Phase-space description of the thermal relaxation of a (2J+1)-level system

The master equation describing the relaxation of a ($2J+1$)-level system is analyzed in the diagonal representation for the density operator using the coherent atomic states as a basis. The $c$-number quasiprobability functions corresponding to the density operator are found to satisfy a second-order partial differential equation on the surface of the Bloch sphere. This equation is solved in the steady-state limit, and a few moments of interest of the atomic operators are calculated in terms of classicallike integrals in the phase space of the atomic variables. In the high-temperature approximation the partial differential equation is solved exactly for all time by a simple eigenfunction expansion procedure. For the special case of a two-level system an exact solution is also available for arbitrary values of the reservoir temperature.