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 Schr\"odinger 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.