Observation of a generalized Gibbs ensemble in photonics

TL;DR

This work demonstrates the experimental emergence of a generalized Gibbs ensemble in a photonic system governed by the defocusing nonlinear Schrödinger equation. By propagating partially coherent waves in a long optical fiber, the authors directly measure the density of states $\rho(\lambda)$ (rapidity) and verify its conservation, establishing the integrable nature of the dynamics. Using the measured $\rho(\lambda)$, they predict and confirm that the intensity PDF relaxes to the GGE form $P_{\text{GGE}}(|\psi|^2)$, with clear differences between weak and strong nonlinear regimes. The study highlights photonics as a powerful platform for probing integrability, bridging classical and quantum perspectives, and offering precise access to conserved quantities and two-time correlations in out-of-equilibrium many-body systems.

Abstract

In generic classical and quantum many-body systems, where typically energy and particle number are the only conserved quantities, stationary states are described by thermal equilibrium. In contrast, integrable systems showcase an infinite hierarchy of conserved quantities that inhibits conventional thermalization, forcing relaxation to a Generalized Gibbs Ensemble (GGE) -- a concept first introduced in quantum many-body physics. In this study, we provide experimental evidence for the emergence of a GGE in a photonic system. By investigating partially coherent waves propagating in a normal dispersion optical fiber, governed by the one-dimensional defocusing nonlinear Schroedinger equation, we directly measure the density of states of the spectral parameter (rapidity) to confirm the time invariance of the full set of conserved charges. We also observe the relaxation of optical power statistics to the GGE's theoretical prediction, obtained using the experimentally measured density of states. These complementary measurements unambiguously establish the formation of a GGE in our photonic platform, highlighting its potential as a powerful tool for probing many-body integrability and bridging classical and quantum integrable systems.

Figures (4)

• Figure 1: The GGE from integrable dynamics in optical fibers.--- Panel (a): Experimental realization of the NLS through light propagation in nonlinear optical fibers. Panel (b): Evolution of a representative field configuration in the strongly nonlinear regime. Physical time and fiber lengths in the experiment are respectively proportional to space and time in the adimensional NLS equation \ref{['eq_nls']}.
• Figure 2: Extracting the GGE from experimental data.--- Panel $(a)$: We show the profile of the norm and phase of one random field configuration after the weak and strong nonlinear evolution. To avoid boundary detection problems, we consider a central window (shadowed area $x\in[-8,8]$) and compute the density of states within it, shown in the rightmost panel in the respective colors. Panel $(b):$ We show the conservation of the density of states by reporting its average values before and after the evolution, for weak and strong nonlinearities. When averaging, we consider $10^3$ experimental samples: the confidence interval (shaded areas) is obtained by considering the maximum spreading of partial averages done on $250$ samples each.
• Figure 3: Relaxation of the intensity PDF to the GGE.--- We show the PDF of the intensity field $|\psi|^2$ extracted from the initial data on the initial conditions and after evolution (red line and blue line respectively), in linear (a,b) and log (c,d) scale. We compare PDF data with the GGE result (black line) \ref{['eq_GGE_PDF']} obtained from the density of states extracted from experimental data, see Fig. \ref{['fig_exp']}. We compare with the exponential decay from Gaussian independent plane waves as reference (dashed line). As before, we show the average over $10^3$ independent samples, the confidence interval is the maximum spreading from partial averages of 250 samples each. The experimental PDF is extracted from the same window of data $x\in[-8,8]$ used for the density of states in Fig. \ref{['fig_exp']}, and the bin size is 0.03.
• Figure 4: An example of the experimentally recorded heterodyne time microscope frame and corresponding power and phase profile extraction. (Top) A representative 2-D snapshot recorded experimentally with the heterodyne time microscope, captured by the sCMOS camera. The spatial interference fringes arise from the non-collinear interaction of signal and reference beams at the detection plane, encoding both amplitude (brightness) and phase (relative fringe position) information in the spatial domain. (Middle) Corresponding temporal power profile extracted from the snapshot, obtained by integrating the intensity distribution along the vertical coordinate. (Bottom) Reconstructed temporal phase profile derived through spatial Fourier analysis of the recorded fringe pattern, which contains the relative phase information of the signal beam, with the reference beam’s phase assumed constant across the measurement window.