Topological Resonances in Scattering on Networks (Graphs)

Abstract

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 non-trivial 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 wavefunction 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.

Chaotic scattering on individual quantum graphs

“…[21][22][23] where many of its properties were displayed with the help of numerical simulations. We benefit from the developments in these papers and from the recent discovery of "topological resonances" in the chaotic scattering on graphs [24]. We take advantage of the fact that chaotic quantum graphs are easier to handle than general dynamical chaotic systems because the propagator amplitudes are plane waves, and the semiclassical approximation is exact in that case.…”

Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

“…Indeed, it has been observed [19] that nonlinear effects such as multistability in quantum graphs occur generically already when the incoming flow is very low due to a generic mechanism for narrow resonances, the so-called topological resonances [20]. We refer to [20] for a more detailed discussion of topological resonances in linear quantum graphs and to [19] for a numerical analysis how topological resonances magnify nonlinearities and lead to multistability for arbitrarily small incoming flows. Here, we want to describe this mechanism briefly for two example graphs.…”

Universality in the spectral and eigenfunction properties of random networks

“…With this averaging procedure we get rid off any individual network characteristic (such as scars [72], which in turn produce topological resonances [73]) that may lead to deviations from RMT predictions used here as a reference (see also [74]). More specifically, we choose this network model to retrieve well known random matrices in the appropriate limits -remember that a diagonal random matrix is obtained for α = 0, when the vertices are isolated, whereas a member of the GOE is recovered for α = 1, when the network is fully connected.…”