Quantum manifestations of classical chaos in a Fermi accelerating disk

Abstract

We study the classical and quantum mechanics of a free particle that collides elastically with the walls of a circular disk with the radius varying periodically in time. The quasi-energy spectral properties of the model are obtained from evaluation of finite-dimensional approximations to the time evolution operator. As the scaled hbar is changed from large to small, the statistics of the Quasienergy Eigenvalues (QEE) change from Poisson to circular orthogonal ensemble (COE). Different statistical tests are used to characterize this transition. The transition of the Quasienergy Eigenfunctions (QEF) is also studied using the chi-squared test with nu degrees of freedom, which goes over to the Porter-Thomas distribution for nu=1. We find that the integrable regime is associated with exponentially localized QEF whereas the eigenfunctions are extended in the chaotic, COE regime. We change the representation of the model to one in which the boundary is fixed and the Hamiltonian acquires a quadratic forcing term. We then carry out a successful comparison between specific classical phase space solutions and their corresponding QEFs in the Husimi representation.