We analyze the quantum efficiency of a microwave photon detector based on acurrent-biased Josephson junction. We consider the Jaynes-Cummings Hamiltonianto describe coupling between the photon field and the junction. We then takeinto account coupling of the junction and the resonator to the environment. Wesolve the equation of motion of the density matrix of the resonator-junctionsystem to compute the quantum efficiency of the detector as a function ofdetection time, bias current, and energy relaxation time. Our results indicatethat junctions with modest coherence properties can provide efficient detectionof single microwave photons, with quantum efficiency in excess of 80%.
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