Use of Transmission and Reflection Complex Time Delays to Reveal Scattering Matrix Poles and Zeros: Example of the Ring Graph

We identify the poles and zeros of the scattering matrix of a simple quantum graph by means of systematic measurement and analysis of Wigner, transmission, and reflection complex time delays [Lei Chen and S. M. Anlage, Phys Rev E 105, 054210 (2022)]. We examine the ring graph because it displays both shape and Feshbach resonances, the latter of which arises from an embedded eigenstate on the real frequency axis. Our analysis provides a unified understanding of the so-called shape, Feshbach, electromagnetically induced transparency, and Fano resonances on the basis of the distribution of poles and zeros of the scattering matrix in the complex frequency plane [Lei Chen, S. M. Anlage, and Yan V. Fyodorov, Phys Rev E 103, L050203 (2021)]. It also provides a first-principles understanding of sharp resonant scattering features and associated large time delay in a variety of practical devices, including photonic microring resonators, microwave ring resonators, and mesoscopic ring-shaped conductor devices. We also create a two-channel microwave graph realization of the Aharonov–Bohm ring, and demonstrate non-reciprocal transport in the presence of finite de-phasing/loss, in the intermediate regime between purely quantum and purely classical transport.