Statistics of Complex Time Delay of Lossy Microwave Networks*

Quantum graphs were introduced over 80 years ago to model electrons in molecules and have since been applied to myriad other problems with great success. Due to the direct analogy between their governing equations, microwave networks (graphs) have been used to simulate quantum graphs, as microwave graphs are far easier to both assemble and measure. Complex time delay, defined as the energy derivative of the scattering matrix, is a measure of how long a particle/wave interacts with the scattering region before leaving through a channel coupled to the system. It has also been experimentally demonstrated to be a practical method to recover the complex poles and zeros of a scattering matrix. We present experimental time delay ensemble statistics of a disordered microwave graph, with comparisons to circuit model and random matrix theory simulations. The experiments and simulations are conducted for a system with time reversal invariance (TRI) and compared to previous results for systems exhibiting broken TRI. The dependence of the complex reflection time delay statistics on the number of scattering channels and uniform absorption is explored, and a potential distribution function for the TRI case is suggested.