Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short Correlations in Singlets and Doublets

Billiards are flat cavities where a particle is free to move between elastic collisions with the boundary. In chaos theory these systems are simple prototypes, their conservative dynamics of a billiard may vary from regular to chaotic, depending only on the border. The results reported here seek to shed light on the quantization of classically chaotic systems. We present numerical results on classical and quantum properties in two bi-parametric families of Billiards, Elliptical Stadium Billiard (ESB) and Elliptical-$C_3$ Billiards (E-$C_3$B). Both are elliptical perturbations of chaotic billiards with originally circular sectors on their borders. Our numerical calculations show evidence that the elliptical families can present a mixed classical phase space, identified by a parameter $\rho_\text{c} < 1$, which we use to guide our analysis of quantum spectra. We explored the short correlations through nearest neighbor spacing distribution $p(s)$, which showed that in the mixed region of the classical phase space, $p(s)$ is well described by the Berry-Robnik-Brody (BRB) distributions for the ESB. In agreement with the expected from the so-called ergodic parameter $\alpha = t_\text{H}/t_\text{T}$, the ratio between the Heisenberg time and the classical diffusive-like transport time signals the possibility of quantum dynamical localization when $\alpha < 1$. For the E-$C_3$B family, the eigenstates can be split into singlets and doublets. BRB describes $p(s)$ for singlets as the previous family in the mixed region. However, the $p(s)$ for doublets are described by new distributions recently introduced in the literature but only tested in a few cases for $\rho_\text{c} < 1$. We observed that as $\rho_\text{c}$ decreases, the $p(s)$'s tend to move away simultaneously from the GOE (singlets) and GUE (doublets) distributions.