Università degli Studi di Trento

Facoltà di Scienze Matematiche Fisiche e Naturali

Tesi di Dottorato di Ricerca in Fisica

QUANTUM MONTE CARLO

STUDY

OF ULTRACOLD GASES

$$\begin{array}{p{6cm}p{3cm}p{6cm}} \bf Relatori:&&\bf Candida... ...khartchik}\\ {\bf\large Prof. Lev P. Pitaevskii}&& \end{array}$$

Dottorato di Ricerca in Fisica, XVII Ciclo

15 Dicembre 2004

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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
    • General approach
    • Scattering on a hard sphere potential
    • Scattering on a soft sphere potential
  • One-dimensional scattering problem
    • General approach
    • Scattering on a pseudopotential
    • Scattering on a 1D square well potential
    • Scattering on a hard-rod potential
  • Pseudopotential
    • The pseudopotential method
  • Resonance scattering
    • Scattering on a square-well potential
    • Scattering on a modified Pöschl-Teller potential
• 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
    • Tonks-Giradeau wave function
    • Hard-rod wave function (exact)
    • Hard-rod wave function (approximate)
    • Wave function of the Lieb Liniger gas
    • Phonon trial wave function ($\delta$-potential)
    • Super-Tonks trial wave function (attractive $\delta$-potential)
    • Scattering on the resonance state of a Bose gas
  • Three-dimensional wave functions
    • Hard sphere trial wave function
    • Soft sphere trial wave function
    • Trial wave function of 3D zero range potential
    • Scattering on the resonance state of a Bose gas
• 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
    • Matching to a constant
• Measured quantities
  • Local energy
    • Local kinetic energy and the drift force
    • Exponentiation
  • 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
• $N$ 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
    • Dragg force
    • Effective mass
  • One-dimensional system. Mean-field theory
    • Dragg force
    • Effective mass
    • Density profile
  • 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 $g_1(r)$

Acknowledgements

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x2html243" HREF="node155.html">Conclusions

Conclusions

Bibliography

Bethe ansatz solutions

• Lieb-Liniger equations
• Attractive Fermi gas
• Repulsive Fermi gas
• Numerical solution
• Expansions

Obtaining the momentum distribution from $g_1(r)$

Acknowledgements

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