Introduction to Quantum Optics Hector Manuel Moya-Cessa Francisco Soto-Eguibar Rinton Press, Inc. a Isabel y Leonardo a Conchis, Sofi y Quique © 2011 Rinton Press, Inc. 565 Edmund Terrace Paramus, New Jersey 07652, USA [email protected] http://www.rintonpress.com All right reserved. No part of this book covered by the copyright hereon may be reproduced or used in any form or by any means-graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval system - without permission of the publisher. Published by Rinton Press, Inc. Printed in the United States of America ISBN 978-1-58949-061-1 Preface Quantum optics is a topic that has recently acquired a great deal of attention, not only for its theoretical and experimental contributions to the understanding of the quantum world, but also for the perspectives of its use in many sophisticated applications. Among them, quantum optical devices are particularly promising tools for quantum information processing applications. The development of a quantum optics research group at the Instituto N a­ cional de Astrofisica, Optica y Electr6nica, INAOE, (National Institute of As­ trophysics, Optics and Electronics) has led to the creation of several quantum optics courses for graduate students. This book is the result of those courses, and the topics presented here are directly related with research activities undertaken by our group. This book intends to teach graduate and postgraduate students several meth­ ods used in quantum optics. Therefore, it is mainly about doing calculations. Throughout the book we have emphasized the "hows" over the "whys". Field quantization will not be studied in this book, as it has already been reviewed in many quantum optics textbooks. Instead, we will treat the case of the time de­ pendent harmonic oscillator by applying invariant techniques (Lewis-Ermakov). In Chapter 1, the harmonic oscillator states and the needed operator algebra are introduced. In Chapter 2, the quasiprobability phase space distributions are reviewed, along with their properties and relations. In Chapter 3, we deal with the time dependent harmonic oscillator, which takes us to the notion of squeezed states. The very important subject of the interaction of light and atoms is the central issue of Chapter 4. The Master Equation (ME) for a real cavity is treated in Chapter 5, where it is solved for a lossy cavity at zero temperature with different initial conditions by means of superoperator techniques. In Chap­ ter 6, the pure states and the statistical mixtures are analyzed by reviewing the concepts of entropy and purity. The reconstruction of quasiprobability distribu- Preface tions in phase space is addressed in Chapter 7 and applied to the measurement of field properties. Here it is used an "inverse" spectroscopic approach: instead of passing light through matter in order to find out about the quantum nature of the matter, we pass matter (two-level atoms) through quantized light, learning Contents about the quantum structure of the field inside a cavity. In Chapter 8, we study the ion-laser interaction, reviewing the type of traps that exist and analyzing the motion of an ion in a Paul trap. The time independent and time dependent trap frequency cases are also studied. In this chapter, the most complete solu­ tion for this system is given, as we, besides treating the low intensity regime, also approach the medium and high intensity regimes. Finally, in Chapter 9, we introduce the Suskind-Glogower nonlinear coherent states; we use two meth­ ods to construct them and we analyze their properties by means of the photon distribution number, the Mandel parameter and the Husimi function. vii Several appendices are added, which we believe enrich the contents of the Preface book. Particularly important is Appendix A, where we study the Master Equa­ Chapter 1 Operator algebra and the harmonic oscillator tion, describing phase sensitive processes for a cavity filled with a Kerr medium. 1 1.1 Introduction . Being the J aynes-Cummings model such an important tool in quantum optics, 4 1.2 von Neumann equation .. · · · · · · we provide different methods for solving it in Appendix B. The interaction of 4 1.3 Baker-Hausdorff formula . · . · · many fields is treated in Appendix C. Appendix D proposes a quantum phase 6 1.4 Quantum mechanical harmonic oscillator formalism, while Appendix E studies a series of Bessel functions, needed to de­ 7 1.4 .1 Ladder operators termine the non-classical character of non-linear coherent states described in 8 1.4.2 Fock states . Chapter 9. 10 1.4.3 Coherent states . We would like to thank the many people who have been essential for the ex­ 14 1.4.4 Displaced number states . istence of this book, especially several generations of students. Finally, we want 14 1.4.5 Phase states to thank our families for the understanding and the patience they have given to 16 1.5 Ordering of ladder operators us during the time we dedicated to the writing of this book. 17 1.5.1 Normal ordering ... 19 1.5.1.1 Lemma 1 .. Hector Manuel Moya Cessa y Francisco Soto Eguibar 20 Santa Maria Tonantzintla, Puebla, Mexico. June 15, 2011. 1.5.2 Anti-normal ordering 20 1.5.2.1 Lemma 2 . 21 1.5.3 Coherent states. 21 1.5.4 Fock states .... 23 Chapter 2 Quasiprobability distribution functions 23 2.1 Introduction ........ · . · · · · · · · 23 2.2 Wigner function ..... · . · · 28 2.2.1 Properties of the Wigner function .. · · · · · · · ·. · · 28 2.2.2 Obtaining expectation values from the Wigner functwn 29 2.2.3 Symmetric averages ... · · · · · · · · · · · 30 2.2.4 Series representation of the Wigner function . Contents Contents 2.3 Glauber-Sudarshan P-function . 32 88 2.4 Husimi Q-function . 33 6.4.2 Atomic entropy operator . · · · 2.5 Relations between quasiprobabilities 34 6.4.3 Field entropy operator 89 2.5.1 Differential forms . 34 6.4.4 Entropy operator from orthonormal states . 91 2.5.2 Integral forms . 37 6.5 Entropy of the damped oscillator: Cat states ... 93 2.6 The Wigner function as a tool to calculate divergent (or not) series . 37 Chapter 7 Reconstruction of quasiprobability distribution func- 2.7 Number-phase Wigner function . 39 tions 97 2.7.1 Coherent state . 40 7.1 Reconstruction in an ideal cavity . 97 2. 7.2 A special superposition of number states . 41 7.1.1 Direct measurement of the Wigner function 97 7.1. 2 Fresnel approach . 98 Chapter 3 Time Dependent Harmonic Oscillator 43 3.1 Time dependent harmonic Hamiltonian 43 7.2 Reconstruction in a lossy cavity . 99 Quasiprobabilities and losses 102 3.1.1 Minimum uncertainty states. 46 7.3 3.1.2 Step function . 46 7.4 Measuring field properties 105 3.2 More states of the field . 49 7.4.1 Squeezing .... 105 7.4.2 Phase properties 3.2.1 Squeezed states . 50 107 3.2.2 Schrodinger cat states 54 Chapter 8 Ion-laser interaction 109 3.2.3 Thermal distribution . 57 8.1 Paul trap ................... 111 8.1.1 The quadrupolar potential of the trap 111 Chapter 4 (Two-level) Atom-field interaction 59 4.1 Semiclassical interaction 59 8.1.2 Oscillating potential of the trap ... 114 4.2 Quantum interaction . 62 8.1.3 Motion in the Paul trap ....... 115 8.1.4 Approximated solution to the Mathieu equation 116 4.2.1 Atomic inversion 63 120 4.3 Dispersive interaction . 65 8.2 Ion-laser interaction in a trap with a frequency independent of time 8.2.1 Interaction out of resonance and low intensity ... 121 4.4 Mixing classical and quantum interactions 66 8.3 Ion-laser interaction in a trap with a frequency dependent of time 126 4.5 Slow atom interacting with a quantized field . 69 8.3.1 Linearization of the system ....... 128 Chapter 5 A real cavity: Master equation 73 8.4 Adding vibrational quanta . 129 5.1 Cavity losses at zero temperature . 73 8.5 Filtering specific superpositions of number states 133 5.1.1 Coherent states . 75 Chapter 9 Nonlinear coherent states for the Susskind-Glogower 5.1.2 Number states . 75 operators 137 5.1.3 Cat states . 76 9.1 Approximated displacement operator ..... 139 5.2 Master equation at finite temperature 77 9.2 Exact solution for the displacement operator 140 Chapter 6 Pure states and statistical mixtures 81 9.3 Susskind-Glogower coherent states analysis 142 6.1 Entropy . 81 9.3.1 The Husimi Q-function ... 143 6.2 Purity . 82 9.3.2 Photon number distribution . 143 6.3 Entropy and purity in the atom-field interaction 83 9.3.3 Mandel Q-parameter ..... 145 6.4 Some properties of reduced density matrices . 84 9.4 Eigenfunctions of the Susskind-Glogower Hamiltonian 147 1 9.4.1 Solution for IO) as initial condition . 148 6.4.1 Proving p'JJ+ = TrA{fJ(t)pA(t)} by induction 87 9.4.2 Solution for lm) as initial condition 151 Contents 9.5 Time-dependent Susskind-Glogower coherent states analysis 154 9.5.1 Q function ......... 155 Chapter 1 9.5.2 Photon number distribution . 155 9.5.3 Mandel Q-parameter . 155 Operator algebra and the harmonic 9.6 Classical quantum analogies .. 160 oscillator Appendix A Master equation 163 A.1 Kerr medium ......... 163 A.2 Master equation describing phase sensitive processes 164 Appendix B Methods to solve the Jaynes-Cummings model 167 B.1 A naive method ........ 167 B.2 A traditional method ...... : : : : : : : : ..... 168 1.1 Introduction Appendix C Interaction of quantized fields 169 ' C.1 Two fields interacting: beam splitters 169 In this chapter, we revise briefly the Dirac notation, some of the algebra that C.2 Generalization ton modes .