  Among the most exciting recent advances in the field of superconducting
quantum circuits is the ability to coherently couple microwave photons in
low-loss cavities to quantum electronic conductors (e.g.~semiconductor quantum
dots or carbon nanotubes). These hybrid quantum systems hold great promise for
quantum information processing applications; even more strikingly, they enable
exploration of completely new physical regimes. Here we study theoretically the
new physics emerging when a quantum electronic conductor is exposed to
non-classical microwaves (e.g.~squeezed states, Fock states). We study this
interplay in the experimentally-relevant situation where a superconducting
microwave cavity is coupled to a conductor in the tunneling regime. We find the
quantum conductor acts as a non-trivial probe of the microwave state; in
particular, the emission and absorption of photons by the conductor is
characterized by a non-positive definite quasi-probability distribution. This
negativity has a direct influence on the conductance of the conductor.
