Dynamic analysis of the propagation of parallel light in a two-dimensional nonlinear optical cavity

We consider the Bose–Einstein condensation of parallel light in a two-dimensional nonlinear optical cavity, and study the propagation and conversion of the parallel light both in classical and quantum treatments. First, by combining the classical propagation equation of electric field and the Schrödinger equation of photons we derive a coupled wave equation which can be used to describe the conversion law of the parallel light in the condensate phase. It is found that in the condensate phase there exists a phase transition. Over the phase transition point, the fundamental wave can all be converted into the second harmonic. The interaction between the parallel light and the nonlinear medium is also investigated in the quantum framework. It is found that in the microcavity the traditional process of second harmonic generation can be explained as the coupling of two massive photons with each other to form a photon molecule. The conversion relation between photons and photon molecules is investigated. The results reveal the existence of a quantum phase transition in the ground state of the two-mode system, and it is consistent with the description using the classical-field method. Further investigation also shows that for the common condensate state the harmonic conversion rate is closely related to the order parameter or the initial imbalance of the system. The dependency relation between the harmonic conversion rate and the effective photon–photon interaction is also discussed.