Signal-to-noise-ratio and SNR-max detection statistics in template bank searches for exotic physics transients with networks of quantum sensors

Signal-to-noise ratio (SNR) detection statistic has wide-spread applications. A potential event is recorded when the SNR from a specific template exceeds a threshold set by a desired false positive rate. In template bank searches, the generalization of the SNR statistic is the SNR-max statistic, defined as the maximum of the absolute value of SNRs from individual template matching. While individual SNR realizations are Gaussian distributed, SNR-max probability distribution is non-Gaussian. Moreover, as the individual template-bank SNRs are computed using the same network data streams, SNRs become correlated between templates. Cross-template correlations have sizable effect on the SNR-max probability distribution, and the threshold SNR-max values. Computing threshold SNR-max values for large banks is computationally prohibitive and we develop analytic approaches to computing properties of SNR-max statistic. This is done for nearly orthogonal template banks and for banks with cross-template correlation coefficients "squeezed" about the most probable cross-template correlation value. Since cross-template correlation coefficients quantify similarity of templates, increasing correlations decrease SNR-max thresholds for specific values of false positive rates. Increasing the number of templates in the bank increases the SNR-max thresholds. Our derivations are carried out for networks that may exhibit colored noise and cross-node correlations. Specific applications are illustrated with a dark matter search with atomic clocks and a "toy" planar network with cyclic rotational symmetry.