Bose einstein condensation-in_dilute_gases_-_pethick_c.j.__smith_h.
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Bose–Einstein Condensation in Dilute GasesIn 1925 Einstein predicted that at low temperatures particles in a gas couldall reside in the same quantum state. This peculiar gaseous state, a Bose–Einstein condensate, was produced in the laboratory for the ﬁrst time in 1995using the powerful laser-cooling methods developed in recent years. Thesecondensates exhibit quantum phenomena on a large scale, and investigatingthem has become one of the most active areas of research in contemporaryphysics. The study of Bose–Einstein condensates in dilute gases encompasses anumber of diﬀerent subﬁelds of physics, including atomic, condensed matter,and nuclear physics. The authors of this textbook explain this excitingnew subject in terms of basic physical principles, without assuming detailedknowledge of any of these subﬁelds. This pedagogical approach thereforemakes the book useful for anyone with a general background in physics,from undergraduates to researchers in the ﬁeld. Chapters cover the statistical physics of trapped gases, atomic properties,the cooling and trapping of atoms, interatomic interactions, structure oftrapped condensates, collective modes, rotating condensates, superﬂuidity,interference phenomena and trapped Fermi gases. Problem sets are alsoincluded in each chapter. christopher pethick graduated with a D.Phil. in 1965 from theUniversity of Oxford, and he had a research fellowship there until 1970.During the years 1966–69 he was a postdoctoral fellow at the Universityof Illinois at Urbana–Champaign, where he joined the faculty in 1970,becoming Professor of Physics in 1973. Following periods spent at theLandau Institute for Theoretical Physics, Moscow and at Nordita (NordicInstitute for Theoretical Physics), Copenhagen, as a visiting scientist, heaccepted a permanent position at Nordita in 1975, and divided his timefor many years between Nordita and the University of Illinois. Apartfrom the subject of the present book, Professor Pethick’s main researchinterests are condensed matter physics (quantum liquids, especially 3 He,4 He and superconductors) and astrophysics (particularly the properties ofdense matter and the interiors of neutron stars). He is also the co-author ofLandau Fermi-Liquid Theory: Concepts and Applications (1991). henrik smith obtained his mag. scient. degree in 1966 from theUniversity of Copenhagen and spent the next few years as a postdoctoralfellow at Cornell University and as a visiting scientist at the Institute forTheoretical Physics, Helsinki. In 1972 he joined the faculty of the University
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iiof Copenhagen where he became dr. phil. in 1977 and Professor of Physics in1978. He has also worked as a guest scientist at the Bell Laboratories, NewJersey. Professor Smith’s research ﬁeld is condensed matter physics andlow-temperature physics including quantum liquids and the properties ofsuperﬂuid 3 He, transport properties of normal and superconducting metals,and two-dimensional electron systems. His other books include TransportPhenomena (1989) and Introduction to Quantum Mechanics (1991). The two authors have worked together on problems in low-temperaturephysics, in particular on the superﬂuid phases of liquid 3 He, superconductorsand dilute quantum gases. This book derives from graduate-level lecturesgiven by the authors at the University of Copenhagen.
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Bose–Einstein Condensation in Dilute Gases C. J. Pethick Nordita H. Smith University of Copenhagen
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published by the press syndicate of the university of cambridge The Pitt Building, Trumpington Street, Cambridge, United Kingdom cambridge university press The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc´n 13, 28014, Madrid, Spain o Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org c C. J. Pethick, H. Smith 2002 This book 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 2002 Printed in the United Kingdom at the University Press, Cambridge Typeface Computer Modern 11/14pt. System L TEX 2ε [dbd] A A catalogue record of this book is available from the British Library Library of Congress Cataloguing in Publication Data Pethick, Christopher.Bose–Einstein condensation in dilute gases / C. J. Pethick, H. Smith. p. cm. Includes bibliographical references and index. ISBN 0 521 66194 3 – ISBN 0 521 66580 9 (pb.) 1. Bose–Einstein condensation. I. Smith, H. 1939– II. Title. QC175.47.B65 P48 2001 530.4 2–dc21 2001025622 ISBN 0 521 66194 3 hardback ISBN 0 521 66580 9 paperback
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ContentsPreface page xi1 Introduction 11.1 Bose–Einstein condensation in atomic clouds 41.2 Superﬂuid 4 He 61.3 Other condensates 81.4 Overview 10 Problems 13 References 142 The non-interacting Bose gas 162.1 The Bose distribution 16 2.1.1 Density of states 182.2 Transition temperature and condensate fraction 21 2.2.1 Condensate fraction 232.3 Density proﬁle and velocity distribution 24 2.3.1 The semi-classical distribution 272.4 Thermodynamic quantities 29 2.4.1 Condensed phase 30 2.4.2 Normal phase 32 2.4.3 Speciﬁc heat close to Tc 322.5 Eﬀect of ﬁnite particle number 352.6 Lower-dimensional systems 36 Problems 37 References 383 Atomic properties 403.1 Atomic structure 403.2 The Zeeman eﬀect 44 v
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vi Contents3.3 Response to an electric ﬁeld 493.4 Energy scales 55 Problems 57 References 574 Trapping and cooling of atoms 584.1 Magnetic traps 59 4.1.1 The quadrupole trap 60 4.1.2 The TOP trap 62 4.1.3 Magnetic bottles and the Ioﬀe–Pritchard trap 644.2 Inﬂuence of laser light on an atom 67 4.2.1 Forces on an atom in a laser ﬁeld 71 4.2.2 Optical traps 734.3 Laser cooling: the Doppler process 744.4 The magneto-optical trap 784.5 Sisyphus cooling 814.6 Evaporative cooling 904.7 Spin-polarized hydrogen 96 Problems 99 References 1005 Interactions between atoms 1025.1 Interatomic potentials and the van der Waals interaction 1035.2 Basic scattering theory 107 5.2.1 Eﬀective interactions and the scattering length 1115.3 Scattering length for a model potential 1145.4 Scattering between diﬀerent internal states 120 5.4.1 Inelastic processes 125 5.4.2 Elastic scattering and Feshbach resonances 1315.5 Determination of scattering lengths 139 5.5.1 Scattering lengths for alkali atoms and hydrogen 142 Problems 144 References 1446 Theory of the condensed state 1466.1 The Gross–Pitaevskii equation 1466.2 The ground state for trapped bosons 149 6.2.1 A variational calculation 151 6.2.2 The Thomas–Fermi approximation 1546.3 Surface structure of clouds 1586.4 Healing of the condensate wave function 161
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Contents vii Problems 163 References 1637 Dynamics of the condensate 1657.1 General formulation 165 7.1.1 The hydrodynamic equations 1677.2 Elementary excitations 1717.3 Collective modes in traps 178 7.3.1 Traps with spherical symmetry 179 7.3.2 Anisotropic traps 182 7.3.3 Collective coordinates and the variational method 1867.4 Surface modes 1937.5 Free expansion of the condensate 1957.6 Solitons 196 Problems 201 References 2028 Microscopic theory of the Bose gas 2048.1 Excitations in a uniform gas 205 8.1.1 The Bogoliubov transformation 207 8.1.2 Elementary excitations 2098.2 Excitations in a trapped gas 214 8.2.1 Weak coupling 2168.3 Non-zero temperature 218 8.3.1 The Hartree–Fock approximation 219 8.3.2 The Popov approximation 225 8.3.3 Excitations in non-uniform gases 226 8.3.4 The semi-classical approximation 2288.4 Collisional shifts of spectral lines 230 Problems 236 References 2379 Rotating condensates 2389.1 Potential ﬂow and quantized circulation 2389.2 Structure of a single vortex 240 9.2.1 A vortex in a uniform medium 240 9.2.2 A vortex in a trapped cloud 245 9.2.3 Oﬀ-axis vortices 2479.3 Equilibrium of rotating condensates 249 9.3.1 Traps with an axis of symmetry 249 9.3.2 Rotating traps 251
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viii Contents9.4 Vortex motion 254 9.4.1 Force on a vortex line 2559.5 The weakly-interacting Bose gas under rotation 257 Problems 261 References 26210 Superﬂuidity 26410.1 The Landau criterion 26510.2 The two-component picture 267 10.2.1 Momentum carried by excitations 267 10.2.2 Normal ﬂuid density 26810.3 Dynamical processes 27010.4 First and second sound 27310.5 Interactions between excitations 280 10.5.1 Landau damping 281 Problems 287 References 28811 Trapped clouds at non-zero temperature 28911.1 Equilibrium properties 290 11.1.1 Energy scales 290 11.1.2 Transition temperature 292 11.1.3 Thermodynamic properties 29411.2 Collective modes 298 11.2.1 Hydrodynamic modes above Tc 30111.3 Collisional relaxation above Tc 306 11.3.1 Relaxation of temperature anisotropies 310 11.3.2 Damping of oscillations 315 Problems 318 References 31912 Mixtures and spinor condensates 32012.1 Mixtures 321 12.1.1 Equilibrium properties 322 12.1.2 Collective modes 32612.2 Spinor condensates 328 12.2.1 Mean-ﬁeld description 330 12.2.2 Beyond the mean-ﬁeld approximation 333 Problems 335 References 336
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Contents ix13 Interference and correlations 33813.1 Interference of two condensates 338 13.1.1 Phase-locked sources 339 13.1.2 Clouds with deﬁnite particle number 34313.2 Density correlations in Bose gases 34813.3 Coherent matter wave optics 35013.4 The atom laser 35413.5 The criterion for Bose–Einstein condensation 355 13.5.1 Fragmented condensates 357 Problems 359 References 35914 Fermions 36114.1 Equilibrium properties 36214.2 Eﬀects of interactions 36614.3 Superﬂuidity 370 14.3.1 Transition temperature 371 14.3.2 Induced interactions 376 14.3.3 The condensed phase 37814.4 Boson–fermion mixtures 385 14.4.1 Induced interactions in mixtures 38614.5 Collective modes of Fermi superﬂuids 388 Problems 391 References 392Appendix. Fundamental constants and conversion factors 394Index 397
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PrefaceThe experimental discovery of Bose–Einstein condensation in trappedatomic clouds opened up the exploration of quantum phenomena in a qual-itatively new regime. Our aim in the present work is to provide an intro-duction to this rapidly developing ﬁeld. The study of Bose–Einstein condensation in dilute gases draws on manydiﬀerent subﬁelds of physics. Atomic physics provides the basic methodsfor creating and manipulating these systems, and the physical data requiredto characterize them. Because interactions between atoms play a key rolein the behaviour of ultracold atomic clouds, concepts and methods fromcondensed matter physics are used extensively. Investigations of spatial andtemporal correlations of particles provide links to quantum optics, whererelated studies have been made for photons. Trapped atomic clouds havesome similarities to atomic nuclei, and insights from nuclear physics havebeen helpful in understanding their properties. In presenting this diverse range of topics we have attempted to explainphysical phenomena in terms of basic principles. In order to make the pre-sentation self-contained, while keeping the length of the book within reason-able bounds, we have been forced to select some subjects and omit others.For similar reasons and because there now exist review articles with exten-sive bibliographies, the lists of references following each chapter are far fromexhaustive. A valuable source for publications in the ﬁeld is the archive atGeorgia Southern University: http://amo.phy.gasou.edu/bec.html This book originated in a set of lecture notes written for a graduate-level one-semester course on Bose–Einstein condensation at the Universityof Copenhagen. We have received much inspiration from contacts with ourcolleagues in both experiment and theory. In particular we thank GordonBaym and George Kavoulakis for many stimulating and helpful discussionsover the past few years. Wolfgang Ketterle kindly provided us with the xi
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xii Prefacecover illustration and Fig. 13.1. The illustrations in the text have beenprepared by Janus Schmidt, whom we thank for a pleasant collaboration.It is a pleasure to acknowledge the continuing support of Simon Capelinand Susan Francis at the Cambridge University Press, and the careful copy-editing of the manuscript by Brian Watts.Copenhagen Christopher Pethick Henrik Smith
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1 IntroductionBose–Einstein condensates in dilute atomic gases, which were ﬁrst realizedexperimentally in 1995 for rubidium [1], sodium [2], and lithium [3], provideunique opportunities for exploring quantum phenomena on a macroscopicscale.1 These systems diﬀer from ordinary gases, liquids, and solids in anumber of respects, as we shall now illustrate by giving typical values ofsome physical quantities. The particle density at the centre of a Bose–Einstein condensed atomiccloud is typically 1013 –1015 cm−3 . By contrast, the density of moleculesin air at room temperature and atmospheric pressure is about 1019 cm−3 .In liquids and solids the density of atoms is of order 1022 cm−3 , while thedensity of nucleons in atomic nuclei is about 1038 cm−3 . To observe quantum phenomena in such low-density systems, the tem-perature must be of order 10−5 K or less. This may be contrasted withthe temperatures at which quantum phenomena occur in solids and liquids.In solids, quantum eﬀects become strong for electrons in metals below theFermi temperature, which is typically 104 –105 K, and for phonons belowthe Debye temperature, which is typically of order 102 K. For the heliumliquids, the temperatures required for observing quantum phenomena are oforder 1 K. Due to the much higher particle density in atomic nuclei, thecorresponding degeneracy temperature is about 1011 K. The path that led in 1995 to the ﬁrst realization of Bose–Einstein con-densation in dilute gases exploited the powerful methods developed over thepast quarter of a century for cooling alkali metal atoms by using lasers. Sincelaser cooling alone cannot produce suﬃciently high densities and low tem-peratures for condensation, it is followed by an evaporative cooling stage, in1 Numbers in square brackets are references, to be found at the end of each chapter. 1
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2 Introductionwhich the more energetic atoms are removed from the trap, thereby coolingthe remaining atoms. Cold gas clouds have many advantages for investigations of quantum phe-nomena. A major one is that in the Bose–Einstein condensate, essentially allatoms occupy the same quantum state, and the condensate may be describedvery well in terms of a mean-ﬁeld theory similar to the Hartree–Fock theoryfor atoms. This is in marked contrast to liquid 4 He, for which a mean-ﬁeldapproach is inapplicable due to the strong correlations induced by the inter-action between the atoms. Although the gases are dilute, interactions playan important role because temperatures are so low, and they give rise tocollective phenomena related to those observed in solids, quantum liquids,and nuclei. Experimentally the systems are attractive ones to work with,since they may be manipulated by the use of lasers and magnetic ﬁelds. Inaddition, interactions between atoms may be varied either by using diﬀerentatomic species, or, for species that have a Feshbach resonance, by changingthe strength of an applied magnetic or electric ﬁeld. A further advantageis that, because of the low density, ‘microscopic’ length scales are so largethat the structure of the condensate wave function may be investigated di-rectly by optical means. Finally, real collision processes play little role, andtherefore these systems are ideal for studies of interference phenomena andatom optics. The theoretical prediction of Bose–Einstein condensation dates back morethan 75 years. Following the work of Bose on the statistics of photons [4],Einstein considered a gas of non-interacting, massive bosons, and concludedthat, below a certain temperature, a ﬁnite fraction of the total number ofparticles would occupy the lowest-energy single-particle state [5]. In 1938Fritz London suggested the connection between the superﬂuidity of liquid4 He and Bose–Einstein condensation [6]. Superﬂuid liquid 4 He is the pro-totype Bose–Einstein condensate, and it has played a unique role in thedevelopment of physical concepts. However, the interaction between heliumatoms is strong, and this reduces the number of atoms in the zero-momentumstate even at absolute zero. Consequently it is diﬃcult to measure directlythe occupancy of the zero-momentum state. It has been investigated ex-perimentally by neutron scattering measurements of the structure factor atlarge momentum transfers [7], and the measurements are consistent with arelative occupation of the zero-momentum state of about 0.1 at saturatedvapour pressure and about 0.05 near the melting curve [8]. The fact that interactions in liquid helium reduce dramatically the oc-cupancy of the lowest single-particle state led to the search for weakly-interacting Bose gases with a higher condensate fraction. The diﬃculty with
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Introduction 3most substances is that at low temperatures they do not remain gaseous,but form solids, or, in the case of the helium isotopes, liquids, and theeﬀects of interaction thus become large. In other examples atoms ﬁrst com-bine to form molecules, which subsequently solidify. As long ago as in 1959Hecht [9] argued that spin-polarized hydrogen would be a good candidatefor a weakly-interacting Bose gas. The attractive interaction between twohydrogen atoms with their electronic spins aligned was then estimated tobe so weak that there would be no bound state. Thus a gas of hydrogenatoms in a magnetic ﬁeld would be stable against formation of moleculesand, moreover, would not form a liquid, but remain a gas to arbitrarily lowtemperatures. Hecht’s paper was before its time and received little attention, but hisconclusions were conﬁrmed by Stwalley and Nosanow [10] in 1976, when im-proved information about interactions between spin-aligned hydrogen atomswas available. These authors also argued that because of interatomic inter-actions the system would be a superﬂuid as well as being Bose–Einsteincondensed. This latter paper stimulated the quest to realize Bose–Einsteincondensation in atomic hydrogen. Initial experimental attempts used ahigh magnetic ﬁeld gradient to force hydrogen atoms against a cryogeni-cally cooled surface. In the lowest-energy spin state of the hydrogen atom,the electron spin is aligned opposite the direction of the magnetic ﬁeld (H↓),since then the magnetic moment is in the same direction as the ﬁeld. Spin-polarized hydrogen was ﬁrst stabilized by Silvera and Walraven [11]. Interac-tions of hydrogen with the surface limited the densities achieved in the earlyexperiments, and this prompted the Massachusetts Institute of Technology(MIT) group led by Greytak and Kleppner to develop methods for trappingatoms purely magnetically. In a current-free region, it is impossible to createa local maximum in the magnitude of the magnetic ﬁeld. To trap atoms bythe Zeeman eﬀect it is therefore necessary to work with a state of hydrogenin which the electronic spin is polarized parallel to the magnetic ﬁeld (H↑).Among the techniques developed by this group is that of evaporative coolingof magnetically trapped gases, which has been used as the ﬁnal stage in allexperiments to date to produce a gaseous Bose–Einstein condensate. Sincelaser cooling is not feasible for hydrogen, the gas is precooled cryogenically.After more than two decades of heroic experimental work, Bose–Einsteincondensation of atomic hydrogen was achieved in 1998 [12]. As a consequence of the dramatic advances made in laser cooling of alkaliatoms, such atoms became attractive candidates for Bose–Einstein conden-sation, and they were used in the ﬁrst successful experiments to producea gaseous Bose–Einstein condensate. Other atomic species, among them
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4 Introductionnoble gas atoms in excited states, are also under active investigation, andin 2001 two groups produced condensates of metastable 4 He atoms in thelowest spin-triplet state [13, 14]. The properties of interacting Bose ﬂuids are treated in many texts. Thereader will ﬁnd an illuminating discussion in the volume by Nozi`res and ePines [15]. A collection of articles on Bose–Einstein condensation in varioussystems, prior to its discovery in atomic vapours, is given in [16], whilemore recent theoretical developments have been reviewed in [17]. The 1998Varenna lectures describe progress in both experiment and theory on Bose–Einstein condensation in atomic gases, and contain in addition historicalaccounts of the development of the ﬁeld [18]. For a tutorial review of someconcepts basic to an understanding of Bose–Einstein condensation in dilutegases see Ref. [19]. 1.1 Bose–Einstein condensation in atomic cloudsBosons are particles with integer spin. The wave function for a systemof identical bosons is symmetric under interchange of any two particles.Unlike fermions, which have half-odd-integer spin and antisymmetric wavefunctions, bosons may occupy the same single-particle state. An order-of-magnitude estimate of the transition temperature to the Bose–Einsteincondensed state may be made from dimensional arguments. For a uniformgas of free particles, the relevant quantities are the particle mass m, thenumber density n, and the Planck constant h = 2π . The only energythat can be formed from , n, and m is 2 n2/3 /m. By dividing this energyby the Boltzmann constant k we obtain an estimate of the condensationtemperature Tc , 2 n2/3 Tc = C . (1.1) mkHere C is a numerical factor which we shall show in the next chapter tobe equal to approximately 3.3. When (1.1) is evaluated for the mass anddensity appropriate to liquid 4 He at saturated vapour pressure one obtainsa transition temperature of approximately 3.13 K, which is close to thetemperature below which superﬂuid phenomena are observed, the so-calledlambda point2 (Tλ = 2.17 K at saturated vapour pressure). An equivalent way of relating the transition temperature to the parti-cle density is to compare the thermal de Broglie wavelength λT with the2 The name lambda point derives from the measured shape of the speciﬁc heat as a function of temperature, which near the transition resembles the Greek letter λ.
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1.1 Bose–Einstein condensation in atomic clouds 5mean interparticle spacing, which is of order n−1/3 . The thermal de Brogliewavelength is conventionally deﬁned by 1/2 2π 2 λT = . (1.2) mkTAt high temperatures, it is small and the gas behaves classically. Bose–Einstein condensation in an ideal gas sets in when the temperature is so lowthat λT is comparable to n−1/3 . For alkali atoms, the densities achievedrange from 1013 cm−3 in early experiments to 1014 –1015 cm−3 in more re-cent ones, with transition temperatures in the range from 100 nK to a fewµK. For hydrogen, the mass is lower and the transition temperatures arecorrespondingly higher. In experiments, gases are non-uniform, since they are contained in a trap,which typically provides a harmonic-oscillator potential. If the number ofparticles is N , the density of gas in the cloud is of order N/R3 , where thesize R of a thermal gas cloud is of order (kT /mω0 )1/2 , ω0 being the angu- 2lar frequency of single-particle motion in the harmonic-oscillator potential.Substituting the value of the density n ∼ N/R3 at T = Tc into Eq. (1.1),one sees that the transition temperature is given by kTc = C1 ω0 N 1/3 , (1.3)where C1 is a numerical constant which we shall later show to be approx-imately 0.94. The frequencies for traps used in experiments are typicallyof order 102 Hz, corresponding to ω0 ∼ 103 s−1 , and therefore, for parti-cle numbers in the range from 104 to 107 , the transition temperatures liein the range quoted above. Estimates of the transition temperature basedon results for a uniform Bose gas are therefore consistent with those for atrapped gas. In the original experiment [1] the starting point was a room-temperaturegas of rubidium atoms, which were trapped and cooled to about 10 µKby bombarding them with photons from laser beams in six directions –front and back, left and right, up and down. Subsequently the lasers wereturned oﬀ and the atoms trapped magnetically by the Zeeman interactionof the electron spin with an inhomogeneous magnetic ﬁeld. If we neglectcomplications caused by the nuclear spin, an atom with its electron spinparallel to the magnetic ﬁeld is attracted to the minimum of the magneticﬁeld, while one with its electron spin antiparallel to the magnetic ﬁeld isrepelled. The trapping potential was provided by a quadrupole magneticﬁeld, upon which a small oscillating bias ﬁeld was imposed to prevent loss
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6 Introductionof particles at the centre of the trap. Some more recent experiments haveemployed other magnetic ﬁeld conﬁgurations. In the magnetic trap the cloud of atoms was cooled further by evapora-tion. The rate of evaporation was enhanced by applying a radio-frequencymagnetic ﬁeld which ﬂipped the electronic spin of the most energetic atomsfrom up to down. Since the latter atoms are repelled by the trap, they es-cape, and the average energy of the remaining atoms falls. It is remarkablethat no cryogenic apparatus was involved in achieving the record-low tem-peratures in the experiment [1]. Everything was held at room temperatureexcept the atomic cloud, which was cooled to temperatures of the order of100 nK. So far, Bose–Einstein condensation has been realized experimentally indilute gases of rubidium, sodium, lithium, hydrogen, and metastable heliumatoms. Due to the diﬀerence in the properties of these atoms and theirmutual interaction, the experimental study of the condensates has revealeda range of fascinating phenomena which will be discussed in later chapters.The presence of the nuclear and electronic spin degrees of freedom addsfurther richness to these systems when compared with liquid 4 He, and it givesthe possibility of studying multi-component condensates. From a theoreticalpoint of view, much of the appeal of Bose–Einstein condensed atomic cloudsstems from the fact that they are dilute in the sense that the scatteringlength is much less than the interparticle spacing. This makes it possible tocalculate the properties of the system with high precision. For a uniformdilute gas the relevant theoretical framework was developed in the 1950s and60s, but the presence of a conﬁning potential – essential to the observationof Bose–Einstein condensation in atomic clouds – gives rise to new featuresthat are absent for uniform systems. 1.2 Superﬂuid 4 HeMany of the concepts used to describe properties of quantum gases weredeveloped in the context of liquid 4 He. The helium liquids are exceptions tothe rule that liquids solidify when cooled to suﬃciently low temperatures,because the low mass of the helium atom makes the zero-point energy largeenough to overcome the tendency to crystallization. At the lowest temper-atures the helium liquids solidify only under a pressure in excess of 25 bar(2.5 MPa) for 4 He and 34 bar for the lighter isotope 3 He. Below the lambda point, liquid 4 He becomes a superﬂuid with many re-markable properties. One of the most striking is the ability to ﬂow throughnarrow channels without friction. Another is the existence of quantized vor-
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1.2 Superﬂuid 4 He 7ticity, the quantum of circulation being given by h/m (= 2π /m). Theoccurrence of frictionless ﬂow led Landau and Tisza to introduce a two-ﬂuiddescription of the hydrodynamics. The two ﬂuids – the normal and thesuperﬂuid components – are interpenetrating, and their densities dependon temperature. At very low temperatures the density of the normal com-ponent vanishes, while the density of the superﬂuid component approachesthe total density of the liquid. The superﬂuid density is therefore generallyquite diﬀerent from the density of particles in the condensate, which for liq-uid 4 He is only about 10 % or less of the total, as mentioned above. Near thetransition temperature to the normal state the situation is reversed: herethe superﬂuid density tends towards zero as the temperature approaches thelambda point, while the normal density approaches the density of the liquid. The properties of the normal component may be related to the elemen-tary excitations of the superﬂuid. The concept of an elementary excitationplays a central role in the description of quantum systems. In an ideal gasan elementary excitation corresponds to the addition of a single particle ina momentum eigenstate. Interactions modify this picture, but for low ex-citation energies there still exist excitations with well-deﬁned energies. Forsmall momenta the excitations in liquid 4 He are sound waves or phonons.Their dispersion relation is linear, the energy being proportional to themagnitude of the momentum p, = sp, (1.4)where the constant s is the velocity of sound. For larger values of p, thedispersion relation shows a slight upward curvature for pressures less than18 bar, and a downward one for higher pressures. At still larger momenta, (p) exhibits ﬁrst a local maximum and subsequently a local minimum. Nearthis minimum the dispersion relation may be approximated by (p − p0 )2 (p) = ∆ + , (1.5) 2m∗where m∗ is a constant with the dimension of mass and p0 is the momen-tum at the minimum. Excitations with momenta close to p0 are referredto as rotons. The name was coined to suggest the existence of vorticityassociated with these excitations, but they should really be considered asshort-wavelength phonon-like excitations. Experimentally, one ﬁnds at zeropressure that m∗ is 0.16 times the mass of a 4 He atom, while the constant∆, the energy gap, is given by ∆/k = 8.7 K. The roton minimum occurs ata wave number p0 / equal to 1.9 × 108 cm−1 (see Fig. 1.1). For excitation
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8 Introduction ∆ p p 0Fig. 1.1. The spectrum of elementary excitations in superﬂuid 4 He. The minimumroton energy is ∆, corresponding to the momentum p0 .energies greater than 2∆ the excitations become less well-deﬁned since theycan decay into two rotons. The elementary excitations obey Bose statistics, and therefore in thermalequilibrium the distribution function f 0 for the excitations is given by 1 f0 = . (1.6) e (p)/kT −1The absence of a chemical potential in this distribution function is due to thefact that the number of excitations is not a conserved quantity: the energy ofan excitation equals the diﬀerence between the energy of an excited state andthe energy of the ground state for a system containing the same number ofparticles. The number of excitations therefore depends on the temperature,just as the number of phonons in a solid does. This distribution functionEq. (1.6) may be used to evaluate thermodynamic properties. 1.3 Other condensatesThe concept of Bose–Einstein condensation ﬁnds applications in many sys-tems other than liquid 4 He and the clouds of spin-polarized boson alkaliatoms, atomic hydrogen, and metastable helium atoms discussed above. His-torically, the ﬁrst of these were superconducting metals, where the bosonsare pairs of electrons with opposite spin. Many aspects of the behaviour ofsuperconductors may be understood qualitatively on the basis of the ideathat pairs of electrons form a Bose–Einstein condensate, but the properties
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1.3 Other condensates 9of superconductors are quantitatively very diﬀerent from those of a weakly-interacting gas of pairs. The important physical point is that the bindingenergy of a pair is small compared with typical atomic energies, and at thetemperature where the condensate disappears the pairs themselves break up.This situation is to be contrasted with that for the atomic systems, wherethe energy required to break up an atom is the ionization energy, which is oforder electron volts. This corresponds to temperatures of tens of thousandsof degrees, which are much higher than the temperatures for Bose–Einsteincondensation. Many properties of high-temperature superconductors may be understoodin terms of Bose–Einstein condensation of pairs, in this case of holes ratherthan electrons, in states having predominantly d-like symmetry in contrastto the s-like symmetry of pairs in conventional metallic superconductors.The rich variety of magnetic and other behaviour of the superﬂuid phasesof liquid 3 He is again due to condensation of pairs of fermions, in this case3 He atoms in triplet spin states with p-wave symmetry. Considerable exper-imental eﬀort has been directed towards creating Bose–Einstein condensatesof excitons, which are bound states of an electron and a hole [20], and ofbiexcitons, molecules made up of two excitons [21]. Bose–Einstein condensation of pairs of fermions is also observed exper-imentally in atomic nuclei, where the eﬀects of neutron–neutron, proton–proton, and neutron–proton pairing may be seen in the excitation spec-trum as well as in reduced moments of inertia. A signiﬁcant diﬀerencebetween nuclei and superconductors is that the size of a pair in bulk nu-clear matter is large compared with the nuclear size, and consequently themanifestations of Bose–Einstein condensation in nuclei are less dramaticthan they are in bulk systems. Theoretically, Bose–Einstein condensationof nucleon pairs is expected to play an important role in the interiors ofneutron stars, and observations of glitches in the spin-down rate of pul-sars have been interpreted in terms of neutron superﬂuidity. The possibilityof mesons, either pions or kaons, forming a Bose–Einstein condensate inthe cores of neutron stars has been widely discussed, since this would havefar-reaching consequences for theories of supernovae and the evolution ofneutron stars [22]. In the ﬁeld of nuclear and particle physics the ideas of Bose–Einsteincondensation also ﬁnd application in the understanding of the vacuum as u ¯a condensate of quark–antiquark (u¯, dd and s¯) pairs, the so-called chiral scondensate. This condensate gives rise to particle masses in much the sameway as the condensate of electron pairs in a superconductor gives rise to thegap in the electronic excitation spectrum.
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10 Introduction This brief account of the rich variety of contexts in which the physics ofBose–Einstein condensation plays a role, shows that an understanding ofthe phenomenon is of importance in many branches of physics. 1.4 OverviewTo assist the reader, we give here a ‘road map’ of the material we cover.We begin, in Chapter 2, by discussing Bose–Einstein condensation for non-interacting gases in a conﬁning potential. This is useful for developing un-derstanding of the phenomenon of Bose–Einstein condensation and for ap-plication to experiment, since in dilute gases many quantities, such as thetransition temperature and the condensate fraction, are close to those pre-dicted for a non-interacting gas. We also discuss the density proﬁle and thevelocity distribution of an atomic cloud at zero temperature. When the ther-mal energy kT exceeds the spacing between the energy levels of an atom inthe conﬁning potential, the gas may be described semi-classically in terms ofa particle distribution function that depends on both position and momen-tum. We employ the semi-classical approach to calculate thermodynamicquantities. The eﬀect of ﬁnite particle number on the transition temperatureis estimated, and Bose–Einstein condensation in lower-dimensional systemsis discussed. In experiments to create a Bose–Einstein condensate in a dilute gas theparticles used have been primarily alkali atoms and hydrogen, whose spinsare non-zero. The new methods to trap and cool atoms that have beendeveloped in recent years make use of the basic atomic structure of theseatoms, which is the subject of Chapter 3. There we also study the energylevels of an atom in a static magnetic ﬁeld, which is a key element in thephysics of trapping, and discuss the atomic polarizability in an oscillatingelectric ﬁeld. A major experimental breakthrough that opened up this ﬁeld was the de-velopment of laser cooling techniques. In contrast to so many other proposalswhich in practice work less well than predicted theoretically, these turnedout to be far more eﬀective than originally estimated. Chapter 4 describesmagnetic traps, the use of lasers in trapping and cooling, and evaporativecooling, which is the key ﬁnal stage in experiments to make Bose–Einsteincondensates. In Chapter 5 we consider atomic interactions, which play a crucial rolein evaporative cooling and also determine many properties of the condensedstate. At low energies, interactions between particles are characterized bythe scattering length a, in terms of which the total scattering cross section
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1.4 Overview 11at low energies is given by 8πa2 for identical bosons. At ﬁrst sight, onemight expect that, since atomic sizes are typically of order the Bohr radius,scattering lengths would also be of this order. In fact they are one or twoorders of magnitude larger for alkali atoms, and we shall show how this maybe understood in terms of the long-range part of the interatomic force, whichis due to the van der Waals interaction. We also show that the sign of theeﬀective interaction at low energies depends on the details of the short-rangepart of the interaction. Following that we extend the theory to take intoaccount transitions between channels corresponding to the diﬀerent hyper-ﬁne states for the two atoms. We then estimate rates of inelastic processes,which are a mechanism for loss of atoms from traps, and present the theoryof Feshbach resonances, which may be used to tune atomic interactions byvarying the magnetic ﬁeld. Finally we list values of the scattering lengthsfor the alkali atoms currently under investigation. The ground-state energy of clouds in a conﬁning potential is the subject ofChapter 6. While the scattering lengths for alkali atoms are large comparedwith atomic dimensions, they are small compared with atomic separationsin gas clouds. As a consequence, the eﬀects of atomic interactions in theground state may be calculated very reliably by using a pseudopotential pro-portional to the scattering length. This provides the basis for a mean-ﬁelddescription of the condensate, which leads to the Gross–Pitaevskii equation.From this we calculate the energy using both variational methods and theThomas–Fermi approximation. When the atom–atom interaction is attrac-tive, the system becomes unstable if the number of particles exceeds a criticalvalue, which we calculate in terms of the trap parameters and the scatteringlength. We also consider the structure of the condensate at the surface ofa cloud, and the characteristic length for healing of the condensate wavefunction. In Chapter 7 we discuss the dynamics of the condensate at zero temper-ature, treating the wave function of the condensate as a classical ﬁeld. Wederive the coupled equations of motion for the condensate density and ve-locity, and use them to determine the elementary excitations in a uniformgas and in a trapped cloud. We describe methods for calculating collectiveproperties of clouds in traps. These include the Thomas–Fermi approxima-tion and a variational approach based on the idea of collective coordinates.The methods are applied to treat oscillations in both spherically-symmetricand anisotropic traps, and the free expansion of the condensate. We showthat, as a result of the combined inﬂuence of non-linearity and dispersion,there exist soliton solutions to the equations of motion for a Bose–Einsteincondensate.
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12 Introduction The microscopic, quantum-mechanical theory of the Bose gas is treated inChapter 8. We discuss the Bogoliubov approximation and show that it givesthe same excitation spectrum as that obtained from classical equations ofmotion in Chapter 7. At higher temperatures thermal excitations deplete thecondensate, and to treat these situations we discuss the Hartree–Fock andPopov approximations. Finally we analyse collisional shifts of spectral lines,such as the 1S–2S two-photon absorption line in spin-polarized hydrogen,which is used experimentally to probe the density of the gas, and lines usedas atomic clocks. One of the characteristic features of a superﬂuid is its response to ro-tation, in particular the occurrence of quantized vortices. We discuss inChapter 9 properties of vortices in atomic clouds and determine the criti-cal angular velocity for a vortex state to be energetically favourable. Wealso calculate the force on a moving vortex line from general hydrodynamicconsiderations. The nature of the lowest-energy state for a given angularmomentum is considered, and we discuss the weak-coupling limit, in whichthe interaction energy is small compared with the energy quantum of theharmonic-oscillator potential. In Chapter 10 we treat some basic aspects of superﬂuidity. The Landaucriterion for the onset of dissipation is discussed, and we introduce the two-ﬂuid picture, in which the condensate and the excitations may be regardedas forming two interpenetrating ﬂuids, each with temperature-dependentdensities. We calculate the damping of collective modes in a homogeneousgas at low temperatures, where the dominant process is Landau damping.As an application of the two-ﬂuid picture we derive the dispersion relationfor the coupled sound-like modes, which are referred to as ﬁrst and secondsound. Chapter 11 deals with particles in traps at non-zero temperature. Theeﬀects of interactions on the transition temperature and thermodynamicproperties are considered. We also discuss the coupled motion of the con-densate and the excitations at temperatures below Tc . We then presentcalculations for modes above Tc , both in the hydrodynamic regime, whencollisions are frequent, and in the collisionless regime, where we obtain themode attenuation from the kinetic equation for the particle distributionfunction. Chapter 12 discusses properties of mixtures of bosons, either diﬀerentbosonic isotopes, or diﬀerent internal states of the same isotope. In theformer case, the theory may be developed along lines similar to those fora single-component system. For mixtures of two diﬀerent internal statesof the same isotope, which may be described by a spinor wave function,
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1.4 Overview 13new possibilities arise because the number of atoms in each state is notconserved. We derive results for the static and dynamic properties of suchmixtures. An interesting result is that for an antiferromagnetic interactionbetween atomic spins, the simple Gross–Pitaevskii treatment fails, and theground state may be regarded as a Bose–Einstein condensate of pairs ofatoms, rather than of single atoms. In Chapter 13 we take up a number of topics related to interference andcorrelations in Bose–Einstein condensates and applications to matter waveoptics. First we describe interference between two Bose–Einstein condensedclouds, and explore the reasons for the appearance of an interference patterneven though the phase diﬀerence between the wave functions of particles inthe two clouds is not ﬁxed initially. We then demonstrate the suppressionof density ﬂuctuations in a Bose–Einstein condensed gas. Following thatwe consider how properties of coherent matter waves may be investigatedby manipulating condensates with lasers. The ﬁnal section considers thequestion of how to characterize Bose–Einstein condensation microscopically. Trapped Fermi gases are considered in Chapter 14. We ﬁrst show thatinteractions generally have less eﬀect on static and dynamic properties offermions than they do for bosons, and we then calculate equilibrium prop-erties of a free Fermi gas in a trap. The interaction can be important ifit is attractive, since at suﬃciently low temperatures the fermions are thenexpected to undergo a transition to a superﬂuid state similar to that for elec-trons in a metallic superconductor. We derive expressions for the transitiontemperature and the gap in the excitation spectrum at zero temperature,and we demonstrate that they are suppressed due to the modiﬁcation ofthe interaction between two atoms by the presence of other atoms. We alsoconsider how the interaction between fermions is altered by the addition ofbosons and show that this can enhance the transition temperature. Finallywe brieﬂy describe properties of sound modes in a superﬂuid Fermi gas,since measurement of collective modes has been proposed as a probe of thetransition to a superﬂuid state. ProblemsProblem 1.1 Consider an ideal gas of 87 Rb atoms at zero temperature,conﬁned by the harmonic-oscillator potential 1 V (r) = mω0 r2 , 2 2
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14 Introductionwhere m is the mass of a 87 Rb atom. Take the oscillator frequency ω0 to begiven by ω0 /2π = 150 Hz, which is a typical value for traps in current use.Determine the ground-state density proﬁle and estimate its width. Find theroot-mean-square momentum and velocity of a particle. What is the densityat the centre of the trap if there are 104 atoms?Problem 1.2 Determine the density proﬁle for the gas discussed in Prob-lem 1.1 in the classical limit, when the temperature T is much higher thanthe condensation temperature. Show that the central density may be written 3as N/Rth and determine Rth . At what temperature does the mean distancebetween particles at the centre of the trap become equal to the thermal deBroglie wavelength λT ? Compare the result with the transition temperature(1.3).Problem 1.3 Estimate the number of rotons contained in 1 cm3 of liquid4 He at temperatures T = 1 K and T = 100 mK at saturated vapour pressure. References [1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995). [2] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [3] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 78, 985 (1997). [4] S. N. Bose, Z. Phys. 26, 178 (1924). Bose’s paper dealt with the statistics of photons, for which the total number is not a ﬁxed quantity. He sent his paper to Einstein asking for comments. Recognizing its importance, Einstein translated the paper and submitted it for publication. Subsequently, Einstein extended Bose’s treatment to massive particles, whose total number is ﬁxed. [5] A. Einstein, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse (1924) p. 261; (1925) p. 3. [6] F. London, Nature 141, 643 (1938); Phys. Rev. 54, 947 (1938). [7] E. C. Svensson and V. F. Sears, in Progress in Low Temperature Physics, Vol. XI, ed. D. F. Brewer, (North-Holland, Amsterdam, 1987), p. 189. [8] P. E. Sokol, in Ref. [16], p. 51. [9] C. E. Hecht, Physica 25, 1159 (1959).[10] W. C. Stwalley and L. H. Nosanow, Phys. Rev. Lett. 36, 910 (1976).[11] I. F. Silvera and J. T. M. Walraven, Phys. Rev. Lett. 44, 164 (1980).[12] D. G. Fried, T. C. Killian, L. Willmann, D. Landhuis, S. C. Moss, D. Kleppner, and T. J. Greytak, Phys. Rev. Lett. 81, 3811 (1998).[13] A. Robert, O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. Westbrook, and A. Aspect, Science 292, 461 (2001).
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References 15[14] F. Pereira Dos Santos, J. L´onard, J. Wang, C. J. Barrelet, F. Perales, E. e Rasel, C. S. Unnikrishnan, M. Leduc, and C. Cohen-Tannoudji, Phys. Rev. Lett. 86, 3459 (2001).[15] P. Nozi`res and D. Pines, The Theory of Quantum Liquids, Vol. II, e (Addison-Wesley, Reading, Mass., 1990).[16] Bose–Einstein Condensation, ed. A. Griﬃn, D. W. Snoke, and S. Stringari, (Cambridge Univ. Press, Cambridge, 1995).[17] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).[18] Bose–Einstein Condensation in Atomic Gases, Proceedings of the Enrico Fermi International School of Physics, Vol. CXL, ed. M. Inguscio, S. Stringari, and C. E. Wieman, (IOS Press, Amsterdam, 1999).[19] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001). ´ u[20] K. E. O’Hara, L. O S´illeabh´in, and J. P. Wolfe, Phys. Rev. B 60, 10 565 a (1999).[21] A. Mysyrowicz, in Ref. [16], p. 330.[22] G. E. Brown, in Ref. [16], p. 438.
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2 The non-interacting Bose gasThe topic of Bose–Einstein condensation in a uniform, non-interacting gasof bosons is treated in most textbooks on statistical mechanics [1]. In thepresent chapter we discuss the properties of a non-interacting Bose gas in atrap. We shall calculate equilibrium properties of systems in a semi-classicalapproximation, in which the energy spectrum is treated as a continuum.For this approach to be valid the temperature must be large compared with∆ /k, where ∆ denotes the separation between neighbouring energy levels.As is well known, at temperatures below the Bose–Einstein condensationtemperature, the lowest energy state is not properly accounted for if onesimply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is thesingle-particle density of states, which is a key ingredient in the calculationsof thermodynamic properties. Calculations of the transition temperatureand the fraction of particles in the condensate are described in Sec. 2.2. InSec. 2.3 the semi-classical distribution function is introduced, and from thiswe determine the density proﬁle and the velocity distribution of particles.Thermodynamic properties of Bose gases are calculated as functions of thetemperature in Sec. 2.4. The ﬁnal two sections are devoted to eﬀects notcaptured by the simplest version of the semi-classical approximation: cor-rections to the transition temperature due to a ﬁnite particle number (Sec.2.5), and thermodynamic properties of gases in lower dimensions (Sec. 2.6). 2.1 The Bose distributionFor non-interacting bosons in thermodynamic equilibrium, the mean occu-pation number of the single-particle state ν is given by the Bose distribution 16
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2.1 The Bose distribution 17function, 1 f 0( ν ) = , (2.1) e( ν −µ)/kT −1where ν denotes the energy of the single-particle state for the particulartrapping potential under consideration. Since the number of particles isconserved, unlike the number of elementary excitations in liquid 4 He, thechemical potential µ enters the distribution function (2.1). The chemicalpotential is determined as a function of N and T by the condition thatthe total number of particles be equal to the sum of the occupancies ofthe individual levels. It is sometimes convenient to work in terms of thequantity ζ = exp(µ/kT ), which is known as the fugacity. If we take the zeroof energy to be that of the lowest single-particle state, the fugacity is lessthan unity above the transition temperature and equal to unity (to withinterms of order 1/N , which we shall generally neglect) in the condensed state.In Fig. 2.1 the distribution function (2.1) is shown as a function of energyfor various values of the fugacity. At high temperatures, the eﬀects of quantum statistics become negligible,and the distribution function (2.1) is given approximately by the Boltzmanndistribution f 0( ν ) e−( ν −µ)/kT . (2.2)For particles in a box of volume V the index ν labels the allowed wavevectors q for plane-wave states V −1/2 exp(iq·r), and the particle energy is = 2 q 2 /2m. The distribution (2.2) is thus a Maxwellian one for the velocityv = q/m. At high temperatures the chemical potential is much less than min , theenergy of the lowest single-particle state, since the mean occupation numberof any state is much less than unity, and therefore, in particular, exp[(µ − min )/kT ] 1. As the temperature is lowered, the chemical potential risesand the mean occupation numbers increase. However, the chemical potentialcannot exceed min , otherwise the Bose distribution function (2.2) evaluatedfor the lowest single-particle state would be negative, and hence unphysical.Consequently the mean occupation number of any excited single-particlestate cannot exceed the value 1/{exp[( ν − min )/kT ] − 1}. If the totalnumber of particles in excited states is less than N , the remaining particlesmust be accommodated in the single-particle ground state, whose occupationnumber can be arbitrarily large: the system has a Bose–Einstein condensate.The highest temperature at which the condensate exists is referred to as theBose–Einstein transition temperature and we shall denote it by Tc . As we
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18 The non-interacting Bose gasFig. 2.1. The Bose distribution function f 0 as a function of energy for diﬀerentvalues of the fugacity ζ. The value ζ = 1 corresponds to temperatures below thetransition temperature, while ζ = 0.5 and ζ = 0.25 correspond to µ = −0.69kTand µ = −1.39kT , respectively.shall see in more detail in Sec. 2.2, the energy dependence of the single-particle density of states at low energies determines whether or not Bose–Einstein condensation will occur for a particular system. In the condensedstate, at temperatures below Tc , the chemical potential remains equal to min , to within terms of order kT /N , which is small for large N , and theoccupancy of the single-particle ground state is macroscopic in the sense thata ﬁnite fraction of the particles are in this state. The number of particlesN0 in the single-particle ground state equals the total number of particles Nminus the number of particles Nex occupying higher-energy (excited) states. 2.1.1 Density of statesWhen calculating thermodynamic properties of gases it is common to replacesums over states by integrals, and to use a density of states in which detailsof the level structure are smoothed out. This procedure fails for a Bose–Einstein condensed system, since the contribution from the lowest state isnot properly accounted for. However, it does give a good approximation
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2.1 The Bose distribution 19to the contribution from excited states, and we shall now calculate thesesmoothed densities of states for a number of diﬀerent situations. Throughout most of this book we shall assume that all particles are inone particular internal (spin) state, and therefore we generally suppress thepart of the wave function referring to the internal state. In Chapters 12–14we discuss a number of topics where internal degrees of freedom come intoplay. In three dimensions, for a free particle in a particular internal state, thereis on average one quantum state per volume (2π )3 of phase space. Theregion of momentum space for which the magnitude of the momentum isless than p has a volume 4πp3 /3 equal to that of a sphere of radius p and,since the energy of a particle of momentum p is given by p = p2 /2m, thetotal number of states G( ) with energy less than is given by 4π (2m )3/2 21/2 (m )3/2 G( ) = V =V , (2.3) 3 (2π )3 3π 2 3where V is the volume of the system. Quite generally, the number of stateswith energy between and + d is given by g( )d , where g( ) is the densityof states. Therefore dG( ) g( ) = , (2.4) dwhich, from Eq. (2.3), is thus given by V m3/2 1/2 g( ) = . (2.5) 21/2 π 2 3For free particles in d dimensions the corresponding result is g( ) ∝ (d/2−1) ,and therefore the density of states is independent of energy for a free particlein two dimensions. Let us now consider a particle in the anisotropic harmonic-oscillator po-tential 1 V (r) = (K1 x2 + K2 y 2 + K3 z 2 ), (2.6) 2which we will refer to as a harmonic trap. Here the quantities Ki denotethe three force constants, which are generally unequal. The corresponding 2classical oscillation frequencies ωi are given by ωi = Ki /m, and we shalltherefore write the potential as 1 V (r) = m(ω1 x2 + ω2 y 2 + ω3 z 2 ). 2 2 2 (2.7) 2
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20 The non-interacting Bose gasThe energy levels, (n1 , n2 , n3 ), are then 1 1 1 (n1 , n2 , n3 ) = (n1 + ) ω1 + (n2 + ) ω2 + (n3 + ) ω3 , (2.8) 2 2 2where the numbers ni assume all integer values greater than or equal tozero. We now determine the number of states G( ) with energy less than agiven value . For energies large compared with ωi , we may treat the nias continuous variables and neglect the zero-point motion. We thereforeintroduce a coordinate system deﬁned by the three variables i = ωi ni , interms of which a surface of constant energy (2.8) is the plane = 1 + 2 + 3 .Then G( ) is proportional to the volume in the ﬁrst octant bounded by theplane, − − 1− 3 1 1 2 G( ) = 3ω ω ω d 1 d 2 d 3 = . (2.9) 1 2 3 0 0 0 6 3 ω1 ω2 ω3Since g( ) = dG/d , we obtain a density of states given by 2 g( ) = . (2.10) 2 3 ω1 ω2 ω3For a d-dimensional harmonic-oscillator potential, the analogous result is d−1 g( ) = d . (2.11) (d − 1)! i=1 ωi We thus see that in many contexts the density of states varies as a powerof the energy, and we shall now calculate thermodynamic properties forsystems with a density of states of the form α−1 g( ) = Cα , (2.12)where Cα is a constant. In three dimensions, for a gas conﬁned by rigidwalls, α is equal to 3/2. The corresponding coeﬃcient may be read oﬀ fromEq. (2.5), and it is V m3/2 C3/2 = . (2.13) 21/2 π 2 3The coeﬃcient for a three-dimensional harmonic-oscillator potential (α = 3),which may be obtained from Eq. (2.10), is 1 C3 = 3ω . (2.14) 2 1 ω2 ω3For particles in a box or in a harmonic-oscillator potential, α is equal to halfthe number of classical degrees of freedom per particle.
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2.2 Transition temperature and condensate fraction 21 2.2 Transition temperature and condensate fractionThe transition temperature Tc is deﬁned as the highest temperature at whichthe macroscopic occupation of the lowest-energy state appears. When thenumber of particles, N , is suﬃciently large, we may neglect the zero-pointenergy in (2.8) and thus equate the lowest energy min to zero, the minimumof the potential (2.6). Corrections to the transition temperature arising fromthe zero-point energy will be considered in Sec. 2.5. The number of particlesin excited states is given by ∞ Nex = d g( )f 0 ( ). (2.15) 0This achieves its greatest value for µ = 0, and the transition temperatureTc is determined by the condition that the total number of particles can beaccommodated in excited states, that is ∞ 1 N = Nex (Tc , µ = 0) = d g( ) . (2.16) 0 e /kTc −1When (2.16) is written in terms of the dimensionless variable x = /kTc , itbecomes ∞ xα−1 N = Cα (kTc )α dx = Cα Γ(α)ζ(α)(kTc )α , (2.17) 0 ex − 1where Γ(α) is the gamma function and ζ(α) = ∞ n−α is the Riemann n=1zeta function. In evaluating the integral in (2.17) we expand the Bose func- ∞tion in powers of e−x , and use the fact that 0 dxxα−1 e−x = Γ(α). Theresult is ∞ xα−1 dx x = Γ(α)ζ(α). (2.18) 0 e −1Table 2.1 lists Γ(α) and ζ(α) for selected values of α. From (2.17) we now ﬁnd N 1/α kTc = . (2.19) [Cα Γ(α)ζ(α)]1/αFor a three-dimensional harmonic-oscillator potential, α is 3 and C3 is givenby Eq. (2.14). From (2.19) we then obtain a transition temperature givenby ω N 1/3 ¯ kTc = ≈ 0.94 ω N 1/3 , ¯ (2.20) [ζ(3)]1/3where ω = (ω1 ω2 ω3 )1/3 ¯ (2.21)
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22 The non-interacting Bose gas Table 2.1. The gamma function Γ and the Riemann zeta function ζ for selected values of α. α Γ(α) ζ(α) 1 √ 1 ∞ 1.5 π/2 = 0.886 2.612 2 √ 1 π 2 /6 = 1.645 2.5 3 π/4 = 1.329 1.341 3 √ 2 1.202 3.5 15 π/8 = 3.323 1.127 4 6 π 4 /90 = 1.082is the geometric mean of the three oscillator frequencies. The result (2.20)may be written in the useful form f¯ Tc ≈ 4.5 N 1/3 nK, (2.22) 100 Hz ¯ ¯where f = ω /2π. For a uniform Bose gas in a three-dimensional box of volume V , corre-sponding to α = 3/2, the constant C3/2 is given by Eq. (2.13) and thus thetransition temperature is given by 2π 2 n2/3 2 n2/3 kTc = ≈ 3.31 , (2.23) [ζ(3/2)]2/3 m mwhere n = N/V is the number density. For a uniform gas in two dimensions,α is equal to 1, and the integral in (2.17) diverges. Thus Bose–Einsteincondensation in a two-dimensional box can occur only at zero temperature.However, a two-dimensional Bose gas can condense at non-zero temperatureif the particles are conﬁned by a harmonic-oscillator potential. In that caseα = 2 and the integral in (2.17) is ﬁnite. We shall return to gases in lowerdimensions in Sec. 2.6. It is useful to introduce the phase-space density, which we denote by .This is deﬁned as the number of particles contained within a volume equalto the cube of the thermal de Broglie wavelength, λ3 = (2π 2 /mkT )3/2 , T 3/2 2π 2 =n . (2.24) mkTIf the gas is classical, this is a measure of the typical occupancy of single-particle states. The majority of occupied states have energies of order kT
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2.2 Transition temperature and condensate fraction 23or less, and therefore the number of states per unit volume that are oc-cupied signiﬁcantly is of order the total number of states per unit volumewith energies less than kT , which is approximately (mkT / 2 )3/2 accordingto (2.3). The phase-space density is thus the ratio between the particledensity and the number of signiﬁcantly occupied states per unit volume.The Bose–Einstein phase transition occurs when = ζ(3/2) ≈ 2.612, ac-cording to (2.23). The criterion that should be comparable with unityindicates that low temperatures and/or high particle densities are necessaryfor condensation. The existence of a well-deﬁned phase transition for particles in a harmonic-oscillator potential is a consequence of our assumption that the separation ofsingle-particle energy levels is much less than kT . For an isotropic harmonicoscillator, with ω1 = ω2 = ω3 = ω0 , this implies that the energy quantum ω0 should be much less than kTc . Since Tc is given by Eq. (2.20), thecondition is N 1/3 1. If the ﬁniteness of the particle number is taken intoaccount, the transition becomes smooth. 2.2.1 Condensate fractionBelow the transition temperature the number Nex of particles in excitedstates is given by Eq. (2.15) with µ = 0, ∞ α−1 1 Nex (T ) = Cα d . (2.25) 0 e /kT −1Provided the integral converges, that is α > 1, we may use Eq. (2.18) towrite this result as Nex = Cα Γ(α)ζ(α)(kT )α . (2.26)Note that this result does not depend on the total number of particles.However, if one makes use of the expression (2.19) for Tc , it may be rewrittenin the form T α Nex = N . (2.27) TcThe number of particles in the condensate is thus given by N0 (T ) = N − Nex (T ) (2.28)or α T N0 = N 1 − . (2.29) Tc
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24 The non-interacting Bose gasFor particles in a box in three dimensions, α is 3/2, and the number ofexcited particles nex per unit volume may be obtained from Eqs. (2.26) and(2.13). It is 3/2 Nex mkT nex = = ζ(3/2) . (2.30) V 2π 2The occupancy of the condensate is therefore given by the well-known resultN0 = N [1 − (T /Tc )3/2 ]. For a three-dimensional harmonic-oscillator potential (α = 3), the numberof particles in the condensate is 3 T N0 = N 1 − . (2.31) TcIn all cases the transition temperatures Tc are given by (2.19) for the ap-propriate value of α. 2.3 Density proﬁle and velocity distributionThe cold clouds of atoms which are investigated at microkelvin temperaturestypically contain of order 104 –107 atoms. It is not feasible to apply the usualtechniques of low-temperature physics to these systems for a number of rea-sons. First, there are rather few atoms, second, the systems are metastable,so one cannot allow them to come into equilibrium with another body, andthird, the systems have a lifetime which is of order seconds to minutes. Among the quantities that can be measured is the density proﬁle. Oneway to do this is by absorptive imaging. Light at a resonant frequencyfor the atom will be absorbed on passing through an atomic cloud. Thusby measuring the absorption proﬁle one can obtain information about thedensity distribution. The spatial resolution can be improved by allowing thecloud to expand before measuring the absorptive image. A drawback of thismethod is that it is destructive, since absorption of light changes the internalstates of atoms and heats the cloud signiﬁcantly. To study time-dependentphenomena it is therefore necessary to prepare a new cloud for each timepoint. An alternative technique is to use phase-contrast imaging [2, 3]. Thisexploits the fact that the refractive index of the gas depends on its density,and therefore the optical path length is changed by the medium. By allowinga light beam that has passed through the cloud to interfere with a referencebeam that has been phase shifted, changes in optical path length may beconverted into intensity variations, just as in phase-contrast microscopy.
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2.3 Density proﬁle and velocity distribution 25The advantage of this method is that it is almost non-destructive, and it istherefore possible to study time-dependent phenomena using a single cloud. The distribution of particles after a cloud is allowed to expand dependsnot only on the initial density distribution, but also on the initial velocitydistribution. Consequently it is important to consider both density andvelocity distributions. In the ground state of the system, all atoms are condensed in the lowestsingle-particle quantum state and the density distribution n(r) reﬂects theshape of the ground-state wave function φ0 (r) for a particle in the trap since,for non-interacting particles, the density is given by n(r) = N |φ0 (r)|2 , (2.32)where N is the number of particles. For an anisotropic harmonic oscillatorthe ground-state wave function is 1 e−x /2a1 e−y /2a2 e−z /2a3 , 2 2 2 2 2 2 φ0 (r) = (2.33) π 3/4 (a1 a2 a3 )1/2where the widths ai of the wave function in the three directions are givenby a2 = i . (2.34) mωiThe density distribution is thus anisotropic if the three frequencies ω1 , ω2and ω3 are not all equal, the greatest width being associated with the lowestfrequency. The widths ai may be written in a form analogous to (2.22) 1/2 100 Hz 1 ai ≈ 10.1 µm, (2.35) fi Ain terms of the trap frequencies fi = ωi /2π and the mass number A, thenumber of nucleons in the nucleus of the atom. In momentum space the wave function corresponding to (2.33) is obtainedby taking its Fourier transform and is 1 e−px /2c1 e−py /2c2 e−pz /2c3 , 2 2 2 2 2 2 φ0 (p) = (2.36) π 3/4 (c1 c2 c3 )1/2where ci = = m ωi . (2.37) aiThe density in momentum space corresponding to (2.32) is given by N e−px /c1 e−py /c2 e−pz /c3 . 2 2 2 2 2 2 n(p) = N |φ0 (p)|2 = (2.38) π 3/2 c1 c2 c3
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26 The non-interacting Bose gasSince c2 /m = ωi , the distribution (2.38) has the form of a Maxwell distri- ibution with diﬀerent ‘temperatures’ Ti = ωi /2k for the three directions. Since the spatial distribution is anisotropic, the momentum distributionalso depends on direction. By the uncertainty principle, a narrow spatialdistribution implies a broad momentum distribution, as seen in the Fouriertransform (2.36) where the widths ci are proportional to the square root ofthe oscillator frequencies. These density and momentum distributions may be contrasted with thecorresponding expressions when the gas obeys classical statistics, at temper-atures well above the Bose–Einstein condensation temperature. The densitydistribution is then proportional to exp[−V (r)/kT ] and consequently it isgiven by N e−x /R1 e−y /R2 e−z /R3 . 2 2 2 2 2 2 n(r) = (2.39) π 3/2 R 1 R2 R3Here the widths Ri are given by 2 2kT Ri = 2, (2.40) mωiand they therefore depend on temperature. Note that the ratio Ri /ai equals(2kT / ωi )1/2 , which under typical experimental conditions is much greaterthan unity. Consequently the condition for semi-classical behaviour is wellsatisﬁed, and one concludes that the thermal cloud is much broader thanthe condensate, which below Tc emerges as a narrow peak in the spatialdistribution with a weight that increases with decreasing temperature. Above Tc the density n(p) in momentum space is isotropic in equilibrium,since it is determined only by the temperature and the particle mass, andin the classical limit it is given by n(p) = Ce−p 2 /2mkT , (2.41)where the constant C is independent of momentum. The width of the mo-mentum distribution is thus ∼ (mkT )1/2 , which is ∼ (kT / ωi )1/2 timesthe zero-temperature width (m ωi )1/2 . At temperatures comparable withthe transition temperature one has kT ∼ N 1/3 ωi and therefore the factor(kT / ωi )1/2 is of the order of N 1/6 . The density and velocity distributionsof the thermal cloud are thus much broader than those of the condensate. If a thermal cloud is allowed to expand to a size much greater than its orig-inal one, the resulting cloud will be spherically symmetric due to the isotropyof the velocity distribution. This is quite diﬀerent from the anisotropic shapeof an expanding cloud of condensate. In early experiments the anisotropy of
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2.3 Density proﬁle and velocity distribution 27clouds after expansion provided strong supporting evidence for the existenceof a Bose–Einstein condensate. Interactions between the atoms alter the sizes of clouds somewhat, as weshall see in Sec. 6.2. A repulsive interaction expands the zero-temperaturecondensate cloud by a numerical factor which depends on the number ofparticles and the interatomic potential, typical values being in the rangebetween 2 and 10, while an attractive interaction can cause the cloud tocollapse. Above Tc , where the cloud is less dense, interactions hardly aﬀectthe size of the cloud. 2.3.1 The semi-classical distributionQuantum-mechanically, the density of non-interacting bosons is given by n(r) = fν |φν (r)|2 , (2.42) νwhere fν is the occupation number for state ν, for which the wave function isφν (r). Such a description is unwieldy in general, since it demands a knowl-edge of the wave functions for the trapping potential in question. However,provided the de Broglie wavelengths of particles are small compared withthe length scale over which the trapping potential varies signiﬁcantly, it ispossible to use a simpler description in terms of a semi-classical distributionfunction fp (r). This is deﬁned such that fp (r)dpdr/(2π )3 denotes the meannumber of particles in the phase-space volume element dpdr. The physicalcontent of this approximation is that locally the gas may be regarded ashaving the same properties as a bulk gas. We have used this approximationto discuss the high-temperature limit of Boltzmann statistics, but it mayalso be used under conditions when the gas is degenerate. The distributionfunction in equilibrium is therefore given by 0 1 fp (r) = fp (r) = . (2.43) e[ p (r)−µ]/kT −1Here the particle energies are those of a classical free particle at point r, p2 p (r) = + V (r), (2.44) 2mwhere V (r) is the external potential. This description may be used for particles in excited states, but it is inap-propriate for the ground state, which has spatial variations on length scalescomparable with those over which the trap potential varies signiﬁcantly.Also, calculating properties of the system by integrating over momentum
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28 The non-interacting Bose gasstates does not properly take into account the condensed state, but prop-erties of particles in excited states are well estimated by the semi-classicalresult. Thus, for example, to determine the number of particles in excitedstates, one integrates the semi-classical distribution function (2.43) dividedby (2π )3 over p and r. The results for Tc agree with those obtained by themethods described in Sec. 2.2 above, where the eﬀect of the potential wasincluded through the density of states. To demonstrate this for a harmonictrap is left as an exercise (Problem 2.1). We now consider the density of particles which are not in the condensate.This is given by dp 1 nex (r) = . (2.45) (2π )3 e[ p (r)−µ]/kT −1We evaluate the integral (2.45) by introducing the variable x = p2 /2mkTand the quantity z(r) deﬁned by the equation z(r) = e[µ−V (r)]/kT . (2.46)For V (r) = 0, z reduces to the fugacity. One ﬁnds ∞ 2 x1/2 nex (r) = √ dx , (2.47) πλ3 T 0 z −1 ex − 1where λT = (2π 2 /mkT )1/2 is the thermal de Broglie wavelength, Eq. (1.2).Integrals of this type occur frequently in expressions for properties of idealBose gases, so we shall consider a more general one. They are evaluated byexpanding the integrand in powers of z, and one ﬁnds ∞ ∞ ∞ xγ−1 dx = dxxγ−1 e−nx z n 0 z −1 ex − 1 n=1 0 = Γ(γ)gγ (z), (2.48)where ∞ zn gγ (z) = . (2.49) nγ n=1For z = 1, the sum in (2.49) reduces to ζ(γ), in agreement with (2.18). The integral in (2.47) corresponds to γ = 3/2, and therefore g3/2 (z(r)) nex (r) = . (2.50) λ3TIn Fig. 2.2 we show for a harmonic trap the density of excited particles inunits of 1/λ3 for a chemical potential equal to the minimum of the potential. T
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2.4 Thermodynamic quantities 29Fig. 2.2. The spatial distribution of non-condensed particles, Eq. (2.50), for anisotropic trap, V (r) = mω0 r2 /2, with R = (2kT /mω0 )1/2 . The dotted line is a 2 2Gaussian distribution, corresponding to the ﬁrst term in the sum (2.49).This gives the distribution of excited particles at the transition temperatureor below. For comparison the result for the classical Boltzmann distribution,which corresponds to the ﬁrst term in the series (2.49), is also exhibited forthe same value of µ. Note that in the semi-classical approximation thedensity has a cusp at the origin, whereas in a more precise treatment thiswould be smoothed over a length scale of order λT . For a harmonic trapabove the transition temperature, the total number of particles is related tothe chemical potential by 3 kT N = g3 (z(0)) , (2.51) ω ¯as one can verify by integrating (2.45) over space. 2.4 Thermodynamic quantitiesIn this section we determine thermodynamic properties of ideal Bose gasesand calculate the energy, entropy, and other properties of the condensedphase. We explore how the temperature dependence of the speciﬁc heatfor temperatures close to Tc depends on the parameter α characterizing thedensity of states.
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30 The non-interacting Bose gas 2.4.1 Condensed phaseThe energy of the macroscopically occupied state is taken to be zero, andtherefore only excited states contribute to the total energy of the system.Consequently in converting sums to integrals it is not necessary to include anexplicit term for the condensate, as it is when calculating the total numberof particles. Below Tc , the chemical potential vanishes, and the internalenergy is given by ∞ α−1 E = Cα d = Cα Γ(α + 1)ζ(α + 1)(kT )α+1 , (2.52) 0 e /kT −1where we have used the integral (2.18). The speciﬁc heat C = ∂E/∂T istherefore given by1 E C = (α + 1) . (2.53) TSince the speciﬁc heat is also given in terms of the entropy S by C =T ∂S/∂T , we ﬁnd C α+1E S= = . (2.54) α α TNote that below Tc the energy, entropy, and speciﬁc heat do not depend onthe total number of particles. This is because only particles in excited statescontribute, and consequently the number of particles in the macroscopicallyoccupied state is irrelevant for these quantities. Expressed in terms of the total number of particles N and the transitiontemperature Tc , which are related by Eq. (2.19), the energy is given by ζ(α + 1) T α+1 E = N kα α , (2.55) ζ(α) Tcwhere we have used the property of the gamma function that Γ(z + 1) =zΓ(z). As a consequence, the speciﬁc heat is given by α ζ(α + 1) T C = α(α + 1) Nk , (2.56) ζ(α) Tcwhile the entropy is α ζ(α + 1) T S = (α + 1) Nk . (2.57) ζ(α) TcLet us compare the results above with those in the classical limit. At high1 The speciﬁc heat C is the temperature derivative of the internal energy, subject to the condition that the trap parameters are unchanged. For particles in a box, C is thus the speciﬁc heat at constant volume.
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