The Moyal Equation for open quantum systems

TL;DR

This paper extends the Moyal equation to open quantum systems interacting with reservoirs, showing how noise influences phase space observables and simplifying the reservoir's effect for Markovian cases.

Contribution

It introduces a generalized Moyal equation for open quantum systems, incorporating reservoir-induced noise and reducing to a simpler form for Markovian reservoirs.

Findings

Phase space observables become functionals of noise forces.

The generalized Moyal equation is a functional differential equation.

Reservoir effects are characterized by a single parameter in Markovian cases.

Abstract

We generalize the Moyal equation, which describes the dynamics of quantum observables in phase space, to quantum systems coupled to a reservoir. It is shown that phase space observables become functionals of fluctuating noise forces introduced by the coupling to the reservoir. For Markovian reservoirs, the Moyal equation turns into a functional differential equation in which the reservoir's effect can be described by a single parameter.