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OXFORD MASTER SERIES IN PHYSICS OXFORD MASTER SERIES IN PHYSICS The Oxford Master Series is designed for final year undergraduate and beginning graduate students in physics and related disciplines. It has been driven by a perceived gap in the literature today. While basic undergraduate physics texts...

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OXFORD MASTER SERIES IN PHYSICS

OXFORD MASTER SERIES IN PHYSICS The Oxford Master Series is designed for final year undergraduate and beginning graduate students in physics and related disciplines. It has been driven by a perceived gap in the literature today. While basic undergraduate physics texts often show little or no connection with the huge explosion of research over the last two decades, more advanced and specialized texts tend to be rather daunting for students. In this series, all topics and their consequences are treated at a simple level, while pointers to recent developments are provided at various stages. The emphasis is on clear physical principles like symmetry, quantum mechanics, and electromagnetism which underlie the whole of physics. At the same time, the subjects are related to real measurements and to the experimental techniques and devices currently used by physicists in academe and industry. Books in this series are written as course books, and include ample tutorial material, examples, illustrations, revision points, and problem sets. They can likewise be used as preparation for students starting a doctorate in physics and related fields, or for recent graduates starting research in one of these fields in industry.

CONDENSED MATTER PHYSICS 1. 2. 3. 4. 5. 6.

M. T. Dove: Structure and dynamics: an atomic view of materials J. Singleton: Baud theory and electronic properties of solids A. M. Fox: Optical properties of solids S. J. Blundell: Magnetism in condensed matter J. F. Annett: Superconductivity R. A. L. Jones: Soft condensed matter

ATOMIC, OPTICAL, AND LASER PHYSICS 7. 8. 9. 15.

C. J. Foot: Atomic Physics G. A. Brooker: Modern classical optics S. M. Hooker, C. E. Webb: Laser physics A. M. Fox: Quantum optics: an introduction

PARTICLE PHYSICS, ASTROPHYSICS, AND COSMOLOGY 10. D. H. Perkins: Particle astrophysics 11. Ta-Pei Cheng: Relativity, gravitation, and cosmology

STATISTICAL, COMPUTATIONAL, AND THEORETICAL PHYSICS 12. M. Maggiore: A modern introduction to quantum field theory 13. W. Krauth: Statistical mechanics: algorithms and computations 14. J. P. Sethna: Entropy, order parameters, and complexity

Quantum Optics An Introduction

MARK FOX Department of Physics and Astronomy University of Sheffield

1

3 Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c Oxford University Press 2006


The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Fox, Mark (Anthony Mark) Quantum optics : an introduction/Mark Fox. p. cm. — (Oxford master series in physics ; 6) Includes bibliographical references and index. ISBN-13: 978–0–19–856672–4 (hbk. : acid-free paper) ISBN-10: 0–19–856672–7 (hbk. : acid-free paper) ISBN-13: 978–0–19–856673–1 (pbk. : acid-free paper) ISBN-10: 0–19–856673–5 (pbk. : acid-free paper) 1. Quantum optics. I. Title. II. Series. QC446.2.F69 2006 535′ .15—dc22 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Antony Rowe, Chippenham ISBN 0–19–856672–7 978–0–19–856672–4 ISBN 0–19–856673–5 (Pbk.) 978–0–19–856673–1 (Pbk.) 10 9 8 7 6 5 4 3 2 1

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Preface Quantum optics is a subject that has come to the fore over the last 10–20 years. Formerly, it was regarded as a highly specialized discipline, accessible only to a small number of advanced students at selected universities. Nowadays, however, the demand for the subject is much broader, with the interest strongly fuelled by the prospect of using quantum optics in quantum information processing applications. My own interest in quantum optics goes back to 1987, when I attended the Conference on Lasers and Electro-Optics (CLEO) for the first time. The ground-breaking experiments on squeezed light had recently been completed, and I was able to hear invited talks from the leading researchers working in the field. At the end of the conference, I found myself sufficiently interested in the subject that I bought a copy of Loudon’s Quantum theory of light and started to work through it in a fairly systematic way. Nearly 20 years on, I still consider Loudon’s book as my favourite on the subject, although there are now many more available to choose from. So why write another? The answer to this question became clearer to me when I tried to develop a course on quantum optics as a submodule of a larger unit entitled ‘Aspects of Modern Physics’. This course is taken by undergraduate students in their final semester, and aims to introduce them to a number of current research topics. I set about designing a course to cover a few basic ideas about photon statistics, quantum cryptography, and Bose–Einstein condensation, hoping that I would find a suitable text to recommend. However, a quick inspection of the quantum optics texts that were available led me to conclude that they were generally pitched at a higher level than my target audience. Furthermore, the majority were rather mathematical in their presentation. I therefore reluctantly concluded that I would have to write the book I was seeking myself. The end result is what you see before you. My hope is that it will serve both as a useful basic introduction to the subject, and also as a tasty hors d’oeuvre for the more advanced texts like Loudon’s. In developing my course notes into a full-length book, the first problem that I encountered was the selection of topics. Traditional quantum optics books like Loudon’s assume that the subject refers primarily to the properties of light itself. At the same time, it is apparent that the subject has broadened considerably in its scope, at least to many people working in the field. I have therefore included a broad range of topics that probably would not have found their way into a quantum optics text 20 years ago. It is probable that someone else writing a similar text

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Preface

would make a different selection of topics. My selection has been based mainly on my perception of the key subject areas, but it also reflects my own research interests to some extent. For this reason, there are probably more examples of quantum optical effects in solid state systems than might normally have been expected. Some of the subjects that I have selected for inclusion are still developing very rapidly at the time of writing. This is especially true of the topics in quantum information technology covered in Part IV. Any attempt to give a detailed overview of the present status of the experiments in these fields would be relatively pointless, as it would date very quickly. I have therefore adopted the strategy of trying to explain the basic principles and then illustrating them with a few recent results. It is my hope that the chapters I have written will be sufficient to allow students who are new to the subjects to understand the fundamental concepts, thereby allowing them to go to the research literature should they wish to pursue any topics in more detail. At one stage I thought about including references to a good number of internet sites within the ‘Further Reading’ sections, but as the links to these sites frequently change, I have actually only included a few. I am sure that the modern computer-literate student will be able to find these sites far more easily than I can, and I leave this part of the task to the student’s initiative. It is a fortunate coincidence that the book is going to press in 2005, the centenary of Einstein’s work on the photoelectric effect, when there are many articles available to arouse the interest of students on this subject. Furthermore, the award of the 2005 Nobel Prize for Physics to Roy Glauber “for his contribution to the quantum theory of optical coherence” has generated many more widely-accessible information resources. An issue that arose after receiving reviews of my original book plan was the difficulty in making the subject accessible without gross oversimplification of the essential physics. As a consequence of these reviews, I suspect that some sections of the book are pitched at a slightly higher level than my original target of a final-year undergraduate, and would in fact be more suitable for use in the first year of a Master’s course. Despite this, I have still tried to keep the mathematics to a minimum as far as possible, and concentrated on explanations based on the physical understanding of the experiments that have been performed. I would like to thank a number of people who have helped in the various stages of the preparation of this book. First, I would like to thank all of the anonymous reviewers who made many helpful suggestions and pointed out numerous errors in the early versions of the manuscript. Second, I would like to thank several people for critical reading of parts of the manuscript, especially Dr Brendon Lovett for Chapter 13, and Dr Gerald Buller and Robert Collins for Chapter 12. I would like to thank Dr Ed Daw for clarifying my understanding of gravity wave interferometers. A special word of thanks goes to Dr Geoff Brooker for critical reading of the whole manuscript. Third, I would like to thank Sonke Adlung at Oxford University Press for his support and patience

Preface

throughout the project and Anita Petrie for overseeing the production of the book. I am also grateful to Dr Mark Hopkinson for the TEM picture in Fig. D.3, and to Dr Robert Taylor for Fig. 4.7. Finally, I would like to thank my doctoral supervisor, Prof. John Ryan, for originally pointing me towards quantum optics, and my numerous colleagues who have helped me to carry out a number of quantum optics experiments during my career. Sheffield June 2005

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Contents List of symbols List of abbreviations

xv xviii

I Introduction and background

1

1 Introduction 1.1 What is quantum optics? 1.2 A brief history of quantum optics 1.3 How to use this book

3 3 4 6

2 Classical optics 2.1 Maxwell’s equations and electromagnetic waves 2.1.1 Electromagnetic fields 2.1.2 Maxwell’s equations 2.1.3 Electromagnetic waves 2.1.4 Polarization 2.2 Diffraction and interference 2.2.1 Diffraction 2.2.2 Interference 2.3 Coherence 2.4 Nonlinear optics 2.4.1 The nonlinear susceptibility 2.4.2 Second-order nonlinear phenomena 2.4.3 Phase matching

8 8 8 10 10 12 13 13 15 16 19 19 20 23

3 Quantum mechanics 3.1 Formalism of quantum mechanics 3.1.1 The Schr¨ odinger equation 3.1.2 Properties of wave functions 3.1.3 Measurements and expectation values 3.1.4 Commutators and the uncertainty principle 3.1.5 Angular momentum 3.1.6 Dirac notation 3.2 Quantized states in atoms 3.2.1 The gross structure 3.2.2 Fine and hyperfine structure 3.2.3 The Zeeman effect 3.3 The harmonic oscillator 3.4 The Stern–Gerlach experiment 3.5 The band theory of solids

26 26 26 28 30 31 32 34 35 35 39 41 41 43 45

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4 Radiative transitions in atoms 4.1 Einstein coefficients 4.2 Radiative transition rates 4.3 Selection rules 4.4 The width and shape of spectral lines 4.4.1 The spectral lineshape function 4.4.2 Lifetime broadening 4.4.3 Collisional (pressure) broadening 4.4.4 Doppler broadening 4.5 Line broadening in solids 4.6 Optical properties of semiconductors 4.7 Lasers 4.7.1 Laser oscillation 4.7.2 Laser modes 4.7.3 Laser properties

48 48 51 54 56 56 56 57 58 58 59 61 61 64 67

II Photons

73

5 Photon statistics 5.1 Introduction 5.2 Photon-counting statistics 5.3 Coherent light: Poissonian photon statistics 5.4 Classification of light by photon statistics 5.5 Super-Poissonian light 5.5.1 Thermal light 5.5.2 Chaotic (partially coherent) light 5.6 Sub-Poissonian light 5.7 Degradation of photon statistics by losses 5.8 Theory of photodetection 5.8.1 Semi-classical theory of photodetection 5.8.2 Quantum theory of photodetection 5.9 Shot noise in photodiodes 5.10 Observation of sub-Poissonian photon statistics 5.10.1 Sub-Poissonian counting statistics 5.10.2 Sub-shot-noise photocurrent

75 75 76 78 82 83 83 86 87 88 89 90 93 94 99 99 101

6 Photon antibunching 6.1 Introduction: the intensity interferometer 6.2 Hanbury Brown–Twiss experiments and classical intensity fluctuations 6.3 The second-order correlation function g (2) (τ ) 6.4 Hanbury Brown–Twiss experiments with photons 6.5 Photon bunching and antibunching 6.5.1 Coherent light 6.5.2 Bunched light 6.5.3 Antibunched light 6.6 Experimental demonstrations of photon antibunching 6.7 Single-photon sources

105 105 108 111 113 115 116 116 117 117 120

Contents xi

7 Coherent states and squeezed light 7.1 Light waves as classical harmonic oscillators 7.2 Phasor diagrams and field quadratures 7.3 Light as a quantum harmonic oscillator 7.4 The vacuum field 7.5 Coherent states 7.6 Shot noise and number–phase uncertainty 7.7 Squeezed states 7.8 Detection of squeezed light 7.8.1 Detection of quadrature-squeezed vacuum states 7.8.2 Detection of amplitude-squeezed light 7.9 Generation of squeezed states 7.9.1 Squeezed vacuum states 7.9.2 Amplitude-squeezed light 7.10 Quantum noise in amplifiers

126 126 129 131 132 134 135 138 139

8 Photon number states 8.1 Operator solution of the harmonic oscillator 8.2 The number state representation 8.3 Photon number states 8.4 Coherent states 8.5 Quantum theory of Hanbury Brown–Twiss experiments

151 151 154 156 157

III Atom–photon interactions

165

9 Resonant light–atom interactions 9.1 Introduction 9.2 Preliminary concepts 9.2.1 The two-level atom approximation 9.2.2 Coherent superposition states 9.2.3 The density matrix 9.3 The time-dependent Schr¨ odinger equation 9.4 The weak-field limit: Einstein’s B coefficient 9.5 The strong-field limit: Rabi oscillations 9.5.1 Basic concepts 9.5.2 Damping 9.5.3 Experimental observations of Rabi oscillations 9.6 The Bloch sphere

167 167 168 168 169 171 172 174 177 177 180

10 Atoms in cavities 10.1 Optical cavities 10.2 Atom–cavity coupling 10.3 Weak coupling 10.3.1 Preliminary considerations 10.3.2 Free-space spontaneous emission

194 194 197 200 200 201

139 142 142 142 144 146

160

182 187

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10.3.3 Spontaneous emission in a single-mode cavity: the Purcell effect 10.3.4 Experimental demonstrations of the Purcell effect 10.4 Strong coupling 10.4.1 Cavity quantum electrodynamics 10.4.2 Experimental observations of strong coupling 10.5 Applications of cavity effects

202 204 206 206 209 211

11 Cold atoms 11.1 Introduction 11.2 Laser cooling 11.2.1 Basic principles of Doppler cooling 11.2.2 Optical molasses 11.2.3 Sub-Doppler cooling 11.2.4 Magneto-optic atom traps 11.2.5 Experimental techniques for laser cooling 11.2.6 Cooling and trapping of ions 11.3 Bose–Einstein condensation 11.3.1 Bose–Einstein condensation as a phase transition 11.3.2 Microscopic description of Bose–Einstein condensation 11.3.3 Experimental techniques for Bose–Einstein condensation 11.4 Atom lasers

216 216 218 218 221 224 226 227 229 230

233 236

IV Quantum information processing

241

12 Quantum cryptography 12.1 Classical cryptography 12.2 Basic principles of quantum cryptography 12.3 Quantum key distribution according to the BB84 protocol 12.4 System errors and identity verification 12.4.1 Error correction 12.4.2 Identity verification 12.5 Single-photon sources 12.6 Practical demonstrations of quantum cryptography 12.6.1 Free-space quantum cryptography 12.6.2 Quantum cryptography in optical fibres

243 243 245 249 253 253 254 255 256 257 258

13 Quantum computing 13.1 Introduction 13.2 Quantum bits (qubits) 13.2.1 The concept of qubits 13.2.2 Bloch vector representation of single qubits 13.2.3 Column vector representation of qubits

264 264 267 267 269 270

230 232

Contents xiii

13.3 Quantum logic gates and circuits 13.3.1 Preliminary concepts 13.3.2 Single-qubit gates 13.3.3 Two-qubit gates 13.3.4 Practical implementations of qubit operations 13.4 Decoherence and error correction 13.5 Applications of quantum computers 13.5.1 Deutsch’s algorithm 13.5.2 Grover’s algorithm 13.5.3 Shor’s algorithm 13.5.4 Simulation of quantum systems 13.5.5 Quantum repeaters 13.6 Experimental implementations of quantum computation 13.7 Outlook 14 Entangled states and quantum teleportation 14.1 Entangled states 14.2 Generation of entangled photon pairs 14.3 Single-photon interference experiments 14.4 Bell’s theorem 14.4.1 Introduction 14.4.2 Bell’s inequality 14.4.3 Experimental confirmation of Bell’s theorem 14.5 Principles of teleportation 14.6 Experimental demonstration of teleportation 14.7 Discussion

270 270 272 274 275 279 281 281 283 286 287 287 288 292 296 296 298 301 304 304 305 308 310 313 316

Appendices A

Poisson statistics

321

B

Parametric amplification B.1 Wave propagation in a nonlinear medium B.2 Degenerate parametric amplification

324 324 326

C

The density of states

330

D

Low-dimensional semiconductor structures D.1 Quantum confinement D.2 Quantum wells D.3 Quantum dots

333 333 335 337

E

Nuclear magnetic resonance E.1 Basic principles E.2 The rotating frame transformation E.3 The Bloch equations

339 339 341 344

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F

Bose–Einstein condensation F.1 Classical and quantum statistics F.2 Statistical mechanics of Bose–Einstein condensation F.3 Bose–Einstein condensed systems

346 346 348 350

Solutions and hints to the exercises

352

Bibliography

360

Index

369

List of symbols The alphabet only contains 26 letters, and the use of the same symbol to represent different quantities is unavoidable in a book of this length. Whenever this occurs, it should be obvious from the context which meaning is intended. a ˆ a ˆ† a
a a0 A Aij
b B Bij B′ ci C CV d dij D D Dp E Eg EX E E0 f f (T ) fij F F FFano FP g g(E) ...