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The development of laser technology allowing precision control over the temporal and spatial profile of ultrafast, highly intense laser pulses, together with the advent of novel means of manipulating static material properties such as layering of 2D materials and tuning cavity confined photon interactions, present substantial opportunities for the engineering of exotic, technologically desirable properties through coherent manipulation of quantum degrees of freedom. However, due to being developed within an equilibrium, perturbative or steady-state context, the theoretical capacity to predict and interpret the rich diversity of ultrafast, far-from equilibrium phenomena observed in experiments in extended systems is oftentimes lacking beyond a coarse phenomenological explanation, or a perturbative approach which oftentimes fails for such strong driving. This is particularly true when assessing one of the most fundamental interactions in matter: the electron-nuclear interaction. While the simulation of strongly driven, non-equilibrium, electron-nuclear dynamics can be done with near exactness for small molecular systems, these quantum chemistry methods face substantial challenges in being applied to extended systems. This thesis collects research done by the author and collaborators to develop and extend simulation methods originating from quantum chemistry which can capture strongly driven electron-nuclear dynamics without dependence on the Born-Oppenheimer framework to extended systems. By transitioning away from the constraints of Born-Oppenheimer, our goal is to develop robust, scalable simulation protocols which can replicate and predict experimental observations in a first-principles, \textit{ab-initio} manner, across a broad range of dynamical regimes and material phases. After giving an overview of some of the experimental phenomena we are trying to address, we give a brief introduction to some aspects of the existing theoretical framework for addressing electron-nuclear interactions as embodied in the Born-Oppenheimer approximation as well as the perturbative and non-perturbative real time-dynamics approaches used to calculate material properties. In the subsequent sections we contextualize the papers presented with a discussion of the primary questions addressed by each, the progress made by others in the field and the contribution made by the author. We first discuss our development of a unique, real-space, grid based ab-intio wavefunction dynamics approach which is capable of exactly capturing electron-nuclear and electron-electron correlation effects in both equilibrium and laser driven regimes across a variety of physical systems and discuss the developments which would be required in order to make this method scalable and competitive. We next apply a semi-classical dynamics method, Multi-trajectory Ehrenfest (MTEF), based on an ensemble of nuclear trajectories which can exactly recover the initial quantum nuclear state, while capturing the electron-nuclear dynamics at the mean-field level. The equations of motion underlying this method are universally implemented throughout real-time ab-initio dynamics code bases, making this approach instantly accessible to the broader community. We find that in combination with a real-space basis treatment of the electronic degrees of freedom, MTEF is able to recover quantum nuclear effects on the equilibrium absorption spectrum of molecules, and demonstrate the ease with which this method can be incorporated into existing simulation protocols. Being semi-classical in nature, MTEF allows for scaling to very large system sizes, while the real space representation of the electronic system allows arbitrarily strong laser driving and the dynamical treatment of the ions allows significant nuclear rearrangement of the nuclear system. We subsequently apply MTEF for the first time to realistic periodic systems in a manner which is generically applicable to any material in order to simulate the sub-30 fs phonon-mediated relaxation of valley selectively excited charge carriers in hexagonal Boron Nitride (hBN) across temperatures spanning $2000$ K. We are able to use MTEF to simulate arbitrarily strong pump-probe measurements of the carrier relaxation and directly replicate a recent `light-wave engineering' experiment on hBN \textit{in-silico}. We find that MTEF constitutes a natural extension of static methods which are widely used to calculate the phonon renormalized equilibrium properties of materials, and that for the far-from-equilibrium phenomena studied, our method converges with a very small number of trajectories. Thus we ultimately develop and present a method which is simple to use, accurate, rapidly convergent, and which is capable of capturing strongly driven electron-phonon dynamics in periodic systems under arbitrary pump-probe setups. We conclude with a discussion on the research which will follow on the basis of our providing this much needed tool to simulate the strongly driven dynamics of quantum materials.
