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We propose a class of quantum simulators for antiferromagnetic spin systems,based on coupled photonic cavities in presence of two-photon driving anddissipation. By modeling the coupling between the different cavities through ahopping term with negative amplitude, we solve numerically the quantum masterequation governing the dynamics of the open system and determine itsnon-equilibrium steady state. Under suitable conditions, the steady state canbe described in terms of the degenerate ground states of an antiferromagneticIsing model. When the geometry of the cavity array is incommensurate with theantiferromagnetic coupling, the steady state presents properties which bearfull analogy with those typical of the spin liquid phases arising in frustratedmagnets.

Simulating frustrated antiferromagnets with quadratically driven QED cavities