Josephson junctions, in appropriate configurations, can be excellent candidates for detection of single photons in the microwave frequency band. Such possibility has been recently addressed in the framework of galactic axion detection. Here are reported recent developments in the modelling and simulation of dynamic behaviour of a Josephson junction single microwave photon detector. For a Josephson junction to be enough sensitive, small critical currents and operating temperatures of the order of ten of mK are necessary. Thermal and quantum tunnelling out of the zero-voltage state can also mask the detection process. Axion detection would require dark count rates in the order of 0.001 Hz. It is, therefore, is of paramount importance to identify proper device fabrication parameters and junction operation point.
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