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

physical review. e

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