Polaritonic linear dynamic in Keldysh formalism

We study the dynamic of polaritons in the Keldysh functional formalism. Dissipation is considered through the coupling of the exciton and photon fields to two independent photonic and excitonic baths. As such, this theory allows to describe more intricate decay mechanisms that depend dynamically on the state of the system, such as a direct upper-polariton lifetime, that is motivated from experiments. We show that the dynamical equations in the Keldysh framework otherwise follow the same Josephson--like equations of motions than the standard master equation approach, that is however limited to simple decay channels. We also discuss the stability of the dynamic and reconsider the criterion of strong coupling in the presence of upper polariton decay.