Superconductivity, Superfluids and Condensates PDF

OXFORD MASTER SERIES IN CONDENSED MATTER 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 undergr...

OXFORD MASTER SERIES IN CONDENSED MATTER 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 in 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.M. T. Dove: Structure and dynamics: an atomic view of materials 2. J. Singleton: Band theory and electronic properties of solids 3. A. M. Fox: Optical properties of solids 4. S. J. Blundell: Magnetism in condensed matter 5. J. F. Annett: Superconductivity, superfluids, and condensates 6. R. A. L. Jones: Soft condensed matter ATOMIC, OPTICAL, AND LASER PHYSICS 7. C. J. Foot: Atomic physics 8. G. A. Brooker: Modern classical optics 9. S. M. Hooker, C. E. Webb: Laser physics PARTICLE PHYSICS, ASTROPHYSICS, AND co5ividioGy 10. D. H. Perkins: Particle astrophysics 11.T. R Cheng: Relativity and cosmology  A PL5 0,6 ob (7 iiperconductivity, Superfluids, and Condensates JAMES F. ANN ETT Department of Physics University of Bristol OXFORD UNIVERSITY PRESS OXFORD lJNIVERSITY PRESS Great Clarendon Street, Oxford 0X2 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 Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Taipei Tokyo Toronto 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 © Oxford University Press 2004 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2004 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 this same condition on any acquirer A catalogue record for this title is available from the British Library Library of Congress Cataloging in Publication Data (Data available) ISBN 0 19 850755 0 (Hbk) ISBN 0 19 850756 9 (Pbk) 10 9 8 7 6 5 4 3 2 1 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Antony Rowe, Chippenham, Wiltshire Preface Ever since their original discovery nearly 100 years ago, superconductors and superfluids have led to an incredible number of unexpected and surprising new phenomena. The theories which eventually explained superconductivity in met- als and superfluid 4He count among the greatest achievements in theoretical many-body physics, and have had profound implications in many other areas, such as in the construction of the "Higgs mechanism" and the standard model of particle physics. Even now there is no sign that the pace of progress is slowing down. Indeed recent years have seen renewed interest in the field in following the 1986 discov- ery of cuprate high temperature superconductivity and the 1995 announcement of Bose—Einstein condensation (BEC) in ultra-cold atomic gases. These break- throughs have tremendously widened the scope of the area of "low temperature physics" from 165 K (only about —100°C, a cold day at the North Pole) the highest confirmed superconducting transition temperature ever recorded, to the realm of nano-Kelvin in laser trapped condensates of atomic gases. Further- more an incredibly wide range of materials is now known to be superconducting. The field is no longer confined to the study of the metallic elements and their alloys, but now includes the study of complex oxides, carbon-based materials (such as fullerene C60), organic conductors, rare earth based compounds (heavy fermion materials), and materials based on sulphur and boron (MgB 2 supercon- ductivity was discovered in 2001). Commercial applications of superconducting technology are also increasing, albeit slowly. The LHC ring currently (in 2003) being installed at the CERN particle physics center is possible only because of considerable recent advances in superconducting magnet technology. But even this uses "traditional" superconducting materials. In principle, even more powerful magnets could be built using novel high temperature superconducting materials, although these materials are difficult to work with and there are many technical problems still to be overcome. The goal of this book is to provide a clear and concise first introduction to this subject. It is primarily intended for use by final year undergraduates and beginning postgraduates, whether in physics, chemistry, or materials sci- ence departments. Hopefully experienced scientists and others will also find it interesting and useful. For the student, the concepts involved in superfluidity and superconductivity can be difficult subject to master. It requires the use of many different elements from thermodynamics, electromagnetism, quantum mechanics, and solid state physics. Theories of superconductivity, such as the Bardeen Cooper Schrieffer (BCS) theory, are also most naturally written in the mathematics of quantum field theory, a subject which is well beyond the usual undergraduate physics curriculum. This book attempts to minimize the use of these advanced math- ematical techniques so as to make the subject more accessible to beginners. vi Preface Of course, those intending to study superconductivity at a more advanced level will need to go on to the more advanced books. But I believe most of the key concepts are fully understandable using standard undergraduate level quan- tum mechanics, statistical physics, and some solid state physics. Among the other books in the Oxford Master Series in Condensed Matter, the vol- umes Band theory and electronic properties of solids by John Singleton (2001), and Magnetism in condensed matter by Stephen Blundell (2001) contain the most relevant background material. This book assumes an initial knowledge of solid state physics at this level, and builds upon this (or equivalent level) foundation. Of course, there are also many other books about superconductivity and superfluids. Indeed each chapter of this book contains suggestions for further reading and references to some of the excellent books and review articles that have been written about superconductivity. However, unlike many of these earl- ier books, this book is not intended to be a fully comprehensive reference, but merely an introduction. Also, by combinining superconductivity, superfluids and BEC within a single text, it is hoped to emphasize the many strong links and similarities between these very different physical systems. Modem topics, such as unconventional superconductivity, are also essential for students studying superconductivity nowadays and are introduced in this book. The basic framework of the earlier chapters derives from lecture courses I have given in Bristol and at a number of summer and winter schools elsewhere over the past few years. The first three chapters introduce the key experimen- tal facts and the basic theoretical framework. First, Chapter 1 introduces BEC and its experimental realization in ultra-cold atomic gases. The next chap- ter introduces superfluid 4He and Chapter 3 discusses the basic phenomena of superconductivity. These chapters can be understood by anyone with a basic understanding of undergraduate solid state physics, quantum mechan- ics, electromagnetism, and thermodynamics. Chapter 4 develops the theory of superconductivity using the phenomenological Ginzburg—Landau theory devel- oped by the Landau school in Moscow during the 1950s. This theory is still very useful today, since it is mathematically elegant and can describe many complex phenomena (such as the Abrikosov vortex lattice) within a simple and powerful framework. The next two chapters introduce the BCS theory of superconductivity. In order to keep the level accessible to undergraduates I have attempted to minimize the use of the mathematical machinery of quantum field theory, although inevitably some key concepts, such as Feynman dia- grams, are necessary. The effort is split into two parts: Chapter 5 introducing the language of coherent states and quantum field operators, while Chapter 6 develops the BCS theory itself. These two chapters should be self-contained so that they are comprehensible whether or not the reader has had prior experi- ence in quantum field theory techniques. The final chapter of the book covers some more specialized, but still very important, topics. The fascinating proper- ties of superfluid 3 He are described in Chapter 7. This chapter also introduces unconventional Cooper pairing and is based on a series of review articles in which I discussed the evidence for or against unconventional pairing in the high temperature superconductors. For a teacher considering this book for an undergraduate or graduate level course, it can be used in many ways depending on the appropriate level for the students. Rather than just starting at Chapter 1 and progressing in linear fashion,  Preface vii one could start at Chapter 3 to concentrate on the superconductivity parts alone. Chapters 3-6 would provide a sound introduction to superconductivity up to the level of the BCS theory. On the other hand, for a graduate level course one could start with Chapters 4 or 5 to get immediately to the many-body physics aspects. Chapter 7 could be considered to be research level or for specialists only, but on the other hand could be read as stand-alone reference by students or researchers wanting to get a quick background knowledge of superfluid 3He or unconventional superconductivity. The book does not attempt to cover comprehensively all areas of modern superconductivity. The more mathematically involved elements of BCS and other theories have been omitted. Several more advanced and comprehensive books exist, which have good coverage at a much more detailed level. To really master the BCS theory fully one should first learn the full language of many-particle quantum field theory. Topics relating to the applications of superconductivity are also only covered briefly in this book, but again there are more specialized books available. Finally, I would like to dedicate this book to my friends, mentors, and col- leagues who, over the years, have shown me how fascinating the world of condensed matter physics can be. These include Roger Haydock, Volker Heine, Richard Martin, Nigel Goldenfeld, Tony Leggett, Balazs GyOrffy, and many others too numerous to mention. James F. Anneal 1 I will be happy to receive any comments University of Bristol, March 2003 and corrections on this book by Email to [email protected] Contents 1 Bose—Einstein condensates 1 1.1 Introduction 1 1.2 Bose—Einstein statistics 2 1.3 Bose—Einstein condensation 6 1.4 BEC in ultra-cold atomic gases 10 Further reading 17 Exercises 18 2 Superfluid helium-4 21 2.1 Introduction 21 2.2 Classical and quantum fluids 22 2.3 The macroscopic wave function 26 2.4 Superfluid properties of He II 27 2.5 Flow quantization and vortices 30 2.6 The momentum distribution 34 2.7 Quasiparticle excitations 38 2.8 Summary 43 Further reading 43 Exercises 44 3 Superconductivity 47 3.1 Introduction 47 3.2 Conduction in metals 47 3.3 Superconducting materials 49 3.4 Zero-resistivity 51 3.5 The Meissner—Ochsenfeld effect 54 3.6 Perfect diamagnetism 55 3.7 Type I and type II superconductivity 57 3.8 The London equation 58 3.9 The London vortex 62 Further reading 64 Exercises 64 4 The Ginzburg—Landau model 67 4.1 Introduction 67 4.2 The condensation energy 67 4.3 Ginzburg—Landau theory of the bulk phase transition 71 x Contents 4.4 Ginzburg-Landau theory of inhomogenous systems 74 4.5 Surfaces of superconductors 76 4.6 Ginzburg-Landau theory in a magnetic field 77 4.7 Gauge symmetry and symmetry breaking 79 4.8 Flux quantization 81 4.9 The Abrikosov flux lattice 83 4.10 Thermal fluctuations 89 4.11 Vortex matter 93 4.12 Summary 94 Further reading 94 Exercises 95 5 The macroscopic coherent state 97 5.1 Introduction 97 5.2 Coherent states 98 5.3 Coherent states and the laser 102 5.4 Bosonic quantum fields 103 5.5 Off-diagonal long ranged order 106 5.6 The weakly interacting Bose gas 108 5.7 Coherence and ODLRO in superconductors 112 5.8 The Josephson effect 116 5.9 Macroscopic quantum coherence 120 5.10 Summary 123 Further reading 124 Exercises 124 6 The BCS theory of superconductivity 127 6.1 Introduction 127 6.2 The electron-phonon interaction 128 6.3 Cooper pairs 131 6.4 The BCS wave function 134 6.5 The mean-field Hamiltonian 136 6.6 The BCS energy gap and quasiparticle states 139 6.7 Predictions of the BCS theory 143 Further reading 145 Exercises 146 7 Superfluid 3 He and unconventional superconductivity 147 7.1 Introduction 147 7.2 The Fermi liquid normal state of 3 He 148 7.3 The pairing interaction in liquid 3 He 152 7.4 Superfluid phases of 3 He 154 7.5 Unconventional superconductors 158 Further reading 166  Contents xi A Solutions and hints to selected exercises 167 A.1 Chapter 1 167 A.2 Chapter 2 169 A.3 Chapter 3 171 A.4 Chapter 4 174 A.5 Chapter 5 176 A.6 Chapter 6 178 Bibliography 181 Index 185 Bose-Einstein condensates 1.1 Introduction 1.1 Introduction 1 Superconductivity, superfluidity, and Bose—Einstein condensation (BEC) 1.2 Bose—Einstein are among the most fascinating phenomena in nature. Their strange and often statistics 2 surprising properties are direct consequences of quantum mechanics. This is 1.3 Bose—Einstein why they only occur at low temperatures, and it is very difficult (but hopefully condensation 6 not impossible!) to find a room temperature superconductor. But, while most 1.4 BEC in ultra-cold atomic other quantum effects only appear in matter on the atomic or subatomic scale, gases 10 superfluids and superconductors show the effects of quantum mechanics acting Further reading 17 on the bulk properties of matter on a large scale. In essence they are macroscopic Exercises 18 quantum phenomena. In this book we shall discuss the three different types of macroscopic quantum states: superconductors, superfluids, and atomic Bose—Einstein condensates. As we shall see, these have a great deal in common with each other and can be described by similar theoretical ideas. The key discoveries have taken place over nearly a hundred years. Table 1.1 lists some of the key discoveries, starting in the early years of the twentieth century and still continuing rapidly today. The field of low temperature physics can be said to have its beginnings in 1908, where helium was first liquified at the laboratory of H. Kammerling Onnes in Leiden, The Netherlands. Very soon afterwards, superconductivity was discovered in the same laboratory. But the theory of superconductivity was not fully developed for until nearly forty years later, with the advent of the Table 1.1 Some of the key discoveries in the history of superconductivity, superfluidity, and BEC 1908 Liquefaction of 4 He at 4.2 K 1911 Superconductivity discovered in Hg at 4.1 K 1925 Bose—Einstein condensation (BEC) predicted 1927 )L. transition found in 4 He at 2.2 K 1933 Meissner—Ochsenfeld effect observed 1938 Demonstration of superfluidity in 4 He 1950 Ginzburg—Landau theory of superconductivity 1957 Bardeen Cooper Schrieffer (BCS) theory 1957 Abrikosov flux lattice 1962 Josephson effect 1963-4 Anderson—Higgs mechanism 1971 Superfluidity found in 3 He at 2.8 mK 1986 High temperature superconductivity discovered, 30-165 K 1995 BEC achieved in atomic gases, 0.5 OK 2 Bose—Einstein condensates i Some offshoots of the development of super- Bardeen Cooper Schrieffer (BCS) theory.' In the case of BEC it was the theory conductivity have had quite unexpected con- that came first, in the 1920s, while BEC was only finally realized experimentally sequences in other fields of physics. The Josephson effect leads to a standard relation- as recently as 1995. ship between voltage, V, and frequency, v: Despite this long history, research in these states of matter is still develop- V = (h/2e) v, where h is Planck's constant ing rapidly, and has been revolutionized with new discoveries in recent years. and e is the electron charge. This provides the most accurate known method of measur- At one extreme we have a gradual progression to systems at lower and lower ing the combination of fundamental constants temperatures. Atomic BEC are now produced and studied at temperatures of hle and is used to determine the best values of nano-Kelvin. On the other hand high temperature superconductors have been these constants. A second surprising discov- ery listed in Table 1.1 is the Anderson—Higgs discovered, which show superconductivity at much higher temperatures than mechanism. Philip Anderson explained the had been previously believed possible. Currently the highest confirmed super- expulsion of magnetic flux from supercon- conducting transition temperature, Tc , at room pressure is about 133 K, in the ductors in terms of spontaneous breaking of gauge symmetry. Applying essentially the compound HgBa2Ca2Cu308 ±8. This transition temperature can be raised to a same idea to elementary particle physics Peter maximum of about 164 K when the material is subjected to high pressures of Higgs was able to explain the origin of mass order 30 GPa, currently the highest confirmed value of T, for any supercon- of elementary particles. The search for the related Higgs boson continues today at large ducting material. Superconductivity at such high temperatures almost certainly accelerators such as CERN and Fermilab. cannot be explained within the normal BCS theory of superconductivity, and the search for a new theory of superconductivity which can explain these remarkable materials is still one of the central unsolved problems of modern physics. This book is organized as follows. In this chapter we start with the simplest of these three macroscopic quantum states, BEC. We shall first review the concept of a BEC, and then see how it was finally possible to realize this state experimentally in ultra-cold atomic gases using the modern techniques of laser cooling and trapping of atoms. The following two chapters introduce the basic phenomena associated with superfluidity and superconductivity. Chapters 4-6 develop the theories of these macroscopic quantum states, leading up to the full BCS theory. The final chapter goes into some more specialized areas: superfluidity in 3 He and superconductors with unconventional Cooper pairing. 1.2 Bose—Einstein statistics In 1924 the Indian physicist S.N. Bose wrote to Einstein describing a new method to derive the Plank black-body radiation formula. At that time Einstein was already world-famous and had just won the Nobel prize for his quan- tum mechanical explanation of the photoelectric effect. Bose was a relatively unknown scientist working in Dacca (now Bangladesh), and his earlier letters to European journals had been ignored. But Einstein was impressed by the 2 For some more historical details see novel ideas in Bose's letter, and helped him to publish the results. 2 The new "The man who chopped up light" (Home and idea was to treat the electromagnetic waves of the black-body as a gas of iden- Griffin 1994). tical particles. For the first time, this showed that the mysterious light quanta, introduced by Planck in 1900 and used by Einstein in his 1905 explanation of the photo-electric effect, could actually be thought of as particles of light, that is, what we now call photons. Einstein soon saw that the same method could be used not only for light, but also for an ideal gas of particles with mass. This was the first proper quantum mechanical generalization of the standard classi- cal theory of the ideal gas developed by Boltzmann, Maxwell, and Gibbs. We know now that there are two distinct quantum ideal gases, corresponding to either Bose—Einstein or Fermi—Dirac statistics. The method of counting quan- tum states introduced by Bose and Einstein applies to boson particles, such as photons or 4 He atoms.  1. 2 Bose—Einstein statistics 3 The key idea is that for identical quantum particles, we can simply count the number of available quantum states using combinatorics. If we have N, identical bose particles in M, available quantum states then there are 1 2 3 (N, +M5 — 1)! Ws = (1.1) Fig. 1.1 Ns boson particles in Ms available NAM, — 1)! quantum states. We can count the number of possible configurations by considering that available ways that the particles can be distributed. To see how this factor arises, the Ns identical particles and the M,— 1 walls imagine each available quantum state as a box which can hold any number of between boxes and can be arranged along a identical balls, as sketched in Fig. 1.1. We can count the number of arrangements line in any order. For bosons each box can hold any number of particles, 0, 1, 2.... by seeing that the N, balls and the M, — 1 walls between boxes can be arranged in any order. Basically there are a total of N, +M, — 1 different objects arranged in a line, N, of those are of one type (particles) while M, — 1 of them are of another type (walls between boxes). If we had N, + M5 — 1 distinguishable objects, we could arrange them in (N, + M, — 1)! ways. But the N, particles are indistinguishable as are the M, — 1 walls, giving a reduction by a factor N5 ! (M5 — 1)!, hence giving the total number of configurations in Eq. 1.1. We now apply this combinatoric rule to the thermodynamics of an ideal gas of N boson particles occupying a volume V. Using periodic boundary conditions, any individual atom will be in a plane-wave quantum state, 1 2eik-r, *(r) = (1.2) V1/ where the allowed wave vectors are ( 27n, 27ny 27n , k- (1 3). Lx L, and where L , 4„ and L are the the lengths of the volume in each direction. The total volume is V = Lx Ly L,, and therefore an infinitessimal volume d3k = Fig. 1.2 A thin shell of states of wave vector dkx clky dk, of k-space contains between k, and ks +Sks. The shell has volume 47t-k8k 5 and so there are 47rks2 31c5 V/(270 3 V quantum states in the shell. d- k (1.4) (2703 quantum states.' 3 The volume Band Theory and Electronic Each of these single particle quantum states has energy Properties of Solids, by John Singleton (2001), in this Oxford Master Series in Con- h2 k2 densed Matter Physics series explains this Elt = (1.5) point more fully, especially in Appendix B. 2m where m is the particle mass. We can therefore divide up the available single particle quantum states into a number of thin spherical shells of states, as shown in Fig. 1.2. By Eq. 1.4 a shell of radius k, and thickness 8k5 contains V = 471‘,28k, (1.6) (27)3 single particle states. The number of available states between energy es and g(e) Es + 8e 5 is therefore vm3/2 6 1/2 Ms = Be,, ,s/27 2 h3 = Vg(e5)8e5, (1.7) where 3/2 g(e) = e 1/2 (1.8) ,47 2 h3 Fig. 1.3 The single particle density of states, g(E), of a three dimensional gas of particles. is the density of states per unit volume, shown in Fig. 1.3.  The fundamental principles of statistical mechanics tell us that the total entropy of the gas is S = kB ln W, where kB is Boltzmann's constant and W is the number of available microstates of a given total energy E. To deter- mine W we must consider how the N atoms in the gas are distributed among the k-space shells of states of different energies. Suppose that there are Ns atoms in shell s. Since there are Ms quantum states in this shell, then we can calculate the total number of available quantum states for this shell using Eq. 1.1. The total number of available microstates for the whole gas is simply the product of the number of available states in each k-space shell, (Ns +M5 — 1)! w=l1ws=F1 NAM,. — 1)! (1.9) Using Stirling's approximation, ln N! N ln N — N, and assuming that Ni, M5 » 1, we have the entropy S = kB ln W = kB E + Ms) in (Ns + Ms) — Ns in Ns — Ms ln M51 (1.10) In thermal equilibrium the particles will distribute themselves so that the numbers of particles in each energy shell, Ns , are chosen so as to maximize this total entropy. This must be done varying Ns in such a way as to keep constant the total number of particles, N= E Ns and the total internal energy of the gas U E esNs. (1.12) Therefore we must maximize the entropy, S, with the constraints of fixed N and U. Using the method of Lagrange multipliers, this implies that au as —k B p— aN = 0, (1.13) aNs aNs aNs where the Lagrange multiplier constants have been defined as kB13 and —kB for reasons which will be clear below. Carrying through the differentiation we find ln (Ns + Ms ) — 111N5 — Pes = 0. (1.14) Rearranging to find Ns we find the result first obtained by Bose and Einstein, 1 Ns = Ms. (1.15) efl(es — A) — 1 The average number of particles occupying any single quantum state is Ns I Ms, and therefore the average occupation number of any given single particle states of energy Ek is given by the Bose—Einstein distribution 1 fB E ( e) = (1.16) efi(€ — A) — 1' In this formula we still have not properly identified the two constants, fi and bt, which were introduced simply as Lagrange multipliers. But we can easily  1.2 Bose—Einstein statistics 5 find their correct interpretation using the first law of thermodynamics for a gas of N particles, dU = T dS — P dV + dN, (1.17) where T is the temperature, P is the pressure, and p, is the chemical potential. Rearranging gives dS = (dU + P dV — ,u,dN). (1.18) The entropy is given by S = kB ln W calculated from Eq. 1.10 with the values of Ns taken from Eq. 1.15. Fortunately the differentiation is made easy using a shortcut from Eq. 1.13. We have dS =E aS aNs dNs, ay aN) d_Ns from Eq. 1.13, s ctINI aNs = kB (dU — dN). (1.19) Comparing with Eq. 1.18, we see that 1 = kBT (1.20) and the constant p, which we introduced above is indeed just the chemical potential of the gas. The method we have used above to derive the Bose—Einstein distribution formula makes use of the thermodynamics of a gas of fixed total particle num- ber, N, and fixed total energy U. This is the microcanonical ensemble. This ensemble is appropriate for a system, such as a fixed total number of atoms, such as a gas in a magnetic trap. However, often we are interested in systems of an effectively infinite number of atoms. In this case we take the thermody- namic limit V —> oo in which the density of atoms, n = NIV, is held constant. In this case it is usually much more convenient to use the grand canonical ensemble, in which both the total energy and the particle number are allowed to fluctuate. The system is supposed to be in equilibrium with an external heat bath, maintaining a constant temperature T, and a particle bath, maintaining a constant chemical potential p. If the N-body quantum states of N particles have energy Er) for i = 1, 2,..., then in the grand canonical ensemble each state occurs with probability 31 exp [—/3 (Er) — P(N) (i) = — !IN)] , (1.21) where the grand partition function is defined by Z = E exp [_$ (E7 — AN)]. (1.22) All thermodynamic quantities are then calculated from the grand potential (T , V , ,u) = —kBT ln (1.23) 4 For the derivation of the Bose—Einstein and using Fermi—Dirac distribution functions by this dS2 = —SdT — PdV — Nd,u. (1.24) method, see standard thermodynamics texts listed under further reading, or Appendix C It is quite straightforward to derive the Bose—Einstein distribution using this in the volume Band Theory and Electronic framework, rather than the microcanonical method used above.4 Properties of Solids (Singleton 2001). 1.3 Bose—Einstein condensation Unlike the classical idea gas, or the Fermi—Dirac gas, the Bose—Einstein ideal gas has a thermodynamic phase transition, Bose—Einstein condensation. In fact it is quite unique, since it is a phase transition occurring for non-interacting particles. The phase transition is driven by the particle statistics and not their interactions. At the phase transition all the thermodynamic observables have an abrupt change in character. This defines the critical temperature, T. The term "con- densation" is used here by analogy with the normal liquid—gas phase transition (such as in the van der Waals theory of gases), in which liquid drops condense out of the gas to form a saturated vapor. In the same way, below the critical temperature T, in BEC "normal gas" particles coexist in equilibrium with "con- densed" particles. But, unlike a liquid droplet in a gas, here the "condensed" particles are not separated in space from the normal particles. Instead, they are separated in momentum space. The condensed particles all occupy a single quantum state of zero momentum, while the normal particles all have finite momentum. Using the Bose—Einstein distribution, 1.16, the total number of particles in the box is (1.25) In the thermodynamic limit, V —>- oc, the possible k values become a continuum and so we should normally expect to be able to replace the summation in Eq. 1.25 with an integration E->jr (2177) 3d3k. If this is valid, then Eq. 1.25 becomes V 1 N d3k (1.26) (270 3 j e fi(6k—o) — 1 ' and so the particle density is 1 1 n= d3 k (1.27) (27) 3 e fi(ek—A) _ or, in terms of the density of states per unit volume g(e) from Eq. 1.8, 1 n = 1 g(E) de. (1.28) f3 efi( 6-11) — This equation defines the particle density n(T, pc) as a function of the temper- ature and chemical potential. But, of course, usually we have a known particle density, n, and wish to find the corresponding chemical potential z. Therefore we must view Eq. 1.28 as an equation which implicitly determines the chemical potential, it (T, n), a function of temperature and the particle density n.  1.3 Bose—Einstein condensation 7 Rewriting Eq. 1.28 in terms of the dimensionless variables z = e)61-` (called the fugacity), and x = /3e gives (mkB 7) 3/2 f oe ze — x n= x 1/2 dx. (1.29) N,h-n-2 h3 Jo 1 — ze —x To calculate this integral we can expand ze—x = ze —x ( 1 ze —x z 2 e-2x — Ze —x =E zp e —px. (1.30) 13=1 This expansion is clearly convergent provided that z is smaller than 1. Inserting this into Eq. 1.29 we can now carry out the integral over x using p00 fo e —pxx c° 1/2 = l e —y y 1/2 dy, 3/2 P 0 1/ p 3/2 2 where the dimensionless integral is a special case of the Gamma function, co F(t) = f y r—i e —y dy (1.31) with the value F (3/2) = N5/2. Combining the numerical constants, the particle density is therefore given as a function of the fugacity, z, by (mkBT )3/2 n= g3/2(Z), (1.32) 27h2 where the function g1 (z) is defined by the series co 7P g3/2(Z) = (1.33) In order to evaluate the particle density in Eq. 1.32, we must consider the shape of the function g1 (z). Using the ratio test for convergence, one can easily show that the series Eq. 1.33 converges when Izi < 1, but diverges if Izi > 1. At z = 1 the series is just convergent, co 1 3 g3/2( 1 ) = n3/2 2 ) = 2.612, (- (1.34) P= 1 r where (s) p=1 Ps is the Riemann zeta function. On the other hand, the function has infinite derivative at z = 1, since dg3/2(z) = 1 t — o e (1.35) dz z p1 2 ' 171 =- 7 Fig. 1.4 The function g3/2(z) as defined in which diverges at z = 1. With these limiting values we can make a sketch of Eq. 1.33. At z = 1 the function is finite but its function g1 (z) between z = 0 and z = 1, as shown in Fig. 1.4. derivative is infinite. 8 Bose-Einstein condensates Equation 1.32 gives the density, n in terms of g1 (z). Turning it around, we can say that the value of z, and hence the chemical potential p, is determined by 27rh2 312 g312 (efi t ) = I J n. (1.36) mkB T If we are at high temperature T or low density n, then the right-hand side of this equation is small, and we can use the small z expansion g3/2(Z) Z ± to obtain, 3 mkB T ,u -- kBT ln (1.37) 2 27rh2n213 )' This gives a negative chemical potential, as sketched in Fig. 1.5. However, on cooling the gas to lower temperatures the value of Z gradually increases until it eventually equals one. At this point the chemical potential, p, becomes zero. The temperature where this happens (for fixed density, n), defines the critical temperature, Te , 27t h 2 ( n ) 2/3 Fig. 1.5 Chemical potential, pt, of a Bose gas T, (1.38) kBm 2.612 as a function of temperature, T. At T = 0 all the particles are in the condensate and no = n. where g3/2(z) has reached its maximum finite value of 2.612. This 7', is the On the other hand, above the critical temper- ature 11. all the particles are in the normal BEC temperature. component, and no = O. But what happens when we cool the gas below Tc ? Einstein realized that as soon as the chemical potential p, becomes zero the number of particles in the lowest energy quantum state becomes infinite. More precisely we can say that out of a total of N particles in the gas, a macroscopic number No occupy the one quantum state with ek = 0. By a "macroscopic number" we mean that No is proportional to the system volume, so that there is a finite fraction of all of the particles, No/N, are in the one quantum state. Recall that we are working in the thermodynamic limit, V -> oc. The Bose-Einstein distribution predicts an occupation of the Ek = 0 state of 1 No = (1.39) - Rewriting Eq. 1.39 we obtain 1 1 p, -kBT ln (1 + — ) -kB T N0 —. (1.40) No If a finite fraction of the particles are in the ground state, then as V -4 oc we will have No -> oc and hence p, 4 0. Therefore below the BEC temperature - - Tc , the chemical potential is effectively zero, as shown in Fig 1.5. Below 7', we must take the k = 0 point into account separately, and so we must replace Eq. 1.25 by 1 N = N0 + (1.41) ePEk - ko where the chemical potential p, is zero. If we again replace the k summation by an integral (excluding the one point k = 0), the density of particles is now (mkBT) 3/2 e' T1/2 n = no + (1.42) 7r 2 h3 1 - e -x -  1.3 Bose—Einstein condensation 9 The definite integral can be evaluated by the same methods as before, and is equal to F (3/2 ) (3/2) and we finally obtain for T < Tc , ( mkB T ) 3/2 n = no + 2.612 (1.43) 27r h2 The particle density, n, can therefore be divided into a condensate density, no and a remainder (or normal density) nn , n = no + nn. (1.44) The fraction of particles in the condensate can be written compactly no T 312 — — 1 — (— 5 (1.45) T, as illustrated in Fig. 1.6. It is clear from this expression that at T = 0 all of the particles are in the ground state, and hence no = n, but at higher temperatures no no gradually decreases. no eventually equals zero at the critical temperature Tc , and is zero above T. Using these results, other thermodynamic quantities of the Bose gas can also be calculated exactly. For example, the total internal energy of the gas is I. co E U=V efite — /-0 — 1 g (E) roo Fig. 1.6 BEC density, no, as a function of rn3 1 2 ze' temperature, T. = V (kBT)5/2 x3/2 dx. (1.46) ,47r 2h3.10 - ze- The average energy per particle can be calculated by dividing by the particle number obtained previously, giving for T> u 3 u = — = — kBT g512(z) (1.47) N 2 g3/2(Z) and for T < 3 T5/2 g5/2(1) u= kB (1.48) 2 11.3 /2 g3/2 (1) Here the function g5/2 (z) is defined by co =1 pz5P/2 g5/2 (Z) = pE (1.49) and the numerical constant g5/2 (1) equals (5/2) = 1.342. In the limit where T is much larger than T, we have a normal Bose gas, and 3 U —kBT (1.50) 2 (since both g1 (z) z and g3 1 2 (z) z when z is small) Obviously, this result is exactly the same as the energy per particle in a classical monatomic ideal gas. Physically it implies that the Bose—Einstein statistics of the particles becomes irrelevant at high temperatures T >> T. By examining the heat capacity of the gas, we can see that the temperature Tc. represents a true thermodynamic phase transition. The heat capacity can  be obtained by differentiating the internal energy while keeping the density n constant, cv = — (1.51) aT per particle. We obtain Cv 3kB /2 for T » T, just as for a classical ideal gas T. and 15 g5/2 (1) ( T ) 3/2 Fig. 1.7 Heat capacity of a Bose—Einstein Cv = kB (1.52) ideal gas as a function of temperature, T. The 4 g312 (1) ) cusp at T, implies that BEC is a thermody- below T. This is sketched in Fig 1.7. At T, the heat capacity has a cusp, or a namic phase transition. discontinuity in slope. This implies that the free energy is not analytic at Tc , 5 The BEC phase transition is usually inter- showing that BEC is indeed a thermodynamic phase transition.' Other ther- preted as a first-order phase transition. The modynamic quantities, such as the entropy or pressure can also be calculated. two phases are a gas of normal particles with nonzero momentum, and a "condensate" of See Exercise (1.6) or texts on themodynamics, such as Huang (1987), for more zero momentum particles. At any nonzero details. temperature below K. we have a mixture of Finally, it is interesting to examine the origin of the BEC a little more care- these two distinct phases. The proof that the full thermodynamic behavior is consis- fully. Originally we replaced the whole sum over the discrete k plane wave tent with that expected for such a two-phase states by a continuum integral. Then we realized that it is neccessary to treat mixture is developed below in Exercise (1.6). the k = 0 state specially, but the rest of the states were still represented by a continuum. Why is this justified? Consider the occupation number of the first states with finite wave number k. In a cubic box of side length L, the very first low energy states have k 2r /L, and hence Ek h2 /mL2 = v-2/3 h2 i m. The occupation of these states is 1 1 = = 0(V213 ) eP(Ek —1 efiv-2/3h2 /rn — 1 where the notation 0(n) means "of order n" in the limit V —> oc. We can see that, although the occupations of the finite k states grow with V, they grow much more slowly than the value of No. In fact NI. /N0 = 0(V -113 ) 0 as V —> oc. Therefore in the infinite size system limit, the occupations of any individual single particle plane wave state k is negligible compared with the one special state at k = O. In the thermodynamic limit we can quite correctly make the continuum approximation for all of the k states except k = 0 with no significant error. 1.4 BEC in ultra-cold atomic gases In the late 1930s, quite soon after Einstein's prediction of BEC, it was discovered that liquid 4He becomes a superfluid below the lambda point at about 2.2 K. Since a 4 He atom contains two electrons, two protons, and two neutrons, it is, as a whole, a boson (i.e. it has total spin zero). Therefore it was natural to postulate a link between BEC and the superfluidity of 4 He. Interestingly, if one uses the density of 4 He of p c 145 kg 111 -3 and the atomic mass m 4mp , to find the particle density n = plm one obtains a value of 71. of about 3.1 K using Eq. 1.38, which is quite close to the superfluid transition of 4 He. Unfortunately the BEC theory, as described earlier, was for the ideal-gas, and completely neglected any interactions between the particles. But in the case of liquid helium the particle density is fairly high and the particle interactions cannot be neglected. Therefore we cannot really view helium as a suitable test case for the concept of BEC. Indeed there are many differences between the  1.4 BEC in ultra-cold atomic gases 11 properties of superfiuid 4He, as described in the next chapter, and the predictions for the ideal Bose gas. In fact only in 1995 were actual physical examples of a BEC finally realized. These were provided not in helium, but in very dilute gases of alkali metal atoms. The techniques for trapping and cooling atoms in magnetic and laser traps had been developed and improved gradually over the preceding two decades. At first sight it may seem surprising that one can achieve the conditions of temperature and density such that BEC can occur in such systems. The densities of atoms in the traps are typically of order 10 11 -10 15 cm-3 , which is many orders of magnitude less than the atomic density of 4He, which is about n — 2 x 1022 cm-3. Furthermore the atomic masses of the alkalis are very much higher than for 4He, especially for heavy alkali atoms such as 87Rb. Using Eq. 1.38 we would expect T, values perhaps 10-6 -10-8 times smaller than for the parameters of 4 He. In other words we expect T, values of order 10 nK — 1 K. It is remarkable that the techniques for cooling and trapping atoms with lasers and magnetic traps can now achieve such incredibly low temperatures in the laboratory. Explaining in detail exactly how this is done would take us far from the main topics of this book. Here, we shall only give a brief outline of some of the fundamental principles involved. First, how can we view a single large object, such as a rubidium atom as a boson? In quantum mechanics a particle will be a boson if it has an integer spin. Alkali metal atoms are in the first column of the periodic table, and so they have a single valence electron in the outermost s-orbital, for example 2s for lithium (Li), 3s for sodium (Na), 4s for potassium (K), or 5s for rubidium (Rb). The other electrons are in completely full shells of quantum states, and as a result they have a total orbital angular momentum and total spin of zero. The only other contribution to the total spin of the atom is the nuclear spin. If the nuclear isotope is one with an odd number of protons and neutrons it will have a net half-integer spin. For example, 7 Li, 23 Na, and 87 Rb all have S = 3/2 nuclei. In this case the total spin of the atom will be the sum of the nuclear spin and the valence electron spin, which will be an integer. Recall that in quantum mechanics adding two spins Si and S2 leads to the possible values of the total spin, S = Si - S21, 1S1 S21 ± 1, ,Si +S2 — 1, S1 +52. (1.53) The spin S = 3/2 nucleus and the S = 1/2 valence electron spin combine to give states with a total spin of either S = 2 or S = 1. If we can prepare the gas so that only one of these types of states is present, then this will be a gas of particles each with an integer spin. Therefore we can view this as a gas of Bose particles. On the other hand, if atoms in both S = 1 and S = 2 quantum states are present in the gas, then this is effectively a mixture of two different species of bosons, since the two types of atoms are distinguishable from each other. In order to see how these atoms can be trapped by a magnetic field we must consider the energy levels of the atom and how they are affected by a magnetic field. For definiteness, let us assume that the alkali atom has an S = 3/2 nucleus. It is helpful to first find the explicit spin wave functions for the different quantum states. First we find the states with maximum total spin, S = 2. For this total spin there are five different states, corresponding to z-components of total spin given by quantum numbers Ms -= 2, 1, 0, —1, 2. The wave function corresponding to the maximum value, Ms = 2, can be represented as 3 1\ IS = 2,Ms = 2) =: 2' 2/ ' where we use the notation Imsi , 11'42) to denote the state where the nucleus is in state mi and the electron is in state ms2. The other Ms quantum states with total spin S = 2 can be found by acting with the spin lowering operator = S + S. Using the identity 3- 1m) = N/s(s + 1) — m(m — 1)Im —1) (1.54) successively we obtain the five quantum states Ms = 2, 1, 0, —1, —2, 31 IS = 2, Ms = 2) = 11\ IS = 2,Ms = 1) -- 21 (.,/ 1(1 1\ 1 (,\/- 1 1\ IS = 2,Ms = —1} — 2 2' 2 / 2' 21 3 1\ IS = 2, Ms = —2} = (1.55) 2' 2 / The three states with total spin S = 1 and Ms = 1,0, —1 must be orthogonal to the corresponding S = 2 states, and this requirement determines them uniquely to be 1,114s = 1) =. ( 1 1 31 IS= 2'2) 3 1 1\ 1 1 IS = 1,Ms =- 0) = 2 2/ 2' 20 31 IS = 1,M = —1) = 2 2/ 2'2 (1.56) In zero magnetic field the S = 2 and S = 1 states have slightly different energy due to the weak hyperfine interaction between the nucleus and the outermost unpaired valence electron. In zero magnetic field all five of the S = 2 states are degenerate with each other, as are the three S = 1 states, as shown in Fig. 1.4. But a magnetic field leads to a Zeeman splitting of these degenerate states. To a good approximation, we can write the relevant effective Hamiltonian as, fi = Al §2 2 AB§2zBz , (1.57) where J is the hyperfine interaction between the nuclear and valence electron spins, 2p,B is the magnetic moment of the valence electron (p,B = eh/2me , the Bohr magneton) , and Bz is the magnetic field which is assumed to be in the z direction. In writing this simplified Hamiltonian we have ignored the magnetic moment of the nucleus, which is very much smaller than that of the valence  1.4 BEC in ultra-cold atomic gases 13 5 =2 = 3/2 S2 = 1/2 s=1 It Fig. 1.8 Energy levels of a typical alkali metal atom with an S = 3/2 nucleus, as a function of a weak external magnetic field, B. At B, = 0 the S = 2 and S = 1 total spin states are separated by the nuclear hyperfine coupling. The weak magnetic field Zeeman further splits these S = 2 and S = 1 hyperfine levels into their different Ms states as shown. electron. It is not difficult to find the eigenstates of this Hamiltonian, especially in the limit of small magnetic fields, B. If Bz = 0 we can use the identity §i §2 = —1 ((§1 §2) 2 (1.58) 2 to find the two energy levels 3 5 E2 = +— J E1 = — —J (1.59) 4 4 corresponding to S = 2 and S = 1, respectively, as sketched in Fig. 1.4. In a small magnetic field we can use perturbation theory in the term = 211B §2zB z the magnetic dipole moment of the valencecorespndigt electron. First order perturbation theory gives the energy shifts 6,E -= (k), which are linear in Bz , as shown in Fig. 1.4. This magnetic field dependence of the quantum state energies is exploited in a magnetic atom trap. The trap is constructed by producing a region of space in which the magnetic field has a local minimum. At first sight it is surprising that it is possible to produce a local minimum in magnetic field, because the field must also obey Maxwell's equations for a region of free space 6 Interestingly one can prove that it is impos- V B = 0, (1.60) sible to find a magnetic field B(r) for which IB(r)I has a local maximum. This result is V x B = O. (1.61) known as Earnshaw's theorem. Fortunately the theorem does not rule out the existence In fact it is indeed possible to both obey these and to have a local minimum of a local mimimum in the field magnitude IB (01.6 Now, if we prepare an atom in a quantum state such as S = 2 Ms = 2 in Fig. 1.4, then it will lower its energy by moving toward a region of smaller magnetic field. It will therefore be attracted into the magnetic trap, which will appear to the atom as a local minimum in potential energy. This potential is only a local minimum, as shown in Fig. 1.9. Atoms which have too much kinetic energy, that is, are too "hot," will not be bound by the trap and will escape. While atoms which have less kinetic energy, that is, are "cold," will be bound by the local minimum in potential energy. Comparing with the energy level diagram in Fig. 1.4 one can see that several different quantum states lower their energy by moving to a region of lower field, and so could be bound by a magnetic trap. In Fig. 1.4 the S = 2, Ms = 2, and Fig. 1.9 A magnetic trap provides a local Ms = 1 states and the S = 1, Ms = —1 state could be trapped. Indeed one minimum in energy. Atoms which are too could also have a mixture of atoms in these states bound together in the trap. But energetic can escape, while atoms with lower because the atoms in different quantum states would be distinguishable from kinetic energy are trapped. Also atoms in a one another, in the case of a mixture there would be effectively two different quantum state Ms whose energy decreases with IBI will see a local maximum in poten- types of boson particles in the trap. In such a case the two distinct boson gases tial energy, not a minimum, and so will be are in thermal equilibrium with each other. expelled from the trap.  The effective potential energy of the trap in Fig. 1.9 also gives rise naturally to a simple mechanism for cooling the trapped gas. Individual atoms which have high kinetic energies will escape over the barrier and carry away their energy. The remaining atoms will be cooler on average. This is simply cooling by evaporation! By controlling the barrier height, one can control the cooling rate, and hence the final temperature of the system. In this way, temperatures below 1 pK can be achieved. Unlike the ideal Bose gas we studied earlier in this chapter, the alkali atoms in a magnetic trap do in fact interact with each other. In fact the interactions can be quite strong, since the atoms strongly repel each other at short distances. Also, at large interatomic distances there is a van der Waals attraction force between the atoms. These interactions would eventually lead to the atoms in the trap binding strongly together into atomic clusters. But fortunately the rate at which this happens is very slow. The reason is that collisions between the atoms will be almost entirely two-body collisions between pairs of atoms. Since these are elastic collisions, no binding can take place. Binding would only be possible in a three-body collision in which one pair of atoms could form a bound state, while the excess kinetic energy is carried off by the third atom. The rate of three body collisions will be tiny when the density of atoms in the trap, n, is sufficiently low. Typically the length scale of the interatomic interaction is of order 0.2-0.3 nm, while the density of atoms in the magnetic trap is of order n 10 11 — — 10 15 cm-3 , corresponding to a typical interatomic spacing of rs — 50-600 nm where r, is defined by n = 1/(4m r/3). Therefore the probability of three particles colliding simultaneously is low, and it is possible to maintain the atoms in the trap for a reasonably long time (seconds, or perhaps a few minutes) to perform experiments. On the other hand the two-body collisions between particles are not entirely negligible. First, it is important to note that the two-body collisions do not allow transitions between the different hyperfine quantum states in Fig. 1.4. This will be the case for the S = 2, M = 2 or the S = 1,Ms = —1 states in Fig. 1.4. Therefore particles prepared in one of these low field seeking quantum states will remain in the same state. Second, the interparticle interactions contribute to the overall potential energy of the atoms in the trap, and cannot be neglected. Pairwise interparticle interactions are also necessary so that thermal equilib- rium is established during the timescale of the experiment. Pairwise collisions lead to redistribution of energy and are necessary for the system to achieve thermal equilibrium. We can treat the interatomic pair interactions approximately as follows. Since the interaction only acts of a very short length scale compared with the typical interparticle separations we can approximate the pair interaction by a Dirac delta-function V(ri — r2) gb (ri — r2). (1.62) The interaction can therefore be characterized by a single constant, g. Using scattering theory this can also be expressed as a two-body s-wave scattering length, a„, defined by 4n- ash2 g = (1.63) Usually g and a, are positive, corresponding to a net repulsive interaction. On average the effects of this interactions can be represented as an additional  1.4 BEC in ultra-cold atomic gases 15 potential felt by each particle, resulting from the average interaction with the other particles. This mean-field contribution to the potential can be written Veff (r) = gn(r), (1.64) where n(r) is the density of atoms at point r in the trap. In this approximation the atoms in the magnetic trap obey an effective Schiidinger equation h2 j--v 2 ± Vt„p (r) ± Veff (r) Vii(r) = EiVii (r), (1.65) 2m where Vtrap (r) is the effective potential of the magnetic trap, as in Fig. 1.9, including both the magnetic field energy and gravity. Equation 1.65 is effectively a nonlinear Schnalinger equation, since the potential depends on the particle density which in turn depends on the wave functions via the Bose—Einstein distribution n(r) = E es ( 1 i i*i(r)1 2. (1.66) As usual, the chemical potential p, is determined from the constraint of the constant total number of atoms in the trap, N, N = f n(r) d3 r = E ep(,,,t)1 _ 1 (1.67) These equations (Eqs 1.64-1.67) are a closed set, which must be solved self- consistently. At zero temperature all of the particles are in the condensate, and n(r) = NI*0(r)1 2 , (1.68) where *0 (r) is the ground state wave function. These coupled equations must be solved self-consistently to find the wavefunctions, density n(r) and the effective potential Veff (r). They are known as the Gross Pitaevskii equations. — Solving this coupled set of nonlinear equations must be done numerically. Nevertheless the solution again shows a form of BEC. If the lowest energy state in the potential well has energy E0, then at some critical temperature 71 the occupation of this one quantum state, No, suddenly increases from a small number (of order 1) to a large number (of order N). From the rules of statistical mechanics we cannot, strictly speaking, call this a thermodynamic phase transition, since the total number of particles is finite, and there is no thermodynamic limit Despite this, the total number of atoms in the trap can be large (104 — 106 ) and so in practice the critical temperature T, is quite sharp and well defined. Bose—Einstein condensation in trapped ultra-cold gases was first observed in 1995. The 2001 Nobel prize for physics was awarded to Cornell, Ketterle, and Weiman for this achievement. This discovery followed decades of work by many groups in which the technology of trapping and cooling atoms with magnetic and laser traps was developed and refined. In 1995 three different groups of researchers achieved BEC, using the different alkali atoms 87 Rb, 23 Na, and 7 Li. The different traps used in the experiments combined magnetic trapping methods (as described previously) with laser trapping and cooling methods (which we will not describe here). The temperatures at which BEC were observed were of order 0.5 — 2 p,K, depending on the alkali atom used and the atomic density achieved in the trap. The dramatic results of one such 16 Bose—Einstein condensates Fig. 1.10 Velocity distribution of atoms in a BEC. On the left we see the broad Maxwell—Boltzmann type distribution in the normal gas above T. In the center we see the gas just below 71, with some fraction of the atoms beginning to appear in the sharp condensate peak at zero velocity. On the right we see the velocity distribution at temperature well below I', where almost all atoms are condensed into the zero-velocity peak. The intrinsic width of the peak is governed by the harmonic oscillator-like ground state wave function of the atoms in the trap. Reprinted figure with permission from Ketterle (2002). Copyright (2002) by the American Physical Society. experiment are shown in Fig. 1.10. The figure shows the velocity distribution of the atoms in the gas, first at temperatures above I', (left), just below T, (centre), and then well below I', (right). The sharp peak at the center can be identified with the atoms in the condensate, No, while the remaining atoms which are not in the condensate N No have a broad distribution of velocities typical — of a normal gas. The picture is actually obtained by suddenly turning off the trap magnetic field and then allowing the atoms to move freely for a certain interval of time. The distance the atoms have moved in this time interval can be measured when the atoms are illuminated with laser light at the frequency of a strong optical absorbtion line of the atom. The spatial distribution of the light absorbed by the atoms then shows their positions, and hence their velocities at the moment the trap was turned off. Interestingly the central condensate peak seen in Fig. 1.10 is not infinitely narrow. This is simply a consequence of the trapping potential, Vtrap (r) Veff (r) in Eq. 1.65. The condensate occurs in the ground state wave function, Vro (r), and this single state has a finite width momentum distribution as can be seen in the momentum representation of the wave function, vfo (r) 1 f Ak edi rd3k. (2 7 ) 3 (1.69) The finite width of the momentum distribution in the ground state can be simply estimated from the uncertainty principle, as Ap h/ Ax , where Ax is the effective width of the ground state wave function in the trap potential minimum of Fig. 1.9. In contrast the width of the velocity distribution of the noncondensed atoms in the trap, the broad background in Fig. 1.10, can be estimated from  Further reading 17 the usual Maxwell—Boltzmann velocity distribution of a gas. A crude estimate is ( vx2r) = 1-k BT, from the equipartition of energy theorem. Hence for the broad normal component A v (kB TIm) 1 /2 , compared with Av hl (in Ax) for the sharper condensate peak. Following the achievement of BEC in atoms gases, many different experi- ments have been performed. The system is ideal for many experiments, since all of the physical parameters of the experiment can be controlled and it is pos- sible to manipulate the BEC in many different ways. It has also been possible to do similar experiments on atoms with Fermi—Dirac statistics, but of course these do not have an analog of BEC. It turns out that the interatomic interaction, Eq. 1.62, is very important in the properties of the atomic BEC, and so strictly one is studying a system of weakly interacting bosons, and not an ideal Bose gas. These weak two-body interactions are very significant. In particular, it can be shown that an ideal BEC is not a superfluid. When there are no interactions the critical velocity for superfluid flow (discussed in the next chapter) is zero. It is only when the interactions are finite that it becomes possible to sustain a true superfluid state, for example, to have a liquid that can flow with zero vis- cosity or which can sustain persistent currents which are unaffected by external perturbations. In the atomic BEC, which have been obtained experimentally the small, but finite, residual interactions between the particles mean that these are indeed effectively true superfluids. Persistent currents and even superfluid vortices have been observed. Finally, experiments have also explored the consequences of macroscopic quantum coherence in BEC, showing quantum superpositions and interfer- ence between systems with macroscopic (10 5 or 106 ) numbers of particles. Such macroscipic superposition states can be used as an actual physical real- ization of the Schredinger cat problem in quantum measurement theory! Just as the SchOdinger cat is placed in a quantum superposition of "dead" and "live" cat quantum states, in exactly the same way the BEC can be placed in a superposi- tion of two quantum states each of which differ from each other in a macroscopic number of particle coordinates. Further reading For a short article on the life of S.N. Bose and the discovery of Bose—Einstein statistics from New Sceintist see: "The man who chopped up light" by Home and Griffin (1994). The fundamental princples of themodynamics in classical and quantum sys- tems which we have used in this chapter are described in many books, for example: Mandl (1988), or Huang (1987). These books also discuss in detail the ideal bose gas, and Bose—Einstein condensation. In writing the section on BEC in atomic gases I have made considerable use of the detailed reviews by: Leggett (2001) and Dalfovo et al. (1999). The Nobel prize lecture given by Ketterle (2002) also describes the events leading up to the 1995 discovery of BEC. The review by Phillips (1998) describes the principles of laser atom traps, which operate on somewhat different principles from the magnetic traps described above in Section 1.4. Two books with comprehensive discussions about all aspects of BEC are Pethick and Smith (2001) and Pitaevskii and Stringari (2003). 18 Bose-Einstein condensates Exercises (1.1) Extend the counting argument given above for the num- (b) Show that the corresponding density of states g(E) ber of ways to place N, particles in /l4", available quantum is constant, states for the case of fermion particles. Show that if only g(6) = 2zr h one particle can be placed in each quantum state, then 2 Eq. 1.1 becomes (1.5) For the two-dimensional gas from Exercise 1.4, show that M! the analog of Eq. 1.29 is W, = N,!(M, -Ni)! mkB T f" ze' (1.2) For the case of the Fermi-Dirac state counting given n= 271- h2 Jo 1 - ze -x dx. above in Exercise 1.1, maximize the total entropy S = kB In W, and hence show that for fermions Do the integral exactly (using an easy substitution) to obtain 1 mkBT el3 (Es -IL) - n= ln (1 - z). 27r h2 the Fermi-Dirac distribution. Hence write down an explicit expression for g as a func- (1.3) The Bose-Einstein distribution can be derived easily tion of n and T, and show that p, never becomes zero. using for formalism of the grand canonical ensemble This proves that the two-dimensional ideal Bose gas never as follows. becomes a BEC! (1.6) Find the entropy per particle in the three-dimensional (a) Write the energies the many-particle quantum states in the form ideal Bose gas in the temperature range 0 < T < Tc , using E = Enkek T cv s(T) = f — where nk = 0, 1,.... Hence show that the grand o partition function for the bose gas can be written as (obtained from ds = dulT and du = Cv dT). From your a product over all the k states result, show that the total entropy of N particles obeys z=n Zk S(T) = Nos(0) N ns(T,), where N = No + N. This result shows that we can con- and show that sider the state below 71. as a statistical mixture (like a 1 liquid-gas saturated vapor) of No particles in the "con- 2k= densate" with entropy per particle s(0), and N 1- fluid particles with entropy s(Te ). This implies that we (b) Show that the average particle number in state k is can view BEC as a first-order phase transition. a ln Zk (1.7) Using the wave functions of the different S = 2 states for (nk ) = -kB T Ms = 2, 1, 0, -1, -2 given in Eq. 1.55 show that the first and hence confirm that this equals the Bose- order in B, the magnetic field changes the energy levels by Einstein distribution. 1 LE = +- ABMsBz (1.4) It is possible to realize a two-dimensional Bose gas by 2 trapping helium atoms on the surface of another material, For the S = 1 energy levels in Eq. 1.56 show that such as graphite. (a) Show that for a two-dimensional gas of area, A, the number of quantum states is Show that this is consistent with the energy level scheme A sketched in Fig. 1.4. d2k (27r ) 2 (1.8) (a) Approximating the trapping potential in a magnetic per unit volume in k-space. trap by a three-dimensional harmonic oscillator  Exercises 19 potential states is _1 m 60 2 (x 2 + y2 ± z2 E2 Vtrap(r) = g(c) = 2 2(hoi) 3 in Eq. 1.65 and ignoring Veff, show that the single when c is large. particle quantum states have energies (c) Write down the analog of Eq. 1.29 for this density en,ny n, = ha) (nx fly nz. of states, and show that the BEC temperature for N (b) Find the total number of quantum states with ener- atoms in the trap is of order 7', N 113 hco/kB when gies less than c, and thus show that the density of N is large. Superfluid helium-4 2.1 Introduction 2.1 Introduction 21 There are only two superfluids which can be studied in the laboratory. These 2.2 Classical and quantum are the two isotopes of helium: 4He and 3 He Unlike all other substances they fluids 22 are unique because they remain in the liquid state even down to absolute zero 2.3 The macroscopic wave in temperature. The combination of the light nuclear masses and the relatively function 26 weak van der Waals interactions between the particles means that at low pressure 2.4 Superfluid properties they do not freeze into a crystalline solid. All other elements and compounds of He II 27 eventually freeze into solid phases at low temperatures. 2.5 Flow quantization and Despite the identical electronic properties of the 4He and 3He atoms, they vortices 30 have completely different properties at low temperatures. This is not due to the 2.6 The momentum difference in nuclear mass, but arises because 4He is a spin zero boson, while distribution 34 3 He is a spin 1/2 fermion (due to the odd number of spin 1/2 constituents in 2.7 Quasiparticle the nucleus). As we shall see below, although they both form superfluid phases, excitations 38 the origin and physical properties of these superfluids is completely different. 2.8 Summary 43 Superfluidity in 4He occurs below 2.17 K, and was first discovered and studied Further reading 43 in the 1930s. 1 0n the other hand, the superfluidity of 3 He only occurs below Exercises 44 about 2 mK, three orders of magnitude lower in temperature than for 4He. Another surprising difference is that 3He has more than one distinct superfluid Kapitsa eventually firmly established the phase. Two of the superfluid transitions in 3 He were first observed by Osheroff, existence of superfluidity in 1938, receiving Richardson, and Lee in 1972, and they received the 1996 Nobel prize in physics the Nobel prize for this work over 40 years for this work. later in 1978, which is perhaps one of the longest ever delays in the award of a Nobel In this chapter we shall focus on the properties of 4He. We shall postpone prize! Surprisingly, Kapitsa shared the 1978 discussion of the properties of 3He until later, since it is more closely related to Nobel prize with Penzias and Wilson who exotic superconductors than 4He. In this chapter we first discuss the concept of a discovered the cosmic microwave backround. Sharing the prize like this was unusual, since quantum fluid, and discuss why helium is unique in its ability to remain liquid there is no obvious connection between their down to absolute zero in temperature. Next we shall introduce the main physical work and Kapitsa's. Perhaps the connection properties of superfluid 4He. Then we discuss the concept of the macroscopic is simply that the microwave backround of the universe is at 2.7 K, which happens to be wavefunction or off diagonal long range order, which provides the main close to the superfluid temperature of helium link to the previous chapter on Bose-Einstein condensation (BEC). Finally, we at 2.17 K! discuss Landau's quasiparticle theory of 4He which shows the very important consequencs of the strong particle—particle interactions. It it these interactions which make the theory of superfluid 4He much more difficult than that of BEC in atomic gases. For completeness we should also note that superfluids other than helium are at least theoretically possible. Inside neutron stars one has a dense "fluid" of neutrons. These are neutral spin half fermions, and so could have super- fluid phases rather like 3 He. Another possibility might be for superfluid states 22 Superfluid helium-4 of molecular or atomic hydrogen. At normal atmospheric pressures hydrogen condenses into a solid phase and not a superfluid. Applying pressure a number of different