  Bosons with non-zero spin exhibit a rich variety of superfluid and insulating
phases. Most phases support coherent spin oscillations, which have been the
focus of numerous recent experiments. These spin oscillations are Rabi
oscillations between discrete levels deep in the insulator, while deep in the
superfluid they can be oscillations in the orientation of a spinful condensate.
We describe the evolution of spin oscillations across the superfluid-insulator
quantum phase transition. For transitions with an order parameter carrying
spin, the damping of such oscillations is determined by the scaling dimension
of the composite spin operator. For transitions with a spinless order parameter
and gapped spin excitations, we demonstrate that the damping is determined by
an associated quantum impurity problem of a localized spin excitation
interacting with the bulk critical modes. We present a renormalization group
analysis of the quantum impurity problem, and discuss the relationship of our
results to experiments on ultracold atoms in optical lattices.
