Bose-Einstein Condensation and Superfluidity

567 Pages·2016·5.86 MB·English

by  Pitaevski, Lev, Stringari, Sandro

INTERNATIONAL SERIES OF MONOGRAPHS
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International Series of Monographs on Physics
165. T.C.Choy:Effectivemediumtheory,Secondedition
164. L.Pitaevskii,S.Stringari:Bose–Einsteincondensationandsuperfluidity
163. B.J.Dalton,J.Jeffers,S.M.Barnett:Phasespacemethods fordegeneratequantumgases
162. W.D.McComb:Homogeneous,isotropicturbulence–phenomenology,renormalizationandstatistical
closures
161. V.Z.Kresin,H.Morawitz,S.A.Wolf:Superconducting state–mechanismsandproperties
160. C.Barrab`es,P.A.Hogan:Advanced generalrelativity –gravitywaves,spinningparticles, andblack
holes
159. W.Barford:Electronicandopticalproperties ofconjugatedpolymers,Secondedition
158. F.Strocchi:Anintroduction tonon-perturbativefoundationsofquantumfieldtheory
157. K.H.Bennemann,J.B.Ketterson:Novelsuperfluids,Volume2
156. K.H.Bennemann,J.B.Ketterson:Novelsuperfluids,Volume1
155. C.Kiefer:Quantumgravity,Thirdedition
154. L.Mestel:Stellarmagnetism,Secondedition
153. R.A.Klemm:Layeredsuperconductors, Volume1
152. E.L.Wolf:Principlesofelectrontunnelingspectroscopy, Secondedition
151. R.Blinc:Advancedferroelectricity
150. L. Berthier, G. Biroli, J.-P. Bouchaud, W. van Saarloos, L. Cipelletti: Dynamical heterogeneities in
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149. J.Wesson:Tokamaks,Fourthedition
148. H.Asada,T.Futamase,P.Hogan:Equationsofmotioningeneralrelativity
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146. B.McCoy:Advancedstatisticalmechanics
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141. W.T.Grandy:Entropyandthetimeevolutionofmacroscopic systems
140. M.Alcubierre:Introductionto3+1numericalrelativity
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138. I.M.Vardavas,F.W.Taylor:Radiation andclimate
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129. W.Barford:Electronicandopticalproperties ofconjugatedpolymers
128. R.E.Raab,O.L.deLange:Multipoletheoryinelectromagnetism
127. A.Larkin,A.Varlamov:Theoryoffluctuationsinsuperconductors
126. P.Goldbart,N.Goldenfeld,D.Sherrington:Stealingthegold
125. S.Atzeni,J.Meyer-ter-Vehn:Thephysicsofinertialfusion
123. T.Fujimoto:Plasmaspectroscopy
122. K.Fujikawa,H.Suzuki:Pathintegralsandquantumanomalies
121. T.Giamarchi:Quantumphysicsinonedimension
120. M.Warner,E.Terentjev:Liquidcrystalelastomers
119. L.Jacak,P.Sitko,K.Wieczorek,A.Wojs:QuantumHallsystems
117. G.Volovik:TheUniverseinaheliumdroplet
116. L.Pitaevskii,S.Stringari:Bose–Einsteincondensation
115. G.Dissertori,I.G.Knowles,M.Schmelling:Quantumchromodynamics
114. B.DeWitt:Theglobalapproachtoquantumfieldtheory
113. J.Zinn-Justin:Quantumfieldtheoryandcriticalphenomena,Fourthedition
112. R.M.Mazo:Brownianmotion–fluctuations,dynamics,andapplications
111. H.Nishimori:Statisticalphysicsofspinglassesandinformation processing–anintroduction
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104. D.Bardin,G.Passarino:TheStandardModelinthemaking
103. G.C.Branco,L.Lavoura,J.P.Silva:CPViolation
102. T.C.Choy:Effectivemediumtheory
101. H.Araki:Mathematical theoryofquantumfields
100. L.M.Pismen:Vorticesinnonlinearfields
Bose–Einstein Condensation
and Superfluidity
Lev Pitaevskii
Department of Physics, University of Trento and National Institute of Optics,
INO-CNR, Italy
Kapitza Institute for Physical Problems, RAS, Moscow, Russia
Sandro Stringari
Department of Physics, University of Trento and National Institute of Optics,
INO-CNR, Italy
3
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Preface
This is an extended and updated version of the book Bose–Einstein Condensation,
publishedbyOxfordUniversityPressin2003.Itswritingwasstimulatedbycontinuous
and exciting developments in the field of ultracold atoms, which started after the
first experimental realization of Bose–Einstein Condensation (BEC) in atomic gases,
achieved in 1995, and are still involving several hundreds of scientists around the
world. While the basic theory of BEC had been developed before 1995, its application
to the new configurations realized with magnetic and optical trapping is more recent
andhasrevealednumerousunexpectedfeatures,stimulatingfurtherexperimentaland
theoretical work.
In the first part of this volume we focus on the key theoretical concepts under-
lying the physics of Bose–Einstein condensation, in connection with the fundamental
developmentsinthetheorywhichtookplacebefore1995.Inthesecondpartthemain
emphasisisinsteadgiventotheconsequencesofBEConatomicBosegasescooledand
confined in traps. These systems are highly inhomogeneous and consequently exhibit
novel features which are presently the object of intense research activity. The third
part of the volume deals with the physics of ultracold Fermi gases, whose experimen-
tal investigation has shown a terrific development in the last ten years, giving a new
insightinmanyproblemsofcondensedmatterphysicswithdeepconnectionswiththe
physics of Bose–Einstein condensation and superfluidity. The last part covers topics
ofjointinterestforthestudyofBoseandFermigases,likethenewphenomenaexhib-
ited in optical traps and in low dimensional configurations, the properties of quantum
mixtures, and the consequences of long-range dipolar interactions.
In writing this book we have profited from stimulating discussions and collabor-
ations with many colleagues around the world. It is a special pleasure to thank the
friends of the Trento group and, in particular, Iacopo Carusotto, Franco Dalfovo,
Gabriele Ferrari, Stefano Giorgini, Giacomo Lamporesi, Chiara Menotti, and Alessio
Recati. We are also grateful to Giovanni Martone for his help in the final preparation
of the book.
Trento L.P.
July, 2015 S.S.

Contents
1 Introduction 1
PART I
2 Long-range Order, Symmetry Breaking, and Order Parameter 9
2.1 One-body density matrix and long-range order 9
2.2 Order parameter 13
3 The Ideal Bose Gas 15
3.1 The ideal Bose gas in the grand canonical ensemble 15
3.2 The ideal Bose gas in the box 19
3.3 Fluctuations and two-body density 26
4 Weakly Interacting Bose Gas 29
4.1 Lowest-order approximation: ground state energy
and equation of state 29
4.2 Higher-order approximation: excitation spectrum
and quantum fluctuations 33
4.3 Particles and elementary excitations 37
5 Nonuniform Bose Gases at Zero Temperature 42
5.1 The Gross–Pitaevskii equation 42
5.2 Thomas–Fermi limit 47
5.3 Vortex line in the weakly interacting Bose gas 48
5.4 Vortex rings 51
5.5 Solitons 55
5.6 Small-amplitude oscillations 60
6 Superfluidity 65
6.1 Landau’s criterion of superfluidity 65
6.2 Bose–Einstein condensation and superfluidity 69
6.3 Hydrodynamic theory of superfluids: zero temperature 70
6.4 Quantum hydrodynamics 71
6.5 Beliaev decay of phonons 74
6.6 Two-fluid hydrodynamics: first and second sound 76
6.7 Fluctuations of the phase 81
6.8 Rotation of superfluids 85
7 Linear Response Function 89
7.1 Dynamic structure factor and sum rules 89
7.2 Density response function 94
viii Contents
7.3 Current response function 98
7.4 General inequalities 100
7.5 Response function of the ideal Bose gas 105
7.6 Response function of the weakly interacting Bose gas 107
8 Superfluid 4He 110
8.1 Elementary excitations and dynamic structure factor 110
8.2 Thermodynamic properties 118
8.3 Quantized vortices 121
8.4 Momentum distribution and Bose–Einstein condensation 125
9 Atomic Gases: Collisions and Trapping 130
9.1 Metastability and the role of collisions 130
9.2 Low-energy collisions and scattering length 132
9.3 Low-energy collisions in two dimensions 141
9.4 Zeeman effect and magnetic trapping 143
9.5 Interaction with the radiation field and optical traps 148
PART II
10 The Ideal Bose Gas in the Harmonic Trap 153
10.1 Condensate fraction and critical temperature 153
10.2 Density of single-particle states and thermodynamics 156
10.3 Density and momentum distribution 158
10.4 Thermodynamic limit 161
10.5 Release of the trap and expansion of the gas 161
10.6 Bose–Einstein condensation in deformed traps 163
10.7 Adiabatic formation of BEC with non-harmonic traps 164
11 Ground State of a Trapped Condensate 168
11.1 An instructive example: the box potential 168
11.2 Interacting condensates in harmonic traps: density
and momentum distribution 170
11.3 Energy, chemical potential, and virial theorem 173
11.4 Finite-size corrections to the Thomas–Fermi limit 175
11.5 Beyond-mean-field corrections 180
11.6 Attractive forces 182
12 Dynamics of a Trapped Condensate 184
12.1 Collective oscillations 184
12.2 Repulsive forces and the Thomas–Fermi limit 187
12.3 Sum rule approach: from repulsive to attractive forces 193
12.4 Finite-size corrections to the Thomas–Fermi limit 196
12.5 Beyond-mean-field corrections 196
12.6 Large-amplitude oscillations 198
12.7 Expansion of the condensate 200
12.8 Dynamic structure factor 202
12.9 Collective versus single-particle excitations 213
Contents ix
13 Thermodynamics of a Trapped Bose Gas 217
13.1 Role of interactions, scaling, and thermodynamic limit 217
13.2 The Hartree–Fock approximation 220
13.3 Shift of the critical temperature 223
13.4 Critical region near T 226
c
13.5 Below T 228
c
13.6 Equation of state and density profiles 233
13.7 Collective oscillations at a finite temperature 236
14 Superfluidity and Rotation of a Trapped Bose Gas 238
14.1 Critical velocity of a superfluid 238
14.2 Moment of inertia 241
14.3 Scissors mode 245
14.4 Expanding a rotating condensate 247
14.5 Rotation at higher angular velocities 248
14.6 Quantized vortices 252
14.7 Vortices, angular momentum, and collective oscillations 259
14.8 Stability and precession of the vortex line 266
14.9 Quantized vortices and critical velocity in a toroidal trap 269
15 Coherence, Interference, and the Josephson Effect 272
15.1 Coherence and the one-body density matrix 273
15.2 Interference between two condensates 276
15.3 Double-well potential and the Josephson effect 284
15.4 Quantization of the Josephson equations 290
15.5 Decoherence and phase spreading 296
15.6 Boson Hubbard Hamiltonian 297
PART III
16 Interacting Fermi Gases and the BCS–BEC Crossover 305
16.1 The ideal Fermi gas 305
16.2 Dilute interacting Fermi gases 307
16.3 The weakly repulsive Fermi gas 307
16.4 Gas of composite bosons 309
16.5 The BCS limit of a weakly attractive gas 311
16.6 Gas at unitarity 313
16.7 The BCS–BEC crossover 319
16.8 The Bogoliubov–de Gennes approach to the BCS–BEC crossover 323
16.9 Equation of state, momentum distribution, and condensate
fraction of pairs 328
17 Fermi Gas in the Harmonic Trap 333
17.1 The harmonically trapped ideal Fermi gas 333
17.2 Equation of state and density profiles 336
17.3 Momentum distribution 339