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

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.