Mean-Field Approximation to the Master Equation for Sympathetic Cooling of Trapped Bosons

We use the mean-field approximation to simplify the master equation for sympathetic cooling of Bosons. For the mean single-particle occupation numbers, this approach yields the same equations as the factorization assumption introduced in an erlier paper. The stationary or equilibrium solution of the resulting master equation for the one-body density matrix shows that the mean-field approximation breaks down whenever the fraction of condensate Bosons exceeds ten percent or so of the total. Using group-theoretical methods, we also solve the time-dependent master equation for the one-body density matrix. Given the time dependence of the mean single-particle occupation numbers, this solution is obtained by quadratures. It tends asymptotically towards the equilibrium solution.