  We study nonlinear cavity arrays where the particle relaxation rate in each
cavity increases with the excitation number. We show that coherent parametric
inputs can drive such arrays into states with commensurate filling that form
non-equilibrium analogs of Mott insulating states. We explore the boundaries of
the Mott insulating phase and the transition to a delocalized phase with
spontaneous first order coherence. While sharing many similarities with the
Mott insulator to superfluid transition in equilibrium, the phase-diagrams we
find also show marked differences. Particularly the off diagonal order does not
become long range since the influence of dephasing processes increases with
increasing tunneling rates.
