Impedance statistics of cable networks that model quantum graphs

Abstract

We present the theoretical framework required to describe the statistics of microwave networks that serve to model quantum graphs. The networks are described by impedance and admittance matrices relating the voltages and currents at the network's ports. As we show, these matrices can be calculated in a number of ways. Normal modes of the network are characterized by a discrete set of wave numbers corresponding to the propagation constants on the network's bonds for which the determinant of the admittance matrix vanishes. The distribution of the spacings between adjacent eigenmode wave numbers is found to depend on the nature of the way bonds are connected at nodes. The critical quantity is the reflection coefficient presented at a node to a wave on a bond. As the reflection coefficient increases, the spacing distribution changes from one characteristic of the spacing of eigenvalues of a Gaussian orthogonal ensemble matrix to a Poisson distribution. The effect of loss is studied, and the scaling of the variance of the impedance values on network size, degree distribution, and other parameters is characterized. We attempted to find universal scaling relations for the distribution of impedance values for networks of different sizes. Finally, we compare the distribution of impedance values predicted by the model with those measured in a network of cables.