Out-of-equilibrium phases in driven-dissipative coupled resonator arrays

Coupled QED resonator arrays have been shown to exhibit interesting many-body physics including Mott and Fractional Hall states of photons. One of the main differences between these photonic quantum simulators and their cold atoms counterparts is in the dissipative nature of their photonic excitations. The natural equilibrium state is where there are no photons left in the cavity. Pumping the sys- tem with external drives is therefore necessary to compensate for the dissipation and realize non-trivial states. The external driving here can in easily be tuned to be inco- herent, coherent or quantum, opening the road for exploration of many body regimes beyond the reach of other approaches. In this chapter, we review some of the physics arising in driven-dissipative coupled resonator arrays including photon fermioniza- tion, crystallization, as well as photonic quantum Hall physics out of equilibrium. We start by briefly describing possible experimental candidates to realize coupled resonator arrays along with the two theoretical models that capture their physics, the Jaynes-Cummings Hubbard and Bose Hubbard Hamiltonians, highlighting the different regimes of applicability for each. A brief review of the analytical and so- phisticated numerical methods required to tackle these systems is included.