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Quantum Electronics

Temperature dependence of condensate fraction of weakly nonideal trapped Bose gas

В. А. Алексеев, D. D. Krylova

Published May 31, 2000• 3 citations
Publisher

Abstract

Condensation of the weakly nonideal gas in an anisotropic parabolic trap was considered. The first order (with respect to the interaction parameter) correction Δ 0 = 0 ( T )— 0 0 ( T ) to the temperature dependence of the ground state population 0 0 ( T ) of the ideal gas atoms was found using thermodynamic perturbation theory. The correction found 0 proved to be much larger than the one calculated within the mean field theory [see Giorgini S, Pitaevskii L P, Stringari S Phys. Rev. A 54 R4633 (1996)] at temperatures below critical. This correction tends to zero as the temperature approaches the critical one; i.e., the interaction of the atoms does not lead to variation in the critical temperature. A simple analytic expression was found that describes the smooth variation of 0 0 ( T ) as it approaches zero in the vicinity of the critical temperature. A new criterion for applicability of the thermodynamic limit is given.

DOI
10.1070/qe2000v030n05abeh001738

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Condensation of the weakly nonideal gas in an anisotropic parabolic trap was considered. The first order (with respect to the interaction parameter) correction Δ 0 = 0 ( T )— 0 0 ( T ) to the temperature dependence of the ground state population 0 0 ( T ) of the ideal gas atoms was found using thermodynamic perturbation theory. The correction found 0 proved to be much larger than the one calculated within the mean field theory [see Giorgini S, Pitaevskii L P, Stringari S Phys. Rev. A 54 R4633 (1996)] at temperatures below critical. This correction tends to zero as the temperature approaches the critical one; i.e., the interaction of the atoms does not lead to variation in the critical temperature. A simple analytic expression was found that describes the smooth variation of 0 0 ( T ) as it approaches zero in the vicinity of the critical temperature. A new criterion for applicability of the thermodynamic limit is given.
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.
The critical temperature and condensate fraction of a trapped interacting Bose gas are investigated when both atom-atom interaction and finite-size effects are taken into account. A canonical ensemble is used to obtain the equations on the condensate fraction for the trapped interacting Bose gas near and below the critical temperature. In our approaches corrections due to atom-atom interaction and finite-size effects are obtained simultaneously for the critical temperature and condensate fraction of the system. An analytical high-order correction to the condensate fraction is given.
In this paper the thermodynamical parameters of a condensed Boson gas are calculated from the partial derivative of the grand potential. In particular, the analytical expressions for some important parameters, such as the condensed fraction, specific heat, critical temperature, effective size, and release energy are investigated. The mean effects which can be altered the ideal Bose gas, such as finite size, highly anisotropic of the external potential and interatomic interaction effects are considered simultaneously. Some new characteristics of the trapped interacting Bose gases in a highly anisotropic trap are revealed. The calculated result for the condensed fraction is compared with the ongoing Stuttgart experiment for 52Cr (Griesmaier et al., Phys. Rev. Lett. 94, 160401(2005)) directly. Good agreement between both the theoretical and experimental data are obtained.
The thermodynamical properties of interacting Bose atoms in a harmonic potential are studied within the mean-field approximation. For weak interactions, the quantum statistics is equivalent to an ideal gas in an effective mean-field potential. The eigenvalue of the Gross–Pitaevskii equation is identified as the chemical potential of the ideal gas. The condensation temperature and density profile of atoms are calculated. It is found that the critical temperature Tc decreases as the interactions increase. Below the critical point, the condensation fraction exhibits a universal relation of N0/N = 1-(T/Tc)γ, with the index γ ≈ 2.3 independent of the interaction strength, the chemical potential, as well as the frequency of the confining potential.
The distribution function W0(n0) of the number n0 of particles in the condensate of an ideal Bose gas confined by a trap is found. It is shown that at the temperature below the critical temperature Tc this function has a Gaussian shape and depends on the trap potential via two parameters only. The center of this function shifts to larger values of n0 with decreasing temperature and its width tends to zero, which corresponds to the suppression of fluctuations. In the narrow vicinity of the critical temperature ∣T−Tc∣⩽Tc/N\left| {T - {T_c}} \right| \leqslant {T_c}/\sqrt N ∣T−Tc​∣⩽Tc​/N​ , where N is the number of particles in the trap, the distribution function changes and at the temperature above the critical one it takes the usual form W0(n0) = [1 − exp(μ)]exp(μn0), where μ is the chemical potential in temperature units. In the limit N→∞, this change occurs at a jump.