Wigner function in the polariton phase space

TL;DR

This paper introduces a novel Wigner function framework in polariton phase space for atom-photon systems, providing a differential equation and propagator that enhance understanding of quantum lattice dynamics.

Contribution

It develops a new Wigner function formalism in polariton phase space and derives a general propagator applicable to hybrid quantum models.

Findings

Derived a closed differential equation for the Wigner function

Mapped atom-photon interactions into a lattice model with polariton states

Provided a general propagator for the Wigner function in this context

Abstract

The Wigner function of a dynamical infinite dimensional lattice is studied. A closed differential equation without diffusion terms for this function is obtained and solved. We map atom-photon interaction systems, such as the Jaynes-Cummings model, into this lattice model, where each dressed or polariton state corresponds to a point in the lattice and the conjugate momenta are described by the eigenvalues of the phase operator. The corresponding Wigner function is defined by these two conjugate variables in what we name the polariton phase space. We derive a general propagator of the Wigner function, which is also valid for other hybrid models.