Frequency-Disorder Mediated Sync Resonance

• The paper demonstrates that intrinsic frequency disorder can induce non-monotonic synchronization resonance in semiconductor laser arrays by optimizing coupling strength.
• It employs a reduction from delayed Lang–Kobayashi equations to a Kuramoto-type gradient flow, elucidating effective thermodynamic potentials that align oscillator phases.
• Experimental results show that intermediate coupling (≈0.4 ns⁻¹) maximizes synchrony (≈0.84), highlighting practical strategies for coherent photonic systems.

Frequency-disorder mediated synchronization resonance refers to a class of phenomena in coupled oscillator systems where the presence of intrinsic frequency disorder—random detuning between oscillators—induces a non-monotonic, resonance-like dependence of the collective synchrony on the coupling strength. This concept challenges the canonical view that disorder merely degrades coherence, showing instead that optimal disorder can enhance synchronization, especially in regimes and platforms where identical, disorder-free systems either do not synchronize or display only trivial behavior. Disorder-induced enhancements and resonances are now observed in diverse physical implementations, with the underlying mechanisms rigorously explained in terms of effective thermodynamic potentials, scaling relations, and their associated gradient flows (Ye et al., 9 Sep 2025).

1. Disorder-Enhanced Synchronization in Semiconductor Laser Arrays

The interplay of disorder and synchronization is exemplified in arrays of coupled external-cavity semiconductor diode lasers where each laser exhibits intrinsic random frequency detuning $\Delta_i$ (Ye et al., 9 Sep 2025). In the disorder-free limit, all-to-all coupled lasers achieve perfect phase synchronization as soon as couplings are nonzero. However, as disorder is introduced, identical synchronization is lost at small coupling, and increasing coupling first enhances, and then suppresses, the overall synchrony measure $\langle S\rangle$; this nonmonotonicity—a resonance—is maximal at an optimal coupling $\kappa^*$. Notably, this effect strictly vanishes if there is no disorder, confirming that it is genuinely disorder-mediated.

The synchronization is quantified by

$$\langle S \rangle = \left\langle \frac{\left| \sum_{i=1}^M E_i(t) \right|^2}{M \sum_{i=1}^M |E_i(t)|^2} \right\rangle,$$

where $E_i(t)$ is the complex field of the $i$-th laser, and $M$ is the number of lasers.

2. Theoretical Mechanism: Effective Thermodynamic Potential

The physical mechanism for disorder-mediated synchronization resonance operates via the phase dynamics of the lasers, described by the delayed Lang–Kobayashi equations, which can be reduction-mapped to a generalized, time-delayed Kuramoto-type gradient flow for the delayed phase difference $\eta_i(t)$: $$\dot{\eta}_i(t) = -\frac{1}{\tau} \frac{dU(\eta_i)}{d\eta_i},$$ with the effective potential,

$$U(\eta_i) = \eta_i^2 - 2\tau \Delta_i\, \eta_i - 2\tau\sqrt{1+\alpha^2}\, k^{\mathrm{in}} \cos \left( \eta_i + \tan^{-1}\alpha + \omega_0 \tau \right).$$

Here,

• $\tau$ is the external cavity delay,
• $\alpha$ is the linewidth enhancement factor,
• $k^{\mathrm{in}} = \kappa(M-1)$ is the total incoming coupling,
• $\Delta_i$ is the random frequency detuning,
• The cosine term arises from delayed feedback and amplitude–phase coupling.

Disorder introduces a linear term $-2\tau\Delta_i\,\eta_i$ that shifts the minima of $U(\eta_i)$ for each oscillator. For moderate disorder and optimal coupling, the local minima of the effective potentials for all oscillators nearly coincide, aligning their steady-state phase differences and maximizing synchrony.

3. Experimental System and Parameter Dependencies

The experimental implementation uses a spatial light modulator (SLM) to achieve all-to-all coupling among $M$ single-transverse, single-longitudinal-mode semiconductor lasers (Ye et al., 9 Sep 2025). Each laser's intrinsic frequency is drawn from a Gaussian distribution via $\Delta_i = \sigma_{\Delta} \mathcal{N}(0,1)$.

• Key parameters: carrier loss $\gamma_n = 0.5\,\mathrm{ns}^{-1}$, cavity loss $\gamma=500\,\mathrm{ns}^{-1}$, external delay $\tau=3\,\mathrm{ns}$, linewidth enhancement factor $\alpha=5$, coupling strength per link $\kappa$, and disorder strength $\sigma_{\Delta}$.
• The optimal coupling $\kappa^*$ scales as $(M-1)^{-1}$, maintaining constant total incoming coupling $k^{\mathrm{in}}$.

Observed synchronization is maximal at intermediate coupling $\kappa^* \approx 0.4\,\mathrm{ns}^{-1}$ for $M=24$, with the synchrony measure peaking at $\langle S \rangle \approx 0.84$.

4. Scaling, Regime Boundaries, and Comparison to Disorder-Free Systems

The resonance phenomenon—a peak in synchrony at finite coupling—vanishes in the disorder-free ($\Delta_i \equiv 0$) limit, where any nonzero coupling suffices for perfect synchrony. With disorder, very weak coupling produces insufficient phase locking due to broad frequency detunings, while excessively strong coupling, in the presence of non-negligible amplitude–phase coupling ($\alpha$), can destabilize the synchronized state, leading to chaotic or multistable dynamics. This distinct maximum in $\langle S \rangle$ as a function of $\kappa$ is a signature of genuine disorder-mediated resonance.

5. Mathematical Structure and Gradient Flow

The steady-state condition for each oscillator's delayed phase difference follows: $$\frac{dU}{d\eta_i} = \eta_i - \tau \Delta_i + \tau\sqrt{1+\alpha^2}\,k^{\mathrm{in}}\sin(\eta_i + \tan^{-1} \alpha + \omega_0 \tau) = 0$$ The optimal resonance is obtained when, for a particular coupling $\kappa^*$, the collection of equations for all oscillators admits solutions $\{\eta_i^*\}$ that are narrowly distributed around a common value. The width of this distribution directly relates to the observed synchrony.

6. Broader Implications and Applications

These findings overturn the conventional expectation that frequency heterogeneity is detrimental to synchrony, showing instead that engineered disorder can be a control variable for maximizing global coherence (Ye et al., 9 Sep 2025). Practical implications include robust design of coherent laser arrays for optical information transfer, beam combining, and photonic computation. The experimental platform is capable of generalizing to other oscillator networks, such as vertical-cavity surface-emitting laser (VCSEL) arrays and nonlinear photonic or electronic systems.

Additionally, the underlying framework—mapping delayed coupled phase oscillator networks with disorder to effective thermodynamic potentials—may inspire new approaches for optimizing synchronization in complex networks where intrinsic or engineered disorder is unavoidable or desirable.

7. Concluding Comparison Table

System	Disorder-Free Behavior	Disorder-Mediated Resonance
Coupled diode lasers	Perfect synchrony at any $\kappa>0$ (in weak-coupling)	Synchrony peaks at intermediate $\kappa^*$; vanishes for $\kappa\to0$ or $\kappa\gg\kappa^*$
General oscillator networks	Monotonic synchrony with coupling	Non-monotonic, with an optimal synchrony as a function of coupling and disorder

In summary, frequency-disorder mediated synchronization resonance is a robust collective phenomenon whereby optimal frequency heterogeneity, combined with tunable coupling, generates a maximum in the global synchrony of coupled nonlinear oscillators. This effect is rigorously established in semiconductor laser arrays using both experimental and theoretical approaches, with broader applicability to a range of complex physical and engineered systems (Ye et al., 9 Sep 2025).