Dinâmica do pacote de onda em caminhadas quânticas discretas com três estados

Abstract

Currently, there is a great interest in studying quantum computation and quantum information theory. Recent studies show that these computers can realize computational calculations much more faster than any classical computer, in addition to serving to understand fundamental properties of quantum systems. Among this new studies, the quantum walks serves as an analogue of the classical random walks, which is widely studied in computation science and in the description of physical systems. In this study, we will analyze the properties of discrete-time quantum walks in linear and nonlinear mediums. Being more specific, we focused on the three-state quantum walk, a model that posses an intrisic localization around the initial position. We show that, in the linear case, relevant physical quantities in the quantum walk literature satisfies universal dynamical scaling laws in the vicinity of the point where occur the transition between localized and delocalized states. Further, we obtain an analytical expression to the participation ratio for the configuration where the walker are completely delocalized in the lattice. This expression is also valid for others quantum walks that posses the same characteristics. When we consider a nonlinear dynamics, we will see that there is an emergence of a new behavior in the quantum walk literature, the irradiation of the localized portion of the wave-packet. It is display that this irradiation follows an specific power-law, which is independent of the initial state and also of the nonlinearity parameter. We hope that the results obtained through this monography brings new ideas to be implemented in a near future in the context of quantum walks, in such a way to be useful in the development of new studies.