Chaos in the Diamond-Shaped Billiard with Rounded Crown

Abstract

We analyse the classical and quantum behaviour of a particle trapped in a diamond-shaped billiard with rounded crown. We defined this billiard as a half stadium connected with a triangular billiard. A parameter ξ smoothly changes the shape of the billiard from an equilateral triangle (ξ = 1) to a diamond with rounded crown (ξ = 0). The parameter ξ controls the transition between the regular and chaotic regimes. The classical behaviour is regular when the control parameter ξ is one; in contrast, the system is chaotic when ξ = 1 even for values of ξ close to one. Several quantities such as Lyapunov exponent and the entropy of the distribution of the incident angle are used to characterize the chaotic behaviour of the classical system. The average information preserved by the classical trajectories increases rapidly as ξ is decreased from 1 and the Lyapunov exponent remains positive for ξ < 1. The Finite Difference Method was implemented in order to solve the quantum counterpart of the billiard. The energy spectrum and eigenstates were numerically computed for different values of ξ < 1. The spacing distribution between adjacent eigenvalues is analysed as a function of ξ, 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. On the other hand, the results found for the quantum billiard are in agreement with the Bohigas-Giannoni-Schmit conjecture and exhibits the standard features of chaotic billiards such as the scarring of the wavefunction.Keywords: quantum chaos, quantum billiards, random matrices, finite difference method.Caos en el billar de forma de diamante y corona redondeada Resumen Se estudia el comportamiento de una partícula en el interior de un billar triangular donde uno de sus lados toma de medio estadio que se llamó billar diamante con corona redondeada o DSRC por su siglas en inglés. Se definió un parámetro ξ que cambia suavemente la forma la frontera partiendo de un billar triangular ξ = 1 a un billar DSRC ξ = 1. Dicho parámetro controla la transición entre el régimen regular y caótico. Clásicamente, el sistema es regular cuando ξ = 1. Por otro lado, el sistema se torna caótico para ξ = 1 incluyendo valores próximos a 1. Se calcula el coeficiente de Lyapunov y la entropía media de la distribución de losángulos de incidencia para caracterizar el comportamiento caótico del sistema. Se observó un rápido crecimiento de la información de las trayectorias hasta saturar la entropía al cambiar levemente la frontera del billar triangular original. A su vez el coeficiente de Lyapunov se mantuvo positivo durante este proceso una vez que ξ se alejaba de 1. Se implementó el método de diferencias finitas FDM para obtener el espectro y los estados propios de la contraparte cuántica del sistema. La distribución de espaciamiento entre primeros vecinos para varios valores de ξ fue construida numéricamente para diferentes valores de ξ encontrando u...

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