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Introduction | |

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Ultracold Atomic Quantum Gases | |

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Outline | |

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Part 1 | |

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Part 2 | |

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Part 3 | |

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Gaussian Integrals | |

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The Gaussian Integral over Real Variables | |

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Generating Function | |

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Multi-Dimensional Gaussian Integral | |

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Complex Analysis | |

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Differentiation and Contour Integrals | |

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Laurent Series and the Residue Theorem | |

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Gaussian Integrals over Complex Variables | |

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Grassmann Variables | |

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Problems | |

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Quantum Mechanics | |

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Hilbert Spaces | |

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Observables | |

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Schrï¿½dinger vs. Heisenberg Picture | |

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Bosonic Harmonic Oscillator | |

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Creation and Annihilation Operators | |

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Three-Dimensional Harmonic Oscillator | |

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Coherent States | |

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Fermionic Harmonic Oscillator | |

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Spin | |

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Perturbation Theory | |

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Scattering Theory | |

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Many-particle Quantum Mechanics | |

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Problems | |

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Statistical physics | |

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Legendre Transformations | |

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Statistical Physics | |

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Spin Chain | |

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Canonical Ensemble | |

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Grand-Canonical Ensemble | |

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Ideal Gases | |

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Ideal Maxwell-Boltzmann Gas | |

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Ideal Bose Gas: Bose-Einstein Condensation | |

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Ideal Fermi Gas | |

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Density Matrix | |

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Problems | |

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Path Integrals | |

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Functionals and Functional Derivatives | |

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Principle of Least Action | |

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Phase-Space Representation | |

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The Feynman Path Integral | |

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Continuum Limit and Fluctuation Expansion | |

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Gel'fand-Yaglom Method | |

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Matrix Elements and Time Ordering | |

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Quantum-Mechanical Partition Function | |

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Expectation Values | |

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Hubbard-Stratonovich Transformation | |

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Problems | |

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Second Quantization | |

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Many-Body Hamiltonian | |

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Fock Space | |

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Creation and Annihilation Operators | |

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Second-Quantized Hamiltonian | |

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Field Operators | |

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Equivalence of First and Second Quantization | |

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Coherent States | |

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Problems | |

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Functional Integrals | |

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Grand-Canonical Partition Function | |

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Ideal Quantum Gases | |

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Semiclassical Method | |

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Matsubara Expansion | |

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Green's Function Method | |

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Wick's Theorem | |

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Problems | |

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Interactions and Feynman Diagrams | |

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Quasiparticles | |

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The Lehmann Representation | |

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The Spectral-Weight Function | |

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Collective Excitations | |

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Perturbation Theory | |

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Dyson's Equation | |

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Hartree-Fock Approximation | |

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Variational Approach | |

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Hubbard-Stratonovich Transformation | |

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Hartree Theory | |

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Fock Theory | |

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Hartree-Fock Theory for an Atomic Fermi Gas | |

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The Jellium Model | |

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Field-Theory Approach | |

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Effective Action | |

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Dispersion and Screened Coulomb Interaction | |

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Problems | |

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Landau Theory of Phase Transitions | |

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Ising Model in d Dimensions | |

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Landau Approach | |

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Hubbard-Stratonovich Transformation | |

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Gaussian Fluctuations | |

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Spontaneous Symmetry Breaking | |

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Problems | |

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Atomic Physics | |

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Atomic Structure | |

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Fine Structure | |

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Hyperfine Structure | |

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Zeeman Effect | |

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Two-body Scattering in Vacuum | |

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Two-Body Transition Matrix | |

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Partial-Wave Expansion | |

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Scattering from a Square-Well Potential | |

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Two-Body Scattering in a Medium | |

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Physical Regimes | |

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Problems | |

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Bose-Einstein Condensation | |

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Definitions for a Bose-Einstein Condensate | |

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Superfluidity | |

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Landau Criterion | |

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Superfluid Density | |

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Field-Theory Approach | |

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Bogoliubov Theory and the Gross-Pitaevskii Equation | |

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Dyson Equation | |

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Quasiparticle Dispersion | |

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Thermodynamic Potential for Bosons | |

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Bogoliubov-de Gennes Equation | |

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Popov Theory | |

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Hydrodynamic-Like Approach | |

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Time-Dependent Gross-Pitaevskii Equation | |

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Collective Modes | |

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Rotating Bose-Einstein Condensates | |

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Attractive Interactions | |

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Effective Action | |

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Breathing Mode | |

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Metastability of the Condensate | |

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Problems | |

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Condensation of Fermionic Pairs | |

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Introduction | |

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Thouless Criterion | |

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Hubbard-Stratonovich Transformation | |

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Bardeen-Cooper-Schrieffer Theory | |

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Critical Temperature | |

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Gap Equation | |

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Thermodynamic Potential for Fermions | |

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The BEC-BCS Crossover | |

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Theoretical Results | |

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Comparison with Experiment | |

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Problems | |

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Symmetries and Symmetry Breaking | |

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Effective Actions | |

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Noether's Theorem | |

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Ward Identities | |

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Hugenholtz-Pines Theorem | |

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Bragg Scattering | |

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RF Spectroscopy | |

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Second-Order Perturbation Theory | |

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Ladder Summations | |

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Absence of Clock Shift | |

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Absence of Vertex Corrections | |

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Phase Diffusion | |

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Problems | |

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Renormalization Group Theory | |

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The Renormalization-Group Transformation | |

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Scaling | |

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Interactions | |

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Quantum Effects | |

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Interactions | |

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Nonuniversal Quantities | |

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Renormalization Group for Fermions | |

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Renormalization-Group Equations | |

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Extremely-Imbalanced Case | |

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Homogeneous Phase Diagram | |

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Problems | |

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Low-Dimensional Systems | |

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Modified Popov Theory | |

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Phase Fluctuations | |

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Many-Body T Matrix | |

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Long-Wavelength Physics | |

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Comparison with Popo Theory | |

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One Dimension | |

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Two Dimensions | |

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Three Dimensions | |

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Vortices in Two Dimensions | |

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Kosterlitz-Thouless Phase Transition | |

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Trapped Bose Gases | |

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Density Profiles | |

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Phase Fluctuations | |

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Comparision with Exact Results | |

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Problems | |

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Optical Lattices | |

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Introduction | |

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Coupling between Atoms and Light | |

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Two-Level Approximation | |

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Fine Structure | |

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Band Structure | |

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Hubbard Models | |

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Superfluid-Mott Insulator Transition | |

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Bogoliubov Approximation | |

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Decoupling Approximation | |

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Hubbard-Stratonovich Transformation | |

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Fluctuations | |

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Mott Insulator | |

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Superfluid Phase | |

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Bragg Spectroscopy | |

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Problems | |

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Feshbach Resonances | |

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Example of a Feshbach Resonance | |

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Bare Atom-Molecule Theory | |

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Ladder Summations | |

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Effective Atom-Molecule theory | |

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Scattering Properties | |

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Bound-State Energy | |

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Molecular Density of States | |

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Bogoliubov Theory for the Bose-Einstein Condensed Phase | |

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Experiments | |

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Josephson Frequency | |

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Problems | |

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References | |

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Index | |