From Disordered Bosons to Dipolar Fermions - Theoretical Studies in Ultracold Atoms

We use numerical simulation to study ultracold, quantum degenerate, atomic gases. In the first part of the thesis we study the effects of disorder, introduced via a bichromatic optical lattice, in one and two dimensional systems. We employ the Aubry-Andr\'{e} model and use time-dependent numerical simulations to investigate the disorder dependent transition to strong localisation present in the model. Weak s-wave interactions are added to the model and we observe the interaction between localisation and interaction induced self-trapping. We then add a tilted lattice potential to the model. In the homogeneous model this induces Bloch oscillations. While one might expect that a strong enough force will break the strong localisation or self-trapping, within the bounds of the single-band model, the trapping effect of the Bloch oscillations reinforces both of the other effects leading to increased confinement, albeit lacking the clear single frequency oscillation signature of pure Bloch oscillations. Along with the two dimensional bichromatic optical lattice we add a term to the Hamiltonian equivalent to that of a uniform external magnetic field on charged particles. Since the experimental realisation of this model would employ neutral atoms, the magnetic field is synthetic, the equivalent effect being produced by an appropriate set of lasers and magnetic fields. We show that in the ballistic regime (weak bichromatic disorder) the system displays positive magnetoresistance. Conversely in the strong localisation regime the system exhibits negative magnetoresistance. In the latter part of the thesis we use density functional theory to calculate the ground-state density of a harmonically trapped dipolar Fermi gas. We then use these to calculate the lowest energy collective mode oscillation frequencies under the hydrodynamic approximation. We find that increasing the strength of the dipoles has the effect of increasing the mode frequencies. The increase saturates for large dipole strengths. We verify this analytically and show that such is due to the local nature of the two dimensional energy functional and not dependent on the specific equation of state. We employ an average density approximation to construct an energy functional for the inhomogeneous, 2D degenerate Fermi gas. The ground-state densities for a cylindrically symmetric harmonic trap are compared to the Kohn-Sham results, showing extremely good agreement in the tail region and good agreement with the exact ground-state energy. We then do the same for higher order polynomial traps and obtain improved agreement for higher degree