Stochastically averaged master equation for a quantum-dynamic system interacting with a thermal bath

The methods of nonequilibrium density-matrix and coarse-temporal conception are used to obtain the kinetic equation for the parameters ${\gamma }_{\mathit{n}\mathit{m}}$(t)=Sp[ρ^(t)‖n〉〈m‖] of a quantum-dynamic system (QDS) interacting with a thermal bath and external stochastic field. It is important that the stochastic field is taken exactly into consideration. For diagonal QDS parameters ${\gamma }_{\mathit{n}\mathit{n}}$(t) this equation is reduced to the generalized Pauli equation (GPE) with stochastic time-dependent coefficients ${\mathit{w}}_{\mathit{n}\mathit{m}}$(t). Special attention is given to the procedure of averaging over stochastic processes. It is shown that after averaging over energy fluctuations affected by the stochastic field, in the first cumulant approximation in terms of stochastic processes ${\mathit{w}}_{\mathit{n}\mathit{m}}$(t), the GPE is transformed to the Pauli equation for the QDS state population ${\mathit{P}}_{\mathit{n}}$(t)=〈${\gamma }_{\mathit{n}\mathit{n}}$(t)${\mathrm{〉}}_{\mathit{f}}$. As an example, the relaxation behavior of a two-level system interacting with a dichotomous field (dichotomous Markovian process of kangaroo type) and a harmonic oscillator coupled with a thermal bath is considered. It is shown that the probability of relaxation transitions between energy levels may be changed by several orders of magnitude under the influence of the dichotomous field.