  Exciton polaritons in two-dimensional semiconductors inside microcavities are
powerful platforms to explore hybrid light-matter quantum systems. Here, we
study the Bose-Einstein condensation of polaron-polaritons, which are
quasiparticles formed by the dressing of exciton polaritons by particle-hole
excitations in a surrounding electron gas. Using a non-perturbative many-body
theory to describe exciton-electron correlations combined with a
non-equilibrium theory for the condensate, we show that the electrons strongly
affect the condensate. This stems from the dependence of the polaron-polariton
energy and the interaction between them on the electron density, which leads to
highly nonlinear effects. We identify stable and unstable regimes of the
condensate by calculating its excitation spectrum, and show that they result in
prominent hysteresis effects when the electron density is varied. Our results
should be readily observable using present experimental technology.
