Density Matrix of a Bose--Einstein Condensate: Steady--State versus Mean--Field Approach

We compare the equilibrium solution for the condensate obtained in the mean–field approximation to the master equation for sympathetic cooling with the one obtained by Scully for a system in contact with a heat bath with the help of an analogy with the laser. While the mean–field approach yields analytical formulas for the approach towards equilibrium and for the equilibrium solution, it neglects the correlations between occupation numbers of different single–particle states which are approximately kept in Scully’s approach. Such neglect is admissible as long as the fraction of Bosons in the condensate does not exceed a few percent or so.