Model for the description of the relaxation of quantum-mechanical systems with closely spaced energy levels

Existing Markov models of the relaxation of quantum-mechanical systems are applicable in the approximations where the difference between eigenfrequencies is either much larger than the relaxation rates (global approach) or much lower than these rates (local approach). In this work, a model has been proposed to describe relaxation in systems where the difference between some eigenfrequencies is close to the relaxation rates. It has been shown that the master equation for the density matrix contains coefficients explicitly depending on the time in the interaction representation. Equations for the occupation numbers of the modes of the system have been derived by example of two interacting oscillators. It has been shown that occupation numbers in the proposed approach in the limits of small and large differences between eigenfrequencies compared to the relaxation rates asymptotically match the occupation numbers in the local and global approaches, respectively.