Collective excitations of a strongly-correlated non-equilibrium photon fluid across the Mott/superfluid phase transition

We develop a Gutzwiller theory for the non-equilibrium steady states of a strongly-interacting photon fluid driven by a non-Markovian incoherent pump. In particular, we explore the collective excitation modes across the out-of-equilibrium Mott/superfluid transition, characterizing the diffusive Goldstone mode in the superfluid phase and the particle/hole excitations in the insulating one. Observable features in the pump-and-probe optical response of the system are highlighted. Our results appear as experimentally accessible to state-of-the-art circuit-QED devices and open the way for driven-dissipative fluids of light as quantum simulators of novel many-body scenarios.