Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States

TL;DR

This paper develops a differential formalism to analyze the evolution of Gaussian states, including nonunitary dynamics and invariant states, providing new insights into their behavior under quadratic Hamiltonians.

Contribution

It introduces a differential approach to describe Gaussian state evolution, including nonunitary processes, and identifies new invariant and quasi-invariant states.

Findings

Derived differential equations for Gaussian state parameters.

Identified invariant states for specific quantum systems.

Analyzed nonunitary evolution of subsystems.

Abstract

In a differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian $\hat{H}$. We also present the corresponding differential equations which describe the nonunitary evolution of the subsystems. The resulting nonlinear equations are used to solve the dynamics of the system instead of the Schr\"odinger equation. The formalism elaborated allows us to define new specific invariant and quasi-invariant states, as well as states with invariant covariance matrices, i.e., states were only the mean values evolve according to the classical Hamilton equations. By using density matrices in the position and in the tomographic-probability representations, we study examples of these properties. As examples, we present novel invariant states for the two-mode frequency converter and quasi-invariant states for the bipartite parametric amplifier.