Interacting quantum degenerate atomic gases provide a playground for
creating novel strongly correlated phases in an easily studied context.
While many new phases have been observed in recent years, few challenge the
Landau-Ginzburg framework of order in the way quantum- or
topologically-ordered systems can. One such system, invoking fractional
quantum Hall (FQH) physics familiar from two-dimensional electron systems,
has been predicted to appear in a gas of interacting bosons confined to a
rapidly rotating trap. I will describe previous experiments observing
strong correlations in few-body clusters of atoms confined in an optical
lattice with rotating lattice sites, consistent with the bosonic analog of
fractional hall states. Extensions of these experiments with a
single-site-resolved optical microscope allowing for occupancy resolved
measurements will be described, as well as coupled FQH systems and new
techniques for their production and interrogation using Feshbach resonance.
New methods to create topologically-ordered states in macroscopic quantum
gases will also be described, using an analogy with the quantum anomalous
Hall effect as a starting point.