Study of Ultracold Fermi Gases in the BCS-BEC Crossover: Quantum Monte Carlo Methods, Hydrodynamics and Local Density Approximation.

In this Thesis we will theoretically address some issues concerning the Physics of ultracold Fermi gases, all of them with experimental relevance. The field of ultracold gases, and more recently of ultracold Fermi gases, is gathering a lot of experimental and theoretical interest for two main reasons: the great control on the relevant parameters of the problem and the relative simplicity of the minimal theory which is able to correctly describe the system. Building upon this solid background, ultracold gases provided an ideal laboratory for testing more refined theories and also for addressing fundamental issues of quantum mechanics as well as for simulating more complex physical systems such as those encountered in condensed matter. Even if the effective hamiltonian of ultracold gases can be simple, due to the diluteness of the system and the low temperature, which imply low energy physics, the solution of the quantum mechanic equations governing the state of these systems is not always simple. For the static properties usually mean-field solutions exist or perturbative expansions can be produced in some regimes. However Quantum Monte Carlo (QMC) techniques provide more accurate results especially in the strongly interacting regimes. For confined systems it is possible to use QMC only for a few particles, so that, for large number of particles, a fruitful combined use of Density Functional Theory (DFT) and QMC is necessary. The study of the dynamics of ultracold gases has received little attention with QMC techniques, due to the intrinsic computational difficulty of the many-body problem, so that general hydrodynamic equations are often used for studying the propagation of smoothly varying perturbations. In this Thesis we use QMC techniques for studying the problem of ferromagnetism in repulsive or effectively repulsive ultracold gases without a lattice and the problem of the Bardeen-Cooper-Schrieffer to Bose-Einstein-Condensation (BCS-BEC) crossover in two dimensions. We use DFT in the Local Density Approximation (LDA) for calculating the density profiles of ultracold Fermi gases in harmonic magneto-optical traps, starting from QMC equations of state. We study the propagation of first and second sound in ultracold Fermi gases in cylindrical geometry, using the hydrodynamic equations of superfluids