Participation Ratio as a Quantum Probe of Hierarchical Stickiness

TL;DR

The paper addresses how hierarchical stickiness in mixed classical phase space imprints itself on quantum dynamics in the kicked top. It uses the participation ratio $PR$ of coherent states in the Floquet basis as a local quantum probe and introduces Gaussian coarse graining of finite-time Lyapunov exponents (GFTLE) with $h_{eff}=1/J$ to match semiclassical resolution, establishing a quantitative quantum–classical correspondence. The findings show that $PR$ reveals a multimodal, layer-by-layer structure mirroring the classical FTLE distribution, with maximal agreement at an intermediate time window and stronger correspondence as $J$ increases, confirming a semiclassical origin. The work provides a practical framework for diagnosing hierarchical transport and anomalous localization in driven quantum systems and can be extended to higher-dimensional or many-body Floquet settings.

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.

Figures (4)

• Figure 1: Hierarchical stickiness in the kicked top for $k=3$. (a1) Finite-time Lyapunov exponent (FTLE) distribution over the chaotic sea in classical phase space. (b1–d1) Phase space partitioned according to intervals of FTLE values. (e1) Multimodal histogram of the FTLE density. (a2) Distribution of the participation ratio (PR) of coherent states in phase space for $J=500$. (b2–d2) Phase space partitioned according to intervals of PR values. (e2) Multimodal histogram of the PR density.
• Figure 2: (a)-(c) GFTLE distribution using a Gaussian smoothing width $\sigma^2=1/J$ for time windows $\tau=1, 100$, and $5000$. (d)-(f) PR distribution of coherent states for $J=500, 1000$, and $1500$. (g) Pearson correlation coefficient between the GFTLE and the PR.
• Figure 3: (a)-(c) GFTLE probability distributions for $\tau=1, 100$, and $5000$. (d)-(f) PR probability distributions for $J=500, 1000$, and $1500$. (g) Jensen--Shannon distance between the GFTLE and PR as a function of the time window $\tau$ (logarithmic scale). Inset: magnified view of the region around the minimum, in linear scale.
• Figure 4: (a–c) Classical phase space for increasing kick strength $k=1, 4$, and $10$, showing the progressive destruction of invariant tori and the growth of the chaotic sea. (d–f) Quantum counterparts obtained from the participation ratio of coherent states evaluated over phase space for the same parameter values. (g) Fraction of chaotic phase-space area relative to the regular region as a function of the kick strength. (h) Level-spacing parameter characterizing the spectral transition from regular to chaotic statistics. (i) Phase-space–averaged participation ratio as a function of the kick strength.