  We complement previous studies of an ion coupled with an optical cavity in
the dispersive regime, for a model which exhibits bistability of different
configurations in the semiclassical description. Our approach is based on a
truncated evolution in phase space and is intended to explore an especially
interesting parameter region where the fully quantum-mechanical solution
becomes hard but the crudest semiclassical approach fails to capture essential
phenomena. We compare the results of our techniques with the ones from
numerical diagonalization of the quantum evolution and find that although the
treatment leads to a smoothening and a slight shift of the transitions in the
system, it still provides a clear improvement over localized semiclassical
approximations.
