  Hall viscosity is a non-dissipative viscosity occurring in systems with
broken time-reversal symmetry, such as quantum Hall phases and $p+ip$
superfluids. Despite Hall viscosity's expected ubiquity and past observations
in classical soft matter systems, it has yet to be measured experimentally in
any quantum phase of matter. Toward this end, we describe the observable
effects of Hall viscosity in a simple family of rotating Bose-Einstein
condensates of electrically neutral bosons, in which all of the bosons condense
into a single LLL orbital. Such phases are accessible to current cold atom
experiments, and we dub them lowest Landau level (LLL) superfluids. We
demonstrate that LLL superfluids possess a non-universal Hall viscosity,
leading to a range of observable consequences such as rotation of
vortex-antivortex dipoles and wave-vector dependent corrections to the speed of
sound. Furthermore, using a coherent state path integral approach, we present a
microscopic derivation of the Landau-Ginzburg equations of a LLL superfluid,
showing explicitly how Hall viscosity enters.
