Dynamics of ultra – cold gases
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Date
2013
Authors
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Journal ISSN
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Publisher
BENAROUS MOHAMED
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.
Description
Memoire de doctorat en physique / Option : physique
Keywords
Dynamics, ultra – cold