Thomas-Fermi approximation for Bose-Einstein condensates in traps

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.