Smoothed-Cubic Spin-Glass Model of Random Lasers

TL;DR

The study investigates the equilibrium glassy behavior of a dense, multimode random laser model with four-body quenched interactions under a realistic smoothed-cubic gain-saturation constraint. By formulating a mode-locking network via the Frequency Matching Condition and performing GPU-accelerated Parallel Tempering Monte Carlo, the authors identify a spin-glass transition whose critical behavior aligns with the Random Energy Model universality class. The smoothed-cubic constraint prevents intensity condensation, enabling simulations to large system sizes and showing energy spread across many modes in the glassy phase, as evidenced by the inverse participation ratios and Parisi overlap distributions. These results establish a robust, scalable framework for studying glassy random lasers with self-starting mode-locking and point to extensions to sparse networks and reintroduction of linear gain terms to connect with lasing thresholds and emission spectra.

Abstract

We study the equilibrium glassy behavior of a multimode random laser model with nonlinear four-body quenched disordered interactions and a global smoothed-cubic constraint on mode intensities. This constraint, which provides a more realistic representation of gain saturation than the commonly used spherical constraint, prevents intensity condensation while preserving the dense, long-range interaction structure characteristic of many multistate random lasers. The model effective Hamiltonian is a function of mode amplitudes with random frequencies and is defined on a complete mode-locked graph. Using large-scale GPU-accelerated Monte Carlo simulations with the Parallel Tempering algorithm, we analyze systems of varying sizes to probe their thermodynamic-limit behavior. Finite-size scaling of the specific heat, of the Parisi overlap distributions, and of the inverse participation ratio's reveals a spin-glass transition, with critical exponents matching the mean-field Random Energy Model universality class. The smoothed-cubic constraint produces broad, non-condensed intensity distributions, avoiding the pseudo-condensation seen in spherical models on the same interaction graph. Our results show that more realistic gain-saturation constraints preserve spin-glass characteristics while enabling simulations of larger, more dilute systems, providing a robust framework for studying glassy random lasers with self-starting mode-locking.

Figures (14)

• Figure 1: Distribution of the system intensity in a $p=4$ multi-mode-coupling model with a smoothed-cubic global constraint \ref{['SmoothedCubic']} at different (photonic) temperatures.
• Figure 2: Intensity spectrum $I(\lambda)$, for a single realization of the quenched disorder of a smoothed cubic random laser with $N=96$ modes at different temperatures (from top to bottom) $T=0.2,0.25,0.32,0.44, 0.67,1.06$. Notice the narrowing of the central part of the spectrum, because of FMC, as $T$ decreases. The dashed lines are the average spectra over $100$ distinct realizations of the quenced disordered couplings. The values of $\lambda$ are arbitrary and obtained as $\lambda = 1/(\omega_0 + r * (\omega_1-\omega_0))$ with $\omega_0 = 1.55\times 10^8$ and $\Delta \omega = 0.25 \times 10^8$ and $r\in[0,1]$. They are chosen to mimic the spectral domain in nm of typical random lasers.
• Figure 3: Scheme of the network connectivity scaling in relationship with the possibility of condensation. The case studied in the present work is the smoothed cubic one ($\rho=4$) with $p=4$ multi-spin interaction.
• Figure 4: Specific heat per spin of the $4$-phasor model on complete ML graph. for sizes $N\in[18,112]$. In the inset $c_V/N^{\alpha/\nu_{\rm eff}}$ is displayed for the largest sizes vs the rescaled reduced temperature, showing the collapse along a unique curve $\hat{f}_{C_V}(N^{1/\nu_{\rm eff}}t_N)$, cf. Eq. (\ref{['eq:cv_scaling']}).
• Figure 5: Finite size behavior of the estimate of the specific heat peaks, $T_c(N)$. Finite size scaling Eq. (\ref{['eq:FSS']}) has been performed including the $4$ largest sizes, on order to avoid pre-asymptotic effects from too smal $N$.