Ultracold atoms in optical lattices are a versatile tool for precisely-controlled quantum simulation of a range of condensed matter phenomena. This thesis describes work to apply this tool in two areas: the simulation of quasicrystalline materials with a novel lattice geometry, and the use of Floquet driving to engineer behaviours qualitatively different from static systems. In particular I will present our work using periodic driving to make a continuous quantum phase transition become discontinuous, which to our knowledge has never been done before.

Quasicrystals are a fascinating but still relatively under-explored class of materials existing as an intermediate between periodic and disordered systems. This makes them an ideal context for studying non-ergodic behaviour, especially with the tunable interactions allowed by ultracold atom physics. At the same time quasicrystals have an intriguing link to higher dimensions, so that a two-dimensional quasicrystal can be used to simulate systems with more than three spatial dimensions. I will describe two experiments performed by our group that use a quasicrystalline optical lattice to explore higher dimensions and non-ergodic states respectively, which we hope are a stepping stone towards future work with many-body localisation and higher-dimensional topological effects.

The other part of the story is Floquet physics: the study of time-periodic Hamiltonians. Time dependence allows us to break many of the usual rules of quantum systems, adding a whole new set of control parameters and allowing qualitatively different behaviours. In this thesis I will discuss the application of Floquet driving to a periodic optical lattice and present our experimental observation of a discontinuous form of the well-known Mott insulator to superfluid quantum phase transition. While interesting in itself, we view this work too as a stepping stone towards Floquet engineering with the full optical quasicrystal.