Алгебраический подход к динамике открытых многомодовых бозонных систем

We develop the method based on analysis of suitably generalized algebra of superoperators and apply it as an efficient tool to study open quantum (bosonic) multi-mode systems and their dynamics. More specifically, we consider the multi-mode Liouvillian superoperator that describes the relaxation dynamics of a quantum system and involves both thermalization and intermode couplings. The algebraic structure of superoperators that enter the Liouvillian in combination with their algebraic properties allow us to simplify the block structure of the multi-mode Liouvillian and derive its spectrum. We utilize our general results to estimate the squared speed of evolution for a single-photon state with arbitrary (linear) polarization. As another application of the method, we introduce the approximation for time-evolution superoperator that linearly depends on the mean number of thermal (environmental) photons. A useful feature of the approximation is that, when the initial state belongs to a finite dimensional Fock subspace, the dimension of the subspace describing the evolved state remains finite. Such approximation may thus be employed to simplify analysis of entanglement dynamics problems. In order to perform analysis in the Heisenberg picture, we have also deduced the expression for the conjugate of Liouvillian which is applicable for derivation of multi-time correlation functions.