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Bogoliubov excitations of a polariton condensate in dynamical equilibrium with an incoherent reservoir

The classic Bogoliubov theory of weakly interacting Bose gases rests upon the assumption that nearly all the bosons condense into the lowest quantum state at sufficiently low temperatures. Here we develop a generalized version of Bogoliubov theory for the case of a driven-dissipative exciton-polariton condensate with a large incoherent uncondensed component, or excitonic reservoir. We argue that such a reservoir can consist of both excitonic high-momentum polaritons and optically dark superpositions of excitons across different optically active layers, such as multiple quantum wells in a microcavity. In particular, we predict interconversion between the dark and bright (light-coupled) excitonic states that can lead to a dynamical equilibrium between the condensate and reservoir populations. We show that the presence of the reservoir fundamentally modifies both the energy and the amplitudes of the Bogoliubov quasiparticle excitations due to the non-Galilean-invariant nature of polaritons. Our theoretical findings are supported by our experiment, where we directly detect the Bogoliubov excitation branches of an optically trapped polariton condensate in the high-density regime. By analyzing the measured occupations of the excitation branches, we extract the Bogoliubov amplitudes across a range of momenta and show that they agree with our generalized theory.