Experimental observation of ballistic correlations in integrable turbulence

TL;DR

The study reports the first direct observation of ballistic two-time correlations in an integrable, photonic system governed by the focusing nonlinear Schrödinger equation, realized in a recirculating optical fiber loop. By recording full space–time dynamics of partially coherent waves and extracting the intensity DOS via inverse scattering, the authors test generalized hydrodynamics (GHD) in a classical integrable PDE context. The measured correlators exhibit ballistic scaling and agree quantitatively with parameter-free GHD predictions when the DOS is accurately reconstructed, validating GHD as a predictive framework for integrable turbulence in waves. This work provides a parameter-free, experimental benchmark for GGE and DOS-based hydrodynamics in an optical platform and opens avenues to study transient approaches to ballistic scaling, diffusion onset, and higher-order correlation phenomena. The combination of high-fidelity field measurements and rigorous IST/GHD analysis demonstrates a powerful route to probe fundamental questions in integrable dynamics and statistical mechanics of soliton gases.

Abstract

Unequal-time correlation functions fundamentally characterize emergent statistical properties in complex systems, yet their direct measurement in experiments is challenging. We report the experimental observation of two-time, ballistic correlations in a photonic platform governed by the focusing nonlinear Schrödinger equation. Using a recirculating optical fiber loop with heterodyne field detection, we acquire the full space-time dynamics of partially coherent optical waves and extract the intensity correlator in stationary states of integrable turbulence. The correlators collapse under ballistic rescaling and quantitatively agree with predictions from Generalized Hydrodynamics evaluated using the density of states obtained via inverse scattering analysis of the recorded fields. Our results provide a direct, parameter-free test of GHD in an integrable waves system.

Figures (12)

• Figure 1: Experimental setup--- The initial conditions are PCWs generated by an ASE source. The signal is spectrally filtered, amplified by an Erbium Doped fiber Amplifier (EDFA). A 1µ s-long section of the signal is sliced by using an Acousto-Optic Modulator (AOM) before being launched into the 5km-long single-mode fiber loop. The signal is recorded at each round trip using both direct and heterodyne detection of the field with photodetectors (PD). Losses are compensated by using a contrapropagating Raman amplifier.
• Figure 2: Ballistic correlation functions in optical fibers.--- (a) Spatio-temporal dynamics of the intensity $|\psi(t, x)|^2$ of a typical field configuration evolving from PCWs. Solitons form at early time and propagate undergoing elastic scattering. $(z, T)$ in physical units correspond to the dimensionless variables $(t,x)$. (b) Evolution of the kurtosis $\kappa_4(t)$. After a short transient, the kurtosis reaches a stationary value (for $t>5$), hinting at the relaxation of the system to a GGE. (c) Correlation function of the intensity $C(\Delta t,\Delta x)=\langle|\psi(t, x)|^2|\psi(t_0, x_0)|^2\rangle-\langle|\psi(t, x)|^2\rangle\langle|\psi(t_0, x_0)|^2\rangle$ with $\Delta t=t-t_0$ and $\Delta x= x- x_0$ for $t_0=7.5$, and its ballistic rescaling $C(\Delta t, \Delta x)=\frac{1}{\Delta t}\mathcal{C}(\Delta x/\Delta t)$ (inset). All figures are plotted for $\Delta k = 1.80$.
• Figure 3: Density of states (DOS) measurement. (a) IST-extracted DOS $\rho(\lambda)$ in the complex $\lambda$-plane at $t=t_0=7.5$ (initial spectral width of PCWs $\Delta k=1.80$) suppmat. Since $\rho(\lambda)$ diverges as $\lambda\to\pm\infty+i0$, we plot $\Im(\lambda)\rho(\lambda)$, which remains finite Koch2022. (b,c) DOS marginals $P_{\Re}=\int d\Im(\lambda)\,\Im(\lambda)\rho(\lambda)$ and $P_{\Im}=\int d\Re(\lambda)\,\Im(\lambda)\rho(\lambda)$. Curves at $t=0$ (orange, solid) and $t=7.5$ (black, dotted) demonstrate DOS conservation.
• Figure 4: Comparison between experimental and theoretical correlations--- Theoretical rescaled ballistic correlations computed from Eq. (\ref{['eq_GHD_C']}) (black solid lines) and average experimental correlations (solid green line); shaded green area: root mean square. (a) Strong nonlinearity $\Delta k=1.80$ (b) Weak nonlinearity $\Delta k=2.55$. Also shown for comparison: the correlation computed without dressing or effective velocities in Eq. \ref{['eq_GHD_C']} (dashed black lines, insets)
• Figure S1: Detailed experimental setup. ASE: Amplified Spontaneous Emission. EDFA: Erbrium Doped Fiber Amplifier. VOA: Variable Optic Attenuator. OPM: Optical Power Meter. AOM: Acousto-Optic Modulator. RF Amp: Radio-frequency amplifier. AFG: Arbitrary Function Generator. FISO: fiber Isolator. WDM: Wave Demultiplexer. PD: Photodetector.