  Motivated by recent interest in implementing circuit quantum electrodynamics
with semiconducting quantum dots, we consider a double quantum dot (DQD)
capacitively coupled to a superconducting resonator that is driven by the
microwave field of a superconducting transmission line. We analyze the DQD
current response using input-output theory and show that the resonator-coupled
DQD is a sensitive microwave single photon detector. Using currently available
experimental parameters of DQD-resonator coupling and dissipation, including
the effects of $1/f$ charge noise and phonon noise, we determine the parameter
regime for which incident photons are completely absorbed and near unit
$\gtrsim$ 98\% efficiency can be achieved. We show that this regime can be
reached by using very high quality resonators with quality factor $Q\simeq
10^5$.
