Sticky eigenstates in systems with sharply-divided phase space

TL;DR

This paper probes quantum signatures of classical stickiness in non-KAM systems with sharply divided phase space, using two piecewise-linear maps and quantization on a torus. By combining the overlap index and the entropy localization length, it classifies eigenstates into regular, chaotic, and mixed, and identifies sticky eigenstates localized near boundary structures formed by MUPOs or quasi-periodic orbits. A random-matrix model shows dynamical tunneling contributes as $\sim \hbar\exp(-b/\hbar)$, while the sticky-state fraction scales as $f_{sticky} \sim \hbar^{1-1/\gamma}$ with $\gamma=2$ for MUPO boundaries (oscillating around this value for quasi-periodic boundaries), linking quantum signatures to classical stickiness. The results generalize hierarchical-state predictions from KAM systems to sharp-boundary, non-KAM contexts and offer a practical framework for identifying stickiness-related eigenstates in quantum maps.

Abstract

We investigate mixed eigenstates in systems with sharply-divided phase space, using different piecewise-linear maps whose regular-chaotic boundaries are formed by marginally unstable periodic orbits (MUPOs) or by quasi-periodic orbits. With the overlap index and the entropy localization length, we classify mixed eigenstates and show that the contribution from dynamical tunneling scales as $\sim \hbar\, \exp(-b/\hbar)$, with $b>0$ associated with the relative size of the regular region. The dominant fraction of states that remain sticky to the boundaries, referred to as sticky eigenstates, scales as $\hbar^{1/2}$ in the MUPO case and oscillates around this algebraic behavior in the quasi-periodic case. This behavior generalizes established predictions for hierarchical states in KAM systems, which scale as $\hbar^{1 - 1/γ}$, with $γ$ set by the corresponding classical stickiness reflected in the algebraic decay of cumulative RTDs $t^{-γ}$. For the piecewise-linear maps studied here, $γ= 2$. These results reveal a clear quantum signature of classical stickiness in non-KAM systems.

Figures (13)

• Figure 1: Illustration of two piecewise-linear functions alongside the standard map for comparison.
• Figure 2: Comparison between the Poincaré sections in panels (a1)–(d1) and the SALI plots in panels (a2)–(d2), for maps (i) with $k = 1.5$ and (ii) with $k = 2,3.5,4$, from left to right. Panels (a1)-(d1) show a single chaotic orbit evolved for $10^4$ iterations. The SALI values, shown on a logarithmic scale in the lower panels, are computed up to 60 iterations. We plot $- 0.5\le p \le 0.5$ in (a) and $- 0.5 \le x,p \le 0.5$ in (b)-(d) for visualization convenience.
• Figure 3: Lagrangian descriptors for the two maps with $a=0.5$, shown from left to right, are presented as in Fig. \ref{['fig:cp1']}, where the top and bottom panels correspond to integration times $M=10$ and $M=60$, respectively.
• Figure 4: 3D representation of the normalized coherent state $|\alpha\rangle$ centered at $(0.25,0)$, showing the surface together with its projection onto the $(x,p)$-plane, for system sizes $N=100$ in (a) and $N=1000$ in (b).
• Figure 5: Examples of Husimi functions $H_n(x,p)$ for two maps, presented in the same manner as the previous figures, with $N=500$. The upper panels (a1–c1) correspond to eigenstates with overlap index $\omega \simeq -1$, while the lower panels (a2-c2) show states with $\omega \simeq 1$. All data are shown in a linear scale. Bright colors indicate regions of large $H_n(x,p)$, and dashed lines in panels (a) and (b) mark the boundaries between regular and chaotic regions.