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arXiv:cond-mat/0701668 · Physical Review A

Bose-Einstein condensate in a harmonic trap decorated with Dirac δ functions

Haydar Uncu, Devrim Tarhan, Ersan Demiralp, Özgür Esat Müstecaplıoğlu

Published Jul 18, 2007• 37 citations
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Abstract

We study Bose-Einstein condensation in a harmonic trap with a dimple potential. We specifically consider the case of a tight and deep dimple potential which is modeled by a Dirac $\ensuremath{\delta}$ function. This allows for simpler, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schr\"odinger equation is used instead of the Gross-Pitaevski equation. Calculating the atomic density, chemical potential, critical temperature, and condensate fraction, the role of the relative depth of the dimple potential with respect to the harmonic trap in large condensate formation at enhanced temperatures is clearly revealed.

DOI
10.1103/physreva.76.013618

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We study Bose-Einstein condensation in a harmonic trap with a dimple potential. We specifically consider the case of a tight and deep dimple potential which is modeled by a Dirac \ensuremathδ\ensuremath{\delta}\ensuremathδ function. This allows for simpler, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schr"odinger equation is used instead of the Gross-Pitaevski equation. Calculating the atomic density, chemical potential, critical temperature, and condensate fraction, the role of the relative depth of the dimple potential with respect to the harmonic trap in large condensate formation at enhanced temperatures is clearly revealed.
We study Bose—Einstein condensation in a linear trap with a dimple potential where we model dimple potentials by Dirac δ function. Attractive and repulsive dimple potentials are taken into account. This model allows simple, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schrödinger equation is used instead of the Gross—Pitaevski equation. We calculate the atomic density, the chemical potential, the critical temperature and the condensate fraction. The role of the relative depth of the dimple potential with respect to the linear trap in large condensate formation at enhanced temperatures is clearly revealed. Moreover, we also present a semi-classical method for calculating various quantities such as entropy analytically. Moreover, we compare the results of this paper with the results of a previous paper in which the harmonic trap with a dimple potential in 1D is investigated.
We investigate Bose-Einstein condensation of noninteracting gases in a harmonic trap with an offcenter dimple potential. We specifically consider the case of a tight and deep dimple potential, which is modeled by a point interaction. This point interaction is represented by a Dirac delta function. The atomic density, chemical potential, critical temperature and condensate fraction, and the role of the relative depth and the position of the dimple potential are analyzed by performing numerical calculations.
We investigate Bose-Einstein condensation of noninteracting gases in a harmonic trap with an off-center dimple potential. We specifically consider the case of a tight and deep dimple potential which is modelled by a point interaction. This point interaction is represented by a Dirac delta function. The atomic density, chemical potential, critical temperature and condensate fraction, the role of the relative depth and the position of the dimple potential are analyzed by performing numerical calculations.
We study the truncated parabolic function and demonstrate that it is a representation of the Dirac δ function. We also show that the truncated parabolic function, used as a potential in the Schrödinger equation, has the same bound state spectrum, tunneling and reflection amplitudes as the Dirac δ potential, as the width of the parabola approximates to zero. Dirac δ potential is used to model dimple potentials which are utilized to increase the phase-space density of a Bose–Einstein condensate in a harmonic trap. We show that a harmonic trap with a δ function at the origin is a limiting case of the harmonic trap with a symmetric truncated parabolic potential around the origin. Hence, the truncated parabolic is a better candidate for modeling the dimple potentials.
We study the formation of a Bose-Einstein condensate in a cigar-shaped three-dimensional harmonic trap, induced by the controlled addition of an attractive ``dimple'' potential along the weak axis. In this manner we are able to induce condensation without cooling due to a localized increase in the phase-space density. We perform a quantitative analysis of the thermodynamic transformation in both the sudden and adiabatic regimes for a range of dimple widths and depths. We find good agreement with equilibrium calculations based on self-consistent semiclassical Hartree-Fock theory describing the condensate and thermal cloud. We observe that there is an optimal dimple depth that results in a maximum in the condensate fraction. We also study the nonequilibrium dynamics of condensate formation in the sudden turn-on regime, finding good agreement for the observed time dependence of the condensate fraction with calculations based on quantum kinetic theory.
Abstract In two-dimensional traps, since the theoretical study of Bose–Einstein condensation will encounter the problem of divergence, the actual contribution of the divergent terms is often estimated in some indirect ways with the accuracy to the leading order. In this paper, by using an analytical continuation method to solve the divergence problem, we obtain the analytical expressions of critical temperature and condensate fraction for Bose gases in a 2D anisotropic box and harmonic trap, respectively. They are consistent with or better than previous studies. Then, we further consider the nonvanishing chemical potential, and obtain the expressions of chemical potential and more precise condensate fraction. These results agree with the numerical calculation well, especially for the case of harmonic traps. The comparison between the grand canonical and canonical ensembles shows that our calculation in the grand canonical ensemble is reliable.