The last two decades have witnessed incredible progress in realising quantum phenomena, from Bose-Einstein condensation to quantum teleportation. The future holds the exciting prospect of applying this knowledge to technologies based on real materials such as transistors, light emitting diodes and solar cells. The central impediment to this is that real materials are disordered; characterised by crystalline imperfections, atomic vibrations and amorphous microstructures. The effect of disorder on a quantum system can generally be thought of as a perturbation that disrupts the material's underlying structure on some length and timescale, causing the electrons to change their behaviour. This change in the behaviour can range from trivial (trapping on a low energy site or a loss of quantum coherence), to rich phenomena (quantum interference suppressed propagation), possibly even violating basic laws of thermodynamics (systems that never reach thermal equilibrium). It is precisely this duality of richness and unpredictability of the physics of electrons in the presence of disorder that this dissertation investigates, bridging theory, experiment and new techniques.

In the first part of this dissertation, we develop the techniques needed to investigate such processes through quantitative ultrafast pump-probe microscopy. These chapters serve as the background and theory chapters typical to most dissertations, but cover original work. In our endeavour to reveal quantum coherent processes in disordered materials we need to solve three key questions - can we tie ultrafast spectral features to the underlying electronic structure of the material, can we image these processes on their native length and timescales, and what do we expect to see when imaging processes at the intersection of quantum coherence and stochastic physics?

We address these questions by first establishing a quantitative spectroscopic approach to studying femtosecond (10^-15 s) processes, with which one can unambiguously determine the optical effects of a photoinduced perturbation on the electronic structure of a material system. By leveraging the information gained through this quantitative spectroscopy, we build the full optical theory of femtosecond optical microscopy, enabling us to image quantum phenomena with sub-10 nm precision in all three dimensions for the first time. Having demonstrated that we can capture quantum processes in a pump-probe microscope in three-dimensions, we then predict theoretical signatures of quantum coherent transport and the crossover through scattering into a stochastic regime, which we subsequently experimentally observe in hybrid light-matter states measured in our pump-probe microscope.

Using this new toolkit, we go on to explore the ultrafast physics of a few material systems in detail, unveiling hitherto inaccessible quantum processes. First, we examine the well-studied photophysical process of singlet fission with our new three-dimensional ultrafast microscope, where we are able to visualise the first three-dimensional picture of the quantum coherent formation of entangled electronic states. Next, in a polycrystalline semiconductor, we find that structural disorder can drive the formation of photoexcited spin domains through local inversion symmetry breaking, exemplifying the unexpected and rich physics that can arise in disordered material systems. In the vein of exploring the coupling of optical excitations to magnetism, we then investigate a new two-dimensional material, monolayer nanoribbons of black phosphorus, where we find that the ribbons show signatures of strong ground state magnetism and on picosecond timescales, the photoexcitation couples to the magnetic edges.

Finally, we examine the role of `dynamic disorder' that manifests through the ever-present lattice vibrations, an area that has historically been challenging to study. We approach this problem computationally using first principle density functional theory (DFT) by capturing the empirically well-established phenomenology of an exponential sub-gap absorption tail known as the Urbach tail. By laying the foundational principles of capturing Urbach tails using DFT in two well-studied semiconductors, we retrieve the role of particular phonon modes, the localisation of Urbach tail states, the role of long wavelength phonon vibrations and finally, a surprising Urbach tail even at 0 K due to zero-point phonons, forcing a serious reconsideration of the physicality of an Urbach tail at 0 K.