Contents

• • Notation and abbreviations
  
  
• Introduction
• Tools
  • Introduction
  • Correlation functions and related quantities
    • Correlation functions: second quantization form
    • Correlation functions: first quantization form
    • Homogeneous system
    • Momentum distribution and static structure factor
    • Trapped system
  • The scattering problem
    • Introduction
    • Three-dimensional scattering problem
    • One-dimensional scattering problem
    • Pseudopotential
    • Resonance scattering
  • Energy of the TG and HR gas
    • Energy of the Tonks-Girardeau gas
    • Hard-rod gas
  • Gross Pitaevskii Equation
    • Variational derivation of the GPE
    • Coupling constant in quasi one- and two- dimensional systems
  • Local Density Approximation
    • General method
    • Exact solution for 1D ``perturbative'' equation of state
    • Exact solution for 3D ``perturbative'' equation of state
    • Static structure factor of a trapped Tonks-Girardeau gas
  • Correlation functions in a Luttinger liquid
    • Stationary density-density correlation function
    • Time-dependent density-density correlation function
    • Calculation with non-logarithmic accuracy
    • Dynamic form factor
    • Popov's coefficient
  
  
• Quantum Monte Carlo technique
  • Introduction
  • Variational Monte Carlo
    • Variational principle
    • Applications
    • Implementation
  • Diffusion Monte Carlo
    • Schrödinger equation
    • Green's function
    • Primitive algorithm
    • Higher-order algorithm
  • Fixed-node Diffusion Monte Carlo method
  • Construction of trial wave functions: system of Bosons
    • Introduction
    • Bijl-Jastrow wave function
    • One-body Bijl-Jastrow term in an anisotropic trap
    • One-dimensional wave functions
    • Three-dimensional wave functions
  • Construction of trial wave functions: system of Fermions
    • Trial wave function in the BCS limit
    • Kinetic energy
    • Calculation of the tail energy
    • Bijl-Jastrow term
    • Trial wave function: zero energy scattering state
  • Measured quantities
    • Local energy
    • Static structure factor
    • One body density matrix in a homogeneous system
    • One body density matrix in a harmonic trap
    • Pair distribution
    • Pure estimators and extrapolation technique
  
  
• 3D-1D crossover of a trapped Bose gas
  • Introduction
  • Theory
    • Model Hamiltonian
    • Relevant parameters and DMC approach
    • Mean-field approach
    • 1D system: local density approximation
    • 1D system: beyond local density approximation
  • Results
    • Small system, medium scattering length
    • Small system, small scattering length
    • Small system, large scattering length
    • Large system
    • Radial size of the system
  • Conclusions
  
  
• Quasi 1D Bose gases with large scattering length
  • Introduction
  • Two Bosons under quasi-one-dimensional confinement
  • bosons under quasi-one-dimensional confinement
  • Energetics of quasi-one-dimensional Bose gases
    • Two-body system
    • N-body system
  • Stability of quasi-one-dimensional Bose gases
  • Conclusions
  
  
• Ground state properties of a one-dimensional Bose gas
  • Introduction
  • Lieb-Liniger Hamiltonian
  • Quantum Monte Carlo Method
  • Homogeneous system
  • Trapped system
  • Conclusions
  
  
• Beyond Tonks-Girardeau: super-Tonks gas
  • Introduction
  • The model and method
  • Energy
  • One-body density matrix and static structure factor
  • Collective modes
  • Conclusions
  
  
• Motion of a heavy impurity through a Bose-Einstein condensate
  • Introduction
  • Three-dimensional system
    • Perturbed solution
    • Total energy
    • Effective mass and normal fraction
    • Drag force and energy dissipation
  • Low dimensional systems
    • Two-dimensional system
    • One-dimensional system. Mean-field theory
    • One-dimensional system. Bethe-ansatz theory
  • Conclusions
  
  
• Interacting fermions in highly elongated harmonic traps
  • Introduction
  • Model
  • Homogeneous system
  • Trapped system
  • Conclusions
  
  
• BEC-BCS crossover
  • Introduction
  • Model
  • Results
  • Conclusions
  
  
• Conclusions
• Bibliography
• Bethe ansatz solutions
  • Lieb-Liniger equations
  • Attractive Fermi gas
  • Repulsive Fermi gas
  • Numerical solution
  • Expansions
  
  
• Obtaining the momentum distribution from
• Acknowledgements
• About this document ...