🌀 nonlinear sciences

Experimental observation of ballistic correlations in integrable turbulence

This paper reports the first experimental observation of ballistic correlations in integrable turbulence within a photonic fiber loop system, demonstrating that the measured intensity correlators quantitatively match parameter-free predictions from Generalized Hydrodynamics.

Original authors: Elias Charnay, Adrien Escoubet, Francois Copie, Stephane Randoux, Thibault Bonnemain, Alvise Bastianello, Pierre Suret

Published 2026-01-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Elias Charnay, Adrien Escoubet, Francois Copie, Stephane Randoux, Thibault Bonnemain, Alvise Bastianello, Pierre Suret

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Watching a Crowd of Ghosts

Imagine you are standing on a busy highway at night. You see thousands of cars (waves) moving in both directions. Usually, when cars crash or swerve, they scatter chaotically, creating traffic jams that spread out randomly over time. This is like normal "turbulence."

However, this paper is about a very special kind of highway where the cars are ghosts. These ghosts have a magical rule: when they bump into each other, they don't crash or slow down. Instead, they pass right through each other, keeping their exact speed and shape, as if the collision never happened. In physics, this is called an integrable system.

The researchers wanted to answer a simple question: If you watch these ghost cars for a while, how do they remember where they came from?

The Experiment: A Loop of Fiber Optics

To study this, the scientists didn't use real cars. They used a fiber optic cable (the kind that brings internet to your house) that was coiled into a giant loop about 5 kilometers long.

The Setup: They shot a pulse of light into this loop. But instead of a single, clean beam, they used "partially coherent waves." Think of this as a crowd of people walking into a room with slightly different speeds and directions, rather than a marching band in perfect step.
The Magic Loop: The light travels around the loop. Every time it goes around, the scientists peeked in (using a special camera that can see both the brightness and the "phase" or timing of the light) and recorded what happened. They did this hundreds of times, effectively watching the light travel thousands of kilometers in a short time.
The Result: As the light traveled, it broke apart into distinct "packets" or solitons. These are like self-contained bubbles of light that act like particles. Because of the special physics of the fiber, these bubbles bounced off each other elastically (like billiard balls) but kept their identity.

The Discovery: The "Ballistic" Memory

The researchers measured how the brightness of the light at one spot was related to the brightness at another spot, but at a different time.

Normal Traffic (Diffusion): In a normal system (like smoke spreading in a room), if you wait longer, the smoke spreads out slowly and gets blurry. The connection between where it started and where it is now fades away quickly.
Ghost Traffic (Ballistic): In this experiment, the connection didn't fade. Instead, the "memory" of the light traveled in a straight line at a constant speed.

The paper calls this ballistic correlation. It's like throwing a ball across a room. If you throw it, you know exactly where it will be in one second, two seconds, or ten seconds, because it travels in a straight line without slowing down. The researchers found that the "ghost light" packets did exactly this. They moved in straight lines, carrying their information perfectly, even after bouncing off each other many times.

The Proof: Matching the Crystal Ball

The scientists didn't just guess this would happen. They used a powerful mathematical theory called Generalized Hydrodynamics (GHD).

Think of GHD as a super-advanced crystal ball that predicts how these ghost particles should behave based on the "density of states" (a fancy way of counting how many different types of ghost particles are in the mix and how fast they are going).

The Measurement: The team used a special technique (heterodyne detection) to measure the exact "fingerprint" of the light waves. From this, they calculated the density of the solitons (the ghost particles).
The Prediction: They fed this data into the GHD crystal ball to predict what the correlations should look like.
The Match: When they compared the crystal ball's prediction with what they actually saw in the fiber loop, they matched perfectly.

Why This Matters

This is a big deal because:

It's a Direct Test: Usually, these theories are tested on tiny quantum computers or cold atoms, where measuring things is very hard and often destroys the experiment. Here, they used light in a fiber cable, which is robust and easy to measure.
It Confirms the Theory: They proved that the math describing these "ghost" systems works exactly as predicted, even in a real-world experiment.
No Guessing: They didn't have to tweak any numbers to make the theory fit the data. The theory predicted the result exactly as it was observed.

In short: The scientists built a loop of light, watched it turn into a swarm of ghost-like particles that bounce off each other without slowing down, and proved that these particles travel in perfectly straight lines, carrying their history with them, just as a complex mathematical theory predicted they would.

Technical Summary: Experimental Observation of Ballistic Correlations in Integrable Turbulence

Problem Statement
Correlation functions are fundamental to understanding complex systems, encoding the spectrum of matter constituents and the foundations of field theory. In strongly interacting or highly nonlinear systems, these functions capture intricate interplays of excitations. While many systems exhibit diffusive scaling ( $\xi = 1/2$ ) at large scales, integrable models—characterized by infinitely many conservation laws—are predicted to exhibit ballistic scaling ( $\xi = 1$ ) due to the stability of solitons and the breakdown of Fick's law. These systems relax to Generalized Gibbs Ensembles (GGE) rather than standard thermal states.

Despite theoretical predictions and numerical evidence supporting ballistic correlations in integrable turbulence, direct experimental observation has remained a major challenge. This difficulty stems from the requirement to access full spatio-temporal dynamics, which is often unavailable or destructive in platforms like cold-atom experiments. Consequently, while Generalized Hydrodynamics (GHD) has been tested in cold atoms, ballistic correlations have never been directly observed in integrable systems.

Methodology
The authors utilize a photonic platform governed by the focusing one-dimensional nonlinear Schrödinger (fNLS) equation, a universal integrable partial differential equation. The experimental setup employs a Recirculating Optical Fiber Loop (ROFL) to simulate the propagation of light over extended distances.

System Generation: The experiment initiates with Partially Coherent Waves (PCWs) generated from a filtered Amplified Spontaneous Emission (ASE) source. These waves, characterized by a finite spectral bandwidth, serve as initial conditions that evolve into a statistically stationary state of integrable turbulence.
Data Acquisition: A 5 km single-mode fiber loop recirculates the signal. At each round trip, 10% of the power is tapped and measured using ultrafast photodetectors and a high-bandwidth oscilloscope. Crucially, the system employs heterodyne field detection, allowing for the reconstruction of both the amplitude and phase of the optical field. This enables the full space-time dynamics of the intensity $|\psi(t,x)|^2$ to be recorded over hundreds of round trips.
Analysis:
- Stationarity: The system's relaxation to a stationary state is monitored via the kurtosis of the intensity distribution.
- Correlation Measurement: The unequal-time connected correlation function of the intensity, $C(\Delta t, \Delta x)$ , is computed from the recorded space-time data.
- Theoretical Comparison: To compare with GHD predictions, the Density of States (DOS) of the solitons is extracted from the experimental fields using the Inverse Scattering Transform (IST) (specifically solving the Zakharov-Shabat problem). The DOS is then used in the GHD framework to calculate theoretical correlation functions without adjustable parameters.

Key Contributions and Results

Experimental Observation of Ballistic Scaling: The study reports the first direct experimental observation of two-time, ballistic correlations in an integrable wave system. The measured intensity correlator $C(\Delta t, \Delta x)$ exhibits a collapse under ballistic rescaling, following the form $C(\Delta t, \Delta x) \simeq \Delta t^{-1} \mathcal{C}(\Delta x / \Delta t)$ . This confirms the ballistic scaling exponent $\xi = 1$ predicted for integrable models.
Parameter-Free Validation of GHD: The authors provide a quantitative, parameter-free test of Generalized Hydrodynamics. By extracting the DOS from the experimental data and inputting it into the GHD correlation formula (derived from hydrodynamic projections), the theoretical predictions match the experimental data with high precision. This agreement holds for two different nonlinearity strengths ( $\Delta k = 1.80$ and $\Delta k = 2.55$ ).
DOS Conservation: The experiment demonstrates the conservation of the Density of States during propagation, a hallmark of integrability. The DOS, extracted at the initial time and at the stationary state, remains invariant, confirming that the system operates very close to the integrable limit.
Methodological Advancement: The work establishes a robust protocol for measuring unequal-time correlations in nonlinear optics, overcoming the limitations of destructive measurements found in other physical platforms.

Significance
The paper claims that these results provide a direct, parameter-free test of GHD in an integrable wave system, bridging the gap between theoretical predictions for quantum integrable models and experimental reality in classical nonlinear optics. By confirming that the exact predictions of GHD describe the stationary properties of integrable turbulence, the study validates the applicability of GHD and soliton gas theory to classical systems.

The authors position this work as a unique probe into the statistical mechanics and hydrodynamics of integrable models. They note that while two-time correlations remain difficult to access in quantum experiments, this photonic approach opens the door to experimental investigations of fundamental problems, such as the transient approach to ballistic scaling, the onset of diffusion from convection, multipoint correlators, and large deviations of charge fluctuations. The successful observation of ballistic correlations confirms that the emergent hydrodynamics of integrable systems are distinct from standard diffusive behavior, driven by the elastic collisions of stable solitons.

The Big Picture: Watching a Crowd of Ghosts

The Experiment: A Loop of Fiber Optics

The Discovery: The "Ballistic" Memory

The Proof: Matching the Crystal Ball

Why This Matters

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