Dynamical Variational Approach for Trapped Gases at Finite Temperature

We analyze the static TDHFB equations in the Thomas-Fermi limit for a gas of bosons in a harmonic trap. These equations
naturally generalize the Gross-Pitaevskii equation.
We first build a simple enough method that allows for the determination of the various density profiles. At zero temperature,
we obtain familiar expressions for the chemical potential and the condensate radius. The standard Thomas-Fermi profile for
the condensate density is also recovered. For finite temperatures and above the transition, we derive analytical expressions for
the condensate radius, the chemical potential, the number of condensed atoms and the depletion as functions of the
temperature. We observe that the condensate radius and the column density are surprisingly very slow functions of the
temperature. Furthermore, the non-condensed density, although being quite uniform inside the trap, grows rapidly with the
temperature. These facts imply therefore that the condensed atoms are very strongly attached and exhibit a certain robustness
against ’’decondensation’’. Moreover, the transition to the non condensed phase seems to be much more controlled by the
thermal cloud which rapidly grows from the borders toward the centre of the trap.

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