  We theoretically and experimentally investigate quantum features of an
interacting light-matter system from a multidisciplinary perspective, unifying
approaches from semiconductor physics, quantum optics, and quantum information
science. To this end, we quantify the amount of quantum coherence that results
from the quantum superposition of Fock states, constituting a measure of the
resourcefulness of the produced state for modern quantum protocols. As an
archetypal example of a hybrid light-matter interface, we study a polariton
condensate and implement a numerical model to predict its properties. Our
simulation is confirmed by our proof-of-concept experiment in which we measure
and analyze the phase-space distributions of the emitted light. Specifically,
we drive a polariton microcavity across the condensation threshold and observe
the transition from an incoherent thermal state to a coherent state in the
emission, thus confirming the build-up of quantum coherence in the condensate
itself.
