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Acta Physica Sinica

Numerical research on critical temperature of Bose-Einstein condensation for gas with interaction in harmonic trap

Chong Wang, Yan Ke-zhu, 曲阜师范大学物理系,曲阜 273165

Published Jan 1, 2004• 8 citations
Publisher

Abstract

In this article we study BoseEinstein condensation of the gas with interactions in a harmonic trap using a numerical method. The distribution of the energy states was given, by which the author calculated the shift of the critical temperature and made a comparison with the situation without interactions. We have obtained a positive shift of critical temperature for systems with repulsive interaction, and a negative one for systems with attractive interaction.

DOI
10.7498/aps.53.1284

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In this article we study BoseEinstein condensation of the gas with interactions in a harmonic trap using a numerical method. The distribution of the energy states was given, by which the author calculated the shift of the critical temperature and made a comparison with the situation without interactions. We have obtained a positive shift of critical temperature for systems with repulsive interaction, and a negative one for systems with attractive interaction.
By using a mean-field approach, based on the Popov approximation, we calculate the temperature dependence of the condensate fraction of an interacting Bose gas of N atoms confined in an anisotropic harmonic trap. For systems interacting with repulsive forces we find a significant decrease of the condensate fraction and of the critical temperature with respect to the predictions of the noninteracting model with the same value of N. An analytic result for the shift of the critical temperature holding to first order in the scattering length is also derived. \textcopyright{} 1996 The American Physical Society.
One of the most interesting properties of boson gases is that under special conditions, there is a possibility of a phase transition, in a critical temperature below which all bosons condensate into the ground state. This phenomenon is called Bose – Einstein Condensation (BEC). In this paper, we investigate BEC in a harmonic oscillator trap. We conclude that, in contrast to a free boson gas, there is no critical temperature for phase transition in a harmonic oscillator trap. However , by numerical and analytical calculation, it is possible to obtain a temperature at which the heat capacity is maximum. We call this the critical temperature . Possible explanation for all these features will be explained in this paper.
We consider a dilute gas of dipole moments in an arbitrary harmonic trap and treat both the short-range, isotropic $\ensuremath{\delta}$ interaction and the long-range, anisotropic dipole-dipole interaction perturbatively. In this way, we calculate the leading shift of the critical temperature with respect to that of an ideal gas as a function of the relative orientation of the dipole moments with respect to the harmonic trap axes. In particular, we determine those magic angles, where the dipolar shift of the Bose-Einstein condensation temperature vanishes. Furthermore, we show for the parameters of the ongoing 52Cr^{52}\mathrm{Cr}52Cr BEC experiment of Pfau and co-workers that this dipolar shift can be enhanced by increasing the number of particles, the geometrical mean trap frequency, and the anisotropy of the trap.
We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency ω{\omega}ω , the temperature T, and the particle number N are related by N∼(T/ω)3→∞{N \sim (T / \omega)^{3} \to\infty}N∼(T/ω)3→∞ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.
We discuss Bose-Einstein condensation in harmonic traps where the confinement has undergone a splitting along one direction. We mostly consider three-dimensional potentials consisting of two cylindrical wells separated a distance 2a2a2a along the z axis. For ideal gases, the thermodynamics of the confined bosons has been investigated, performing exact numerical summations to describe the major details of the transition and comparing the results with the semiclassical density-of-states approximation. We find that for large particle number and increasing well separation, the condensation temperature evolves from the thermodynamic limit value Tc(0)(N){T}_{c}^{(0)}(N)Tc(0)​(N) to Tc(0)(N/2).{T}_{c}^{(0)}(N/2).Tc(0)​(N/2). The effects of adding a repulsive interaction between atoms has been examined resorting to the Gross-Pitaevskii-Popov procedure, and it is found that the shift of the condensation temperature exhibits different signs according to the separation between wells. In particular, for sufficiently large splitting, the trend opposes the well-known results for harmonic traps, since the critical temperature appears to increase with growing repulsion strength.
We discuss certain specific features of the calculation of the critical temperature of a dilute repulsive Bose gas. Interactions modify the critical temperature in two different ways. First, for gases in traps, temperature shifts are introduced by a change of the density profile, arising itself from a modification of the equation of state of the gas (reduced compressibility); these shifts can be calculated simply within mean field theory. Second, even in the absence of a trapping potential (homogeneous gas in a box), temperature shifts are introduced by the interactions; they arise from the correlations introduced in the gas, and thus lie inherently beyond mean field theory – in fact, their evaluation requires more elaborate, non-perturbative, calculations. One illustration of this non-perturbative character is provided by the solution of self-consistent equations, which relate together non-linearly the various energy shifts of the single particle levels k . These equations predict that repulsive interactions shift the critical temperature (at constant density) by an amount which is positive, and simply proportional to the scattering length a ; nevertheless, the numerical coefficient is difficult to compute. Physically, the increase of the temperature can be interpreted in terms of the reduced density fluctuations introduced by the repulsive interactions, which facilitate the propagation of large exchange cycles across the sample.