Instantons and Rarefaction Pulses as Pathways to Global Phase Coherence in Gain-Based Optical Networks

We investigate how to reliably remove unwanted global phase windings in gain-based optical oscil-lator networks, thereby ensuring convergence to the true synchronized configuration correspondingto the XY Hamiltonian’s global minimum. Focusing on one-dimensional rings and two-dimensionaltoroidal lattices, we show that two key strategies greatly enhance the probability of reaching thedefect-free state. First, operating at a low effective injection rate just above threshold, exploits theamplitude degree of freedom, allowing the system to form transient zero-amplitude holes: instantonsin one dimension, or vortex-antivortex rarefaction pulses in two-dimensional space, that enable phaseslips. Second, preparing the initial condition with amplitude or phase inhomogeneities can directlyseed these amplitude collapses and prevent the system from getting trapped in higher-energy stateswith nonzero winding. Using the Stuart–Landau and Ginzburg–Landau equations as models forfast-reservoir lasers, we derive analytic and numerical evidence that even relatively minor amplitudedips can trigger global unwinding. We further demonstrate that the pump’s slow annealing favoursthese amplitude-driven events, leading to improved success in finding the globally coherent groundstate. These findings highlight the critical role of amplitude freedom in analogue solvers for XYoptimization problems, showing how local amplitude suppression provides a direct route to ejectingtopological defects.