  We derive an effective Hamiltonian that describes a cross-Kerr type
interaction in a system involving a two-level trapped ion coupled to the
quantized field inside a cavity. We assume a large detuning between the ion and
field (dispersive limit) and this results in an interaction Hamiltonian
involving the product of the (bosonic) ionic vibrational motion and field
number operators. We also demonstrate the feasibility of operation of a phase
gate based on our hamiltonian. The gate is insensitive to spontaneous emission,
an important feature for the practical implementation of quantum computing.
