Classical and Quantum Chaos in the Diamond Shaped Billiard

TL;DR

This paper investigates the transition from regular to chaotic behavior in a diamond-shaped billiard system, analyzing classical chaos indicators and quantum spectral statistics to understand their correlation.

Contribution

It introduces a parameterized billiard shape that transitions from regular to chaotic regimes and provides numerical analysis of classical and quantum chaos indicators.

Findings

Classical chaos increases as the shape parameter decreases from 1.

Quantum energy spectra follow Poisson and GOE distributions in regular and chaotic regimes.

Quantum wavefunctions exhibit scars consistent with classical periodic orbits.

Abstract

We analyse the classical and quantum behaviour of a particle trapped in a diamond shaped billiard. We defined this billiard as a half stadium connected with a triangular billiard. A parameter $\xi$ which gradually change the shape of the billiard from a regular equilateral triangle ($\xi=1$) to a diamond ($\xi=0$) was used to control the transition between the regular and chaotic regimes. The classical behaviour is regular when the control parameter $\xi$ is one; in contrast, the system is chaotic when $\xi \neq 1$ even for values of $\xi$ close to one. The entropy grows fast as $\xi$ is decreased from 1 and the Lyapunov exponent remains positive for $\xi<1$. The Finite Difference Method was implemented in order to solve the quantum problem. The energy spectrum and eigenstates were numerically computed for different values of the control parameter. The nearest-neighbour spacing distribution is analysed as a function of $\xi$, finding a Poisson and a Gaussian Orthogonal Ensemble(GOE) distribution for regular and chaotic regimes respectively. Several scars and bouncing ball states are shown with their corresponding classical periodic orbits. Along the document the classical chaos identifiers are computed to show that system is chaotic. On the other hand, the quantum counterpart is in agreement with the Bohigas-Giannoni-Schmit conjecture and exhibits the standard features for chaotic billiard such as the scarring of the wavefunction.