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arXiv:cond-mat/0510265 · Physical Review A

Modified semiclassical approximation for trapped Bose gases

Vyacheslav I. Yukalov

Published Sep 12, 2005• 60 citations
arXivHTMLPDF

Abstract

A generalization of the semiclassical approximation is suggested allowing for an essential extension of its region of applicability. In particular, it becomes possible to describe Bose-Einstein condensation of a trapped gas in low-dimensional traps and in traps of low confining dimensions, for which the standard semiclassical approximation is not applicable. The result of the modified approach is shown to coincide with purely quantum-mechanical calculations for harmonic traps, including the one-dimensional harmonic trap. The advantage of the semiclassical approximation is in its simplicity and generality. Power-law potentials of arbitrary powers are considered. The effective thermodynamic limit is defined for any confining dimension. The behavior of the specific heat, isothermal compressibility, and density fluctuations is analyzed, with an emphasis on low confining dimensions, where the usual semiclassical method fails. The peculiarities of the thermodynamic characteristics in the effective thermodynamic limit are discussed.

DOI
10.1103/physreva.72.033608

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A generalization of the semiclassical approximation is suggested allowing for an essential extension of its region of applicability. In particular, it becomes possible to describe Bose-Einstein condensation of a trapped gas in low-dimensional traps and in traps of low confining dimensions, for which the standard semiclassical approximation is not applicable. The result of the modified approach is shown to coincide with purely quantum-mechanical calculations for harmonic traps, including the one-dimensional harmonic trap. The advantage of the semiclassical approximation is in its simplicity and generality. Power-law potentials of arbitrary powers are considered. The effective thermodynamic limit is defined for any confining dimension. The behavior of the specific heat, isothermal compressibility, and density fluctuations is analyzed, with an emphasis on low confining dimensions, where the usual semiclassical method fails. The peculiarities of the thermodynamic characteristics in the effective thermodynamic limit are discussed.
Abstract. Using a field-theoretic approach, we systematically generalize the usual semiclassical approximation for a harmonically trapped ideal Bose gas in such a way that its range of applicability is essentially extended. With this we can analytically calculate thermodynamic properties even for small particle numbers. In particular, it now becomes possible to determine the critical temperature as well as the temperature dependence of both heat capacity and condensate fraction in low-dimensional traps, where the standard semiclassical approximation is not even applicable. PACS. 03.65.Sq Semiclassical theories and applications – 03.75.Hh Static properties of condensates; thermodynamical, statistical, and structural properties – 05.70.Ce Thermodynamic functions and equations of state 1
We investigate ideal quantum gases in D-dimensional space and confined in a generic external potential by using the semiclassical approximation. In particular, we derive density of states, density profiles and critical temperatures for Fermions and Bosons trapped in isotropic power-law potentials. From such results, one can easily obtain those of quantum gases in a rigid box and in a harmonic trap. Finally, we show that the Bose–Einstein condensation can set up in a confining power-law potential if and only if D/2+D/n>1, where D is the space dimension and n is the power-law exponent.
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.
The critical temperature for Bose-Einstein condensation, ground-state fraction, and heat capacity of an ideal gas of Bose particles that are confined by an n-dimensional generic power-law potential trap are derived. The conditions for Bose-Einstein condensation and the discontinuous conditions of the heat capacity at critical temperature are obtained. All these quantities and conditions are found to vary markedly with the dimensionality of space, characteristics of particles, and shape of the external potential. The results obtained here are universal.
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We calculate the quantum corrections of the thermodynamic quantities of a system of confined bosons at finite temperature. Systematically quantum corrections are written in a series of $\ensuremath{\Elzxh},$ which is convergent when kT\mathrm{kT}kT is much larger than the spacing between energy levels of the system. We apply this method to the different physical systems. First we calculate the thermodynamic potential of spin-1 bosons confined in a general power-law potential, and calculate the paramagnetic susceptibility of the gas including the quantum corrections. Local-density approximation for charged bosons in the weak magnetic field is improved by the semiclassical expansion. Diamagnetic susceptibility of a system of confined charged bosons is calculated using this method. Finally we apply this method to calculate analytically the thermodynamic potential of a weakly interacting Bose-gas confined in at three-dimensional harmonic-oscillator potential. For a large number of particles, quantum corrections become small, and contribute to the finite-size corrections to scaling.