Solution of the master equation for quantum systems weakly coupled to reservoirs and far from thermal equilibrium

The Liouville operator of the master equation is decomposed into a part describing the proper system and another operator describing the interaction of the proper system with a set of reservoirs. If the motion of the proper system is described by a Hamiltonian, the corresponding solutions of the density matrix equation are spanned by all conserved quantities and are thus highly degenerate. We demonstrate how this degeneracy may be lifted by applying some sort of perturbation theory for degenerate systems. If there is only one relevant conserved quantity and if the heatbaths connect the quantum numbers of this conserved quantity by nearest-neighbour steps, the solution of the problem can be found explicitly. Otherwise the solution of the original master equation is reduced to a much simpler master equation of orderN, whereN is the number of conserved quantities. An application to a new type of parametric processes in nonlinear optics is given.