Graduate Texts in Contemporary Physics

Ser ies Editors:

R. Stephen Berry Joseph L. Birman Jeffrey W. Lynn Mark P. Silverman H. Eugene Stanley Mikhail Voloshin

Springer Science+Business Media, LLC

Graduate Texts in Contemporary Physics

S.T. Ali, J.P. Antoine, and J.P. Gazeau: Coherent States, Wavelets and Their Generalizations

A Auerbach: Interacting Electrons and Quantum Magnetism

N. Boccara: ModeUng Complex Systems

T.S. Chow: Mesoscopic Physics of Complex Materials

B. Felsager: Geometry, Particles, and Fields

P. Di Francesco, P. Mathieu, and D. senechal: Conformal Field Theories

A Gonis and W.H. Butler: Multiple Scattering in Solids

K. Gottfried and T -M. Yan: Quantum Mechanics: Fundamentals, 2nd Edition

K. T. Hecht: Quantum Mechanics

J.H. Hinken: Superconductor Electronics: Fundamentals and Microwave Applications

J. Illadik: Spinors in Physics

Yu.M.lvanchenko and AA Lisyansk:y: Physics of Critical Fluctuations

M. Kaku: Introduction to Superstrings and M-Theory, 2nd Edition

M. Kaku: Strings, Conformal Fields, and M-Theory, 2nd Edition

H.V. Klapdor (ed.): Neutrinos

R.L. Liboff (ed): Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, 3rd Edition

J.W. Lynn (ed.): High-Temperature Superconductivity

HJ. Metcalf and P. van der Straten: Laser Cooling and Trapping

R.N. Mohapatra: Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics, 3rd Edition

V.P. Nair: Quantum Field Theory: A Modem Perspective

R.G. Newton: Quantum Physics: A Text for Graduate Students

H. Oberhummer: Nuclei in the Cosmos

G.D.J. Phillies: Elementary Lectures in Statistical Mechanics

R.E. Prange and S.M. Girvin (eds.): The Quantum Hall Effect

(continued after index)

Harold J. Metcalf Peter van der Straten

Laser Cooling and Trapping

With 115 Illustrations

, Springer

Harold J. Metcalf Department of Physics SUNY at Stony Brook

Peter van der Straten Debye Institute

Stony Brook, NY 11794-3800 USA hmetcalf@notes.cc.sunysb.edu

Faculty ofPhysics and Astronomy Princetonplein 5 3584 ee Utrecht The Netherlands P. vanderStraten@phys.uu.nI

Series Editors R Stephen Berry Department of Chemistry University of Chicago Chicago, IL 60637 USA

Mark P. Silverman Department of Physics Trinity College Hartford, CT 06106 USA

Joseph L. Birman Department of Physics City College of CUNY New York, NY 10031 USA

H. Eugene Stanley Center for Polymer Studies Physics Department Boston University Boston, MA 02215 USA

Library of Congress Cataloging-in-Publication Data Metcalf, Harold J.

Laser cooling and trapping / H.J. Metcalf, P. van der Straten. p. cm. - (Graduate texts in contemporary physics)

Includes bibliographical references and index.

Jeffrey W. Lynn Department of Physics University of Maryland College Park, MD 20742 USA

Mikhail Voloshin Theoretical Physics Institute Tate Laboratory of Physics University of Minnesota Minneapolis, MN 55455 USA

ISBN 978-0-387-98728-6 ISBN 978-1-4612-1470-0 (eBook) DOI 10.1007/978-1-4612-1470-0 1. Laser manipulation. 2. Laser cooling. 1. Van der Straten, P.

II. Title. III. Series. QC689.5.L35 M47 1999 539.7-dc21 98-55408

Printed on acid-free paper.

© 1999 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1999 AII rights reserved. This work may not be translated or copied in whole or in part without the written pennission ofthe publisher (Springer Science+Business Media, LLC ), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodo­logy now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe fonner are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

(AUIMVY)

9 8 7 6 543

ISBN 978-0-387-98728-6

springeron/ine.com

To the memory of my parents

Voor Wilma, Lieke en Anouk

Foreword

When Hal Metcalf and I began to work on laser cooling of neutral atoms in about 1979, we found ourselves in a field that was nearly unoccupied by other researchers, or by any real understanding of what the problems and possibilities were. While the study of laser cooling of trapped ions was well under way, only two other groups had ventured into laser manipulation of neutral atoms, one in Moscow and one at Bell Labs (although the latter had temporarily dropped this line of research). Today, laser cooling and its applications represent one of the major subfields of atomic, molecular and optical physics, with over one hundred active groups around the world. Laser cooling has been the enabling technolgy for a wide range of new endeavors. These range from a new generation of atomic clocks, which are operating or under construction in many of the world's major standards laboratories, to the achievement of Bose-Einstein condensation in atomic alkali vapors, one of the fastest growing fields of basic research at the close of the twentieth century. From the highly practical to the very fundamental, laser cooling has become an important part of many research programs.

With this explosion of interest in laser cooling came the obvious question of writing a book about it. The "right time" to write a book on a new subject is a delicate thing. On the one hand, the subject needs to be well enough understood and developed so one can produce a text that stands the test of time, a text that will not be outdated in a few years. On the other hand, one wants the subject to be exciting and of current interest. Metcalf and Van der Straten are two of the finest scientists in the field of laser cooling and they have written the right book at the right time. Certainly, Laser Cooling and Trapping will serve as a valuable reference for researchers working in this field. More importantly it will serve to introduce young people to this exciting field. Now, when someone asks me how to

viii Foreword

start learning about laser cooling, I'll tell them to read Metcalf and Van der Straten. I fully expect that some of the most exciting developments yet to come will come from researchers who begin their study of laser cooling with this book.

April 1999 William D. Phillips Gaithersburg, MD

Preface

The purpose of this book is to introduce students to the dramatic developments in electromagnetic control of atomic motions that has emerged since the 1980s. The book evolved from lectures and courses given by each of us at Stony Brook and Utrecht to advanced undergraduates and beginning graduate students. Its three parts have quite different purposes: Part I serves to review, but not teach, those elements of quantum mechanics and atomic physics that are applicable to the material that follows. Its last chapter addresses certain topics in kinetic theory and statistical physics. Part II introduces the experimental tools and techniques that have been used for electromagnetic control of atomic motion. The first such topic is simply slowing down atoms, usually called laser cooling. But trapping them with magnetic or optical fields (or both), focussing and steering them, and other kinds of manipulation are discussed. The theoretical methods developed in Part I are integrated into these descriptions. Part III discusses some of the manifold applications of the spectacular new tools provided to physicists by these technologies. It is divided-inte-twoMll:!-j><Uts: those topics for which the traditional classical description of atoms moving as classical p6ii1Tpartieles-iuppr~priate, and those topics for which this view must be abandoned and the center-of-mass motion of the atoms must be described quantum mechanically. It is here where some of the most dramatic progress has occurred: atoms in optical lattices, de Broglie wave optics, Bose-Einstein condensation, and the fascinating Schrodinger cat states.

Although there are 50 year-old articles suggesting optical cooling, the topic at­tracted serious attention with proposals for cooling trapped ions and neutral atoms, as well as trapping neutral atoms, at the end of the 1970s. The experimental aspects really began with trapped ions in the late 1970s and the beam slowing demonstra­tions in Troitsk and Gaithersburg in the early 1980s. Then in 1985 the first neutral

x Preface

atom traps were demonstrated: trapping in velocity space was done with optical molasses at AT&T Bell Labs, and in configuration space with purely magnetic fields at Gaithersburg. Simultaneous cooling and trapping was demonstrated us­ing the magneto-optical trap at AT&T Bell Labs in 1987. Then there followed an explosion of interest in this field that was culminated by the award of the Nobel Prize in Physics in 1997 to three of its earliest practitioners.

The text is written from our experimentalists' perspective. There are no long, formal derivations, and most of the theoretical material is presented in a conver­sational rather than formal manner. It is our goal to inspire the readers with some of the beautiful "finger physics" pictures that have evolved in this new field, even though there have been quite elegant formalisms developed by many theorists. Any book intended as a complete, up-to-date, thorough treatment would be obsolete in a few years, and for this lack of completeness we apologize to those whose work may have been slighted or omitted. Instead, we have intended this book to be a guide for students learning the basic elements of the field.

Both of us are indebted to the generations of students and postdocs who have passed through our laboratories, and whom we have encountered in visits to other laboratories, who have taught us so very much. Their fresh approach to the new problems posed by this expanding field have made our research careers fascinating and our time a great pleasure. We also wish to thank Hanneke de Vries and the staff at Springer, including external readers, for all their work on the manuscript.

June 1999 Harold J. Metcalf Peter van der Straten

Contents

Foreword vii

Preface ix

I Introduction 1

1 Review of Quantum Mechanics 3 1.1 Time-Dependent Perturbation Theory ............... 3 1.2 The Rabi Two-Level Problem. . . . . . . . . . . . . . . . . . .. 4

1.2.1 Light Shifts . . . . . . . . . . . . . . . . . . . . . . . .. 7 1.2.2 The Dressed Atom Picture . . . . . . . . . . . . . . . .. 9 1.2.3 The Bloch Vector . . . . . . . . . . . . . . . . . . . . .. 11 1.2.4 Adiabatic Rapid Passage . . . . . . . . . . . . . . . . .. 12

1.3 Excited-State Decay and its Effects. . . . . . . . . . . . . . . .. 14

2 The Density Matrix 17 2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 2.2 Spontaneous Emission ....................... 20 2.3 The Optical Bloch Equations . . . . . . . . . . . . . . . . . . .. 23 2.4 Power Broadening and Saturation ................. 24

3 Force on Two-Level Atoms 3.1 Laser Light Pressure ..

29 ...................... 29

xii Contents

3.2 A Two-Level Atom at Rest 31 3.3 Atoms in Motion . . . . 34

3.3.1 Traveling Wave . 34 3.3.2 Standing Wave 35

4 Multilevel Atoms 39 4.1 Alkali-Metal Atoms. . . . . . 39 4.2 Metastable Noble Gas Atoms . 43 4.3 Polarization and Interference . 45 4.4 Angular Momentum and Selection Rules . 47 4.5 Optical Transitions in Multilevel Atoms . 50

4.5.1 Introduction............ 50 4.5.2 Radial Part . . . . . . . . . . . . 51 4.5.3 Angular Part of the Dipole Matrix Element 52 4.5.4 Fine and Hyperfine Interactions . . 53

5 General Properties Concerning Laser Cooling 57 5.1 Temperature and Thermodynamics in Laser Cooling 58 5.2 Kinetic Theory and the Maxwell-Boltzmann Distribution . 61 5.3 Random Walks ............... . . . . 63 5.4 The Fokker-Planck Equation and Cooling Limits 66 5.5 Phase Space and Liouville's Theorem .. . . . . . 68

II Cooling & Trapping

6 Deceleration of an Atomic Beam 6.1 Introduction.......... 6.2 Techniques of Beam Deceleration

6.2.1 Laser Frequency Sweep . 6.2.2 Varying the Atomic Frequency: Magnetic Field Case 6.2.3 Varying the Atomic Frequency: Electric Field Case . 6.2.4 Varying the Doppler Shift: Diffuse Light 6.2.5 Broadband Light . 6.2.6 Rydberg Atoms.. . ...... .

6.3 Measurements and Results ....... . 6.4 Further Considerations .. ....... .

6.4.1 Cooling During Deceleration . . . . 6.4.2 Non-Uniformity of Deceleration . . . 6.4.3 Transverse Motion During Deceleration 6.4.4 Optical Pumping During Deceleration .

71

73 73 74 76 77 77 78 79 79 80 83 83 84 85 86

7 Optical Molasses 87 7.1 Introduction............................. 87 7.2 Low-Intensity Theory for a Two-Level Atom in One Dimension . 88

7.3 Atomic Beam Collimation ........... . 7.3.1 Low-Intensity Case ........... . 7.3.2 Experiments in One and Two Dimensions

7.4 Experiments in Three-Dimensional Optical Molasses

8 Cooling Below the Doppler Limit 8.1 Introduction................... 8.2 Linear -1 Linear Polarization Gradient Cooling

8.2.1 Light Shifts . . . . . . . . . . 8.2.2 Origin of the Damping Force.

8.3 Magnetically Induced Laser Cooling . 8.4 0'+-0'- Polarization Gradient Cooling 8.5 Theory of Sub-Doppler Laser Cooling 8.6 Optical Molasses in Three Dimensions . 8.7 The Limits of Laser Cooling ..... .

8.7.1 The Recoil Limit ....... . 8.7.2 Cooling Below the Recoil Limit

8.8 Sisyphus Cooling . . . . . . . . . . 8.9 Cooling in a Strong Magnetic Field. 8.10 VSR and Polarization Gradients

9 The Dipole Force 9.1 Introduction. 9.2 Evanescent Waves. . . . . . . . . . . . . . . . . . 9.3 Dipole Force in a Standing Wave: Optical Molasses

at High Intensity ............... . 9.4 Atomic Motion Controlled by Two Frequencies

9.4.1 Introduction............ 9.4.2 Rectification of the Dipole Force. 9.4.3 The Bichromatic Force ..... 9.4.4 Beam Collimation and Slowing

10 Magnetic Trapping of Neutral Atoms 10.1 Introduction . . . . . . . . . . . .

Contents xiii

90 90 92 95

99 99

100 101 102 104 106 107 III 113 113 114 116 118 120

123 123 124

126 128 128 129 131 135

137 137

10.2 Magnetic Traps . . . . . . . . . . . . . . . . . . . . . . . 138 10.3 Classical Motion of Atoms in a Magnetic Quadrupole Trap 140

10.3.1 Simple Picture of Classical Motion in a Trap 140 10.3.2 Numerical Calculations of the Orbits .. 141 10.3.3 Early Experiments with Classical Motion . . 143

10.4 Quantum Motion in a Trap . . . . . . . . . . . . . . 145 10.4.1 Heuristic Calculations of the Quantum Motion

of Magnetically Trapped Atoms . . . . . . 146 10.4.2 Three-Dimensional Quantum Calculations. 146 10.4.3 Experiments in the Quantum Domain . . . 147

xiv Contents

11 Optical Traps for Neutral Atoms 149 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 149 11.2 Dipole Force Optical Traps . . . . . . . . . . . . . . . . . . 150

11.2.1 Single-Beam Optical Traps for Two-Level Atoms . . 150 11.2.2 Hybrid Dipole Radiative Trap . 152 11.2.3 Blue Detuned Optical Traps . 153 11.2.4 Microscopic Optical Traps . 155

11.3 Radiation Pressure Traps . . . . 156 11.4 Magneto-Optical Traps . . . . . . . . 156

11.4.1 Introduction. . . . . . . . . . 156 11.4.2 Cooling and Compressing Atoms in a MOT . . 158 11.4.3 Capturing Atoms in a MOT ... . 159 11.4.4 Variations on the MOT Technique . . . . . . . 162

12 Evaporative Cooling 165 12.1 Introduction . . . . 165 12.2 Basic Assumptions . 166 12.3 The Simple Model . 167 12.4 Speed and Limits of Evaporative Cooling . 171

12.4.1 Boltzmann Equation . . 171 12.4.2 Speed of Evaporation . . 171 12.4.3 Limiting Temperature . 174

12.5 Experimental Results . . . . . . 175

III Applications

13 Newtonian Atom Optics and its Applications 13.1 Introduction . 13.2 Atom Mirrors ....... . 13.3 Atom Lenses ....... .

13.3.1 Magnetic Lenses .. 13.3.2 Optical Atom Lenses.

13.4 Atomic Fountain ...... . 13.5 Application to Atomic Beam Brightening

13.5.1 Introduction. . . . . . . . . . . . 13.5.2 Beam-Brightening Experiments . 13.5.3 High-Brightness Metastable Beams

13.6 Application to Nanofabrication 13.7 Applications to Atomic Clocks .

13.7.1 Introduction ...... . 13.7.2 Atomic Fountain Clocks

13.8 Application to Ion Traps .... 13.9 Application to Non-Linear Optics

177

179 · 179 · 180 · 181 · 181 · 184 · 185 · 186 · 186 · 188 · 189 · 190 · 192 · 192 · 193 · 194 · 195

14 Ultra-cold Collisions 14.1 Introduction . . . 14.2 Potential Scattering . . . 14.3 Ground-state Collisions . 14.4 Excited-state Collisions .

14.4.1 Trap Loss Collisions 14.4.2 Optical Collisions . 14.4.3 Photo-Associative Spectroscopy

14.5 Collisions Involving Rydberg States

15 deBroglie Wave Optics 15.1 Introduction .. . 15.2 Gratings .... . 15.3 Beam Splitters .. 15.4 Sources .... . 15.5 Mirrors .... . 15.6 Atom Polarizers . 15.7 Application to Atom Interferometry

16 Optical Lattices 16.1 Introduction 16.2 Laser Arrangements for Optical Lattices 16.3 Quantum States of Motion .... 16.4 Band Structure in Optical Lattices 16.5 Quantum View of Laser Cooling

17 Bose-Einstein Condensation 17 .1 Introduction . . . . . 17.2 The Pathway to BEC .. 17.3 Experiments . . . . . . .

17.3.1 Observation of BEC 17.3.2 First-Order Coherence Experiments in BEC . 17.3.3 Higher-Order Coherence Effects in BEC . 17.3.4 Other Experiments ............. .

18 Dark States 18.1 Introduction . . . . . . . . . 18.2 VSCPT in Two-Level Atoms 18.3 VSCPT in Real Atoms ...

18.3.1 Circularly Polarized Light 18.3.2 Linearly Polarized Light .

18.4 VSCPT at Momenta Higher Than ±hk . 18.5 VSCPT and Bragg Reflection . 18.6 Entangled States. . . . . . . . . . . . .

Contents xv

199 · 199 .200 .204 .207 .207 .209 .213 · 218

219 .219 .220 .223 .224 .225 .226 .227

231 · 231 .232 .235 .238 .239

241 · 241 .243 .244 .244 .246 .248 .249

251 · 251 .252

· .. 254 · .. 255

.257 · .. 258 · .. 259 · .. 261

xvi Contents

IV Appendices 263

A Notation and Definitions 265

B Review Articles and Books on Laser Cooling 269

C Characteristic Data 273

D Transition Strengths 279

References 291

Index 317