Emergent Equilibrium in All-Optical Single Quantum-Trajectory Ising Machines

We investigate the dynamics of multimode optical systems driven by two-photon processes and subject to nonlocal losses, incorporating quantum noise at the Gaussian level. Our findings show that the statistics retrieved from a single Gaussian quantum trajectory exhibits emergent thermal equilibrium governed by an Ising Hamiltonian, encoded in the dissipative coupling between modes. The system's effective temperature is set by the driving strength relative to the oscillation threshold. Given the ultrashort timescales typical of all-optical devices, our Letter demonstrates that such multimode optical systems can operate as ultrafast Boltzmann samplers, paving the way toward the realization of efficient hardware for combinatorial optimization, with promising applications in machine learning and beyond.