Modern Foundations of tatwn iptics www.com Modern Foundations of Oudntum Dptics by VLATKO VEDRAL University of Leeds, UK iMt Imperial College Press www.com Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H9HE Distributed by World Scientific Publishing Co. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. MODERN FOUNDATIONS OF QUANTUM OPTICS Copyright © 2005 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 1-86094-531-7 1-86094-553-8 (pbk) Printed in Singapore by World Scientific Printers (S) Pte Ltd www.com Dedicated to Ivona and Michael www.com Preface This book represents the lecture notes for the course I gave at the Imperial College London for three years in a row between 2001 and 2004. I have edited the notes to make them more suitable for pub- lication, but at the same time I have tried to change as little as possible in order to stay close to the spirit and style of the lectures which were an optional course for third and fourth year physics un- dergraduate studies.
The course consisted of 26 lectures and three extra special topic lectures. The extra topics were intended to cover very recent advances in and applications of quantum optics. I fo- cused on experiments on Rabi oscillations in cavity QED, on the achievement of atomic Bose—Einstein condensation and on quan- tum teleportation. These recent advancements — some of which have resulted in several recent Nobel prizes — show that quantum optics is a very exciting and important subject to learn.
The reader will see that in addition to the modern application, I have tried to present many topics in an original way, always keeping in mind modern developments and understanding. Of course, there are many standard derivations in my notes that can also be found in many other textbooks, some of them covered in much more detail in these other books. I pretend neither to have written a detailed nor a complete exposition of the subject. The choice of topics reflects very much my personal bias, my research interests and preferences.
For example, I discuss the topic of Maxwell's demon and how the wave and particle nature of light can possibly be used to violate the second law of thermodynamics. I also discuss the notion of phase in quantum mechanics, the difference between dynamical and geometrical phases, as well as some very basic ideas behind the gauge principle and how electromagnetism can be derived from the Schrodinger equation. These additional topics, not traditionally covered by conventional texts, were intended to show that quantum optics is not an isolated subject, but that it is very intimately vii www.com viii Modern Foundations of Quantum Optics related to other areas of physics. They were also intended to break the monotony of the routine of only going through the, frequently tedious, background material.
I wanted to show my students how exciting and lively the subject can be even at this introductory level, and that they can actively participate in it from the very start. The order in which the notes are written is sometimes histori- cal, sometimes didactic, frequently neither. More frequently than not they are written in the order of increasing complexity — which does not always coincide with the historical development. The logic of the course was to present different levels of our understanding of light — and quantum optics is the most sophisticated such under- standing we have — through its interaction with matter.
Loosely speaking, there are four levels in the notes: the classical, the old quantum, the semi-classical and the fully quantum level. I moti- vate some of the more traditional topics with examples that are both technologically and conceptually challenging. For example, I introduce the Mach-Zehnder interferometer with single photons at the very start to show not only that photons behave like particles and waves at the same time, but also that this can be exploited to perform operations that are unimaginable in classical physics — such as the interaction-free measurement. I have included five sets of problems and solutions.
These are taken mainly from my three exam papers and are meant for the students to test their under- standing of the presented material. Problem solving is, as always, crucial for understanding of any subject. The notes end at the point where the field theory proper should begin. One could say — perhaps somewhat misleadingly — that quantum optics is the lowest order approximation to the full quan- tum field theory.
From my experience in teaching, it seems that learning quantum optics first is a much better way of understanding the field theory than the usual second quantization formalism. Finally, I had great fun working with students at Imperial Col- lege London, who not only taught me the subject, but also taught me how to teach. I hope you enjoy reading the notes as much as I enjoyed teaching the course! V.com Acknowledgements I would like to thank Artur Ekert for initially encouraging me to publish this book and for being supportive during the key stages of the publication process. The support and encouragement of Imperial College Press, especially Laurent Chaminade, is gratefully acknowledged.
I would like to thank all the third and fourth year students at Imperial College London between the years 2001 and 2004 for correcting many "typoes" and improving my notes a great deal by telling me what points need to be clarified. In particular thanks to William Irvine (now at Santa Barbara) for reading and revising a very early version of my notes (back in 2000). I am also grateful to Luke Rallan for his help with a very early version of the book. I acknowledge Peter Knight, who proposed the first course on Quantum Optics at Imperial College London and whose syllabus I have modified only a bit here and there when I taught it myself.
Very special thanks goes to Caroline Rogers for preparing the manuscript for the final submission to Imperial College Press. She has redrawn many of the figures, as well as corrected and clarified some parts of the book. Her hard work was essential for the final preparation, which otherwise may have taken a much longer time to complete. My deepest gratitude goes to my family, Ivona and Michael, who provide a constant source of inspiration and joy.com Contents Preface vii Acknowledgements ix 1.
Prom Geometry to the Quantum 1 2. Introduction to Lasers 13 2.1 Normal Modes in a Cavity 14 2.2 Basic Properties of Lasers 17 3. Properties of Light: Blackbody Radiation 19 3.1 Planck’s Quantum Derivation 20 3.2 The Proper Derivation of Planck’s Formula 24 3.3 Fluctuations of Light 26 3. Interaction of Light with Matter I 37 4.1 Stimulated and Spontaneous Emission 39 4.2 Optical Excitation of Two Level Atoms 41 4.3 Life-Time and Amplification 43 5.
Basic Optical Processes — Still Classical 45 5.1 Interference and Coherence 45 5.4 Amplification: Three Level Systems 53 5.5 Classical Treatment of Atom-Light Interaction .2 Radiation damping 57 xi www.com xii Modern Foundations of Quantum Optics 5. More Detailed Principles of Laser 63 6.1 Basic Theory: Classical Electrodynamics 63 6.3 Non-linear Optics 70 6. Interactions of Light with Matter II 81 7.3 Time Dependent Perturbation Theory 87 7.4 Alternative Derivation of Perturbation 92 7.5 The Wigner–Weisskopf Theory 94 7.6 Digression: Entropy and the Second Law 97 7.8 Multiphoton Processes Revisited 102 8. Two Level Systems 105 8.1 Operator Matrix Algebra 105 8.2 Two Level Systems: Rabi Model 107 8.3 Other Issues with Two Level Systems 114 8.4 The Berry Phase 116 8.2 The Bloch sphere 119 8.4 Generalization of the phase 124 8.1 Quantum Harmonic Oscillator 133 9.2 What Are Photons? 137 9.3 Blackbody Spectrum from Photons 139 9.4 Quantum Fluctuations and Zero Point Energy .6 Composite Systems — Tensor Product Spaces .1 Beam splitters 147 www.com Contents xiii 9.2 Generation of coherent states 150 9.7 Bosonic Nature of Light 151 9.8 Polarization: The Quantum Description 153 9.1 Unpolarized light — mixed states 154 10.
Interaction of Light with Matter III 157 10.1 Fully Quantized Treatment 157 10.2 Jaynes–Cummings Model 158 10.3 Spontaneous Emission — At Last 163 10.4 The Lamb Shift 164 10.5 Parametric Down Conversion 166 10.6 Quantum Measurement: A Brief Discussion 167 11. Some Recent Applications of Quantum Optics 171 11.1 Bose–Einstein condensation 173 11.2 Quantum Information Processing 176 11. Problems and Solutions 183 13.1 Problem and Solutions 1 183 13.2 Problem and Solutions 2 190 13.3 Problems and Solutions 3 197 13.4 Problems and Solutions 4 203 13.5 Problems and Solutions 5 210 13.2 Solutions 5 212 Bibliography 217 Index 219 www.com Chapter 1 From Geometry to the Quantum According to one legend, Lucifer was God's favorite angel before stealing light from him and bringing it to mankind. For this, to us a generous act, Lucifer was expelled from heaven and subsequently became the top angel in hell.
Most of us are not able to steal pos- sessions from God, but we can at least admire his most marvellous creation — light. Quantum optics is the theory describing our most sophisticated understanding of light. This book intends to acquaint you with the basic ideas of how physics describes the interaction of light and matter at three dif- ferent levels: classical, semi-classical and quantum. You will be able to understand basic principles of laser operation leading to the ideas behind non-linear optics and multiphoton physics.
You will also become familiar with the ideas of field quantization (not only the electromagnetic field, but also a more general one), nature of photons, and quantum fluctuations in light fields. These ideas will bring you to the forefront of current research. At the end of this book, I not only expect you to understand the basic methods in quantum optics, but also to be able to apply them in new situ- ations — this is the key to true understanding. The notes contain five sets of problems, which are intended for your self-study.
Being able to solve problems is definitely crucial for your understanding, and a great number of problems have been chosen from the past exam papers at Imperial College London set by me. I also hope — and this is I believe really very important — that the book will teach you to appreciate the way that science has developed within the last 100 years or so and the importance of the basic ideas in optics in relation to other ideas and concepts in science in general. The book contains a number of topics from thermodynamics, sta- tistical mechanics and information theory that will illustrate that quantum optics is an integral part of a much larger body of scien- tific knowledge. I hope that at the end of it all, and this is really 1 www.com 2 Modern Foundations of Quantum Optics my main motivation, you will appreciate how quantum description of light forms an important part of our cultural heritage.
Optics itself is an ancient subject. Like any other branch of science, its roots can be found in Ancient Greece, and its develop- ment has always been inextricably linked to technological progress. The ancient Greeks had some rudimentary knowledge of geomet- rical optics, and knew of the laws of reflection and refraction, al- though they didn't have the appropriate mathematical formalism (trigonometry) to express these laws concisely.