Master equation for sympathetic cooling of trapped particles

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Abstract
A model for cooling a system of bosons in a harmonic trap via their interactions with a thermal bath of other particles is studied. The master equation describing the evolution of the system is derived for an arbitrary number of spatial dimensions. This equation is characterized by transition rates between trap levels. We present an analytic approximation for these rates and compare it with exact formulas, derived for the case of an even number of spatial dimensions. Analytic expressions show very good agreement with the exact ones for a wide range of parameters. We also discuss the cooling dynamics in terms of the approximated rates.