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ATOM INTERFEROMETRY

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ATOM INTERFEROMETRY
Edited by
Paul R. Berman
PHYSICS DEPARTMENT UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN

ACADEMIC PRESS

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London Sydney

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Copyright 91997 by Academic Press
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Library of Congress Cataloging-in-Publication Data

Atom interferometry / edited by Paul R. Berman.

p. cm.

Includes bibliographical references and index.

ISBN 0-12-092460-9

1. Interferometry. 2. Interferometers. 3. A t o m s a O p t i c a l

properties. I. Berman, Paul R.

QC411.A86 1997

96-9899

539.7mdc20

CIP

Printed in the United States of America 96 97 98 99 00 BB 9 8 7 6 5 4 3 2 1

Contents

CONTRIBUTORS

ix

PREFACE

xiii

Optics and Interferometry with Atoms and Molecules

J. Schmiedmayer, M. S. Chapman, C. R. Ekstrom, T. D. Hammond, D. A. Kokorowski, A. Lenef, R. A. Rubenstein, E. T. Smith, and D.E. Pritchard

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

2

II. Beam Machine . . . . . . . . . . . . . . . . . . . . . . . . .

4

III. Optics for Atoms and Molecules . . . . . . . . . . . . . . . .

9

IV. Interferometry with Atoms and Molecules . . . . . . . . . . .

18

V. A t o m Interferometry Techniques . . . . . . . . . . . . . . . .

30

VI. Measuring Atomic and Molecular Properties ..........

39

VII. Fundamental Studies . . . . . . . . . . . . . . . . . . . . . .

51

VIII. Inertial Effects . . . . . . . . . . . . . . . . . . . . . . . . .

65

IX. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

Appendix: Frequently Used Symbols . . . . . . . . . . . . . .

76

References . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Classical and Quantum Atom Fringes

H. Batelaan, S. Bernet, M. K. Oberthaler, E. M. Rasel, J. Schmiedmayer, and A. Zeilinger

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

85

II. Experimental Apparatus . . . . . . . . . . . . . . . . . . . . .

86

III. Classical Atom Fringes: The Moir6 Experiment ........

90

IV. Quantum Fringes: The Interferometer . . . . . . . . . . . . . .

100

V. C o m p a r i n g Classical and Q u a n t u m Fringes: The Classical

Analog to an Interferometer . . . . . . . . . . . . . . . . . .

108

VI. Atoms in Light Crystals . . . . . . . . . . . . . . . . . . . .

112

References . . . . . . . . . . . . . . . . . . . . . . . . . . .

118

Generalized Talbot-Lau Atom Interferometry

J. F. Clauser and S. Li

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

121

II. SBE Interferometry . . . . . . . . . . . . . . . . . . . . . . .

122

vi

Contents

III. GTL Interferometry vs. SBE Interferometry ..........

123

IV. What Happens When Frauenhofer Diffraction Orders

Overlap? . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

V. Historical Development of the Generalized Talbot

Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130

VI. Spatial Properties of the Generalized Talbot Effect

"Image" . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132

VII. Wavelength Dependence of the Spatial Spectrum of the Fringe

Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

VIII. The Lau Effect . . . . . . . . . . . . . . . . . . . . . . . . .

135

IX. The Talbot Interferometer . . . . . . . . . . . . . . . . . . . .

136

X. Generalized Lens-Free Talbot-Lau Interferometers .......

136

XI. Fresnel Diffraction and the Talbot Effect with a Spatially

Varying Potential . . . . . . . . . . . . . . . . . . . . . . . .

138

XII. GTL Atom Interferometry Experiments with K and Li2 .....

140

XIII. Talbot Interferometer Using Na .................

143

XIV. "Heisenberg Microscope" Decoherence GTL Atom

Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . .

144

XV. Conclusions and Future Applications . . . . . . . . . . . . . .

147

Appendix: Kirchoff Diffraction with Spatially Varying V(r)...

148

References . . . . . . . . . . . . . . . . . . . . . . . . . . .

150

Interferometry with Metastable Rare Gas Atoms

F. Shimizu

Io Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .

153

II. Atomic Beam Source . . . . . . . . . . . . . . . . . . . . . .

153

III. Young's Double-Slit Experiment ................

158

IV. Holographic Manipulation of Atoms . . . . . . . . . . . . . .

161

V. Two-Atom Correlation . . . . . . . . . . . . . . . . . . . . .

164

References . . . . . . . . . . . . . . . . . . . . . . . . . . .

169

Classical and Nonclassical Atom Optics

C. Kurtsiefer, R. J. C. Spreeuw, M. Drewsen, M. Wilkens, and J. Mlynek

I~ Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .

171

II. Models and Notation . . . . . . . . . . . . . . . . . . . . . .

173

III. Atom Focusing and Applications ................

177

IV. Correlation Experiments with Atoms and Photons . . . . . . .

190

V. Scheme for an Atomic Boson Laser . . . . . . . . . . . . . .

205

References . . . . . . . . . . . . . . . . . . . . . . . . . . .

214

Contents

vii

Atom Interferometry and the Quantum Theory of Measurement

H.-J. Briegel, B.-G. Englert, M. O. Scully, and H. Walther

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

217

II. Fundamental Physics and Atom Interferometers ........

219

III. The Stern-Gerlach Interferometer . . . . . . . . . . . . . . .

240

IV. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

253

References . . . . . . . . . . . . . . . . . . . . . . . . . . .

253

Matter-Wave Interferometers: A Synthetic Approach

C. J. Bord~

I. Physics of the Generalized Beam Splitter . . . . . . . . . . . .

257

II. Architecture of Interferometers . . . . . . . . . . . . . . . . .

276

III. Sensitivity to Gravitational and Electromagnetic Fields:

A Unified Approach through the Dirac Equation ........

281

IV. Conclusions and Directions of Future Progress . . . . . . . . .

288

References . . . . . . . . . . . . . . . . . . . . . . . . . . .

290

Atom Interferometry Based on Separated Light Fields

U. Sterr, K. Sengstock, W. Ertmer, F. Riehle, and J. Helmcke

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

293

II. Theoretical Framework . . . . . . . . . . . . . . . . . . . . .

299

III. Discussion of Different Types of Interferometers ........

312

IV. Experimental Realization of Bord6 Interferometry . . . . . . .

318

V. Precision Determination of Physical Quantities . . . . . . . . .

331

VI. Geometrical and Topological Phases . . . . . . . . . . . . . .

339

VII. Influence of the Quantum-Mechanical Measurement Process in

the Interferometer . . . . . . . . . . . . . . . . . . . . . . . .

349

VIII. Applications of Atom Interferometry in Optical Frequency

Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351

IX. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .

358

References . . . . . . . . . . . . . . . . . . . . . . . . . . .

358

Precision Atom Interferometry with Light Pulses

B. Young, M. Kasevich, and S. Chu

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

363

II. Interferometer Theory . . . . . . . . . . . . . . . . . . . . .

365

viii

Contents

III. Multiphoton Transitions . . . . . . . . . . . . . . . . . . . .

375

IV. Inertial Force Measurements . . . . . . . . . . . . . . . . . .

389

V. Photon-Recoil Measurement . . . . . . . . . . . . . . . . . .

395

VI. Experimental Techniques . . . . . . . . . . . . . . . . . . . .

398

VII. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .

404

References . . . . . . . . . . . . . . . . . . . . . . . . . . .

405

Atom Interference Using Microfabricated Structures

B. Dubetsky and P. R. Berman

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

407

II. Qualitative Considerations . . . . . . . . . . . . . . . . . . .

413

III. Talbot Effect . . . . . . . . . . . . . . . . . . . . . . . . . .

417

IV. Shadow Effect with Microfabricated Structures . . . . . . . . .

424

V. T a l b o t - L a u Effect . . . . . . . . . . . . . . . . . . . . . . .

437

VI. Talbot and Talbot-Lau Effects in a Thermal Atomic

Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

453

VII. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .

461

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . .

463

References . . . . . . . . . . . . . . . . . . . . . . . . . . .

467

INDEX

469

Contributors
Numbersin parenthesesindicatethe pageson whichthe authors'contributionsbegin.
H. BATELAAN (85), Institut fiir Experimentalphysik, Universit~it Innsbruck,
Innsbruck, Austria P. R. BERMAN (407), Physics Department, University of Michigan, 500 E.
University, Ann Arbor, Michigan 48109-1120 S. BERNET(85), Institut fur Experimentalphysik, Universit~it Innsbruck, Innsbruck,
Austria C. J. BORDI~ (257), Laboratoire de Physique des Lasers, URA/CNRS 282,
Universit6 Paris-Nord, Villetaneuse, France; and Laboratoire de Gravitation et Cosmologies Relativistes, URA/CNRS 769, Universit6 Pierre et Marie Curie, Paris, France H.-J. BRIEGEL (217), Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138
M. S. CHAPMAN (2), Department of Physics and Research Laboratory for
Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; present address, Caltech, Mail Code 12-33, Pasadena, California 91125 S. CHU (363), Physics Department, Stanford University, Stanford, California 94305 J. E CLAUSER (121), Department of Physics, University of Californian Berkeley, Berkeley, California 94720
M. DREWSEN(171), MIC, Building 345 East, Technical University of Denmark,
DK-2800 Linby, Denmark B. DUBETSKY (407), Physics Department, University of Michigan, 500 E.
University, Ann Arbor, Michigan 48109-1120 C. R. EKSTROM (2), Department of Physics and Research Laboratory for
Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; present address, US Naval Observatory, 3450 Mass. Ave., NW, Washington, DC 20392
B.-G. ENGLERT(217), Max-Planck-Institut ftir Quantenoptik, Hans-Koptermann-
Strasse 1, D-85748 Garching, Germany

x

Contributors

W. ERTMER(293), Institut fiir Quantenoptik Universit~it Hannover, Welfengarten 1, D-30167 Hannover, Germany
T. D. HAMMOND (2), Department of Physics and Research Laboratory for Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
J. HELMCKE (293), Physikalisch-Technische Bundesanstalt, Braunschweig, Bundesallee 100, D038116 Braunschweig, Germany
M. KASEVICH (363), Physics Department, Stanford University, Stanford, California 94305
D. A. KOKOROWSKI(2), Department of Physics and Research Laboratory for Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
C. KURTSIEFER(171), Fakult~it Physik, Universit~it Konstanz, 78434 Konstanz, Germany
A. LENEF(2), Department of Physics and Research Laboratory for Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; present address, Lighting Research Center, OSRAM SYLVANIA INC., 71 Cherry Hill Dr., Beverly, Massachusetts 01915
S. LI (121), Department of Physics, University of Califomia~Berkeley, Berkeley, California 94720
J. MLYNEK (171), Fakult~it Physik, Universit~it Konstanz, 78434 Konstanz, Germany
M. K. OBERTHALER(85), Institut ftir Experimentalphysik, Universit~it Innsbruck, Innsbruck, Austria
D. E. PRITCHARD (2), Department of Physics and Research Laboratory for Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
E. M. RASEL (85), Institut fiir Experimentalphysik, Universit~it Innsbruck, Innsbruck, Austria
F. RIEHLE (293), Physikalisch-Technische Bundesanstalt Braunschweig, Bundesallee 100, D-38116 Braunschweig, Germany
R. A. RUBENSTEIN(2), Department of Physics and Research Laboratory for Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
J. SCHMIEDMAYER(2, 85), Department of Physics and Research Laboratory for Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; Institut fur Experimentalphysik, Universit~it Innsbruck, A-6020 Innsbruck, Austria

Contributors

xi

M. O. SCULLY(217), Department of Physics, Texas A & M University, College Station, Texas 77843-4242
K. SENGSTOCK (293), Institut fur Quantenoptik Universit~it Hannover, Welfengarten l, D-30167 Hannover, Germany
E SnnvllZU(153), Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan
E. T. SMITH(2), Department of Physics and Research Laboratory for Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
R. J. C. SPREEUW(171), Van der Walls-Zeeman Institute, University of Amsterdam, NL- 1018XE Amsterdam, Netherlands
U. STERR(293), Institut fiir Quantenoptik Universit/~t Hannover, Welfengarten l, D-30167 Hannover, Germany
H. WALTHER(217), Max-Planck-Institut fur Quantenoptik, Hans-KoptermannStrasse 1, D-85748 Garching, Germany
M. WILKENS (171), Fakult~it Physik, Universit~it Konstanz, 78434 Konstanz, Germany
B. YOUNG(363), Physics Department, Stanford University, Stanford, California 94305
A. ZEILINGER (85), Institut fur Experimentalphysik, Universit/~t Innsbruck, Innsbruck, Austria

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Preface
A few years ago, in my role as a member of the editorial board of Advances in Atomic, Molecular, and Optical Physics, I proposed that a special volume in that series be devoted to the rapidly emerging field of Atom Interferometry. This suggestion was met enthusiastically by the editors of that series, Benjamin Bederson and Herbert Walther. With their encouragement, I started to solicit contributions for this volume in the spring of 1994. Since I was fortunate enough to obtain commitments from many of the researchers who were instrumental in the development of atom interferometry, a decision was made to go ahead with the publication of a special volume in the Advances series. Somewhere along the line, the publishers at Academic Press, with the consent of Bederson and Walther, decided that it would be better for this book to be published as a standalone volume rather than as a special supplement to the Advances series. Be that as it may, the contributions to this book were written in the spirit of Advances articles, that is, reasonably long contributions summarizing recent accomplishments of the authors.
When I was on the faculty at New York University, I developed a course for nonscience majors entitled 20th Century Concepts of Space, Time, and Matter, which I now teach at the University of Michigan. An important component of that course, as well as any introductory physics sequence, is an appreciation of the fact that both electromagnetic radiation and matter exhibit wave-like properties. The wave nature of electromagnetic radiation is often illustrated using some form of Young's double slit apparatus, which produces interference fringes that are explained in terms of constructive and destructive interference of the radiation that has traveled different optical path lengths to the screen on which the pattern is displayed. The wave nature of matter is often illustrated using electron diffraction patterns.
Although the equations that govern the propagation of electromagnetic radiation (Maxwell's equations) and nonrelativistic matter waves (Schrrdinger's equation) are not the same, many of the basic wave-like properties of electromagnetic waves and matter waves are quite similar. Thus, it is possible for both electromagnetic radiation and matter to exhibit particle-like behavior if the wavelength of the radiation or matter waves is much smaller than all the relevant length scales in the problem, such as the size of obstacles that are scattering the waves. On the other hand, both electromagnetic waves and matter exhibit wavelike properties when the wavelength of the radiation or matter waves is comparable with the dimensions of the obstacles that are scattering the waves.
xiii

xiv

Preface

An interferometer is a device that exploits the wave nature of light. Typically, an interferometer contains a beam splitter that separates an incident beam of radiation or matter into two or more mutually coherent outgoing beams. The beams then recombine on a screen and exhibit interference fringes. Sometimes, additional beam splitters or mirrors are used to recombine the beams. Radiation and matter interferometers work on the same principle mthey differ only in the wavelength of the working medium (radiation or matter) and the nature of the beam splitters and mirrors that are needed. The wavelength associated with matter waves is typically 100 to 1000 times smaller than the wavelength of visible light.
Interferometers using visible light as the working medium date to the 19th century. The wave nature of electrons was demonstrated by Davisson and Germer in 1927 by scattering electrons from nickel crystals. Crystals, microfabricated slits, and electric fields can be used as "optical" elements for scattering electron waves. Electron interferometers were constructed using electron biprisms in the 1950s [for a review, see M611enstedt, G. and Lichte, H., in Neutron Interferometry, edited by Bonse, U. and Rauch, H. (Clarendon Press, Oxford, 1979), pps. 363-388]. Neutron interferometers were developed first in the 1960s using refraction from biprisms and Bragg scattering from crystals, but major advances in the field occurred following the use of interferometers cut from single Si crystals [for reviews, see, for example, Neutron Interferometry, edited by Bonse, U. and Rauch, H. (Clarendon Press, Oxford, 1979) and Neutron Optics by Sears, V. F. (Oxford University Press, New York, 1989)]. With the development of ultra-cold neutron sources, the de Broglie wavelength could be increased from a characteristic value of about 1.0 Afor thermal neutrons to tens of Afor ultra-cold neutrons, enabling one to use slits as optical elements [for reviews, see, for example, Ultra-Cold Neutrons by Golub, R., Richardson, D., and Lamoreaux, S. K. (Adam Hilger, Bristol, 1991) and G~ihler, R. and Zeilinger, A., Am. J. of Phys. 59, 316-324 (1991)].
The major stumbling block in the development of atom interferometers has been development of atom optics, that is, beam splitters and mirrors for atom matter waves of which the de Broglie wavelengths are typically a fraction of an angstrom. In the past five years, however, significant advances have been made in atom interferometry. Many of the research groups that contributed to these advances and pioneered the field of atom interferometry are represented in this volume.
The chapter by Schmiedmayer, Chapman, Ekstrom, Hammond, Kokorowski, Lenef, Rubenstein, Smith, and Pritchard reviews many of the important contributions to atom interferometry made by this group. The atom interferometers are constructed using a beam of sodium atoms as the matter wave and microfabricated structures as the "optical" elements. Diffraction from a single grating has allowed them to distinguish between sodium atoms and sodium dimers in their

Preface

xv

beam. Using a three grating Mach-Zehnder atom interferometer, they were able to measure the electric polarizability of the ground state of sodium and the index of refraction of the sodium matter waves in a buffer gas environment. A key feature of their measurements was the physical separation of the matter waves in the two arms of the interferometer. They were also able to monitor the loss of atomic coherence resulting from scattering of radiation from the matter waves in the interferometer.
Batelaan, Bernet, Oberthaler, Rasel, Schmiedmayer, and Zeilinger also report on an atom interferometer of the Mach-Zehnder type, but with standing-wave fields rather than microfabricated gratings used as the optical elements. The matter wave used in their experiments was metastable argon, and different transitions could be used to study the effect of spontaneous emission on the interference signals. In addition, they carried out an experiment using three microfabricated structures to scatter the metastable argon atoms, in which the atoms' center-of-mass motion could be treated classically. They show that the "shadow" or moir6 pattern that is formed when atoms pass through the gratings can be used to measure the value of the acceleration of gravity and the Sagnac effect (modification of the fringe pattern resulting from rotation of the apparatus). Finally, they study scattering from standing wave light fields in the Bragg scattering limit.
Clauser and Li compare interferometers in which the scattered matter waves are separated physically within the interferometers with those in which the scattered waves overlap within the interferometer. Both the Talbot and Talbot-(Ernst) Lau interferometers are examples of the latter class of interferometers. Clauser and his colleagues were the first to stress that Talbot-Lau interferometry had important potential applications in atom interferometry. In this article, Clauser and Li discuss the basic features of both the Talbot and Talbot-Lau interferometers, and present results from experiments in which potassium atoms were used as the matter waves in a three (microfabricated) grating Talbot-Lau interferometer. Applications discussed include Sagnac and electric polarizability measurements, as well as interferometric studies of matter wave decoherence produced by light scattering.
The chapter by Shimizu includes a review of his work on two-slit interference patterns using metastable neon atoms released from a magneto-optical trap. This is followed by a description of a method for creating a binary hologram. When such a hologram is fabricated on a SiN film and illuminated with a matter wave of neon, the original object is reconstructed. Also included in this contribution is a report of a measurement of the second order correlation function associated with a matter wave.
Kurtsiefer, Spreeuw, Drewsen, Wilkens, and Mlynek explore several aspects of atom optics in their contribution. They begin by reviewing the interaction of atoms with radiation fields. For a beam of atoms scattered by a standing-wave

xvi

Preface

optical field that is detuned from the atomic transition frequency, the radiation field can be considered as a lens for the matter waves. Methods for correcting the various aberrations associated with this type of lens are discussed, as well as applications to atom lithography and surface probes. The authors then describe an experiment using metastable helium atoms scattered by a resonant standingwave field; as a result of spontaneous emission following the atom-field interaction, the visibility of the atom interference pattern is reduced. The visibility can be restored by measuring only those atomic events that are correlated with specific spontaneous emission modes. Additional methods are described for preparing entangled states involving the atoms and one or more photons. Finally a proposal for an atomic boson laser is set forth in which spontaneous emission into bound states of an optical lattice is stimulated by identical atoms already in that state.
The subject matter shifts slightly with the article by Briegel, Englert, Scully, and Walther. They begin a discussion of atom interferometry in which intemal state labels of the atoms take on an important role. The first part of their chapter is devoted to a study of internal state atomic interference for atoms passing through modified versions of the Young's double slit experiment. They discuss complementarity and the importance of "which path" information in establishing interference patterns. The use of micromaser cavities in such experiments and the role played by the quantized field modes in the cavities is emphasized. The second half of the chapter contains a critical assessment of the possibility to recombine different spin states of atoms that have been split by a Stem-Gerlach magnet.
In his contribution, Bord6 presents a general discussion of the theory of atom interferometers, including those employing either microfabricated slits or standing-wave fields as beam splitters and combiners. The role played by the internal states of the atoms is stressed. He studies Bragg scattering in the limit of off-resonance excitation and also uses a wave packet approach in analyzing the interferometers. Bord6 presents a unified approach to matter-wave interferometry in which the atoms, represented by Dirac .fields, are coupled to the electromagnetic field and to inertial fields. Effects such as the recoil splitting, gravitational shift, Thomas precession, Sagnac effect, Lense-Thirring effect, spin-rotation effect, and topological phase effects emerge naturally from this treatment.
The next chapter by Sterr, Sengstock, Ertmer, Riehle, and Helmcke contains contributions from groups at the University of Hannover and the PhysikalischTechnische Bundesanstalt. The atom interferometers studied by these groups use either calcium or magnesium atoms as the active element and optical fields as the beam splitters and combiners. Internal state labeling plays an important role in these interferometers, in which the scattered waves overlap within the interferometer. Experiments are carded out for a geometry corresponding to a Ramsey-Bord6 interferometer using both continuous wave (cw) and pulsed optical fields. For the cw experiments, thermal or laser-cooled atomic beams are sent

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xvii

through several field regions where state-dependent scattering occurs. In the pulsed experiments, the interferometer is constructed in the time domain rather than the spatial domain. Among the measurements discussed are those of de Stark shift and polarizability, ac Stark effect, Aharonov-Bohm effect, AharonovCasher effect, and the Sagnac effect. Also included are applications of the interferometers as frequency standards.
The chapter by Young, Kasevich, and Chu also discusses Ramsey-type interferometers, although the working atomic transition is one between different ground state sublevels rather than between a ground state level and a long-lived excited state level as in the case of calcium and magnesium. A review of the theory of the Ramsey interferometer is given, including effects of atomic recoil. The beam splitters and combiners used by Young, Kasevich, and Chu are based on single or multiple Raman pulses of counterpropagating optical fields, or on adiabatic transfer between the ground state sublevels. Both of these methods are reviewed. Atom interferometric measurements of the acceleration of gravity, variations in the acceleration of gravity, and the fine structure constant are reported, and the potential use of the interferometer as a gyroscope is discussed.
The article by Dubetsky and myself returns to calculations of Talbot and TalbotLau interferometry using microfabricated slits as scatterers for the matter waves. Scattering in the classical and Fresnel diffraction (Talbot and Talbot-Lau effects) limits is interpreted in terms of the recoil that atoms undergo when they are scattered from the microfabricated gratings. It is shown that it is possible to produce atomic density profiles having periods that are a fraction of the periods of the microfabricated structures in both the classical and Fresnel diffraction limits. Moreover, it is shown that Talbot effect fringes can be produced even when the atomic beam has a thermal longitudinal velocity distribution. Processes that lead to modulation of the atomic density profile are classified into those that rely critically on quantization of the atoms' center-of-mass motion and those that do not.
Of course, in a volume of this size it is impossible to present chapters from all the individuals and groups who have made important contributions to atom interferometry. In particular, material directly related to atom interferometers has been included somewhat at the expense of research focused in the areas of atom optics and atom lithography. Moreover, since atom interferometry is a rapidly developing field, many new and important contributions will have appeared between the planning stage and publication date of this volume. Readers are referred to the chapters of this book for additional references as well as the following journal volumes, which are special issues devoted to atom interferometry:
9Applied Physics, Volume B 54, Number 5, May, 1992
9Journal de Physique H, Volume 4, Number 11, November, 1994
9Quantum and Semiclassical Optics, Volume 8, Number 3, June, 1996.

xviii

Preface

Finally, I would like to thank each of the contributors for their cooperation in preparing this volume. I am aware of the amount of work that goes into writing chapters of this nature and also understand that all of the participants are heavily burdened with other demands on their time. I would also like to thank Zvi Ruder and Abby Heim at Academic Press for their help, encouragement, and patience.
Paul R. Berman Ann Arbor, 1996

ATOM INTERFEROMETRY
OPTICS AND INTERFEROMETRY WITH ATOMS AND MOLECULES
JORG SCHMIEDMAYER, MICHAEL S. CHAPMAN, CHRISTOPHER R. EKSTROM, TROY D. HAMMOND, DAVID A. KOKOROWSKI, ALAN LENEF, RICHARD A. RUBENSTEIN, EDWARD T. SMITH, and DAVID E. PRITCHARD Departmentof Physicsand ResearchLaboratoryfor Electronics, MassachusettsInstituteof Technology, Cambridge,Massachusetts

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

2

II. Beam Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

A. Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

B. Supersonic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

C. Atomic Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

D. Molecular Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

E. Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

III. Optics for Atoms and Molecules . . . . . . . . . . . . . . . . . . . . . . . . .

9

A. Nanofabrication Technology . . . . . . . . . . . . . . . . . . . . . . . . .

11

B. Diffraction of Atoms and Molecules . . . . . . . . . . . . . . . . . . . . .

13

C. Near Field Atom Optics: The Talbot Effect . . . . . . . . . . . . . . . . . .

14

D. Rabi Oscillations Observed Using Momentum Transfer . . . . . . . . . . . .

15

IV. Interferometry with Atoms and Molecules . . . . . . . . . . . . . . . . . . . .

18

A. Three Grating Mach-Zehnder Atom Interferometer . . . . . . . . . . . . . .

18

B. Phase and Contrast Measurement . . . . . . . . . . . . . . . . . . . . . .

21

C. Optimizing Contrast and Signal to Noise . . . . . . . . . . . . . . . . . . .

22

D. Grating Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

E. Sensitivity to Motions of Gratings . . . . . . . . . . . . . . . . . . . . . .

24

F. Interaction Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

G. Molecular Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . .

28

V. A t o m Interferometry Techniques . . . . . . . . . . . . . . . . . . . . . . . .

30

A. Significance of Phase Shifts . . . . . . . . . . . . . . . . . . . . . . . . .

30

B. Averaging over the Velocity Distribution: The Coherence Length . . . . . . . .

31

C. Contrast Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . .

33

D. Velocity Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

E. Measuring Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

VI. Measuring Atomic and Molecular Properties . . . . . . . . . . . . . . . . . . .

39

A. Electric Polarizability of Na . . . . . . . . . . . . . . . . . . . . . . . . .

39

B. Refractive Index for Na Matter Waves . . . . . . . . . . . . . . . . . . . .

42

VII. Fundamental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

A. Particle Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

B. Coherence Loss Due to Scattering a Single Photon--Discussion . . . . . . . .

53

C. Coherence Loss Due to Scattering a Single Photon--Experiment . . . . . . . .

55

D. Coupling to the Environment . . . . . . . . . . . . . . . . . . . . . . . .

59

E. Regaining Entangled Coherence by Selective Observations . . . . . . . . . . .

61

E Scattering a Single Photon off an Atom in Two Interferometers . . . . . . . . .

63

Copyright 91997 by Academic Press, Inc. All fights of reproduction in any form reserved.
ISBN 0-12-092460-9

2

Jrrg Schmiedmayer et al.

VIII. Inertial Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

IX. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

A. Atomic and Molecular Physics . . . . . . . . . . . . . . . . . . . . . . . .

71

B. Fundamental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

C. Berry's Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

D. Relativistic Effects in Electromagnetic Interactions . . . . . . . . . . . . . .

73

E. Differential Force Interferometry . . . . . . . . . . . . . . . . . . . . . .

74

E Inertial Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

G. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Appendix: Frequently Used Symbols . . . . . . . . . . . . . . . . . . . . . .

76

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

I. Introduction
In the 19th century, the work of Fizeau (1853), Michelson (1881), Rayleigh (1881), and Fabry and Perot (1899) with light interferometers established a tradition of beautiful experiments and precise measurements that continues to this day. Shortly after the de Broglie 1924 proposal that every particle should exhibit wavelike behavior, atomic diffraction was observed in scattering from crystal surfaces (Estermann and Stern 1930). Subsequently, matter wave interferometry with electrons (Marton, 1952; Marton et al., 1954; Mollenstedt and Duker, 1955) and neutrons (Maier-Leibnitz and Springer 1962; Rauch et al., 1974) was demonstrated. Today, neutron and electron interferometry are invaluable tools for probing fundamental physics, for studying quantum mechanical phenomena, and for making new types of measurements. For an overview of matter wave interferometry, see Bonse and Rauch (1979) and Badurek et al. (1988).
The scientific value of interferometry with atoms, and even molecules, has long been recognized. In fact, the concept of an atom interferometer was patented in 1973 (Altschuler and Franz 1973) and has been extensively discussed (Chebotayev et al., 1985; Bordr, 1989; Special Issue Atom Optics, 1992, 1994). Atom interferometry offers great richness, stemming from the varied internal structure of atoms, the wide range of properties possessed by different atoms (e.g., mass, magnetic moment, absorption frequencies, and polarizability), and the great variety of interactions between atoms and their environment (e.g., static E-M fields, radiation, and other atoms).
The technology for the production and detection of beams of atoms and molecules of many different species is well developed. Even a sophisticated supersonic molecular beam machine like the one used in our experiments (and described in Section II) can be built largely of commercially available components at moderate expense, unlike the nuclear reactor required for neutron beams. Hence, the delay in the development of atom interferometers can be attributed to the lack of suitable optical elements for coherently manipulating atomic and molecular de Broglie waves. Therefore, it is appropriate that our review address

OPTICS AND INTERFEROMETRY

3

(in Section III) the recent advances in atom optics that have allowed the development of atom interferometers. We concentrate mostly on atom optics techniques based on nanofabrication technology, since this is the type employed in our atom and molecule interferometer.
These advances have allowed us to construct a versatile three grating Mach-Zehnder atom and molecule interferometer (Keith et al., 199 l a; Schmiedmayer et al. 1993; Chapman et al., 1995a). Over the past few years our work at MIT has focused on the development of new techniques for atom and molecule interferometry (Schmiedmayer et al., 1994a; Hammond et al., 1995; Chapman et al., 1995a) and especially on the application of atom interferometers to the three classes of scientific problems for which they are ideally suited: study of atomic and molecular properties (Ekstrom et al., 1995; Schmiedmayer et al., 1995a), investigation of fundamental issues (Chapman et al., 1995c, Schmiedmayer et al., 1995b), and measurement of inertial effects (Lenef et al., 1996).
In this chapter, we provide an overview of the recent accomplishments in atom and molecular optics and interferometry at MIT. We begin with a discussion of the details of our experimental apparatus (Section II) and give an overview of our recent~accomplishments in atom and molecular optics (Section III). We then describe our atom and molecule interferometer, which is unique in that the two interfering components of the atom wave are spatially separated and can be physically isolated by a metal foil (Section IV), and give an overview of atom interferometry techniques (Section V).
Our interferometer is especially well suited to the study of atomic and molecular properties (Section VI), as it enables us to apply different interactions to each of the two components of the wave function, which in turn permits spectroscopic precision in the study of interactions that shift the energy or phase of a single state of the atom. We describe an experiment in which we have used this capability to determine the ground state polarizability of sodium to 0.3%man order of magnitude improvementmby measuring the energy shift due to a uniform electric field applied to one component of the wave function. In a different experiment, we measured the index of refraction seen by sodium matter waves traveling through a gas sample, thus determining previously unmeasureable collisional phase shifts, which we interpreted to reveal information about the form of the long-range interatomic potential.
Our studies of fundamental issues (Section VII) have both investigated and taken advantage of effects arising from the internal structure of atoms. In particular, we have addressed the limitations to interferometry due to complexity of the interfering particles and conducted experiments investigating the loss of coherence due to the scattering of a single photon from each atom passing through the interferometer. As a probe of the basic process of measurement in a quantum system, we performed a correlation experiment in which the lost coherence was regained. Finally, as a demonstration of the application of atom interferometers

4

Jrrg Schmiedmayer et al.

as inertial sensors, we demonstrated both the accuracy and the sensitivity of out interferometer to rotation (Section VIII).

II. Beam Machine
All the experiments described in this review were carried out in a 3-meter-long beam machine using a Na or Na 2 beam seeded in a supersonic noble carrier gas. After an overview of the apparatus, we will outline the various techniques used to prepare and detect atomic and molecular beams. The reader more interested in atom interferometry and its applications may wish to skip this section and return later for more information on the experimental details.
A. VACUUM SYSTEM
The vacuum envelope of our atom beam machine consisted of five differentially pumped chambers (Fig. 1). The first chamber, which enclosed the supersonic beam source, was pumped by a special high throughput 4 in Stokes ring jet

Ist Differential

Source Pumping Chamber 2nd Differential

Chamber -

Pumping Chamber

Interferometer Chamber

Detector Chamber

Bellows for vibration isolation~

10" Stokes Booster Pump

Diffusion Pump

4" Diffusion Pump

4" Diffusion Pump

80 l/s Turbo Pump

Skimmer
l I Sesd.i.u.m... . . ./ [i . J]

Stern Gerlach Magnet

1 1st Atom Grating,~, x,x. \\ - ~ 2nd Atom Grating 3rd Atom Grating

1 J

.

gz::::::~

~[~ ' O p t i c a l ~

] /t 'I ~ / ~ 7

PRumegDiion~n

L ,\ ~ First Slit

/t . fKorni"fBe rEodogme "

Seco d "

~]~[~~" ~ L ~

"ght nterferometer

Hot Wire Detector

FIG. 1. The vacuum chamber of our atomic beam machine. The lower figure gives a top view, showing the paths of both the atom interferometer and the laser intefferometer (which is used to measure the relative positions of the atom diffraction gratings).

OPTICS AND INTERFEROMETRY

5

booster diffusion pump with a pumping speed of 100 liter/sec at 0.015 torr. The gas load from the source into this chamber was about 0.5 torr-liter/sec, resulting in a typical pressure of a few mtorr. A conical skimmer removed the central portion of the expanding gas from the oven and also formed the aperture into the second chamber, allowing roughly 0.3% of the gas load from the source into this differential pumping region.
The second and third chambers provided access for beam preparation, collimation, and manipulation as well as the differential pumping needed to achieve a good vacuum in the main experimental chamber. In the second chamber the pressure was maintained at 2 x 10-6 torr by an unbaffled 10 in diffusion pump (4200 liter/sec). The beam traveled only about 20 cm in this chamber, then entered the third chamber through the first of two collimation slits. The third chamber was held at a pressure near 5 x 10-7 torr by a 4 in diffusion pump (800 liter/sec) hung from a water cooled elbow. The aperture between the third chamber and the main chamber was 1 cm in diameter and could be sealed with a transparent Plexiglas gate valve that allowed optical alignment within the evacuated main chamber with the source chamber open to air.
The 150 cm long main chamber contained all of the interferometer components except the detector. Pumped by a 4 in baffled diffusion pump, the pressure in this chamber was typically 3 X 10-7 torr, good enough to limit losses from scattering to below 10%. A 1 cm aperture with another Plexiglas gate valve separated the main chamber from the detector chamber.
The detector chamber required the lowest possible pressure, to reduce false counts in the detector due to residual background gas. This chamber was pumped with a turbo pump and in addition had a liquid nitrogen pumping surface, a combination yielding pressures of 2 to 5 X 10-8 torr. The turbo pump was hung on a 6 in vacuum bellows to isolate its vibrations from the rest of the machine.
To further reduce vibrations in the machine, the roughing pumps were situated several meters from the machine and mounted on vibration isolation pads. Roughing lines were mounted solidly to a wall and connected to the machine with flexible vibration isolating lines. In addition, we could lift up the whole vacuum system and hang it from the ceiling, thereby significantly reducing the higher frequency vibrations transmitted by the building floor.

B. SUPERSONICSOURCE
The atomic and molecular sodium beams were produced in an inert gas supersonic expansion seeded with sodium vapor. The most important feature of this source was that it produced an intense beam with a narrow longitudinal velocity distribution (<5% rms), which was necessary for most of the experiments that we undertook. Sodium metal, contained in a stainless steel reservoir, was heated to temperatures as

6

J~rg Schmiedmayer et al.

high as 800~ (>5 torr vapor pressure of sodium) and mixed with a noble gas at high pressure (typically >2 atm). The noble gas/sodium mixture flowed through a 70/xm diameter nozzle into vacuum, producing an isenthalpic supersonic expansion. The result was a very cold beam. The 500/xm skimmer orifice leading to the first differential pumping chamber was located inside this expansion, allowing the cooler core of the supersonic beam to propagate down the rest of the machine with low probability of further collisions. The total detected brightness of the sodium beam was as large as 1021atoms str-1 sec -1 cm -2.
Clogging of the 70/xm nozzle was prevented by carefully purifying the inert carrier gas before transferring it into the source. We used two gas purifier stages between the gas handling system and the source, yielding residual water and oxygen impurities of a few ppb. With the gas purifier system in place, we were able to operate our source continuously for several weeks.
An additional feature of this source is that the velocity (and hence the de Broglie wavelength) of the atomic or molecular sodium is (to within a few percent) determined by v = ~/5kbT/mcarder, where mcarrier is the mass of the inert carrier gas (Scoles, 1988). Hence, the velocity of the source could be varied by changing the carrier gas (Table I). With a source temperature of 700~ and argon as the cartier gas, our supersonic sodium beam had a mean velocity of 1000 m/sec, which corresponds to a de Broglie wavelength of hdB " - 0 . 1 7 ~. By mixing carrier gases, we were able to vary the beam velocity continuously from 650 m/sec using pure xenon to 3300 m/sec using pure helium (Table I).
The velocity distribution of the source also varies according to changes in carrier gas pressure. We have produced velocity distributions with a FWHM ranging from Av/v = 70% with no carrier gas to Av/v < 8% at 3 atm of argon, the latter figure corresponding to an rms velocity width of 3.4%, and a (longitudinal) translational temperature of 1.6 K. Narrower velocity distributions with lower final temperature are obtainable with helium, in part because it does not heat the expansion by forming dimers.

TABLE I TYPICAL PARAMETERSFOR OUR N a AND N a 2 SOURCE

Seed gas Ve1ocity (m/sec)

Kr

Ar

Ne

750

1000

1700

Na

)tdB

(/~)

0.23

0.17

0.10

Separation (/,~m)

75

55

34

Na2

AdB

(,/k)

0.125

0.085

0.05

Separation (/xm)

38

28

17

He 3300
0.05 18
0.025 9

OPTICS AND INTERFEROMETRY

7

C. ATOMICBEAM
After exiting the source chamber, the supersonic sodium beam passed through a series of operations preparing it for use in the interferometer. In general, our experiments required that the beam be fairly monochromatic and well collimated, and sometimes that the ensemble of sodium atoms be prepared in a particular quantum state.
Beam collimation was provided by two slits, 20 (or 10)/xm wide, the second of which could be rotated under vacuum for proper alignment. The slits were spaced 87 cm apart, yielding a ribbon-shaped beam up to 3 mm high with a typical beam divergence of 23 (12) /xrad FWHM. For a 1000 m/sec atomic Na beam, this collimation represents a transverse velocity of about 2 (1) cm/sec, or 23( 89of the recoil velocity induced by a single photon, and a corresponding transverse "temperature" of 0.5 (0.25)/xK.
To prepare the Na atoms in a single hyperfine state, we optically pumped them to the 3S1/2F = 2, m F = 2 state using a circularly polarized laser beam tuned to the F = 2 ~ F' = 3 transition of the sodium D 2 line. An additional sideband tuned to the F = 1 ~ F ' = 2 transition pumped atoms out of the F = 1 ground state. A standing wave dye laser (Coherent 599) generated the resonant F - 2 ~ F' = 3 light. This light was directed through an electro-optical modulator to generate sidebands at 1713 MHz and then transferred to the beam machine via single mode, polarization preserving optical fiber. We employed a locking technique described in Gould et al., (1987) to select a specific atomic state and to achieve long-term frequency stability of the laser (McClelland and Kelley, 1985). This technique is based on the fact that the transverse position of the fluorescent spot formed when the laser intersects a diverging atomic beam depends on the laser frequency due to the spatially varying doppler shift. The differential signal obtained by imaging this spot onto a split photodiode provides the error signal for laser frequency locking (Gould, 1985).
Because atoms optically pumped in this manner necessarily have scattered many photons, these atoms are deflected relative to the other beam constituents. For this reason, we chose to optically pump the atoms in the first vacuum chamber, before the first collimation slit, so that we still obtained good beam collimation and so that by optimizing the positions of the collimation slits we could greatly reduce the background of Na2 molecules and unpolarized atoms. Weak (--~4 Gauss) magnetic guide fields provided a quantization axis for the optically pumped atoms, and maintained the atomic polarization throughout the interferometer. The direction and strength of these fields were variable, allowing us to select the orientation of the atomic spins in our experiments.
Due to the rapid transit of atoms through the collimation slits, any residual magnetism in these slits can cause nonadiabatic transitions, also known as Marjorana flops, with resulting loss of polarization. We observed significant depolarization when using stainless steel slits, even after they had been demagnetized,

8

J6rg Schmiedmayer et al.

so we elected to fabricate our own slits out of silicon. Employing the silicon slits, we achieved better than 95% polarization, as determined by a two-wire Stern-Gerlach magnet (Ramsey 1985), located 30 cm after the second collimating slit, used to measure the state-dependent deflections of the atomic beam.

D. MOLECULARBEAM
To prepare a pure beam of molecules for molecular optics and interferometry experiments, a number of additional steps were necessary. By heating the sodium reservoir to 800~ (Na vapor pressure ---6.5 torr), we were able to enhance the population of sodium dimers in the beam to as much as 30% of the detected beam intensity. To obtain a pure beam of molecules, we deflected atoms out of the beam using resonant laser light applied halfway between the two collimating slits (Fig. 2) (this required less than 2hk of transverse momentum). Sodium molecules are not resonant with the deflecting laser beam and therefore were unaffected [the X ~Eg----~A ~s transition to the first excited dimer state lies around 680 nm (Herzberg, 1950)]. A knife edge, positioned directly upstream from the laser beam, blocked atoms that could have been deflected back into the now purely molecular beam. At a carder gas pressure of 2000 torr, our Na 2 beam had only 3.5% rms longitudinal velocity spread, corresponding to a (longitudinal) translational temperature of 2 K.

E. DETECTOR
In the detection chamber, individual sodium atoms and molecules were ionized on the surface of a 50/xm rhenium wire heated to approximately 850~ and detected by a channeltron electron multiplier. To reduce background

FIG.2. Productionof a pure molecularbeam by removingthe sodiumatoms.The deflectinglaser imparts a transversemomentumto the sodiumatoms, deflectingthemaway fromthe second collimation slit.The knife edgepreventsscatteringof sodiumatoms back into the molecularbeam.

OPTICS AND INTERFEROMETRY

9

noise the wire was cleaned thoroughly by baking it out at temperatures >2500~ Furthermore, we employed specially designed electric fields so that only ions from the hot wire were collected, and thermally emitted electrons were prevented from ionizing the residual gas in the detector chamber. To achieve high efficiency, we grew an oxide layer on the wire by periodically oxidizing it at a low pressure (10 -4 torr 02) and high temperature (again >2500~ for 10 sec. Typical performance characteristics of the Na atom detector were a response time of 1 msec, and background count rate of less than 50 counts per second (cps).
The use of this detector for Na2 dimers raises the interesting question of whether an Na2 dimer will produce one or two detector counts (Parrish and Herm 1969). If the molecule dissociates on the hot surface and each atom gets ionized independently, then we might see two separate counts. Using our molecular beam, we measured the time correlation function of neighboring counts. For single counts arriving randomly, the correlation exhibited an exponential decay, reflecting the average count rate. A pair of counts from a single dimer that was thermally dissociated before ionization had a faster correlation decay, reflecting the average ionization time scale. To use this difference to study the degree of thermal dissociation, we chose a higher than normal operating temperature of the hot wire to make the ionization time fast enough that a correlated ion pair from a Na2 molecule could be distinguished from the random counts. From the correlation data (Fig. 3), we find the probability of detecting a pair of disassociated ions to be at least 5%. This figure includes an estimated single ion detection efficiency of 20%. From this data we can also estimate a lower boundary of the ionization efficiency of the hot Re surface to be >50%, and a probability of the Na2molecule to break up at the surface to be larger than 66%.

III. Optics for Atoms and Molecules
The field of atom/molecule interferometry has been opened up by recent advances in atom optics. Optical elements based on both the mechanical forces of light (Gould et al., 1986; Bord6, 1989; Kasevich and Chu, 1991; Riehle et al., 1991; Sterr et al., 1992) and nanofabricated structures (Keith et al., 1988, 1991a; Carnal and Mlynek, 1991; Shimizu et al., 1992) allow sufficiently coherent manipulation of de Broglie waves that atomic/molecular interferometers can now be built and used in a variety of different experimental applications. These two types of optical elements for atoms and molecules are complimentary in many respects: nanofabricated optics are inexpensive, rugged, reliable, and species insensitive, whereas light-based optics are species and state selective, require light from stabilized single-mode lasers, are highly precise, and do not clog up if used with high intensity atom beams.

10

4

~

2

Jarg Schmiedmayer et al.

I

I

I

I

loo

~

9

)

r~

6

~ 0

4

-4

I ....

I

I

I

t-

O

1

2

3

4

5

Delay Time (ms)
FIG. 3. Histogram of the time between successive counts obtained if Na 2 molecules impinge on our Re hot wire detector. Two different time scales can be distinguished. The first is an exponential decay that matches our constant count rate of 240 counts per second. The second feature is a steeper decay at short time delays. We attribute this to two sodium ions being produced from a single sodium dimer and infer that the time constant of the decay reflects the time response of the detector wire. Subtracting our background of 125 counts/sec we can estimate that for about 5% of the Na2 molecules we see two counts.

We predict that both will see wide future application, perhaps combined in the same experiment as we have done for molecules (see Sections III.B and IV.G). Many of the light force based developments in atom optics are reviewed in special issues of JOSA-B. (Special Issue Mechanical Effects of Light, 1985, 1989; Special Issue Atom Optics, 1992, 1994).
Our group was instrumental in the development of atom and molecule diffraction gratings, elements used in practically all atom interferometers, using both of the major approaches described previously: light forces (Moskowitz et al., 1983; Gould et al., 1986; Martin, et al. 1988) and nanofabricated optical elements (Keith, et al. 1988, 1991b, Ekstrom et al., 1992 and references therein). The earlier atom optics work performed by our group, which was concerned primarily with light forces, is covered in several previous articles and reviews (Moskowitz et al., 1983; Gould et al., 1986, 1987b, Martin et al., 1988; Oldaker et al., 1990; Gould and Pritchard, 1996) and will not be discussed here. Therefore, in this section, we shall concentrate on nanofabricated atom/molecule elements such as those used in our interferometer.

OPTICS AND INTERFEROMETRY

11

A. NANOFABRICATIONTECHNOLOGY
The major enabling factor for our atom/molecule interferometer was the development of nanofabricated diffraction gratings, consisting of thin (100-200 nm) low-stress silicon nitride membranes with precisely patterned holes (see Fig. 4). These structures are used as diffractive optical elements for atoms and molecules. The fabrication process has been described in detail in Keith et al. (1991b) and Ekstrom et al. (1992), we will give only a quick overview here (see Fig. 5).
Our procedure for fabricating atom optics devices begins with the deposition of low-stress silicon nitride by low-pressure chemical vapor deposition on both sides of a standard double polished < 1 0 0 > silicon wafer 250 /xm thick. We then apply a layer of optical photoresist on which a pattern of windows is exposed. Each window is etched entirely through the silicon, leaving a suspended nitride "window pane" on the front of the wafer. We next apply a 120-210 nm layer of PMMA (polymethyl methacrylate) to the front side of the wafer, on which is evaporated a thin layer of gold to prevent distortions due to the accumulation of charge from the electron beam. Afterward, the desired pattern is written into the PMMA using electron beam lithography.
To make diffraction gratings suitable for use in the interferometer, great care has to be taken that the pattern is written with positional accuracy below a small fraction of the grating period (typically a few tens of nanometers). Since the electron beam writer must piece together many (80/xm square) fields to write a large area pattern such as our gratings, "stitching" errors can occur. To prevent

FIG. 4. Transmissionelectron microscopepicture of a 140 nm period grating. The orthogonal supportstructurehas a 4/zm period.

12

Jarg Schmiedmayer et al.

FIG. 5. Construction steps to produce a patterned, free-standing silicon nitride membrane (after Keith et al., 1991b).
this misalignment of the many small fields, we write markers on the chip that subsequently are used to realign the translation stage prior to writing each small area (Rooks et al., 1995).
The areas in the PMMA exposed by the e-beam writer are washed away with a mixture of methyl isobutyl ketone (MIBK) and isopropanol (IPA). The exposed pattern is then directly transferred onto the silicon nitride window using a specially developed reactive ion etching gas mixture (Keith et al., 199 l b), leaving a free-standing pattern of slots in the silicon nitrate membrane (Fig. 4). Using this method, we can fabricate gratings possessing better than 10 nm accuracy over areas as large as 0.8 x 0.8 mm.

OPTICS AND INTERFEROMETRY

13

B. DIFFRACTIONOF ATOMS AND MOLECULES
We have investigated atomic and molecular diffraction by directing our Na and
Na 2 beams through nanofabricated gratings with various periods (Keith et al., 1988; Chapman et al., 1995a). Diffraction patterns for a pure Na 2 beam and a
mixed Na-Na 2 beam, obtained using a 100 nm grating, are shown in Fig. 6. Note that the various atomic diffraction orders are sufficiently separated to permit easy identification of the intermediate molecular diffraction peaks at half the atomic diffraction angle (Fig. 6b). This is exactly as we would expect since atoms and molecules in the argon-seeded supersonic beam have nearly identical velocities, while their unequal masses result in a factor of 2 difference in de Broglie wavelength. Further comparison of the two patterns in Fig. 6 reveals that our pure Na 2 beam contains residual Na contamination of less than 2%.
These diffraction patterns were powerful tools for analyzing atoms and molecules in our supersonic expansion. Knowing the diffraction angle, we deter-
mined an average beam velocity using 0diff = AdB/dg = (h/mv)(1/dg) where dg is the grating period and Ada = 27r/k o is the de Broglie wavelength. Further, we ex-
tracted the width of our beam's velocity distribution from the broadening of

,

, w

,

4 (a)

' ....

2

Na2

i

i

,

,

i

,

,

,

i

' Na+Na2

(b) " '

'

'

'

l

'

l

,

,

I

'

'

'

'

I

'

'

'

'

~/ l

1048

Pure Na2
-

6

4

2

1038

6

4

~!

2

,

,

1 ,

,

,

,

I

,

,

,

,

I

,

-500

0

500

,

i

l

i

I

l

l

i

-500

0

500

Detector Position (~tm)

Detector Position (~tm)

FIG. 6. Diffraction of 750 m/sec sodium atoms and molecules (Kr as a carrier gas) by a 100 nm period nanofabricated diffraction grating: (a) Diffraction of the mixed atom-molecule beam (de-

flecting laser off). One can clearly distinguish the atoms from the Na2 molecules by their different diffraction angle. A fit to the combined diffraction pattern (thin solid line) indicates 16.5% of the intensity is molecules. The thick solid line is the fit to the Na 2 diffraction pattern in (b). For this mea-

surement the deflecting laser was on. The fits determine the grating open fraction to be 30% and are a very good measurement (<. 1%) of the de Broglie wavelength (velocity) of the atomic/molecular beam.

14

Jgrg Schmiedmayer et al.

higher order diffraction peaks. In experiments with mixed beams, we were able to determine the mean fraction of dimers in the beam, as well as the center and width of the velocity distributions for both atoms and dimers. We observed a velocity slip between the atoms and slower moving molecules of as much as 3.5(6)% at low source pressures (400 torr). At a more typical source pressure of 1500 torr, the slip was less than 1% (Scoles, 1988).
A good fit to the measured diffraction pattern also provides information about the open fraction (ratio of slit width to grating period) and homogeneity of the grating. Due to imperfections in the fabrication process, the width of the grating bars, and hence the open fraction, is not uniform. We modeled this nonuniformity as an incoherent sum of diffraction patterns with a distribution of various open fractions. Fits to diffraction patterns from many different gratings suggest that our grating bars are uniform to within + 10 nm.
Using these and other diffraction techniques to investigate the properties of atomic and molecular beams has the advantage of being non-destructive. Indeed, our method and gratings have recently been used to produce unequivocal evidence for the existence of the weakly bound Van der Waals molecule He2 (and of higher He n clusters as well) (Sch611kopf and Toennies, 1994).

C. NEARFIELDATOM OPTICS: THE TALBOTEFFECT
To more directly measure the homogeneity of the gratings, one could place two gratings close to each other in an atomic beam and examine the moir6 pattern due to the "shadow" of the first grating falling on the second. The difficulty is that the "shadow" quickly blurs downstream from the first grating due to diffraction. However, further downstream, the shadows remarkably return at discrete distances from the first grating. These "self-images" of the first grating are known as Talbot images (Talbot, 1836), and in this section we discuss our measurement of these images using atom waves. This effect is well-known in classical optics and has many applications in image processing and synthesis, photolithography, optical testing, and optical metrology (Patorski, 1989).
Classical wave optics recognizes two limiting cases, near and far field. In the far-field limit, the intensity pattern of the beam is characterized by Fraunhofer diffraction, in which the curvature of the atom wave fronts is negligible. However, in the near-field limit the curvature of the wave fronts must be considered. In this case, the intensity pattern of the beam is characterized by Fresnel diffraction. Our study of the Talbot effect is one example of near-field atom optics, the self-imaging of a periodic structure (Chapman et al., 1995b).
We can understand the Talbot effect by considering the image formed by the interference of three plane waves: the 1st, 0th, and - 1 s t diffracted orders from a grating. At a characteristic distance beyond the grating known as the Talbot
(dg length, LTalbot = 2dZ/AdB is the grating period, AdB is the wavelength of the in-

OPTICS AND INTERFEROMETRY

15

cident wave), the three diffraction orders are laterally displaced from their initial positions by an even number of grating periods and interfere to form a grating self-image. At ~1 LTalbotan image identical to the grating is formed, which is laterally shifted by half a period. Images of both grating and shifted self-images appear repetitively further downstream, spaced one Talbot length apart. Other selfimages with smaller periods dg/n (n = 2,3,4 . . . . ) are produced at intermediate distances (Cowley and Moodie, 1957; Rogers, 1964; Winthrop and Worthington, 1965; Clauser and Reinisch, 1992) if diffraction into higher orders is significant. A full treatment of the problem, including the other diffracted orders and more detailed predictions of the positions and contrast of the subperiod images, requires solving the Fresnel diffraction problem with more formal techniques (Patorski, 1989; Clauser and Reinisch, 1992; Clauser and Li, 1994).
We investigated these successive self-images with atom waves (Schmiedmayer et al., 1993; Chapman et al., 1995b), using transmission gratings with two different periods, 200 and 300 nm, which yield Talbot lengths of 4.7 and 10.6 mm, respectively, for our atomic beam. The Talbot self-images were detected by masking them with a second transmission grating placed downstream (see inset of Fig. 7). When the second grating, whose period exactly matched that of the image, was scanned laterally across the self-image, the total transmitted intensity measured by the detector behind the grating revealed a high-contrast moir6 fringe pattern.
In our experiment, we varied the separation between the gratings from 3.5 to 13.5 mm, and the contrast of the moir6 fringe pattern was determined as a function of grating separation. Experimental results for both the 200 and 300 nm gratings are shown in Fig. 7. The contrast of the images damps out for larger grating separations, primarily because of the transverse incoherence of our atom beam as determined by the imperfect collimation of the source.
An especially promising application of Talbot (or Lau) imaging with atoms is atom lithography (Timp et al., 1992). It should be possible to write small features using the reduced period intermediate images discussed earlier. These images have been used successfully in x-ray lithography to write half-period gratings (Flanders et al., 1979). Grating self-images may also be used in quantum optics experiments to produce a periodic atom density in an optical resonator.

D. RABIOSCILLATIONSOBSERVEDUSING MOMENTUMTRANSFER
If an atom traverses a running light wave that is focused to a narrow waist so that the traversal time is smaller than the radiative decay time, then damping by spontaneous emission is negligible and the state of the atom after the traversal is determined by the coherent interaction with the light field. The probability for resonant excitation in a two-state system (ignoring damping), is given by the Rabi formula P(g---~e)=sin 2 (toRt/2) (here toR = 27r. 10 MHz ~ / I / ( 1 2 m W / c m 2)

16

J6rg Schmiedmayer et al.

200 nm
4O
~" 30
ra~
~ 20
9
~ 10

Na C ~ t s

source r--._

~.

Gratings Hot w i r e ~ t o r

~ ~

-H.

~---85 c m - ~ 7 5 cite -- z - ~ 100cm--~

300 nm
40
v 30
= 20 r,0..)
10

0.4

0.6

0.8

1.0

1.2

1.4

Grating Separation z (cm)
FIG. 7. The experimental data and calculations showing the contrast of the Talbot self-image as a function of grating separation for 200 nm gratings (above) and 300 nm gratings (below). The insert shows a schematic of the experimental apparatus. The distance between the two gratings, z, can be varied from 0.35 to 1.35 cm. The lateral position of the second grating is scanned using a PZT.

for the 3P3/2 transition in Na and I is the intensity of the excitation light). The oscillations of the probability are called Rabi oscillations.
We observed these predicted Rabi oscillations, corresponding to the coherent exchange of photons, in our atomic beam. Rabi oscillations correspond to the alternate absorption and (stimulated) emission of one photon from the laser beam. Since the transferred momentum is respectively lhk and - l h k , there is a corresponding oscillation in the transverse momentum of the atoms. Excited atoms were identified by the deflection imparted to them by the absorbed photon. An atom exiting the laser field in the excited state will have received lhk of momentum in the direction of propagation of the laser and the subsequent spontaneous

OPTICS AND INTERFEROMETRY

17

photon will transfer another l hk of momentum in a random direction. Therefore, excited atoms will be deflected with momentum around l hk.
For this experiment, the atoms were first prepared in the F = 2,m F = 2 ground state by optical pumping (---95% efficiency) with a o-+ polarized laser beam (see Section II.C). They were then excited from this state to the F' = 3 m' = 3
' F
excited state (this constitutes a closed two-level system) using resonant o-+ polarized laser light focused to a---15 /xm waist (FWHM of the field) along the atom propagation direction. A cylindrical lens was used to defocus the beam in the direction perpendicular to the atomic beam to ensure uniform illumination over the full height of the beam (---1 mm). Using 3000 m/sec atoms from a He driven expansion, the transit time through the waist (5 ns) was smaller than the lifetime of the excited state (16 nsec), and hence the probability for resonant excitation in the two-state system showed weakly damped Rabi oscillations as a function of laser power. Data taken with the detector wire displaced from the atomic beam axis by a distance corresponding to a single photon recoil are shown in Fig. 8.
In conducting our single-photon scattering experiments (see Section VII), we used this effect as a tool to align our laser beam relative to the atomic beam and to adjust the laser intensity to produce a rr-pulse, ensuring as nearly as possible that exactly one photon was scattered by each atom.

I

I

I

I

I 1

300-

: - - .. ;:._:.
"..
2 0 0 - .9,9....:..,..:9,....:-...:...-.:.':,v..~...;;.-."-..~.:.;~:,.-~:::,..:.: .

..... -:~......

.. 9' .:9:-"..:.~:,~.e~.i~y-~~-..;.', . . . _ 9

..,:,r

.."-

o
(..)

.~" . . . . . . . .9. . ."'". , : :"~,.:,-'q,,I,..,..~.s.., :.. :,. . -

" ~:,~;::~ :7.~;. ~'~r ' :. " -: . : L ~ . ~

1 O0

r9,-~ ~...:.,~:-~g~:~ 9

::,~:~

,, . . . . . . .

..

9

. . ....:r ",:.....,

o

i.

I -t

I

I-

0.1

0.2

0.3

0.4

0.5

Laser Field Strength (arb. units)
FIG. 8. Observing Rabi flops in m o m e n t u m transfer. The detector is displaced from the collimation axis by one photon recoil, and we measure the count rate as a function of laser intensity. As the power increases, the atoms have an oscillatory probability of being excited that is given by the Rabi formula. To scatter a single photon, we set the power to the value at the first maximum of these oscillations, w h i c h c l o s e l y corresponds to a 7r pulse.

18

J6rg Schmiedmayer et al.

IV. Interferometry with Atoms and Molecules
In this section we will outline the theoretical and design principles underlying the construction of our atom/molecule interferometer.

m. THREE GRATINGMACH--ZEHNDER ATOM INTERFEROMETER
The MIT atom interferometer (Keith et al., 1991a; Schmiedmayer et al., 1993) was built with a Mach-Zehnder geometry employing three 200 nm period nanofabricated transmission gratings (Keith et al., 199 l b; Ekstrom et al., 1992) mounted on separate translation stages inside the vacuum chamber (Fig. 9). The first grating diffracts the atomic beam primarily into the diverging orders -1, 0, and + 1. T h e 0th and 1st orders are diffracted through the second grating a distance L downstream. The second grating diffracts a portion of each of the two incoming beams toward each other. These diffracted beams, which are the -1st and + 1st orders of the two incident beams, respectively, overlap after traveling another distance L, forming a standing matter wave pattern, just upstream of the third grating, whose crests are parallel to the longitudinal axis of the interferometer. This standing wave pattern propagates along the longitudinal axis through

Translation Stages
Collimation Slits

< ~ - - Reference

Photodiode

Photodiode Signal

I I

Na/Ar Source

10 gm Copper Foil

Interaction Region

Hot Wire Detector

He-Ne Laser

i <

0.6 m

> <

0.6 m

>

FIG. 9. A schematic, not to scale, of our atom interferometer (thick lines are atom beams). The 0th and 1st order beams from the first grating strike the middle grating where they are diffracted in the 1st and - 1 st orders. These orders form an interference pattern in the plane of the third grating, which acts as a mask to sample this pattern. The detector, located beyond the third grating, records the flux transmitted through the third grating. The 10 cm long interaction region with the 10 /zm thick copper foil between the two arms of the interferometer is positioned behind the second grating. An optical interferometer (thin lines are laser beams) measures the relative position of the 200 nm period atom gratings (which are indicated by vertical dashed lines).

OPTICS AND INTERFEROMETRY

19

the third grating, which then acts as a mask, with its transverse position relative to the interference pattern determining the total transmitted flux. This flux is then measured by the detector (a 50/zm wire, which is much wider than a grating period). Uniform translation of either the standing wave pattern or the grating results in a periodic change in the transmitted intensity, creating an observable fringe pattern. The diamond shaped pattern of the interfering beams forms the classic Mach-Zehnder interferometer. We have observed atomic interference patterns with up to 50% contrast (Fig. 10) and obtained maximum interfering amplitudes of more than 50000 counts/sec at slightly lower contrast.
An interferometer geometry employing three equally spaced transmission gratings but without the collimation that would restrict it to the Mach-Zehnder geometry just described also creates a robust interferometer (Chang et al., 1975). Like the Mach-Zehnder geometry, it is white fringe, with phase and period of the interference pattern independent of the wavelength, wavelength spread, width, and initial direction of the input beam. This second geometry obviously offers the advantage of greatly enhanced signal, and we have applied it in studies where it is not necessary to physically isolate the two interfering atom waves.
An added benefit of both geometries just discussed is that the grating periodicity (200 nm) determines the scale of transverse dimensional stability required, rather than the much smaller de Broglie wavelength of sodium atoms (16 pm) or

3000 _ I

I

I

_11

I _

2500 or 2000

1500

1000
o
500

!-I

I

I

I

I-I

-400

-200

0

200

400

Position (nm)
FIG. 10. Interferencepattern from 30 sec of data (1 sec per point). The contrastis 49% and the phase uncertaintyis < 10mrad.

20

Jrrg Schmiedmayer et al.

molecules (8 pm). Requirements on the longitudinal spacing are much less restrictive (Turchette et al., 1992).
To understand the principles of our interferometer, it is helpful to review the theoretical foundations of matter wave interferometry using semi-classical physics. We begin by considering the difference in phase between two possible paths F 1 and F2 through the interferometer from source to detector, since this difference determines the phase of the interference pattern. The difference between the phases accumulated along each path can be expressed in terms of the classical actions along these paths S~,2 (Feynman and Hibbs 1965; Storey and Cohen-Tannoudji, 1994):

1

= ~ (s, - s2).

(1)

The classical action is defined in terms of the Lagrangian, which is (for a onedimension system with a position-dependent potential)

L(x, Yc) = ~ mYc2 _ V(x)

(2)

for a particle with mass m in a potential V(x). The classical action along each path then becomes (for i = 1,2)

Si - ~ L[x(t),Yc(t)] dt
aFi

= , mv--~- V(x) +-~ dt

(3)

= I- (p dx - H dt)
aF i
where H is the Hamiltonian governing the classical motion of the particle. In a time-independent problem, H is constant and the phase difference accumulated along the classical paths can be written as

~= fr kl(x) dx - fr kz(x)dx

(4)

where k(x) = ~V/2m(E - V(x)) is the local k vector. To answer the question "What will the interference pattern look like?" we
must consider in detail (Turchette, 1991; Turchette et al., 1992) the superposition of both contributing paths in the interferometer [Eq. (1)]. In doing this, we discover that the phase of the interference pattern can be attributed to two separate terms: a term dependent on the paths through the interferometer and a term dependent on any interaction that alters the de Broglie wavelength along these

OPTICS AND INTERFEROMETRY

21

paths" that is, q~= ~position +- m ~ . The first phase contribution, hereafter referred to as the position phase, is a function of the relative transverse grating positions, x i, given by

27r

%osition -- d - (xl - 2x2 + x3) -- kg(xl -- 2x2 + x3)

(5)

g

where kg = 27r/dg is the lattice vector of the grating. The second or interaction

phase shift (Aq~) arises from the difference between the interactions along the

two paths"

A=-hlfroL[x(t),k(t)] dt-lfr-h oL[x(t),.;c(t)] dt

(6)

where F ~ and F ~ now denote classical paths through the interferometer with x~ = x3 = 0 and with no applied interaction. This split is allowed because the action is stationary with respect to small perturbations of the paths. By splitting the observed phase in this manner, we can focus our attention on analyzing the phase difference between just the two paths F ~ and F ~ rather than solving the full path integral problem (Feynman and Hibbs, 1965). It is important to note here that Aq~ is 0 when the action along both paths are equal; that is, only a difference in the applied potential V(x) along the two paths will lead to an interaction phase shift Aq~.

B. PHASE AND CONTRAST MEASUREMENT

The near field detection scheme discussed above, in which the third grating masks the interference pattern, gives rise to oscillations in the total transmitted flux as the grating is translated with respect to the pattern. This method gives interference fringes like those shown in Fig. 10.
The detected intensity, I, from the portion of the two interfering beams passing through the third grating is

i - I,Vl 2 -- z 2 + Z 2 + 2A1A2 cos (q~)

(7)

= (I) (1 + C cos(q~))

where A~, A 2 are the amplitudes of these interfering beams and q~is their phase difference. The second preceding equation has been reformulated in terms of the mean intensity (I) = a] + a 2 and contrast:

C ~ I max - I min = 2A 1A2

(8)

/ a x + / i n a 2 + a 2"

The output of the interferometer signal is fitted to Eq. (7) to determine the phase difference, q~, the contrast, C, and the mean intensity, (I). Since the atoms

22

Jrrg Schmiedmayer et al.

in the interferometer do not interact with each other, the contrast is independent of the source intensity. Moreover, if the intensity of one of the interfering beams is attenuated by some factor, the contrast is reduced by only the square root of this factor. Thus, an interference pattern with 1% contrast may be obtained even if one beam is attenuated by as much as 10-4 (Rauch et al., 1990; Schmiedmayer et al., 1995a).

C. OPTIMIZINGCONTRASTAND SIGNALTO NOISE

One of our primary goals in designing this experiment was to be able to determine the phase of an interference pattern as accurately as possible. Neglecting systematic errors and assuming Poissonian counting statistics, the theoretical limit on the rms error in the phase measurement (Rauch et al., 1990; Dowling and Scully, 1993) is given by

1

o-= V~C

(9)

where N is the total number of atoms contributing to the recorded interference pattern and C is the observed contrast. The quantity V~C depends strongly on the open fractions ~i of the three amplitude gratings. For example, the third grating alone contributes a reduction in the observed fringe amplitude of sin(Tr/33)/Tr, and a reduction in contrast of sin(Tr/33)/7r/33.
The problem of determining the ideal open fractions can be split into two parts: optimizing the interference pattern and optimizing the open fraction of the third grating for near field detection. Taking into account all possible paths through the interferometer, the largest interference signal at the position of the third grating is obtained for/3~ = 0.56 and/32 = 0.5. Maximizing V~C for the third grating yields ~3 = 0.37 as the best value. With these open fractions, we expect a maximum contrast of 67% and a maximum detected signal of 1% of the initial beam. Note that higher contrasts (up to 100%) can be obtained with small first and third grating open fractions, but only at the expense of a reduced transmitted intensity.

D. GRATINGALIGNMENT
Parallel alignment of the axes of the three gratings with respect to each other was essential to the production of high-contrast fringes in our experiment. Roughly, the gratings had to be aligned to better than half a grating period over the beam height. For our interferometer, this corresponds to about 0.1 mrad (100 nm/1 mm). An expression for the contrast reduction due to rotational misalign-

OPTICS AND INTERFEROMETRY

23

ment results from assuming an extended incoherent source and an extended de-

tector in the plane of the third grating. The total interference pattern is then an

incoherent sum over all possible interferometers located in all allowed planes.

The total contrast depends on the relative rotations of the gratings a 1 = 01 - 02

and a3 = 03 - 02 according to E k s t r o m (1993):
( L)

( Lso3))

sin k h o L + 2 L al sin kgh3 0ll + L s + 2 L

h 3) C(a l, a 3, h o,

---

L
/. L
'~g"o Ls + 2 L Ol 1

LsO3t )
8k9h3 al + L s + 2 L

(lO)

where h0 and h3 are the source and detector heights respectively, L s is the distance from the source to the first grating, and L the separation between gratings. Figure 11 illustrates the relationship b e t w e e n the contrast and the rotational alignment of the third grating.

25-
h;
20-

15-

O
L)

10-

._,
I
0

I

I

I

I

1

2

3

4

I

I

5

6

Third Grating Rotation (mrad)
FIG. 11. The dependence of the contrast in our interferometer on grating rotation. The data shown is for the rotational alignment of the third grating. The insert illustrates the geometric arrangement discussed in the text and Eq. (10).

24

JSrg Schmiedmayer et al.

E. SENSITIVITY TO MOTIONS OF GRATINGS

In our discussions up to this point, we have assumed that the gratings were fixed in an inertial frame, although both vibrations and overall acceleration of the apparatus cause this assumption to fail. To account for the time-dependent displacements of the gratings from an inertial reference frame Eq. (5) must be generalized. The time-dependent phase of the interference depends on the position of each grating when the atom passed through it:

%osition(t) = kg(Xl(t- 27-) - 2 x 2 ( t - 7") -+- x3(t))

(11)

where z - L/v is the time it takes a particle with velocity v to travel between two gratings separated by a distance L. It is convenient to rewrite Eq. (11) as

qgposition(t) = qggrating(t- T) q- kg(Xl(t- 2"/') - X l ( t - T)) -k- kg(X3(t ) - x 3 ( t - T)) (12)

where the first term, which is called the grating phase and is given by qggrating(t) -- kg(Xl(t ) - 2Xz(t ) + Xa(t)), is the position phase [Eq. (5)] at the instant when the particle passed through the middle grating, while the other two terms describe the effects of grating motion during the free flight of the particle through the interferometer.
If the changes in position of the gratings are due to acceleration and rotation of the interferometer as a whole, we can derive expressions for phase shifts due to these non-inertial motions. Assuming that changes in the rotation rate and acceleration occur over much longer time periods than the particles' transit time through the interferometer, we express the time-dependent grating positions that determine the observed phase [Eq. (12)] in terms of the velocity and acceleration of the interferometer, then rearrange terms to reflect these specific non-inertial motions:

qgposition(t) ~ qggrating(t- 7") -k- qgrotation(t- 7") -F ~acceleration(t- 7"). (13) Here, the phase from rotation

qgrotation(t) = k g(Jc3(t) - Jcl(t))7" = kgLl~ 7"

(14)

is determined by the difference in the velocities of the first and last gratings,

which follows from the rotation rate, 1~; and the phase from acceleration

q~acceleration(t) =

kg

1 ~(Xl

(t)

+

.~3(t))7- 2

(15)

is given by the average acceleration of the first and last gratings. Because of these time-dependent phase shifts, vibrations of the gratings can
wash out the fringe pattern if it is not observed on a sufficiently short time scale. Since the gratings are mounted on independent stages on three different flanges, a reasonable model to assess the contrast loss due to vibrations invokes the assumption of independent, random, Gaussian distributed positions for each of the

OPTICS AND INTERFEROMETRY

25

three gratings, each with variance O-~x.Following Eq. (5) we separately consider

the positions at the time of passage through the middle grating and the subse-

quent displacements A x = x(t - z) - x(t) of the gratings during the passage of a

particle through the interferometer. The displacements zLr are also assumed to

T

be Gaussian distributed (variance or2with or orx O)vibT/V2 for ('Ovib'/"<< 1). Aver-

T

T

aging over a time larger than the characteristic time scale of the displacements,

we find that this model predicts that the contrast of the interferometer will be re-

duced to

C = COe x p ( - 1(or2(~grating) + or2(~inertial)))

= COexp(-kg2(3o-~ + or~))

(16)

--- COexp( -~k1" 2O'x2(6 + r 2 T2)).

The first term in the exponent (or2(~0grating) --6k2orx 2) comes from the random grating phase ~grating(t -- T), and the second term (org2(~inertial) -- 20"2) comes from the random movements of the gratings during the flight of the atoms through the interferometer (inertial noise). Equation (16) implies significant (70%) reduction of the contrast at rms displacement amplitudes of or~~ 1/10 of the grating period, or ---20 nm. Thus, vibration reduction represents a serious experimental challenge.
The best way to prevent grating motion from reducing contrast would be to isolate the whole interferometer and mount it on a stable inertial platform. This was not practical in our experiment, as it is very hard to isolate the whole vacuum chamber. We therefore adopted a combination of passive isolation of the apparatus from sources of vibration, active grating control, and digital data processing that corrected for the vibrational misalignment. Both of the latter remedies required knowledge of the relative positions of the gratings, which was provided by a light interferometer formed by three 3.3 /xm phase gratings rigidly connected to our atom optics gratings (see Fig. 9).
Our active control system assured long-term alignment of the gratings by servoing the second grating to stay at a given position relative to the other two gratings. This point was picked to ensure that the light interferometer was always near its maximum sensitivity point for position measurements. As an added bonus, the servo allowed us to apply a well-defined grating phase to the interferometer by deliberately shifting the second grating.
The position information from the light interferometer also made it possible to digitally correct our data after it had been collected: the light interferometer measured the grating phase qggrating in real time, allowing us to make suitable corrections for the dominating first term in Eq. (16). During each sampling period At, the readings from the light interferometer were recorded and stored along with the rest of the associated experimental data. During analysis, the data were sorted according to the measured relative grating positions (which corresponds

26

J&g Schmiedmayer et al.

tO sorting with respect to q~grating). Typically data were taken over a range of q)grating' and a fit to this data (a plot of atom counts as a function of qggrating) of the form of Eq. (7) was made. Recalling that ~p= ~position -Jr"A(~ and plugging in Eq. (13), we can express the total phase of the interference pattern as
~O= (~rotation q- ~0acceleration q- ~grating ~- m(~). Thus, this fit determines the sum of the phases due to interaction and non-inertial motion, the variables we observed in our experiments.
The combined effect of the servo and plotting atom intensity versus q)grating is to effectively remove the contribution of the first term of Eq. (16) to the noise. The residual average grating motion, after this correction, corresponded to an effective rms displacement 0.(x1 - 2x2 + x3,At), of typically 10-30 nm during the sampling period, resulting in a typical rms grating phase 0.(qggrating,At)of 0 . 3 - 0 . 9 rad for a 200 nm grating period. The best operating conditions were achieved by keeping the position servo relatively loose 0.x ~ 300 nm to suppress the higher frequency components caused by a tight lock. The typical contrast reduction was then about 25%.
While this method was effective in reducing noise due to independent grating vibrations, the problem remained of collective rotations or accelerations of the whole interferometer as expressed by 0"T2. The effect of this collective motion was found to be significantly smaller than that of the vibrations and was most evident when the interfering particles were moving relatively slowly (e.g., in experiments using Kr as a carrier gas). For our slowest atoms (700 m/sec) we observe a 25% contrast reduction due to residual accelerations and rotations. This could be corrected for by measuring the rotation and acceleration directly with a pair of accelerometers mounted close to the first and third gratings. In the future, we will employ this technique to improve the performance of the interferometer when operating at lower velocities.

F. INTERACTIONREGION
A unique feature of our interferometer is that the two interfering beams have been physically isolated by inserting a foil or "septum" between them where they are spatially separated immediately behind the second grating. To fit between the beams, whose centers are separated by only 55/xm, this septum must be thin andvery flat (<30/xm peak to peak ripple over its whole length).
Our interaction region was 10 cm long and the stretched foil was held symmetrically between two side electrodes. The foil was spaced from the side plates with insulating 2 mm thick precision ground alumina spacers (Fig. 12). We cut the foil in a "butterfly" shape, then pulled all wrinkles out of the area that was used in the final interaction region by clamping it in a special jig that stretched the edges away from the center and flattened the foil. The stretched foil was then carefully clamped between the spacers and side plates using a mounting clamp.

OPTICS AND INTERFEROMETRY

27

FIG. 12. Explodedview of the interactionregion. The foil is black. The insulating alumina spacers are shown in white,and the aluminumside plates are gray.The split atomicbeams of the interferometerenterfrom the front (lowerleft) and pass on either side of the foil.
We have made good septa using both 10/xm thick copper foil and 12/xm thick metalized mylar.
The interaction region was mounted behind the second grating on a stack of manipulators. These provided transverse translation to move the foil in and out of the beam line and rotation about both the vertical axis and the beam axis to align the plane of the foil parallel to the ribbon-shaped atomic beam. A typical 10 cm long septum, aligned to the atomic beam, cast a shadow on the detector which was 2 0 - 3 0 / x m wide. This is wider than the nominal 10/xm foil thickness due to overall deviations from planarity, waviness, and material rolled over at the cut ends. With the septum carefully positioned using precision translator and rotators between the beams in the interferometer, we have observed fringes with 23% contrast and an interference amplitude of more than 2800 counts/sec (Fig. 10).
This conducting physical barrier between the separated beams allows the application of different interactions to the two paths in the interferometer and measurement of the resulting differential phase shift. The sensitivity of this phase shift measurement is set by the interaction time. The intrinsic line width is 10 kHz for a 1050 m/s beam and 10 cm long interaction region. In an typical experiment we can determine the phase of the interference pattern with a precision of 5 mrad in one min, which corresp_onds to a sensitivity to energy shifts of roughly 3 x 10 -14 eV/m~/-~n or 8 Hz/Vmin.

28

J6rg Schmiedmayer et al.

G. MOLECULAR INTERFEROMETRY
Combining our pure Na2 beam (described in Section II.D) with our three grating atom interferometer, we constructed a Mach-Zehnder interferometer for molecules (Chapman et al., 1995a) (Fig. 13a). With 200 nm gratings, our molecular beam produced high-contrast fringes (Fig. 13b). Molecular and atomic fringes in our interferometer have the same period, since the period is independent of de Broglie wavelength in our white fringe interferometer. Therefore, we used two different methods to verify that the observed interference was actually from molecules:
9We introduced a (decoherence) laser that destroyed the atom interference pattern by scattering resonant light from the split atomic wave function inside the interferometer. 9We checked that the molecular interference signal (or fringe height) was maximum at a smaller detector offset from the beam axis than the atomic interference signal. Since the de Broglie wavelength of Na 2 is smaller than Na, the molecules diffract at smaller angles and pass through the interferometer on different paths than the atoms (Fig. 13a).
The results from a study combining these two methods together with turning on and off the laser used to deflect the atoms out of the molecular beam are shown in Fig. 13b. We observed the largest interference signal from the combined atom and molecule fringe pattern (both deflection and decoherence lasers

(a)

Decoherence

laser

I ~ Na~u~a2 -

Interaction region
-

Na 2 /

(b) 2.5x103 -
2.0 "d 1.5 ~ ~

12x103 --I,- _ _ _l
~8 ~6
4~

o 1.0

~N~

~

0.5

nm200

,c 0.6 m

>, I,~ 0.6 m

~,

0.0 I i I ~ I ~

I

0 40 80 120 160

3rdGrating Position (~tm)

FIG. 13. Interferometry with molecules: In (a), we show a schematic of our three grating interferometer displaying the different paths of Na and Na 2. In (b), the variation of the interfering signal vs. the third grating offset from the collimation axis is shown for the mixed N a - N a 2 beam (o = no laser on) and the pure Na 2 beam (• = decoherence laser on, A = deflecting laser on, # = both lasers on). Calculated curves are discussed in the text. The inset shows the interference fringe data for the mixed Na-Na 2 beam (e) and the pure Na 2 beam (0) observed for a third grating offset of - 10/xm.

OPTICS AND INTERFEROMETRY

29

off). The amplitudes of the interference fringes were reduced by the same amount with either the deflection or decoherence laser beam on, suggesting that only molecules contribute to the interference in both of these cases. This interpretation was confirmed by the fact that the fringe amplitude did not decrease further when both lasers were on simultaneously. The maximum interference signal for the predominately atomic beam was observed at 55/xm from the collimation axis, as expected from the diffraction angle for Na in Ar carrier gas, whereas the molecular interference signal maximized much closer to this axis. In Fig. 13b, the data are compared with curves calculated in the far-field limit using a convolution of the trapezoidal beam profiles with the 50-~m acceptance of the third grating. The upper curve is normalized to the maximum observed interference signal and the lower curve follows from the known fraction of molecules (27% of the detected signal).
For both the mixed interferometer and the purely molecular interferometer, the maximum observed contrast was nearly 30% and was the same to within 1%. We observed no degradation in interference signal despite the plethora of close lying rotational-vibrational states in the molecules. This is not very surprising since the first order interference observed in an interferometer involves only the interference of each particle with itself. The fact that two nearby molecules are very unlikely to have the same quantum numbers for both the rotational-vibrational state and total angular momentum projection is irrelevant. Although the 300 K thermal background photon energies typically exceed the internal level spacing of molecules (---1 cm-~ for rotations and --- 100 cm-~ for vibrations), decoherence effects due to transitions between vibrational or rotational levels or spontaneous emission are minimized because electric dipole transitions between rotational-vibrational levels in the same electronic state are not allowed in a homonuclear diatomic molecule (Herzberg 1950). Scattering of the molecules on the nanofabricated diffraction gratings could also cause rotational or vibrational transitions, since a beam velocity of 1000 m/sec and a grating thickness of 200 nm produces Fourier components up to 5 GHz (or 0.17 cm-~). However, this is less than the smallest allowed rotational transition 4B (B is the rotational constant) of 0.61 cm -~ and much smaller than the vibrational spacing of 159 cm-~. The fact that we did not observe any contrast reduction places only a weak bound on the probability of these transitions.
Using Kr as the carrier gas, our Na2 interferometer produced a beam separation of 38/xm at the second grating. This just exceeded the beam width at that position and allowed insertion of an interaction region with a thin foil barrier between the interfering beams. The foil cast a shadow 20/xm wide, which partially blocked the edges of the two beams and reduced the contrast from 19% without the foil to 7% with the foil. The lower observed contrast with Kr as the carrier gas (even without the inserted foil) is attributed to the slower beam velocity, which enhanced the inertial sensitivity of the interferometer, making it more vul-

30

J6rg Schmiedmayeret al.

nerable to vibrations of the entire apparatus. A similar contrast reduction was observed with atoms when using Kr as a carrier gas.

V. Atom Interferometry Techniques
We are now in a position to examine how interferometric techniques can be used to obtain useful physical information. We concentrate first on what information can be extracted from the phase of the interference pattern and the limits to the accuracy with which the phase can be determined. Then we describe how the contrast may be exploited to infer properties of the interaction even though the various unselected internal states of the atoms or molecules have different phase shifts. In the last section, we discuss a new technique that will greatly reduce the systematic errors and contrast loss arising from velocity averaging.

A. SIGNIFICANCEOF PHASE SHIFTS

We can learn a great deal about various interactions by measuring the phase shift

of an interference pattern caused by the applied interaction. In most of our ex-

periments, we exploited the ability to physically separate the two arms of our in-

terferometer by applying a time-independent interaction potential V(x) to one

arm only (classical path F~ Since the other arm of the interferometer has no po-

tential applied, the interaction induces a relative phase difference between the

two arms (see Eq. (6)).

We recall that the interaction phase Aq~ from Eq. (6) is 0 for the case of no ap-

plied interaction to either area. Hence, if one arm is unshifted, the overall phase

shift of the interference pattern is given by the difference between the phase ac-

cumulated along the shifted arm with the interaction on and the phase accumu-

lated along this arm if there were no applied interaction.

Thus, the phase shift induced by the potential is of the form:

A,p(ko)=(fkr(x)-ko(x))dX=Akf(rx) dx (17)

o

o

where k0 = 1 / h ~ and k(x)= 1/hV'2m(E- V(x)) are the unperturbed and
perturbed k vectors, respectively. If the potential V is much smaller than the energy of the atom E (as is the case for all of the work described here), the phase
shift can be expanded to first order in V/E:

A'P(k~ = ~ k~f -v-(Ux) dx = -h---l~f v(x) _ - - Ul f V(t) dt (18)
a
where, in the integral over the time, we used the fact that the potential is time

OPTICS AND INTERFEROMETRY

31

independent and one can apply the paraxial approximation using t = x/v. Alternatively, we can think of the potential V(x) as giving rise to a refractive index:

k

V

n=~ = 1 2E

(19)

where the phase shift can now be expressed as Aq~(k) = kf(n(k) - 1) dx. We can see from Eq. (18) that the phase shift due to a constant scalar potential applied over a length Lint is Atp(k0)--(-m/h2ko)VLinc In our interferometer with a 10 cm long interaction region and 1000 m/sec Na atoms, an applied potential of V = 6.6 x 10 -12 eV corresponds to a refractive index of
I1 nl = 2.7 x 10 -11 and gives a phase shift of 1 rad. Note that positive V -
corresponds to a repulsive interaction that reduces k in the interaction region, giving rise to an index of refraction less than unity and a negative phase shift.
Equation (18) shows that the phase shift associated with a constant potential depends inversely on velocity and therefore is dispersive (it depends linearly on the de Broglie wavelength). If, on the other hand, the potential has a linear velocity dependence, as in the Aharonov-Casher effect (Aharonov and Casher, 1984), the phase shift becomes independent of atomic velocity. Similarly, a potential applied to all particles for the same specific length of time, rather than over a specific distance, will produce a velocity-independent phase shift Aq~= 1/hfV(t)dt, the scalar Aharonov-Bohm effect (Allman et al., 1993; Badurek et al., 1993).

B. AVERAGINGOVER THE VELOCITY DISTRIBUTION: THE COHERENCE LENGTH
Real experiments are not performed with monochromatic beams, and since atom sources tend to have relatively large velocity spreads, velocity averaging is an important consideration in the analysis of our experiments (our velocity spread is typically about 4% rms). In our previous analysis, we have not discussed the fact that the observed phase shift, Aq~, and contrast, C, result from weighted averages over the different velocity components present in the beam.
In general, one can represent the averaged interference pattern by an averaged phase vector C e ia~~ in the complex plane. Velocity (momentum) averaging is calculated by integrating over the normalized initial atomic k vector (velocity) distribution f(k):

-C e iA~ = k)Co(k) e iA~k) dk

(20)

where we take into account a possible dependence of the original contrast of the interferometer, Co(k), on the wave vector k. The average phase shift Aq~ and con-

32

J6rg Schmiedmayer et al.

trast C are the argument and magnitude of the averaged phase vector, respectively.
In the simplified case, when the contrast of the interference fringes is independent of the velocity of the atoms, then -C e ia~ = C Off(k)e iA~k) dk and one finds for the observed phase shift A~o, and contrast, C:

9 { f f ( k ) sin(Aq~(k)) dk

A~o = arctan -9~

}

(21 )

\ f f ( k ) cos(Aq~(k)) d-k

-C = Co~/[ff(k ) sin(Aq~(k)) dk] 2 + [ f f ( k ) cos(Aq0(k)) dk] 2 = CoP(Aq~) (22)

where Co is the contrast at zero phase shift and p(Aq~) is the relative retained contrast as a function of the applied phase.
Due to the non-linear dispersion of the phase shift (1/ko for a scalar potential, 1/k 2 for a deflection in a potential gradient), Aq~ is not simply the phase shift for particles traveling at the mean velocity. This nonlinear dispersion causes systematic phase shifts that depend on the width, and to a lesser extent on the form, of the velocity distribution. For precision measurements these shifts must be accounted for in the analysis. For a 4% Gaussian velocity spread, the contrast is reduced to 28% of its initial value and the observed phase [given by Eq. (21)] differs from the applied phase q~(vo) by 0.20% at Aq~ = 40 rad. This phase error can be avoided by the velocity multiplexing technique described later.
In our experiments, the reduction of the contrast can be parametrized by the coherence length defined by l~oh= 1/tr, of the beam. In the case of linear dispersion, iff(k) is a Gaussian distribution with rms width tr, centered at k0, the above equation then reduces to

C = Co exp - ~[Aq~(ko)]2 ,-D-: .

These assumptions are reasonable because seeded supersonic beams give a flux density that varies approximately as v3 exp(-(v VO)2/2Av2), which is quite
-
Gaussian when Av << v0, the mean velocity. Furthermore, the true 1/v dispersion is well represented by a linear (negative) dispersion over our narrow velocity distribution. Our contrast data were fit within estimated total errors by this expression when the differential phase shift was supplied by a constant potential due to an electric field (see Fig. 14).
It is important to note that, unlike photons in vacuum, the coherence length and wave packet size for matter waves are not the same, except perhaps at specific points in time. This is because the vacuum is dispersive for matter waves. In our beam, the coherence length is only 0.65 A (1.6 ./~ FWHM) at the source. But, by the time a minimum uncertainty wave packet that could be created at the source reaches the third grating (where the interference "occurs"), its length

OPTICS AND INTERFEROMETRY

33

30 -

25 -

20-

15-

0

-

~ 10-

5

0 ~- ~
, , , , I , , ~ , I ....

-100

-50

0

I
I ~, , , I

50

100

Phase (radians)

FIG. 14. Reduction of the interference contrast with applied phase. From the width of the contrast curve, we calculated a coherence length of 0.65(3)/~ in good agreement with a determination of the velocity distribution from a measurement of the diffraction pattern.

would be on the order of 10 cm FWHM, an increase by a factor of 108 (Klein et al., 1983; Kaiser et al., 1983).
C. CONTRASTINTERFEROMETRY
If all atoms in our interferometer have the same interaction potential and the same velocity, their interference patterns will all be in phase and will combine to give an observed interference pattern of maximum contrast. Observations of decreased contrast therefore allow investigation of the differences among the interfering atoms (Schmiedmayer et al., 1994a). Of particular interest is the case when the internal states of atoms or molecules in our beam respond differently to some applied interaction. As we shall now discuss, this can result in a periodic degradation and revival of the contrast of fringes in our interferometer. This effect can be employed to gain new, highly accurate information from measurements of the contrast--we call this contrast interferometry.

34

J6rg Schmiedmayer et al.

We consider now the case in which each internal state interacts differently. Atoms in each state therefore form independent interference patterns, and the observed intensity is the incoherent sum of all these individual patterns

/observed = ~ C0(l -k-L" Cos(Aq~i))

(24)

i

where f,. is the fraction of atoms in the ith state, A q~;is the phase shift of atoms in

that state, and COis the contrast of atoms in a pure state. Both the phase and the

contrast of the interference pattern thus reflect this average over internal states.

The key point is that, if there are a finite number of internal states, one can

expect destructions and revivals of the total contrast, especially if the phase

shifts of these states are regularly spaced. The presence of revivals gives contrast

interferometry the potential for high-accuracy measurements.

As a demonstration of contrast interferometry, we studied the interactions of a

magnetic field with the ground level of sodium atoms (Schmiedmayer et al.,

1994a). The sodium ground state, 2S1/2, consists of two hyperfine levels with to-

tal angular momentum F = 1, 2, respectively. In the presence of a weak mag-

netic field, B, each atom experiences a potential V ( x ) = - I x " B = gFtZemFB

(Zeeman splitting), where gFtZBF is the atom's total magnetic moment and

--grl.t,BmF is the projection of the magnetic moment in the direction of the field.

This interaction splits these levels into eight magnetic substates, each with one

of five possible magnetic moment projections: gFl~BmF--- ( - 2 , - 1 , 0, 1, 2)/xJ2 with associated f / - 81, 41, 41, 41, 81-

By applying field magnitudes differing by AB(x) to the two arms of our inter-

mf ferometer, we introduced a relative phase shift Aq~,which is given by

A~k~ = ~ o gFI,~BmFZ~kB(x) fix.

(25)

For a 1000 m/sec Na atom in a F = 2, m F -- +- 2 state in a magnetic field of 0.01 G, we found a phase shift of 8 rad corresponding to a refractive index of (1 - n) ~ 2.5 • 10-10.
Since the relative phase shift given by the preceding equation differs for the
five possible values of gFmF, the total interference pattern is an incoherent sum
of five individual patterns averaged over the incoming velocity distribution. The contrast of the interference pattern, as a function of the velocity averaged phase shift A~o2 of the (F = 2, m F = 2) stretched state, therefore is

C(Aq92)=-C~[P(Aq92)cos(Acp2)+2P(~~-)cos(~2)+ 1] (26)

where Co is the 'initial' contrast and p(Aq~2) is defined in Eq. (22). The main feature of Eq. (26) is a rapid decrease in contrast with rising phase shift and later revivals of the contrast at specific values of A~o2, where the interference patterns rephase. The
nth revival occurs when [A~o21= 4nlr for the ImFI= 2 states, IAq~21= 2nzr for the

OPTICS AND INTERFEROMETRY

35

[mFI = 1 states, and 0 for the m F -- 0 states. If mq)2 is small (Aq~2 < k/ok), the average over the velocity distribution tends to diminish and broaden the rephasing resonances. As the applied phase shift becomes larger, (~lq~2 >> k/ok), such averaging reduces the contrast to 0 for all atoms except those in m F = 0 states, which experience no Zeeman shift. The total contrast in the large phase shift limit thus is decreased to one quarter of the original contrast. Data from a typical rephasing experiment are shown in Fig. 15, together with a fit from Eq. (26).
Contrast interferometry has various applications. If a particular interaction is known accurately, as in the case of the magnetic interaction just described, it be-

25 20
,~ 15

, ~ias

5

0

"" i I / ~1

I

i

I

i

i

,

, I

0

~ ~]~_o._ -..e~T - - - J, 6 ~ - - ~ a . _ ~ - , , _ . - - . 0
, ~ ~I -* Tt--'~' " "~t"r "" . . . . .

___~

9 "---

-

_//- gsep tum O

-5 , 0

9

~-

x

I

,

I

200 400

t

--, - - , - - i , t . ,

,

I

600

~_-Boia s

,

I

,

T

800 1000

Current (mA)
FIG. 15. Contrast and phase of the interference pattern versus septum current Is. The upper graph shows the contrast revivals from constructive rephasing of the independent interference patterns of the eight different magnetic substates of the sodium atom. The lower graph shows the phase (in rad) of the observed interference pattern. The inset shows the septum geometry, currents, and magnetic fields as used in the magnetic rephasing experiment. The upper inset shows a schematic of the metal septum separating the two interfering beams, the current connections, and the current flowing through the septum. For clarity, the side plates of the interaction region were omitted. The lower inset shows a detail of the interaction region and the magnetic fields for the rephasing experiment. The dark arrows are magnetic fields, and the light arrows represent the atomic beams.

36

Jiirg Schmiedmayer et al.

comes possible to measure the beam's velocity distribution. Altematively, if rephasing techniques are applied to systems where the interaction is unknown, they may reveal new information about it. One such system is a molecule having a tensor polarizability, which causes an orientational dependence of the polarizability. States with different projections of the total angular momentum [mj]have different interactions, leading to variations in the contrast that may be used to infer the tensor polarizability (the phase shift basically is determined by the isotropic polarizability).
Another use of contrast techniques in interferometry is to isolate the interference pattern of atoms in a single state by destroying the contrast of interference patterns of all other states. This may be achieved by applying a large dispersive state-independent phase shift to one arm of the interferometer and then selectively regaining the contrast in the desired state by applying a state-dependent interaction to the other arm whose magnitude cancels the dispersion only in the desired state. We demonstrated this idea by using a Stark phase shift to compen-
sate for a large magnetic phase shift (Atp >> k/crk) with the same dispersion properties (i.e., Atp ~ 1/v) but opposite sign. Contrast was regained for one spe-
cific magnetic substate at a time, allowing experiments with polarized atoms even though the atomic beam was unpolarized (Fig. 16).

30

i] 11[I[lllllllU/

20 10-

, R~_ Bseptum Bbias
I

0 -

mF[ ---1

-10 -

600

800

1000

Septum Current (mA)

FIG. 16. Magnetic phase shift for Irnrl = 2 and ImFI= 1 states as observed with an unpolarized beam in a "magnetic rephasing" experiment. An additional dc Stark phase shift of 65 rad is applied in one arm of the interferometer. For 400-700 mA septum current, one of the ImFI = 2 states and, between 700 and 1050 mA, two of the IrnFI = 1 states are shifted back in coherence. The two slopes of the phase correspond to different magnetic moment projections being within their coherence length. The insert shows a schematic for the field configurations of the E - B balance experiment.

OPTICS AND INTERFEROMETRY

37

D. VELOCITY MULTIPLEXING
We have proposed a rephasing technique that recovers the contrast lost due to velocity averaging (Eq. 22) with a constant potential (Hammond et al., 1995). The basic idea is to make the velocity distribution discontinuous, selecting a discrete set of velocities such that the acquired phase shifts are all multiples of an applied phase. At some applied phase, the acquired phase shifts will all be multiples of 27r and the interference patterns of atoms in all velocity classes will rephase. This creates a contrast revival analogous to those just discussed for contrast interferometry. Another view of this is that those velocities that do not add constructively to the final interference pattern are filtered out.
The desired velocity distribution can be formed by two fast choppers (beam shutters) a distance L c apart that are periodically and simultaneously opened for a time fAt, where f is the open fraction and At is the period. These cut the original velocity distribution of the beam into a comb of velocities. The transmitted atoms will have a velocity distribution with peaks at velocities v = Lc/t , = L~/nAt. The integer n is the number of shutter cycles that occur during the traversal time t, = nAt between the two shutters for a particle with velocity v n. For an interaction V = htoint applied to one arm of the interferometer over an interaction region length Lint, the applied phase shift for velocity class v, is
~n --~ ('Oint/int = ('Oint (Lint/tc) n A t . The phase shifts of the different velocity classes will be equally spaced and the mth rephasing will occur when tOint(Lint/Lc) At =
2mTr. All velocity classes will then have accumulated phase shifts that are a multiple of 2rr and will be in phase, resulting in a contrast revival.
This rephasing technique will allow us to apply much larger phase shifts without losing contrast due to the velocity dependence of the phase. More important, the nonlinear relation between phase shift and velocity that causes pulling of the averaged phase from its center value acts only within the individual narrow velocity slices; consequently the phase pulling is small enough to permit an overall fractional uncertainty of less than 1 part in 105.
Velocity multiplexing combines the advantage of very narrow velocity slices with the high intensity of an unchopped beam. The optimum parameters for best phase determination using this velocity multiplexing technique show a broad maximum around open fraction f = 0.375, where 14% of the original beam is transmitted. Increasing the chopping frequency widens the spacing between contrast revivals and permits a more accurate determination of V. Numerical calculations show that high precision phase measurements can be made even with a thermally effusive beam with a velocity width of 100% (Fig. 17).

E. MEASURING DEFLECTIONS
Another potential application of matter wave interferometers utilizes their high spatial resolution to measure small deflections resulting from the application of a uniform potential gradient applied across the region traversed by the atoms. Ap-

38

J6rg Schmiedmayer et al.

n (tP(Vo)/2rc)

20

40

60

80 100 120 140 160

1.0

I

I

I

I

I

I

I

I

I

I

I

35

4p

45

0.6

G)
0.4

0.8
0.6 G) r,.O)
0.4
0.2
0.0

il

(") 0.2

Ii

0.0
, ,

3 2

i !

, ~

I ! ~ ,

-1 o"

'A;~ ~
! ,

F ;a 0.2
9- 0.0

-2 -3

9"-~ -0.2
I
220
' I I . ' i I
;i
; i
;; A " ,i

!

I

I

240

260

280

g~(Vo) in radians

open fraction f

-- - 25%
37.5%
...... 50%

/x,

,:,.--.,X

9

0

200

400

600

800

1000

tp(vo) in radians

FI6. 17. Revivals in the contrast are shown as a function of applied phase ~v0) and n (top axis) for three velocity distributions. Revivals occur at n = 40 (m = 1), n = 80 (m = 2), etc. The inset shows a detailed study of the first revival in contrast for an open fraction of f = 0.375. The contrast (top) and the progression of the observed phase (middle) are shown as a function of applied phase ~v0). A phase 0 in %bs coincides with the contrast maximum to better than 1 part in 105. A vertical dashed line is drawn through the contrast maximum to guide the eye. The generation of the lower graph is described in the text. It shows that the phase difference is very linear in the region of the contrast maximum.

plying a potential gradient across the entire interferometer leads to a phase shift between the interfering paths proportional to the difference of the potentials on the two paths traversed, as described in the previous section.
The primary difference between applying a uniform potential gradient and uniform (but different on the two arms) potentials is that, in an interferometer with diffractive beam splitters (like ours), the separation between the interfering paths depends on the de Broglie wavelength and hence on the velocity of the

OPTICS AND INTERFEROMETRY

39

atom. When passing through a potential gradient, slower atoms therefore will have a bigger separation between the two arms and see a bigger potential difference than fast atoms. This adds one additional power of 1/v to the dispersion, resulting in a total velocity dependence of 1/v2 in experiments where a constant, velocity independent potential gradient is applied (the other power of 1/v comes from Eq. (18). This means that averaging over realistic velocity distributions will give more blurring of the fringes when using a potential gradient rather than a stepwise uniform potential.
Not only the 1/v2 dispersion, but also the absolute amount of phase shift of the atoms in the potential gradient may be calculated from the classical displacement Ax of the atoms' trajectories in passing through the potential gradient (i.e., force), converted to a phase A~o = Ax kg by multiplying by the grating vector kg. The interference pattern, the envelope of the fringe pattern, and the fringes themselves all move as a unit in a potential gradient, following the classical trajectory of the atoms (Ehrenfest's theorem).

VI. Measuring Atomic and Molecular Properties
Atom interferometers will find wide application in the study of atomic and molecular interactions, particularly through measurements of the phase shifts due to differential interactions applied to the arms of the interferometer. A separated beam atom interferometer has the important advantage that one can investigate ground state atomic properties and interactions with spectroscopic precision, even in cases where atomic beam resonance techniques (Ramsey, 1985) are inapplicable because all the sublevels are shifted by the same amount. We discuss next the first application of an atom interferometer in this manner, a precise measurement of the polarizability of atomic Na.
When the observed phase shift results from the time integral of some applied interaction potential, as described in Section V.A, the interferometer is essentially measuring energy level shifts, and it is unlikely that the phenomenon under study cannot be studied by some sort of spectroscopy. However, a separated beam interferometer can also directly investigate phase shifts associated with interactions like collisions with other atoms or surfaces, which are often not accessible by other techniques. Such a novel application will be discussed later in this section--the measurement of the index of refraction of a gas for atomic matter-waves.
A. ELECTRICPOLARIZABILITYOF NA
We have used our separated beam atom interferometer to perform a high-accuracy measurement of the electric polarizability, a, of the Na atom (Ekstrom, 1993; Ekstrom et al., 1995). The dramatic increase in accuracy achieved here

40

Jrrg Schmiedmayer et al.

came from two sources: our ability to apply a very well-controlled interaction characterized by a uniform electric field and our ability to gain precise knowledge of the interaction time by measuring the beam velocity using single-grating diffraction patterns (Section Ill.B). Previous methods relied on deflection of an

atomic beam in a potential gradient, and were limited by the uncertainties in the

characterization of the applied gradient and the velocity distribution of the

atomic beam (Molof et al., 1974).

In our experiment, we applied a uniform electric field, E, to one of the sepa-

rated atomic beams, shifting its energy by the Stark potential V = -aEe/2. The

resulting induced phase shift is quadratic in the applied potential and given by

(Eq. 18):

(~

Aq~=~v ~aEZ(x) dx --~v~a ~ Le.

(27)

where v is the mean velocity of the atomic beam, 9is the voltage applied to one

side of the interaction region across a distance D, the spacer width, and Leff is the

effective interaction region length defined as

(.)2 fLetf ---- E 2 dx.

(28)

In our interferometer with a 10 cm long interaction region (Letf ~ 10 cm) and a beam velocity of v-- 1000 m/sec, an electric field of 280 V/cm produces a phase shift of 1 rad. A typical measurement of the Stark phase shift is shown in Fig. 18. The phase shift for various voltages was measured with respect to the phase with no voltage applied. To correct for drifts and fluctuations of the 0 phase, we took frequent measurements of the 0 reference phase. We found the Stark phase shift to be a quadratic function of the applied voltage whose coefficient we measured with a statistical uncertainty of typically 0.2%.
For an accurate determination of the electric polarizability, the crucial elements are the knowledge of the magnitude of the applied fields, the exact geometry of the interaction region, and the width and mean of the velocity distribution of the Na atoms. The main contributor of uncertainties in the electric field and Leff were the spacer thicknesses D and the fringing fields near the ends of the septum. The spacer thicknesses D were measured to 0.05% with a dial indicator calibrated with precision gauge blocks. The electric fields around the ends of the interaction region were calculated numerically using standard relaxation methods, and the results were parameterized by an effective length Letf.
We performed polarizability measurements with three different interaction regions, displaying different field configurations. The first and second interaction regions had foils with lengths 10 cm and 7 cm, but no guard electrodes. The third interaction region had guard electrodes located at the ends of the side plates and spaced 6 cm apart (Fig. 18 insert), which were held at the same potential as

OPTICS AND INTERFEROMETRY

41

(a)

60

Interaction region Hot wire

40-

I

Na beam _

_~,

detector

20-

0

I~ 0.6m -,- 0.6m ~-]

-20 -
.=

Detail of interaction region

T 2 m-m ' ~~
N ~ I_ ~

-- ~ f
lO~Otmm8 0 tm m~ 60 mm -1

_i_
4 mm

-40 -
-60 -
"=io.. 0.5-
0.0 ~, -0.5 -
0

i

I _

-

"

-" "t

I

1

I

I

100 200 300 400 50O

Voltage Applied (volts)

FIG. 18. Measurement of the electric polarizability of Na: (a) shows a schematic of the separated beam interferometer with the interaction region installed behind the second grating; (b) shows the measured phase shifts vs. applied voltage. The two different signs of the phase shift stem from the voltage being applied on either the left (open circles) or the right (filled circles) side of the interaction region (arm of the interferometer). The fit is to a quadratic and the residuals are shown in the lower graph.

the foil to minimize the fringing fields. In this interaction region, the fringe fields had a much smaller contribution to Leff. We also performed polarizability measurements with voltages applied to the right side or the left side of the interaction region (see Fig. 18), using both the left and the right interferometer. In addition we measured the asymmetry of the interaction region (it was 0.1%) by applying a voltage to the septum with the side plates grounded.
The mean velocity and velocity width of the Na beam were determined to 0.15% from a fit to the diffraction pattern produced by the first grating (period 200 +__0.1 nm) (see Section Ill.B). The velocity distribution of our beam complicated this simple analysis. The rms width of the velocity distribution was on the order of 3-5%, and we had to average over the actual velocity distribution to extract the polarizability accurately from the phase of the interference pattern (see Eq. (21) and the discussion in Section V.B).
Additional systematic shifts can arise, because the velocity distribution contributing to the interference pattern may differ from the velocity distribution of the atomic beam as determined from the diffraction pattern. This can be caused by blocking of atoms by the septum or variation in the detector position, both of which are velocity selective because faster atoms have a smaller diffraction angle and therefore travel closer to the axis than slower ones. These effects constituted a correction of about 0.4% and were measured by changing the positions of the interaction region and the detector. These data were found to agree with a

42

Jrrg Schmiedmayer et al.

model obtained with a ray tracing algorithm. We estimated that these corrections introduced an additional uncertainty of 0.15% into our determination of the polarizability.
Taking all corrections and all sources of errors into account, we found the Stark shift of the ground state of sodium to be 40.56(10)(10) kHz/(kV/cm) a, which corresponds to an electric polarizability of a = 24.11(6)(6) • 10-24 cm 3, where the first error is statistical and the second is systematic (Ekstrom et al., 1995). Our systematic error was dominated by uncertainties in the geometry of the interaction region and uncertainties in the determination of our velocity distribution, and our statistical error was dominated by the short-term stability of the phase reference in our experiment and to a lesser extent by counting statistics.
Our measurement represents a nearly 30-fold improvement on the best previous direct measurement of the polarizability of sodium 24.4(1.7) • 10-24 cm 3 (Hall and Zorn 1974), a 7% result. The currently accepted value 23.6(5) • 10 -24 cm 3, with a 2% uncertainty (Molof et al., 1974), comes from a measurement of the Na polarizability with respect to that of the 23S1 metastable state of He, which is calculated (Chung and Hurst, 1966). Using our measurement together with the experiment of (Molof et al., 1974) allows us to determine the polarizability of the 3S~ metastable state of He to be 47.7(1.0) x 10-24 cm 3, in good agreement with the calculated value 46.77 • 10-a4 cm 3, (Chung and Hurst, 1966). The error in the experimental value is dominated by the experimental error of Molof, et al. (1974).
Significant improvements in our technique would result from an interaction region whose spacing was determined more accurately (e.g., with light interferometry) and from finding a better way to determine the velocity of the interfering atoms. Better determination of the velocity distribution can be accomplished by a magnetic or radio frequency rephasing experiment (Schmiedmayer et al., 1994a; see also Section V.C) or by using our velocity multiplexing scheme (Hammond et al., 1995; see also Section V.D). With these improvements it seems feasible to perform polarizability measurements with uncertainties in the 10 -4 range.

B. REFRACTIVEINDEXFOR NA MATTERWAVES
In this section, we discuss a study of an atomic property that was inaccessible to measurement before the advent of atom intefferometers - - the index of refraction seen by atomic de Broglie matter waves traveling through a gas sample (Schmiedmayer et al., 1995a). This effect is the direct counterpart to the wellknown index of refraction found in optical physics, in which an optical wave is phase shifted (and possibly attenuated) while passing through a dispersive medium. In the case of atomic de Broglie waves, the index of refraction arises

OPTICS AND INTERFEROMETRY

43

from the collision-induced phase shift between the ground state Na atoms and

the molecules in the gas (Schmiedmayer et al., 1993). Our studies of the phase

shift in collisions add significant information to long-standing problems, such as

solving ambiguities in the inversion of the scattering problem to find the poten-

tial (Chadan and Musette 1989), the attempts to interpret other data sensitive to

the form of the long-range interatomic potential (Bagnato et al., 1993; Lett et

al., 1993; Cline et al., 1994; Walker and Feng, 1994) and to collective effects in

a weakly interacting gas (Stoof, 1991; Moerdijk et al., 1994; Moerdijk and Ver-

haar, 1994; Stwalley et al. 1994).

From the perspective of wave optics, the evolution of the wave function, ~,

propagating through a medium in the x direction for a distance x is given by

27r

27r

xIt(x) = xlt(O) eikox ei ~ Nx Re(f(k,'mO))e - ~ Nx tm(f(k,.m'O)).

(29)

Here k0 is the wave vector in the laboratory frame, k cm is the wave vector in the center of mass frame of the collision, N is the density of the medium and f(kcm,O ) is the forward scattering amplitude. To measure the index of refraction, we introduce a gas in the path of one arm of the interferometer. The phase shift of 9 on the arm with the medium relative to the arm with no medium is then given by

27/"

A(#(X) = K~cm Nx Re(f(kcm,O))

(30)

which is proportional to the real part of the forward scattering amplitude. In addition, the amplitude of 9is attenuated in proportion to the imaginary part of the forward scattering amplitude, which is related to the total scattering cross section by the optical theorem

471"

Otot = kcm Im(f(kcm,O)).

(31)

In analogy to light optics, one defines the complex index of refraction

27]"

n= 1+

N "f(kcm,O).

(32)

k0kcm

We elected to use the ratio of the real and imaginary parts of the forward scattering amplitude, Re[f(k,O)]/Im[f(k,O)], as the primary variable to be measured and compared with theory. This ratio proves to be a more natural theoretical variable with the advantage that it gives quite "orthogonal" information to the previously studied total scattering cross section. In addition, it has the experimental advantage of being independent of our knowledge of the absolute pressure in the scattering region, which is known less accurately than the 3% accuracy with which we were able to determine this ratio by measuring the slope of

44

J6rg Schmiedmayer et al.

the observed phase shift plotted as a function of the log of the fringe amplitude for each particular gas density (see also Fig. 19c)"

-Aq~(N)

Re[f(k,O)]

(33)

ln[A(N)/A(O)] Im[f(k,0)]"

Since both A and Aq~ could be determined from the same interference scan, this method did not rely on a pressure measurement at all. This procedure also took advantage of the fact that the interference amplitude decreases only as the square root of the intensity in the attenuated beam (Rauch et al. 1990) and therefore is easier to measure at high target gas densities, where the intensity of the beam passing through the gas-filled side of the interaction region is strongly reduced.
We have used our separated beam atom/molecule interferometer to measure the ratio Re[f(k,O)]/Im[f(k,O)] for the scattering of Na atoms on various monatomic rare gases He, Ne, Ar, Kr, and Xe and the molecular gases N2, CO 2, NH 3, and H20 (Table II) (Schmiedmayer et al., 1995a). In addition, we have measured both the phase shift and attenuation of Na2 de Broglie waves that pass

(a)
200 gm

~

_)__2mm

,00cm

(b)

(c)

/

N

~ 4

?

/

~-)~i

~

Ne

-

3

Ar

-

-

~ 2 "T/"

e -

x: 1

"

He

o

'

,

I

,

I

,

0.00

0.02

0.04

, , ,,,,,I

, ~ i,,,,,I

0.01

0.1

, ~ ~ . . . . ,-71

Gas Density (1016atoms/cm2)

Fringe Amplitude

Fro. 19. Experiment to measure the refractive index for Na matter waves when passing through a dilute gas: (a) The detail of the interaction region shows the 10 p,m mylar foil suspended between the side plates. The side plates that form the gas cell are indicated in black at both ends. (b) The phase shift of Na matter waves passing through He, Ne, Ar, and Xe gas as a function of the estimated gas density in the cell. (c) The phase shift of Na matter waves plotted vs. the interfering amplitude when passing through He, Ne, and Ar in the gas cell. The slope of the fitted line is a direct measurement of
the ratio Re[f(k,O)]/Im[f(k,O)].

OPTICS AND INTERFEROMETRY

45

TABLE II PHASE SHIFTAqg,REFRACTIVEINDEXn, ANDTHE RATIORe[f(k,O)]/Im[f(k,O] FOR 1000 M/SECNa
ATOMSPASSINGTHROUGHVARIOUSGASESAT 300 K AND 1 MTORRPRESSURE

Experiment

Calculations

A~p mtorr-l

( n - 1) 10 l~ mtorr-1

Re (f)/Im (f)

(6-8) Potentials

(6-12) Potentials

General Potentials

He

0.50

0.14 _+ 1.18i

0.12(2)

Ne

2.0

0.55 _+0.56i

0.98(2)

1.24

Ar

3.9

1.07 -+- 1.81i

0.59(3)

Kr

5.4

1.51 _+ 2.45i

0.62(6)

0.75

Xe

6.5

1.81 _ 2.49i

0.73(3)

0.76

N 2

4.7

0.91 __+1.39i

0.60(2)

NH 3

3.3

1.30 _ 2.16i

0.65(4)

CO 2

5.0

1.37 __+2.21i

0.62(2)

H20

6.2

1.71 __+2.40i

0.72(3)

0.26

1.1

0.69

0.65

0.73

0.73

Note." The data are compared to JWKB calculations using (6-8) (Gottscho et al. 1981) and (6-12) (Duren et al. 1972 and Barwig et al. 1966) potentials and in the last column for potentials given by J. Pascale (He) (1983) and Tang and Toennies (Ne, Ar) (1977).

through Ne gas in one path of the interferometer (Chapman et al., 1995a). To perform these experiments, we modified the interaction region so that a gas target could be inserted in one arm of the interferometer. An inlet was added to the center of one side plate for the introduction of gas, and end tabs were added to restrict the openings at the entrance and exit to only 200/xm (Fig. 19a). This allowed us to send one portion of the atom wave through a gas with pressure of "~ 10 -3 torr without noticeably attenuating the atom wave passing on the other side of the septum. By changing the carrier gas used in our source (see Section 11.2 and Table I), and hence the velocity of our atomic beam, we also measured the velocity dependence off(k,0) for the rare gases (Fig. 20) or equivalently the dispersion of the refractive index.
Our experimental procedure was to determine both the amplitude reduction A(N)/A(O), proportional to exp(-27r/km)N Im[f(kcm,O)]), and the phase shift Aq~(N), proportional to -(2rr/kcm)N Re[f(kcm,O)] , from a fit to the observed interference fringes with and without gas in one arm of the atom interferometer. We positioned the interaction region with the stronger 0th order beam passing through the gas sample so that the absorption at first equalized the amplitude of the two interfering beams, resulting in higher contrast for the observed fringes. Interference patterns were recorded alternately: first while leaking gas into the interaction region and then with zero gas flow. This procedure provided the reference amplitudes and 0 phase points. The gas flow into the interaction region

46

J6rg Schmiedmayer et al.

"~ 4
t~
~9 0

-4

10-

8-

.~ 6 -

4-

2-

0-

~~l'""'"""g'"-

2

4

6

8

10

12

14

impact parameter in Angstroms

F~G.20. Integrandsfor Re[f] and Irn[f] for Na-Ar collisions at 1000rn/sec. The rapidly oscillating behaviorat small impact parameteraveragesto 0 for Re[f] and to a positive value for hn[f].

was varied, and interference patterns were recorded with amplitude reductions varying by over a factor of 30, corresponding to beam attenuations through the gas cell of more than a factor of 1000. The corresponding pressures in the main chamber were also recorded.
In a separate experiment, the absorption of a well-collimated Na or Na 2 beam was measured. This allowed us to verify that the amount of gas that caused a factor of b reduction in the amplitude of the interference fringes caused a factor b2 attenuation of the transmitted beam intensity, and also allowed us to measure the relative attenuations for Na or Na 2 in Ne.
The measured phase shift was found to be a linear function of the pressure rise (Fig. 19b). It is noteworthy that the measured phase shifts/torr vary by a factor of 13, whereas the total scattering cross sections vary by only a factor of 4. Comparing the measured attenuations for Ar, Kr, and Xe to the cross sections calculated from the potentials (Buck and Pauly, 1968; Dfiren et al., 1968, 1972) allowed us to estimate the column density of the gas in our interaction region.
The refractive index of matter for de Broglie waves has been demonstrated in electron holography (Lichte, 1988) and extensively studied in neutron optics (Sears, 1990), especially using neutron interferometers (Badurek et al., 1988). In neutron optics, scattering is dominantly s wave and measuring the refractive index gives information about the scattering length.
In contrast, many partial waves (typically ~max--Xr/}tdB = a few hundred)

OPTICS AND INTERFEROMETRY

47

contribute to scattering in the present study because the range of the interatomic potential x r between two atoms is much larger than the de Broglie wavelength (0.17 ]k for 1000 m/s Na atoms). This results in the differential cross section
having a considerable angular structure at the scale 1/~max,which is a few milli-
radians. Fortunately, the angular acceptance of our interferometer (30/xrad) is much smaller than the size of this structure, so we are exclusively sensitive to f(k,O) as assumed previously.
Using the standard partial wave treatment for central potential scattering, we find the real and imaginary parts of the scattering amplitude in the forward direction:

1

Re[f(k,0)] = ~ ~ (2f + 1) sin 26 e

(34)

~ ~- 0

1

Im[f(k,0)] = ~ e ~~ (2e + 1)2 sin26e

(35)

where 6e is the phase shift of the partial wave with angular momentum e. For a typical interatomic potential with a reasonably deep attractive well, the rapidly oscillating sin 26 e term in Eq. (34) averages to zero at most impact parameters (this is the random phase approximation), and the main contribution to Re[f(k,0)] comes from large impact parameters beyond the potential minimum, where the phase shift is on the order of 7r or less. In contrast, the sin2 6 e term on Eq. (35) averages to ~1 for impact parameters inside the point at which 6e = rr where the random phase approximation is valid. The value of Im[f(k,0)] and therefore the total cross section basically is determined by the location of this point. Figure 20 shows a typical calculation for the phase shifts 6e and their contribution to Re[f(k,0)] and Im[f(k,0)].
To make comparisons with our data, any theory must be averaged over the velocity distribution of the target gas (Schmiedmayer et al., 1995a). This averaging is best done by calculating the mean scattering amplitude as seen by the atoms. This velocity averaging is very strong in the case of N a - H e scattering (the mean velocity of the He atoms is comparable to the beam velocity) and gets less and less for heavier target atoms.
Our measurements show that Re[f(k,0)] varies substantially more than Im[f(k,0)] with the collision system. The theoretical models discussed by Schmiedmayer et al. (1995a) show that Re[f(k,0)] gives new information about the shape of the long range potential. In the following paragraphs, we will summarize these calculations and give some simple illustrative examples.
In the case of a hard sphere with radius r H, the sum over all partial wave phase shifts can be evaluated numerically. We have shown that the constraint that the wave function vanish at rH affects partial waves whose classical impact parameter b = ((+ 1/2)/k is smaller than r, and, due to tunneling through the centrifugal barrier, also slightly beyond r/4. The numeric sum can be approximated by R e [ f ( k , O ) ] / I m [ f ( k , O ) ] ~ - 1 / V ~ r H, roughly equal to the inverse of the square

48

Jrrg Schmiedmayer et al.

root of the number of partial waves contributing to the scattering process. Fur-

thermore, the ratio of real to imaginary parts of the scattering amplitude is al-

ways negative, reflecting the repulsion of the partial waves affected by tunneling.

In the case of a pure long-range attractive interaction potential of the form

Cnr -n, it is possible to predict analytically the ratio Re[f(k,O)]/Im[f(k,O)]. Cal-

culating the partial wave phase shifts in the Eikonal approximation and convert-

ing the partial wave sums in Eqs. (34) and (35) into an integral over the impact

( 1) parameter (the semi-classical approximation) we find, for an r -n potential,

F 1

1 F +--

Re[f(k,0)] - -
-
Im[f(k,0)l

2 n- 1

F

F

n- 1

( 7r ~

= +tan

(36)

n-1

n-1

where the upper signs are for attractive potentials. This ratio is independent of both the strength of the potential (i.e., of Cn) and de Broglie wavelength, which follows because no length scale is defined by the potential. It depends strongly on n (Table III). For neutral atoms in s-states one would expect the long range tail of the actual potential to be a van der Waals interaction, Vdw(r) - -C6 r-6, in which case we would expect Re[f(k,O)]/Im[f(k,O)] ~-0.72. This approximation fails at small energies (i.e., for ultracold atomic collisions) because there are not enough partial waves to justify the semi-classical approximation. Applying it to real systems requires that the actual potential be reasonably well-represented by a power law into small enough distances to apply the random phase approximation; consequently it will certainly not be applicable to potentials whose wells are not deep enough to generate a phase shift of several ~.
To make a more detailed comparison of our results with theory we used the JWKB approximation to calculate the phase shifts 6e in Eqs. (34) and (35) leading to the forward scattering amplitude for modified 6 - 1 2 potentials (Buck and Pauly 1968; Dtiren et al. 1968) and 6 - 8 potentials (Barwig et al., 1966; Dtiren et al., 1972; Gottscho et al., 1981) as well as more sophisticated dispersion potentials from Tang and Toennies (1977, 1984, 1986), and Proctor and Stwalley (1977) determined from scattering and spectroscopic data. Based on these calcu-

TABLE III THE RATIO Re[f(k,O)l/Im[f(k,O)] CALCULATEDFOR A LONG-RANGE 1/r n POTENTIAL

n

5

6

7

8

Re[f(k,0)]/Im[f(k,0)]

1.00

0.72

0.58

0.48

OPTICS AND INTERFEROMETRY

49

lations, we can give the following basic characteristics of the phase shifts and
the ratio Re[f(k,O)]/Im[f(k,O)] in atom-atom collisions.
9The biggest contribution to the real part of the scattering amplitude and therefore to the observed phase shift A~o stems from regions of the potential where the partial wave phase shifts t~e are slowly varying and smaller than 7r. This is certain to be the case for large interatomic separation. In contrast, all partial waves contribute to the imaginary part of the scattering amplitude and the total
cross section. Therefore, the phase shift A~oand the ratio Re[f(k,O)]/Im[f(k,O)]
carry new information about the long-range part of the scattering amplitude.
9The phase shift A~o and the ratio Re[f(k,O)]/Im[f(k,O)] both show glory os-
cillations similar to those seen in the total cross section, but shifted. In the experiment, these oscillations were reduced by the velocity averaging, but indications of them can be seen in our 1993 data (Fig. 21) as described by
Schmiedmayer et al. (1994b). These glory oscillations have also been predicted by Audouard et al. (1995).
9For the light gases with a weak interaction potential (t~e< 7r at the potential
minimum), the ratio Re[f(k,O)]/Im[f(k,O)] is very sensitive to the minimum
of the potential and therefore also to the form of the inner core. 9For the heavy gases with a strong interaction potential, the ratio
Re[f(k,O)]/Im[f(k,O)] tends to approach the simple pure long range limit
[Eq. (36)].
Using these basic characteristics, we can say several qualitative things about the collisional phase shifts we have observed. We find that the collision-induced phase shift for sodium atom waves passing through a variety of target gases is much more strongly dependent on the collision partner than the previously measured cross sections (Diaren, 1980). Semi-classical calculations of this phase
shift show that it is very sensitive to the shape of the interatomic potential at in-
teratomic separations beyond where the random phase approximation is valid. As a general trend, we see that the depth of the minimum of the interatomic
potential varies considerably from He to Xe. Helium has the weakest long range attraction, a very shallow minimum, and it behaves most like a hard sphere. The
ratio Re[f(k,O)]/Im[f(k,O)] for N a - H e scattering is very small, but its positive
sign gives clear evidence of an attractive long-range interaction. The large ratio of real to imaginary part for Na-Ne results from the fact that the maximum of the phase shift near the potential minimum is never larger than 1 rad, generating a large contribution to the sum in Eq. (34). The Na-Xe potential, on the other hand, has a well deep enough to generate many radians of phase, and so the longrange part of the potential should dominate. Its ratio comes closest to the value expected for a long-range r -6 interaction. The values measured for the other gases N a - K r and N a - A r deviate progressively further from this ratio as the well depth decreases (which it does monotonically with decreasing mass of the rare gas).

50

J6rg Schmiedmayer et al.

For the scattering of Na 2 molecules on Ne gas (Chapman et al., 1995a), we find the ratio Re[f(k,O)]/Im[f(k,O)] = 1.413). For better comparison with the Na atom, we have separately measured the total absorption of Na 2 by Ne (i.e., (47r/k)Im[f(k,O)]) to be 57(2)% larger than the corresponding absorption of Na. These measurements are in qualitative agreement with N a - N e potentials from Tang and Toennies (1977, 1984) if extended to Na2 using combination rules from Tang and Toennies (1986).
Significant discrepancies remain between our experiments and the predictions based on potentials obtained by standard scattering experiments (Dfiren, 1980; Table II), especially for the velocity-dependent measurements for Na-He, Na-Ne, and Na-Ar (Fig. 21). This shows the power of refractive in-

0.4~ He . . . .

0.3 /[--. . . .

P~c=l~11973)

o / ..§... ........ /

.2

P , ~996) .......

o0t sL.........

-'--l.P.... ]e' q

--"

-

GP~6l ttingen

d l

....C...t.k. 1

1

-4

1.2

.-~ ................e...9.~C5~1(~77)~I

0.8/ ~ - z . . .

..../

0.9f
[
~ I U.~ r

t I I t
Tang and roennies 11977)
- I

I t I t1
]
Buck und Pauly (1968) --

0.5~

Sr

t' ~ t
0.91 Kr

11972)
o I
(

0.5

Schlierctal (1972

'

IaI, I, I 0.9 Xe
Buck arid Pauly (1968)

0.8

y

0.7

0.6

~

0.5

i

=

, i

,

i

800 1200 1600 2000

velocity Ira/s]

FIG. 21. Velocity dependence of the ratio Re[f(k,O)]/Im[f(k,O)] for Na matter waves passing
through He, Ne, Ar, and Xe gas. The data for the heavy gases (Ar, Kr, Xe) show indication of glory oscillations. The calculated curves stem from various potentials found the literature.

OPTICS AND INTERFEROMETRY

51

dex measurements to test existing potentials, and indicates an opportunity to refine these potentials using this new technique. Considerably more effort is required to understand both the velocity-selective data and the molecular data.
We think that information from phase shift experiments may add significant understanding to atom-atom and atom-molecule interactions and, we hope, will allow us to learn more about the interatomic potentials in these simple systems. We also hope to study the effects of inelastic processes and excitations in forward scattering. These should cause a reduction in the ratio of Re[f(k,O)]/Im[f(k,O)] if they occur at large impact parameters (but no evidence for this is seen here).

VII. Fundamental Studies
Interferometers of all types have had application to fundamental problems and precision tests of physical theories, especially in quantum mechanics, and atom interferometers are sure to continue this tradition (see also other contributions in this volume; for example, the measurement of h/m for Cs by Young et al.). In this section, we focus our attention on the fundamental question, "What limits do the size and complexity of particles place on the ability of their center of mass motion to exhibit interference effects?"
The quantum mechanical treatment of the center of mass motion of increasingly complex systems is an important theme in modern physics. This issue is manifest theoretically in studies of the transition from quantum through mesoscopic to classical regimes and experimentally in efforts to coherently control and manipulate the external spatial coordinates of complex systems, as exemplified by the wide interest in matter wave optics and interferometry. As described earlier, matter wave optics and interferometry have been extended to atoms and molecules, systems characterized by many degenerate and non-degenerate internal quantum states. In this section, we will investigate if and where there might be limits, in theory or practice, to coherent manipulation of the center of mass motion of larger and more complex particles. We shall first consider the effect of particle size and mass, showing that the minimum transit time needed in the interferometer varies as a high power of the particle size. We then will consider the interaction of radiation with the atom as it is passing through the interferometer, actually performing a gedanken experiment suggested by Feynman. This will lead naturally to an understanding of the limitations to observing interference with macroscopic objects posed by their coupling to the environment. Whereas internal state coherences in complex molecules have long been cleverly manipulated in spectroscopy in both the radio (Ramsey 1985) and optical frequency domains (Bord6 et al., 1994; Chebotayev et al., 1985), we re-emphasize our concern

52

Jgrg Schmiedmayer et al.

here with the limits to the coherent manipulation of the center of mass motion. Finally, we shall consider the question of what happens to the coherence lost when the particle passing through the interferometer interacts with radiation, demonstrating that it becomes entangled with the scattered radiation-and showing that this coherence can be regained by selectively detecting particles that scatter this radiation into a subspace of possible scattered photon directions.

A. PARTICLESIZE

First we will concentrate on particle size and complexity and their influence on interference. Our experiments described previously demonstrate that interference fringes can be observed when the size of the particle (--~6 & for Na 2 molecules) is considerably larger than both its de Broglie wavelength (0.16 &) and its coherence length, typically 1/~ (Schmiedmayer et al., 1993; Chapman et al., 1995a).
If the particle's size relative to its de Broglie wavelength or coherence length pose no fundamental limits to matter wave interferometry, large mass or physical size may limit the ability to observe interference fringes in a more practical sense. To produce interference fringes in a grating interferometer with particles of large mass, the single slit diffraction angle Oa~fe-~Ada/d~ from the first grating must be large enough to include at least two adjacent slits in the second grating. This implies that L >> dg2/Aaa > S2/AdB, where L is the spacing between the first and second gratings, dg is the grating period, and s is the particle diameter. (The last inequality follows from the requirement that the particle must be able to pass between the grating bars.) The quantity L = d2/Ada is exactly half the Talbot length (see Section Ill.C). Fulfilling this condition will allow interferometry in the Talbot-Lau regime (but not with separated beams). For heavier particles, the diffraction angle would be reduced further and there would be no opportunity for two different paths to interfere. The pattern observed in a three-grating geometry would then be a classical moir6 fringe pattern (Batelaan et al., 1997 in this volume and Oberthaler et al., 1996) and not interference fringes.
We can rewrite this limit in terms of the mass density p of the (assumed spherical) particle and the transit time between gratings r = L/v as

ps 5

~">> - - .

(37)

h

For example, an interferometer with a 1 sec transit time (between gratings) would be able to interfere particles with a diameter smaller than 70 nm (typically clusters of about 8 • 107 sodium atoms). Even if we waited one year, we could not expect to observe interference from composite particles with a diameter exceeding 2/xm, corresponding to an atomic weight of about 1013; this is the size of a large bacterium.

OPTICS AND INTERFEROMETRY

53

B. COHERENCELOSS DUE TO SCATTERINGA SINGLE PHOTON--DISCUSSION
The principle that a system can be in a coherent superposition of different states and exhibit interference effects is a fundamental element of quantum mechanics. Immediately, the question arises as to what happens to the interference if one tries to determine experimentally which state the system is in. This is the basis of the famous debate between Bohr and Einstein, in which they discussed Welcher Weg ("which way") information in the Young's double slit experiment (Bohr, 1949; Feynman et al., 1965; Wooters and Zurek, 1979; Zeilinger, 1986). In a more recent gedanken experiment suggested by Feynman, a Heisenberg light microscope is used to provide Welcher Weg information in a Young's twoslit experiment with electrons (Feynman et al., 1965) or atoms (Sleator et al., 1991; Tan and Walls, 1993). In this section, we will discuss our experimental realization of this gedanken experiment using our atom interferometer. Since the contrast of the fringes is a measure of the amount of the atomic coherence, complementarity suggests that the fringes must disappear when the slit separation (more generally the path separation at the point of measurement) is large enough that, in principle, one could detect through which slit the particle passed (Scully et al., 1991) using a Heisenberg microscope. This explains why scattering photons from the atoms in our interferometer at a location, where the separation of the paths was many wavelengths of light, completely destroyed the atomic interference fringes (Section IV.G).
Since the loss of contrast is caused by the measurement of the atom's position by the photon, it is necessary to consider a quantum treatment of the measurement process. The measurement interaction here is the elastic scattering of the photon by the atom that causes their initially separable wave function to evolve into an entangled state (Schrrdinger 1935)--a sum of separable wave functions, each one of which conserves the total momentum and energy of the system, that no longer can be written as a product of separate atom and photon wave functions. This entanglement can result in a loss of atomic coherence when information about the scattered photon is disregarded. The effects of such entanglement is an important issue in contemporary quantum mechanics, particularly with regards to EPR-type correlations and for understanding the measurement process and the loss of coherence in the passage from quantum to classical mechanics. The details of the loss of coherence of one system due to entanglement with another can be studied directly in interferometry experiments like the one discussed here by scattering a probe particle off an interfering superposition of the observed system.
In this section, we discuss experiments we performed to measure the loss of atomic coherence due to scattering single photons from the atoms inside our three grating Mach-Zehnder interferometer (see Fig. 22). Our experiments (Chapman et al., 1995c) demonstrate that the loss of coherence may be attributed to the random phase imprinted by the scattering process and that it depends

54

J~rg Schmiedmayer et al.

I

- --"

I IIA E ....

Bl"r"

t

..----

~,.

Slits

G1

~

G2

G3

O.P. Laser

Excitation Laser

FIG. 22. A schematic, not to scale, of our atom interferometer. The original atom trajectories (dashed lines) are modified (solid lines) due to scattering a photon (wavy lines). The inset shows a detailed view of the scattering process.

on the spatial separation of the interfering waves at the point of scattering compared with the wavelength of the scattering probe.
Our experiments also address the questions of where the coherence is lost to and how it may be regained. Although the elastic scattering of a photon produces an entangled state, it is not per se a dissipative process and may be treated with Schr6dinger's equation without any ad hoc dissipative term. Therefore, the coherence is not truly lost but rather becomes entangled with the scattered photon, which may be considered a simple reservoir, consisting of only the vacuum radiation modes accessible to it. We show that this indeed is the case by demonstrating that selective observation of atoms which scatter photons into a restricted part of the accessible phase space results in fringes with regained contrast.

OPTICS AND INTERFEROMETRY

55

C. COHERENCE LOSS DUE TO SCATTERING A SINGLE PHOTON m EXPERIMENT

To study the effects of photon scattering on the atomic coherence as a function of the interfering path separation, single photons were scattered from the atoms within the interferometer. The contrast and the phase of the interference pattern were measured as a function of the separation of the atom paths at the point of
scattering (Chapman et al., 1995c).
In the absence of scattering, the atom wave function at the third grating may
be written ~(x) = ul(x) + ei~u2(x) eik~x, where Ul, 2 are (real) amplitudes of the upper and lower beams, respectively; kg = 27r/dg, where dg is the period of the grat-
ings, and q~is the phase difference between the two beams. To describe the effects of scattering within the interferometer, we first consider an atom within the inter-
ferometer elastically scattering a photon with well-defined incident and final (measured) momenta, k i and kf with [ki[ = [kf[-- kphoton.After this well-defined
scattering event, the atomic wave function becomes

XII"(X) oc Ul(X -- ~ f ) + ei~u2(x -- ,~f) eikRx+A~.

(38)

The resulting atomic interference pattern shows no loss in contrast but acquires a phase shift (Bordr, 1989; Storey and Cohen-Tannoudji, 1994):

Aq~ = A k . d = Akxd

(39)

where Ak = k f - k i, and d is the relative displacement of the two arms of the interferometer at the point of scattering. Equation (38) shows that there is a spatial shift of the envelope of the atomic fringes due to the photon recoil given by Ax = (2L - 7,)Akx/katom , where katom = 277"/AdB, and (2 L - z) is the distance from the point of scattering to the third grating.
In the case that the photon is disregarded, the atom interference pattern is given by an incoherent sum of the interference patterns with different phase
shifts (Stem et al., 1990) corresponding to different final photon directions (i.e.,
a trace over the photon states):

C' cos(kgX + Aq~') = f d(Akx)P(Akx)C o cos(kgX + Akxd)

(40)

where P(Akx) is the probability distribution of transverse momentum transfer
and COis the original contrast or visibility of the atomic fringe pattern. For scat-
tering a single photon (shown in the insert to Fig. 23), P(Ak x) is given by the ra-
diation pattern of an oscillating dipole. The average transverse momentum trans-
fer is hAk x = l hk (the maximum of 2hk occurs for backward scattering of the incoming photon and the minimum of Ohk occurs for forward scattering). Due to
the average over the angular distribution of the unobserved scattered photons, there will be a loss of contrast (C' -< C0) and a phase shift Aq~' of the observed atomic interference pattern. It follows from Eq. (40) that the measured contrast (phase) of the interference pattern as a function of the separation d of the atom

56

J6rg Schmiedmayer et al.

,of,"rl'+ 1 . . . . ~

! ....

I ....

I ....

I-

0.8
~

]

0.6

'~ 0.4

--~ 0.2

0.0

~

3

~

2

".9... ",,,,

i..

",

.-
",,

k)
i
- +
"~

~

1

~

0

a.

f-1 i

I

i

i

I

I

I

i

I

I

I

!

~

~

,

~

I

~

,

~

~

" -
I"

0.0

0.5

1.0

1.5

2.0

d/~photon
FIG. 23. Relative contrast and phase shift as a function of the separation of the interferometer arms at the point of scattering. The inset shows the angular distribution of spontaneously emitted photons projected onto the x axis. The dashed curve corresponds to purely single-photon scattering, and the solid curve is a best fit that includes contributions from atoms that scattered no photons (4%) and two photons ( 14%).

waves will vary as the magnitude (argument) of the Fourier transform of P(Akx). Equation (40) is equivalent to the theoretical results obtained for the two-slit gedanken experiment (Sleator et al. 1991; Tan and Walls 1993) (in which case d is the slit separation), even though explicit which-path information is not necessarily available in our Mach-Zehnder interferometer, in which the atom wave functions can have a lateral extent (determined by the collimating slits) much larger than their relative displacement, d.
We arranged to scatter a single photon from each atom within the interferometer by using a short interaction time and a laser field strength most likely to re-

OPTICS AND INTERFEROMETRY

57

suit in the scattering of a single photon (Fig. 22). The scattering cross section was maximized using o'+ polarized laser light tuned to the D2 resonant line of
Na ('~photon-- 589 nm) connecting the F = 2, m F = 2 ground state to the F' = 3,
m F' = 3 excited state. This ensured that the scattering left the atom in the same hyperfine state. The atomic beam was prepared in the F = 2, m F = 2 state by optical pumping with a tr+ polarized laser beam intersecting the atomic beam before the first collimating slit (see Section II.C). We typically achieved ---95% optical pumping, as measured (to within a few percent) by a two-wire Stern-Gerlach magnet, which caused state-dependent deflections of the atomic beam.
The excitation laser beam was focused to a --~15 /xm waist (FWHM of the field) along the atom propagation direction. A cylindrical lens was used to defocus the beam in the y direction to ensure uniform illumination over the full height of the atomic beam (---1 mm). The transit time through the waist was smaller than the lifetime of the excited state, hence the probability for resonant scattering in the two-state system showed weakly damped Rabi oscillations, which we observed by measuring the number of atoms deflected from the collimated atom beam as a function of laser power (see Section III.D and Fig. 8.). To achieve one photon scattering event per atom, we adjusted the laser power to the first maximum of these oscillations, closely approximating a zr pulse.
The contrast and phase of the measured atomic interference patterns are shown in Fig. 23 for different path separations. The contrast was high when the separation d at the point of scattering was much less than ,~photon/2 (corresponding to Akxd << zr), but fell smoothly to zero as the separation was increased to about half the photon wavelength, at which point A k x d ~- 7r. (This would occur exactly at d = ,~photon/2 if the scattered photon angular distribution were isotropic.) As d increased further, a periodic rephasing of the interference gave rise to partial revivals of the contrast and to a periodic phase modulation.
The observed behavior of the contrast (Chapman et al. 1995c) is consistent with the complementarity principle. Considering the photon scattering as a position measurement of the atom, complementarity suggests that fringe contrast must disappear when the path separation is approximately half the wavelength of the scattered light, since this is the smallest distance that can be resolved by a perfect optical microscope for this wavelength. At larger separations, we see not only the general suppression of the fringe contrast expected from complementarity, but also several subsequent revivals of the fringe contrast. These contrast revivals reflect the inability (because of diffraction) of an optical system to spatially localize the atom using a single scattered photon. If light were scattered from an atom localized on one side of the interferometer and imaged with a lens, this image would have diffraction tings determined by the wavelength of the scattered photon (even in the limit of an infinitely large lens). Thus, if a single photon is recorded where it would be expected if it had scattered from an atom

58

J6rg Schmiedmayer et al.

localized on the upper arm of the interferometer, it may actually have come from an atom on the lower arm if one of the diffraction rings coincides with the position of the recorded photon. Under these circumstances there is significant uncertainty as to which side the atom that emitted the photon really traversed; consequently the fringe contrast can be (and is) revived to some extent.
While the contrast generally decreases as d increases, the phase shift A q~ of the fringes exhibits a sawtooth oscillation that is damped by the finite resolution of the machine. Starting at d = 0, it increases linearly, with slope 27r. This is the slope expected for momentum transfer of l hk, which is the average momentum transfer of the symmetrical distribution of momentum transfer (Fig. 23). After each 0 of the contrast, the sign of the interference pattern is reversed, subtracting 7r from the phase and resulting in the observed sawtooth pattern.
In studying the decoherence and phase shift, we used a 50/xm detector wire, which is larger than the deflection Ax of the atom beam that results from the recoil momentum imparted by the scattered photon. The finite collimation of the atomic beam further degrades the overall momentum resolution of the apparatus. The result of this lack of resolution is that the measured interference patterns are averaged evenly over all values of Ax, which can be as big as 40/xm in our experiment-corresponding to displacement of the envelope of the fringe pattern by ---100- 200 fringes.
These numbers highlight the distinction between the expectation value of the atom's classical transverse position (the peak of the fringe envelope) and the phase of the fringes (which are never shifted by more than half a fringe). In fact, moving the point of scattering further downstream reduces the displacement of the fringe envelope for a given kI, while monotonically increasing the corresponding phase shift. Therefore, the measured loss of fringe visibility cannot simply be understood as resulting from the transverse deflections of the atom at the detection screen (in our case, the third grating) due to the photon momentum transfer, as it can be for the two-slit gedanken experiments. We point out that Ax (or equivalently the x component of the photon momentum transfer) is precisely what is measured in determining the transverse momentum distribution of an atomic beam after scattering a photon. These distributions have been measured for diffraction of an atomic beam passing through a standing light wave and undergoing a single (Pfau et al., 1994) or many (Gould et al., 1991) spontaneous emissions, as well as for a simple collimated beam excited by a traveling light wave (Oldaker et al., 1990). These results are usually discussed using a simple classical argument: the recoil momentum from spontaneous emission produces random angular displacements that smear the far-field pattern, a viewpoint also applicable to two-slit gedanken experiments. Clearly the quantum phase shift measured in our experiment is distinct from the "deflection" of the atom Ax due to the photon recoil. It reflects the phase difference of the photon wave function where it intersects the two arms of the interferometer.

OPTICS AND INTERFEROMETRY

59

These results also are interesting as a contrast to the gedanken experiments recently proposed in which loss of contrast in an atom interferometer occurs after emission of a photon by the atom, even though there is insufficient momentum transfer to the wave function to explain this loss on the basis of dephasing (Englert et al., 1994). In our experiment the opposite occurs: there is sufficient momentum transfer to the atom by the emitted photon to be easily detected, but the interference pattern is not destroyed for small separations. In both experiments the interaction with the radiation adds insignificant relative phase difference between the two arms of the interferometer. The crucial difference is that in the gedanken experiment of Englert et al., 1994), the photon emitted by the atom is retained in one of two cavities located symmetrically on the two sides of the interferometer and can be used to determine which path the atom traversed (assuming the cavities were initially in number states), whereas in our experiment the scattered photon scatters without constraint and no subsequent measurement can determine which path was traversed by the atom (for the case considered here). Indeed, if a metal foil were interposed between the two sides of our interferometer, and a beamsplitter and mirrors added so the laser beam was split and excited both sides with well determined relative phase, detection of the scattered photon would then determine which side of the foil the atom traversed, and would destroy the interference pattern even though the phase shift imparted to the atoms was negligible, just as in (Englert et al., 1994).

D. COUPLING TO THE ENVIRONMENT
An important limitation to matter wave interferometry is posed by the interaction of the interfering particle with the environment, of which the most troublesome aspect is interaction with thermal radiation. The mechanism of dephasing will then be the scattering of blackbody photons from the interfering particle as just discussed, their absorption by the particle resulting in a change of internal state or possibly the emission of spontaneous thermal radiation by the particle.
In general, the interference pattern will be destroyed if the interaction with the environment will allow, even in principle, the path of the interfering particle to be determined with certainty (Scully et al., 1991). For interference that results from internal coherences, as in Ramsey type experiments where the particles travel in different states, this implies that any scattering that can differentiate which state the particle is in (e.g., by frequency of absorption or polarization sensitivity of the scattering) will destroy the interference. This is illustrated by the necessity of working at temperatures below 4 K in separated oscillatory resonance experiments using high n Rydberg atoms.
The formation of interference fringes requires that the final internal states of the particle wave along the interfering paths be the same (or at least nonorthogonal). If the particle arrives in different, orthogonal states along the two paths, no interference will be observable. This makes an interferometer the ultimate state-

60

Jrrg Schmiedmayer et al.

sensitive detector: Any change of internal state induced by radiation confined to
one side of the interferometer will completely destroy the interference pattern. This would allow the detection of low-frequency transitions that cause reorientation or rotational state changes in molecules, for example. Even if the absorption or spontaneous emission is accompanied by the same change of state on both paths, interference will not be observed if the radiation on the two sides is uncoupled (e.g., a barrier is present to shield the fields) so that a subsequent measurement of the fields, in principle, could determine which side the atom was on when it underwent the transition.
In interferometers in which the particles are initially in the same state in both arms, the absorption, spontaneous emission, or scattering of a photon is not necessarily sufficient to destroy the interference pattern if no barrier is present. While such an event can be exploited to indicate that a particle passed through the interferometer and possibly to determine its initial and final internal states, it
must be able to provide Welcher Weg (Scully et al., 1991) information in order to
destroy the interference. Even if the separation of the paths at the point of scattering is several half wavelengths of the radiation, some coherence may be retained as shown in our experiment earlier.
The dephasing experiment discussed in the preceding section of this review shows the extent to which the scattering of a single photon from an atom in an interferometer destroys the interference. This work shows that the destruction of interference due to scattering of a single photon may be regarded as being caused by a random dephasing of the interfering wave function's external phase caused by the uncertainty in the direction of the scattered photon. In the likely event that a strongly interacting particle like a polar molecule or a Rydberg atom interacts with a number of photons from a thermal radiation field, one can use a random phase diffusion model to evaluate the destruction of spatial coherence between two paths separated by a distance d. Each interaction with a background
radiation photon kphotonwill imprint a random phase -kphotond< Aq~< kphotond.
Since thermal photons have typical k vectors on the order of about 500 cm -1 the typical imprinted phase for separations on the order of 1 mm is 0.05 rad. Consequently, many scattering events will be needed to destroy the coherence.
In this limit, we expect that, for N isotropic scattering events, the contrast will be reduced to

C = (sin<kphotond))N

exp[- N(kphotond)2/ 6].

(41)

Co \ ~photodn

Assuming a scattering rate of n photons per second, the interference contrast will be destroyed after a characteristic time %:

6 ( ;o. )2

(42)

% ~ n kph d "

OPTICS AND INTERFEROMETRY

61

This clearly shows that the coherence in the external motion can be preserved much longer than internal coherences. Thus, a large particle with dense internal levels coupling to a thermal radiation field does not have to be completely isolated from the environment to exhibit interference.
For ground state atoms, this dephasing by thermal radiation poses no problem, since the photon density at typical excitation energies of 2 eV are negligibly small. However, the same does not hold for slowly moving more massive particles like molecules or clusters, or especially Rydberg atoms.

E. REGAININGENTANGLEDCOHERENCEBY SELECTIVEOBSERVATIONS
Returning to the loss of coherence by scattering of single photons from atoms in the interferometer, we now address the questions of where the coherence is lost to and how it may be regained. We performed an experiment (Chapman et al., 1995c) to show that selective observation of atoms that scatter photons into a restricted part of the accessible phase space results in fringes with regained contrast. This demonstrates that the coherence is not truly lost, but becomes entangled with the scattered photon, which may be considered as a simple reservoir consisting of only the vacuum radiation modes.
In this experiment, we observed atoms that are correlated with photons scattered in a narrow range of final directions. In principle, this could be achieved by detecting the photons scattered in a specific direction in coincidence with the detected atoms. With Akxd now being a known quantity, the fringe shift is predicted to be the same for all the atoms; consequently, the fringe patterns of this restricted set of atoms would line up and no coherence would be lost. Unfortunately, such an approach is not feasible in our experiment for a number of technical reasons--principally the slow response of our atom detector and the inefficiency of photon detectors.
Fortunately, an alternative experimental approach is made possible by the fact that the change of momentum of the photon, Ak x, is imparted to the atom and can be measured by the atom's deflection 2ix. Hence, a measurement of an atom's 2tx gives the Akx of its scattered photon. Furthermore, it is easily possible to measure 2tx in our three grating interferometer, because (since we scatter the photons close to the first grating) the deflection of the envelope of the atomic fringes for a particular Ak x is 100 times larger than the associated fringe shift, Akxd. In practice, this approach is superior to a correlation experiment because no inefficiencies or accidental coincidences are introduced by the measurement of the scattered photon: the measurement of an atom's position reliably indicates the momentum transferred to that particular atom.
We have performed an experiment based on this technique to demonstrate the recovery of the entangled coherence. By using very narrow beam collimation in conjunction with a smaller detector, we can selectively detect only those atoms

62

Jrrg Schmiedmayer et al.

correlated with photons scattered within a limited range of Akx. This restricts the possible final photon states and results in a narrower distribution P'(Akx) in Eq. (40). We performed experiments with recoil distributions centered on three different photon momenta. Figure 24 shows three different realizations (referred to as Cases I-III) with the corresponding momentum transfer distributions, P~(Akx), i = I, II, III. The contrast is plotted as a function of d for Cases I and III, where we preferentially detect atoms that scattered photons in the forward and

1.0 ,~t
~ ,..,~ 0.8
o u.r'.':
0
N 0.4

I /

' ' ' '

' ' ' ' '

I II

0 . 2 ~'I~I I~ ./L ~ ' " / /""

0.0

"/

-I'
': ~

~-

/

. 9

~

. 9,9

,~o.~

,~9

o~176

I ............. "III

r~

!

I

!

o

l

2

I

Y'

I

"

9

oO~

1

o~176 oO

0

!,. o0""'"

" 9.o~ ....

0.0

0.5

1.0

1.5

2.0

d/~photon

FIG. 24. Relative contrast and phase shift of the interferometer as a function of d for the cases in which atoms are correlated with photons scattered into a limited range of directions. The dashed curve is for the uncorrelated case. The inset shows the acceptance of the detector for each case compared to the original distribution (dotted line). Case I corresponds to predominantly forward-scattered photons (minimal transfer of momentum), Case III corresponds to backward-scattered photons (transfer of 2 photon momenta), and Case II lies in between.

OPTICS AND INTERFEROMETRY

63

backward directions, respectively. The contrast for Case II is similar to Case I and is not shown. The measured contrasts in this figure were normalized to the d = 0 (laser on) values, since a different number of atoms was detected with the laser off due to the absence of the deflection by the photons.
Our results show that the contrast falls off much more slowly than previo u s l y - i n d e e d , we have regained over 60% of the lost contrast at d ~ A/2. The contrast falls off more rapidly for the faster beam velocity (Case III, Vbeam = 3200 m/sec) than the slower beam velocity (Cases I and II, Vb~am= 1400 m/sec) because the momentum selectivity for the final photon states is correspondingly lower.
The phase shift is plotted as a function of d for the three cases in the lower half of Fig. 24. The slope of Case III is nearly 47r, indicating that the phase of the interference pattern is determined predominantly by the backward-scattering events. Similarly, the slope of Case I asymptotically approaches a small constant value due to the predominance of forward-scattering events. Case II is an intermediate case in which the slope of the curve, --~37r, is determined by the mean
accepted momentum transfer of 1.Shk. The lower inset shows the transverse mo-
mentum acceptance of the detector for each of the three cases (i.e., the functions
P~(Akx)), which we determined using the known collimator geometry and beam
velocity. The fits for the data in Fig. 24 were calculated using Eq. (40) and the
modified distributions P~(Akx) and include effects of velocity averaging as well
as the effects of those few atoms that scattered no or two photons.

E SCATTERINGA SINGLE PHOTON OFF AN ATOM IN Two INTERFEROMETERS
Our atom interferometer employs thin diffraction gratings that split the incident
beam into many diffracted beams. As a consequence, in the absence of addi-
tional collimation, there are always at least two or more equivalent interferometers in which the atom can be found. By positioning the detector at specific points between two interferometers, one can selectively detect atoms that were in a superposition of more than two states (and are in more than one interferometer). This is especially interesting if one considers photon scattering off of an atom in such states. In general, the picture will be more complicated than the simple two path case discussed previously. Different final photon momenta can now cause the atom from different interferometers to be found at the same detector position (see inset in Fig. 25). Photons in different final photon momentum states leading to the atom being scattered into the detector are distinguishable and therefore carry information about which of the two interferometers the atom went through. This dramatically alters the observed contrast patterns.
In Fig. 25, we show measurements for the simplest case in which the detector (third grating) is centered between the deflected profiles of the two principle interference orders. The atom has two possibilities to be detected: The atom either

64

J6rg Schmiedmayer et al.

"<~1 0 . 4 ~" 0.2

0.0 0.0
1.0 _ 0.8 :i\
o6

' I 1.0 2.0 3.0
Akx/k

~,

-100

0

100

Position (~m)

4

. ~. , . , 3

', i

~

2 g

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

d/)~photon

d/~photon

FIG. 25. Experimentaldata (solid markers) for the detector centered between the deflectedprofiles of the two central interferenceorders as shown in the upper right graph. The solid curves in the lower graph are calculatedwith the modifieddistribution shown in the upper left graph, and are compared with the original (dashed).

comes from the first interferometer, scattering a photon with nearly maximal momentum transfer Akatom -" 2Akphoton, or it comes from the other interferometer, scattering the photon in the forward direction with nearly no momentum transfer Akatom-- 0. These photons are distinguishable, and the two interference patterns add incoherently, with a differential phase shift determined by the beam separation d. This will lead to a "beating" between the two interference patterns, showing strong contrast revivals with an envelope given by the single interferometer contrast. As predicted by the calculation, the contrast showed striking revivals (Fig. 25). The first revival was twice as high as for the uncorrelated case. The agreement with the calculation is quite good for the contrast data. For the phase data, the agreement is very good up until d = A and less satisfactory thereafter, which we attribute to contributions from interferometers containing higher diffraction orders.
This information about from which interferometer the atom was emitted can be erased by building a cavity around the scattering point and mixing the two photons. This opens up the possibility of combining cavity QED experiments, which focus on the quantum states of the radiation field, with atomic interference experiments probing the coherence of the center of mass motion of the atoms.

OPTICS AND INTERFEROMETRY

65

VIII. Inertial Effects
Phase shifts that arise in accelerating frames have been discussed by many authors in both nonrelativistic and relativistic contexts (see, e.g., Colella et al., 1975; Anandan, 1977; Greenberger and Overhauser, 1979; Werner et al., 1979; Clauser, 1988). Because such phase shifts increase with the mass of the interfering particle, atom interferometers are especially sensitive to inertial effects and may be developed into accelerometers, rotation sensors, gravimeters, and gradiometers.
The inertial sensitivity of an atom interferometer arises because the freely propagating atoms form fringes with respect to an inertial reference frame. These fringes appear shifted if the interferometer moves with respect to this inertial frame while the atoms are in transit. To illustrate this, we now present a simple calculation of the fringe shift that results from acceleration a of a three grating interferometer in a direction perpendicular to both the grating bars and the atomic beam axis.
In the time z = L/v that it takes an atom moving with velocity v to travel the distance L between adjacent gratings (Fig. 26), the interferometer moves a distance a~'2/2 = D/2. Atoms moving with an initial transverse velocity Vtran s = at~2 = aL/2v with respect to the interferometer axis at the time the atoms passed

a

t

9

"

a "

a
t= 2x-~
l
"__L_ 2~.

r

L

~,

L

T, = L / v
FIG. 26. The interferometer in motion under the influence of a transverse acceleration. The atomic beam travels from left to right in the laboratory frame but interacts with the progressively displaced gratings of the moving apparatus. Because a centerline (short dash) between the atom beam paths passes through the middle of the first grating at t = 0, and is offset by a transverse velocity, Vtr~s = 1/2a'r, it also passes through the middle of the displaced second grating at t = z. The dashed curve (long dash) represents the displacement of the interferometer due to acceleration. The centerline of the accelerating interferometer is shown (short-long dash) at t = 0 and t = 2~', where fringes have a relative displacement of -D.

66

J6rg Schmiedmayer et al.

through the first grating therefore will pass through the first and second gratings of the accelerating interferometer at the same positions as would on-axis atoms in the case of a nonaccelerated interferometer. These atoms continue with this transverse velocity, forming a fringe pattern at the third grating that is displaced by Vtrans T = aT 2= D from the original axis. When these atoms now reach the third grating, the interferometer will have moved a(2z)2/2 = 2D from its original position, resulting an apparent fringe shift of - D . This is observed as a phase shift:

(~acceleration~--- 2rr

d

a

h2 a

(43)

where dg is the period of the gratings, A~B = h/mv is the de Broglie wavelength for an atom with mass m and velocity v, and A = L2(AdJdg) is the area enclosed by the two arms of the interferometer. This expression can also be obtained directly from Eq. (15) and the discussion about the sensitivity of our interferometer to vibrations (see Section IV.E). It should be noted that the phase shift in our three grating white light geometry is independent of the mass of the particle, and was derived using classical physics.
The phase shift due to rotation of the interferometer (called the Sagnac effect) follows by noting that rotation with angular rate fl gives rise to a Coriolis acceleration a -- 2v • 1~, allowing one to use Eq. (43) to calculate the phase shift due to rotation:

rotation

44'

where we call the bracketed factor the rotational response factor. This expression can also be directly obtained from Eq. (14) in the discussion about the sensitivity of our interferometer to vibrations (see Section IV.E).
The results of these simple derivations agree with the nonrelativistic phase contributions derived by various more sophisticated methods (Greenberger and Overhauser 1979; Werner et al., 1979; Clauser, 1988). Relativistic contributions to the phase shift caused by accelerations and rotations are of the o r d e r Eldn/mC2 smaller than the nonrelativistic terms (Anandan, 1977) and are unresolvable in our experiments.
The Sagnac effect is not dispersive per se and is independent of the velocity of the particle in an interferometer in which the area is constant (as it would be for an interferometer employing conventional beam-splitters). However, since most atom interferometers employ diffractive beam splitters (as does ours), their rotational response factors will exhibit 1/v dependence. This dependence arises from the variation of the enclosed area, which in turn results from the variation of the diffraction angle with velocity. In contrast to rotations, the phase shift due to linear accelerations [see Eq. (43)] varies with velocity as 1/v 2. Thus atom in-

OPTICS AND INTERFEROMETRY

67

terferometers that use slow atoms will be relatively more sensitive to acceleration than to rotation.
Phase shifts due to rotation and acceleration, as well as shifts due to gravitational fields (which give the same response factor as acceleration due to the equivalence principle), have been observed in many kinds of matter wave interferometers. Accelerations were measured using neutron interferometers (Colella et al. 1975; Werner et al., 1979) and using atoms (Kasevich and Chu, 1991, Oberthaler et al., 1996). The Sagnac phase shift for matter-waves has been verified with accuracy on the order of 1% for neutrons (Werner et al., 1979; Atwood et al., 1984) and electrons (Hasselbach and Nicklaus, 1993), and to about 10% for atoms using both interferometers (Bord6, 1989; Riehle et al., 1991) and classical moire regime atom optics (Oberthaler et al., 1996).
In view of the numerous demonstrations of the sensitivity of matter wave interferometers to noninertial motion, the motivation for such experiments is principally technological: Can such devices become the sensors of choice in practical applications or can they demonstrate such high sensitivity that they open up new scientific possibilities? With these considerations in mind, the observation that the rotation-induced phase shift in an atom interferometer exceeds the Sagnac phase for light of frequency to by an amount mc2/hto (typically 101~ suggests the tremendous potential of atom interferometer rotation sensors (Clauser, 1988).
We now estimate the minimum angular velocities and accelerations detectable by our atom interferometer using the atomic velocities and signal intensities achieved in our apparatus. We assume that only Poissonian detection statistics degrade the signal-to-noise ratio, which is therefore proportional to C V~, where C is the fringe contrast and N is the total number of counts (see Section IV.C). The response factor and corresponding (purely statistical) rotational noise are summarized in Table IV.

TABLE IV RESPONSEFACTORSANDROTATIONALSHOTNOISEFORINERTIALSENSITIVITYMEASUREMENTS

Response Factor

Rotational Noise

Atom interferometer:rotation

1.86 rad/Oe

5.8 x 10-4 ~'~7~ r

Commercial laser gyroscope: rotation

---2 rad/~ e

1.7 X 10-4 ~-~e] V ~

Atom interferometer:acceleration

116 rad/g

9.2 • 10-6g/V~

Note: In obtaining these estimates,we used actual values for our interferometer,which are an average beam velocity of 1075 m/sec, a contrast of 12.9%, and an rms rate of 29 kcounts/sec for data taken in the reproducibilityexperiment(one earthrate is 7.3 X 10-5 rad/sec).

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J6rg Schmiedmayer et al.

We performed experiments to measure both the response factor and the reproducibility for rotations of our interferometer, comparing the response factor with the predictions of Eq. (44) and reproducibility with the noise predicted in Table IV. Both measurements were made by suspending the interferometer by a cable from the ceiling and then driving it with a sinusoidally varying force applied at some distance from the center of mass, thereby giving the interferometer a rotation rate of

fl(t) = ~0 sin(2"n'ft).

(45)

The rotation rate ~0 was typically several earth rates (11e = 7.3 X 10-5 rad/sec)
for the response factor measurement, and about ~-~e/10 for the noise measure-
ments. For the response measurements, f was chosen just over 1 Hz in order to minimize deformations of our interferometer (which has several prominent mechanical resonances in the 10-30 Hz frequency range). For the noise measurements, f was around 4.6 Hz, where the measured residual rotational noise spectrum of the apparatus had a broad minimum.
Our procedure was to measure the acceleration of the suspended interferometer using accelerometers at the sites of the first and third gratings. While modulating the grating phase, q)grating' with a sawtooth pattern at a frequency just less than 1 Hz, we recorded accelerations (from both accelerometers), (~grating' and the atom counts each millisecond. Readings from the accelerometers allowed us to infer the atom phase expected from the acceleration and rotation rate of the interferometer using equations (14, 15). We called this predicted inertial phase
~predicted" To study the magnitude and constancy of the response factor, we binned these
data according to the ~predicted predicted from the accelerometer readings after suitable correction for their known frequency response. Since the frequency of the sawtooth modulation of q~gratingwas chosen to be incommensurate with f, the data in a bin with a particular value of ~predicted had a variety of values q~grating' allowing us to make a fit using Eq. (7) to determine the inertial phase contribution
to Eq. (1), which is (~rotation"A plot of (4~rotationVS. ~Opredicted is shown in Fig. 27 from a combination of 20 sec runs totaling ---400 sec (i.e., ---10 sec of data in each
(4)predicted bin). The data reveal a linear response and an average response factor
within error (0.8%) of that predicted from Eq. (44). To study the reproducibility of our interferometer we employed a phase mod-
ulation technique to immediately convert atom counts into ~Oot~t~on()t. This was accomplished by scanning the second grating (~grating) at 1 Hz to produce a carrier modulation on the atom count rate in Eq. (7). The rotation of the interferom-
eter introduced a phase modulation qgrotation()t onto the carrier that was demodu-
lated by homodyne detection, using the sine of 27r/dg times the grating position signal from the optical interferometer as the local oscillator.
From each of 21 data sets 32 sec long, we analyzed samples of different

OPTICS AND INTERFEROMETRY

69

Rotation Rate Inferred From Accelerometers (~e)

-2

-1

0

1

2

4

e2 2
? o

9 Averaged Data (-10 sec/point) ] Fit to Data (slope = 1.008_+0.01)1

-4

,~ 0.2
"~ 0.0 -0.2

i

i

i

i

-4

-2

0

2

4

Phase Inferred F r o m Accelerometers, (I)predicted (rad)

FIG. 27. A plot of the measured interferometer phase, q0rotation , versus the inferred phase from the accelerometer readings, qOpredicted, from a combination of 20 sec runs totaling approximately 400 sec of data ( - 1 0 sec of data per point). There is a 0.8% difference between these measurements with a total error of 1%.

sizes to find the rotation rate ~-~measured(t)measured from the rotation phase qgrotation(/) of the interferometer. The samples were taken from the middle of
each data set and ranged in duration, T, from 0.66 to 10.66 sec. Each sample was Fourier transformed and the magnitude of the amplitude of the rotation at drive frequency f found. The rms fluctuations in the amplitudes for given sample lengths were then determined for the various averaging times, T. In Fig. 28, they are plotted and compared to the shot noise limit computed in Table IV.
For averaging times up to 2 sec, the reproducibility is close to the predicted

70

Jrrg Schmiedmayer et al.

0. l-

~

8~

~

6~

"~ 4

~ 2

. . . . . Shot Noise

]

9 Reproducibility at Do= O.l~e

~

-9 0 . 0 1 -

O

81

"~...

~'" " " I

'" "

.......

i

"*, .... *'.b.

.......

9

1

10

100

Integration Time (secs)

FIG. 28. Reproducibility of the rotation measurement while driving the interferometer by a 4.6 Hz sinusoidally varying force. The standard deviation of the fluctuations in the measured rotation of the 4.6 Hz spectral peak is plotted versus integration time (circles). This is compared to the shot noise limited error verses integration time (dashed line).

statistical noise; for longer intervals, the observed fluctuations rise above this limit. Since long-term zero drift of the interferometer does not contribute to the amplitude of the rotation at 4.6 Hz, we think this irreproducibility reflects contributions from drive amplitude fluctuations, residual 4.6 Hz rotational noise of the interferometer, and possibly other unidentified sources of rotation such as fluctuations in cooling water pressure.
We regard these results as highly encouraging for the future of inertial sensors using atom interferometers. Our interferometer was designed for separated beam interferometry, not inertial sensing. This resulted in restricting the usable area of our small 1 mm x 200/zm gratings by a combined factor of 100 for both ends of the machine. Furthermore, the vacuum envelope, with heavy diffusion pumps hung at odd angles, had numerous low frequency mechanical resonances. Despite these difficulties, we verified the rotational response factor to better than 1%, indicating that atom interferometric rotation sensor~s_perform as predicted. Moreover, we achieved reproducibility at the 10 mf~e/Vhr level. This is about three orders of magnitude more sensitive than previous rotation measurements using atom interferometry (Riehle et al., 1991) and approaches the sensitivities of much more difficult neutron interferometry measurements that required integration times of many minutes per point (Werner et al., 1979). Clearly, a dedicated rotation sensor using atom interferometric techniques would perform many orders of magnitude better than ours and should considerably exceed the performance of laser gyroscopes.

OPTICS AND INTERFEROMETRY

71

IX. Outlook
In documenting both scientific and technical advances made with atom interferometers, this chapter demonstrates that atom interferometers have progressed beyond the "demonstration" phase and now must be considered as instruments for performing a variety of scientific and technological measurements. Even without new developments, which are expected in atom optics and slow atom technology, atom interferometers can be expected to make further scientific advances in the three basic areas we have discussed: atomic and molecular physics, fundamental studies, and inertial sensing. We shall now discuss several applications in these areas, enumerating in each several developments that we forsee.

A. ATOMICAND MOLECULARPHYSICS
We expect to see increased application of atom and molecule interferometers to atomic and molecular physics. It is certainly possible to use such devices to make significant advances in the accuracy of atomic and molecular measurements. Techniques such as velocity multiplexing (Section V.D) can produce atomic or molecular beams with narrow velocity widths, making it feasible to perform measurements with uncertainties in the 10 -4 range. This level of precision would be particularly welcome for the measurement of the polarizability of cesium, since a better determination of this quantity would help constrain the atomic structure theories used to determine the Weinberg angle from measurements of parity violation in this system (Noecker et al., 1988; Wood et al., 1995).
A qualitatively new application of atom and molecule interferometers would be to measure the separate parallel and perpendicular components of the polarizability of a dimer molecule such as Na 2 in the ground state manifold (or else the anisotropy of polarizability of an atom with ground state electron spin greater than ~l ). This could be done using our technique of contrast interferometry (Section V.C) to determine the anisotropy of the polarizability, while simultaneously determining the weighted average of the ground state polarizability from the average phase shift, which is a function of electric field squared (see Section VI.A). An even more elegant technique would be to apply the same magnitude of electric field to both sides of the interferometer, but in orthogonal directions. Then the average polarizability would cause the same phase shift on both sides of the interferometer, giving no average phase shift, while the contrast would be reduced at a rate proportional to the anisotropy of the polarizability.
Our measurement of the index of refraction of a gas for matter-waves (Section VI.B) has been shown to provide unique information about the long-range shape of the potential interaction between two colliding atoms or molecules. By varying the relative velocity of the two colliding particles, a more detailed study of the index of refraction of various gases for matter-waves could be obtained.

72

J6rg Schmiedmayer et al.

This is a relatively straightforward measurement in our apparatus, since the velocity of the Na beam can be varied simply by changing the mass of the cartier gas in our beam source.
It is an open question whether atoms will bounce coherently from a surface (although they are known to bounce specularly from some surfaces, suggesting that there should be some coherence). Investigation of this phenomenon may be possible by studying the phase shift for bounces of atoms from surfaces using atom interferometry. Perhaps the most interesting investigation here would be to vary the surface and observe the consequent phase change.

B. FUNDAMENTALSTUDIES
We predict that the most fruitful area for scientific application of atom interferometers will be to the study of a wide variety of fundamental processes. While interferometry experiments with neutrons will continue to illuminate many fundamental points, atoms possess many advantages. Atoms have much larger magnetic dipole moments, very large electric polarizabilities, strong interactions with laser light, and easily accessible (with RF or laser radiation) internal structure. The availability of species with either Fermi or Bose statistics also presents interesting opportunities for experiments. Furthermore, atom interferometers are in a rapid stage of development and great increases in signal strength can be anticipated.
In the following brief overview of fundamental experiments that we foresee, we will first discuss geometric phases (e.g., Berry's phase), then phases that come from the interaction of the atoms with B or E fields induced by the motion of the atom through E or B fields, respectively, then address the contentious issue of whether these induced fields produce potentials whose derivative causes observable forces on the particles.

C. BERRY'S PHASE
When a quantum system evolves (even adiabatically) around a cyclic pat?, in phase space, it gains an additional phase, as first described by Berry (Berry, 1984, 1990; Wilczek and Shapere, 1989). Various demonstrations of this effect have been performed with photons (Chiao and Wu, 1986; Kwiat and Chiao, 1991), spin rotation experiments with neutrons (Bitters and Dubbers, 1987; Wilczek and Shapere, 1989), and atomic hydrogen (Miniatura et al., 1992). With our separated beam interferometer, we can measure the geometric phase resulting from transport of the state vector of the atoms through two different paths in phase space back to their starting points (Schmiedmayer et al., 1993). For example, atoms with a definite spin projection could traverse a magnetic field configuration arranged such that the spin (which would follow the field adiabatically) evolves through different paths in the two arms of the interferometer. If atoms on

OPTICS AND INTERFEROMETRY

73

the two different arms see the same total integrated magnetic field, and hence acquire the same total dynamic phase, the only phase shift observed will be the purely geometric phase shift, Aq~geometric = mFm~-~evolution, given by the (easily varied) difference between the solid angle that the magnetic field and hence the state vector subtends on the two arms. Measurements with even and odd m F have not previously been performed; in addition, the adiabaticity can be varied by changing the total field strengths, thus probing nonadiabatic geometrical phases. It is interesting to point out that an experiment like this can also be seen as a demonstration of geometric forces (Stem, 1992; Aharonov and Stem, 1992). The phase shift has the same relation to the geometric force as the Aharonov-Bohm phase shift has to the Lorentz force.

D. RELATIVISTIC EFFECTS IN ELECTROMAGNETIC INTERACTIONS

Atoms are so sensitive to electric and magnetic fields that even the small fields generated by their motions (i.e., E = (v/c) X B and B = -(v/c) X E) produce observable effects, some of which are especially interesting because the extra linear power of v cancels the usual 1/v dispersion of phase shifts in the interferometer, producing an effect independent of velocity. An example of this is the Aharonov-Casher (AC) effect. In its simplest form, this effect manifests itself as a phase shift of the interference pattern of a particle possessing a magnetic moment whose interfering paths form a loop around a line of charge (Anandan, 1982; Aharonov and Casher, 1984). The fundamental importance of the AC effect, as well as the Aharonov-Bohm effect (Aharonov and Bohm, 1959), is the prediction of a phase shift of the atom wave even though the classical force on the particle vanishes. The first measurement of the AC phase was made with a neutron interferometer (Cimmino et al., 1989). Several recent measurements have employed single-beam Ramsey atom interferometers to measure the phase shift to within a few percent (Sangster et al., 1994; Zeiske et al., 1995; G6flitz et al., 1995). An atom interferometer like ours could measure the AC phase shift to better than 1% using a geometry that would also allow examination of the hitherto unstudied dependence of the effect on the relative orientation of the magnetic moment and the line charge.
An effect complimentary to the AC effect involves the electric fields induced by motion through magnetic fields. The small electric fields produced by such motion may be sensitively detected by exploiting the fact that a polarizable atom responds quadratically to the applied electric field. If a large dc electric field, Eheterodyne, is applied oppositely to the two arms of the interferometer, the total phase shift would then have a form like

[

v

v ]

A~motion --

Of

(Eheterodyne

+

C

B) 2

-

(--Eheterodyne

+ -C- B)2

/2

(46)

V = O fc- Eheterodyne B9

74

Ji~rg Schmiedmayer et al.

assuming v and B are perpendicular. It was recently proposed that this effect should be observable (Wei et al. 1995). Finally, motion of an electric dipole (even one induced by an electric field) through a magnetic field can also cause a geometrical phase analogous to the Aharonov-Casher topological phase (Wilkens 1994).

E. DIFFERENTIALFORCEINTERFEROMETRY
Any matter-wave interferometer is very sensitive to differences in longitudinal force applied between the two arms because these change the relative momenta and hence the relative energy of the two particle waves that are recombined, resulting in time-varying phase shifts. The great sensitivity of atoms to electromagnetic fields makes it easy to construct a region where differential forces result from differences in the field gradients on the two sides of the interferometer. Assuming that the atoms entering such a region are initially in a mixture of pure momentum states (as is the case for an atomic beam emitted by an effusive oven), a small differential force will destroy the time-averaged interference contrast. A larger differential force could be studied by means of its ability to rephase a momentum correlation that was induced in the atom beam upstream by some modulation process (e.g., a fast beam chopper).
Using the intrinsically time-varying methods just described, a generalization of the scalar Aharonov-Bohm effect (Aharonov and Bohm, 1959) might be performed. Instead of measuring the phase accumulated by the interaction of the atom's magnetic dipole moment with a uniform magnetic field present during some fixed interval, one could apply a magnetic field gradient during this interval.
Differential force interferometer experiments could address two separate controversies that have recently arisen in discussions of the AC effect and related theoretical issues involving the interaction of the dipole with a time-varying magnetic field induced by the motion of the atom in a time-varying electric field. The first concems whether a motion-induced magnetic field can be treated exactly like an applied magnetic field, as Boyer claims in his analysis of the AC phase shift (Boyer 1987). If this is the case, then there will be a longitudinal force on the dipole as it enters or leaves the gradient at the ends of the applied electric field. Varying the electric field while the atom is present in the interaction region would then make the gradient experienced by the atom exiting the interaction different from the one it experienced on entering, causing a differential momentum that could be detected as described earlier. If, as others (Aharonov et al., 1988; Goldhaber, 1989; Casella, 1994) claim, a full classical analysis finds the net force in the rest frame of the spin equal to 0 (in the special case when the spin is along the direction of the line charge) then no differential force will be observed and at most a fixed relative phase shift would result.

OPTICS AND INTERFEROMETRY

75

A second controversy concerning forces on moving dipoles centers on the observability of the force term

F = V(IX. B)

1 c

I X x 0-OEt

+

2/x -~c

E

X

(IXX B)

(47)

derived by Anandan (1989) for a magnetic dipole moving through crossed electric and magnetic fields. Casella and Wener (1992) claim that for a spm9-~1 particle the last term in this expression is unobservable in principle, but Anandan's group disagrees (Anandan and Hagen 1994). Atom interferometers could resolve this controversy by applying a spatially varying electric field to atoms whose magnetic moments are precessing about a parallel magnetic field at a rate chosen so that the force on the atoms due to the E x (Ix x B) term keeps the same sign throughout the interaction but is opposite on the two sides of the interaction region.

E INERTIALMEASUREMENTS
In view of the smallness of relativistic contributions to the response of atom interferometers to inertial motion, there is little likelihood that atom interferometers will uncover any flaws in the theories presented in section VIII. Therefore, we argue that performance and performance per unit cost are the parameters by which atom interferometric inertial sensors ultimately must be judged. Since their cost is unlikely to be low, their best opportunity lies in spectacular performance.
We first discuss rotation sensing. We have calculated that a 1 m long atom interferometer with three microfabricated matter gratings 1 cm2 in area would give rotational noise of 2 x 10-8 fle/V~r (earthrate per square root hour) with a consumption rate of 1 g of cesium per hour. Slightly improved performance could be achieved using light gratings instead of matter gratings, owing to the larger overall throughput. This performance is four orders of magnitude better than commercially available laser gyros and might have interesting applications to study of the geophysics of earthquakes and other short-term phenomenon. Zero drift of such an instrument would result from long-term mechanical misalignment of the gratings and would constitute a formidable problem, even if overall drifts in grating alignment are corrected for by running atoms in two directions through the interferometer. Overall sensitivity and stability could be improved by using slow atoms, but only at the price of increasing sensitivity to accelerations.

G. CONCLUSION
The work presented in the bulk of this chapter, together with the suggestions for the future just presented, show that atom interferometers have considerable prospect for future study of fundamental physical phenomena and atomic and

76

Jrrg Schmiedmayer et al.

molecular properties and as tools for measurement of noninertial motion. While this much can be anticipated with fair certainty, we hope that there will be exciting developments for applications not anticipated here.

Acknowledgments
The atom gratings used in this work were made in collaboration with Mike Rooks at the Cornell Nanofabrication Facility at Cornell University (Rooks et al., 1995). We are grateful for the existence and assistance of this facility and also for help with nanofabrication from Hank Smith and Mark Schattenburg at M.I.T. The substantial work of David Keith, as well as contributions of Stefan Wehinger, Quentin Turchette, and Bridget Tannian to these experiments also are gratefully acknowledged. This work was made possible by support from the Army Research Office contracts DAAL0389-K-0082 and AASERT 29970-PH-AAS, the Office of Naval Research contract N00014-89-J-1207, NSF contract 9222768-PHY, the Joint Services Electronics Program Contract DAAL03-89-C-0001, and the Charles Stark Draper Laboratory Contract DL-H-484775 9. T. D. H. and E. T. S. acknowledge the support of National Science Foundation graduate fellowships. J. S. acknowledges the support of an Erwin Schrrdinger Fellowship of the Fond zur Frrderung der Wissenschaftilchen Forschung in Austria and an APART fellowship of the Austrian Academy of Sciences.

A A A~, A2 B B C C C O Co(k) C n D D E E Eheterodyne Ekin F G H (I) I /max'I m i n lob.... d L

Appendix: Frequently Used Symbols
area of interferometer fringe amplitude amplitudes for beam paths 1, 2 rotational constant of molecular energy level magnetic field contrast observed contrast initial contrast (before taking into account vibrations) original contrast of interferometer coefficient of 1/r-n potential spacer width transverse distance moved by interferometer energy electric field heterodyne electric field kinetic energy total angular momentum gauss Hamiltonian mean detected intensity laser intensity maximum, minimum intensity observed intensity grating separation

OPTICS AND INTERFEROMETRY

77

L Talbot Lc
Left Lint Ls N p(Acp)
e(Ak x) P(g---~ e) S Si T T V
V~dw a
b b c d
f f
f,.
f(k) f(kcm,O) gF ho hs k ko
kg
katom kh
kcm k;,kI
kphoton /cob e ~ max m
m E
mcarrier n p
F H
S
t
t n
Ul,2 12

Talbot length shutter separation effective length of interaction region length of applied potential region, length of interaction region source to 1st grating distance number of counts, density of medium relative retained contrast as a function of applied phase probability distribution of transverse momentum transfer probability of transition from ground to excited state action action along classical path average time temperature (in Kelvins) potential Van der Waals potential acceleration (transverse to gratings and atomic beam) interferometer fringe amplitude reduction factor classical impact parameter speed of light separation of two arms of interferometer at point of scattering grating period frequency of rotation of interferometer shutter open fraction fraction of atoms in ith state atomic k vector (velocity) distribution forward-scattering amplitude g factor source height detector height wave vector initial wave vector of atoms lattice vector of grating atom wave vector Boltzmann constant wave vector in center of mass frame initial and final photon momentum magnitude of photon momentum coherence length angular momentum highest contributing partial wave mass of atom magnetic quantum number mass of carrier gas atom index of refraction, number of shutter cycles during traversal time momentum hard sphere radius particle diameter time traversal time amplitudes of upper and lower beams in interferometer velocity of atomic beam

78
v n Vtrans X X i
X r
F
r0r0
A~ Aq~
a~
Aq~2
A CPgeometric
Aq~
A~'),evolutio n
AB(x) Ak
Akx Ak
At Av Ax Ax Ax
z
lq 0
'~measured ~~rotation O~ Oti
6 e 0diff
oi
Ada Aphoton
It P~B P 0-+ 0-k
O'to t 0 -x2 7"
%
%
q~
q)acce|eration

Jrrg Schmiedmayer et al.
particle velocity transverse velocity particle position transverse position of grating i range of interatomic potential path paths with x I -- x3 = 0 dynamic phase phase shift induced by a potential observed phase shift phase shift for stretched state geometric phase shift phase shift of atoms in a particular state difference in solid angle subtended by state vectors on two paths differential magnetic field between two arms of the interferometer change in photon momentum during scattering event x component of change in photon momentum average value of Akx sampling period, shutter period velocity width classical displacement deflection due to photon recoil displacements of grating (due to random vibrations) electric field voltage (potential) wave function rotation rate of interferometer base rotation rate dynamic phase contribution rotation rate measured electric polarizability relative grating rotations open fraction of grating i phase shift of the partial wave diffraction angle rotation angle of grating i de Broglie wavelength photon wavelength magnetic moment Bohr magneton mass density circularly polarized light rms width of Gaussian k vector distribution of atoms total scattering cross section variance of grating position (due to random vibrations) transit time between gratings decoherence time rms error in phase measurement phase difference between two paths phase due to interferometer accelerations

OPTICS AND INTERFEROMETRY

79

qPgrating ~inertial ~%osition qPposition(t) @predicted ~rotation O) wR
O-)vib

phase due to relative grating positions inertial phase grating phase measured phase rotational phase of interferometer predicted by the accelerometers phase due to interferometer rotations angular frequency of light Rabi frequency vibration frequency of gratings

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