  We analyze the quantum efficiency of a microwave photon detector based on a
current-biased Josephson junction. We consider the Jaynes-Cummings Hamiltonian
to describe coupling between the photon field and the junction. We then take
into account coupling of the junction and the resonator to the environment. We
solve the equation of motion of the density matrix of the resonator-junction
system to compute the quantum efficiency of the detector as a function of
detection time, bias current, and energy relaxation time. Our results indicate
that junctions with modest coherence properties can provide efficient detection
of single microwave photons, with quantum efficiency in excess of 80%.
