et al., 1995a; Kylstra et al., 1995, Latinne et al., 1995; Cyr et al., 1997; Kylstra, 1997a; Fearnside, 1998). For example, in the case of two-photon ionization, due to the dipole selection rules, resonances can in general occur between the ground state and the members of two different Rydberg series. Now, when two Rydberg states, each belonging to a different Rydberg series and lying close in energy, are resonantly coupled to the ground state by a single photon transition, interference between the two resonant pathways will occur. This

HIGH-INTENSITY LASER-ATOM PHYSICS l

i

0.6

0.8

"

'

l

I "='

"

1 .o

'

I

"

'

l

"

1

n=21

1.2

26 1

1

1.4

Photoelectron Energy (a.u.) FIG.16. Two-photon ionization rate of He versus photo-electron energy at an intensity of lo'* W cm?. The positions of the He+(n = 1) and He+@ = 2) thresholds are indicated by the arrows. From Purvis et al. (1993).

interference, which depends on the laser intensity, will in turn modify the ionization rate of the ground state. This effect has been demonstrated in neon, using the RMF theory to study resonant two-photon ionization via the 5s and 4d Rydberg states (Kylstra et al., 1995). A spectacular effect that has been predicted by the RMF theory is the occurrence of laser-induced degenerate states (LIDS) involving autoionizing states in complex atoms (Latinne et a/., 1995). To understand this phenomenon, we first recall that autoionizing states of atoms produce characteristic resonance structures in the photo-electron yield, not only in (one-photon) photo-ionization, but also in multiphoton ionization (see Fig. 16). At low intensities, these structures can be reproduced by using perturbation theory to treat the interaction of the atom with the radiation field. By contrast, for the case of atoms in intense fields, a perturbative description of the ground-stateautoionizing-state coupling will fail when the intensity is large enough so that the laser-induced width of the ground state becomes comparable to the width of the autoionizing state near resonance (Lambropoulos and Zoller, 1981; Rzazewski and Eberly, 1981). As an example, we show in Fig. 17 the results of a RMF calculation (Latinne el al., 1995) in which the influence of a strong laser-induced coupling between the ground state and the 3s3p64p 'P autoionizing state of Ar has been studied. The trajectories of the complex quasi-energiesof the ground state and the autoionizing state are plotted in the complex energy plane, for intensities I

C. J. Joachain, M.Don; and N. Kylstra

262

0.987

0.0000

-0.0005 n

? -0.001 0 0

W

n

W

W

E -

: 0.884

-0.001 5

-0.0020

-0.0025

-a

-0.003

-0.580

-0.578

-0.576

Re(E) (a.u.)

FIG. 17. Trajectories of the complex Floquet quasi-energies for the ground state and the 3s3p64p ‘P autoionizing state of argon, for intensities varying from o to 5 x 1 0 ’ ~Wcm-’. The values of the angular frequency o are indicated next to the trajectories. The small dots correspond to values of the intensity increasing in steps of 9 x 10’’ W cm-’. For each angular frequency q there are two trajectory curves: one correspondingto the ground state and the other to the autoionizing state. From Latinne et al. (1995).

ranging from 0 to 5 x l O I 3 W cm-2, and for fixed values of the angular frequency o,chosen in the vicinity of 0.99 a.u., corresponding to a one-photon resonance. The zero-field position of the ground state on the real axis is Eg = -0.57816 a.u., whereas the energy of the autoionizing state is shifted by -a. Thus the zero-field position of the autoionizing state (denoted by the big circles) changes with o and is at the complex energy - o 0.40936 a.u. -i0.00119a.u. =-a + E, - ira/2,where Fa is the field-free width of the autoionizing state. We see from Fig. 17 that for a fixed angular frequency there are two curves: one that, in the limit of small intensities, is connected to the zero-field position of the ground state and the other, which is connected to the zero-field position of the autoionizing state. The detuning from resonance is defined as 5 = E, - Eg - o. At large values of I SI (e.g., o = 0.984a.u. or o = 0.991 a.u.), the autoionizing state does not move much from its position, whereas the width of the ground state increases with intensity. At very small values of 16I (e.g., 6.1= 0.987 a.u.), just the opposite happens: the curve connected to the autoionizing state increases in width with intensity, whereas the ground state is “trapped” close to the real axis. For intermediate detunings, both on the positive and on the negative side, two structures are visible, about which the curves of the ground state and of the autoionizing state exchange their roles. At the center of each of these two structures, there is a critical point (to which correspond a critical intensity and angular frequency) such that the two complex quasi-energies are exactly degenerate, i.e., where laser-induced

+

HIGH-INTENSITYLASER-ATOM PHYSICS

263

degenerate states (LIDS) occur. Thus, owing to the existence of LIDS, for a fixed angular frequency lying between the two critical angular frequencies, the rate of ionization of the ground state first increases with intensity and then exhibits a “stabilization” behavior, namely a decrease of the ionization rate with increasing intensity. The existence of LIDS is a general phenomenon, which has been observed in RMF calculations for multiphoton transitions (Latinne et al., 1995; Kylstra, 1997a; Cyr et al., 1997; Kylstra et al., 1998b) and understood by constructing models that retain the essential ingredients of the full RMF calculations. Through an adiabatic path in the frequency and intensity parameter space, one can in principle complete a circuit around the degeneracy, as discussed for degeneracies occurring in atomic hydrogen in a two-color field (Pont et al., 1992). In this sense, LIDS constitute an interesting extension of the work of Berry (1984), where the adiabatic passage around degeneracies in a parameter space was described, and which has attracted considerable interest, particularly with respect to the associated geometric phase. In the case discussed above, the parameter space is two-dimensional and is characterized by the laser intensity and angular frequency. The RMF theory has also been applied to study multiphoton processes in negative ions (Purvis et al., 1993; Dorr et al., 1995a; van der Hart,1996b; Fearnside, 1998; Glass et al., 1998)and to the calculation of harmonic generation (Gebarowski et al., 1997a, 1997b; Bensaid et al., 1999). The first application of the RMF theory to laser-assisted collisions has been a study (Dorr et al., 1995b) of electron-proton scattering in a laser field of the type described by Eq. (24). Total and differential cross sections have been calculated for laser-assisted “elastic” scattering. The laser field induces resonances (Dimou and Faisal, 1987) due to the temporary capture of the projectile electron into atomic hydrogen-bound states, and structures corresponding to different sublevels can also appear. This is illustrated in Fig. 18, where the ratio of the differential cross section (TO for laser-assisted elastic electron-proton scattering with no net exchange of photons to the field-free (Coulomb) differential cross section (T, is shown, for a laser field of angular frequency w = 0.074a.u. and intensity Z = 10l2Wcm-2. We note the onephoton resonances with the n = 3 manifold (the s and do angular momentum components are mixed, but the dominant component is indicated) and twophoton resonances with the n = 2 manifold. The RMF theory has been applied to electron-argon scattering in a laser field, where the argon target was represented as a model potential (Chen and Robicheaux, 1996), and to the two-electron system of electron-atomic hydrogen scattering (Charlo et al., 1998).Work is currently in progress to apply the RMF theory to laser-assisted electron-helium scattering, where other theoretical results (Francken and Joachain, 1990a;Joachain, 1994; Ehlotzky et al., 1998)and experimental data (Mason, 1993; Sanderson and Newell, 1997) are available.

264

C. J. Joachain, M.Dorr, and h? Kylstra

Energy (a.u.1 FIG. 18. The ratio of the differential scattering cross sections Q / O , for electron-proton scattering in a laser field of angular frequency o = 0.074a.u. and intensity I = 10I2Wcm-2. The incoming electron angle 0, = 0 with respect to the unit polarization vector i . The outgoing electron angles are 0f = go", 140",and 175". From Dorr et al. (1995b).

We now consider briefly applications of the RMF theory to two-color processes. The extension of the method for atoms in two laser fields with incommensurable frequencies has been used to analyze light-induced continuum structures (LICS)in helium (van der Hart, 1996a; Kylstra el al,, 1998a),as well as doubly and triply resonant multiphoton processes involving autoionizing resonances in magnesium (Kylstra et af., 1998b). Within the context of these multiply resonant processes, coherent control of the ionization can be exercised in the sense that by changing the laser parameters, the degree of interaction between the resonant processes can be varied. In addition, the RMF calculationsperformed for the case of magnesium (Kylstra el al., 1998b) predict the occurrence of laser-induced degenerate states (LIDS) between autoionizing levels, at laser intensities and frequencies that are accessible to experimental studies.

HIGH-INTENSITYLASER-ATOM PHYSICS

265

To conclude this section on the R-matrix-Floquet theory, we remark that recent reviews of this method and its applications have been given by Joachain (1997), Dorr (1997), and Joachain et al. (1997). F. LOW-FREQUENCY METHODS When the laser period is much longer than the typical “orbital period” of the bound electron, the laser frequency can be characterized as being in the “lowfrequency” regime. Thus, most experiments using short, intense pulsed lasers on noble gas atoms or (positively charged) ions fall into this category, because the typical ground state binding energies are of the order of the atomic unit, whereas the corresponding photon energy is typically an order of magnitude smaller. The continuous passage from multiphoton ionization to static field (“tunnel”) ionization has been investigated theoretically (Dorr et al., 1990b; Shakeshaft et al., 1990) and observed experimentally (Mevel et al., 1993). Thus at sufficiently low frequency and moderately high intensities, a tunnel formula (Keldysh, 1965; Perelomov et al., 1966; Perelomov and Popov, 1967) describes the ionization rate very well (Larochelle et al., 1998). The Keldysh approach, generalized by Faisal (1973) and Reiss (1980), is called the KFR theory. A useful formula for general atoms based on quantum defects has been given by Ammosov et al. (1986). In the limit of low intensities, the ionization process is always a multiphoton one, and the intensity scaling agrees with lowest-order perturbation theory even for very high orders. At high intensities the tunnel formulas break down, because the electron is ionized “over the barrier.” Extensions of ionization formulas from the tunnel regime into the OTB regime have been proposed (Delone and Kraimov, 1998). However, the concept of an ionization rate becomes questionable because ionization occurs during a time comparable to the field period and the orbital period (Bauer and Mulser, 1998). While total ionization rates at not too high intensities can be adequately described by a tunnel formula, some of the significant finer details of the ionization process must include a refinement, namely processes which involve the further interaction of the ionizing electron with the residual core (Corkum et al., 1992; Corkum, 1993; Kulander et al., 1993).As discussed in the Section II.A, the interaction of the laser-driven quasi-free electron with the core can lead to single or multiple ionization or to harmonic generation. The latter has been analyzed in detail by Protopapas et al. (1996b). A semiclassical formulation of the three-step recollision model has been successfully applied to analyze high harmonic generation (Lewenstein et al., 1994; Salibes et al., 1999). This approach considers only a single bound state of the atom and also neglects the influence of the atomic potential on the

266

C. J. Joachain, M. Dorr, and N. Kylstra

motion of the electron driven by the laser field. Corrections to this last approximation have been considered by several authors (see Ivanov et al., 1996; Lohr et al., 1997; Milosevic and Ehlotzky, 1998). The semiclassical theory of Lewenstein et al. (1994) is particularly useful at low frequencies and high intensities, because it does not suffer from one of the difficulties arising when solving the time-dependent Schrodinger equation, namely the presence of the large excursion amplitude a0 which is proportional to Eo. This difficulty is avoided in a simple way by using the analytic semiclassical form for the electron propagator in the field once the electron has ionized. Calculations of ionization and harmonic generation have also been performed for a zero-range (or &-function)potential (Becker et al., 1990, 1994) within the Floquet formalism. A detailed comparison of the zero-range potential results and the three-step model results (Becker et al., 1997) has shown that the two agree well and in fact can even be reduced to almost the same form. The part that is missing in the three-step model results, compared to the full zero-range potential Hoquet results, can be termed continuumcontinuum transitions. These do not play an important role in harmonic generation (see also Faria er al., 1998b). In a recent comparison between the three-step model results and harmonic generation results from the numerical solution of the time-dependent Schrodinger equation for the hydrogen atom, good qualitative agreement has been reported (de Bohan et al., 1998). We note that the semiclassical model has also been widely employed by several groups to study macroscopic coherence properties of harmonic generation (Salikres et al., 1999). In the spirit of the low-frequency approach, a theory of double ionization has been proposed by Faisal and Becker (Faisal and Becker, 1997; Becker and Faisal, 1997). This approach, termed many-body S-matrix theory, identifies the dominant interaction diagrams for the double electron ejection process, based on a description of the free electrons as Gordon-Volkov waves in the field and incorporating two or more two-body interactions between the electrons and the nucleus. Good agreement has been obtained with several experiments, for different frequencies, intensities, and atomic species. G. NUMERICAL SOLUTION OF THE TIME-DEPENDENT SCHRODINGER EQUATION

The nonperturbative Hoquet and R-matrix-Floquet methods as well as the low-frequency approximations considered so far are based on the assumption that the Hamiltonian of the atomic system in the laser field is periodic in time. Although this is not true for a realistic laser pulse, it is still possible to incorporate pulse shape effects into the Hoquet or R-matrix-Hoquet calculations for laser pulses that are very short, even down to a few laser cycles. In

HIGH-INTENSITY LASER-ATOM PHYSICS

267

time (atomic unite)

FIG. 19. Ionized fraction versus time (in am) for H(1s) in linearly polarized laser fields with peak electric field strengths €0 = 1,5, and 10 a.u., indicated next to the curves. The thin lines give the corresponding Floquet results. From Latinne er al. (1994).

particular, if the variation of the laser intensity is slow enough, the atom will remain in the Floquet eigenstate adiabatically connected to the initial state. Numerical studies indicate that this adiabaticity condition is robust for nonresonant, multiphoton ionization in short laser pulses (Latinne et af., 1994; Dorr et al., 1995c; Zakrzewski and Delande, 1995; Piraux and Potvliege, 1998). In Fig. 19 we show such a comparison, for a hydrogen atom in a high-frequency pulse, where no resonances between bound states are important. It can be seen that the agreement between the solution of the time-dependent Schrodinger equation (TDSE) and the single Floquet state approximation is good. Even at resonances - that is, crossings of Floquet quasi-energies - the Floquet energies and widths can be used to calculate photo-electron spectra (Potvliege and Shakeshaft, 1992; Dorr et al., 1990a; Rottke et af., 1994). However, because in general atomic population can be transferred between various Floquet states, these multiple populations must be taken into account in the full time evolution (Potvliege and Day, 1997). Population transfer at avoided crossings from the viewpoint of Floquet theory has been studied by Breuer, Dietz, and Holthaus (see Connerade et af.,1995 and references therein). For ultrashort pulses, typically in the femtosecond range, one must in general obtain information about the multiphoton processes by direct numerical integration of the TDSE. This approach, pioneered by Kulander (1987, 1988), has the advantage that no restrictions need to be imposed on

268

C. J. Joachain, M. Don; and N. Kylstra

the type of laser pulse and that solutions can, in principle, be obtained for all regimes of intensity and frequency. It has the disadvantage that it is computationally very intensive. A straightforward way of reducing the computational load is to study onedimensional models. Because the one-dimensional models are relatively easy to solve numerically, it is possible to conduct “numerical experiments” by investigating a large range of parameters. However, one-dimensional calculations present a number of disadvantages due to the oversimplification of the problem. Nevertheless, a number of interesting results have been obtained in one-dimensional studies of the TDSE. These are reviewed in Eberly et al. (1992) and Protopapas et al. (1997a). We also note the one-dimensional,timedependent, R-matrix calculation of Burke and Burke (1997). The dependence of multiphoton ionization and harmonic generation on the ellipticity of the laser field has been studied by Protopapas et al. (1997b) and Pate1 et al. (1998) using a two-dimensional model. In what follows, we shall describe results obtained by solving the TDSE for one- and two-electron systems.

I . Single Active Electron Advances in computer technology over the past ten years have made possible the numerical integration of the TDSE for atoms or ions with one single active electron (SAE)in laser fields. These single-electron calculations are “exact” for hydrogenic systems. However, for atoms or ions with more than one electron, dynamic electron correlations are neglected so that in the SAE model, a single active electron moves in an effective time-independent (but possibly nonlocal) potential under the influence of the laser field (Kulander et al., 1992). As an illustration, we show in Fig. 20 the probability density of the continuum part of the SAE wavefunction for neon, after exposure to one laser cycle. Effects of rescattering and interference can be clearly seen. An example of a TDSE calculation performed for atomic hydrogen is shown in Fig. 21, where a high-order AT1 spectrum, obtained by solving the TDSE numerically (Cormier and Lambropoulos, 1997) is displayed. These calculations confirm the existence of a plateau in the spectra, observed in experiments (Paulus et al., 1994b, 1996), which can be understood qualitatively by using the recollision picture mentioned in Section I1.A. In Fig. 22 we show the two-color photo-electron spectrum obtained by Taieb et al. (1996) when hydrogen atoms are submitted to an intense radiation pulse containing the fundamental of a Ti:Sapphire laser operated at h a L = 1.55eV (ie., W L = 0.057 a.u.) and a weaker 13th harmonic having an angular frequency O H = 0.741 a.u. (in the UV range) high enough so that the atom can be ionized by a single photon. Interferences arise between

HIGH-INTENSITY LASER-ATOM PHYSICS

269

FIG.20. Probability density of the continuum part of the wave function of a neon atom, after being exposed to one cycle of a 810-nm linearly polarized laser field at an intensity of 6 x 1014W cm-’. The position of the nucleus is indicated by the arrow. The dominant part of the wave function resides in the bound states, near the nucleus. This part has been subtracted in order to exhibit only the continuum part. Note the large spatial extent of the wave function. Courtesy of K. Kulander.

multiphoton “above-threshold ionization” (ATI) and laser-assisted singlephoton ionization (LASPI). These interference effects can lead to a partial coherent control of the photo-ionization process (Vkniard et al., 1995; Taieb et af., 1996). A related process, laser-assisted Auger transitions, has been observed by Schins et af. (1994) in xenon. In Section III.D.3, the adiabatic stabilization of atoms in intense, highfrequency laser fields was discussed from the point of view of the Floquet theory, in particular the high-frequency Floquet theory (HFFT). Demonstrations of stabilization in short, intense laser pulses were given by Su et al. (1990) in one dimension and for hydrogen by Kulander et al. (1991) and also by Pont and Shakeshaft (1991) and Latinne et al. (1994). Here stabilization manifests itself as an increase of the survival probability with increasing laser intensity at some fixed, high frequency. For very short pulses, not only the ground state of the static, dressed potential Vo(ct0,r) [see Eq. (53)] is populated, but also many excited states. For tllis reason, in the Kramers

270

C, J. Joachain, M. Dorr; and N. Kylstra 1oo 10''

lo']

f

10.'

80

16'

= a

10'3

10"

10' 10"O

0

1

2

3

5

4

6

7

8

9 1 0 1 1 1 2 1 3 1 4 1 5

Electron Energy (UJ FIG. 21. AT1 spectrum of atomic hydrogen in a linearly polarized laser pulse of 25 fs = 2 x 1014W cm?. From Cormier and Lambropoulos (1997).

FWHM,ho = 2eV, and I

I

g

L

v)

-g8 u

w

I

1

I

10

15

20

I0 . ' lo-* 10-9

10-10

lo-"

0

g Io-lD

10-l~

0

5

Energy (eV) FIG. 22. Effect of the presence of the 13th harmonic of a Ti:Sapphire laser on the photoelectron spectrum of atomic hydrogen. The fundamental laser photon energy is h w = ~ 1.55 eV and its intensity is I L = 10 l 3 W cm-2. The thick line corresponds to a 13th harmonic of intensity I H = 3 x lo8W cm-*, the thin line to an intensity I H = 0. From Ta'ieb et al. (1996).

frame the dichotomy shown in Fig. 11 will not be present. However, the wavefunction will remain localized within the region -010 to uo along the polarization axis. It is worth pointing out that the high-frequency condition requires that the photon energy be large compared with the ionization potential of the dressed states in the laser (Marinescu and Gavrila, 1996).

HIGH-INTENSITYLASER-ATOM PHYSICS

27 1

As an example of stabilization in a pulsed laser field, we show in Fig. 19 the ionized fraction as a function of time, for H(1s) in high-frequency (o= 2a.u.), very intense laser fields that are turned on rapidly, with a twocycle ramp, as obtained by Latinne et al. (1994). The Floquet results at the corresponding field strengths are also displayed. We also see from Fig. 19 that both the Floquet and the time-dependent calculations exhibit the stabilization behavior, the ionized fraction corresponding to l o = 10 a.u. being inferior to that for Eo = 5 a.u. The extra beat structure visible for the curve corresponding to &O = 10 a.u. in Fig. 19 is due to interference between the populations of the 1s and the higher lying Floquet states (mainly the 2s state) that are populated during the turn-on of the field. Detailed comparisons between Floquet and “exact” TDSE calculations for atomic hydrogen in very strong, ultrashort, high-frequency laser pulses of various shapes have been made by Dorr et al. (1995c, 1997), who have also studied the population transfer from the ground state to excited states, for one- and two-color linearly polarized fields. A number of calculations have been carried out in one dimension in order to gain further insight into various aspects of stabilization dynamics using the “soft”-core Coulomb potential (Eberly et al., 1992). In addition, stabilization with various types of short-range potentials have been investigated (Su et al., 1996). The influence on the stabilization dynamics of excited states that are populated during rapid laser pulse turn-ons has been analyzed (Reed et al., 1991; Vivirito and Knight, 1995). Grobe and Fedorov (1992, 1993) demonstrated that for very large excursion amplitudes, the initial evolution of the wavepacket can be modeled by the spreading of a free wavepacket in the Kamers frame, as long as the value of a0 during the turn-on is larger than the width of the spreading wavepacket. Classical Monte Carlo simulations of stabilization in one and three dimensions have been performed (Grochmalicki et al., 1991; Grobe and Law, 1991; Minis et al., 1992; Gajda et al., 1992; Keitel and Knight, 1995). Classical versus quantum effects in the stablization dynamics have been investigated by Watson et al. (1995a) by examining quantum and classical phase space distributions. The issue of stabilization of atoms in laser pulses has been considered for asymptotically large electric field strengths by Fring et al. (1996, 1997) and Faria et al. (1998a). They have remarked that for ultrashort pulses, the total momentum transfer to the electron after the pulse is not necessarily zero, and thus in the limit of large intensities in general the electron will be ejected by this “kick” in the polarization direction. Evidence exists that, for laser pulses of sufficiently large intensity, no population will survive. This has been seen in classical Monte Carlo simulations (Keitel and Knight, 1995), as well as in one-dimensional calculations (Su et al., 1996; Kylstra, 1997b). Furthermore, at ultrahigh intensities stabilization must be modified by relativistic effects,

272

C. J. Joachain, M. Dorr, and N . Kylstra

including the presence of the magnetic field and the momentum of the photon in the laser propagation direction, as will be discussed in Section 1II.H. The present experimental drive towards shorter, and thus more intense, laser pulses in investigations of high harmonic generation calls for numerical modeling using explicit time-dependent methods. Many features of harmonic generation have been studied by solving the TDSE (Sanpera et al., 1995; Watson et al., 1995b; Antoine et al., 1996b; Preston et al., 1996). For more detailed reviews on this subject, we refer the reader to Salikres et al. (1999) and Platonenko and Strelkov (1998).

2. Several Electrons In order to model multiphoton processes in complex atoms, most TDSE calculations have been performed using the SAE approximation. Solving accurately the TDSE for atoms or ions with two or more electrons in a short, intense laser pulse presents a great computational challenge. The first insight into the role of correlation in the multiphoton ionization of atoms in short, intense laser pulses was gained by investigating the simplest dimensionally reduced model in which each of two electrons moves in one dimension (Pindzola et al., 1991; Grobe and Eberly, 1992). Using this model, effects of correlation have been explored by comparing with the results of one-dimensional SAE calculations (Lappas et al., 1996) as well as density functional calculations (Lappas and van Leeuwen, 1998). The theoretical study of two-electron atoms or ions in a short, intense laser pulse requires the solution of the TDSE in five spatial dimensions (taking into account the conservation of the total magnetic quantum number M).Using a massively parallel computer, Parker et al. (1996, 1998) and Taylor et al. (1997) have studied multiphoton processes in helium by expanding the total wave function in terms of products of single-particle angular momentum eigenstates and solving the resulting, coupled radial equations. The numerical integration of these radial equations is facilitated by restricting the Coulomb interaction between the two electrons to a few multipoles in order to reduce interprocessorcommunication.They have studied the effects of correlation on harmonic generation spectra and on ionization probabilities. A computationally more tractable approximation is gained when the basis is tailored to the particular atom or process under study. Employing a correlated, Hylleraas-type basis and complex scaling, Scrinzi and Piraux (1997, 1998) have computed ionization and double excitation of helium in very short, intense laser pulses of relatively high frequency. The TDSE is solved by propagating in time the coefficients associated with the field-free basis states. This approach has been applied to analyze single ionization, double excitation, and harmonic generation in helium and the negative hydrogen ion.

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We also note that a Floquet calculation involving a Hylleraas-type basis has been used for calculating harmonic generation in He (Moiseyev and Weinhold, 1997).The calculation was restricted to two angular momenta, and it was optimized to yield harmonic generation rates but not ionization rates. Building on earlier work (Rudolph et al., 1991;Tang et al., 1991),calculations of the photo-electron spectra for helium (Zhang and Lambropoulos, 1995)and magnesium (Zhang and Lambropoulos, 1996)have been performed using B-splines to construct field-free, one-electron basis states. In particular, the influence of the presence of open channels associated with excited states of the remaining ion has been investigated. Similar to the inner-region basis used in R-matrix-Floquet calculations, a two-electron basis consisting of a set of suitably chosen orbitals for one of the electrons and a complete set of basis functions for the second (ionizing) electron was employed. Using a basis of field-free states obtained from multiconfiguration HartreeFock calculations, Mercouris et al. (1997)have calculated ionization rates for helium which they have compared with the recent experimental findings of Charalambidis et af. (1997). Multiphoton ionization from the metastable helium ls2s ‘S state has also been investigated (Nicolaides et al., 1996). A test for theories of multiphoton processes including electron correlation effects is to calculate accurately double-ionization yields for two-electron systems in intense laser fields. These quantities have been measured in a number of experiments for different atoms (Walker et al., 1994;Larochelle et al., 1998).A striking feature of these experimental results is the existence of two distinct intensity regimes: one in which the double-ionization process proceeds predominantly sequentially (at higher intensities) and one in which it is mostly non-sepuential (lower intensities). This phenomenon has been analyzed (Watson et al., 1997) by a “semiindependent” electron approach requiring the .solution of two single-activeelectron problems, the second incorporating the results from the first and thus subject to interelectronic correlation. This calculation reproduces the large enhancement, due to the non-sepuential (NS) mechanism, of the double electron ejection at low intensity. The results are shown in Fig. 23. Onedimensional calculations also qualitatively reproduce the “knee structure” found in the double-ionization yields (Watson et al., 1997;Bauer, 1997). A review of the current state of the theory of two-electron atoms in intense laser fields has been written by Lambropoulos et af. (1998). Using the above-mentioned methods, the TDSE can be solved for more than two active electrons only on a restricted basis, built on judiciously chosen orbitals for each atomic system under consideration and combined with a “complete” basis for one or maximally two electrons. Several other methods, however, have been employed. In particular, time-dependentdensity functional theory allows the time-dependent problem to be formulated

274

C. J. Joachain, M. Dorr, and

N. Kylstra

a t-

Intensity (w/crn2) FIG.23. Comparison of the He+ (dotted line) and He2+ (solid line) yields predicted by the model of Watson et al. (1997) with the experimental results (+ and x ) of Walker et al. (1994). The broken line gives the SAEi (sequential) results. From Watson et al. (1997).

in terms of single-particle orbitals evolving in an effective, local potential (Ullrich and Gross, 1997; Erhard and Gross, 1997; Tony and Chu, 1998). In practice, the exchange and correlation potentials for the interelectronic interaction must be treated within some approximation in order to become computationally tractable. Other methods include a time-dependent approach based on the Thomas-Fermi model (Brewczyk et al., 1995) and a classical model using momentum-dependent potentials (Lerner et al., 1994).

H. RELATIVISTIC EFFECTS Relativistic effects arise when atoms interact with ultrastrong laser fields. Such effects are expected to become important when the “quiver” velocity of

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the electron approaches the velocity of light, i.e., when its ponderomotive energy Up is of the order of its rest mass energy mc2. Using Eq. (2), this means that the quantity

must then be of the order of unity. If the electric field strength &O and the angular frequency o are expressed in atomic units (a.u.), we have q = 1.33 x 10-5(&~/0)2. Thus, if o = 0.043a.u. (corresponding to a Nd-YAG laser), we see that q = 1 when &O = 11.8 a.u. - that is, when the intensity I = 4.9 x lo'* W cm-2. For a laser of higher angular frequency o = 1 a.u., we have q = 1 when &O = 274 a.u., corresponding to the very large intensity I = 2.6 x 1021W cm-2. In the low-frequency or tunneling ionization regime, as introduced in Section ILA, the laser-atom interaction can in first approximation be viewed in terms of the quasi-static model in which the electron moves in an instantaneous effective potential given by the Coulomb potential and the instantaneous electric field. Therefore, at very high intensities, relativistic effects will essentially manifest themselves in the dynamics of the free, laser-driven (relativistic) electron wavepackets. In this high-intensity, low-frequency regime, classical Monte Car10 simulations have been carried out (Kyrala, 1987; Keitel et al., 1993; Schmitz et al., 1998), as well as studies within the framework of the KFR theory (Krainov and Shokri, 1992; Crawford and Reiss, 1994). The situation is quite different for the case of a high-frequency laser, i.e., in the stabilization regime. As reviewed in Sections 1II.D and III.G, studies of the stabilization of atoms in intense, high-frequency laser fields have, with a few exceptions, relied on the nonrelativistic quantum theory. For sufficiently high intensities, a number of interrelated issues arise concerning the validity of a nonrelativistic approach and hence the degree of stabilization of atoms. These include the modification of the electron's quiver motion by the magnetic field component and retardation effects, both of which are not present in the dipole approximation,relativistic effects which involve the dressing of the mass of the electron due to its relativistic motion in the laser field and spin effects. The validity of the dipole approximation for atomic hydrogen has been tested by Bugacov et al. (1993), who included multipole terms beyond the dipole approximation, and Latinne et al. (1994) who solved numerically the Schrodinger equation with retardation. At high frequencies, only small modifications in the ionization probabilities of the ground state were found, compared to calculations using the dipole approximation,up to an intensity of I = 2.5 x 1019W cm-2. The possible influence of retardation on stabilization had been pointed out earlier by Katsouleas and Mori (1993).

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Within the framework of the high-frequency Floquet theory in the Kramers frame, a relativistic high-frequency approximation has been considered by Kaminski (1985, 1990)and Krstit and Mittleman (1990). This generalizes the high-frequency theory described in Section III.D.3 to the relativistic domain. Ermolaev (1998) has studied atomic states of one-electron atoms in superintense laser fields, using a one-dimensional version of the high-frequency approximation of Krstii and Mittleman. Relativistic, classical Monte Car10 simulations of ionization and harmonic generation have been performed by Keitel and Knight (1995) for hydrogen atoms with emphasis on the stabilization regime. The magnetic field component of the laser field may induce a significant motion of the electron in the propagation direction of the field. As a result, they found that this leads to additional ionization, a possible breakdown of stabilization, and modified harmonic generation. Ultimately, issues concerning the stabilization of atoms in ultra-intense laser pulses can be addressed by numerically solving the time-dependent Dirac equation. This is a formidable task because computationally the problem scales approximately as &: /a4.For this reason, nearly all the quantum-mechanical calculations of laser-atom interactions in the relativistic domain have been restricted until now to lower-dimensionaltreatments. In particular, Protopapas et al. (1996a) have solved the time-dependent relativistic Schrodinger equation in the Kramers frame, and Kylstra et al. (1997) have used a B-spline expansion in momentum space to solve numerically the time-dependent Dirac equation. In both cases, the calculations were performed for a model, one-dimensional atom in the high-frequency, high-intensity (stabilization) regime. Magnetic field and retardation effects are clearly not included in one-dimensional model calculations, because their description requires an additional spatial dimension. On the other hand, relativistic effects due to the dressing of the electron mass by the laser field (Brown and Kibble, 1964) can be investigated. Protopapas et al. (1996a) found differences between the relativistic and nonrelativistic results, with the stabilization of the relativistic model atom slightly improved with respect to its nonrelativistic counterpart. They attributed this difference to the mass-shift of the electron in the laser. In one-dimension, the influence of the negative energy states can also be investigated. Kylstra et al. (1997) found that for a peak electric field strength &O = 175 a.u. and an angular frequency o = 1 a.u. (such that q = 0.4), relativistic effects become apparent. Even under these extreme conditions, however, the Dirac wave function remains localized in a superposition of field-free bound states and low-energy continuum states. We show in Fig. 24 the results of Kylstra et al. (1997) for the Dirac and Schriidinger probability densities (the latter being obtained from the numerical solution of the time-dependent nonrelativistic Schrodinger equation) at the end of the ninth laser cycle, when the electric

277

HIGH-INTENSITY LASER-ATOM PHYSICS

x (a.u.) FIG.24. The Dirac (solid line) and Schrodinger (dashed line) probability densities, at the end of the ninth cycle, for a laser pulse with a four-cycle sin2 turn-on, an angular frequency w = 1 a.u., and a peak electric field strength €0 = 175 a.u. From Kylstra et al. (1997).

10-19

10-1

100

10’ 102 103 Electron energy (0.u.)

104

RG.25. The ejected electron energy distribution obtained from the Dirac wavefunction after the laser turn-off, for a laser pulse with a four-cycle sin’ turn-on, an angular frequency w = I a.u., and a peak electric field strength €0 = 175 a.u. From Kylstra et al. (1997).

field is maximum. The peak in the Dirac probability density corresponds to the relativistic “classical” excursion amplitude, xo = 124 a.u. Likewise, the peak in the Schrodinger probability density occurs at xg = 175 a.u., the nonrelativistic classical excursion amplitude. At the end of the pulse, the ionization probabilities are, respectively, 0.52 for the Dirac wave function and 0.58 for the Schrodinger wave function, indicating that the Dirac wave function is slightly more stable against ionization. In Fig. 25, the energy distribution of the ionized electrons, obtained from the Dirac wave function at

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N. Kylstra

the end of the pulse, is shown. In accordance with the above discussion, it is found that the majority of the electrons are emitted with very low energies, a result also obtained by Faisal and Radozycki (1993a, 1993b), who considered an exactly solvable (separable potential) model of a bound Klein-Gordon particle interacting with an intense laser field. Rathe et af. (1997) have studied the relativistic wavepacket and spin dynamics of a single electron initially bound by a smoothed Coulomb potential in two dimensions, using the Dirac equation, and Szymanowski et al. (1999) have investigated harmonic generation as a function of increasing nuclear charge using the one-dimensional Dirac equation. Finally, we note that a number of other interesting phenomena involving super-intense laser fields have been studied recently. These include nonlinear Compton scattering (Bula et af.,1996;Hartemann and Kerman, 1996; Salamin and Faisal, 1997), laser-assisted Mott scattering (Szymanowski et al., 1997), and positron production in multiphoton light-by-light scattering (Burke et af., 1997).

IV. Conclusions and Future Developments The availability of very intense laser pulses whose electric field strength approaches or even exceeds the Coulomb binding field within atoms has made possible, in recent years, the discovery of new phenomena in laser-atom interactions. These include the “above-threshold ionization” process in multiphoton ionization, the stabilization of atoms at super-high intensities, the emission by atoms of very-high-orderharmonics of the exciting laser light, and multiphoton processes in laser-assisted collisions. In this article we have given a survey of these phenomena and of the main theoretical methods which have been used to analyze them. A similar review could be written about molecules in intense laser fields, where new effects such as Coulomb explosions, bond softening, and coherent control of dissociation have been discovered (see Bandrauk, 1993; Codling and Frasinski, 1994; Giusti-Suzor et al., 1995). The interaction of clusters of atoms with strong laser fields constitutes a new area of multiphoton physics, where enhanced yields of high harmonics and the generation of very energetic ionization fragments have been observed (Ditmire et al., 1997; Tisch et al., 1997). Solid targets and plasmas interacting with intense laser pulses also exhibit a wide range of interesting phenomena and potential applications, such as electron acceleration to relativistic energies and recombination X-ray lasers (Perry and Mourou, 1994). As seen from Fig. 26, the peak power of lasers has increased by twelve orders of magnitude since the sixties. The petawatt ( lOI5 W) laser now being developed at Livermore (Perry et al., 1996) will yield intensities of the order of lo2’Wcm-*. At such

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HIGH-INTENSITY LASER-ATOM PHYSICS I"

a,

u 104

10'2 0

a

a,

4

6

E

103

109

a

106 103 100

1960

1970

1980 Year

1990

2000

FIG.26. Evolution since 1960 of peak laser power (from Perry and Mourou, 1994) and of computing performance.

intensities, the electric field strength &o 11 lo'* Vcm-' is about two hundred times larger than the atomic unit ( E , = 5.1 x 109Vcm-'), the radiation pressure P = I / c reaches the enormous value of 300 Gbar, and the quiver motion of the ejected electrons (for an angular frequency o = 0.043 a.u. of a Nd-YAG laser) is fully relativistic. Much work will be required to understand the phenomena occurring in this regime, which is relevant for studies of the Fast Ignitor concept of inertial confinement fusion, as well as for astrophysical applications(Mourou et al., 1998).As shown in Fig. 26, this will be facilitated by the emergence of a new generation of massively parallel supercomputers.

V. Acknowledgments It is a pleasure to thank our colleagues for many fruitful discussions, particularly P. L. Knight, C. H. Keitel, A. M. Ermolaev, and W. Becker for their comments on a preliminary version of this manuscript. This work has been supported by the European Commission HCM program and by the Belgian IISN.

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ADVANCES IN ATOMIC. MOLECULAR.AND OFTICAL PHYSICS. VOL. 42

COHERENT CONTROL OF ATOMIC. MOLECULAR. AND ELECTRONIC PROCESSES MOSHE SHAPIRO Department of Chemical Physics. The Weizmann Institute. Rehovot. Israel

PAUL BRUMER Chemical Physics Theory Group. Department of Chemistry. University of Toronto Toronto. Canada

.

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Preparation and Dynamics of a Continuum State . . . . . . . . . . . . . . . I11. Bichromatic Control of a Superposition State . . . . . . . . . . . . . . . . . IV. The Coherent Control Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Weak-Field Coherent Control: Unimolecular Processes . . . . . . . . . . . A . Interference Between N-Photon and M-Photon Routes . . . . . . . . . 1. One-Photon versus Three-Photon Interference . . . . . . . . . . . . 2. One-Photon versus Two-Photon Interference . . . . . . . . . . . . . 3 . Two-Photon versus Wo-Photon Interference . . . . . . . . . . . . . 4 . Polarization Control of Differential Cross Sections . . . . . . . . . B. Pump-Dump Control: Two-Level Excitation . . . . . . . . . . . . . . . . C. Symmetry Breaking and the Generation of Chirality . . . . . . . . . . VI . Strong-Field Incoherent Interference Control . . . . . . . . . . . . . . . . . . A. Theory of Incoherent Interference Control . . . . . . . . . . . . . . . . . B. Computational and Experimental Demonstration . . . . . . . . . . . . . VII. Coherent Control of Bimolecular Processes. . . . . . . . . . . . . . . . . . . A. Degenerate em Superpositions .......................... B. Sculpted Imploding Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Optimized Bimolecular Scattering: Enhancement and Total Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287 289 296 304 304 304 305 309 311 313 315 321 325 326 330 332 334 337 339 342 343 343

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I Introduction One of the central questions in the physical sciences is the extent to which the present determines the future. Quantum mechanics. although a probabilistic theory. gives a deterministic answer to this question: given the wavefunction of an isolated system in the present. the system wavefunction in the future is 287

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completely determined. This is a consequence of the fact that the Schrodinger equation is a first-order differential equation in the time variable. Thus, if we wish to predict future probabilities, all we need to do is numerically solve the time-dependent Schrodinger equation, propagating from the present to the future. In spite of the obvious practical difficulties in applying such a program to many-body problems, there is in principle no reason why this cannot be done. Indeed, current methods enable the numerical solution of the time-dependent Schrodinger equation of many (Kosloff, 1988, 1994; Hammerich et al., 1994; Zhang and Miller, 1989; Manolopoulos et al. 1991) few- (three- and four-) particle systems. Although a buildup of integration errors with propagation time does occur (Leforestier et al., 1991), the errors are not expected to grow at a faster-than-polynomial rate. This fact is in sharp contradistinction, for example, to the situation in classical mechanics in the chaotic region. In that case, the exponential growth of integration error prevents the numerical determination of the state of the system to acceptable accuracy after the elapse of a sufficiently long time unless the initial phase-space coordinates are known to infinite accuracy. Given then that the integration of the Schrodinger equation is possible, and that given the present we are able to predict the future probabilities, a more ambitious question can be asked: If we know the initial wavefunction, what dynamics (e.g., what Hamiltonian) guarantees a desirable outcome (“objective”) in the future? This question constitutes the essence of the field now called quantum control. In practice, one can modify the Hamiltonian by introducing external fields (e.g., laser light) to alter the dynamics. It is then possible to answer the above question in a “trial-and-error” fashion; we guess a Hamiltonian, propagate the initial wavefunction into the future, compare the result with the desirable objective, and correct the guess for the Hamiltonian until satisfactory agreement with the objective is reached. Indeed, a systematic way of executing this procedure is the subfield called optimal control (Gordon and Rice, 1997; Tannor and Rice, 1985; Tannor et al., 1986; Tannor and Rice, 1988; Kosloff et af., 1989; Shi et al., 1988; Shi and Rabitz, 1989; Peirce et a f . , 1991; Jakubetz et al., 1990; Warren et al., 1993; Yan et al., 1993; Krause et al., 1993; Kohler et al., 1995). This trial-and-error method is very time-consuming, requiring the repeated solution of the time-dependent Schrodinger equation. Further, by its very nature it often leads to solutions that provide little physical insight. When the explicit time-dependent terms in the Hamiltonian serve only to prepare a state that then evolves in the absence of an external field, or when its explicit time dependence can be treated adiabatically, there exists a more elegant method, called coherent control (CC) (Brumer and Shapiro, 1986;Shapiro and Brumer,

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1987; Brumer and Shapiro, 1992; Shapiro and Brumer, 1993,1997; Bandrauk et al., 1992; Ivanov et al., 1995; Muller et al., 1990; Potvliege and Smith, 1992; Schafer and Kulander, 1992; Charron et af., 1993), which requires the solution of the (time-independent)Schrodingerequation only once. Moreover, in that case, the CC solution allows for the simultaneous exploration of other possible future outcomes (and not just a single “desirable” objective), which results from different preparations of the initial wavefunction, provides physical insight into the nature of the control solution and provides analytic formulas for control that are useful experimentally. The coherent control method is the subject of the present review. This review is organized as follows: Section I1 describes the basic principles behind the preparation and subsequent dynamics of a state excited by laser irradiation to the dissociative continuum. Section 111 then extends this approach to the excitation of a bound superposition state to show that quantum interference allows for control over dissociative dynamics. This idea, the principle of coherent control, is summarized in Section IV. Section V then describes a number of weak-field coherent control scenarios, including the demonstration that coherent control can be used to break symmetry and to generate chirality. In Section VI we introduce control methods in the strong-field limit, resulting in a powerful method (incoherent interference control) for the control of unimolecular processes. Section VII addresses the application of coherent control to collisional processes, and Section VIII provides a brief summary.

11. Preparation and Dynamics of a Continuum State The desire to attain control over natural processes is of greatest significance if the control objectives involve permanent changes. Transitory objectives, which, once reached, exist only over a fleeting moment in time, are of academic interest but are of very little practical use. Thus, the types of systems we shall review here are those where events, such as bond breaking (dissociation),ionization, or particle exchange, take place over a small region in configuration space (the “interaction zone”). As the system departs from the interaction zone, its constituents decouple from one another and cease to change thereafter. Under the above circumstances one is invariably dealing with continuous energy spectra. This is so because for bound states, which give rise to discrete spectra, decoupling at the end of the process is not possible. By its very definition, the constituents of a bound system remain close together at all times. Therefore, these constituentsrarely reach a configuration where they decouple and cease to interact with one another.

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Given that we are dealing with continuous spectra, the CC method utilizes some of the formal properties of the solutions of the multichannel scattering problem. In order to understand how and why these properties work, we first review the theory of the coherentpreparation of an initial state composed of a continuum of scattering eigenfunctions. Consider the case in which we prepare a wavepacket consisting of a superposition of scattering eigenstates IE, m*) of the material Hamiltonian, HM, where

subject to the normalization

Here E is the (continuous) energy and m designates all additional quantum numbers of relevance, e.g., the identity of the collision partners after the collision and all of the internal quantum numbers associated with each partner. The f notation differentiates between outgoing (+)and incoming (-) boundary conditions (Levine, 1969), as explained in detail below. As a specific method of preparation, we consider excitation of an initial bound state I El ) by a laser pulse of the form dwF(w)exp(-iot)

(3)

where e ( t ) is the pulse’s electric field vector, i is the polarization direction, and E(w) is the Fourier transform of E ( t ) at angular frequency o. We wish to solve the time-dependent Schrodinger equation,

where H is the total Hamiltonian in the presence of the laser field,

with d being the transition-dipole operator and d 2 its projection on the light’s polarization direction. Assuming that the radiation-free eigenstates that predominate are the initial state I El ) and a set of continuum states I E, m* ),

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29 1

we expand 9 as

I 9 ( t )) = bi ( t )I El ) exp( -iEl t / h ) +

N n= 1

bi(t)lEi) exp(-iElt/h)

1

dEbE,n(t)I E , n* ) exp( -iEt/h)

+ IQe(t))

(6)

where I \Tle(t)),defined by Eq. (6), is the excited portion of the wavefunction (the “excited wavepacket”). In order to calculate bl and bE,n we substitute Eq. (6) in Eq. (4) to obtain a set of coupled first-order differential equations,

dr

= (i/h)bl(t)exp(ioE,lr)E(r)(E,m’Id. ;[El)

(7)

where OEJ = (E - E l ) / h . In the presence of a sufficiently weak pulse, we can use first-order perturbation theory, according to which bl (t) w bl (t = 0) = 1, and bE,, at the end of the pulse Eq. (7) is (Shapiro, 1993)

After the pulse is over, the excited wavepacket is therefore given as

We wish now to investigate the long-time properties of Eq. (9). To do so we need to relate the eigenstates of HM to the eigenstates that describe the freely moving fragments at the end of the process. As an example, consider a triatomic molecule ABC, which breaks apart at the end of the process to yield the A BC channel (denoted q = 1) or the B AC channel (denoted q = 2). Below we explicitly discuss the formalism for the A BC product. However, the structure is the same for the B A C channel, with obvious substitutions in the equations. The Hamiltonian HM can be written as composed of three parts:

+

+

+

+

Moshe Shapiro and Paul Brumer

292

Here R is the radius vector between A and the B - C center of mass, r is the B - C separation, W(R, r) is the total potential energy of A, B, and C, and

-h2 2P

-h2 Kr=-V: 2m

KR=-v~

are the kinetic energy operators in R and r, and p and m are the reduced masses,

If v(r) denotes the asymptotic limit of W(R, r) as A departs from B - C ,

~ ( r=) lim W(R,r) R-iw

it is clear that the A - BC interaction potential, defined as

V(R, r) = W(R, r) - v(r) vanishes as R + as

00,

i.e., lirnR,,

(14)

V(R, r) = 0. Defining the B - C Hamiltonian

the triatomic Hamiltonian of Eq. (10) can now be rewritten, using Eq. (14), as

We see that it is the interaction potential V(R, r) that couples the motion of the A atom to the motion of the BC diatomic. In its absence, the two free fragments A and BC described by the free Hamiltonian

move independently of one another. Because HO is a sum of two independent terms, its eigenstates, I E, m;0 ) ,

are given as products

COHERENT CONTROL OF ELECTRONIC PROCESSES

293

with em being the internal (electronic, vibrational, rotational) energy of the B - C diatomic and with I km) satisfying

describing the free (translational) motion of A relative to BC. The solution of Eq. (21), written in the coordinate representation

describes an energy-normalized plane wave of kinetic energy E - em,

where

is the wavevector of the free motion of A relative to the BC center of mass. Because the free solutions assume the same continuous energy spectrum as do the full solutions I E, n*), they too satisfy the continuous spectrum normalization,

The eigenstates, I E, n*), of the fully interacting Hamiltonian HM are related to 1 E, n;0 ) via the Lippmann-Schwinger equation (Taylor, 1972)

where the plus solution is known as the outgoing solution and the minus solution as the incoming solution. Each solution is an independent (though not mutually orthogonal) eigenstate of the full Schrodinger equation [Eq.(l)].

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Moshe Shapiro and Paul Brumer

We now use the Lippmann-Schwinger equation to explore the long-time behavior of the wavepacket q e ( t ) created with the laser pulse. Either the outgoing or the incoming set of solutions can be used as the basis set for expanding Q e ( t ) .In what follows we shall see which type of solution is best for which purpose. Substituting Eq. (26) in Eq. (9), we obtain that

Using the spectral resolution of [E f if - Ho]-’, we have from Eq. (27) that the probability amplitude of finding a free state 1 E’, m;0) at time t is given as

Using the normalization of the free states [Eq. (25)], we have that

In the t -+ cx) limit the integration over E can be performed analytically by contour integration. Note first that in that limit, the integrand on a large semicircle of radius R in the lower part of the complex E plane is zero, because when E = Reie, with 8 < 0, exp(-iEt/h) = exp(-iReiet/h) = exp (-iR cos 8 t p )exp (R sin 8t/h),,,

+0

(30)

Thus, the result of the real E integration remains unchanged by supplementing it with integration along a large semicircle in the lower half E-plane. Because in the -if case, the integrand has a pole at E = E’ ie that is outside the

+

COHERENT CONTROL OF ELECTRONIC PROCESSES

295

closed contour, the whole integral is zero. We obtain that (d a i ( E 1 ) lim(E’,rn;O19,(t)) = (271i/A)E(0~’,1)exp(-iE’t/fi)(E’,m-

t+c€

(31)

Hence the coefficients of expansion of the full wavepacket of Eq. (9), in terms of the I E, m- ) states give the probability of observing states 1 E , m; 0) in the distant future. If instead of the incoming states we use the outgoing states, the closedcontour integration encirclesa pole at E = E‘ - k . Hence the integration yields

where the Sn,m(E’) matrix,

is known as the S-matrix or scattering matrix. The form of Eq. (32) appears more complicated than that of Q. (31) because each (E,m; 0 I Q e ( t ) ) component appears to be made up of contributions from all degenerate 1 E, n; 0) states. Why use it at all then? The reason is that in ordinary scattering events, we want to use states whose past is well known to us. These are the outgoing states because when t --+ -00, it is the contour on the semicircle in the upper half of the complex E-plane that vanishes. Supplementing the real-E integration by such a contour keeps the E = E‘ - k pole out of the contour, and we obtain that lim (E’,m;O19,(t)) = (2ni/fi)E(o~/,l)exp(-iE’t/fi)(E’,m+ Id. ; [ E l )

t+-w

(34) In contrast, the t -+ --oo limit appears more complicated with the incoming-states expansion because now the E = E’ if pole is enclosed by the contour, and we obtain that

+

x E S , ; , ( E ’ ) ( E ’ , n - Id il El ) n

(35)

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Moshe Shapiro und Paul Brumer

where the S , , ( E ’ ) matrix is defined as

In the case of the optical pulse excitation, we use the incoming solutions because the origin of the system in the remote past is known to be I El ) and our interest is in the fate of the system in the distant future.

111. Bichromatic Control of a Superposition State Consider now how the laser field can be made to modify the outcome of the photo-dissociation process. As seen above, the probability of populating a “free” state I em)l km) at any given time is

Using the expansion of 1 * ( t ) ) [Eq. ( 6 ) ] ,we have that

Assuming that (emIEl) = 0, (e.g.. the two states belong to different electronic states), it follows from Eq. (38) that in the long-time limit,

Because

it follows that

We see that the relative probabilities of populating different asymptotic states at a fixed energy E are independent of the laser power and pulse shape. This

COHERENT CONTROL OF ELECTRONIC PROCESSES

297

result, which coincides with that of perturbation theory, holds true irrespective of the laser power, provided that only one initial state I El ) is coupled to the continuum. In order to affect the long-time outcome, we must therefore extend the treatment beyond the use of a single initial bound state. For example, starting from a linear superposition of two initial states

we have that

In first-order perturbation theory, bl(t) and b2(t) are constant, so in the weak-field regime,

Recognizing that E(o)is complex, we can write

and transform Eq. (44)into

Then, the probability of observing product state n at infinite time is given as,

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Moshe Shapiro and Paul Brumer

We see that the pulse attributes have been "imprinted" onto the material matrix elements. As a result, by changing the pulse attributes, we can change the branching ratios into different product channels. The properties we wish to control are often the branching ratios to different chemical products. Note, however, that the approach advocated below, and indeed any CC scenario, can be readily modified to control probabilities of populating individual product states. Realizing that any chemical process such as

involves a multitude of internal fragment states (I en)) in each chemical channel, we calculate the total probability to produce products in one of the q channels as

Here we have modified our notation so that n denotes all quantum numbers other than the arrangement channel label q. By expanding the square, the above expression transforms to

where

where

COHERENT CONTROL OF ELECTRONIC PROCESSES

299

The P i ) ( E ) and P $ ) ( E ) terms are the probabilities of photo-dissociating levels I El ) and I E2), respectively. The P $ ) ( E ) is the interference term. It is the only term influenced by the relative phase 012 between the T ( o E , ~ ) and E(oE,J) field modes. In order to make the structure of the probability expression Eq. (5 1) clearer, we write the complex amplitude pi) as

where is the so-called “molecular” or “material” phase, and define the phase a12 via

With these definitions, the interference term assumes the form

+k)

Because of the dependence of on q, this term can be positive (“constructive interference”) or negative (“destructive interference”) with respect to one q channel and the opposite with respect to the other. Hence, by tuning the external phases a12 or 012, we can make the sign of this term negative with respect to one q chemical channel and positive with respect to another. In this way, by changing an external phase factor that is indifferent to the final channels, we attain selectivity (discrimination) between the final channels. The magnitude of this effect can be enhanced by varying the ratio

The method of controlling the final outcome of a process in this way is at the heart of coherent control. The structure of the CC equations is most transparent when we write Eq. (50) and Eq. (51) as

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Moshe Shapiro and Paul Brumer

In order to attain maximum control over Rdq, we can set P(q)(E)= 0 for one of the q channels. Both P(q)(E)and pfi) x2&) of Eq. (56) are positive, so the only way we can minimize P(q)(E)is by making the interference term h l p $ ) I cos[$$) - 0112 - 0121 as negative as possible, which means setting the external phases such that

+

a12

+ 012 = R - ($12( 4 )

(57)

Under these circumstances, cos[($$) - 0112 - 0121 = - 1 and Eq. (56) becomes

This is a quadratic equation in x that has a solution x = Ipk)I/p$, if and only (4) 2 - (4) (4) if I P n I - PI1 PZZ * In-order to see the circumstances under which this condition is fulfilled we write,

where

and we have used the fact that the projection operator P satisfies PP = P. Hence the ,)!p pg),and lp$)[ matrix elements are related as scalar products of the lEld . i P ) and IPd . iE2) state vectors. By the Schwarz inequality,

301

COHERENT CONTROL OF ELECTRONIC PROCESSES

with the equality holding only when lEld E^ P) and IPd. EIE2) are parallel to one another. If P is a projection onto a single state - i.e., no n summation need be performed - then the equality in Eq. (61) holds. That is, by definition, the case, because by definition,

In all other cases, because of the existence of many n internal states, the strict inequality holds and the solution of Eq. (58) can never be realized. Nevertheless, numerous computational studies, some discussed below, show that control is extensive. In general, experiments measure energy-averaged quantities such as the probability Pqof forming product in channel q, and the ratio Rqrq!of product in each channel:

because products are not distinguished on the basis of total energy. For the case considered above, the photo-dissociation of a superposition state, Pq(E) is nonzero at three energies: E = El ROE, 1 = E2 h 0 ~ , 2E, = El h 0 ~ , 2 , and E = E2 AWE,]. The contribution from the first of these energies, Pq(E= El h~~,l),isgiveninEq. (49), wheretheremainingcontributionsare

+ +

+

+

+

Thus, the overall Pq for N = 2 is given by

The latter two terms correspond to traditional photo-dissociation terms without associated interference contributions and provide uncontrollable photo-dissociation terms that we call “satellites.” In this and all coherent

302

Moshe Shapiro and Paul Brumer

control scenarios discussed below, it is important to attempt to reduce the relative magnitude of the satellite terms in order to increase overall controllability. As the first example of coherent control, consider the bichromatic control of the photo-dissociation of methyl iodide in the A band (near 266 nm): CH3 +I*(2f'1/2)

+

CH31+ CH3 +I(2P3/2)

(65)

The control objective is the formation of ground-state iodine [I( 2P3/2), denoted I] versus excited-state iodine [1(2Pl/2),denoted I*]. The computation considered a rotating collinear model in which the H3 center of mass, the C, and the I groups are assumed to lie on a line (Shapiro and Brumer, 1987). All satellite terms are included. The bound states I E i ) are characterized in this case by v , J , and M J , the vibrational, rotational, and magnetic quantum numbers. Figure 1 shows contour plots of the yield of CH3 +I* (i.e.. the fraction of product that is CH3 I*), as a function of 012 and s = x2/[1 x2] in the photo-dissociation of CH3I out of an equal superposition of the I q = 0, JI = 2) and I v2 = 1,J2 = 2) states at two different frequencies, WE,^ = 39,639cm-' and WEJ = 42,367cm-'. The MJ magnetic quantum number is averaged over, and all satellite terms are included. Clearly, control is extensive, ranging from 0.3 to 0.75 as the control parameters are varied. Note also that a comparison of Figs. l(a) and l(b), which correspond to results at different excitation frequencies, shows that there is considerable dependence of the control contour topology on frequency. This bichromatic scenario has been extended theoretically in a number of ways. For example, we have considered (Shapiro and Brumer, 1992) the extension of this scenario to a superposition of N bound states excited by N laser frequencies and demonstrated total control over the dynamics under certain conditions. We have also considered the two-level approach in the condensed phase (Shapiro and Brumer, 1989) in order to examine the effect of collisions and dephasing on control. In particular, the CC scenario described above was extended in the following way. The initial superposition state [Eq. (42)] was assumed prepared by two-photon absorption in the presence of collisions, modeled by a Bloch equation with appropriate T I ,T2 relaxation times. This transition was assumed saturated, establishing a time-independent density matrix describing this two-level system. This superposition was then pumped to dissociation by a pulsed laser whose width exceeds the spacing between the pair of bound levels. Thus, the pump laser contains both frequencies WE,1, WE,2 necessary to excite the superposition to the same continuum energy E. The resultant branching in CH3 I [@. (65)] was examined, and control was found to survive over a substantial temperature range. This model computation motivates applications of CC in the condensed phase, as do the experiments discussed in Section VIII.

+

+

COHERENT CONTROL OF ELECTRONIC PROCESSES

303

FIG. 1 . Contour plot of the yield of I' (i.e., fraction of I' as product) in the bichromatically controlled photo-dissociation of CH31 starting from an M-averaged initial state. (a) O E , I = 39,639cm-I, (b) OEJ = 42,367cm-I. (From Shapiro and Brumer, 1987.)

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Moshe Shapiro and Puul Brumer

IV. The Coherent Control Principle Photo-dissociation of a superposition state, the scenario described above, will be seen to be just one particular implementation of a general principle of coherent control, i.e., that coherently driving a state with phase coherence through multiple optical excitation routes to the samefinal state allows for the possibility of control. This procedure has a well-known analogy: the interference between paths as a beam of either particles or light passes through a double slit. In that case, interference between two coherent beams leads to spatial patterns of enhanced or reduced probabilities on an observation screen. In the case of coherent control, the overall coherence of a pure state plus laser source allows for the constructive or destructive manipulation of probabilities in product channels. Active control results because the excitation process explicitly imparts experimentally controllable phase and amplitude information to the molecule. It is important to note that, in general, quantum-interference-based control occurs only between energetically degenerate states. To see this, note that if the excitation creates product states of energy E and E l , then the interference term [e.g., Eq. (55)] would carry the phase exp [i(E- E’)t/h].Thus, this term, as well as the interference term, would average to zero over a small time interval, and control would be lost. Further, it is worth noting that CC scenarios often lead to simple analytic expressions for reaction probabilities in terms of a few molecular parameters and a few control parameters. Hence, the entire dependence of product probabilities on the control parameters can be easily generated experimentally once the molecular terms are determined from a fit of the control expression to a small number of experimentally determined yields. Numerous scenarios can be designed that rely on the essential coherent control principle. Several are discussed in the following sections.

V. Weak-Field Coherent Control: Unimolecular Processes A. INTERFERENCE BETWEEN N-PHOTONAND M-PHOTONROUTES Rather than starting with a nonstationary superposition state, as above, we can achieve CC by photo-dissociating a single stationary state via two optical paths (Shapiro et al., 1988). Such paths can consist, for example, of an N-photon process and an M-photon process satisfying NON = MOM, with ON and O M being the optical frequenciesof each path. The numbers N and M can be of the same parity or of opposite parity. It turns out that the latter allows for control over the photo-dissociation differential cross sections, whereas the former

COHERENT CONTROL OF ELECTRONIC PROCESSES

305

allows for control over both the integral and the differential cross sections. For simplicity we focus here on the three lowest-order cases (N, M) = (1,3), (N, M) = (1,2), and (N, M) = (2,2). Other cases, such as the (N, M) = (2,4) case (Bandrauk et af., 1992; Chelkowski and Bandrauk, 1991), and strongfield extensions (Charron et af.,1995; Szoke et af., 1991; Zuo and Bandrauk, 1996) have been discussed in the literature.

I . One-Photon versus Three-Photon Inte$erence We consider (Shapiroet af.,1988) a molecule, initially in state I g ) I E; ), where I g ) denotes the ground electronic state, subjected to two co-propagating CW fields of frequencies 01 and 0 3 with w3 = 301. The total Hamiltonian is given by

H = HM -2d.Re(t3F3exp(-iw3t)+i1F,exp(-iwlt)]

(66)

where C; = S(wi). We assume the following physics: (a) dipole transitions within electronic states are negligible compared to those between electronic states; (b) the fields are sufficiently weak to allow the use of perturbation theory; and (c) E; 2Rwl is below the dissociation threshold, with dissociation occumng in the I e)-excited electronic manifold. Given the above assumptions, the lowest-order expression for the onephoton or three-photon dissociation amplitude A,,,(E = E; hwl) is

+

+

where de,g= ( e Id. iI g ) and T denotes the three-photon transition operator, given in third-order perturbation theory as

Because all light sources have a finite frequency width, we can integrate over this width to obtain

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Moshe Shapiro and Paul Brumer

where the one-photon photodissociation probability is

the three-photon photodissociation probability is

and the one-photodthree-photon interference term is

As in our discussion of the photo-dissociation of a superposition state, we I and a “molecular” phase define a “molecular” interference-amplitude 6 y 3 )as

exp (

C ( Ei 1 T 1 E , 4 , m- ) ( E , 4 , m- I de, I Ei ) g

m

(73)

Recognizii-g that Ei may be complex, Ei = ICilei4i,we can write the above interference term as

The branching ratio Rqq/(E)for channels q and 4’ can then be written as

where

COHERENT CONTROL OF ELECTRONIC PROCESSES

307

and

where €0 is defined as a single unit of electric field. x is therefore a dimensionless parameter. The numerator and denominator of Eq. (75) both display the canonical form for coherent control, i.e., a form similar to Eq. (50) in which there are independent contributions from more than one route, modulated by an interference term. Because the interference term is controllable through variation of laboratory parameters, so too is the product ratio Rqql(E).Thus, the principle on which this control scenario is based is the same as that in Section III, but the interference is introduced in an entirely different way. With the qualitative principle of interfering pathways exposed, we demonstrate the quantitative extent to which the one-photon versus three-photon scenario alters the yield ratio in a realistic system by considering the photodissociation of IBr: I

+ Br

t

IBr + I

+ Br'

(78)

where Br = Br( 2P3p) and Br' = Br( 2P1p). Reliable IBr potential curves were used throughout the calculation. Computational results on this system were obtained (Chan et al., 1991) for two different cases: excitation from states of fixed Mi (the projection of the diatomic angular momentum Ji along the z-axis) and for the average over initial M i . Results, in the form of a contour plot of the Br' yield for excitation from 21 = 0, Ji = 1,Mi = 0, and Ji = 42 (Miaveraged) are shown in Figs. 2 and 3 as a function of s = x2/( 1 x2) and of the relative laser phase (43- 34q). The range of control in each case is impressive, with essentially no loss of control due to M averaging. The three-photon versus one-photon scenario has been experimentally realized in atoms (Chen et al., 1990) and by Gordon and coworkers (Park et al., 1991;Lu etal., 1992;Kleimanetaf., 1995;Zhu etal., 1995,1997) in a series of experimentson HCl, H2 S, and CO. In the case of HCl, the molecule was excited to an intermediate 3C- ( W vib-rotational ) resonance, using a combination of three o1(hl = 336nm) photons and one 0 3 (h3 = 112nm) photon. The 0 3 beam was generated from an 01 beam by tripling in a Kr gas cell. Ionization of the intermediate state takes place by absorption of one additional 01 photon. Similar demonstrations in ammonia, trimethylamine, triethylamine, cyclooctatetraene, and 1,l-dimethylhydrazine have been reported by Bersohn and coworkers (Wang et al., 1996b) and in Na by Cavalieri et al. (1997). Later

+

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Moshe Shapiro and Paul Brumer

3 rr, I

d?

0.2

0

0.4

0.8

0.6

I .o

J

FIG.2. Contour plot of the yield of Br’ (percentage of Br* as product) in the photodissociationof IBr by the one-photonversus three-photon scenario.The abscissa is the amplitude parameters = x2/(1 x 2 ) , and the ordinate is 03 - 301. Here 01 = 6657.5 cm-I. (From Chan et al., 1991).

+

studies (Zhu el aZ., 1995, 1997) demonstrated control over the production of different channels, specificallythe HI+ versus the H I channels, in the photoexcitation of HI. This result is highly significant, showing the ability to control the relative yield of products in photo-dissociation. In all of these experiments, control over Rqqf(E)was obtained by varying the phase difference ($3 - 34q) and the parameter x. In doing so, the experiments used co-propagating 01 and 0 3 beams with wavevectors suitably “phase-matched” so that Eq. (75) no longer contains the spatial coordinate z, and the interference term is independent of the position in space. It is also possible to use the one-photon versus three-photon (indeed any N-photon versus M-photon) control scenario to control differential cross sections. To see this, consider rewriting Eqs. (70) through (73) so that they apply to the probability of observing the product in channel q, but at a fixed scattering angle. Then the sum on m no longer includes an integral over scattering angles. The resultant interference term P t 3 )is nonzero, so varying the properties of the lasers will indeed alter the differential cross section into channel q.

+

COHERENT CONTROL OF ELECTRONIC PROCESSES

309

6

m I

a?!

144-

3

FIG.3 . As in Fig. 2 but for v = 0, Ji = 42,

MJ.(From Chan et al., 1991.)

01

= 6635.0cm-' with an average over initial

2. One-Photon versus Two-Photon Interference Although scenarios for simultaneous absorption of N plus M photons, where N ,M are of the same parity, allow for control over both the differential and integral photo-dissociation cross sections, this is not the case when N ,M are of different parity. In this case, only control over the differential cross section is possible. To see this, we consider the case of simultaneous one-photon versus two-photon absorption. The Hamiltonian for a molecule irradiated with two frequencies 01 and 0 2 , with 0 2 = 201,is

H = HM - 2 d . Re[i2Fzexp (--i02t)

+

i1Z1 exp (-iolt)]

(79)

+

Assuming that Ei ha1 is below the threshold for photo-dissociation and that absorption of 0 1 is via an intermediate electronic state that is dipole accessible, we obtain, in complete analogy to the one-photon versus three-photon case, that the probability P q ( E , k ) of photo-dissociation into channel q at recoil angles k = (&, &) is given by

+

+

Pq(E,k) = P!)(E, k) P!2)(E, k) P f ' ( E , k)

(80)

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Moshe Shapiro and Paul Brumer

where

Here, all channel indices m (which can be readily included) other than the final direction k have been suppressed for clarity. The interference-amplitude lFF2)(k)land molecular-phase SF2)(k) are defined by

where D is the two-photon transition operator, given in lowest-order perturbation theory as

The interference term P y 2 ) ( E , k )is generally nonzero, so control over the differential cross section is possible. Consider, however, the integral cross section into channel q,

and focus explicitly on the contribution from P y 2 )(E,k). That is, consider

where we have explicitly inserted the angular momentum characteristics of the initial state, which is of energy Ei, angular momentum J i , and projection Mi. Using the definition of D [Eq. ( 8 5 ) ] and inserting unity in terms of the

COHERENT CONTROL OF ELECTRONIC PROCESSES

311

states I Ej, J j , Mi) of the intermediate electronic state we get,

The above expression must be zero because it embodies two contradictory requirements: Dipole selection rules as applied to (Ei,Ji, Mi 1 dg,et 1 E,, J j , M,) require that Jj - Ji = f l , whereas by the same rules, (E’, Jj, Mil dd,eI E, k,q- ) x(E,k,qIEi,Ji,Mi)isnonzeroonlyifJ, - Ji=f2,0.HencePF2)(E) is zero; that is, coherent control over integral cross sections is not possible using the one-photon versus two-photon scenario. However, as noted above, control over the differential cross section is possible. A similar conclusion obtains for any N-photon versus M-photon process where N and M are of different parities. Experimental implementations of the one-plus-two photon absorption scenario have taken a variety of forms (Baranova et al., 1990; Tin et al., 1992, 1995; Dupont, 1995; Sheeny et al., 1995). For example, Corkum and coworkers (Dupont et al., 1995) have canied out one-photon versus two-photon absorption in crafted quantum wells to demonstrate control over the directional motion of the excited electron. Sipe, van Driel, and coworkers (Hache et al., 1997) have extended this work to the complex case of bulk semiconductors. Following the theoretical work of Charron et al. (1995), Sheeny et al. (1995) have used this scenario to control product directionality in HD+ dissociation to H D+ and H+ D.

+

+

3. Two-Photon versus Two-Photon Intet$erence

Here we show that by considering (resonantly enhanced) two-photon versus two-photon dissociation, it is possible to maintain control in a molecular system in thermal equilibrium. The resonant character of the excitations is important because in this way only one state, out of the ensemble of thermally populated molecular states, participates in the photo-dissociation. As shown below, it is also possible in this way to overcome phase jitter in the laser sources. Consider first photo-dissociation due to the absorption of two photons, of frequency 01 and 02, where the first photon is assumed resonant with an intermediate bound level. In this process a molecule, initially in a state [Ei,Ji,Mi), is photo-dissociated because of a combination of two CW fields,

+ El1C1 exp(-iolt)]

c(t) = 2Re[i2~2exp(-io2t)

(89)

to yield a number of different product channels labeled by q. The nearresonance condition means that absorption of the first photon, of frequency

312

Moshe Shapiro and Paul Brumer

w1 ,lifts the system to a region close to an intermediate bound state IE,J, M,). The second photon, of frequency 0 2 , further excites the system to a dissociating state IE, k, q-). The probability-amplitude Dkq(E, EiJiMi, w 2 , 0 1 ) for resonant two-photon (01 0 2 ) dissociation is given by (Chen et al.,1993b)

+

Dkq(E, EiJiMi, 0 2 , W l ) =

+

+

Here E = Ei ( 0 1 02)h, and A, and ,?I are respectively the radiative shift and width of the intermediate state. As a consequence of the form of the denominator in Eq. (90), the photodissociationprobability, given by the square of &q, is greatly enhanced by the inverse square of the detuning A = Ei ti01 - Em - A, irrn/2. Hence, only the levels closest to the resonance A = 0 contribute significantly to the dissociation probability. Ideally, this allows us selectively to photo-dissociate a single state from a thermal bath. This holds true as long as the line width ,?I is less than A and the spacings between neighboring transitions are smaller than the laser bandwidth. Consider now the simultaneous excitation by two resonant two-photon routes using three interrelated frequencies, 0 0 , a+,a- with associated field amplitudes and phases denoted as Eg, 4, C, and 80,8+, 8-, respectively, where wo and a+ are chosen resonant with intermediate bound-state levels. Choosing the three frequencies such that 200 = o+ 0-,we can make the absorption of 200 photons (pathway “a”) interfere with the absorption of an a+ and an a- photon (pathway “b”), because Ej 2 0 0 = Ei (a+ 0-)= E. The probability of photo-dissociation at energy E into arrangement channel q is therefore given as the square of the sum of the two-photon amplitude of pathway “a” and the two-photon amplitude of pathway “b”:

+

+

+

+

+

+

Pq(E,EiJiMi;00,0+,0-)

+

+

Pq(a) Pq(b) P,(ab)

(91)

Here Pq(a) and P,(b) are the independent photo-dissociation probabilities associated with routes a and b respectively and Pq(ab)is the interference term between them,

COHERENT CONTROL OF ELECTRONIC PROCESSES

313

where the interference-amplitude IF,(&) I and the molecular phase difference (6: - 6,") are defined via the equation

We see that control over the the quantity of interest, the channel branching ratio R,,!, can be attained, as in previous scenarios, by varying such quantities as the (280 - 8+ - 8-) phase difference and the ( x = IZ+E-/Cil) amplitude ratio. Besides the ability to work in a thermal environment,the great advantage relative phase term allows of this control scenario is that the (280 - 8+ - €I-) for the cancellation of individual phase jitters arising in each laser source. For example, one can generate the o+ and 0-frequencies by frequency summing o+ = 00 + 6 and frequency differencing o- = 00 - 6 with a third source of frequency 6. Because a+ and a- are generated in this way, the phase difference 280 - 8+ - 8- between path (a) and path (b) vanishes (Schubert and Wilhelmi, 1986). Thus, fluctuations in 80 or 6 cancel and have no effect on the interference term. To demonstrate the range of control afforded by this scheme, consider the photo-dissociation of the Na2 molecule in 'ui = Ji = 0 [Fig. 41 to form the Na(3s) Na(3p) and Na(3s) Na(4s) products. As the I E m J m M m ) intermediate resonance we choose vib-rotational states belonging to the spin-orbit coupled A'C; and b3111,electronic manifolds (Chen et al., 1993b). Despite the multitude of electronic states involved in the process, the predominant contributions to the products Na(3s) Na(3p) and Na(3s) Na(4s) are found to come (Chen et al., 1993b) from the 'IIg and ' E l states. Typical control results are shown in Fig. 5 , which shows a contour plot of the Na(3s) + Na(4s) yield as a function of the ratio of the laser amplitudes x , and of the relative laser phase 68 = 280 - 8+ - 0-. We show the results of photo-dissociation with wavelengths ho = 594.505 nm, h+ = 582.057 nm, and 1- = 607.498nm (corresponding to wg,o+, and 0 - ) excited via the CT v, = 13 and 18, Jm = 1 intermediate states. The range of control is considerable, with the Na(4s) yield varying from 10% to 5 1% as 68 is varied. Experimental demonstrations of this scenario for the ionization of atomic Ba (Wang et al., 1996b) and NO (Pratt, 1996) have been reported.

+

+

+

+

'

4. Polarization Control of Differential Cross Sections

Rather than attempting coherent control with two different frequencies, it would seem that the use of two different polarizations of the same frequency would be much easier to implement experimentally. It turns out that this

314

Moshe Shapiro and Paul Brumer

. ...

. ..

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

0.12

--.

__________-----*-

0.09 0

v

-

v)

0

'=c 0.06 a,

+.

0

a $0.03

z

0.00

-0.03

6

4

8

10

12

14

16

18

R (Bohr)

FIG.4. Na2 potential energy surfaces relevant to the two-photon versus two-photon control scenario. The arrows indicate the resonant two-photon versus two-photon pathways included in the computation discussed here. (From Chen et aL, 1993a.)

scenario is akin to the one-photon versus two-photon control in the sense that integral control is not possible. Further, though differential cross sections can be controlled (Asaro et al., 1988), there is no breaking of the forwardbackward symmetry in this case. In order to see this, we consider the photo-dissociation of a single bound state [ E l ) by a single CW source of the type 6

+

= €1 exp (ia1)il €2 exp (ia2)i2

where 21 and & are two orthonormal vectors. We can regard the two components 21 and i 2 as inducing two independent excitation routes. Choosing il and 22 parallel and pevendicular to the quantization (z) axis, respectively, the differential cross section is composed of three terms; one correspondsto photo-dissociation of (El)by the 2, component, one corresponds to photo-dissociation of [El ) by the 22 component, and one is the cross term between these two contributions. Excitation by the parallel

COHERENT CONTROL OF ELECTRONIC PROCESSES

3 15

2Tr

68 4 0 n .--/

0

0 5

FIG.5. Contours of equal Na(4s) yield in the controlled photo-dissociation of Nap, initially in vibrational state v = 10. The ordinate is the relative laser phase, and the abscissa is the field intensity ratio s. (From Chen et al., 1993a.)

component allows AMJ = 0 transitions, whereas excitation by the perpendicular i 2 component allows AMJ = f l transitions. The interference term is therefore comprised of a product of two bound-continuum matrix elements, where the two continua differ in MJ by f1. If this cross term is nonzero, then control over the differential cross section is possible. However, producing the integral cross section necessitates integrating the differential cross section over k,and under these circumstances, the cross term vanishes. Contrary to the one-photon versus two-photon case, the states comprising the I E, k- ) state are of the same parity. Thus the backward-forward symmetry is not broken. The control manifests itself in our ability to sharpen or broaden the angular distribution about a given recoil direction. Polarization control in intense fields has been proposed as a means of generating subfemtosecond pulses (Ivanov et al., 1995), but it has yet to be demonstrated experimentally.

B. PUMP-DUMP CONTROL: TWO-LEVEL EXCITATION Control of the dynamics via a pump-dump scenario was first introduced by Tannor and Rice (1985). These authors emphasized the localized wavepacket aspects of pump-dump control, entailing the excitation of, and interference

316

Moshe Shapiro and Paul Brumer

between, many levels. In this section, we consider excitation of only two levels. It can be regarded as the pulsed analog of the bichromatic control with a superposition state outlined in Section 111. Consider a molecule, initially (t = 0) in an eigenstate 1 El ) of the molecular Hamiltonian HM,which is subjected to two transform-limited light ) of two temporally separated pulses pulses. The electric field ~ ( tconsists E ( t ) = cx(t)

+ Ed(?)

(94)

The pump pulse ex(?) induces a transition to a linear combination of the eigenstates I E i ) of the excited electronic state. Though the pump pulse may be chosen to encompass any number of states, here we choose it sufficiently narrow to excite only a superposition of two states I E2) and I E 3 ) . The dump pulse cd(r) dissociates the molecule by further exciting this superposition state to the continuous part of the spectrum. Both fields are chosen sufficiently weak for perturbation theory to be valid. For convenience we use Gaussian pulses that peak at t = tx and t d , respectively. In particular, the excitation pulse is of the form

The associated frequency profile is given by

Ex(o)= (fi/2)cX~,exp[+(ox - w)tx]exp [-Z;(O~

-

0)~/4]exp(-iS~)

(96) By writing Ex(o)= IEX(w)lei4(o), we see that $(o)= (o- ox)tx - 6,. The analogous quantities e d ( t ) and Z d ( o ) are defined similarly, with the parameters td and od replacing tx and ox,etc. Because the two pulses are temporally distinct, it is convenient to deal with each of their effects independently. The superposition state prepared after the ex(?) pulse is over is given in first-order perturbation theory as

I $ ( t ) ) = I El )e-iE1rlK + b21 E2 )e-iE2rlh+ b3l E3 )e- iE3rlh

(97)

where [Eq. (8)] bk = (271i/h)(Ek Id *

El

)EX(ok,l),

k = 2,3

(98)

= (Ek - E l ) / A . with ok,] After a delay time of z = td - tX.the system is subjected to the E d ( ? ) pulse. It follows from Eq. (97) that at that time, each preparation coefficient has

COHERENT CONTROL OF ELECTRONIC PROCESSES

317

picked up an extra phase factor of e-iEkslh,k = 2,3. Hence, the phase of b2 relative to b3 at that time increases by [-(E2 - E3)z/fi = 03,2z]. Thus the natural two-state time evolution controls the relative phase of the two terms, replacing the externally controlled relative laser phase of the bichromatic control scenario of Section 111. After the conclusion of the Ed(t) pulse, the system wavefunction is given as

The probability of observing the q product at total energy E in the remote future is therefore given by

where oEEk = ( E - Ek)/h, bk is given by Eq. (98), and Fd(0EEK)is given via an expression analogous to Eq. (96). Expanding the square gives

Integrating Eq. (101) over E to encompass the full width of the second pulse yields the final expressions for the quantities we wish to control: Pq, the probability of forming channel q, and R , , ! , the ratio of product probabilities into q versus q'.

318

Moshe Shapiro and Paul Brumer

Examination of Eq. (101) makes clear that R , , t can be varied by changing the delay time r = (td - t,) or the ratio x = lb2/b3I; the latter is most conveniently done by detuning the initial excitation pulse. Note that once again, as in the scenarios above, the z dependence of Pq vanishes because of cancellation between the excitation and dump steps. In addition, the phases 6,, &jdo not appear in the final result, so the relative phases of the two pulses do not affect the result. To gain insight into the control afforded by this scenario, we initially applied (Seideman et al., 1989) it to a model collinear branching photodissociation reaction with masses of D and H, i.e., H +HD

+-

DH2 + D

+ H2

(103)

in which one uses the first pulse to excite a pair of states in an electronic state supporting bound states and the second pulse to dissociate the system by deexciting it back to the ground state, above the dissociation threshold. Qpical control results (Seideman et al., 1989) are displayed in Fig. 6, which shows contours of equal DH yield as a function of Ex - EAVand z. Here (Ex - EAV) measures the deviation of the central excitation energy of the pump pulse from the average energy EAV of the pair of bound states it excites. The DH yield is shown to vary significantly, from 16% to 72%, as the control parameters are varied. This is an extreme range of control, especially if one considers that the product channels differ by only a mass factor. It is highly instructive to examine the nature of the superposition state prepared in the initial excitation, [Eq. (97)] and its time evolution during the delay between pulses. An example is shown in Fig. 7, where we plot the wavefunction for a collinear model of DH2 photo-dissociation [Eq. (103)l. Specifically,the axes are the H HD reaction coordinate S and its orthogonal conjugatex. The wavefunction is shown evolving over half of its total possible period. An examination of Fig. 6 in conjunction with Fig. 7 shows that deexciting this superposition state at the time of panel (b) would result in a substantially different product yield than de-exciting at the time of panel (e). However, Fig. 7 shows that there is clearly no particular preference of the wavefunction for either large positive or large negative S at these particular times, which would be the case if the reaction control were a result of some spatial characteristics of the wavefunction. Rather, the results make clear that the essential control characteristics of the wavefunction are encrypted in the quantum amplitude and phase of the created superposition state. The pump-dump scheme has also been applied (Abrashkevich et al., 1998b) (computationally) to the photo-dissociation of a fully realistic representation of Liz to control the cross sections for production of Li(2s) + Li(2p), Li(2s) Li(3p) and Li(2s) Li(3s). In particular, a CW laser was

+

+

+

COHERENT CONTROL OF ELECTRONIC PROCESSES

319

>

0 W I X

W

FIG.6. Contour plot of the DH yield as a function of the detuning of the exciting pulse Ex - E,, and the delay variable T. In this case the time between pulse centers is AT = (8.44 2.11n)ps + 7,ensuring nonoverlappingpulses and allowing for arbitrary positive integer n. Here the initially created superposition state is between levels 56 and 57 (El = 0.323 849a.u.. E2 = 0.323 968 a.u.) of the G1 surface. The letters H and L denote the positions of the absolute maxima and minima, whose magnitudes are explicitly shown. (From Seideman et al., 1989.)

+

FIG.7. Time evolution of the square of the wavefunction for a superposition state comprised of levels 56 and 57. The probability is shown as a function of S and its orthogonal coordinatex at times (a) 0, (b) 0.0825 ps. (c) 0.165 ps, (d) 0.33 ps, (e) 0.495 ps, and (f) 0.66 ps, which comespond to equal fractions of one half of the period 211/02,1. (From Seideman et al., 1989.)

320

Moshe Shapiro and Paul Brumer h,=805.6 nm, +lo45

nm, A,=95

cm-'

100

9 a

h

20

N

Y

2l

0

FIG.8. Li(2p) and Li(3p) yields in the pump-dump controlled photo-dissociation of Liz as a function of the delay between pulses. Wavelengths of the two pulses, as well as the frequency width of the second pulse, are indicated. (From Abrashkevich et al., 1998b.)

used to prepare a single rovibrational state of the A' C rf electronically excited state. Subsequent pump-dump excitation allowed extensive control over product yields with, for instance, Li(2s) Li(2p) ranging from 2% to 82% as the time delay between pulses is varied over 1 ps. Simultaneously, as seen in Fig. 8, the Li(2s) Li(3p) product is exactly out of phase so that we have almost total control over the Li(3p) to Li(2p) ratio. Computations have also been done on the control of the polyatomic system:

+

+

D + OH +- HOD --t H + OD via the B-state of HOD (Shapiro and Brumer, 1993). Note, finally, that control is sensitive to the degree of laser coherence. If the pump laser is only partially coherent (Jiang and Brumer, 1991), then control can be significantly degraded (Jiang et al., 1996). This is not the case for the dump pulse, where a significant degree of laser incoherence can be tolerated. Experimental studies of pump-dump control fall into two categories: (a) the large number of pump-dump experiments where the dump pulse is used as a probe of the previously prepared dynamics (Potter er al., 1992; Baumert et af., 1991) but that can be interpreted as demonstrations of pump-dump control; and (b) those expressly designed to demonstrate coherent control. The latter category includes work by Wilson (Bardeen et al., 1997), Silberberg (Yelin er al., 1997), and Gerber (Assion er al., 1998). For example, Gerber et al. controlled the ratio of CpFeCOCl+/ FeCl+ products in the photo-fragmentation

COHERENT CONTROL OF ELECTRONIC PROCESSES

32 1

of CpFe(C0)2Cl using pulsed and chirped femtosecond sources, opening the way for laser control of large molecular systems. C. SYMMETRY BREAKING AND THE GENERATION OF CHIRALITY

Symmetry breaking occurs in nature whenever a system undergoes a (spontaneous or forced) transition to a nonsymmetric eigenstate (i.e., states that do not belong to any of the representations of the symmetry group) of the Hamiltonian. Such nonsymmetric eigenstates occur if there exist several degenerate eigenstates, each belonging to a different irreducible symmetry representation, because a linear combination of eigenstates of different symmetry will in general be nonsymmetric. Nonsymmetric eigenstates of a symmetric Hamiltonian occur naturally in the continuous spectrum of a BAB type molecule. It is clear that the 1 E, m, R - ) state, which correlates asymptotically with the dissociation of the right B group, must be degenerate with the I E, m, L- ) state, which correlates with the departure of the B group on the left-hand side. Hence, any experiment performed in the asymptotic B + A B or BA B regions must, by necessity, measure the probability of populating a nonsymmetric state. It is also possible to form symmetric I E, m, s- ) and antisymmetric I E, m, a- ) eigenstates of the same Hamiltonian by taking the plus and minus combinations of nonsymmetric states. However, symmetric and antisymmetric states are not directly observable in the asymptotic regime. We may say that the very act of observation of the dissociated molecule entails the collapse of the system to one of the nonsymmetric states. Because the probability of collapse to the I E, m,R - ) state is equal to the probability of collapse to the I E, m, 15- ) state, the collapse to a nonsymmetric state does not lead to a preference of R over L in an ensemble of molecules. The above collapse is due to (random) factors that are not in our control. Coherent control techniques do not change this b‘spontaneoussymmetry breaking” aspect of quantum mechanics. Rather, as we show below, it allows us to bias the apn’ori probability of producing the R or the L form. One of the most important cases of symmetry breaking arises when the two B groups (now denoted as B and B’) are not identical but are enantiomers of each other. (fro molecules are said to be enantiomers of each other if one is the mirror image of the other. If these groups are also “chiral”, i.e., if they lack a center of inversion symmetry, then the two enantiomers are distinguishable and can be detected through the distinctive direction of rotation of linearly polarized light). The existence and role of enantiomers are recognized as one of the fundamental broken symmetries in nature (Barron, 1982; Woolley, 1975; Walker, 1979). It has motivated a long-standing interest in asymmetric synthesis, i.e.,

+

322

Moshe Shapiro and Paul Brumer

processes that preferentially produce specific chiral species. Contrary to the prevailing belief (Barron, 1986)that asymmetric synthesis must involve either chiral reactants or chiral external system conditions such as chiral crystalline surfaces, we have shown (Shapiro and Brumer, 1991) (and review below) that preferential production of a chiral photofragment can occur even though the parent molecule is not chiral. In particular, two results have been demonstrated (Shapiro and Brumer, 1991): (1) ordinary photo-dissociation, using linearly polarized light, of a BAB’ “pro-chiral” molecule can yield different cross sections for the production of right-handed (B) and left-handed ( B ’ ) products if the projection of the angular momentum (mj) of the products is selected; and (2) this natural symmetry breaking may be enhanced and controlled using coherent control. To treat this problem, consider the pump-dump scenario described in Section V.B, with attention focused on control of the relative yield of two product arrangement channels, where the product angular momentum projection m, is fixed. That is, we consider P J E ; m,)with q labeling either the right- (q = R) or the left- (q = L ) handed product. As above, the product ratio Rqqt = P R ( E ;m,)/PL(E;mi) is a function of the delay time z = ( t d - t x ) and the ratio x = lcl/c21. the latter by varying the energy of the initial excitation pulse. Active control over the products B AB’ versus B’ AB, i.e., a variation of Rq4t with z and x , and hence control over left- versus right-handed products, will result only if P R ( E ;mi)and PL(E;m,) have different functional dependences on the control parameters x and z, To show that PR(E;m,) may differ from PL(E;mi) for the B‘AB case, note first that this molecule belongs to the C, point group. This group possesses only one plane of symmetry, denoted 0,which is defined as the collection of points satisfying the requirement that the B - A distance equals the A - B‘ distance. Furthermore, we shall focus upon transitions between electronic states of the same representations, e.g., A’ to A’ or A” to A” (where A’ denotes the symmetric representation and A” the antisymmetric representation of the C,group). We further assume that the ground vibronic state belongs to the A’ representation. To obtain control, we wish to choose the intermediate state I E3) to be symmetric and the intermediate state 1 E2 ) to be antisymmetric with respect to reflection in the o-plane. Hence we first demonstrate that it is possible optically to excite, simultaneously, both the symmetric I E3) and the antisymmetric I E2) states. Using Eq. (98) we see that this requires the existence of both a dipole component that is symmetric, denoted dz,,g,and a component that is antisymmetric, denoted d:l,g,because by the symmetry properties of I & ) and IE2),

+

( E 3 I d e l , g I E ~ )= (E31d:f,glE1)1( E 2 J & , g l E 1 ) = (E21dzj,glE1)

+

(104)

COHERENT CONTROL OF ELECTRONIC PROCESSES

323

Both dipole-momentcomponentsdo occur in A' + A' electronic transitions whenever a bent B' --A --B molecule deviates considerably from the equal-distance CzV geometries (where d a = 0). The effect is non-FranckCondon in nature because the dipole moment must vary with the nuclear configurations. In the terminology of the theory of vibronic transitions, both symmetric and antisymmetric components can be nonzero because of a Herzberg-Teller intensity borrowing (Hollas, 1982) mechanism. It is therefore the case that the excitation pulse can create a I E3 ) , I E2 ) superposition consisting of two states of different reflection symmetry, a state that is therefore nonsymmetric. We now show that the nonsymmetry created by this excitation of nondegenerate bound states translates to a nonsymmetry in the probability of populating the degenerate I E , m, R- ), I E , m, L- ) continuum states. To do so we examine the properties of the bound-free transition matrix elements (E,q, m-ld,,,flEk) that enter into the probability of dissociation [Eq. (loo)]. Note first that although the continuum states I E , q, m - ) are nonsymmetric, we can define symmetric and antisymmetric continuum eigenfunctions IE, m, s-) and IE, m, a - ) via the relations

Note that IS( E , m, R- ) = I E , m, L- ). Using the fact that 1 E3 ) is symmetricand 1 E2) is antisymmetric, and adopting the notation as2 = ( E ,m, s- Idt,,,l E2), s,3 = (E,rn,~-IdS,,,,IE3),etc.,wehave[seeEq.(102)],

where the plus sign applies for q = R, the minus sign applies for q = L, and ~ ~ ( 2=3 p)i ( 3 2 ) . Equation (109) displays two noteworthy features: 1 . pL(kk) # pR(kk),k = 2 , 3 . That is, the system displays natural syrnrnerry breaking in photo-dissociation from state I E3) or state I E2), with right- and left-handed product probabilities differing by

324

Moshe Shapiro and Paul Brumer

4C, Re(~:3ua3) for excitation from I E3) and by 4C, Re(us2~:2) for

excitation from I E 2 ) . Note that these symmetry-breaking terms exist only if the transition dipole operator possesses both a symmetric and an antisymmetric component, which can occur only if the FranckCondon approximation breaks down. 2. ~ ~ ( 2 #3 ~) ( ~ ( 2 3If )the . Franck-Condon approximation holds, there is no “natural” symmetry breaking of the type discussed above. However, even when the FC approximation holds, ~ ~ ( 2 #3 ~) ~ ( 2 3 ) and laser-controlled symmetry breaking according to Eq. (101) is possible. To demonstrate the range of expected control, we consider a model of enantiomer selectivity (Shapiro and Brumer, 1991), i.e., HOH photo-dissociation in three dimensions, where the two hydrogens are assumed distinguishable:

HO

+H

+ HOH + H

+ OH

+

+

The computation of Rqq!,the HO H (as distinct from the H OH) product for polarized OH fragments, was done using the formulation and computational methodology of Segev and Shapiro (1982) and Balint-Kurti and Shapiro (1981). Figure 9 shows the result of first exciting the superposition of symmetric plus antisymmetric vibrational modes [( 1, 0, 0) + (0, 0, l)] with rotational quantum numbers Ji = J k = 0 in the ground electronic state,

260

360

400

T ( f sec)

560

660

FIG.9. Contour plot of percent HO+H (as distinct from H+OH) in HOH photodissociation. The ordinate- is the detuning from E,,, = (& - E1)/2, and the abscissa is the time delay between pulses. (From Shapiro and Brumer, 1991.)

COHERENT CONTROL OF ELECTRONIC PROCESSES

325

followed by dissociation at 70,700 cm-’ to the B state using a pulse width of 200 cm-’. Results show that varying the time delay between pulses allows for controlled variation of Rqst from 61% to 39%! Finally, we discuss symmetrybreaking and chirality control with unpolarized OH fragments where a summation over the magnetic quantum number mj is performed. It can be shown that summing over mj eliminates all contributions to Eq. (109) that involve both I E , m, a- ) and I E , m,s- ), and as a result,

That is, natural symmetry breaking is lost upon mj summation, both channels q = R and q = L having equal photo-dissociation probabilities, and control over the enantiomer ratio is lost because the interference terms, which still exist, no longer distinguish the q = R and q = L channels.

VI. Strong-Field Incoherent Interference Control In this section we discuss both the theory and an experiment of an elegant strong-field laser control scenario. As we saw above, the quantum nature of weak-field CC manifests itself in the sensitivity of the outcome to a change in an external phase. In contrast, under some strong-field situations, the interference term may become independent of the phase of the light sources involved, which therefore no longer need be coherent. Instead of the phase, the interferenceterm now becomes sensitive to the relativefrequency between the two light sources. We call the resulting control scenario “incoherent interference control” (IIC) (Chen ef al., 1995). The above two features are very favorable from the experimental point of view because one can use conventional non-transform-limited lasers and molecules in thermal environments. Indeed, an experimental realization of IIC has already been reported (Shnitman et af., 1996). In general terms, the IIC scenario operates as follows: Consider a molecule in an initial bound state I Ei ) that absorbs two photons of frequency 01 and, in doing so, is excited to a continuum state I E , q, m-) via a resonant intermediate state 1 Ej, ). The outcome of this photo-dissociation process can be controlled by applying a “control laser” w2 that couples initially unpopulated bound states I Ejz) to the same continuum. With both lasers on, dissociation to I E , q, m-) occurs via numerous dissociation pathways. To lowest order,

326

Moshe Shapiiv and Paul Brumer

thesearetheroutesIEi)-.IE,,)--tIE,q,m-)aswellasIEi)--tIEj,)--t 1 El, q’, m’- ) I Ej2) -+ I E, q, m- ), etc. Contributions from these multiple --f

pathways to the product in a given channel q at energy E interfere (either constructively or destructively) with one another. Varying the frequency and intensity of the control laser alters the interference and hence the dissociation line shape and the yield of product into a given channel. The IIC scenario may be viewed as the multichannel extension of the “laser-induced continuum structure” (LICS) phenomenon (see, for example, Knight er al., 1990; Faucher et al., 1993; Cavalieri et al., 1998; and references therein). According to this view, the excitation by the 0 2 photon embeds an unpopulated bound state, I Ejl ), in the I E , q, m- ) continuum. As a result, the continuum becomes structured while the 1 Ejl ) state becomes an unstable resonance. The structured continuum then interferes with the two-ol -photon dissociation of the populated state I Ei). The main new feature of the IIC theory presented below is the discovery that this interference effect may be of a different nature for different final channels. A. THEORY OF INCOHERENT INTERFERENCE CONTROL

The equations governing the IIC scenario are most easily derived by treating both the light and the matter quantum-mechanically. The molecule, whose Hamiltonian is denoted by HM, interacts with a quantized radiation field with radiative Hamiltonian HR through the dipolar interaction term HMR.The total Hamiltonian H is then given by

The photo-dissociation process is characterized, as for classical light, by a transition from JEi), a bound eigenstate of H M , to JE,q,m-), a continuum eigenstate of the same Hamiltonian, which, as in the classical field case, correlates in the infinite future with the noninteracting I product IE state. When describing the radiation pulse, it is convenient to work with (multimode) number states INk),defined as the eigenstates of H R ,

where EN^ is the total radiation energy. The letters k = i andf are used to label the initial and final states, respectively. The eigenstates of HO = HM + HR are a direct product of the molecular and photon states; e.g., I(E,q,m-)N’) =

COHERENT CONTROL OF ELECTRONIC PROCESSES

327

( E ,q, m - ) IN’). We call these states “partially” interacting because they encompass interaction via the material part of the Hamiltonian only. The molecule-radiation interaction HMRis given in the dipole approximation (Cohen-Tannoudji et al., 1992) as HMR= -d

.E,

with

E

=i

- ;;af) 1

where d is the electric dipole operator, E is the amplitude of the radiation electric field, €1 = ( 2 n h ~ l / L ~ )il, ” 01~ , are the polarizationvector and angular frequency of mode I, respectively, and al, a! are photon annihilation and creation operators. The dynamics of photo-dissociation is completely described by the fully interacting state l ( E , q , m - ) , N i ) , which is an eigenstate of the total Hamiltonian H ,

The additional minus superscript on Nk indicates that the state [ ( E ,q, m-)N;) becomes the partially interacting state I (E, q, m - ) N k ) when the radiative interaction HMRis switched off. The [ ( E ,q, m - ) N , ) states satisfy an augmented Lippmann-Schwinger equation [whose purely material analogue is Eq. (26)] of the form

where the resolvent G ( z ) = l/(z - H ) . If the system is initially in the partially interacting bound state JEi,N;) = (EJINJ, and the radiation-matterinteraction is switched on suddenly, then the photo-dissociation amplitude to form the partially interacting state IE, q, m-) IN’) is given by ((E,q, m-)Ni IEi, Ni).Because ((E,q,m-)Nf(Ei,Nj)= 0, it follows from Eq. (116) that

+

Two quantities, derived from the ((E,q, m-)N’lHMRG(E+ ENf)IEj,Nj) matrix elements, which can be computed numericallyby a variety of techniques (Chen et al., 1994,1995; Shapiro and Bony, 1985; Balint-Kurti, 1986; Brumer and Shapiro, 1986b. Bandrauk et al., 1989), are of interest: P ( E , q,NflEi,Ni), the probability to obtain products in channel q at a given photon number state

328

Moshe Shapiro and Paul Brumer

distribution { N f } and total material energy E,

and the total dissociation probability to channel q,

Focusing on the case of a molecule in the presence of just two field modes, of electric field vectors €1 and €2 and frequencies 01 and w2, we can write the initial state as I E i , N ; ) = I Ei, n1,nz), where nl and n2 are the initial occupation numbers of the two field modes, and the molecule-radiation interaction HMRas

The two frequencies are chosen such that hwl M E,, - Ei, i.e., 01 is in resonance with the excitation frequency to state I Ej, ). 0 2 is chosen such that E; 2hwl M Ej, h02, which means that 2hol - h02 is in (2-1) resonance with the transition from I E i ) to 1 E,,) (see Fig. 1 for the application to Na2). Performing a perturbative expansion of Eq. (1 17) with HMR as given in Eq. (120), and retaining the two lowest-order terms, we obtain that

+

+

COHERENT CONTROL OF ELECTRONIC PROCESSES

329

describe transitions between I Ej, ) and I EjZ), accompanied by the absorption of one wIphoton and the stimulated emission of an 02 photon. Other sequential absorption and emission terms result from the higher-order contributions to Eq. (1 17). The term A in Eq. (122) describes the direct resonant two-photon dissociation path I E ; ) + I E , q, m- ) via the intermediate states I Ej, ) (path A). The term B describes the dissociation path I E ; ) + I E', q', m'-) --t I Ej2) + 1 E , q, m- ) induced by 01 plus 0 2 (path B). It is important to note that the relative sign of the terms A and B depends on the frequency 0 2 , resulting in a sensitivity of the final probability to the frequency of the control laser. Equation (121) describes the photon fields by number states. However, a complete analysis of interference between the A and B paths necessitates an understanding of the role of the photon phase. Hence we sketch the same argument using multimode coherent states I a ) = I a1 ) @ I az), where I al) and I a2) are coherent states of the 01 and 02 modes:

The quantity cti is related to the average photon number iii and to the phase c$i of the mi laser as a; = f i e x p (ic$i). ReplacingEq.(121)by ( a l ( E , m -IHMRG(E++E~,)lEi>la ) gives,within the rotating wave approximation,

where A and B are of the same form as A and B in Eq. (121) but with the photon numbers n1 and n:!replaced by the average photon numbers f i l and ji2. Hence, the leading terms in the dissociationprobability I ( a I ( E ,m- IHMRG(E) I Ei)lct) l2 can be obtained from Eq. (121) by replacing the photon numbers ni by the average photon number iii. Most significantly, we see that the photodissociation probability is independent of the laser phase. Examination of the higher-order terms in the perturbative expansion of Eq. (117) (within the rotating wave approximation)shows a similar cancellation of the laser phase. Thus, the interference between path A and path B exists even for incoherent light. Extensive control of the detailed and total probability to form a given q channel using this approach is demonstrated in the next subsection, where computations and experiments pertaining to Na2 photo-dissociation are discussed. We note that in the actual calculations, it was in fact easier

330

Moshe Shapiro and Paul Brumer

to compute ( ( E , q , m - ) , n l- 2,n2 IHMRG(E' + E N , ) I E ~ , ~ [Eq. I , ~ (117)l ~) directly, using such methods as the high-field extension of the artificial channel method (Bandrauk and Atabek, 1989; Chen et al., 1995; Shapiro and Bony, 1985)rather than the HMRperturbative expansion. Note also that an alternative, complementary perspective on IIC has been advanced by Kobrak and Rice (1998a), based on photoselective adiabatic passage (Kobrak and Rice, 1998b). Their approach provides useful insights into incoherent interference control in the high-intensity limit and successfully reproduces the Na2 results described below.

B. COMPUTATIONAL AND EXPERIMENTAL DEMONSTRATION Incoherent interference control has been demonstrated both computationally and experimentally for the case of the two-photon dissociation of the Na2 molecule. The photo-dissociation process we have examined is,

Na2+{

Na(3s) Na(3s) Na(3s)

+ Na(3p) + Na(4s) + Na(3d)

(127)

The control objective is to produce preferentially the Na(3d) or Na(3p) product. In the IIC scenario this is done by varying one of the frequencies, either 01 or 0 2 . In the IIC experiment (Shnitman et al., 1996) (see Fig. lo), two dye lasers pumped by a frequency-doubled Nd-Yag laser were used. One dye laser, whose frequency wzwas tuned between 13,312cm-I and 13,328cm-' , was used to dress the continuum with a I Ej2) vib-rotational state of the A1C,/311, electronic manifold (Chen et al., 1993a).The other dye laser, whose frequency colwas fixed at 17,474.12cm-', was used to induce a two-photon dissociation of the I Eu=5,+37) ground state of Na2, through intermediate resonances (assigned as v = 35, J = 38 and = 35, J = 36) of the A'C,/311, manifold. The 01 and 0 2 pulses, both about 5 ns in duration with the stronger of them (a2) having an energy of -3.5mJ, were made to overlap in a heat pipe containing Na vapor at 370410°C. Spontaneous emission from the excited Na atoms [Na(3d).+ Na(3p) and Na(3p) + Na(3s)l resulting from the Na2 photo-dissociation, was detected and dispersed in a spectrometer and a detector with a narrow-bandpass filter. Figures 11 and 12 display the observed and computed Na(3d) and Na(3p) emission signal as a function of 0 2 at a fixed 01. We see that the computed and experimental results are in excellent agreement. Clearly, as the Na(3d) yield dips, the Na(3p) yield peaks, in accordance with theoretical calculations (Chen et al., 1995). The controlled modulation of the Na(3p)/Na(3d)

COHERENT CONTROL OF ELECTRONIC PROCESSES I

.20 .15 -

e

I

I

I

I

33 1

I

3s+3d

\

T

3

4 .10 v

v l * m

3 .05

B

4

a

c - .05

I

I

I

6 8 R(Na-Na)

4

I

10 (a,..)

I

I

12

14

1

FIG.10. Incoherent interference control (IIC) scheme and potential energy curves for N q . In this scheme a two-wl-photon excitation interferes with an photon. The two-photon process proceeds from an initial state, assigned here as (v = 5, J = 37), via the v = 35, J = 36,38 levels, belonging to the interacting A' C,/ 311u electronic states, acting as intermediate resonances. The w 2 photon dresses the continuum with the (initially unpopulated) v = 93, J = 36 and 'u = 93, J = 38 levels of the A1C,/311, electronic states.

I

I

Na,-Na+Na(Sd);

I

I

Theory a n d experiment

h

v)

a,

2h 125

115

4

.I

v

I-

\ I

I

t

\I

-V

theory (a, -1.5 crn-')

+-

I

I

az (cm-')

+

FIG.1 1. Comparison of the experimental and theoretical Na2 --t Na(3s) Na(3d) yields as a function of 02. In the calculation, an intermediate v = 33, J = 31,33 resonance is used, and w~ is fixed at 17,720.7cm-I. The intensities of the two laser fields are I(w1) = 1.72 x 10' W/cm2 and 1 ( 0 2 ) = 2.84 x lo8W/cm2. The 02 frequency axis of the calculated results was shifted by - 1.5 cm-I so that we could better compare the predicted and measured lineshapes.

332

Moshe Shapim and Paul Bnrrner

70

Naz4Na+Na(3p); Theory and experiment

h

m

5 d

68

9 l d

66

v

4

64

.d

h

3 62 m

v

60 58 12

5

I

13320

I

13325 o2 (cm-')

I

13330

FIG. 12. Comparison of the experimental and theoretical Na2 + Na(3s) a function of y , with parameters as in Fig. 11.

13;

+ Na(3p) yields as

branching ratio is seen to exceed 30%. It is important to note that this degree of experimental control was attained in a rather hostile environment, i.e., a heat pipe with ongoing molecular collisions, and using lasers with only partially coherent light.

VII. Coherent Control of Bimolecular Processes The results discussed above deal with control of unimolecular processes, i.e., processes that begin with a single molecule that subsequently undergoes excitation and dynamics. Control has been described in two interlinked stages: (a) the use of multiple excitation routes to create controllable superposition states in the continuum and (b) the effect of the controllable interference term on the final outcome of the process. In this section we generalize coherent control to bimolecular collision processes, i.e., collisions of the type

where A , D , F , G are, in general, molecules. Here F and G can be identical to A and D (nonreactive scattering) or different from A and D (reactive scattering). Unlike unimolecular processes, we first demonstrate that coherent control is possible in bimolecular scattering if one creates a superposition of energetically

COHERENT CONTROL OF ELECTRONIC PROCESSES

333

degenerate scattering states; then we discuss methods for experimentally preparing such states. Consider then Eq. (128), where we label A D as arrangement q and F G as arrangement q’. Below we focus on atom (A) plus diatom ( D = B - C) scattering, although the results are easily generalized to polyatomic scattering. Eigenfunctions of the asymptotic Hamiltonian, where A D are widely separated are given, in accordance with Eq. (18), by I E, q, m; 0), where we have made the channel label q explicit. States of the product are similarly denoted I E, q’,n; 0), and I E, q’, n- ) denotes the incoming solutions associated with product in I E, q’,n; 0). The probability PE(n,q’; m, q ) of forming I E, q’,n; 0), having initiated the scattering in I E, q, m; 0), is given by (Taylor, 1972) as

+

+

+

PEb, 4’;m, 4 ) = I@, q’,n; 0 IS1E , 4 , m;0)l2

(129)

where S is the scattering matrix. In analogy with Eq. (36), one can rewrite the probability in terms of (E,q, n- I Vql E , q, m; 0), where Vq is defined, in analogy with Eq.(14), as the component of the total potential that vanishes as the distance A to D becomes arbitrarily large. It is traditional in scattering theory, however, to compute the cross section, given by

rather than the probability. In addition to the above state-to-state cross section, the cross section of forming product in arrangement q’ independent of the internal state n,

is also of interest. Assorted other cross sections may be defined, depending upon which of the elements of n are summed over. Of relevance below are (a) oE(q’, 0,+;m, q ) ,corresponding to scattering into the q’ product channel and into scattering angles (0, +), and (b) the traditional differential cross section oE(q’,e;m,q)= Jd+oE(q’,Qr+;m,q). In order to control bimolecular cross sections, we now consider scattering from an initial superposition state I E, q, {cm} ) comprised of N energetically degenerate states I E, q, m;0):

334

Moshe Shapiro and Paul Brumer

The cross section associated with using EQ. (132) as the initial state, obtained by substituting Eq. (132) in Eq. (130), is

where o(n, q’;m’,m, q ) is defined via Eq. (133). The total cross section is given by

Note that Eq. (133) and hence Eq. (134) are now of the standard coherent control form, i.e., direct contributions from each member of the superposition, proportional to lcm12,plus interference terms that are proportional to c m c ~ , . Hence, by controlling the c,, we can control the scattering cross section. Note also that both the direct and the interference amplitudes are composed of the same amplitudes. As such, we can even expect control to be effective far from the onset of reaction at reaction thresholds, a feature that overcomes the limitationsof previously proposed bimolecular control methods (Krause et al., 1990). A. DEGENERATE eM SUPERPOSITIONS

The easiest way of implementing bimolecular control in the lab is to start with a superposition of scattering states having the same incoming translational wave function 1 k m ) . This means, however, by Eq. (132), that in order to maintain the degeneracy of the total continuum energy E, we must build the superposition state from degenerate internal states I em) :

m

m

335

COHERENT CONTROL OF ELECTRONIC PROCESSES

In atom-diatom scattering, the most obvious candidate for degenerate I em ) are the I J , MJ ) states associated with fixed diatomic rotational angular momentum J. However, in this case the interference terms o(n,4'; m',m,q ) contain the J M>)$] factors (Abrashkevich ef aZ., cylindrically asymmetric exp [ ~ ( M1999), which average out upon integration over the azimuthal angle $. As a result, although control over the fully differential oE(q', 8, $; {Cm}, q) cross section is possible, control over the +-averaged differential cross section oE(q', 8; {c,}, q ) andthe totalcrosssection o E ( q ' ; {c,}, q ) cannotbe achieved in this way. To examine the extent of control afforded in this way, we performed detailed computations on one of the most widely studied prototype exchange reactions, that of D+Hz+H+HD Our computationswere carried out for E = 1.25eV and H2 in the v = 0, J = 2 vib-rotational state. Both the reactive and the nonreactive differential cross sections a $ = 0 [denoted o"(0)and oNR(0)] were examined. Figure 13 ON"(@)for the linear superposition c1 I E, J , ~1 = 2) c21 E, J, shows oR(0)/ 2 2 2 ~2 = 0 ) forvariousvaluesofs = lcll /[1c11 lczl ] with$12= arg(a2/al)= 157". Here s = 0 correspondsto scattering out of the initial state ZI = 0, J = 2, K = 2; and s = 1 correspondsto scattering from v = 0, J = 2, K = 0. The ratio of controlled differential cross sections is seen to be considerably different at from the uncontrolled ratio. For example, the controlled oR(0)/oNR(O) 0 = 91" at s = 0.748 is approximately twice as large as the uncontrolled ratios. Analysis of Fig. 13 shows that the maxima and minima of the controlled ratio in the region between 50" and 120" are positioned at the corresponding minima and maxima of the uncontrolled ratios. Exactly the opposite behavior is seen in the outer regions of 8. Hence, coherent control changes both the magnitude and the structure of the differential cross section. Results for the ratio of the reactive versus nonreactive cross sections at a fixed 0, J value are shown in Fig. 14 for scattering from cil v = 0, J = 2, K = 2) c 2 ( v = 0, J = 2, K = 1), as a function of the control parameters + , 2 and s. The ratio is seen to vary from 0.032 to 0.1 13, showing maxima and minima that are well outside the range of results for scattering from a single K state. This approach can be extended to include all available energetically 1, . . . ,J), with concomitant improvedegenerate ~i states ( ~ =i -J, -J ment in the ratio of reactive to nonreactive product. In addition, one can optimize this ratio (Abrashkevichef al., 1999) as a function of the coefficients c,. For the J = 2 case, the maximum o R / o N=R0.143, a 20% improvement over the results for a superposed pair of ~i states.

+

+

+

+

+

336

Moshe Shapiro and Paul Brumer 1.0

0.8 h

m

0.6

b

1 0.4 b

0.2

0.0

FIG. 13. Dependence of the ratio of the reactive to nonreactive differential cross section oR(e)/oNR(0) in controlled D HZon the scatteringangle 8 at $,2 = 157" for four values of s: s = 0,1,0.252, and 0.748. (From Abrashkevich et al., 1998a.)

+

FIG. 14. Contour plot of the ratio of reactive to nonreactive cross section o R / o N( R x lo3) in controlled D H2 as a function of $,* and s, at a fixed 0, J value.

+

It remains to ask how such a superposition of helicity states may be prepared. One possibility is a precursor step consisting of the coherently controlled photo-dissociation of a molecule to produce a diatomic product in a controlled superposition of K states relative to an incoming partner. For example, in the D+H2 case we can subject H2S to a coherently controlled photo-dissociation step to produce H2 in a superposition such that aiming the

COHERENT CONTROL OF ELECTRONIC PROCESSES

337

D atom exactly antiparallel to the direction of the H2 motion will produce the desired initial I El q, {cm}). B. SCULPTED IMPLODING WAVES Rather than using a superposition of internal states [ e m ) ,as in the above discussion, it is possible to effect bimolecular control by using a superposition of translational wavefunctions 1 km) (Frishman et al., 1999). In order to examine a superposition of incident plane waves, we first perform a partialwave decomposition of each of the plane waves, assumed directed along the Z-axis:

where cl = i1(21+ 1) andjl(kmR), Pl(cos 0) are the spherical Bessel function and Legendre polynomial, respectively. We see that each incoming plane wave is in fact a superposition of energetically degenerate states with fixed coefficients cl. This suggests the possibility of altering the c1 to produce modified states (RIkmod) that will display different quantum interferences, hence altering the product cross sections. Thus, in this instance, the initial ) E , q ,{cm})isgivenby/E,q,{cm})= le,)Ik,~),where)k,d),a"Sculpted incoming wavepacket," is parametrized by coefficients {q}. The effect of changing the structure of the incident wavepacket cannot, however, be measured using the standard definition of the cross section, because that definition relies on a constant flux from one direction (Taylor, 1972). Rather, we consider the magnitude of Fqt, the outgoing flux into the with the constraint that the incident product channel, as a function of the Iq(, wavefunction is normalized to the usual Dirac delta function (for alternative choices of constraints, see Frishman et a/., 1999). As an example of the control afforded by this approach, consider rotational excitation in a model of the Ar H2(Jl M J ) -+Ar H~(J'MJ~) collision. Optimizing the phases xl of CI = lcll exp ( i X r ) allows a direct study of the effect of varying the interferences between partial-wave components on F,!. Typically, altering xr allowed for considerable control. For example, with J = 2, MJ = 0, J' = 0, it was possible to change Fqt by two orders of magnitude, from 5.1 x to 3.8 x lo-*, just by varying the x1. These values are to be compared to Fqt = 1 x associated with scattering from an incident plane wave. Real and imaginary parts of the incident wave functions leading to these maximum and minimum values of the outgoing flux are

+

+

338

Moshe Shapiro and Paul Brumer

shown in Fig. 15. They are distinctly different from one another and from a plane wave. It remains to establish viable methods to prepare such sculpted imploding matter waves experimentally. Once again, we anticipate doing so via a

Ar-H, I m(d:,)

Ar-H2 Re($:,) 0.6

0.6

... :

a

.':

0

$

' > :,.:::c

........... .............. ............... .:,, .....................

2:

0

>

0

, c

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

<

E

0

$

-0.6

xpr m

0.3

0

-0.6

Ar-H2 Re($:,) 0.6

0.6

SE

sE o

0 1.

-0.f

1.3

I

Xpr

0.3 m

-0.6

-0.3

xpr m

Oa3

FIG. 15. Real and imaginary parts of the incident wave function leading to maximum and minimum outgoing flux for Ar+Hz(v = 0, j = 2, rn, = 0). (From Frishman et al., in preparation.)

COHERENT CONTROL OF ELECTRONIC PROCESSES

339

pre-reactive photo-dissociation step, possibly in conjunction with matter interferometry techniques. C. OPTIMIZED BIMOLECULAR SCATTERING: ENHANCEMENT AND TOTALSUPPRESSION We now extend the treatment of Section VI1.A to the case of a superposition composed of more than two states. On the basis of Eqs. (130) and (131), we introduce in this subsection a scheme that optimizes o(q’;{ C m } , q ) or o(q’,n; {c,}, q ) as a function of c, for an arbitrary number of states. One of the most interesting outcomes of this procedure is that it leads to a strong analytic result: if the number of initial open states in the reactant space exceeds the number of open states in the product space, it is possible to find a particular set of {c,} such that one can totally suppress reactive scattering. This result is proven below and applied to display the total suppression of tunneling (Frishman et al., 1998). Consider scattering from incident state I E, q, n;0) to final state [ E ,q’,m; 0). To simplify the notation, we specialize the treatment to the case where the m and n labels pertain to just a single quantum number, denoted i andf, with associated free states I E, q, i; 0) and I E, q’, f;0). In accordance with Eq. (129), the probability P ( f , q ’ ; i , q ) of producing product in final state I f , E , q ’ ) , having started in the initial state I i, E, q ) , is

where S, = (E,f,q’ IS1 E, i, q ) and where S is the scattering matrix for the process. The total probability P(q’;i, q ) of scatteringinto arrangement channel q’, assuming m open product states, is given by

f=l

To simplify the notation, we have not carried an E label in the probabilities: fixed energy E is understood. If we now consider scattering from an initial state I E, q, { c i } ) comprised of a linear superposition of k states, [i.e., Eq. (146) with = k]. Then the probability of forming I E, q’, f ) from this initial state is

340

Moshe Shapiro and Paul Brumer

and the total reactive scattering probability into channel q‘, P(q‘;c, q), is

To simplify the notation, we introduce the matrix o = SL! S,! with elements Si S,, which allows us to rewrite Eq. (140) as

00 =

zT=,

P ( q ’ ;c, q) = ctoc

(141)

Here t denotes the Hermitean conjugate, and the q’ subscript on the S indicates that we are dealing with the submatrix of the S matrix associated with scattering into product channel q’. One can optimize scattering into arrangement channel q’, with the normalization constraint C L1I cil2 = 1, by requiring

where h is a Lagrange multiplier. Explicitly taking the derivative gives the result that the optimized coefficients ck satisfy the eigenvalue equation oc). = hc).

(143)

Additional labels may be necessary to account for degeneracies of the eigenvectors ck. We first note that if h = 0 is an eigenvalue of Eq. (143) with eigenfunctions q,then by inserting Eq. (143), into Eq. (141) we have that P ( q ’ ;c0,q) = 0. That is, if h = 0 is a solution to Eq. (143), then the coefficients Q completely suppress reaction into arrangement channel q’. Clearly, h = 0 is a solution if det(a) = det(SitS:) = 0

(14.4)

which is the case if the number of initial states k participating in the initial superposition is greater than the number m of open product states, a situation that invariably occurs for endoergic processes. To see this result, note that under these circumstances, cr is a matrix of order k x k and S,t is of order k x m. If k > m, we can construct a k x k-order matrix A: by adding a submatrix of

COHERENT CONTROL OF ELECTRONIC PROCESSES

34 1

(k - rn) rows of zeros to the lower part of S,!. Then

det(o) = det(Si,S,t) = det(Ai,A,,) = det(Ai,)det(A,!) = 0

(145)

The last equality holds because the determinants of A,! and A:, are zero. As an example of the kind of results that are possible, consider the optimization of a barrier penetration problem modeled by a set of multichannel Schrodinger equations of the type 2P (E - V)Q(r) Sr”(r) = - 3 A where p is the relevant mass, V is the potential matrix, E is a diagonal matrix with elements E - ei, and k is the total number of open channels in arrangement q. Sample scattering results for k = 4 using a potential matrix constructed from Eckart potentials, i.e., Vj&) = --aie‘

1-5

-

~

(1-6)

+tie,

i,t = 1 , . . . , 4

(147)

where E, = -exp (2nr/t),with t a distance potential parameter, are shown in Fig. 16. Reactivity is shown as a function of energy for the case where the number of populated initial states j is less than k; here j = 3. The curves labeled Pi correspond to the standard P ( q ’ ;i, q ) ,i.e., total reaction probability from each of the individual initial states. The quantities P I and P3, which are open asymptotically at all energies, show a gradual rise with increasing energy, whereas P2, which is closed on the product side until E,h(3) = 0.008a.u., stays rather small until E = Elh(3),where it displays a very rapid rise to near unity. Total reaction probability reaches unity above Erh(4)= 0.010a.u., the threshold for the opening of the fourth channel. Of particular relevance here are the solid curves in Fig. 16, which show the maximum and minimum reactivity obtained from the optimal solutions to Eq. (143). The maximum reactivity is seen to be substantially larger than any of the individual Pi and to reach unity at significantly lower energies than any of these solutions. Minimal reactivity, as predicted by the argument presented above, is seen to be zero for E < E,h(3) because the total number of states ( j= 3) in the superposition exceeds the number of open product states (rn = 2). At E > E,h(3) a third product channel opens so thatj = rn and the minimal solution is no longer zero. Note also that the minimum reactivity curve in Fig. 16 reflects a variety of different interesting behaviors, depending on the particular energy. Specifically, below the maximum of V11 at 0.005 a.u., the zero minimum corresponds

342

Moshe Shapim and Paul Brumer

FIG.16. Reactivity shown as a function of energy in a model system. Dashed curves labeled Pi correspond to the total reactivity from each of the three individual initial states in the prepared superposition. Solid curves with crosses denote the reactivity obtained by solving Eq.(143) for the optimal solutions. The two arrows indicate the threshold energies for opening of the third and fourth channels. The dot-dash curve shows the minimum reactivity resulting from a separate computation that includes four states in the initial superposition.

to suppression of tunneling through that barrier. Above 0.005 a.u., the zero minimum corresponds to suppression of the reactive scattering that occurs above the banier. Thus it is clear that the ability to superimpose degenerate scattering states has great potential for the control of scattering processes. Note also that, as an obvious extension, similar results hold for tunneling in bound systems if the total number of initial degenerate states at the energy of interest exceeds the number of accessible final states at that energy.

VIII. Summary Coherent control has been demonstrated formally, computationally, and experimentally to be a viable method for controlling the outcome of isolated atomic molecular and electronic processes that form products in the continuum. It is a method that takes advantage of the quantum nature both of matter and of the incident light to encode quantum interference information into the molecular dynamics. That is, molecular reaction dynamics is intimately linked to the wavefunction phases that are controllable through coherent optical phase excitation. The result is a powerful method to control the dynamics of atomic, molecular, and electronic processes.

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343

M.Acknowledgments We are grateful to the U.S.Office of Naval Research for support of the vast majority of this research.

X. References Abrashkevich, A., Brumer, P., and Shapiro, M. (1999). In preparation. Abrashkevich, A., Shapiro, M., and Brumer, P. (1998a). Phys. Rev. Lett. 81, 3789. Abrashkevich,D., Shapiro, M., and Brumer, P. (1998b). J. Chem. Phys. 108, 3585. Asaro, C., Brumer, P., and Shapiro, M. (1988). Phys. Rev. Lett. 60,1634. Assion, T., Baumert, T., Bergt, M., Brixner, T., Kiefer, B., Seyfried,V., Strehle, M., and Gerber, G. (1998). Science 282,919. Baht-Kurti, G. G., and Shapiro, M. (1981). Chem. Phys. 61, 137. Balint-Kurti, G. G., and Shapiro, M. (1986). Adv. Chem. Phys. 60,403. Bandrauk, A. D., and Atabek, 0. (1989). Adv. Chem Phys. 73, 823. Bandrauk, A. D., Gauthier, J.-M., and McCann, J. F. (1992). Chem. Phys. Lett. 200, 399. Baranova, B. A., Chudinov, A. N., and Ya Zel’dovitch, B. (1990). Opt. Comm. 79, 116. Bardeen, C. J., Yakovlev, V. V.,Wilson, K. R., Carpenter, S. D., Weber, P. M., and Warren, W. S. (1997). Chem. Phys. Lett. 280, 151. Barron, L. D. (1982). Molecular light scattering and optical activify. Cambridge University Press (Cambridge, UK). Barron, L. D. (1986). Chem. Soc. Rev. 15, 189. Baumert, T., Grosser, M., Thalweiser, R., and Gerber, G. (1991). Phys. Rev. Lett. 67, 3753. Brumer, P., and Shapiro, M. (1986a). Chem. Phys. Lett. 126, 541. Brumer, P., and Shapiro, M. (1986b). Adv. Chem. Phys. 60,371. Brumer, P., and Shapiro, M. (1992). Ann. Rev. Phys. Chem. 43, 257. Cavalieri, S., Eramo, R., and Fini, L. (1997). Phys. Rev. A 55, 2941. Cavalieri, S., Eramo, R., Fini, L., Materazzi, M., Faucher, O., and Charalambidis, D. (1998). Phys. Rev. A 57, 2915. Chan, C. K., Brumer, P.,and Shapiro, M. (1991). J. Chem. Phys. 94,2688. Charron, E., Guisti-Suzor, A., and Mies, F. H. (1993). Phys. Rev. Lett. 71, 692. Charron, E., Giusti-Suzor, A., and Mies, F. H. (1995). Phys. Rev. Lett. 75,2815. Chelkowski, S., and Bandrauk, A. D. (1991). Chem. Phys. Lett. 186,284. Chen, C., Yin, Y.-Y., andElliott, D. S. (1990). Phys. Rev. Lett. 64,507; Phys. Rev. Lett. 65,1737. Chen, Z., Shapiro, M., and Brumer, P.(1993a). J. Chem. Phys. 98,6843. Chen, Z., Shapiro, M., and Brumer, P.(1993b). J. Chem. Phys. 98, 8647. Chen, Z., Shapiro, M., and Brumer, P.(1994). Chem. Phys. Lett. 228,289. Chen, Z., Shapiro, M., and Brumer, P. (1995). J. Chem. Phys. 102,5683. Cohen-Tannoudji,C., J. Dupont-Roc,J., and G. Grynberg, G. (1992). Atom-phoron interactions. Wiley (New York). Dupont, E., Corkum, P. B., Liu, H. C., Buchanan, M., and Wasilewski, Z. R. (1995). Phys. Rev. Lett. 74, 3596. Faucher, O.,Charalambidis, D., Fotakis, C., Zhang, J., and Lambropoulos, P. (1993). Phys. Rev. Lett. 70,3004. Frishman, E., Shapiro, M., and Brumer, P. (1998). J. Chem. Phys. In press. Frishman, E., Shapiro, M., and Brumer, P.(1999). In preparation. Gordon, R. J., and Rice, S. A. (1997). 48, 595.

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Hache, A., Kostoulas, Y.. Atanasov, R., Hughes, J. L. P., Sipe, J. E., and van Driel, H.M. (1997). Phys. Rev. Lett. 78, 306. Hammerich, A. D., Manthe, U., Kosloff, R., Meyer, H. D., and Cederbaum, L. S. (1994). J. Chem. Phys. 101,5623. Hollas, J. M. (1982). High resolution spectroscopy. Butterworths (London). Ivanov, M. Y., Corkum, P. B., and Dietrich, P. (1993). Laser Physics 3, 375. Ivanov, M., Corkum, P. B., Zuo, T., and Bandrauk, A. (1995). Phys. Rev. Lett. 74,2933. Jakubetz, W., Manz, J., and Schreier, H. J. (1990). Chem. Phys. Lett. 165, 100. Jiang, X.-P., and Brumer, P. (1991). J. Chem. Phys. 94, 5833; Chem. Phys. Lett. 180, 222. Jiang. X.-P., Shapiro, M., and Brumer. P. (1996). J. Chem. Phys. 104, 607. Kleiman, V. D., Zhu, L., Li, X., and Gordon, R. J. (1995). J. Chem. Phys. 102, 5863. See, for example, Knight, P. L., Lauder, M. A., and Dalton, B. J. (1990). Phys. Rep. 190, 1 . Kobrak, M. N., and Rice, S. A. (1998a). J. Chem. Phys. 109, 1. Kobrak, M. N., and Rice, S. A. (1998b). Phys. Rev. A 57,2885. Kohler, B., Krause, J. L., Raski, F., Wilson, K. R., Yakovlev, V. V., Whitnell, R.M., and Yan, Y. (1995). Accr. Chem. Res. 28, 133. Kosloff, R. (1988). J. Phys. Chem. 92, 2087. Kosloff, R. (1994). Ann. Rev. Phys. Chem. 45, 145. Kosloff, R., Rice, S. A., Gaspard, P., Tersigni, S., Tannor, and D. J. (1989). Chem. Phys. 139,201. Krause, J., Shapiro, M., and Brumer, P. (1990) J. Chem Phys. 92, 1126. Krause, J. L., Whitnell, R. M., Wilson, K. R., Yan, Y., and Mukamel, S. (1993). J. Chem. Phys. 99, 6562. Leforestier, C., Bisseling, R., Cerjan, C., Feit, M., Friesner, R., Guldberg, A., Hammerich, A. D., Julicard, G., Karrlein, W., Dieter Meyer, H., Lipkin, N., Roncero, 0.. and Kosloff, R. (1991). J. Comp. Phys. 94, 59. Levine, R. D. (1969).Quantum mechanics of molecular rate processes. Clarendon (Oxford, UK). Lu, S.-P., Park, S. M., Xie, Y., and Gordon, R. J. (1992). J. Chem. Phys. 96, 6613. Manolopoulos, D. E., D’Mello, M., and Wyatt, R. E. (1991). J. Chem. Phys. 93,403. Muller, H. G., Bucksbaum, P. H., Schumacher, D. W., and Zavriyev, A. (1990). J. Phys. E 23, 2761. Park, S. M., Lu, S.-P., and Gordon, R. J. (1991). J. Chem. Phys. 94, 8622. Peirce, A. P., Dahleh, M. A., and Rabitz, H. (1988). Phys. Rev. A 37, 4950; ibid. (1990). 42, 1065, Shi, S., and Rabitz, H. (1991). Comp. Phys. Comrn. 63, 71. Potter, E. D., Herek, J. L., Pedersen, S., Liu, Q., and Zewail, A. H. (1992). Nature 355, 66. Potvliege, R. M., and Smith, P. H. G. (1992). J. Phys. E 25, 2501. Pratt, S. T. (1996). J. Chem. Phys. 104,5776. Schafer, K. J., and Kulander, K. C. (1992). Phys. Rev. A 45, 8026. Schmidt, I. (1987). Ph.D. Thesis, Kaiserslautern University. Schubert, M., and Wilhelmi, B. (1986). Nonlinear optics and quantum electronics. Wiley (New York). Segev, E., and Shapiro, M. (1982). J. Chem. Phys. 77,5604. Seideman, T., Shapiro, M., and Brumer, P. (1989). J. Chem. Phys. 90, 7132. Shapiro, M. (1993). J. Phys. Chem. 97,7396. Shapiro, M., and Bony, H. (1985). J. Chem. Phys. 83, 1588. Shapiro, M., and Brumer, P. (1987). Faraday Disc. Chem. SOC.82, 177. Shapiro, M., and Brumer, P. (1989). J. Chem. Phys. 90, 6179. Shapiro, M., and Brumer, P.(1991). J. Chem. Phys. 95,8658. Shapiro, M., and Brumer, P. (1992). J. Chem. Phys. 97,6259. Shapiro, M., and Brumer, P. (1993). Chem. Phys. Left. 208, 193. Shapiro, M., and Brumer, P. (1993). J. Chem. Phys. 98, 201.

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Shapiro, M., and Brumer, P. (1997). Trans. Farad. SOC.93, 1263. Shapiro. M., Hepburn, J. W., and Brumer, P. (1988). Chem. Phys. Lett. 149,451. Sheeny, B., Walker, R. B.. and DiMauro, L. F. (1995). Phys. Rev. Lett. 74,4799. Shi, S., Woody, A., and Rabitz, H. (1988). J. Chem. Phys. 88, 6870. Shi, S., and Rabitz, H. (1989). Chem. Phys. 139, 185. Shnitman, A., Sofer, I., Golub, I., Yogev, A., Shapiro, M., Chen, Z., andBrumer, P. (1996). Phys. Rev. Lett. 76, 2886. Szoke, A., Kulander, K. C., and Bardsley, J. N. (1991). J. Phys. B. 24, 3165. Tannor, D., and Rice, S. A. (1985). J. Chem. Phys. 83,5013. Tannor, D., Kosloff, R., and Rice, S. A. (1986). J. Chem. Phys. 85,5805. Tannor, D. J., and Rice, S. A. (1988). Adv. Chem. Phys. 70,441. Taylor, J. R. (1972). Scattering theory. Wiley (New York). Walker, D. C., ed. (1979). Origins of optical activity in nature. Elsevier (Amsterdam). Wang, F., Chen, C., and Elliott, D. S . (1996). Phys. Rev. Lett. 77,2416. Wang, X . , Bersohn, R., Takahashi, K., Kawasaki, M., and Kim, H. L. (1996). J. Chem. Phys. 105,2992. Warren, W. S., Rabitz, H., and DahIeh, M. (1993). Science 259, 1581. Yan, Y., Gillilan, R. E., Whitnell, R. M., and Wilson, K. R. (1993). J. Phys. Chern. 97, 2320. Woolley, R. G. (1975). Adv. Phys. 25, 27. Yelin, D., Meshulach, D., and Silberberg, Y. (1997). Opt. Letr. 22, 1793. Yin, Y.-Y., Chen, C., Elliott, D. S., and Smith, A. V. (1992). Phys. Rev. Lett 69, 2353. Yin, Y.-Y., Shehadeh, R., Elliott, D., and Grant, E. (1995). Chem. Phys. Lett. 241, 591. Zhang, J. Z. H., and Miller, W. H. (1989). J. Chem. Phys. 91, 1528. Zhu, L., Kleiman, V.D., Li, X., Lu,S., Trentelman, K., andcordon, R.J. (1995). Science 270,77. Zhu, L., Suto, K., Fiss, J. A., Wada, R., Seideman, T., and Gordon, R. J. (1997). Phys. Rev. Lett. 79, 4108. Zuo, T., and Bandrauk, A. D. (1996). Phys. Rev. A 54,3254.

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ADVANCES IN ATOMIC, MOLECULAR. AND OITCAL PHYSICS. VOL. 42

RESONANT NONLINEAR OPTICS IN PHASE-COHERENT MEDIA M. D. LUKIN Department of Physics, Texas A&M University, College Station, Texas Max-Planck-Institut fur Quantenoptik, Garching, Germany Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA

P. R. HEMMER Department of Physics, Texas A&M University, College Station, Texas Max-Planck-Institut fur Quantenoptik, Garching, Germany

M. 0. SCULLY Department of Physics, Texas A&M University, College Station, Texas Max-Planck-Institut fin? Quantenoptik, Garching, Germany

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Review of Atomic Coherence Studies. . . . . . . . . . . . . . . . . . . . . . . 111. Resonant Enhancement of Nonlinear Optical Processes: The Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Analysis of Nonlinear Optical Enhancement . . . . . . . . . . . . . . . . . . V. Resonant Enhancement of Four-Wave Mixing Processes . . . . . . . . . . VI. Physical Origin of Nonlinear Enhancement. . . . . . . . . . . . . . . . . . . VII. Optical Phase Conjugation in Double-A Systems . . . . . . . . . . . . . . . VIII. Correction for Optical Aberrations in Double-A Medium . . . . . . . . . IX. Nonlinear Spectroscopy of Dense Coherent Media . . . . . . . . . . . . . X. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347 350 353 357 363 367 371 375 376 382 384 384

Abstract: We review some recent theoretical and experimental studies on resonantly enhanced nonlinear interactions in phase-coherent media (“phaseonium”). Basic physics of resonant enhancement and applicationssuch as efficient optical phase conjugation and nonlinear laser spectroscopy are discussed.

I. Introduction Ever since the development of the first laser, researchers have been intrigued by the possibility of nonlinearly mixing optical signals in the same way that electronic signals are routinely mixed in circuits. However, because optical wavelengths are closer to the scale of atomic dimensions than to circuit 347

Copyright 8 Zoo0 by Academic Ress All rights of reproduction in any form reserved. ISBN 0-12-00384241ISSN 1049-250X/OO$30.00

348

M. D.Lukin, l? R. Hemmer; and M. 0. Scully

component dimensions, it has been necessary to accept the significant restrictions imposed by the optical properties of atomic-scale systems. In particular, this led to an initial requirement of very large electric field strength of the optical beams (Franken et al., 1961). This is why the development of the laser was crucial for the success of the first optical mixing experiments. Once the door was opened, researchers began to think of clever ways to use the coherence properties of lasers and the bulk optical properties of materials to reduce intensity thresholds and increase conversion efficiencies. The first breakthrough was to use phase matching to compensate for the fact that input and output optical frequencies generally propagate at different speeds in bulk materials (Giordmaine, 1962). Without phase matching, the output fields rapidly dephase with the polarizations produced by the input fields, resulting in destructive interference and low conversion efficiency. Phase matching made possible much longer interaction lengths, which significantly enhanced conversion efficiencies. Since the early days of nonlinear optics, there has also been substantial interest in utilizing resonant atomic and molecular systems for efficient nonlinear optical processes (Reintjes, 1984;Boyd, 1992).The most successful applications of resonant systems involved two-photon rather than singlephoton transitions. For example, two-photon Raman resonances in molecular liquids were found to enhance conversion efficiencies for input and output laser fields having difference frequencies near that of the two-photon transition (Eckhardt et al., 1962). Replacing the molecular resonances with acoustic or Brillouin resonances in crystals also proved successful (Chiao et al., 1964). However, to realize the full potential of resonant systems, the input laser@)must also be single-photon resonant. Indeed, the magnitude of the two-photon coherence needed for nonlinear generation decreases rapidly with the laser detuning from a given single-photon transition. Unfortunately, attempts to use this resonance enhancement have been frustrated by problems associated with resonant absorption, phase shifts, and unwanted nonlinearities leading to, for example, self-focusing and beam distortion. For these reasons it was generally recognized that even though the nonlinear susceptibility is enhanced in the vicinity of resonant transitions, in practice it is extremely difficult to take advantage of such resonant enhancement. Theoretical and experimental work of the past few years has challenged this paradigm. It was shown that the properties of optical materials change drastically in systems where superpositions of the quantum states are coherently excited. Such a material consisting of a phase-coherent ensemble of atoms has been aptly named “phaseonium” to emphasize that it is really a new state of matter displaying many new effects. Examples of such properties include the modification of absorptive properties resulting in electromagnetically induced transparency (Harris et al., 1990; Boller et al., 1991) and lasing without

RESONANT NONLINEAR OPTICS IN PHASE-COHERENT MEDIA 349 TABLE I NONLINEAR OPTICALPROPERTIES OF PHASEON~UM COMPARED TO OTHERRESONANT MEDIA. _ _ _ _ ~

Phase matching Coherence magnitude Sign of nonlinear coherence Excited-state population

~

Previous Resonant Media

Phaseonium

Determined by medium Very small

As in vacuum

Finite

0

+

112

-

population inversion (Kocharovskaya and Khanin, 1988a; Harris, 1989; Scully et al., 1989; Zibrov et al., 1995), as well as the modification of dispersive properties to give a resonantly enhanced index of refraction accompanied by vanishing absorption (Scully et al., 1991; Zibrov et al., 1996). In a medium displaying quantum coherence and interference effects, it is also possible to suppress absorption efficiently, control refractive index, and, at the same time, utilize resonantly large nonlinearities. As a result, efficient laser frequency conversion or large parametric gain is possible within a propagation distance as short as a single coherence length (or, sometimes, a single absorption length) with an unusually low input power (sometimes corresponding to that of the few light quanta). The nonlinear optical properties of coherent media are compared to previous resonant systems in Table I adapted from (Harris et al., 1997). Recently, the use of fully resonant double-A systems was found to endow resonant coherent media with surprisingly superior nonlinear optical properties (Hemmer et al., 1995; Lukin et al., 1998). It is this new scheme, the physical mechanisms making it possible, and its various applications that are the main topic of this paper. We begin in Section 11by reviewing the important early work, as well as more recent work, on atomic coherence in linear and nonlinear optics. We next turn to a detailed discussion of the nonlinear interactions in the Raman double-A type systems in Section III. Here, we analyze the physics of efficient frequency conversion in coherent atomic media. In Section IV we consider the process of parametric amplification in double-A media. Four-wave mixing and the possibility of spontaneous buildup of the atomic coherence grating in such a medium is discussed in Section V. The physical origin of resonant enhancement in a double-A parametric amplifier is the subject of Section VI.Experiments on efficient and fast optical phase conjugation utilizing resonant four-wave mixing in doubleA systems are reviewed in Section VII. An important practical application of this scheme for the correction of high-speed optical aberrations is discussed in Section VIII. Experiments demonstrating a new type of nonlinear laser spectroscopy based on dense coherent media are discussed in Section IX. Finally, we outline the current status of several research directions based on

350

M. D.Lukin, P R. Hemmer, and M. 0.Scully

nonlinear interactions in coherent atoms and molecules. The recent results indicate that an entire new domain of quantum nonlinear optics is emerging from these studies.

II. Review of Atomic Coherence Studies Effects of atomic coherence in quantum electronics have a long history. One of the first examples was the Hanle effect (Hanle, 1924): Excitation by polarized light creates atomic coherence that can evolve in a magnetic field, causing a change of polarization of the resulting fluorescence. Another important example is the interference of decay processes, first suggested by Fano (1961). The first schemes involving atomic coherence generated by strong coupling lasers in Raman-like systems were introduced by Javan (1957). Coherent interaction of two fields in such systems and its influence on the absorption spectrum were studied in the late 1960sby Hinsch and Toschek (1969,1970) by Beterov and Chebotaev (1969), and by Feld and Javan (1969). A general theory of coherent spectroscopy of multilevel systems was developed by Popov and colleagues (1970) (see also, Popov and Rautian, 1996). The effect that lies at the foundation of current interest in coherence effects, coherent population trapping (CPT), was experimentally discovered and theoretically explained by the group in Pisa (Alzetta et al., 1976). It has been studied extensively in the following years both theoretically and experimentally (see Arimondo, 1996, for a review). A dense-media analog of coherent population trapping (Gorny et al., 1989), electromagnetically induced transparency, was first predicted and demonstrated by Boller et al. (1991). It was later studied by the Stanford group and the others both theoretically and in a series of impressive experiments (Harris, 1993;Harris, 1994a;Grobe et al., 1994;Eberly et al., 1994; Kasapi et al., 1995). (See especially the review in Hanis, 1997.) The nonreciprocity of absorption and emission in interfering systems was pointed out by hvkhipkin and Heller (1983). The concept of lasing without inversion (LWI) was theoretically formulated at the end of the 1980s (Kocharovskaya and Khanin, 1986b; Harris, 1989; Scully et al., 1989) and was extensively studied thereafter (see the reviews in Kocharovskaya, 1992; Scully, 1992; Scully and Fleischhauer, 1994; and Mandel, 1994). Several experiments (Fry et al., 1993; Nottelmann et al., 1993; Lange et al., 1994) produced temporary amplification of an external signal in the absence of population inversion. Proof-of-principle experiments demonstrating cw inversionless amplification and laser oscillation have been reported by Zibrov et al. (1995) and Padmabandu et al. (1996). The concept of modification and resonant enhancement of the refractive index with vanishing absorption was proposed by Scully (1991). Harris

RESONANT NONLINEAR OPTICS IN PHASE-COHERENT MEDIA 35 1

FIG. 1. An EIT-based scheme for resonant enhancement of nonlinear optical processes proposed by Harris and coworkers in (Harris et al., 1990). Strong driving field R causes destructive interference for absorption of the generated field a. Nonlinearities are resonantly enhanced and interfere constructively.

(1994b) has pointed out the possibility of refractive index control with strong fields. Several experiments (Harris et al., 1992; Xiao et al., 1995)have demonstrated the large dispersion of the index of refraction accompanying EIT, and refractive index enhancement (Zibrov et al., 1996). Effective control of the nonlinear refractive index, particularly relevant for resonant nonlinear optics, has been demonstrated by Jain et al. (1995). Applications of atomic coherence and interference effects in nonlinear optics were pioneered by Tewari and Agarwal (1986) and by Harris and coworkers (1990); see Fig. 1. In particular, the latter work demonstrated, for the first time, that cancellation of linear susceptibility due to quantum interference is not mirrored in the nonlinear part of the polarization and hence can result in substantial enhancement of nonlinear optical efficiency. Nonlinear generation based on this principle was experimentally demonstrated in the beautiful experiments of Hakuta, Stoicheff and coworkers (1991; Zhang et al., 1993) in atomic hydrogen; see Fig. 2. Rathe and coworkers (1993) have analyzed nonlinear index enhancement via quantum coherence, showing that the possibility of combining absorption cancellation with resonantly large linear and nonlinear refractive index can result in unusually large Kerr nonlinearities. The possibility to enhance V W generation by suitable use of control fields was pointed out by Agarwal and Tewari (1993). In another

352

M. D.Lukin, P. R. Hemmer, and M. 0.Scully

4'

DC

- - - - _.2P

*--

-L----

I I I

I I I

I I

A

a

I

I I

I I I

b FIG.2. Atomic-level scheme used for a first demonstration of resonant enhancement of nonlinear frequency conversion in atomic hydrogen (Hakuta et al., 1991). DC electric field is used to generate coherent superposition of *S and ' P states.

important work Jain and coworkers (1993) have demonstrated the importance of atomic coherence for phase matching in near-resonant frequency conversion. This early work has now been extended in many theoretical and experimental studies (see Hemmer et al., 1995; Li and Xiao, 1996; Jain et al., 1996; Babin et al. 1996; Popov and Rautian, 1996; Petch et al., 1996; Harris and Jain, 1997; Grove et aL, 1997; Lukin et al., 1997; Hakuta et al., 1997; Lukin et al., 1998; Lii et al., 1998; Popov and Baev, 1999; Babin et al., 1999; and the review in Harris, 1997). Futhermore, EIT was extended to the control of twophoton absorption and dispersion (Agarwal and Harshawardhau, 1996). The pioneering work on the Raman double-A system, on which we focus below, involved studies of amplification of laser fields without inversion (Scully, 1990; Zhu et al., 1992; Kocharovskaya and Mandel, 1990) and noninversion laser oscillation (Kocharovskaya et al., 1990). In the field of nonlinear optics, this system was first used by Hemmer and coworkers for experiments on optical phase conjugation (1995) in Na. Efficient frequency conversion utilizing the double-A system with maximal coherence was experimentally demonstrated by Jain et al. (1996). Important later work includes demonstration of an efficient and fast performance of double-A phase conjugators (Sudarshanan et al., 1997), applications to high-resolution laser

RESONANT NONLINEAR OPTICS IN PHASE-COHERENT MEDIA 353

FIG.3. Parametric conversion process of weak fields ul an open double-A system.

+ u2

mediated by strong fields in

spectroscopy (Lukin et al., 1997), and efficient coherent Raman scattering in hydrogen (Hakuta et al., 1997). A variety of different proposals related to this basic system are currently being discussed, and several experiments are under way in the laboratories around the world. Some of these most recent developments are discussed in the concluding paragraphs of this article.

111. Resonant Enhancement of Nonlinear Optical Processes: The Concept We now examine a specific scheme in which the substantial enhancement of nonlinear efficiency becomes apparent. This is a so-called double-A scheme shown in Fig. 3. Here four optical waves are tuned to the vicinity of the corresponding optically allowed transitions of a four-level atomic system. To illustrate the concept of resonant enhancement, we consider, first, the situation relevant to recent experiments by Jain et al. (1996), in which efficient frequency conversion with maximal coherence was studied. In addition to being potentially useful for various applications, this technique has a clear physical origin. Here two strong fields of frequencies 01 and 0 2 mediate the parametric conversion from weaker field of frequency 0 3 into a field of frequency 0 4 . In essence, two strong fields generate a grating, which is a running wave with a k-vector k2 - kl. A weak signal can scatter off this grating, resulting in parametric amplification of the field with frequency 0 4 . We first restrict our analysis to the cw case and assume an infinitely longlived coherence on the dipole-forbidden transition bz + bl .We further assume

354

M.D.Lukin, P. R. Hemmel; and M. 0.Scully

that both of the strong driving fields with complex Rabi frequencies R 1,2 are tuned to exact resonance with corresponding single-photon transitions. Weak fields with Rabi frequencies a 1,2 are detuned from the respective single-photon transitions by A . In this case, we can describe the response of the system using a wavefunction approach. The equations of motions can be written as

Let us consider the case in which the atoms are prepared in a coherent superposition of two lower states:

d m .

where Ro = Clearly, this state corresponds, in the limit a1,2 + 0, to a vanishing amplitude in the state u2. Hence, an atom prepared in such a coherent superposition of two lower states becomes decoupled from the drive fields. This is the essence of the so-called “dark state.” Let us consider the case of approximately equal absolute values of drive fields’ Rabi frequencies 101 1 1R2) R / d . In this case, the coherence between ground-state sublevels is maximal. Solving the first equation of the system, Eq. (4),in steady state yields N

N

.Elh

a1 = 1

+ a2b2

y-iA

(7)

Using lower-level probability amplitudes, we find that the off-diagonal matrix elements proportional to the weak-field polarizations Pal,b(12) are given by

RESONANT NONLINEAR OPTICS IN PHASE-COHERENTMEDIA 355

which indicate that, for the present system, linear susceptibility and nonlinearity are of the same order. This is in sharp contrast to the conventional nonlinear optics, where nonlinearities are typically much smaller than the resonant linear susceptibilities (Boyd, 1992). The implications of such large nonlinearities for parametric conversion have also been demonstrated in Jain et al. (1996). A medium with maximal coherence can essentially serve as an efficient atomic local oscillator. To illustrate this point, we consider the case when all three fields a1, R 1,2 are introduced in a nonlinear medium of Fig. 3. A new field a2 is generated by the frequency-mixing process. Let us assume that all fields are continuous waves and are propagating collinearly and that we can use again a small-signal approximation ( lal,2 I << IR 1,2I). In this case, strong drive fields will propagate with constant intensities and phases, and evolution of the weak field is governed by

which can be easily solved with initial conditions q ( 0 ) = a, a2(0) = 0. The solution is given by

where we have defined complex propagation constants 51,2by

It is clear that this solution implies that there is a parametric energy transfer from the field a1to the field a2.It is instructive to distinguish two regimes of parametric conversion. Let us first assume that fields are in exact resonance A = 0, in which case both propagation constants are real. From Eqs. (12) and (13) it follows that after the propagation distance on the order of c2)-l, both fields approach the steady state and than propagate as in free space. Hence the evolution of field components here is in the spirit of pulse matching

(el +

356

M.D.Lukin, R R. Hemmel; and M. 0.Scully

'ti

FIG.4. Normalized field intensities I I , ~ 1a1,2I2as a function of propagation distance in parametric converter based on population trapped atoms. (a) Near-resonant propagation (A = 0). (b) Off-resonantpropagation (A >> y). In both cases it is assumed that 6, = = 5. N

c2

discussed in Harris (1993). It is illustrated in Fig. 4(a), where field amplitudes are plotted as a function of propagation distance. Let us now turn to the case of large detunings IAl >> y. Here propagation constants become purely imaginary, and Eqs. (12) and (13) imply that the energy is transferred back and forth between two fields (Fig. 4b) with a half-period LO = 2n/14, + which is a quantity on the order of the coherence length. Note that in this case, the present system can act as an ideal converter by transferring all photons of the field a1 into the field up. Before concluding, we note that there exists a simple interpretation of the present results in terms of the normal modes corresponding to a pair of propagating fields a1 and u2. One of these normal modes is coupled to the populated dark superposition of states (see, for example, Fig. 3,and another normal mode is coupled to the empty bright superposition and propagates as in free medium. Near single-photon resonance, the normal mode coupled to

c21,

RESONANT NONLINEAR OPTICS IN PHASE-COHERENT MEDIA 357

I I

\ \

-a1 I /

I I

b-

- Is > - Ib2>

\

- IS ’+ IbL ’

FIG. Normal mode interpretation of efficient conversion via maximal cc --:rence. Normal modes coupled to the dressed “dark” and “bright” states are shown. This figure corresponds to the dressed-state picture of the system shown in Fig. 3 and discussed for 01 = 02 = fin,.

the dark state is absorbed in a distance on the order of the absorption length; hence, a steady-statesolution of the type shown in Fig. 4(a) corresponds to the “bright” normal mode (Harris, 1994a). In the case of off-resonant propagation, the normal mode coupled to the dark state experiences a phase shift, whereas the other normal mode does not. This leads to oscillation of the type shown in Fig. 4(b).

IV. Analysis of Nonlinear Optical Enhancement In this section, a different nonlinear process will be studied, in which a pair of correlated Stokes and anti-Stokes fields can be generated from infinitesimally small initial values. We consider the case in which two drive fields of frequencies 01 and 0 3 (Rabi frequencies R1,2)are parametrically converted into Stokes and anti-Stokes fields w2 and 0 4 with complex amplitudes &1,2 and Rabi frequencies a1,2. As will be discussed below, such a process is of particular importance for such applications as optical phase conjugation and high-resolution laser spectroscopy. This system is depicted in Fig. 6. One of the key results of the present work can be understood by analyzing expressions for the polarizations Pi induced by the weak anti-Stokes El and Stokes E2 fields with respective Rabi frequencies a ~a2, in a system (Fig. 6) dressed by two strong fields El and E 2 with respective Rabi frequencies R 1,R2. In the simple case of a homogeneously broadened system with symmetric decay rates, equal drive Rabi-frequencies (Ifill = 1!22( = R),

358

M. D.Lukin, I? R. Hernrnel; and M. 0.Scully

.9

a2

FIG. 6. Parametric amplification and generation of fields c q , in ~ a closed double-A system.

weak saturation of optical transitions, vanishing relaxation of the lower-level coherence (y, + 0), and at the point where the detunings (A) of all four fields are equal, these polarizations take the form

Here we have_defined I’i = yQ+ in,where yo is the decay rate of the optical coherence, 6k = kd2 - kl - i 2 is the geometrical phase mismatch, and k, are the wave vectors of the drive, anti-Stokes ( j= l), and Stokes ( j= 2) fields. Here and below, all polarizations, field strengths, and Rabi frequencies are slowly varying parts of the corresponding oscillating quantities. An interesting aspect of the dressed polarizations given by Eq. (15) is that none of them (e.g., P I )depends on the optical field driving the corresponding transition (e.g., E l ) . At the same time, the weak field acting on the adjacent transition induces a polarization that resembles the linear response of a usual two-level system. Hence, according to Eq. (15), the medium of Fig. l(a) can be transparent for each of the Stokes and anti-Stokes fields in the absence of the other. These fields, however, interact strongly via a large cross-coupling nonlinearity, resulting in an efficient parametric amplification. In order to quantify the conditions for efficient parametric interactions, we proceed with the steady-state density matrix analysis of the closed four-level atomic system (Fig. 6). In the following, it is always assumed that a “closedloop” condition for optical frequencies is satisfied - that is, cod1 a d 2 = 01 + 0 2 . Only fields satisfying these conditions can interact via induced nonlinearity. The equations of motion describing the atomic response of this

+

+

RESONANT NONLINEAR OPTICS IN PHASE-COHERENT MEDIA 359

system can be written as

The rest of the equations are obtained by interchanging of indices 1, 2 and a, b and by complex conjugation. Here we have defined complex relaxation

rates

+

+

A d 2 is two-photon detuning. and 6 = -A 1 A d 1 = - A 2 The relevant set of equations can be solved analytically, assuming weak probe fields a 1 , 2 << R1,2.First, we are interested in the response of the ) , which case the polarizations on the dressed system to weak cw fields ( E I , ~ in probe transitions can be written as

+

PI

=

~2

=~

+

g

x

+~ g x ~ ~ e x p ( i 6 i ~ ) ~ ; + exp(i&)E; ~

~

~

2

(29) (30)

where 6; = $dl k d 2 - $1 - $2 describes the phase mismatch, xii are linear susceptibilities, and xij (i # j) describe dressed x3-type susceptibilities. To calculate optical polarizations, we have solved the equations (16) through (23) in steady state, neglecting terms higher than the first order in ( ~ 1 1 , ~ )Polarizations . are related to the off-diagonal elements of the density matrix by = Npl Palbl and P 2 = N 6 3 2 pazb. The resulting susceptibilities

360

M. D.Lukin, P. R. Hemmer, and M. 0.Scully

have the following general form:

In the derivation of the above equations we have assumed, for simplicity, that 101 I = 1Qd1 = Q and that the radiative decay rates on probe transitions are equal to y. A and B are the population differences given by

puu = p a l a l = paZa2, and D is defined as

We now analyze in detail some special cases. First, let us focus on the case when one of the driving fields, say R 1, is tuned to the vicinity of the respective single-photon resonance, while another ( 0 2 ) is detuned far away from the single-photon resonance. Resonant enhancement is achieved when weak-field E l is close to the two-photon resonance with a driving field 0 1. while weak field &2 is close to the corresponding resonance with a driving field 0 2 . We note that in the limit of a very large detuning, the second driving field ( 0 2 ) can be disregarded. In this limit we should recover the usual result for a three-state driven system - that is, electromagnetically induced transparency. At the same time, in this limit, the nonlinearity and hence the crosscoupling between fields will be vanishingly small. As the detuning of the second driving field decreases, three principal effects should take place. First, cross-coupling nonlinearity will increase. Second, this field will lead to light

RESONANT NONLINEAR OPTICS IN PHASE-COHERENT MEDIA 361

shifts, and finally, it will cause effective leakage out of the trapped state, which can be viewed as an additional contribution to the relaxation of groundstate coherence. Clearly, the last two effects are expected to destroy the original transparency. Assuming large single-photon detunings for fields R2,b12 so that most of atoms remain in the state bl and A M 0, B M 1, we find

x22

=0

(41)

The assumptions leading to the above expressions are A , x A d 2 = A , IAl >> y i , l r a l b , 1, J r a l b 2 1 .We have neglected all contributions scaling with orders higher than linear in l/lAl. We note that in this approximation, the effects of the additional relaxation caused by optical pumping are implicitly neglected. From Eq. (38) it follows that in the presence of the second driving field (fl2), the medium is no longer transparent at the point of two-photon resonance, even in the limit of vanishing ground-state relaxation r b l b 2 + 0. Moreover, at this point the linear susceptibility is always larger than the corresponding nonlinearities. The transparency, however, can be recovered by a small detuning,

6 = 60 := (RI2/A

(42)

required to compensate the light shift.We also note that such a small detuning does not affect the cross-coupling nonlinearity as long as lAl >> IFalbl I [i.e., within the validity domain of Eqs. (38)-(41)]. Hence, at the ac-Stark-shifted transparency point, nonlinear cross-coupling exceeds the residual linear susceptibility when

> IAlYblb2

(43)

indicating that an efficient enhancement of nonlinear optical processes is possible. As the cross-coupling scales inversely with detuning of the field Cl2, it is advantageous to have it as small as possible. However, it is obvious that the above analysis breaks down when optical pumping out of the state bl becomes significant. This happens when Y a z b l02 Yblb 2 1 ~ 1 2 .

362

M. D. Lukin, R R. Hemmel; and M. 0. Scully

To illustrate the influence of the population redistribution effect, let us consider the case when the detunings of all four fields are equal. In this case, both of the driving fields have similar rates of optical pumping, and as a result, population is distributed evenly among the metastable levels. That implies that the populations of the dark and the orthogonal (bright) combinations of states are approximately equal and that ground-state coherence is very small (I pblb2 I << d-). The linear susceptibilities take the following form:

We have defined ro = yo in,where yo is the decay rate of the optical coherences (assumed to be identical on each of the optical transitions) and prr(uu)is the population of the lower (upper) levels, and we have assumed a small dephasing rate of the lower-level coherence (yblb2<< yo). The above equation indicates that the diagonal elements xii vanish in the limit yblb2(r~12 << yo a’. Note that, in the present case, susceptibility vanishes exactly at the point of two-photon resonance, and the resonant frequency is not light-shifted. This is so because in the case of symmetric laser tunings, the shifts induced by the drive fields compensate each other. Let us now turn to the off-diagonal elements of the matrix xii. Under the conditions of vanishing absorption,the nonlinearitiesbecome resonantly large:

+

Note that in the limit yblb211’~12<< yon2, x12resembles the familiar expression for the linear susceptibility of a usual two-level system. That is, the nonlinear polarization excited under these conditions is of the same order as the “bare” resonant polarization. This is illustrated in Fig. 7, where the quantities are plotted for different Rabi frequencies of the drive field. When the latter exceeds the threshold value

the nonlinear susceptibilityexceeds the linear absorption at the line center and becomes comparable with the “bare” absorption at resonance [curve ii in Fig. 7(b)]. Note that for large detunings, the latter condition is identical to Eq. (43) - that is, even in the absence of complete coherent population trapping, we have similar conditions for efficient parametric gain. Before concluding, we emphasize another important feature of dressed polarizations. The large dispersion of the refractive index near resonance

RESONANT NONLINEAR OPTICS IN PHASE-COHERENT MEDIA 363

1.0

x2

I

I

I

I

0

1

0.0 -0.5 -2 -1

2

A170 FIG.7. Susceptibility spectrum for the resonantly driven closed system ofFig. 6. (a) Im(xii) (i) and Re (xii)(ii). The dashed curve corresponds to R 1 = R2 = 0. (b) - Im (xu) (i) and Re ( x u ) (ii). The parameters are yd = OSy, and 0: = 0;= 0 . 2 ~Note ~ . that xll = xZ2 and xI2= xzl in this case.

(Fig. 7) is very important here, because it can be used to eliminate any residual phase mismatch that may arise, for example due to non-colinear propagation geometry or from noncompensated light shifts. Hence, phase matching can almost always be achieved for the present system by a small two-photon detuning.

V. Resonant Enhancement of Four-Wave Mixing Processes To illustrate the significance of the present system for nonlinear parametric processes, we consider propagation of the coupled waves in a nonlinear medium of Fig. 6. In a typical experimental situation, two reference waves (drive fields &l,d2) are introduced into a nonlinear medium together with a

M. D.Lukin, F! R. Hemmer; and M. 0.Scully

364

signal wave E l . A new field E2 is generated by a four-wave mixing process. Below, we will consider two distinct cases. In the first case, two driving fields are assumed to propagate in similar directions; in the second, they are supposed to be nearly counter-propagating.In each case, a weak field (&i) is assumed to be nearly collinear with its respective drive field (Szi). A simplified model used for the description of such a process neglects the effects of depletion and absorption of the driving fields and treats the signal and generated fields only to the first order. The evolution of the signal and the generated field is described by Maxwell's equations

where P1,2 are given by (30), z is the direction of signal wave propagation, and a "-"sign should be taken in the case of propagation in the negative z direction. Let us first consider the case when all of the fields are co-propagating. The evolution of Stokes and anti-Stokes components is described by Eqs. (30) and (47) and can be represented in the following form:

Solving this system with proper boundary conditions El (0) = &', &2(0)= 0, we find a El(L) = E0exp(6aL)[cosh(E,L) +- sinh(5L)I

5

E ~ ( L )= * EOexp(tiaL)% sinh(5L)

(50)

(51)

5

+

+

Here alj = i k l x l j / 2 , a2j = i k 2 ~ ; ~ / 6a 2 , = (all - a22 iAkz)/2,a = (all a22 iAk,)/2, and 5 y J-u12u21 u2. A k , is the z-component of the total phase mismatch A k , which includes the phase shifts of the driving fields. From Eq. (51) it follows that if the phase matching condition is satisfied (Rea = 0), which can always be accomplished by small two-photon detuning, and nonlinearity exceeds linear absorption, both of the weak fields experience exponential growth with gain coefficient on the order of ( k i l ~ ~ . The typical example of collinear propagation in a medium of double-A atoms

+

+

RESONANT NONLINEAR OPTICS IN PHASE-COHERENTMEDIA 365

0.00

I o-~ (a>

0

20

10

30

50

40

60

1

2.98

ZJL

3.00

3.02

Ah

FIG.8. Intensity gain of the probe (solid) and the conjugate (dashed) as a function of (a) propagation distance and (b) the probe field detuning for the co-propagating fields.

is illustrated in Fig. 8. Here the results of numerical simulation of a complete system of equations (with propagation of all fields included) are presented. Clearly, both Stokes and anti-Stokes fields exhibit exponential growth with a gain coefficient on the order of a resonant absorption length. This initial period of exponentialgrowth is typically followed by saturation,after which all four fields propagate as in free space. Hence, the system can ultimately evolve toward establishing a trapping state, as in matched-pulse propagation [29]. A quite different evolution occurs in the case of counter-propagating fields. In this case the evolution of Stokes and anti-Stokes components is described by

a

1

E2 = - k az 2

--

1

~

~

+? ~ k~

+

&2

exp(i6kT)E; 2~ ~ ~

(53)

This is a boundary-value problem with boundary conditions 1 I (0) = &', E2(L) = 0. The solution of this linear problem yields the following expression for the amplified probe and generated phase-conjugate field:

366

M. D.Lukin, P. R. Hemmer; and M. 0. Scully

Upon inspection of Eq. (55), one finds that the amplitude of the generated field becomes large and diverges when the resonance condition exp(26L) = (6u + q)/(Su - q) is fulfilled. In general, this condition gives rise to two real equations for the single variable L and can be fulfilled only under special circumstances. In particular, such a solution exists when (i) 6u is real, (ii) q is purely imaginary, and (iii) the length of the cell is equal to L, = tan-’ (lql/Su)/1q1. Physically, (i) is just the phase-matching condition, (ii) determines the threshold condition for the drive intensity, and (iii) defines the resonant cell length (L,). For the present scheme at the point described by Eqs. (44)and (49,we find that if 6k, = 0, then 6a is real, q is imaginary for R > Rm, and L, can be as small as the “bare” absorption length La = 2hy,/ (kp2).Note that by properly choosing the directions of propagation of the drive fields, 6 k , can be made to vanish. That is, a large parametric gain and mirrorless oscillation is possible within a very short optical path with a drive intensity much smaller than the optical saturation intensity. Such a striking difference between co-propagating and counter-propagating geometries is due to the fact that in the latter case, x3-type nonlinearity provides, in addition to nonlinear gain, an effective feedback mechanism. That is, Stokes and anti-Stokes fields propagating in the opposite directions scatter into each other, thus forming an effective cavity. Combined with large nonlinear amplification, such feedback can lead to very large gain, or even to oscillation. Also note that in such a case, no evolution toward establishing a general trapping state occurs. It is also important to note that threshold driving power given Eq. (46) corresponds, in the limit of a long-lived ground-state coherence, to that of only a few interacting light quanta. Assuming that both of the driving fields are near the respective single-photon resonances and that the lifetime of the coherence is on the order of the driving-pulse duration (r), we find

where A is the cross-sectional area of the driving beams. These predictions were verified by numerical simulations wherein the effects of saturation and driving-field propagation were included. We simulated the situation typical for OPC (Hemmer, 1995) when the frequencies of all four fields are similar, for which case 6k, = 0. The optical polarizations were calculated by solving the density matrix equations numerically for steady state. The results are shown in Fig. 9. They clearly indicate that close to the optimal conditions described above, there is a large amplification of the signal and conjugate fields. In general, the resulting parametrically generated intensity is a substantial fraction of the initial drive intensity. Numerical simulations also

RESONANT NONLINEAR OPTICS IN PHASE-COHERENT MEDIA 367 80000 60000

2 40000 10'

Y

20000

0

0

0.2

0.4

0.6

0.8

I"

1

1.2

1.4

z/L,

-0.05 0.00 0.05

w

FIG.9. Intensity gain of the probe (solid) and the conjugate (dashed) as a function of the probe field detuning for the parameters of Fig. 7 and a: = 10-4yp, a: = 0, L = 1.44L,, and Y b l b2

= 10-4Yp.

showed that, in order to describe correctly the onset of mirrorless parametric oscillation, dissipative absorption of the driving should always be taken into account. In general, oscillations become possible when nonlinear gain exceeds the driving-field absorption coefficient.

VI. Physical Origin of Nonlinear Enhancement We here discuss the origin of the resonant enhancement of nonlinear optical processes described in the previous sections. The above analysis suggests that this origin can be quite different from the coherent population trapping that accounts for efficient parametric conversion in Jain et al. (1996). In the case when one of the driving fields detuned is sufficiently far from resonance while another is resonant with the corresponding optical transition, it can be argued that the trapping state is relatively weakly perturbed by the detuned field. However, this additional field leads to a light shift, which results in a large linear susceptibility for the probe tuned to exact two-photon resonance. That is, even such weak perturbation leads to coupling of the fields to the dark state. Such coupling, and the corresponding light shifts, can be compensated by small two-photon detunings. Hence, even in this case, population trapping can only partially account for efficient nonlinear conversion. Moreover, in the cases when population leakage out of the trapping state due to the presence of a second driving field becomes significant and causes substantial population redistribution within ground-state levels, efficient parametric interaction occurs under conditions when complete CIT does not take place. The particular mechanism making this possible can be understood

368

M. D.Lukin, I? R. Hemmer, and M . 0.Scully

2

4

6

8

1

0

2

4

6

FIG. 10. (a) A generic double-A system. Decays are outside of the system for an open system model and into the lower metastable levels for a closed system. (b) Different elementary processes leading to linear (phase-insensitive) absorption and amplification: (i) Usual linear absorption. (ii) Absorption in the presence of only one drive field R 1 . (iii) Absorption in the presence of two drive fields. (iv) Two-photon amplification of the field 1x2. (c) Evolution of linear gain and absorption as a function of interaction time for atoms injected in the state I b l ) . Curves i-iv correspond to the elementary processes depicted in Fig. lO(b). (d) Curve v shows the linear absorption for the case of equal injection into lower states (the net result of processes iii and iv). Curve vi is the parametric amplification for an atom initially in either of the lower states.

by considering the response of a simple double-A scheme (Fig. 10) on weak probe fields with Rabi frequencies ct1,2. When most atoms are in the metastable lower states I b1,2),transitions usually result in absorption and phase shifts for these fields. It is not so if such a system is coherently prepared by two resonant drive fields (Rabi frequencies R 1,2). Here, interference of absorption paths affects various elementary processes such that the stimulated absorption [(i) in Fig. 10(b)] is reduced and, at the same time, two-photon stimulated amplification (iv) is possible. For appropriately chosen parameters, the two processes cancel each other while the near-resonant nonlinearity is enhanced, resulting in a large parametric gain. The processes causing large nonlinear gain in the present system can be illustrated by tracing the evolution of an atom injected into one of the lower

RESONANT NONLINEAR OPTICS IN PHASE-COHERENTMEDIA 369

states of an open system [Fig. 10(a)]. Let a;, bi be the atomic probability amplitudes in the states la;), IbJ. The evolution equations in a rotating frame for these amplitudes are

+

where i , j = 1,2 (i # j ) and r = y i A . We have included the decay out of the upper levels (y) and assumed the same detuning (A) for all fields from the respective transitions. The ensemble-averaged induced probe polarizations can be calculated from the atomic wave function by averaging over the injection times (tin):

+

where T = Jb [Ci=1,2 [ail2 Ib;12]dtinis the normalization related to the average interaction time of the atom and N is the atomic density. For simplicity, we assumed equal matrix elements (p) on the probe transitions. Let us begin by injecting an atom into the state Ibl). The linear steady-state susceptibilities ( f i i ) are

These solutions are valid provided that 52 1,2 # 0. To examine the dynamics leading to these steady-state results, we first consider a completely resonant configuration (A = 0), for which the evolution of the polarizations corresponding to different processes is shown in Fig. 10. The case corresponding to the usual linear absorption [curve i in Fig. lO(c)] is obtained in the long-time limit by setting R 1 = 0, R 2 0 in Eq. (59). The driving field 01 causes a suppression of the absorption after the time ~i = y/R: necessary for establishing quantum interference (curve ii). In the long-time limit, absorption vanishes, as can be seen by setting 0 2 0, RI # 0 in Eq. (59). In this situation, EIT is achieved by accumulating the population in a coherent superposition of the states I b l ) and I b2) (dark state), which is decoupled from the optical fields. When the intensity of the second coherent drive field (R2) increases, the coherent superpositioncorrespondingto the decoupled state is partly destroyed. --f

--f

370

M. D.Lukin, l? R. Hemmer, and M. 0.Scully

Quantum interference, however, leads to a reduction of absorption in this case as well: in the case when R l = R 2, for example, absorption of a1 is reduced to half of its value in the absence of the drive fields [curve iii and Eq. (59)]. At the same time, however, the probe field 012 undergoes a two-photon Raman-like amplificationprocess involving the drive field R2 [Fig. lo@)]. From Eqs. (59) and (60) it follows that, in general, the rate of residual absorption of a1 is precisely equal to the two-photon amplification rate of c12. Physically this implies that the two are affected equally by a partial distortion of the “dark state.” It is the competition between the residual loss and the two-photon gain that mitigates the influence of linear absorption on parametric interactions. This can be seen most easily in a case when the atoms are injected into both of the lower states. Then the probabilities of these processes subtract, yielding vanishing absorption for a properly chosen injection configuration [see curve v in Fig. 10(d)]. As we show below, this situation is realized in a closed system with equal decay rates on the probe transitions. Here the drive fields cause population redistribution between Ibl) and I b2), which is equivalent to such an injection and leads to complete elimination of the linear susceptibility. Let us now turn to the interaction of the probe fields via a x3-type nonlinearity. The relevant steady-state susceptibilities for an atom initially in state I ~ I ) <%$,

indicate that for a properly chosen phase between the fields, a closed-loop parametric process is possible [Fig. 10(a)], resulting in equal amplification of both weak fields. This process is completely resonant and is enhanced rather than suppressed by the quantum interference: after the time ~ i the , resulting parametric gain is on the order of the “bare” linear absorption [curve vi in Fig. 10(d) and Eq. (61)]. We note that for an injection configuration yielding vanishing linear polarizations, the nonlinear susceptibilities add. Thus, under the conditions of suppressed absorption, the nonlinearity is resonantly large, resulting in efficient parametric gain. The implication of Eqs. (59) and (60) for a phase-matching condition should not be overlooked. It can be seen from these equations that the refractive index resulting from single-photon transitions is completely eliminated and that the residual contribution due to the two-photon transitions is zero when R1 = 0 2 . The latter, however, is not a necessary condition for phase matching. It follows from Eqs. (59) and (60) that the residual refractive index is identical for the two weak fields, which can lead to perfect phase matching at arbitrary drive intensities. We emphasize that this conclusion is not

RESONANT NONLINEAR OWICS IN PHASE-COHERENT MEDIA 371

restricted to systems with symmetric decay rates and equal drive intensities. As can be seen, for example, by substituting Eqs. (59), (60),and (61) into the resonance conditions,efficient processes can occur on resonance with unequal drive Rabi frequencies and with asymmetric injection into the lower states. The symmetric case considered here is most favorable because of the maximal nonlinearity. Before concluding the theoretical discussion, we note that Doppler broadening frequently plays an important role in experiments involving resonant interactions with coherent atomic media. In general, the consequences of Doppler broadening can be divided into two groups. First of all, Doppler broadening on the dipole-forbidden two-photon transitions (bl -, b2) leads to washout of the (typically narrow) transparency window. This effect is especially important when the wavelengths of the Raman fields (e.g., R 2 , a 2 andR1,al)differbyalargemargin Ikl - k d l l Ikll,Ik2 - k d 2 1 Ik21.This undesirable effect of inhomogeneous broadening can be mitigated by increasing the Rabi frequencies of the driving fields: when the latter become on the order of the two-photon inhomogeneous broadening, transparency can be restored. This approach has been used in Pb experiments of the Stanford group (see Jain et al., 1996).At the same time, if the frequencies of the Raman fields are similar and if fields in each Raman pair propagate nearly colinearly, two-photon Doppler broadening can be very small. This is the case for the experiments described in the present article. In this case, transparency resonance is not seriously affected by atomic motion. However, the values of generated nonlinearity become smaller due to effective broadening of optical transitions. Furthermore, because of atomic motion, nonlinear efficiency becomes sensitive to the angle between the Raman fields, which is a drawback for applications such as optical phase conjugation. In conclusion, we have discussed various mechanisms for resonant enhancement of nonlinear optical interaction. We have shown that this effect makes it possible to have efficient parametric processes at much lower laser intensities and optical path lengths than required for conventional nonlinear optics. In particular, a double-A system driven by strong optical fields can yield a very large parametric gain or even oscillation in a frequency-mixing process. We have also shown that, in general, the mechanism making such efficient processes possible is somewhat different from coherent trapping and is due to a more general effect of atomic interference. N

N

VII. Optical Phase Conjugation in Double-A Systems The use of double A-like systems in resonant nonlinear optics can provide long-sought-afterperformance by allowing operation much closer to resonance

372

M. D.Lukin, P. R. Hemmer; and M. 0.Scully

LASER AIO

1

I

LENS

CA PINHOLE DET

. .

m

FIG. 1 1 . (a) Sodium energy-level diagram with approximate laser detunings referenced to zero-velocity atoms. (Diagram not to scale.) (b) Schematic of experimental setup for optical phase conjugation. The lens and helium jet apply only to aberration correction experiments.

than was possible previously. Demonstrated performance enhancements in sodium vapor include an order-of-magnitude improvement in optical phase conjugate gain with a two-order-of magnitude reduction in optical pump intensity (Hemmer et al., 1995).These enhancements have made it possible to demonstrate the correction of high-speed optical aberrations (Sudarshanam et al., 1997), large optical gain with minimal distortion (Grove et al., 1997), and classical noise correlation (Grove et al., 1997). For the most part, these proof-of-principle demonstrations have been performed in sodium vapor because of the ease of comparison with a large number of prior experiments (Leite et al., 1986; Vallet and Grynberg, 1991). Nonetheless, preliminary experiments have also been performed in rubidium vapor (Grove et al., 1998), which has the advantage that it can be excited by semiconductor lasers, and spectral holeburning solids (Ham et al., 1997), which are free from the problem of grating washout due to atomic motion. Representative experimental conditions for optical phase conjugation in sodium vapor are illustrated in Fig. 11. Here, Fig. ll(a) shows the hyperfine sublevels of sodium with approximate laser detunings. Ground-state hyperfine sublevels F = 1 and

RESONANT NONLINEAR OPTICS IN PHASE-COHERENTMEDIA 373

F = 2 correspond to levels bl and b2, respectively, and excited-state hyperfine sublevels F = 1 and F = 2 correspond to levels a1 and u2, respectively. The forward and backward pump beams, F and B, correspond to 01and 02, respectively, and the signal and conjugate beams, S and C, correspond to El and E 2, respectively. The excited-state detunings shown correspond to optimal performance for aberration correction. In general, the frequency separation between lasers F and C is in the range of 300400MHz in frequency, with laser F tuned slightly below the F = 1 excited state. Here, it should be noted that the detunings shown are referenced to zero-velocity atoms. For moving atoms, the detunings from excited levels F = 1 and F = 2 depend on Doppler shifts, which are opposite for counter-propagating fields. The Doppler width of sodium in the typical operating temperature range of 180-220°C is 1GHz, so that all laser beams are tuned well inside the Doppler-broadened absorption line. Thus, Fig. 1 I(a) applies only to a small subset of atoms in the vapor and should be considered only approximate. Finally, the excited-state hyperfine splitting is also not resolved in sodium vapor, and hence the assignments given for levels a1 and a2 are somewhat arbitrary. The experimental setup for optical phase conjugation is illustrated in Fig. Il(b), where the lens and helium jet apply only to the aberration correction experiments. As shown, laser beams F and B are derived from separate dye lasers, for convenience. To suppress the effects of laser jitter on the Raman transition, laser beam S is generated from F using an acousto-opticfrequency shifter (A/O), driven with a stable microwave source. This ensures that the frequency jitters of F and S are correlated. Laser beams F and S (and therefore B and C) intersect at an angle of about 5 m a d inside the sodium cell to avoid Doppler broadening of the two-photon (Raman) transition. The polarizations of beams F and S are linear and orthogonal. Beam B has the same polarization as F, and beam C is found to have the same polarization as S. Typical pump intensities (F and B) are on the order of 5 W/cm2 with spot sizes in the range of 1.3-1.5 mm in diameter (FWHM) inside the sodium cell. The signal (and conjugate) beams are somewhat smaller, typically l.0mm in diameter. Under optimal conditions, pump depletion is large and the pump beams exiting the cell have visibly dark centers. The sodium cell is a heat pipe oven with an active length of about 5cm and a background pressure of about 10 mtorr, and it is magnetically shielded to better than 100 mG. Under the conditions illustrated in Fig. 11, it is easy to observe phase conjugate (and transmitted signal) gains greater than 100 and sometimes in excess of 1000. However, for many applications, these very large values of gain are not usable because of optical distortions such as self-focusing and filamentation. To determine experimentally the largest gain that can be achieved without distortion, profiles of the transmitted signal beam S are recorded for different values of optical gain in Fig. 12. The transmitted signal

M. D.Lukin, P. R. Hemmer; and M. 0. Scully

374

1

h

8"

Y

-1

0

1

X POSITION (mm)

X POSITION (mm)

FIG. 12. Laser beam profiles for transmitted signal beam. Contour plots are shown on the left. Contours at 30%. 60%. and 85%. Line traces (near Y = 0) are shown on the left. (a) Unamplified probe beam. (b) Optical gain of 6. (c) Optical gain of 30. Because of pump depletion, it was necessary to attenuate the input signal beam by 10 to achieve this gain.

beam is monitored, rather than the conjugate beam, because it is a more sensitive measure of self-focusing and distortion. Figure 12(a) shows the profile of the unamplified S at a distance of 1.2 m from the cell with all lasers tuned far from resonance. In Fig. 12(b), S is shown when the optical gain is 6. As illustrated, the transmitted signal is only slightly self-focused under these conditions. At a gain of 30 [see Fig. 12(c)], spatial distortions are just beginning to appear, and for many applications this would be the maximum usable gain. To illustrate the large improvement obtained using the double-A-like system, these results can be compared with previous four-wave-mixing experiments in sodium vapor that reveal strong distortion and self-focusing effects for gain values above 2.5 (Leite et al., 1986). Classical intensity noise correlations are also observed between the amplified signal S and conjugate C beams (not shown). The noise is measured by monitoring the power in S and C with a photodiode and a spectrum analyzer. Noise correlations are measured by subtractingthe photocurrent signals, generated by S and C, before inputting to the spectrum analyzer. In particular, a noise correlation experiment is performed for an optical power gain of 20. At a noise frequency of 300kHz, the noise in S (and C) is about 3 dB higher than in a laser beam of equal intensity taken directly from the dye laser output. When the difference current

RESONANT NONLINEAR OPTICS IN PHASE-COHERENTMEDIA 375

is monitored, the noise drops by 15dB, indicating a high degree of noise correlation. An attempt was also made to measure quantum noise correlations, but the dye laser classical noise proved too large.

VIII. Correction for Optical Aberrations in Double-A Medium To demonstrate the ability of phase conjugation in the double-A system to correct for high-speed optical aberrations, the helium jet and imaging lens shown in Fig. 1l(b) are used. The lens (focal length of 17cm) is arranged to image the jet into the active region of the sodium cell, ensuring that all the signal beam light fits within the aperture determined by the F and B pump beam diameters. The jet is produced by forcing helium gas through a rectangular nozzle (4 mm x 0.125 mm) aligned with the 4-mm side along the path of S. The average helium flow velocity at the nozzle exit is estimated to be about 170m/s. The nozzle is located about 2 mm below the signal beam path. To provide a large-amplitude high-speed aberration, the jet is driven acoustically at the flow resonance frequency of 18kHz,using a piezo-electric transducer (PZT). Aberration correction with gain is shown in Fig. 13. Here,

FIG.13. Aberration correction demonstration. Top row shows (overexposed) video images. Middle row shows a horizontalline trace, taken near the vertical center of a properly exposed video image. Bottom row shows temporal aberrations obtained using a pinhole detector. (a) Aberrated signal beam after a single pass through the jet and propagating 80cm. (b) Corrected conjugate beam after passing back though the jet and propagating 65 cm. Optical power gain is 30.

376

M. D.Lukin, P R. Hemmer; and

M. 0.Scully

Fig. 13(a) shows a video image and line trace (top and middle, respectively) of the aberrated signal beam after passing once through the turbulent jet and propagating 80cm. The video camera could not respond fast enough to image the 18-kHz oscillations in the turbulent flow, and hence a photodiode with a pinhole aperture 0.2mm in diameter is used. The temporal signal from the pinhole detector is shown in Fig. 13(a) (bottom) for the aberrated signal beam. There is a large-amplitude signal at the PZT drive frequency, indicating large-amplitude flow oscillations. The corrected phase conjugate beam is shown in Fig. 13(b) after passing back through the jet and propagating 65 cm. The spatial aberrations are well corrected. The conjugate spot looks nearly identical to that seen when the flow is turned off (not shown). The temporal signal from the pinhole detector shows only a small residual signal at the flow drive frequency, indicating a factor of 8 reduction high-speed aberrations. The optical power in the conjugate beam of Fig. 13(b) is 30 times larger than in the input signal beam. Without the turbulent flow, the optical power gain is 45. Hence, very little performance is lost, even with the high-speed aberrator in place. The significance of this power gain is best appreciated by comparing with the highest previously observed cw phase conjugate gain in sodium vapor, 2.5 (Leite er al., 1986). The enhancement is over an order of magnitude. Moreover, the 5 W/cm2 of pump power used is more than two orders of magnitude smaller than the 1 kW/cm2 used in earlier experiments. Thus, altogether, a three-order-of-magnitude improvement in phase conjugate performance is achieved by using atomic coherence. Finally, to verify the role of long-lived ground-state coherences, such as those expected in the double-A system, the phase conjugate gain is monitored as a function of two-photon (Raman) detuning. Experimentally,this is accomplished by tuning the microwave frequency applied to the A/O [see Fig. 1 l(b)] while compensating for angular deviations. At a gain of 30, with the highspeed aberrator present, the two-photon linewidth was measured to be 2 MHz. Similar two-photon linewidths were also obtained with no aberrator present for gain values between 1 and 30. This width is much less than the 10-MHz natural width of the sodium excited state, and it is taken as evidence of ground-state hyperfine coherences produced by the double-A interaction.

IX. Nonlinear Spectroscopy of Dense Coherent Media In conventional methods of resonant spectroscopy, optically thin ensembles of atoms or molecules are probed with lasers of limited intensity (Vanier and Audoin, 1989). Low intensities are used in order to avoid power broadening and frequency shifts. In general, as the Rabi frequency of the

RESONANT NONLINEAR OPTICS IN PHASE-COHERENT MEDIA 377

resonant electromagnetic field exceeds the natural width of the transition under consideration,the width of the resonance increases, resulting in a rapid loss of resolution. This effect is especially profound when transitions between long-lived states are considered, in which case the necessity of low-intensity light fields results in an unfavorably low signal-to-noiseratio. A resonant twophoton transition between metastable levels in a A-type atomic configuration is an example of such a system. In what follows, we show that an optically thick ensemble of coherently prepared A atoms exhibits quite different spectroscopic properties. In particular, measurements carried out in such media can result in extremely narrow resonance widths, which are no longer restricted by the single-atom power-broadening limit. This allows one to combine the advantages of a narrow atomic response with those of large signal-to-noise ratio. The physics of this new type of spectroscopy can be understood by considering a simple three-level A-system. A strong driving field with Rabi frequency R produces Autler-Townes dressed states of the transition b2 + a. When the driving field is tuned to the center of this resonance, the dressed states are symmetric and separated by 2fl. The interference of the dressed states causes a drastic reduction of absorption of the probe field when it is tuned to the center of the bl + a transition. The width of the transparency window, or dark resonance, generated this way can be estimated by noting that absorption reaches a maximum when the probe field is tuned to the resonance with either of the Autler-Townes components. Thus, we arrive at the conclusion common to all resonant detection schemes that the dark resonance is power-broadened and its spectral width scales with R. This conclusion, which describes correctly a single-atom response, is not valid in the case when transmission through an optically thick atomic ensemble is considered. In this regime, only spectral components that are very close to the center of the transparency window are transmitted, so the coherent atomic system displays a density-dependent spectral narrowing of the transparency window. The narrowing of the dark resonances with increasing density can easily be understood from an idealized three-level model in which probe and drive interact solely with the respective transitions a + b and u + c. For simplicity let us assume, in addition, a spatially homogeneous Rabi frequency of the drive field. The intensity transmission of a monochromatic probe of wavelength h through a vapor cell of length L is described by

Here k = 2n/h and X" is the imaginary part of the susceptibility x for a A medium.

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M.D.Lukin, P. R. Hemmer; and M. 0.Scully

In the low-density limit ( q k L << l), the spectral features of the transmission coincide exactly with those of x”. In the high-density limit, however, only spectral components that are very close to the center of the transparency are transmitted. In order to estimate the effective width of the EIT window, we expand x” around the point of maximum transmission, assuming the usual EIT conditions.The resulting transmission spectrum is a Gaussian function of the detuning. Its characteristic half-width,

scales inversely with the square root of the density-length product and can become substantially smaller than the single-atom power-broadened width. However, this is still not the ultimate limit of resolution that is attainable. Narrower spectral features can be achieved by using dispersion spectroscopy, that is, when an interferometric measurement of phase shifts is carried out. In such a measurement the signal is typically given by

where 4 is a phase shift of a probe field in the cell. Because the index of refraction close to the point of maximum transmission is a linear function of the probe detuning, the phase shift of the transmitted field is also linear in A . For a large density-length product, the phase shift accumulated over the cell length can easily surpass 271 already for a very small detuning from the twophoton resonance. We define a characteristic width AWdis,

as the detuning from the line center at which the phase of a probe laser shifts by n / 2 . This expression scales inversely with the density-length product itself instead of its square root as in Eq. (63).Thus we conclude that in dense media, the width of interferometric fringes (“dispersive” width A a d i s ) can be considerably smaller than the characteristic width of the EIT-transmission equation, Eq. (63). Let us consider the implications of the compensation of power broadening in dense media. To this end we estimate the theoretical limit of this compensation by considering an ideal three-level A-type medium with a large density-length product, such that the probe-field intensity is attenuated

RESONANT NONLINEAR OPTICS IN PHASE-COHERENTm I A 379 at line center by 1/e. In this case, which corresponds to a maximum signal-tonoise ratio at the output, the “absorptive” and “dispersive” widths approach

which implies that under idealized conditions, the effect of power broadening can be completely compensated in a dispersive measurement (Scully and Fleischhauer, 1992). Thus, an intense laser field can be used without penalty in resolution. As a result, the potential sensitivity of EIT-based spectroscopy can substantially exceed that of the conventional techniques. In the experiments, the 5 S l p + 5P1/2 transition (the D1 line) of 87Rbwas used to realize a three-level A system. n o extended-cavity diode lasers, a drive laser, and a probe laser were phase-locked with a frequency offset ( 0 0 x oms = 6.83 GHz) determined by a tunable microwave-frequency synthesizer. The locking of the two lasers was sufficient to keep the linewidth of the beat note to below 1 Hz. Two collinear laser beams were passed through a cell 5 cm long containing natural Rb, and the transmitted power was detected by a fast photo-detector (PD). Both beams were of identical circular polarization. The powers of the drive and probe beams in the cell ranged from 5 to lOmW and from 0.05 to 0.1 mW with spot sizes of 5 mm and 3 111111, respectively. The frequency of the , = 2 + Pip, F = 2 transition drive laser was tuned to the center of the S 1 / 2F of the Rb D1 line, and the frequency of the probe laser was scanned across the S112,F = 1 + P1/2, F = 2 transition by tuning the synthesizer frequency. In such a configuration, probe and drive fields create a A-type system within the Rb D 1 manifold (see Fig. 14).

MG 6.8GHz

F=2’

F=l’

F=2

F=I

(a)

@)

FIG. 14. (a) Energy levels in the DI line of 87Rb.(b) Experimental setup (schematic). ECDLl and ECDL2 are extended-cavity diode lasers (probe and driving lasers, respectively). PD: photo-detector, P polarizer, MG: microwave synthesizer, and SA:spectrum analyzer. The Rb cell is inside magnetic shields.

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M.D.Lukin, l? R. Hemmer; and M. 0.Scully

probe detuning

probe detuning

-

FIG.15. Measured signal as a function of a relative laser detuning for various atomic densities. (a) Curves i-iv correspond to atomic densities N 6 x lo'', 1 x l o 1 ' ,2 x 10". and 6 x 10" cm-3 respectively. Estimated Rabi frequencies are 10MHz and 2MHz for drive and probe. (b) Extra resonances in optically dense Rb vapor. Drive-field Rabi frequency is 13 MHz, and atomic density is 10l2 Scan time was a few seconds. The estimated time-of-flight broadening in the probe and drive beams is ~ 1 lcHz 5 and 8 kHz, respectively. There is no external magnetic field. Natural linewidth of the Rb Dl line is 5.4MHz. Doppler broadening is 500 MHz.

-

-

N

In the present experiment the signal was recorded by monitoring the amplitude of the time-dependent component of the detector current at frequency 0 0 = cop - cod corresponding to the difference frequency between probe and drive fields. In the case of a dilute medium, the heterodyne signal at the photo detector is proportional to the transmitted probe field amplitude. At relatively low values of the atomic density, the observed signal shows the usual absorptive features corresponding to EIT (Fig. 15). When the beat-note frequency of the lasers matches the resonant frequency of the b -+ c transition, the medium becomes transparent, displaying an EIT-type resonance [curve i in Fig. 15(a)] with a nearly symmetric line-shape. The peculiar feature of this resonance is that its width decreases with the atomic density. At higher densities the transparency window becomes substantially asymmetric. Within this window, additional resonances a few kilohertz wide [Fig. 15(b)] were clearly resolved. Their characteristics such as position, width, and amplitude depend on the strength of the drive field and on the atomic density. Furthermore, their position was found to be very sensitive to atomic-level shifts, induced, for example, by weak magnetic fields. Under the conditions of dense media, resonances as narrow as 3 kHz were observed with a high signal-to-noiseratio. For the data presented here, the estimated Rabi frequencies of drive and probe fields were correspondingly on the order of 10MHz and 2 MHz. We now turn to the origin of the novel resonances. These resonances are due to a new field generated by resonantly enhanced coherent Raman

RESONANT NONLINEAR OPTICS IN PHASE-COHERENTMEDIA 381

scattering of the type discussed in Section IV,and they reflect the dispersive properties of the medium. The mechanism of this process can be understood if we note that the strong driving field couples the excited state to both groundstate hyperline levels. The interaction of the drive and probe field with the resonant transitions generates coherence between the ground states. In the presence of this coherence, the interaction of the drive with the off-resonant transition S112,F = 1 + Pip, F = 2 leads to a nonlinear generation of a new field E n with frequency w, = 2wd - cop. This process is very similar to the frequency conversion in coherent atomic media discussed in the previous section. For a large density-length product, the output amplitude of the new field E n can be of the same order as that of the probe field, E,. At the photo-detector, the new field gives rise to an additional component to the beat note at frequency 00. Taking this contribution into account, we find that the signal power at 0 0 in the present heterodyne detection scheme is given by

To understand the nature of the additional resonances, in particular their relatively narrow width, let us consider again the simplified three-level model discussed before. We now, however, take the nonresonant couplings and the associated parametric process into account. Let us further assume that the probe and “new” field are much weaker than the drive field, and that the drive field (Ad = 0) is undepleted. These assumptions (which will not be used in the later numerical simulations) allow for a simplified discussion of the relevant physics. In this case we arrive at the system of equations

The coefficient a l l is proportional to the linear absorption and dispersion of the resonant field E,, and a22 to that of the off-resonant field E n . The crosscoupling coefficients a12 and a21 describe the resonantly enhanced type nonlinearity. Under near-resonance conditions and in the absence of Doppler broadening, they are proportional to susceptibilities given by Eqs. (38) through (41). Solving this linear propagation problem and substituting the solution into Eq. (68), we find that the beat note at the photo-detector contains an interference term with an amplitude depending on the probe laser detuning , modulates the usual transmission via the phase shift A 4 = ~ ’ k L / 2which profile and gives rise to extra resonances. Hence the characteristics of the new resonances are determined by the dispersive properties of the EIT medium.

382 0.9

b

3 0.6

..-( am

I’

100 kHz

m

0.3 0 probe detuning

probe detuning

FIG. 16. Calculated signal [EQ.(68)] as a function of probe detuning. (a) Curves i-iv correspond to atomic densities of Fig. lS(a). (b) Extra resonances under the conditions corresponding to those of Fig. IS. Relaxation rate of ground-state coherences is taken to be 15 kHz,and relaxation rate of ground-statepopulations is 8 kHz. Signal is normalized such that unity corresponds to undepleted probe and drive fields.

As was shown above, in dense media, the width of interferometric fringes (“dispersive” width Aodi,) can be considerably smaller than the characteristic width of the EIT-transmission equation, Eq. (63). To make a detailed comparison with the experiment, we consider a theoretical model in which the three-wave mixing process is included, together with Doppler broadening and the full (hyperfine) level structure of Rb87.The propagation equations for the three fields were solved numerically. The atomic polarization was calculated from the density matrix equations with a Floquet ansatz. To truncate the resulting hierarchy, the beat-note frequency 0 0 was assumed to be large compared to the relevant decay rates and Rabi frequencies. As shown in Fig. 16, the result of this calculation is in good agreement with the experimental data. In particular, the additional narrow resonances indeed have a width close to the “dispersive” width in a dense medium, Eq. (65).

X. Outlook The main goal of this concluding section is to point out that much fascinating physics involving nonlinear interactions in coherent media remains to be learned, and many practical applications in this area remain to be discovered. This conclusion is motivated by the most recent developments in the field, indicating several promising new proposals as well as exciting experimental results.

RESONANT NONLINEAR OPTICS IN PHASE-COHERENT MEDIA 383

On the theoretical side, Harris and Jain (1997) have suggested an efficient optical parametric oscillator based on population trapped atoms. Such an oscillator has very remarkable properties because it combines high efficiency with wide tunability. For example, in dense Pb vapor, the estimated gain bandwidth is on the order of central frequency. A different study suggested a technique for sub-femtosecond pulse generation based on coherent population trapping-like effects in molecules (Harris and Sokolov, 1998). An EIT-based scheme yielding large Kerr nonlinearity has been suggested by Imamoglu and coworkers, who suggested the possibility of using this scheme for manipulation of single photons (1997). A related scheme has been analyzed in the context of few-photon quantum control in Dunstan et al., 1998. Quantum correlations and substantial suppression of quantum noise in nonlinear interactions based on four-wave mixing in a double-A system has been predicted by Lukin et al. (1999).This work suggested that efficient nonlinear interactions in phase coherent media can be extended into domains involving just a few interacting light quanta at a time. This indicates that an entire new domain of quantum nonlinear optics is emerging from these studies (see also, Harris and Hau, 1999). In a closely related work, Harris and Yamomoto (1998) suggested a quantum switch in which cross-coupling nonlinearity is so large that a few photons can switch the transmission for a probe field from complete transparency to complete absorption. The possibility of the interference effects induced by interactingdark resonances in a similar system have been studied in Lukin et al. (1999b). On the experimental side, the ideas of atomic coherence have been extended to solid-state materials. In particular, a new regime of stimulated Raman scattering has been demonstrated in solid hydrogen (Hakuta et al., 1997), and multiwave mixing has been observed in Pr-doped crystals (Ham el al., 1997). Furthermore, spontaneous parametric oscillation has been observed in a double-A-like system in a Rb vapor (Zibrov et al., unpublished-a). Such oscillation does not require an optical cavity, occurs at very low driving intensities, and manifests itself as a comb of Stokes and anti-Stokes components separated by a two-photon Raman transition frequency. Spontaneous oscillation of this kind can be viewed as a spontaneousformation of atomic coherence grating. It is of direct relevance to the theoretical studies noted in the previous paragraph. Furthermore, it can be used to produce a narrow-band beat signal at a frequency corresponding to that of hyperfine splitting of the Rb ground state, which is of interest for frequency standards and optical magnetometry. In a similar vein, Hau and coworkers (1999) demonstrated a group velocity reduction of a light pulse to 17m/s and reported large Kerr nonlinearities in an atomic Bose-Einstein condensate (see also, Kash et al., 1999). Finally, we note that unusually efficient frequency up-conversion using atomic coherence in a cascade system has been observed (Zibrov et al., unpublished-b).

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XI. Acknowledgments The authors warmly thank L. Hollberg, S. Harris, B. Ham, E. Fry, M. Fleischauer, M. Loeffler, Yu. Rostovtsev, S. Shahriar, A. Sokolov, V. Sautenkov, V. Velichansky, S. Yelin, and A. Zibrov for their contributions and useful discussions. Special thanks are due to Stephen Harris for useful comments and suggestions on the manuscript. We gratefully acknowledge the support from the Office of Naval Research, the National Science Foundation, the Welch Foundation, the Texas Advanced Research and Technology Program, and the Air Force Laboratories.

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Scully M. O., and Fleischhauer, M. (1994). Science 263, 337. Sudarshanam, V. S., Cronin-Golomb, M., Hemmer, P. R., and Shahriar, M. S. (1997). Oprics Letters 22, 1141. Tewari, S.P., and Agarwal, G. S. (1986). Phys. Rev. Lett. 56, 1811. Vallet, M. P.M., and Grynberg, G. (1991). Opt. Comm. 81,403. Vanier, J., and Audoin, C. (1989). The quanrumphysics ofatomicfrequency standards. A. Hilger (Bristol). Xiao, M., Li, Y.,Jin, S., and Gea-Banacloche, J. (1995). Phys. Rev. Lett. 74, 666. Zhang, G. Z., Hakuta, K., and Stoicheff, B. P. (1993). Phys. Rev. Lerr. 71, 3099. Scully, M. O., Fearn, H., and Narducci, L. M. (1992). Z. Phys. D22, 483. Zhu, S.-Y., Zibrov, A. S., et al. (1995). Phys. Rev. Lett. 75, 1499. Zibrov, A. S., et al. (1996). Phys. Rev. Lett. 76, 3935. Zibrov, A. S., et al. (Unpublished-a). Zibrov, A. S., Lukin, M. D., Hollberg, L., and Scully, M. 0.. (Unpublished-b).

ADVANCES W ATOMIC, MOLECULAR,AND OPTICAL PHYSICS, VOL. 42

THE CHARACTERIZATION OF LIQUID AND SOLID SURFACES WITH METASTABLE HELIUM ATOMS H. MORGNER Institut f i r Experimentalphysik, Universitat Witten-Herdecke, Stockumer Strape 10, Witten, ER. Germany Wilhelm-Ostwald-lnstitut fur Physikulishe und Theoretishe Chemie, Universitat Leipzig, Linnestr. 2, Leipzig

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Reactions of Metastable Helium Atoms with Matter: General Survey. . A. Gas Phase Reactions of He* ........................... B. Interaction of He' Atoms with Surfaces . . . . . . . . . . . . . . . . . . . 111. Quantitative Evaluation of MIES Data ...................... A. Homogeneous Surface of Known Material . . . . . . . . . . . . . . . . . B. Surfaces of Composite Materials: Series of Spectra. . . . . . . . . . . IV. Discussion of Selected Systems ........................... A. Liquid Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Solid Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Summary.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

387 393 393 41 1 438 438

444 449 449 454 482 483 483

I. Introduction Surface science is a natural domain of electron spectroscopy because the limited mean free path of electrons discriminates strongly against signal from the bulk. The observation depth is governed by the energy-dependent mean free path of electrons in condensed matter, which ranges from a few tenths of a nanometer to several nanometers. There is, however, one particular electron spectroscopy whose surface sensitivity is perfect and does not rely on a small mean free path. This is MIES (MetastableInduced Electron Spectroscopy). Acronyms used by other authors for the same technique are MDS (Metastable Deexcitation Spectroscopy), MQS (Metastable Quenching Spectroscopy), and MAES (Metastable Atom Electron Spectroscopy). The method consists in colliding a beam of electronically excited metastable particles with the investigated surface. In most cases the projectiles are helium atoms in a metastable state that carries 387

Copyright 0 Zoo0 by Academic Press All rights of nproduction in any form reserved. ISBN 0-12-003842-01ISSN 1049-25OX100 $30.00

388

H. Morgner

excitation energy of about 20 eV. The transfer of this energy to the surface is very efficient and causes electron emission with high probability. After the reaction with the surface, the helium atoms have returned to their ground state. Thus, there is no danger that they contaminate the surface. Further, the kinetic energy being in the thermal energy range, the helium projectiles do not penetrate the surface, nor can they modify the surface by means of their impact energy. This is important because it is a desirable property of an analytical tool that it not induce changes in the surface investigated. The only risk for influencing surface properties could originate from the transfer of electronic excitation energy. This possibility can indeed not be excluded once and for all. We note, however, that even very sensitive surfaces like self-assembled monolayers of alkanethiols do not show any effect even after exposure to a beam of metastable helium atoms for a few hours. It is, of course, very important to control carefully the quality of the beam of metastable helium atoms so as to avoid the presence of fast neutral helium atoms. In conclusion, we state that electron spectroscopy based on the excitation energy of metastable helium atoms has an excellent potential for surface analysis. One feels tempted to ask why only a small minority of surface scientists make use of the inherent perfect surface sensitivity of MIES. This is all the more surprising because in technical applications,the thickness of films becomes smaller and smaller. Any spectroscopy with the potential to monitor exclusively the top layer of a sample without blending by substrate signal should be welcomed as ideal support in research or applications like quality control. One answer to this question might be that the interpretation of MIE spectra in terms of surface properties is more complicated than for other electron spectroscopies, e.g., for photo-electron spectroscopy. In the full range of photon energies employed for photoelectron spectroscopy, the emission of electrons is governed by the familiar dipole operator. This leads to a close and well-understood correlation between the properties of the investigated sample and the measurable properties of the ejected electrons, such as kinetic energy, wave vector, and spin orientation. The widespread use of X P S (X-ray Photoelectron Spectroscopy) for elemental and chemical analysis via core electron spectroscopy and of UPS (Ultraviolet Photo-electron Spectroscopy) for chemical analysis via valence state spectroscopy reflects this fact. This is in particular true for the angular resolved versions of both techniques (ARXPS and ARUPS), which are indispensable for destruction-free depth profiling, for determination of molecular orientation, and for band structure investigation. The situation is somewhat different for the reaction of metastable rare gas atoms with other matter, be it atoms, molecules, or the surface of condensed matter. The mechanism leading to electron emission is not unique as in the

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

389

case of photo-electron spectroscopy. So far three different mechanisms have been identified. Their occurrence depends on the nature of the surface, and it may even well be that more than one mechanism contributes to one and the same spectrum. The interpretation of MIE spectra sometimes requires careful analysis in order to separate contributions caused by different ways of energy transfer. An alternative perspective on the problem is the following: in the microscopic world there can never exist a mere measurement of sample properties. One always changes the properties of the investigated object. The reason is that one observes - strictly speaking - not simply the investigated object but the interaction between the object and the particles employed to carry out the measurement. One merit of photo-electron spectroscopy is that although the probing agent (the photons) alter the state of the sample, the sample has virtually no effect on the photons. Thus, calculating back from the observed reaction to the sample properties as they were before carrying out the measurement is often not trivial, but it is essentially straightforward. The reaction of excited rare gas atoms with other matter is burdened by the fact that the interaction between target and projectile leads to mutual changes. This is easily understood if one recalls that a metastable rare gas atom is not simply the carrier of a certain amount of excitation energy but is an atom capable of undergoing chemical interactions. Its valence shell structure is described by an excited s-electron that moves in the field of a singly charged ionic core. Thus, it is a particle that resembles very much the ground state alkali atoms of the adjacent row of the periodic system. Indeed, a close similarity between the behavior of metastable rare gas atoms and alkali atoms has been observed. In consequence, we note that the rare gas atoms in their metastable state as employed in MIES are far from being inert particles but are, rather, reactive as long as they have not transferred their excitation energy to the target. Accordingly, one expects a behavior that can be predicted from the behavior of alkali atoms. Indeed, the full spectrum of reactions known from alkalis has been observed for metastable rare gas atoms. Very important is the notion that metastable rare gas atoms can experience a sizable attraction by other matter. If so, part of the electronic excitation energy of the metastable is converted into kinetic energy of its nucleus and thus lost for driving the electron emission. As a consequence, the energy of the emitted electron may be substantially lower then expected from the excitation energy of the isolated rare gas atom. Further, the harpooning mechanism for which alkali atoms are famous in the presence of targets with high electron affinity has been observed to happen with large probability for metastable rare gas atoms if energetically allowed. Obviously, the electron-emitting process occurs then between the target and a rare gas ion rather than between the target and an excited neutral rare gas atom. Because the interaction potential is generally more attractive

390

H. Morgner

for an ion than for a neutral, the above remark on the conversion of electronic energy into kinetic energy applies a forten'ori in this case. The present understanding of MIES derives from two fields. The first one consists of all those activities that attempt to elucidate the reactions of metastable rare gases in the gas phase. The origin of this kind of work can unambiguously be traced back to F. M. Penning. In 1927 he reported - in a paper of less than half a page - that the breakdown voltage in rare gas cells is lowered considerably in the presence of impurity gases (Penning, 1927). He found that even very small concentrations of foreign atoms or molecules with sufficientlylow ionization potential reproducibly cause the effect. He concluded that at the lowered voltage, the mean energy of accelerated electrons was too low to ionize the rare gas atoms but sufficient to bring them into an excited state. The energy thus stored in the electronic degree of freedom of the rare gas could then be used to ionize the impurity species. Two further ideas completed the picture: the lifetime of the excited rare gas atoms had to be long enough to allow an encounter with one of the spurious foreign atoms, and the transfer of energy had to be very efficient. This resulted in the concept of the ionizing collision between a metastable rare gas atom Rg' and a target species B:

This equation has explicitly been written down by Penning (1927) except for the electron, which he apparently did not care for. Today, where reaction (1) is employed as the basis for an electron spectroscopy named PIES (Penning Ionization Electron Spectroscopy), everyone would consider equation (1) incomplete without the electron. Reaction (1) takes place with high probability provided that it is energetically allowed, i.e. if the excitation energy of the metastable particle exceeds the ionization potential of the target. The actual ionization cross section was measured later on and found always to lie in the range typical for gas kinetic collisions (Schut and Smit, 1943). Because all reaction partners involved in the process of Penning ionization have only thermal kinetic energy, it was obvious from the beginning that the ionizing mechanism had to be electronic in nature. The attempt to understand the details of the process was furthered by the application of electron spectroscopy to studying Penning ionization. The pioneers in this field are V. Cerm& (Cerma, 1996) and A. Niehaus (Fuchs and Niehaus, 1968). In particular A. Niehaus began to study the role of the interaction potentials in the entrance channel V [Rg*+ B] as well as in the exit channel V [Rg + B+] and their influence on the shape of the energy spectrum of the emitted electrons (Hotop and Niehaus, 1969). A number of techniques have been developed to study Penning ionization in more and more detail. Recently, substantial progress in

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

391

this line of research has been achieved by H. Hotop, who made it possible to characterize the influence of nuclear motion and of electronic mechanisms separately on the basis of highly sophisticated experiments (Merz et al., 1989, Merz et al., 1990). The second area on which present MIES is based has been shaped by Homer Hagstrum. More than 40 years ago, he began to study surface properties by recording spectra of electrons released under the influence of impinging rare gas ions (Hagstrum, 1954). All the concepts that Hagstrum has developed over decades of studying ion-surface reactions can be applied to MIES of metal surfaces. The reason is that metastable rare gas atoms approaching a surface can lose their excited electron, provided that unoccupied states in the surface are in resonance with the excited electron of the projectile. In front of metal surfaces with sufficiently high work function, this condition is always fulfilled, leading to RI (Resonance Ionization) of the metastable rare gas. Consequently, a rare gas ion finally arrives at the surface and causes electron emission via a process termed IN (Ion Neutralization) or AN (Auger Neutralization) by Hagstrum. The electron-emitting process of Auger neutralization is a genuine two-electron event. Hagstrum has stated (1966) that many body effects do not influence the experimental results so much that the interpretation of the AN spectra would require the treatment of more than two electrons. This is the standard notion today and as such is the underlying assumption in evaluating electron energy spectra obtained from Auger neutralization of ions. If the metastable is not ionized but arrives intact at the surface, then the electronemitting process is named AD (Auger Deexcitation). In the following we will use the term MIES for the characterization of an experimental situation in which a beam of metastable rare gas atoms is directed to the investigated surface. On the other hand, if we talk about an electron spectroscopic experiment with a primary beam of ions, we will employ the term INS (Ion Neutralization Spectroscopy). Neither term is meant to allude to the mechanism of the electron-emitting process. This is done by the expressions AN (Auger Neutralization), which points to the above-mentioned two-electron process, and AD (Auger Deexcitation), which formally is a two-electron process as well but leads to spectra whose structure closely resembles UPS. In order to avoid confusion, the terms MIES and INS will always be reserved for the experimental situation, whereas AN and AD are meant to describe the reaction mechanism, which is always subject to the interpretation of the experimental data. It is a familiar notion that in MIES both mechanisms AN and AD can occur. However, both mechanisms can be found even in INS. Whereas the presence of AN hardly needs a comment, the occurrence of AD is well documented: Hagstrum has pointed out that helium ions can pick up an electron in front of a potassium-covered metal surface and hence cause a typical AD spectrum (Hagstrum, 1979).

H.Morgner

392 PROPERTIES OF

Atom He Ne

Ar Kr Xe

TABLE I METASTABLE STATES OF RAREG A S ATOMS.~

Metastable State

Excitation Energy, E' (eV)

Ionization Energy, I* (eV)

Lifetime,bT ( S )

lS2S3SI ls2s 'so 2ps 3s 3P2 2p53s 'Po 3p54s P2 3ps 4s Po 4p5 5s 3 P* 4p5 5s 3Po 5p5 6s P1 5p5 6s 3P0

19.8196 20.6158 16.6191 16.7154 1 1 S484 1 1.7232 9.9182 10.5624 8.3153 9.4472

4.7678 3.9716 4.9454 4.849 1 4.21 12 4.0364 4.0844 3.4372 3.8145 2.6826

7870; 9090 & 30% 0.0196; 0.0197(10) 24.4; >0.8 430 55.9; 38:; 44.9 85.1; 39:: 0.488 149.5;42.9(9) 0.078; (128+:i2) x ~ _ _ _ _

a

All values taken from Hotop ( 1 996). Conflicting values of the lifetime are from different sources compiled by Hotop (1996).

At first glance, one would not recognize any merit of MIES over INS in studying metal surfaces. However, it has been argued (Sesselmann et al., 1987) that using metastables rather than ions is advantageous in the following respect: on one side the spectra from INS tend to get broadened with increasing collision energy (Hagstrum, 1954), and on the other side lowering of the kinetic energy of the ions is possible only at the expense of intensity. Accordingly, it is of interest that high-intensity beams of metastable atoms can easily be produced experimentally with thermal kinetic energy. Thus, in the AN spectra obtained from MIES rather than from INS, the broadening effect should be minimized. Experiments in the gas phase as well as at surfaces have been carried out with all metastable rare gas atoms. Their properties are listed in Table I. The species with the shortest lifetime is He(2'S) with T x 20 ms. For the purpose of atomic beam studies this is still sufficient, because at thermal velocity the traveling time in a normal-sized vacuum machine is of the order of only several hundred microseconds. In the present paper we will concentrate mainly on the reactions of metastable helium atoms. The metastable states of helium store the largest excitation energy, which makes the accessible spectral range broader than is the case for the other metastable rare gas atoms. Further, the body of information is largest for helium because most studies have been carried out with this metastable species. Another important aspect is that only with helium can we select the spin state of the metastable, and in some cases this gives interesting insight into the projectile-target interaction.

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

393

The goal of the present work is not to compile a complete list of all activities in the field of MIES. This would be superfluous, because an excellent review article covering the works with MIES on many inorganic and organic surfaces has appeared (Harada et al., 1997). The authors summarize, at the end of their paper, that “the information obtained by MAES [their preferred abbreviation for the same technique] is unique but rather qualitative compared to other electron spectroscopic techniques.” We share this point of view with respect to the status of the field up to recently. We feel, however, that we can offer now some progress toward a quantitative assessment of information from MIES. The paper is organized as follows: In Section II.A, the gas phase reactions of metastable helium atoms are described. The goal of the discussion is only partly to portray the gas phase properties as such. We rather try to provide insight that could also be fruitful in dealing with surface reactions. The present status of MIES as a surface probe is discussed in Section 1I.B. Recent advances with respect to the quantitative evaluation of MIES data are described in Section 111, and some applications are presented in Section IV.

11. Reactions of Metastable Helium Atoms with Matter: General Survey A. GASPHASEREACTIONSOF HE* 1. Elastic Scattering

Collisions of metastable helium atoms in the gas phase under single-collision conditions lead to elastic scattering with high probability. In general, the elastic channel (2) dominates by far the inelastic processes (3) and (4). Rg* + B Rg* + B

+ Rg*+B + Rg

+ B+ + e-

Rg*+B+Rg+B**

(2) (3) (4)

The quantities describing the elastic channel are the differential elastic cross section % (9) and the total elastic cross section oel =J% (9)dS-l. Mathematically, the latter is simply the integral over the differential cross section. Experimentally, the situation is not as easy. The reason is that the differential cross section is usually not measured in absolute units but reflects only the shape of the dependence on the polar angle 9. Therefore, the determination of

394

H. Morgner

the total elastic cross section is carried out in a rather indirect fashion. Careful evaluation of the differential elastic cross section allows the determination of the interaction potential between the collision partners. Once the interaction potential is known, the total elastic cross section can be calculated. The differential elastic cross section depends on the interaction potential between Rg' and B and, conversely, must contain information on this quantity. Scattering theory (Child, 1974)provides a link between the differential elastic cross section and the interaction potential. One complication arises from the fact that the mere presence of the inelastic channels (3) and (4) influences the outcome of elastic scattering. Multichannel scattering theory (Child, 1974) teaches how to cope with this situation: it is possible to avoid the explicit incorporation of the numerous inelastic channels and to restrict the treatment to the elastic channel, but the price paid for this formal simplificationconsists in the replacement of the real-valued potential curve by a nonlocal, energydependent operator, called the optical potential. Fortunately, it could be shown that for ionizing reactions of metastable rare gas atoms, this unpleasant operator can again be replaced by a local and energy-independent,even though complex-valued potential (Morgner, 1990)

To zeroth order, one can relate the real part V (r) to the shape of the elastic cross section and the imaginary part to the size of the inelastic cross section. However, present experiments are so accurate that this simplification cannot be applied in the interpretation of experimental data. The fact that the relation between elastic scattering data and the local complex potential, Eq. (3,is provided by theory for the center-of-mass system of the collision partners, whereas the experimental results are obtained for scattering angles in the laboratory fixed frame, imposes an additional complication but no hindrance. Excellent agreement between calculated and measured differential cross sections has been obtained for many combinations of metastable rare gas atoms and atomic or even small molecular targets, resulting in the accurate determination of the local complex potentials (Altpeter et al., 1977). It has become standard practice to reproduce the experimental data not only from elastic but also from inelastic channels. Accordingly, the determined interaction potentials have a high degree of reliability. A review article on this activity has been published by Haberland et al. (1981) describing in detail experimental methods, data evaluation, and results. Even though it is state of the art to treat the nuclear motion of the system Rg' + B quantum-mechanically when addressing elastic scattering, we will now make use of classical trajectories to gain some insight into the situation.

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

395

In the simple classical picture, the real part V ( r )of the potential governs the trajectory, whereas the imaginary part -il?(r) controls the survival probability of the system in the initial - i.e., in the elastic - channel. The concept of a classical trajectory is related to a well-defined impact parameter. In Fig. 1 14

-

12

--

l o --

E

G

8

--

6

--

4 -..

2;;

. aP

0 --

N

..

0

4

2

6

10

8

12

x l Angstrom I T

0.5 f 0.4

'E 0.3

7 0.2 0.1 Y

0

2

4

6

8

1

0

12

impact parameterI Angstrom

FIG. 1. The upper panel shows classical trajectories for different impact parameters. The interaction potential is chosen so as to model the system He(Z3S) Ar. The collision energy is set to 100meV. The center of mass of the system is located at x = 0, z = 0. The lower panel displays the survival probability as a function of the impact parameter.

+

396

H. Morgner

we show the trajectories for several different impact parameters. The potential curve is selected to reproduce the collision between He(23S) and Ar (Brutschy et al., 1976);the collision energy is set at 100meV, which is a typical value in the thermal energy range. The survival probability along the trajectories is indicated in the lower panel as a function of the impact parameter. We observe that trajectories with small impact parameters lead to close proximity between the collision partners. This causes reflection into backward direction as well as a low survival probability, i.e., a large probability for inelastic processes. The latter finding is easily understood because the quantity r ( r ) increases strongly with decreasing separation. We observe that a high survival probability occurs only for large impact parameters that are related to forward scattering. It is obvious in which way this picture has bearing for surface reactions of metastable rare gas atoms. The target atom density at a surface renders the large impact parameters insignificant: a trajectory that would have a large impact parameter with respect to one atom necessarily has a small impact parameter with respect to a neighboring surface atom. Accordingly, the elastic cross section for metastable rare gas atoms in front of a surface must be much smaller compared to inelastic reactions than is the case in the gas phase. We will see, however, that this observation is still not sufficient to explain the extremely low survival probability of metastable rare gas atoms in collisions with surfaces. 2. Excitation Transfer into Bound and Continuum States

In the preceding section we noted that electronic excitation transfer from a metastable rare gas atom to a target species can result in ionization or in the formation of an electronically excited neutral target:

+

Rg* B -, Rg + B+ + eRg* + B -+ Rg B**

+

(3) (4)

Experimentally, both processes appear to be very different because (3) manifests itself in electron spectroscopy whereas (4) usually requires optical spectroscopy for its detection. Physically, however, these channels are closely related. In particular, if one views the excited state B** as a member of a Rydberg series, the similarity is obvious. In both cases the target takes on the electronic configuration of the ion B + , whereas one electron is excited to a state whose wavefunction is determined by the field of the ionic core B + . For one and the same ionic core, the primary excitation process is hardly affected by the excitation energy being below or above the ionization threshold. This is well known from photo-excitation. A beautiful example has

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

15.73

15.75

15.n

15.79

15.81

15.83

15.85

15.87

15.89

15.91

15.93

397

15.95

photon e m g y I SV

FIG.2. Yield of Arf ions from the photo-excitation of Ar atoms as a function of the photon energy. The data are taken from Radler and Berkowitz (1979).

been presented by Radler and Berkowitz (1979), who have studied the excitation of Ar atoms by varying the photon energy in a range below and above the ionization thresholds of Ar+(2P3/2, 2P1/2). Their yield of Ar+ ions is shown as a function of photon energy in Fig. 2. The photo-excitation into bound states, i.e., by photons with an energy below the Ar+(2P3/2) threshold, can be monitored via Ar+ detection because of collision-induced ionization of highly excited Ar+[(2P3/2)nI]Rydberg states. This secondary process is the more effective the smaller the energy gap between the Rydberg rA yield monitors the formation level and the ionization threshold. Thus, the ' of Rydberg states best at the series limit. Irrespective of these details, it is obvious that the probability for photo-excitation (normalized to the energy interval) passes smoothly from the quasi-continuum of the Rydberg family through the threshold into the true continuum of free electron states. Above the Arf(2P32) threshold one observes the superposition of direct ionization + e- continuum and the excitation into the Rydberg into the Ar ( 2P3/2) series converging to Ar+(2P1/2). Because autoionizationof the Rydberg states is fast, the Ar+ yield monitors both channels with like probability. Before the Ar+(2P1/2) threshold is reached, the resolution becomes insufficient to resolve the individual Rydberg states. Hence, the Ar+ yield takes on a smooth behavior as a function of photon energy and approaches the Ar+(2P1/2) threshold with an almost constant value. Apparently, the difference between the two reactions (3) and (4)does not lie in the primary mechanism of excitation transfer but rather in the ensuing development of the target, which is governed - for one and the same ionic core B+ - only by the amount of energy transferred. We will discuss now for metastable helium atoms the possible occurrence of processes (3) and (4).We discuss the reaction of He* with a molecular target CD. If the molecule has a positive electron affinity, the harpooning

4

398

H. Morgner

reaction He*

+ CD + He' + CD-

(6)

may take place at relatively large separation and thus precede any further reaction. If this process does not happen, the by far dominating reaction is direct electron emission:

+

He* CD + He

+ CD' + e-

(7)

The primary energy transfer into a bound target state He*

+ CD + He + CD** CD" CD' + et

has been observed, too. Because the primary process must conserve energy, the excited species CD**always lies in its autoionization continuum. Thus, the occurrence of reaction (7) is observed via the emitted electron. One example for this rare reaction has been identified for the system He*p 3 S ) 0 2 (Leisin, 1982). Hotop (1980) has reported that the reaction of laser-excited Ne* (3D3) with Nz leads to features that can unambiguously be traced to process (8). Because processes (7) and (8) are both measured in the electron energy spectrum and lead to similar final states, the distinction between the processes requires accurate data and careful data evaluation. The situation changes distinctly if the formation of an intermediate ion pair state via reaction (6) takes place. In this case the Coulomb interaction makes the potential curve strongly attractive. The well depth may reach values of several eV. In consequence the electronic energy of the system varies by this amount during the collision. The potential curves are schematically sketched in Fig. 3. Once in the ion pair channel He' CD-, the potential energy drops e- and passes through below the threshold of the ionized state He CD' the Rydberg series converging to this state. Then resonant energy transfer into the Rydberg states He CD**is enabled. A theoretical analysis of the situation shows that no particular phenomena occur at threshold and that the cross section for excitation transfer per unit energy interval passes smoothly through threshold (Miller and Morgner, 1977). A well-studied system is He + Clz. The positive electron affinity of the target molecule leads to the formation of the ion pair channel Hef* + Cl;. The small survival probability in the covalent channel He* Clz results only from symmetry conservation, which applies in selected collision geometries (Leisin et al., 1985). From the ion pair channel the system can proceed either via

+

+ +

+

+

+

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM 21 20

19

T

399

He'+CD

I c

18 --

17 --

Heo+CD" I

10 --

154 0

: 1

: 2

I

I

3

4

: 5

: 6

: 7

: 8

i

9

FIG.3. Potential curves for the interaction between metastable helium atom and target molecule with positive electron affinity.

ionization He+

+ Cl,

--+

He

+ Cl: + e-

(9)

or by population of bound states of the target molecule He'

+ Cl;

-+

He

+C l r

(10)

The Cl;* thus formed dissociates readily, because the chlorine atoms gain a fair amount of kinetic energy while in the Cl; state. Hence, the excitation transfer process (10) is identified via its atomic dissociation products, which either autoionize Cl;*

--+

C1+ C1**[3p4('D)nl'] -,Cl+[3p4(3P)]+ e-

(11)

+

(12)

or emit light Cl;*

--+

C1 +C1**[3p4(3P)nl]+ C1t[3p5(2P3/2,1/2)] hv

depending on their excitation energy.

400

H. Morgner

Apparently, the link between the measured data and the primary process of excitation transfer is much closer if the final states are in the continuum (9) than if bound states (10) are populated. Therefore, it required several experimental techniques, such as electron spectroscopy and optical spectroscopy (Leisin et al., 1985), electron-ion-coincidence measurements (Kischlat and Morgner, 1983), and computer simulation (Benz and Morgner, 1986) in order to disentangle the various reaction pathways. Similarly detailed studies under single-collision conditions exist for the triatomic molecules NO2 (Goy et al., 1981; Leisin et al., 1985a), SO2 (Goy etal., 1981a; Leisin et al., 1985b), and CS2 (Benz et a1.,1985) and for several halogenhydrides (Yencha et al., 1989; Yencha et al., 1991; Yencha et al., 1994). The evidence from gas phase reactions that electronic excitation transfer processes into continuum and into bound target states are closely related may have some impact on the detailed understanding of the behavior of metastable rare gas atoms at surfaces. One can hardly avoid the conclusion that both types of processes occur at surfaces as well. The difference, however, consists in the fact that excited states in condensed matter may not be observable because they are quenched too swiftly. More important is the insight from the gas phase that these processes must exist if one wishes to establish a detailed balance between incoming and outgoing channels for surface reactions of metastable rare gas atoms. 3. Ionization as the Most Important Reaction Channel Role of Spin Conservation. Penning ionization in the collision between two metastable helium atoms in the triplet state has been investigated by Hill et al. (1972) to study the role of spin orientation:

He(23S)

+ He(23S)

+

He

+ He' + e-

(13)

They found that the cross section for this process is zero within the experimental-error limits if the spins of the two metastable atoms are oriented parallel to each other by optical pumping. In this case the system forms a quintet state in the initial state, whereas the ionized channel can take on only a total spin of S = 0 or S = 1, i.e., can be only in a singlet or triplet state. Similar arguments hold for the reactions H€03S)

+ X ( 2 S 1 p ) + He + Xf('So) + e-

(14)

with X = H or X = alkali atom. Here the final state has doublet spin symmetry, whereas the initial state can be either doublet or quartet. The potential curve

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

401

of the doublet state is strongly attractive, giving rise to a very broad energy distribution, as actually observed (Ruf et al., 1987; Waibel et al., 1988; Merz et al., 1990). The quartet state is repulsive, and the corresponding electron energy distribution would be very narrow, giving a significant contribution in a small energy band on top of the broad doublet contribution. Accordingly, even a relatively small ionization cross section out of the quartet entrance state should be visible. Its absence clearly indicates that the quartet state is not active in process (14). For discussing spin conservation in Penning ionization, it has been pointed out that it is worthwhile to distinguish three steps (Morgner, 1982): 1. the approach of Rg* and B 2. the electronic transition from the initial configuration into the configuration with one electron in a continuum state 3. the separation of Rg and B f

Step 2 occurs on a time scale of atomic length of time, i.e. 2.4 . It has been argued by Keliher et al. (1975) that this is far too short to allow spin orbit or spin-molecular rotation interaction to noticeably influence the electronic spin state. One readily finds examples of Penning ionizing systems in which either step 1 or step 3 does not conserve total spin or the orientation of spin components in space, so it is obvious that spin conservation is not a general feature of Penning ionization if referred to the whole process, including step 1 and step 3. On the other hand, if spin conservation is meant only with respect to step 2, it is self-evident on the basis of common knowledge in molecular physics. In the two reactions (13) and (14) discussed above, spin is a good quantum number during steps 1 and 3, and therefore spin conservation for both processes holds in toto. Often one is interested only in steps 1 and 2, e.g., if the spin of He(23S) is oriented before a collision and the spin of the emitted electron is related to the initial spin orientation of He(23S). Dunning and Walters have carried out experiments of this kind on several target atoms and target molecules. They have applied spin orientation of metastable helium atoms HeQ3S) in different ways. On one side they have used the efficiency of optical pumping of He(23S)to create an intense beam of spin-polarizedelectrons (Rutherford et al., 1990),and on the other side they have studied the degree of spin orientation of the released Penning electrons and have drawn conclusions about the dynamical behavior of the collision system during the approach which corresponds to step 1 in the above nomenclature (Rutherford et al., 1992). Role of lnteraction Potential. According to the potential curve model of Penning ionization (Hotop and Niehaus, 1970), the electron energy spectrum depends on the shape of the potential curves in the entrance channel

H.Morgner

402

+

V [Rg*+ B] and in the exit channel V [Rg B+].As a general rule of thumb, one finds that the exit channel potential is flat and gets repulsive at short separation. This holds in particular if the rare gas Rg is helium. The only drastic exception to the rule is found for B = atomic hydrogen, which after ionization is a proton with no electrons and therefore does not feel the Pauli repulsion of the closed shell of helium. The well depth of the system amounts to &[He - H+] = 2.025 eV (Kolos and Peek, 1976). In all other cases the attraction of the Rg - B + potential is so small that the electron energy spectrum is predominantly determined by the interaction potential between the collision partners before electron emission occurs. Hence, this interaction potential can be employed to distinguish different categories among Penning ionization systems. First, we will discuss the situation if the target atom has a closed electron shell. Because the electron affinity of closed-shell atoms is small or negative, the formation of an intermediate ion pair He+ B- is virtually prohibited in this case, and electron emission results from the direct process

+

He+X+He+X++e-

+

For the series of earth alkalis (X = Mg, Ca, Sr, Ba) the well depth D,[He* X] scales reasonably well with the c6 coefficient as derived from the SlaterKirkwood formula (Ruf et al., 1996), which relates the c6 coefficient to TABLE I1 INTERACTION POTENTIAL OF CLOSED-SHELL SPECIES WITH LI AND METASTABLE RAREGAS ATOMS. He(2 * S)

He(23S) Target Species

L~(~s,*s) D,/meV

Mg Ca

170f 270

re/A

3.73

Sr Ba Hz0

5969

Dc/meV 130 f25 250 f25 243 240 f25 260 f 25 610 f 5OC 606

Experimental data from Ruf et al. ( 1996). Ab inirio result from Ruf er al. (1996). Experimental data from Haug et al. (1985). Ab initio result from Haug er al. (1985). Experimental data from Haug (1980). Theoretical result from Jones (1980). g Theoretical result from Trenary and Schaefer, III (1978). a

rc/A

3.53

D,/meV 300f25a 570 f 25 a 544b 550 f 25 a 670 f 35 a 910

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

403

+

the polarizability of both collision partners. Values for De [He* X] are listed in Table 11.One finds that systematically the attraction is stronger for He(2IS) compared to He(23S) by a factor between 2.28 and 2.58 (Ruf et al., 1996). This corresponds well to the fact that the ratio of the polarizabilities is 01 [He(2'S)]/u [He(23S)]= 2.53. The experimental values of De[He* XI are extracted from Penning electron spectra in the following way: an attractive interaction potential V [He*+ XI leads to a Penning electron energy spectrum that is shifted to energies smaller than the nominal value, which is defined as

+

Eo = E*[He*]- P [ X ] At the low-energy side a peak maximum can typically be identified, cf. Fig. 4. If the energy position that corresponds to 0.44of the maximum intensity is denoted by E d , , , the potential well depth can be evaluated via (Miller, 1970) N

&[He* - XI = Eo

-Ed,,

as long as the influence of the final-state potential is negligible (Ruf et al., 1987). The same recipe has been used for molecular targets. As an example, data on the water molecule are given in Table II. A molecular target differs from an atom in that the interaction potential may depend not only on the

1.o

0.44

Electron Energy FIG.4. Schematic shape of Penning electron energy spectrum for system with strongly attractive interaction potential.

404

H.Morgner

distance between He* and X but also on the orientation of X with respect to the projectile He*. This is indeed the case of H20 (Haug et al., 1985). He* experiences a strong attraction if it approaches the oxygen end of the molecule out of plane. This geometry is reminiscent of the favored directions for hydrogen bonding. Indeed, the similarity between the hydrogen-bonding properties of a molecule and its behavior toward Hep3S) has been noted by several authors (Ohno et al., 1983; Keller et af., 1986). The maximum attraction of -0.6eV (cf. Table 11) is the right order of magnitude to support this concept. Again we find that metastable helium in the singlet state experiences a much stronger attraction, owing to its greater polarizability. In some cases the interaction potential between He* and closed-shell targets is essentially repulsive with a shallow well depth at large separation of a few meV. This is the case for targets with no dipole moment and a very small polarizability. Rare gas atoms in their ground state and small symmetric molecules like H2, N2, and saturated hydrocarbons belong to this category. Entirely different phenomena are encountered if the target has a finite positive electron affinity. Often this goes along with an open-shell structure, which is the case for hydrogen atoms, alkalis, atomic oxygen, and atomic halogens and for the stable molecular radicals N0(211), N 0 2 ( 2 A , ) , and 0,(3C,). But closed-shell molecules can have a sizable electron affinity as well, the halogen molecules being an important example. The occurrence of a positive electron affinity on the side of the target, together with the small ionization potential of metastable rare gas atoms (cf. Table I), makes the harpooning reaction and thereby the formation of an intermediate ion pair state very probable: He* + X

-t

Hef + X -

If no symmetry requirement keeps the colliding particles He* and X in their covalent electronic configuration,the above process occurs and is the dominant reaction path. At first, we consider open-shell target atoms with symmetry X( 2 S ) ,which holds for alkalis and hydrogen. Figure 5 displays schematically the diabatic covalent potential curves for He(2'S) + X( 2 S ) and He(23S)+ X( 2 S ) , which are crossed by the ion pair curve He+(*S) X-( IS).Two of these configurations can take on only doublet spin symmetry, but Hep3S)+ X( 2 S ) can be in a quartet state as well. As pointed out above, this quartet state does not lead to Penning ionization, so we omit it from the further discussion. We consider the doublet subset of potential curves. The diabatic curves are represented by dashed lines. After switching on the coupling between the three configurations, we observe the appearance of avoided crossing. The He(23S)+ X( 2S)state follows a strong attractive potential curve with a large well depth. The higher-lying He(2'S) + X( 2 S ) channel develops into the ion pair configuration but is shortly thereafter forced to go back to a covalent

+

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

405

-

He(2%,2'S) X('Sln)

"T

t

'*

I

17

I

0

2

4

6

8

10

intemudear distance /Angstrom

+

FIG. 5. Schematic potential curves for He(23S,2's) X(*S). Parameters are chosen to reproduce potential wells for X = atomic hydrogen.

+

configuration. Clearly, the effective well depth of He(2'S) X( 'S) is much smaller than found for He(z3S) X( 'S). The experimental findings support this schematic concept, as can be seen in Table 111. Apparently, the situation for open-shell atomic targets is very different from that for closed-shell target atoms (cf. Table 11). The picture changes again if we consider molecular targets that enable the formation of an intermediate ion pair

+

He*

+ CD + He' + CD-

Although transition into the ion pair state at the avoided crossings occurs similarly to the case of an atomic target, the situation has changed if the system that started out as He(2'S) X('S) reaches the second crossing. A transition back into the covalent channel will generally not take place. The reason is the time development within the negative molecular ion CD-. In most cases it has left its original CD conformation. A renewed electron transfer

+

He'

+ CD-

4

He*

+ CD(v)

H.Morgner

406

TABLE III INTERACTION POTENTIAL BETWEEN METASTABLE RAREGASATOMSAND OPEN-SHELL SPECIES. Target

He(23S)

/A

Species

re

H

1.75 1.85 2.9 a 3.1 a 3.5 a 3.6' 3.8 a

Li Na K Rb

cs

He(2'S) D,/meV

re/A

D,/meV

2260 2284 868 f 20 740 f 25 591 f 24 546 f 18 533 f 18

3.2 3.7 3.9b 4.3 b 4.4b 4.6

460 f50 426 g 3 3 0 f 17' 277 f 24' 202 f 23 ' 219 f 18' 277 f 18'

4.

Uncertainty -0.2 Uncertainty -0.4 A. Experimental data from Ruf et al. (1987). The well depth for He(23S)+ X(' S ) with X = alkali is always close to 80% of the well depth for Li X('S). Movre and Meyer (1997). Experimental data from Morgner and Niehaus (1979). Hotop et al. (1971). g Movre et al. (1994). a

+

would require the formation of a highly vibrationally excited neutral CD(V), and the energy for this process is lacking. Therefore, the system is bound to remain in the ion pair channel. Hence, in this case the effective well depth for He(2'S) X(2S) tends to be larger than for Hep3S) X(2S). The resulting electron energy spectrum depends little on whether the metastable helium atom enters the collision in its singlet or triplet state. The gross features of both spectra are the same, but small differences are found because the dynamics of the collision is somewhat different. Because the excitation energy of the He(2lS) is 0.8 eV greater than that of He(23S), the total energy of the system He+ CD- differs by this amount. The influence of this difference in total energy can be assessed by comparison with the effective well depth of the interaction potential, which often amounts to several eV. The electron energy spectra for He(2lS), Hep3S) NO2 are shown in Fig. 6. The He(23S) NO2 spectrum displays features that result from the direct Penning ionization out of the covalent entrance channel. The remainder of the spectrum originates from ionization out of the ion pair channel, as is the case of He(2'S) - N02. The similarity is striking. The effective well depth of the interaction potential is about 5 eV, as can be read off the potential-curve diagram in Fig. 7. The well depth of the system He(23S,2's) CS2 is much smaller, and one observes fairly large differences between electron spectra from the ion pair entrance channel, depending on whether the metastable helium was originally in the singlet or the triplet state (Benz ef al., 1985).

+

+

+

+

+

+

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

407

I

I

I

1

1

I

I

I

I

1

I

I

0

1

2

3

4

5

6

1

8

9

Elecmn Energy I eV

F k . 6. Electron energy spectra for He(2'S), He(Z3S) (1985a).

+ NO2. Adapted Erom Leisin er al.

The Spatial Part ofthe Wavefunction. The properties of the spatial part of the wavefunction influence the outcome of a Penning ionization collision in two ways. First, we will discuss the impact on the transition into the intermediate ion pair state

He*

+ CD + He' + CD-

H.Morgner

408

*l

i

doublet

He'(23S)-No2?Al1

"1

He-NG'tBz)

v

He-NG'CAl) He-No2'(3B~) spinforbidden Ill

' a C

H l5I

t

l3

llL I

He-NG+('Al)

,

9

0

2

4

6

8

10

He-N@ distance /Angstrom

FIG. 7. Potential-curve diagram for He(2'S), Hep3S)

+ NOz. Adapted from Goy et al.

(1981).

Provided that this transition is energetically allowed, which is the case for targets with positive electron affinity, it can be prohibited only by symmetry restrictions. The role of spin symmetry has been discussed already, the system He* NO:! being a prototype. There are, however, several systems where the symmetry of the spatial part of the wavefunction can keep the system in its covalent channel. As an example we consider the molecule SO2

+

He(23S,2'S) +SOz('A,)

-+

He+(2S)+ S O ; ( * B )

Because the ion pair state can be singlet and triplet, no restriction by spin symmetry is imposed. The spatial symmetry of the system depends on the relative position of the helium with respect to the molecular plane of S02. We consider a situation where the metastable helium is placed in the molecular

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

409

plane. The total system then belongs to the C, symmetry group. The covalent configuration has A' symmetry whereas the ion pair configuration has A'' symmetry. The obvious conclusion is that the electron exchange is symmetry forbidden if the He* atom finds itself in the molecular plane of SO2 when it reaches the separation where the ion pair channel crosses. He* approach outside this plane is not subject to a symmetry restriction. Goy et al. (1981a) have modeled this situation by a coupling matrix element that depends on the . cos 8, the molecular plane polar angle 8 of the He* position V12(8)= being defined by 8 = $. Similar considerations, backed by inspection of Penning electron energy spectra, have been published for other systems, e.g., the halogen molecules (Leisin et al., 1985; Morgner and Seiberle, 1994). The second aspect of spatial symmetry is its role in the electron-emitting process. The two possibilities (namely, electron emission out of the covalent entrance channel and out of the ion pair channel) have to be considered separately. We start with the discussion of direct Penning ionization. It is established that this process can be described as tunneling of a target electron into the 1s hole of helium and the concomitant emission of the former excited 2s electron of the helium metastable. The question to be raised is only whether this two-electron process can be characterized any further. In case of rare gases as target atoms, the analysis of experimentaldata lead to a description of the process in terms of one-electron properties. The collision system +eHe*(ls2s) +Ar(3p6) + He(ls2) +A~+(~P',~P) can be characterized by the same symmetries that apply for the electronic wavefunction of linear molecules. The initial state is clearly C,whereas the ionized system can be either C or II. In order to conserve total symmetry, the symmetry of the emitted electron with respect to the nuclear axis must be a or 7c, respectively. The angular distribution of the emitted electrons, as well as the slightly nonstatistical population of the two Ar+(2P3/2,2P1/2)fine structure states, point out that the ionized system He(ls2) Ar+(2P) is formed with C symmetry, even though the statistical expectation would favor 7c symmetry by a factor of 2 (Morgner and Hoffmann, 1979). Accordingly, the electron that tunnels from the target to the helium 1s hole leaves a (T hole in the Ar 3p shell and thus originates from a 3 p 0 orbital. Because the 1s helium orbital has (T symmetry as well, we observe that the tunneling electron keeps its symmetry during tunneling. This means symmetry conservation with respect to individual electrons holds and is thereby much stricter than conservation of symmetry for the entire electronic wavefunction. This finding has been extended into the independent-particle model of Penning ionization (Morgner, 1988). According to this model, which is based on many experimental findings, the two electrons that are active in Penning

+

410

H. Morgner

ionization are coupled to each other only by total energy conservation: within experimental accuracy they do not exchange angular momentum or momentum. The angular distribution of Penning electrons can be understood via projecting of the wavefunction of the excited helium electron at the instant of ionization onto the wavefunction of those continuum states that have the energy required by total energy conservation. On the other hand, if one concentrates on the behavior of the electron that tunnels from the target to the He 1s hole, one ends up with a very simple picture: the tunneling probability is proportional to the electron density of the target orbital in the range of the He 1s hole orbital. In view of the small diameter of this orbital, this is almost identical to saying that the probability for Penning ionization of a given target orbital is proportional to the electron density at the nucleus of the metastable helium. In this formulation, one has a generally valid guideline to describe Penning ionization even in cases where the complete collision system, e.g., He*+H20 in arbitrary geometry, cannot be characterized any more by symmetry operators (Haug et al., 1985). The validity of this simple concept of Penning ionization has been demonstrated in one case even for an atomic target. For the system He*(23S)+ Yb(4f146s2'S~), ionization from both target orbitals, 4f and 6s. is energetically possible. It turns out, however, that the Yb+((4f)-') peaks are hardly visible in the electron spectrum (Hotop ef al., 1990). The authors point out that the radial electron densities of both orbitals are very different in that the 4f orbital is concentrated around the nucleus, whereas the 6s orbital extends much further out. The turning point of the interaction potential is too far out to allow a substantial overlap between the He 1s hole and the Yb 4f orbital. This finding demonstrates at a very simple target a concept that has been successfully applied to large molecules: for any target, one defines a surface that is the sum of all turning points (at thermal collision energy) for He*. The activity of any target orbital in the Penning ionization process is then given (except for energy requirements) by the fraction of the orbital electron density that is outside this surface (Ohno et al., 1983a; Harada ef al., 1983; Ohno ef al., 1994; Masuda and Harada, 1992; Aoyama et al., 1989;Nakanishi ef al., 1989; Masuda er al., 1990; Aoyama et al., 1989a). It is quite obvious that (and how) this concept is meaningful for the interpretation of He* reactions at surfaces (Harada, 1990). We proceed now to discussing the case of electron emission out of the intermediate ion pair state: Hef

+ CD-

+ He

+ CDf + e-

Here, both electrons that play an active role must be supplied by the target because it changes its charge state by 2 units. One of the electrons is emitted, and the other tunnels into the He 1s hole. Obviously, the target-helium

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

41 1

tunneling process is always present in the process of electron emission. It is responsible for the sensitivity of the Penning process for local electron target density. The electron to tunnel most readily to the helium is the electron in the affinity level, because this is the most diffuse orbital with the highest electron density near the collision partner. Unlike for the direct Penning ionization, the two active electrons do not move independently. Their interaction can be described by the Coulomb operator. Both electrons are at the same location, so the monopole operator should be the leading term in the expansion of the Coulomb operator. If so, the two electrons active in the ionization process are expected to preferentially have the same symmetry within the frame of the target molecule. Indeed, the little information available seems to point in this direction (Benz et al., 1985). We will see later that in surface reactions of He*, a similar observation seems to hold in the case of AN (Auger neutralization), which is similar to the above reaction in that the recombination energy of the He+ ion is used to remove two electrons from the surface material. B. INTERACTIONOF HE* ATOMSWITH SURFACES 1. Electronically Elastic Processes

The probability for the reflection of intact metastable rare gas atoms from surfaces is very low. The survival probability of up to 50% calculated by Dunning et al. (1991) for metastable helium atoms near the turning point at the Cu( 1 00) surface is clearly not supported by experimental data; cf. Table IV. At the end of Section II.A.l we pointed out that the survival probability for excited atoms at the surface should be lower than in the gas phase, because (effectively) only collisions with small impact parameters can occur. This argument would predict a survival probability of a few percent, i.e., still not yielding an explanation for the actually observed reflectivities (Table IV). A rationalization for the small reflectivities has been offered by Conrad et al. (1982a) under the keyword “dimensionality effect”: the density of states for electronically elastic scattering is compared to the density of states for inelastic reaction channels. Because the Born-Oppenheimer approximation is assumed to hold, electronic and nuclear degrees of freedom are decoupled. The situation for the electronic degrees of freedom is considered to be essentially the same for gas phase and for surface reactions. However, the claim is made that for the interaction of He* with the surface, the degrees of freedom for nuclear motion were strongly affected. The phase space open to the He nucleus after electronically inelastic reactions is considered to be large, whereas the phase space for an elastically reflected He* atom is claimed to shrink by several orders of magnitude, namely to the specular direction. This line of argument is difficult to follow: the nature of the reflected particle is not explicitly involved, so the

412

H.Morgner

argument should - if valid - also have bearing on the reflection of light with the consequence that all mirrors should have an extreme low reflectivity provided their material is not perfectly free of any absorption. Another counterargument could be drawn from the fact that the low He* reflectivity of metal surfaces should sensitively depend on the flatness of the surface, and this has not been observed. In the attempt to find alternative reasons for the low reflectivity of He* at surfaces, I have carried out two types of model calculations. The first one treats the approach of a He*p3S)atom to the surface of an Ar crystal. In this study the Ar atoms are fixed in space. Accordingly, the investigation focuses on the electronic degree of freedom. The interaction potential between the metastable atom and the surface is constructed as pairwise interaction between He(23S) and the Ar atoms, which in turn is taken to be identical to the potential employed for the trajectory calculations in Section 1I.A.1.; cf. Fig. 1. The potential-energy curve of Hep3S) depends strongly on whether it approaches an isolated Ar atom or an Ar atom embedded in the crystal surface. The well depth, being only 3.72meV in the first case, increases to almost 50 meV in the case of the surface due to the attraction of the neighboring Ar atoms. Further, the distance of closest approach decreases by a few tenths of an angstrom, causing a sizable increase in the ionization probability and, thus, a decrease in the survival probabilityp,,,. The survival probability of He(23S) along a trajectory can be written as

with

1

00

Z =

W(t)dt

-00

where the integral is carried out along the trajectory and where W ( t )denotes the ionization probability per unit time at a given point of the trajectory. Inspection of the calculated trajectories shows that the integral Z is larger if Hep3S) approaches the Ar crystal surface compared to the encounter with a single Ar atom by a factor c Z(surface) = c . Z(sing1e atom) which lies in the range of c = 1.5-3.5. The corresponding survival probabilities are then related via psW (surface) = psu, (single atom)c

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

413

It is obvious that the survival probability of He* at the surface can get very low, in particular if psw (single atom) is small. For example, if the survival probability for a central collision with a single atom is 5%, then the reflectivity from the surface would be only 1.25. lop4if c takes on a value of c = 3. The second study carries out trajectory calculationsas well, but it leaves out the explicit treatment of the ionization channel. Instead, the surface is composed of atoms that are free to move and, thus, are able to exchange energy with the projectile. Accordingly, a second reason for the low reflectivity of He* turns up along the lines that hold for any particle approaching a surface at low velocity. The interaction of the impinging particles with the individual atoms in the surface causes with high probability a transfer of momentum perpendicular to the surface into momentum parallel to the surface. Then the particle cannot escape but carries out several jumps on the surface before it acquires sufficient perpendicular momentum to leave the surface. I have carried out a MD computer simulation for He* atoms impinging upon a gold surface that has been employed in a previous calculation (Morgner, 1997). The interaction between the He* atoms and the Au atoms has been described by a LJ-potential, the parameters being chosen so as to reproduce a typical He* metal interaction; cf. Table V. The probability for direct reflection of the He* atom depends strongly on the kinetic energy. At = 0.00125 eV, only four out of 37 trajectories led to direct reflection of the He* atom. In the other cases numerousjumps occurred, keeping the He* atom for a long time close to the surface, which would cause deexcitation of the metastable with almost unit probability. At a kinetic energy of 50 meV, multiple jumps occurred only in four out of 20 cases. The conclusion from these computer studies seems to be that the observed low reflectivity of He* atoms from surfaces cannot be attributed to one single reason, but that several mechanisms contribute to the effect. The interaction potential between a metastable helium atom and a metallic surface strongly depends on whether the helium remains in its original state or is converted into a helium ion via resonance ionization. In the latter case, the well depth of the potential curve is attractive due to the image force and becomes repulsive only at small distances. The image force potential is vima,(z>

=

14.3997 eV 4 . 2

if the distance from the image plane is measured in angstroms. The similar gas phase process of ion pair formation,

414

H.Morgner

leads to an interaction potential governed by the Coulomb attraction, vco”l(r) = -

~

14.3997 eV r

if the separation of the collision partners is given in angstroms. If one equates these distances, i.e., z = r, the Coulomb potential is four times stronger than the image force potential. Because the repulsive forces set in at comparable distances, the ionpair potential can develop large well depths of up to 6eV, whereas the well depth of the Hef metal surface potential is restricted to smaller values, typically 2 to 3 eV. Once the He* atom is resonantly ionized in front of a metal surface, it cannot escape anymore except as ground state He”. The reason is that the electron transferred to the surface is not localized on the target, as in the gas phase, but is lost in the large phase space of the unoccupied metal states. However, even in gas phase reactions, the electron transfer process can - in general -be reversed only in the case of atomic targets. For molecular targets the degrees of freedom of internal nuclear motion prohibit the system from separating again into its original configuration. Consequently, it is virtually impossible to observe a scattered He* atom once the projectile has lost its excited electron to the surface or to a molecular target. Therefore, we will concentrate now on those collision events during which the metastable helium atoms keep their excited electron. Even though only a very small fraction of -lop6 of He* metal surface collisions lead to a survival of the He* atom and thus to an electronically elastic scattering event, there is distinct interest in these processes. One of the reasons is the observation that the survival probability of the He* at a surface can vary by one to two orders of magnitude if the chemical nature of the surface is changed, e.g., due to oxidation or adsorption. Data on several systems are compiled in Table IV. Fouquet (1996) points out that the He* survival probability may be a very precise monitor of surface properties in selected cases. One example presented by Fouquet (1996) is the process of metallization of an adsorbate layer of Cs atoms on Cu( 100) as a function of coverage. The claim is that metallization and He* reflectivity are very directly related, whereas access to the same surface properties via electron spectroscopy was obscured by the interplay of several different electron emission mechanisms. Another hope connected with He* scattering off surfaces is related to the observation of diffraction patterns. Based on the fact that the survival probability rises upon adsorption, one could hope to develop He* scattering into a sensitive diffraction method for adsorbate superstructures. A further aspect has been concerned with the two spin states of the metastable helium.

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

415

TABLE IV HE*-&FLECTIVITY. Sample

Angle of Incidence

He(2'S)

Pd( 1 10) clean

45" 15" 15" 15" 45" 75" 45" 15" 45" 45" 45" 45" 45" 45" 45" 45"

4.2. 4.8.10-58 5.1.10-4a 3.0.10-38 6.0.10-58 1.5.10-48 3.0.10-38 3.0.10-38 < 1.5.

Pd( 1 10) ox. Pd( 1 10) CO (sat.) W(poly) clean

+

w ox. Cu( 1 1 0) clean Cu(100) c s / c u ( i 00)e 0.25 8-1 NiO( 1 00) cleaved in UHV c(2 x 2) CO/Cu( 1 00) LiF(0 0 1) NaCl(00 I )

-

a

He(23S)

(1.6f0.8).10-6b 8 . 10-5b -5.10-6b (1.9 0.9) . 1 0 - 5 b - 7 . 10-6b

+

2.1.10-4c 1.2.

Conrad et al. (1982a) Fouquet (1996) Conrad et al. (1982b).

Because beams of He* atoms in the singlet as well as triplet state can be produced with ease, magnetic surface properties could be investigated. A recent study exploring the feasibility of the He* diffraction came to the conclusion that the He* atom interacts strongly with the phonon bath of the surface,which destroys the coherence necessary for diffractionstudies (Fouquet, 1996; Fouquet et al., 1998).One must conclude that at present, the prospects for a diffraction technique based on excited rare gas atoms appears to be rather bad. The careful evaluation of the loss of kinetic energy upon He* scattering resulted nonetheless in interesting information. The well depth of the He* metal surface could be assessed. It was determined to -0.1 eV. This constitutes an independent confirmation of theoretical predictions for several metals, which are collected in Table V. Further, information derived from electron spectroscopy points to the same size of the well depth. Even though only a small fraction of He* atoms reach a metallic surface without undergoing RI (Resonance Ionization) in all MIE spectra of metals, one observes a small signal near the energy position Eel that corresponds to Eel - EF = E*[He*] the position of the Fermi energy EF being gauged by UPS. A distinct step appears in metallic MIE spectra, which is within the experimental energy

H.Morgner

416

TABLE V HE*-METAL WELLDEPTH. Equilibrium Distance blA He' fjellium a He'/Pd(lll)b He'/Cs(lOO)C

Well Depth D,leV

3.1 N

0.1 0.09 0.1

a Dunning et al. (1991). bTrentini and Doyen (1985). Doyen (private communication, 1995).

resolution at the correct position to represent the Fermi edge. Because electron spectrometersin these experiments are set to resolutions of A E M 0.1 eV, one must conclude that the actual energy supplied in the ionization events agrees with the asymptotic excitation energy E*[He*] within A E. This finding corresponds to the expectation that the well depth between He* and the metallic surface is 0.1 eV or less.

2. Electronic Excitation Transfer General Remarks. If allowed with respect to energy and symmetry, the dominant process is the resonant transfer of the excited electron of the metastable atoms into unoccupied states at surfaces. The accepted term for this process is resonant ionization (RI), introduced by Hagstmm (1954). The ensuing process of electron emission, called Auger neutralization (AN), involves two electrons from the surface, one electron tunneling into the 1s hole of the helium ion while the other is emitted. The requirement of total energy conservation couples the initial and final states of both electrons. Auger neutralization leads to electron energy spectra that are not related in a simple manner to the target density of states. This will be discussed below. Several reasons can suppress RI and thus electron emission via AN: 0

lack of unoccupied target orbitals in resonance with the excited electron. This condition must hold in the range of distances between He* and th? surface where RI is likely to occur, which typically is between z 5 5 A and z 5 8 A. Consideration of the energy levels at asymptotic distances is usually not meaningful as a criterion of suppression of RI. Candidates for surfaces that usually do not favor RI are wide band gap insulators (e.g., oxides or halogen halides) or metals with very low work function (alkalis).

LIQUID/SOLD SURFACES WITH METASTABLE HELIUM 0

0

417

the overlap between the excited metastable rare gas atom and the unoccupied surface states is small due to adsorbed spacer molecules that do not offer an energetically close-lying unoccupied orbital (e.g., saturated alkanes). Not every adsorbed molecule is suited to avoid RI. Drakova and Doyen (1994) argue that a molecule with an affinity level above the Ferrni edge of the metallic substrate might even enhance the resonant ionization of the approaching metastable atom. RI is suppressed due to a symmetry mismatch between the excited electron of the metastable atom and the unoccupied surface states. So far only one well-investigated example exists for this situation. It has been observed for the system 2lS) graphite (Masuda et al., 1990; Masuda et al., 1990a).

+

If RI + AN does not occur, the process of Auger deexcitation leads to electron emission. Again an electron from the surface tunnels into the hole orbital of the metastable atom, while the excited electron of the metastable is emitted. The resulting spectra show in general the same bands as UPS, even though the relative activities of the different target orbitals are different between MIES and UPS. The explanation for this is the same as for gas phase Penning ionization out of the covalent channel. Again the electron density of an orbital outside the surface defined via the turning points plays the key role for its activity in the AD process (Harada, 1990). According to the typical interaction potential between He* and surfaces, the AD process samples the electron density approximately 2 to 4 angstroms above the outermost atomic layer of a surface. In some particular cases, a third path to electron emission has been described. It is the capture of a surface electron into the 2s orbital of helium, giving rise to intermediateformation of the negative projectile He*- ( 1s2s2,*S) which then decays by autoionization (Hemmen and Conrad, 1991). The spectral features that motivated postulating this mechanism are found only on metal surfaces partially covered with alkali atoms (Hemmen and Conrad, 1991; Maus-Friedrichs et al., 1991). On the basis of ab initio calculations, Crisa and Doyen (1987) and Doyen (1995) argued that the very same spectral features could be explained without involving a temporal He*- species but by considering the behavior of the adsorbed alkali affinity level under the influence of the approaching metastable helium atom. Hemmen and Conrad (1991) point out, in favor of their model, that it also provides an explanation for the so-called singlet-triplet conversion that has been observed experimentally by several authors (Lee et al., 1985a; Woratschek er al., 1985). Electron Yield. In a review article, Hotop (1996) describes efforts to determine the number of electrons ejected from a surface per incident metastable rare gas atom. The electron yield is found to depend on the rare gas as well

418

H. Morgner

as on the surface probed. For metastable helium atoms at thermal kinetic energy, the yield is approximately -0.3 for atomically clean metallic surfaces and -0.45-0.95 for contaminated metals. A contaminated metal can according to our own experience with many MIES measurements - be understood as surface covered predominantly with a film of saturated hydrocarbons the thickness of which is equivalent to one or a few monolayers. As outlined above, this coverage reliably prohibits resonance ionization leaving AD as main electron-emitting mechanism. Thus, the consistently smaller electron yield for atomically clean metals compared to contaminated metals means the following: whenever RI and AN take over, the electron yield is -0.3 and thereby smaller than if electron emission is caused by AD. In view of the fact that survival of He* - even though different by two orders of magnitude between the two cases - never exceeds lop3,one is compelled to ask why the e- yield is not always close to unity. Niehaus has discussed the situation for clean metals where electron emission is dominated by the process of AN. In this case the emitted electrons start within the surface. Decisive for their escape probability are their energy and their angular distribution, the latter being further influenced by the surface barrier. Assuming an initial isotropic angular distribution of the electrons, Niehaus has computed a yield of 20% from a W(110) surface in the limit of vanishing kinetic energy of the He+ projectile. This value agrees reasonably well with Hotop’s experimental result of -0.3 (Hotop, 1996). In case of AD that is active at contaminated surfaces, the emitted electrons start from the He* atom in front of the surface. Assuming again an isotropic angular distribution, one would expect a maximum yield of 0.5. The values close to unity found by Hotop (1996) indicate that the angular distribution of the emitted electrons must be pointed away from the surface, i.e., directed toward the He* side of the collision system. This is reminiscent of the angular distribution found in gas phase reactions of He* with closed-shell target atoms (Hotop and Niehaus, 1971; Ebding and Niehaus, 1974; Ebding, 1976). In addition to the angular distribution, the shape of the electron spectrum has bearing on the emission yield. This can easily be understood if one recalls the situation found for gas phase reactions: the transfer of the electronic excitation energy from the metastable can lead to continuum as well as bound states. The same holds for surface reactions, even though the population of excited bound states can hardly be observed directly due to immediate quenching. The formation of bound states furnishes an additional explanation for the yield being smaller for AN compared to AD. Inspection of spectra shows that the average electron energy is generally larger for AD spectra than for AN spectra. This simply means that for AD processes, a larger fraction of the spectrum lies above the vacuum level and thus is visible to electron

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

419

spectroscopy. This can be exemplified by comparison of the M E spectra of pure Au( 1 1 1) and of alkanethiol-covered Au( 1 1 1) (Chenakin er al., 1998). The same principle applies even if one compares different AN spectra. Electron emission from Ni(100) drops by more than 30% if the surface is covered by atomic oxygen. The atomic oxygen adds density of states at a binding energy of 6 eV but blocks access to the Fermi level. In consequence, the average energy of excited electrons decreases, and a larger percentage of events ends with excitation energies below the vacuum level. That the work function increases during oxygen adsorption supports the effect further. N

Auger Neutralization. Following Hagstrum (1954), the expression for the probability that an electron with kinetic energy Eel is emitted in an Auger neutralization process involving two electrons with energies E’ = E - 6 and E” = E 6 below the Fermi edge EF can be written as

+

where N ( E )denotes the electronic density of states as a function of energy and Hif denotes the matrix element for the two-electron process leading to the neutralization of the ion and to the emission of an electron. If the effective recombination energy of the ion is given by Zeff and the work function of the surface by 4, the relationship between the various energies is

Here, the electron energy is measured with respect to the vacuum energy of the surface, as usual. An alternative formulation that refers the electron energy to the Fermi level is written as

The electron energy takes on its maximum value for E = 0:

Thus, the determination of the effective recombination energy Zeff of the Hef ion in front of the surface can - in principle - be evaluated in a straightforward way. In practice, however, this is very difficult because the AN spectra do not show a sharp high-energy edge but approach zero intensity rather gradually.

420

H. Morgner

The expression (17) and (18) are applicable even in cases of inhomogeneous surfaces, provided the work function is understood as a local work function +lot rather than a global work function. We will see later that this concept is useful in treating AN data from a Ni( 1OO)surfacepartially covered by atomic oxygen. The above expression for the electron energy distribution cannot be evaluated in terms of the density of states because the operator Hif is not given in a simple way as, for instance, the dipole operator in UPS. In order to go beyond mere data recording, Hagstrum (1954) proposed to introduce the simplified expression

with U ( E )as so-called “weighted density of states,” which means that U ( E ) incorporates all dependencies introduced by the operator H if. The advantage of the simplified ansatz rests in the fact that the quantity U ( E ) can be evaluated - at least in principle - in a straightforward manner. The hope is, of course, that U ( E )retains the key features of N ( E ) when obtained by deconvolution of equation (19). Some recent progress in numerical data treatment will be presented below. A more ambitious theoretical concept has been devised by Hood et al. (1985). They proposed to employ an estimate of H i f ( E - 6, E 6, z) as a function of binding energies of the two active electrons and of the distance z between He+ ion and the surface and then to determine the density of states N ( E ) itself by comparison to experimental data rather than the “weighted” density of states U ( E ) .For the system He*/Ni( 1 1 1) they demonstrate that U ( E )and N ( E )differ significantly. It is obvious that the model dependencies chosen for Hif have bearing on the outcome of the evaluation. If, on the other hand, Hif were supplied by a reliable calculation, the same calculation would easily compute the density of states as well. Yet another approach has been chosen by Keller ef af. (1986a). They used experimental information on the density of states N ( E ) from UPS data and In their treatment, this quantity is factorized as concentrated on fitting

+

The first factor Ql (E’) describes the tunneling of one electron into the He 1s hole orbital, whereas the second factor controls the concomitant emission of the other electron. The experimental data for the system He* Hgliquid are satisfactorily reproduced if eland Q2 are modeled by exponentials. The paper also reveals a notable propensity for both active electrons to originate

+

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

421

from orbitals of the same symmerty. A similar propensity has been observed in gas phase reactions if ionization occurs out of the intermediate ion pair state. In evaluating AN spectra, it is usually assumed that the effectiverecombination energy Zeff - cf. Eqs. (16) and (17) - is a constant even though it is well known that this does not fit into the accepted concept. The projectile, once resonantly ionized, experiences the attractive image force, gets accelerated toward the surface, and thus converts eletronic into nuclear kinetic energy. In consequence, the effective recombination energy takes on a band of values rather than a single value. This aspect has been thoroughly studied by A. Niehaus quite generally for the interaction of atomic projectiles with metallic surfaces (Eeken et al., 1992; Niehaus, 1993; Niehaus, 1994). Niehaus has explicitly taken into account the variation of all relevant quantities with the distance between projectile and surface. In order to provide a generally valid scheme, he has devised a description that involves the properties of the surface and of the projectile atom and induction and dispersion forces, but no specific chemical interaction. The planar potential V ( z ) that governs the trajectory depends on the atomic numbers of the projectile and of the surface atoms and on the areal density of the surface atoms. The variation of the atomic energy levels of the projectile E k ( z ) is described by the image potential, by the estimated screening, and by second-order effects like image charge-induced dipole. Finally, the transition rates between different states G&) are conceived to depend on the density of states p(&) at the surface and the binding energy of the involved projectile orbital. A prefactor of the G,(z) has to be determined via comparison with experiment. Niehaus has designed his theoretical approach in order to describe electron emission from metallic surfaces interacting with impinging ions in different charge states and with a large range of impact energies and angles. The effect of metastable helium atoms at thermal kinetic energies has not been the focus of his interest. In the following, we will see that the application to MIES yields reasonable agreement with experimental data. For the system He* Ni( loo), Niehaus (1993a) has calculated for a kinetic energy of 30 meVof the projectile the survival probability of the approaching metastable as a function of separation from the surface, the formation of He+ by resonance ionization, and the conversion into ground state He". The probabilities for He* + He' conversion by resonance ionization and for He' +He" conversion dominantly accompanied by electron emission are shown as a function of helium surface distance in Fig. 8a. These probabilities can be referred to the energy level of the He' by means of the image force potential; cf. Fig. 8b. The average energy of the He+ after resonance ionization is -0.8 eV below the vacuum level, which underlines again that it cannot escape other than by excitation transfer, which turns He+ into ground state He".

+

H.Morgner

422

1

2

3 Q 0.5 m

2 n

0

-2

0

2

4

6

8

10

distance from image plane I Angstrom FIG. 8. He* approach to Ni(l00). Top: probabilities for existence of He*, He+, and He". Bottom: potential curves. Data from Niehaus (1993a).

More relevant to the present discussion is the distribution of He+ energies at the instant of Auger neutralization. The mean value lies 1.9 eV below the asymptotic energy - i.e., the energy of the free He+ ion - the FWHM of the distribution being 1.5 eV. We will now try to derive the distribution of l e from ~ experimental data. For this purpose we assume the ideal shape of the high-energy part of the AN spectrum - i.e., the hypothetical shape that the spectrum would take on in ) to be the case of g(Zeff)= S(Zeff - l e ~ , 0N

N

If the unknown distribution of Zeff is denoted as g(5) with 5 = leff the measured spectrum is to be equated with the expression

- Zeff,o, then

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

This leads to

and

The distribution g ( c ) can then be evaluated directly from experimental data. The measured AN spectrum for He* Ni(100), the assumed ideal shape, and the evaluated distribution of Zeff are displayed in Fig. 9. The comparison with the prediction of Niehaus (1993a) for the same system is shown in Fig. 10. The agreement is not perfect, but it is quite satisfactory in view of the scattering of the experimental data. With a newly developed algorithm, the

+

MlES Ni(l00) W

6.E94 E

8 v)

a

c 0

22 !wf

spectrum

'o b

:

C

4 0

0

O.E+OO

1

14

15

16

17

18

19

20

electron energy I 0V FIG.9. High-energy part of He'-Ni(100) AN-spectrum. The dashed line indicates the assumed shape for one single value of fca. The evaluated distribution of fee is shown.

424

H. Morgner dislributionof effective ionization potential of He+ in front of Ni(100) svface 1*4

T

-0-

1

from calculation fit to ew.

~

lS2 1

*

0.8 --

.2 n

0.6 --

F 5 0

n

g 0.4 -Q

0.2 --

0 --0.2 ! -1.5

I 1

I

,

I

-1

-0.5

0

0.5

1

1.5

2

k/eW FIG. 10. He*-Ni(1 00) AN-spectrum. The distribution of as derived from experiment in comparison with the same quantity derived from data calculated by Niehaus (1993a).

AN spectrum has been deconvoluted according to Eq. (19). The high-energy part of U ( E )is shown in Fig. 11. The resulting step function is assumed to reflect the Fermi edge broadened by the fact that one has - unlike in UPS or in the AD process - not a single energy for the process but the distribution of energies Zeff discussed above. The idealized Fermi edge is indicated. Its position can be determined within &0.05eV, which is a very precise result compared to previous attempts. The effective convolution function can easily be evaluated as the first derivative of U ( E ) .A Gaussian fitted to the convolution function has a FWHM of 0.53 eV. One further observation is noteworthy. Once the position of the Fermi edge in Fig. 11 is chosen, one can calculate the corresponding maximum electron energy in the AN spectrum. An alternative way to determine Eel,max is the fit of the idealized shape given in Eq. (21). We note that both ways to determine Eel,maxare entirely independent of each other but agree within 0.1 eV. In Table 6 the mean effective recombination energies Zeff of the helium ion and their deviation Ale, from the asymptotic value are given for a number of pure metals and for Ni( 10 0) covered by chemisorbed atomic oxygen.

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

Ni(100)

I

-1.5

1

m

-1

-0.5

0

0.5

425

1

1.5

2

Energy - EFERM I eV FIG.1 1 . Deconvolution of He*-Ni(100) AN-spectrum. The part of U ( E )near the Fermi edge is shown. The idealized F e d step function is indicated.

Auger Deexcitution. Auger deexcitation leads to electron energy spectra which resemble UP spectra in the sense that the same simple relation holds between the energy of the emitted electron Eel and the binding energy Ebind of the orbital from which an electron is removed:

where E * [He*]denotes the excitation energy of the metastable helium atom. This equation holds because of conservation of electronic energy even though two electrons are involved in the process of AD. The tunneling of an electron from either of the target orbitals into the 1s hole at the metastable is the decisive step accompanied by emission of the former 2s electron of the metastable helium atom. The activity of a particular orbital in the ionization process is governed by the tunneling probability, and this in turn depends on the electron density of the orbital at the position of the He 1s hole orbital. Thus, the 1s hole orbital acts as a local detector of electron density. Based on the experience from gas phase reactions, one must conclude that the symmetry of the involved target orbital has no influence on its activity in AD beyond its influence on the local electron density.

426

H. Morgner

TABLE VI EFFECTIVE RECOMBINATION ENERGYI,, OF HE+ IN FRONTOF A METALSURFACE. GIVENIS THE DECREASE WITH RESPECT TO VACUUM VALUE Al,, = 24.587 E V - I E ~VALUES . ARE DETERMINED BY TWO METHODS: FROM DECONVOLUTED SPECTRA AND DIRECTLY VIA STRAIGHT LINE FITTED TO THE HIGH-ENERGY PART OF THE SHAPE OF THE

M E SPECTRUM DOES NOT ALLOW

M E SPECTRA. IN SOME CASES, THE

THE FIT OF A STRAIGHT LINE.

FURTHER

EXPLANATION IN THE TEXT.

A d 1 11) Au(l11) FY( 1 1 I ) Cr( 1 10) O/Cr( 1 10) Ag(1 10) Cu( 1 1 1) FeSi( 100) Ni( 100) O/Ni( 100) isolated Ohs O/Ni( 100) maximum coverage Pd( 1 1 1) Pd( 1 10) Cu( 1 10) w (POlY)

are,

are,

Work Function (eV)

(eV) Deconvolution

Direct'

4.67 5.35 5.63 5.13 5.54 4.37 4.97 4.88 5.19 5.51

1.7 2.13 2.35 1.64 1.69 1.92 1.99 2.19 1.67 1.74

1.8 2.1 2.38 1.55 1.92 2.1 1.91 2.12

5.52

2.34

2.3

5.95 g 5.26 4.48 g

2.3 2.0h 2.lh 2.2h

(ev)

Spectra from a

a

h C

C

d e

f C

C

Heinz and Morgner ( 1997). Briickner et al. (1994). Kubiak (1998). HofXmecht, A. (1996). unpublished result. Heinz B. (1996), unpublished result. S. Chenakin and R. Kubiak (1995). unpublished result. g Weast (1987). Sesselmann et al. (1987). Reference spectra from Section IV.B.4.

a

This property of metastable rare gas atoms can be used to distinguish experimentally different orientations of a molecule at a surface. The effect has been demonstrated for the first time by Kuchitsu and his group for an evaporated biphenyl film. Adsorbed at a surface temperature of 109K, the molecules began to reorient upon increasing the temperature to 170 K, which could be followed by M E S (Kubota, et al., 1980). In the meantime the sensitivity of MIES for the orientation of adsorbed moelcules has been employed for studying a large number of organic molecules (Harada et al., 1984; Harada

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

427

et al., 1984a; Kubota et al., 1984; Suzuki etal., 1985;Harada and Ozaki, 1987; Ozaki and Harada, 1987; Ozaki and Harada, 1990a; Harada and Hayashi, 1989; Briickner et al., 1994; Heinz and Morgner, 1997) and of small inorganic molecules (Yencha et al., 1981; Conrad et al., 1982; Bozso et al., 1984; Arias et al., 1985; Lee et al., 1985; Lee et al., 1985a). Even the orientation at liquid surfaces has been characterized by MIES (Keller et al., 1986). A further important application of MIES is the monitoring of chemical changes at surfaces by coadsorption (Lee et al., 1983; Lee et aL,1983a; Maus-Friedrichs et al., 1992) and by interaction with the substrate (Bozso et al., 1984; Woratschek et al., 1986; Woratschek et al., 1987; Huang et al., 1992; Dehnbostel et al., 1992; Canepa et al., 1995). Radiation-induced surface modifications have been identified by MIES as well: Ohno et al. (1982) have investigated photo-oxidation of naphtacene and rubrene; the effect of electron bombardment on the electronic structure of alkali halide films in comparison to temperature modifications was studied by Dieckhoff et al. (1992); and the reduction of a metal oxide under ion bombardment has been studied by Kubiak et al. (1994). So far, we have discussed the characterization of adsorbates. But even on bare surfaces, some interesting observations have been made concerning the behavior of MIES. The (100)-face of NiO is - within a few hundredths of an angstrom - an atomically flat surface. The M E spectrum taken with He*p3S) is shown in Fig. 12. The energies of the O(2p) and Ni(3d) bands are indicated according to the assignment by Kuhlenbeck et al. (1991). It is obvious that the Ni(3d) band is rather weak compared to the O(2p) band. This finding fits well the known properties of both orbitals: the 3d orbital at the Ni2+ site is very localized, whereas the 2p orbitals at the doubly negatively charged oxygen are very diffuse, as evidenced by the large dispersion of the corresponding band (Kuhlenbeck et al., 1991). Thus, it is fully in line with the notion that the activity of an orbital in MIES reflects its electron density at the position of the He nuclei whose turning points are about 3 angstroms in front of the nuclei forming the top surface layer. A similar high probability for detecting the diffuse orbitals of a negatively charged species has been observed at the surface of halogen halides. Investigating several of these salt surfaces, Munakata et al. (1980) have consistently found that the np band of the halide dominates the M E spectrum. Of course, a preference for the halide bands can be recognized only if their signal can be compared to the signal from other orbitals. The alkali mp bands would be the natural candidates for such a comparison. Unfortunately, for most alkalis X(mp6(m l)s), the ionization energy of the mp orbitals is too large to be ionized by metastable helium. One exception is the surface of CsI, which has been studied by Dieckhoff et al. (1992). They have been able to identify emission from the Cs(5p)orbital in their MIE spectrum. The corresponding intensity is - after

+

428

H. Morgner

NiO(100) thin film

or 0.5

2

I r

L

l

f

1.5

sg

2

-MIES -10

-16

-14

-12

x

-UPS -10

a

6

4

-2

o

2

E - E ~ e [ev] d FIG.12. NiO(lO0). HeI-UPS and He(2’S) MIES.The upper panel shows the bandstructure of NiO. Data points are from ARUPS (Kuhlenbeck er al., 1991).Full lines are values calculated by Borstel as quoted by Hiifner era\. (1991).

subtracting the background of secondary electrons - distinctly smaller than the intensity of the l ( 5 p ) band. We proceed now to discussing the surface of another ionic crystal, Cr2O3 (000 1). which - in contrast to NiO( 100) - is far from being atomically flat. The electrostatic instability of the (000 1) face causes strong relaxation that has recently been characterized by Rohr et al. (1997, 1997a). The surface appears to be terminated by chromium ions that amount to the equivalent of half a monolayer sitting on top of a closed monolayer of oxygen. In Fig. 13 the spectra of Cr203(OO01) as obtained from UPS and MIES are compared. Photo-electron spectroscopy shows the O(2p) as well as the Cr(3d) bands. On the other hand, in MIES only the Cr(3d) band is clearly discernible. This is surprising because the general rule found for NiO and alkali halides is that the diffuse orbitals located at the negatively charged species should be more active in MIES than the localized orbitals at the positively charged ions. Obviously,

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

t

-20

429

\ UPS

-18

-16

-14

12

-10

8

4

4

-2

0

2

Mndlfg rmr~ywllhnfanrcaLo Fmmi enrgy I aV

FIG. 13. Crz03(0001). HeI-UPS and He(Z3S) MIES. From Kubiak (1998).

the top position of chromium fully ovemdes this general rule, which apparently holds only for atomically flat surfaces. So far we have discussed surfaces that prohibit resonance ionization of the approaching metastable helium atom and thus lead to pure AD spectra. However, even at metal surfaces, a very small fraction of metastable helium atoms survives and impinges on the surface as intact metastable species. The existence of a very small fraction of metastables escaping RI is the prerequisite for observing scattered metastable helium atoms (cf., Section II.B.1). He* atoms that come sufficiently close to a metal surface cause Auger deexcitation.The resulting spectra are UPS-like and accordingly give rise to a high-energy edge of the spectrum that reflects the F e d energy. This feature allows us to identify the occurrence of AD unambiguously. In Fig. 14 the M E spectrum from He*(23S) Cu( 1 1 1) is shown. The high-energy part is blown up in order to make the AD contribution visible. The range between the electron energy E,, = E*[He*]- E F corresponding to the Fermi energy and 0.75eV below this energy can safely be ascribed to AD with no significant contribution from the RI and AN process. In order to get a measure for the relative importance of AD, we evaluate the ratio between the signal ZAD in the and the signal integrated over the full spectrum, interval [E,, - 0.75 eV, Eup] which can be equated with the AN signal IAN. The ratio ZAD/ZAN is listed in Table VII for several metallic surfaces. This ratio is always below 1. lob4for pure metals. If one corrects for the small energy interval of 0.75 eV used for evaluation of the AD signal, one has to multiply by a factor of -20 because

+

430

H. Morgner MlES of Cu(ll1) 160 140

x 300

120

1w 80

60 40

$I

20 0

4

6

8

10

12

14

16

18

20

22

eledronenergyleV

FIG. 14. Cu(1 1 1). ME-spectrum taken with He'p3S). The high-energy part is blown up i n order to make the AD contribution visible.

TABLE VII THERATIOOF AUGERDEEXCITATION TO AUGERNEUTRALIZATION IN REACTIONS OF HE(z3S) WITH METALSURFACES. AD: INTEGRAL OVER 0.75-EV INTERVAL AT FEW EDGE; IN: INTEGRAL OVER FULL SPECTRUM. THEFULL SPECTRUM EXTENDS OVER ABOUT 15 EV. THEREFORE, THE AD VALUE SHOULD BE MULTIPLIED BY A FACTOR OF ABOUT 20 IN ORDER TO GET AN ESTIMATE OF THE TRUE RATIO.

Ratio AD/AN

Surface clean clean clean clean clean clean O(0 = 0.38)/

A d 1 11) Au( 1 1 1) CU(1 1 1) Cu( 1 1 I )

Pt(I 1 1)

Ni( 100) N i ( 100)

4.3E-05 8.2E-05 6.6E-05 7.8E-05 9.3E-05 4.88-05 6.48-05

the AD spectrum can be about 15eV wide. This would bring the ratio of AD to AN to ~ 2 . 1 0 -If~ one . interprets this value as a rough measure for the survival probability in the He* channel near the turning point of the trajectory, the probability of the metastable to avoid resonance ionization along the whole trajectory would be the square of this value. The probability for AD

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

431

at a metallic surface can be enhanced by adsorbates, which represent a local barrier for RI. This has been found for NaCl islands on W(1 lo), a system that is described in Section IV.B.l, and for saturated alkanes adsorbed on Ni(lO0) (Heinz, 1997a). Experimental Distinction Between AD and AN. Earlier in this section, we discussed how the nature of the surface influences the probability for FU and AN on one side and AD on the other side. It may well be, however, that in a practical experiment, the properties of an investigated surface are not known sufficiently well to predict with certainty which mechanism leads predominantly to electron emission. One way to check on the dominant mechanism is the comparison between MIE spectra taken with He(23S) and He(2'S). In RI and AN the spectra are almost identical, as discussed in the section on Auger neutralization. If, on the other hand, the dominant contribution to the spectra is caused by the AD mechanism, the spectra should be different because of the higher excitation energy of He(2'S) compared to Hep3S). In particular, if both spectra show similar features but shifted by the energy difference A E = E * [ H e ( 2 1 S ) ] - E * [ H e ( 2 3 S= ) ]0.80eV, one can safely conclude that AD is the dominant or even the only process leading to electron emission. We have, however, encountered surfaces that show more complicated behavior. In Fig. 12 the Hep3S) M E spectrum of a NiO( 1 0 0) film epitaxially grown on Ni( 100) is shown together with the Hel-UP spectrum. As expected from a wide-band semiconductor, the NiO( 100) surface shows the feature of an AD spectrum, the 0(2p,) band being clearly discernible. That all other bands known from photo-ionization are strongly suppressed is a consequence of the AD mechanism and has been discussed in Section II.B.2. On the basis of the above arguments, one would not hesitate to predict the He(2'S)-MIE spectrum to be similar in shape but shifted to higher electron energies by -0.8eV. Therefore, the actually measured spectrum in Fig. 15 is surprising. The 0(2p,) band is clearly missing, and the center of gravity of the spectrum is by no means shifted to higher electron energies. The structureless shape of the spectrum is reminiscent of an AN spectrum rather than an AD spectrum. In order to clarify the situation, an INS experiment using helium ions with a kinetic energy of 28 eV perpendicular to the surface was carried out (Kubiak, 1998). The spectrum is shown in Fig. 15. Its similarity to the He(2'S) M E spectrum points out unambiguously that He(2'S) atoms in front of the NiO(100) surface react via Auger neutralization. This requires that the He(2'S) metastables lose their 2s electron during their approach to the surface while the metastables in the triplet state survive. At first glance this seems to contradict the generally accepted reaction model. However, a closer look at the situation reveals that this finding is easily explained within the standard concept. In Fig. 16 the density of states of NiO is shown, and the positions of

H.Morgner

432

INS He' 50eV -2

0

2

4

0

12

10

8

14

16

electron energy I eV FIG. 15. NiO(l00). h4IE spectrum taken with He(2'S). IN spectrum taken with He+ at kinetic energy perpendicular to surface of about 30 eV. Kubiak (1998).

NiO 7

--

8

-.5 ~.4

Emf

- - -2

-- 3 -- 4 0

density of states [ h u . ]

2

4

8

8

1

0

di6tmce [Ang.]

FIG. 16. Density of states of NiO. The energy of the 2s orbital is indicated for both metastable helium atoms as a function of distance from surface. The density of states is from Sawatzky and Allen (1984).

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

433

the asymptotic energy levels of both metastable helium atoms are indicated. If one takes into account that the effective ionization potential of the helium atoms decreases with decreasing distance from the surface, one recognizes that metastables in the higher-lying singlet state can lose their excited electron by means of resonance ionization. This may have the consequence that electron emission is dominated by the AN process if the metastable is initially in the singlet state. The above observation has been made by accident, but obviously it could be turned into a systematic investigation. Apparently, the position of the bottom of the conduction band in NiO in relation to the energy level of the metastable helium controls whether RI and AN or AD is dominant. Because the respective spectra are very different, they can serve as an indicator for the occurrence of either process. Let us assume now that if we were able to tune the energy level of the metastable atom continuously up and down, we would obtain a clear indication of whether the atomic energy level is below or above the onset of the conduction band. Obviously, such an experiment would not be practical, because a continuous tuning of an atomic level over the required energy range is hardly feasible. However, one is not restricted to a single rare gas. The ionization potentials of all metastable rare gas atoms span a range from 4.9454eV down to 2.6826eV (cf. Table I). Further, one could make use of lasers and excite the metastable rare gas atoms in front of the surface, thereby modifying not only the ionization potential of the projectile but also the symmetry of the excited electron and thus studying the overlap between the unoccupied orbitals at the surface and the orbital of the excited electron in the atom as a function of the symmetry of the latter. A new class of experiments could emerge from this scheme. Homogeneous and Inhomogeneous Sulfates. So far we have tacitly assumed that all surfaces considered are homogeneous. In a strict sense, homogeneous surfaces do not exist except as a model concept like the jellium model. Whether the concept of a homogeneous surface is meaningful for the purpose of understanding experimental data depends on the type of experiment. Even an atomically flat metal surface appears to be corrugated if investigated by scanning tunneling microscopy; otherwise, the STM would be of little use. The same holds for ISS (Ion Scattering Spectroscopy) and NICISS (Neutral Impact Collision ISS). This method exploits the fact that ions of an energy of several keV follow trajectories that sensitively depend on the lateral position of a surface. The ion may hit a surface atom with small impact parameter and be directly reflected, or it may follow a trajectory between surface atoms and enter the bulk of the sample, the latter being by far the more probable event (Niehaus, 1992). The apparent corrugation for the method TEAS (Thermal Energy Atom Scattering)lies between 0.02 A for atomically flat metals and several angstroms for surfaces with defects and adsorbate-covered surfaces (Poelsema, 1989).

434

H. Morgner

The question is now whether it would be a fruitful decision to call an atomically flat surface inhomogeneous on the grounds that several analytical methods experience corrugation. If so, the term homogeneous would be discarded from the discussion of surfaces because any surface would be “inhomogeneous” by definition. In consequence, we were obliged to invent a new term to characterize surfaces that are partially oxidized, partially adsorbate-covered,and so forth. In this paper we adopt the following nomenclature: we reserve the term homogeneous for all surfaces that are as homogeneous as physically possible, e.g., flat metal surfaces, surfaces fully covered by adsorbates or with closed epitaxially grown overlayers, and surfaces of pure liquids. This decision is justified in general by the above considerations, but it gains additional weight when we are dealing with MIES. In order to explain the argument, we first address other methods. When investigating a metallic surface with the STM, it is possible to distinguish experimentally between different lateral positions. In an NICISS experiment, one can relate different spectral features to different trajectories of the scattered particle, i.e., the lateral position at which the projectile hits the surface is reflected in the measured spectrum. Now we consider MIES. In view of the fact that in STM as well as in MIES, the tunneling of an electron between the surface and an atom in front of the surface is involved, there can be little doubt that the MIE spectrum depends on the lateral position of the metastable atom. However, at present nobody has been able to make visible such dependencies, nor has anybody tried to do so or even discussed the possibility. In other words, we have to content ourselves with the result that different positions on a flat metallic surface cannot be distinguished experimentally by MIES. We now consider an inhomogeneous surface if different parts of the surface yield different MIE spectra that can be experimentally distinguished. A prototype of an inhomogeneous surface is the surface of a binary mixture of miscible liquids (Morgner et al., 1991;Morgner and Wulf, 1995). Both species are found in the top layer of the surface and cause a characteristic spectrum. The measured M E spectrum can often be interpreted as a linear combination of the spectra of the pure substances. Liquid surfaces partially covered by tensides belong to the same category of inhomogeneous surfaces (Morgner et al., 1991a; Morgner et al., 1992; Morgner et al., 1993; Morgner and Oberbrodhage, 1995). Another type of an inhomogeneous surface is represented by a partially oxidized metallic surface. The system NiO/Ni( 1 0 0 , l l 1) has been investigated with emphasis on the composition of the top layer (Morgner and Tackenberg, 1994). In this case one finds that the presence of one surface species influences the interaction of He* with the other species. Again, new phenomena were discovered in the M E spectra taken by Dieckhoff et al. (1992a) during epitaxial growth of a NaCl layer on W( 1 10). At intermediate coverage, new surface properties turn up that cannot be

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

435

prepared homogeneously on the surface. The methods for data evaluation that have made this discovery possible are presented in Section III, and the system is discussed in Section IV. When we are dealing with MIES data from a surface that is suspected to be inhomogeneous, a practical problem of data analysis arises. Different parts of the surface lead to different spectra with the consequence that the measured MIE spectrum must be considered a superposition of different contributions.In order to take full advantage of the perfect surface sensitivity of MIES, one would like to have a systematic way to disentangle such composite spectra and to analyze the shape of the different spectral contributions and their relative weight. In particular, this would be important in the case of surfaces composed of different areas, some of which cause the RI and AN mechanism of electron emission and some of which react via Auger deexcitation. The partially oxidized surface of a metal could serve as an example. The oxidized areas will in general prohibit resonance ionization, whereas the still-metallic areas support this reaction. It is obvious that in such a case the data processing by means of deconvolution cannot be applied to the whole spectrum. Even if the procedure were formally carried out, the result could not be identified with any meaningful physical quantity. Deconvolution could be applied only if the AN contribution had been separated from the rest of the spectrum. Even if Auger neutralization is the dominating mechanism of electron emission everywhere on the surface, caution discourages naive application of the deconvolution. As an example, we name Ni( 100) partially covered by chemisorbed oxygen, a system that will be discussed in detail in Section IV. It is commonly accepted (and supported by the analysis in Section IV) that adsorbed oxygen atoms do not prohibit resonance ionization. At maximum coverage (0 x 0.4)just before the onset of oxidation, the surface can be considered as homogeneously covered by chemisorbed oxygen. Compared to the pure metal, the surface density of states has changed, which clearly shows when comparing the MIE spectra. In both cases the deconvolution of the measured spectra is meaningful, and the influence of the oxygen atoms on the SDOS can be assessed. However, it appears doubtful that the same procedure remains meaningful if only a small part of the surface is covered by chedsorbed oxygen. The reason is that the SDOS is modified by an adsorbate only in a finite neighborhood. If the helium lands within this range, it will experience - in the terms of our discussion of Auger ) deviates from that of the neutralization - a density of state U ~ ( Ethat unperturbed metal surface U1 ( E ) . Accordingly, a projectile hitting the surface in a still-uncovered area will cause a contribution to the spectrum that can be written as

436

H.Morgner

where the bracket stands for the operation of self-convolution.Helium near an adsorbed oxygen atom at the instant of electron emission will contribute a spectral shape that is given by

If 01 is the fraction of the surface whose SDOS is modified by the adsorbates, the spectrum of the inhomogeneous surface can be written as

What would happen if this spectrum were deconvoluted as a whole? One would formally obtain a result U ‘ ( E )with

U ’ ( E )could be considered meaningful only if a simple relation to the surface densities U1(E) and U ~ ( Ecould ) be established. Because self-convolution is clearly a nonlinear operation, one cannot expect a linear relationship between U’(E) and UI(E), U2(E). If we assume

we get

Comparison of coefficientsleads to P2 = 1- ct and y 2 = 01 and consequently to ( U2 I U1) = ( U1 IU2 ) = 0, which is not the case in general. A possible relationship between U ‘ ( E )and the physically meaningful quantities U ~ ( EU) ,~ ( E had ) to be nonlinear and thus loses any appeal for interpretation of the data. We draw the conclusion that for any inhomogeneous surface, one has to separate the spectral contributions before any further attempt can be made to derive surface properties. A successful strategy to this end is presented below.

The Role of Electronic Spin. As in gas phase work, experiments with spinpolarized He(23S)-atoms have been pioneered by Dunning and Walters and their group (Keliher et al., 1975). Their experimental setup is distinguished in that they not only employ spin-polarized projectiles but also analyze the spin

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

437

orientation of the emitted electrons. In this way, meaningful information on the role of electronic spin can be gained even for nonmagnetic surfaces. The quantity derived from the measurement is the degree of polarization of the emitted electrons normalized to the polarization of the projectiles. The kinetic energy of the electrons is determined as well, which means that the spin polarization of the emitted electrons can be measured as a function of their kinetic energy (Hart et al., 1989). An interesting observation has been made for atomically clean Cu( 100). Although the degree of spin polarization of the emitted electrons is only of the order of 20% when averaged over the whole spectrum, it reaches a value of 70% at the high-energy end of the spectrum. The fact that a positive electron polarization is observed at all is interesting and had not been expected for metal surfaces where the process leading to electron emission is Auger neutralization. In this case the emitted electron originates from the surface rather than from the polarized projectile, and therefore the possibility of zero spin polarization among the emitted electrons has even been discussed (Hagstrum, 1954). Obviously, the finite - though small - degree of polarization characterizes the interaction between the two electrons active in the process. The electron tunneling into the He 1s hole orbital has to accommodate its spin antiparallel to the 1s electron, whose spin orientation is assumed to be identical to the original polarization of the metastable atom. A positive polarization of the emitted electron points out that the two active electrons preferentially form a singlet rather than a triplet state. This propensity becomes even more pronounced the higher the energy of the emitted electron is (Hart et al., 1989). If one extrapolates the degree of polarization from the measured values to the maximum energy Eel,maxin the spectrum, one finds that even 100% polarization is not excluded by experiment, the smallest value compatible with the experiment being N 75%. If the emitted electron has the maximum energy E,,,,,, both active electrons originate from the Fermi edge and thus have in general the same spatial symmetry with respect to their local environment. It is reminiscent of the Pauli principle that in this situation the two electrons are preferentially, if not exclusively, in a singlet state. An alternative explanation for the fairly high degree of electron polarization was discussed by the same group in the context of theoretical work (Dunning et al., 1991). The claim was that a much larger fraction of metastable helium atoms than generally assumed would reach the metal surface and thus lead to electron emission via the AD mechanism, which is characterized by conservation of spin polarization. As outlined in the preceding section, this interpretation is not in line with measured electron energy spectra. Further, Dunning and Walters have supported their original interpretation convincingly with experimental data (Or0 et al., 1992). They showed that the electron energy spectrum of Cu( 100) is identical, within experimental uncertainty, N

N

438

H. Morgner

irrespective of whether He(23S),He(2'S), or laser-excited He(z3P) is used as projectile. This proves that the process of RI and AN dominates electron emission. In the case of metal surfaces covered by a sufficiently thick adsorbate film, resonance ionization of the metastable projectile is prohibited. Accordingly, the emission of electrons occurs via Auger deexcitation, which conserves spin polarization to the same extent as does direct Penning ionization out of the covalent entrance channel in gas phase reactions. This point of view has been demonstrated by Or6 et al. (1992) and Butler et al. (1992) to be true for several systems. In the considerations described above, the spin-labeling technique has been used to investigate the dynamics of the electronic interaction between the metastable atom and the surface. Because the spin polarization of the projectile and the spin polarization of the emitted electrons are controlled, the surface does not need to have a preferred spin orientation. Another approach to spin-dependent studies involves magnetic surfaces. In this case the polarization of the emitted electrons is often not monitored, which reduces the experimental effort considerably. Instead, the number of emitted electrons is recorded as a function of the relative orientation between He(z3S) spin and magnetization of the sample. In particular, the evaluation of electron intensity near Eel,mais meaningful. It monitors the spin density a few angstroms above the top layer of the surface and reacts sensitively to changes at the surface by adsorption (Hammond et al., 1992; Steidl and Baum, 1996). The acronym used by the authors for this experimental scheme is SPMDS (Spin Polarized Metastable Deexcitation Spectroscopy).

111. Quantitative Evaluation of MIES Data A. HOMOGENEOUS SURFACE OF KNOWN MATERIAL 1. Clean Sur$ace Reacting via Auger Deexcitation

MIES occurs via Auger deexcitation at the surface of insulators like alkali halides and oxides, but a semimetal like graphite belongs to this category as well. Further, the surfaces of metals with very low work function like alkalis are known to favor the AD mechanism. In this section we will consider which kind of quantitative information could be of interest and be derived from MIES data. The mechanism of AD allows the conclusion that the He* projectile probes the electron density in front of the surface, the separation being of the order of 2-4 angstroms. This means that MIES is sensitive to the same property as the STM at positive tip voltage. The difference between these techniques is that STM is distinguished

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439

by its lateral resolution but is restricted to a limited range of binding energies, whereas MIES monitors the laterally averaged electron density over a wide range of binding energies. We will see later that MIES indirectly acquires some lateral sensitivity at inhomogeneous surfaces, but at homogeneous surfaces and with present techniques, the lateral resolving power of MIES is several orders of magnitude below that of STM; cf. Harada et al. (1994). The knowledge of the electron density as a function of energy is quite generally of interest for the understanding of surface adsorbate interactions, e.g., in catalysis. MIES offers an experimental access to this quantity. Even if one concedes that the inhomogeneous surface is a more realistic model for applications, it is still true that the analysis of data from homogeneous surfaces provides a good testing ground for the attempts to establish quantitative evaluation of MIES. It has been found for the reaction of Hep3S) with a surface of HOPG that the electron density experienced by He* can be related in a simple way to the electronic properties of the bulk (Heinz and Morgner, 1998). For the bulk band structure, the expression P ( E , Z ) shall describe the probability for the occurrence of E = E ( $ ) . The bulk density of states can be calculated from this expression as

J

DOS(E) = dk3P(E,Q

The MIE spectrum and, thus, the appropriate surface density of states can be fitted by

J

SDOS (E) = dk3P(E,Z) . W(Z)

where W ( Z )is a i-vector-dependent weighting function. As argued by Heinz and Morgner (1998), the analytical form of W ( Z ) depends on the symmetry of the electronic band considered. The It-band of graphite can be treated by the ansatz W ( Q = W(kx,ky,k,) = W(k,)

*

W(k)

& 0

Jk=

with W(k,) a exp( - y k k ) and W(k,) a sin2 ,where k, = is the component parallel to the surface and k, parallel to the surface normal. The weighting function favors the center of the Brillouin zone. It turns out that a point in k-space at the border of the Brillouin zone contributes to the surface density of states only with a few percent of the r-point contribution.

440

H . Morgner 2 D bandstructureof Graphite

12 10 8

6 4

a -10 -12 -14

-16

K

r

M

K

Fw. 17. Two-dimensional band structure of graphite. From Heinz and Morgner (1998).

In principle, the weighting function should be given an explicit energy dependence. The experience with HOPG, however, indicates clearly that the k-dependence ovemdes any possible E-dependence: the point of the 7[: band lies as much as 9 eV below the energy at the zone boundary; cf. Fig. 17. From general considerations about the extension of orbitals as a function of binding energy, one must conclude that the states near the r point are much less diffuse than those at the BZ boundary and thus should contribute less to the electron density a few angstroms in front of the surface. The opposite is the case, indicating the dominance of the k-dependence; cf. Fig. 18. A similar propensity for the r point of the Brillouin zone to yield electron density in front of the surface has been found for the alkali halides LiF and NaCl (Morgner, 1998) and for MgO (Morgner, 1998a). Again, the BZ boundary is suppressed by almost two orders of magnitude. In the course of time, more systems will be analyzed in this way. It is hoped that these studies will make it possible to derive reliable estimates for the surface electron density as a function of binding energy, provided that the electronic band structure of the bulk is known. For a large number of materials, the bulk band structure is available from the literature (e.g., Hellwege and Olsen, 1981). Another aspect of general interest concerns the localization of the hole state that is created in

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

-16

-14

-12

-10

EF-I

8

S

-4

-2

441

0

- Energy I eV

FIG. 18. M E spectrum of HOPG taken with Hep3S) under normal emission. The density of states of the A bands is plotted as a solid line. The best fit of the A band’s contribution to the measured spectrum is shown by open circles. Further explanation in the text. From Heinz and Morgner (1 998).

the ionization process. The mechanism of AD suggests that MIES creates a hole in the top atomic layer in the first place. It is obvious that after a certain time this hole will diffuse into the bulk, or else the hole will be filled again by bulk electrons. However, whether the decay of the hole is still felt by the emitted electron or whether the electron experiences only the hole state in the top layer, cannot be answered in general by experimental data. The surfaces of the alkali halides constitute an interesting exception. The Madelung energy causes a substantial energy difference between a hole in the top surface layer and a hole in the bulk. In this case the question of the localization of the hole in the direction of the surface normal can be answered experimentally: indeed, the energy of the emitted electron indicates that the AD mechanism of MIES leads to a hole in the top layer and that this hole remains stable during the relevant interaction time between surface and emitted electron (Morgner, 1998). 2. Clean Sugaces Reacting via RI and AN

The MIE spectra obtained from metal surfaces with high work function are due to the mechanism of Auger neutralization (AN) following the resonance ionization (RI)of the metastable rare gas atom. As discussed in Section II.B.2, the relationship between the AN spectra and the electronic properties of the sample is not a simple one. Following Hagstrum (1954), one can consider the

442

H. Morgner

measured spectrum P(Eel) as self-convolution of the so-called “weighted density of states”U(E) of the sample:

where E is the binding energy with respect to the Fermi level. The task of data evaluation consists then in carrying out the deconvolution procedure in order . the spectrum P(Eel) were exactly known, the solution of to determine U ( E ) If Eq. (27) would be unique. In fact, P(Eel) is given as a set of discrete data points P ( j ) = P(E,) with experimental uncertainty. The energies Ej are usually spaced equidistant on the electron energy scale. Correspondingly, the solution of the problem is sought in the form of another set of data U ( i ) = U ( E ~equidistant ), on the binding energy scale. In terms of Eqs. (17) and (18), one identifies

If one gives both sets of data the same energy channel width, i.e., Ej one gets the set of equations

- Ej+l =

Ei+l - Eir

P ( 1 ) = U; P(2) = 2U(l)U(2)

P(3) = 1!l(2)~+ 2U(l)U(3) (28) which can be solved iteratively in a straightforward way. Even though this algorithm is fast, it is hardly used. It requires heavy smoothing of the data before they can be subjected to the procedure, and even then it is unstable and produces negative values of U ( i ) .Mathematically, U ( E )could be negative, but the concept of a density of states - even though weighted - for U ( E )makes negative values undesirable. The goal will always be to find a positive definite function U ( E )that reproduces the data P(Eel) within experimental uncertainty. Formally, the solution can be obtained by transforming into Fourier space, making use of Noether’s theorem, but the disadvantage of instability with respect to small variation of input data and the appearance of negative values for U ( i ) make this scheme impractical. An efficient and stable algorithm has been presented by Dose et af. (1981). Its strategy is as follows: the unknown data U ( i ) are formally fitted to an analytical function that is piece-wise composed of cubic splines. The

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

443

self-convolution can then formally be carried out. The problem is solved by fitting the unknown parameters U ( i ) under the side condition that U has minimum curvature and that the experimental data P ( j ) are met as accurately as possible. A strategic parameter weighs the two criteria with respect to each other. The optimum value of this parameter depends on the problem and can be set only after some trial calculations. In general, the outcome of the algorithm by Dose et al. (1981) is very satisfying. A weakness that can be observed in some cases concerns the handling of steep onsets (Fermi edge) of U ( E )by this algorithm: the slope is sometimes less steep than expected and/or negative values are found before it saturates at zero. Therefore, we have tried out yet another method. The problem is posed with energy channel widths such that ~ i + l - ~i = 0.5. (E’ - E j + l ) . The set of equations then reads P( 1) = U (

+

P (2) = V ( 2 ) 2 U(l)U(3)

c k-

P(j)=

+

U ( j - k ) U ( j k ) , k,,

=j -1

k=O

and for N data P ( j ) ,one has 2N - 1 unknowns U ( i ) .The unknowns are varied until the experimental data points P ( j ) are approximated satisfactorily. The solution U ( i )is improved, starting from a random guess, by means of a genetic algorithm (Eschen et al., 1994),which is surprisinglyefficient in selecting sets of U ( i ) ,i = 1,2, . . . , 2 N - 1 that satisfy the above equations with excellent accuracy. The advantages of the procedure are that one can easily restrict the algorithm to pick only positive numbers for U ( i ) , thus avoiding the main problem left, and that no smoothing of U ( E )or of data P ( j ) is applied. Of course, the problem is entirely underdetermined. Therefore, one gets a large number of solutions U ( i ) .Out of these, we simply pick the smooth ones, and, further, we average them. The averaged solution U ( i ) is convoluted via the above equation in order to verify that the average is indeed a satisfying solution. The outcome has strictly non-negative numbers and the slope at the Fermi edge appears particularly steep. In many cases this procedure yields results that are very similar to the outcome of the much more time-efficient algorithm by Dose et al. (1981), but sometimes it leads to a definite improvement. 3. Molecular Sugaces

In this section we deal with surfaces that are composed of molecules. Monolayers and multilayers of adsorbed molecules on solid substrates, but

444

H.Morgner

also the surfaces of molecular liquids, are this type of surface. If only one species of molecules is involved (as is usually the case if we talk about a homogeneous surface), then the main information one can gain is the orientational behavior. The molecules in the outermost layer may be perfectly oriented, may have a preferred orientation, or may be randomly distributed. That MIES is sensitive to molecular orientation has been outlined already in Section II.B.2. Here we will consider how much quantitative information can be gained. One approach makes use of the shape of the valence orbitals. Electron structure calculations provide this information for small and medium-sized molecules with satisfying precision. Comparison with photo-electron spectra in the gas phase allows one to judge the reliability of the theoretical information (Kimura et aL, 1981). In MIES the valence orbitals can be identified via their binding energy as in U P S . The relative peak intensities can be interpreted as being caused by the accessibility of the respective orbitals to the impinging metastable helium atoms. This in turn should allow one to evaluate the orientation or the distribution of molecular orientations. So far as we know, this last step has never been carried out, but qualitative orientation has been derived, and changes of Orientation due to sample treatment like heating or further adsorption of molecules have been observed. For other systems another strategy, which has been found to work well for long-chain alkanes, may be more suitable. There, the possibility of continuously varying orientation is conceptually replaced by a small number of discretized orientations. Under this restriction, quantitative data processing allows one to evaluate the relative probability to find the molecules in either of a limited number of orientations. Formally, molecules in different orientations are treated as different species. Therefore, this scheme is identical to the procedures discussed for inhomogeneous surfaces. B. SURFACES OF COMPOSITE MATERIALS: SERIES OF SPECTRA I . Surj-ace Composed of 7bo Species

Let us assume a surface composed of two species that yield different characteristic spectra in MIES. We denote by S ( E ) the MIE spectrum of this surface and by R,(E) the spectra that would be obtained from a surface homogeneously covered by either species j ( j = 1,2). If the two species do not influence each other too much, one can represent the spectrum S ( E ) as a linear combination of the two reference spectra R,(E).

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

445

This equation has been found to hold for several systems: surfaces of the binary mixture of liquids, surfaces of solutions with surface-active solvents, solid surfaces partially covered by adsorbates, and the early states of heteroepitaxy. There is no a priori necessity that the above equation be valid. The presenceof one speciesmay well influencethe behavior of the second. Inview of the sensitivity of MIES for the orientation of molecules, one might even expect that combining the spectra of the pure substances would often fail to reproduce the spectrum of a mixture. However, in many cases the above equation holds with good accuracy. If so, one can conclude that both species react with He* in the same way as they would react in the pure state. One expects that

a1 +a2 = 1

(31)

must be fulfilled. The coefficients a, can be understood as a measure of the fraction of surface that is covered by species j . The computation of the coefficients u, is straightforward: the spectra are given as finite number N of data points and can be considered as vectors in an N-dimensional linear vector space. The scalar product is then defined in a natural way as N n= 1

From Eq. (30) one derives

which is solved by (34) The solution of Eq. (33) does not guarantee a physically meaningful result. The following conditions must be fulfilled: The spectrum S must be reproduced within experimental accuracy. The coefficient a, must be non-negative in order to represent a fraction of the surface. 0 The coefficients must sum to unity; see Eq.(31). (35) Often one has not only spectrum S, but a series Si(E) with i = 1,2, . . . ,I . As an example, one might think of a solution of a surface-active solute the surface of which is investigated by MIES. For every bulk concentration, one 0

0

446

H. Morgner

gets a characteristic spectrum S i ( E ) .If Eq. (30) and conditions (35) hold for the whole series, one is able to monitor the composition of the fraction. If more than two species are present at the surface, the concept can easily be generalized, provided that the reference spectra Rj ( j = 1,2, . . . J > 2) are known. However, if one wishes to analyze a series of spectra without knowing the reference spectra or without knowing even the number of necessary reference spectra, one must start with a basic analysis of the situation. This is described in the following section. )

2. Sulface Composed of Unknown Number of Species: SVD Algorithm Let us assume that a series of spectra has been obtained by varying a physical parameter like surface temperature, length of exposure to an adsorbate, or duration of ion bombardment. In general, we have no prior knowledge whether this series of spectra can be represented as a linear combination of a small number of reference spectra, how many reference spectra have to be chosen, or how the reference spectra look. Fortunately, well-known mathematical tools are available. The algorithm called SVD (Singular Value Decomposition) allows a unique decomposition of any matrix A whose number of rows exceeds or equals its number of columns into the product of three matrices (Press et al., 1994): A = U , W ,VT

(36)

The matrix U has orthonormal column vectors, W is diagonal with positive or zero elements (singular values), and V Tis the transpose of an orthogonal matrix V. The dimension of W and V is equal to the number of columns of A and U . For our purpose we identify the column vectors of A with the spectra to be analyzed:

where i = 1,2,. . . ) I numbers the spectra in the series and n = 1,2,. . . ,N counts the energy channels of the spectra. We define basis spectra Bi via &(En) = &(En) . wi

(38)

Since the spectra Bi are orthogonal, they contain in general negative values, and thus, they cannot be considered as physically meaningful spectra. Only their linear combination

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

447

can generate physically meaningful spectra. Because the elements W k , k = 1,2,.. . , I are ordered according to size, the importance of the basis spectra Bk declines with increasing index k. Now we can try out how many basis spectra Bk (k = 1 , 2 , ... ,I/)are needed in order to reproduce the series of spectra Si within experimental uncertainty. If it turns out that the whole number of basis spectra is needed, i.e., I’ = I, then this is a clear signal that the spectra Si - if understood as vectors in a linear vector space - span a subspace whose dimension equals the number of spectra in the series. This means that the attempt to analyze the series of spectra as a linear combination of a small number of reference spectra has failed. However, in many cases we have found that a small number I’ of basis spectra is sufficient to describe the whole series of measured spectra. Leaving out the basis spectra Bk with k > I‘ reduces noise but does not diminish any meaningful information. Thus, if one deals with a series of spectra and not with a single spectrum, it appears possible to get rid of noise without smoothing. We concentrate now on the analysis of the reconstructed spectra Si obtained via Eq. (39). They can be understood as column vectors of matrix A’, which has the same number of rows and columns as the original matrix A but has by definition the rank Z’. Consequently, the vector space spanned by the Si can be described by I’ linear independent spectra. One set of spectra are the Bk, k = 1,2,. . . ,1’.However, the Bk are mathematical objects without physical meaning. We are seeking reference spectra Rk, k = 1,2, . . . ,I’ that span the same subspace but have the following properties: They must have non-negative values. If used to describe the series Si, their weighting coefficients ajk must be non-negative. The weighting coefficients must sum to unity:

The search for the suitable reference spectra will be described in the following section. 3. Determination of Reference Spectra

Once the number I’ of linear independent contributions to a series of spectra Si with i = 1,2,. . . ,I > I’ is determined, the task remains to find meaningful reference spectra. Sometimes, this may not require much effort. For example,

448

H.Morgner

consider the surface of a ternary liquid mixture. If a series of MIE spectra of this surface with a large variation of compositions, including the spectra of the pure liquids, can be analyzed by SVD to contain I’ = 3 independent spectra, it appears obvious that the reference spectra Ri (i = 1,2,3) can be identified with the spectra of the pure substances. Any spectrum Si of the series can be expanded in a unique way as

where the au measure the fraction of the surface occupied by speciesj. In other cases, there may be only two identified species at the surface, but still the SVD analysis yields a dimension of I’ = 3. One has to conclude that the two species interact with each other in such a way that the electronic surface structure in the neighborhood of the contact zone is more than a mere superposition. Hence, we have to accept the zone of contact as a third species at the surface that requires its own characteristic reference spectrum. In this case, we would employ the two spectra obtained from the surface fully covered by either species as two reference spectra, whereas the third reference spectrum must be constructed from the series of spectra. The search for this third reference spectrum may appear futile at first glance, but we make use of the low dimensionality I‘ = 3 of the problem. The unknown spectrum can be written as linear combination the three basis spectra r=3 k= 1

Thus, we have to vary only a small number of parameters U3kr k = 1,2,. . . , I’ = 3. Any chosen set of a3k defines a trial shape for the spectrum R3. R3 is an acceptable spectrum if

holds for all spectra of the series and if the conditions (40) are fulfilled. It may be that the solution for R3 is not unique. The bandwidth of solutions defines error bars for the third reference spectrum and, hence, error bars for the weighting coefficients uij in Eq. (43). Often, however, the variety of possible solutions for R3 is rather narrow. It is straightforward to generalize the method to higher dimensions, i.e., a larger number of reference spectra. As an example, we discuss the possibility

LIQUID/SOLLD SURFACES WITH METASTABLE HELIUM

449

that the SVD algorithm determines I’ = 4 while nominally only two different components have been brought together, indicating the formation of two different compounds. Consequently, we have to determine two reference spectra rather than only one.

r=4 k= 1

and

r=4 k= I

This time we have to vary eight parameters instead of three. The criteria for acceptable solutions are the same as before. It is, of course, less probable than before that a unique solution can be identified. The situation encountered during the analysis of a series of spectra can be even more demanding than is the determination of two unknown reference spectra. In one case (spectra of alkanes adsorbed in arbitrary orientation on a substrate; see below), I found by SVD that four reference spectra were needed. However, as long as some of the measured spectra were employed as reference spectra, the criteria (40) could not be fulfilled. Finally, I had to determine all four reference spectra at the same time. The sixteen parameters aik(i,k = 1,2, . . . ,4) seem to offer manifold solutions, prohibiting a unique solution. As it turned out, however, it was laborious to find one solution at all. For this type of quantitativedata evaluation,one needs accurately measured spectra. If the experimental uncertainty is too large, then the determination of the effective rank of matrix A defined in Eq. (37) can be wrong; in general, it will be too small. If so, this does not necessarily invalidate the entire procedure. It may be that the presence of one compound escapes detection and that its weight is distributed among weighting coefficients of the other species.

IV. Discussion of Selected Systems A. LIQUIDSURFACES 1. Pure Liquid: Orientation of Molecules in Topmost Luyer

We show as an example a molecule whose structure is known, which is easy to handle, and which has been studied by MIES in the gas phase, in the liquid phase, and as adsorbate in mono- and multilayers.

450

0

H.Morgner

2

6 8 Electron Energy / eV

10

12

0

2

4 6 8 Elecbon Energy I eV

4 6 6 Electron EnerW I eV

10

12

0

2

4 8 8 Electron E n e r I~eV

4

10

12

10

12

residuum x 5 -+--t-

0

2

FIG. 19. He(23S)-MIESof formamide in different states of aggregation: gas phase, liquid phase, multilayer adsorbed on graphite at 115 K and monolayer adsorbed on graphite at 190 K. Data of gas and liquid phase from Oberbrodhage (1992). Data of adsorbed phase from Heinz (1997).

The He*-induced spectra of formamide (FA) in different states of aggregation are shown in Fig. 19. In the gas phase the peak due to the OC=O orbital (90) dominates because this end of the molecule is attractive to He*atoms (Keller et af., 1986a). The situation changes drastically when the molecules form an ordered monolayer on a surface of HOPG at a temperature of 190 K. The 9o(C=O) orbital is hardly populated anymore, whereas the two 7c orbitals cany most of the ionization activity. LEED data of this sample show lateral order with a pattern that supports the notion that FA molecules are adsorbed with their molecular plane parallel to the substrate surface. At lower temperature (1 15 K) a multilayer can be adsorbed. Inspection of the spectrum and evaluation of peak areas indicate that the order is reduced compared to the

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

45 1

MIES of Formamide

t

'C

.-0

FIG.20. MIES of formamide. The relative ionization probabilities of the four orbitals o(no),2n, 1n, and o(C0) with the lowest binding energies are shown, normalizedto the sum of the ionization probabilities of all four orbitals.

monolayer. The M E spectrum of the liquid surface indicates further decay of order. By fitting Gaussians to the bands in the spectra, it is possible to obtain a measure for the ionization probability of individual orbitals. In particular, the four orbitals with the lowest binding energies, o(n0),2n,In, and o(CO), have been evaluated. Figure 20 shows the results of spectrum analysis in terms of contributions of individual orbitals. Common to all spectra pertaining to the condensed phase is the absence of the dominant 9 0 peak. Together with details on the energetic positions (Keller et al., 1986),this indicates that the condensed phase hardly exposes free -C=O ends but involves the C=O group in hydrogen bonds with the H2N group of neighboring molecules. This fits well the result of MD computer simulations (Oberbrodhage et al., 1997; Dietter, 1997). The above results are certainly interesting and convincing. However, in a strict sense, these considerations do not constitute a quantitative measure of molecular orientation. Below, we will discuss the orientation of long-chain alkanes in a different way: alkanes in different characteristic orientations give different spectra. It turns out that all alkane spectra taken so far can indeed be understood as linear combination of these reference spectra. This means that the possibility of continuouslyvarying orientationis replaced by a small number of discretized orientations. Under this restriction, quantitative data evaluation

452

H.Morgner

is possible. Formally, alkanes in different orientations are treated as different species. Therefore, this scheme is discussed in the section on inhomogeneous surfaces.

2. Sugace Composition of Binary Liquid Mixture As discussed in Section III.B.1, the relative abundance of two species at

a surface can be determined by MIES. The requirement is only that the two reference spectra are established. In the ultra-high vacuum used for studying solid surfaces, this is a simple task. However, the molecules evaporating from liquid surfaces will interact with the beam of He* atoms and thus weaken the He* intensity. This effect can prohibit the exact intensity gauging between the reference spectra, which in turn influences the determination of the weighting coefficients u1, a2. We have developed a technique to cope with this situation (Morgner and Wulf, 1995). The vapor pressure around the liquid is varied under control of a mass spectrometer by varying mildly the temperature of the liquid surface. Extrapolation to zero pressure makes it possible to construct the M E spectrum of the liquid in the absence of vapor. In all cases investigated, the shape of the spectra has been found to be independent of the vapor pressure. Once the reference spectra of the pure liquids are established, a mixture can be measured without applying the vapor pressure correction to every measured spectrum, because the weakening of the He* beam affects both species in a mixture in the same way. For the mixture hydroxipropionitrile (HPN)/formamide (FA), the surface fraction of HPN given by UHPN is plotted in Fig. 21 as a function of the bulk molar fraction CWN. The surface molar fraction defined by

where n denotes the number density, is shown for comparison. The latter quantity would yield a straight line in case of an ideal mixture. As expected from the lower surface tension of hydroxipropionitrile (48mNm-I at 20°C),compared with formamide (58mNm-' at 20°C)we observe a mild but distinct segregationof HPN at the surface. Similar plots are familiar from physical chemistry textbooks that show the thermodynamically defined surface excess versus bulk composition. It is noteworthy to point out that the surface excess is conceptually a quantity that is integrated over a layer of unknown thickness, whereas the surface fraction in Fig. 21 measures the chemical composition of the topmost layer, a property inaccessible to conventional methods. As the chemical behavior of a liquid surface (e.g., droplets in

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

453

Binary mixture HPWFA 1

0.8

0.6

0.4

0.2

0

FIG.21. Liquid mixture of HPN/FA.The surface fraction of HPN u r n is plotted as a function of the bulk molar fraction of HPN. Also shown is the surface molar fraction c g Nthat would yield a straight line for an ideal mixture. The data are from Morgner and Wulf (1995).

the air) is determined by the outermost layer, the possibility of measuring its composition can be of widespread interest. The surface composition of the mixtures of formamide (FA)/polyethyleneglycol (PEG) and hydroxipropionitrile (HPN)/polyethyleneglycol (PEG) as a function of the bulk molar fraction is shown in Fig. 22 (Stietz, 1996).The surface tension of PEG (52 mNm-') lies between the values of FA and PEG. This should lead to segregation of PEG in the mixture with FA and depletion in the mixture with HI". This expectation is fulfilled, as seen in Fig. 22. Detailed knowledge of the composition of the topmost molecular layer creates the wish to determine the liquid composition below the surface and to learn at which separation from the surface the liquid takes on its bulk properties. For this purpose, MIES must be complemented by other methods like angular resolved photo-electron spectroscopy (ARUPS or A R X P S ) (Eschen et al., 1995) or ion scattering spectroscopy in the form of NICISS (Andersson and Morgner, 1998).

454

H. Morgner

Binary liquid mixtures of PEG 1 I

.O 0.6 --

E

0

0

0

0.4

rn

3

u)

rn

0

U

4

8

0

C

0

0 0.2 -_ 0

PEG / HPN

8 8 8

rn8

0 0

0.2

0.4

0.6

0.8

1

bulk molar fraction (PEG) FIG. 22. Surface composition measured by MIES as a function of the bulk molar fraction for the binary mixtures PEG/FA and PEG/HF". The bulk molar fraction of PEG is computed on the basis of the -0CH2CH2- unit of polyethyleneglycol.

B. SOLIDSURFACES 1. Fonnation of NaCl Layer on W(ll0); Identification of a Third Chemical Species; Evaluation of Topological Information A few years ago, Dieckhoff et al. (1992a) investigated the formation of a layer

of NaCl on a tungsten substrate. They exposed W( 1 10) to vapor of sodium chloride and monitored the changes by MIES. In the series of spectra shown in Fig. 23, the lowest one refers to clean tungsten, and the top spectrum corresponds to a surface fully covered by a closed layer of NaC1. At first glance, it seems that the development is simply described by a continuous decrease of the metal contribution and likewise a continuous increase of the salt contribution which saturates at -1 ML coverage. However, a rigorous data analysis with S V D proves that the series of spectra cannot be explained as a linear combination of only two spectra, but that three significantly different contributions build up the series (Heinz and Morgner, 1998a).The first and last spectra

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

455

MIES NaCl I W(110) (Dleckhoff et al. 1992)

4

6

8

10

12

14

16

18

20

electron energy I eV FIG.23. NaCI/W. Selection of 7 spectra out of a series of 15 MIE spectra measured by Dieckhoff et al. (1992a). The coverage of the selected spectra is 0 = 0 (bottom), 0.06,O.1.0.14, 0.18, 0.6, 1 (top).

from Fig. 23, i.e., the spectrum of pure metal and the spectrum of a closed layer of NaC1, are natural choices as reference spectra. Using the algorithm from Section B.III.3, the shape of the third reference spectrum can be reconstructed as well. It is shown in Fig. 24. Figure 25 displays the development of the relative contributions of the three reference spectra through the series: the metallic contribution drops sharply with coverage. If extrapolated linearly, it would reach zero at 8 x 0.2. The salt contribution rises steadily and almost linearly with coverage. It is interesting to note, however, that this signal shows a delayed onset at 0 x 0.04. During the initial stages

H.Morgner

456

NaCl / W(110) MlES 3.reference spectrum

3

5

7

9

11

13

15

17

19

21

electron energy I eV FIG. 24. Third reference spectrum determined from the series of NaCI/W(lIO) M E spectra.

1

0.9 0.8

0.7 0.6

0.6 0.4

0.3 0.2 0.1

0

0

0.2

0.4

0.6

0.8

1

1.2

coverage FIG.25. NaCI/W( 1 10). Weighting coefficients for the three reference spectra through the series of NaCI/W( 110) M E spectra.

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

I

457 I

He*

He*

He* He' He'

Na'

~

tungsten ~~

FIG.26. Schematic situation at the surface of tungsten partially covered by NaCI.

of NaCl adsorption, the metal contribution loses in favor of the third, i.e., the still unknown, contribution. The next task is identifying the chemical species that causes a MIE spectrum of this shape. It has been argued by Heinz and Morgner (1998a) that the third spectrum must be assigned to the interaction of He* with the border range around an NaCl island, i.e., a range that is characterized in M E S neither exclusively by metal nor by salt, but by a combined influence of both and thus can be considered as a third,independent species. Figure 26 illustrates the situation on the surface at intermediate coverage. There are still areas that are unaffected and that yield a spectrum via the RI and AN mechanism typical for the metallic surfaces. If the He* atom approaches an already formed NaCl island, then resonance ionization is prohibited and the emitted electron contributes to the typical AD spectrum of the halogen band. One has little choice but to assign the third spectrum to the interaction of He* with the border range around a NaCl island. This assumption could be backed by identifying the individual features of the third reference spectrum in Fig. 24 (Heinz and Morgner, 1998a). The peak at 13.54eV is caused by ionization of C1- via Auger deexcitation. Because this peak is distinctly more narrow than the spectrum caused by the fully developed C1- band (top spectrum in Fig. 23), it supports the notion that the chlorine ions that contribute to this spectrum do not belong to bulk NaC1. They either sit on the border of a NaCl domain or belong to an isolated NaCl molecule. The energy scale is gauged to place the Fermi energy at the excitation energy of He*Q3S) of 19.8eV. The intensity found between 14eV and 19.8eV in the third

458

H.Morgner

reference spectrum has been attributed to the metal conduction band via Auger deexcitation (AD) as the ionization mechanism. A small contribution of AD is always found in MIES of metals; cf. Section II.B.2. The fairly strong contribution of AD in the third reference spectrum is explained in the following way: if a He* atom follows a trajectory that points toward the border of an island, the atom sees on the average half of the surface covered by NaCl, whose presence prohibits resonance ionization. This enhances the chance of the projectile to reach, as intact neutral He*, not only NaCl but the nearby metallic surface as well. In other words, the approach in the “shadow” of an NaCl island increases the survival probability of He* to an extent that enhances, at the metal, Auger deexcitation that is strongly suppressed at a pure metal surface. The intensity between -7eV and 13eV in the third reference spectrum could be assigned to the RI and AN process (Heinz and Morgner, 1998a).This assumption could be backed by experimental data from Dieckhoff et al. (1992a), who have investigated the system NaC1/ W( 1 1 0) from pure metal to a fully covered surface by INS (Ion Neutralization Spectroscopy) using He+ ions of 50eV kinetic energy under grazing incidence. Again, the series of spectra could be shown by SVD to be composed of three independent reference spectra. The IN spectrum of the pure metal and of the halogen band of the closed NaCl layer are as expected (Heinz and Morgner, 1998a). The third reference spectrum of the INS series is attributed to the border range around NaCl islands. It shows strong intensity in the energy interval between -7 eV and 13eV as the MIES reference spectrum in Fig. 24. Thus all features of this spectrum are satisfactorily explained as arising from the border around NaCl islands. So far we have described the finding of a third species with electronic properties that are different from those of both the bare tungsten and the closed NaCl layer. We have pointed out that this third species can be identified with the transition range between metal and salt. In Fig. 25 we observe that near a coverage of 8 M 0.2, about 65% of the MIES signal is caused by this third species. If we assume that the MIES signal strength is roughly proportional to the fraction of the area covered by a species, we may conclude that at intermediate coverage the transition range between metal and salt dominates the surface. In addition to the mere identification of the transition range, one can try to use it for obtaining topological information. With the assumption that the NaCl islands have circular shapes, one can assess the growth of the island radius R with coverage. Let R be the averaged radius of the NaCl islands and AR the width of the island border that represents the transition range between salt and metal. If the weighting coefficients for the unaffected metal and for ~, we obtain the the NaCl area are denoted as wmeta1 and W N ~ Crespectively,

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

459

equation (Morgner and Tackenberg, 1994)

which leads to

We have no clue whether the NaCl domains form individual and even circular islands or whether they take on rather different, e.g., dendritic, shapes. But under the above assumption, Eq. (45) yields the island radius in units of the width A R . The number n of islands per area can be evaluated on the same level as n=

WNaCl

1 (46)

( R / A R ) *'

The quantities R I A R and n are shown in Fig. 27. Up to 8 M 0.2, the number of islands as well as their size increases. Beyond this coverage the islands begin to coalesce, as can be read in the decrease of their number.

NaCl islands 30

- 0.15

-

r4

a

Q

- c\:

;

.-

5

b E

--

0

.-3

0.2 0.4 0.6 0.8

1

0 1.2

cowrage FIG.27. Assumption that NaCl islands grow in circular shape. Evaluation of averaged radius R of islands and of number of islands per area. Definitions are given in the text.

460

H.Morgner

2. Saturated Alkanes Adsorbed on Solid Substrates

The M E spectra of adsorbed alkanes represent a clear example for the orientation dependence of the method. Alkane chains that are extended parallel to the surface, and thus expose their -CH2- groups to the metastable atoms, yield spectra that differ markedly from those spectra taken from alkanes that stand upright and expose their methyl end groups (Harada and Hayashi, 1989; Ozaki and Harada, 1990; Heinz and Morgner, 1997). In the quoted works it was tacitly assumed that only two different orientations (lying and standing) were to be distinguished and were correlated to two characteristic spectra. In a more recent paper (Heinz and Morgner, 1998) it was found that a third spectrum could be identified. In the meantime, the number of reference spectra for adsorbed alkanes has grown to four: I have evaluated a large number of M E spectra of long-chain alkanes (number of C atoms 2 16) adsorbed on solid surfaces under very different conditions: monolayers of alkanes on graphite at 300 K, multilayers of alkanes on graphite at T 2 115 K, and alkanethiol monolayers on Au(ll1) and Ag(ll1) at temperatures ranging from T = 120K to T = 450K. All spectra were taken repeatedly in the same machine over a period of about three years (Heinz, 1997). The shape of the transmission function appeared to be very stable; the intensity of the He*Q3S) beam and the electron detection probability were carefully controlled. Altogether, a set of about 100 spectra was analyzed. The SVD algorithm (Section III.B.2) showed that four linear independent reference spectra were needed to reproduce the whole set of spectra within the experimental uncertainty. The effort to identify some of the measured spectra as reference spectra in the sense of Section III.B.2 failed. It turned out that all four reference spectra had to be determined. Only one solution could be found. Thus, a unique set of four reference spectra has been obtained. In Fig. 28 the four reference spectra are shown. They are considered to correspond to the following situation: A: Alkanes lying in all-trans conformation with the C-C-C plane parallel to the substrate surface B: Alkanes standing in all-trans conformation and exposing the -CH3 group C: Alkanes lying in all-trans conformation with the C-C-C plane perpendicular to the substrate surface D: Alkanes in gauche conformation. In principle, many different molecular structures with gauche conformation are conceivable. Since the data analysis led to only one reference spectrum in addition to the spectra of the three orientations in all-trans conformation, one must assume that if gauche conformers turn up at all, they are present with the same variety of conformations.

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

TIl 0.2 0

o

z

4

e

--gauche

I

1 0 1 2 1 4

461

A B C D

D

-HDT/Ag 45W

Electmn Energy I eV

A B C D

FIG.28. Four reference spectra that make it possible to describe all measured MIE spectra of adsorbed alkanes. Directly measured spectra that are closest in shape to the reference spectra are shown for comparison. Their composition in terms of the reference spectra is given in the small figures on the right. A: reference spectrum for alkanes in all-trans conformation with the C-C-C plane parallel to the substrate surface, thin line: monolayer of hexadecane on HOPG at 300K B: reference spectrum for standing alkanes with the -CH3 group exposed, thin line: hexadecanethiol film/Ag( 1 1 1) at 110K C: reference spectrum for alkanes in all-trans conformation with the C-C-C plane perpendicular to the substrate surface, thin line: multilayer of hexadecane on HOPG at 1 15 K D: reference spectrum for alkanes in gauche conformation, thin line: hexadecanethiol film/Ag( 1 1 1) at 450 K.

462

H. Morgner

The identification of the reference spectra with a particular physical situation at the surface is taken from the comparison with directly measured spectra; cf. Fig. 28. The interpretation of reference spectrum D as representing the contribution of gauche conformers is taken from the observation that in all series of spectra, the weight of reference spectrum D correlates positively with temperature. For the three reference spectra A, B, and C, which are conceived as being due to alkanes in all-trans conformation, a comparison with theoretically determined spectra is instructive. The first attempt relies on an ab initio calculation of hexadecane in all-trans conformation (Richter, 1998) with the STO-3G basis set (Hehre et al., 1969). The M E spectra are evaluated as laterally averaged electron density as a function of binding energy in planes in front of the hexadecane molecule. These planes are meant to represent the positions of closest approach of the metastable helium atom and, thus, the location of highest reaction probability. In order to simulate the reference spectrum A, the “reaction” plane has been chosen parallel to the C-C-C molecular plane. The separation between both planes has little influence on the shape of the simulated spectrum. The reference spectrum C is determined within a “reaction” plane perpendicular to the C-C-C-plane. The reference spectrum B for standing alkanes has been simulated by collecting the electron density in front of the three H atoms of the methyl end group. The three simulated spectra obtained by convoluting the electron density as a function of binding energy by a Gaussian of FWHM = 0.8 eV are compared to the determined reference spectra in Fig. 29. It is obvious that the absolute peak positions in the reference spectra A, B, and C are not always met by the simulation. For example, the band gap in the simulation is too wide and the separation between the peaks at 6eV in spectrum B and at 5 eV in spectrum A is almost doubled in the simulated spectra. Apparently, far more refined calculations were needed in order to achieve quantitative agreement. However, the variation of spectral shape with molecular orientation is reproduced remarkably well. Lying alkanes have a peak maximum between 9 eV and 10eV electron energy (A, C), whereas standing molecules have less prominent structure in this energy range. On the other hand, the peak maximum at 6 eV in spectrum B is absent in A and weakly present in spectrum C. Spectrum C has a double peak structure (5 eV and 6 eV), whereas spectra A and B have strong peak maxima at either of these energies. Qualitatively, these features are fairly well reflected in the simulated spectra. A more quantitative description of the three reference spectra A, B, and C can be given within a semiempirical theory that has been used by Zubragel (1995) in order to evaluate the band structure of polyethylene from ARUPS data. The theory is called equivalent orbital theory and dates back to LennardJones (1949) and Hall (1951, 1958). The following short description of the theoretical concept follows the work of Zubragel(l995). The alkane chain is

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

463

bidllrp emw/eV

MIE8 nhnm(or .Ymlylw prpndlcu*r

e*dronemfgy/eV 4

IJ-

.l

0

n1 .m1

1

4

8

8

.171

.IS(

.la1

1

111

0

1

.B+

2

(

4

.I(

bidirp anew / eV

FIG.29. M E reference spectra of alkanes in all-trans conformation. Comparison between reference spectra evaluated from experimental data (thick line) and laterally averaged electron density evaluated from STO-3G ab initio calculation (open circles). The binding-energy scale from the ab initio calculation is adjusted with respect to the experimental energy scale in order to let the peak in spectrum B at 6 eV electron energy coincide with the corresponding peak in the simulated spectrum.

464

H. Morgner

schematically represented as a sequence of CH2- groups: H1

I

-c I

H2

HI’

H1”

l

l

-C’

-C”

l

l

H2’

H2”

HI”’

I

-C”’-

I

H2’”

The electronic structure is built up with localized orbitals that describe the bond between two atoms. Here one has the three different orbitals ICHl), 1 CH2), and 1 CC) and their equivalent orbitals along the hydrocarbon chain. The interaction matrix elements between these orbitals are the parameters of the semiempirical treatment. Two diagonal matrix elements exist: a:= (CHIHolCH) c:= (CC’IHolCC’) The off-diagonal matrix elements are b: = (CHlIVICH2) d: = (CHIVICC’) e: = (CC’ IV I C’CC”) f: = (CH1 IVIC’H2’)

g : = (CHI IVIC’Hl’) h:= (CC’lVlC”C’’’) k: = (CH’ I V I C’C’’)

Due to the zig-zag arrangement of the carbon atoms in alkanes, the unit cell contains two CH2- groups. With 1 being the length of this unit cell, Zubragel (1995) defines a phase 0 that is related to the wave vector k via 8 = k . The secular equation for a given phase 0 is then derived as

A.

C+

2e cos 8

+ 2h cos 20-

E 2d COS!

+ 2kcos?

+~

~ C 0O-SE

2dcos:+2kcos$ 2dcos~+2kcos~

u

b + 2 f Cose

2d C O S !

+ 2k COSY

b+2fcose u 2g cos 0 - E

+

=o

The solution of this equation yields three eigenvalues Ei(0) with 0 E [0,2x]. In view of the two CH2- groups per unit cell, it is common to fold the bands Ei(0) back at 0 = x into the first Brillouin zone, which leads to six eigenstates. The ARUPS data measured by Zubragel (1995) are given in an extended zone scheme. The fit of the bands Ei(0) to these data determines the nine parameters u, . . . ,k. Because of apparent misprints in the work of Zubragel

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

465

TABLE VIIl MATIUXELEMENTS WlTHlN a

-16.486

b

-2.261

THE EQUIVALENT ORBlTAL THEORY, DESCRIBING THE VALENCE BANDSOF POLYETHYLENE.

e

f

-1.326

0.611

d

C

-16.635

-2.092

g

-0.556

h

k

0.563

-0.488

(1993, I have repeated the fitting procedure. The parameters are given in Table VIII, and the fitted bands are shown in Fig. 30, together with the experimental data by Zubragel (1995). In addition to the eigenvalues Ei(0), one can determine the analytical expressions for the roots, which are given as amplitudes for orbitals CHI, CH2, CC' and denoted by aFc'(0),aFH'(0),

afH2 (0). With the definition that El (0) represents the C2po band, Ez(8) the C2pn band, and E3 (0) the C2so band, we obtain for the coefficients a the following expressions. For i = 2, i.e., for the C2pn band, the result is particularly simple:

a y

= 0;

For the two o-bands, labeled i = 1 and i = 3, one gets

aFH1(0) =

aCH2

(e) =nOm1

where m12

0 30 =2dcos-+2kcos2 2 = a 2gcos0

+ m23 = b + 2f cos 0 m22

The calculation of the M E spectra within the above sketched equivalentorbital theory follows the procedure from Heinz and Morgner (1998). For

H.Morgner

466

Band Structure of Polyethylene

...-._. Pi o

-C(2s)

sigm ARUPS

-1 1

-13

-15

3 \

-17

ocal

r

-21

-23

-25

-27 4

ri

I

x1

l-2

x2

FIG. 30. Valence bands of polyethylene. The lines represent the analytical expressions within the equivalent orbital theory. The parameters are fitted to the experimental ARUPS data (open circles) by Zubragel(l995).

every band i = 1,2,3, the probability P i ( E ,0) for the occurrence of energy E at phase 8 is constructed. The band-specific density of states is then given as DOS~(E)=

J

2rr

d e Pi(E,e> 0

The MIE spectra are then constructed by multiplying P i ( E , @ ) with a @-dependentweighting factor Wi (0) that takes into account the interference

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

467

between the amplitudes from different sites (Heinz and Morgner, 1998). In this paper the x band of graphite has been treated within a model that would be built up by only one type of orbital within the equivalent-orbital theory. Here we have to deal with three orbitals, which requires some attention. The CC' orbitals are buried inside the alkane molecules, whereas the CH1 and CH2 orbitals are pointing to the outside. Therefore, one has to project out only the contributions from the latter orbitals in order to get the M E spectra. All three bands contain CH orbitals, and thus, they all contribute to the M E spectra. This finding is in contradiction to a paper by Heinz and Morgner (1997), which is based on the assumption that the band between 3 eV and 11eV originates only from ionization out of the C2pn band. This notion must obviously be corrected according to both theoretical treatments. Because the amplitudes of CHI and CH2 always have the same absolute value, it suffices to consider CH1. Of course, the interference between CHI and CH2 will be different for the C2pn band (a;"' = -agH2)compared to the CT bands (a:"' = -a:"'). For the present purpose this is unimportant, because the @dependenceof the weighting factors Wi(0)reflects the interference not within CH2- groups but between different CH2- groups. The part of the MIE spectra of alkanes above an electron energy of 3 eV is composed of contributions from the C2pn and C2po bands. The spectrum of alkanes lying with their C-C-C plane parallel can be simulated as

and the same expression for alkanes lying with their C-C-C plane perpendicular to the substrate surface reads

Following Heinz and Morgner (1998), the @dependence of the weighting factors is described by

w(e)

O(

exp(-y'.

e2)

For both spectra, SA and SC, two parameters y' can be varied in order to achieve agreement with the reference spectra A and C. Figure 3 1 shows that this attempt is successful. The values of the parameters y' are shown in Table IX. The data fit well into the concept developed by Heinz and Morgner

H.Morgner

468

electron energy I eV 4

-

2

0

2

4

6

8 1 0 1 2 1 4

Reference spectrum A

-25.3 -23.3 -21.3 -19.3 -17.3 -15.3 -13.3 -11.3 -9.3

-7.3

binding energyIeV electronenergy I eV 4

-

2

0

2

4

6

8

I 0 1 2 1 4

t Reference spectrum C

-25.3 -23.3 -21.3 -19.3 -17.3 -15.3 -13.3 -11.3 -9.3

-7.3

bindingenergyIeV

FIG.31. Comparison between the reference spectra A (alkanes with C-C-C plane parallel to surface) and (alkanes with C-C-C plane perpendicular to surface) shown as thick lines and the fitted spectra S, and SC plotted as open circles. The energy positions of all peaks are met with good accuracy. The contributions of the C2pn band (thin line) and the C2po band (dotted line) are shown separately.

(1998) that y ’ measures the degree of interference between neighboring site orbitals. y’ is expected to increase with the size of the site orbitals. Because the average binding energy of the C 2 p o band is smaller than found for the C 2 p x band, one expects the site orbitals contributing to C2pa to be slightly more diffuse, leading to larger values of y‘. y’ is further influenced by the distance between site orbitals. The smaller their spacing, the stronger should be the effect of interference, and thus, the

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

TABLE IX VALUES OF PARAMETERS y’

DETERMINED FROM

FIT TO REFERENCE SPECTRA

469

A AND c.

Reference Spectrum

C2px Band

C2pa Band

A: Alkanes lying with C-C-C plane parallel to surface C: alkane lying with C-C-C plane perpendicular to surface

y‘ = 0.35 y‘ = 0

y’ = 0.72

y‘ = 0.42

FIG.32. Alkanes lying flat on a substrate surface. If the C-C-C plane is perpendicular to the surface, the spacing between neighboring site orbitals is more than 60% larger than that for the C-C-C-plane parallel to the surface.

larger y’ should be. This is indeed born out by the values obtained for reference spectra A and C.The y’ values for spectrum A are noticeably larger, indicating enhanced interference compared to spectrum C.This observation correlates well with the spacing between neighboring site orbitals; cf. Fig. 32. The reference spectrum for standing alkanes, spectrum B, can be simulated within the equivalent-orbital theory as well. The matrix elements fitted to the ARWS data from Zubragel(l995) (cf. Table VIII) are employed to build up the energy matrix of an alkane of finite length, namely hexadecane. A finite alkane has a terminating methyl group that carries a third H atom. Thus, a new orbital has to be introduced as CH3.The diagonal matrix for this orbital is (CHlIHolCHl)=(CH2IHoICH2). This chosen to be (CH3)HoICH3):= leaves only one unknown off-diagonal matrix element m:= (CH3 V I C’C”).

H.Morgner

470

-25.3 -23.3 -21.3 -19.3 -17.3 -15.3 -13.3 -11.3

-9.3

-7.3

binding energy I eV FIG. 33. Simulation of MIE spectrum of standing alkanes within the equivalent-orbital theory (open circles). Comparison with reference spectrum B (thick line) indicates good agreement.

Inspection of the topology of the molecule in all-trans conformation suggests that this matrix element should have a value that lies between matrix elementsf := (CH1 I V I C’H2’) and h := (CC’ I V I C”,”‘). I have employed m = (f . h) 1/2 as a trial value. Variation of this matrix element by up to 60% did not change essential features of the simulated spectrum. Thus the value

m := (CH3 I V I C’C’’) = (f . h)’I2 = 0.586 eV has been used. The spectrum has been evaluated by taking the sum of the squares of the ~ ~ ~IuCH3l2 ~ at the methyl end group. After conamplitudes I c I ~1ctCH2I2 volution with a Gaussian of FWHM = 0.8eV, the simulated spectrum is compared to the reference spectrum C in Fig. 33. The most marked features of spectrum C are reproduced: the peak maximum at 6 eV, the steep descent on its low-energy side, and the absence of a peak between 2 eVand 3 eV. Only the energy range above 10 eV is not fed with intensity. In summary, the comparison to two differnt theoretical treatments indicates that the interpretation of the first three reference spectra A, B, and C as being due to three characteristic orientations of alkane molecules in all-trans conformation appears justified. At the same time, this supports the interpretation of reference spectrum D as being caused by alkanes in gauche conformation.

+

+

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

471

It is worthwhile to note that neither of the reference spectra is the result of a direct measurement but that all four reference spectra have been determined by advanced mathematical data processing of a large number of measured spectra. Once the four reference spectra are established, it is possible to check how perfect the best experimental preparation of samples has been. The most marked deviation is found between spectrum B, the reference spectrum for methyl-terminated standing alkanes, and the spectrum of hexadecanethiol/ Ag( 1 1 1) at a temperature of 110K, which had been thought - on the basis of careful sample preparation - to represent the spectrum of standing alkanes with great accuracy. The analysis of the measured spectrum with the aid of the four reference spectra shows that only 80% of the surface is actually covered by upright -CH3 groups. The four reference spectra can be used to characterize quantitatively the top layers of alkane films as long as no additional component is present and the substrate does not shine through. As an example, the behavior of a self-assembled monolayer of hexadecanethiol (HDT) on Ag(ll1) is monitored under variation of temperature (Heinz, 1997).The freshly prepared film is let into the UHV chamber at room temperature (300K). M E spectra of the sample are taken during cooling down to 110K, during warming up to 370 K, and renewed lowering of the temperature down to 110 K. The composition of the topmost layer as evaluated from M E S is shown in Fig. 34. The fraction of the surface contributing to reference spectrum B (standing alkanes, terminated by upright methyl groups) is largest at 110 K, but only after previous annealing to 370 K. The fraction of the surface covered by alkanes in gauche conformation (reference spectrum D) correlates reversibly with temperature. A different behavior is observed for an HDT film on Au(ll1); cf. Fig. 35. The same temperature program has been MIES of HDTIAg(ll1). Influenceof temperature

"'T

0.1

Od

0

05

FIG.34. S A M of the HDT/AG(l 1 1). Evaluation of series of MIE spectra during temperature variation. The largest contribution of upright - CH3 groups and, thus, the most perfectly ordered film surface is obtained at 1 10 K only after annealing to 370 K. The data are from Heinz ( 1997).

H.Morgner

472

MIES of HDT/Au(lIl). Influence of temperature

I Oe

T

O'

0.2

04

0.1

03

0

02

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

Y

~ H ~ R ~ 8 Q P ~ a W R ~ H R 4 ~ 8 ~ 8 FIG.35. SAM of HDT/Au(lll). Evaluation of a series of M E spectra taken during temperature variation. Exceeding T = 325 K leads to an irreversible change at the film surface. The data are from Heinz (1997).

applied as for the previous film (Heinz, 1997). From the start, the amount of gauche conformations is rather large. Upon exceeding the temperature of 325 K, the amount of gauche conformations in the topmost layer increases irreversibly to almost 40%. Concomitantly, the surface area covered with upright CH3 groups drops. The last spectrum, even though taken at llOK, shows almost the smallest contribution from reference spectrum B within the whole series. The difference in behavior between HDT/Ag( 1 11) and HDT/Au(1 1 1) can be explained by the different desorption temperature. The X P S derived layer thickness of HDT films on both substrates is compared in Fig. 36. Obviously, the maximum temperature during film treatment is fairly close to the desorption temperature of HDT/Au( 1 1 1) but safely apart from the desorption of HDT/Ag( 1 1 1). Thus, a temperature of about 370 K leads to annealing of HDT/Ag(l 1 l), but to loss of material in the case of HDT/Au(l 1 l), which in turn causes disorder at the topmost layer.

3. Eflect of Ion Bombardment on a Self-Assembled Alkanethiolate Film; Ident$cation of Reaction Products Modification of surface properties by ion bombardment has been used with inorganic and organic films (Fuchs et al., 1991). In a recent study (Chenakin et al., 1998), the ion-induced changes at the surface of a hexadecanethiol film on Ag( 1 1 1) have been followed by several techniques, including MIES. At the end of the process, the alkanes are almost completely sputtered away, exposing the silver substrate with some sulfur contamination. Figure 37 shows a series of MIE spectra taken during sputtering by 800-eV He+ ions for a dose range below 7 1014ion/cm2, i.e., at doses sufficiently large to visibly modify

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

473

Effect of temperature on HDT films 2.5

f 2 0

L1.5 \

3 $ 1

3

'

0.5 0 275

375

475

575

temperature I K FIG. 36. Thickness of HDT/Ag( 1 1 1) and HDT/Au( 1 1 1) derived from XPS data (Heinz 1997). The desorption temperature of the HDT/Au( 1 1 1) film is lower by about 50 K compared to the film on silver. From Heinz (1997).

the organic film but sufficiently small to leave the surface entirely covered by hydrocarbons. I have tried to analyze this series of spectra by means of the four spectra A, B, C, and D established as reference for the three possible orientations of saturated alkanes in trans conformation and in gauche conformation. The linear combination of the four reference spectra is not able to reproduce the measured spectra except for the first one, i.e., the spectrum of the intact hexadecanethiol film. Indeed, the SVD algorithm reveals that a fifth contribution is hidden in the series. The unknown fifth spectrum can be determined in the way described in Section III.B.3. The resulting spectrum is shown as upper curve in Fig. 38. The uncertainty computed by the algorithm is indicated by error bars. The question is now whether one can assign this spectrum to a chemical species at the surface. In addition to the intensity between 4 eV and 11 eV, familiar for alkanes, one observes in the spectrum two peaks: at about 3 eV (which falls into the band gap of saturated hydrocarbons) and near 11eV (which lies at an energy where the intensity of saturated hydrocarbons begins to vanish). These peaks are familiar from spectra of alkanes with double bonds. For comparison, a MIE spectrum of decadiene (gray line) is shown in the lower part of Fig. 38. In order to emphasize even more markedly the features that are specific for double bonds, we have calculated the difference between the spectrum of a monolayer of decadiene and 80% of the spectrum of a monolayer of octane: S[decadiene] - 0.8 S[octane]. The prefactor of S[octane] has been chosen as large as possible without causing negative

474

H. Morgner HexadecanethioVAg(ll1) ion bombardment

0 sec 140 sec 419 sec 529 sec 790 sec I I

0

2

,

4

,

,

6

.

I

.

,

,

,

,

t 1052 sec

8 1 0 1 2 1 4

Electron Energy I eV

FIG. 37. Effect of ion bombardment on a hexadecanethiol film on A g ( l l 1 ) measured by MIES. The He+ ions have an energy of 800eV, the current density being 0.1 1 pAm-2. Data from Chenakin et al. ( 1998).

values in the difference spectrum, hence reducing the contribution from single bonds substantially. The only prominent features in the difference spectrum are the two peaks at 3.1 eV and 11.3eV, very close to the energy positions of the peaks found in the fifth spectrum. Even though deviations between the spectra are clearly recognizable, it appears safe to assume that the fifth spectrum represents the occurrence of double bonds in the organic film in the course of ion bombardment. This is not unexpected; the preferential sputtering of hydrogen from organic films is known (Wittmaack et al., 1987). Removal of hydrogen leads to the formation of radicals, which then relax chemically via double-bond formation. It is interesting to note that triple bonds can be distinguished in MIES from double bonds in that they display a peak near 11 eV but no peak near 3eV (Heinz, 1997). One may conclude that in the ion dose range studied,

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

0

2

4

8

8

475

1 0 1 2 1 4

EleCtrOflEmrgyleV

FIG.38. Fifth spectrum determined from the MIES series of sputtered hexadecanethiol film (top). The bottom spectra show the M E spectrum of decadiene (gray line) and the difference spectrum S[decadiene] - O.I*S[octane]. The comparison suggests that the fifth spectrum originates from hydrocarbon double bonds. Further explanation in the text. From Heinz and Morgner (1998a).

the formation of double bonds at the surface of the hexadecanethiol film is dominant with only small probability for triple-bond formation. Figure 39 shows how the contributions of some of the reference spectra vary as a consequence of ion bombardment. The intact film ( t = 0) is analyzed to be 76% standing alkanes, i.e., the film is terminated mostly by -CH3 groups. Through the series this contribution decreases continuously. Gauche conformations appear at the surface at small ion dose but die out under further ion bombardment in favor of double bonds. At the end of the series, the surface appears to be predominantly covered by double-bonded hydrocarbon groups. The amount of signal due to lying alkanes rises swiftly from 15% in the intact film to about 35% at 3 . 1014ion/cm2 and then remains fairly constant. Thus, the main ion-induced effect at the top surface layer in the dose range above 3 lOI4 ions/cm2 appears to be the formation of double bonds.

H.Morgner

476

MlES of HDT/Ag(l 11)

mder He'4on bombardment o'8 0.7

A

-m-

standing

4- double bond

P 0.3

'g

.F

g

0.2

0.1

0

-I

0

2E+14

4€+14

8E+14

E+14

ions per cmz FIG.39. Effect of ion bombardment on a hexadecanethiol film on Ag(l11) measured by MIES. The current density is 0.11 pA crn-'. The development of the relative contributions of the reference spectra of standing alkanes (spectrum B), of gauche conformers (spectrum D), and of double bonds (spectrum E) is given. The contribution from lying alkanes (spectra A and C) growths from about 15%to about 35%. From Heinz and Morgner (1998a).

4. Adsorption of Oxygen on Ni(lO0) A Ni( 100) surface that is exposed to oxygen at ambient or elevated temperature covers itself with chemisorbed atomic oxygen up to a coverage of 0 M 0.4. Concomitantly, the sticking coefficient drops to very low values. Only if the oxygen exposure is raised by an order of magnitude or more does the sticking coefficient go up again and the further uptake of oxygen lead to oxidation (Brundle and Broughton, 1990). Therefore, it is easily possible to experimentally prepare a Ni( 100) surface with maximum coverage by chemisorbed atomic oxygen, with no onset of oxidation. Figure 40 shows selected spectra from a series of 30 M E spectra taken during chemisorption of oxygen on Ni(lO0) at T = 200°C (Kubiak, 1998). The SVD analysis yields the result that three linear independent spectra contribute. to the series. The natural choice for two of the reference spectra is obvious: the spectrum of pure Ni( 100) (reference spectrum A) and the spectrum of maximum coverage (reference spectrum B). The third reference spectrum C as determined by Kubiak (1998) is shown in Fig. 41. The whole series Si can be reproduced as a linear combination of the three reference spectra A, B, and C, Si = ~

l * iA

+ pi B + yi *

*

C

(47)

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM MIES of OMi(lO0) T=200°C

\ e0.38 1

:

:

:

:

r

5

9

:

:

11

:

:

:

;

13

:

:

16

:

:

17

I

19

energyIeV

FIG. 40. Selected spectra from MIES series of OINi(lO0) taken at T = 200°C.

Reference Spednm C

T

4

0

8

10

12

14

10

ElmbunEmtgy/aV

FIG.41. Reference spectrum C for MIES series of O/Ni(lOO).

18

a0

477

478

H. Morgner MIES of O/Ni(100) T=200°C 1.o

1

'

0.8

0.6

A : Ni(lO0) meas.

6 :OIFS(100) meas. 0.4

A

C :O/Ni(lOO) reconstr.

0.2

0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.90

0.35

0.40

cowregee pJIy

FIG.42. Development of the weighting coefficients of reference spectra A, B, and C with oxygen coverage.

within experimental accuracy. The development of the weighting coefficient a;, pi,and 7; with oxygen coverage is given in Fig. 42. Their behavior is well described by the analytical expressions

m(8) = exp(-u. 8 - b .€Ic)with u = 8.6, b = 446, c = 3.2 p(8) = 1 - exp(-u. 8 - b . O C ) with u = 0.445, b = 120, c = 3.0 r(8)= 1 - 4 8 ) - P(9) The initial slope of the coefficient of the metallic contribution is

1%

(8 = 0)I

= 8.6. This indicates that at low coverage, any additional oxygen atom blocks

the direct contact between the impinging He*p3S)atom and about 9 Ni atoms. Now the question arises how to identify the species that causes spectrum C. The steep onset of the related coefficient y(8) at low coverage indicates that spectrum C must be associated with chemisorbed oxygen in an isolated position, whereas spectrum B reflects a situation in which adsorbed oxygen atoms are close to each other. In the spirit of Section III.B, it would be desirable to assign a specific local situation at the surface to every one of the three reference spectra. In order to obtain a better insight into the lateral arrangement of oxygen atoms chemisorbed on Ni(l00), Kubiak (1998) has carried out a computer simulation. The adsorption sites in the neighborhood of a chemisorbed 0 atom are denoted by numbers; cf. Fig. 43. The adsorbed oxygen whose

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

479

FIG.43. The site of the adsorbed oxygen atom is named 0. Site 1 represents the next neighbor, which is strictly forbidden.Occupationof sites 3 and 5 without neighborson sites 1,2, and 4 yields a p ( 2 x 2) pattern in LEED, whereas occupation of sites 2, 3, and 5 leads to a c(2 x 2) LEED pattern.

neighborhood is described sits in site 0. In the simulation, site 1 is considered as strictly forbidden, in agreement with experimental evidence. The energies of lateral interaction with 0 atoms on sites 2 and 3 are employed as parameters in the calculation. These two parameters have been varied (Kubiak, 1998) in order to reproduce the development of the experimental LEED data through the whole range of coverage and the site occupation distribution determined by Kopatzki and Behm (1991) for 8 = 0.16 with STM. Kubiak (1998) got a best fit to the experimental data with the interaction energies of -0.020 eV for site 3 and 0.1 10eV for site 3. The simulation shows that at 8 = 0.25 the surface is overwhelmingly covered by 0 atoms in a p ( 2 x 2) pattern, i.e., with neighbors in sites 3 and 5 and without neighbors in sites 1, 2, and 4. The fraction of 0 atoms with this neighborhood is close to unity. Deviations are mainly due to antiphase boundaries. Once the lateral interaction parameters between chemisorbed oxygen atoms are established, it is possible to calculate the probability for the occurrence of any local pattern of oxygen atoms one is interested in. Kubiak (1998) has searched for characteristic local arrangements whose probability develops with coverage as closely as possible to any of the coefficients a, p, and y. He found the following way to describe characteristic adsorption patterns: the squares between four Ni atoms, whose area represents one unit cell of the Ni(l00) surface, are inspected, including their immediate neighborhood of eight adjacent unit cells. Kubiak has identified the characteristic situations shown in Fig. 44. The patterns are labeled by the same letters as the corresponding reference spectra.

A: The central site and the eight neighboring sites are unoccupied by oxygen. Then the central site contributes to the reference spectrum of pure metal.

480

H. Morgner

A

M C

B

FIG.44. Local arrangementson O/Ni( 100) surface that contributeto reference spectra A, B, and C. The filled circles indicate the position of Ni atoms; the open circles represent adsorbed oxygen atoms. The arrangements shown and their equivalents represent the respective definitions (48).

C: Only one adsorbed 0 atom is found either on the central or on the adjacent sites. Then the area of the central site contributes to reference spectrum C. This is the more precise definition of the situation of an isolated oxygen atom. B: At least two adsorbed 0 atoms are found on the nine sites considered. Then the area of the central site contributes to reference spectrum B. The contribution that fulfills this requirement saturates long before the c(2 x 2) coverage is completed, in agreement with experimental data; (48) cf. Fig. 42. The comparison between the weighting coefficients from Fig. 42 and their counterparts determined with the above definitions from the computer simulation is given in Fig. 45. The above considerations on the MIES series of O N ( 100) have provided new insight into the feasibility of quantitative data evaluation. The determination of the reference spectra A, B, and C and their respective weighting coefficients ui,pi. and yi has been straightforward and, thus, satisfying from a mathematicalpoint of view. However, the identification of the species that are to be related to the reference spectra required some effort. Only the incorporation of additional information (experimental LEED data and the related computer simulation) allowed us to come up with a satisfying definition such that all reference spectra can keep their physical meaning throughout the whole series of MIE spectra. The main result consists in establishing the

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

48 1

MIES of O/Ni(lOO) I simulation 1.0

0.8

f

0.6

A : Ni(lO0) meas.

8

a B :oMi(lO0)meas.

0.4

0.2

0.0 0.w

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

commQef3[ML]

FIG.45. Comparison between the weighting coefficients evaluated from MIES (symbols) and the probabilities (full lines) that the sites on the surface fulfill the definitions (48). From Kubiak (1998).

relation between the shape of the MIE spectra and the related situations at the surface. It is interesting to devote some more attention to the results. The question of what is the minimum size of an uncovered area in order that it retain its metallic surface density of states is answered by pattern A in Fig. 44. A more precise definition of an isolated oxygen adatom is found in pattern C. Our initial expectation that the membership of adatoms to p(2 x 2) or c(2 x 2) domains would largely govern their behavior in MIES has apparently not been confirmed by the data analysis. The p(2 x 2) LEED pattern takes on its maximum value at 8 = 0.25, but this is not accompanied by a noticeable feature in MIES; cf. Fig. 42. Obviously, the long-range order investigated by LEED and the local density of states probed by MIES are not necessarily closely related to each other. The MIES spectra pertaining to specific surface situations can be considered as being due to a homogeneous surface in the sense of Section II.B.2. Hence, it is justified to deconvolute these spectra in order to monitor the variation of the corresponding surface density of states. The “weighted” density of states U ( E )according to Eq. (19) is displayed in Fig. 46 for all three reference spectra. For clean Ni(100), one observes a steep onset at the position of the Fermi edge. The “weighted” DOS U ( E )in the case of high coverage displays strong intensity between 4 eV and 8 eV binding energy which is assigned to the oxygen band. For low coverage, the intensity near the

H.Morgner

482

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

Energy - EFermi / eV FIG.46. MLES of OINi(l00). “Weighted” density of states U ( E )computed via deconvolution of the reference spectra A, B, and C. From Kubiak (1998).

Fermi edge is clearly diminished, which indicates that the adsorbed oxygen atoms prohibit the interaction between the helium atom and the metal states. On the other hand, there is no clear hint that the oxygen orbitals are involved in the AN process.

V. Summary MIES (Metastable Induced Electron Spectroscopy) combines spectroscopic information with perfect surface sensitivity. The ability to spin-polarize the metastable atoms by optical pumping opens access to the characterization of magnetic surface properties. In spite of its advantages, the method is used only by a small minority of surface scientists. This may be caused by the difficulty of turning the experimental data into quantitative information on

LIQUID/SOLID SURFACES WITH METASTABLE HELIUM

483

the surface. The quantitative treatment of data could have two goals: (1) the explicit relation between measured spectra and the electronic structure of the sample and (2) the use of reference spectra as fingerprints for known species at the surface and, thus, the quantitative analysis of the surface composition. In the present contribution, we see that with respect to both directions, some improvement in quantitative evaluation of MIES data has been made in the last few years.

VI. Acknowledgments This article would not have been written without the persistent encouragement by Mitio Inokuti. Discussions with A. Niehaus and V.Staemmler were very helpful in clarifying aspects of electronic reaction mechanisms. H. Hotop and A. Niehaus have suggested several improvements. The author owes a lot to the members of his research group, in particular to B. Heinz, R. Kubiak, and J. Oberbrodhage. The work described has been made possible by financial support from the Deutsche Forschungsgemeinschaft via specific projects, via the Graduiertenkolleg “Dynamik an Festkorperobedachen - Adsorption, Reaktion und Katalyse”, the Graduiertenkolleg “Struktur-Dynamik-Wechselwirkung an mikrostrukturierten Systemen” and via the Schwerpunkt “Transportprozesse an fluiden Phasengrenzen” and by the support from the Bundesministerium fur Forschung und Technologie.

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484

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Brutschy, B., Haberland, H., and Schmidt, K. (1976). J. Phys. B 9, 2693. Butler, W. H., Or6, D. M., Dunning, F. B., and Walters, G. K. (1992). Surf: Sci. 261, 382. Canepa, M., Cantini, P., Mattera, L., Narducci. E., Salvietta, M., and Terreni, S. (1995). Surf: Science 322, 27 1. Cerrnik, V. (1966). J. Chem. Phys. 44,3781. Chenakin, S . P., Heinz, B., and Morgner, H. (1998). Surf: Science 397, 84. Child, M. S. (1974). Molecular collision themy. Academic Press (New York). Ching, W. Y., Gan, F., and Huang, M.-Z. (1995). Phys. Rev. B52, 1596. Conrad, H.. Ertl, G., Kiippers, J., Sesselmann, W.. and Haberland, H. (1982). Surf: Science 121, 161. Conrad, H., Doyen, G., Ertl, G., Kiippers, J., Sesselmann, W., and Haberland, H. (1982a). Chem. Phys. Lett. 88, 281. Conrad, H. G.. Ertl, G., Kiippers, J., Sesselmann, W., Woratschek, B., and Haberland, H. (1982b). Surf: Science 117,98. Cris& N., and Doyen, G. (1987). Surf: Science 189/190,984. Dehnbostel, C. P., Ludviksson, A., Huang, C., Jghsch, H. J., and Martin, R. M. (1992). Surf: Science 265, 305. Dieckhoff, S., Maus-Friedrichs, W., and Kempter, V. (1992). Nucl. Instr: & Merh. in Phys. Res. B65,488. Dieckhoff, S., Miiller, H., Maus-Friedrichs, W., Brenten, H., and Kempter, V. (1992a). Surf: Science 279, 233. Dietter, J. (1997). Ph.D. thesis, University Witten-Herdecke. Dose, V., Fauster, Th., and Gossmann, H.-J. (1981). J. Comp. Phys. 41, 34. Doyen, G. (1995). Private communication. Drakova, D., and Doyen, G. (1994). Phys. Rev. B 49, 13787. Dunning, F. B., Nordlander, P., and Walters, G. K. (1991). Phys. Rev. B 44, 3246. Ebding, T. (1976). Ph.D. thesis, University Freiburg. Ebding, T., and Niehaus, A. (1974). Z. Phys. 270.43. &ken, P., Fluit, J. M., Niehaus, A., and Urazgil’din, I. (1992). Surf: Science 273, 160. Eschen, F., Heyerhoff, M., Morgner, H., and Vogt, J. (1995). J. Phys.: Condens. Matter 7,1961. Fouquet, Peter (1 996). Diplom-Thesis. Fouquet, P., Day, P. K., and Witte, G. (1998). Surf: Science 400,140. Fuchs, V., and Niehaus, A. (1968). Phys. Rev. Lett. 21, 1136. Fuchs, H., Ohst, H., and Prass, W. (1991). Adv. Matel: 3, 10. Goy, W., Kohls, V., and Morgner, H. (1981). J. Electr: Spectr: Rel. Phen. 23, 383. Goy, W., Morgner, H., and Yencha, A. J. (1981a). J. Elect,: Spectr: Rel. Phen 24, 77. Haberland, H., Lee, Y. T., and Siska, P. E. (1981). In J. W. McGowan, (Ed.), The excited state in chemical physics. Part 2 (pp. 487-585). Wiley (New York). Hagstrum, H. D. (1954). Phys. Rev. 96,336. Hagstrum, H. D. (1966). Phys. Rev. 150,495. Hagstrum, H. D. (1979). Phys. Rev. Lett. 43, 1050. Hall, G. G. (1951). Proc. Roy. Soc. A205,541. Hall, G. G. (1958). Philos. Mag. 3, 429. Hammond, M. S., Dunning, F. B., Walters, G. K.,and Prinz, G. A. (1992). Phys. Rev. B 45, 3674. Harada, Y. (1990). Pure & Appl. Chem. 62,457. Harada, Y.,and Hayashi, H. (1989). Thin Solid F i l m 178,305. Harada, Y., and Ozaki. H. (1987). Jap. J. Appl. Phys. 26, 1201. Harada, Y., Ohno, K.. and Mutoh, H. (1983). J. Chem. Phys. 79, 3251. Harada, Y., Ozaka, H., and Ohno, K. (1984). Surf: Science 147, 356.

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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL, PHYSICS, VOL. 42

QUANTUM COMMUNICATION WITH ENTANGLED PHOTONS HARALD WEINFURTER Sektion Physik, LMU Miinchen, Miinchen, Germany Max-Planck-lnstitut fir Quantenoptik, Garching, Germany

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Entanglement: Basic Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Quantum Communication via Entangled States . . . . . . . . . . . . . . . . A. Quantum Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Quantum Dense Coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Quantum Teleportation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. The Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.SomeRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. The Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Entangled Pairs of Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Parametric Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Entanglement Produced by Parametric Down-Conversion. . . . . 2. Polarization Entangled Pairs from Qpe-Il Down-Conversion . . 3. Pulsed Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Interferometric Bell-State Analysis . . . . . . . . . . . . . . . . . . . . . . 1. The Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Bell-State Analysis of Independent Photons . . . . . . . . . . . . . . D. Transformation, Manipulation, and Detection of Single Photons. . V. The Quantum Communication Experiments . . . . . . . . . . . . . . . . . . A. Quantum Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Quantum Dense Coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Quantum Teleportation of Arbitrary Quantum States . . . . . . . . . VI. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

490 492 494 495 497 498 498 50 1 503 505 505 506 508 512 513 513 516 518 519 519 520 524 528 530 530

Abstract: Quantum entanglement lies at the heart of new proposals for quantum communication and computation. For a long time it was seen just as one of those fancy features that make quantum mechanics so counterintuitive,but, recently, the new field of quantum information theory showed the tremendous importance of quantum correlations for the formulation of new methods of information transfer and for algorithms exploiting the capability of quantum computers.Whereas the latter applicationrequires entanglement among a large number of quantum systems, the basic quantum communication schemes rely only on entanglement between the members of a pair of particles, directly pointing at a possible realization of such schemes by means of correlated photon pairs as produced by parametric down-conversion. In the present work we report on the fmt experimental realizations of quantum communication schemes using entangled photon pairs. We describe how to make 489

Copyright 0 Zoo0 by Academic Press AU rights of reproduction in any form r e ~ e r ~ e d . ISBN 0-12-0038424lISSN 1049-25OXl00$30.00

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Harald Weinfurter communication secure against eavesdropping by using entanglement-based quantum cryptography, how to increase the information capacity of a quantum channel by quantum dense coding, and finally how to communicate quantum information itself in the process of quantum teleportation.

I. Introduction Quantum mechanics is probably the most successful physical theory of this century. It provides powerful tools, which form one of the cornerstones of scientific progress and which are indispensable for understanding such omnipresent technical devices as the transistor, semiconductor chips, and the laser. The most important areas where those devices are used are all kinds of modem communication and information-processing technologies. But until now, quantum mechanics has been used only to construct these devices; quantum effects are absolutely avoided in the representationand manipulation of information. Rather than using single photons, one still uses strong light pulses to send information along optical high-speed connections, or instead of single electrons, one relies on electrical currents in semiconductorlogic chips. Of course, the inherent stochastic character of quantum effects does not at a first glance recommend their use. Quantum information theory shows us, in more and more examples, how fundamental quantum effects can add to the power and features of classical information processing and transmission (Bennett, 1995). For example, quantum computers outperform conventional computers, and quantum cryptography makes truly secure communication possible for the first time. Whereas quantum cryptography, in principle, can already be performed with single quantum particles, all the other proposals utilize entanglement between two or more particles - for example, to enhance communication rates or to allow the teleportation of quantum states. Entanglement between quantum systems is a pure quantum effect. It is closely related to the superposition principle and describes correlations between quantum systems that are much stronger and richer than any classical correlation could be. Originally this property was introduced by Einstein, Podolsky, and Rosen (EPR) (1935), and by Schrodinger (1935) and Bohr (1935) in the discussion on the completeness of quantum mechanics and by von Neumann (1932) in his description of the measurement process. After Bohm introduced entanglement for spin-1/2 particles, Bell showed that measurements of such systems should be correlated, according to local, realistic theories, in a different way than predicted by quantum mechanics (1964). He gave a clear criterion with which to determine the validity of the different theories experimentally. A number of experiments have confirmed

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the quantum predictions. Of course, local realistic theories are not violated as long as the experiments still rely on certain assumptions. But after the recent Bell-experiment with space-time-separated observers (Weihs et al., 1998) showed that any causal influences between the observations can be excluded, it is only the detection loophole (Pearle, 1970) that has to be closed for a final disproof of one or the other theory (Clauser, 1978). Such loopholes also hold for tests of the Greenberger-Home-Zeilinger (GHZ) argument (1989) and of Hardy’s example (1993), which show that contradictionscan arise for the observation of single triples or pairs of particles. However, up to the usual detection loophole, the very recent observation of three-particle entanglement (Bouwmeester et al., 1999) and the fist GHZ experiment (Bouwmeester ef al., submitted) give strong indications of the validity of the various theories. Quantum information is not concerned with the fundamental issues (it actually assumes the validity of standard quantum mechanics). Instead, it applies the characteristic features of entangled systems to devise powerful new schemes for communication and computation. Entanglement among a large number of quantum systems facilitates very efficient computations. In particular, the factorizationalgorithm by Shor (1994) and the search algorithm by Grover (1997) (together with the increasing numbers of algorithms derived from one or the other) show how entanglementand the associated interference between entangled states can boost the power of quantum computers. Quantum communication exploits entanglementbetween only two or three particles. As will be seen in the following sections, the often counterintuitive features of such small entangled systems make powerful communication methods possible. After introducing the very basic properties of pairs of entangled particles (Section II), in Section 111 we give an overview of the general ideas behind three important quantum communication schemes: entanglement-based quantum cryptography enables secret key exchange and thus truly secure communication (Ekert, 1991); using quantum dense coding, one can send classical information more efficiently (Bennett and Wiesner, 1992); and with quantum teleportation, one can transfer quantum information - that is, the quantum state itself - from one quantum system to another (Bennett ef al., 1993). The tools for the experimental realization of those quantum communication schemes are presented in Section IV. In particular, we show how to produce polarization-entangled photon pairs by parametric down-conversion (Kwiat et al., 1995) and how to observe these nonclassical states by interferometric Bell-state analysis (Weinfurter, 1994). In Section V we describe the first experimental realizations of the basic quantum communication schemes. In the experiments performed during the last years at the University of Innsbruck, we could realize entanglement-based quantum cryptography with randomly switched analyzers and with a

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separation of the two users by more than 400m (Jennewein et al., submitted); we demonstrated the possibility of transmitting 1.58 bit of classical information by encoding trits on a single 2-state photon (Mattle et al., 1996); and we could transfer a qubit - in our case the polarization state from one photon to another by quantum teleportation (Bouwmeester et al., 1997) and entanglement swapping (Pan et al., 1998).

11. Entanglement: Basic Features For a long time, entanglement was seen merely as one of these counterintuitive features of quantum mechanics, important just within the realm of the EPR paradox. Only lately has quantum information started to exploit these features for new types of information transmission and processing. Recent literature offers a thorough discussion of all the various properties of entangled systems (Clauser, 1978; Greenberger et al., 1989; Peres, 1993). In this review, we concentrate on those features that form the foundation of the basic quantum communication schemes. At the heart of entanglement lies another fundamental feature of quantum mechanics, the superposition principle: If we look at a classical, two-valued system, such as a coin, we find it in either one of its two possible states - that is, either head or tail. Its quantum-mechanical counterpart,a two-state quantum system, however, can be found in any superposition of two possible basis states, e.g., in I*) = l / d ( ( O ) 11)). Here we denote the two orthogonal basis states as I 0) and I 1), respectively. [This notation should not be confused with the description of an electromagneticfield (vacuum or single-photon state) in second quantization. Here we only use the notations of first quantization to describe the properties of two-state systems.] This generic notation can stand for any of the properties of various two-state systems - for example, for ground 1s) and excited ) . 1 state of an atom or, as will be the case in our experiments, for horizontal I H )and vertical IV) polarization of a photon. In the classical world, we find two coins with head/head, head/tail, tail/ head, or tail/tail. We can identify these four possibilitieswith the four quantum states \O)l10)2, lO),l 1)2, 11)110)2,or I 1)111j2, describing two 2-state quantum systems. But the superposition principle also holds for more than one quantum system. Thus the two quantum particles are no longer restricted to the four “classical” basis states but rather can be in any superposition thereof - for example, in the entangled state

+

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Of course, one is restricted neither to two particles nor to such maximally entangled states. During the last decade, enormous progress was made in the theoretical studies of quantum features of multiparticle systems. One will observe even more stunning correlations between three or more entangled particles (Greenberger et al., 1989; 1996); one can generalize to the observation of interference and entanglement between multistate particles (Zukowski et al., 1997) and to entanglement for mixed states. There is also the possibility of purifying entanglement (Bennett et al., 1996) and one even found twoparticle systems, which are actually not entangled, but a local observer cannot distinguish them from entangled states (Bennett et al., 1999). For the basic quantum communication schemes and experiments, we can concentrate on the particular properties of maximally entangled two-particle systems. Considering two 2-state particles, we find a basis of four orthogonal, maximally entangled states, the so-called Bell-states basis:

The name Bell-states was assigned because these states maximally violate a Bell inequality (Braunstein et al., 1992). This inequality is deduced in the context of so-called local realistic theories, and it gives a range of possible results for certain statistical tests on identically prepared pairs of particles. Quantum mechanics predicts different results if the measurements are performed on entangled pairs. If the two particles are not correlated, i.e., are described by a product state, the quantum-mechanical prediction is also within the range given by Bell’s inequality. The remarkably nonclassical features of entangled pairs arise from the fact that the two systems can no longer be seen as independent but have to be seen as one combined system, where observation of one of the two will change the possible predictions for measurement results obtained for the other (Schrodinger, 1935; Bohr, 1935). Formally, this mutual dependence is reflected by the fact that the entangled state can no longer be factored into a product of two states for the two subsystems. If one looks at only one of the two particles, one fmds it with equal probability in state 10) or in state I 1). One has no information about the particular

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outcome of a measurement to be performed. However, the observation of one of the two particles determines the result of a measurement of the other particle. This holds not only for a measurement in the basis I O)/l 1 ) but also for any arbitrary superposition - that means for any arbitrary orientation of the measurement apparatus. Particularly for the state I@-), we will find the two particles always in orthogonal states, no matter how the two measurement apparata are oriented. For the case of polarization-entangled photons, this means that photon 2 has vertical polarization if we found horizontal polarization for photon 1 but also that photon 2 will be left-circular-polarized if we observed right-circular polarization for photon 1. Another important feature of the four Bell-states is that a manipulation of only one of the two particles suffices to transform any Bell-state to any of the other three states. This is not possible for the basis formed by the products. For example, to transform lo), I 0), into I 1) I 1), ,one has to flip the state of both particles. These three features are the ingredients for the fundamental quantum communication schemes described here:

,

0

0

0

Different statistical results for measurements on entangled or unentangled pairs Perfect correlations between the observations of the two particles of a pair, although the results for the measurements on the individual particles are fully random The possibility of transforming between the Bell-states by manipulating only one of the two particles

In Section 111 we will see how these fundamental properties enable one to guarantee secure communication via quantum cryptographic key exchange, how they can be used to enhance channel capacity for data transmission, and how quantum states can be transferred by quantum teleportation. Section V then explains how we could realize these schemes experimentally with entangled photon pairs and two-photon interferometry.

111. Quantum Communication via Entangled States Quantum communication methods utilize fundamental properties of quantum mechanics to enhance the power and feasibilities of today’s communication systems. The first step toward quantum information processing is the generalization of the classical digital encoding using the bit values “0” and “ 1.” Quantum information associates two distinguishable, orthogonal states

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of a 2-state system with these bit values. We thus directly translate the two values “0” and “1” to the two basis states 10) and I 1). In extension to classical communication,the quantum system can be in any superposition of the two basis states. To distinguish such a quantum state and the information contained in it from the classical bit, it was called a qubit (Schumacher, 1995). The general state of a qubit is

+

2 where a0 and a 1 are complex amplitudes (with la0 1’ la1 I = 1). A measurement of the qubit projects the state onto either 10) or I 1) and therefore cannot give the full quantum information of the state. Evidently, because in order to avoid errors, we have to restrict ourselves to sending only basis states,just one bit of classical information can be sent with a single qubit. Thus the new features do not seem to offer additional power. (We will see in Section 1II.B how this limit can be surpassed when we employ entanglement.) However, by provoking errors, the security of quantum cryptography (Bennett and Brassard, 1984) just relies on the fact that an eavesdropper cannot unambiguously read the state of a single quantum particle transferred from Alice to Bob. A potential eavesdropper induces errors, which allow Alice and Bob to check the security of their quantum key generation. By using two-particle systems, entanglement enhances the superiority of quantum over classical communication systems. During recent years, several proposals have suggested how to exploit the basic features of entangled states in new quantum communication schemes. Here we will see how entangled pairs make possible a new formulation of quantum cryptography, how we can surpass the limit of transmitting only one bit per qubit, and how entanglement allows us to transfer quantum information from one particle to another in the process of quantum teleportation.

CRYPTOGRAPHY A. QUANTUM

Let us first discuss how quantum cryptography can profit from the fascinating properties of entangled systems (Ekert, 1991). Suppose that Alice and Bob want to exchange secret messages and thus first have to perform secure key exchange (Shannon, 1949). Suppose furthermore that Alice and Bob receive particles that are in pairs entangled with each other from an EPR source (Fig. 1). Beforehand, Alice and Bob agreed on some preferred basis, here called again I O)/ll), in which they start to perform measurements. Because of the entanglement of the particles, the measurement results of Alice and Bob will be perfectly correlated or, for the case when the source produced the pairs in the I ) states, perfectly anticorrelated. For each instance where Alice obtained,

*-

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FIG. 1. Scheme for entanglement-based quantum cryptography (Ekert, 1991).

say 0, she knows that Bob observed 1, and if she got the result 1, she knows that Bob had 0. Alice and Bob can use these results to establish a random key, ideal for encoding messages. But how can they be sure that no eavesdropper intercepted the key exchange? In standard quantum cryptography schemes, they have to use a second basis and have to change between the two basis systems randomly. An eavesdropper, not knowing the actual basis, causes errors because he cannot determine the quantum state without knowing the preferred basis. Thus, Alice and Bob can find out, by communication over a classical, public channel, whether or not their key exchange was attacked by sacrificing key material when checking whether or not key bits are different. Entangled systems are very fragile against measurements. Any attack an eavesdropper might perform reduces the entanglement and allows Alice and Bob to check the security of their quantum key exchange. As described in Section 11, measurements on entangled pairs obey statistical correlations and will violate a Bell inequality. It can be shown that the inequality is less violated the more knowledge the eavesdropper gained when intercepting the key exchange. How much a Bell inequality is violated is thus an ideal measure of the security of the key. Alice and Bob therefore measure the entangled particles not only in the basis l O ) / l l ) but also along some other directions given by the Bell inequality used. A particularly simple form of a Bell inequality, one particularly well suited for experimental application, is the version deduced by E. P. Wigner (1970):

Here N(l a , l p ) stands for the rate of pairs where Alice obtains the result “1” from her measurement apparatus oriented according to the parameter u, and Bob observes “1” with his analyzer oriented at p, etc. Both Alice and Bob

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perform their measurements with the setting 0, which we therefore identify with our standard basis 1 O ) / l I ) . The other directions (aat Alice’s side and y for Bob) are then given by 116) = sin@ - 6)lO) +cos(p - S)l1)

106) = cos(p - 6)(0)- sin@ - 6)(1)

where 6 = ct or 6 = y. From quantum mechanics we obtain, for example, N ( I,, lp) = 1/2 NOsin2(ct- p), etc. The inequality is maximally violated for (a- p) = (p - y) = 30”. A practical scheme therefore can run as follows. Alice randomly switches her analyzer between 0” (for the lo)/(1) basis) and a = 30°, whereas Bob randomly analyzes at 0” and y = -30”. Afterward they communicate via a classical channel the orientations they used when they observed a photon. If these happened to be the same, they can use their results for the key. The subset of events where the orientation differed is used for the statistical test according to Wigner’s Bell-inequality. The amount they violate the inequality tells Alice and Bob about the security of their key exchange and allows them to send secret messages securely. B. QUANTUM DENSECODING When encoding a message, one uses distinguishablesymbols and writes them on some physical entity that then is transmitted to the receiver. If one wants to send one bit of information, one uses, for example, the binary values “0” and “1”. If one wants to send two bits of information, one consequently has to repeat the process twice - that is, one has to send two such entities. As mentioned above, in quantum information one identifies the two binary values by the two orthogonal basis states 10) and 1 1 ) of the qubit. In order to send a classical message to Bob, Alice will use particles all produced in the same state by some source. Alice translates the bit values of the message either to leaving the state of the qubit unchanged or to flipping to the other, orthogonal state, and Bob consequently will observe the particle in one or the other state. That means that Alice can encode one bit of information in a single qubit. Obviously, she cannot do better, because in order to avoid errors, the states arriving at Bob have to be distinguishable,which is guaranteed only when using orthogonal states. In this respect, they do not gain anything by using qubits compared to classical bits. Also, if she wants to communicate two bits of information, Alice has to send two qubits. C. H. Bennett and S. Wiesner (1992) found a clever way to circumvent the classical limit and showed how to increase the channel capacity by utilizing entangled particles. Suppose the particle that Alice obtained from the source

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FIG.2. Scheme for the efficient transmission of classical information by quantum dense coding (Bennett and Wiesner, 1992) (BSM: Bell-state measurement;U: unitary transformation).

is entangled with another particle, which was directly sent to Bob (Fig. 2). The two particles are in one of the four Bell states, say I 9 - ). Therefore, Alice can use the particular feature of the Bell-basis that manipulation of one of the two entangled particles suffices to transform to any other of the four Bell-states. Thus Alice can perform one out offour possible transformations - that is, do nothing, shift the phase by x,flip the state, or flip and phase-shift the state to transform the two-particle state of their common pair to another one. After Alice has sent the transformed two-state particle to Bob, he can read the information by performing a combined measurement on both particles. He will make a measurement in the Bell-state basis and can identify which of four possible messages was sent by Alice. Thus it is possible to encode two bits of classical information by manipulating and by transmitting a single two-state system. Entanglement enables one to communicate information more efficiently than any classical system could do. The preceding examples show how quantum information can be applied for secure and efficient transmission of classical information. But can one also transmit quantum information - that is, the state of a qubit? Obviously quantum mechanics provides a number of obstacles to this intention, above all the problem of measuring quantum states, which, however, could be utilized by quantum cryptography.

C. QUANTUMTELEPORTATION 1. The Idea

It is an everyday task, in our classical world, for Alice to send some information to Bob. Consider fax machines. Alice might have some message,

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written on a sheet of paper. For the fax machine the actual written information does not matter; in fact, it reduces to just a sequence of white and black pixels. For the transmission, the machine scans the paper pixel by pixel. It measures whether a pixel is white or black and sends this information to Bob’s machine, which writes the state of each pixel onto another sheet of paper. In classical physics, by definition, one can make the measurements with arbitrary precision, and Bob’s sheet can thus become an ideal copy of Alice’s original sheet of paper. If Alice’s pixels get smaller and smaller, in reality they sooner or later are encoded on single molecules or atoms. If we again confine ourselves to coding in the basis states, we surely could measure and transfer the binary value of even such a pixel. Now, imagine not only that Alice has classical binary values encoded on her system, but that she has to send quantum information to Bob. She has a qubit encoded on some quantum system and wants a quantum system in the hands of Bob to represent this qubit at the end of the transmission. Evidently, Alice cannot read the quantum information - that is, measure the state of the quantum object with arbitrary precision. All she would learn from her measurement is that the amplitude of the observed basis state was not zero. But this is indeed not enough information for Bob to reconstruct the qubit on his quantum particle. Another limitation, which definitely seems to bring the quest for perfect transfer of the quantum information to an end, is the no-cloning theorem (Wootters, 1982).The state of a quantum system cannot be copied onto another quantum system with arbitrary precision. Thus, how could Bob’s quantum particle obtain the state of Alice’s particle? In 1993 Charles Bennett, Giles Brassard, Claude Crepeau, Richard Josza, Asher Peres, and Bill Wootters found the solution to this task. In their scheme a chain of quantum correlations is established between the particle carrying the initial quantum state and Bob’s particle. They dispense with measuring the initial state and actually avoid gaining any knowledge about this state at all! To perform quantum teleportation, initially, Alice and Bob share an entangled pair of particles 2 and 3, which they obtained from some source of entangled particles, say, in the state lXV-)2,3 (Fig. 3). As mentioned before, we cannot say anything about the state of particle 2 on its own. Nor do we know the state of particle 3. But whatever these two states are, we know for sure that they are orthogonal to each other. Next, particle 1, which carries the state to be sent to Bob, is given to Alice. She now measures particle 1 and 2 together by projecting them onto the Bellstate basis. After projecting the two particles into an entangled state, she can no longer infer anything about the individual states of particles 1 and 2. However, she knows about correlations between the two. Let us assume she This tells her that whatever the two states of obtained the result I

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FIG.3 . Scheme for teleporting the quantum state from one system to another one (Bennett et al., 1993).

particles 1 and 2 were, they were orthogonal to each other. But from this, Alice already knows that the state of particle 3 is equal to the state of particle 1 (up to a possible overall phase shift). This follows because the state of particle 1 was orthogonal to 2, and, due to the preparation of particles 2 and 3, the state of particle 2 was orthogonal to 3. All Alice has to do is to tell this to Bob to let him know that, in this particular case, the state of his particle 3 is already the same that particle 1 had initially. Of course, because there are four orthogonal Bell-states, there are four equally probable outcomes for Alice’s Bell-state measurement. If Alice obtains another result, the state of Bob’s particle is again related to the initial state of particle 1, up to a characteristic unitary transformation. This stems from the fact that a unitary transformation of only one of two entangled particles can transform from any Bell-state to any other. Therefore, Alice has to send the result of her Bell-state measurement (i.e., a number between 0 and 3, equivalently 2 bit of information) via a classical communication channel to Bob. He then can restore the initial quantum state of particle 1 on his particle 3 by the correct unitary transformation. Formally, we first describe the initial state of particle 1 by I x ) = a I H ) b l V ) , and the state of the EPR pair 2 and 3 by IQ-)2,3. Therefore, the joint three-photon system is in the product state

,

which can be decomposed into

+

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One easily sees that after the Bell-state measurement of particles 1 and 2 and after the corresponding unitary transformation of particle 3, the state of particle 1 is transferred to particle 3. 2. Some Remarks

The principle of quantum teleportation incorporates all the characteristic features of entangled systems and, in an astounding manner, profits from the obstacles imposed by quantum mechanics. It should be emphasized that quantum teleportation is well within the concepts of conventional physics and quantum mechanics. Let us briefly discuss a few common misunderstandings. First, the no-cloning theorem is not violated. The state of particle 1 can be restored on particle 3 only if the measurement performed by Alice does not give any information about the state! After Alice’s Bell-state measurement, particle 1 is in a mixed state that is absolutely uncorrelated with the initial state of particle 1. Therefore, the particular quantum state that is teleported can be attributed only to one particle at a time, never to two. Second, there is no faster-than-light communication achieved in quantum teleportation. Even if Alice knows, right after her measurement, whether Bob’s particle is already in the correct state or not, she has to send this information to Bob. The classical information sent to Bob is transmitted, according to the theory of relativity, at most with the speed of light. Only after receiving the result and after performing the correct unitary transformation can Bob restore the initial quantum state. Without knowing the result of Alice’s measurement, Bob’s particle is in a mixed state that is not correlated at all with the initial state. Thus quantum information, the qubit, cannot be transferred faster than classical information. And third, there is no transfer of matter or energy (besides the transmission of classical information). The particle is made up of its properties, described by the quantum state. For example, the state of a free neutron defines its momentum and its spin. If one transfers the state onto another neutron, this particle obtains all the properties of the fist one - in fact, it becomes the initial one. We leave it to the science fiction writers to apply the scheme to bigger and bigger objects. The question of whether this idea will help some Captain Kirk to get back to his space ship cannot be answered here. And a lot of other problems need to be solved as well (Gauss, 1995). It is appropriate to cite some generalizations of the principle of quantum teleportation. It is not necessary that the initial state that has to be teleported be a pure state. In fact it can be any mixed state, or even the undefined state of an entangled particle. This is best demonstrated by entanglement swapping (Zukowski ef al., 1998). Here, the particle to be teleported (1) is entangled with yet another one (4) (Fig. 4). The state of 1 on its own is a mixed state,

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FIG.4. Scheme for entangling particles that never interacted by the process of entanglement swapping (Zukowski et al., 1998).

but it will be also determined by the observation of particle 4. Quantum teleportation allows us to transfer the state of 1 onto particle 3. Because quantum teleportation works for any arbitrary quantum state, 3 thus becomes entangled with 4.Note that particles 3 and 4 do not come from the same source, nor did they ever interact with each other. Still, it is possible to entangle them by swapping the entanglement in the process of quantum teleportation. Quantum teleportation is not confined to transferring two-state quantum systems. If Alice and Bob share an entangled pair of N-state particles, they can teleport the state of an N-dimensional quantum system (Vaidman, 1994). As before, Alice performs projection onto the N2-dimensional basis of entangled states spanning the product space of particles 1 and 2. The result, one out of N2 equally probable, has to be communicated to Bob, who then can again restore the initial state of particle 1 by the corresponding unitary transformation of his particle 3. If Alice and Bob share a pair of particles entangled in the original sense of EPR - that is, for continuous variables or oo-dimensional states - they also can teleport properties like position and momentum of particles or the phase and amplitude of electromagnetic fields (Furusawa et al., 1998). A considerable simplification of quantum teleportation, especially in terms of experimental realization, transfers not the quantum state of a particle but rather the manipulation performed on the entangled particle that is given to Alice (Popescu, 1995) (Fig. 5). Again, we first distribute an entangled pair to Alice and Bob. But before Alice gets hold of her particle 1 and can perform measurements on it, the state of this particle is manipulated in another degree of freedom. We cannot longer talk about a two-state system. Rather, particle 1 now is described in a four-dimensional Hilbert space, spanned by the original degree of freedom and the new one. Formally, however, this mimics the two 2-state particles given to Alice in the standard quantum teleportation scheme. Consequently, a measurement in the four-dimensional Hilbert space of particle 1, which perfectly erases the quantum information by mixing the two

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FIG. 5. Remote state preparation of Bob’s particle 2, by a manipulation (M) of particle 1.

degrees of freedom, is performed. This gives Bob the information he needs to perform the correct unitary transformation on his particle. This way, the originally mixed state of particle 2 can be turned into a pure state that depends on the manipulation that was initially performed on particle 1. Using such a scheme one can remotely prepare particle 3 in any pure quantum state. Thus it is not necessary to send two real numbers to Bob if one wants him to have a certain, pure quantum state prepared on his particle. If he is provided with one of a pair of entangled particles Alice simply has to transmit 2 bit of classical information to Bob.

IV. The Building Blocks Before turning to the fascinating applications of entangled systems, let us review how to produce, how to manipulate, and how to measure such quantum systems. Recent years saw incredible progress of the experimental techniques for handling various quantum systems. However, there are additional challenges in working with entangled systems, especially the careful control of interactions and decoherence of the quantum systems. In their seminal work, Einstein, Podolsky, and Rosen considered particles that interacted with each other for a certain time and thereafter exhibited the puzzling, nonclassical correlations. To maximize such correlations and to achieve optimal entanglement, the interaction needed to entangle a pair of particles is the one von Neumann had in mind when describing the measurement process. Ideally it couples two quantum systems in such a way that if the first system is in one out of a set of distinguishable (orthogonal) states, the second system will change into a well-defined corresponding state. Let us

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look at such a coupling for the most simple case of two 2-state systems. As before, the two basis states are denoted as 10) and 1 1 ). The coupling is such that if system 1 is in state 10) system 2 will remain in its initial state, say I O)*, whereas if system 1 is in state I 1 ), , system 2 will flip to the orthogonal state, i.e., to I 1)2. The nonclassical features arise if system 1 is in a superposition of its basis states. Then, coupling it with the second system results in an entangled state:

Although this basic principle for producing entangled states has been known since the very beginning of quantum mechanics, until recently there was no physical system where the necessary coupling could be realized. The progress in cavity-QED (Hagley et al., 1997) and ion-trap experiments (Turchette et al., 1998) allowed the first observation of entanglement between two atoms or two ions. These experiments are of great importance for the further development of experimental quantum computation. However, for quantum communication, one needs to transfer the entangled particles over reasonable distances. Thus photons (with wavelengths in the visible or near infra-red) are clearly the better choice. For entangling photons via such a coupling, various methods have been proposed and partially realized (Hood et al., 1998; Imamoglu et al., 1997; Franson, 1998) but they still need to be investigated more thoroughly. Fortunately, the parametric down-conversion offers an ideal source for entangled photon pairs without the need for strong coupling (see Section 1V.A). To perform Bell-state analysis, one first has to transform the entangled state into a product state. This is necessary because two particles can be analyzed best if they are measured separately. Otherwise one would need to entangle the two measurement apparata, each of which analyzes one of the two particles - clearly an even more challenging task. In principle, the disentangling transformation can be performed by reversing the entangling interaction described above. However, as long as such couplings are not achievable yet, one has to find replacements. In the following it is shown how two-particle interference can be employed for partial Bell-state analysis (see Section 1V.B). Because the manipulations and unitary transformations have to be performed on one quantum particle at a time, this does not impose new obstacles. Such operations are often routine.

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A. ENTANGLED PAIRSOF PHOTONS Entanglement between photons cannot yet be generated by coupling them via some interaction. However, there are several emission processes, like atomic cascade decays and parametric down-conversion, where the properties of two emitted photons become entangled because of the conservation of energy and of linear or angular momentum. Historically, entanglement was first observed in measurements of the polarization correlation between y+y- emissions in positron annihilation (Wu and Shaknov, 1950), soon after Bohr’s proposal to observe EPR phenomena for spin- 1/2 systems. After Bell’s discovery that contradicting predictions between quantum theories can be actually observed, a series of measurements was performed mostly with polarization-entangled photons from a two-photon cascade emission in calcium (Freedman and Clauser, 1972; Aspect et al., 1982). In these experiments, the two photons are in the visible spectrum and thus can be manipulated and controlled by standard optical techniques. Of course, this is a great advantage compared to the positron annihilation source. However, the two photons are now no longer emitted in opposite directions, because the emitting atom carries away some randomly determined momentum. This again makes experimental handling more difficult and also reduces the brightness of the source. The process of parametric down-conversion offers possible means of efficiently generating entangled pairs of photons (Kwiat et al., 1995). B. PARAMETRIC DOWN-CONVERSION When light propagates through an optically nonlinear medium, with secondorder nonlinearity ~ ( (possible ~ 1 only in non-centro-symmetric crystals), the conversion of a light quantum from the so-called pump field into a pair of photons in the “idler” and “signal” modes can occur. In principle, it can be seen as the inverse of the frequency doubling process in nonlinear optics (Boyd, 1992). Energy conservation and momentum conservation determine the correlations between the emitted photons. We will see in the following section how these conservation laws give rise to momentum and time-energy entanglement. However, according to the Heisenberg uncertainty relation, the interaction time and volume will determine the sharpness of the observed correlations, which are formally obtained by integration of the interaction Hamiltonian (Gosh et al., 1996). The interaction time is given by the coherence time 7, of the W-pump light, the volume by its extension and spatial distribution in the nonlinear crystal. The relative orientations of pump beam direction and polarization and the optical axis of the crystal determine the actual direction of the emission of a

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FIG.6. Different correlation of the emission directions for type-I and type41 downconversion.

certain wavelength. We distinguish two possible alignment types (Fig. 6). For type-I down-conversion, the pump is extraordinary-polarized, and idler and signal beams have the same (ordinary) polarization. The different colors are emitted into cones centered on the pump beam; particularly, for an angle of 90" between optical axis and pump beam, the angle depends almost linearly on the wavelength. In type-I1down-conversionthe pump is extraordinary-polarized,and in order to fulfill the momentum conservation inside the crystal (phase-matching),the two down-converted photons have different, for most directions orthogonal, polarization, offering the possibility of a new source of polarization-entangled photon pairs.

1. Entanglement Produced by Parametric Down-Conversion The conservation of momentum, energy, and angular momentum can give rise to various types of correlations between the emitted photons. If there are two or more possibilities for the single photons to be emitted, we obtain a superposition of all possible states for the photon pair. In general, one then selects two such states from the manifold generated by the emission to obtain the desired entanglement.

However, one has to keep in mind that as a random process, the emission obeys Poissonian statistics. Thus there is always a certain probability that two pairs, which are uncorrelated with each other, could be observed together, then in an unentangled state. Yet if the rate of down-conversion into a specific pair of modes (typically lOOOOs-') is small compared to the inverse of the time resolution of the detectors (typically 1 ns), there is only a negligible probability for the registration of two pairs during this interval. Registering only pairs that are coincident within a short interval thus avoids the observation of different pairs and guarantees a high degree of observed entanglement.

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Entanglement After Selection of Detection Events. Sometimes the experimental realization becomes considerably easier if one uses the high time correlation and two-photon interferenceto generate entanglement by selection of the events, rather than the entanglement produced by the source. To generate polarization correlations from type-I down-conversion, Shih and Alley (1988) overlapped the two degenerate modes (equal wavelength) on a beamsplitter with the polarization in one of the arms rotated from vertical to horizontal. In such a case, the two photons are distributed randomly into the two outputs, yielding the product state

,

where, for example, IH) I V), describes that the horizontally polarized photon is in output 1 and the vertically polarized in output 2. If one now only detects coincidences between a detector in arm 1 and a detector in arm 2, one selects an entangled subset of all possible detection events. Figure 7 shows configurations where such a selection leaves various forms of entanglement. Actually, these configurations are equivalent in terms of the manipulation of the respective degree of freedom. In all cases, the subset selected by coincidence between detectors on each side form entangled states, but the cases where two photons are detected by one detector (Fig. 7a and b) or outside the time interval AT (Fig. 7c), (which are as many!) have to be neglected. Such schemes have been realized in a number of experiments (Reid and Walls, 1986) because they are much easier to perform and very often

FIG.7. Various schemes for observing entanglement from initially unentangled pairs by conditioned detection.

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FIG.8. Scheme of the experiment by Rarity and Tapster (1990). utilizing mode entanglement as produced by the down-conversion source.

equally well suited for EPR-Bell experiments (Popescu et al., 1997). For quantum communication experiments, however, the selection step should be avoided, and sources that directly produce entangled pairs are needed. Entanglement as Produced by the Down-ConversionProcess. In fact, there are only a few experiments where no additional selection of events is necessary. The first one, using momentum entanglement, was performed by Rarity and Tapster (1990) (Fig. 8). Here, only two directions have been allowed for each of the photons, A and D for the first and B and C for the second photon. This results in the state

where consequently [A),, I B ) , are the two possible states for photon one, IC),, ID)*for photon two. Similarly, mode entanglement can be obtained from two coherently pumped down-conversion processes, either in two crystals or from different points in one crystal (Ou et al., 1990). Although entanglement in any degree of freedom is equally good in principle, polarization is often much easier to deal with in practice because of the availability of high-efficiency polarization-control elements and the relative insensitivity of most materials to birefringent thermally induced drifts. Several methods employing two down-conversioncrystals have been proposed (Klyshko, 1988), but noncollinear degenerate type-I1 phase matching offers a much simpler technique.

2. Polarization-Entangled Pairs from T y p e 4 Down-Conversion In type-I1 down-conversion,polarization-entangled states are produced directly out of a single nonlinear crystal [BBO (beta-barium borate)], with no need for extra beam splitters or mirrors and no requirement of discarding detected pairs. Verifying the correlations produced by this source, one observes strong

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violations of Bell’s inequalities (modulo the typical auxiliary assumptions), within a short measurement time. Moreover, using two extra birefringent elements, one can easily produce any of the four orthogonal Bell-states. As mentioned above, with type41 phase matching, the down-converted photons are emitted into two cones, one ordinary-polarized, the other extraordinary-polarized. Because of conservation of transverse momentum, the photons of each pair must lie on opposite sides of the pump beam. In the collinear situation, the two cones are tangent to one another on exactly one I , , line, namely the pump beam direction (Shih and Sergienko, 1994). If € the angle between the crystal optic axis and the pump beam, is decreased, the two cones will separate from each other entirely. However, if the angle is increased, the two cones tilt toward the pump, causing an intersection along two lines (see Fig. 6 right, and Fig. 9) (Kwiat et af.,1995; Kwiat, 1993). Along the two directions (“1” and “2”) where the cones overlap, the light can be essentially described by an entangled state:

where the relative phase u arises from the crystal birefringence, and an overall phase shift is omitted. Using an additional birefringent phase shifter (or even slightly rotating the down-conversion crystal itself), the value of u can be set as desired, e.g., to the

FIG.9. Photons emerging from type-Il down-conversion. The photons are always emitted with the same wavelength but orthogonal polarization. At the intersection points, their polarizations are undefined but different, resulting in entanglement.

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values 0 or 71. Such a net phase shift of 71 may be obtained by rotation of a quarter-wave plate in one of the two paths by 90" from the vertical to the horizontal direction. Similarly, a half-wave plate in one path can be used to change horizontal polarization to vertical, and vice versa. Thus one can easily produce any of the four Bell-states. The birefringent nature of the down-conversion crystal complicates the actual entangled state produced, because the ordinary and the extraordinary photons have different velocities inside the crystal and propagate along different directions,even though they become collinear outside the crystal. The resulting longitudinal and transverse walk-off between the two polarizations in the entangled state are maximal for pairs created near the entrance face of the crystal, which consequently acquire the greatest time delay and relative lateral displacement. Thus the two possible emissions become, in principle, distinguishable by the order in which the detectors would fire, or by their spatial location, and no entanglement will be observable. Yet the photons are produced coherently along the entire length of the crystal. One can thus completely compensate for the longitudinal and partially for the transverse walk-off by using two additional crystals, one in each path (Rubin et aL, 1994). The experimental setup is shown in Fig. 10: the 351.1-nm pump beam (1 50 mW) from a single-mode argon ion laser, followed by a dispersion prism to remove unwanted laser fluorescence (not shown). Our 3-mm-long BBO crystal was nominally cut at €Ipm = 49.2"to allow collinear degenerate operation when the pump beam is precisely orthogonal to the surface. The optical axis was oriented in the vertical plane, and the entire crystal was tilted (in the plane containing the optic axis, the surface normal, and the pump beam) by 0.72", thus increasing the effective value of 0,,, inside the crystal to 49.63". The two cone-overlap directions, selected by irises before the detectors, were consequently separated by 6.0". Each polarization analyzer consisted of two stacked polarizing beamsplitters preceded by a rotable half-wave plate. The detectors were cooled silicon avalanche photo-diodes operated in the Geiger

FIG. 10. Experimental setup for the observation of entanglement produced by the type-I1 down-conversion source. The additional birefringent crystals are needed to compensate the birefringent walk-off effects from the first crystal.

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mode. Coincidence rates C(01,02) were recorded as a function of the polarizer settings 01 and 02. In this experiment the transverse walk-off (0.3 mm) was small compared to the coherent pump beam width (2 mm), so the associated labeling effect was minimal. However, it was necessary to compensate for the longitudinal walkoff, because the 3.0-mm BBO crystal produced a time delay that was about the same as the coherence time of the detected photons (% 390 fs, determined by interference filters with a width of 5 nm at 702 nm). As discussed above, we used an additional BBO crystal (1.5 mm thick) as compensator in each of the paths, preceded by a half-wave plate to exchange the roles of the horizontal and vertical polarizations. Under such conditions, one now obtains routinely a coincidence fringe visibility (as polarizer 2 is rotated, with polarizer 1 fixed at - 45") of more than 97% (Fig. 11). The high quality of this source is crucial for the overall performance of our experiments, in quantum dense coding (Mattle et al., 1996) in quantum cryptography, and in tests of Bell's inequality (Weihs et al., 1998). For the later experiments, the photons are coupled into single-mode fibers, to bridge long distances of the order of 400m. To reach a high coupling, the pump beam should be slightly focused into the BBO crystal to match optimally with the microscope objectives used. Because the compensation crystals also partially compensate for the transverse walk-off, the focusing down to 0.2mm is not crucial. Visibilities of more than 98% have been obtained this way, with an overall collection and detection efficiency of 10%. Such a source has a number of distinct advantages. It seems to be relatively insensitive to larger collection irises, an important feature in experiments

polarizer angle 8 (forcp=450) FIG. 1 1 . Coincidence fringes for the Bell-states I q ' )

and

I*-).

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where high count rates are crucial. In addition, thanks to its simplicity, the source is much quicker to align than other down-conversion setups, and it is remarkably stable. One of the reasons is that phase drifts are not detrimental to a polarization-entangled state unless they are birefringent, i.e., polarization-dependent. This offers a clear advantage over experiments with momentum-entangled or energy-time-entangled photon pairs. 3. Pulsed Down-Conversion As we will see in the description of the interferometric Bell-state analysis (Section IV.C), for quantum teleportation and the following experiments, one has to interfere photons produced by independent sources, such as two downconversion crystals. In practical terms, this is possible only if the time of the creation of the photons is significantly shorter than their coherence time. The low brightness of down-conversion sources makes it necessary to accept photons within a wide frequency or wavelength distribution. Filtering reduces the count rates below any acceptable level. Usually, one therefore uses photons within a distribution with a width of 3 to 5 nm and a center wavelength between 700nm and 800nm. This results in a coherence time of about 800fs, which means that the duration of the down-conversionprocess should be below 200 fs. In the experiments, we therefore used frequency-doubled pulses of a modelocked Ti :Saphire laser as the pump source for our type-I1 cut BBO crystal. The infra-red pulses with a mean wavelength of 788 nm have a peak power of almost 20 kW and a pulse length of about 150fs at a pulse repetition rate of 76 MHz. From the frequency doubler (type-I phase-matched LBO crystal), we achieved an efficiency of up to 40% and obtained up to 800mW at 394nm averaged UV power. However, to reduce the effect of transversal walk-off in the LBO crystal, we had to reduce the focusing and finally arrived at 500 mW UV power. Before launching the light onto the BBO crystal (crystal length of 1.5 mm), we used focusing mirrors and a cylindrical telescope to obtain a reasonable beam spot (about 500pm) at the crystal. This is necessary for optimal coupling into single-mode fibers, which served both as spatial filters and as collection optics for the fiber-pigtailed single-photon detectors. Also for this setup we used the compensation crystals behind a half-wave plate. Here we can only compensate the transverse walk-off. A necessary condition for the compensation of the longitudinal walk-off was that the separation of the polarization components be less than the coherence time of the pump beam. Even if we assume an ideal, bandwidth-limited UV pulse, we have a coherence time equal to the pulse duration of only 150fs, which is less than the (maximal) separation of 190fs caused by the birefringence of the BBO crystal. However, when the bandwidth of the detected photons is limited to 4 nm, the coherence time of these photons is 520 fs and thus longer

QUANTUM COMMUNICATION WITH ENTANGLED PHOTONS

5 13

than any separation between H- and V-polarized photons. One therefore can obtain reasonable entanglement from pulsed down-conversion. The narrow filtering, though, causes a reduction in the collection efficiency of the photon pairs. From the principles of down-conversion, it follows that the sum of the frequencies of the two down-converted photons is equal to the frequency of the UV pump. Assume that we detect one of the photons behind a filter with a bandwidth significantly narrower than the pump bandwidth. Then the relative bandwidth of the correlated photon is about as large as the relative bandwidth of the pump beam. If we now select this second photon also by a narrow-band filter, we consequently lose a certain amount of coincident detections. This results in a reduction of the collection efficiency by a factor of 4 compared with similar cw-experiments. Detrimental for the observation of interference and entanglement would be the emission of two pairs from one source. If these two emissions are independent of each other, we expect Poissonian statistics - that is, the probability of a two-fold emission is p 2 / 2 , where p is the probability for emission of one pair in a pulse. In our experiments, the probability of one pair-creation per pulse was very low, on the order of lop4.Still, the probability that two independent sources fire is equal to the probability that one of the two sources creates two pairs. In order to avoid those cases, we need detectors that enable us to distinguish between the detection of one and two photons. Because the detectors used in the experiments (silicon-avalanchephoto-diodes operated in the Geiger mode) do not offer this possibility, we confined our experiments to the detection of all photons coming from the desired down-conversion processes. For example, for quantum teleportation, the effects of two emissions from one source have been suppressed, because they would have contributed, as third- or fourth-order processes, with much smaller probability .

C. INTERFEROMETRIC BELL-STATE ANALYSIS At the heart of Bell-state analysis of a pair of particles is the transformation of an entangled state to an unentangled, product state. The necessary coupling, however, has not been achieved for photons yet. But it turns out that interference of two entangled particles (and thus the photon statistics behind beamsplitters) depends on the entangled state the pair is in (Weinfurter, 1994; Braunstein, 1995);first experiment by Michler, 1996).

I . The Principle Let us first discuss the generic case of two interfering particles. If we have two otherwise indistinguishableparticles in different beams and overlap these two beams at a beamsplitter, we ask ourselves, “What is the probability of finding

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FW.12. Interference of two particles at a beamsplitter. The observation of coincident detection - that is, detection of one particle at each of the two detectors - is sensitive to the symmetry of the spatial component of the quantum state of the combined system.

the two particles in different output beams of the beamsplitter?" (See Fig. 12.) In other words, what is the probability that two detectors, one in each output beam, detect one photon each? If we perform this experiment with fermions, then we obviously expect the two fermions to arrive in different output beams. This is necessary, because according to the Pauli principle, the two particles cannot be in the same quantum state - that is, they cannot exist in the same output beam. Interfering bosons on a beamsplitter will result in both bosons in one output beam. For a symmetric 50/50 beamsplitter, it is fully random whether the two bosons will be detected in the upper or lower detector, but they will be always detected by the same detector. The reason for the different behavior lies in the different symmetry of the wavefunction describing bosonic or fermionic particles. There are four different possibilities for how the two particles could propagate from the input to the output beams of the beamsplitter. We obtain one particle in each output if both particles are reflected or both particles are transmitted, and we observe both particles at one detector if one particle is transmitted and the other reflected or vice versa. For the antisymmetric states of fermions, the two possibilities for both particles being transmitted and both being reflected interfere constructively, resulting in firing of each of the two detectors. For the symmetric state of bosons, these two amplitudes interfere destructively,giving no simultaneous detection in different output beams (Loudon, 1990). For photons (that means for bosons), this interference effect has been known since the experiments of Hong, Ou, and Mandel (1987), but up to now, it has not been observed for fermions. If we interfere two polarization-entangled photons at a beamsplitter, the Bell-state only describes the internal degree of freedom. Inspection of the four Bell-states shows that the state I@-') is antisymmetric, whereas the

QUANTUM COMMUNICATION WITH ENTANGLED PHOTONS

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other three are symmetric. However, in two-particle interferometry, all that matters is the spatial part of the wavefunction. The symmetry of it is determined by the requirement that for two photons, the total state has to be symmetric again. We therefore obtain, for the total state of two photons in the antisymmetric Bell-state falling from two beams a and b at the beam splitter,

9 -), we also have an antisymmetric spatial part This means that for the state 1 of the wavefunction and thus expect a different detection probability - that is, coincidences between the two detectors, compared with the other three Bell-states (Fig. 13). We therefore can discriminate the state IS-) from all the other states. It is the only one that leads to coincidences between the two detectors in the output beams of the beamsplitter. Can we also identify the other Bell-states? If two photons are in the state IS+), they will both propagate in the same output beam, but with orthogonal polarization in the H/V basis, whereas two photons in the state I@+)or in the state I@ -), which also both leave the beamsplitter in the same output arm, have the same polarization in this basis. Thus we can further discriminate between the state IS + ) and the states I@*) by a polarization analysis in the H/V basis and observing either coincidences between the outputs of a two-channel polarizer or both photons again in only one output (Fig. 14). Note that reorientation of the polarization analysis allows us to 0000

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FIG. 14. Bell-state analyzer for identifying the Bell-states Ilk+) and Ilk-) by observing different types of coincidences. The other two Bell-states I@*) exhibit the same detection probabilities(both photons detected by one detector) for this set-up and cannot be distinguished.

separate any one of these three states from the other two, but it is not possible to distinguish among all of them simultaneously (Vaidman, 1998). If the photons were entangled in yet another degree of freedom, one could also discriminatebetween the states I@+) and I@-) (Kwiat and Weinfurter, 1998). But, up to now, no quantum communication scheme seems to profit from this fact. Summarizing, we conclude that two-photon interference can be used to identify two of the four Bell-states, with the other two giving a third detection result. Thus we cannot perform complete Bell-state analysis by these interferometric means, but we can identify three different settings in quantum dense coding, and for teleportation the identification of one of the Bell-states is already sufficient to transfer any quantum state from one particle to another, though only in a quarter of the trials.

2. Bell-State Analysis of Independent Photons The above description of how to apply two-photon interference for Bell-state analysis can only give some hint on the possible procedures. One intuitively feels that the necessary joint detection of the two photons has to be “in coincidence.” But what really are the experimental requirements for the two

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photons to interfere? The coincidence conditions can be obtained with the use of a more refined analysis that takes into account the multimode nature of the states involved (Zukowski et al., 1995). For interference to occur, the contributing possibilities of finding one photon in each output have to be indistinguishable.If the two photons come from different sources or, as is the case in the experiments, from different down-conversion emissions, there might be some timing information (in our case, detection of the second photon of each down-conversion) that might render the possibilities distinguishable. For example, if we detect one photon behind the beamsplitter at almost the same time as one of the additional downconversion photons, we can infer the origin of the photon to interfere. However, if the time difference of the detection events of the two interfering photons - that is, the overlap at the beamsplitter - is much less than their coherence time, then the detection of any other photon cannot give any additional information about their origin. This ultra-coincidence condition requires the use of narrow filters in order to make the coherence time as long as possible. Nevertheless, even if we consider using state-of-the-art interference filters yielding a coherence time of about 3ps, no detectors fast enough exist at present. And an even stronger filtering by Fabry-Perot cavities (to achieve the necessary coherence time of about 500ps) results in forbiddingly low count rates. An alternative approach is not to try to detect the two photons simultaneously, but rather to generate them with a time definition much better than their coherence time. Consider two down-conversion processes pumped by pulsed UV beams (either two crystals or, as is the case in our experiments, one crystal pumped by two passages of a UV beam). Again we attempt to observe interference of two photons, one from each down-conversion process. Then, without any narrow filters in the beams, the tight time correlation of photons coming from the same down-conversion permits one again to associate simultaneously detected photons with each other. This provides path information and hence prohibits interference. We thus now insert filters before (or behind) the beamsplitter. With standard filters, and thus also with high enough count rates, one easily achieves coherence times on the order of 1 ps. And it is also possible to pump the two down-conversion processes with UV pulses with a duration shorter than 200 fs. Thus it follows that the photons detected behind the beamsplitter carry practically no information anymore on the detection times of their twin photons, and conversely, detection of those latter photons does not give which-path information, which would destroy the interference. The “coincidence time” for registering the photons now can be very long; it should just be shorter than the repetition time of the UV pulses, which is on the order of lOns for commercially available laser systems. One thus can

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expect very good interference visibility and precision of the Bell-state analysis. D. TRANSFORMATION, MANIPULATION, AND DETECTION OF SINGLE PHOTONS

For polarization-entangled photons, the unitary transformations between the four Bell-states can be done with standard half- and quarter-wave retardation plates. These birefringent quartz plates have different indices of refraction for polarization parallel and orthogonal to the optical axis and, in accordance with their name, cause retardation of one polarization component by either a half or a quarter of the optical wavelength relative to the other polarization component. If the incoming polarization is at 45" relative to the optical axis of a half-wave plate, it is rotated by 90"; i.e., horizontal polarization is changed to vertical polarization, and vice versa. This corresponds to the unitary transformation CT, and can be used to change from the Bell-state I*+) to I@+), or from I*-) to I@-), and so on. The unitary transformation o, needs a phase shift of IT between the two components, which effectively is just what the half-wave plate does if it is oriented parallel to the respective polarization. However, then to obtain 0 phase shift, i.e., the unity transformation, one would have to remove the halfwave plate out of the beam. Because this might change the optical path lengths and thus the alignment of the experiment, one has to use another solution. If one inserts a quarter-wave plate into the beam, with optical axis parallel to V, the V polarization is advanced by a quarter-wave. If one now rotates this plate by 90", such that the optical axis is parallel to H, the H polarization is advanced by a quarter-wave, resulting in a net phase change of exactly n. Therefore, in order to have maximum freedom in setting any of the Bellstates, one inserts one half-wave and one quarter-wave plate into the beam. Precompensating the additional quarter-wave shifts by the compensator plates of the EPR source, one obtains at the output of this transformation plates the state I@-) if both optical axes are aligned along the vertical direction. Rotation of only the quarter-wave plate to the horizontal direction transforms this state to I*+), rotation of only the half-wave plate to 45" gives I@-). Finally, rotating both plates by 90" and 45", respectively, one obtains I@'). For the first experimental realizations of the quantum communication ideas, such static polarization manipulations are sufficient. However, for quantum cryptography, and also for practical applications of the other schemes, one would like to switch the unitary transformation rapidly to any position. This can be achieved by fast Pockels cells. Depending on the applied voltage, these devices have different indices of refraction for two orthogonal polarization components and can be used similarly to the quartz retardation plates described above (Weihs et al., 1998).

QUANTUM COMMUNICATION WITH ENTANGLED PHOTONS

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Manipulations of the spatial modes of the photons, a possible extra degree of freedom, can be done with Mach-Zehnder interferometers and extra phase shifters. The Mach-Zehnder interferometer will change the amplitude of a photon being in mode a or in mode b, depending on the internal phase. Formally, this is equivalent to the usage of the half-wave plate for polarized photons. In a similar way one can also generalize to other degrees of freedom, such as different arrival times, etc. Detection of the single photons was done using silicon avalanche photodiodes operated in the Geiger-mode. The diodes used have a detection efficiency of about 40%. (Because of losses in the interference filters and other optical components, the overall detection efficiency of a photon emitted from the source was around 10% in the cw-experiments. For experiments using the pulsed source, we arrived at an efficiency of only about 4%.) In many interference experiments, a good definition of the transverse mode structure of the beams is necessary. Therefore, an ideal solution to achieve high interference contrast is to couple the output arms of a beamsplitter into single-mode fibers and connect these fibers to pigtailed avalanche photo diodes. The single-mode fiber acts as a very good spatial filter for the transverse modes and couples the light efficiently to the diodes. We avoided using fiber couplers instead of the standard beamsplitters and polarizers. This is because of additional complications due to the birefringence of the fibers and because most fiber components are not regularly commercially available for the wavelengths used in the experiments.

V. The Quantum Communication Experiments After all the basic building blocks for our quantum communication experiments have been devised, they can be put together to make possible the first realizations of the basic principles. The handling of entangled photons, their production and their detection, and (above all) the Bell-state analysis require significant improvements to bring quantum communication further into the realm of practical application. At this stage, however, quantum communication with entangled photons has proven its power and its additional features in first proof-of-principle experiments. A. QUANTUM CRYPTOGRAPHY The first experiments (Townsend et al., 1993) concentrated on the distribution of pairs of entangled photons over large distances, rather than also including fast and random switching. In order to minimize the losses in optical fiber, one

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Harald Weinfurter

of the photons was chosen with a wavelength of h = 1300nm, the other in the near infra-red for optimal detection efficiency (here the down-conversion was pumped by a krypton ion laser at 460nm). Time-energy entanglement was used with asymmetric interferometers at the observer stations and selection of true coincidences (see Section IV.B.l). Such a scheme allows the correlated photons to have a wide frequency distribution, and thus relatively high intensity, because the visibility of the interference effects depends on the monochromaticity of pump laser light. In a more recent experiment, both photons have been produced with a wavelength around 1300nm. Here, for the first time, laser diodes (h = 650nm) were used for pumping the downconversion, in contrast with the expensive laser systems used in other experiments. This allowed the demonstration of nonclassical correlations between two observers separated by more than lOkm in the Geneva area (Tittel et al., 1998).Standard optical telecom fibers connecting offices of the Swiss telecom were used to send the photons to two interferometers, where phase modulation served to set the analysis parameters. The robustness of the source, together with the high degree of quantum entanglement, opens new prospects for this secure-communicationtechnique. As this manuscript is being written, preparation of the first realistic demonstration of quantum cryptography with entangled photons is just getting under way in our labs. It is a further development of the long-distance Bellexperiment with independent observers (Weihs et al., 1998). Two observers, located in buildings at opposite ends of the campus of the science faculty of the University of Innsbruck, are separated by a straight distance of 400m. They are fully independent of each other, and each is equipped with a physical random-number generator controlling the orientation of the polarization analysis by a rapid switching Pockels cell and with a Rb-atomic clock for precise, independent recording of the time of the detection events. Both are connected via glass fibers to a source of polarization-entangled photons. As a next step, the Pockels cells will be oriented in such a way as to analyze the security of our quantum key exchange with Wigner's Bell-inequality. For the first time, this achieves fully random setting of the analyzer direction for each of the detected photons (the analyzer direction is set according to the value of the random-number generator at a frequency of 10MHz). In full analogy with the original idea of quantum cryptography, data are recorded over a certain time interval (here 1 s, which yields about 4000 detected pairs), and after this, the public discussion is performed to obtain the secret key. B. QUANTUM DENSECODING The experiment consists of three distinct parts (Fig. 15): the EPR source generating entangled photons in a well-defined state; Alice's station for encoding

QUANTUM COMMUNICATION WITH ENTANGLED PHOTONS

52 1

FIG. 15. Experimental set-up for quantum dense coding. Because of the nature of the Siavalanche photodiodes, the extension shown in the inset is necessary for identifying two-photon states in one output.

the messages by a unitary transformation of his particle; and finally, Bob’s Bell-state analyzer to read the signal sent by Alice. The polarization-entangled photons were produced by degenerate noncollinear type-11 down-conversion in a nonlinear BBO crystal. A UV beam (h = 351 nm) from an argon ion laser is down-converted into pairs of photons (h = 702 nm) with orthogonal polarization. We obtained the entangled state I\E - ) after compensation of birefringence in the BBO crystal along two distinct emission directions (carefully selected by 2-mm irises 1.5 m away from the crystal). One beam was first directed to Alice’s encoding station, the other directly to Bob’s Bell-state analyzer. In the alignment procedure, optical trombones were employed to equalize the path lengths to well within the coherence length of the down-convertedphotons (l x 100 pm) in order to observe the two-photon interference. As mentioned before, for polarization encoding, the necessary transformations of Alice’s particle were performed using a half-wave retardation plate for changing the polarization and a quarter-wave plate to generate the polarizationdependent phase shift. The beam manipulated in this way in Alice’s encoding station was then combined with the other beam at Bob’s Bell-state analyzer. It consisted of a single beamsplitter followed by two-channel polarizers in each of its outputs and proper coincidence analysis between four single-photon detectors. The experiments were performed by first setting the output state of the source such that the state I@-) left Alice’s encoder when both retardation

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plates were set to vertical orientation, and the other Bell-states could then be generated with the respective settings. To characterize the interference observable at Bob’s Bell-state analyzer, we varied the path length difference A of the two beams with the optical trombone. If the path length difference is larger than the coherence length, no interference occurs, and one obtains classical statistics for the coincidence count rates at the detectors. For optimal path length tuning, interference enables one to read the encoded information. Figures 16 and 17 show the dependence of the coincidence rates CHV(e) and CHVI(0)on the path length difference for I@’+)and I@-), respectively (the rates CHjV,and CHjVdisplay analogous behavior; we use the notation C Afor ~ the coincidence rate between detectors DA and DB). For perfect path length tuning, CHVreaches its maximum for I@+) (Fig. 16) and vanishes (aside from noise) for I@-) (Fig. 17). CHV/displays the opposite dependence and clearly signifies I@-). The results of these measurements imply that if both photons are detected, we can identify the state I@+)with a reliability of 95% and the state I@-) with 93% reliability. The performance of the dense coding transmission is influenced not only by the quality of the interference alignment but also by the quality of the states sent by Alice. In order to evaluate the latter, the beamsplitter was translated out of the beams. Then an Einstein-Podolsky-Rosen Bell-type correlation measurement analyzed the degree of entanglement of the source as well as the quality of Alice’s transformations. The correlations were only 1-2% higher than the visibilities with the beamsplitter in place, which means that the

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quality of this experiment is limited more by the quality of the entanglement of the two beams than by that of the achieved interference. When one is using silicon avalanche diodes in the Geiger-mode for singlephoton detection, a modification of the Bell-state analyzer is necessary, because then one also has to register the two photons leaving the Bell-state analyzer for the states I@*) via a coincidence detection. One possibility is to avoid interference for these states entirely by introducing polarizationdependent delays before Bob's beamsplitter. Another approach is to split the incoming two-photon state at an additional beamsplitter and to detect it (with 50% likelihood) by a coincidence count between detectors in each output (inset of Fig. 15). For the purpose of this proof-of-principle demonstration, we put such a configuration in place of detector DH.Figure 18 shows the increase of the coincidence rate C,- (0) for zero path length difference, with the other rates at the background level, when Alice sends the state I@-). Because we now can distinguish the three different messages, the stage is set for the quantum dense coding transmission. Figure 19 shows the various coincidence rates (normalized to the respective maximum rate of the transmitted state) when sending the ASCII codes of "KM"" (i.e., codes 75, 77, 179) in only 15 trits instead of 24 classical bits. From this measurement one also obtains a signal-to-noise ratio by comparing the rates signifying the actual state with the sum of the two other registered rates. The ratios for the transmission of the three states varied because of the different visibilities of the respective interferences and were

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about 14% and 9%. The achieved signal-to-noise ratio results in an actual channel capacity of 1.13 bits per transmitted (and detected) two-state photon and thus clearly exceeds the channel capacity achievable with classical communication. C. QUANTUM TELEPORTATION OF ARBITRARY QUANTUM STATES

In this experiment, polarization-entangled photons were produced again by type-I1 down-conversion in a nonlinear BBO crystal (see Fig. 20), but here the

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FIG.20. Quantum Teleportation: A UV pulse passing through a nonlinear crystal creates a pair of photons, one of which will be prepared in the initial state of photon 1 to be teleported; the other one serves as a trigger indicating that a photon to be teleported is under way. Before retroflection through the crystal, the UV pulse creates the ancillary entangled pair 2 and 3. Alice then looks for coincidences after a beamsplitter where the initial photon and one of the ancillaries are superposed. Bob, after receiving the classical information that Alice obtained a coincidence count identifying the I*-) Bell-state, knows that his photon 3 is in the initial state of photon 1, which then can be checked using polarization analysis.

UV beam was pulsed to obtain the high time definition of the pair creation (pulses with a duration of about 200fs and h = 394nm). The entangled pair of photons 2 and 3 is produced in the first passage of the W pulse through the nonlinear crystal, and the pair 1 and 4 is produced after reflecting the pulse at a mirror back through the crystal. Mirrors and beamsplitters (BS)are used to steer and to overlap the light beams. Polarizers (Pol) and polarizing beamsplitters (PBS), together with half-wave plates (h/2),prepare and analyze the polarization of the photons. All single-photon detectors indicated (silicon avalanche photo-diodes operated in the Geiger mode) are equipped with narrow-band interference filters, and the detectors of Alice’s Bell-state analyzer are also equipped with single-mode fiber couplers for spatial filtering. For the first demonstration of quantum teleportation (Bouwmeester et al., 1997), we prepared particle 1 in various nonorthogonal polarization states, using polarizer and quarter-wave plate (not shown). Behind Bob’s “receiver,” polarization analysis is performed to prove the dependence of the polarization of photon 3 on the polarization of photon 1. (In this case we used the registration of photon 4 only to define the appearance of photon 1.) To prove that any arbitrary quantum state can be transferred, we used the fact that we can also obtain entanglement between photons 1 and 4 (Fig. 21). After we removed the polarizer from arm 1 and put it into arm 4, the state of 1

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FIG. 21. Experimental setup for swapping the entanglement initially between particles 1 and 4 and between particles 2 and 3 to the new pair of particles 3 and 4. 22500

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was not defined anymore but still could be teleported to photon 3, which we proved by showing that now the entanglement is swapped to photons 3 and 4. The first task now is to prove that no information on the state of photon 1 is revealed during the Bell-state measurement of Alice. Figure 22 shows the coincidence rate between detectors f l and f2 when the overlap of photons 1 and 2 at the beamsplitter is varied (for this we changed the position of the mirror reflecting the pump beam into the crystal). The characteristic interference effect, a reduction of the coincidence rate, occurs only around zero delay. Outside this region, which is on the order of the coherence length of the detected photons, no reduction occurs, and the two photons are detected in coincidence with 50% probability. Besides statistics, there is no difference between the two data sets, although particle 1 was prepared in two mutually

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orthogonal states (+45” and -45”). Obviously, Alice has no way to determine which of the two states particle 1 was in after the projection into the Bell-state basis. Figure 23 shows the polarization of photon 3 after the teleportation protocol is performed, again when the delay between photon 1 and 2 is varied. Once interference occurs at the beamsplitter, the polarization of photon 3 is given by the settings for photon 1. The reduction in the polarization to about 65% is due to the limited degree of entanglement between photons 2 and 3 (85%) and by the reduced contrast of the interference at the beamsplitter that results from the relatively short coherence time of the detected photons. Of course, better beam definition by narrow pinholes and more stringent filtering could improve this value, but that would cause further, unacceptable loss for the four-fold coincidence rates. Each of the polarization data points shown was obtained from about 100 four-fold coincidence counts in 4000 s. Finally, in order to prove it is possible to teleport any arbitrary quantum state of a single particle, entanglement was also adjusted between photons 1 and 4 (also roughly 85%) and the polarizer was moved from arm 1 to arm 4. This enabled us to demonstrate that entanglement between particles 1 and 4 can be swapped to particles 3 and 4 [24]. Figure 24 verifies the entanglement between photon 3 and 4, conditioned on coincidence detection of photon 1 and 2. Varying the angle 8 of the polarizer in arm 4 causes a sinusoidal variation of the count rate, here with the analyzer of photon 3 set to f45”. These experiments present the first demonstration of quantum teleportation - that is, the transfer of a qubit from one 2-state particle to another. In the meantime, further steps have been taken, particularly the remote state preparation of Bob’s photon (sometimes also called “teleportation”) (Boschi et al., 1998) and, especially important, the teleportation of the state of the electro-magnetic field (Furusawa et al., 1998). The latter is the first example of the teleportation of continuous variables based on the original EPR

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Harald Weinfurter I

0 (degrees) FIG.24. Verification of the entanglement between photons 3 and 4. The sinusoidal dependence of the four-fold coincidence rate on the orientation 8 of the polarizer in arm 4 for f45" polarization analysis of photon 3 confirms the possibility of teleporting any arbitrary quantum state.

entanglement. The first experiment demonstrated the feasibility of transferring the fluctuations of a coherent state from one light beam to another. Although the experiment was limited to a narrow bandwidth of 100kHz,these are only technical limitations that result from the detection electronics, the modulators, and the bandwidth of the source of EPR-entangled light beams. In principle, it soon should be possible to transfer nonclassical states of light, such as squeezed light and number states.

VI. Conclusion Quantum communication with entangled photons has proven its power and its fascinating features. Our experiments, where realistic entanglement-based quantum cryptography is performed, where the capacity of communication channels is increased beyond classical limits, and where the polarization state of a photon was transferred to another one by means of quantum teleportation, are only first steps toward the exploitation of new resources for communication and information processing. Quantum communication can offer a wealth of further possibilities, especially when combined with simple quantum logic circuitry. Quantum computers have to operate on large numbers of qubits to demonstrate their

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power fully. But quantum communication schemes already profit from combining only a few qubits and entangled systems. Quantum logic operations with several particles have already been used in examples on the quantum coding theorem (Schumacher, 1995), but they showed their importance especially in the proposal of entanglement purification (Bennett et al., 1996). Any realistic transmission of quantum states will suffer from noise and decoherence along the line. If one wants to distribute entangled pairs of particles to, say, Alice and Bob, the entanglement between the received particles will be considerably degraded, which would prevent successful quantum teleportation, for example. If Alice and Bob now combine the particles of several such noisy pairs on each side by quantum logic operations, they can improve the quality of entanglement by the proposed distillation process. These ideas are closely related to quantum error correction for quantum computers and have recently been implemented in a proposal for efficient distribution of entanglement via so called quantum repeaters (van Enk et al., 1997). Such systems might one day form the core of quantum networks (Grover, 1997); allowing quantum communication over large distances. Of course, one always should keep in mind the obstacles that decoherence of quantum states poses (Landauer, 1996). Yet quantum communication schemes should be significantly more stable because of the much lower number of quantum systems involved. Once entangled particles have been distributed, various quantum communication protocols could be implemented. Besides those described in the preceding sections, there are some recent proposals giving a new twist to quantum information processing. Quantum gambling (Goldenberg et al., 1999) and quantum games (Eisert et al., 1998), such as a “quantized” version of the prisoner’s dilemma, bring the field of game theory to the quantum world and demonstrate new strategies in well-known classical games. But the new ideas and thoughts might be quite useful for other types of communicationproblems. For example, the quantum version of “Chinese whisper” (Hardy, 1999) can also be seen as a special type of error correction scheme. Errors in the classical communication, due to the whispering, can be more efficiently corrected if sender and receiver have been provided with entangled pairs of particles. New possibilities arise when we contemplate using entangled triples of particles. For certain tasks, the communication among three or more parties becomes less complex, and thus more efficient, if the parties share entanglement initially (Cleve and Buhrman, 1997). Furthermore, schemes for quantum cloning (Buzek and Hillery, 1996) the state of a qubit become feasible with entangled triples (Bruss et al., 1998). After significant improvements of down-conversion sources (Brendel et al., 1999) and the first observation of three-particle entanglement (Bouwmeester et al., in press), the realization of those schemes with entangled pairs - and even with entangled triples - comes within the reach of future experiments.

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For realizing entanglement purification and similar schemes, the experiments immediately become much more involved. It first has to be determined which methods can be used to perform quantum logic operations on photons and also which types of photon sources should be used. However, progress in improving experimental techniques and in better understanding the principles of quantum information theory makes the more complicated schemes feasible. Quantum cryptography was the first to venture beyond the shielded environment of quantum physics laboratories (Buttler et d.,1998) and to become a promising candidate for commercial exploitation. The future will show the enormous potential and benefits of using other quantum communication methods, such as the distribution of entanglement over large distances and the transfer of quantum information in the process of quantum teleportation.

VII. Acknowledgments This chapter describes research performed in recent years by our group at the Institut fuer Experimentalphysik, University of Innsbruck, Austria. It could not have been achieved without the significant input and work of my collegues and friends Marek Zukowski, Klaus Mattle, Paul Kwiat, Manfred Eibl, Dik Bouwmeester, Jian-Wei Pan, and Matthew Daniel. In particular I want to thank Anton Zeilinger for being the stimulating spirit in our group and for making this work possible.

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Index A Aberration correction in double-A scheme, 375-376 Above-threshold ionization (ATI), 227-232,268 ac Stark shift, 37, 103, 148, 228, 249, 36 1 Adiabatic expansion, 113 Adiabatic passage by light-induced potentials (APLIPs), 205 Adiabatic stabilization, 252, 255 Alkali atoms, 104-108 Alkanes adsorbed on surfaces, saturated, 460-472 Atom holography applications, 73 atomic beam source, 74-77 Bragg, 91 conclusions, 92 Fraunhofer hologram by cell subdivision, 8 1-84 generation of simple pattern, 8 1 imaging function and, 84-86 gray-scale, 86-89 optical reconstruction of interference, 9 1-92 phase, 90 quality evaluation, 89-90 thin-film design, 77-89 Atomic beams coherent flux, 74-75 metastable neon beam source example, 76-77 source and detector, 75-76 Atomic coherence studies, 350-352 Atomic polarizability, 99-100 Atom interferometers, wave-particle duality in atomic beam scheme, 33-34 atom interferometer, 42-48 Bragg reflection, 36-42 conclusions, 69 defined, 29 delayed choice, 48-49 experimental results, 48 experimental setup, 33-36 fluorescence detection, 36 535

historical background, 29-33 interferometer with which-way information, 54-57 internal state preparation, 35 microwave field, 34-35 plane wave theory, 42-45 quantum erasure, 57-59 standing light wave, 35 wave packet theory, 45-48 which-way detector, storing information, 49-54 which-way information, incomplete, 60-65 Wigner function, 65-68 Atoms, cooling and trapping of, 176-177 Atoms, experiments based on compared with photon-based experiments, 21-23 entanglement in the micromaser, 9-12 two-particle entanglement based on photo-dissociation, 12-21 Auger deexcitation (AD), 39 1, 425433,438-441 Auger neutralization (AN), 391, 416-424,431-433,441-443 Autler-Townes dressed states, 377

B Bell-Clauser-Home (BCH) inequality, 3,4-5 Bell inequalities, 3,4-5,493 Wigner, 496-497 Bell-states basis, 493-494,498 interferometric, 5 13-5 18 Bichromatic control of a superposition state, 296-303 Bimolecular processes coherent control and, 332-334 degenerate superpositions, 334-337 optimized bimolecular scattering, 339-342 sculpted imploding waves, 337-339 Birefringent phase shifter, 5 10 Boltzmann factor, 119

536

INDEX

Bose-Einstein condensation (BEC), 75, 112, 118, 121, 174, 176, 184 in focused-beam traps, 129- 131 of molecules, 187-190 Bound-bound emissions, 206 Bound-free emissions, 206 Bragg hologram by a standing light wave, 91 Bragg reflection, 36-42

C Chirality, symmetry breaking and generation of, 321-325 Chirped pulse amplification (CPA) scheme, 226 Circular polarization, 107, 108 Classical Monte Car10 simulations, 27 1 Coherence studies, atomic, 350-352 Coherent control (CC) benefits of, 288-289 bichromatic control of a superposition state, 296-303 conclusions, 342-343 preparation and dynamics of continuum state, 289-296 principle of, 304 Coherent control, bimolecular processes and, 332-334 degenerate superpositions, 334-337 optimized bimolecular scattering, 339-342 sculpted imploding waves, 337-339 Coherent control, weak-field interference between N-photon and M-photon routes, 304-3 15 pump-dump control, 315-321 symmetry breaking and generation of chirality, 321-325 Coherent flux, 74-75 Coherent optical manipulation, 205 Coherent population trapping (CPT), 350,367 Coherent preparation, 290 Cold, use of term, 172 Cold molecules, applications of Bose-Einstein condensation of molecules, 187-190 cold collisions, 185- 186 degenerate Fermi gases of molecules, 190

manipulation of molecules, 180- 185 molecule laser, 190- 191 nucleation and metastability, 186-1 87 S ~ ~ C ~ ~ O S178-180 COPY, Cold molecules, formation by photo-association proposed stimulated processes, 215-2 18 spontaneous decay, 206-2 15 Collinear propagation, 364-365 Collisions cold, 185-186 description of, in optical dipole traps, 121-123 in focused-beam traps, 126 laser-assisted electron-atom, 236-237,244-247,255-256 Communication loophole, 6,7-8,21 Complex polarizability, 98 Computer holography, 74 Conical atom trap, 153-155 Cooling Doppler, 109 evaporative, 112- 113 optical dipole traps and, 109-1 13 polarization-gradient, 109-1 10 Raman, 110-1 11 resolved-sideband Raman, 1 11-1 12 Cooling and trapping of atoms, 176-177 Cooling of molecules See also Cold molecules, applications of; Cold molecules, formation by photo-association near-resonance radiation, 175, 176 traditional methods, 175-176 Cooling techniques, nonoptical helium buffer-gas cooling, 192 helium cluster cooling, 174, 191- 192 three-body processes including resonances, 194- 195 trap compression and evaporative cooling, 192-1 93 Cooling techniques, optical coherent, 205 incoherent, 196-205 Counter propagation, 365-366 Crossed-beam traps, 138-141

D Dark state, 354, 369-370

537

INDEX Degenerate Fermi gases of molecules, 190 Degenerate superpositions, 334-337 Detection efficiency loophole, 6-7 Dipole force, 99, 101 -102 Doppler cooling, 109 Double-A scheme,. 353.. 368 optical aberrations correction in, 375-376 optical phase conjugation in, 371-375 Doughnut-beam trap, plugged, 151-152 Down-conversion parametric, 505-5 13 pulsed, 5 12-5 13

E Einstein, Albert, 2, 490, 503 EIT-based (electromameticallv induced transpiirency) scKeme 35 i, 369, 378-382 Elastic collisions, 123 Elastic scattering, 393-396 Electronic spin, 436-438 Electro-optic modulators (EOM), 21 Enforcement of locality (communication) loophole, 6, 7-8, 21 Entanglement after event detection, 506-507 of atoms in a micromaser, 9-12 basic features, 492-494 Bell-state analysis, 513-518 between photons, 504-505 development of, 490-491 parametric down-conversion, 505-5 13 polarization-entangled states, 508-5 12 produced by down-conversionprocess, 508 pulsed down-conversion,5 12-5 13 transformation. maniwlation. and detection of-single'photons, 5 18-519 two-particle, based on photodissociation, 12-21 Entanglement, quantum communication and, 494 conclusions, 529-530 quantum cryptography, 495-497, 520-52 1

quantum dense coding, 497-498, 521-525 quantum teleportation, 498-503, 525-529 EF'R gedankenexperiment, 2 Equivalent orbital theory, 462,465-467, 469 Essential states, 243-244 Evanescent-wave traps, 155- 162 Evaporative cooling, 112- 113 crossed-beam traps and, 139-140 trap compression and, 192-193

F Fair sampling assumption, 6 Fare-detuned light, 97 Far off-resonancetraps (FORTS), 126, 176, 177, 183 Fast Fourier transform (FFT) algorithm, 80, 86, 89 Feynman's light microscope, 30 Floquet-close-coupling (FCC) equations, 255 Floquet-Fourier expansion, 248 Flwuet Hamiltonian. 248.249 Fliuet-Lippmann-Schwinger equation, 255-256 Floquet theory basic equations, 247-249 high-frequency, 25 1-255,269-270 for laser-assisted electron-atom collisions, 255-256 R-matrix-, 256-265 Sturmian-Hoquet method, 249-251 Fluorescence detection, 36 Focused-beam traps, 125-132 Four-wave mixing processes, enhancement of, 363-367 Fraunhofer approximation, 46 Fraunhofer hologram by cell subdivision, 81-84 Free-free transitions (FFT), 236-237

G Gedankenexperiments, 2,25,30-3 1 Gordon-Volkov wave-function, 245-246 Gravito-optical cavities, 156-1 58

538

INDEX

Gravito-optical surface trap (GOST), 158- 162 Gray-scale hologram, 86-89 Greenberger-Home-Zeilinger states, 4, 23-24,491 Green's operator, 242, 243-244 Ground-state light shifts and optical potentials, 102-104

H Hanle effect, 350 Harmonic generation, 232-236 Heating, optical dipole traps and, 113-1 17 Helium buffer-gas cooling, 192 Helium cluster cooling, 174, 191-192 Hidden variables, (HV), 2 High-frequency Floquet theory (HFFT), 25 1-255,269-270 Hollow-beam traps, 150-155 Holography See also Atom holography computer, 74 matter-wave, 74 Homogeneous versus inhomogeneous surfaces, 433-436

I Imaging function, hologram and, 84-86 Incoherent interference control (IIC), strong-field, 325 computational and experimental demonstration, 330-332 theory of, 326-330 Incoherent optical cooling of molecules, 196-205 Inhomogeneous versus homogeneous surfaces, 433-436 Interaction potential, 98 metastable helium atoms and, 40 I -406 Interference crossed-beam traps and, 140-141 standing-wave traps and, 135-136 Interference between photons, 304 one- versus three-photon, 305-309 one- versus two-photon, 309-3 11 polarization control of differential cross sections, 313-315

two- versus two-photon, 3 11-313 Interferometer with which-way information, 54-57 Ion bombardment on self-assembled alkanethiolate film, 472-476 Ionization, atom, 18 above-threshold and multiphoton, 227-232 Ion neutralization (IN), 391 Ion neutralization spectroscopy (INS) compared with MIES, 392 use of term, 391 Ion scattering spectroscopy (ISS), 433

K Kirchhoff's diffraction theory, 80 Kramers frame, 240-241,25 1,258

L Lamb-Dicke regime, 112, 133 Land6 factor, 106 Laser-assisted Auger transitions, 269 Laser-assisted electron-atom collisions, 236-237,244-247 Floquet theory for, 255-256 Laser-assisted single-photon ionization (LASPI), 268 Laser-atom interaction, high-intensity above-threshold and multiphoton ionization, 227-232 basic equations for studying, 239-241 conclusions and future work on, 278-279 Floquet theory, 247-265 harmonic generation, 232-236 laser-assisted electron-atom collisions, 236-237,244-247,255-256 laser frequency, 238 laser intensity, 238-239 laser pulse duration, 238 low-frequency methods, 265-266 numerical solution of TDSE, 266-274 perturbation theory, 24 1-243 relativistic effects, 274-278 R-matrix-Floquet theory, 256-265 semiperturbative methods, 243-247 Sturmian-Floquet method, 249-25 1 Laser-induced degenerate states (LIDS), 261-263

539

INDEX Lasing without inversion (LWI), 350 Lattices, 141- 145 quasi-electrostatic, 137-138 Light-induced continuum structures (LICS), 264 Light-induced degenerate states (LIDS), 264 Light-induced states (LIS), 250-251 Light-sheet traps, 146-150 Light shift, 37, 103, 361, 367 Linear polarization, 107, 108 Lippmann-Schwingerequation, 293-294,327 Floquet-, 255-256 Liquid surfaces composition of binary, 452-453 pure, 449-452 Local hidden-variable (LHV) theory, 2-3 Loopholes, 5-8, 21 Lowest-order perturbation theory (LOP"), 227,241-243,253 Low-temperature molecules traditional techniques for formation of, 173-175

M Mach-Zehnder interferometers, 5 19 Magnetic traps, 96 Magneto-optical traps (MOTS), 33-34, 117-119, 176 Many-body S-matrix theory, 266 Matrix isolation spectroscopy, 175 Matter-wave holography, 74 Maxwell's equations, 364 Metastable atom electron spectroscopy (MAES), 387 Metastable deexcitation spectroscopy (MDS), 387 Metastable helium atoms Auger deexcitation (AD), 391, 425-433 Auger neutralization (AN), 391, 416-424,431-433 elastic scattering, 393-396 electronically elastic processes, 41 1-416 electronic excitation transfer, 416-438

electronic spin, 436-438 electron yield, 417-424 excitation transfer into bound and continuum states, 396-400 gas phase reactions of, 393-41 1 homogeneous versus inhomogeneous surfaces, 433-436 interaction potential, 401-406 interaction with surfaces, 41 1-438 properties of, 392 liquid surfaces and, 449-453 solid surfaces and, 454-482 spatial part of the wavefunction, 407-4 11 spin conservation, 400-401 Metastableinduced electron spectroscopy (ME9 Auger deexcitation and, 438-441 Auger neutralization (AN) and, 441-443 compared with INS,392 conclusions, 482-483 description of, 387-393 molecular surfaces, 443-444 other acronyms for, 387 properties of rare gas atoms, 392 resonance ionization (RI)and, 441-443 surfaces of composite materials,

444-449

use of term, 391 Metastable neon beam source example, 76-77 Metastable quenching spectroscopy (MQS), 387 Micromaser, entanglement of atoms in a, 9-12 Molecule(s) See also Cold molecules, applications of; Cold molecules, formation by photo-association; Cooling of molecules laser, 190-191 Multiphoton ionization (MPI), 227-232 Multiphoton processes above-threshold and multiphoton ionization, 227-232 harmonic generation, 232-236 laser-assisted electron-atom collisions, 236-237

540

INDEX

Multiple single-frequency (MSF) laser, 201-202,205

N Neutral impact collision ISS (NICISS), 433,434 Nonoptical cooling techniques helium buffer-gas cooling, 192 helium cluster cooling, 174, 191- 192 three-body processes including resonances, 194-195 trap compression and evaporative cooling, 192-193

0 Optical cooling techniques coherent, 205 incoherent, 196-205 Optical dipole traps advantages of, 162 applications, 96-97 atoms, number of, 119 collisions, 121- 123 conclusions, 162-165 cooling, 109-1 13 experimental techniques, 117-121 heating, 113- 117 historical background, 97 internal distribution, 120- 121 magnetic traps compared to, 96 multilevel atoms, 102-108 oscillator model, 98- 102 radiation-pressure traps compared to, 96 temperature, 119- 120 trap loading, 117-1 18 Optical dipole traps, blue-detuned, 145 evanescent-wave, 155-162 hollow-beam, 150-155 light-sheet, 146-150 Optical dipole traps, red-detuned, 123 crossed-beam, 138- 141 focused-beam, 125-132 lattices, 141-145 standing-wave, 133-138 Optical potential, 394

Optical reconstruction of atomic interference, 9 1-92 Optimal control, 288 Oscillator model, 98-102

P Parametric down-conversion, 505-5 13 Pendellosung frequency, 40 Penning ionization electron spectroscopy (PIES), 390-391,400-401 spatial part of the wavefunction, 407-4 11 Perturbation theory, 241-243,297 Phase hologram, 90 Phase matching, 348 Phaseonium, 348 Photo-association formation of cold molecules proposed stimulated processes, 215-2 18 spontaneous decay, 206-215 Photo-dissociation, 301-304 See also Coherent control; Interference between photons Photons, experiments based on, 8-9 compared with atom-based experiments, 21-23 Photons, transformation, manipulation, and detection of single, 518-519 Photon-scattering rate, 107-1 08 Plane wave theory, 42-45 Podolsky, Boris, 2,490,503 Polarization-entangled states, 508-5 12 Polarization-gradient cooling, 109- 110 in focused-beam traps, 128-129 Pulsed down-conversion, 5 12-5 13 Pump-dump control, 315-321

Q Quantum communication, entanglement and, 494 Bell-state analysis, 5 13-5 18 conclusions, 529-530 parametric down-conversion, 505-5 13 quantum cryptography, 495-497, 520-521

541

INDEX quantum dense coding, 497-498, 521-525 quantum teleportation, 498-503, 525-529 transformation, manipulation, and detection of single photons, 518-5 19 Quantum control, 288 Quantum cryptography, 495-497, 520-521 Quantum dense coding, 497-498, 52 1-525 Quantum erasure, 57-59 Quantum information theory, 490 Quantum mechanics tests applications, 24 Bell inequalities, 3,4-5 comparison between photon and atom-based experiments, 21-23 experiments based on atoms, 9-21 experiments based on photons, 8-9 future outlook, 25 Greenberger-Horne-Zeilingerstates, 4,23-24 historical overview, 1-4 1ooDholes. 5-8, 21 Quanbm teleporktion, 498-503, 525-529 Quasi-electrostatictraps (QUESTS), 131-132, 177, 184 lattices, 137-138 Qubit, 495

R Radiation-pressuretraps, 96 Raman cooling, 110-1 11 Refractive index control, 350-351, 362-363 Relativistic effects, 274-278 Rescattering rings, 232 Resolved-sideband Raman cooling, 111-112 Resonance ionization (RI),39 1, 416-417,441-443 Resonances, three-body recombination and, 194-195 Resonant multiphoton ionization (RFMPI), 243 Resonant nonlinear optics analysis of, 357-363

description of enhancement, 353-357 development of, 347-350 double4 scheme, 353, 368 double4 scheme, optical aberrations correction in, 375-376 double-A scheme, optical phase conjugation in, 371-375 enhancement of four-wave mixing processes, 363-367 origin of, 367-371 outlook for, 382-383 review of atomic coherence studies, 350-352 spectroscopyof dense coherent media, 376-382 R-matrix-Floquet 0theory, 256-265 Rosen, Nathan, 2,490,503 Rotating-wave approximation, 101,329 Rydberg states, 396,397 S

Scanning tunneling microscopy (STM), 433,434,438-439 Scattering, optimized bimolecular, 339-342 Scattering rate, 99, 101-102 Scattering theory, 394 S c h w a inequality, 300-301 Sculpted imploding waves, 337-339 Semiclassical approach, 100 Semiperturbativemethods essential states, 243-244 laser-assisted electron-atom collisions, 244-247 Single active electron (SAE) model, 268-272 Single-beam trap, 152-153 Single-particle loss coefficient, 121 Singular value decomposition (SVD) algorithm, 4 4 6 4 7 , 4 4 8 , 4 4 9 Sisyphus effect, 110, 159 Solid surfaces adsorption of oxygen on Ni( loo), 476-482 formation of NaCl layer on tungsten substrate, 454-459 ion bombardment on self-assembled allranethiolate film,472-476

542

INDEX

saturated alkanes adsorbed on, 460-472 Spatial correlation loophole, 5, 6 Spatial part of the wavefunction, 407-41 1 Spectroscopy of dense coherent media, 376-382 high-resolution molecular, 178-180 matrix isolation, 175 Spin analysis and detection of atoms, 15-21 Spin conservation, 400-401 Spin manipulation, standing-wave traps and, 136-137 Spin-polarized helium, 174-175 Spin polarized metastable deexcitation spectroscopy (SPMDS), 438 Spin relaxation, in focused-beam traps, 126- 128 Spontaneous decay, 206-2 15 Standing light wave, 35 Bragg hologram by, 91 Standing-wave traps, 133-138 Stark effect, second-order, 90 Stern-Gerlach methods, 120, 136, 137 Stimulated Raman adiabatic passage (STIRAF') process, 214-217 Stokes and anti-Stokes fields, 357-363, 364-366 Strong-field incoherent interference control. See Incoherent interference control (IIC), strong-field Sturmian-Floquet method, 249-25 1 Superposition principle, 492 Supersonic molecular beams, 174-175 Surfaces of composite materials, 444-449 homogeneous, of known material, 438-444 homogeneous versus inhomogeneous, 433-436 liquid, 449-453 solid, 454-482 Symmetry breaking, 321-325

T Thermal energy atom scattering (TEAS), 433 Thin-film hologram design, 77-89

Three-body loss coefficient, 122- 123 Three-body processes including resonances, 194-195 Time-dependent Schrodinger equation (TDSE), 234,239-240, 243-244, 255,256,257 numerical solution of, 266-274, 288 Time-of-flight method, 120 Trap compression and evaporative cooling, 192-1 93 Trapping of atoms, 176-177 Two-body loss coefficient, 121

U Ultraviolet photoelectron spectroscopy (UPS),388 Unimolecular processes, weak field coherent control and interference between N-photon and M-photon routes, 304-3 15 pump-dump control, 3 15-321 symmetry breaking and generation of chirality, 321-325

W Wave packet theory, 45-48 Wave-particle duality, in atom interferometers atomic beam scheme, 33-34 atom interferometer, 42-48 Bragg reflection, 36-42 conclusions, 69 defined, 29 delayed choice, 48-49 experimental setup, 33-36 fluorescence detection, 36 historical background, 29-33 interferometer with which-way information, 54-57 internal state preparation, 35 microwave field, 34-35 quantum erasure, 57-59 standing light wave, 35 which-way detector, storing information, 49-54 which-way information, incomplete, 60-65

543

INDEX Wigner function, 65-68 Weak-field coherent control. See Coherent control, weak-field Which-way detector, storing information, 49 beamsplitter for two internal states, 50-52 combined with microwave field, 52-54 Which-way information incomplete, 60-65

interferometer with, 54-57 Wigner function, 65-68

X X-ray photoelectron spectroscopy ( X P S ) , 388 Z

Zeeman effect, 90

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Contents of Volumes in This Serial Radiofrequency Spectroscopy of Stored Ions I: Storage, H. G. Dehmelt Optical Pumping Methods in Atomic Spectroscopy, B. Eudick Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C. Worf 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 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. Z Amos Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, B. H. Branrden The Production of Rotational and Vibrational Transitions in Encountersbetween Molecules, K. Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J. €? Toennies High-Intensity and High-Energy Molecular Beams, J. B. Anderson, R. €? Andres, and J. B. Fen

Volume 4

H.S. W. Massy-A Sixtieth Birthday Tribute,

Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and U! D. Davison Thermal Diffusion in Gases, E. A. Mason, R. J. Mum, and Francis J. Smith Spectroscopy in the Vacuum Ultraviolet, U! R. S Carton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and K Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, E J. de Heer Mass Spectrometry of Free Radicals, S.N. Foner

Volume 3 The Quantal Calculation of Photoionization Cross Sections, A. L Stewart

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. €! A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I. C. Percival Born Expansions, A. R. Holt and B. L Moiselwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionizations, C. E. 0. Mohr Recent Measurements on Charge Transfer, J. B. Hasted Measurementsof Electron Excitation Functions, D, U! 0. Heddle and R. G. U! Keesing Some New Experimental Methods in Collision Physics, R. E 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. E Boyd

545

546

CONTENTS OF VOLUMES IN THIS SERIAL

Volume 5 Flowing Afterglow Measurements of IonNeutral Reactions, E. E. Ferguson, E C. Fehsenfeld and A. L Schmelrekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions II: 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-Reuven The Calculation of Atomic Transition Probabilities, R. J. S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations s,s',, pq, C. D. H. Chisholm, A. Dalgarno, and E. R. Innes Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle

Volume 6 Dissociative Recombination,J. N. Eardsley and M. A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A. S. Kaufmnn The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa The Diffusion of Atoms and Molecules, E. A. Mason and I: 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. F! Schermann, and F! Griver Molecular Wave Functions: Calculations and Use in Atomic and Molecular Processes, J. C. Bmwne

Localized Molecular Orbitals, Hare2 Weinstein, Ruben Pauncz, and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J. Gerrarr Diabatic States of Molecules-Quasi-Stationary Electronic States, Thomas E O'Malley Selection Rules within Atomic Shells, B. R. Judd Green's Function Technique in Atomic and Molecular Physics, Gy. Csanak. H. S. Tayloz and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Narhan Wiser and A. J. Greenfreld

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. E Chen and Augustine C. Chen Photoionization with Molecular Beams, R. B. Cairns, Halsread Harrison, and R. I. Schoen The Auger Effect, E. H. S. Eurhop 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 Oka The Differential Cross Section of Low-Energy 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

CONTENTS OF VOLUMES IN THIS SERIAL

547

Volume 10

Volume 13

Relativistic Effects in the Many-Electron Atom, Lloyd Annstmng, Jr and Serge Feneuille The First Born Approximation, K. L Bell and A. K. Kingston Photoelectron Spectroscopy,M? C. Price Dye Lasers in Atomic Spectroscopy, M? Lunge, J. Luther; and A. Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcen A Review of Jovian Ionospheric Chemistry, Wesley Z Huntress, Jr:

Atomic and Molecular Polarizabilities-A Review of Recent Advances, Thomas M. Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R. Bennan Collision Experiments with Laser-Excited Atoms in Crossed Beams, 1. V Herrel 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.Nesber Microwave Transitions of Interstellar Atoms and Molecules, W B. Somerville

Volume 11 The Theory of Collisions between Charged Particles and Highly Excited Atoms, I. C. Percival and D. Richads Electron Impact Excitation of Positive Ions, M. J. Seaton The R-Matrix Theory of Atomic Process, I? G. Burke and M? D. Robb Role of Energy in Reactive Molecular Scattering: An Information-TheoreticApproach, R. B. Bemstein and R. D. Levine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Griem Chemiluminescencein Gases, M. E 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, I? Lambmpoulos Optical Pumping of Molecules, M. Broyer; G. Goudedard, J. C. Lehmann, and J. Kgu’ Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid

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 E. Stewar? (e, 2e) Collisions, Erich Weigold and Ian E. McCarrhy ForbiddenTransitions in One- and no-Electron Atoms, Richard Manus and Petel; J. Mohr Semiclassical Effects in Heavy-Particle Collisions, M. S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in IonAtom Collisions, S. V Bobashev Rydberg Atoms, S. A. Edelstein and i? E Gallagher UV and X-Ray Spectroscopy in Astrophysics, A. K. Dupree

Volume 15 Negative Ions, H.S. W Massey Atomic Physics from Atmospheric and Astrophysical Studies, A. Dalgamo Collisions of Highly Excited Atoms, R. E Stebbings

548

CONTENTS OF VOLUMES IN THIS SERIAL

Theoretical Aspects of Positron Collisions in Gases, J. W Hwnbersron Experimental Aspects of Positron Collisions in Gases, I: C. Gri@h Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein Ion-Atom Charge Transfer Collisions at Low Energies, J. B. Hasted Aspects of Recombination, D. R. Bares The Theory of Fast Heavy Particle Collisions, B. H.Bramden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H. B. Gilbody 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-Fwk Theory, M. Cohen and R. P McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. Diiren Sources of Polarized Electrons, R. J. Celorta and D. Z Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopy of Laser-Produced Plasmas, M. H. Key and R. J. Hurcheon 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. E Cmrhers

Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot Superfluorescence,M. E H. S c h u u m n s , Q.H. E 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. Hursr, and G. W Folrz Inner-Shell Vacancy Production in Ion-Atom Collisions, C. D. Lin and Parrick Richard Atomic Processes in the Sun, R L Dufton and A. E. Kingston

Volume 18 Theory of Electron-Atom Scattering in a Radiation Field, Leonard Rosenberg Positron-Gas Scattering Experiments, Talbert S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, J. Morellec, D. N o m n d , and G. Petite Classical and SemiclassicalMethods in Inelastic Heavy-Particle Collisions, A. S. Dickimon andD. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B. R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N. Anderson and S. E. Nielsen Model Potentials in Atomic Structure, A. Hibben Recent Developments in the Theroy of Electron Scattering by Highly Polar Molecules, D. W Norcmss and L. A. Collins Quantum Electrodynamic Effects in FewElectron Atomic Systems, G. W E Dmke

Volume 19 Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B. H. Bmnsden and R. K. Janev Interactions of Simple Ion-Atom Systems, J. I: Park High-Resolution Spectroscopy of Stored Ions, D. J. Wneland, Wayne M. Itano. and R. S. V m Dyck JI:

CONTENTS OF VOLUMES IN THIS SERIAL Spin-DependentPhenomena in Inelastic Electron-Atom Collisions, K. BIum and H. Kleinpoppen The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, E JenE The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N.A. Cherepkov

Volume 20 Ion-Ion Recombination in an Ambient Gas, D. R. Bates Atomic Charges within Molecules, G. G. Hall Experimental Studies on Cluster Ions, Z D. Mark and A. W Castleman, J,: Nuclear Reaction Effects on Atomic Inner-Shell Ionization, W E. Meyerhof and J.-E Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bortcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstmng On the Problem of Extreme UV and X-Ray Lasers, I. L Sobel'man and A. c! Vinogradov Radiative Properties of Rydberg State, in Resonant Cavities, S.Hamche 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 F? 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

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. Zurrone 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, Z C. Grifirh Doubly Excited States, Including New Classification Schemes, C. D. f i n Measurementsof 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, Edwanl Pollack and Y h p Hahn Relativistic Heavy-Ion-Atom Collisions, R. Anholt and Harvey Gould Continued-FractionMethods in Atomic Physics, S. Swain

Volume 23 Vacuum Ultraviolet h e r Spectroscopy of Small Molecules, C. R. Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian l? Grant and Harry 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 h e r Radiation, E. J. Wuilleumier;D. L Ederer; and J. L Picque'

550

CONTENTS OF VOLUMES IN THIS SERIAL

Volume 24 The Selected Ion Flow Tube (SIDT): Studies of Ion-Neutral Reactions, D. Smith and N. G. A d a m Near-ThresholdElectron-Molecule Scattering, Michael A. Morrison Angular Correlation in Multiphoton Ionization of Atoms, S. J. Smith and G. Leuchs Optical Pumping and Spin Exhange in Gas Cells, R. J. Knize, Z Wu, and W Happer Correlations in Electron-Atom Scattering, A. Cmwe

Volume 25 Alexander Dalgarno: Life and Personality, David R,Bates and George A. Victor Alexander Dalgamo: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgamo: Contributions to Aeronomy, Michael B. McElmy 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. E Stebbings Atomic Excitation in Dense Plasmas, Jon C. Weisheit Pressure Broadeningand Laser-Induced Spectral Line Shapes, Kenneth M. Sand0 and Shih-I Chu Model-Potential Methods, G. Loughlin 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 Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Borrcher

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 E Drake and S. P: Goldman Dissociation Dynamics of Polyatomic Molecules, I: Uzer Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate I? Kirby and Ewine E van Dishoeck The Abundances and Excitation of Interstellar 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, Mineo Kimura and Neal E Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V Sidis Associative Ionization: Experiments, Potentials, and Dynamics, John Weiner; Franqoise Masnou-Sweeuws, and Annick Giusti-Suzor On the Decay of lE7Re:An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen. Leonard Rosenberg, and Larry Spruch Progress in Low Pressure Mercury-Rare Gas Discharge Research, J. Maya and R. Lagushenko

Volume 27 Negative Ions: Structure and Spectra, David R. Bates Electron Polarization Phenomena in ElectronAtom Collisions, Joachim Kessler

CONTENTS OF VOLUMES IN THIS SERIAL 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, I! I. Lengyel 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 Developmentsin 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 MultiphotonIonization of Atoms, M. I! Ammosov, N. B. Delone, M. Yu. Ivanov, I. I. Bondar; and A. I!Masalov Collision-InducedCoherences in Optical Physics, G. S. Aganval Muon-Catalyzed Fusion, Johann Rafelski and Helga E. Rafelski Cooperative Effects in Atomic Physics, J. t? Connerade Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J. H. McGuire

Volume 30 Differential Cross Sections for Excitation of Helium Atoms and Helium-Like Ions by Electron Impact, Shinobu Nakaznki

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Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Tmjmar and J. C. Nickel The DissociativeIonization of Simple Molecules by Fast Ions, Colin J. Latimer Theory of Collisions between Laser Cooled Atoms, t?S. Julienne, A. M. Smirh, and K. Burnett Light-Induced Drift, E. R. Eliel Continuum Distorted Wave Methods in IonAtom Collisions, Derrick S. E Cmrhers and Louis J. Dub6

Volume 31 Energies and Asymptotic Analysis for Helium Rydberg States, G. W E Drake Spectroscopy of Trapped Ions, R. C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H. Walrher Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, RudolfDuren, and Jacques Robert Atomic Physics and Non-Maxwellian Plasmas, Michtle Lamoureur

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, I? G. Burke Terrestrial and Extraterrestrial H3+, Alexander Dalgarno Indirect Ionization of Positive Atomic Ions, K. Dolder Quantum Defect Theory and Analysis of HighPrecision Helium Term Energies, G. W! E Drake Electron-Ion and Ion-Ion Recombination Processes, M. R. FInnnery Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H. B. Gilbody

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CONTENTS OF VOLUMES IN THIS SERIAL

Relativistic Electronic Structure of Atoms and Molecules, I. F! 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 Humbersron How Perfect are Complete Atomic Collision Experiments?, H. Kleinpoppen and H. Handy Adiabatic Expansions and Nonadiabatic Effects, R. McCarmll 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 AfterglowLangmuir Technique, David Smith and Parrik Spanil Exact and Approximate Rate Equations in Atom-Field Interactions, S.Swain Atoms in Cavities and Traps, H. Walther Some Recent Advances in Electron-lmpact Excitation of n = 3 States of Atomic Hydrogen and Helium, J. E Williamsand 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. t i n , L. W Andersen, and J. W McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Analysis of Scattered Electrons, S. Trajmar and J. W McConkey 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. Cilbody The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Barry I. Schneider Analytic Representation of Cross-Section Data, Mitio Inokuti, Mineo Kimura, M. A. Dillon, and Isao Shimamura

Electron Collisions with N2, 02 and 0: What We Do and Do Not Know, Yukikazu Itiknwa Need for Cross Sections in Fusion Plasma Research, Hugh P Summers Need for Cross Sections in Plasma Chemisby, M. Capitelli, R. Celiberro, and M. Cacciatore 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. 1.: Chiao, t? G. Kwiat, and A. M. Steinberg Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andreas Buchleitner 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 Photoionizationof Molecules,N.A. Cherepkov Role of Two-Center Electron-Electron Interaction in Projectile Electron Excitation and Loss, E. C. Montenegm, W E. Meyerhof; and J. H. McCuire 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 E DiMaum and I? Agostini Infrared Spectroscopy of Size Selected Molecular Clusters, U.Buck

CONTENTS OF VOLUMES IN THIS SERIAL Femtosecond Spectroscopy of Molecules and Clusters, 'I: Baumer and G. Gerber Calculation of Electron Scattering on Hydrogenic Targets, I. Bray and A. 'I: Stelbovics Relativistic Calculations of Transition Amplitudes in the Helium Isoelectronic Sequence, U? R. Johnson, D. R. Plante, and J. Sapirstein Rotational Energy Transfer in Small Polyatomic Molecules, H. 0. Everin and E C. De Lucia

Volume 36 Complete Experimentsin Electron-Atom Collisions, Nils Overgaad Andersen and Klaus Bartschat Stimulated Rayleigh Resonances and RecoilInduced Effects, J.-L Courtois and G. Grynberg Precision Laser Spectroscopy Using AcoustoOptic Modulators, U? A. van Wijngaaden Highly Parallel Computational Techniques for Electron-Molecule Collisions, Carl Wnsread and Vincent McKoy Quantum Field Theory of Atoms and Photons, Maciej kwenstein and Li You

Volume 37 Evanescent Light-Wave Atom Mirrors, Resonators, Waveguides, and Traps, Jonarhan l? Dowling and Julio Gea-Banacloche Optical Lattices, l? S.Jessen and I. H. Deutsch Channeling Heavy Ions through Crystalline Lattices, Herbert E Krause and Sheldon Datz Evaporative Cooling of Trapped Atoms, Wolfgang Kerterle and N. J. van Druten Nonclassical States of Motion in Ion Traps, J. I. Cirac. A. S. Parkins, R. Blan, and f? Zoller The Physics of Highly-Charged Heavy Ions Revealed by Storage/Cooler Rings, l? H. Mokler and Th. Stohlker

Volume 38 Electronic Wavepackets, Robert R. Jones and L. D. Nooniam

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Chird Effects in Electron Scattering by Molecules, K. Blum and D. G.Thompson Optical and Magneto-Optical Spectroscopy of Point Defects in Condensed Helium, Serguei I. Kanorsky and Antoine Weis Rydberg Ionization: From Field to Photon, G. M. Lankhuijzen and L D. N o o h Studies of Negative Ions in Storage Rings, L. H.Andersen, ZAndersen, and l? Hvelplund Single-Molecule Spectroscopy and Quantum Optics in Solids, W E. Moernes R. M. Dicbon, and D. J. Norris

Volume 39 Author and Subject Cumulative Index Volumes 1-38 Author Index Subject Index Appendix: Tables of Contents of Volumes 1-38 and Supplements

Volume 40 Electric Dipole Moments of Leptons, Eugene D. Commins High-Precision Calculations for the Ground and Excited States of the Lithium Atom, Frederick W King Storage Ring Laser Spectroscopy, Thomas U.Kiihl Laser Cooling of Solids, Carl E. Mungan and i'lmothy R. Gosnell Optical Pattern Formation, L. A. hgiaro, M. Brambilla and A. Gani

Volume 41 Two-Photon Entanglement and Quantum Reality, Yanhua Shih Quantum Chaos with Cold Atoms, Mark G. Raizen Study of the Spatial and Temporal Coherence of High-Order Harmonics, Pascal Saliires, Ann L'Huiller Philippe Antoine, and Maciej Lewenstein

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CONTENTS OF VOLUMES IN THIS SERIAL

Atom Optics in Quantized Light Fields, Manhias Freyburgez Alois M. Herkommez Daniel S. Krahmec Erwin Mayr; and Wolfgang I? Schleich Atom Waveguides, Victor I. Balykin Atomic Matter Wave Amplification by Optical Pumping, Ulf Janicke and Martin Wilkens

Volume 42 Fundamental Tests of Quantum Mechanics, Edward S. Fry and Thomas Walther Wave-Particle Duality in an Atom Interferometer, Stephan Diirr and Gerhard Rempe Atom Holography, Fujio Shimizu

Optical Dipole Traps for Neutral Atoms, Rudolf G r i m Matthias Weidemiiller; and Yurii B. Ovchinnikov Formation of Cold (T 5 1K)Molecules, J. Z Bahns, I? L. Gould, and W C. Stwalley High-Intensity Laser-Atom Physics, C.J. Joachain. M. Don; and N. J. Kylstra Coherent Control of Atomic, Molecular, and Electronic Processes, Moshe Shapiro and Paul Brumer Resonant Nonlinear Optics in Phase Coherent Media, M. D. Lukin. F? Hemmer; and M. 0. Scully The Characterization of Liquid and Solid Surfaces with Metastable Helium Atoms, H.Morgner Quantum Communication with Entangled Photons, Harald Weinfurter

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ISBN 0-12-003842-0