In the last few years experimental progress in the area of out-of-equilibrium exciton-polariton gases [1] gave rise to several questions concerning the physical behaviour of Bose-Einsten condensates under pump and dissipation. Such systems can be theoretically described by a generalised Gross-Pitaevskii Equation (gGPE) in which complex coecients and noise enrich the equilibrium picture. An analytical mapping between gGPE and the Kardar-Parisi-Zhang(KPZ) equation has been demonstrated at long wavelength if the uctuations of the amplitude of the condensate are negligible with respect to the ones of the phase field. Hence one expects that the long-distance properties of driven-dissipative condensates belong to the KPZ universality class and a numerical proof was given in (1+1)D [2]. However an experimental observation of such mapping is still missing. An important feature of experimental set-up is the presence of unavoidable disorder due to cavity imperfections and phonons. In this work we develop a Keldysh field-theoretical approach and derive the gGPE- KPZ mapping taking into account the role of disorder and interactions with an external phonon-reservoir at thermal equilibrium; we furthermore perform numerical simulations to test our predictions.

[1] J. Kasprzak et al., "Bose-einstein condensation of exciton polaritons", Nature, vol. 443, pp. 409-414, Sep 2006.

[2] L. He, L. M. Sieberer, E. Altman, and S. Diehl, "Scaling properties of one-dimensional driven-dissipative condensates," Phys. Rev. B, vol. 92, p. 155307, Oct 2015.