Topological Resonances in Scattering on Networks (Graphs)

We report on a hitherto unnoticed type of resonances occurring in scattering from networks (quantum graphs) which are due to the complex connectivity of the graph—its topology. We consider generic open graphs and show that any cycle leads to narrow resonances which do not fit in any of the prominent paradigms for narrow resonances (classical barriers, localization due to disorder, chaotic scattering). We call these resonances “topological” to emphasize their origin in the nontrivial connectivity. Topological resonances have a clear and unique signature which is apparent in the statistics of the resonance parameters (such as, e.g., the width, the delay time, or the wave-function intensity in the graph). We discuss this phenomenon by providing analytical arguments supported by numerical simulation, and identify the features of the above distributions which depend on genuine topological quantities such as the length of the shortest cycle (girth). These signatures cannot be explained using any of the other paradigms for narrow resonances. Finally, we propose an experimental setting where the topological resonances could be demonstrated, and study the stability of the relevant distribution functions to moderate dissipation.

Figure 1

Tails of the integrated distribution $I(\alpha )$ for the mean intensity $\alpha$. Insets show the underlying graphs. Full lines represent numerical results where $L=1,\dots ,5$ is the number of the attached lead. Note that the data for $L=2$, 3, 4, 5 for the cube consist of more than one line—these correspond to nonequivalent ways to attach $L$ leads. The dashed lines are power-law tails $\sim {\alpha }^{-\mu }$ with indicated exponents.

Figure 2

Left: A cycle of length $C$. Right: Numerically obtained tail of the integrated distribution $I(\alpha )$ of the mean intensity for a lossy beam-splitter setup (see inset).