Experimental observation of ballistic correlations in integrable turbulence

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.

💡 Research Summary

The authors present the first direct experimental observation of two‑time, ballistic correlation functions in a system governed by the focusing one‑dimensional nonlinear Schrödinger equation (fNLS), thereby providing a stringent, parameter‑free test of Generalized Hydrodynamics (GHD) for integrable wave turbulence. The experimental platform consists of a recirculating optical fiber loop (ROFL) of about 5 km length. Partially coherent waves (PCWs) generated from a filtered ASE source serve as the initial condition; their spectral width Δν≈5 GHz and mean power P₀ are tuned to realize two different dimensionless nonlinearity parameters Δk (≈1.80 and 2.55). After each round‑trip, 10 % of the circulating field is extracted and measured by both direct and heterodyne detection, enabling full reconstruction of the complex field ψ(t,x) over hundreds of round‑trips with sub‑100 ps temporal resolution.

The PCWs rapidly evolve under the fNLS dynamics into a statistically stationary state described by a Generalized Gibbs Ensemble (GGE). In this regime the authors compute the unequal‑time intensity correlator C(Δt,Δx)=⟨|ψ(t,x)|²|ψ(t₀,x₀)|²⟩−⟨|ψ|²⟩², with q(t,x)=|ψ|² acting as a locally conserved charge density. Experimental data show that C decays as Δt⁻¹ and its shape collapses onto a universal function of the ratio Δx/Δt, i.e. ballistic scaling with dynamical exponent ξ=1, in stark contrast to the diffusive scaling expected in generic non‑integrable systems.

To compare with theory, the authors extract the density of states (DOS) ρ(λ) of the soliton gas directly from the measured fields via the inverse scattering transform (IST) of the Zakharov‑Shabat problem. The complex spectral parameter λ=R(λ)+i I(λ) encodes each soliton’s velocity v(λ)=−4 R(λ) and peak intensity 4 I(λ). The DOS is shown to be conserved during propagation, confirming the system’s integrability. Within GHD, the DOS obeys a kinetic equation ∂ₜρ+∂ₓ(v_eff ρ)=0, where the effective velocity v_eff(λ) is self‑consistently renormalized by soliton‑soliton scattering shifts Δ(λ,μ)=1/I(λ) log| (μ−λ̄)/(μ−λ) |. The authors use the measured ρ(λ) to solve the integral equation for v_eff(λ) and compute the dressed charge h_dr⁰(λ)=4 I(λ)