The edge switch transformation in microwave networks

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., $ν_{n-r}\leq \tilde ν_{n}\leq ν_{n+r}$, 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 $\{ν_{n}\}_{n=1}^{\infty}$ and $\{\tilde ν_{n}\}_{n=1}^{\infty}$ the spectra of microwave networks before and after the edge switch transformation. We demonstrate that the experimental distribution $P(ΔN)$ of the spectral shift $ΔN$ 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 $P(ΔN)$ of the spectral shift $ΔN$ agrees within the experimental uncertainly with the numerical one.