  We study the non-equilibrium behavior of optically driven dissipative coupled
resonator arrays. Assuming each resonator is coupled with a two-level system
via a Jaynes-Cummings interaction, we calculate the many-body steady state
behavior of the system under coherent pumping and dissipation. We propose and
analyze the many-body phases using experimentally accessible quantities such as
the total excitation number, the emitted photon spectra and photon coherence
functions for different parameter regimes. In parallel, we also compare and
contrast the expected behavior of this system assuming the local nonlinearity
in the cavities is generated by a generic Kerr effect rather than a
Jaynes-Cummings interaction. We find that the behavior of the experimentally
accessible observables produced by the two models differs for realistic regimes
of interactions even when the corresponding nonlinearities are of similar
strength. We analyze in detail the extra features available in the
Jaynes-Cummings-Hubbard (JCH) model originating from the mixed nature of the
excitations and investigate the regimes where the Kerr approximation would
faithfully match the JCH physics. We find that the latter is true for values of
the light-matter coupling and losses beyond the reach of current technology.
Throughout the study we operate in the weak pumping, fully quantum mechanical
regime where approaches such as mean field theory fail, and instead use a
combination of quantum trajectories and the time evolving block decimation
algorithm to compute the relevant steady state observables. In our study we
have assumed small to medium size arrays (from 3 up to 16 sites) and values of
the ratio of coupling to dissipation rate $g/\gamma \sim 20$ which makes our
results implementable with current designs in Circuit QED and with near future
photonic crystal set ups.
