Mapping the XY Hamiltonian onto a Network of Coupled Lasers

Abstract

In recent years there has been a growing interest in the physical implementation of classical spin models through networks of optical oscillators. However, a key missing step in this mapping is to formally prove that the dynamics of such a nonlinear dynamical system is toward minimizing a global cost function which is equivalent with the spin model Hamiltonian. Here, we introduce a minimal dynamical model for a network of dissipatively coupled optical oscillators and prove that the dynamics of such a system is governed by a Lyapunov function that serves as a cost function for the system. This cost function is in general a function of both phases and intensities of the oscillators and depends strongly on the pump parameter. In case of bipartite network topologies, the amplitudes of the oscillators become identical in the steady state and the cost function reduces to the XY Hamiltonian. In the general case for non-trivial network topologies, however, the cost function approaches the XY Hamiltonian only in the strong pump limit. We show that by adiabatically tuning the pump parameter, the network can largely avoid trapping into the local minima of the governing cost function and stabilize into the ground state of the associated XY Hamiltonian. These results show the great potential of laser networks for unconventional computing.

References in corpus (7)

• Double-slit photoelectron interference in strong-field ionization of the neon dimer
• Network of Time-Multiplexed Optical Parametric Oscillators as a Coherent Ising Machine
• A Coherent Ising Machine Based On Degenerate Optical Parametric Oscillators
• Large-scale photonic Ising machine by spatial light modulation
• Realizing the $XY$ Hamiltonian in polariton simulators
• Simulating Ising and Potts models and external fields with non-equilibrium condensates
• Realizing spin-Hamiltonians in nanoscale active photonic lattices

Cited by in corpus (8)

• Experimental Observation of Phase Transitions in Spatial Photonic Ising Machine
• Multidimensional hyperspin machine
• Wavelength-division multiplexing optical Ising simulator enabling fully programmable spin couplings and external magnetic fields
• Machine learning of phase transitions in nonlinear polariton lattices
• Synchronization in coupled laser arrays with correlated and uncorrelated disorder
• Self-organized vortex and antivortex patterns in laser arrays
• 10-GHz-clock time-multiplexed non-degenerate optical parametric oscillator network with a variable planar lightwave circuit interferometer
• Equalized Hyperspin Machine