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In this thesis theoretical models for describing ultrafast dynamics and High Harmonic Generation (HHG) in bulk periodic systems are developed. HHG in bulk solid state systems has been achieved by several groups over the last few years. In this thesis a review of recent results is presented, with attention paid to the development of theoretical models for the HHG process in periodic solids. A closed form expression for a Landau-Dykhne type sub-cycle transition rate between bands of nearest-neighbour tight-binding structures is derived. This rate is used to construct a semi-classical model for HHG in solids. The sub-cycle nature of the transition rate is shown to lead to destructive interference of currents in the conduction band. The time dependent Schödinger equation is employed in the accelerated Bloch basis to study the effect of multiple bands on the HHG process. For mid-IR fields transitions between bands can be sufficiently strong that transitions between the conduction bands suppress the Bragg reflection process. It is shown that such transitions can form an effective nearly parabolic conduction band, and lead to a large reduction in harmonic intensity compared to single conduction band models. The prediction of destructive interference of current in the conduction band of periodic solids is studied in ZnO, using a non-local empirical pseudopotential band structure and matrix elements, in the density matrix formalism, with the inclusion of dephasing effects. It is shown that the quantum destructive interference is present in the density matrix calculation, closely matching semi-classical predictions. The effect of multiple bands of the structure, and variation in the dephasing timescale of the system is also considered.

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