Spatio-temporal equilibrium thermodynamics of guided optical waves at positive and negative temperatures

Optical thermalization has been recently studied in the 2D spatial evolution of (quasi-)monochromatic light waves propagating in multimode waveguides. Here, we investigate the spatio-temporal equilibrium properties of optical waves through the analysis of the (2+1)D Bose-Einstein thermal distribution and the corresponding classical Rayleigh-Jeans approximation. Numerical simulations of the nonlinear Schr\"odinger equation (NLSE) demonstrate relaxation toward the spatio-temporal 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, we make use of a quantum version of the NLSE whose associated kinetic equation describes relaxation toward the spatio-temporal Bose-Einstein equilibrium distribution. The analysis of thermodynamic properties reveals a strong dependence on the dispersion regime. In the anomalous dispersion regime, the system relaxes to positive-temperatures 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 spatio-temporal condensation in the thermodynamic limit. In the normal dispersion regime, the system evolves toward negative-temperature equilibrium states characterized by an inverted spatial modal population. In this regime, we predict 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 and lay the groundwork for the development of spatiotemporal optical thermodynamics.