154 research outputs found

    Quantum Graphs: A model for Quantum Chaos

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    We study the statistical properties of the scattering matrix associated with generic quantum graphs. The scattering matrix is the quantum analogue of the classical evolution operator on the graph. For the energy-averaged spectral form factor of the scattering matrix we have recently derived an exact combinatorial expression. It is based on a sum over families of periodic orbits which so far could only be performed in special graphs. Here we present a simple algorithm implementing this summation for any graph. Our results are in excellent agreement with direct numerical simulations for various graphs. Moreover we extend our previous notion of an ensemble of graphs by considering ensemble averages over random boundary conditions imposed at the vertices. We show numerically that the corresponding form factor follows the predictions of random-matrix theory when the number of vertices is large---even when all bond lengths are degenerate. The corresponding combinatorial sum has a structure similar to the one obtained previously by performing an energy average under the assumption of incommensurate bond lengths.Comment: 8 pages, 3 figures. Contribution to the conference on Dynamics of Complex Systems, Dresden (1999

    Statistical properties of resonance widths for open Quantum Graphs

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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.Comment: 17 pages, 7 figures. submitted to Waves in Random Media, special issue for graph

    Quantum Graphology

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    We review quantum chaos on graphs. We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wavefunction statistics. This operator is the analogue of the classical evolution operator on the graph. It allow us to establish a connection between the corresponding periodic orbits and the statistical properties of eigenvalues and eigenfunctions. Specifically, for the energy-averaged spectral form factor we derived an exact combinatorial expression which illustrate the role of correlations between families of isometric orbits. We also show that enhanced wave function localization due to the presence of short unstable periodic orbits and strong scarring can rely on completely different mechanisms.Comment: 15 pages, 7 figures, Paper presented at the 2nd Workshop on Quantum Chaos and Localisation Phenomenon, May 20th, 2005, Warsaw, Poland. To be Published in Acta Physica Polonica

    Non-perturbative response: chaos versus disorder

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    Quantized chaotic systems are generically characterized by two energy scales: the mean level spacing Δ\Delta, and the bandwidth Δb\Delta_b\propto\hbar. This implies that with respect to driving such systems have an adiabatic, a perturbative, and a non-perturbative regimes. A "strong" quantal non-perturbative response effect is found for {\em disordered} systems that are described by random matrix theory models. Is there a similar effect for quantized {\em chaotic} systems? Theoretical arguments cannot exclude the existence of a "weak" non-perturbative response effect, but our numerics demonstrate an unexpected degree of semiclassical correspondence.Comment: 8 pages, 2 figures, final version to be published in JP

    A Concept of Linear Thermal Circulator Based on Coriolis forces

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    We show that the presence of a Coriolis force in a rotating linear lattice imposes a non-reciprocal propagation of the phononic heat carriers. Using this effect we propose the concept of Coriolis linear thermal circulator which can control the circulation of a heat current. A simple model of three coupled harmonic masses on a rotating platform allow us to demonstrate giant circulating rectification effects for moderate values of the angular velocities of the platform