Dynamics of moments of higher orders in exactly solvable models of the theory of open quantum systems

In this work we consider quantum master equations for which the dynamics can be obtained explicitly. A Leibniz-type formula has been derived which allows one to compute the action of the conjugate Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator on the product of the creation and annihilation operators, up to higher orders. Also, the Heisenberg equations for arbitrary moments of higher orders of the birth and annihilation operators in the case of a generator in the form of GKSL quadratic in terms of the bosonic creation and annihilation operators have been obtained in explicit form. Moreover, solutions of such equations in the case of time-dependent coefficients have been obtained. In addition, the Isserlis-Wick theorem was proved in the notations used in the paper and the consistency of the results obtained earlier with the theorem was demonstrated. On the basis of the Heisenberg equations obtained in the paper, analogous equations were derived for the quantum master equation arising after averaging the dynamics by a quadratic generator over a classical Poisson process. This allowed us to show that the dynamics of arbitrary moments of finite order of the creation and annihilation operators in this case is completely determined by a finite number of linear differential equations.