Cambridge University Press
0521820359 - Introductory Quantum Optics - by Christopher Gerry and Peter Knight
Frontmatter/Prelims



Introductory Quantum Optics




This book provides an elementary introduction to the subject of quantum optics, the study of the quantum-mechanical nature of light and its interaction with matter.

   The presentation is almost entirely concerned with the quantized electromagnetic field. Topics covered include single-mode field quantization in a cavity, quantization of multimode fields, quantum phase, coherent states, quasi-probability distribution in phase space, atom–field interactions, the Jaynes–Cummings model, quantum coherence theory, beam splitters and interferometers, nonclassical field states with squeezing etc., tests of local realism with entangled photons from down-conversion, experimental realizations of cavity quantum electrodynamics, trapped ions, decoherence, and some applications to quantum information processing, particularly quantum cryptography. The book contains many homework problems and a comprehensive bibliography.

   This text is designed for upper-level undergraduates taking courses in quantum optics who have already taken a course in quantum mechanics, and for first- and second-year graduate students.

   A solutions manual is available to instructors via solutions@cambridge.org and a website for this book containing updates and errata has been established. The errata can be accessed via www.imperial.ac.uk/people/p.knight/publications.

CHRISTOPHER GERRY is Professor of Physics at Lehman College, City University of New York. He was one of the first to exploit the use of group theoretical methods in quantum optics and is also a frequent contributor to Physical Review A. In 1992 he co-authored, with A. Inomata and H. Kuratsuji, Path Integrals and Coherent States for SU (2) and SU (1, 1).

PETER KNIGHT is a leading figure in quantum optics, and in addition to being President of the Optical Society of America in 2004, he is a Fellow of the Royal Society. In 1983 he co-authored Concepts of Quantum Optics with L. Allen. When this book was written, he was Chief Scientific Advisor at the UK National Physical Laboratory. He is currently Head of the Physics Department of Imperial College.



Introductory Quantum Optics

Christopher Gerry

Lehman College, City University of New York

Peter Knight

Imperial College London



PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
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CAMBRIDGE UNIVERSITY PRESS
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© C. C. Gerry and P. L. Knight 2005

This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.

First published 2005

Printed in the United Kingdom at the University Press, Cambridge

Typefaces TimesNewRoman 10/13 pt. and Universe System LATEX 2e [TB]

A catalog record for this book is available from the British Library

Library of Congress Cataloging in Publication data
Gerry, C. C. (Christopher C.)
Introductory quantum optics / Christopher Gerry, Peter Knight.
p. cm.
Includes bibliographical references and index.
ISBN 0 521 82035 9 – ISBN 0 521 52735 X (paperback)
1. Quantum optics. I. Knight, Peter (Peter L.) II. Title.
QC446.2.G47 2004
535′.15 – dc22 2004051847

ISBN 0 521 82035 9 hardback
ISBN 0 521 52735 X paperback
The publisher has used its best endeavors to ensure that the URLs for external websites referred to in this book are correct and active at the time of going to press. However, the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate.



C. C. G. dedicates this book to his son, Eric.

P. L. K. dedicates this book to his wife Chris.



Contents




  Acknowledgements page xii
  1 Introduction 1
  1.1  Scope and aims of this book 1
  1.2  History 2
  1.3  The contents of this book 7
          References 8
          Suggestions for further reading 8
  2 Field quantization 10
  2.1  Quantization of a single-mode field 10
  2.2  Quantum fluctuations of a single-mode field 15
  2.3  Quadrature operators for a single-mode field 17
  2.4  Multimode fields 18
  2.5  Thermal fields 25
  2.6  Vacuum fluctuations and the zero-point energy 29
  2.7  The quantum phase 33
          Problems 40
          References 41
          Bibliography 42
  3 Coherent states 43
  3.1  Eigenstates of the annihilation operator and minimum uncertainty states 43
  3.2  Displaced vacuum states 48
  3.3  Wave packets and time evolution 50
  3.4  Generation of coherent states 52
  3.5  More on the properties of coherent states 53
  3.6  Phase-space pictures of coherent states 56
  3.7  Density operators and phase-space probability distributions 59
  3.8  Characteristic functions 65
          Problems 71
          References 72
          Bibliography
  4 Emission and absorption of radiation by atoms 74
  4.1  Atom–field interactions 74
  4.2  Interaction of an atom with a classical field 76
  4.3  Interaction of an atom with a quantized field 82
  4.4  The Rabi model 87
  4.5  Fully quantum-mechanical model; the Jaynes–Cummings model 90
  4.6  The dressed states 99
  4.7  Density-operator approach: application to thermal states 102
  4.8  The Jaynes–Cummings model with large detuning: a dispersive interaction 105
  4.9  Extensions of the Jaynes–Cummings model 107
          Schmidt decomposition and von Neumann entropy for the Jaynes–Cummings model 108
          Problems 110
          References 113
          Bibliography 114
  5 Quantum coherence functions 115
  5.1  Classical coherence functions 115
  5.2  Quantum coherence functions 120
  5.3  Young’s interference 124
  5.4  Higher-order coherence functions 127
          Problems 133
          References 133
          Bibliography 134
  6 Beam splitters and interferometers 135
  6.1  Experiments with single photons 135
  6.2  Quantum mechanics of beam splitters 137
  6.3  Interferometry with a single photon 143
  6.4  Interaction-free measurement 144
  6.5  Interferometry with coherent states of light 146
          Problems 147
          References 149
          Bibliography 149
  7 Nonclassical light 150
  7.1  Quadrature squeezing 150
  7.2  Generation of quadrature squeezed light 165
  7.3  Detection of quadrature squeezed light 167
  7.4  Amplitude (or number) squeezed states 169
  7.5  Photon antibunching 171
          SchrÕdinger cat states 174
  7.7  Two-mode squeezed vacuum states 182
  7.8  Higher-order squeezing 188
  7.9  Broadband squeezed light 189
          Problems 190
          References 192
          Bibliography 194
  8 Dissipative interactions and decoherence 195
  8.1  Introduction 195
  8.2  Single realizations or ensembles? 196
  8.3  Individual realizations 200
  8.4  Shelving and telegraph dynamics in three-level atoms 204
  8.5  Decoherence 207
  8.6  Generation of coherent states from decoherence: nonlinear optical balance 208
  8.7  Conclusions 210
          Problems 211
          References 211
          Bibliography 212
  9 Optical test of quantum mechanics 213
  9.1  Photon sources: spontaneous parametric down-conversion 214
  9.2  The Hong–Ou–Mandel interferometer 217
  9.3  The quantum eraser 219
  9.4  Induced coherence 222
  9.5  Superluminal tunneling of photons 224
  9.6  Optical test of local realistic theories and Bell’s theorem 226
  9.7  Franson’s experiment 232
  9.8  Applications of down-converted light to metrology without absolute standards 233
          Problems 235
          References 236
          Bibliography 237
  10 Experiments in cavity QED and with trapped ions 238
  10.1  Rydberg atoms 238
  10.2  Rydberg atom interacting with a cavity field 241
  10.3  Experimental realization of the Jaynes–Cummings model 246
  10.4  Creating entangled atoms in CQED 249
  10.5  Formation of Schrōdinger cat states with dispersive atom–field interactions and decoherence from the quantum to the classical 250
  10.6  Quantum nondemolition measurement of photon number 254
  10.7 Realization of the Jaynes–Cummings interaction in the motion of a trapped ion 255
  10.8  Concluding remarks 258
          Problems 259
          References 260
          Bibliography 261
  11 Applications of entanglement: Heisenberg-limited interferometry and quantum information processing 263
  11.1  The entanglement advantage 264
  11.2  Entanglement and interferometric measurements 265
  11.3  Quantum teleportation 268
  11.4  Cryptography 270
  11.5  Private key crypto-systems 271
  11.6  Public key crypto-systems 273
  11.7  The quantum random number generator 274
  11.8  Quantum cryptography 275
  11.9  Future prospects for quantum communication
          Gates for quantum computation 281
  11.11  An optical realization of some quantum gates 286
  11.12  Decoherence and quantum error correction 289
          Problems 290
          References 291
          Bibliography 293
  Appendix A The density operator, entangled states, the Schmidt decomposition, and the von Neumann entropy 294
  A.1 The density operator 294
  A.2 Two-state system and the Bloch sphere 297
  A.3 Entangled states 298
  A.4 Schmidt decomposition 299
  A.5 von Neumann entropy 301
  A.6 Dynamics of the density operator 302
  References 303
  Bibliography 303
  Appendix B Quantum measurement theory in a (very small) nutshell 304
  Bibliography 307
  Appendix C Derivation of the effective Hamiltonian for dispersive (far off-resonant) interactions 308
  References 311
  Appendix D Nonlinear optics and spontaneous parametric down-conversion 312
  References 313
  Index 314



Acknowledgements




This book developed out of courses that we have given over the years at Imperial College London, and the Graduate Center of the City University of New York, and we are grateful to the many students who have sat through our lectures and acted as guinea pigs for the material we have presented here.

   We would like to thank our many colleagues who, over many years have given us advice, ideas and encouragement. We particularly thank Dr. Simon Capelin at Cambridge University Press who has had much more confidence than us that this would ever be completed. Over the years we have benefited from many discussions with our colleagues, especially Les Allen, Gabriel Barton, Janos Bergou, Keith Burnett, Vladimir Buzek, Richard Campos, Bryan Dalton, Joseph Eberly, Rainer Grobe, Edwin Hach III, Robert Hilborn, Mark Hillery, Ed Hinds, Rodney Loudon, Peter Milonni, Bill Munro, Geoffrey New, Edwin Power, George Series, Wolfgang Schleich, Bruce Shore, Carlos Stroud Jr, Stuart Swain, Dan Walls and Krzysztof Wodkiewicz. We especially thank Adil Benmoussa for creating all the figures for this book using his expertise with Mathematica, Corel Draw, and Origin Graphics, for working through the homework problems, and for catching many errors in various drafts of the manuscript. We also thank Mrs. Ellen Calkins for typing the initial draft of several of the chapters.

   Our former students and postdocs, who have taught us much, and have gone on to become leaders themselves in this exciting subject: especially Stephen Barnett, Almut Beige, Artur Ekert, Barry Garraway, Christoph Keitel, Myungshik Kim, Gerard Milburn, Martin Plenio, Barry Sanders, Stefan Scheel, and Vlatko Vedral: they will recognize much that is here!

   As this book is intended as an introduction to quantum optics, we have not attempted to be comprehensive in our citations. We apologize to authors whose work is not cited.

   C. C. G. wishes to thank the members of the Lehman College Department of Physics and Astronomy, and many other members of the Lehman College community, for their encouragement during the writing of this book.

   P. L. K. would like especially to acknowledge the support throughout of Chris Knight, who has patiently provided encouragement, chauffeuring and vast amounts of tea during the writing of this book.

   Our work in quantum optics over the past four decades has been funded by many sources: for P. L. K. in particular the UK SRC, SERC, EPSRC, the Royal Society, The European Union, the Nuffield Foundation, and the U. S. Army are thanked for their support; for C. C. G. the National Science Foundation, The Research Corporation, Professional Staff Congress of the City University of New York (PSC-CUNY).



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