Bridging Rayleigh-Jeans and Bose-Einstein condensation of a guided fluid of light with positive and negative temperatures

We consider the free-propagation geometry of a light beam (or fluid of light) in a multimode waveguide. As a result of the effective photon-photon interactions, the photon fluid thermalizes to an equilibrium state during its conservative propagation. In this configuration, Rayleigh-Jeans (RJ) thermalization and condensation of classical light waves have been recently observed experimentally in graded index multimode optical fibers characterized by a two-dimensional parabolic trapping potential. As is well known, the properties of RJ condensation differ substantially from those of Bose-Einstein (BE) condensation: The condensate fraction decreases quadratically with the temperature for BE condensation, while it decreases linearly for RJ condensation. Furthermore, for quantum particles the heat capacity tends to zero at low temperatures and it takes a constant value in the classical particle limit at high temperatures. This is in contrast with classical RJ waves, where the specific heat takes a constant value at low temperatures and tends to vanish above the condensation transition in the normal (uncondensed) state. Here we reconcile the thermodynamic properties of BE and RJ condensation: By introducing a frequency cutoff inherent to light propagation in a waveguide, we derive generalized expressions of the thermodynamic properties that include the RJ and BE limits as particular cases. We extend the approach to encompass negative temperatures. In contrast to positive temperatures, the specific heat does not display a singular behavior at negative temperatures, reflecting the noncritical nature of the transition to a macroscopic population of the highest-energy level. Our work contributes to understanding the quantum-to-classical crossover in the equilibrium properties of light, within a versatile experimental platform based on nonlinear optical propagation in multimode waveguides.