Classical and quantum chaos in bean- and peanut-shaped billiards

TL;DR

The paper investigates classical and quantum chaos in two mixed-curvature billiards (bean and peanut) with no neutral boundary segments to examine the classical–quantum correspondence. It combines classical analyses (billiard flow, billiard map, Poincaré sections) with quantum diagnostics (eigenfunctions, level statistics, spectral staircase) and novel measures (spectral complexity and OTOCs). It finds strong alignment between classical chaos and quantum signatures, including chaotic Poincaré maps, GOE-like level statistics for bean and peanut, and eigenfunction scars whose strength depends on symmetry; peanut exhibits stronger scar formation. The spectral-complexity analysis reveals universal quadratic, linear, and logarithmic growth regimes with earlier saturation in chaotic geometries, while the OTOC demonstrates temperature-dependent scrambling consistent with chaos. Altogether, the work shows that geometry alone can induce chaotic dynamics and provides a versatile framework for probing quantum chaos beyond conventional spectral metrics.

Abstract

The boundary of a billiard system plays a crucial role in shaping its dynamics, which may be integrable, mixed, or fully chaotic. When a boundary has varying curvature, it offers a unique setting to study the relation between classical chaos and quantum behaviour. In this study, we introduce two geometrically distinct billiards: a bean- and a peanut-shaped billiard. These systems incorporate both focusing and defocusing walls with no neutral segments. Our study reveals a strong correlation between classical and quantum dynamics. Analysis of billiard flow diagrams confirms sensitivity to initial conditions-a defining feature of chaos. Poincaré maps further show the phase space intricately woven with regions of chaotic motion and stability islands. We employ both statistical and dynamical measures to characterise quantum chaos. Statistical indicator includes nearest-neighbour spacing distribution, level spacing ratios, and spectral staircase function, while dynamical measures includes out-of-time-order correlators and spectral complexity. We also observe eigenfunction scarring in both the billiards.

Figures (19)

• Figure 1: A schematic representation of two nearby trajectory traced by a particle in a billiard domain ($\varOmega$) with boundary $\mho (= \partial{\varOmega})$. Here, $\hat{i}$, $\hat{r}$, and $\hat{n}$ are the incident vector, reflection vector, and normal vector at the point of collision, respectively. $\theta_{i}$ and $\theta_{r}$ are angle of incidence and angle of reflection, respectively and they follow the relation $\theta_{i}=\theta_{r}$ for every collision with the boundary.
• Figure 2: (a) Bean curve for $a_{1} < b_{1}$. The red and blue points are intercept points on the closed curve, whereas green and orchid points are extrema points. (b) Bean curves as planar sections of a torus. The top row features $2D$ slices of a torus, which take on different shapes from circular to bean-like curves as the parameters $a_{1}$ and $b_{1}$ are adjusted. In the bottom row, the torus is shown in $3D$, with each toroidal shape representing a different combination of $a_{1}$ and $b_{1}$.
• Figure 3: (a) Cassini oval with two foci ($F_1$ & $F_2$) at $(a_{2},0)$ and $(-a_{2},0$), respectively. The pink, magenta and green points are extrema points on the closed curve. (b) A family of Cassini Ovals as planar sections of a torus. The top row features $2D$ slices of a torus, whereas in the bottom row, the torus is shown in $3D$, with each toroidal shape representing a different combination of $a_{2}$ and $b_{2}$.
• Figure 4: Billiard flow diagrams representing real space trajectories (periodic, quasi-periodic and chaotic) for (a) Circle, (b) Bean, (c) Ellipse, and (d) Peanut billiards for different ICs. Here, $\{{\color{green} \CIRCLE}, ~{\color{gold(web)(golden)}\CIRCLE}, ~{\color{red}\CIRCLE}, ~{\color{cyan} \CIRCLE}, ~{\color{orange}\CIRCLE}, ~{\color{blue}\CIRCLE}, ~{\color{magenta!70} \CIRCLE}\}$ represent seven initial positions. The reflection trajectories within circular and elliptical billiards are predictable, periodic, or quasi-periodic. Within the boundaries of a circular billiard, every concentric circle acts as a caustic. Conversely, within the elliptical billiard, all confocal ellipses and hyperbolas are caustics. Chaotic trajectories are the norm for bean- and peanut-shaped billiards, with only a few specific ICs resulting in periodic or quasi-periodic outcomes.
• Figure 5: Caustics produced in (a) Circle, (b) Ellipse, (c) Bean and (d) Peanut billiards. In (a-d), sub-figures labelled (i) correspond to the first sets of ICs, where only the positions differ and sub-figures labelled (ii) represent the second set of ICs characterised by variations in the direction of momenta. Here, the '${\color{red}\CIRCLE}$' represent the ICs. In circular and elliptical billiards, regular geometry (constant curvature) enforces a regular focusing effect, causing the trajectories to trace out recurrent caustics. Whereas, the mixed-curvature breaks the symmetry and continuity of caustic formation, resulting in non-repetitive, irregular trajectories and the destruction of stable caustics. This geometric complexity gives rise to chaotic behaviour.