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Quantum Graphology

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Kottos T.

Acta Physica Polonica A

2006

vol. 109

issue 1

7-21

We review quantum chaos on graphs. We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wave function statistics. This operator is the analogue of the classical evolution operator on the graph. It allows 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

Fidelity in Quasi-1D Systems as a Probe for Anderson Localization

45%

Bodyfelt J., Zheng M., Kottos T., Kuhl U., Stöckmann H.

Acta Physica Polonica A

2009

vol. 116

issue 5

756-764

We analyze the echo dynamics in quasi-one-dimensional random media to investigate how the transition from localization to delocalization is encoded in its temporal decay properties. Our analysis extends from the standard perturbative regime corresponding to small perturbations (with respect to the mean level spacing) in the echo dynamics, out to the Wigner decay regime. On the theoretical side, our results rely on a banded random matrix modeling, and show in the localized regime under small perturbations a novel decay of the fidelity (Loschmidt echo), differing from the typical Gaussian decay seen within both diffusive and chaotic systems. For larger perturbation strengths, typical Wigner exponential decays are observed. Scattering echo measurements are performed experimentally within a quasi-1D microwave cavity randomly populated with point-like scatterers. Agreements are observed between experiments, numerics, and theoretical predictions.

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2 Acta Physica Polonica A

1 2009

1 2006

2 Kottos T.

1 Bodyfelt J.

1 Kuhl U.

1 Stöckmann H.

1 Zheng M.

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Quantum Graphology

Fidelity in Quasi-1D Systems as a Probe for Anderson Localization