Analysis of Josephson junctions switching time distributions for the detection of single microwave photons

We investigate an optimal scheme for the detection of single microwave photons by a Josephson junction through the analysis of its switching times distribution. The proposed analysis is of support for the decision about the existence of the photon field, which is important in the case of rare events. We assume that the cavity and the transmission line are ideal (each photon absorbed to the cavity gives a current pulse as the output of the transmission line) and the photon source is periodic. The employed methodology consists in comparing the switching probabilities of a Josephson junction exposed to a train of current pulses, simulating a weak photon field, with that of the same device in absence of pulses. In both cases, thermal noise can induce thermal activated switchings. The investigation of the unbalance in the number of switching events in the two cases, gives an estimate of the efficiency of the detection. Furthermore, in the assumption of escapes described by Kramers model, it is possible to provide a relationship between the properties of the photons field, the quantum efficiency of the detection process, and the Josephson junctions switching features at finite temperatures.