We present an extensive experimental study of the distributions of the real and imaginary parts of the off-diagonal elements of the scattering matrix $\widehat{S}$ and the Wigner's reaction $\widehat{K}$ matrix for open microwave networks with broken time $\left(T\right)$ 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 $\widehat{S}$ are compared with the theoretical predictions from the supersymmetry random matrix theory [A. Nock, S. Kumar, H.-J. Sommers, and T. Guhr, Ann. Phys. (NY) 342, 103 (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 $\widehat{K}$ matrix obtained within the framework of the Gaussian unitary ensemble of random matrix theory [S. B. Fedeli and Y. V. Fyodorov, J. Phys. A: Math. Theor. 53, 165701 (2020)]. Both theories include losses as tunable parameters and are therefore well adapted to the experimental verification.

中文翻译：

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破碎时间逆不变性网络的散射矩阵和Wigner'sK̂矩阵的非对角线元素分布的实验研究

我们对散射矩阵的非对角元素的实部和虚部的分布进行了广泛的实验研究 $\widehat{\mathrm{小号}}$ 和威格纳的反应 $\widehat{ķ}$ 时间破损的开放式微波网络的矩阵 $（Ť）$反转不变性。应用微波法拉第循环器以打破$Ť$不变性。散射矩阵非对角线项的实部和虚部的实验分布$\widehat{\mathrm{小号}}$与超对称随机矩阵理论[A. 诺克（Nock），库马尔（S. Kumar），H.-J。Sommers和T. Guhr，Ann。物理 （NY） 342，103（2014）]。此外，我们表明实验结果与维格纳反应非对角线元素的实部和虚部的最新预测非常吻合$\widehat{ķ}$在随机矩阵理论的高斯unit集合框架内获得的矩阵[S．B.Fedeli和Y.V.Fyodorov，J.Phys。答：数学。理论。 53，165701（2020）]。两种理论都将损耗作为可调参数，因此非常适合实验验证。