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arXiv:cond-mat/9803216 · Physical Review A

Analytical description of a trapped semi-ideal Bose gas at finite temperature

M. Naraschewski, Dan Stamper-Kurn

Published Sep 1, 1998• 86 citations
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Abstract

Present experiments with Bose condensed gases can be largely described by a semi-ideal two-gas model. In this model, the condensate is influenced only by the mean-field repulsion among condensed atoms, while the thermal cloud is considered an ideal gas confined by an effective potential that consists of the external trap and the mean-field repulsion by the condensate. This simple, intuitive model provides explicit analytical expressions for the density distributions of the condensate and the thermal component. It describes the reduction of the condensate fraction relative to that of an ideal gas as a consequence of the positive chemical potential due to interactions in the condensate.

DOI
10.1103/physreva.58.2423

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Present experiments with Bose condensed gases can be largely described by a semi-ideal two-gas model. In this model, the condensate is influenced only by the mean-field repulsion among condensed atoms, while the thermal cloud is considered an ideal gas confined by an effective potential that consists of the external trap and the mean-field repulsion by the condensate. This simple, intuitive model provides explicit analytical expressions for the density distributions of the condensate and the thermal component. It describes the reduction of the condensate fraction relative to that of an ideal gas as a consequence of the positive chemical potential due to interactions in the condensate.
A simple picture describes the results of recent treatments of partially--condensed, dilute, trapped Bose gases at temperature T > 0. The condensate wave function is nearly identical to that of a T = 0 condensate with the same number of condensate atoms, Ν0.The cloud of non-condensed atoms is described by the statistical mechanics of an ideal Bose gas in the com- bined potentials of the magnetic trap and the cloud-condensate interaction.We provide a physical motivation for this result, show how it emerges in the Hartree-Fock-Bogo liubov-Popov approximation, and explore some of its implications for future experiments.
We show that recent treatments of magnetically trapped Bose-Einstein condensates (BECs) at temperature T >0 fit a simple picture: the condensate wavefunction is that of a T = 0 BEC with the appropriate (depleted) number of condensate atoms N 0 , and the surrounding thermal cloud is described by the statistical mechanics of an ideal gas in the combined potentials of the magnetic trap and the cloud-condensate interaction. Even a simple classical description of the cloud reproduces its essential outline, and facilitates easy approximate calculations of BEC properties for T >0.
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 fact that two-dimensional interacting trapped systems do not undergo Bose-Einstein Condensation (BEC) in the thermodynamic limit, though rigorously proved, is somewhat mysterious because all relevant limiting cases (zero temperature, small atom numbers, noninteracting particles) suggest otherwise. We study the possibility of condensation in finite trapped two-dimensional systems. We first consider the ideal gas, which incorporates the inhomogeneity and finite size of experimental systems and can be solved exactly. A semiclassical self-consistent approximation gives us a feel for the temperature scales; diagonalization of the one-body density matrix confirms that the condensation is into a single state. We squeeze a three-dimensional system and see how it crosses over into two dimensions. Mean-field theory, our main tool for the study of interacting systems, prescribes coupled equations for the condensate and the thermal cloud: the condensate receives a full quantum-mechanical treatment, while the noncondensate is described by different schemes of varying sophistication. We analyze the T = 0 case and its approach to the thermodynamic limit, finding a criterion for the dimensionality crossover and obtaining the coupling constant of the two-dimensional system that results from squeezing a three-dimensional trap. We next apply a semiclassical Hartree-Fock approximation to purely two-dimensional finite gases and find that they can be described either with or without a condensate; this continues to be true in the thermodynamic limit. The condensed solutions have a lower free energy at all temperatures but neglect the presence of phonons within the system and cease to exist when we allow for this possibility. The uncondensed solutions, in turn, are valid under a more rigorous scheme but have consistency problems of their own. Path-integral Monte Carlo simulations provide an essentially exact description of finite interacting gases and enable us to study highly anisotropic systems at finite temperature. We find that our two-dimensional Hartree-Fock solutions accurately mimic the surface density profile and predict the condensate fraction of these systems; the equivalent interaction parameter is smaller than that dictated by the T = 0 analysis. We conclude that, in two-dimensional isotropic finite trapped systems and in highly compressed three-dimensional gases, there is a phenomenon resembling a condensation into a single state.
We analyze the results of a recent experiment with bosonic rubidium atoms harmonically confined in a quasi-two-dimensional (2D) geometry. In this experiment a well-defined critical point was identified, which separates the high-temperature normal state characterized by a single component density distribution, and the low-temperature state characterized by a bimodal density distribution and the emergence of high-contrast interference between independent 2D clouds. We first show that this transition cannot be explained in terms of conventional Bose–Einstein condensation of the trapped ideal Bose gas. Using the local density approximation (LDA), we then combine the mean-field (MF) Hartree–Fock theory with the prediction for the Berezinskii–Kosterlitz–Thouless (BKT) transition in an infinite uniform system. If the gas is treated as a strictly 2D system, the MF predictions for the spatial density profiles significantly deviate from those of a recent quantum Monte Carlo (QMC) analysis. However, when the residual thermal excitation of the strongly confined degree of freedom is taken into account, excellent agreement is reached between the MF and the QMC approaches. For the interaction strength corresponding to the experiment, we predict a strong correction to the critical atom number with respect to the ideal gas theory (factor ∼2). Quantitative agreement between theory and experiment is reached concerning the critical atom number if the predicted density profiles are used for temperature calibration.
Abstract This chapter describes the main properties of the ideal Bose gas confined by a harmonic potential. It calculates the temperature dependence of the condensate density and identify the value of the critical temperature in terms of the oscillator frequency and the number of atoms. This identification provides an important temperature scale for experiments with ultracold atomic gases. The chapter also discusses the behaviour of the expanding cloud after the release of the trap and how the anisotropy of the gas evolves in time, reflecting the initial anisotropy of the momentum distribution. It finally discusses the possibility of getting Bose–Einstein condensation through adiabatic transformations, working with non-harmonic trapping potentials or with mixtures of two atomic species trapped by different harmonic potentials.