Mapping the XY Hamiltonian onto a Network of Coupled Lasers

TL;DR

This paper demonstrates that a network of dissipatively coupled optical oscillators can be modeled with a Lyapunov function that acts as a cost function, which in certain conditions reduces to the XY Hamiltonian, enabling potential for unconventional computing.

Contribution

The authors introduce a minimal dynamical model for optical oscillator networks and prove the connection between their Lyapunov-based cost function and the XY Hamiltonian.

Findings

The cost function governs the system dynamics and depends on phases, intensities, and pump parameter.

In bipartite networks, the steady state amplitudes become identical, simplifying the cost function to the XY Hamiltonian.

Adiabatic tuning of the pump parameter allows the network to reach the XY Hamiltonian ground state, avoiding local minima.

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.