BOSE-CONDENSED GASES AT FINITE TEMPERATURES The discovery of Bose Einstein condensation (BEC) in trapped ultracold atomic gases in 1995 has led to an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first treatment of BEC at finite temperatures, this book presents a thorough account of the theory of two-component dynamics and nonequilibrium behaviour in superfluid Bose gases. It uses a simplified microscopic model to give a clear, explicit account of collective modes in both the collisionless and collision-dominated regions. Major topics such as kinetic equations, local equilibrium and two-fluid hydrodynamics are introduced at an elementary level, before more detailed treatments and microscopic derivations of the underlying equations are given. Explicit predictions are worked out and linked to experiments. Providing a platform for future experimental and theoretical studies on the finite temperature dynamics of trapped Bose gases, this book is ideal for researchers and graduate students in ultracold atom physics, atomic, molecular and optical physics, and condensed matter physics. Allan Griffin is Professor Emeritus of Physics at the University of Toronto, Canada. His research has been on superfluid helium, superconductivity and the theory of ultracold matter and quantum gases. He is co-editor of Bose Einstein Condensation, and the author of Excitations in a Bose-condensed Liquid (1996, 2005, both from Cambridge University Press). Tetsuro Nikuni is Associate Professor at the Tokyo University of Science, Japan. His research has been on the theory of quantum antiferromagnets and more recently on BEC at finite temperatures and in optical lattices. Eugene Zaremba is Professor of Physics at Queen s University, Canada. He has a wide range of interests in theoretical condensed matter physics, including surface physics, density functional theory of electronic structure, and mesoscopic physics. His current research is on the theory of ultracold matter.
BOSE-CONDENSED GASES AT FINITE TEMPERATURES ALLAN GRIFFIN University of Toronto, Canada TETSURO NIKUNI Tokyo University of Science, Japan EUGENE ZAREMBA Queen s University, Canada
cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521837026 c A. Griffin, T. Nikuni and E. Zaremba 2009 This publication 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 2009 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN 978-0-521-83702-6 hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents Preface page ix 1 Overview and introduction 1 1.1 Historical overview of Bose superfluids 9 1.2 Summary of chapters 12 2 Condensate dynamics at T =0 19 2.1 Gross Pitaevskii (GP) equation 20 2.2 Bogoliubov equations for condensate fluctuations 28 3 Coupled equations for the condensate and thermal cloud 32 3.1 Generalized GP equation for the condensate 33 3.2 Boltzmann equation for the noncondensate atoms 39 3.3 Solutions in thermal equilibrium 43 3.4 Region of validity of the ZNG equations 46 4 Green s functions and self-energy approximations 54 4.1 Overview of Green s function approach 54 4.2 Nonequilibrium Green s functions in normal systems 58 4.3 Green s functions in a Bose-condensed gas 68 4.4 Classification of self-energy approximations 74 4.5 Dielectric formalism 79 5 The Beliaev and the time-dependent HFB approximations 81 5.1 Hartree Fock Bogoliubov self-energies 82 5.2 Beliaev self-energy approximation 87 5.3 Beliaev as time-dependent HFB 92 5.4 Density response in the Beliaev Popov approximation 98 v
vi Contents 6 Kadanoff Baym derivation of the ZNG equations 107 6.1 Kadanoff Baym formalism for Bose superfluids 108 6.2 Hartree Fock Bogoliubov equations 111 6.3 Derivation of a kinetic equation with collisions 115 6.4 Collision integrals in the Hartree Fock approximation 119 6.5 Generalized GP equation 122 6.6 Linearized collision integrals in collisionless theories 124 7 Kinetic equation for Bogoliubov thermal excitations 129 7.1 Generalized kinetic equation 130 7.2 Kinetic equation in the Bogoliubov Popov approximation 135 7.3 Comments on improved theory 143 8 Static thermal cloud approximation 146 8.1 Condensate collective modes at finite temperatures 147 8.2 Phenomenological GP equations with dissipation 157 8.3 Relation to Pitaevskii s theory of superfluid relaxation 160 9 Vortices and vortex lattices at finite temperatures 164 9.1 Rotating frames of reference: classical treatment 165 9.2 Rotating frames of reference: quantum treatment 170 9.3 Transformation of the kinetic equation 174 9.4 Zaremba Nikuni Griffin equations in a rotating frame 176 9.5 Stationary states 179 9.6 Stationary vortex states at zero temperature 181 9.7 Equilibrium vortex state at finite temperatures 184 9.8 Nonequilibrium vortex states 187 10 Dynamics at finite temperatures using the moment method 198 10.1 Bose gas above T BEC 199 10.2 Scissors oscillations in a two-component superfluid 204 10.3 The moment of inertia and superfluid response 220 11 Numerical simulation of the ZNG equations 227 11.1 The generalized Gross Pitaevskii equation 228 11.2 Collisionless particle evolution 231 11.3 Collisions 237 11.4 Self-consistent equilibrium properties 248 11.5 Equilibrium collision rates 252
Contents vii 12 Simulation of collective modes at finite temperature 256 12.1 Equilibration 257 12.2 Dipole oscillations 260 12.3 Radial breathing mode 263 12.4 Scissors mode oscillations 270 12.5 Quadrupole collective modes 279 12.6 Transverse breathing mode 286 13 Landau damping in trapped Bose-condensed gases 292 13.1 Landau damping in a uniform Bose gas 293 13.2 Landau damping in a trapped Bose gas 298 13.3 Numerical results for Landau damping 303 14 Landau s theory of superfluidity 309 14.1 History of two-fluid equations 309 14.2 First and second sound 312 14.3 Dynamic structure factor in the two-fluid region 317 15 Two-fluid hydrodynamics in a dilute Bose gas 322 15.1 Equations of motion for local equilibrium 324 15.2 Equivalence to the Landau two-fluid equations 331 15.3 First and second sound in a Bose-condensed gas 339 15.4 Hydrodynamic modes in a trapped normal Bose gas 345 16 Variational formulation of the Landau two-fluid equations 349 16.1 Zilsel s variational formulation 350 16.2 The action integral for two-fluid hydrodynamics 356 16.3 Hydrodynamic modes in a trapped gas 359 16.4 Two-fluid modes in the BCS BEC crossover at unitarity 370 17 The Landau Khalatnikov two-fluid equations 371 17.1 The Chapman Enskog solution of the kinetic equation 372 17.2 Deviation from local equilibrium 377 17.3 Equivalence to Landau Khalatnikov two-fluid equations 387 17.4 The C 12 collisions and the second viscosity coefficients 392 18 Transport coefficients and relaxation times 395 18.1 Transport coefficients in trapped Bose gases 396 18.2 Relaxation times for the approach to local equilibrium 405 18.3 Kinetic equations versus Kubo formulas 412
viii Contents 19 General theory of damping of hydrodynamic modes 414 19.1 Review of coupled equations for hydrodynamic modes 415 19.2 Normal mode frequencies 418 19.3 General expression for damping of hydrodynamic modes 420 19.4 Hydrodynamic damping in a normal Bose gas 424 19.5 Hydrodynamic damping in a superfluid Bose gas 428 Appendix A Monte Carlo calculation of collision rates 431 Appendix B Evaluation of transport coefficients: technical details 436 Appendix C Frequency-dependent transport coefficients 444 Appendix D Derivation of hydrodynamic damping formula 448 References 451 Index 459
Preface Since the creation of Bose Einstein condensation (BEC) in trapped atomic gases in 1995, there has been an enormous amount of research on ultracold quantum gases. However, most theoretical studies have ignored the dynamical effect of the thermally excited atoms. In this book, we try to give a clear development of the key ideas and theoretical techniques needed to deal with the dynamics and nonequilibrium behaviour of trapped Bose gases at finite temperatures. By limiting ourselves from the beginning to a relatively simple microscopic model, we can concentrate on the new physics which arises when dealing with the correlated motions of both the condensate and noncondensate degrees of freedom. This book also sets the stage for the future generalizations that will be needed to understand the coupled dynamics of the superfluid and normal fluid components in strongly interacting Bose gases, where there is significant depletion of the condensate even at T =0. The core of this book is based on a long paper published by the authors (Zaremba, Nikuni and Griffin, 1999). In the last decade, together with our coworkers, we have extended and applied this work in many additional papers. The starting point of our approach is not original, in that it consists of combining the Gross Pitaevskii equation for the condensate with a Boltzmann equation for the noncondensate atoms. The kinetic equation for trapped superfluid Bose gases we use was first developed and studied in a pioneering series of papers by Kirkpatrick and Dorfman in 1985 on a uniform Bose gas at finite temperatures. In the initial phase of writing this book, we intended to refer to our coupled equations for the condensate and the thermal cloud by some name other than ZNG, but nothing we came up with was particularly descriptive or natural and we finally gave up the attempt. Therefore, we simply refer to them as the ZNG coupled equations, following the custom in the current literature of the last few years. In the introductory chapter we discuss in more detail how our starting ix
x Preface equations and general approach for dealing with a Bose-condensed gas at finite temperature are built on the work of previous theorists over the last five decades. On a longer time scale, our book makes use of methods for solving kinetic equations for Bose gases that were originally developed for classical gases by Boltzmann, Enskog and Chapman over a century ago. This book should be of interest to all graduate students and researchers working in the area of ultracold dilute quantum gases and BEC. We hope that it will be particularly useful to experimental researchers, who may find the original theoretical literature difficult to understand, not to mention widely dispersed among many separate papers. It should also appeal to those in the wider theoretical and mathematical physics community who are interested in nonequilibrium problems in general, in quantum transport theory, in many body theory or in quantum field theory at finite temperatures. We have made no attempt to repeat the material on BEC in trapped atomic gases discussed in the two excellent books by Pethick and Smith (2008) and by Pitaevskii and Stringari (2003), and we hope these books will be referred to as needed by the reader. The book includes a detailed treatment of the collision-dominated twofluid region. In a graduate course on ultracold atoms, an introduction to the two-fluid hydrodynamics in trapped superfluid gases could be based on Chapters 2, 3, 14 and 15 as well as Section 8.1 of Chapter 8. Originally our plan was to discuss the connections with other theoretical approaches used to deal with trapped Bose gases at finite temperatures. However, once we found that a clear exposition and development of our approach and the physics underlying it already filled 450 pages, our plan to review other formalisms in any depth was not feasible. We give references to other approaches at various places in the book. In addition, excellent reviews given at a recent workshop are available online (see footnote 2 in Chapter 1). Allan Griffin would like to acknowledge the work of his students on the theory of interacting Bose gases, both before and after the discovery of BEC in 1995. In particular, Edward Taylor has been helpful in editing Chapters 14 and 16. Tetsuro Nikuni thanks Satoru Konabe and Takashi Inoue for valuable discussions and useful suggestions. We thank our colleague Alexander (Sandy) Fetter for a critical reading of Chapter 9 on vortices, and our postdoctoral associates Brian Jackson and Jamie Williams for their important contributions to some of the topics covered in this book. We specifically acknowledge Brian Jackson for his suggestions on Chapters 11 and 12 and for a careful reading of the earlier chapters.
Preface xi We owe special thanks to Helen Iyer at Toronto for her work over several years in preparing this book for publication. Helen did much of the manuscript preparation using the Cambridge LaTex macro and also coordinated the endless changes to various chapter files on our book s webpage. We are extremely grateful to Edward Taylor for his crucial assistance in checking and revising the tex file and figures, as well as preparing the index. We thank Hui Hu for help with preparing the cover illustration. The English in several chapters has been greatly improved by the editorial work of Christine McClymont. It is a pleasure to thank Simon Capelin at Cambridge University Press for his continuing support over many years. We still are amazed by the care Susan Parkinson took in copy-editing our difficult manuscript. Susan s detailed thoughtful analysis of every line of text, equation and figure eliminated many errors and ambiguities; this has led to a much more readable book. The research on ultracold quantum gases of Allan Griffin and Eugene Zaremba has been supported by grants from NSERC of Canada over the last decade. Tetsuro Nikuni has been supported by grants from JSPS of Japan.