Scope and Method of Study. The purpose of this research was to experimentally study quantum dynamics of systems whose classical dynamics are chaotic. Quantum δ-kicked systems such as kicked rotor and kicked accelerator were used. The cold non condensed atoms were kicked first to realize the kicked accelerator. Among the objectives were the realization of resonances of the kicked accelerator and associated phenomena of quantum accelerator modes using a Bose-Einstein Condensation (BEC). One of the major achievements of the work in this thesis was the creation of the quantum δ-kicked rotor and its associated resonances to realize a quantum ratchet. The properties of the ratchet were studied in detail.

Findings and Conclusions. The Quantum Accelerator Modes (QAM) were realized using both thermal samples of atoms and a BEC. Multiple micro optical traps were accidentally observed and in order to understand their behavior a theory was developed using spherical aberration of a lens. The maps produced by an effective classical theory were studied using the QAM. The resonances of the δ-kicked accelerator were observed for the first time and the theory was developed. One of the models that describes the QAM using rephasing of momentum states was observed in the experiments. The ratchet was realized using the resonances of the kicked rotor and accelerator where the diffusion in the case of classical ratchets was replaced by chaos in the quantum ratchet mechanism.