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

1. Problem Statement

The paper addresses a fundamental question in the statistical physics of non-integrable Hamiltonian systems: How do random waves evolve toward thermodynamic equilibrium, and under what conditions can this irreversible process be inhibited by reversible dynamics?

• Context: In weakly nonlinear regimes, random waves typically undergo thermalization, an irreversible evolution toward a maximum entropy state (Rayleigh-Jeans distribution). This is traditionally described by Wave Turbulence (WT) theory and the associated kinetic equation (KE), which assumes the neglect of phase correlations between waves.
• The Gap: While phase correlations (or "anomalous correlators") are known to drive coherent structures like solitons in strongly nonlinear regimes, their role in the weakly nonlinear regime of random dispersive waves is less understood. Recent theoretical work suggested that phase-invariant scalar systems cannot develop anomalous correlations. However, the behavior of vector systems (coupled waves) with phase invariance remains an open question.
• Objective: To investigate whether a conservative (Hamiltonian) system of coupled nonlinear waves can exhibit two distinct regimes: standard irreversible thermalization and a novel regime of fast, reversible dynamics driven by the spontaneous emergence of anomalous phase correlations.

2. Methodology

The authors employed a combined theoretical, numerical, and experimental approach using temporally incoherent light propagating in a weakly birefringent optical fiber.

• Theoretical Model:

  • The system is governed by two coherently coupled Nonlinear Schrödinger Equations (NLSE) describing the polarization evolution ($u_x, u_y$) of light.
  • Standard WT Approach: They derived a kinetic equation (KE) assuming statistical homogeneity and neglecting anomalous correlators ($m=0$). This predicts irreversible thermalization.
  • Anomalous Correlator Approach: They derived a generalized set of equations (Anomalous Correlator Kinetic Equation, AC-KE) that tracks the evolution of both normal correlators ($n_\mu$) and anomalous correlators ($m$, representing cross-polarization phase correlations).
  • Stability Analysis: A linear stability analysis was performed on the AC-KE to determine the conditions under which anomalous correlations grow exponentially from an initially uncorrelated state.

• Numerical Simulations:

  • Direct numerical integration of the coupled NLSEs.
  • Numerical integration of the reduced WT KE and the AC-KE to compare predictions against the full wave dynamics.

• Experimental Setup:

  • Source: A temporally incoherent optical pulse (100 ps duration, ~1.93 THz bandwidth) at 558 nm.
  • Generation: Incoherence was generated via the Raman beam clean-up effect in a multimode fiber, followed by spectral filtering.
  • Propagation: The pulse was split into two orthogonal linear polarizations with a tunable power imbalance and a time delay ($\Delta \tau \approx 15$ ps) exceeding the coherence time to ensure initial decorrelation ($|M| \approx 0$).
  • Medium: A 6.2-meter-long weakly birefringent silica fiber (spun fiber).
  • Measurement: Optical Spectrum Analyzer (OSA) and a custom polarimeter measured the power spectra, power imbalance ($\Delta N$), and the complex anomalous correlator ($M$) at the output.

3. Key Contributions

1. Identification of Two Distinct Turbulent Regimes:
   The study reveals that a single Hamiltonian system can exhibit two fundamentally different behaviors depending on initial conditions and parameters:

   • Regime I: Irreversible Thermalization. When phase correlations are negligible, the system follows standard WT theory, evolving irreversibly toward equilibrium with entropy growth ($H$-theorem).
   • Regime II: Reversible Oscillatory Dynamics. Under specific conditions, anomalous phase correlations ($m$) spontaneously emerge and grow exponentially, driving a fast, reversible exchange of energy between polarization components.

2. Derivation of the Anomalous Correlator Kinetic Equation (AC-KE):
   The authors derived a new set of kinetic equations (Eqs. 4-5 in the text) that couple normal and anomalous correlators. Unlike the standard KE, the AC-KE is formally reversible in the propagation variable $z$. It describes the dynamics as a rotation of a Stokes vector on the Poincaré sphere, conserving the degree of polarization.

3. Theoretical Proof of Instability in Phase-Invariant Vector Systems:
   Contrary to previous findings in scalar systems (where phase invariance precludes anomalous correlations), the authors proved that in vector systems with coherent coupling, anomalous correlations can emerge spontaneously even if the Hamiltonian is phase-invariant. They derived an instability criterion (Eq. 7):
   $$\alpha L_{nl} < \frac{2}{3} \frac{\Delta N_0}{N}$$
   where $\alpha$ is the birefringence, $L_{nl}$ is the nonlinear length, and $\Delta N_0$ is the initial power imbalance.

4. Experimental Validation:
   This work provides the first experimental evidence of reversible turbulent dynamics mediated by anomalous correlations in a conservative optical system, contrasting with the well-known irreversible thermalization.

4. Key Results

• Irreversible Thermalization (Fig. 1 & 3):

  • When the initial power imbalance is small or the system parameters do not satisfy the instability criterion, the anomalous correlator $M$ remains negligible.
  • The system undergoes slow thermalization: power transfers irreversibly between polarization axes, and the degree of polarization $P$ increases until it matches the power imbalance ($P \approx \Delta N/N$).
  • The spectral tail follows a power law $\sim \omega^{-2}$, characteristic of Rayleigh-Jeans equilibrium.
  • Experimental data matches the standard WT KE predictions perfectly.

• Reversible Dynamics (Fig. 2 & 4):

  • When the instability criterion is met (or when a large initial anomalous correlation is artificially induced), $M$ grows exponentially.
  • The system enters a fast, oscillatory regime where the normal correlator ($\Delta N$) and anomalous correlator ($M$) exchange energy periodically.
  • The degree of polarization $P$ remains constant throughout the evolution (unlike in thermalization), as the system evolves on a fixed-radius Poincaré sphere.
  • The AC-KE accurately predicts this oscillatory behavior, even in regimes where the nonlinearity is strong ($|E/U| \approx 1$), demonstrating the robustness of the kinetic description.

• Comparison of Timescales:

  • Thermalization: Driven by cubic nonlinearities (four-wave mixing), it is a slow process.
  • Reversible Dynamics: Driven by quadratic nonlinearities in the AC-KE, it is a fast process.

5. Significance and Impact

• Fundamental Physics: The work challenges the assumption that all weakly nonlinear Hamiltonian systems inevitably thermalize. It demonstrates that phase correlations can act as a mechanism to arrest thermalization and sustain reversible, coherent-like dynamics even in random wave systems.
• Optical Thermodynamics: It expands the emerging field of optical thermodynamics by showing that "temperature" and "equilibrium" are not the only possible end-states for light in fibers; reversible self-organization is also possible.
• Unified Theory: The paper highlights the need for a generalized theory that unifies irreversible thermalization (standard WT) and reversible correlation-driven dynamics. It suggests that the interplay between these two regimes is a critical area for future research.
• Experimental Benchmark: By successfully observing these contrasting regimes in a single experimental setup, the study validates the theoretical framework of anomalous correlators in vector wave turbulence, bridging the gap between abstract statistical mechanics and practical nonlinear optics.

In conclusion, the paper establishes that anomalous phase correlations are not merely a feature of strongly nonlinear coherent structures but can spontaneously emerge in weakly nonlinear random waves, fundamentally altering the thermodynamic evolution of the system from irreversible thermalization to reversible oscillatory dynamics.