Simulating frustrated antiferromagnets with quadratically driven QED cavities

We propose a class of quantum simulators for antiferromagnetic spin systems based on coupled photonic cavities in the presence of two-photon driving and dissipation. By modeling the coupling between the different cavities through a hopping term with negative amplitude, we solve numerically the quantum master equation governing the dynamics of the open system and determine its nonequilibrium steady state. Under suitable conditions, the steady state can be described in terms of the degenerate ground states of an antiferromagnetic Ising model. When the geometry of the cavity array is incommensurate with the antiferromagnetic coupling, the steady state presents properties which bear full analogy with those typical of the spin-liquid phases arising in frustrated magnets.