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Statistical properties of resonance widths for open quantum graphs

Tsampikos Kottos, Holger Schanz

Published Jan 1, 2003• 9 citations
OpenAlex

Abstract

We connect quantum compact graphs with infinite leads, and turn them into scattering systems.We derive an exact expression for the scattering matrix, and explain how it is related to the spectrum of the corresponding closed graph.The resulting expressions allow us to get a clear understanding of the phenomenon of resonance trapping due to over-critical coupling with the leads.Finally, we analyze the statistical properties of the resonance widths and compare our results with the predictions of Random Matrix Theory.Deviations appearing due to the dynamical nature of the system are pointed out and explained.

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We connect quantum compact graphs with infinite leads, and turn them into scattering systems.We derive an exact expression for the scattering matrix, and explain how it is related to the spectrum of the corresponding closed graph.The resulting expressions allow us to get a clear understanding of the phenomenon of resonance trapping due to over-critical coupling with the leads.Finally, we analyze the statistical properties of the resonance widths and compare our results with the predictions of Random Matrix Theory.Deviations appearing due to the dynamical nature of the system are pointed out and explained.
Quantized, compact graphs are excellent paradigms for quantum chaos in bounded systems. Connecting them with leads to infinity, we show that they display all the features which characterize quantum chaotic scattering. We derive exact expressions for the scattering matrix, and an exact trace formula for the density of resonances, in terms of classical orbits, analogous to the semiclassical theory of chaotic scattering. A statistical analysis of the cross sections and resonance parameters compares well with the predictions of random matrix theory. Hence, this system is proposed as a convenient tool to study the generic behavior of chaotic scattering systems and their semiclassical description.
We report on high-precision measurements that were performed with superconducting waveguide networks with the geometry of a tetrahedral and a honeycomb graph. They consist of junctions of valency three that connect straight rectangular waveguides of equal width but incommensurable lengths. The experiments were performed in the frequency range of a single transversal mode, where the associated Helmholtz equation is effectively one-dimensional and waveguide networks may serve as models of quantum graphs with the joints and waveguides corresponding to the vertices and bonds. The tetrahedral network comprises T junctions, while the honeycomb network exclusively consists of Y junctions, that join waveguides with relative angles 90^{∘} and 120^{∘}, respectively. We demonstrate that the vertex scattering matrix, which describes the propagation of the modes through the junctions, strongly depends on frequency and is nonsymmetric at a T junction and thus differs from that of a quantum graph with Neumann boundary conditions at the vertices. On the other hand, at a Y junction, similarity can be achieved in a certain frequency range. We investigate the spectral properties of closed waveguide networks and fluctuation properties of the scattering matrix of open ones and find good agreement with random matrix theory predictions for the honeycomb waveguide graph.
Resonance trapping appears in open many-particle quantum systems at high level density when the coupling to the continuum of decay channels reaches a critical strength. Here a reorganization of the system takes place and a separation of different time scales appears. We investigate it under the influence of additional weakly coupled channels as well as by taking into account the real part of the coupling term between system and continuum. We observe a saturation of the mean width of the trapped states. Also the decay rates saturate as a function of the coupling strength. The mechanism of the saturation is studied in detail. In any case, the critical region of reorganization is enlarged. When the transmission coefficients for the different channels are different, the width distribution is broadened as compared to a ${\ensuremath{\chi}}_{K}^{2}$ distribution where KKK is the number of channels. Resonance trapping takes place before the broad state overlaps regions beyond the extension of the spectrum of the closed system.
Quantum walks that depend smoothly on a small parameter ε≥0\varepsilon\ge0ε≥0 are considered on directed graphs. The asymptotic behavior of the scattering matrix of the quantum walk as ε→+0\varepsilon\to+0ε→+0 is investigated. It is shown that, in this limit, the scattering matrix does not converge to that for ε=0\varepsilon=0ε=0 at points in the essential spectrum (the unit circle) that are asymptotically approached by a quantum resonance. Furthermore, a phenomenon resembling and extending the resonant tunneling effect is observed by analyzing this discrepancy through resonant states.
Reflection of particles from a disordered or chaotic medium is characterized by a scattering matrix that can be represented as a superposition of resonances. Each resonance corresponds to an eigenstate inside the medium and has a width related to the decay time of this eigenstate. We develop a general approach to study the distribution function of these resonance widths based on the nonlinear sigma model. We derive an integral representation of the distribution function that works equally well for systems of any symmetry and for any type of coupling to the measuring device. From this integral representation we find explicit analytic expressions for the distribution function in the case of disordered metallic grains. We also compare the analytic results to large-scale numerical simulations and observe their perfect agreement.
We derive an explicit expression for the coupling constants of individual eigenstates of a closed billiard that is opened by attaching a waveguide. The Wigner time delay and the resonance positions resulting from the coupling constants are compared to an exact numerical calculation. Deviations can be attributed to evanescent modes in the waveguide and to the finite number of eigenstates taken into account. The influence of the shape of the billiard and of the boundary conditions at the mouth of the waveguide are also discussed. Finally we show that the mean value of the dimensionless coupling constants tends to the critical value when the eigenstates of the billiard follow random-matrix theory.