Quantum chaos has emerged in the half of the last century with the notorious
problem of scattering of heavy nuclei. Since then, theoreticians have developed powerful
techniques to approach disordered quantum systems. In the late 70's, Casati
and Chirikov initiated a new field of research by studying the quantum counterpart
of classical problems that are known to exhibit chaos. Among the several quantum-classical chaotic systems studied, the kicked rotor stimulated a lot of enthusiasm in
the scientific community due to its equivalence to the Anderson tight binding model.
This equivalence allows one to map the random Anderson model into a set of fully
deterministic equations, making the theoretical analysis of Anderson localization considerably
simpler. In the one-dimensional linear regime, it is known that Anderson
localization always prevents the diffusion of the momentum. On the other hand, for
higher dimensions it was demonstrated that for certain conditions of the disorder parameter,
Anderson localized modes can be inhibited, allowing then a phase transition
from localized (insulating) to delocalized (metallic) states. In this thesis we will numerically and theoretically investigate the properties of
a multidimensional quantum kicked rotor in a nonlinear medium. The presence of
nonlinearity is particularly interesting as it raises the possibility of having soliton
waves as eigenfunctions of the systems. We keep the generality of our approach
by using an adjustable diffusive nonlinearity, which can describe several physical
phenomena.
By means of Variational Calculus we develop a chaotic map which fully describes
the soliton dynamics. The analysis of such a map shows a rich physical scenario that
evidences the wave-particle behavior of a soliton. Through the nonlinearity, we trace
a correspondence between quantum and classical mechanics, which has no equivalent
in linearized systems.
Matter waves experiments provide an ideal environment for studying Anderson
localization, as the interactions in these systems can be easily controlled by Feshbach
resonance techniques. In the end of this thesis, we propose an experimental realization
of the kicked rotor in a dipolar Bose Einstein Condensate.
| Identifer | oai:union.ndltd.org:kaust.edu.sa/oai:repository.kaust.edu.sa:10754/225111 |
| Date | 05 1900 |
| Creators | Brambila, Danilo |
| Contributors | Fratalocchi, Andrea, Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division, Bagci, Hakan |
| Source Sets | King Abdullah University of Science and Technology |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Rights | 2013-05-30, At the time of archiving, the student author of this thesis opted to temporarily restrict access to it. The full text of this thesis became available to the public after the expiration of the embargo on 2013-05-30. |
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