  We introduce a novel frequency-dependent incoherent pump scheme with a
square-shaped spectrum as a way to study strongly correlated photons in arrays
of coupled nonlinear resonators. This scheme can be implemented via a reservoir
of population-inverted two-level emitters with a broad distribution of
transition frequencies. Our proposal is predicted to stabilize a
non-equilibrium steady state sharing important features with a zero-temperature
equilibrium state with a tunable chemical potential. We confirm the efficiency
of our proposal for the Bose-Hubbard model by computing numerically the steady
state for finite system sizes: first, we predict the occurrence of a sequence
of incompressible Mott-Insulator-like states with arbitrary integer densities
presenting strong robustness against tunneling and losses. Secondly, for
stronger tunneling amplitudes or non-integer densities, the system enters a
coherent regime analogous to the superfluid state. In addition to an overall
agreement with the zero-temperature equilibrium state, exotic non-equilibrium
processes leading to a finite entropy generation are pointed out in specific
regions of parameter space. The equilibrium ground state is shown to be
recovered by adding frequency-dependent losses. The promise of this improved
scheme in view of quantum simulation of the zero temperature many-body physics
is highlighted.
