As compared to the equilibrium state of matter, very little is known about the non-equilibrium state of matter at ultrafast time scales (attoseconds to femtoseconds). A special role in this emerging field is thus played by ab-initio theory. While this approach has begun to show signs of great promise, it has been plagued by an unavoidable mismatch between experimental and theoretical observables. The only numerically feasible ab-initio theory is the time dependent extension of density function theory (TDDFT), and in this theory the natural observables are charge and spin densities and their corresponding currents. In experiments at ultrafast time scales, however, the only practical way to deduce transient electronic states and spin structures is via spectral information. The primary purpose of this thesis is to bridge this divide between state-of-the art theory and experiment. Our approach to solving this problem combines two flavours of TDDFT: real-time TDDFT (RT-TDDFT) and linear-response TDDFT (LR-TTDFT). The former is employed to solve the fundamental time dependent problem, while the latter serves to, at each time step of interest, determine the MCD spectra. In Chapter 4 we have used this approach to study transient magnetic circular dichroism (tr-MCD) in the extreme ultraviolet spectral range (XUV) in bulk- Co and CoPt, which shows excellent agreement between theoretical and experimental spectral data. Deploying our approach, we also show how these complex spectra can be efficiently decoded revealing a wealth of details of the underlying spin dynamics of complex alloy systems at ultrafast time scales. In particular, we compare our theoretical prediction for the tr-MCD for CoPt with experimental measurements and find excellent agreement at different frequencies. In the second part of the research of this thesis we have explored the question of how to probe quasi-particle excitations within TDDFT. The time dependent electronic structure has, in contrast to its ground state counterpart, no resemblance to the quasi-particle excitations of the system. To circumvent this problem we project the transient state of the system onto the Kohn- Sham ground state, and subsequently integrate over energy to find the charge excited at each crystal momenta k, a quantity we denote Nex(k). As we show in Chapter 5 this object provides, in the case of single layer graphene, a picture of momentum space excitations consistent with the two dimensional transient Electron Momentum Density (2D tr-EMD). This both verifies the Nex(k) as providing a good description of momentum space excitations, as well as demonstrating that 2D tr-EMD, which can be obtained experimentally via Compton tomography, forms an experimental basis for investigating momentum space in such 2D materials. Using both these approaches we probe the phenomena of Landau-Zener-Stückelberg (LZS) interference patterns in the first Brillouin zone (1BZ) of graphene, carefully demonstrating how various laser pulse parameters can be tuned to enhance the patterns. Finally in Chapter 6 we deploy the tool of Nex(k) to explore momentum space excitations in monolayer WSe2. The combination between valley selection rule and the large (and opposite, by time reversal symmetry) spin splitting at each valley leads to selection by helical light of valley-spin excitation: spin-valley locking. Taking this further we show that hybrid pulses of circular and infra-red linear laser light can be used to selectively excite spin throughout the Brillouin zone – taking back control from the valleys that have, to date, dominated the ultrafast physics in this material. Such hybrid pulses combine two canonical optical effects: the Bloch acceleration theorem and the valley selection rule, and employing the Nex(k) we show how this combination of fundamental phenomena leads to the desired full control over momentum space. We thus establish a route from laser light to local control over excitation in reciprocal space, opening the way to the preparation of momenta specified excited states at ultra-fast time scales.
