Fundamentals and New Frontiers of Bose Einstein Condensation
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1 Contents Preface v 1. Fundamentals of Bose Einstein Condensation Indistinguishability of Identical Particles Ideal Bose Gas in a Uniform System Off-Diagonal Long-Range Order: Bose System Off-Diagonal Long-Range Order: Fermi System U(1) Gauge Symmetry Ground-State Wave Function of a Bose System BECandSuperfluidity Two-Fluid Model Fragmented Condensate Two-state model Degenerate double-well model Spin-1 antiferromagnetic BEC Interference Between Independent Condensates FeshbachResonance Weakly Interacting Bose Gas Interactions Between Neutral Atoms Pseudo-PotentialMethod Bogoliubov Theory Bogoliubov transformations Bogoliubov ground state Low-lying excitations and condensate fraction Properties of Bogoliubov ground state xi
2 xii Fundamentals and New Frontiers of Bose Einstein Condensation 2.4 Bogoliubov Theory of Quasi-One-Dimensional Torus Case of BEC at rest: stability of BEC Case of rotating BEC: Landau criterion Ground state of BEC in rotating torus Bogoliubov de Gennes (BdG) Theory Method of Binary Collision Expansion Equationofstate Cluster expansion of partition function Ideal Bose and Fermi gases Matsubaraformula Trapped Systems Ideal Bose Gas in a Harmonic Potential Transition temperature Condensate fraction Chemical potential Specific heat BEC in One- and Two-Dimensional Parabolic Potentials Density of states Transition temperature Condensate fraction Semiclassical Distribution Function Gross Pitaevskii Equation Thomas FermiApproximation Collective Modes in the Thomas Fermi Regime Isotropic harmonic potential Axisymmetric trap Scissorsmode VariationalMethod Gaussian variational wave function Collective modes Attractive Bose Einstein Condensate Collective modes Collapsing dynamics of an attractive condensate Linear Response and Sum Rules Linear Response Theory Linear response of density fluctuations
3 Contents xiii Retarded response function Sum Rules Longitudinal f-sum rule Compressibility sum rule Zero energy gap theorem Josephson sum rule Sum-Rule Approach to Collective Modes Excitation operators Virial theorem Kohntheorem Isotropictrap Axisymmetric trap Statistical Mechanics of Superfluid Systems in a Moving Frame Transformation to Moving Frames Elementary Excitations of a Superfluid Landau Criterion Correlation Functions at Thermal Equilibrium Normal Fluid Density Low-Lying Excitations of a Superfluid Examples Ideal Bose gas Weakly interacting Bose gas Spinor Bose Einstein Condensate Internal Degrees of Freedom General Hamiltonian of Spinor Condensates Spin-1 BEC Mean-field theory of a spin-1 BEC Many-body states in single-mode approximation Superflow, spin texture, and Berry phase Spin-2 BEC Vortices Hydrodynamic Theory of Vortices Quantized Vortices Interaction Between Vortices VortexLattice
4 xiv Fundamentals and New Frontiers of Bose Einstein Condensation Dynamics of vortex nucleation Collective modes of a vortex lattice Fractional Vortices Spin Current Fast Rotating BECs Lowest Landau level approximation Mean field quantum Hall regime Many-body wave functions of a fast rotating BEC Fermionic Superfluidity Ideal Fermi Gas Fermi Liquid Theory Cooper Problem Two-body problem Many-body problem Bardeen Cooper Schrieffer (BCS) Theory BCS BEC Crossover at T = Superfluid Transition Temperature BCS BEC Crossover at T Gor kov Melik Barkhudarov Correction Unitary Gas Imbalanced Fermi Systems P-Wave Superfluid Generalized pairing theory Spin-triplet p-wavestates Low-Dimensional Systems Non-interacting Systems Hohenberg Mermin Wagner Theorem Two-Dimensional BEC at Absolute Zero Berezinskii Kosterlitz Thouless Transition Universaljump Quasi long-range order Renormalization-group analysis Quasi One-Dimensional BEC Tonks Girardeau Gas Lieb Liniger Model
5 Contents xv 10. Dipolar Gases Dipole Dipole Interaction Basic properties Order of magnitude and length scale D-wave nature Tuning the dipole dipole interaction Polarized Dipolar BEC Nonlocal Gross Pitaevskii equation Stability Thomas Fermi limit Quasi two-dimensional systems Spinor-Dipolar BEC Einstein de Haas effect Flux closure and ground-state circulation Optical Lattices Optical Potential Optical trap Optical lattice Band Structure Bloch theorem Brillouin zone Bloch oscillations Wannier function Bose Hubbard Model Bose Hubbard Hamiltonian Superfluid Mott-insulator transition Phase diagram Mean-field approximation Supersolid Topological Excitations Homotopy Theory Homotopic relation Fundamental group Higher homotopy groups Order Parameter Manifold Isotropy group
6 xvi Fundamentals and New Frontiers of Bose Einstein Condensation Spin-1 BEC Spin-2 BEC Classification of Defects Domains Linedefects Point defects Skyrmions Influence of different types of defects Topological charges Appendix A Order of Phase Transition, Clausius Clapeyron Formula, and Gibbs Duhem Relation 321 Appendix B Bogoliubov Wave Functions in Coordinate Space 323 B.1 Ground-State Wave Function B.2 One-Phonon State Appendix C Effective Mass, Sound Velocity, and Spin Susceptibility 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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