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Physical Review A

Thomas-Fermi approximation for Bose-Einstein condensates in traps

Peter Schuck, X. Viñas

Published Mar 8, 2000• 32 citations
Publisher

Abstract

Thomas-Fermi theory for Bose-Einstein condensates in inhomogeneous traps is revisited. The phase-space distribution function of the condensate in the Thomas-Fermi limit $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0)$ is ${f}_{0}(\mathit{R},\mathit{p})\ensuremath{\propto}\ensuremath{\delta}(\ensuremath{\mu}\ensuremath{-}{H}_{\mathrm{cl}})$ where ${H}_{\mathrm{cl}}$ is the classical counterpart of the self-consistent Gross-Pitaevskii Hamiltonian. Starting from this distribution function the Thomas-Fermi kinetic energy is calculated for any number of particles. Good agreement between the Gross-Pitaevskii and Thomas-Fermi kinetic energies is found even for low and intermediate particle numbers N. Application of this Thomas-Fermi theory to the attractive case and to the calculation of the frequencies of the monopole and quadrupole excitations in the sum rule approach yields conclusive results as well. The difference with the usual Thomas-Fermi approach to the Bose-Einstein condensates (large-N limit) is discussed in detail.

DOI
10.1103/physreva.61.043603

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Thomas-Fermi theory for Bose-Einstein condensates in inhomogeneous traps is revisited. The phase-space distribution function of the condensate in the Thomas-Fermi limit (\ensuremath\Elzxh\ensuremath→0)(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0)(\ensuremath\Elzxh\ensuremath→​0) is f0(R,p)\ensuremath∝\ensuremathδ(\ensuremathμ\ensuremath−Hcl){f}_{0}(\mathit{R},\mathit{p})\ensuremath{\propto}\ensuremath{\delta}(\ensuremath{\mu}\ensuremath{-}{H}_{\mathrm{cl}})f0​(R,p)\ensuremath∝\ensuremathδ(\ensuremathμ\ensuremath−Hcl​) where Hcl{H}_{\mathrm{cl}}Hcl​ is the classical counterpart of the self-consistent Gross-Pitaevskii Hamiltonian. Starting from this distribution function the Thomas-Fermi kinetic energy is calculated for any number of particles. Good agreement between the Gross-Pitaevskii and Thomas-Fermi kinetic energies is found even for low and intermediate particle numbers N. Application of this Thomas-Fermi theory to the attractive case and to the calculation of the frequencies of the monopole and quadrupole excitations in the sum rule approach yields conclusive results as well. The difference with the usual Thomas-Fermi approach to the Bose-Einstein condensates (large-N limit) is discussed in detail.
Thomas-Fermi theory for Bose condesates in inhomogeneous traps is revisited. The phase-space distribution function in the Thomas-Fermi limit is f0(R,p)f_0(\bold{R},\bold{p})f0​(R,p) ααα δ(μ−Hcl)δ(μ- H_{cl})δ(μ−Hcl​) where HclH_{cl}Hcl​ is the classical counterpart of the self-consistent Gross-Pitaevskii Hamiltonian. No assumption on the large N-limit is introduced and, e.g the kinetic energy is found to be in good agreement with the quantal results even for low and intermediate particle numbers N. The attractive case yields conclusive results as well.
Bose-Einstein condensates of 10485Rb{10}^{4}{}^{85}\mathrm{Rb}10485Rb atoms in a cylindrical trap are studied using a recently proposed approach based on a local-density approximation. Since the existence of a Feshbach resonance allows for widely tuning the scattering length of the atoms, values of the peak gas parameter xpk{x}_{\mathrm{pk}}xpk​ of the order of ${10}^{\ensuremath{-}2}$ may be attained and the standard analysis based on Gross-Pitaevskii and/or Thomas-Fermi equations may result in being questionable. Energy functionals derived from the correlated basis functions theory and the low-density expansion for a homogeneous hard-spheres gas are used to estimate corrections to the Gross-Pitaevskii equation. The two functionals give similar results, both showing large differences with respect to the Thomas-Fermi and Gross-Pitaevskii ones. The column densities at z=0z=0z=0 may differ by as much as $\ensuremath{\sim}30%$ and the half maximum radius by $\ensuremath{\sim}20%.$ The scattering lengths estimated by fitting the half maximum radius may differ by $\ensuremath{\sim}40%$ from those given by a Thomas-Fermi-based analysis of the experimental data.
Abstract We study the mean-field phase-space distribution of the ground state of an anisotropic, harmonically trapped Bose–Einstein condensate at zero temperature. First, we derive the Fourier transform of the ground-state wave function in the Thomas–Fermi limit in d dimensions. Second, we obtain analytical approximations for the Thomas–Fermi Wigner function in one and three dimensions with a factorization ansatz of the coherence function. These approximate expressions are compared with exact numerical simulations of the Gross–Pitaevskii equation.
By incorporating the zero-point energy contribution we derive simple and accurate extensions of the usual Thomas-Fermi (TF) expressions for the ground-state properties of trapped Bose-Einstein condensates that remain valid for an arbitrary number of atoms in the mean-field regime. Specifically, we obtain approximate analytical expressions for the ground-state properties of spherical, cigar-shaped, and disk-shaped condensates that reduce to the correct analytical formulas in both the TF and the perturbative regimes, and remain valid and accurate in between these two limiting cases. Mean-field quasi-one-dimensional (quasi-1D) and -2D condensates appear as simple particular cases of our formulation. The validity of our results is corroborated by an independent numerical computation based on the 3D Gross-Pitaevskii equation.
Bose-Einstein condensation in trapped gases of 85 Rb atoms can now be obtained in the large gas-parameter region in presence of Feshbach resonance and the gas density which is characterized by peak gas parameter x pk can attain the value ~ 10 -2 . For such large gas-parameter, the first correction term to the Bogoliubov energy functional is considered to give modified Gross-Pitaevskii (MGP) equation. In this work analytical expressions for the chemical potential, the density of particles and the excitation frequency of the monopole mode have been derived in Thomas-Fermi(TF) approximated approach to the MGP equation (modified TF approach) following method of higher order iteration for Bose-Einstein condensate (BEC) of atoms in spherically symmetric trap. We have also solved the MGP equation by using Runge-Kutta method for integration of nonlinear differential equation considering spherically symmetric trap. The analytical results for chemical potential and density of particles for BEC of 10 3 and 10 4 85 Rb atoms in a spherically symmetric trap are in very good agreement with the numerical results from MGP equation at low as well as at large scattering length i.e, at large gas-parameter region. The frequency of monopole mode calculated using the density of particles in the ground state (found out in higher order iterative method) are also in very good agreement with the existing numerical results. The difference between the analytical and numerical results decreases as the value of scattering length i.e, gas-parameter is increased.
We derive a description of the spatially inhomogeneous Bose-Einstein condensate, treating the system locally as a homogeneous Bose gas. This approach, similar to the Thomas-Fermi model for the inhomogeneous many-particle fermion system, is well suited to describe the atomic Bose-Einstein condensates that have recently been obtained experimentally through atomic trapping and cooling. In this paper, we confine our attention to the zero-temperature case, although the treatment can be generalized to finite temperatures, as we shall discuss elsewhere. Several features of this approach, which we shall call the Thomas-Fermi-Bogoliubov description, are very attractive. (i) It is simpler than the Hartree-Fock-Bogoliubov technique. We can obtain analytical results in the case of weakly interacting bosons for quantities such as the chemical potential, the local depletion, pairing, pressure, and density of states. (ii) The method provides an estimate for the error due to the inhomogeneity of the Bose-condensed system. This error is a local quantity so that the validity of the description for a given trap and a given number of trapped atoms can be tested as a function of position. We see, for example, that at the edge of the condensate the Thomas-Fermi-Bogoliubov theory always breaks down. (iii) The Thomas-Fermi-Bogoliubov description can be generalized to treat the statistical mechanics of the Bose gas at finite temperatures.