Experimental Investigation of Reflection Coefficient and Wigner’s Reaction Matrix for Microwave Graphs

Quantum graphs of connected one-dimensional wires were introduced by Pauling seven decades ago [1]. The same idea was used a decade later by Kuhn [2] to describe organic molecules by free electron models. Quantum graphs can be considered as idealizations of physical networks in the limit where the lengths of the wires greatly exceed their widths assuming that propagating waves remain in a single transverse mode. Among the systems modeled by quantum graphs are, e.g., electromagnetic optical waveguides [3, 4], mesoscopic systems [5, 6], quantum wires [7, 8] and excitation of fractons in fractal structures [9, 10]. Recent works have shown that quantum graphs provide an excellent system for studying quantum chaos [11–21]. Quantum graphs with external leads (antennas) have been analyzed in detail in [16, 17]. Quantum graphs with absorption, more realistic but more complicated systems, have been recently studied in [20, 21]. In this paper we present the results of the experimental study of the distribution P (R) of the reflection coefficient R and the distributions of Wigner’s reaction matrix [22] (also called the K matrix [23]) for microwave networks that correspond to graphs with time reversal symmetry (β = 1 symmetry class of random matrix theory [24]) in the presence of absorption. Our experimental results are compared with the exact theoretical predictions of Savin et al. [25].