  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.
