Density-matrix approach for an interacting polariton system

Using the Lindblad approach we develop a general formalism for theoretical description of a spatially inhomogeneous bosonic system with dissipation provided by the interaction of bosons with a phonon bath. We apply our results to model the dynamics of an interacting one-dimensional polariton system in real space and time, analyzing in detail the role of polariton-polariton and polariton-phonon interactions.