Elements of Quantum Optics

Pierre Meystre Murray Sargent lll

Elements of Quantum Optics

Second Edition

With 123 Figures

Springer-Verlag Berlin Heidelberg GmbH

Professor Dr. Dr. Rer. Nat. Habil. Pierre Meystre Professor Murray Sargent Ill, Ph. D. Optical Sciences Center, University of Arizona, Thcson, AZ 85721, USA

ISBN 978-3-662-11656-2

Library of Congress Cataloging-in-Publication Data Meystre, Pierre. Elements of quantum optics I Pierre Meystre, Murray Sargent III.- 2nd ed. p.cm. Includes bibliographical references and index. ISBN 978-3-662-11656-2 ISBN 978-3-662-11654-8 (eBook) DOI 10.1007/978-3-662-11654-8 1. Quantum optics. I. Sargent, Murray. II. Title. QC446.2.M48 1991 535-dc20 91-21949

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C Springer-Verlag Berlin Heidelberg 1990, 1991 Originally published by Springer-Verlag Berlin Heidelberg New York in 1991 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a spe­cific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typesetting: Camera ready by author This text was prepared using the PS™ Technical Word Processor 54/3140-543210- Printed on acid-free paper

To

Ren& and Pierre-Andr~

Helga, Nicole, and Tina

Preface

This book grew out of a 2-semester graduate course in laser physics and quantum optics. It requires a solid understanding of elementary electro­magnetism as well as at least one, but preferably two, semesters of quantum mechanics. Its present form resulted from many years of teaching and research at the University of Arizona, the Max-Planck-Institut fiir Quanten­optik, and the University of Munich. The contents have evolved signifi­cantly over the years, due to the fact that quantum optics is a rapidly chang­ing field. Because the amount of material that can be covered in two semes­ters is finite, a number of topics had to be left out or shortened when new material was added. Important omissions include the manipulation of atomic trajectories by light, superradiance, and descriptions of experiments.

Rather than treating any given topic in great depth, this book aims to give a broad coverage of the basic elements that we consider necessary to carry out research in quantum optics. We have attempted to present a vari­ety of theoretical tools, so that after completion of the course students should be able to understand specialized research literature and to produce original research of their own. In doing so, we have always sacrificed rigor to physical insight and have used the concept of "simplest nontrivial exam­ple" to illustrate techniques or results that can be generalized to more com­plicated situations. In the same spirit, we have not attempted to give exhaustive lists of references, but rather have limited ourselves to those papers and books that we found particularly useful.

The book is divided into three parts. Chapters 1 through 3 review various aspects of electromagnetic theory and of quantum mechanics. The material of these chapters, especially Chaps. 1 and 3, represents the mini­mum knowledge required to follow the rest of the course. Chapter 2 intro­duces many nonlinear optics phenomena by using a classical nonlinear oscil­lator model, and is usefully referred to in later chapters. Depending on the level at which the course is taught, one can skip Chaps. 1 through 3 totally or, at the other extreme, give them considerable emphasis.

Chapters 4 through 11 treat semi-classical light-matter interactions. They contain more material than we have typically been able to teach in a one-semester course. Especially if much time is spent on the Chaps. 1 - 3,

vii

viii PREFACE

some of Chaps. 4 - 11 must be skipped. However, Chap. 4 on the density matrix, Chap. 5 on the interaction between matter and cw fields, Chap. 6 on semi-classical laser theory, and to some extent Chap. 8 on nonlinear spec­troscopy are central to the book and cannot be ignored. In contrast one could omit Chap. 7 on optical bistability, Chap. 9 on phase conjugation, Chap. 10 on optical instabilities, or Chap. 11 on coherent transients.

Chapters 12 through 17 discuss aspects of light-matter interaction that require the quantization of the electromagnetic field. They are tightly knit together and it is difficult to imagine skipping one of them in a one-semes­ter course. Chapter 12 draws an analogy between electromagnetic field modes and harmonic oscillators to quantize the field in a simple way. Chapter 13 discusses simple aspects of the interaction between a single mode of the field and a two-level atom. Chapter 14 on reservoir theory is essen­tial for the discussion of resonance fluorescence (Chap. 15) and squeezing (Chap. 16). These chapters are strongly connected to the nonlinear spec­troscopy discussion of Chap. 8. In resonance fluorescence and in squeezing the quantum nature of the field appears mostly in the form of noise. We conclude in Chap. 17 by giving elements of the quantum theory of the laser, which requires a proper treatment of quantum fields to all orders.

In addition to being a textbook, this book contains many important for­mulas in quantum optics that are not found elsewhere except in the original literature or in specialized monographs. As such, and certainly for our own research, this book is a very valuable reference. One particularly gratifying feature of the book is that it reveals the close connection between many seemingly unrelated or only distantly related topics, such as probe absorp­tion, four-wave mixing, optical instabilities, resonance fluorescence, and squeezing.

We are also indebted to the many people who have made important con­tributions to this book: they include first of all our students, who had to suffer through several not-so-debugged versions of the book and have helped with their corrections and suggestions. Special thanks to S. An, B. Capron, T. Carty, P. Dobiasch, J. Grantham, A. Guzman, D. Holm, J. Lehan, R. Morgan, M. Pereira, G. Reiner, E. Schumacher, J. Watanabe, and M. Watson. We are also very grateful to many colleagues for their encourage­ments and suggestions. Herbert Walther deserves more thanks than anybody else: this book would not have been started or completed without his con­stant encouragement and support. Thanks are due especially to the late Fred Hopf as well as to J. H. Eberly, H. M. Gibbs, J. Javanainen, S. W. Koch, W. E. Lamb, Jr., H. Pilloff, C. M. Savage, M. 0. Scully, D. F. Walls, K. Wod­kiewicz, and E. M. Wright. We are also indebted to the Max-Planck-lnstitut fiir Quantenoptik and to the U.S. Office of Naval Research for direct or in­direct financial support of this work.

PREFACE ix

The text and figures of this book were prepared and printed using the Scroll Systems PS ... Technical Word Processor.

Tucson, August 1989

Pierre Meystre

Murray Sargent m

Preface to the Second Edition

This edition contains a significant number of changes designed to improve clarity. We have also added a new section on the theory of resonant light pressure and the manipulation of atomic trajectories by light. This topic is of considerable present interest and has applications both in high resolution spectroscopy and in the emerging field of atom optics. Smaller changes in­clude a reformulation of the photon-echo problem in a way that reveals its relationship to four-wave mixing, as well as a discussion of the quantization of standing-waves versus running-waves of the electromagnetic field. Finally, we have also improved a number of figures and have added some new ones.

We thank the readers who have taken the time to point out to us a number of misprints. Special thanks are due to Z. Bialynicka-Birula, S. Haroche, K. Just, S. LaRochelle, E. Schumacher, and M. Wilkens.

Tucson, February 1991

P.M. M.S.m

Readers interested in the latest electronic version of this book can contact Scroll Systems, Inc., 5530 N. Camino Escuela, Tucson, AZ 85718.

X PREFACE

0.8

0.7

0.6

Re{o:1) 0.5

O:o

0.4

0.3

0.2 -3 -1.5 0 1.5 3

fl.

Probe absorption in an inhomogeneously broadened two-level medium versus pump-probe beat frequency fl. for the dipole decay time T2 = 1, the population difference decay time T 1 = T2, ••• , 64T2, and dimensionless pump intensity / 2 = 4. The physics illustrated in this figure is dicussed in Sec. 8-3. o:1 is given by Eq. (8.28) as approximated by Eqs. (8.48) and (5.52). The reduction of absorption for lll.l <the resonant Rabi frequency ~0 (= 2 for T1 = T 2 = 1) occurs due to the addition of curves like those in Figs. 8-5 and 8-7. which acquire appreciable values only at and beyond the generalized Rabi frequency~-

CONTENTS

L Classical Electromagnetic Fields . ................... . 1-1. Maxwell's Equations in a Vacuum .................. 2 1-2. Maxwell's Equations in a Medium .................. 5 1-3. Linear Dipole Oscillator ........................ II 1-4. Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1-5. Free-Electron Lasers .......................... 26 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2. Classical Nonlinear Optics ........................ 41

2-1. Nonlinear Dipole Oscillator ...................... 41 2-2. Coupled-Mode Equations . . . . . . . . . . . . . . . . . . . . . 45 2-3. Cubic Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2-4. Four-Wave Mixing with Degenerate Pump Frequencies . . . . 51 2-5. Nonlinear Susceptibilities . . . . . . . . . . . . . . . . . . . . . . . 56 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3. Quantum Mechanical Background . . . . . . . . . . . . . . . . . . 60

3-1. Review of Quantum Mechanics ................... 61 3-2. Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . 73 3-3. Atom-Field Interaction for Two-Level Atoms . . . . . . . . . . 82 3-4. Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 93 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4. Mixtures and the Density Operator. . . . . . . . . . . . . . . . . 101

4-1. Level Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 02 4-2. The Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 106 4-3. Vector Model of Density Matrix . . . . . . . . . . . . . . . . . 117 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Xl

xii CONTENTS

5. CW Field Interactions 129 5-1. Polarization of Two-Level Medium . . . . . . . . . . . . . . . 130 5-2. Inhomogeneously Broadened Media . . . . . . . . . . . . . . . 137 5-3. Counterpropagating Wave Interactions . . . . . . . . . . . . . . 143 5-4. Two-Photon Two-Level Model . . . . . . . . . . . . . . . . . . 148 5-5. Polarization of Semiconductor Gain Media . . . . . . . . . . . 156 5-6. Light Forces and Atomic Motion . . . . . . . . . . . . . . . . . 164 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6. Introduction to Laser Theory . . . . . . . . . . . . . . . . . . . . . 178 6-1. The Laser Self -Consistency Equations . . . . . . . . 179 6-2. Steady-State Amplitude and Frequency . . . . . . . . . . . . . 182 6-3. Standing-Wave, Doppler-Broadened Lasers . . . . . . . . . . . 190 6-4. Two-Mode Operation and the Ring Laser . . . . . . . . . . . 197 6-5. Mode Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6-6. Single-Mode Semiconductor Laser Theory . . . . . . . . . . . 205 6-7. Transverse Variations and Gaussian Beams . . . . . . . . . . . 2ll References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7. Optical Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7-1. Simple Theory of Dispersive Optical Bistability . . . . . . . . 225 7-2. Absorptive Optical Bistability . . . . . . . . . . . . . . . . . . . 230 7-3. Ikeda Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8. Saturation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 238 8-1. Probe-Wave Absorption Coefficient . . . . . . . . . . . . . . . 239 8-2. Coherent Dips and the Dynamic Stark Effect . . . . . . . . . 246 8-3. Inhomogeneously Broadened Media . . . . . . . . . . . . . . . 255 8-4. Three-Level Saturation Spectroscopy . . . . . . . . . . . . . . . 259 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

9. Three- and Four-Wave Mixing . . . . . . . . . . . . . . . . . . . . 267 9-1. Phase Conjugation in Two-Level Media . . . . . . . . . . . . 268 9-2. Two-Level Coupled-Mode Coefficients . . . . . . . . . . . . . 271 9-3. Modulation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 275 9-4. Nondegenerate Phase Conjugation by Four-Wave Mixing . . 279 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

CONTENTS

10. Time-Varying Phenomena in Cavities 10-1. Relaxation Oscillations in Lasers . . . . . . . . . . . . . . . . . 10-2. Stability of Single-Mode Laser Operation .......... . 10-3. Multimode Mode Locking .................... . I 0-4. Single-Mode Laser and the Lorenz Model . . . . . . . . . . . References ................................. . Problems .................................. .

xiii

283 284 288 292 296 300 301

1L Coherent Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 11-1. Optical Nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 11-2. Free-Induction Decay . . . . . . . . . . . . . . . . . . . . . . . 307 11-3. Photon Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 11-4. Ramsey Fringes ........................... 311 11-5. Pulse Propagation and Area Theorem . . . . . . . . . . . . . . 313 11-6. Self-Induced Transparency . . . . . . . . . . . . . . . . . . . . 317 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

12. Field Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 12-1. Single-Mode Field Quantization . . . . . . . . . . . . . . . . . 323 12-2. Multimode Field Quantization . . . . . . . . . . . . . . . . . . 326 12-3. Single-Mode Field in Thermal Equilibrium . . . . . . . . . . 329 12-4. Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 12-5. Coherence of Quantum Fields . . . . . . . . . . . . . . . . . . 338 12-6. P(a) Representation . . . . . . . . . . . . . . . . . . . . . . . . . 341 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

13. Interaction between Atoms and Quantized Fields . . . . . . 347 13-1. Dressed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 13-2. Jaynes-Cummings Model . . . . . . . . . . . . . . . . . . . . . 353 13-3. Spontaneous Emission in Free Space . . . . . . . . . . . . . . 358 13-4. Quantum Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

14. System-Reservoir Interactions. . . . . . . . . . . . . . . . . . . . 374 14-1. Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 14-2. Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . 387 14-3. Langevin Equations . . . . . . . . . . . . . . . . . . . . . . . . . 391 14-4. Quantum Regression Theorem and Noise Spectra . . . . . . 396 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

xiv CONTENTS

15. Resonance Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . 406 15-1. Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 15-2. Langevin Equations of Motion . . . . . . . . . . . . . . . . . . 410 15-3. Scattered Intensity and Spectrum . . . . . . . . . . . . . . . . 414 15-4. Connection with Probe Absorption . . . . . . . . . . . . . . . 421 15-5. Photon Antibunching . . . . . . . . . . . . . . . . . . . . . . . . 427 15-6. Off-resonant Excitation . . . . . . . . . . . . . . . . . . . . . . 428 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

16. Squeezed States of Light . . . . . . . . . . . . . . . . . . . . . . . . 436 16-1. Squeezing the Coherent State . . . . . . . . . . . . . . . . . . . 437 16-2. Two-Sidemode Master Equation . . . . . . . . . . . . . . . . . 442 16-3. Two-Mode Squeezing . . . . . . . . . . . . . . . . . . . . . . . 446 16-4. Squeezed Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . 450 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

17. Quantum Theory of a Laser . . . . . . . . . . . . . . . . . . . . . 457 17-1. The Micromaser . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 17-2. Single-Mode Laser Master Equation . . . . . . . . . . . . . . 469 17-3. Laser Photon Statistics and Linewidth . . . . . . . . . . . . . 474 17-4. Quantized Sidemode Buildup . . . . . . . . . . . . . . . . . . . 483 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488