Casimir-Polder shifts on quantum levitation states

An ultracold atom above a horizontal mirror experiences quantum reflection from the attractive Casimir- Polder interaction, which holds it against gravity and leads to quantum levitation states. We analyze this system by using a Liouville transformation of the Schrodinger equation and a Langer coordinate adapted to problems with a classical turning point. Reflection on the Casimir-Polder attractive well is replaced by reflection on a repulsive wall, and the problem is then viewed as an ultracold atom trapped inside a cavity with gravity and Casimir-Polder potentials acting, respectively, as top and bottom mirrors. We calculate numerically Casimir-Polder shifts of the energies of the cavity resonances and propose an approximate treatment which is precise enough to discuss spectroscopy experiments aimed at tests of the weak-equivalence principle on antihydrogen. We also discuss the lifetimes by calculating complex energies associated with cavity resonances.