Quantum signatures of classical multifractal measures

Abstract

A clear signature of classical chaoticity in the quantum regime is the fractal Weyl law, which connects the density of eigenstates to the dimension $D_0$ of the classical invariant set of open systems. Quantum systems of interest are often {\it partially} open (e.g., cavities in which trajectories are partially reflected/absorbed). In the corresponding classical systems $D_0$ is trivial (equal to the phase-space dimension), and the fractality is manifested in the (multifractal) spectrum of Rényi dimensions $D_q$. In this paper we investigate the effect of such multifractality on the Weyl law. Our numerical simulations in area-preserving maps show for a wide range of configurations and system sizes $M$ that (i) the Weyl law is governed by a dimension different from $D_0=2$ and (ii) the observed dimension oscillates as a function of $M$ and other relevant parameters. We propose a classical model which considers an undersampled measure of the chaotic invariant set, explains our two observations, and predicts that the Weyl law is governed by a non-trivial dimension $D_{\mathrm{asymptotic}}<D_0$ in the semi-classical limit $M\to \infty$.

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• Hierarchical Fractal Weyl Laws for Chaotic Resonance States in Open Mixed Systems
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Cited by in corpus (13)

• Statistics of Chaotic Resonances in an Optical Microcavity
• Resonance eigenfunction hypothesis for chaotic systems
• Structure of resonance eigenfunctions for chaotic systems with partial escape
• Short periodic orbits theory for partially open quantum maps
• Localization of Chaotic Resonance States due to a Partial Transport Barrier
• Universal intensity statistics of multifractal resonance states
• Multifractality of open quantum systems
• The role of short periodic orbits in quantum maps with continuous openings
• Local random vector model for semiclassical fractal structure of chaotic resonance states
• Resonance spectra for quantum maps of kicked scattering systems by complex scaling
• Resonance states of the three-disk scattering system
• Counting statistics of chaotic resonances at optical frequencies: theory and experiments
• Self-Similarity Among Energy Eigenstates