Chaos on quantum graphs with Andreev scattering

The present thesis investigates the spectral statistics of superconducting-normalconducting hybrid systems. These hybrid systems are formed by a normal-metal non-integrable billiard being placed adjacent to a superconductor. The symmetry classification scheme of such systems due to Altland and Zirnbauer is at the basis of the thesis. For the mesoscopic systems described above, we give a semiclassical interpretation of the random-matrix theory prediction (by Altland and Zirnbauer) using periodic-orbit theory. Periodic-orbit theory links the quantum spectrum of a system with its classical periodic orbits. The model of choice for the treatment of the hybrid systems are quantum graphs. For an implementation of the hybrid character, the so-called Andreev scattering process is incorporated on the vertices of the graph. After an introduction to the concepts and methods used (chapter 1), a numerical treatise shows us how to generate an ensemble of graphs by appropriately choosing random scattering conditions at the vertices (chapter 2). The spectrum of these graphs coincides perfectly with the random-matrix theory predictions in the limit of large graphs. Models of Andreev graphs with symmetries of the classes C, CI, D, and DIII are formulated with the aid of Andreev star graphs (chapter 3). By the use of periodic-orbit theory, the short-time behaviour of the spectral form factor (the Fourier transform of the spectral density) is calculated semiclassically and shows excellent agreement with the predictions of Altland and Zirnbauer. All analytical calculations are supplemented with numerical results which are in perfect agreement with the analytical results and the random-matrix theory predictions. For symmetry classes C and CI, the approximation schemes developed with the help of quantum graphs have been carried over to the original physical system of Andreev billiards