Theoretical and Numerical Estimate of Signal-to-Noise Ratio in the Analysis of Josephson Junctions Lifetime for Photon Detection

The performances of a Josephson junction employed to reveal a train of pulses (a rough model for single photon detection) are analyzed with a theoretical estimate that exploits an index employed in statistical decision theory, the Kumar-Carroll index. The approximate analysis compares the numerically simulated performances of the device through the receiver operating characteristics, that offer an overview of the rate of false detection, as well as the probability to miss a signal (in this case, a pulse train). It is thus demonstrated the usefulness and the limits of the succinct Kumar-Carroll parameter. On the first side, it is proven that an increase of the parameter corresponds to an improvement of the detection. However, on the side of the limitations, the expected performances are not quite accurate, for the actual performances are systematically worse than the theoretical estimates. The results may be relevant to characterize Josephson junctions as detectors of weak signals, as those stemming from axions.