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

TL;DR

The work investigates two fundamentally different turbulent regimes in a conservative, Hamiltonian wave system described by coherently coupled vector NLSEs for optical-fiber polarization. It develops a reduced wave turbulence kinetic equation (WT KE) under statistical homogeneity to describe irreversible thermalization, and simultaneously derives an anomalous-correlator kinetic equation (AC-KE) that captures phase-correlations and a fast reversible dynamics, revealing a competition between regimes. Linear stability analysis yields a criterion $\alpha L_{nl}<\frac{2}{3}\frac{\Delta N^0}{N}$ for phase-correlations to emerge from uncorrelated initial states (with growth rate $\lambda(0)=2\sqrt{\alpha(\frac{2\gamma}{3}\Delta N^0-\alpha)}$), while a nonzero initial anomalous correlator drives reversible oscillations described by a Stokes-vector rotation on a Poincaré sphere with radius $S_0={\cal P}N$. Experiments in a 6.2 m weakly birefringent fiber observe both regimes and corroborate WT predictions and AC-KE dynamics, demonstrating that phase-correlations can mediate rapid reversible turbulence in a closed Hamiltonian wave system and suggesting a path toward a unified theory of reversible and irreversible turbulent evolution.

Abstract

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.

Figures (8)

• Figure 1: Irreversible thermalization. Evolution during the propagation of $N_{x,y}(z)$ (a), and corresponding degree of polarization ${\cal P}(z)$ [Eq.(\ref{['eq:P']})] (b), obtained from the simulation of the NLSE (\ref{['eq:nls']}) (5 realizations, dark and light blue lines), and the simulation of the WT KE (\ref{['eq:wtkin']}) (red lines). Parameters: $\alpha L_{nl}=1, \Delta N^0=0, \sigma \tau_0=0.8 \pi, \kappa=2/3$, with $L_{nl}=1/(\gamma N)$ the nonlinear length and $\tau_0=\sqrt{|\beta| L_{nl}}$ the 'healing time' ($|E/U| \simeq (\sigma \tau_0)^2$).
• Figure 2: Reversible turbulent dynamics. (a) Exponential growth of the anomalous correlator starting from initially uncorrelated random waves $|M(z=0)| \simeq 0$: Simulations of the NLSE (\ref{['eq:nls']}) (100 realizations, black lines), and corresponding average (green line). The red line reports the theoretical prediction Eq.(\ref{['eq:lambda_0']}). (b) Nonlinear dynamics on the Poincaré sphere: Simulation of the NLSE (\ref{['eq:nls']}) (red line), and corresponding theoretical prediction from the simulation of AC-KE (\ref{['eq:s_dyn']}) (black line), showing the periodic exchange among the normal and anomalous correlators. The blue line reports the homoclinic orbit emanating from the unstable fixed point $S_1=+S_0$, i.e., $|M|=0$. Parameters: $\alpha L_{nl}=0.2, \sigma \tau_0=4\pi, \Delta N^0/N=0.6, \kappa=2/3, L_{nl}=1/(\gamma N), \tau_0=\sqrt{|\beta| L_{nl}}$ ($|E/U| \simeq (\sigma \tau_0)^2$).
• Figure 3: Observation of optical repolarization. (a) Experimental measurements of the normalized power difference $\Delta N/N$ (blue dots), and degree of polarization ${\cal P}$ (red crosses), vs input power: In the thermalization regime, the anomalous correlator $M$ is negligible, and ${\cal P} \simeq \Delta N/N$, see Eq.(\ref{['eq:P']}). The measured repolarization is in good agreement with the simulations of NLSE (\ref{['eq:nls']}) (dashed black lines). (b) Measured power spectrum at the input (blue line), and output (orange line), of the fiber. The experimental spectra agree well with the simulation of NLSE (\ref{['eq:nls']}) (black lines), and exhibit a power-law signature of light thermalization $\sim \omega^{-2}$ in the spectral tail.
• Figure 4: Observation of the reversible turbulent regime. Measurements, at the fiber output and as a function of the input power, of (a) the power difference $\Delta N/N$ and (b) the real and imaginary parts of the normalized anomalous correlator $M/N$. The insets in (a)-(b) show the dynamics along the fiber length $L=6.2$m for $\Delta N/N$ and $M^i/N$ with an input power of $50\ \rm{ W}$ ($L_{nl}\approx 0.83$ m) in the strongly nonlinear regime $|E/U| \approx 1$: The blue and red lines report the simulations of NLSE (\ref{['eq:nls']}) and AC-KE (\ref{['eq:s_dyn']}), respectively. (c) Corresponding field evolution on the Poincaré sphere for the NLSE (blue) and the AC-KE (\ref{['eq:s_dyn']}) (red). Note in the insets (a)-(b) and in panel (c) that the AC-KE (\ref{['eq:s_dyn']}) (valid in the weakly nonlinear regime $|E/U| \gg 1$) still provides a qualitative description of the oscillatory dynamics even in the strongly nonlinear regime $|E/U| \approx 1$.
• Figure 5: Irreversible thermalization. Numerical simulations of the NLSE (\ref{['eq:nls_x']}-\ref{['eq:nls_y']}) (blue and light-blue lines), and the wave turbulence KE (\ref{['eq:red_wtkin_x']}-\ref{['eq:red_wtkin_y']}) (red lines): (a) Evolution of the kinetic $E_k(z)$, and coherent $E_c(z)$, contribution to the energy $E=E_k(z)+E_c(z)$ in Eq.(\ref{['eq:E_lin']}), the inset shows the corresponding evolution of the entropy $S(z)$. Spectra of the waves $n_x(\omega)$ (b), and $n_y(\omega)$ (c) at $z=3000 L_{nl}$. The thermalization process is characterized by a growth of the kinetic energy $E_k(z)$, which entails a decrease of $E_c(z)$: The irreversible transfer of power (particles) from $N_x$ to $N_y$ entails the repolarization effect (increase of ${\cal P}(z)$), see Fig. 1 (main text). An average over 5 realizations of the initial condition (i.e., initial random spectral phases) has been considered for the NLSE simulations.