Dynamics of ultra – cold gases

Abstract

The first part of this thesis devoted to establish a system of nonlinear differential equations describing the dynamics of the order parameter (the condensate wave function), the thermal cloud and the anomalous density, these equations often known as TDHFB (Time- Dependent-Hatree-Fock-Bogoliubov). The TDHFB equations are time-dependent variational equations derived using the Balian and Vénéroni principle. They are valid for any Hamiltonian and in any dimensions. We present the numerical solution of such equations in the static case in the presence of a spherical confining potential beyond the Thomas-Fermi approximation (taking into account the kinetic energy). We then examine the profiles of the different densities as function of the temperature and the interactions and compare them with the theoretical results existing in the literature, especially with those obtained by the approximation of HFB-BdG. In order to shed some light on the anomalous density, remind us that the presence of this quantity in a Bose system indicates that the atoms are correlated. We again use our theoretical approach and varying interactions and temperature. Our results are in good agreement with those of HFB-BdG in a system of large number of particles. For a gas of weak interactions, our results start to deviate from the HFB-BdG, where we observe the disappearance of the "dip". The study of the anomalous correlation function in two dimensional Bose gas was also among of our subjects in this thesis. We found that this quantity has a finite value in the limit of weak interactions at zero temperature. The effects of the anomalous density on some thermodynamic quantities such as the chemical potential (equation of state), the depletion and the superfluid fraction have been also highlighted. Our results show good agreements with Monte Carlo simulations and perturbative analytical methods. On other hand, by using the density-phase fluctuation, we found that the anomalous density does not exist at finite temperature in analogous manner with the true condensate. In the second part of this thesis we turn to discuss effects of the rotonization in two dimensional dilute dipolar Bose gas. We have calculated in particular the density-density correlation function, fluctuation corrections to the chemical potential, compressibility, and the normal (superfluid) fraction. We have shown that the presence of the roton strongly enhances fluctuations of the density, and we establish the validity criterion of the Bogoliubov approach. At zero temperature the condensate depletion becomes significant if the roton minimum is sufficiently close to zero. At finite temperatures in the dilute regime where ������������ ���� ���� 1, exceeding the roton energy and the interaction between particles, the effect of thermal fluctuations is stronger and it may lead to a large normal fraction of the gas which may significantly reduce the superfluid fraction even at temperatures well below the Kosterlitz-Thouless transition temperature.