153 research outputs found
Quantum Graphs: A model for Quantum Chaos
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
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
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
Quantized chaotic systems are generically characterized by two energy scales:
the mean level spacing , and the bandwidth . 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
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
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