Gaussian dynamics equation in normal product form

TL;DR

This paper derives a new Gaussian dynamical equation in normal product form for multimode Gaussian states, providing an alternative mechanism for understanding their time evolution in quantum systems.

Contribution

It introduces a novel Gaussian dynamical equation in normal product form and explores its feasibility and advantages over existing methods.

Findings

Derived the correlation equation between kernel matrices R and G.

Established the Gaussian dynamical equation R=i(RJH-HJR).

Demonstrated the feasibility of the new mechanism for Gaussian dynamics.

Abstract

In this paper, we discuss the normal product form of the density operator of multimode Gaussian states, and obtain the correlation equation between the kernel matrix R of the Gaussian density operator in the normal product form and its kernel matrix G in the standard quadratic form. Further, we explore the time evolution mechanism of R and obtain the Gaussian dynamical equation under the normal product R=i(RJH-HJR). Our work is devoted to searching for another mechanism for Gaussian dynamics. By exploring the description of the normal ordered density matrix under the coherent state representation, we find that our mechanism is feasible and easy to operate.