Experimental investigation of distributions of the off-diagonal elements of the scattering and the Wigner's hat K matrices for networks with broken time reversal invariance

We present an extensive experimental study of the distributions of the real and imaginary parts of the off-diagonal elements of the scattering matrix hat S and the Wigner's reaction hat K-matrix for open microwave networks with broken time (T) reversal invariance. Microwave Faraday circulators were applied in order to break T-invariance. The experimental distributions of the real and imaginary parts of the off-diagonal entries of the scattering matrix hat S are compared with the theoretical predictions from the supersymmetry random matrix theory \[A. Nock, S. Kumar, H.-J. Sommers, and T. Guhr, Annals of Physics {\bf 342}, 103-132 (2014)\]. Furthermore, we show that the experimental results are in very good agreement with the recent predictions for the distributions of the real and imaginary parts of the off-diagonal elements of the Wigner's reaction hat K-matrix obtained within the framework of the Gaussian unitary ensemble of random matrix theory (RMT) \[S. B. Fedeli and Y. V. Fyodorov, J. Phys. A: Math. Theor. {\bf 53}, 165701 (2020)\]. Both theories include losses as tunable parameters and are therefore well adapted to the experimental verification.