  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.
