Ultracold two-level atom in a quadratic potential

•We study an ultra-slow two-level atom interacting with a quantum field.•We use a right unitary operator approach to give the time evolution of the system.•We report an evolution operator for an atom–field quadratic potential coupling.•A type of initial entangled states produce squeezing in the atomic motion state.•We present examples of evolution changes due to the position-dependent coupling.

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.