Cooling of a small sample of Bose atoms with accidental degeneracy

A system of bosons in a harmonic trap is cooled via their interactions with a thermal reservoir. We derive the master equation that governs the evolution of the system and may describe diverse physical situations: laser cooling, symphatetic cooling, etc. We investigate in detail the dynamics of the gas in the Lamb-Dicke limit, whereby the size of the trap is small in comparison to the de Broglie wavelength of the reservoir quanta. In this case, the dynamics is characterized by two time scales. First, an equilibrium is reached on a fast time scale within the degenerated subspaces of the system. Then, an equilibrium between these subspaces is reached on a slow time scale.