Irreversible thermalization vs reversible dynamics mediated by anomalous correlators: Wave turbulence theory and experiments in optical fibers

We theoretically and experimentally investigate spontaneous self-organization in a conservative (Hamiltonian) turbulent wave system, operating far from thermodynamic equilibrium. Our system is governed by two coherently coupled nonlinear Schrödinger equations, describing the polarization evolution of light in a dispersive nonlinear optical fiber. The analysis reveals the emergence of two fundamentally distinct turbulent regimes. In a first regime, the waves undergo a slow, irreversible thermalization process, which is accurately described by the wave turbulence kinetic equation and the associated H-theorem of entropy growth. In stark contrast with this expected irreversible process, we identify a second different regime, where strong phase-correlations spontaneously emerge, giving rise to a fast reversible oscillatory dynamics of the normal correlator and anomalous phase-correlator. Experimental observations confirm the occurrence of both irreversible thermalization and reversible dynamics mediated by the anomalous correlated fluctuations.

💡 Research Summary

In this work the authors investigate two fundamentally different turbulent regimes that can arise in a conservative (Hamiltonian) wave system described by two coherently coupled nonlinear Schrödinger equations (NLSEs). The model captures the polarization dynamics of light propagating in a weakly birefringent silica fiber, with the propagation distance playing the role of an evolution “time”. The equations contain linear dispersion, a weak birefringent coupling term (α), and nonlinear interaction coefficients (γ, κ, ρ). The system conserves total power N and the Hamiltonian H, allowing a clean separation between reversible and irreversible processes.
The first regime corresponds to the standard weak‑wave‑turbulence (WT) picture. Assuming statistical homogeneity and the absence of phase correlations between the two polarization components, the authors derive a kinetic equation (KE) for the averaged spectra nₓ(ω) and n_y(ω). This KE contains cubic collision integrals that describe four‑wave interactions and obey an H‑theorem: the entropy S monotonically increases with propagation distance, driving the system toward a Rayleigh‑Jeans equilibrium. Numerical simulations of the full NLSE with initially uncorrelated, equal‑power Gaussian spectra show excellent agreement with the KE predictions. In this regime the anomalous correlator M remains negligible, the degree of polarization P is essentially given by the power imbalance ΔN/N, and the dynamics is a slow, irreversible thermalization.
The second regime emerges when the assumption of vanishing phase correlations is relaxed. Starting from the same uncorrelated initial condition, the authors retain the anomalous correlator m(ω,t,z) in the statistical description and derive a set of coupled equations for the normal and anomalous correlators (Eqs. 4‑5), which they term the anomalous‑correlator kinetic equation (AC‑KE). Linear stability analysis of the AC‑KE reveals a growth rate λ(Ω) for the anomalous correlator; the most unstable mode is homogeneous (Ω = 0) with a growth rate λ₀ = 2α(2γΔN₀/3 − α). The instability condition α L_nl < (2/3)ΔN₀/N shows that a sufficient initial power imbalance is required for the anomalous correlations to develop. Once the anomalous correlator grows, the normal and anomalous components form a Stokes vector S = (ΔN, 2M_r, 2M_i)ᵀ that precesses on a Poincaré sphere. This precession conserves the total “polarization length” S₀, i.e., ΔN² + 4|M|² = const, leading to periodic, reversible exchange between the power imbalance and the anomalous correlator. Numerical NLSE simulations confirm exponential growth of |M| followed by sustained oscillations, in perfect agreement with the AC‑KE predictions.
Experimentally, the authors generate 100‑ps, temporally incoherent pulses centered at 558 nm with a 1.93 THz bandwidth, split them into two orthogonal polarizations with controllable power imbalance, and launch them into a 6.2 m weakly birefringent fiber. By measuring output spectra, powers, and the anomalous correlator using an optical spectrum analyzer and a polarimeter, they observe both regimes. When the input powers are equal (ΔN₀≈0), the output degree of polarization follows P≈ΔN/N and the anomalous correlator remains below detection, confirming irreversible thermalization. When a sizable power imbalance (ΔN₀/N≈0.6) is introduced, the anomalous correlator grows rapidly, the degree of polarization stays constant, and ΔN and |M| oscillate periodically, evidencing the reversible dynamics predicted by the AC‑KE. The measured spectra exhibit the expected power‑law tails and match the NLSE simulations quantitatively.
Overall, the paper demonstrates that in a Hamiltonian wave system the conventional wave‑turbulence kinetic equation accurately describes irreversible thermalization under phase‑uncorrelated conditions, while the spontaneous emergence of phase (anomalous) correlations can suppress thermalization and give rise to a distinct, fast, reversible turbulent dynamics. This duality enriches our understanding of far‑from‑equilibrium self‑organization in conservative systems and opens avenues for exploring similar phenomena in multimode, higher‑dimensional, or quantum wave systems.