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Microwave photon detection E. Il’ichev Leibniz Institute of Photonic Technology, P.O. Box 100239, D-07702 Jena, Germany We demonstrate the narrow switching distribution of an underdamped Josephson junction from the zero to the finite voltage state at millikelvin temperatures. The width of the switching distribution at a nominal temperature of about 20mK was 4.5 nA, which corresponds to an effective noise temperature of the device below 60 mK. We argue that such junctions can be used as ultrasensitive detectors of the single photons in the GHz range, operating close to the quantum limit: a given initial (zero voltage) state can be driven by an incoming signal to the finite voltage state. The Hamiltonian for a λ/4 resonator shunted by a current-biased Josephson junction is derived and it explicitly includes a quantum treatment of the switching dynamics. The full quantum dynamics of the system are analyzed numerically. Results of the first experimental realization of this system are shown. References C. Andersen et.al. Phys.Rev. A 87, 052119 (2013) G. Oelsner et al., Appl. Phys. Lett. 103, 142605 (2013). C. Andersen et.al. Phys.Rev. A 89, 033853 (2014)