Multimode quantum master equations: linear dynamics beyond quadratic generators

We present several examples of quantum master equations such that the dynamics of moments up to any finite order of bosonic creation and annihilation operators is (exactly) closed and linear.
Namely, we consider the quantum master equations which occur in the case of averaging of unitary dynamics with quadratic generators with respect to classical Levy fields, in particular with respect to Poisson and Wiener stochastic processes. We show that the dynamics of the moments of any fixed order is described by the closed system of ordinary linear differential equations.
We discuss some recent physical applications of these results.
The talk is based on the works:
T. Linowski, A. Teretenkov, L. Rudnicki, Physical Review A, 106 (2022), 052206.
D. D. Ivanov, A. E. Teretenkov, Math. Notes, 112:2 (2022), 318–322.
A. E. Teretenkov, Math. Notes, 107:4 (2020), 695–698.