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.