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

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.