Studies of chaos in two-dimensional billiards

A billiard is defined here as a two-dimensional system of particles confined by infinitely hard boundary as an idealization of the system of electrons in low-temperature small-scale semiconductor heterostructures. We only focus on one-particle systems by ignoring interactions between particles. We study open and closed billiards, and we use classical mechanics and quantum mechanics to solve the system. In closed billiards, chaos, which is characterized by strong dependence on initial conditions, arises when the system is nonintegrable in classical mechanics. There are two kinds of chaos, soft chaos and hard chaos. Hard chaos is the special case having global chaos and ergodicity. We see manifestations of underlying classical chaos in quantum mechanics using the quantum web and the Husimi plot using two kinds of simple billiards. We also study open billiards by attaching leads to closed billiards. For an open classical billiard, we observe that fractal-like behaviors arise when the underlying closed system is chaotic by finding out from which lead the particle comes out when the particle is injected with an angle. In observing manifestations of chaos, we find that soft chaos and hard chaos show different behaviors. In an open quantum circular billiard, we study a circular billiard with a point flux at the center. The underlying classical closed billiard is non-chaotic, and the system shows Aharonov-Bohm effect and resonances with the corresponding closed billiard when we compute the conductance as the energy of a particle varies.