Fundamentals and New Frontiers Bose-Einstein Condensation

FUNDAMENTALSNEW FRONTIERSBOSE-EINSTEINCONDENSATION

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N E W J E R S E Y L O N D O N S I N G A P O R E B E I J I N G S H A N G H A I H O N G K O N G TA I P E I C H E N N A I

World Scientific

FUNDAMENTALSNEW FRONTIERSBOSE-EINSTEINCONDENSATION

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Masahito UedaUniversity of Tokyo, Japan

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British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

For photocopying of material in this volume, please pay a copying fee through the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission tophotocopy is not required from the publisher.

ISBN-13 978-981-283-959-6ISBN-10 981-283-959-3

Desk Editor: Ryan Bong

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FUNDAMENTALS AND NEW FRONTIERS OF BOSE–EINSTEIN CONDENSATION

Ryan - Fundamentals and New Frontiers.pmd 6/28/2010, 4:27 PM1

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June 28, 2010 14:0 World Scientific Book - 9in x 6in NewFrontiers

Preface

Experimental realization of Bose–Einstein condensation (BEC) of diluteatomic gases [Anderson, et al. (1995); Davis, et al. (1995); Bradley,et al. (1995, 1997)] has ignited a virtual explosion of research. Theunique feature of the atomic gas BEC is its unprecedented controllabil-ity, which makes the previously unthinkable possible. Almost all parame-ters of the system such as the temperature, number of atoms, and evenstrength and sign (attractive or repulsive) of interaction can be variedby several orders of magnitude. The interaction between atoms is usu-ally considered to be an immutable, inherent property of individual atomicspecies. In alkali and some other Bose–Einstein condensates, we can notonly control the strength of interaction but also switch the sign of inter-action from repulsive to attractive and vice versa [Inouye, et al. (1998);Cornish (2000)]. The atomic-gas BEC may thus be regarded as an ar-tificial macroscopic matter wave that act as an ideal testing ground forthe investigation of quantum many-body physics. The atomic-gas BECmay also be regarded as an atom laser because the condensate provides aphase-coherent, intense atomic source with potential applications for pre-cision measurement, lithography, and quantum computation. Fermionicspecies may also undergo BEC by forming molecules or Cooper pairs. Bothmolecular condensates [Greiner, et al. (2003); Zwierlein, et al. (2003)] andBardeen–Cooper–Schrieffer-type resonant superfluids [Regal, et al. (2004);Zwierlein, et al. (2004)] have been realized using alkali fermions, openingup the new research field of strongly correlated gaseous superfluidity. Thisbook is intended as an introduction to this rapidly developing, interdisci-plinary field of research.

Most phase transitions occur due to interactions between constituentparticles. For example, superconductivity occurs due to effective interac-

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vi Fundamentals and New Frontiers of Bose–Einstein Condensation

tions between electrons, and ferromagnetism is caused by the exchangeinteraction between spins. In contrast, BEC is a genuinely quantum-statistical phase transition in that it occurs without the help of interaction(Einstein called it “condensation without interaction” [Einstein (1925)]).The fundamentals of noninteracting BECs are reviewed in Chapter 1.

In a real BEC system, interactions between atoms play a crucial rolein determining the basic properties of the system. Neutral atoms have ahard core that is short-ranged (∼ 1 A) and strongly repulsive. At a longerdistance (∼ 100 A), the atoms are attracted to each other because of the vander Waals force. When two atoms collide, they experience both these forces,and the net interaction can be either repulsive or attractive depending onthe hyperfine and translational states of the colliding atoms. Under normalconditions, a dilute-gas BEC system can be treated as a weakly interactingBose gas. The Bogoliubov theory of a weakly interacting Bose gas andrelated topics are described in Chapter 2.

One of the remarkable aspects of a dilute gas BEC system is the greatsuccess of the mean-field theory governed by the Gross–Pitaevskii (GP)equation [Gross (1961); Pitaevskii (1961)]. The GP equation describes themean-field ground state as well as the linear and nonlinear response ofthe system. Various nonlinear matter-wave phenomena including four-wavemixing [Deng, et al. (1999); Rolston and Phillips (2002)] and topological ex-citations such as solitons [Denschlag, et al. (2000)] and vortices [Matthews,et al. (1999); Madison, et al. (2000)], have been successfully described bythe GP equation. This remarkable success of the mean-field theory is dueto the high (> 99%) degree of condensation of bosons into a single-particlestate, which in turn originates in an extremely low density (∼ 1011 − 1015

cm−3) of the system operating at ultralow temperatures (! 10−6 K). TheGross–Pitaevskii theory together with its various applications is discussedin Chapter 3.

The linear response theory provides a general theoretical framework toinvestigate collective modes of Bose–Einstein condensates and superfluids.A sum-rule approach is also very useful for this purpose because the groundstate for a dilute-gas Bose–Einstein condensate can be obtained very accu-rately. These subjects are discussed in Chapter 4.

Superfluidity manifests itself as a response of the system to its movingcontainer. A statistical-mechanical theory to tackle such problems andsome basic properties of superfluidity are described in Chapter 5.

Alkali atoms have both electronic spin s and nuclear spin i, and thesetwo spins interact with each other via the hyperfine interaction. When the

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Preface vii

energy of the hyperfine coupling exceeds the electronic and nuclear Zeemanenergies as well as the thermal energy, the total spin f = s + i, which iscalled the hyperfine spin, is a conserved quantum number. When atomsare confined in a magnetic potential, the spin of each atom points in thedirection of an external magnetic field. The spin degrees of freedom aretherefore frozen and the mean-field properties of the system are describedby a scalar order parameter. When the system is confined in an opticaltrap, the frozen degrees of freedom are liberated, yielding a rich variety ofphenomena arising from the magnetic moment of the atom. Since the mag-netic moments of alkali atoms originate primarily from the electronic spin,this system’s response to an external magnetic field is much greater thanthat of superfluid helium-3. We can expect interesting interplay betweensuperfluidity and magnetism with the possibility of new ground states, spindomains, and vortex structures. Spinor condensates are discussed in Chap-ter 6.

When the rotational speed of the container of the system is faster thanthe critical frequency, vortices enter the system and form a vortex lattice.The direct observation of vortex lattice formation [Madison, et al. (2000);Abo-Shaeer, et al. (2001)] has attracted considerable interest in the equilib-rium and nonequilibrium dynamics of condensates. The effect of rotationon neutral particles is equivalent to that of a magnetic field on chargedparticles. Therefore, the properties of a vortex lattice of neutral particlesare similar to those of superconductors. Furthermore, it is pointed out thatin systems containing neutral bosons that are subject to very fast rotation,the vortex lattice melts, and a new vortex liquid state similar to the Laugh-lin state in the fractional quantum Hall system may be realized. A briefoverview of these subjects is presented in Chapter 7.

Almost every bosonic atom has its fermionic counterpart. Fermions andbosons of the same species exhibit the same properties at high temperature,but they exhibit remarkably different behavior when quantum degeneracysets in. Bosons undergo BEC below the transition temperature; in con-trast, fermions become degenerate below the Fermi temperature, wherealmost every quantum state below the Fermi energy is occupied by onefermion and most quantum states above the Fermi energy are empty. Ateven lower temperatures, fermionic systems may exhibit superfluidity byforming Cooper pairs via the Bardeen–Cooper–Schrieffer transition. Thisis a rapidly developing field that has relevance to high-temperature super-conductivity. We describe the basics and some of the recent developmentsof ultracold fermionic systems in Chapter 8.

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viii Fundamentals and New Frontiers of Bose–Einstein Condensation

It is known that BEC does not occur at finite temperature in one-or two-dimensional infinite systems because thermal fluctuations destroythe off-diagonal long-range order (ODLRO). In one-dimensional systems,BEC does not occur even at absolute zero because quantum fluctuationswash out the ODLRO. However, confined low dimensional systems canexhibit BEC because long-wavelength fluctuations are cut off by confine-ment. We may thus investigate interesting phenomena associated with low-dimensional BEC, such as solitons and the Berezinskii–Kosterlitz–Thoulesstransition. These subjects are discussed in Chapter 9.

Atoms with magnetic moments and polar molecules undergo dipole–dipole interactions, which are long-ranged and anisotropic and yield awealth of novel phenomena. The magnetic dipole–dipole interaction is byfar the weakest of the relevant interactions in cold atom systems; yet itplays a dominant role in forming spin textures and magnetic ordering andproduces a spectacular effect in the course of the collapsing dynamics. Theelectric dipole–dipole interaction between polar molecules, in contrast, isvery strong and may cause instabilities of the system; at the same time,it has the potential to yield several exotic phases and for use in quantuminformation processing. Some basic properties of the dipolar condensatesare reviewed in Chapter 10.

An optical lattice is a periodic potential created by interference betweentwo counterpropagating laser beams. Atoms in an optical lattice behave likeelectrons in a crystal. An optical lattice can host bosons as well as fermions,and it offers an ideal testbed to simulate quantum many-body physics andquantum information processing. Chapter 11 provides a brief overview ofsome basic properties of this artificial condensed matter system.

Superfluids host a rich variety of topological defects such as vortices,monopoles, and skyrmions. Those topological excitations are best describedby the homotopy theory. Chapter 12 is devoted to an introduction of thehomotopy theory, classfication of topological excitations, and an account ofam how to calculate various topological charges.

Fifteen years after its first experimental realization, the field of ultracoldatomic gases is still growing at a remarkable speed, such that coverage ofevery topic of importance far exceeds the range of this or perhaps anybook. Rather, I have chosen a small number of important issues and triedto discuss their physical aspects as engagingly as possible. Many of thephenomena that have been observed in the past decade and those thatwill possibly be observed in the near future are of fundamental importancebecause of the very fact that they are being “seen” on a macroscopic scale.

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Preface ix

If this book succeeds in conveying even a portion of the fascination inherentin this field, it will have well served its intended purpose.

This book derives from a set of lecture notes delivered at several univer-sities over the past decade or so. I have benefited greatly from students andcolleagues who actively participated in the class and collaboration. Specialthanks are due to Rina Kanamoto, Yuki Kawaguchi, Michikazu Kobayashi,Tony Leggett, Hiroki Saito, and Masaki Tezuka. I would like to thank all ofthem for their questions, comments, and criticisms that helped me clarifymy thoughts and improve the presentation of the material in this book. Iam grateful to A. Koda, Y. Ookawara, and A. Yoshida for their efficientediting and preparation of the figures.

March 2010TokyoMasahito Ueda

Revisions and corrections will be posted on:http://cat.phys.s.u-tokyo.ac.jp/~ueda/E_kyokasyo.html/

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Contents

Preface v

1. Fundamentals of Bose–Einstein Condensation 1

1.1 Indistinguishability of Identical Particles . . . . . . . . . . 11.2 Ideal Bose Gas in a Uniform System . . . . . . . . . . . . 31.3 Off-Diagonal Long-Range Order: Bose System . . . . . . 61.4 Off-Diagonal Long-Range Order: Fermi System . . . . . . 101.5 U(1) Gauge Symmetry . . . . . . . . . . . . . . . . . . . . 111.6 Ground-State Wave Function of a Bose System . . . . . . 131.7 BEC and Superfluidity . . . . . . . . . . . . . . . . . . . . 151.8 Two-Fluid Model . . . . . . . . . . . . . . . . . . . . . . . 201.9 Fragmented Condensate . . . . . . . . . . . . . . . . . . . 23

1.9.1 Two-state model . . . . . . . . . . . . . . . . . . . 231.9.2 Degenerate double-well model . . . . . . . . . . . 251.9.3 Spin-1 antiferromagnetic BEC . . . . . . . . . . . 27

1.10 Interference Between Independent Condensates . . . . . . 281.11 Feshbach Resonance . . . . . . . . . . . . . . . . . . . . . 31

2. Weakly Interacting Bose Gas 33

2.1 Interactions Between Neutral Atoms . . . . . . . . . . . . 332.2 Pseudo-Potential Method . . . . . . . . . . . . . . . . . . 362.3 Bogoliubov Theory . . . . . . . . . . . . . . . . . . . . . . 40

2.3.1 Bogoliubov transformations . . . . . . . . . . . . . 402.3.2 Bogoliubov ground state . . . . . . . . . . . . . . 452.3.3 Low-lying excitations and condensate fraction . . 482.3.4 Properties of Bogoliubov ground state . . . . . . . 50

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xii Fundamentals and New Frontiers of Bose–Einstein Condensation

2.4 Bogoliubov Theory of Quasi-One-Dimensional Torus . . . 542.4.1 Case of BEC at rest: stability of BEC . . . . . . . 552.4.2 Case of rotating BEC: Landau criterion . . . . . . 562.4.3 Ground state of BEC in rotating torus . . . . . . 59

2.5 Bogoliubov–de Gennes (BdG) Theory . . . . . . . . . . . 602.6 Method of Binary Collision Expansion . . . . . . . . . . . 65

2.6.1 Equation of state . . . . . . . . . . . . . . . . . . 652.6.2 Cluster expansion of partition function . . . . . . 662.6.3 Ideal Bose and Fermi gases . . . . . . . . . . . . . 672.6.4 Matsubara formula . . . . . . . . . . . . . . . . . 69

3. Trapped Systems 73

3.1 Ideal Bose Gas in a Harmonic Potential . . . . . . . . . . 733.1.1 Transition temperature . . . . . . . . . . . . . . . 753.1.2 Condensate fraction . . . . . . . . . . . . . . . . . 763.1.3 Chemical potential . . . . . . . . . . . . . . . . . 773.1.4 Specific heat . . . . . . . . . . . . . . . . . . . . . 77

3.2 BEC in One- and Two-Dimensional Parabolic Potentials . 793.2.1 Density of states . . . . . . . . . . . . . . . . . . . 793.2.2 Transition temperature . . . . . . . . . . . . . . . 793.2.3 Condensate fraction . . . . . . . . . . . . . . . . . 80

3.3 Semiclassical Distribution Function . . . . . . . . . . . . . 813.4 Gross–Pitaevskii Equation . . . . . . . . . . . . . . . . . . 833.5 Thomas–Fermi Approximation . . . . . . . . . . . . . . . 843.6 Collective Modes in the Thomas–Fermi Regime . . . . . . 88

3.6.1 Isotropic harmonic potential . . . . . . . . . . . . 893.6.2 Axisymmetric trap . . . . . . . . . . . . . . . . . 913.6.3 Scissors mode . . . . . . . . . . . . . . . . . . . . 92

3.7 Variational Method . . . . . . . . . . . . . . . . . . . . . . 933.7.1 Gaussian variational wave function . . . . . . . . 943.7.2 Collective modes . . . . . . . . . . . . . . . . . . . 96

3.8 Attractive Bose–Einstein Condensate . . . . . . . . . . . . 983.8.1 Collective modes . . . . . . . . . . . . . . . . . . . 993.8.2 Collapsing dynamics of an attractive condensate . 102

4. Linear Response and Sum Rules 105

4.1 Linear Response Theory . . . . . . . . . . . . . . . . . . . 1054.1.1 Linear response of density fluctuations . . . . . . 105

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Contents xiii

4.1.2 Retarded response function . . . . . . . . . . . . . 1084.2 Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.2.1 Longitudinal f -sum rule . . . . . . . . . . . . . . 1104.2.2 Compressibility sum rule . . . . . . . . . . . . . . 1124.2.3 Zero energy gap theorem . . . . . . . . . . . . . . 1144.2.4 Josephson sum rule . . . . . . . . . . . . . . . . . 115

4.3 Sum-Rule Approach to Collective Modes . . . . . . . . . . 1204.3.1 Excitation operators . . . . . . . . . . . . . . . . . 1214.3.2 Virial theorem . . . . . . . . . . . . . . . . . . . . 1224.3.3 Kohn theorem . . . . . . . . . . . . . . . . . . . . 1234.3.4 Isotropic trap . . . . . . . . . . . . . . . . . . . . 1244.3.5 Axisymmetric trap . . . . . . . . . . . . . . . . . 127

5. Statistical Mechanics of Superfluid Systems in a Moving Frame 129

5.1 Transformation to Moving Frames . . . . . . . . . . . . . 1295.2 Elementary Excitations of a Superfluid . . . . . . . . . . . 1315.3 Landau Criterion . . . . . . . . . . . . . . . . . . . . . . . 1335.4 Correlation Functions at Thermal Equilibrium . . . . . . 1345.5 Normal Fluid Density . . . . . . . . . . . . . . . . . . . . 1365.6 Low-Lying Excitations of a Superfluid . . . . . . . . . . . 1405.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.7.1 Ideal Bose gas . . . . . . . . . . . . . . . . . . . . 1415.7.2 Weakly interacting Bose gas . . . . . . . . . . . . 143

6. Spinor Bose–Einstein Condensate 145

6.1 Internal Degrees of Freedom . . . . . . . . . . . . . . . . . 1456.2 General Hamiltonian of Spinor Condensates . . . . . . . . 1466.3 Spin-1 BEC . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.3.1 Mean-field theory of a spin-1 BEC . . . . . . . . . 1536.3.2 Many-body states in single-mode approximation . 1576.3.3 Superflow, spin texture, and Berry phase . . . . . 161

6.4 Spin-2 BEC . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7. Vortices 171

7.1 Hydrodynamic Theory of Vortices . . . . . . . . . . . . . 1717.2 Quantized Vortices . . . . . . . . . . . . . . . . . . . . . . 1747.3 Interaction Between Vortices . . . . . . . . . . . . . . . . 1807.4 Vortex Lattice . . . . . . . . . . . . . . . . . . . . . . . . 181

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xiv Fundamentals and New Frontiers of Bose–Einstein Condensation

7.4.1 Dynamics of vortex nucleation . . . . . . . . . . . 1817.4.2 Collective modes of a vortex lattice . . . . . . . . 183

7.5 Fractional Vortices . . . . . . . . . . . . . . . . . . . . . . 1867.6 Spin Current . . . . . . . . . . . . . . . . . . . . . . . . . 1877.7 Fast Rotating BECs . . . . . . . . . . . . . . . . . . . . . 189

7.7.1 Lowest Landau level approximation . . . . . . . . 1897.7.2 Mean field quantum Hall regime . . . . . . . . . . 1927.7.3 Many-body wave functions of a fast

rotating BEC . . . . . . . . . . . . . . . . . . . . 194

8. Fermionic Superfluidity 197

8.1 Ideal Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . 1978.2 Fermi Liquid Theory . . . . . . . . . . . . . . . . . . . . . 2008.3 Cooper Problem . . . . . . . . . . . . . . . . . . . . . . . 205

8.3.1 Two-body problem . . . . . . . . . . . . . . . . . 2058.3.2 Many-body problem . . . . . . . . . . . . . . . . . 209

8.4 Bardeen–Cooper–Schrieffer (BCS) Theory . . . . . . . . . 2118.5 BCS–BEC Crossover at T = 0 . . . . . . . . . . . . . . . . 2158.6 Superfluid Transition Temperature . . . . . . . . . . . . . 2198.7 BCS–BEC Crossover at T = 0 . . . . . . . . . . . . . . . . 2218.8 Gor’kov–Melik–Barkhudarov Correction . . . . . . . . . . 2258.9 Unitary Gas . . . . . . . . . . . . . . . . . . . . . . . . . . 2288.10 Imbalanced Fermi Systems . . . . . . . . . . . . . . . . . . 2318.11 P-Wave Superfluid . . . . . . . . . . . . . . . . . . . . . . 234

8.11.1 Generalized pairing theory . . . . . . . . . . . . . 2348.11.2 Spin-triplet p-wave states . . . . . . . . . . . . . . 238

9. Low-Dimensional Systems 241

9.1 Non-interacting Systems . . . . . . . . . . . . . . . . . . . 2419.2 Hohenberg–Mermin–Wagner Theorem . . . . . . . . . . . 2439.3 Two-Dimensional BEC at Absolute Zero . . . . . . . . . . 2469.4 Berezinskii–Kosterlitz–Thouless Transition . . . . . . . . . 247

9.4.1 Universal jump . . . . . . . . . . . . . . . . . . . . 2479.4.2 Quasi long-range order . . . . . . . . . . . . . . . 2499.4.3 Renormalization-group analysis . . . . . . . . . . 250

9.5 Quasi One-Dimensional BEC . . . . . . . . . . . . . . . . 2529.6 Tonks–Girardeau Gas . . . . . . . . . . . . . . . . . . . . 2569.7 Lieb–Liniger Model . . . . . . . . . . . . . . . . . . . . . . 258

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Contents xv

10. Dipolar Gases 261

10.1 Dipole–Dipole Interaction . . . . . . . . . . . . . . . . . . 26110.1.1 Basic properties . . . . . . . . . . . . . . . . . . . 26110.1.2 Order of magnitude and length scale . . . . . . . 26310.1.3 D-wave nature . . . . . . . . . . . . . . . . . . . . 26410.1.4 Tuning the dipole–dipole interaction . . . . . . . . 265

10.2 Polarized Dipolar BEC . . . . . . . . . . . . . . . . . . . . 26610.2.1 Nonlocal Gross–Pitaevskii equation . . . . . . . . 26610.2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . 26710.2.3 Thomas–Fermi limit . . . . . . . . . . . . . . . . . 26910.2.4 Quasi two-dimensional systems . . . . . . . . . . . 271

10.3 Spinor-Dipolar BEC . . . . . . . . . . . . . . . . . . . . . 27310.3.1 Einstein–de Haas effect . . . . . . . . . . . . . . . 27410.3.2 Flux closure and ground-state circulation . . . . . 274

11. Optical Lattices 277

11.1 Optical Potential . . . . . . . . . . . . . . . . . . . . . . . 27711.1.1 Optical trap . . . . . . . . . . . . . . . . . . . . . 27711.1.2 Optical lattice . . . . . . . . . . . . . . . . . . . . 280

11.2 Band Structure . . . . . . . . . . . . . . . . . . . . . . . . 28311.2.1 Bloch theorem . . . . . . . . . . . . . . . . . . . . 28311.2.2 Brillouin zone . . . . . . . . . . . . . . . . . . . . 28511.2.3 Bloch oscillations . . . . . . . . . . . . . . . . . . 28611.2.4 Wannier function . . . . . . . . . . . . . . . . . . 287

11.3 Bose–Hubbard Model . . . . . . . . . . . . . . . . . . . . 28811.3.1 Bose–Hubbard Hamiltonian . . . . . . . . . . . . 28811.3.2 Superfluid–Mott-insulator transition . . . . . . . . 28911.3.3 Phase diagram . . . . . . . . . . . . . . . . . . . . 29111.3.4 Mean-field approximation . . . . . . . . . . . . . . 29211.3.5 Supersolid . . . . . . . . . . . . . . . . . . . . . . 295

12. Topological Excitations 297

12.1 Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . 29712.1.1 Homotopic relation . . . . . . . . . . . . . . . . . 29712.1.2 Fundamental group . . . . . . . . . . . . . . . . . 29912.1.3 Higher homotopy groups . . . . . . . . . . . . . . 302

12.2 Order Parameter Manifold . . . . . . . . . . . . . . . . . . 30312.2.1 Isotropy group . . . . . . . . . . . . . . . . . . . . 303

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xvi Fundamentals and New Frontiers of Bose–Einstein Condensation

12.2.2 Spin-1 BEC . . . . . . . . . . . . . . . . . . . . . 30412.2.3 Spin-2 BEC . . . . . . . . . . . . . . . . . . . . . 305

12.3 Classification of Defects . . . . . . . . . . . . . . . . . . . 30612.3.1 Domains . . . . . . . . . . . . . . . . . . . . . . . 30612.3.2 Line defects . . . . . . . . . . . . . . . . . . . . . 30612.3.3 Point defects . . . . . . . . . . . . . . . . . . . . . 31112.3.4 Skyrmions . . . . . . . . . . . . . . . . . . . . . . 31312.3.5 Influence of different types of defects . . . . . . . 31612.3.6 Topological charges . . . . . . . . . . . . . . . . . 318

Appendix A Order of Phase Transition, Clausius–ClapeyronFormula, and Gibbs–Duhem Relation 321

Appendix B Bogoliubov Wave Functions in Coordinate Space 323

B.1 Ground-State Wave Function . . . . . . . . . . . 323B.2 One-Phonon State . . . . . . . . . . . . . . . . . 327

Appendix C Effective Mass, Sound Velocity, and SpinSusceptibility of Fermi Liquid 329

Appendix D Derivation of Eq. (8.155) 333

Appendix E f -Sum Rule 335

Bibliography 337

Index 347

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FUNDAMENTALS AND NEW FRONTIERS OF BOSE-EINSTEIN CONDENSATION

© World Scientific Publishing Co. Pte. Ltd.

http://www.worldscibooks.com/physics/7216.html


Chapter 1

Fundamentals of Bose–EinsteinCondensation

1.1 Indistinguishability of Identical Particles

Quantum statistics is governed by the principle of indistinguishability ofidentical particles. Particles with integer (half-integer) spin (in multiplesof !, where ! is the Planck constant divided by 2π) are called bosons(fermions). Bosons obey Bose–Einstein statistics in which there is no re-striction on the occupation number of any single-particle state. Fermionsobey Fermi–Dirac statistics in which not more than one particle can occupyany single-particle state. The many-body wave function of identical bosons(fermions) must be symmetric (antisymmetric) under the exchange of anytwo particles. This symmetry requirement drastically reduces the numberof available quantum states of the system, resulting in highly nonclassicalphenomena at low temperature.

To understand this, let us suppose that we obtain a wave functionΦ(ξ1, ξ2) of a two-particle system by solving the Schrodinger equation,where ξ1 and ξ2 represent the space and possibly spin coordinates of thetwo particles. For identical bosons (fermions), the symmetrized (antisym-metrized) wave function is given by

Ψ(ξ1, ξ2) =1√2

!Φ(ξ1, ξ2) ± Φ(ξ2, ξ1)

", (1.1)

where the plus (minus) sign indicates bosons (fermions). The joint proba-bility of finding the two particles at ξ1 and ξ2 is given by

|Ψ(ξ1, ξ2)|2 =12|Φ(ξ1, ξ2)|2 + |Φ(ξ2, ξ1)|2

± 2Re[Φ∗(ξ1, ξ2)Φ(ξ2, ξ1)], (1.2)

where Re denotes the real part. Because of the last interference term inEq. (1.2), the probability of finding the two identical bosons at the same

1





2 Fundamentals and New Frontiers of Bose–Einstein Condensation

coordinate, |Ψ(ξ, ξ)|2, is twice as high as |Φ(ξ, ξ)|2, which gives the corre-sponding probability for distinguishable particles. In contrast, for fermions,|Ψ(ξ, ξ)|2 vanishes in accordance with Pauli’s exclusion principle.

Such a bunching effect of bosons becomes increasingly pronounced whenthe number of bosons is large. For N number of bosons, the symmetrizedwave function is given by

Ψ(ξ1, ξ2, · · · , ξN ) =1√N !

!

(i1,i2,··· ,iN )

Φ(ξi1 , ξi2 , · · · , ξiN ), (1.3)

where the summation over i1, i2, · · · , iN is to be taken over all N ! per-mutations of 1, 2, · · · , N . The joint probability of finding all N bosonsat the same coordinate is thus N ! times that for distinguishable bosons,|Φ(ξ, ξ, · · · , ξ)|2, due to the constructive interference of the permuted prob-ability amplitudes:

|Ψ(ξ, ξ, · · · , ξ)|2 = N !|Φ(ξ, ξ, · · · , ξ)|2. (1.4)The constructive interference of the probability amplitudes is effective

only when the wave packets of bosons overlap each other. At temperatureT , each wave packet has a spatial extent of ∆x ∼ !/

√MkBT , where M

is the mass of the boson and kB is the Boltzmann constant. By setting∆x equal to the average interparticle distance n− 1

3 , where n is the particlenumber density, we can estimate the transition temperature T0 of Bose–Einstein condensation (BEC) to be

kBT0 ∼ !2

Mn

23 . (1.5)

Because of the large enhancement factor of N ! in Eq. (1.4), a large num-ber of particles suddenly begin to condense into a single-particle state belowT0. When N is macroscopic, the onset of this condensation becomes promi-nent, endowing BEC with a conspicuous trait of quantum phase transition.Substituting n = N/V , where V is the volume of the system, in Eq. (1.5)gives

kBT0 ∼ !2

MV23N

23 . (1.6)

Here, !2/(MV23 ) gives an estimate of the energy gap between the ground

state and the first excited state. Classical particles would condense intothe ground state below the corresponding temperature Tcl ∼ !2(kBMV

23 ).

Equation (1.6) shows that BEC occurs at a considerably higher tempera-ture; further, the large enhancement factor N

23 can be attributed to the

interference effect as discussed above. A more quantitative treatment de-scribed in Sec. 1.2 will validate Eq. (1.5).





Fundamentals of Bose–Einstein Condensation 3

1.2 Ideal Bose Gas in a Uniform System

The grand partition function Ξ of a system of particles with the HamiltonianH and particle-number operator N is given by

Ξ = Tre−β(H−µN), (1.7)

where β ≡ (kBT )−1, Tr denotes a trace operation, and the chemical poten-tial µ serves as a Lagrange multiplier that is to be determined so as to fixthe average number of particles to a prescribed value.

For ideal (i.e., noninteracting) identical bosons with the dispersion re-lation ϵk = !2k2/2M , H − µN is given by

H − µN =!

k

(ϵk − µ)nk, (1.8)

where nk denotes the number operator of particles with wave vector k.Substituting Eq. (1.8) in Eq. (1.7) gives

Ξ ="

k

∞!

nk=0

(eβ(µ−ϵk))nk . (1.9)

For the geometric series in Eq. (1.9) to converge, eβ(µ−ϵk) must be less thanone. It follows from ϵk ≥ 0 that

µ < 0. (1.10)

Then, Eq. (1.9) gives

Ξ ="

k

11 − eβ(µ−ϵk)

.

The thermodynamic potential Ω is defined in terms of Ξ as

Ω ≡ − 1β

ln Ξ =1β

!

k

ln(1 − eβ(µ−ϵk)) =!

k

Ωk, (1.11)

where

Ωk =1β

ln(1 − eβ(µ−ϵk)). (1.12)

The average number of particles with wave vector k is given by

nk = −∂Ωk

∂µ=

1eβ(ϵk−µ) − 1

, (1.13)

which is referred to as the Bose–Einstein distribution function. The averagetotal number of bosons is expressed in terms of the chemical potential µ as

N =!

k

1eβ(ϵk−µ) − 1

. (1.14)

For a given N , µ is determined such that it satisfies Eq. (1.14).






In the thermodynamic limit in which both N and V are made infinitewith the particle number density N/V held constant, the sum over k maybe replaced with the following integral1:

!

k

→ V

(2π)3

"d3k. (1.15)

Equation (1.14) then gives

N

V=

1(2π)3

"d3k

1eβ(ϵk−µ) − 1

. (1.16)

When the temperature is reduced while maintaining N/V constant, µ in-creases and eventually becomes zero at a certain temperature T0. Substi-tuting µ = 0 and ϵk = !2k2/2M in Eq. (1.16) yields

N

V=

(MkBT0)3/2

√2π2!3

" ∞

0

√x

ex − 1dx

= ζ

#32

$#MkBT0

2π!2

$ 32

= 2.612#

MkBT0

2π!2

$ 32

, (1.17)

where ζ(x) is the Riemann zeta function, and the following formulae areused:

" ∞

0

xa−1

ex − 1dx = Γ(a)ζ(a), ζ

#32

$= 2.612, Γ

#32

$=

√π

2. (1.18)

From Eq. (1.17), the transition temperature T0 of BEC is given by

kBT0 =2π

%ς(3/2)

&2/3

!2

M

#N

V

$ 23

= 3.31!2

M

#N

V

$ 23

, (1.19)

which is in agreement with Eq. (1.5). For T < T0, a nonzero fraction ofbosons should therefore remain in the ground state; i.e., they condense intothe lowest-energy state. For T < T0, the replacement of the sum with theintegral (Eq. 1.15) is applicable only for particles with positive energy ϵ > 0,since the particles with ϵ = 0 cannot contribute to the integral in Eq. (1.17)because of the factor

√x in the integrand. From Eq. (1.17), we find that

T0 is related to the particle-number density N/V through the relation

N

V= ζ

#32

$#MkBT0

2π!2

$ 32

. (1.20)

1In the presence of spin multiplicity g = 2S + 1, where !S is the spin of a boson, thefollowing results hold true if we replace V by gV .






For T < T0, we haveNϵ>0

V= ζ

!32

"!MkBT

2π!2

" 32

. (1.21)

From Eqs. (1.20) and (1.21), it follows thatNϵ>0

N=!

T

T0

" 32

. (1.22)

This quantity is referred to as the normal fraction. Hence, the condensatefraction is given by

Nϵ=0

N= 1 −

!T

T0

" 32

. (1.23)

BEC occurs when the de Broglie waves of individual bosons begin tooverlap, i.e., when the quantum degeneracy sets in. The thermal de Broglielength λth is conventionally defined as

kBT =1

2πM

!h

λth

"2

→ λth =h√

2πMkBT. (1.24)

Here, λth characterizes the spatial extension of a wave packet of an indi-vidual boson. Substituting this in Eq. (1.20) gives

nλ3th = ζ

!32

"≃ 2.612 at T = T0. (1.25)

Thus, the thermal de Broglie length at the transition temperature is onthe order of the average interparticle distance n− 1

3 . The quantity nλ3th is

referred to as phase-space density. Equation (1.25) shows that an ideal Bosegas undergoes BEC at a phase-space density of 2.612.

At the transition point, the specific heat of an ideal Bose gas at con-stant volume is continuous and its derivative is discontinuous. Therefore,the BEC of an ideal Bose gas at constant volume is a third-order phase tran-sition2. The specific heat of liquid 4He shows a discontinuous jump at thelambda point (T = 2.17 K), which indicates that the superfluid transitionof liquid 4He is a second-order phase transition. By studying the similaritybetween the behavior of the specific heat of liquid 4He near the lambdapoint and that of an ideal Bose gas, Fritz London found that BEC plays anessential role in both superfluidity and superconductivity [London (1938)].In the several decades after London’s seminal work, the physics communityhas gradually acknowledged the special role of BEC in superfluidity.2If we consider the state of the system as a function of pressure P and volume at

constant temperature, i.e., the isotherm, P becomes constant below a certain volume,where the Bose–Einstein condensate coexists with the normal component in a manneranalogous to gas-liquid transition. In this situation, BEC may be considered to be afirst-order phase transition. Refer to K. Huang, Statistical Mechanics, 2nd edition (JohnWiley & Sons, New York, 1987).






1.3 Off-Diagonal Long-Range Order: Bose System

The essence of BEC is the off-diagonal long-range order (ODLRO). Wefirst explain the concept of the ODLRO using a simple example and thenprovide its general definition.

A system is said to possess an ODLRO if the single-particle densitymatrix

ρ1(r, r′) ≡ Trρψ†(r)ψ(r′) ≡ ⟨ψ†(r)ψ(r′)⟩ (1.26)

has a large eigenvalue, i.e., an eigenvalue proportional to the total num-ber of particles N , where ρ is the density operator of the system andψ†(r) (ψ(r′)) is the field operator that creates (annihilates) a particle atr (r′). Since ρ is Hermitian, ρ1(r, r′) is a Hermitian matrix. When thesystem is in a pure state |Φ⟩, Eq. (1.26) reduces to

ρ1(r, r′) = ⟨Φ|ψ†(r)ψ(r′)|Φ⟩. (1.27)

This expression implies that the single-particle density matrix gives theprobability amplitude that the quantum state of the system remains un-perturbed if a particle is removed from the system at r′ and added to it at r.Under normal conditions, ρ1(r, r′) decreases exponentially with increasing|r − r′| [see Eq. (1.48)]. When the system undergoes BEC, the de Brogliewaves of individual bosons overlap, and thus a particle at r′ becomes in-distinguishable from a particle at r. As a consequence, ρ1(r, r′) does notvanish over a long distance |r−r′|. If this condition holds, the system is saidto maintain spatial coherence over a long distance. As shown below, thesystem possesses an ODLRO when ρ1(r, r′) remains on the order of N/Vas |r − r′| increases. This shows that a particle can travel a long distancewithout disturbing the state of the system. In this respect, the ODLRObears a close relation with superfluidity. When r = r′, Eq. (1.26) gives theparticle number density.

Let us first consider a spatially uniform system. In this case, it is conve-nient to expand the field operator ψ(x) in terms of plane waves as follows:

ψ(x) =1√V

!

k

akeikx, (1.28)

where ak is the annihilation operator of bosons with wave vector k. Weassume that ak satisfies the boson commutation relation

"ak, a†

q

#= δkq. (1.29)






The single-particle density matrix is then given by

ρ1(x,y) =1V

!

k,q

⟨a†kaq⟩e−i(kx−qy). (1.30)

Here, we note that ⟨a†kaq⟩ = δkq⟨a†

kak⟩ holds for a translationally invariantsystem. To prove this, let us calculate the commutation relation betweena†kaq and the momentum operator P =

"p !pa†

pap, given by#P, a†

kaq

$=!

p

!p#a†pap, a†

kaq

$= !(k − q)a†

kaq. (1.31)

Taking the thermal average of this quantity, we obtain%#

P, a†kaq

$&= Z−1Tr

'e−βH

(Pa†

kaq − a†kaqP

)*

= Z−1Tr'e−βHPa†

kaq − Pe−βH a†kaq

*, (1.32)

where Z ≡ Tre−βH and the cyclic property of the trace,

Tr(AB) = Tr(BA),

is used in deriving the last equality. Since [H, P] = 0 for a spatially uniformsystem, the last term in Eq. (1.32) vanishes; thus,

%#P, a†

kaq

$&= 0.

Hence,

(k − q)⟨a†kaq⟩ = 0, (1.33)

which implies that

⟨a†kaq⟩ = δkq⟨a†

kak⟩ = δkq⟨nk⟩, (1.34)

where nk ≡ a†kak is the number operator.

Substituting Eq. (1.34) in Eq. (1.30) gives

ρ1(x,y) =1V

!

k

⟨nk⟩e−ik(x−y) =⟨n0⟩V

++

d3k

(2π)3⟨nk⟩e−ik(x−y). (1.35)

The last term vanishes in the limit |x − y| → ∞ because of the rapidlyoscillating term e−ik(x−y) (Riemann–Lebesgue lemma). Consequently,

ρ1(x,y) → ⟨n0⟩V

as |x − y| → ∞. (1.36)

This result shows that the system exhibits the off-diagonal (i .e., x = y)long-range order in the thermodynamic limit if and only if an extensive






number of bosons (proportional to the volume) condense into a state withzero momentum. This shows that the ODLRO implies BEC.

Single-particle energy levels are not well defined in the presence of an in-terparticle interaction. However, the following reduced single-particle den-sity operator is well defined in this case:

ρ1 ≡ Tr2,3,···,N ρ, (1.37)

where Tr2,3,···,N denotes the trace over particles 2, 3, · · ·, N ; it should benoted that because bosons are identical, we can choose arbitrary N − 1particles without loss of generality. Let nM be the maximum eigenvalue of

σ1 ≡ N ρ1. (1.38)

The condition for a Bose–Einstein condensate to exist can be formulatedas follows [Penrose and Onsager (1956)]3:

nM

N= eO(1). (1.39)

This definition is applicable irrespective of the presence or absence of in-teractions. It is also applicable when the system is not uniform. Thesingle-particle density matrix can also be expressed in terms of the reducedsingle-particle density operator ρ1 as follows:

ρ1(x,y) = ⟨ψ†(x)ψ(y)⟩ =!

dz⟨z|ρ1ψ†(x)ψ(y)|z⟩ (1.40)

= ⟨y|ρ1ψ†(x)|0⟩ = ⟨y|ρ1|x⟩, (1.41)

where ψ(y)|z⟩ = δ(y − z)|0⟩ was used in deriving the third equality.When the system is spatially uniform, the momentum is a good quantum

number. Therefore,

⟨p|ρ1|p′⟩ ∝ δ(p − p′). (1.42)

In this case, the condition of BEC is that a macroscopic number of particlesoccupy the same single-particle momentum state. When the system is notspatially uniform, the condition is given by

ρ1(x,y) −→ ψ∗(x)ψ(y) as |x − y| → ∞, (1.43)

where ψ(x), to a very good approximation, is an eigenfunction of the single-particle reduced density matrix ρ1(x,y):

!dxρ1(x,y)ψ(x) ≃ nMψ(y), nM =

!dx|ψ(x)|2. (1.44)

3eO(1) is a positive number of the order of unity.






Here, ψ(x) in Eq. (1.43) is often referred to as the condensate wave functionor the order parameter and nM is the number of condensed bosons. Theratio nM/N is referred to as the condensate fraction. We note that if ψ(x)is a solution to the eigenvalue equation (1.44), ψ(x)eiφ is also a solution toit, where φ is an arbitrary real number. The global phase of the condensatewave function is therefore arbitrary.

Comparing Eq. (1.43) with Eq. (1.36), we find that the condensate wavefunction is a thermodynamic quantity that appears in the thermodynamiclimit. The significance of BEC is thus to generate a new thermodynamicvariable that is a macroscopic wave function representing the ODLRO. Withthe macroscopic wave function, it is possible to describe coherent propertiesof a many-body system without referring to the microscopic details of thesystem.

For comparison, let us consider the classical limit of Eq. (1.35). In thethermodynamic limit at T > T0, we have ⟨n0⟩/V = 0. Hence, Eq. (1.35)reduces to

ρ1(x,y) =!

d3k

(2π)3⟨nk⟩e−ik(x−y), (1.45)

where

⟨nk⟩ ≃ eβ(µ−!2k2/2M).

The chemical potential µ is determined such that ρ1(x,x) gives the totalparticle-number density n:

n =!

d3k

(2π)3⟨nk⟩ =

eβµ

(2π)3

"!dke−β !2k2

2M

#3

= eβµ

"M

2πβ!2

# 32

. (1.46)

Hence,

µ =1β

ln$nλ3

th

%. (1.47)

Substituting this in Eq. (1.46), we obtain

ρ1(x,y) =eβµ

(2π)3

!d3ke−

βk22M −ik(x−y) = ne

−M|x−y|2

2β!2 ; (1.48)

we thus find that only the diagonal (i.e., x = y) order remains nonvanishingin the high-temperature limit (β → 0).






1.4 Off-Diagonal Long-Range Order: Fermi System

In the case of a Fermi system, the commutator on the left-hand side of(1.29) is replaced with the anti-commutator as follows:

ckσ, c†qσ′ ≡ ckσ c†qσ′ + c†qσ′ ckσ = δkqδσσ′ , (1.49)

where c†qσ′ (ckσ) is the creation (annihilation) operator of a fermion withwave number q(k) and spin σ′(σ). As in the case of bosons [see Eq. (1.34)],it can be shown that for a translation-invariant system

⟨c†kσ cqσ′⟩ = δkqδσσ′ ⟨c†kσ ckσ⟩. (1.50)

The single-particle reduced density matrix of a Fermi system is defined as

ρ1(rσ, r′σ′) = ⟨ψ†σ(r)ψσ′ (r′)⟩, (1.51)

where ψσ(r) is the field operator of a fermion with spin σ at position r.Substituting the Fourier expansion of ψσ(r),

ψσ(r) =1√V

!

k

ckσeikr, (1.52)

into (1.51) and using (1.50), we obtain

ρ1(rσ, r′σ′) ="

d3k

(2π)3⟨c†kσ ckσ′⟩e−ik(r−r′). (1.53)

Due to the Pauli exclusion principle, the occupation number of any single-particle state cannot exceed unity, and therefore, the term correspondingto the first term on the right-hand side of (1.35) does not appear in (1.53).At absolute zero,

⟨c†kσ ckσ′⟩ = δσσ′θ(kF − |k|), (1.54)

where kF is the Fermi wave number. Substituting (1.54) into (1.53), weobtain

ρ1(rσ, r′σ′) =δσσ′

2π2r3(sin kFr − kFr cos kFr), (1.55)

where r ≡ |r− r′|. Since ρ1 vanishes for r → ∞, the single-particle densitymatrix does not show ODLRO but decays algebraically.

The Fermi system may show ODLRO at the two-particle level. Toverify this, let us consider the two-particle reduced density matrix of aFermi system:

ρ2(r1σ1, r2σ2; r′1σ′1, r

′2σ

′2) ≡ ⟨ψ†

σ1(r1)ψ†

σ2(r2)ψσ′

2(r′2)ψσ′

1(r′1)⟩. (1.56)






Substituting (1.52) into (1.56), we have

ρ2(r1σ1, r2σ2; r′1σ′1, r

′2σ

′2) =

1V 2

!

k,k′K

⟨c†k+K

2 ,σ1c†−k+K

2 ,σ2c−k′+K

2 ,σ′2ck′+K

2 ,σ′1⟩

×ei[k′r′−kr+K(R−R′)], (1.57)

where r ≡ r2 − r1, r′ ≡ r′2 − r′1, R ≡ r1+r22 , and R′ ≡ r′1+r′2

2 . In the limitof |R − R′| → ∞, all terms except for the K = 0 term vanish due to therapidly oscillating factor eiK(R−R′). Thus,

lim|R−R′|→∞

ρ2(r1σ1, r2σ2; r′1σ′1, r

′2σ

′2) =

1V 2

!

k,k′

⟨c†k,σ1c†−k,σ2

c−k′,σ′2ck′,σ′

1⟩

×ei(k′r′−kr). (1.58)

In terms of the Cooper-pair operator defined as

Ψσ2σ1(r) ≡1V

!

k

c−k,σ2 ck,σ1eikr, (1.59)

Eq. (1.58) is expressed as

lim|R−R′|→∞

ρ2(r1σ1, r2σ2; r′1σ′1, r

′2σ

′2) = ⟨Ψ†

σ2σ1(r)Ψσ′

2σ′1(r′)⟩. (1.60)

If the left-hand side of Eq. (1.60) does not vanish, its right-hand side maybe written as

⟨Ψ†σ2σ1

(r)Ψσ′2σ′

1(r′)⟩ = Ψ∗

σ2σ1(r)Ψσ′

2σ′1(r′). (1.61)

Thus, two-particle correlations are essential for a Fermi system to showODLRO [Gor’kov (1958); Yang (1962)], and if ODLRO occurs, the quantityΨσ2σ1(r) defined in Eq. (1.61) is referred to as the order parameter or themacroscopic wave function of Cooper pairs.

1.5 U(1) Gauge Symmetry

The concept of order parameter plays a key role in our understanding ofthe second-order phase transition, which is accompanied by a change insymmetry. As a typical example, let us consider the case of a ferromagnet,where the order parameter is magnetization, which is an observable repre-senting a collective order of microscopic spins, and is coupled to anotherobservable — the magnetic field. In contrast, BEC is unique in that theorder parameter is a macroscopic wave function that is complex and is notan observable per se, because the phase of the wave function is arbitrary.






The arbitrariness of the phase reflects the symmetry, called the U(1) gaugesymmetry, that results from the conservation of the particle number.

In the mean field theory, it is convenient to break the U(1) symmetryand let the order parameter have a definite phase φ. A mathematical trickto implement this is to add to the Hamiltonian a term H ′ that establishescorrelations among phases of states having different particle numbers:

H ′ = ϵ

! "e−iφψ(r) + eiφψ†(r)

#dr. (1.62)

The phase of the order parameter is thus coupled to the non-Hermitianfield operators ψ† and ψ. If we operate the perturbation given by theintegrand of H ′ at a local point in a normal fluid, the phase at that pointis fixed, but the phase at a point located at a distance greater than thecorrelation length becomes random. However, when the temperature islowered below the transition temperature, the phase of the system becomesspatially uniform. If we change the phase over space, the energy of thesystem increases by an amount κ(∇φ)2, where the positive coefficient κ isenhanced by a repulsive interaction. We may consider that the stability of asuperfluid is a consequence of the rigidity of the macroscopic wave functiondue to the repulsive interaction.

While breaking of the U(1) gauge symmetry greatly simplifies calcu-lations of physical quantities, there is one conceptual difficulty here. Thesymmetry-breaking perturbation (1.62) brings the system in a superposi-tion of states having different particle numbers. However, for massive parti-cles, such a superposition state is precluded by the superselection rule [Haag(1996)]. Fortunately, we can understand BEC and superfluidity withoutbreaking the U(1) symmetry [see Secs. 1.3 and 1.7]; moreover, theorieswith and without the U(1) gauge theories virtually yield the same resultsin the thermodynamic limit. However, significant differences may arise inthe mesoscopic regime in which the number of particles is finite.

The condition for BEC is often stated as follows:ρ1(x,y)

|x−y|→∞−−−−−−→ ⟨ψ†(x)⟩⟨ψ(y)⟩. (1.63)Here, the symbol ⟨···⟩ should be interpreted, to a good approximation, as theexpectation value between states in which only the numbers of condensateparticles, nM, differ by one and all the other quantum numbers representedby ξi remain unaltered:4

⟨ψ†(x)⟩ = ⟨nM, ξi|ψ†(x)|nM − 1, ξi⟩,⟨ψ(y)⟩ = ⟨nM − 1, ξi|ψ(y)|nM, ξi⟩. (1.64)

4If we would interpret ⟨ψ†(x)⟩ as an expectation value over a state with a definiteparticle number, we would get ⟨ψ†(x)⟩ = 0. A nonzero value of ⟨ψ(x)⟩ might be obtained






To show this, let us expand the field operator as

ψ(x) = a0u0(x) +!

ξ

axuξ(x),

where u0(x) is the mode function of the condensate and uξ(x)’s are non-condensate modes. Then,

ψ(x)|nM, ξi⟩ =√

nMu0(x)|nM − 1, ξi⟩+!

ξ

√nξuξ(x)|nM, · · · , ξi − 1, · · · ⟩.

Since nM ≫ nξ, Eqs. (1.64) hold to a good approximation. Physically, thisresult implies that when a single-particle state is macroscopically occupied,it plays the role of a particle reservoir that can absorb or emit a particlewith negligible (∼ N−1) influence upon itself.

Let us now consider the expectation value of the time-dependent fieldoperator ψ(x, t) = eiHt/!ψ(x)e−iHt/! in the Heisenberg representation,where H |nM, ξi⟩ = EN |nM, ξi⟩ with N being the total number of par-ticles. Then,

⟨nM − 1, ξi|ψ(x, t)|nM, ξi⟩ = ⟨ψ(x)⟩e−i(EN−EN−1)t/!. (1.65)

Since for N ≫ 1

EN − EN−1 ≃ ∂EN

∂N= µ, (1.66)

where µ is the chemical potential, the time dependence of the condensatewave function is governed by the chemical potential:

ψ(x, t) = ψ(x)e−iµt/!. (1.67)

1.6 Ground-State Wave Function of a Bose System

The most fundamental property of the ground-state wave function (GSWF)Ψ0 of a Bose system is that it can be taken to be real, nodeless, andnondegenerate.

To show that Ψ0 can be taken to be real, we decompose Ψ0 into ampli-tude |Ψ0| and phase χ:

Ψ0(x1,x2, · · · ,xN ; t) = |Ψ0(x1,x2, · · · ,xN ; t)|eiχ(x1,x2,··· ,xN ;t). (1.68)if we would assume that the system is in a superposition of states having different particlenumbers. However, this assumption, which is often made in literature, runs counter tothe superselection rule, as described in the preceding paragraph.






If χ is constant, we can consider it as zero since the overall phase can bechosen arbitrarily. If it varies in space, the system shows a mass flow. Toverify this, let us express the density of particles at position r in terms ofΨ0:

ρ(r, t) =!

dx1 · · · dxN |Ψ0(x1, · · · ,xN ; t)|2N"

k=1

δ(r − xk). (1.69)

The mass current density j(r, t), which, together with ρ(r, t), satisfies theequation of continuity

∂ρ

∂t+ ∇j = 0, (1.70)

is given by

j(r, t) =!

2Mi

!dx1 · · ·dxN

N"

k=1

(Ψ∗0∇kΨ0 − Ψ0∇kΨ∗

0)δ(r − xk), (1.71)

where ∇k denotes differentiation with respect to xk. Substitution ofEq. (1.68) into Eq. (1.71) yields

j(r) =!M

!dx1 · · ·dxN |Ψ0|2

N"

k=1

δ(r − rk)∇kχ. (1.72)

When χ depends on coordinates, j is, in general, nonzero; therefore, Ψ0

cannot be the ground state. Thus, the GSWF of a Bose system can beconsidered as real.

To show that Ψ0 is nodeless, let us recall that the GSWF can be deter-mined from the variational principle: Ψ0 is determined so as to minimizethe energy functional

F [Ψ] =!

dx1 · · · dxN

#!2

2M

N"

k=1

(∇kΨ)2 + V (x1, · · · ,xN )Ψ2

$, (1.73)

where the potential function V is assumed to be finite everywhere, and Ψis assumed to be real and subject to the normalization condition

!dx1 · · ·dxNΨ2 = 1. (1.74)

With a Lagrange multiplier E, the variational principle

δ

δΨ

%F − E

!dx1 · · ·dxNΨ2

&= 0 (1.75)






leads to the Schrodinger equation!− !2

2M

N"

k=1

∇2k + V (x1, · · · ,xN )

#Ψ = EΨ. (1.76)

It is noteworthy that Eqs. (1.73) and (1.74) are invariant under Ψ → −Ψ;therefore, if Ψ is a solution to Eq. (1.76), |Ψ| is also a solution that has thesame energy E. Now suppose that the GSWF Ψ0 has a node and changesits sign at a certain point; then, |Ψ0| must have a cusp at that point, andits derivative must be discontinuous there. We can then construct a newwave function Ψ′ from |Ψ0| by smoothing out the cusp of |Ψ0| over aninfinitesimal region. However, the energy of Ψ′ would then be smaller thanthat of |Ψ0| because Ψ′ has no such cusp that costs the kinetic energy butotherwise coincides with |Ψ0|. By reductio ad absurdum, the GSWF of aBose system has no node and can therefore be taken to be non-negative.

Let us assume that there are two such non-negative solutions Ψ1 andΨ2. By the linearity of the Schrodiger equation, Ψ1 −Ψ2 is also a nodelesssolution. This implies either Ψ1 ≥ Ψ2 or Ψ1 ≤ Ψ2, which, however, iscompatible with the normalization condition (1.74) if and only if Ψ1 = Ψ2.Thus, the GSWF is nondegenerate.

A corollary of the uniqueness of the GSWF of a Bose system is that atabsolute zero, the thermodynamic properties of a Bose system are the sameas those of a Boltzmann gas.

1.7 BEC and Superfluidity

There is no unique relationship between BEC and superfluidity. An ideal-gas Bose–Einstein condensate shows no superfluidity and a two-dimensionalsuperfluid shows no BEC. However, there are many cases in which BEC andsuperfluidity do occur simultaneously. Under such circumstances, a genericargument can be made, which offers insight into the interplay between BECand superfluidity.

Consider a nonequilibrium situation in which the state of a systemchanges in time. Since ρ1 is Hermitian, we can consider the representa-tion in which the single-particle density operator is diagonal at all times:

ρ1(r, r′; t) = ⟨ψ†(r, t)ψ(r′, t)⟩ ="

ν

nν(t)ψ∗ν(r, t)ψν(r′, t). (1.77)

We denote the mode in which BEC occurs as ν = 0, that is, n0 = O(N). IfBEC occurs only in the ν = 0 mode, nν =0 = O(1). In the limit |r−r′| → ∞,






contributions other than the BEC mode do not add up but rather canceleach other, since ψν =0’s are orthogonal to each other. Thus, in the limit|r − r′| → ∞, we have

⟨ψ†(r, t)ψ(r′, t)⟩ → n0ψ∗0(r, t)ψ0(r′, t)

≡ Ψ∗(r, t)Ψ(r′, t), (1.78)

We may interpret

Ψ(r, t) ≡ √n0ψ0(r, t) (1.79)

as the condensate wave function (or the order parameter) that is applicableto a nonequilibrium situation. The density of the condensed bosons isdefined as

ρ(r, t) ≡ Ψ∗(r, t)Ψ(r, t). (1.80)

It follows from Eq. (1.80) and the continuity equation∂

∂tρ(r, t) + divj(r, t) = 0 (1.81)

that the current density of particles is given by

j(r, t) =!

2Mi[Ψ∗(r, t)∇Ψ(r, t) − Ψ(r, t)∇Ψ∗(r, t)]. (1.82)

Let us decompose Ψ(r, t) into the amplitude and phase as

Ψ(r, t) = A(r, t)eiφ(r,t). (1.83)

In terms of A and φ, ρ and j can be expressed as

ρ(r, t) = A2(r, t),

j(r, t) = A2(r, t)!M

∇φ(r, t).

The superfluid velocity vs(r, t) is defined as the ratio of j to ρ.

vs(r, t) ≡j(r, t)ρ(r, t)

=!M

∇φ(r, t) (1.84)

Thus, the phase of the condensate wave function plays the role of the veloc-ity potential in superfluidity. The equation of motion for vs is given fromEq. (1.84) as

d

dtvs = ∇Ω, Ω ≡ !

M

∂φ

∂t. (1.85)

Because the superfluid velocity is the gradient of a scalar function, its ro-tation vanishes identically.

rotvs = 0. (1.86)






Thus, a superfluid is irrotational. However, when we rotate a containerholding a superfluid, the surface of the superfluid shows a parabolic menis-cus on the periphery, as in the case of a normal fluid. This shows that thevalue of the surface integral of vorticity,

!rotvsdS, is nonzero in appar-

ent contradiction with Eq. (1.86). Onsager [Onsager (1949)] resolved thisparadox by assuming that rotvs is nonzero only within microscopic regionswhere the liquid is not a superfluid. These singular regions are where vor-tex lines penetrate, as shown in Fig. 1.1. The integral of vs along a closed

Fig. 1.1 Vortex lines. A rotating superfluid becomes normal within narrow regionswhere vortex lines penetrate. The size of each vortex line is on the order of the healinglength, which is on the order of a few angstroms for 4He and on the order of a fewmicrometers for alkali Bose–Einstein condensates.

contour gives "vsdl =

!M

"∇φ(r, t)dl. (1.87)

Because of the single-valuedness of the condensate wave function (or theorder parameter),

#∇φ(rt)dl is equal to an integer multiple of 2π, leading

to the celebrated quantization of circulation"vsdl =

h

Mn, (1.88)

where n is an integer. Thus, the line integral of the superfluid velocity vs

is quantized in units of

κ0 ≡ h

M≃$

9.97 × 10−4cm2/s for 4He;4.59 × 10−5cm2/s for 87Rb,

where κ0 is called the quantum of circulation. When the radius of a con-tainer is R and the angular frequency of rotation is ω, we have"

vsdl = ωR · 2πR = 2πR2ω = nκ0.






The observed meniscus on the periphery can be explained if the vortex linesare distributed with density n/(πR2) = 2ω/κ0. For the case of ω/2π =100 Hz, we have 2ω/κ0 ≃ 0.013/µm2 for 4He and 0.3/µm2 for 87Rb.

A persistent current in a ring geometry is attenuated if a quantized vor-tex having circulation with the same (opposite) sign crosses the ring from in-side (outside) to outside (inside). The decay of a persistent current is causedby thermal excitations or quantum tunneling of quantized vortices. It isgenerally held that turbulent flow in a superfluid is induced by an entan-glement of vortices. However, it is difficult to judge whether attenuation ofa persistent current is caused by nucleation of a new vortex or by entangle-ment of already existing vortices. The study of turbulence using a gaseousBose–Einstein condensate is expected to shed new light on this problem,because the dynamics of vortices can be visually tracked in real time.

Fig. 1.2 Decay of persistent current in a torus. This decay occurs stepwise becausea quantized vortex with the same (opposite) sense of circulation crosses the ring frominside (outside) to outside (inside).

The wave function of identical bosons must be symmetric under ex-change of any two particles. The ground-state wave function of a Bosesystem does not have a node and can take only non-negative real numbers.The wave function of a low-lying excited state is expected to be similar tothat of the ground state but modulated slightly depending on the nature ofthe excitation. One of the candidates that satisfy the requirement of Bosesymmetry is given by

ψ(r1, r2, · · ·, rN ; t) = exp

⎡

⎣iN∑

j=1

φ(rj , t)

⎤

⎦ψ(r1, r2, · · ·, rN ). (1.89)

Then, the eigenvector for ρ1(r, r′) in Eq. (1.26) for a uniform system changesfrom a constant to a constant times eiφ(r,t).






When φ is a real number, Eq. (1.89) can describe a state of superflow,because the superfluid velocity

vs(r, t) =!M

∇φ(r, t) (1.90)

is nonzero if φ changes spatially. In particular, when the system is incom-pressible (i.e., |ψ| = const.), the equation of continuity dictates that thepotential function φ must satisfy Laplace’s equation ∇2φ = 0.

When φ is a complex number, Eq. (1.89) describes the state of a soundwave accompanied by density modulations. Let us consider, for simplicity,a one-dimensional case. Taking the velocity of the sound wave as vx =c cos(kx − ωt), the equation of continuity

∂ρ

∂t+

∂

∂x(ρ0vx) = 0

leads to the following density modulation:

ρ = ρ0

(1 +

ck

ωcos(kx − ωt)

).

The wave function that describes such an excited state can be expressed byinspection as [Thouless (1998)]

ψ = exp

⎡

⎣iMc

!k

∑

j

sin(kxj − ωt) +ck

2ω

∑

j

cos(kxj − ωt)

⎤

⎦ψ0. (1.91)

It follows that the current density is given by

jx =!

2Mi

(ψ∗ ∂

∂xjψ − ψ

∂

∂xjψ∗)

= c cos(kxj − ωt)eckω

∑j cos(kxj−ωt)ψ2

0 .

Since the density of particles is given by

ρ = |ψ|2 = eckω

∑j cos(kxj−ωt)ρ0, ρ0 = ψ2

0 ,

we have jx = vxρ. We can obtain the many-body wave function for the sys-tem with one phonon excitation by linearizing the amplitude in Eq. (1.91)as

ψ =1√N

∑

j

eikxjψ0.

The generalization of this argument to higher dimensions is straightforward.






1.8 Two-Fluid Model

Let us consider a metastable state in which a superfluid moves along a wallwith velocity vs. As the mass current j vanishes for vs = 0, it is reasonableto assume a linear relation between j and vs when vs is small [Leggett(1991)]:

j(r) =!

Ks(r, r′)vs(r′)dr′, (1.92)

where Ks depends on the microscopic details of the system. When thespatial variation of vs(r) is smooth, we can assume the following localrelation:

j(r) = ρs(r)vs(r), ρs(r) ≡!

Ks(r, r′)dr′. (1.93)

While the superfluid velocity vs defined in Eq. (1.84) is a microscopic quan-tity, the superfluid density ρs is a hydrodynamic concept, as can be observedfrom the definition in Eq. (1.93). The normal fluid density ρn is defined asρ− ρs.

A normal fluid may be considered as an assembly of quasiparticle exci-tations in a superfluid. When the entire system moves with velocity vs, themass current is ρvs. In addition, when there are quasiparticle excitations,they contribute to the mass current density by an additional amount

ρn(vn − vs), (1.94)

where ρn is the normal density and vn is the normal velocity. The totalmass current density is then given by

j = ρvs + ρn(vn − vs) = (ρ− ρn)vs + ρnvn = ρsvs + ρnvn. (1.95)

The superfluid component shows a vortex-free or irrotational potential flow,whereas the normal fluid component shows a viscous flow.

The field operators of bosons follow the canonical commutation relations

[ψ(r), ψ†(r′)] = δ(r − r′), [ψ(r), ψ(r′)] = 0, [ψ†(r), ψ†(r′)] = 0. (1.96)

In analogy with Eq. (1.83), let us formally decompose ψ(r) as

ψ(r) ="ρ(r)eiφ(r). (1.97)

For Eq. (1.97) to be consistent with Eq. (1.96), it is sufficient to assumethe following commutation relations between ρ and φ:

[ρ(r), φ(r′)] = iδ(r − r′), [ρ(r), ρ(r′)] = 0, [φ(r), φ(r′)] = 0. (1.98)






To show this, we first note that

ρ(r) = iδ

δφ(r), φ(r) = −i

δ

δρ(r).

We can use these relations to obtain

[eiφ(r), ρ(r′)] = eiφ(r)δ(r − r′), [ρ(r), eiφ(r′)] = −eiφ(r′)δ(r − r′).

Furthermore, noting that (see the end of this section for details)

[eiφ(r), f [ρ(r′)]] ≃ [eiφ(r), ρ(r′)]f ′[ρ(r′)] = eiφ(r)δ(r − r′)f ′[ρ(r′)],

we obtain the following relations:!eiφ(r),

"ρ(r′)

#≃ eiφ(r) 1

2"ρ(r′)

δ(r − r′),

!"ρ(r), e−iφ(r′)

#≃ e−iφ(r′) 1

2"ρ(r)

δ(r − r′).

Thus,!ψ(r), ψ†(r′)

#=!"

ρ(r)eiφ(r), e−iφ(r)"ρ(r′)

#

="ρ(r)e−iφ(r′)

!eiφ(r),

"ρ(r′)

#+!"

ρ(r), e−iφ(r′)#"

ρ(r′)eiφ(r)

=

$"ρ(r)e−iφ(r′)eiφ(r) 1

2"ρ(r′)

+ e−iφ(r′) 12"ρ(r)

"ρ(r′)eiφ(r)

%δ(r − r′)

= δ(r − r′).

Let us define the phase operator φ and the total number operator N as

φ ≡ 1V

&φ(r)dr, (1.99)

N ≡&ρ(r)dr. (1.100)

It is easy to show that

[N , φ] = i. (1.101)

From Eq. (1.101), it follows that

φ = −i∂

∂N.

We can use this result to obtain

∂φ

∂t=

i

! [H, φ] = −1!∂H

∂N. (1.102)






Taking the expectation value on both sides of Eq. (1.102), we have

∂φ

∂t= −1

!∂E

∂N= −1

!µ, (1.103)

where µ is the chemical potential of the system. We substitute this intoEq. (1.85) and find that vs satisfies the following Euler equation:

d

dtvs =

∂vs

∂t+ (vs ·∇)vs = − 1

M∇µ. (1.104)

Thus, the superfluid velocity is accelerated by the gradient of the chemicalpotential. On the other hand, the total mass current density j is driven bythe pressure gradient,

d

dtj = −∇P, (1.105)

where P is the pressure. Substituting Eq. (1.95) into Eq. (1.105), we obtain

ρsvs + ρnvn = −∇P. (1.106)

Substituting Eq. (1.104) for vs into Eq. (1.106) gives

ρnvn =ρs

M∇µ −∇P. (1.107)

The Gibbs–Duhem relation [see Appendix A] holds in a system at thermalequilibrium:

SdT − V dP + Ndµ = 0. (1.108)

When there is no temperature variation in the system, i.e., dT = 0, wehave

ρnvn =1M

(ρs − ρ)∇µ. (1.109)

When the system is stationary at absolute zero, we have vn = 0. There-fore, we find that the superfluid density is equal to the total density(ρs = ρ) [Leggett (1991)]. (It must be noted, however, that N0 = N .)

Finally, let us examine how the approximation

[eiφ(r), f [ρ(r′)]] ≃ [eiφ(r), ρ(r′)]f ′[ρ(r′)] (1.110)

is justified. To analyze the validity of this approximation, let us start withthe following operator equation:

[A, Bn] = Bn−1[A, B] + Bn−2[A, B]B + Bn−3[A, B]B2

+ · · · +[A, B]Bn−1. (1.111)






The approximation (1.110) is obtained if the right-hand side of Eq. (1.111)is approximated by [A, B](Bn)′:

[A, Bn] ≃ [A, B](Bn)′. (1.112)Then,

[A, f(B)] =∞!

n=0

1n!

[A, f (n)(B = 0)Bn] ≃ [A, B]∞!

n=0

1n!

f (n)(0)(Bn)′

= [A, B]f ′(B);hence, Eq. (1.110) is proved. The following approximation is essential forderiving Eq. (1.112):

Bn−k[A, B]Bk−1 = [A, B]Bn−1 + [Bn−k, [A, B]]Bk−1.It is clear that if we ignore the last double commutator, Eq. (1.112) follows.As observed from Eq. (1.98), each commutation relation yields an extraunit density compared to ρ(r). Therefore, if the particle-number densityis considerably higher than 1, we can ignore the double commutator. Thisrequirement is satisfied when the system is Bose–Einstein condensed.

1.9 Fragmented Condensate5

When the one-particle density matrix has k (≥2) eigenvalues that are ofthe same order as the total number of particles, the system comprises kcondensates whose relative phases are random, and the system as a wholeis called a fragmented Bose–Einstein condensate (fragmented condensate).The concept of a fragmented BEC was first discussed by Nozieres and SaintJames [Nozieres and James (1982); Nozieres (1995)] and has witnesseda remarkable resurgence of interest, particularly in systems of gaseousBose–Einstein condensates [for reviews, see Castin and Herzog (2001) andMueller, et al. (2006)]. The fragmented condensate is obtained when thesystem possesses an exact symmetry. Here, we illustrate this concept usinga few examples.

1.9.1 Two-state model

First, let us consider a model in which N spinless bosons occupy only twostates with k and −k. The corresponding field operator is given by

ψ(r) =1√V

"akeikr + a−ke−ikr

#, (1.113)

5This section may be skipped on first reading.






and the second-quantized Hamiltonian is given by

H =!

dr ψ(r)"−!2∇2

2M

#ψ(r) +

gV

2

!dr ψ†(r)ψ†(r)ψ(r)ψ(r)

= ϵkN +g

2N(N − 1) + gNkN−k, (1.114)

where ϵk ≡ !2k2

2M , N±k ≡ a†±ka±k, and N ≡ Nk+N−k. Since N is constant,

the ground state of the system is determined by the last term in Eq. (1.114).In the case of a repulsive interaction g > 0, the ground state is

|Nk = N, N−k = 0⟩ or |Nk = 0, N−k = N⟩, (1.115)

and it is doubly degenerate. The single-particle density matrix

ρ1 =

$⟨a†

kak⟩ ⟨a†ka−k⟩

⟨a†−kak⟩ ⟨a†

−ka−k⟩

%(1.116)

for each ground state in Eq. (1.115) is"

N 00 0

#or

"0 00 N

#. (1.117)

Thus, the eigenvalues of ρ1 are N and 0 and only one condensate exists forboth cases.

In the case of an attractive interaction g < 0, the ground state of theHamiltonian (1.114) is given by the Fock state:

|F ⟩ =&&&Nk =

N

2, N−k =

N

2

', (1.118)

where we assume that N is even. Then, ρ1 becomes

ρ1 ="

N2 00 N

2

#; (1.119)

thus, the system is fragmented. From this example, we find that fragmen-tation occurs when the system undergoes BEC to form several degeneratesingle-particle states.

It is instructive to compare the Fock state (1.118) with the coherentstate

|φ⟩ =1√

2NN !

(a†ke−

i2 φ + a†

−kei2 φ)N

|vac⟩. (1.120)

The single-particle density matrix (1.116) for this state is

ρ1 ="

N2

N2 eiφ

N2 e−iφ N

2

#, (1.121)






and the eigenvalues are found to be N and 0. Thus, |φ⟩ describes a sin-gle condensate. The expectation value of gNkN−k over |φ⟩ is calculatedto give gN(N − 1)/4. Thus, the energy of the fragmented state (1.118),which is gN2/4, is lower than that of the coherent state (1.120) by |g|N/4.Interestingly, the average of |φ⟩ over φ gives |F ⟩:

! 2π

0

dφ

2π|φ⟩ =

"N !

2N#$

N2

%!&2'''N

2,N

2

(∝ |F ⟩. (1.122)

Thus, |F ⟩ is the equal-weighted average of the family of degenerate states|φ⟩.

1.9.2 Degenerate double-well model

Next, let us consider a degenerate double-well model [see Fig. 1.3] [Leggett(2001)], where the Hamiltonian is given by

t!a b!

Fig. 1.3 A degenerate double-well model. Bosons can hop between the two wells withthe transfer energy t. In each well, bosons undergo contact interactions.

H = −t)a†b + b†a

*+

g

2

)a†2 a2 + b†

2b2*, (1.123)

where t is the transfer energy and g is the strength of the contact interactionin each well. Assuming that the total number of particles is constant (N =Na + Nb = N), Eq. (1.123) can be rewritten as

H = −t)a†b + b†a

*+

g

4N(N − 2) +

g

4

)Na − Nb

*2. (1.124)

In the case of a repulsive interaction g > 0, a candidate for the groundstate is the Fock state:

'''Na =N

2, Nb =

N

2

(, (1.125)

EFock =g

4N(N − 2). (1.126)






As shown in Eq. (1.119), this is a fragmented condensate. Another can-didate is the coherent state in which all N bosons condense into a single-particle state described by the creation operator (a†e−

i2 + b†e

i2 φ)/

√2:

|φ⟩ =1√

2NN !

!a†e−

i2 φ + b†e

i2 φ"N

|0⟩. (1.127)

Calculating the expectation value of Eq. (1.123) over the state (1.127), weobtain

Ecoherent =g

4N(N − 1) − tN cosφ. (1.128)

Hence, if |t| > g/4, the coherent state (1.127) can have lower energy thanthe Fock state (1.126). The single-particle density matrix for the coherentstate is given by

ρ1 =

#⟨a†a⟩ ⟨a†b⟩⟨b†a⟩ ⟨b†b⟩

$=%

N2

N2 eiφ

N2 e−iφ N

2

&. (1.129)

The eigenvalues of ρ1 are determined from

det%

N2 − λ N

2 e−iφ

N2 e−iφ N

2 − λ

&= λ(λ− N) = 0 (1.130)

to be N and 0. The state (1.127) is, therefore, a single condensate.On the other hand, if |t| < g/4, the Fock state (1.125) has lower energy

than the coherent state (1.128). Note that the critical value tc = g/4 issmaller than the interaction energy per particle ∼ gN/4 by a factor of N .Thus, a tiny symmetry-breaking perturbation, which is induced by the hop-ping term, makes the fragmented state unstable against the coherent state.The physical reason for the vulnerability of the fragmented condensate liesin the fact that the fragmented state (1.125) has two macroscopically occu-pied components that enhance the matrix element of the hopping term bya factor of N [see the last term in Eq. (1.128)]. Thus, for the fragmentedstate to survive, the symmetry-breaking perturbation must be suppressedto below 1/N . This is why a single condensate is so robust in the thermo-dynamic limit; yet, it is possible to realize a fragmented condensate in amesoscopic regime.

In the case of an attractive interaction g < 0, the last term in Eq. (1.124)attains the minimum energy when the number difference Na − Nb is max-imal. When t = 0, the minimal energy is attained by a Schrodinger catstate:

|cat⟩ =1√2

(|N, 0⟩ + |0, N⟩) . (1.131)






If the double-well potential has an asymmetry, the single condensate thathas the lower energy, i.e., |N, 0⟩ or |0, N⟩, will be favored. When the twowells are degenerate, the cat state will be favored, because atoms tendto bunch up due to the attractive interaction; however, because of thedegeneracy of the two possible states, the system evolves, resulting in asuperposition state (1.131). The major symmetry-breaking perturbationfor the cat state is the asymmetry between the two wells. If the energies ofthe wells are degenerate, the hopping term plays the role of the symmetry-breaking perturbation in favor of the coherent state (1.127).

The single-particle density matrix of the cat state is given by Eq. (1.119)and has two extensive eigenvalues. Thus, we find that two-state systemshave two possible candidates for a fragmented condensate: the Fock statein the case of a repulsive interaction and the Schrodinger cat state in thecase of an attractive interaction.

1.9.3 Spin-1 antiferromagnetic BEC

Finally, let us consider a system of N spin-1 bosons described by the Hamil-tonian

H = JS · S, (1.132)

where

S =1!

m,n=−1

(S)mna†man. (1.133)

Here, S = (Sx, Sy, Sz) is a vector of spin-1 matrices, and a†m and an are the

creation and annihilation operators of the bosons with magnetic sublevelsm = 0,±1 [see Chap. 6 for details]. For simplicity, we assume that N iseven. In the case of an antiferromagnetic interaction J > 0, the groundstate is a spin-singlet state (S = 0), as described in Sec. 6.3.2. Introducingthe spin operators in the Cartesian coordinates

A ="Ax, Ay, Az

#≡$− a1 − a−1√

2,a1 + a−1√

2, a0

%, (1.134)

the spin-singlet state is expressed as

|S = 0⟩ =1&

(N + 1)!

"A† · A†

#N2 |0⟩. (1.135)

In this example, ρ1 is a 3 × 3 matrix, and its (i, j) component (i.e.,i, j = x, y, z) is given by ⟨A†

i Aj⟩. It can be shown [Koashi and Ueda (2000);






Ho and Yip (2000)] that ρ1 has triply degenerate eigenvalues equal to N/3and is therefore fragmented. The physics behind the fragmentation is therotational invariance of the spin-singlet state, which transforms as a scalarunder rotation; it follows from Schur’s lemma that the irreducible represen-tation of ρ1 must be proportional to the identity matrix. Since Trρ1 = N ,all the eigenvalues must be N/3.

The state that breaks the spherical symmetry is given by

|n⟩ =

!n · A†

"N

√N !

|0⟩. (1.136)

The single-particle density matrix for this state has eigenvalues N , 0, and 0,and therefore, |n⟩ describes a single condensate. If we calculate the averageof Eq. (1.136) over the direction of n, we reproduce the fragmented statein Eq. (1.135) [Mueller, et al. (2006)]:

#dΩ4π

|n⟩ =12

# 1

−1d(cos θ)

1√N !

!$A† · A† cos θ

"N

|0⟩

=1

(N + 1)√

N !

!A† · A†

"N2 |0⟩, (1.137)

where dΩ is an element of the solid angle. Thus, a fragmented state isinterpreted as the average of the symmetry-broken state over the directionof n.

1.10 Interference Between Independent Condensates

An interference experiment involving two independent Bose–Einstein con-densates that was demonstrated by the MIT group [Andrews, et al. (1997)]provided rather surprising results. Independently prepared condensateshave no definite relative-phase relationship; yet, they exhibit interferencepatterns whose peak positions exhibit shot-to-shot fluctuations [Hadzibabic,et al. (2004)].

To understand this phenomenon, we begin by recalling Young’s double-slit experiment, as shown in Fig. 1.4(a). An atomic wave emanating fromsource A passes through slits 1 and 2 and recombines on the screen toproduce an interference pattern. It is important to recognize that indepen-dently and identically prepared single atoms can exhibit such interference.A single-atom state that passes through the double slits is expressed as

|one atom⟩ = cos θ|1, 0⟩ + sin θ|0, 1⟩, (1.138)






where |1, 0⟩ (|0, 1⟩) describes the state in which an atom passes throughslit 1 (slit 2) at position r1 (r2). The probability P (r) that the atom isdetected at position r is given by

P (r) = c⟨one atom|Ψ†(r)Ψ(r)|one atom⟩, (1.139)

where c is a constant and Ψ(r), the field operator at position r; it can beexpressed in terms of the annhilation operators of the atom at slits 1 and2, a1 and a2, as

Ψ(r) = a1u1eik1(r−r1) + a2u2e

ik2(r−r2), (1.140)

where u1 and u2 are geometrical factors that depend on the experimentalconditions. Substituting Eq. (1.140) into Eq. (1.139), we obtain

P (r) = I1 + I2 +!

I1I2 cos(k1r1 − k2r2 + φ), (1.141)

where I1 ≡ c|u1|2 cos2 θ, I2 ≡ c|u2|2 sin2 θ, and φ ≡ arg(u∗1u2).

This example shows that the necessary and sufficient conditions forsingle atoms to exhibit Young’s interference pattern are that (i) one cannottell, in principle, the slit (1 or 2) through which each atom passes and that(ii) every atom is prepared in an identical state (1.138). The first condition(i) leads to the superposition of the two possibilities (1.140) and the secondcondition (ii) allows the state of the atoms to be represented by the samestate vector (1.138).

A A B(a) (b)

Bscreen

double slits1

k1

kA1 kA2 kB2kB1

k2

2

1 2

Fig. 1.4 (a) Young’s double slit experiment, and (b) two-particle interference experi-ment.

Next, let us consider an interference experiment involving two conden-sates, as shown in Fig. 1.4(b). The initial atomic state is

|AB⟩ = |NA⟩|NB⟩, (1.142)






where NA and NB are the atomic numbers of condensates A and B, re-spectively. The fact that they are Bose-Einstein condensed is implicit inthe assumption that they are represented by single-mode Fock states. Be-cause the two condensates are assumed to be independent, |Ψ⟩ is the directproduct of their state vectors with no relative-phase relationship betweenthem.

Now, suppose that the two atomic waves emanating from condensatesA and B arrive at various positions on the screen. The field operator atposition ri (i = 1, 2) on the screen is expressed in terms of the annihilationoperators aA and aB of the two condensates as

Ψ(ri) = aAuAeikAi(ri−rA) + aBuBeikBi(ri−rB). (1.143)

The joint probability distribution that one atom is detected at r1 and an-other atom is detected at r2 is given by

P (r1, r2) = c′⟨AB|Ψ†(r1)Ψ†(r2)Ψ(r2)Ψ(r1)|AB⟩, (1.144)

where c′ is a constant. Substituting Eqs. (1.142) and (1.143) in this equa-tion, we obtain

P (r1, r2) = 2c′NANB|uAuB|2(1 + cosϕ) + constant, (1.145)

where

ϕ = kA1(r1 − rA) + kB2(r2 − rB) − kA2(r2 − rA) − kB1(r1 − rB)

≃ k

L|r1 − r2||rA − rB|. (1.146)

Here, L is the distance between the source and screen, and we assume|kAi| = |kBi| = k (i = 1, 2) to obtain the last equality. Equations (1.145)and (1.146) show that the two independent condensates do exhibit an in-terference pattern. Because ϕ does not depend on the individual positionsr1 and r2 but on the distance between them |r1 − r2|, the interference pat-tern exhibits shot-to-shot fluctuations with regard to where the peaks andvalleys appear on the screen. Averaging over many events, the interferencepattern disappears; in fact, the ensemble average of the one-particle den-sity ⟨Ψ†(r)Ψ(r)⟩ over the state (1.142) shows no interference pattern. Theseresults indicate that the interference pattern observed by the MIT groupis a two-particle effect similar to the Hanbury Brown–Twiss experimentfor photons [Hanbury Brown and Twiss (1956)]. The two-particle interfer-ence may also be interpreted as a conquence of the emergence of coherenceby the action of a sequence of measurements [Javanainen and Yoo (1996);Naraschewski, et al. (1996); Castin and Dalibard (1997)].






1.11 Feshbach Resonance

Feshbach resonance [Feshbach (1958); Fano (1961)] is a resonance of amany-body system between a scattering state in one collision channel anda bound state in another. Suppose that two particles with total energyE in the center-of-mass frame interact via a molecular potential Vo(r) [seeFig. 1.5]. Such a collision path of a scattering state is referred to as anopen channel. We assume that there exists another collision path, referredto as a closed channel, that supports a bound molecular state with energyEb. A Feshbach resonance occurs when the incident particles come intoresonance with the bound state. The presence of such a bound state as anintermediate state in a scattering process alters the energy of the scatteringstate. According to the second-order perturbation theory, the energy shift∆E is given by

∆E =|⟨b|Hint|i⟩|2

E − Eb, (1.147)

where |i⟩ and |b⟩ are the incident and bound states, respectively, and Hint

is the interaction Hamiltonian that couples the closed and open channels.For ultracold collisions, we may set E = 0 in Eq. (1.147) to obtain

∆E = − |⟨b|Hint|i⟩|2

Eb. (1.148)

A magnetic Feshbach resonance utilizes the fact that the magnetic momentof a pair of atoms changes from µo in the open channel to µc in the closed

closed channel

open channel

bound state

Ener

gy

0

Vo(r)

Vc(r)

Interatomic distance r

Eb

Fig. 1.5 Schematic illustration of a Feshbach resonance, where Vc(r) and Vo(r) areinteratomic potentials for closed and open channels, respectively. The closed channel isassumed to support a bound state with energy Eb.






channel. In the presence of an external magnetic field B, the energy of thebound state relative to the open-channel threshold changes as

Eb = δµ(B − B0), (1.149)

where δµ ≡ µo − µc and B0 is the value of B at which Eb vanishes. Sub-stituting Eq. (1.149) in Eq. (1.148), we obtain

∆E = − |⟨b|Hint|i⟩|2

δµ(B − B0). (1.150)

As discussed in the next section, the interaction between ultracold atomsis proportional to the scattering length a [see Eq. (2.25)]. Therefore, neara Feshbach resonance, the scattering length depends on B as

a = abg

!1 − ∆

B − B0

", (1.151)

where abg is the background scattering length and ∆ ∝ (δµ)−1, the widthof the resonance. The magnitude and sign of the scattering length can thusbe tuned by an external magnetic field [Inouye, et al. (1998); Courteille, etal. (1998); Cornish (2000)].

A scattering state may be brought into resonance with a bound stateby other means. If it is implemented optically, it is called an optical Fesh-bach resonance. If the resonance is caused by a quasibound state thatis supported by an open-channel potential Vo rather than Vc, it is called ashape resonance. Comprehensive surveys of Feshbach resonances have beenreported, e.g., in [Kohler, et al. (2006); Chin, et al. (2008)].





Chapter 2

Weakly Interacting Bose Gas

2.1 Interactions Between Neutral Atoms

The interactions between neutral atoms are strongly repulsive when theinteratomic distance is of the order of the size of the atoms. The origin ofthis strong repulsion is the Coulomb interaction between electrons; how-ever, this interaction is effective only at atomic-scale distances because atlonger distances, electric charges are screened and atoms tend to behavelike neutral particles with electronic polarizability.

When the distance between atoms is of the order of 100 A, an attrac-tive force called the van der Waals force (or fluctuating dipole force) actsbetween the atoms. A neutral atom has an instantaneous electric dipolemoment p1 that points randomly and averages to zero: p1 = 0. An in-stantaneous dipole moment p1 of an atom induces the dipole moment p2

of another atom at a distance of r as

p2 = αE ∝ −αp1

r3,

where α is the atomic polarizability. The interaction energy V (r) betweenthe dipole moments at distance r is given by

V (r) ∝p1p2

r3∝ −

αp21

r6;

this does not vanish upon time averaging!

p21 = 0

"

and causes a weak

attractive force between the atoms.Two colliding atoms experience both attractive and repulsive parts of

the potential, and the net force can be attractive or repulsive. Accordingto experimental results, 1H, 4He, 23Na, 41K, and 87Rb undergo repulsiveinteractions, whereas 7Li, 39K, 85Rb, and 133Cs undergo attractive ones atzero magnetic field.

33






From a theoretical point of view, the presence of a singular short-rangedpotential makes it difficult to calculate the physical properties of a systemfrom first principles, because simple perturbative approaches such as theBorn approximation break down. In fact, the scattering amplitude V (k) fora potential V (r) is given in the Born approximation by its Fourier transform

V (k) =

!

drV (r)e−ikr (2.1)

that becomes infinite if V (r) diverges. It is therefore of interest to determinewhether it is possible to replace a real potential with an effective potentialto which perturbation theory is applicable. This is indeed possible whenthe gas is dilute, i.e., when the average interparticle distance is much longerthan the range of the interaction, as discussed in the next section.

As shown above, the van der Waals potential, which acts between theelectric dipoles of atoms, is of the form −C/r6. Let r0 be a characteristiclength scale over which the interaction between atoms acts at ultralow tem-peratures. Since at ultralow temperatures, the total energy of two collidingatoms is nearly zero as compared to that at the other energy scales, weobtain

total energy =!2

Mr20

−C

r60

= 0 → r0 =

"CM

!2

# 1

4

, (2.2)

where M is the mass of the atom. The coupling constant C is determinedfrom a simple dimensional analysis; the relevant length scale is the Bohrradius

aB =4πϵ0!2

e2me≃ 5.29 × 10−11 m, (2.3)

where me is the mass of the electron, and the relevant energy scale is theCoulomb energy e2/(4πϵ0aB). Thus, C can be expressed in terms of thesequantities as

C = C6 ·e2

4πϵ0aB· a6

B =C6e2

4πϵ0a5B = C6

!2

mea4B, (2.4)

where C6 is a dimensionless constant that has a value of the order of a fewthousands for alkali atoms. Substituting Eq. (2.4) in Eq. (2.2) gives

r0 =

"

C6M

me

# 1

4

aB. (2.5)

The mass of the atom M is typically 10,000 times that of the electron me.Considering this and using a typical value C6 ∼ 104, we obtain r0 ∼ 100aB





Weakly Interacting Bose Gas 35

as a rough estimate of the s-wave scattering length of an alkali atom or atypical extent of the least bound state.

A minimal model that takes into account the two important propertiesof the interatomic potential—strong repulsion at short distance and the vander Waals attraction at long distance—is given by

V (r) =

!

∞ if r < R0;−C6

r6 otherwise.(2.6)

The scattering problem with this potential can be solved exactly [Gribakinand Flambaum (1993)] with the result

a = a

"

1 − tan#

Φ −3

8π$%

, (2.7)

where a is the average scattering length and Φ =√

2MC6/(2!R0) is theWKB phase shift that the incoming wave experiences due to the poten-tial (2.6). Assuming that Φ distributes randomly from one atomic speciesto another, we find from Eq. (2.7) that the probability of finding the signof the scattering length of a given atomic species to be positive is 3/4.

Atoms are composite particles comprising electrons, protons, and neu-trons. They have internal structures that are revealed by their discreteexcitation spectra. It is noteworthy that atoms behave as structurelessbosons or fermions at ultralow temperatures. This can be explained as fol-lows. The ground-state and excited-state energies of particles confined toa sphere with radius r are of the order of

ϵ ∼!2

2Mr2.

When the temperature T of the system satisfies the condition

kBT ≪ ϵ, (2.8)

the system cannot get excited, and therefore, the internal structures ofthe composite particles do not manifest themselves. If we consider thetransition temperature of BEC as a typical temperature of the system,

kBT0 ∼!2

M

&N

V

'2/3

=!2

M

1

R2,

where R is the average interatomic distance, the condition kBT ≪ ϵ isreplaced by

kBT0

ϵ∼

( r

R

)2≪ 1. (2.9)






Condition (2.8) is thus satisfied if R ≫ r, that is, if the gas is dilute. Inthe case of alkali BEC, typically, R ∼ 1000 A and r ∼ 1 A, and thus,(r/R)2 ∼ 10−6. Condition (2.9) is thus well satisfied.

It is also noteworthy that gaseous BECs can be realized for alkali atomswhose ground states are known to be metals. The first step in the solidi-fication of atomic gases is for these gases to form molecules. However, fora molecule to be formed, at least three atoms must collide for the laws ofenergy and momentum conservations to be obeyed; then, two of the threeatoms form a molecule, while a third one carries away extra energy andmomentum that are released upon molecular formation. In a tenuous gas,the probability of three atoms being found within the range of interactionis very small, and therefore, BECs can actually be formed in an excitedmany-body state with a sufficiently long lifetime (typically a minute). Aslong as three-body inelastic collisions are negligible, BECs with repulsiveinteractions can exist in a thermodynamically stable state. However, BECswith attractive interactions can only exist in a metastable state even if theinelastic collisions are negligible, as it is prone to collapse due to dynamicalinstability, as discussed in Sec. 3.8.

2.2 Pseudo-Potential Method

If the range of the interaction is much shorter than the mean interatomicdistance, it is not necessary to know the details of the short-distance be-havior of the interaction potential to describe the low-energy properties ofthe system. These properties will then be described only by the knowledgeof the asymptotic behavior of the wave function before and after scatteringcharacterized by the phase shift of the scattered wave.

Consider the scattering of two particles via a hard-sphere potential ofradius a. The wave function of the two particles with respect to the relativecoordinate r satisfies the Schrodinger equation@

(∇2 + k2)ψ(r) = 0 (r > a) (2.10)

subject to the boundary condition

ψ(r) = 0 (r ≤ a). (2.11)

The hard-core boundary condition (2.11) makes it difficult to calculate themany-body wave function. Huang and Yang [Huang and Yang (1957)]

pointed out that boundary condition (2.11) can be effectively replaced with






a pseudo-potential. In particular, when only the s-wave scattering is im-portant and as long as the solution with r ≥ a is concerned, Eq. (2.10) withboundary condition (2.11) is equivalent to the following equation:

(∇2 + k2)ψ(r) = 4πtan(ka)

kδ(r)

∂

∂r(rψ(r)) (r ≥ 0). (2.12)

That is, for r ≥ a, the solution of Eq. (2.12) coincides with that of Eq. (2.10)with boundary condition (2.11).

To prove this, we first note that the general solution of Eq. (2.10) canbe expanded as

ψ(r) =∞!

l=0

l!

m=−l

Alm[jl(kr) − tan ηl nl(kr)]Ylm(Ω), (2.13)

where jl(x) and nl(x) are the spherical Bessel function and the sphericalNeumann function, respectively, and Ylm(Ω) is the spherical harmonic func-tion. Here, l denotes the relative angular momentum of the two scatteringatoms and m is the magnetic quantum number. The boundary conditionψ(r) = 0 at |r| = a is satisfied if

tan ηl =jl(ka)

nl(ka). (2.14)

For s-wave scattering (i.e., l = 0),

j0(x) =sin x

x, n0(x) = −

cosx

x, r

and hence, Eq. (2.14) leads to

η0 = −ka.

Substituting these results in Eq. (2.13) gives

ψ(r) ∝ j0(kr) − tan η0n0(kr) =1

cos ka

sin k(r − a)

kr. (2.15)

This solution satisfies the Schrodinger equation (2.10) for |r| ≥ a; however,it cannot formally be extrapolated to |r| < a because it is singular at r = 0.This singularity can be removed if we add an appropriate term proportionalto the delta function to the right-hand side of Eq. (2.10). In fact,

∇2 sin k(r − a)

kr=

sin k(r − a)

k∇2 1

r+

1

r∇2 sin k(r − a)

k−

2

r2cos k(r − a)

= 4πδ(r)sin ka

k− k2 sin k(r − a)

kr, (2.16)






where

∇2 1

r= −4πδ(r) (2.17)

is used. Hence,

(∇2 + k2)sin k(r − a)

kr= 4πδ(r)

sin ka

k= 4πδ(r)

tan(ka)

kcos k(r − a)

= 4πtan(ka)

kδ(r)

∂

∂r

!

rsin k(r − a)

kr

"

,

and Eq. (2.12) is reproduced.Equation (2.12) can be rewritten as

−!2∇2

2(M/2)ψ(r) + V pseudo(r)ψ(r) = Eψ(r), (2.18)

where M/2 is the reduced mass,

E =!2k2

M, (2.19)

and

V pseudo(r)ψ(r) ≡4π!2

M

tan(ka)

kδ(r)

∂

∂r(rψ(r)). (2.20)

We thus find that the hard-sphere boundary condition at r = a can bereplaced with the pseudo-potential defined in Eq. (2.20). We also see fromEq. (2.15) that the use of the pseudopotential is equivalent to imposing thefollowing boundary condition

1

rψ

d

dr(rψ)

####r=0

= −k cot(ka) (2.21)

which in the low-energy limit reduces to [Bethe and Peierls (1935)]

limk→0

1

rψ

d

dr(rψ)

####r=0

= −1

a. (2.22)

Note that V pseudo(r) does not satisfy hermiticity. To show this, we takea regular function φ(r) and a function ψ(r) that has a 1/r-singularity atr = 0 but is regular elsewhere, i.e., ψ(r) = ψ(r) + A/r, where ψ(r) is aregular part of ψ(r). The hermiticity condition of V is

$

φ∗(V pseudoψ)dr?=

$

ψ(V pseudoφ)∗dr.

However, it can be shown by a straightforward calculation that the integralon the left-hand side is proportional to φ∗(0)ψ(0), which is finite, whereasthe one on the right-hand side is proportional to (ψ+ A/r)φ∗|r=0, which is






infinite. Thus, V pseudo(r) is not hermitian. Because of this non-hermiticity,V pseudo(r) introduced in Eq. (2.20) is called as a pseudo-potential.

When the energy of the system is so small or when the temperature isso low that ka ≪ 1, we have

tan(ka)

k= a +

1

2(ka)2reff + · · · = a + O(a3), (2.23)

where reff = 23a is an effective range of the interaction. When the terms of

the order of O(a3) are ignored, pseudo-potential (2.20) reduces to

V pseudo(r) = U0δ(r)∂

∂rr, (2.24)

where

U0 =4π!2a

M. (2.25)

Moreover, if ψ(r) does not diverge at the origin, the pseudo-potential be-comes a simple delta-function potential

V (r) = U0δ(r). (2.26)

In fact, if ψ(r) is regular at the origin, we have

V pseudo(r)ψ(r) = U0δ(r)∂

∂r(rψ(r)) = U0δ(r)

!

ψ(r) + r∂ψ(r)

∂r

"

= U0δ(r)ψ(r).

However, if ψ(r) has a 1/r-singularity, the operator ∂∂r r eliminates it.

It is straightforward to generalize the two-body problem to the N -bodyproblem, where the corresponding Hamiltonian is given by

H = −N#

i=1

!2

2M∇2

i +4π!2a

M

#

i<j

δ(ri − rj)∂

∂rijrij , (2.27)

where rij ≡ |ri − rj |. It is known that the δ-function potential in threedimensions does not scatter waves. Therefore, the eigenspectrum that isobtained by exactly diagonalizing Eq. (2.27) without the term (∂/∂rij)rij

should exactly agree with that of the corresponding free-particle system.However, we can accurately calculate the ground-state energy up to termsof the order of a3 by using the pseudo-potential as a perturbation, because(i) it is exact to replace tan(ka)/k with a because the wave number in theground state is k = 0, and (ii) the symmetry requirement of bosons excludesthe contribution of the p wave (l = 1) that is of the order of a3.






The ground-state wave function does not diverge at r = 0. Therefore,when one is interested in the ground-state properties, Hamiltonian (2.27)can be simplified to

H = −N!

i=1

!2

2M∇2

i +4πa!2

m

!

i<j

δ(ri − rj). (2.28)

The pseudo-potential that describes a three-body collision can be describedas

!2K

M

!

i<j<k

δ(ri − rj)δ(rj − rk), (2.29)

where by the dimensional analysis, K is of the order of a4.The wave function corresponding to the s-wave scattering is given from

Eq. (2.15) as

ψ(r) =A

cos ka

sin k(r − a)

kr; (2.30)

in the low-energy limit (k → 0), this reduces to

ψ(r) = A"

1 −a

r

#

. (2.31)

For a > 0, the wave function is suppressed around r = 0 as comparedto that in the non-interacting case with a = 0 [see Fig. 2.1 (a)]. Thiscan be interpreted as being due to repulsive interaction. For a < 0, thewave function is enhanced around r = 0 as compared to that in the non-interacting case [see Fig. 2.1 (b)]. This can be interpreted as being due toattractive interaction.

2.3 Bogoliubov Theory

2.3.1 Bogoliubov transformations

The many-body Hamiltonian (2.27) is expressed in the second-quantizedform as

H =

$

dr ψ†(r)

%

−!2∇2

2M

&

ψ(r)

+U0

2

$$

dr1dr2ψ†(r1)ψ

†(r2)δ(r1 − r2)∂

∂r12

'

r12ψ(r1)ψ(r2)(

, (2.32)

where r12 ≡ |r1−r2|, and the field operators ψ and ψ† satisfy the canonicalcommutation relations

'

ψ(r), ψ†(r′)(

= δ(r − r′),'

ψ(r), ψ(r′)(

= 0,'

ψ†(r), ψ†(r′)(

= 0. (2.33)






0 a

1

no interaction (a = 0)

r

(a) a > 0

(r)ψ

(b) a < 0

0 |a|

1

(r)

r

ψ

Fig. 2.1 Behavior of the wave function for (a) a > 0 (repulsive interaction) and (b)a < 0 (attractive interaction).

In Eq. (2.32), the operator δ(r1 − r2)(∂/∂r12)r12 can be replaced withδ(r1 − r2), unless the divergence of the 1/r form appears in the integral.With this proviso, Eq. (2.32) reduces to

H =

!

d3rψ†(r)

"

−!2∇2

2M

#

ψ(r) +U0

2

!

drψ†(r)2ψ(r)2. (2.34)

In Eq. (2.34), by substituting the Fourier expansion of the field operator

ψ(r) =1√V

$

k

akeikr, (2.35)

where V is the volume of the system, we find that

H =$

k

!2k2

2Ma†kak +

U0

2V

$

pqk

a†pa†

qap+kaq−k, (2.36)

where the creation and annihilation operators are assumed to satisfy thecanonical commutation relations:

%

ak, a†k′

&

= δkk′ ,%

ak, ak′

&

= 0,%

a†k, a†

k′

&

= 0. (2.37)






Suppose that most of the particles are Bose–Einstein condensed in the k = 0state, that is,

n0 ≈ N,!

k

′

nk ≪ N, (2.38)

where nk is the number of particles in the k = 0 state; N , the total num-ber of particles; and

"′k, the summation over k excluding k = 0. In the

Bogoliubov approximation, we retain terms up to the second order in a0 inEq. (2.36):!

pqk

a†pa†

qap+kaq−k

∼= a†2

0 a20 + a†2

0

!

k

′

aka−k +!

k

′

a†ka†

−ka20 + 4a†

0a0

!

k

′

a†kak

∼=

#

a†0a0 +

!

k

′

a†kak

$2

+ 2a†0a0

!

k

′

a†kak + a†2

0

!

k

′

aka−k + a20

!

k

′

a†ka†

−k

= N2 + 2a†0a0

!

k

′

a†kak + a†2

0

!

k

′

aka−k + a20

!

k

′

a†ka†

−k.

For the same order of approximation, we may replace a†0a0, a†2

0 , and a20 in

the last line with N . Then, Eq. (2.36) reduces to

H =U0n

2N

+!

k

′

%

(ϵk + U0n) a†kak +

1

2U0n

&

a†ka†

−k + a−kak

'(

, (2.39)

where ϵk = !2k2/2M and n = N/V .To diagonalize this Hamiltonian, we perform the canonical transforma-

tions known as Bogoliubov transformations:

ak =bk − αk b†−k)

1 − α2k

, a†k =

b†k − αk b−k)

1 − α2k

, (2.40)

where bk and b†k obey the canonical commutation relations*

bk, b†k′

+

= δkk′ ,*

bk, bk′

+

= 0,*

b†k, b†k′

+

= 0. (2.41)

We show that b†k and bk are the creation and annihiliation operators ofa quasiparticle called Bogolon. Substituting Eq. (2.40) in Eq. (2.39) andputting

x =

,ϵk

U0n, (2.42)






we have

(ϵk + U0n) a†kak +

1

2U0n

!

a†ka†

−k + a−kak

"

=U0n

1 − α2k

#

(x2 + 1 − αk)b†kbk +$

(x2 + 1)α2k − αk

%

b−kb†−k

−!

b†kb†−k + b−kbk"&'

x2 + 1(

αk −1 + α2

k

2

)*

. (2.43)

With the requirement that the coefficient of b†kb†−k + b−kbk vanishes, wehave

αk = 1 + x2 − x+

x2 + 2, (2.44)

where the minus sign is taken because αk < 1 [see Eq. (2.40)]. Then, theHamiltonian becomes

H =U0n

2N +

,

k

′+

ϵk(ϵk + 2U0n)b†kbk

−1

2

,

k

′-

U0n + ϵk −+

ϵk(ϵk + 2U0n).

. (2.45)

The k-summation in the last term diverges. To examine the nature of thisdivergence, we expand the last term in Eq. (2.45):

U0n + ϵk −+

ϵk(ϵk + 2U0n) =(U0n)2

2ϵk+ · · · . (2.46)

Thus, the divergence is of the 1/k2 form which changes to the 1/r formupon Fourier transformation to a real space. This term should be ignoredbecause it is to be eliminated by the action of the operator δ(r)(∂/∂r)r inthe pseudo-potential. By subtracting the corresponding term, the last termin Eq. (2.45) converges, giving

−,

k

′

&

U0n + ϵk −+

ϵk(ϵk + 2U0n) −(U0n)2

2ϵk

)

= −!2

4M

V

2π2g

5

2

/ ∞

0dxx2

0

1 + x2 − x+

x2 + 2 −1

2x2

1

=U0n

2N

128

15

2

na3

π, (2.47)

where g ≡ 8πan and/ ∞

0dx x2

0

1 + x2 − x+

x2 + 2 −1

2x2

1

= −8

15

√2






is used to obtain the last equality. Substituting Eq. (2.47) in Eq. (2.45),we obtain

H =2πa!2N2

MV

!

1 +128

15

"

na3

π

#

+$

k

′ !2k

2M

%

k2 + 16πna b†kbk. (2.48)

The first two terms on the right-hand side give the ground-state energyE0. Including higher-order corrections [Wu (1959); Hugenholtz and Pines(1959); Sawada (1959); Braaten, et al. (2001)], we obtain

E0 =2πa!2N2

MV

&

1 +128

15

"

na3

π+ 8

'4π

3−√

3

(

na3 ln(na3)

+Cna3)

, (2.49)

where n = N/V . Note that the expansion parameter of the theory isna3. The coefficient C of the last term is contributed from a shape-dependent parameter of the s-wave scattering as well as the p-wave scat-tering, both of which are proportional to a3. The last term in Eq. (2.49)involves nonuniversal effects that are sensitive to 3-body physics [Braaten,et al. (2001)]; other terms are universal in that they are insensitiveto the short-range physics and characterized by the scattering lengthalone. The first two terms on the right-hand side were obtained byLee and Yang [Lee and Yang (1957); Lee, et al. (1957)], and the thirdterm was obtained by Wu [Wu (1959); Hugenholtz and Pines (1959);Sawada (1959)]. The appearance of the logarithmic term in the expansionimplies that it is an asymptotic expansion.

The last term in Eq. (2.48) describes the Bogoliubov excitations calledBogoliubov phonons and the dispersion relation is given by

Ek =!2k

2M

%

k2 + 16πan. (2.50)

In the long-wavelength limit k → 0, Eq. (2.50) reduces to

Ek =!2k

2M

√16πan ∝ k. (2.51)

Thus, the excitation spectrum at the low-energy limit (k → 0) is identicalto that of a phonon Ek = !ck, where the sound velocity c is given by

c =!

M

√4πan. (2.52)

At the high-energy limit (k → ∞), on the other hand, Eq. (2.50) reducesto

Ek =!2k2

2M+

4π!2a

Mn, (2.53)






implying that excitations move freely in a uniform Hartree potential4π!2an/M . As Eq. (2.50) suggests, the crossover between the phononspectrum (2.51) and the single-particle spectrum (2.53) occurs at k = ξ−1,where

ξ =1√

8πan(2.54)

is known as the healing length.The sound velocity can also be calculated from the compressibility κ =

−(1/V )(∂V/∂P0) of the ground state:

c =1√

Mnκ=

!

−V

M

∂P0

∂V, (2.55)

where

P0 = −∂E0

∂V(2.56)

is the pressure of the ground state. Considering the first two terms inEq. (2.49), we obtain

c =!

M

√4πan

"

1 + 16

!

na3

π

#

. (2.57)

The first term on the right-hand side is equivalent to Eq. (2.52), and the sec-ond term gives the Lee–Yang correction to it. Experiments on a zero-soundwave were performed for a sodium BEC [Andrews, et al. (1997,1998)].

2.3.2 Bogoliubov ground state

The eigenfunction of the diagonalized Hamiltonian (2.48) can be expandedin terms of the Fock state |nk1

, nk2, ···⟩ in which there are nk1

quasiparticleswith wave vector k1, nk2

quasiparticles with wave vector k2, etc. Thenumber operator of quasiparticles with wave vector k is given by b†kbk andsatisfies the eigenvalue equation

b†kbk| · ··, nk, · · ·⟩ = nk| · ··, nk, · · ·⟩. (2.58)

The Bogoliubov ground state |ψ0⟩ ≡ |0, 0, · · ·⟩ is the state in which noquasiparticles are present and therefore satisfies

bk|ψ0⟩ = 0 (2.59)

for all k. Since the Bogoliubov transformations (2.40) connect only thestates with wave vectors k and −k, |ψ0⟩ may be expanded in terms of the






states |n1, m1; n2, m2; · · ·⟩ in which ni particles are present in the ki stateand mi particles in the −ki state:

|ψ0⟩ =∞!

n1,m1=0

∞!

n2,m2=0

· · ·"

Ck1

n1m1Ck2

n2m2· · ·

#

|n1, m1; n2, m2; · · · ⟩. (2.60)

Substituting this in Eq. (2.59) and noting that bk is related to the creationand annihiliation operators of original particles as

bk =1

$

1 − α2k

%

ak + αka†−k

&

, (2.61)

we have

0 =∞!

n,m=0

Cknm

'√n|n − 1, m⟩ + αk

√m + 1|n, m + 1⟩

(

=∞!

n,m=0

'

Ckn+1,m

√n + 1 + Ck

n,m−1αk√

m(

|n, m⟩,

where we define Ckn,−1 ≡ 0. From the orthogonality of the basis states, we

obtain

Ckn+1,m

√n + 1 + Ck

n,m−1αk√

m = 0. (2.62)

Substituting m = 0, we have Ckn+1,0

√n + 1 = 0; hence, Ck

n+1,0 = 0 forn ≥ 0. We also obtain Ck

0,n+1 = 0 by following a similar procedure for

b−k. If we assume Ckn,m−1 = 0 for n = m − 1, we have Ck

n+1,m = 0 fromEq. (2.62). By mathematical induction, all the off-diagonal componentsvanish. The diagonal component is determined from Eq. (2.62) as

Ckmm + αkCk

m−1,m−1 = 0 → Ckmm = (−αk)mCk

00. (2.63)

Here, Ck00 is determined from the normalization condition of the wave func-

tion. We thus find that the ground state is a state in which pairs of particleswith wave vectors k and −k are excited. Let |n1, n2, · · · ⟩ be the state inwhich there are n1 pairs of particles with wave vectors k1 and −k1, n2

pairs of particles with wave vectors k2 and −k2, etc. Then, |ψ0⟩ can beexpressed as

|ψ0⟩ = Z∞!

n1=0

∞!

n2=0

· · · [(−α1)n1 (−α2)

n2 · · · ] |n1, n2, · · · ⟩|φ0⟩, (2.64)

where |φ0⟩ is a state with k = 0 [see Eq. (2.80)], and Z is the normalizationfactor, which is determined from

⟨ψ0|ψ0⟩ = 1 = Z2∞!

n1=0

∞!

n2=0

· · · (−α1)2n1 (−α2)

2n2 · · · = Z2)

kx>0

"

1 − α2k

#−1,






where!

kx>0implies that the product is to be taken over all ky and kz and

all positive kx. Hence,

Z ="

kx>0

#

1 − α2k = exp

$

−4

9(3π − 8)N

%

na3

π

&

. (2.65)

Note that Z is the overlap integral between the ground states with andwithout interaction. Equation (2.65) shows that the interacting groundstate becomes orthogonal to the noninteracting ground state for N → ∞.To prove the last equality in Eq. (2.65), we note that

"

kx>0

#

1 − α2k = exp

$

1

2

'

kx>0

ln(

1 − α2k

)

&

.

Here, the sum in the exponent can be converted into an integral as

1

2

V

(2π)32π

* ∞

0k2dk ln

(

1 − α2k

)

=V

8π2

* ∞

0dk

k3

3

2αk

1 − α2k

dαk

dk

=V

12π2

* ∞

0dkk3 αk

1 − α2k

dαk

dk= −

V

12π2

* 1

0dαkk3 αk

1 − α2k

. (2.66)

We then eliminate k in favor of αk using the relation

k =√

8πan1 − αk√

2αk. (2.67)

The last term in Eq. (2.66) then becomes

−V

12π2(8πan)

3

21√8

* 1

0dα

(1 − α)3

α3/2

α

1 − α2

= −2N

3

%

na3

π

* 1

0dα

1√α

(1 − α)2

1 + α. (2.68)

Substituting α = t2, the integral can be evaluated as

2

* 1

0tdt

1

t

(

1 − t2)2

1 + t2=

2

3(3π − 8).

We thus obtain Eq. (2.65).The Bogoliubov ground state (2.64) can also be expressed in a second-

quantized form as

|ψ0⟩ = Zexp

$

−'

kx>0

αka†ka†

−k

&

|φ0⟩. (2.69)






To prove this, it is sufficient to consider only the terms with k = k1. Then,we have

exp!

−αk1a†k1

a†−k1

"

|φ0⟩ =∞#

n1=0

(−αk1)n1

n1!

!

a†k1

a†−k1

"n1

|φ0⟩

=∞#

n1=0

(−αk1)n1 |n1⟩|φ0⟩,

which is the same as the corresponding term in Eq. (2.64). Equation (2.69)can also be proved more directly as follows. Substituting Eq. (2.61) inEq. (2.59) gives

bk|ψ0⟩ =1

$

1 − α2k

!

ak + αka†−k

"

|ψ0⟩ = 0. (2.70)

From the commutation relation%

ak, a†k

&

=1, we may formally write ak =

d/da†k; then, Eq. (2.70) is expressed as

'

d

da†k

+ αka†−k

(

|ψ0⟩ = 0. (2.71)

By inspection, we find that Eq. (2.69) is indeed a solution of Eq. (2.71).

2.3.3 Low-lying excitations and condensate fraction

The state in which one Bogoliubov phonon with momentum !k is excitedis given by

|ψk⟩ = b†k|ψ0⟩ =a†k + αka−k$

1 − α2k

|ψ0⟩. (2.72)

Substituting Eq. (2.64) on the right-hand side of this equation gives

a†k + αka−k$

1 − α2k

Z#

[· · · (−αk)nk · · · ] | · · · , nk, · · · ⟩, (2.73)

where |nk⟩ represents the state that has nk pairs of particles with wavevectors k and −k, namely |nk⟩ ≡ |nk, n−k⟩. Noting the fact that nk = n−k,Eq. (2.73) becomes

Z#

· · ·(−αk)nk

$

1 − α2k

)√nk + 1|nk + 1, nk⟩ + αk

√nk|nk, nk − 1⟩

*

= Z#

· · ·(−αk)nk

$

1 − α2k

+

1 − α2k

,√nk + 1|nk + 1, nk⟩ =

-

1 − α2ka†

k|ψ0⟩.






We thus obtain1

|ψk⟩ = b†k|ψ0⟩ =!

1 − α2ka†

k|ψ0⟩. (2.74)

Note that the state (2.74) is normalized to unity for

⟨ψk|ψk⟩ = ⟨ψ0|bkb†k|ψ0⟩ = ⟨ψ0|b†kbk + 1|ψ0⟩ = ⟨ψ0|ψ0⟩ = 1, (2.75)

where we used bk|ψ0⟩ = 0. The one-phonon state (2.74) is therefore formedwhen a particle with momentum !k is added to the Bogoliubov groundstate |ψ0⟩. The cordinate representation of the one-phonon state (2.74) isobtained from Eq. (B.20) in Appendix B as

ψk(r1, r2, · · · , rN ) =N"

n=1

eikrnψ0(r1, r2, · · · , rN ). (2.76)

In the Bogoliubov ground state |ψ0⟩, bosons are virtually excited tostates with k = 0 even at T = 0 due to the interactions. The averagenumber of virtually excited bosons with wave vector k is obtained fromEqs. (2.40) and (2.75) by

⟨nk⟩ = ⟨ψ0|a†kak|ψ0⟩ =

α2k

1 − α2k

(k = 0). (2.77)

Summing Eq. (2.77) over k = 0 gives the total number of virtually excitedbosons:

"

k

′

⟨nk⟩ ="

k

′ α2k

1 − α2k

=V

(2π)34π

# ∞

0k2dk

α2k

1 − α2k

.

1Another proof of Eq. (2.74): From Eq. (2.69) and

eαka

†k

a†−k a−ke

−αa†k

a†−k = −αka†

k+ a−k,

we have

a−k|ψ0⟩ = Za−ke−$

αk a†ka†−k

|φ0⟩

= Ze−$

αk a†ka†−k

%

−αka†k

+ a−k

&

|φ0⟩ = −αka†k|ψ0⟩.

Therefore, we obtain

%

a†k

+ αk a−k

&

|ψ0⟩ ='

1 − α2k

(

a†k|ψ0⟩,

which proves Eq. (2.74).






By performing integrations in a manner similar to Eqs. (2.66)–(2.68), weobtain

!

k

′

⟨nk⟩ =V

3π2

" 1

0dαkk3 αk

(1 − α2k)

2

=V

3π2

#8πaN

V

$ 3

2 1√8

" 1

0dα

(1 − α)3

α3/2

α

(1 − α2)2

=8N

3

%

a3N

πV.

Hence, the total number of virtually excited bosons is given by

!

k

′

⟨nk⟩ =8

3

%

na3

πN, (2.78)

and the fraction of the bosons remaining in the k = 0 state, which is referredto as the condensate fraction, is given by

⟨n0⟩N

= 1 −8

3

%

na3

π. (2.79)

This result shows that for a dilute gas with na3 ≪ 1, the condition (2.38),which is assumed in deriving the effective Hamiltonian (2.39), is satisfied.

2.3.4 Properties of Bogoliubov ground state

To investigate the properties of the Bogoliubov ground state, it is compu-tationally convenient to assume a coherent state for |φ0⟩:

|φ0⟩ = e−|φ0|2

2 eφ0a†0 |vac⟩ = e−

|φ0|2

2

∞!

n0=0

φn0

0√n0!

|n0⟩, (2.80)

where |φ0⟩ is an eigenstate of the annihilation operator

a0|φ0⟩ = φ0|φ0⟩.

Substituting Eq. (2.80) in Eq. (2.69), we obtain the ground-state wavefunction in the Bogoliubov approximation:

|ψ0⟩ = N exp

&

φ0a†0 −

!

kx>0

αka†ka†

−k

'

|vac⟩, (2.81)

where

N = e−|φ|2

2 Z = e−|φ|2

2 exp

&

−4

9(3π − 8)N

%

na3

π

'

. (2.82)






The expectation value of the annihilation operator over |ψ0⟩ does not van-ish; it is given as

⟨a0⟩ = N2⟨vac|eφ∗0a0−

!

kx>0α∗

kaka−k a0eφ0a†

0−!

kx>0αka†

ka†−k |vac⟩

= N2 ∂

∂φ∗0N−2 = φ0, (2.83)

where the last equality is obtained by using Eq. (2.82). The average numberof particles in the k = 0 state is given by

n0 ≡"

a†0a0

#

="

a0a†0 − 1

#

= N2

$∂2

∂φ0∂φ∗0− 1

%

N−2 = |φ0|2. (2.84)

On the other hand, the average number of particles in the k = 0 state isgiven by

nk ≡"

a†kak

#

=&

1 − |αk|2'

⟨vac|eα∗kaka−k a†

kakeαka†ka†−k |vac⟩

=&

1 − |αk|2'

∞(

n,m=0

(−1)n+mα∗nk αm

k

"

nk, n−k|a†kak|mk, m−k

#

=∞(

n=0

n|αk|2n =|αk|2

1 − |αk|2. (2.85)

Solving Eq. (2.85) for |αk|, we obtain

|αk| =

)nk

1 + nk. (2.86)

The expectation value of aka−k, which is referred to as the anomalousaverage, is given by

χk ≡ ⟨aka−k⟩ = N−2

$

−∂

∂α∗k

%

N2

=αk

1 − |αk|2=

αk

|αk|*

nk(1 + nk). (2.87)

Let us assume φ0 and αk to be variational parameters and determinetheir expression by minimizing the expectation value of H − µN , where

H − µN =(

k

(ϵk − µ)a†kak +

1

2

(

qkk′

Vqa†k+qa†

k′−qakak′ . (2.88)






The expectation value of the first term on the right-hand side is∑

k(ϵk −µ)nk, and that of the second term is approximated to give

1

2

∑

qkk′

Vq

⟨

a†k+qa†

k′−qakak′

⟩

∼=1

2V0

∑

kk′

⟨

a†ka†

k′ akak′

⟩

+1

2

∑

q( =0),k

Vq

⟨

a†k+qa†

kakak+q

⟩

+1

2

∑

q( =0),k

Vq

⟨

a†k+qa†

−k−qaka−k

⟩

,

(2.89)

where each term on the right-hand side is obtained by substituting q = 0(Hartree), k′ = k+q (Fock), or k′ = −k (pairing). The Hartree term gives∑

kk′

⟨

a†ka†

k′ akak′

⟩

=∑

k =k′

nknk′ +∑

k

nk(nk − 1) +∑

k

⟨

a†ka†

−kaka−k

⟩

= N(N − 1) +∑

k

⟨

a†ka†

−kaka−k

⟩

; (2.90)

the Fock term gives∑

q( =0),k

Vq

⟨

a†k+qa†

kakak+q

⟩

=∑

q( =0)

Vq

⎛

⎝⟨

a†qaq

⟩

n0 +⟨

a†−qa−q

⟩

n0 +∑

k( =0,−q)

⟨

a†k+qak+qa†

kak

⟩

⎞

⎠

= 2n0

∑

q( =0)

Vqnq +∑

k( =0,−q),q

Vqnk+qnk, (2.91)

where in the first and second terms of the middle equation, we have assumedk = 0 and k = −q, respectively. The pairing term combined with the lastterm in Eq. (2.90) gives∑

kq

Vq

⟨

a†k+qa†

−k−qaka−k

⟩

=∑

q

Vq

(⟨

a†qa†

−q

⟩⟨

a20

⟩

+⟨

a†2

0

⟩⟨

a†−qa†

q

⟩

+∑

k( =0,−q)

⟨

a†k+qa−k−qa†

ka−k

⟩)

=∑

q

Vq

[

χ∗qφ

2 + φ∗2

χq +∑

k

χ∗k+qχk

]

. (2.92)






Using the relations

φ ≡√

n0eiθ0 , χk =

αk

|αk|!

nk(1 + nk) =!

nk(1 + nk)eiθk , (2.93)

we obtain"

qk

Vq

#

a†k+qa†

−k−qaka−k

$

= 2n0

"

q

Vq|χq| cos(2θ0 − θq)+"

kq

Vqχ∗k+qχk.

Adding the above results, we obtain#

H − µH$

="

k

(ϵk − µ)nk +V0

2N(N − 1) + n0

"

q =0

Vqnq

+1

2

"

k,q

Vqnk+qnk + n0

"

q

Vq|χq| cos(2θ0 − θq) +1

2

"

kq

Vqχ∗k+qχk,

(2.94)

where the second term on the right-hand side describes the Hartree energy,the third and fourth terms describe the Fock exchange energy, and the lasttwo terms arise from pair correlation, also called anomalous correlation,which depends on the phase difference between the condensate and virtualexcitations.

For the case of repulsive interaction Vq > 0, the states with k = 0 arenot occupied by a macroscopic number of particles because it costs theexchange interaction n0

%

q =0 Vqnq. If φ0 = 0 and hence n0 = 0, it is notpossible for a pair of states (k,−k) to be macroscopically occupied; this isbecause χk

∼= nkeiθk if nk ≫ 1. Then, the sum of the fourth and last termson the right-hand side of Eq. (2.94) gives

1

2

"

k,q

Vqnk+qnk +1

2

"

k,q

Vqχ∗k+qχk

∼=1

2

"

k,q

Vqnk+qnk [1 + cos (θk+q − θk)] > 0,

which results in a large (extensive) loss of energy.Let us assume that only the k = 0 state is occupied by a macroscopic

number of particles. Then, the dominant term, which depends on the phase,on the right-hand side of Eq. (2.94) is

n0

"

k

Vk|χk| cos(2θ0 − θk).

To minimize this term, the phase θk of the amplitude of pair excitationsmust be locked to the phase θ0 of the condensate so that

θk = 2θ0 + π. (2.95)






Thus, in the Bose–Einstein condensed system, all the particles (condensateand virtual excitations) are correlated in a phase coherent manner. Thisis the reason why the superfluid fraction can be equal to unity at zerotemperature even though the condensate fraction is not so.

2.4 Bogoliubov Theory of Quasi-One-Dimensional Torus

Major consequences of the Bogoliubov theory can be obtained analyticallyby considering a quasi-one-dimensional torus geometry as shown in Fig. 2.2,where the circumference and cross-sectional area of the torus are L ≡ 2πRand S = πr2. It is assumed that r ≪ R and !2/(mr2) ≫ kBT . This systemresides in the lowest radial mode and can be considered to be effectivelyone-dimensional.

rR

Fig. 2.2 Quasi-one-dimensional torus of circumference L = 2πR and cross-sectionalarea πr2 in which N identical bosons are enclosed, where r ≪ R.

A complete set of basis functions is given by

φn(x) =1√L

eiknx, (2.96)

where

kn =2π

Ln (n = 0,±1,±2, · · · ) (2.97)

is the wave vector along the circumference of the torus. The basis functionssatisfy the orthonormal relations

! L

0φ∗n(x)φm(x)dx = δnm, (2.98)

and the completeness relation∞"

n=−∞

φn(x)φ∗n(x′) =1

L

∞"

n=−∞

ei 2πL

n(x−x′) =∞"

n=−∞

δ(x − x′ − Ln)

= δ(x − x′), (2.99)






where the second equality is derived from the Poisson summation formula,and the lost equality is derived using the condition 0 ≤ x, x′ < L. The fieldoperators are expressed in terms of the basis functions as

ψ(x) =∞∑

n=−∞

anφn(x), ψ†(x) =∞∑

n=−∞

a†nφn(x), (2.100)

where a†n and an are the creation and annihilation operators of bosons, re-

spectively, with the wavenumber kn and satisfy the canonical commutationrelations of bosons

[

an, a†m

]

= δnm, [an, am] = 0,[

a†n, a†

m

]

= 0.

It can be shown that the field operators satisfy[

ψ(x),ψ†(x′)]

=δ(x − x′),[

ψ(x), ψ(x′)]

=0,[

ψ†(x), ψ†(x′)]

=0. (2.101)

The kinetic and interaction parts of the Hamiltonian are given by

HKE =

∫ L

0dx ψ†(x)

(

−!2

2M

∂2

∂x2

)

ψ(x) =∑

n

ϵna†nan, (2.102)

Hint =gL

2

∫ L

0dx ψ†2(x)ψ2(x) =

g

2

∑

lmn

a†l a

†mal+nam−n, (2.103)

respectively, where

ϵn ≡!2k2

n

2M=

!2n2

2MR2= !ωcn

2, ωc ≡!

2MR2, (2.104)

g ≡4π!2a

M

1

LS=

2!2a

MRS, (2.105)

and

gN =4π!2a

Mn, (2.106)

gives the mean-field interaction energy per particle.

2.4.1 Case of BEC at rest: stability of BEC

We first consider a case in which the torus is at rest and N0 particlesundergo BEC in the k = 0 state. In the Bogoliubov approximation, weretain terms up to the second order in an and a†

n (n = 0). Then, theinteraction Hamiltonian is approximated as

Hint ≃g

2

⎡

⎣N2 + N0

∑

n=0

(

ana−n + a†na†

−n + 2a†nan

)

⎤

⎦, (2.107)






where N = N0 +!

n=0 a†nan. We may also subsititute N0 ≃ N in the last

term of Eq. (2.107) with the same degree of approximation. We thus obtain

H ≃gN2

2+

∞"

n=1

#

ϵn(a†nan + a†

−na−n) + gN(ana−n + a†na†

−n)$

, (2.108)

where ϵn ≡ ϵn + gN . To diagonalize Eq. (2.108), we use the followingBogoliubov transformations:

an = bn cosh θn − b†−n sinh θn, a†n = b†n cosh θn − b−n sinh θn. (2.109)

The off-diagonal terms vanish if

sinh 2θn =gN

%

ϵn(ϵn + 2gN), cosh 2θn =

ϵn%

ϵn(ϵn + 2gN). (2.110)

Then, the Hamiltonian is diagonalized as

H = E0 +∞"

n=1

!ωcn%

n2 + 2γ (b†nbn + b†−nb−n), (2.111)

where

γ ≡gN

!ωc(2.112)

and

E0 =gN2

2+

∞"

n=1

!ωc(n%

n2 + 2γ − n2 − γ). (2.113)

All excitation energies !ωcn%

n2 + 2γ in Eq. (2.111) must be positive forthe system to be stable against excitations. This is guaranteed for thecase of repulsive interaction (γ > 0). For the case of attractive interaction,the condition 1 + 2γ > 0 must be satisfied, thereby yielding the followingstability condition:

|γ| =|g|N!ωc

<1

2. (2.114)

2.4.2 Case of rotating BEC: Landau criterion

We next consider a case in which a condensate of Nm atoms is rotatingwith the wavenumber km, while the torus is at rest. The kinetic part of theHamiltonian is given by

HKE = ϵmNm +"

n=0

ϵm+na†m+nam+n

= ϵmN +"

n=0

(ϵm+n − ϵm) a†m+nam+n.






Using the Bogoliubov approximation, the interaction Hamiltonian is givenby

Hint≃g

2

!

a†2

m a2m+

"

n=0

#

a†2

m am+nam−n+a†m+na†

m−na2m+4a†

mama†m+nam+n

$%

≃gN2

2+

gN

2

"

n=0

#

am+nam−n + a†m+na†

m−n + 2a†m+nam+n

$

,

where N = Nm +&

n=0 a†m+nam+n. Thus,

H = ϵmN +gN2

2+

∞"

n=1

'

ϵm+na†m+nam+n + ϵm−na†

m−nam−n

+ gN#


m−n

$(

, (2.115)

where

ϵm+n ≡ ϵm+n − ϵm + gN = !ωc

)

n2 + 2mn + γ*

,

ϵm−n ≡ ϵm−n − ϵm + gN = !ωc

)

n2 − 2mn + γ*

. (2.116)

To diagonalize Eq. (2.115), we use the Bogoliubov transformations

am+n = bn cosh θn − b†−n sinh θn,

am−n = b−n cosh θn − b†n sinh θn. (2.117)

The off-diagonal terms vanish if

sinh 2θn =2gN

+

(ϵm+n + ϵm−n)2 − (2gN)2=

γ

n,

n2 + 2γ,

cosh 2θn =ϵm+n + ϵm−n

+

(ϵm+n + ϵm−n)2 − (2gN)2=

n2 + γ

n,

n2 + 2γ,

where θn depends on n, but not on m. We finally obtain

H = Em +∞"

n=1

!ωc

'#

n,

n2 + 2γ + 2mn$

b†nbn

+#

n,

n2 + 2γ − 2mn$

b†−nb−n

(

, (2.118)

where

Em = !ωcm2N +

gN2

2+

∞"

n=1

!ωc

#

n,

n2 + 2γ − n2 − γ$

. (2.119)






When the condensate is at rest (i.e., m = 0), Eqs. (2.118) and (2.119)reduce to Eqs. (2.111) and (2.113), respectively.

For a rotating condensate to be stable, all excitation energies must bepositive. For the case m > 0, the excitation energy is minimal for n = 1. Itfollows from Eq. (2.118) that the stability criterion for a rotating condensateis given by

m <1

2

!

1 + 2γ, i.e., vs <!

2MR

"

1 +2gN

!ωc, (2.120)

where vs is the superfluid velocity:

vs ≡!km

M=

!m

MR. (2.121)

In particular, for the case gN ≫ !ωc, the right-hand side of Eq. (2.120)yields the sound velocity:

vs <

"

gN

M=

!

MR

"

2aRN

S. (2.122)

The condition (2.122) is referred to as the Landau criterion. For the case ofattractive interaction (γ < 0), Eq. (2.120) is not satisfied, even for m = 1.Hence, persistent currents cannot exist stably.

For comparison, let us consider the stability of the system within theHartree–Fock (HF) approximation. The energy of the system, in which Nm

atoms are in the n = m state and N −Nm atoms are in the n = 0 state, is

E(Nm) = ϵmNm + gNm(N − Nm) +gN2

2. (2.123)

Suppose that all the atoms are initialy in the m state. Then,

E(N) = ϵmN +gN2

2. (2.124)

The energy of the system in which one of the atoms in the m state hasdecayed into the n = 0 state is given by

E(N − 1) = ϵm(N − 1) + g(N − 1) +gN2

2≃ ϵm(N − 1) + gN +

gN2

2.

For the persistent current to be stable, the following condition must besatisfied:

E(N − 1) − E(N) = −ϵm + gN > 0 → m <√γ. (2.125)

This result differs from the one given in Eq. (2.120) by a factor of√

2 forγ ≫ 1, indicating that the HF approximation overestimates the stability ofthe persistent current as compared to the Bogoliubov approximation.






2.4.3 Ground state of BEC in rotating torus

We finally consider the case in which the torus is rotating at angular fre-quency ω. The Hamiltonian of the system in the rotating frame of referenceis given by H −ωL, where L is the total angular momentum of the systemgiven by

L =!

m

!ma†mam = !mNm +

!

n=0

!(m + n)a†m+nam+n

= !mN +!

n=0

!na†m+nam+n. (2.126)

From Eqs. (2.126) and (2.115), we obtain

K ≡ H − ωL = !"

ωcm2 − ωm

#

N +gN2

2

+∞!

n=1

$

(ϵm+n − !ωn) a†m+nam+n + (ϵm−n + !ωn) a†

m−nam−n

+ gN%


m−n

&'

. (2.127)

By diagonalizing K in a manner similar to that described in the precedingsubsection, we obtain

K = Em +∞!

n=1

!ωcn

()*

n2 + 2γ + 2

+

m −ω

2ωc

,-

b†nbn

+

)*

n2 + 2γ − 2

+

m −ω

2ωc

,-

b†−nb−n

.

, (2.128)

where

Em = !ωc

+

m2 −mω

ωc

,

N +gN2

2+

∞!

n=1

!ωc

%

n*

n2 + 2γ − n2 − γ&

.

(2.129)

For the system to be stable against excitations, the following conditionshould be satisfied:

///m −

ω

2ωc

/// <

1

2

*

1 + 2γ. (2.130)

This result shows that the stability range is a periodic function of ω with pe-riod 2ωc. In general, the properties of a system in a quasi-one-dimensionaltorus are periodic functions of ω with the same period. Figure 2.3 comparesthe stability region of an attractive condensate (shaded regions) obtainedby the Bogoliubov approximation with that obtained by the HF approxi-mation.






0 1 2 3 4-1-2-3-4

1

12

1m = − 0m = 1m = 2m =

0 1 2 3 4-1-2-3-4

γ

m = -1 m = 0 m = 1 m = 2

c

ωω

m = -2

HF approximation

Bogoliubov approximation

Fig. 2.3 Stability diagram of an attractive condensate under rotation. The dotted(solid) curve show the stability boundary calculated by the HF (Bogoliubov) approxi-mation, where m denotes the value of the quantized circulation in units of h/M

2.5 Bogoliubov–de Gennes (BdG) Theory

Many physical systems are confined in a spatially nonuniform potentialU(r). Hence, it is important to generalize the Bogoliubov theory to suchcases [de Gennes (1989)].

We start with the second-quantized Hamiltonian

H =

!

drψ†(r)h(r)ψ(r) +U0

2

!

drψ†(r)ψ†(r)ψ(r)ψ(r), (2.131)

where the field operators satisfy the canonical commutation relations (2.33)and

h(r) ≡ −!2∇2

2M+ U(r). (2.132)

We decompose the field operator into the condensate part ψ0(r), which weassume to be a c-number, and the noncondensate part φ(r), as follows:

ψ(r) = ψ0(r) + φ(r), (2.133)

where the condensate and noncondensate modes are assumed to be orthog-onal to each other, i.e.,

!

drψ∗0(r)φ(r) =

!

drψ0(r)φ†(r) = 0. (2.134)

The number operator is then expressed as

N =

!

dr"

|ψ0(r)|2 + φ†(r)φ(r)#

. (2.135)

Substituting Eq. (2.133) into Eq. (2.131), we obtain

H =

!

dr ψ∗0(h +

U0

2|ψ0|2)ψ0 +

!

dr"

φ†(h + U0|ψ0|2)ψ0

+ψ∗0(h + U0|ψ0|2)φ

#

+ HB + H ′, (2.136)






where

HB =

!

dr

"

φ†(h + 2U0|ψ0|2)φ+U0

2(ψ∗

02φ2 + φ†2ψ2

0)

#

(2.137)

is the Bogoliubov Hamiltonian and

H ′ =

!

dr

"

U0(ψ0φ†2φ+ ψ∗

0 φ†φ2) +

U0

2φ†2φ2

#

(2.138)

describes higher-order terms.Up to the zeroth order in φ, we take terms in H that involve only ψ0

and ψ∗0 . Assuming H − µN to be stationary with respect to the variations

in ψ∗0 , we obtain the Gross–Pitaevskii equation:

$

h + U0|ψ0|2%

ψ0 = µψ0. (2.139)

Substituting Eq. (2.139) into Eq. (2.136) and using Eq. (2.134), we findthat the terms linear in φ or φ† vanish; thus, we obtain

H =

!

dr

&

µ|ψ0|2 −U0

2|ψ0|4

'

+ HB + H ′. (2.140)

The BdG theory diagonalizes HB by introducing generalized Bogoliubovtransformations [Fetter (1972)]:

φ(r) = eiS(r)(

j

′)

uj(r)αj − v∗j (r)α†j

*

, (2.141)

where the prime symbol (′) indicates that the sum is to be taken over allnoncondensate modes and αj are the Bogoliubov quasiparticle operatorsthat satisfy

+

αj , α†k

,

= δjk,+

αj , αk

,

= 0+

α†j , α

†k

,

= 0. (2.142)

Let S(r) denote the phase of ψ0(r), i.e., ψ0(r) = |ψ0(r)|eiS(r). SubstitutingEq. (2.141) into Eq. (2.137), we obtain

HB =(

jk

′

!

dr

-

α†jαk

"

u∗jLuk −

U0

2|ψ0|2(u∗

jvk + ukv∗j )

#

+ αjα†k

"

vjLv∗k −U0

2|ψ0|2(ujv

∗k + u∗

kvj)

#

− α†jα

†k

"

u∗jLv∗k −

U0

2|ψ0|2(u∗

ju∗k + v∗j v∗k)

#

− αjαk

"

vjLuk −U0

2|ψ0|2(ujuk + vjvk)

#.

, (2.143)






where

L ≡ −!2

2M[∇+i∇S(r)]2 + U(r) + 2U0|ψ0(r)|2. (2.144)

For the diagonalization of HB, uj and vj should satisfy the following BdGequations:

Luj − U0|ψ0|2vj = Ejuj , (2.145)

L∗vj − U0|ψ0|2uj = −Ejvj , (2.146)

where uj and vj are normalized so that they satisfy the condition:!

dr(|uj |2 − |vj |2) = 1. (2.147)

As the BdG equations suggest, uj and vj represent the amplitudes of parti-cles and holes, respectively. It can be shown from Eqs. (2.145)–(2.147) thatthe eigenvalues Ej are real, and uj and vj satisfy the following orthonor-malization conditions:

!

dr (u∗juk − v∗j vk) = δjk, (2.148)

!

dr (ujvk − ukvj) = 0 if Ej + Ek = 0. (2.149)

Thus, the low-energy excitations are described by Bogoliubov quasiparti-cles that are expressed as linear combinations of particles and holes. Equa-tions (2.145) and (2.146) can be used to simplify Eq. (2.143) as

HB =1

2

"

jk

′

!

dr#

(Ej + Ek)(α†j αku∗

juk − αjα†kvjv

∗k)

+(Ej − Ek)(αj αkukvj − α†jα

†ku∗

jv∗k)$

. (2.150)

Because of Eq. (2.149), the last term in Eq. (2.150) vanishes for

"

jk

′

!

dr(Ej − Ek)(αj αkukvj − α†jαku∗

jv∗k)

=1

2

"

jk

′(Ej − Ek)

!

dr%

αjαk(ukvj − ujvk) − α†jα

†k(u∗

jv∗k − u∗

kv∗j )&

= 0.

Using Eq. (2.148), HB is diagonalized as

HB ="

j

′Ejα†jαj −

"

j

′Ej

!

dr|vj(r)|2. (2.151)

Following are a few remarks about the BdG equations:






• It can be seen from Eqs. (2.145) and (2.146) that if (uj , vj) is asolution of the BdG equations with the eigenvalue Ej with posi-tive normalization, (v∗j , u∗

j ) is a solution with eigenvalue −Ej withnegative normalization. For uniform condensates with plane-wavesolutions, the positive normalization always leads to positive eigen-values. However, nonuniform condensates such as one with a vortexcan have physical states with positive normalization and negativeeigenvalues [Fetter and Svidzinsky (2001)]. If a positive-norm so-lution has a negative eigenvalue, the system exhibits the Landauinstability because it can lower the energy of the system by excitingquasiparticles with negative eigenvalues.

• If the system has a complex eigenvalue, the assumed ground stateψ0 is not stable and the system exhibits a dynamical instability.The corresponding solutions (uj , vj) have zero norm. In fact, fromEqs. (2.145) and (2.146), we can show

(Ej − E∗j )

!

dr(|uj |2 − |vj |2) = 0. (2.152)

Hence, if Ej = E∗j , the integral must vanish.

• The substitution of

ψ0(r) =

"

N0

ΩeiS(r)f(r) (2.153)

in Eq. (2.139) gives

Lf(r) − U0|ψ0|2f(r) = 0. (2.154)

Comparing this with Eq. (2.145), we find that

u0(r) = v0(r) = f(r)

is the solution of the BdG equations with the eigenvalue E0 = 0.Thus, the condensate wave function is the zero-energy eigenfunc-tion of the BdG equations.

• From the canonical commutation relation for ψ, it follows that thecompleteness relation

1

Ωf(r)f(r′) +

#

j

′$

uj(r)u∗j (r

′) − v∗j (r)vj(r′)%

= δ(r − r′) (2.155)






holds, where f(r) satisfies the normalization condition

1

Ω

!

dr [f(r)]2 = 1 (2.156)

and!

drf(r)uj(r) =

!

drf(r)vj(r). (2.157)

The last equation can be proved using Eqs. (2.145), (2.146), and(2.154).

As an illustrative example, let us consider a case in which the condensateis moving in a uniform system (i.e., U(r) = 0) with velocity

v =!k

M. (2.158)

The condensate wave function is

ψ0(r) =√

n0eikr, (2.159)

and the chemical potential is obtained from Eq. (2.139) as

µ = n0U0 +M

2v2.

The BdG equations can be solved using the assumptions:

uk(r) =1√Ω

Akeikr, vk =1√Ω

Bkeikr. (2.160)

Substituting Eq. (2.160) in Eqs. (2.145) and (2.146), we obtain coupledequations, which yield two eigenenergies:

E(±)k = !k · v ± Ek, (2.161)

where

Ek ="

ϵk(ϵk + 2n0U0). (2.162)

The corresponding eigenfunctions are given by

u(+)k (r) =

1√Ω

ukeikr, v(+)k =

1√Ω

vkeikr, (2.163)

u(−)k (r) =

i√Ω

vkeikr, v(−)k =

i√Ω

ukeikr, (2.164)

where

uk =

#ϵk + n0U0

2Ek+

1

2

$ 1

2

, vk =

#ϵk − n0U0

2Ek−

1

2

$ 1

2

. (2.165)






2.6 Method of Binary Collision Expansion

Lee and Yang [Lee and Yang (1957)] developed a method of binary collisionexpansion of the partition function based on the “binary kernel,” whichdescribes two-body scattering exactly. Because two-body scattering is thedominant interaction mechanism for dilute gases, this method is a powerfultool for determining their thermodynamic properties. This section providesa brief introduction to this method.

2.6.1 Equation of state

In the following discussions, we shall assume ! = 2M = 1. The Hamiltonianof our system is

H =N!

i=1

Ti +!

i<j

Vij , (2.166)

where Ti = p2i is the kinetic energy of the i-th particle and Vij =

V (|ri − rj |) describes the interaction between the i-th and j-th particles.The partition function ZN is

ZN ≡ Tre−βH =!

m

e−βEm , (2.167)

where β ≡ 1/kBT and Tr indicaates summation over all symmetrized (oranti-symmetrized) states for Bose (or Fermi) systems. When ZN is given,the pressure P of the system can be obtained as

βP = −∂

∂VlnZN , (2.168)

where V is the volume of the system. The grand partition function ZG canbe defined using ZN as

ZG(z) =∞!

N=0

zNZN , (2.169)

where Z = eβµ is the fugacity. The pressure P and the mass density ρ canbe obtained from ZG as

βP =1

VlnZG(z), ρ = z

d

dz

"1

ΩlnZG(z)

#

. (2.170)

Eliminating z from Eq. (2.170) will yield pressure as a function of thedensity n. The energy of the ground state E0 can be obtained from therelation P = P (ρ) at absolute zero as

P = ρ2 d

dρ

E0

N→

E0

N=

$ ρ

0

P

ρ2dρ. (2.171)






2.6.2 Cluster expansion of partition function

Usually, it is not possible to carry out the summation in Eq. (2.167) exactly.We use the systematic method developed by Kahn and Uhlenbeck [Kahnand Uhlenbeck (1938)] to calculate the partition function. Consider aHamiltonian consisting of two parts:

H = H0 + HI . (2.172)

Let Ψn be a complete set of eigenfunctions of H . Then, the partitionfunction can be expanded in terms of Ψn as

ZN = Tr e−β(H0+HI) =!

n

"

dr1 · · · drNΨ∗ne−β(H0+HI )Ψn

=1

N !

"

dr1 · · · drNW SN (1, · · · , N ; 1, · · · , N), (2.173)

where

W SN (1′, · · · , N ′; 1, · · · , N) ≡ N !

!

n

Ψ∗n(1′, · · · , N ′)e−βHΨn(1, · · · , N),

(2.174)

and the superscript S indicates summation over completely (anti-)symmetrized Ψn. In Eqs. (2.173) and (2.174), 1, · · · , N , and 1′, · · · , N ′

in fact denote r1, · · · , rN , and r′1, · · · , r′N , respectively.We decompose W S

N into linked clusters. From Eq. (2.174), we find thatW S

N is a matrix element in which the coordinates of the N -particle systemmake a transition via the interaction e−βH from 1, 2, · · · , N to 1′, 2′, · · · , N ′.In general, W S

N can be decomposed into the product of linked clusters as

W SN =

!

ml

!

D

#

US1 · · ·US

1

$

% &' (

m1

#

US2 · · ·US

2

$

% &' (

m2

· · ·#

USN

$

% &' (

mN

, (2.175)

where m1, m2, · · · , mN denote the numbers of the linked clusters in eachgroup within parentheses and satisfy

)Nl=1 lml = N . Here, D indicates

summation over all combinations of coordinates for a given ml.The partition function can be obtained by substituting r′i = ri in

Eq. (2.175) and integrating the resultant expression over r1, r2, · · · , rN .Here, since the integrals of US

1 (i; i) are all equal, we have

ZN =1

N !

!

ml

N !

m1!m2! · · ·mN !

N*

l=1

+1

l!

"

dr1 · · · drlUSl (1, · · · , l; 1, · · · , l)

,ml

.

(2.176)






Introducing the cluster integration

bSl ≡

λ3

V

1

l!

!

dr1 · · · drlUSl (1, · · · , l; 1, · · · , l), (2.177)

where λ =√

4πβ denotes the thermal de Broglie wavelength, we obtain

ZN ="

ml

δ#l lml,N

N$

l=1

1

ml!

%Ω

λ3bSl

&ml

. (2.178)

The grand partition function can be obtained by substituting Eq. (2.178)in Eq. (2.169):

ZG(z)=∞"

N=0

"

ml

δ#l lml,N

N$

l=1

1

ml!

%Ω

λ3zlbS

l

&ml

=∞$

l=1

∞"

ml=0

1

ml!

%Ω

λ3zlbS

l

&ml

=exp

'

Ω

λ3

∞"

l=1

bSl zl

(

. (2.179)

Therefore, we obtain the expressions for the pressure and density of thesystem from Eq. (2.170) as

βP =1

λ3

∞"

l=1

bSl zl, ρ =

1

λ3

∞"

l=1

lbSl zl. (2.180)

The equation of state of the system can be obtained from the cluster inte-gration bS

l in Eq. (2.180).

2.6.3 Ideal Bose and Fermi gases

We first consider the case of an ideal Bose gas. The grand partition functionZG is given by

ZG =$

k

∞"

nk=0

)

eβ(µ−ϵk)*n

k=

$

k

)

1 − eβ(µ−ϵk)*−1

= exp

'

−"

k

ln)

1 − eβ(µ−ϵk)*(

= exp

'

−V

4π2

%2m

!2

& 3

2! ∞

0dE E

1

2 ln)

1 − eβ(µ−E)*(

= exp

'

V

6π2

%2m

!2

& 3

2

β

! ∞

0dE

E3

2

eβ(E−µ) − 1

(

.






Substituting eβµ = z and eβE = t, we have

ZG = exp

!

V

6π2

"2M

β!2

# 3

2$ ∞

1dt

t3

2

z−1et − 1

%

= exp

"V

λ3g 5

2

(z)

#

, (2.181)

where$ ∞

1dt

t3

2

z−1et − 1= Γ

"5

2

#

, g 5

2

(z) =3√π

4

∞&

l=1

zl

l5

2

. (2.182)

Comparing Eq. (2.181) with Eq. (2.179), we find that

bSl = l−

5

2 . (2.183)

For the case of an ideal Fermi gas, we have

ZG ='

k

1&

nk=0

(

eβ(µ−ϵk))nk

='

k

(

1 + eβ(µ−ϵk))

= exp

!

&

k

ln(

1 + eβ(µ−ϵk))%

= exp

!

V

4π2

"2M

!2

# 3

2$ ∞

0dE E

1

2 ln(

1 + eβ(µ−E))%

= exp

!

−V

6π2

"2M

!2

# 3

2

β

$ ∞

0dE

E3

2

eβ(E−µ) + 1

%

= exp

!

V

λ3

∞&

l=1

(−z)l

l5

2

%

,

where$ ∞

0dt

t3

2

(−z)−1et − 1= Γ

"5

2

# ∞&

l=1

(−z)l

l5

2

. (2.184)


bSl = (−1)ll−

5

2 . (2.185)

As shown in Eqs. (2.183) and (2.185), the cluster integration exists up tol = ∞ even in the absence of interaction between particles because of the(anti)symmetrization of the wave function.

For comparison, let us consider a classical ideal gas which obeys theBoltzmann statistics. Consider a one-particle partition function given as

Z1 =&

k

e−βϵk =V

4π2

"2M

!2

# 3

2$ ∞

0dE E

1

2 e−βE =V

λ3.






Then, the N -particle partition is given by

ZN =1

N !Z1 =

1

N !

!V

λ3

"N

.

We thus obtain

ZG =∞#

N=0

zNZN = exp

!Ω

λ3z

"

. (2.186)


bcll = δl,1 (2.187)

That is, in a classical ideal gas, linked clusters among particles (l ≥ 2) donot exist. This is because of the absence of interaction between particles.However, in quantum systems, the entanglement among many particles ex-ists even in the absence of interactions because of the (anti)symmetrizationof the wave function.

2.6.4 Matsubara formula

The grand partition function can be calculated directly using operators.

ZG = Tr$

e−β(H−µN)%

= Tr$

e−β(H0−µN)% Tr

$

e−β(H0−µN)eβ(H0−µN)e−β(H−µN)%

Tr$

e−β(H0−µN)%

= Z0G⟨S(β)⟩0, (2.188)

where

Z0G ≡ Tr

$

e−β(H0−µN)%

,

S(β) ≡ eβ(H0−µN)e−β(H−µN),

and

⟨· · · ⟩0 ≡Tr

$

e−β(H0−µN)eβ(H0−µN)e−β(H−µN)%

Tr$

e−β(H0−µN)% . (2.189)

The derivative of S(β) is

d

dβS(β) = eβ(H0−µN)(H0 − H)e−β(H−µN)

= −eβ(H0−µN)HIe−β(H0−µN)S(β)

= −HI(β)S(β), (2.190)






where HI = H − H0 and HI(β) = eβ(H0−µN)HIe−β(H0−µN). IntegratingEq. (2.190), we get

S(β) = 1 −! β

0dτHI(τ)S(τ)

= 1 −! β

0dτHI(τ) +

! β

0dτ1dτ2HI(τ1)HI(τ2) − · · ·

= Tτ exp

"

−! β

0dτHI(τ)

#

, (2.191)

where Tτ is the time-ordered operator and

! β

0dτ1dτ2HI(τ1)HI(τ2) =

1

2!Tτ

"! β

0dτHI(τ)

#2

, etc. (2.192)

Taking the average of Eq. (2.191), we have

⟨S(β)⟩0 = 1 +∞$

n=1

(−1)n

n!

! β

0dτ1

! β

0dτ2 · · ·

! β

0dτn

× ⟨TτHI(τ1)HI(τ2) · · · HI(τn)⟩0. (2.193)

Here, second or higher-order terms in the perturbation series include dis-connected diagrams as well as connected diagrams. In detail,

(−1)n

n!

!

0βdτ1 · · ·

!

0βdτn⟨TτHI(τ1) · · · HI(τn)⟩0

=(−1)n

n!

$

ml

δ%l lm1,n

n!

m1!m2! · · ·mn!

&! β

0dτ⟨HI(τ)⟩con0

'm1

×

&

1

2!

! β

0dτ1

! β

0dτ2⟨TτHI(τ1)HI(τ2)⟩con0

'm2

· · ·

=$

ml

δ%l lml,n

n(

l=1

1

ml!Cml

l , (2.194)

where

C1 = −! β

0dτ⟨HI(τ)⟩con0 , C2 =

1

2!

! β

0dτ1

! β

0dτ2⟨TτHI(τ1)HI(τ2)⟩con0

and

Cl ≡(−1)l

l!

! β

0dτ1 · · ·dτl⟨TτHI(τ1)HI(τ2) · · · HI(τl)⟩con0 .






Substituting Eq. (2.194) in Eq. (2.193), we have

⟨S(β)⟩ = 1 +∞!

n=1

!

ml

δ"l lml,n

n#

l=1

1

ml!Cml

l =∞#

l=1

!

ml=0

Cmll

ml!

= e"∞

l=1Cl . (2.195)

Here, we have∞!

l=1

Cl =

$

Tτ exp

%

−β&

0dτHI(τ)

'(con

0

− 1 = ⟨S(β)⟩con0 − 1. (2.196)

We thus obtain

⟨S(β)⟩0 = exp)

⟨S(β)⟩con0 − 1*

. (2.197)


ZG = Z0G exp

)

⟨S(β)⟩con0 − 1*

,

S(β) = Tτ exp

+

−& β

0dτHI(τ)

,

. (2.198)

Therefore, we obtain

Ω = −β−1 lnZG = Ω0 − β−1)

⟨S(β)⟩con0 − 1*

. (2.199)

Equation (2.199) gives the Matsubara formula. The thermodynamic po-tential can be described using two-body temperature Green’s functions.

S(β, g) ≡ Tτ exp

+

−g

& β

0dτHI(τ)

,

(2.200)

Using Eq. (2.200), we obtain

Ω(g)(20)= Ω0 − β−1 ln⟨S(β, g)⟩0. (2.201)

Thus,

∂Ω(g)

∂g= −β−1

⟨ ∂∂gS(β, g)⟩0⟨S(β, g)⟩0

= β−1

& β

0dτ

⟨TτHI(τ)S(β, g)⟩0⟨S(β, g)⟩0

, (2.202)

where Eq. (2.188) was used for obtaining the last equality. Integrating bothsides of Eq. (2.202) over g from 0 to 1 and substituting Ω(g = 0) = Ω0 andΩ(g = 1) = Ω, we obtain

Ω = Ω0 + β−1

& 1

0dg

& β

0dt⟨TτHI(τ)S(β, g)⟩0

⟨S(β, g)⟩0. (2.203)






Assuming two-body interactions,

HI(τ) =1

2

!

dr1dr2Ψ†(r1τ)Ψ

†(r2τ)V (r1, r2)Ψ(r2τ)Ψ(r1τ), (2.204)

Eq. (2.203) gives

Ω = Ω0 +1

2β

! 1

0dg

! β

0dτ

!

dr1dr2V (r1, r2)

×⟨TτΨ†(r1τ)Ψ†(r2τ)Ψ(r2τ)Ψ(r1τ)S(β, g)⟩0

⟨S(β, g)⟩0. (2.205)





Chapter 3

Trapped Systems

3.1 Ideal Bose Gas in a Harmonic Potential

Let us consider a system of N noninteracting particles with mass M con-fined in a harmonic potential. The Hamiltonian of the system is givenby

H =N!

i=1

1

2M

"

p2xi + p2

yi + p2zi

#

+N!

i=1

M

2

"

ω2xx2

i + ω2yy2

i + ω2zz2

i

#

. (3.1)

It follows from the canonical commutation relations

[xi, pxj] = [yi, pyj ] = [zj , pzj ] = i!δij (3.2)

that the energy spectrum of the system is given by

Enxnynz = !(ωxnx + ωyny + ωznz) + E0, (3.3)

where nx,y,z = 0, 1, 2, · · ·, and E0 is the zero-point energy given by

E0 =!

2(ωx + ωy + ωz). (3.4)

At thermal equilibrium, the bosons distribute over the energy levels (3.3)according to the Bose–Einstein distribution

Nnxnynz =1

exp[β(Enxnynz − µ)] − 1,

where β ≡ 1/(kBT ) and µ is the chemical potential that is determined suchthat the average total number of particles is equal to N :

N =∞!

nx,ny,nz=0

1

exp$

β(Enxnynz − µ)%

− 1. (3.5)

73






When kBT is much higher than the discrete energy-level spacings, the sumin Eq. (3.5) can be replaced with the integral

!

nxnynz

· · · → N0 +

" ∞

0dED(E) · · · , (3.6)

where D(E) is the density of states and

N0 =1

exp[β(E0 − µ)] − 1(3.7)

is the number of bosons in the lowest-energy state. Since this number canbe macroscopic as µ approaches E000 below the BE transition temperature,we keep it separate from the integral in Eq. (3.6).

In Eq. (3.6), D(E)!ω denotes the number of lattice points (nx, ny, nz)in the energy range of E−!ω < !(ωxnx+ωyny+ωznz) ≤ E. For simplicity,we assume that the parabolic potential is isotropic, ωx = ωy = ωz = ω, andlet n be the maximum integer that does not exceed E/!ω. Then, D(E)!ωgives the number of lattice points (nx, ny, nz) that satisfy

nx + ny + nz = n (nx,y,z = 0, 1, 2, · · · ).

A combinatorial calculation gives

D(E)!ω = n+2C2 =(n + 2)(n + 1)

2≃

1

2(

E

!ω)2 +

3

2

E

!ω+ 1. (3.8)

The coefficient 3/2 on the right-hand side is modified for an anisotropicpotential. Let us write this coefficient for a general case as γ and ignore1 in Eq. (3.8) by assuming E ≫ !ω; then, the density of states can bedescribed as follows.

D(E) ≃1

2

E2

(!ω)3+ γ

E

(!ω)2, (3.9)

where ω ≡ (ωxωyωz)1

3 . A numerical study suggests that γ is well approxi-mated by [Grossmann and Holthaus (1995)]

γ ≃ωx + ωy + ωz

2ω. (3.10)

Replacing the discrete sum in Eq. (3.5) by the integral using Eqs. (3.6) and(3.9), we obtain

N = N0 +1

2(!ω)3

" ∞

0

E2dE

eβ(E+E0−µ) − 1+

γ

(!ω)2

" ∞

0

EdE

eβ(E+E0−µ) − 1

= N0 +

#

kBT

!ω

$3

g3(z) + γ

#

kBT

!ω

$2

g2(z), (3.11)





Trapped Systems 75

where z ≡ eβ(µ−E0) and gn(z) is the Bose–Einstein distribution (or poly-logarithmic) function defined by

gn(z) ≡1

Γ(n)

! ∞

0

xn−1

z−1ex − 1dx =

∞"

l=1

zl

ln. (3.12)

As shown below, z lies in the range 0 < z < 1 in the classical regime andapproaches 1 as the transition temperature is neared. At z = 1, gn(z)becomes the Riemann zeta function ζ(n):

gn(1) =∞"

l=1

1

ln= ζ(n), (3.13)

where

ζ(2) =π2

6, ζ

#3

2

$

= 2.612, and ζ(3) = 1.202. (3.14)

3.1.1 Transition temperature

The number of bosons that occupy the lowest-energy level is given fromEq. (3.7) and z ≡ eβ(µ−E0) by

N0 =z

1 − z. (3.15)

Solving Eq. (3.15) for z gives

z =N0

N0 + 1. (3.16)

At Tc, we may substitute z = 1 in Eq. (3.11) because N0 ≫ 1, and ignoreN0 in Eq. (3.11) because N ≫ N0. We then obtain

N ≃%

kBTc

!ω

&3

ζ(3) + γ

%

kBTc

!ω

&2

ζ(2).

Solving this for Tc gives

Tc ≃!ω

kB

%

N

ζ(3)

&1/3 '

1 −γ

3

ζ(2)

ζ(3)2/3N− 1

3

(

. (3.17)

We thus find that when N ≫ 1, the transition temperature T0 is given by

T0 =!ω

kB

%

N

ζ(3)

&1

3

≃ 0.94!ω

kBN

1

3 ≃ 4.5

%

ω/2π

100Hz

&

N1

3 nK. (3.18)






For the case of finite N , the deviation of the transition temperature fromT0 is found from Eq. (3.17) to be

Tc − T0

T0≃ −

γ

3

ζ(2)

ζ(3)2/3N− 1

3 ≃ −0.243ωx + ωy + ωz

(ωxωyωz)1/3N− 1

3 . (3.19)

In dilute-gas BEC systems, it is possible to make a detailed comparisonbetween theories and experiments [Ensher, et al. (1996); Giorgini, et al.(1996)]. In the case of a uniform system, the transition temperature risesdue to a repulsive interaction by an amount proportional to an1/3 [Baym,et al. (1999, 2001); Yukalov (2004)]. This is because the repulsive in-teraction suppresses density fluctuations in favor of BEC. In the case of aharmonically trapped system, the transition temperature decreases becausethe repulsive interaction reduces the peak density. A semiclassical estimategives [Giorgini, et al. (1996)]

δTc

T0≃ −1.326

a

d0N

1

6 , (3.20)

where d0 ≡!

!/(Mω) is the width of the ground-state wave function ofa particle in a harmonic trap. The experiment on a harmonically trapped87Rb condensate deviates significantly from the ideal gas formula (3.17) andagrees well with Eq. (3.20) [Gerbier, et al. (2004)]. Critical fluctuations anddiverging behavior of the correlation length are expected near the second-order phase transition. The correlation length was measured just abovethe BEC transition temperature and found to diverge as (T − Tc)−ν withν = 0.67 ± 0.13 [Donner, et al. (2007)] in agreement with the criticalexponent of the 3D XY model.

Equation (3.17) suggests that the thermodynamic limit in the trappedsystem is obtained by taking the following limits:

N → ∞, ω → 0, N ω3 = constant. (3.21)

3.1.2 Condensate fraction

Let us examine the temperature dependence of the condensate fraction,i.e., the ratio of the number of condensate particles to the total particlenumber for T < Tc. Since N0 ≫ 1 for T < Tc, we may substitute z = 1 inEq. (3.11) to obtain

N0

N= 1 −

"

kBT

!ω

#3 ζ(3)

N− γ

"

kBT

!ω

#2 ζ(2)

N. (3.22)





Trapped Systems 77


N0

N= 1 −

!

T

T0

"3

− γζ(2)

ζ(3)2/3

!

T

T0

"2

N− 1

3 , (3.23)

where ζ(2)/ζ(3)2/3 ≃ 1.46. Comparing this result with Eq. (1.23), wefind that the trapped condensate grows more rapidly than the uniformcondensate as a function of temperature.

3.1.3 Chemical potential

Eliminating !ω in Eq. (3.11) in favor of T0 using Eq. (3.18), we obtain

N = N0 +

!

T

T0

"3 N

ζ(3)g3(z) + γ

!

T

T0

"2 ! N

ζ(3)

"2

3

g2(z). (3.24)

When N ≫ 1, we can ignore the last term. Furthermore, because N0 ≈ 0for T > Tc, z = N0/(N0 + 1) ≪ 1 and hence g3(z) ≈ z = eβ(µ−E0). Thus,

N ≃!

T

T0

"3 N

ζ(3)eβ(µ−E0),

and hence,

µ ≃ E0 − 3kBT lnT

T0. (3.25)

The chemical potential increases as the temperature is lowered and it be-comes equal to the zero-point energy E0 of the system at the transitiontemperature.

3.1.4 Specific heat

In the limit of N → ∞, the specific heat exhibits a discontinuous jump atT = T0. To show this, let us consider the internal energy of the system:

E =

# ∞

E0

ϵ

eβ(ϵ−µ) − 1ρ(ϵ)dϵ. (3.26)

Since E changes continuously as a function of temperature, the jump in thespecific heat, ∆C, should arise from the temperature dependence of µ:

∆C ≡ C(T0 + 0) − C(T0 − 0) =

!

∂E

∂µ

"

T

∂µ

∂T

$

$

$

T0+0

T0−0, (3.27)






where T0 +0 (T0−0) represents a temperature that is infinitesimally larger(smaller) than T0, and

!

∂E

∂µ

"

T

=

# ∞

E0

ϵρ(ϵ)∂

∂µ

1

eβ(ϵ−µ) − 1dϵ

=

# ∞

E0

1

eβ(ϵ−µ) − 1

∂(ϵρ)

∂ϵdϵ. (3.28)

In three dimensions and for a large-N limit, ρ(ϵ) = ϵ2/2(!ω)3, and there-fore,

!

∂E

∂µ

"

T

= 3N. (3.29)

Thus,

∆C = 3N∂µ

∂T

$

$

$

T0+0

T0−0. (3.30)

Since µ is constant and equal to E0 below T0, ∂µ/∂T = 0 at T = T0 − 0.On the other hand, ∂µ/∂T at T = T0 + 0 is obtained by differentiatingEq. (3.24) with respect to T . Noting that N0 = 0 for T > T0 and that thelast term in Eq. (3.24) is negligible for N ≫ 1, we have

0 = 3T 2g3(z) + T 3g′3(z)∂z

∂T

= 3T 2g3(z) + T 3g2(z)

%

−µ − E0

kBT 2+ β

∂µ

∂T

&

,

where g′3(z) = g2(z)/z is used. Thus, at T = T0 + 0, where z = 1 andµ = E0, we obtain

∂µ

∂T= −3kB

g3(1)

g2(1)= −3kB

ζ(3)

ζ(2). (3.31)

Substituting this into Eq. (3.30) gives

C(N=∞)

NkB= 9

ζ(3)

ζ(2)≃ 6.577. (3.32)

Thus, the BEC phase transition of a harmonically trapped ideal Bose gasis of the second order. This is in sharp contrast with the uniform sys-tem of noninteracting bosons, where the constant-volume specific heatis continuous at T = T0. When N is finite, the specific heat derivedfrom Eq. (3.26) exhibits a continuous change. However, when N is suffi-ciently large (! 10, 000), the specific heat exhibits an almost discontinuousjump [Grossmann and Holthaus (1995)].





Trapped Systems 79

3.2 BEC in One- and Two-Dimensional Parabolic

Potentials

3.2.1 Density of states

Let us count the number of combinations (nx, ny, nz) that satisfy

E − dE < !(ωxnx + ωyny + ωznz) ≤ E. (3.33)

A one-dimensional (1D) system is defined as the system that satisfies

!ωz ≪ kBT ≪ !ωx, !ωy (3.34)

so that nx = ny = 0. Condition (3.33) then reduces to

E − dE < !ωznz ≤ E.

Since there exists only one state per energy interval dE = !ωz, the densityof states is given by

D(E) =1

!ωz(1D). (3.35)

A two-dimensional (2D) system is defined as the system that satisfies

!ωx, !ωy ≪ kBT ≪ !ωz (3.36)

so that nz = 0. To simplify matters, let us consider an isotropic case inwhich ωx = ωy ≡ ω. Then, we have

E − dE < !ω(nx + ny) ≤ E.

Since there exist E/(!ω) states per energy interval dE = !ω, the densityof states is given by

D(E) =E

(!ω)2(2D). (3.37)

For comparison, in the uniform system, D(E) is constant in 2D and D(E) ∝E− 1

2 in 1D.

3.2.2 Transition temperature

The transition temperature is defined as the temperature at which thechemical potential becomes equal to the zero-point energy of the system.For the 1D case, we have

N =!

nz

1

e!ωznz/kBT0 − 1=

" ∞

!ω2

D(E)

eE/kBT0 − 1dE

= −kBT0

!ωln

#

#

#

#

#

e!ω

2kBT0 − 1

e!ω

2kBT0

#

#

#

#

#

≃kBT0

!ωln

2kBT0

!ω.






N ≃kBT0

!ωln

2kBT0

!ω(1D). (3.38)

The transition temperature can be determined by numerically solvingthis equation. According to the Hohenberg–Mermin–Wagner theorem [seeSec. 9.2], spatially uniform infinite 1D systems do not undergo BEC evenat zero temperature. An important distinction in the harmonically trapped1D system is the constancy of the density of states that makes the conden-sation of bosons in the lowest-energy state easier than that in the uniformcase in which the density of states diverges at E = 0.

Similarly, in two and three dimensions, we obtain

N ≃!

kBT0

!ω

"2

ζ(2) (2D), (3.39)

N ≃!

kBT0

!ω

"3

ζ(3) (3D). (3.40)

Solving these for T0, we obtain

T0 = !ω

!

N

ζ(2)

"1

2

(2D), (3.41)

T0 = !ω

!

N

ζ(3)

"1

3

(3D). (3.42)

3.2.3 Condensate fraction

In general, we have

N = N0 +

# ∞

E0

D(E)

eβE − 1dE. (3.43)

For the 1D case, we substitute Eq. (3.35) into Eq. (3.43), obtaining

N = N0 +1

!ω

# ∞

E0

dE

eβE − 1= N0 −

kBT

!ωln(1 − e−βE0).

Assuming βE0 ≪ 1, we obtain

N0

N≃ 1 −

kBT

!ω

1

Nln

!ω

2kBT. (3.44)

Substituting Eq. (3.38) in Eq. (3.44), we obtain

N0

N≃ 1 −

T

T0

ln !ω2kBT

ln !ω2kBT0

(1D). (3.45)





Trapped Systems 81

Table 3.1 Summary of density of states D(E), critical temperature T0, andcondensate fraction N0/N of harmonically trapped Bose–Einstein condensates,where W (x) is a positive solution to WeW = x and it is approximated asW (x) ≃ ln x − ln ln x + ln ln x/ lnx + · · · for large x.

3D 2D 1D uniform

D(E) E2

2(!ω)3E

(!ω)21

!ω

√

2M3E2π2!3

T0!ωkB

!

Nζ(3)

" 1

3 !ωkB

!

Nζ(2)

" 1

2 !ωNkBW (2N)

2π

[ζ( 3

2)]

23

!2

kBMn

2

3

N0

N1 −

!

TT0

"31 −

!

TT0

"21 −

T ln !ω2kBT

T0 ln !ω2kBT0

1 −

!

TT0

" 3

2

For the 2D case, we substitute Eq. (3.37) into Eq. (3.43), obtaining

N ≃ N0 +

#

kBT

!ω

$2

ζ(2).

Combining this with Eq. (3.39), we obtain

N0

N≃ 1 −

#

T

T0

$2

.

We can reproduce the above results by taking the direct sum of the discretelevels [de Groot, et al. (1950); Ketterle and van Druten (1996)].

The results obtained in this section are summarized in Table. 3.1. Acomprehensive survey of the condensates in harmonic traps is given byMullin [Mullin (1997)].

3.3 Semiclassical Distribution Function

When the thermal energy of the system kBT is greater than the quantum-mechanical zero-point energy, we may use the semiclassical distributionfunction

fp(r) =1

eβ[ϵp(r)−µ] − 1, (3.46)

where

ϵp(r) =p2

2m+ V (r). (3.47)






The function fp(r) gives the number of particles in the phase space ofr ∼ r + dr and p ∼ p + dp:

fp(r)dpdr

(2π!)3. (3.48)

The integral of fp(r) over momentum gives the semiclassical density distri-bution of the particle-number density:

n(r) =

!

dp

(2π!)31

eβ[ϵp(r)−µ] − 1. (3.49)

Assuming z(r) = eβ(µ−V (r)) and changing the variable of integration tox = p2/(2mkBT ), Eq. (3.49) becomes

n(r) =2√πλ3

th

! ∞

0dx

x1/2

z−iex − 1=

1

λ3th

∞"

n=1

zn

n3

2

=1

λ3th

g 3

2

(z), (3.50)

where g 3

2

(z) is defined in Eq. (3.12). If we keep only the n = 1 term in thesum, we obtain the Maxwell–Boltzmann distribution

nMB(r) =Z

λ3th

eβ(µ−V (r)),

where Z is the normalization constant. The semiclassical distribution failsover a length scale shorter than λth.

The integral of n(r) over r gives the total number of particles. In thecase of the harmonic potential V (r) = mω2r2/2, we have

N =

!

dr n(r) =

#

kBT

!ω

$3 ∞"

n=1

%

eβµ&

n3=

#

kBT

!ω

$3

g3(eβµ). (3.51)

The chemical potential µ is determined so as to satisfy this equation. Inparticular, at T = Tc, we have µ = 0, and g3(1) = ζ(3) = 1.202. Equa-tion (3.51) thus gives the transition temperature that is correct up to theleading order in N [cf. (3.17)]:

T0 =!ω

ζ(3)1/3kBN

1

3 . (3.52)

For the harmonic potential, the zero-point energy is 3!ω/2; thus, the semi-classical distribution function (3.46) is applicable if

kBT ≫ !ω. (3.53)

Comparing this with Eq. (3.52), we find that the temperature range overwhich Eq. (3.46) is valid is given by

T ≫ T0N− 1

3 . (3.54)

This condition is consistent with the fact that when N is finite, the transi-tion temperature (3.52) has a correction of the order of N−1/3 [cf. (3.17)].





Trapped Systems 83

3.4 Gross–Pitaevskii Equation

We consider a system of N identical bosons with mass M that are con-fined in a one-body potential V (r) and that interact via the delta-functionpotential. The Hamiltonian of the system is

H =N!

i=1

p2i

2M+

N!

i=1

V (ri) +U0

2

!

i=j

δ(ri − rj), (3.55)

where pi = −i!∇i. Let Ψ(r1, r2, · · · , rN ) be an eigenstate of the Hamilto-nian (3.55) that is normalized to unity and that satisfies the Schrodingerequation

HΨ(r1, r2, · · · , rN ) = EΨ(r1, r2, · · · , rN ). (3.56)

Multiplying Ψ∗(r1, r2, · · · , rN ) from the left and integrating over the coor-dinates gives

E =

"

ψ∗Hψ dr1 · · ·drN . (3.57)

Now, assume that the system undergoes BEC and that a majority of thebosons share the same single-particle state ψ1(r). Then, the many-bodywave function may be approximated by the product of ψ1(ri):

Ψ(r1, r2, · · · , rN ) =N#

i=1

ψ1(ri), (3.58)

where ψ1 is normalized to unity. Substituting Eq. (3.58) in Eq. (3.57) yields

E = N

"

dr ψ∗1(r)

$

−!2

2M∇2 + V (r) +

U0

2(N−1)|ψ1(r)|2

%

ψ1(r). (3.59)

Since the wave function of the condensate is typically normalized to N , wesubstitute

ψ(r) =√

Nψ1(r), (3.60)

and then, Eq. (3.59) becomes

E[ψ] =

"

drψ∗(r)

$

−!2

2M∇2 + V (r) +

U0

2|ψ(r)|2

%

ψ(r), (3.61)

where we replace (1−1/N)U0 with U0 in the last term because N ≫ 1. Thelast term in Eq. (3.61) is the Hartree energy that describes the mean-fieldinteraction between bosons. We refer to Eq. (3.61) as the Gross–Pitaevskii(GP) energy functional. The condensate wave function ψ is determined byrequiring that the GP energy functional (3.61) be extremal subject to the






normalization condition!

|ψ|2dr = N . Introducing the Lagrange multiplierassociated with this normalization condition, we have

δ

δψ∗(r)(E[ψ] − µ

"

dr|ψ|2)

=

#

−!2

2M∇2 + V (r) + U0|ψ(r)|2 − µ

$

ψ(r) = 0, (3.62)

where ψ and ψ∗ must be considered to be independent in functional differ-entiation because ψ is a complex field that has two degrees of freedom (i.e.,real and imaginary parts). We thus find that the condensate wave functionobeys a nonlinear Schrodinger equation known as the GP equation

#

−!2

2M∇2 + V (r) + U0|ψ(r)|2

$

ψ(r) = µψ(r), (3.63)

where µ plays the role of the chemical potential of the system. The dynam-ics of the condensate is governed by the time-dependent GP equation

i!∂

∂tψ(r, t) =

#

−!2

2M∇2 + V (r) + U0|ψ(r, t)|2

$

ψ(r, t). (3.64)

The consistency between Eqs. (3.63) and (3.64) is guaranteed by the factthat the time dependence of the condensate wave function is governed bythe chemical potential [see Eq. (1.67)].

The crucial assumption in the derivation of the GP equation (3.63) isthe mean-field approximation (3.58) for the many-body wave function. Indilute BEC systems in which the condensate fraction is close to unity, thisapproximation is expected to be valid to a high degree, and the deviationsfrom the mean field are well accounted for by the Bogoliubov theory ofweakly interacting bosons. It can also be shown [Lieb, et al. (2000)] thatthe Gross–Pitaevskii energy functional gives the exact ground-state energyand particle density of a dilute Bose gas with a repulsive interaction inthe limit of N → ∞ and a → 0 with Na fixed. For a dilute Bose gaswith a repulsive interaction, the Gross–Pitaevskii equation is shown to beasymptotically exact in the limit of N → ∞ and a → 0 with Na/d0 keptconstant [Lieb, et al. (2000)].

3.5 Thomas–Fermi Approximation

Consider the case of a repulsive interaction which expands the condensateand suppresses density fluctuations. As a consequence, the kinetic energybecomes less important as the number of bosons, N , increases. In fact, as





Trapped Systems 85

shown in Eq. (3.77), the ratio of the kinetic energy to the potential energyscales as N− 4

5 , and therefore, the kinetic energy becomes negligible whenN is very large. This regime is referred to as the Thomas-Fermi (TF) limitin which the energy of the system consists only of the one-body potentialV (r) and the mean-field interaction (Hartree) energy. Then, it follows fromEq. (3.63) that the TF wave function is given by

ψTF(r) =

!

1

U0(µ − V (r)) θ(µ − V (r)), (3.65)

where θ(x) is the unit step function and the TF density is

nTF(r) =1

U0(µ − V (r)) θ(µ − V (r)). (3.66)

For the case of an isotropic harmonic potential

V (r) =Mω2

2r2, (3.67)

Eq. (3.65) gives

ψTF(r) =

"

Mω2

2U0(R2 − r2) θ(R − |r|), (3.68)

where R ≡#

2µ/Mω2 is a characteristic radius of the condensate. Substi-tuting U0 = 4π!2a/M in Eq. (3.68) gives

ψTF(r) =

"

R2 − r2

8πad40

θ(R − |r|). (3.69)

The chemical potential µ is determined from the normalization condition$

ψ2TF(r)dr = N. (3.70)


R =

%

15Na

d0

&1

5

d0. (3.71)

Hence,

µ =Mω2

2R2 =

!ω

2

%

15Na

d0

&2

5

=!ω

2

%

R

d0

&2

. (3.72)

It follows from the relation µ = ∂E/∂N that the total energy is given by

E =5!ω

14

%

15Na

d0

&2

5

N =5

7µN. (3.73)






In the TF approximation, the total energy consists of the potential energyEpot and the interaction energy Eint. The virial theorem then gives 2Epot =3Eint. Thus,

Epot =3

7µN, Eint =

2

7µN, (3.74)

where Eint can be found from the release-energy measurement1[Ensher, et

al. (1996)].The peak density of the condensate is given by

n(0) = ψ2TF(0) =

R2

8πad40

=5

2

N

4πR3/3. (3.75)

It follows from Eqs. (3.72) and (3.75) that

µ =!ω

2

!

R

d0

"2

= 4π!ωad20n(0) = U0n(0). (3.76)

The measurement of the release energy per particle, Eint/N = 2µ/7 =2U0n(0)/7, thus provides direct determination of the peak density and thechemical potential of the condensate.

When the size of the condensate is R, the kinetic energy per particle isof the order of !2/2MR2. The ratio of the kinetic energy to the potentialenergy is estimated to be

kinetic energy

potential energy∼

!2/(2MR2)

Mω2R2/2=

!

d0

R

"4

=

!

15Na

d0

"− 4

5

. (3.77)

Thus, the kinetic energy is negligible if

15Na

d0≫ 1. (3.78)

Because the interaction causes virtual excitations of bosons out of thecondensate, the number of condensate bosons is actually less than the totalnumber of bosons, N , in the system. The ratio of the number of virtuallyexcited bosons to N is called the fraction of the condensate depletion, andit is roughly given by [see Eq. (2.79)]

#

n(0)a3 =

$

15

8π

% a

R

&3N. (3.79)

1The release-energy measurement is performed by switching off the trapping potentialand measuring the total energy of the gas; therefore, the released energy is the sum ofthe kinetic energy and the interaction energy. In the TF limit, the kinetic energy isnegligible, and hence, the release-energy measurement gives Eint.





Trapped Systems 87

Because the GP theory ignores the depletion of the condensate, this frac-tion must be much smaller than unity for both the GP theory and theBogoliubov theory to be valid. This gives the condition

N ≪!

R

a

"3

. (3.80)

For typical parameters R ∼ 1 µm and a ∼ 10 A, it gives N ≪ 109, whichhas been satisfied for most experiments performed thus far. However, it ispossible that condition (3.80) does not hold when a is very large [Papp, et

al. (2008); Pollack, et al. (2009)]. In this case, neither the GP theory northe Bogoliubov theory is valid.

When the system is locally perturbed by external fields, container walls,or by topological defects such as vortices, the system tries to restore its un-perturbed density. A characteristic length scale ξ over which the localdensity restores its equilibrium value is called the healing length or correla-tion length; it is determined by the balance between the zero-point kineticenergy !2/2Mξ2 and the mean-field interaction energy 4π!2an/M :

ξ =1√

8πan. (3.81)

In the case of an isotropic harmonic trap, the healing length at the densitypeak can be obtained from Eqs. (3.75) and (3.81) as

ξ =1

#

8πan(0)=

d20

R. (3.82)

Because R ≫ d0 in the TF regime, the healing length is much shorter thanthe size of the condensate R. However, it is much longer than the meaninterparticle distance because

n1

3 ξ =1

#

8πan1

3

≫ 1. (3.83)

Thus, in the TF regime, we have a clear separation of various length scales:R ≫ d0 ≫ ξ ≫ n−1/3.

It follows from Eqs. (3.72) and (3.76) that

!ω

U0n(0)=

!ω

µ= 2

!

d0

R

"2

= 2

!

15Na

d0

"− 2

5

. (3.84)

When 15Na/d ≫ 1, the mean-field interaction energy per particle, U0n(0),is much larger than the single-particle energy level spacing !ω, implyingthat single-particle energy levels are substantially blurred by the interac-tion. When calculating the properties of BEC, we therefore need not care






about the discreteness of the harmonic potential. It is also worthwhile tonote that the usual definition of BEC as n0/N = O(1), where n0 is thenumber of bosons in the lowest single-particle state, cannot be used. In-stead, we must resort to a more general definition of BEC formulated byPenrose and Onsager [Penrose and Onsager (1956)], which we restate here[see Sec. 1.3 for details]. Irrespective of the presence of interactions, wecan always define the single-particle reduced density operator ρ1. Let thelargest eigenvalue of the reduced density operator be nmax. Then, we maysay that BEC exists if nmax/N = O(1). If there is one and only one eigen-value that is of the order of N , the matrix element ⟨r|ρ1|r′⟩ is shown tohave an asymptotic form

⟨r|ρ1|r′⟩ → ψ(r)ψ∗(r′) for |r − r′| → ∞. (3.85)

Conversely, when condition (3.85) holds, it can be shown that ψ(r) is avery good approximation of the eigenfunction of ρ1 corresponding to thelargest eigenvalue nmax. In this sense, we call ψ(r) the condensate wavefunction and nmax/N , the condensate fraction. As given in Eq. (3.58),the GP wave function is well approximated by ψ(r) in Eq. (3.85) if thecondensate fraction is close to unity.

3.6 Collective Modes in the Thomas–Fermi Regime

In the Thomas–Fermi regime, the frequencies of various collective modescan be calculated analytically. Substituting

ψ(r, t) =!

n(r, t) eiφ(r,t) (3.86)

into Eq. (3.64) and separating the real and imaginary parts, we obtain

∂n

∂t= −∇(nv), (3.87)

M∂v

∂t= −∇

"

µ +M

2v2

#

, (3.88)

where

v =!

M∇φ (3.89)

is the superfluid velocity and

µ = V + U0n −!2

2M√

n∇

2√n (3.90)





Trapped Systems 89

is the chemical potential. The last term in (3.69) is called the quantumpressure term and it may be ignored in the Thomas–Fermi regime. Welinearize n and µ around their equilibrium values:

n(r, t) = nTF(r) + δn(r, t), (3.91)

µ(r, t) = µ + δµ(r, t). (3.92)

Substituting Eqs. (3.91) and (3.92) into Eqs. (3.87) and (3.88) and ignoringthe higher-order terms such as δnv and v2, we obtain

∂δn

∂t= −∇(nTFv), (3.93)

M∂v

∂t= −U0∇δn, (3.94)

where δµ = U0δn is used. Taking the time derivative of both sides ofEq. (3.93) and substituting Eq. (3.94), we obtain

∂2δn

∂t2=

U0

M∇(nTF∇δn). (3.95)

Assuming the time dependence of the collective mode to be δn ∝ e−iΩt, weobtain

Ω2δn(r) = −U0

M∇

!

nTF(r)∇δn(r)"

. (3.96)

By solving this equation for a given potential V (r), we obtain the frequencyΩ of the collective mode.

3.6.1 Isotropic harmonic potential

For an isotropic harmonic potential (3.67) and the TF density (3.66),Eq. (3.96) reduces to

Ω2

ω2δn = r

∂

∂rδn −

1

2(R2 − r2)

#

1

r

∂2

∂r2(rδn) −

ℓ2

r2δn

$

, (3.97)

where we used the polar-coordinate representation of the Laplacian

∇2 =1

r

∂2

∂r2r −

ℓ2

r2(3.98)

with

ℓ2 ≡ −

1

sin θ

∂

∂θ

%

sin θ∂

∂θ

&

−1

sin2 θ

∂2

∂φ2. (3.99)

As a solution to Eq. (3.97), we assume

δn(r) = f(r)rℓYℓm(θ,φ), (3.100)






where Yℓm is the spherical harmonic function that is an eigenfunction of ℓ2

with eigenvalue ℓ(ℓ+ 1):

ℓ2Yℓm = ℓ(ℓ+ 1)Yℓm. (3.101)


x(x − 1)d2f

dx2+

!

2ℓ+ 3

2−

2ℓ+ 5

2x

"

df

dx+

#

Ωω

$2 − ℓ

2f = 0, (3.102)

where x ≡ r2/R2. This equation takes the same form as the differentialequation of the hypergeometric function 2F1(a, b, c; x):

x(x − 1)d2

2F1

dx2+ [c − (a + b + 1)x]

d2F1

dx− ab2F1 = 0, (3.103)

where

2F1(a, b, c; x) = 1 +ab

c

x

1!+

a(a + 1)b(b + 1)

c(c + 1)

x2

2!+ · · ·

≡∞%

n=0

(a)n(b)n

(c)n

xn

n!. (3.104)

Comparing Eq. (3.102) with (3.103), we find that

c =2ℓ+ 3

2, a + b =

2ℓ+ 3

2, and ab =

#

Ωω

$2 − ℓ

2. (3.105)

For the series in Eq. (3.104) to terminate at finite n, a or b must be zeroor a negative integer. Let a = −n (n = 0, 1, 2, · · · ). Then, b = n + ℓ + 3/2and [Stringari (1996)]

Ω = ω&

2n2 + 2nℓ+ 3n + ℓ (n = 0, 1, 2, · · · ) (3.106)

and

f(r) = 2F1

!

−n, n + ℓ+3

2, ℓ+

3

2;

r2

R2

"

. (3.107)

Equation (3.106) is to be compared with the ideal-gas result Ω = ω(2n+ ℓ).The mode with n = 0 and ℓ = 0 oscillates along the radial direction and

it is called the monopole mode or the breathing mode with the frequencygiven by

Ω = ω&

2n2 + 3n. (3.108)

The mode with ℓ = 0 and n = 0 has the frequency

Ω =√ℓ ω (3.109)

and the density oscillation

δn ∝ rℓYℓm(θ,φ)e−iΩt. (3.110)

The amplitude of the oscillation increases for larger r and peaks near thesurface of the condensate. This mode is therefore called the surface mode.





Trapped Systems 91

3.6.2 Axisymmetric trap

For an axisymmetric potential,

V (ρ, z) =Mω2

2(ρ2 + λ2z2), (3.111)

where ρ ≡!

x2 + y2 and λ is the trap aspect ratio. Then, the TF densityis

nTF =Mω2

2U0(R2 − ρ2 − λ2z2)θ(R2 − ρ2 − λ2z2), (3.112)

and Eq. (3.96) reduces to

Ω2

ω2δn =

"

ρ∂

∂ρ+ λ2z

∂

∂z

#

δn −1

2(R2 − ρ2 − λ2z2)∇2δn. (3.113)

We note that the Laplacian is expressed in cylindrical coordinates as

∇2 =∂2

∂ρ2+

1

ρ

∂

∂ρ+

1

ρ2

∂2

∂φ2+

∂2

∂z2. (3.114)

Then, by inspection, we find that a solution to Eq. (3.113) is given by thesurface mode

δn ∝ ρe±iℓφe−iΩt (3.115)

which satisfies ∇2δn = 0 and gives

Ω =√ℓ ω. (3.116)

For an axisymmetric trap, the magnetic quantum number m is con-served, but ℓ is, in general, not conserved. Thus, the quadrupolar modewith ℓ = 2 and m = 0 is coupled with the breathing mode with n = 1and ℓ = m = 0. Since the density oscillations of these modes are givenby δn ∝ r2Y20 ∝ 2z2 − x2 − y2 and δn ∝ x2 + y2 + z2, respectively, it isreasonable to assume that the density oscillation of the coupled mode isgiven by δn = aρ2 +bz2 +c. Substituting this into Eq. (3.113), we find that

Ω = ω

$

2 +3

2λ2 ±

%

4 − 4λ2 +9

4λ4

&1

2

. (3.117)






3.6.3 Scissors mode

The irrotational property of a superfluid gives rise to a unique responseof the system, known as a scissors mode, against a sudden rotation ofthe trapping potential [Guery-Odelin and Stringari (1999); Marago, et al.(2000)]. We consider a general harmonic potential

V (r) =M

2(ω2

xx2 + ω2yy

2 + ω2zz2) (3.118)

and assume density fluctuations of the form

δn = αxye−iΩt, (3.119)

where α is a constant. Substituting Eq. (3.119) into Eq. (3.95), we obtain

Ω2δn = −αU0

M

!

∂nTF

∂xy +

∂nTF

∂y

"

e−iΩt

= α(ω2x + ω2

y)xye−iΩt.

Thus, the frequency of the scissors mode is given by

Ω =#

ω2x + ω2

y. (3.120)

The velocity field of this mode can be found from Eq. (3.94) as

v = −iU0α

MΩ(y, x, 0), (3.121)

which satisfies the irrotationality condition rotv = 0. In contrast, if thecondensate rotated like a rigid body, the velocity field would be proportionalto (−y, x, 0). Instead of a uniform rotation, the scissors mode describes anoscillation of the condensate axes in a manner similar to periodic openingand closing of a pair of scissors, as shown in Fig. 3.1.

Ω

Fig. 3.1 Scissors mode. The axis of the condensate oscillates around an equilibriumposition in a manner similar to periodic opening and closing of a pair of scissors.





Trapped Systems 93

3.7 Variational Method

Variational methods based on the minimal action principle have successfullybeen applied to many phenomena, especially time-dependent ones such ascollective modes. The time-dependent GP equation can be derived fromthe minimal action principle with the action given by

S =

!

dt

!

dr

"

i!ψ∗ ∂ψ

∂t+

!2

2Mψ∗∇2ψ −

Mω2

2(x2 + y2 + λ2z2)ψ∗ψ

−2π!2a

M(ψ∗ψ)2

#

, (3.122)

where we assume an axially symmetric harmonic potential and λ is anasymmetry parameter. In fact, the functional derivative of S with respectto ψ∗ gives

δS

δψ∗= i!

∂ψ

∂t+

!2

2M∇2ψ −

Mω2

2(x2 + y2 + λ2z2)ψ −

4π!2a

Mψ∗ψ2 = 0,

that is

i!∂ψ

∂t=

"

−!2

2M∇2 +

Mω2

2(x2 + y2 + λ2z2) +

4π!2a

M|ψ|2

#

ψ, (3.123)

which is nothing but the time-dependent GP equation.Let us measure the length, time, and ψ in units of d0 ≡

$

!/Mω, ω−1,and

$

N/d30, respectively. This is equivalent to the substitutions

t → ω−1t, r → d0r, and ψ →%

N

d3ψ. (3.124)

Then, the action (3.122) becomes

S = N!

!

dt

!

dr

"

iψ∗∂ψ

∂t+

1

2ψ∗∇2ψ −

1

2(x2 + y2 + λ2z2)ψ∗ψ

−g

2(ψ∗ψ)2

&

, (3.125)

where ψ is normalized to unity'

|ψ|2dr = 1 and

g ≡4πNa

d0(3.126)

is the dimensionless strength of interaction. Separating ψ into amplitudeA and phase φ as

ψ(r, t) = A(r, t)eiφ(r,t), (3.127)






and substituting this into Eq. (3.125), we obtain

S = N!

!

dt

!

dr

"

i

2

∂A2

∂t+

i

2∇(A∇φ) − A2 ∂φ

∂t+

1

2A∇2A −

1

2A2(∇φ)2

−1

2(x2 + y2 + λ2z2)A2 −

g

2A4

#

. (3.128)

Here, the first two terms on the right-hand side play no role when thevariational principle is applied because the first term can be integratedwith respect to time, while the second term can be transformed via Stoke’stheorem into a surface integral.

The requirement that the action be stationary with respect to smallvariations in phase φ yields

∂A2

∂t+ ∇(A2vs) = 0, (3.129)

where

vs = ∇φ (3.130)

is the superfluid velocity, and hence, A2vs describes the mass current.Equation (3.129) is the equation of continuity that guarantees the conserva-tion of the particle number. The requirement that the action be stationarywith respect to small variations in amplitude A leads to

−∂φ

∂t=

1

2v2

s +1

2(x2 + y2 + λ2z2) + gA2 −

1

2A∇2A. (3.131)

This is a quantum version of the Hamilton–Jacobi equation, where the lastterm is referred to as the quantum pressure term or quantum potential.The solution of Eqs. (3.129) and (3.131) is equivalent to the solution of theGP equation

i∂ψ

∂t=

$

−∇2

2+

1

2(x2 + y2 + λ2z2) + g|ψ|2

%

ψ. (3.132)

3.7.1 Gaussian variational wave function

Any variational wave function other than the exact one cannnot satisfyEqs. (3.129) and (3.131) simultaneously. Instead, we minimize the actionS in Eq. (3.125) within the functional subspace of a given variational wavefunction. Let us consider a Gaussian vaiational wave function whose am-plitude is of the form

A(r, t) =

&

1

π3

2 dx(t)dy(t)dz(t)exp

"

−x2

2d2x(t)

−y2

2d2y(t)

−z2

2d2z(t)

#

,

(3.133)





Trapped Systems 95

where dx(t), dy(t), and dz(t) are variational parameters that are madetime-dependent to account for shape oscillations of the condensate in aharmonic trap. The preexponential factor in Eq. (3.133) is chosen to satisfythe normalization condition

!

A2dr = 1. The phase of the wave functionis determined to satisfy the equation of continuity (3.129). SubstitutingEq. (3.133) in Eq. (3.129) gives

f(x, dx) + f(y, dy) + f(z, dz) = 0, (3.134)

where

f(x, dx) = −dx

dx+

2dx

d3x

x2 −2x

d2x

∂φ

∂x+∂2φ

∂x2

= −dx

dx+

2dx

d3x

x2 + ex2

d2x∂

∂x

"

e− x2

d2x∂φx

∂x

#

. (3.135)

Here, the overdot denotes differentiation with respect to t. Since x, y, andz are independent, we may substitute f(α, dα) = aα for α = x, y, z whereaα’s are constants that satisfy ax + ay + az = 0. Integrating f(x, dx) = ax

with respect to x gives

φx =dx

2dxx2 + ax

$ x

0dx′

$ x′

0dx′′e

x′2−x

′′2

d2x + bx

$ x

0dx′e

x′2

d2x + Cx, (3.136)

where bx and Cx are integration constants. Because the global phase isarbitrary, we can substitute Cx = 0. By symmetry, we assume that thesuperfluid velocity at the origin is zero; thus

bx =∂φx

∂x

%

%

%

%

x=0

.

The mass current density is

jx ∝1

i(ψ∗∂xψ − ψ∂xψ

∗) ∝ A2 ∂φ

∂x

∝ e− x2

d2x− y2

d2y− z2

d2z

&

dx

dxx + axe

x2

d2x

$ x

0dx′e

−x′2

d2x

'

.

By requiring that jα should vanish at x → ±∞, we have ax = 0. Thus,f(x, dx) = (dx/2dx)x2 and

φ(r, t) =dx

2dxx2 +

dy

2dyy2 +

dz

2dzz2. (3.137)






Combining Eqs. (3.133) and (3.137) yields [Perez-Garcia, et al. (1996,1997)]

ψ(r, t) =

!

1

π3/2dxdydzexp

"

−x2

2d2x(1 − idxdx)

−y2

2d2y(1 − idydy) −

z2

2d2z(1 − idzdz)

#

; (3.138)

this gives the Gaussian variational wave function subject to the conservationof particle number and to the requirement that the mass current vanishesat the origin and at infinity.

The variational parameters dα are determined so as to make the actionextremal. Substituting Eq. (3.138) in Eq. (3.125) gives

S =N!

4

$

dt

%

&

α=x,y,z

'

−dαdα − d−2α − λ2

αd2α

(

−γ

dxdydz

)

, (3.139)

where λα = 1 for α = x, y and λα = λ for α = z, and

γ ≡g√2π3

=4N√2π

a

d0. (3.140)

Taking the functional derivative of S with respect to dα gives the equationof motion for dα:

dα = −λ2dα + d−3α +

γ

2dxdydzdα≡ −

∂V eff

∂dα(α = x, y, a), (3.141)

where

V eff =1

2

&

α=x,y,z

(d2α + d−2

α ) +γ

2dxdydz(3.142)

is an effective potential for dα.

3.7.2 Collective modes

The equations of motion (3.141) for variational parameters dα′s can be used

to find the frequencies of the collective modes of a condenstate such thatthe widths of the condensate, which are characterized by dα, oscillate intime. We consider a trap potential that is symmetric with respect to thez-axis, and let dr and dz be the equilibrium values of dα along the radialand axial directions, respectively. From Eq. (3.141), we see that they obey

dr = d−3r +

γ

2d3rdz

, (3.143)

λ2dz = d−3z +

γ

2d2rd2

z

. (3.144)





Trapped Systems 97

Expanding Eq. (3.141) up to linear terms from these equilibrium valuesleads to

∆dx = −(

1 + 3d−4r +

γ

d4rdz

)

∆dx −γ

2d4rdz

∆dy −γ

2d3rd2

z

∆dz, (3.145)

∆dy = −γ

2d4rdz

∆dx −(

1 + 3d−4r +

γ

d4rdz

)

∆dy −γ

2d3rd2

z

∆dz, (3.146)

∆dz = −γ

2d3rd2

z

∆dx −γ

2d3rd2

z

∆dy −(

λ2 + 3d−4z +

γ

d2r d3

z

)

∆dz , (3.147)

where ∆dα ≡ dα − dα. Substituting

a ≡ 1 + 3d−4r +

γ

d4rdz

= 3 + d−4r , (3.148)

b ≡ λ2 + 3d−4z +

γ

d2rd

3z

= 3λ2 + d−4z , (3.149)

α ≡γ

2d4rdz

, β ≡γ

2d3rd

2z

, (3.150)

the eigenvalue equation becomes∣

∣

∣

∣

∣

∣

ω2 − a −α −β−α ω2 − a −β−β −β ω2 − b

∣

∣

∣

∣

∣

∣

= (ω2 − a + α)[ω4 − (a + b + α)ω2 + ab + αb − 2β2] = 0.

Hence,

ω =

⎧

⎪

⎪

⎨

⎪

⎪

⎩

2√

1 − γ4d4

rdz,

√

2 + 2λ2 − γ4d2

rd3z±

√

4(

1 − λ2 + γ8d2

rd3z

)2+2

(

γ2d3

rd2z

)2 . (3.151)

For a given strength of interaction γ and asymmetry parameter λ, wecan solve Eqs. (3.143) and (3.144) for dr and dz. Substituting these inEq. (3.151) give the frequencies of the collective modes.

In the TF regime, the kinetic-energy terms d−3r and d−3

z in Eqs. (3.143)and (3.144) can be neglected, so that we have

γ

2d4rdz

= 1,γ

2d2rd

3z

= λ2,γ

2d3rd

2x

= λ.

Substituting these in Eq. (3.151) gives

ωQ =√

2, (3.152)

ωM−Q =

√

2 +3

2λ2 ±

√

4 − 4λ2 +9

4λ4, (3.153)






where ωQ is the frequency of the quadrupole mode and ωM−Q is that of thecombined monopole and quadrupole modes. For the case of an isotropicpotential (λ = 1), Eqs. (3.152) and (3.153) reduce to ω =

√2 (doubly degen-

erate) and√

5. For the case of a prolonged potential (λ≪ 1), Eqs. (3.152)and (3.153) give ω =

√2, 2, and

!

5/2λ. For the case of an oblate trap(λ≫ 1), we have ω =

√2,

!

10/3, and√

3λ.

3.8 Attractive Bose–Einstein Condensate

Attractive bosons are believed not to undergo BEC in a uniform systembecause the system would collapse into a high-density state. However,Bose–Einstein condensates with attractive interactions have been realizedin systems of confined atomic gases [Bradley, et al. (1995, 1997)], andremarkable collapsing dynamics have been observed [Gerton, et al. (2000);Donley, et al. (2001)]. We begin by offering a quantative argument of whyattractive condensates can exist in a confined system.

A confined atomic gas has three characteristic energies. The first is thezero-point kinetic energy that is proportional to d−2, where d is the size ofthe condensate. The second is the energy of a harmonic potential that isproportional to d2. The third is the interaction energy that is proportionalto the density of particles and is therefore proportional to N0/d3, where N0

is the number of condensate bosons. Because the interction is attractive,the interaction energy is negative. The total energy E is the sum of thesethree energies,

E(d) = N0(Ad−2 + Bd2 − CN0d−3), (3.154)

where A, B, and C are positive constants. It follows that the total energyhas a metastable minimum at d = dm, where E′(dm) = 0, if N0 is below acritical value Nc that is determined from E′(dc) = E′′(dc) = 0 by

Nc =8A

15C

"

A

5B

#1

4

. (3.155)

The condensate is believed to be formed at this local energy minimum.However, because of the interaction energy −CN0d−3, E tends to minusinfinity as d tends to zero. This implies that the condensate is not thetrue ground state but it is in a metastable one. The energy barrier sep-arating these two states arises from the term Ad−2 in Eq. (3.154). Thus,the metastability of BEC is ensured by the zero-point kinetic energy thatcounterbalances the attractive interaction. If N0 exceeds Nc, however, the





Trapped Systems 99

energy barrier vanishes and no condensate can exist. Since the zero-pointenergy arises from the confinement of the condensate due to a trapping po-tential, we may say that the BEC with attractive interaction is a mesoscopicphenomenon and it does not exist in the thermodynamic limit.

The above qualitative argument suggests that for the metastable Bose–Einstein condensate to exist, the zero-point energy ∼ !ω must exceed themean-field interaction energy per particle, i.e., !ω > N0U0/V . The criti-cal number of condensate bosons Nc can therefore be estimated from thecondition !ω ∼ Nc|U0|/V . Since the volume of the condensate is roughlygiven by V ∼ 4πd3

0, Nc is estimated to be

Nc ∼d0

|a|. (3.156)

A more quantitative argument below shows that Nc is indeed proportionalto d0/|a|, and the constant of proportionality is of the order of unity. Onemight conclude from (3.156) that Nc can be made infinite by increasing d0.However, this is not the case, since the density of particles n ∼ Nc/d3

0 ∼1/|a|d2

0 decreases with increasing d0. Since n must be larger than λ−3dB for

the system to undergo BEC, d0 cannot be larger than ∼!

λ3th/|a|, and

thus, we have the following fundamental upper limit for Nc:

Nc ≤"

λth

|a|

#3

2

. (3.157)

3.8.1 Collective modes

We consider the case of an isotropic potential. Then, the equation of motionfor the variational parameter dα is obtained from Eq. (3.141) with λ = 1 as

dα = −dα + d−3α +

γ

2dxdydzdα= −

∂V eff

∂dα(α = x, y, z), (3.158)

where

V eff =1

2

$

α=x,y,z

(d2α + d−2

α ) +γ

2dxdydz, (3.159)

γ =g√2π3

=4N√2π

a

d0. (3.160)

The equilibrium value d is the same for all α and it is determined from

d 5 − d −γ

2= 0. (3.161)






Linearizing Eq. (3.158) with respect to ∆dα ≡ dα − d yields

∆dα =!

β

[d−4 − 1 − 2(d−4 + 1)δαβ ]∆dβ . (3.162)

Substituting ∆dα(t) = ∆dα(0)e−iωt leads to the eigenvalue equation

det(ω2δαβ − [d−4 − 1 − 2(d−4 + 1)δαβ])

= (ω2 + d−4 − 5)(ω2 − 2d−4 − 2)2 = 0, (3.163)

and hence we obtain [Stringari (1996)]

ωM ="

5 − d−4, (3.164)

ωQ =#

2(1 + d−4), (3.165)

where ωM is the frequency of the monopole mode and ωQ is that of the dou-bly degenerate quadrupole modes. For a given γ, Eq. (3.161) can be solvedfor d, which then gives the frequencies of the monopole and quadrupolemodes via Eqs. (3.164) and (3.165), respectively. As the strength of theattractive interaction increases, the equilibrium width d of the conden-sate decreases. When d becomes smaller than rc = 5−

1

4 , the frequencyof the monopole mode becomes pure imaginary, which signals the onsetof a dynamical instability of an attractive condensate, i.e., of the collapseof the condensate. The critical strength of interaction is determined fromEq. (3.161) with d = rc as [Baym and Pethick (1996)]

γc = −8

55/4≃ −1.070. (3.166)

The corresponding critical number of condensate bosons is given by

Nc ≃2√

2n

55/4

d0

|a|≃ 0.6705

d0

|a|. (3.167)

This number is somewhat larger than the more precise value of

Nc ≃ 0.575d0

|a|(3.168)

that is obtained from the numerical integration of the GP equation [Dodd,et al. (1996)]. The difference arises from the fact that the variationalmethod overestimates the stable phase. To understand the nature of thecollective modes, let us substitute ∆dα(t) = ∆dα(0)e−iωQt in Eq. (3.162).Then, we have

∆dx + ∆dy + ∆dz = 0. (3.169)





Trapped Systems 101

Thus, the quadrupole oscillation occurs in such a manner that the volumeof the condensate is conserved. For the monopole mode, we substitute∆dα(t) = ∆dα(0)e−iωMt in Eq. (3.162), obtaining

∆dx = ∆dy = ∆dz .

This implies that the condensate alternately expands and contracts inan isotropic manner. We can thus substitute dα(t) = r(t) for all α inEq. (3.139), obtaining

SM =N!

4

!

dt(3r2 − 3r−2 − 3r2 − γr−3), (3.170)

where the constant term that results from the partial integration is dropped.The first term on the right-hand side may be interpreted as arising fromthe collective kinetic motion of the condensate. When transformed back tothe original units, it becomes

N!

4

!

dt3r2 →4!

4

!

d(ωt)3

"

d

d(ωt)

#

R

d0

$%2

=

!

dt1

2M∗R2, (3.171)

where

M∗ =3

2Nm (3.172)

is the effective mass of the condensate for the monopole motion [Ueda andLeggett (1998)].

To understand the behavior of the monopole mode near the criticalpoint, let us assume that N is close to but below Nc, and substitute γ =γc + δγ and r = rc + δr, where γc is given in Eq. (3.166) and rc = 5−

1

4 isobtained from Eq. (3.164) with the condition ωM = 0. Substituting thesein Eqs. (3.161), (3.164), and (3.165) yield

δr

rc=

&

2

5

#

1 −N

Nc

$

, (3.173)

ωM = 4

&

160

#

1 −N

Nc

$

, (3.174)

ωQ =

'

(

(

)12 − 8

&

10

#

1 −N

Nc

$

. (3.175)

We thus find that near the critical point, the frequency of the monopolemode vanishes according to the one-fourth power of 1 − N/Nc.






3.8.2 Collapsing dynamics of an attractive condensate

Let us consider a situation in which the strength of attractive interaction |γ|slightly exceeds its critical value |γc|. Then, d becomes smaller than 5−

1

4 sothat the frequency of the monopole mode (3.164) becomes pure imaginary.The radius of the condensate evolves with time as r(t) ∝ e−|ωM|t, so thatthe condensate implodes upon itself. The Lagrangian of the system is givenfrom Eq. (3.170) as

L =N!

4[3r2 − f(r)], (3.176)

where

f(r) = 3r−2 + 3r2 − |γ|r−3. (3.177)

The energy E of the system, which is a conserved quantity, is given by

E = r∂L

∂r− L =

N!

4[3r2 + f(r)] ≡

N!

4f(rc). (3.178)

Hence,

dr

dt= −

!

1

3[f(rc) − f(r)]. (3.179)

We substitute δγ ≡ |γ|− |γc| and expand f(rc)−f(r) up to the third powerin x ≡ rc − r. Then,

dx

dt=

!

5δγx + 10r−1c δγx2 +

20

3r−1c x3. (3.180)

In an initial stage of the collapse with x ≪ 1, it is sufficient to keep up tothe x2 term in Eq. (3.180), giving

x(t) =rc

4

"

cosh(#

10r−1c δγ t) − 1

$

≃5

4γc

%

N

Nc− 1

&

t2. (3.181)

When x becomes of the order of 1, the last term in Eq. (3.180) becomesdominant because other terms include a small factor δγ. Hence,

x(t) =3 · 5− 5

4

(t0 − t)2. (3.182)

The fact that x(t) diverges at a finite time implies that the collapse occursin a finite time. Here, t0 is determined by an initial slow dynamics governedby the first term in Eq. (3.180). Thus, taking the first and third terms inEq. (3.180), we obtain

dx

dt=

'

ax3 + bx, (3.183)





Trapped Systems 103

where a = 20.51/4/3 and b = 5δγ = (8/51/4)(N/Nc − 1). Integrating thisgives

t =1

4√

ab

! sinh−1√

ab x

0sinh−1 y dy. (3.184)

When N is close to Nc, a/b ≫ 1. Thus, the collapse time is estimated tobe

tcollapse ≃1

4√

ab

! ∞

0sinh− 1

2 ydy =1

4√

ab

Γ(14 )2

2√π

≃ 1.37

"

N

Nc− 1

#− 1

4

. (3.185)

At the final stage of collapse, the implosion is accelerated by higher-orderterms that are neglected in Eq. (3.180). The time timplosion required for thesystem to implode is therefore shorter than tcollapse by a constant amount∆t:

timplosion = tcollapse − ∆t,

where ∆t is numerically evaluated to be 1.83 [Saito and Ueda (2001)].





Chapter 4

Linear Response and Sum Rules

4.1 Linear Response Theory

In this section, we present basic tools to investigate the excitation spectrumof a many-body system. All results obtained in this section are applicableto both bosons and fermions.

4.1.1 Linear response of density fluctuations

The excitation spectrum of quasiparticles can be probed through the in-teraction of a test particle with the system of interest. Let U(r, t) be atime-dependent external potential that couples to the system at position r.In second-quantized language, the corresponding Hamiltonian is given by

Hext(t) =

!

dr U(r, t)ψ†(r)ψ(r), (4.1)

where ψ(r) is the field operator of the system. Substituting Fourier trans-forms

ψ(r) =1√V

"

k

akeikr, (4.2)

U(r, t) ="

k

!

dω

2πU(k,ω)eikre−iωt, (4.3)

into Eq. (4.1) gives

Hext(t) ="

k

!

dω

2πU(k,ω)ρ−ke−iωt. (4.4)

Here,

ρk ≡!

dr n(r)e−ikr ="

p

a†pap+k = ρ†−k (4.5)

105






is the Fourier transform of the number-density operator n(r) ≡ ψ†(r)ψ(r).Let us consider a situation in which the external potential has a single

wave vector k and single frequency ω. Then,

Hext(t) = U(k,ω)ρ−ke−iωt + U∗(k,ω)ρ†−keiωt. (4.6)

The state of the system evolves with time according to the Schrodingerequation

i!∂

∂t|ψ(t)⟩ =

!

H + Hext(t)eϵt"

|ψ(t)⟩, (4.7)

where H is the Hamiltonian of the system and ϵ, an infinitesimal positivenumber. The factor eϵt is introduced to ensure that the external potentialis adiabatically switched off in the remote past. The initial condition isassumed to be

|ψ(−∞)⟩ = |0⟩, (4.8)

where |0⟩ is the ground state of H.In linear response theory, we solve Eq. (4.7) up to first order in U(k,ω).

We expand the state vector in terms of a complete set of eigenstates |n⟩of H:

|ψ(t)⟩ =#

n

cn(t)e−i

!Ent|n⟩, (4.9)

where

H |n⟩ = En|n⟩ (n = 0, 1, 2, · · · ). (4.10)

The initial condition (4.8) is satisfied if the following condition is met:

cn(−∞) = δn0. (4.11)

Substituting Eq. (4.9) in Eq. (4.7), we obtain

cm(t) = −i

!

#

n

cn(t)e(iωmn+ϵ)t⟨m|Hext(t)|n⟩

= −i

!

#

n

cn(t)$

U(k,ω)⟨m|ρ−k|n⟩ei(ωmn−ω−iϵ)t

+U∗(k,ω)⟨m|ρ†−k|n⟩ei(ωmn+ω−iϵ)t

%

, (4.12)

where ωmn ≡ (Em−En)/! and Eq. (4.6) is substituted in the last equation.





Linear Response and Sum Rules 107

Integrating Eq. (4.12) with respect to t from −∞ to t up to the firstorder in U gives

cm(t) = δm0 + (1 − δm0)

!

U(k,ω)⟨m|ρ−k|0⟩!(ω − ωm0 + iϵ)

ei(ωm0−ω−iϵ)t

−U∗(k,ω)⟨m|ρ†−k|0⟩!(ω + ωm0 − iϵ)

ei(ωm0+ω−iϵ)t

"

. (4.13)

The change in density due to Hext is given as

δ⟨ρk(t)⟩ ≡ ⟨ψ(t)|ρk|ψ(t)⟩ − ⟨0|ρk|0⟩. (4.14)

Substituting Eq. (4.9) for |ψ(t)⟩ gives

δ⟨(ρk(t)⟩ =#

n

′ $cn(t)e−iωn0t⟨0|ρk|n⟩ + c∗n(t)eiωn0t⟨n|ρk|0⟩

%

, (4.15)

where&′

n denotes the summation over n except n = 0. SubstitutingEq. (4.13) in Eq. (4.15) and simplifying the result using1

⟨n|ρk|0⟩⟨n|ρ−k|0⟩ = ⟨0|ρk|n⟩⟨n|ρ†−k|0⟩ = 0 (4.16)

leads to

δ⟨ρk(t)⟩ =1

!U(k,ω)e−(iω−ϵ)t

#

n

′'

|⟨0|ρk|n⟩|2

ω − ωn0 + iϵ−

|⟨n|ρk|0⟩|2

ω + ωn0 + iϵ

(

.

(4.17)

We assume that the system possesses the space-inversion symmetry(ρk = ρ−k), so that

⟨n|ρk|0⟩ = ⟨n|ρ−k|0⟩ = ⟨n|ρ†k|0⟩ = ⟨0|ρk|n⟩∗. (4.18)

Then, Eq. (4.17) reduces to

δ⟨ρk(t)⟩ =1

!U(k,ω)e−(iω−ϵ)t

#

n

′|⟨n|ρ†k|0⟩|

2 2ωn0

(ω + iϵ)2 − ω2n0

. (4.19)

Applying the Fourier transform to this equation gives

δ⟨ρ(k,ω)⟩ = U(k,ω)Dret(k,ω), (4.20)

where

Dret(k,ω) =1

!

#

n

′|⟨n|ρ†k|0⟩|

2 2ωn0

(ω + iϵ)2 − ω2n0

. (4.21)

Equation (4.20) gives the linear response of density against the externalperturbation U(k,ω), and the ratio

χρ(k,ω) ≡δ⟨ρ(k,ω)⟩U(k,ω)

= Dret(k,ω) (4.22)

defines the linear susceptibility of density fluctuations.

1Since ρk =&

p a†pap+k annihilates the net momentum k from the system, ⟨n|ρk|0⟩ =

0 only if the total momentum of |n⟩ is −k. Thus, ⟨n|ρk|0⟩⟨n|ρ−k|0⟩ = 0. Similarly, weobtain the second equation using Eq. (4.5).






4.1.2 Retarded response function

The function Dret(k,ω) defined in Eq. (4.21) is referred to as the retardedresponse function or the retarded Green’s function of the density. Takingthe imaginary part of Eq. (4.21) and using

1

x + iϵ= P

!

1

x

"

− iπδ(x), (4.23)

where P#

1x

$

denotes the principal value of 1x , we obtain the dynamic struc-

ture factor

S(k,ω) ≡ −!

πImDret(k,ω) =

%

n

′|⟨n|ρ†k|0⟩|

2[δ(ω − ωn0) − δ(ω + ωn0)],

(4.24)

which gives the excitation spectrum of density, in which the perturbationtransfers energy !ω and momentum !k to the system.

The inverse Fourier transform of Dret(k,ω) is given by

Dret(k, t) =

& ∞

−∞

dω

2πDret(k,ω)e−iωt

=1

!

%

n

′|⟨n|ρ†k|0⟩|

2

& ∞

−∞

dω

2π

!

1

ω + iϵ− ωn0−

1

ω + iϵ+ ωn0

"

e−iωt.

(4.25)

Because of the factor e−iωt, the integration contour in Eq. (4.25) must betaken in the lower (or upper) half of the complex ω-plane if t > 0 (or ift < 0). Since the poles ω = ±ωn0−iϵ lie in the lower-half plane, the integralis nonzero only for t > 0. Hence,

Dret(k, t) = −i

!θ(t)

%

n

′|⟨n|ρ†k|0⟩|

2#

e−iωn0t − eiωn0t$

, (4.26)

where θ(t) is the unit step function. Comparing this with Eq. (4.24), wefind that

Dret(k, t) = −i

!θ(t)

& ∞

−∞

dωS(k,ω)e−iωt. (4.27)

Applying the Fourier transform to this equation with respect to time gives

Dret(k,ω) =1

!

& ∞

−∞

S(k,ω′)

ω + iϵ− ω′dω′. (4.28)






We may use Eq. (4.18) and the completeness relation to eliminate the sumover n in Eq. (4.26); in fact,

!

n

′|⟨n|ρ†k|0⟩|

2e−iωn0t =!

n

⟨0|ei

!Htρke−

i

!Ht|n⟩⟨n|ρ†k|0⟩ = ⟨0|ρk(t)ρ†k|0⟩,

!

n

′|⟨n|ρ†k|0⟩|

2eiωn0t =!

n

|⟨n|ρk|0⟩|2eiωn0t

=!

n

⟨0|ρ†k|n⟩⟨n|ei

!Htρke−

i

!Ht|0⟩ = ⟨0|ρ†kρk(t)|0⟩,

where"

n′ is replaced by

"

n because ⟨0|ρ†k|0⟩ = 0. We thus find that

Dret(k, t) = −i

!θ(t)

#

0$

$

$

%

ρk(t), ρ†k(0)&$

$

$0'

, (4.29)

where

ρk(t) ≡ ei

!Htρke−

i

!Ht. (4.30)

In a special case in which S(k,ω) has a single peak of the form

S(k,ω) = S(k)δ(ω − ωk), (4.31)

we obtain

Dret(k, t) = −i

!θ(t)S(k)e−iωkt (4.32)

from Eq. (4.27) and Eq. (4.28). Applying the Fourier transform to thisequation with respect to time gives

Dret(k,ω) =S(k)

!(ω − ωk + iϵ). (4.33)

Equation (4.33) implies that the pole of the retarded Green’s function ofthe density gives the frequency of the collective mode.

4.2 Sum Rules

When the system is translationally invariant in space, the excitation spec-trum satisfies some exact relations known as sum rules.






4.2.1 Longitudinal f-sum rule

When the system possesses space-translation invariance, the single-particleHamiltonian can be diagonalized with respect to the wave vector k:

H0 =!

k

ϵka†kak, ϵk =

!2k2

2m. (4.34)

The interaction Hamiltonian is expressed in the second-quantized form as

V =1

2

"

dr

"

dr′ψ†(r)ψ†(r′)V (r − r′)ψ(r′)ψ(r). (4.35)

Substituting Eq. (4.2) and

V (r) =!

k

Vkeikr (4.36)

into Eq. (4.30), we obtain

V =1

2

!

p,q,k

Vka†pa†

qaq+kap−k =1

2

!

k

Vk

#

ρ−kρk − N$

, (4.37)

where ρk is given in Eq. (4.5) and

N =!

p

a†pap (4.38)

is the total number operator. The total Hamiltonian is given by

H = H0 + V . (4.39)

A straightforward calculation gives%

ρ−k,%

ρk, H&&

= −!

p

(ϵp+k + ϵp−k − 2ϵp)a†pap. (4.40)

Since ϵk = !2k2/2m,

ϵp+k + ϵp−k − 2ϵp =!2k2

m= 2ϵk,

and thus,%

ρ−k, [ρk, H ]&

= −2ϵkN. (4.41)

Taking the expectation value of the left-hand side of Eq. (4.41) over |0⟩ andinserting the completeness relation, we have

'

0(

(

(

%

ρ−k,%

ρk, H&&

(

(

(0)

= −!

n

!ωn0

*

|⟨n|ρk|0⟩|2 + |⟨n|ρ−k|0⟩|2+

, (4.42)






where !ωn0 = En − E0. Substituting Eq. (4.41) in Eq. (4.42), we obtain!

n

!ωn0

"

|⟨n|ρk|0⟩|2 + |⟨n|ρ−k|0⟩|2#

= 2ϵkN. (4.43)

Equation (4.43) is referred to as the longitudinal f -sum rule. Whenthe system possesses space-inversion symmetry, the relation |⟨n|ρ−k|0⟩| =|⟨n|ρk|0⟩| holds, and thus Eq. (4.43) reduces to

!

n

fn0 = N, (4.44)

where

fn0 ≡!ωn0

ϵk|⟨n|ρk|0⟩|2 (4.45)

is called the oscillator strength. As Eq. (4.44) suggests, the f -sum rulereflects the conservation of the particle number. Equation (4.44) may beregarded as a generalization of the Thomas–Reiche–Kuhn sum rule: for asingle particle,

!

n

(E0 − En)|⟨n|x|0⟩|2 =!2

2M, (4.46)

where H |n⟩ = En|n⟩.In terms of the dynamic structure factor S(k,ω) in Eq. (4.24), the lon-

gitudinal f -sum rule is expressed as$ ∞

0dω!ωS(k,ω) = ϵkN. (4.47)

For the special case of S(k,ω) = NS(k)δ(ω − ωk), Eq. (4.47) gives theenergy of an elementary excitation as [Bijl (1940); Feynman (1954)]

!ωk =!2k2

2mS(k), (4.48)

where the static structure factor S(k) determines the dispersion relation,i.e., the relation between ωk and k.

An extension to finite temperature is straightforward. Multiplying bothsides of Eq. (4.41) by

ρ =e−βH

Z, (4.49)

where Z = Tr e−βH , we obtain Eq. (4.47) with

S(k,ω) =1

Z

!

m,n

e−βEm(|⟨m|ρ†k|n⟩|2 + |⟨m|ρk|n⟩|2)δ(ω − ωnm), (4.50)






where ωnm ≡ (En − Em)/!. In the presense of space-inversion symmetry,Eq. (4.50) reduces to

S(k,ω) =2

Z

!

m,n

e−βEm |⟨m|ρk|n⟩|2δ(ω − ωnm). (4.51)

Under the same assumption, we obtain the detailed balance of the dynamicstructure factor:

S(k,−ω) = e−β!ωS(k,ω). (4.52)

4.2.2 Compressibility sum rule

The compressibility κ measures the degree of volume reduction against anincrease in pressure at a fixed number of particles.

κ = −1

V

"

∂V

∂P

#

N

. (4.53)

The pressure P is defined as the derivative of energy with respect to volume,

P = −"

∂E

∂V

#

N

. (4.54)

Substituting this in Eq. (4.53) gives

κ−1 = V

"

∂2E

∂V 2

#

N

. (4.55)

Noting that V is related to the particle density n through V = N/n, wehave

∂

∂V=∂n

∂V

∂

∂n= −

n2

N

∂

∂n. (4.56)

Substituting E = Nϵg, where ϵg is the ground-state energy per particle, weobtain

κ−1 = nd

dn

"

n2 dϵgdn

#

= n2 d2

dn2(nϵg). (4.57)

On the other hand, the chemical potential µ is given by

µ =

"

∂E

∂N

#

V

=

"

∂(E/V )

∂(N/V )

#

V

=d

dn(nϵg). (4.58)

Comparing Eqs. (4.57) and (4.58), we obtain

κ−1 = n2 dµ

dn. (4.59)






A microscopic expression of the compressibility can be found fromEqs. (4.22) and (4.28):

δ⟨ρ(k,ω)⟩U(k,ω)

= Dret(k,ω) =1

!

! ∞

−∞

S(k,ω′)

ω + iϵ− ω′dω′. (4.60)

Suppose that we first take the limit k → 0 and then take the limit ω →0. Then, the denominator on the left-hand side of Eq. (4.60) provides auniform scalar potential U(k = 0,ω = 0) which may be interpreted as aminus shift in the chemical potential. On the other hand, the numeratorgives the concomitant change in the number of particles. Thus, we obtain

−"

∂N

∂µ

#

V

= limω→0

limk→0

Dret(k,ω) = −1

!

! ∞

−∞

S(k = 0,ω)

ωdω. (4.61)

It follows from Eqs. (4.59) and (4.61) that! ∞

−∞

S(k = 0,ω)

!ωdω = κn2V. (4.62)

This relation is known as the compressibility sum rule.The compressibility gives the isothermal and adiabatic sound velocity

c. In fact, defining the mass density as ρ ≡ mn , we have

c =

$

∂P

∂ρ=

%

1

m

dP

dn. (4.63)

Since

P = −"

∂E

∂V

#

N

= n2 dϵgdn

, (4.64)

from Eq. (4.57), we find that

dP

dn=

d

dn

"

n2 dϵgdn

#

=1

nκ. (4.65)

Thus,

c =1√

mnκ. (4.66)

We may use this relation to obtain another expression for the compressibil-ity sum rule.

! ∞

−∞

S(k = 0,ω)

!ωdω =

N

mc2. (4.67)






The combination of the f-sum rule and the compressibility sum rule givesan upper bound for the static structure factor. The Schwartz inequalitygives

S(k) ≡! ∞

−∞

S(k,ω)dω ≤

"

! ∞

−∞

!ωS(k,ω)dω

! ∞

−∞

S(k,ω)

!ωdω. (4.68)

Substituting Eq. (4.47), we have (note that the range of integration is dou-bled here)

S(k) ≤

"

2ϵkN

! ∞

−∞

S(k,ω)

!ωdω. (4.69)

Taking the limit of k → 0 and using Eq. (4.67), we obtain

S(k) ≤!k

mcN (k → 0). (4.70)

This inequality implies that the density fluctuations of the system with afinite compressibility become negligible in the long-wavelength limit.

4.2.3 Zero energy gap theorem

The zero energy gap theorem holds for translationally invariant systemswith positive compressibility.Theorem. If a system is translationally invariant in space, the excitationspectrum has zero energy gap in the long-wavelength limit as long as thecompressibility is positive.Proof. Let us assume that the excitation spectrum has an energy gap ∆ inthe limit k → 0. Then, S(k = 0,ω) = 0 for !ω < ∆, and therefore, thecompressibility sum rule (4.62) gives

1

2κn2V =

! ∞

∆/!

S(k = 0,ω)

!ωdω ≤

1

∆

! ∞

∆/!

S(k = 0,ω)dω. (4.71)

On the other hand, the f -sum rule (4.47) leads to

!2k2

2mN =

! ∞

∆/!

dω !ω S(k,ω) ≥ ∆

! ∞

∆/!

S(k,ω)dω. (4.72)

Combining Eq. (4.71) and Eq. (4.72), we find

limk→0

!2k2

2mN ≥

∆2

2κn2V. (4.73)

This inequality implies that as long as κ > 0, ∆ must vanish in the long-wavelength limit.

As a special application of this theorem, we find that a spatially uniformBose system with repulsive interaction is gapless in the long-wavelengthlimit.






4.2.4 Josephson sum rule

The condensate density is defined in terms of the eigenfunction of the single-particle density matrix corresponding to a macroscopic (i.e., extensive)eigenvalue and it is a thermodynamic quantity. The superfluid density,on the other hand, is defined in terms of linear response theory and it isa transport quantity. These two quantities therefore belong to differentnotions despite their apparent similarity. However, Josephson reported aninteresting relation between them that is referred to as the Josephson sumrule [Josephson (1966)].

We consider a situation in which a superfluid is flowing with velocityvs through a long container that is at rest with respect to the laboratoryframe. Then, the mass current density operator j is given by

j(r) =1

2i

!

ψ†(r)∇ψ(r) − (∇ψ†(r))ψ(r)"

. (4.74)

The quantum-statistical average of j defines the superfluid mass density ρs

through the relation

⟨j⟩ = ρsvs. (4.75)

The condensate density |ψ0|2 is defined in terms of the eigenfunctionψ0 corresponding to a macroscopic eigenvalue of the single-particle densitymatrix. Because ψ0 is complex, we may decompose it into the amplitudeand the phase

ψ0(r) = A(r)eiφ(r). (4.76)

When the amplitude A may be considered as a constant, a variation in thewave function is related to a change in the phase through

δψ0 = iψ0δφ. (4.77)

Since the spatial variation in φ is related to the superfluid velocity vs

through

vs =!

m∇φ, (4.78)

one may expect to find a relationship between ρs and |ψ0|2 by examiningthe responses ⟨j⟩ and ψ0 to a common external perturbation.

A variation in ψ0 is caused by a Hamiltonian that includes a term con-jugate to ψ. Here, we consider the response of the system to the followingHamiltonian:

Hext(t) =

#

dr$

ξ(k,ω)ei(kr−ωt)ψ†(r) + ξ∗(k,ω)e−i(kr−ωt)ψ(r)%

. (4.79)






The state evolution due to Hext(t) can be found by following a proceduresimilar to the one in Sec. 4.1.1. We expand the state vector in tems of theeigenstates |n⟩ of H as in Eq. (4.9), where the expansion coefficients canbe calculated up to the first order in Hext as

cn(t) = δn0 + (1 − δn0)

!

dr

"

ξ(k,ω)⟨n|ψ†(r)|0⟩!(ω − ωn0 + iϵ)

eikrei(ωn0−ω−iϵ)t

−ξ∗(k,ω)⟨n|ψ(r)|0⟩!(ω + ωn0 − iϵ)

e−ikrei(ωn0+ω−iϵ)t

#

. (4.80)

The response of ψ is given by

δ⟨ψ(r, t)⟩ ≡ ⟨ψ(t)|ψ(r)|ψ(t)⟩ − ⟨0|ψ(r)|0⟩

=$

n

′ %

cn(t)e−iωn0t⟨0|ψ(r)|n⟩ + c∗n(t)eiωn0t⟨n|ψ(r)|0⟩&

. (4.81)

We assume that each state |n⟩ has a fixed number of particles, so that

⟨0|ψ(r)|n⟩⟨n|ψ(r′)|0⟩ = ⟨n|ψ(r)|0⟩⟨n|ψ†(r′)|0⟩ = 0. (4.82)

Substituting cn(t) in Eq. (4.80) into Eq. (4.81) and using Eq. (4.82), weobtain

δ⟨ψ(r, t)⟩ =1

!ξ(k,ω)e−iωt+ϵt

$

n

′!

dr′'

⟨0|ψ(r)|n⟩⟨n|ψ†(r′)|0⟩ω − ωn0 + iϵ

−⟨n|ψ(r′)|0⟩∗⟨n|ψ(r)|0⟩

ω + ωn0 + iϵ

(

eikr′

= −i


$

n

′!

dr′! ∞

0dt′

%

⟨0|ψ(r, t′)|n⟩⟨n|ψ†(r′, 0)|0⟩

−⟨0|ψ†(r′, 0)|n⟩⟨n|ψ(r, t)|0⟩&

ei(ω+iϵ)t′eikr′ , (4.83)

where

ψ(r, t) ≡ ei

!Htψ(r)e−

i

!Ht. (4.84)

Since ⟨0|ψ(r, t)|0⟩ = 0, we may replace the restricted sum)

n′ in Eq. (4.83)

with the unrestricted one)

n. Then, it follows from the completenessrelation

$

n

|n⟩⟨n| = 1 (4.85)






that Eq. (4.83) reduces to

δ⟨ψ(r, t)⟩

= ξ(k,ω)e−iωt+ϵt

!

dr′! ∞

0dt′Gret(r, t′; r′, 0)eikr′ei(ω+iϵ)t′, (4.86)

where we introduced the single-particle retarded Green’s function

Gret(r, t; r′, t′) ≡ −i

!θ(t − t′)

"

0#

#

#

$

ψ(r, t), ψ†(r′, t′)%#

#

#0&

. (4.87)

For convenience in later discussions, let us introduce the spectral densityfunction A(k,ω):

A(k,ω) ≡ i!

!

d(r − r′)e−ik(r−r′)

! ∞

−∞

d(t − t′)eiω(t−t′)Gret(r, t; r′, t′),

(4.88)

Gret(r, t; r′, t′) = −i

!

!

dk

(2π)3eik(r−r′)

! ∞

−∞

dω

2πe−iω(t−t′)A(k,ω), (4.89)

where A(k,ω) satisfies! ∞

−∞

dω

2πA(k,ω) = 1. (4.90)


δ⟨ψ(r, t)⟩ =1

!ξ(k,ω)ei(kr−ωt)+ϵt

! ∞

−∞

dω′

2π

A(k,ω′)

ω + iϵ− ω′. (4.91)

Taking the limit of ω → 0 and ϵ→ 0, we obtain

δ⟨ψ(r)⟩ = −1

!ξ(k, 0)eikr

! ∞

−∞

dω

2π

A(k,ω)

ω. (4.92)

In a similar manner, the response of the mass current density is given by

δ⟨j(r, t)⟩ = −i


!

dr′! ∞

0dt′

×"

0#

#

#

$

j(r, t′), ψ†(r′, 0)%#

#

#0&

eikr′ei(ω+iϵ)t′ . (4.93)

We introduce another spectral density function B(k,ω) of the correlationfunction

"

0#

#

#

$

j(r, t′), ψ†(r, 0)%#

#

#0&

=

!

dk′

(2π)3

!

dω′

2πB(k′,ω′)eik′(r−r′)−iω′t′ ,

(4.94)

B(k,ω) =

!

dr

!

dt"$

j(r, t), ψ†(r′, t′)% &

e−ik(r−r′)+iω(t−t′).

(4.95)






Substituting this in Eq. (4.93) gives

δ⟨j(r, t)⟩ =1

!ξ(k,ω)ei(kr−ωt)+ϵt

! ∞

−∞

dω′

2π

B(k,ω′)

ω + iϵ− ω′. (4.96)

Taking the limit of ω → 0 and ϵ→ 0, we obtain

δ⟨j(r)⟩ = −1

!ξ(k, 0)eikr

! ∞

−∞

dω

2π

B(k,ω)

ω. (4.97)

The right-hand side is proportional to δ⟨ψ(r)⟩, since the Fourier transfor-mation of the continuity equation

∇j(r, t) + m∂ρ(r, t)

∂t= 0 (4.98)

gives

ik · j(k, t) + m∂ρ(k, t)

∂t= 0. (4.99)

Assuming that j(k, t) is proportional to k, we obtain

j(k, t) =imk

!2

∂ρ(k, t)

∂t. (4.100)

Hence,

j(r, t) =

!

dk

(2π)3eikr imk

k2

∂ρ(k, t)

∂t

= im∂

∂t

!

dr′!

dk

(2π)3k

k2eik(r−r′)ρ(r′, t). (4.101)

Substituting Eq. (4.101) in Eq. (4.95), we have

B(k,ω) = im

!

dr

!

dt e−ik(r−r′)+iω(t−t′) ∂

∂t

!

dr′′

×!

dk′

(2π)3k′

k′2eik′(r−r′′)

"#

ρ(r′′, t), ψ†(r′, t′)$ %

=mωk

k2

!

dt eiω(t−t′)

!

dr′′e−ik(r′′−r′)"#

ρ(r′′, t), ψ†(r′, t′)$%

. (4.102)

Substituting this in Eq. (4.97), we obtain

δ⟨j(r)⟩ = −mk

!k2ξ(k, 0)eikr

!

dr e−ik(r−r′)"#

ρ(r, t), ψ†(r′, t)$%

. (4.103)

Using"#

ρ(r, t), ψ†(r′, t)$%

="#

ρ(r), ψ†(r′)$%

="

ψ†(r)#

ψ(r), ψ†(r′)$%

= ⟨ψ†(r)⟩δ(r − r′), (4.104)






we obtain

δ⟨j(r)⟩ = −mk

!k2ξ(k, 0)eikr⟨ψ†(r)⟩. (4.105)

Comparing Eqs. (4.92) and (4.105), we find that the desired relation be-tween δ⟨j⟩ and δ⟨ψ⟩ is given by

δ⟨j(r)⟩ = mk

k2

!" ∞

−∞

dω

2π

A(k,ω)

ω

#−1

⟨ψ†(r)⟩δ⟨ψ(r)⟩, (4.106)

where we substitute ψ0(r) = ⟨ψ(r)⟩ and use Eq. (4.77) to obtain

δ⟨ψ(r)⟩ = δψ0(r) = iψ0(r)δφ(r). (4.107)

Then, Eq. (4.106) may be rewritten as

δ⟨j(r)⟩ = imk

k2

!" ∞

−∞

dω

2π

A(k,ω)

ω

#−1

|ψ0(r)|2δφ(r).

=m

k2

!" ∞

−∞

dω

2π

A(k,ω)

ω

#−1

|ψ0(r)|2∇δφ(r), (4.108)

since δφ ∝ eikr. From Eqs. (4.75) and (4.78), on the other hand, we have

δ⟨j(r)⟩ =!

mρs∇δφ(r). (4.109)

Equating Eqs. (4.108) and (4.109), we finally obtain the relation amongthe superfluid density ρs, condensate density |ψ0|2, and spectral densityfunction A(k,ω) as

" ∞

−∞

dω

2π

A(k,ω)

ω=

m2|ψ0|2

!k2ρs. (4.110)

This relation may be interpreted as a sum rule obeyed by the single-particlespectral density function A(k,ω), and is referred to as the Josephson sumrule.

Another sum rule obeyed by A(k,ω) is

" ∞

−∞

dω

2πA(k,ω) = 1, (4.111)

which can be shown directly from Eq. (4.88).






4.3 Sum-Rule Approach to Collective Modes

We investigate the collective mode of a system described by HamiltonianH . Let |n⟩ and En be a complete set of exact eigenstates and that ofthe corresponding eigenvalues:

H |n⟩ = En|n⟩,

where we assume that E0 ≤ E1 ≤ E2 ≤ · · · . In general, a system willexhibit various types of collective modes characterized by symmetries andexcitation energies. Let F be an excitation operator of the system. WhenF acts on the ground state |0⟩, various states |F1⟩, |F2⟩, · · · can, in general,be excited, where |Fi⟩ belongs to the set |n⟩ and satisfies

H |Fi⟩ = EFi|Fi⟩ (i = 1, 2, 3, · · · ) (4.112)

with EF1≤ EF2

≤ · · · . The following theorem is useful for finding an upperbound of a collective mode.Theorem. An upper bound !ωupper to the minimum excitation energy!ωmin ≡ EF1

− E0 of the states excited by F is given by

!ωupper =

!

m3

m1, (4.113)

where E0 is the ground state energy and

mp ≡"

i

|⟨Fi|F |0⟩|2(EFi− E0)

p (4.114)

is the p-th energy-weighted moment of the excitation.Proof. A straightforward calculation shows that

(!ωupper)2 −#

!ωmin$2

(FF1− E0)3

=

%

i |⟨Fi|F |0⟩|2&

'

EFi−E0

EF1−E0

(3− EFi

−E0

EF1−E0

)

%

i |⟨Fi|F |0⟩|2(EFi− E0)

.

Since (EFi− E0)/(EFi

− E0) ≥ 1, we have !ωmin ≤ !ωupper.When F is Hermitian, m1 and m3 can be rewritten as

m1 =1

2

*

0+

+

+

,

F †,,

H, F--

+

+

+0.

, (4.115)

m3 =1

2

*

0+

+

+

,,

F †, H-

,,

H,,

H, F---

+

+

+0.

. (4.116)

Equations (4.115) and (4.116) can be shown by inserting the completenessrelation

%

n |n⟩⟨n| = 1 and noting that only states |Fi⟩ are connected






to the ground state |0⟩ via F . In fact, calculating the right-hand sides ofEqs. (4.115) and (4.116), we have

1

2

!

0"

"

"

#

F †,#

H, F$$

"

"

"0%

=1

2

&

i

#

|⟨Fi|F |0⟩|2 + |⟨Fi|F †|0⟩|2$

(EFi− E0)

, (4.117)

1

2

!

0"

"

"

##

F †, H$

,#

H,#

H, F$$$

"

"

"0%

=1

2

&

i

#

|⟨Fi|F |0⟩|2 + |⟨Fi|F †|0⟩|2$

(EFi− E0)

3. (4.118)

When F is Hermitian (F = F †), Eqs. (4.117) and (4.118) respectively givem1 and m3, as defined in Eq. (4.114). When F is not Hermitian, as inEq. (4.140), only one among ⟨Fi|F |0⟩ and ⟨Fi|F †

i |0⟩ can be nonzero. Inthis case, Eqs. (4.117) and (4.118) give m1/2 and m3/2, respectively. Informing the ratio, the factor of 1/2 is canceled out, and thus, Eq. (4.113)still holds.

The advantage of the sum-rule approach is that no information concern-ing the excited states is required to find the excitation energies. In par-ticular, given a correct excitation operator F and the exact ground state,!ωupper gives the exact excitation energy.

4.3.1 Excitation operators

Consider an excitation operator

F (n, l, m) =N&

i=1

r2n+li Ylm(θi,φi) (4.119)

that excites a state characterized by radial quantum number n, angular-momentum quantum number l, and magnetic quantum number m, whereN is the number of atoms, and ri, θi, and φi are polar coordinates of thei-th atom, that is,

xi = ri sin θi cosφi, yi = ri sin θi sinφi, zi = ri cos θi. (4.120)

The spherical harmonic function Ylm(θi,φi) is given by

Ylm(θ,φ) = eimφP−ml (cos θ) = (−1)m (l − m)!

(l + m)!eimφPm

l (cos θ), (4.121)

where Pml is the associated Laguerre polynomial defined as

Pml (x) =

(1 − x2)m

2

2ll!

dl+m

dxl+m(x2 − 1)l. (4.122)






The excitation with n = 1 and l = m = 0 is called the monopole mode.In this case, Y00 = 1 and the corresponding excitation operator is givenfrom Eq. (4.119) as

F =∑

i

r2i =

∑

i

(

x2i + y2

i + z2i

)

. (4.123)

The excitations with n = 0 and l = 1 are called dipole modes that areclassified into three types depending on the value of m. In this case, thecorresponding excitation operator is given from Eq. (4.119) as

F =∑

i

riYlm(θi,φi) =

⎧

⎨

⎩

∑

i(xi + iyi) for m = 1,∑

i zi for m = 0,∑

i(xi − iyi) for m = −1.(4.124)

The excitations with n = 0 and l = 2 are called quadrupole modes that areclassified into five types depending on the value of m. The correspondingexcitation operators are given by

F =∑

i

r2i Y2m(θi,φi) =

⎧

⎨

⎩

∑

i (xi ± iyi)2 for m = ±2,

∑

i (xi ± iyi) zi for m = ±1,∑

i

(

x2i + y2

i − 2z2i

)

for m = 0.

(4.125)

4.3.2 Virial theorem

In the following discussions, we shall restrict ourselves to a situation inwhich N particles are confined in a harmonic potential and undergo contactinteractions described by a delta function. The Hamiltonian of our systemis then given by

H =∑

i

p2i

2m+

∑

i

m

2

(

ω2xx2

i + ω2yy2

i + ω2zz2

i

)

+U0

2

∑

i=j

δ(ri − rj). (4.126)

We assume that the state of our system is stationary. Then, the expectationvalue of

∑

i xipix is constant in time:

d

dt

⟨

∑

i

xipix

⟩

= 0, (4.127)

where pix is the x-component of the momentum of the i-th particle. Onthe other hand, Heisenberg’s equation of motion gives

d

dt

∑

i

xipix =i

!

[

H,∑

i

xipix

]

. (4.128)







d

dt

!

i

xipix = 2!

i

p2ix

2m− 2

!

i

mω2x

2x2

i − U0

!

i=j

xi∂δ(ri − rj)

∂xi. (4.129)

Hence, we have

2⟨Tx⟩ − 2⟨Ux⟩ − U0

"

!

i=j

xi∂δ(ri − rj)

∂xi

#

= 0, (4.130)

where Tx and Ux are the x-component of the kinetic energy and the po-tential energy, respectively. The last term in Eq. (4.130) may be rewrittenas

"

!

i=j

xi∂δ(ri − rj)

∂xi

#

=

$

dridrjxi∂δ(ri − rj)

∂xiψ2(ri)ψ

2(rj)

= −$

dri∂%

xiψ2(ri)&

∂xiψ2(ri)

= −$

dri

'

ψ4(ri) +xi

2

∂ψ4(ri)

∂xi

(

= −1

2

$

driψ4(ri)

= −1

2

"

!

i=j

δ(ri − rj))

. (4.131)

We thus obtain

2⟨Tx⟩ − 2⟨Ux⟩ + ⟨V ⟩ = 0, (4.132)

where

V =U0

2

!

i=j

δ(ri − rj). (4.133)

We can obtain equations similar to Eq. (4.132) for the y- and z-components.Summing up the x-, y-, z-components, we obtain

2⟨T ⟩ − 2⟨U⟩ + 3⟨V ⟩ = 0. (4.134)

The relations (4.132) and (4.134) are known as the virial theorem.

4.3.3 Kohn theorem

The collective-mode frequency of the dipole mode (n = 0, l = 1) is indepen-dent of interactions and equal to the frequencies of the trapping potential.This fact is known as the Kohn theorem. To show this, we consider somecases of the axisymmetric harmonic potential

U =!

i

m

2

*

ω2⊥

%

x2i + y2

i

&

+ ω2zz2

i

+

. (4.135)






4.3.3.1 Case of m = 0

The excitation operator with m = 0 is given by

F =!

i

zi. (4.136)

Straightforward calculations give"

F †,"

H, F##

=!2

mN, (4.137)

""

F †, H#

,"

H,"

H, F###

=!4ω2

z

mN. (4.138)

We note that the right-hand sides of Eqs. (4.137) and (4.138) are constants,and independent of the state of the system. Substituting Eqs. (4.137) and(4.138) in Eq. (4.113) gives

!ωupper = !ωz. (4.139)

4.3.3.2 Case of m = ±1

The excitation operators with m = ±1 are given by

F =!

i

(xi ± iyi). (4.140)

Straightforward calculations give"

F †,"

H, F##

=2!2

mN, (4.141)

""

F †, H#

,"

H,"

H, F###

=2!4ω2

⊥

mN. (4.142)

The right-hand sides of Eqs. (4.141) and (4.142) are again independent ofthe state of the system. Substituting Eqs. (4.141) and (4.142) in Eq. (4.113)gives

!ωupper = !ω⊥. (4.143)

Equations (4.139) and (4.143), in fact, give the exact frequencies of thedipole modes.

4.3.4 Isotropic trap

When the trap is isotropic, the Hamiltonian is given by

H =!

i

p2i

2m+

!

i

mω2

2r2

i +U0

2

!

i=j

δ(ri − rj). (4.144)

We discuss two important collective modes.






4.3.4.1 Monopole mode

The excitation operator of the monopole (or breathing) mode with n = 1and l = 0 is given by

F =!

i

r2i =

!

i

"

x2i + y2

i + z2i

#

. (4.145)

Calculating the commutation relations for m1 gives

$

F †,$

H, F%%

=4!2

m

!

i

r2i . (4.146)

Hence,

m1 =4!2

m2ω2⟨U⟩. (4.147)

Calculating the commutation relations for m3 is slightly complicated. Wefirst note that

$

F †, H%

=2i!

m

!

i

&

piri +3

2i!'

, (4.148)

$

H,$

H, F%%

= −4!2

m

&

!

i

p2i

2m−

!

i

mω2

2r2

i −U0

2

!

i=j

ri∂δ(ri − rj)

∂ri

'

.

(4.149)

Hence,

$$

F †, H%

,$

H,$

H, F%%%

=16!4

m2

(

!

i

p2i

2m+

!

i

mω2

2r2

i

+U0

4

!

i=j

(ri ·∇i + rj ·∇j) ri ·∇iδ (ri − rj)

)

. (4.150)

The expectation value of the last term can be evaluated as

A = ⟨!

i=j

(ri ·∇i + rj ·∇j) ri ·∇iδ (ri − rj)⟩

=

*

drdr′ψ2(r)ψ2(r′) (xi∂i + x′i∂

′i)xj∂jδ(r − r′),






where x1 = x, x2 = y, x3 = z, etc. Integration by parts gives

A = −!

drdr′"

ψ2(r′)#

∂i$

xiψ2(r)

%&

xj∂jδ(r − r′)

+ψ2(r)#

∂′i$

x′iψ

2(r′)%&

xj∂jδ(r − r′)'

=

!

dr"

ψ2∂jxj

#

∂i

$

xiψ2%&

+#

∂j

$

xjψ2%& #

∂i

$

xiψ2%&'

=

!

dr"

−$

∂jψ2%

xj

#

∂i

$

xiψ2%&

+#

(∂jxj)ψ2 + xj

$

∂jψ2%& #

∂i

$

xiψ2%&'

= 3

!

drψ2∂i

$

xiψ2%

= 3

!

dr#

ψ4 (∂ixi) + xiψ2$

∂iψ2%'

= 3

!

dr

(

3ψ4 +1

2xi∂iψ

4

)

=9

2

!

drψ4 =9

2

*

+

i=j

δ(ri − rj),

=9

U0⟨V ⟩.

Hence, we have

m3 =8!2

m2⟨T + U +

9

4V ⟩. (4.151)

We may use the virial theorem (4.134) to eliminate ⟨V ⟩ in Eq. (4.151), thusobtaining

m3 =4!2

m2⟨5U − T ⟩. (4.152)

Substituting Eqs. (4.146) and (4.152) in Eq. (4.113) gives

!ωupper = !ω

-

5 −⟨T ⟩⟨U⟩

. (4.153)

In the absence of interactions, the vitial theorem gives ⟨T ⟩ = ⟨U⟩, andhence, we have !ωupper = 2!ω. In the Thomas–Fermi limit, where ⟨T ⟩ = 0,we have !ωupper =

√5!ω.

4.3.4.2 Quadrupole mode

The excitations with n = 0 and l = 2 are called quadrupole modes. In thecase of an isotropic trap, the excitation frequency is independent of m. Itis therefore sufficient to consider the case of m = 0, where the excitationoperator is given by

F =+

i

$

x2i + y2

i − 2z2i

%

. (4.154)






Straightforward calculations give

m1 =8!2

m2ω2⟨U⟩, (4.155)

m3 =16!4

m2⟨T + U⟩. (4.156)

Hence, we obtain

!ωupper = !ω

!

2

"

1 +⟨T ⟩⟨U⟩

#

. (4.157)

In the absence of interactions, !ωupper =√

3!ω, while in the Thomas–Fermilimit, !ωupper =

√2ω.

4.3.5 Axisymmetric trap

When the trap is axisymmetric, the Hamiltonian is given by

H =$

i

p2i

2m+$

i

m

2

%

ω2⊥

&

x2i + y2

i

'

+ω2zz2

i

(

+U0

2

$

i=j

δ (ri − rj) . (4.158)

In this case, the frequency of the quadrupole mode depends on the value ofm. When n = 0, l = 2, and m = ±2, the excitation operators are given by

F =$

i

(xi ± iyi)2 . (4.159)

In this case,

m1 =4!2

m

)

$

i

&

x2i + y2

i

'

*

≡16!2

m2ω2

)

U⊥

*

, (4.160)

m3 =16!4

m2

)

$

i

p2ix + p2

iy

2m+

$

i

mω2

2

&

x2i + y2

i

'

*

≡32!2

m2⟨T⊥ + U⊥⟩.

(4.161)

Hence, we have

!ωupper = !ω⊥

!

2

"

1 +⟨T⊥⟩⟨U⊥⟩

#

. (4.162)

The mode with m = 0 is called the radial breathing mode. When ω⊥ = ωz,the angular momentum is no longer a good quantum number, and the mode

with n = 1, l = m = 0+

F =,

i r2i

-

couples with the mode with n = 0,






l = 2, m = 0!

F ="

i

#

x2i + y2

i − 2z2i

$

%

. The excitation operator for the

coupled mode is, in general, given by

F =&

i

#

x2i + y2

i − αz2i

$

, (4.163)

where α is a variational parameter to be determined later. Calculations ofthe commutation relations give

m1 =4!2

m2ω2⊥

'

2⟨U⊥⟩ +α2

λ2⟨Uz⟩

(

, (4.164)

m3 =8!4

m2

'

2(⟨T⊥⟩+⟨U⊥⟩)+α2(⟨T⊥⟩+⟨Uz⟩)+!

1 −α

2

%2⟨V ⟩

(

, (4.165)

where λ ≡ ωz/ω⊥. Hence, we have

!ωupper =√

2!ω⊥

)

2(⟨T⊥⟩ + ⟨U⊥⟩) + α2(⟨Tz⟩ + ⟨Uz⟩) +#

1 − α2

$2 ⟨V ⟩2⟨U⊥⟩ + α2

λ2 ⟨Uz⟩

*

1

2

.

(4.166)In the Thomas–Fermi limit, we have ⟨T⊥⟩ = ⟨Tz⟩ = 0. It can also be shownthat ⟨U⊥⟩ = ⟨Uz⟩. On the other hand, according to the virial theorem, weobtain ⟨V ⟩ = 2

3 ⟨U⟩ = 2⟨U⊥⟩. Hence, we have

!ωupper = !ωz

+

3α2 − 4α+ 8

α2 + 2λ2. (4.167)

Minimizing this with respect to α gives

ωupper = ω⊥

)

2 +3

2λ2 ±

+

9

4λ4 − 4λ2 + 4

*1

2

. (4.168)

Fundamentals and New Frontiers Bose-Einstein Condensation

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