Microscopic Theory of Photon Bose-Einstein Condensation

Abstract

We introduce a novel approach for analysing light-matter interactions within driven-dissipative environments. Our study centers on the interaction between dye molecule solutions and photonic cavity modes, mediated by Jaynes-Cummings coupling. These dye molecules exhibit discrete electronic and rovibrational energy levels, influenced by the surrounding thermal environment imposed by the solvent. <br /> The principal contribution of this work lies in the development of a mapping that links the discrete-level structure of dye molecules to auxiliary bosons, subject to an additional operator constraint. This mapping facilitates the application of field theory methods, particularly leveraging the Schwinger-Keldysh formalism to address general non-equilibrium scenarios. Especially, this frame-work allows for the exact implementation of the operator constraint. It enables us to consider Markovian and non-Markovian baths coupled to the molecules. Including non-Markovian baths is of significant importance in achieving thermalisation beyond the occupation of levels and allowing to imprint the thermal fluctuation-dissipation relation onto the spectra. <br /> The main goal of this work is to investigate the emergence of phase coherence in the photon field. Our method enables a unified treatment of photon field fluctuations and coherence dynamics, allowing the spontaneous breaking of the U (1) symmetry inherent in the Jaynes-Cummings interaction. The large dye reservoir is incorporated via a simplified Dynamical-Mean-Field Theory leveraging an expansion in the dye molecule density. We investigate the implications of broken U (1) symmetry in an open-driven context and validate the corresponding Ward identity in the phase of the Bose-Einstein condensate. This sheds light on the intricate interplay between coherence, fluctuations, and symmetry in light-matter coupled systems within driven-dissipative environments.