Quantum chaos and semiclassical behavior in mushroom billiards II: Structure of quantum eigenstates and their phase space localization properties

TL;DR

The paper investigates how quantum eigenstates localize in the phase space of the Bunimovich mushroom billiard, a canonical mixed-type system, by systematically varying the stem width to tune stickiness. It employs Poincaré–Husimi representations and two localization measures, entropy and inverse participation ratio, to characterize regular, chaotic, and mixed states, and separates chaotic from regular states using a PH-based overlap criterion. A key finding is that chaotic-state localization follows a beta distribution whose parameters depend on the stem width, while the joint A–IPR statistics reveal a strong, nearly linear correlation. The mixed-state fraction decays as a power law χ(e) ∝ e^{β} with β ≈ −1/3 across geometries, in agreement with PUSC, thereby linking classical stickiness mechanisms to quantum localization and highlighting universal aspects of the approach to the semiclassical limit.

Abstract

We investigate eigenstate localization in the phase space of the Bunimovich mushroom billiard, a paradigmatic mixed-phase-space system whose piecewise-$C^{1}$ boundary yields a single clean separatrix between one regular and one chaotic region. By varying the stem half-width $w$, we continuously change the strength and extent of bouncing-ball stickiness in the stem, which for narrow stems gives rise to phase space localization of chaotic eigenstates. Using the Poincaré-Husimi (PH) representation of eigenstates we quantify localization via information entropies and inverse participation ratios of PH functions. For sufficiently wide stems the distribution of entropy localization measures converges to a two-parameter beta distribution, while entropy localization measures and inverse participation ratios across the chaotic ensemble exhibit an approximately linear relationship. Finally, the fraction of mixed (neither purely regular nor fully chaotic) eigenstates decays as a power-law in the effective semiclassical parameter, in precise agreement with the Principle of Uniform Semiclassical Condensation of Wigner functions (PUSC).

Figures (16)

• Figure 1: Boundary representing the mushroom billiard. For the classical dynamics we use the full boundary and for the quantum calculations the half mushroom boundary represented by the green boundary. The Poincaré-Birkhoff coordinates are $(s,p)$, where $s$ is counted anti-clockwise from the origin $s=0$, while $p$ is the sine of the reflection angle. The origin in configuration space is designated by $O=(0,0)$. It coincides with the re-entrant corner for technical reasons betcke. On the dashed green vertical lines the Dirichlet BC are automatically satisfied by the construction (CAFB). The purple colored dashed curve is the inscribed circle to which MUPOs forming the boundary between the regular and chaotic region must be tangent to.
• Figure 2: (Top row) Typical wavefunction plots for a (a) regular state ($k=213.949$) and (b) a non-localized chaotic state ($k=214.008$) for $w=0.5$. (Bottom row) (c) Poincaré-Husimi function of the regular wavefunction above, (d) Poincaré-Husimi function of the chaotic wavefunction above.
• Figure 3: Example of a mixed-type state (a) wavefunction and (b) PH function ($M=0.51,w=0.5$) showing increased density around the phase space region generated by MUPOs forming the border between the regular and chaotic region (external stickiness) and also the bouncing-ball region (internal stickiness).
• Figure 4: S plots \ref{['S_eq']} for $w=0.1-0.9$, labeled $(a)-(i)$, showing stickiness around bouncing-ball modes for small $w$ and no apparent stickiness on the border between the regular and chaotic region. To enhance visualization, the color scale for $S$ is saturated at $5$ (i.e., values $S>5$ are shown as if $S=5$). Cells in the regular region are rendered white despite their value being S=0 (since $\sigma(\tau)=0$). The chaotic trajectory was iterated $N=10^8$ times.
• Figure 5: Distribution of entropy localization measures for the mushroom billiard with stem width $w=0.1$ for chaotic states in a window centered around an increasing wavenumber $k$ containing in total $N=2000$ PH functions. (a)-(f). The fitting parameters for the beta distribution \ref{['beta_distribution']}$(A_0,a,b)$ are given in each subfigure along with the center of the wavenumber window designated as $k$.