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arXiv:1201.2233 · Modern Physics Letters B

THERMODYNAMICAL PROPERTIES OF A TRAPPED INTERACTING BOSE GAS

SHI-JIE YANG, YUECHAN LIU, SHIPING FENG

Published Mar 14, 2012• 1 citation
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Abstract

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.

DOI
10.1142/s0217984912500534

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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.
Abstract The critical properties of the phase transition from a normal gas to a Bose–Einstein condensate (superfluid) of a harmonically confined Bose gas are addressed with the knowledge of an equation of state of the underlying homogeneous Bose fluid. It is shown that while the presence of the confinement trap arrests the usual divergences of the isothermal compressibility and heat capacities, the critical behavior manifests itself now in the divergence of derivatives of the mentioned susceptibilities. This result is illustrated with a mean-field like model of an equation of state for the homogeneous particle density as a function of the chemical potential and temperature of the gas. The model assumes the form of an ideal Bose gas in the normal fluid while in the superfluid state a function is proposed such that, both, asymptotically reaches the Thomas–Fermi solution of a weakly interacting Bose gas at large densities and low temperatures and, at the transition, matches the critical properties of the ideal Bose gas. With this model we obtain the global thermodynamics of the harmonically confined gas, from which we analyze its critical properties. We discuss how these properties can be experimentally tested.
We investigate the interaction effect between atoms and the finite size effect of a Bose-Einstein gas at finite temperature. Using a mean field approach, we derive the thermodynamic potential on finite systems and obtain the condensate fraction, the chemical potential and the internal energy with the interaction and the finite size effects taken into account. In this approach, we can treat thermodynamic quantities as those of the ideal Bose gas. It is found that the interaction effect shifts the transition temperature by a few percent toward the lower temperature. Above the transition temperature, these quantities agree with the noninteracting Bose gas.
The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point Tc{T}_{c}Tc​ requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length asc,{a}_{\mathrm{sc}},asc​, increases the number of condensate atoms at each temperature, and raises the value of Tc{T}_{c}Tc​ relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near Tc.{T}_{c}.Tc​. Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.
In this paper, we discuss theoretically the behavior of ideal Bose-Einstein gases in a 3D box and in a harmonic trap. We make use of the mean filed theory based on the time dependent Gross-Pitaevskii equation which describes the main features of Bose-Einstein condensation. We investigate and discuss the density profiles and the ground state properties of a trapped condensate confined in harmonic trap. We found out that in an attractive interaction, the gas tends to increase its central density in order to lower the interaction energy.
We have investigated the thermodynamic behaviour of ideal Bose gases with an arbitrary number of particles confined in a harmonic potential. By taking into account the conservation of the total number N of particles and using a saddle-point approximation, we derive analytically the simple explicit expression of mean occupation number in any state of the finite system. The temperature dependence of the chemical potential, specific heat, and condensate fraction for the trapped finite-size Bose system is obtained numerically. We compare our results with the usual treatment which is based on the grand canonical ensemble. It is shown that there exists a considerable difference between them at sufficiently low temperatures, especially for the relative small numbers of Bose atoms. The finite-size scaling at the transition temperature for the harmonically trapped systems is also discussed. We find that the scaled condensate fractions for various system sizes and temperatures collapse onto a single scaled form.
We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number NNN of atoms, harmonic oscillator length, deformation of the trap, sss-wave scattering length) fix its thermodynamic behavior in the large NNN limit through two dimensionless scaling parameters. These are the reduced temperature t=T/Tc0t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T/{T}_{c}^{0}t=T/Tc0​ and the ratio $\ensuremath{\eta}$ between the T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature Tc0{T}_{c}^{0}Tc0​ of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.