Cross-section fluctuations in open microwave billiards and quantum graphs: The counting-of-maxima method revisited

Closed and open superconducting microwave waveguide networks as a model for quantum graphs

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.

Correlation functions and correlation widths in quantum-chaotic scattering for mesoscopic systems and nuclei

We derive analytical expressions for the correlation functions of the electronic conductance fluctuations of an open quantum dot under several conditions. Both the variation of energy and that of an external parameter, such as an applied perpendicular or parallel magnetic fields, are considered in the general case of partial openness. These expressions are then used to obtain the ensemble-averaged density of maxima, a measure recently suggested to contain invaluable information concerning the correlation widths of chaotic systems. The correlation width is then calculated for the case of energy variation, and a significant deviation from the Weisskopf estimate is found in the case of two terminals. The results are extended to more than two terminals. All of our results are analytical. The use of these results in other fields, such as nuclei, where the system can only be studied through a variation of the energy, is then discussed.