Polariton XY-simulators revisited

Arrays of bosonic condensates of exciton-polaritons have emerged as a promising platform for simulating classical XY models, capable of rapidly reaching phase-locked states that may be mapped to arrays of two-dimensional classical spins. However, it remains unclear whether these states genuinely minimize the corresponding XY Hamiltonian and how the convergence time scales with the system size. Here, we develop an analytical model revealing that an array of $N$ condensates possesses $N$ stable phase configurations. The system selectively amplifies a specific configuration dependent on the pump power: at low power, the state with the smallest eigenvalue of an effective XY Hamiltonian is favored, while at high power, the state with the largest eigenvalue prevails. At intermediate pump powers, the system visits all eigenstates of the Hamiltonian. Crucially, the formation rate for any of these phase-locked states remains on the order of 100 ps, independent of the size of the array, demonstrating the exceptional speed and scalability of polariton-based XY simulators.