We propose a photonic quantum simulator for anti-ferromagnetic spin systems based on reservoir engineering. We consider a scheme where quadratically driven dissipative Kerr cavities are indirectly coupled via lossy ancillary cavities. We show that the ancillary cavities can produce an effective dissipative and Hamiltonian anti-ferromagnetic-like coupling between the cavities. By solving the master equation for a triangular cavity configuration, we demonstrate that the non-equilibrium steady state of the system bears full analogy with the ground state of an antiferromagnetic Ising model, exhibiting key signatures of frustration. We show that when the effective photon hopping amplitude is zero, the engineered non-local dissipation alone is capable of inducing antiferromagnetic interaction and frustration. This simple scheme can be generalised to arbitrary lattice geometries, providing a fully controllable recipe for simulating antiferromagnetism and frustration on a controlled quantum optical platform.
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