Algebraic approach for investigation of a multi-mode quantum system dynamics

TL;DR

This paper presents an algebraic method for analyzing the dynamics of multi-mode quantum bosonic systems, enabling spectrum calculation and approximations for time evolution, with applications to entanglement and correlation functions.

Contribution

It introduces an algebraic framework for superoperators in multi-mode quantum systems, facilitating spectrum analysis and finite-dimensional approximations of dynamics.

Findings

Diagonalization of multi-mode Liouvillian spectrum

Linear approximation for time evolution in finite Fock space

Application to entanglement and correlation function analysis

Abstract

We introduce algebraic approach for superoperators that might be useful tool for investigation of quantum (bosonic) multi-mode systems and its dynamics. In order to demonstrate potential of proposed method we consider multi-mode Liouvillian superoperator that describes relaxation dynamics of a quantum system (including thermalization and intermode coupling). Considered algebraic structure of superoperators that form Liouvillian and their algebraic properties allows us to diagonilize multi-mode Liouvillian to find its spectrum. Also it allows to derive linear by mean number of thermal (environmental) photons approximation for time-evolution superoperator that keeps amount of considered dimensions in Fock space finite (assuming initial amount of dimensions finite) that might be helpful regarding entanglement dynamics problems. Conjugate Liouvillian is considered as well in order to perform analysis in Heisenberg picture, it can be implemented for multi-time correlation functions derivation.