INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS SERIES EDITORS J.BIRMAN CityUniversityofNewYork S.F.EDWARDS UniversityofCambridge R.FRIEND UniversityofCambridge M.REES UniversityofCambridge D.SHERRINGTON UniversityofOxford G.VENEZIANO CERN,Geneva 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 glasses,colloids,andgranularmedia 149. J.Wesson:Tokamaks,Fourthedition 148. H.Asada,T.Futamase,P.Hogan:Equationsofmotioningeneralrelativity 147. A.Yaouanc,P.DalmasdeR´eotier:Muonspinrotation,relaxation,andresonance 146. B.McCoy:Advancedstatisticalmechanics 145. M.Bordag,G.L.Klimchitskaya,U.Mohideen,V.M.Mostepanenko:AdvancesintheCasimireffect 144. T.R.Field:Electromagneticscatteringfromrandommedia 143. W.G¨otze:Complexdynamicsofglass-formingliquids–amode-couplingtheory 142. V.M.Agranovich:Excitationsinorganicsolids 141. W.T.Grandy:Entropyandthetimeevolutionofmacroscopic systems 140. M.Alcubierre:Introductionto3+1numericalrelativity 139. A.L.Ivanov,S.G.Tikhodeev:Problemsofcondensedmatterphysics–quantumcoherencephenomena inelectron-holeandcoupledmatter-lightsystems 138. I.M.Vardavas,F.W.Taylor:Radiation andclimate 137. A.F.Borghesani:Ionsandelectronsinliquidhelium 135. V.Fortov,I.Iakubov,A.Khrapak:Physicsofstronglycoupledplasma 134. G.Fredrickson:Theequilibriumtheoryofinhomogeneous polymers 133. H.Suhl:Relaxationprocessesinmicromagnetics 132. J.Terning:Modernsupersymmetry 131. M.Marin˜o:Chern-Simonstheory,matrixmodels,andtopological strings 130. V.Gantmakher:Electronsanddisorderinsolids 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 110. N.B.Kopnin:Theoryofnonequilibriumsuperconductivity 109. A.Aharoni:Introduction tothetheoryofferromagnetism, Secondedition 108. R.Dobbs:Heliumthree 107. R.Wigmans:Calorimetry 106. J.Ku¨bler:Theoryofitinerantelectronmagnetism 105. Y.Kuramoto,Y.Kitaoka:Dynamicsofheavyelectrons 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 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries (cid:2)c LevPitaevskiiandSandroStringari2016 Themoralrightsoftheauthorshavebeenasserted FirstEditionpublishedin2016 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2015947456 ISBN978–0–19–875888–4 Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY 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