EXPANSION METHOD FOR NONLINEAR QUANTUM MASTER-EQUATIONS

We are interested in the solutions of those master equations which appear when we consider a nonlinear coupling between an oscillator and an arbitrary thermal bath. For this purpose we implement a power series expansion in the parameter OMEGA = kT/homega0BAR. After observing that the master equation is of the diffusion type, we obtain a nonlinear Fokker-Planck equation for the probability density. Solving this equation we find that the relaxation becomes non-exponential. Going beyond lowest order in the expansion we deal again with a nonlinear Fokker-Plank equation which is equivalent to the obtained equation to first order in the case of a linear-plus-quadratic coupling. Finally, we transform the obtained equations to Schrodinger's ones and analyze the corresponding eigenvalue spectrum.