The edge switch transformation in microwave networks

TL;DR

This study experimentally and numerically investigates how edge switch operations affect the spectral properties of microwave networks simulating quantum graphs, confirming theoretical predictions about spectral interlacing and distribution consistency.

Contribution

The paper provides the first experimental verification of spectral interlacing and distribution predictions for microwave networks undergoing edge switch transformations, including cases with broken time-reversal symmetry.

Findings

Spectra are level-1 interlaced before and after edge switch.

Experimental spectral shift distribution matches theoretical predictions.

Results hold for networks with preserved and violated time-reversal symmetry.

Abstract

We investigated the spectra of resonances of four-vertex microwave networks simulating both quantum graphs with preserved and with partially violated time-reversal invariance before and after an edge switch operation. We show experimentally that under the edge switch operation the spectra of the microwave networks with preserved time reversal symmetry are level-1 interlaced, i.e., $\nu_{n-r}\leq \tilde \nu_{n}\leq \nu_{n+r}$, where $r=1$, in agreement with the recent theoretical predictions of [M. Aizenman, H. Schanz, U. Smilansky, and S. Warzel, Acta Phys. Pol. A {\bf 132}, 1699 (2017)]. Here, we denote by $\{\nu_{n}\}_{n=1}^{\infty}$ and $\{\tilde \nu_{n}\}_{n=1}^{\infty}$ the spectra of microwave networks before and after the edge switch transformation. We demonstrate that the experimental distribution $P(\Delta N)$ of the spectral shift $\Delta N$ is close to the theoretical one. Furthermore, we show experimentally that in the case of the four-vertex networks with partially violated time reversal symmetry the spectra are level-1 interlaced. Our experimental results are supplemented by the numerical calculations performed for quantum graphs with violated time-reversal symmetry. In this case the edge switch transformation also leads to the spectra which are level-1 interlaced. Moreover, we demonstrate that for microwave networks simulating graphs with violated time-reversal symmetry the experimental distribution $P(\Delta N)$ of the spectral shift $\Delta N$ agrees within the experimental uncertainly with the numerical one.