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Motivated by experimental progress in cold atomic systems, we use and advanceLocalisation Landscape Theory (LLT), to examine two-dimensional systems withpoint-like random scatterers. We begin by showing that exact eigenstates cannotbe efficiently used to extract the localisation length. We then provide acomprehensive review of known LLT, and confirm that the Hamiltonian with theeffective potential of LLT has very similar low energy eigenstates to that withthe physical potential. Next, we use LLT to compute the localisation length forvery low-energy, maximally localised eigenstates and test our method againstexact diagonalisation. Furthermore, we propose a transmission experiment thatoptimally detects Anderson localisation, and demonstrate how one may extract alength scale which is correlated with (and in general smaller than) thelocalisation length. In addition, we study the dimensional crossover from oneto two dimensions, providing a new explanation to the established trends. Theprediction of a mobility edge coming from LLT is tested by direct Schr\"odingertime evolution and is found to be unphysical. Moreover, we investigateexpanding wavepackets, to find that these can be useful in detecting andquantifying Anderson localisation in a transmission experiment, with the onlydisadvantage being the inability to resolve the energy dependence of thelocalisation length. Then, we utilise LLT to uncover a connection between theAnderson model for discrete disordered lattices and continuous two-dimensionaldisordered systems, which provides powerful new insights. From here, wedemonstrate that localisation can be distinguished from other effects by acomparison to dynamics in an ordered potential with all other propertiesunchanged. Finally, we thoroughly investigate the effect of acceleration andrepulsive interparticle interactions, as relevant for current experiments.

Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations