Bose-Einstein condensates (BECs) exhibit strange a surprising phenomena when a small
complication is introduced beyond the usual, weakly-interacting picture. I consider the effects of
a long-range dipolar interaction, where the dipoles interact anisotropically due to their intrinsic
magnetic dipole moments. Here, all the dipoles are aligned by an external field. I first consider an
alternative theoretical framework, known as the hyperspherical formalism, to approach the dipolar
BEC. I show a general correspondence between the hypshperical approach and certain Gaussian
ansatze to the Gross-Pitaevskii equation. I then consider the effects of a weak optical lattice on the
supersolid ground state of a dipolar BEC. In this state, without a perturbing lattice, the BEC forms
arrays of self stable droplets. These arrays often have a hexagonal symmetry, and so I consider a
square optical lattice, whereby the symmetry of the lattice competes with the intrinsic symmetry
of the ground state.
I also consider the effects of suddenly changing the scattering length. In order to reach the
strongly interacting state of a BEC, one must greatly increase the scattering length. If this is done
too slowly, then all the atoms are lost to three-body recombination. If this is done too quickly,
then one does not project strongly onto the ground state here. Instead, I propose a mode-matching
protocol, whereby one can project onto the strongly interacting ground state nearly exactly by a
simultaneous quench of the trap and the scattering length. I also consider the effects of a quench
in a box-trapped BEC, where a quench to exactly four times the initial scattering length can cause
robust production of solitons in a box. This is robust to 3D effects as well as imperfect quenches
and box walls.
