Lectures on Light: Nonlinear and Quantum Optics Using the Density
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Lectures on Light This page intentionally left blank Lectures on Light Nonlinear and Quantum Optics using the Density Matrix Stephen C. Rand University of Michigan 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 Stephen C. Rand 2010 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2010 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 Control Number: 2009942568 Typeset by SPI Publisher Services, Pondicherry, India Printed in Great Britain on acid-free paper by the MPG Books Group, Bodmin and King’s Lynn ISBN 978–0–19–957487–2 13579108642 This book owes its existence to the abiding support of my wife Paula, the soaring spirit of my mother Margaret, and the wisdom of my father Charles Gordon Rand. This page intentionally left blank Contents Preface xi List of Abbreviations xv 1 Basic Classical Concepts 1 1.1 Introduction 1 1.2 Electric and magnetic interactions 3 1.2.1 Classical electromagnetism 3 1.2.2 Maxwell’s equations 3 1.2.3 The wave equation 4 1.2.4 Absorption and dispersion 5 1.2.5 Resonant response 6 1.2.6 The vectorial character of light 7 Supplementary reading 9 2 Basic Quantum Mechanics 10 2.1 Particles and waves 10 2.2 Quantum observables 12 2.2.1 Calculation of quantum observables 12 2.2.2 Time development 13 2.2.3 Symmetry 14 2.2.4 Examples of simple quantum systems 16 2.3 Dynamics in two-level systems 22 2.4 Representations 25 2.4.1 Representations of vector states and operators 25 2.4.2 Equations of motion in different representations 26 2.4.3 Matrix representations of operators 30 2.4.4 Changing representations 32 Problems 35 3 Atom–Field Interactions 40 3.1 The interaction Hamiltonian 40 3.2 Perturbation theory 41 3.3 Exact analysis 46 3.4 Preliminary consideration of AC Stark or Rabi splitting 48 3.5 Transition rates 49 3.6 The density matrix 52 3.6.1 Electric dipole transition moments 52 3.6.2 Pure case density matrix 53 3.6.3 Mixed case density matrix 54 3.7 Decay phenomena 56 viii Contents 3.8 Bloch equations 58 3.9 Inhomogeneous broadening, polarization, and signal fields 63 3.10 Homogeneous line-broadening through relaxation 65 3.11 Two-level atoms versus real atoms 67 Problems 70 4 Transient Optical Response 76 4.1 Optical nutation 76 4.1.1 Optical nutation without damping 76 4.1.2 Optical nutation with damping 79 4.2 Free induction decay 80 4.3 Photon echoes 84 4.3.1 Algebraic echo analysis 85 4.3.2 Rotation matrix analysis 90 4.3.3 Density matrix operator analysis 91 Problems 96 5 Coherent Interactions of Fields and Atoms 101 5.1 Stationary atoms 101 5.1.1 Stationary two-level atoms in a traveling wave 101 5.1.2 Stationary three-level atoms in a traveling wave 104 5.1.3 Stationary two-level atoms in a standing wave 106 5.2 Moving atoms 109 5.2.1 Moving atoms in a traveling wave 109 5.2.2 Moving atoms in a standing wave 113 5.3 Tri-level coherence 118 5.3.1 Two-photon coherence 118 5.3.2 Zeeman coherence 120 5.4 Coherent multiple field interactions 126 5.4.1 Four-wave mixing 126 5.4.2 Pump-probe experiments 130 5.4.3 Higher-order interactions and Feynman diagrams 135 Problems 139 6 Quantized Fields and Coherent States 145 6.1 Quantization of the electromagnetic field 145 6.2 Spontaneous emission 154 6.3 Weisskopf–Wigner theory 157 6.4 Coherent states 160 6.5 Statistics 168 6.5.1 Classical statistics of light 168 6.5.2 Quantum statistics of light 172 6.6 Quantized reservoir theory 175 6.6.1 The reduced density matrix 175 6.6.2 Application of the reduced density matrix 179 Contents ix 6.7 Resonance fluorescence 183 6.7.1 Fluorescence of strongly driven atoms 183 6.7.2 Coherence of strongly driven two-level atoms 191 6.8 Dressed atoms 193 6.8.1 Strong coupling of atoms to the electromagnetic field 193 6.8.2 Dressed state population dynamics 197 Problems 200 7 Selected Topics and Applications 210 7.1 Mechanical effects of light and laser cooling 210 7.1.1 Radiation pressure, dipole forces, and “optical tweezers” 210 7.1.2 Laser cooling via the Doppler shift 212 7.1.3 Magneto-optic trapping 215 7.1.4 Laser cooling below the Doppler limit 216 7.2 Dark states and population trapping 222 7.2.1 Velocity-selective coherent population trapping 222 7.2.2 Laser cooling via VSCPT 226 7.3 Coherent population transfer 228 7.4 Coherent transverse optical magnetism 233 7.5 Electromagnetically induced transparency 241 7.6 Squeezed light 245 7.7 Cavity quantum electrodynamics (QED) 249 7.7.1 Damping of an optical field by two-level atoms 249 7.7.2 Weak coupling regime 251 7.7.3 Strong coupling regime 254 Problems 257 Appendices 263 A Expectation Values 263 B The Heisenberg Uncertainty Principle 264 C The Classical Hamiltonian of Electromagnetic Interactions 266 D Stationary and Time-dependent Perturbation Theory 268 E Second Quantization of Fermions 274 F Frequency Shifts and Decay due to Reservoir Coupling 278 G Solving for Off-diagonal Density Matrix Elements 281 H Irreducible Spherical Tensor Operators and Wigner–Eckart (W–E) Theorem 283 I Derivation of Effective Hamiltonians 294 Index 296 This page intentionally left blank Preface This book, and the course from which it sprang, attempts to bridge the enormous gap between introductory quantum mechanics and the research front of modern optics and other fields that make use of light. This would be an impossibly daunting task, were it not for the fact that most of us love to hear things again and again that we already know. Taking that into account, this book uses a single approach repeatedly to tackle progressively more exciting topics in the science of light, moving systematically and swiftly from very basic concepts to sophisticated topics. The reader should be aware from the outset that the approach taken here is unconventional. It is highly selective instead of encyclopedic, and it teaches the reader only how to use the density matrix on new problems. Nowadays, scientific answers are being sought to an ever-expanding array of problems across numerous disciplines. The trend in textbooks on quantum optics is understandably to cover an increasing number of topics comprehensively, and to familiarize students with an ever- widening array of analytic tools, exhaustively. There are many fine texts that fulfill this function, but this book is not one of them. Instead, important topics and alternative methods of analysis have been omitted here to keep it as brief as possible, with a single-minded pedagogical purpose in mind. The main objective of the book is to provide students and researchers with one reliable tool, and the confidence that comes from practice, to analyze new optical phenomena in their chosen field successfully and rigorously. Thus, the one and only analytic tool developed here for attacking research-level problems in optical science is the density matrix. A systematic procedure is applied to representative problems of “critical subjects” – usually only one example each – to show how virtually any problem can be analyzed with the density matrix. Each succes- sive example adds one system property at a time, with the result that one qualitatively new feature appears in the dynamics each time. By using a systematic “building- up” principle to approach complicated interactions, students begin to recognize what terms in the analysis are associated with particular changes in the dynamics. Following two slow-paced introductory chapters on review material, the text shifts focus to the development of insights as to when atomic motion, or multilevel structure, or coherence effects dominate the behavior of complicated systems. It ends with fast-paced coverage of selected topics and applications. The organization of the book follows the original sequence of lectures on light pre- pared for applied physics graduates with typical undergraduate physics, chemistry, and engineering preparation, expected to handle interdisciplinary research topics during their careers. Present-day graduate students who use light often face important prob- lems that no longer fall into the neat categories of unique models from the early days of quantum mechanics, and require broad perspective and reliable mathematical tools to handle new cross-disciplinary topics quickly.