Table Of ContentElements ofQ uantum Optics
Pierre Meystre Murray Sargent III
Elements
of Quantum Optics
With 113 Figures
Springer-Verlag Berlin Heidelberg GmbH
Professor Dr. Dr. Rer. Nat. Habit. Pierre Meystre
Professor Murray Sargent III, Ph. D.
Optical Sciences Center, University of Arizona, Thcson, AZ 85721, USA
ISBN 978-3-662-07009-3 ISBN 978-3-662-07007-9 (eBook)
DOI 10.1007/978-3-662-07007-9
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Originally published by Springer· Verlag Berlin Heidelberg New York in 1990
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Preface
This book grew out of a 2-semester graduate course in laser physics and
quantum optics. It requires a solid understanding of elementary electro
magnetism as well as at least one, but preferably two, semesters of quan
tum mechanics. Its present form resulted from many years of teaching and
research at the Max-Planck-Institut fi.ir Quantenoptik, the University of
Munich, and the University of Arizona. The contents have evolved signi
ficantly 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 var
iety 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 var
ious aspects of electromagnetic theory and of quantum mechanics. The
material of these chapters, and specially 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,
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 conjuga
tion, 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-sem
ester 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 nonli
near 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 ori
ginal 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 fluo
rescence, 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 espe
cially to the late Fred Hopf as well as to J. H. Eberly, H. M. Gibbs, J. Jav
anainen, 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 in
debted to the Max-Planck-Institut 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
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Tucson Pierre Meystre
August 1989 Murray Sargent III
Contents
L Classical Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . 1
1-1. Maxwell's Equations in a Vacuum .................. 2
1-2. Maxwell's Equations in a Medium .................. 5
1-3. Linear Dipole Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 11
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4-2. The Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 106
4-3. Vector Model of Density Matrix . . . . . . . . . . . . . . . . . 117
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
xu CONTENTS
5. CW Field Interactions 129
5-1. Polarization of Two-Level Medium .............. . 129
5-2. lnhomogeneously 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
References ................................. . 164
Problems .................................. . 165
6. Introduction to Laser Theory . . . . . . . . . . . . . . . . . . . . . 169
6-1. The Laser Self -Consistency Equations . . . . . . . . . . . . . . 170
6-2. Steady-State Amplitude and Frequency . . . . . . . . . . . . . 173
6-3. Standing-Wave, Doppler-Broadened Lasers . . . . . . . . . . . 181
6-4. Two-Mode Operation and the Ring Laser . . . . . . . . . . . 188
6-5. Mode Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6-6. Single-Mode Semiconductor Laser Theory . . . . . . . . . . . 196
6-7. Transverse Variations and Gaussian Beams . . . . . . . . . . . 202
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7. Optical Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7-1. Simple Theory of Dispersive Optical Bistability . . . . . . . . 216
7-2. Absorptive Optical Bistability . . . . . . . . . . . . . . . . . . . 221
7-3. Ikeda Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
8. Saturation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 229
8-1. Probe-Wave Absorption Coefficient . . . . . . . . . . . . . . . 230
8-2. Coherent Dips and the Dynamic Stark Effect . . . . . . . . . 237
8-3. Inhomogeneously Broadened Media . . . . . . . . . . . . . . . 247
8-4. Three-Level Saturation Spectroscopy . . . . . . . . . . . . . . . 250
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
9. Three and Four Wave Mixing. . . . . . . . . . . . . . . . . . . . . 258
9-1. Phase Conjugation in Two-Level Media . . . . . . . . . . . . 259
9-2. Two-Level Coupled Mode Coefficients . . . . . . . . . . . . . 262
9-3. Modulation Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 266
9-4. Nondegenerate Phase Conjugation by Four-Wave Mixing . . 270
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
CONTENTS xiii
10. Time-Varying Phenomena in Cavities 274
10-1. Relaxation Oscillations in Lasers . . . . . . . . . . . . . . . . . 275
10-2. Stability of Single-Mode Laser Operation . . . . . . . . . . . 279
10-3. Multimode Mode Locking . . . . . . . . . . . . . . . . . . . . . 284
10-4. Single-Mode Laser and the Lorenz Model . . . . . . . . . . . 287
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
11. Coherent Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
11-1. Optical Nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
11-2. Free Induction Decay . . . . . . . . . . . . . . . . . . . . . . . 296
11-3. Photon Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
11-4. Ramsey Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
11-5. Pulse Propagation and Area Theorem . . . . . . . . . . . . . . 302
11-6. Self-Induced Transparency . . . . . . . . . . . . . . . . . . . . 306
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
12. Field Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
12-1. Single-Mode Field Quantization . . . . . . . . . . . . . . . . . 312
12-2. Multimode Field Quantization . . . . . . . . . . . . . . . . . . 315
12-3. Single-Mode Field in Thermal Equilibrium . . . . . . . . . . 317
12-4. Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
12-5. Coherence of Quantum Fields . . . . . . . . . . . . . . . . . . 327
12-6. P(a) Representation . . . . . . . . . . . . . . . . . . . . . . . . . 330
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
13. Interaction Between Atoms and Quantized Fields . . . . . . 336
13-1. Dressed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
13-2. Jaynes-Cummings Model . . . . . . . . . . . . . . . . . . . . . 342
13-3. Spontaneous Emission in Free Space . . . . . . . . . . . . . . 347
13-4. Quantum Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
14. System-Reservoir Interactions. . . . . . . . . . . . . . . . . . . . 363
14-1. Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
14-2. Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . 376
14-3. Langevin Equations . . . . . . . . . . . . . . . . . . . . . . . . . 380
14-4. Quantum Regression Theorem and Noise Spectra . . . . . . 385
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
