Simulating Ising Spins in External Magnetic Fields with a Network of Degenerate Optical Parametric Oscillators

Abstract

We report a coherent Ising machine (CIM) that can simulate the Ising model with external magnetic fields. To represent Ising spins, the CIM employs degenerate optical parametric oscillator (DOPO) pulses, which are coupled by a measurement-feedback scheme. By adding to a feedback signal a term that is proportional to a given magnetic field and an absolute value of the measured amplitude of a DOPO pulse, the networked DOPO pulses behave like Ising spins in a magnetic field. Using the CIM, we implement antiferromagnetically coupled one- and two-dimensional networks of the Ising spins in uniform magnetic fields. At the field amplitudes that agree with a theoretical prediction, we experimentally observe a transition from an antiferromagnetic state governed by spin-spin interactions to a ferromagnetic state induced by the external magnetic field. In addition, staggered magnetization measurement reveals that the CIM behaves differently depending on the network dimensionality, which is consistent with the results obtained with the analytical solution and Monte Carlo simulations, but cannot be reproduced with a simple mean-field theory. These results suggest that the CIM clearly simulates the behavior of Ising spin networks better than a mere mean-field solver and will thus become a useful tool for simulating various statistical physics problems based on the Ising model.