Finite Temperature Excitations of a Trapped Bose Gas

Abstract

We present a detailed study of the temperature dependence of the condensate and noncondensate density profiles of a Bose-condensed gas in a parabolic trap. These quantitites are calculated selfconsistently using the Hartree-Fock-Bogoliubov equations within the Popov approximation. Below the Bose-Einstein transition the excitation frequencies have a realtively weak temperature dependence even though the condensate is strongly depleted. As the condensate density goes to zero through the transition, the excitation frequencies are strongly affected and approach the frequencies of a noninteracting gas in the high temperature limit.

Thermodynamics of a trapped Bose-condensed gas

“…show that this depletion is always smaller than one percent of the total number of atoms in the condensate [14]. We have therefore at T = 0, ψ(r) = Φ(r) and n T (r) = m T (r) = 0, whereas at finite temperatures we take n T and m T proportional to the thermal density of quasi-particles…”

“…To obtain the mean-field factorizations (14) and (15) we have neglected the terms proportional to the anomalous non-condensate density m T (r) = ψ (r)ψ(r) , and to its complex conjugate. This approximation, which is discussed in depth in Ref.…”

“…[12], corresponds to the so called Popov approximation [27], and is expected to be appropriate both at high temperatures, where n T ≫ m T , and at low temperatures where the two densities are of the same order, but both negligibly small for the very dilute systems we are considering here. Using decomposition (12) and the mean-field prescriptions (14), and (15) the grand-canonical…”

“…The thermodynamic behaviour of trapped Bose gases has been already the object of several theoretical works, starting from the investigation of the ideal Bose gas [8], the development of the Hartree-Fock formalism [9], the inclusion of interaction effects using the local density approximation [10,11], up to the most recent approaches based on self-consistent mean-field theory [12][13][14][15] and numerical simulations [16].…”

“…These modes have been the object of recent extensive experimental and theoretical work [22][23][24][25]14].…”