Participation Ratio as a Quantum Probe of Hierarchical Stickiness

TL;DR

This paper demonstrates that the participation ratio effectively probes hierarchical stickiness in quantum systems, aligning quantum localization with classical phase space structures, and offers a practical method for diagnosing anomalous localization.

Contribution

It introduces the participation ratio as a sensitive quantum probe for hierarchical transport, bridging quantum localization with classical phase space structures in driven systems.

Findings

Participation ratio resolves classical layered structures.

Quantum and classical indicators show optimal agreement within finite times.

Method provides a practical route for diagnosing anomalous localization.

Abstract

We investigate how quantum localization encodes the hierarchical stickiness that governs transport in mixed classical phase spaces. Using the periodically driven kicked top, we show that the participation ratio (PR) of coherent states in the Floquet eigenbasis resolves the same layered structure that appears classically as a multimodal distribution of finite-time Lyapunov exponents (FTLEs). To establish a quantitative correspondence, we introduce a Gaussian coarse graining of the FTLE matched to the intrinsic semiclassical resolution of coherent states. Both local correlations and global comparisons of probability distributions demonstrate that quantum and classical indicators agree optimally within a finite window of evolution times, where sticky structures are most clearly resolved. Our results promote the participation ratio from a global measure of chaos to a sensitive probe of hierarchical transport and provide a practical route for diagnosing anomalous localization in driven quantum systems.