Non-Markovian Stochastic Gross-Pitaevskii Equation for the Exciton-Polariton Bose-Einstein Condensate

In this paper, a non-Markovian version of the Gross-Pitaevskii equation is proposed to describe the condensate formation in an exciton-polariton system subject to incoherent pumping. By introducing spatially delta-correlated noise terms, we observe a transition from a spatially ordered phase to a disordered one with simultaneous density reduction as the temperature increases. Above the transition temperature, the uniform condensate breaks up into multiple irregularly located separate dense spots. Using the Gabor transform, we demonstrate condensate decoherence with increasing temperature, which is accompanied by the transition from narrow-band to broadband spectral density.