Markovian evolution of Gaussian states in the semiclassical limit

TL;DR

This paper develops an approximate method to describe the evolution of Gaussian quantum states under open system dynamics in the semiclassical limit, using a phase space approach that generalizes Heller's theory.

Contribution

It introduces a Gaussian solution to the Lindblad equation in the semiclassical regime, extending Heller's theory to include environmental effects through a chord representation.

Findings

Provides a system of nonlinear equations for Gaussian state evolution

Connects Lindblad dynamics with phase space methods

Enables direct access to Wigner and position representations

Abstract

We derive an approximate Gaussian solution of the Lindblad equation in the semiclassical limit, given a general Hamiltonian and linear coupling with the environment. The theory is carried out in the chord representation and describes the evolved quantum characteristic function, which gives direct access to the Wigner function and the position representation of the density operator by Fourier transforms. The propagation is based on a system of non-linear equations taking place in a double phase space, which coincides with Heller's theory of unitary evolution of Gaussian wave packets when the Lindbladian part is zero.