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Journal of Physics B Atomic Molecular and Optical Physics

An analytical approach for the nonlinear modified Thomas–Fermi equation to derive the ground-state and dynamic properties of a spherically and cylindrically trapped Bose–Einstein condensate

Moumita Gupta, Krishna Rai Dastidar

Published Sep 10, 2008• 10 citations
Publisher

Abstract

It has been shown previously that the modified Thomas–Fermi (MTF) equation can be solved analytically by using a higher order iteration method (MTF(p)) for a spherically trapped atomic Bose–Einstein condensate (BEC). The ground-state properties, e.g. chemical potential, peak gas parameter and the ground state density of atoms (except in the surface region), thus obtained can successfully reproduce the results obtained by solving the modified Gross–Pitaevskii (MGP) equation numerically. In this paper, we have extended this analytical method for obtaining the ground-state and dynamical properties of a cylindrically trapped atomic BEC. In this analytical approach, the ground-state density of atoms at the surface of the trap fails to reproduce the correct behaviour due to the neglect of kinetic energy in the MTF equation. We have proposed a model function which when fitted with the analytical wavefunction (from the MTF equation) near the boundaries of the trap can reproduce the correct behaviour of the ground-state density of atoms at the surface region. The kinetic energy and the other energy components thus obtained using the modified analytical method (MTF(p)+model function fitting) can successfully reproduce the results obtained by solving the MGP equation numerically for the 85Rb BEC containing 104 atoms both for spherical and cylindrical traps. The excitation frequency thus obtained for the compressional mode of both spherically and axially symmetric traps is in good agreement with the numerical results. For these calculations, the virial relation is satisfied within the limit ≤10−4.

DOI
10.1088/0953-4075/41/19/195302

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It has been shown previously that the modified Thomas–Fermi (MTF) equation can be solved analytically by using a higher order iteration method (MTF(p)) for a spherically trapped atomic Bose–Einstein condensate (BEC). The ground-state properties, e.g. chemical potential, peak gas parameter and the ground state density of atoms (except in the surface region), thus obtained can successfully reproduce the results obtained by solving the modified Gross–Pitaevskii (MGP) equation numerically. In this paper, we have extended this analytical method for obtaining the ground-state and dynamical properties of a cylindrically trapped atomic BEC. In this analytical approach, the ground-state density of atoms at the surface of the trap fails to reproduce the correct behaviour due to the neglect of kinetic energy in the MTF equation. We have proposed a model function which when fitted with the analytical wavefunction (from the MTF equation) near the boundaries of the trap can reproduce the correct behaviour of the ground-state density of atoms at the surface region. The kinetic energy and the other energy components thus obtained using the modified analytical method (MTF(p)+model function fitting) can successfully reproduce the results obtained by solving the MGP equation numerically for the 85Rb BEC containing 104 atoms both for spherical and cylindrical traps. The excitation frequency thus obtained for the compressional mode of both spherically and axially symmetric traps is in good agreement with the numerical results. For these calculations, the virial relation is satisfied within the limit ≤10−4.
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.
A simple analytical ansatz, which has been used to describe the intensity profile of the similariton laser (a laser with self-similar propagation of ultrashort pulses), is used as a variational wave function to solve the Gross-Pitaevskii equation for a wide range of interaction parameters. The variational form interpolates between the noninteracting density profile and the strongly interacting Thomas-Fermi profile smoothly. The simple form of the ansatz is modified for both cylindrically symmetric and completely anisotropic harmonic traps. The resulting ground-state density profile and energy are in very good agreement with both the analytical solutions in the limiting cases of interaction and the numerical solutions in the intermediate regime.
Based on the Gross-Pitaevskii(G-P) energy functional and variational method, we present a new method to find the ground state wave function for Bose-condensed gas in a harmonic trap at zero temperature. With this method we are able to find the analytic expression of ground state wave function and explore the relevant quantities, such as energy, chemical potential and so on. These results agree well with previous ground state numerical solutions of the G-P equation given by Dalfovo et al. Under the large number of particles N limit for repulsive interaction, our results and Thomas-Fermi approximation model approach to reaching unanimity. This new method is simple compared to other methods used to solve numerically the G-P equation, and one can obtained the analytic and reliable results.
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 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.
We propose a method for finding analytical solutions of the Bogoliubov--de Gennes (BdG) equations for the low-lying collective excitations in a harmonically trapped Bose-Einstein condensate beyond the Thomas-Fermi limit. We first use a simple variational wave function for ground state to eliminate the divergence at the boundary layer of the condensate, which appears in the Thomas-Fermi approximation. We then solve the BdG equations analytically and obtain explicit and divergence-free expressions for the eigenvalues and eigenfunctions of the excitations for the traps with spherical and cylindrical symmetries. The solutions of the zero-energy mode of the BdG equations are also presented.