  Strongly coupled light-matter systems can exhibit nonequilibrium collective
phenomena due to loss and gain processes on the one hand and effective
photon-photon interactions on the other hand. Here we study a photonic lattice
system composed of a linear array of driven-dissipative coupled cavities (or
cavity modes) with linearly increasing resonance frequencies across the
lattice. The model amounts to a driven-dissipative Bose-Hubbard model in a
tilted potential without the particle-conservation constraint. We predict a
diverse range of stationary and non-stationary states resulted from the
interplay of the tilt, tunneling, on-site interactions, and the loss and gain
processes. Our key finding is that, under weak on-site interactions, photons
mostly Bose condense into a selected, single-particle Wannier-Stark state,
instead of exhibiting expected Bloch oscillations. As the strength of the
photon-photon interactions increase, a non-stationary regime emerges which is
marked surprisingly by periodic Bloch-type oscillations. These intriguing,
nontrivial effects are a direct consequence of the driven-dissipative nature of
the system.
