Preface page ix 1 Overview and introduction 1 1.1 Historical overview of Bose superfluids 9 1.2 Summary of chapters 12 2 Condensate dynamics at T = 0 19 2.1 Gross-Pitaevskii (GP) equation 20 2.2 Bogoliubov equations for condensate fluctuations 28 3 Coupled equations for the condensate and thermal cloud 32 3.1 Generalized GP equation for the condensate 33 3.2 Boltzmann equation for the noncondensate atoms 39 3.3 Solutions in thermal equilibrium 43 3.4 Region of validity of the ZNG equations 46 4 Green's functions and self-energy approximations 54 4.1 Overview of Green's function approach 54 4.2 Nonequilibrium Green's functions in normal systems 58 4.3 Green's functions in a Bose-condensed gas 68 4.4 Classification of self-energy approximations 74 4.5 Dielectric formalism 79 5 The Beliaev and the time-dependent HFB approximations 81 5.1 Hartree-Fock-Bogoliubov self-energies 82 5.2 Beliaev self-energy approximation 87 5.3 Beliaev as time-dependent HFB 92 5.4 Density response in the Beliaev-Popov approximation 98 6 Kadanoff-Baym derivation of the ZNG equations 107 6.1 Kadanoff-Baym formalism for Bose superfluids 108 6.2 Hartree-Fock-Bogoliubov equations 111 6.3 Derivation of a kinetic equation with collisions 115 6.4 Collision integrals in the Hartree-Fock approximation 119 6.5 Generalized GP equation 122 6.6 Linearized collision integrals in collisionless theories 124 7 Kinetic equation for Bogoliubov thermal excitations 129 7.1 Generalized kinetic equation 130 7.2 Kinetic equation in the Bogoliubov-Popov approximation 135 7.3 Comments on improved theory 143 8 Static thermal cloud approximation 146 8.1 Condensate collective modes at finite temperatures 147 8.2 Phenomenological GP equations with dissipation 157 8.3 Relation to Pitaevskii's theory of superfluid relaxation 160 9 Vortices and vortex lattices at finite temperatures 164 9.1 Rotating frames of reference: classical treatment 165 9.2 Rotating frames of reference: quantum treatment 170 9.3 Transformation of the kinetic equation 174 9.4 Zaremba-Nikuni-Griffin equations in a rotating frame 176 9.5 Stationary states 179 9.6 Stationary vortex states at zero temperature 181 9.7 Equilibrium vortex state at finite temperatures 184 9.8 Nonequilibrium vortex states 187 10 Dynamics at finite temperatures using the moment method 198 10.1 Bose gas above TBEC 199 10.2 Scissors oscillations in a two-component superfluid 204 10.3 The moment of inertia and superfluid response 220 11 Numerical simulation of the ZNG equations 227 11.1 The generalized Gross-Pitaevskii equation 228 11.2 Collisionless particle evolution 231 11.3 Collisions 237 11.4 Self-consistent equilibrium properties 248 11.5 Equilibrium collision rates 252 12 Simulation of collective modes at finite temperature 256 12.1 Equilibration 257 12.2 Dipole oscillations 260 12.3 Radial breathing mode 263 12.4 Scissors mode oscillations 270 12.5 Quadrupole collective modes 279 12.6 Transverse breathing mode 286 13 Landau damping in trapped Bose-condensed gases 292 13.1 Landau damping in a uniform Bose gas 293 13.2 Landau damping in a trapped Bose gas 298 13.3 Numerical results for Landau damping 303 14 Landau's theory of superfluidity 309 14.1 History of two-fluid equations 309 14.2 First and second sound 312 14.3 Dynamic structure factor in the two-fluid region 317 15 Two-fluid hydrodynamics in a dilute Bose gas 322 15.1 Equations of motion for local equilibrium 324 15.2 Equivalence to the Landau two-fluid equations 331 15.3 First and second sound in a Bose-condensed gas 339 15.4 Hydrodynamic modes in a trapped normal Bose gas 345 16 Variational formulation of the Landau two-fluid equations 349 16.1 Zilsel's variational formulation 350 16.2 The action integral for two-fluid hydrodynamics 356 16.3 Hydrodynamic modes in a trapped gas 359 16.4 Two-fluid modes in the BCS-BEC crossover at unitarity 370 17 The Landau-Khalatnikov two-fluid equations 371 17.1 The Chapman-Enskog solution of the kinetic equation 372 17.2 Deviation from local equilibrium 377 17.3 Equivalence to Landau-Khalatnikov two-fluid equations 387 17.4 The C12 collisions and the second viscosity coefficients 392 18 Transport coefficients and relaxation times 395 18.1 Transport coefficients in trapped Bose gases 396 18.2 Relaxation times for the approach to local equilibrium 405 18.3 Kinetic equations versus Kubo formulas 412 19 General theory of damping of hydrodynamic modes 414 19.1 Review of coupled equations for hydrodynamic modes 415 19.2 Normal mode frequencies 418 19.3 General expression for damping of hydrodynamic modes 420 19.4 Hydrodynamic damping in a normal Bose gas 424 19.5 Hydrodynamic damping in a superfluid Bose gas 428 Appendix A Monte Carlo calculation of collision rates 431 Appendix B Evaluation of transport coefficients: technical details 436 Appendix C Frequency-dependent transport coefficients 444 Appendix D Derivation of hydrodynamic damping formula 448 References 451 Index 459
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