Feedback-Driven Ground-State Search in Coupled Laser Arrays

TL;DR

Optimization problems can be mapped to ground-state searches of spin Hamiltonians, but reaching the global minimum is hindered by local minima. The authors propose an intrinsic feedback-driven adaptive annealing mechanism in class-B semiconductor laser arrays, where the interplay of internal coupling $\alpha$ and external coupling $\eta$ creates amplitude fluctuations that act as an effective temperature to reshape the potential and escape local minima. In a 1D ring of $N=20$ lasers, they identify a defect-free regime (region (ii-a)) with near-100% ground-state probability and show a universal Kibble–Zurek–style scaling where defect probability collapses when plotted against the ratio $\langle t_{phase}\rangle/\langle t_{amp}\rangle$. This work establishes feedback-driven annealing as a practical, scalable route to ground-state search in optical spin simulators and is readily testable in VCSEL arrays.

Abstract

Optimisation problems, which appear in numerous fields of science and industry, are challenging to solve even with modern supercomputers. Many such problems can be mapped onto ground-state searches of spin Hamiltonians, implemented on various physical platforms whose intrinsic dynamics are analogous to spin systems. However, the complex energy landscape of spin Hamiltonians often traps the system in local minima, preventing the system from reaching the ground-state (global minimum). We demonstrate an intrinsic feedback-driven annealing mechanism in class-B semiconductor laser arrays arising from the interplay of internal ($α$) and external ($η$) coupling. The instantaneous phase configuration self-modulates amplitude fluctuations, which act as an effective temperature, dynamically reshaping the potential and enabling the system to escape from local minima. Using a one-dimensional ring laser array, we analyze defect formation in the $α$-$η$ parameter space and identify an optimal regime achieving nearly 100% ground-state probability. Although both $α$ and $η$ are essential for the feedback loop, defect suppression results from modifying two competing timescales: amplitude stabilization (t_amp) and phase locking (t_phase), analogous to the Kibble-Zurek mechanism. These timescales can be tuned independently via $α$ or $η$. Identical timescale ratios yield identical defect probabilities, confirming that relative timescales, not specific parameters, govern defect formation. Our findings establish internal feedback-driven annealing as a practical route to ground-state search in semiconductor laser arrays, providing a foundation for efficient and scalable laser-based spin simulators for tackling hard optimization problems.

Figures (12)

• Figure 1: (a) One-dimensional ring array of $N=10$ semiconductor lasers with dissipative coupling $\eta$ between nearest neighbours (external coupling). The factor $\alpha$ provides amplitude-phase coupling within each laser (internal coupling). (b) The corresponding steady-state phase-locked solutions are $\phi_i = i q 2\pi/10$, where $i$ is the laser index and $q$ is the topological charge. Here, $q=0$ represents the in-phase state and $q\neq0$ denotes topological defect states. (c) The schematic loss landscape, where the in-phase state ($q=0$) is the global minimum, while states with topological defects ($q\neq0$) correspond to local minima.
• Figure 2: In a 1-D ring array of $N=20$ lasers, time evolution of order parameter ($\rho$) leading to steady-state (a) ground state ($\rho=1$), and (b) topological defect state ($\rho=0$). The insets show the phase distributions at different times during evolution. The simulation parameters are $\eta=0.005$, $r=2$ and $\alpha$=0.
• Figure 3: Time evolution of $\sigma$ and $\rho$ in a 1-D ring array of 20 lasers (a, b) without feedback $\alpha=0$, and (c, d) with feedback $\alpha=4$. Simulation parameters are $\eta=0.005$ and $r=2$.
• Figure 4: In a 1-D ring array of $N=20$ lasers, the probability of topological defects ($\mathrm{P_d}$) as a function $\alpha$ and $\eta$ at a fixed value of $r=3$, indicating four different key regimes of laser array dynamics. Probability is determined by averaging the results over 2500 different random initial conditions. Note, in the white region the 1D ring array of lasers exhibits unstable behavior and hence no phase-locking is obtained. The insets indicate the representative dynamics of the order parameter $\rho$ and amplitude fluctuations $\sigma$ for each of the regions marked by the arrow.
• Figure 5: Dynamics of the phase difference $\Delta\phi$ for a representative laser pair (lasers 1 and 2) in a 1D ring array of $N = 20$ lasers, described by an effective potential $V_{\mathrm{tot}}(\Delta\phi)$ for $N = 2$. (a) Effective potential without feedback ($\alpha = 0$), shown as the solid blue curve. Vertical lines indicate the phase differences $\Delta\phi = 2\pi/20$ (brown), $0$ (green), and $-2\pi/20$ (pink), corresponding to the $q = +1,~0,~-1$ phase-locked states of the $N = 20$ array, which represent local and global minima in the higher-dimensional phase space of the full array. (b) Effective potential with feedback for different values of $\alpha$. Feedback asymmetrically tilts the potential; the sign of the driving term $\alpha \Delta n$ determines the tilt direction, while its magnitude controls the tilt strength. (c) Time evolution of the order parameter $\rho$ without feedback ($\alpha = 0$). Insets show the instantaneous potential and the effective phase-difference position (red dot) at the indicated times. In the absence of feedback, the potential remains static and the phases become trapped in a local minimum. (d) Time evolution of $\rho$ with feedback ($\alpha = 4$). Strong initial fluctuations in $\rho$ arise from the dynamically evolving, flattened asymmetric potential (insets), enabling the phases to escape from local minima. The later insets show relaxation toward the ground-state ($\rho \rightarrow 1$), where oscillations vanish and the system stabilizes. For panels (b) and (d), the simulation parameters are $\eta=0.005$ and $r=2$.