Ultracold two-level atom in a quadratic potential

We use a right unitary decomposition to study an ultracold two-level atom interacting with a quantum field. We show that such a right unitary approach simplifies the numerical evolution for arbitrary position-dependent atom-field couplings. In particular, we provide a closed form, analytic time evolution operator for atom-field couplings with quadratic dependence on the position of the atom; e.g. a two-level atom near an extremum of a cavity field mode amplitude. Our approach allows us to show that the center of mass wave function may be squeezed by choosing a proper atom-field initial state.