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Dynamical Variational Approach for Trapped Gases at Finite Temperature

Mohamed Benarous

Published Jun 30, 2011• 0 citations

Abstract

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 f or the determination of the various density profile s. At zero temperature, we obtain familiar expressions for the chemical pot ential and the condensate radius. The standard Thomas-Fermi profile for the condensate density is also recovered. For finit e 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 f unctions of the temperature. Furthermore, the non-condensed density, although being quite uniform inside the trap, gro ws rapidly with the temperature. These facts imply therefore that the c ondensed atoms are very strongly attached and exhib it a certain robustness against ''decondensation''. Moreover, the transitio n 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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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 f or the determination of the various density profile s. At zero temperature, we obtain familiar expressions for the chemical pot ential and the condensate radius. The standard Thomas-Fermi profile for the condensate density is also recovered. For finit e 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 f unctions of the temperature. Furthermore, the non-condensed density, although being quite uniform inside the trap, gro ws rapidly with the temperature. These facts imply therefore that the c ondensed atoms are very strongly attached and exhib it a certain robustness against ''decondensation''. Moreover, the transitio n 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.

We rely on a variational approach to derive a set of equations governing a trapped self-interacting Bose gas at finite temperature. In this work, we analyze the static situation both at zero and finite temperature in the Thomas-Fermi limit. We derive simple analytic expressions for the condensate properties at finite temperature. The noncondensate and anomalous density profiles are also analyzed in terms of the condensate fraction. The results are quite encouraging owing to the simplicity of the formalism.

Based on the Gross-Pitaevskii(G-P) energy functional and variational method, we present a new method to find the ground state wave function for Bose-condensed gas in a harmonic trap at zero temperature. With this method we are able to find the analytic expression of ground state wave function and explore the relevant quantities, such as energy, chemical potential and so on. These results agree well with previous ground state numerical solutions of the G-P equation given by Dalfovo et al. Under the large number of particles N limit for repulsive interaction, our results and Thomas-Fermi approximation model approach to reaching unanimity. This new method is simple compared to other methods used to solve numerically the G-P equation, and one can obtained the analytic and reliable results.

We derive a description of the spatially inhomogeneous Bose-Einstein condensate, treating the system locally as a homogeneous Bose gas. This approach, similar to the Thomas-Fermi model for the inhomogeneous many-particle fermion system, is well suited to describe the atomic Bose-Einstein condensates that have recently been obtained experimentally through atomic trapping and cooling. In this paper, we confine our attention to the zero-temperature case, although the treatment can be generalized to finite temperatures, as we shall discuss elsewhere. Several features of this approach, which we shall call the Thomas-Fermi-Bogoliubov description, are very attractive. (i) It is simpler than the Hartree-Fock-Bogoliubov technique. We can obtain analytical results in the case of weakly interacting bosons for quantities such as the chemical potential, the local depletion, pairing, pressure, and density of states. (ii) The method provides an estimate for the error due to the inhomogeneity of the Bose-condensed system. This error is a local quantity so that the validity of the description for a given trap and a given number of trapped atoms can be tested as a function of position. We see, for example, that at the edge of the condensate the Thomas-Fermi-Bogoliubov theory always breaks down. (iii) The Thomas-Fermi-Bogoliubov description can be generalized to treat the statistical mechanics of the Bose gas at finite temperatures.

We derive a unified set of scaling equations for Bose-Einstein condensates in time-dependent harmonic traps, connecting the weakly interacting Gaussian regime to the strongly interacting Thomas-Fermi regime. The exact Gross-Pitaevskii ground-state density is taken as a fixed profile in rescaled coordinates, with its dynamics described by three scaling factors corresponding to compression or expansion along the three spatial directions. The resulting equations contain no adjustable parameter, since their coefficients are determined once from the initial state. They recover analytically the Gaussian variational and Thomas-Fermi scaling equations in their respective limits and satisfy an axis-resolved virial theorem. The approach also yields the global phase and the low-lying collective-mode frequencies of the condensate. We benchmark the model against three-dimensional Gross-Pitaevskii simulations and against measured expansion energies. The model remains accurate across all interaction regimes, in strongly anisotropic traps, and along time-dependent sequences including a relatively fast quench, up to the point where the underlying frozen-profile hypothesis, shared by all three scaling approaches, breaks down.

In this paper, we present a theoretical study of a Bose-Einstein condensate of interacting bosons in a quartic trap in one, two, and three dimensions. Using Thomas-Fermi approximation, suitably complemented by numerical solutions of the Gross-Pitaevskii equation, we study the ground sate condensate density profiles, the chemical potential, the effects of cross-terms in the quartic potential, temporal evolution of various energy components of the condensate, and width oscillations of the condensate. Results obtained are compared with corresponding results for a bose condensate in a harmonic confinement.