  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.
