Lindblad dynamics of Gaussian states and their superpositions in the semiclassical limit

TL;DR

This paper explores the semiclassical evolution of Gaussian states under Lindblad dynamics, revealing non-Hamiltonian flows and interference effects in superpositions, with exact results for linear operators and quadratic Hamiltonians.

Contribution

It introduces a novel phase-space dynamics framework for Gaussian states under Lindblad evolution, including interference effects in superpositions, and derives semiclassical equations for these processes.

Findings

Lindblad terms induce non-Hamiltonian flows in phase space.

The Gaussian approximation is exact for linear Lindblad operators and quadratic Hamiltonians.

Interference in superpositions like cat states can be described semiclassically.

Abstract

The time evolution of the Wigner function for Gaussian states generated by Lindblad quantum dynamics is investigated in the semiclassical limit. A new type of phase-space dynamics is obtained for the centre of a Gaussian Wigner function, where the Lindblad terms generally introduce a non-Hamiltonian flow. In addition to this, the Gaussian approximation yields dynamical equations for the covariances. The approximation becomes exact for linear Lindblad operators and a quadratic Hamiltonian. By viewing the Wigner function as a wave function on a coordinate space of doubled dimension, and the phase-space Lindblad equation as a Schr\"{o}dinger equation with a non-Hermitian Hamiltonian, a further set of semiclassical equations are derived. These are capable of describing the interference terms in Wigner functions arising in superpositions of Gaussian states, as demonstrated for a cat state in an anharmonic oscillator subject to damping.