  Atomic superfluids formed using Bose-Einstein condensates (BECs) in a ring
trap are currently being investigated in the context of superfluid
hydrodynamics, quantum sensing and matter-wave interferometry. The
characterization of the rotational properties of such superfluids is important,
but can presently only be performed by using optical absorption imaging, which
completely destroys the condensate. Recent studies have proposed coupling the
ring BEC to optical cavity modes carrying orbital angular momentum to make
minimally destructive measurements of the condensate rotation. The sensitivity
of these proposals, however, is bounded below by the standard quantum limit set
by the combination of laser shot noise and radiation pressure noise. In this
work, we provide a theoretical framework that exploits the fact that the
interaction between the scattered modes of the condensate and the light reduces
to effective optomechanical equations of motion. We present a detailed
theoretical analysis to demonstrate that the use of squeezed light and
backaction evasion techniques allows the angular momentum of the condensate to
be sensed with noise well below the standard quantum limit. Our proposal is
relevant to atomtronics, quantum sensing and quantum information.
