We study the propagation of strongly interacting Rydberg polaritons through an atomic medium in a one-dimensional optical lattice. We derive an effective single-band Hubbard model to describe the dynamics of the dark state polaritons under realistic assumptions. Within this model, we analyze the driven-dissipative transport of polaritons through the system by considering a coherent drive on one side and by including the spontaneous emission of the metastable Rydberg state. Using a variational approch to solve the many-body problem, we find strong antibunching of the outgoing photons despite the losses from the Rydberg state decay.

We present a method to perform a variational analysis of the quantum master equation for drivendisspative bosonic fields with arbitrary large occupation numbers. Our approach combines the P representation of the density matrix and the variational principle for open quantum system. We benchmark the method by comparing it to wave-function Monte-Carlo simulations and the solution of the Maxwell-Bloch equation for the Jaynes-Cummings model. Furthermore, we study a model describing Rydberg polaritons in a cavity field and introduce an additional set of variational paramaters to describe correlations between different modes.