Quantum signatures of classical multifractal measures

Area of Science:

• Quantum chaos
• Statistical physics
• Dynamical systems

Background:

• The fractal Weyl law links quantum eigenstate density to classical chaoticity in open systems.
• Partially open quantum systems, common in applications, exhibit multifractality in their classical counterparts' phase space.
• Classical systems with multifractality show a spectrum of Rényi dimensions D(q), unlike trivial D(0) in simpler models.

Purpose of the Study:

• Investigate the impact of multifractality on the fractal Weyl law in partially open quantum systems.
• Analyze how the system size and other parameters influence the observed dimension governing the Weyl law.
• Develop a classical model to explain the observed phenomena and predict semiclassical behavior.

Main Methods:

• Numerical simulations using area-preserving maps to model partially open systems.
• Analysis of the Weyl law's dependence on system size (M) and other relevant parameters.
• Development of a classical model based on an undersampled measure of the chaotic invariant set.

Main Results:

• The Weyl law in these systems is governed by a dimension distinct from the classical phase-space dimension (D(0)=2).
• The governing dimension exhibits oscillations as a function of system size (M) and other parameters.
• A proposed classical model successfully explains these observations.

Conclusions:

• Multifractality significantly alters the Weyl law in partially open quantum systems.
• The Weyl law is governed by a non-trivial dimension, D(asymptotic), which is less than D(0) in the semiclassical limit.
• The findings provide a deeper understanding of quantum-classical correspondence in chaotic systems.