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We study the thermalization and the Bose-Einstein condensation of a paraxial,spectrally narrow beam of quantum light propagating in a lossless bulk Kerrmedium. The spatiotemporal evolution of the quantum optical field is ruled by aHeisenberg equation analogous to the quantum nonlinear Schr\"odinger equationof dilute atomic Bose gases. Correspondingly, in the weak-nonlinearity regime,the phase-space density evolves according to the Boltzmann equation.Expressions for the thermalization time and for the temperature and thechemical potential of the eventual Bose-Einstein distribution are found. Afterdiscussing experimental issues, we introduce an optical setup allowing theevaporative cooling of a guided beam of light towards Bose-Einsteincondensation. This might serve as a novel source of coherent light.

Thermalization and Bose-Einstein condensation of quantum light in bulk nonlinear media