Spatiotemporal equilibrium thermodynamics of guided optical waves at positive and negative temperatures

Year: 2026

Authors: Zanaglia L., Garnier J., Michel C., Doya V., Ferraro M., Wabnitz S., Carusotto I., Picozzi A.

Autors Affiliation: Univ Cote Azur, Inst Phys Nice, CNRS, Nice, France; Inst Polytech Paris, Ecole Polytech, CNRS, CMAP, F-91120 Palaiseau, France; Inst Univ France IUF, 1 Rue Descartes, F-75005 Paris, France; Univ Calabria, Dept Phys, Via Pietro Bucci, I-87036 Arcavacata Di Rende, CS, Italy; Sapienza Univ Rome, Dept Informat Engn Elect & Telecommun, Via Eudossiana 18, I-00184 Rome, Italy; Univ Trento, INO CNR Pitaevskii BEC Ctr, I-38123 Povo, Italy; Univ Trento, Dipartimento Fis, I-38123 Povo, Italy; Univ Bourgogne Europe, CNRS, Lab Interdisciplinaire Carnot Bourgogne ICB, UMR 6303, F-21000 Dijon, France.

Abstract: Optical thermalization has been recently studied theoretically and experimentally in the two-dimensional (2D) spatial evolution of (quasi)monochromatic light waves propagating in multimode fibers. In this work, we investigate the spatiotemporal equilibrium properties of incoherent multimode optical waves through the analysis of the (2 + 1)D Bose-Einstein thermal distribution and the corresponding classical Rayleigh-Jeans approximation. In the classical regime, we perform numerical simulations of the nonlinear Schr & ouml;dinger equation and demonstrate relaxation toward the spatiotemporal Rayleigh-Jeans equilibrium state, as described by the corresponding wave turbulence kinetic equation. Remarkable adiabatic cooling phenomena stemming from the high-frequency tails of the Rayleigh-Jeans distribution are discussed and the consequent limitations of the classical approximation are highlighted. To overcome these issues and properly include quantum effects, we make use of a quantum version of the nonlinear Schr & ouml;dinger equation that is obtained from the general quantum theory of light propagating in nonlinear waveguides. The associated kinetic equation describes relaxation toward the spatiotemporal Bose-Einstein equilibrium distribution with a fully regular ultraviolet behavior. The analysis of thermodynamic equilibrium properties reveals a strong dependence on the specific dispersion regime under consideration. In the anomalous dispersion regime, the system relaxes to positive-temperature equilibrium states: as the number of modes of the waveguide increases, the fundamental spatial mode becomes macroscopically populated, while its temporal spectrum undergoes significant narrowing, ultimately leading to complete (2 + 1)D spatiotemporal condensation in the thermodynamic limit. By contrast, in the normal dispersion regime the system evolves toward negative-temperature equilibrium states characterized by a hybrid structure: the spatial equilibrium displays an inverted modal population, whereas the temporal spectrum remains peaked around the fundamental (carrier) optical frequency. In this regime, we predict that spatiotemporal light waves exhibit a phase transition to Bose-Einstein condensation at negative temperatures, which occurs by increasing the temperature above a negative critical value. Our work opens new avenues for future research, including the possibility for a dual spatiotemporal beam cleaning through full spatiotemporal light condensation, and lay the groundwork for the development of spatiotemporal optical thermodynamics.

Journal/Review: PHYSICAL REVIEW A

Volume: 113 (6)      Pages from: 63533-1  to: 63533-22

More Information: Funding was provided by Agence Nationale de la Recherche (Grants No. ANR-23-CE30-0021 and No. ANR-21-ESRE-0040) . Calculations were performed using HPC resources from Universite Cote d’Azur’s Center Azzura, and DNUM-CCUB (Universite de Bourgogne Europe). I.C. acknowledges financial support from the PE0000023-NQSTI project by the Italian Ministry of University and Research, cofunded by the European Union-NextGeneration EU; from Provincia Autonoma di Trento (PAT), partly via the Q@TN initiative. M.F. acknowledges funding from the Italian Ministry of University and Research (MUR) through the Italian Science Fund (FIS2) , project SOFT (Grant No. H53C24001530001) . S.W. acknowledges support from the European Innovation Council, project MULTISCOPE (Grant No. 101185664) .
KeyWords: Bose-einstein Condensation; Statistical-mechanics; Nonlinear Propagation; Multimode Fibers; Thermalization; Turbulence; Boltzmann; Frequency; Systems; Gibbs
DOI: 10.1103/bf7m-chkd