Quantum annealing with ultracold atoms in a multimode optical resonator

A dilutely filled N-site optical lattice near zero temperature within a high-Q multimode cavity can be mapped to a spin ensemble with tailorable interactions at all length scales. The effective full site to site interaction matrix can be dynamically controlled by the application of up to $N(N+1)/2$ laser beams of suitable geometry, frequency, and power, which allows for the implementation of quantum annealing dynamics relying on the all-to-all effective spin coupling controllable in real time. Via an adiabatic sweep starting from a superfluid initial state one can find the lowest-energy stationary state of this system. As the cavity modes are lossy, errors can be amended and the ground state can still be reached even from a finite temperature state via ground-state cavity cooling. The physical properties of the final atomic state can be directly and almost nondestructively read off from the cavity output fields. As an example we simulate a quantum Hopfield associative memory scheme.