Abstract
In this thesis we develop an orthogonalised Hartree-Fock-Bogoliubov (HFB) formalism that has a zero-energy excitation (in contrast with standard HFB). We demonstrate that this formalism satisfies the number and linear/angular momentum conservation laws (as does standard HFB). This formalism is applied to vortices in 2D Bose-Einstein condensates (BECs) in axially-symmetric harmonic traps, where we initially find solutions for on-axis vortices, determining the energy spectrum and hence the lowest core localised state (LCLS) energy. In the T=0 case we identify this with the anomalous mode which gives a zero excitation energy in the frame rotating at this frequency. For this reason the anomalous mode frequency was identified in the earlier literature with the precessional frequency for an off-axis vortex. However the LCLS energy is positive in the finite temperature case. Hence, associating this LCLS energy with the precessional frequency leads to the erroneous conclusion that the vortex precesses in the opposite direction to the T=0 case, which is clearly physically unreasonable. In order to address this problem, we derive an equation for the prediction of vortex precessional frequencies from the continuity equation, and use this equation solved self-consisently with the orthogonal HFB equations in the frame rotating at this predicted frequency to create off-axis vortices at pre-specified positions. Hence we are able to predict the precessional frequencies and show that these are consistent with the T=0 case, and are entirely unrelated to the LCLS energy. We also consider a two-state model and demonstrate that this model is insufficient for the description of single off-axis precessing vortices. We formulate a generalised multi-state model, using the normalisation conditions for the model to derive an equation predicting the precessional frequency of the vortex, and demonstrate equivalence with the continuity equation prediction. We use the time-dependent HFB equations to simulate creation of vortices by stirring the BEC by means of a Gaussian optical potential, finding very good agreement of the measured precessional frequencies of the vortices in stirred BECs with the predicted values. We find the existence of a critical stirring frequency for the creation of vortices in regions of appreciable superfluid density, in qualitative agreement with experiment. We then investigate the consequences of breaking rotational symmetry and find that breaking the axial symmetry of the harmonic trapping potential leads to loss of angular momentum, and hence to the decay of vortices. Finally we develop a finite temperature treatment of the Bose-Hubbard model based upon the Hartree-Fock-Bogoliubov formalism in the Popov approximation to study the effect of temperature upon the superfluid phase of ultracold, weakly interacting bosons in a one-dimensional optical lattice.