Beyond-mean-field corrections to the blueshift of a driven-dissipative exciton-polariton condensate

In the absence of vortices or phase slips, the phase dynamics of exciton-polariton condensates was shown to map onto the Kardar-Parisi-Zhang (KPZ) equation, which describes the stochastic growth of a classical interface. This implies that the coherence of such non-equilibrium quasi-condensates decays in space and time following stretched exponentials, characterized by KPZ universal critical exponents. In this work, we focus on the time evolution of the average phase of a one-dimensional exciton-polariton condensate in the KPZ regime and determine the frequency of its evolution, which is given by the blueshift, i.e. the non-equilibrium analog of the chemical potential. We determine the stochastic corrections to the blueshift within Bogoliubov linearized theory and find that while this correction physically originates from short scale effects, and depends both on density and phase fluctuations, it can still be related to the effective large-scale KPZ parameters. Using numerical simulations of the full dynamics, we investigate the dependence of these blueshift corrections on both noise and interaction strength, and compare the results to the Bogoliubov prediction. Our finding contributes both to the close comparison between equilibrium and non-equilibrium condensates, and to the theoretical understanding of the KPZ mapping.