We describe a novel High Bandwidth Arbitrary Lattice Generator (HiBAL) we’ve

created to skirt limits imposed on monochromatic standing waves of light. With its

current iteration we can phase and amplitude modulate optical lattices over a broad range

of wavevectors simultaneously at MHz frequencies. We characterize the HiBAL with

a multi-Mach-Zehnder interferometer and a 0.5 NA diffraction limited imaging system,

both designed and built in-house. We report lattice phase control to within a few parts in

a thousand.

Disorder plays an important role in the phase diagrams of many materials. Crystal

defects can cause exotic phases to coexist with the mundane in real world systems, and

some phase diagrams are even dominated by the effects of disorder. We report the

trapping and characterization of a Bose gas in an optical field isotropic in two dimensions

and disordered in a third. We evaluate the phase diagram of our system as a function

of temperature and disorder depth, and find favorable comparisons with indications of

an intermediate Griffiths phase predicted by previous Monte Carlo and Renormalization

Group studies separating 2D and 3D superfluid regimes.

Finally, I discuss the possibility of realizing the BKT transition in a non-orientable

space. The BKT phase transition is an infinite order phase transition in two dimensions

from a normal gas to a superfluid mediated by vortices, which are orientable topological

phase defects in two dimensions. I discuss the properties of vortices and their intractions

on a Mobius strip, and describe how a relay-imaged bichromatic optical potential could

be used to form a Mobius strip out of ultracold gases.