Van der Waals and Casimir-Polder Interactions of Hydrogen Atoms in Excited States

We analyze the long-range interactions between two hydrogen atoms. Provided both atoms are in the ground states, the retardation regime is achieved when the interatomic distance, $R$, is larger than $137a_0$, where $a_0$ is the Bohr radius. However, for an excited atom interacting with the ground state atom, the presence of virtually low lying energy levels accessible by a dipole transition from an excited reference state makes the situation different. For excited reference states, we match the scattering amplitude and effective perturbation Hamiltonian of the system. For $a_0\ll R\ll137a_0$, the nonretarded approximation is valid and the interaction energy takes a $R^{-6}$ functional form. For a higher excited states, we obtain a numerically large van der Waals coefficient, e.g., about 24000 in atomic units for $12S$-$1S$ system. As soon as the interatomic separation satisfies $R\ge137a_0$, the interaction energy does not have only the $R^{-6}$ term, but surprisingly it also has (i) attractive/repulsive oscillatory terms, in addition to (ii) the familiar Casimir-Polder $R^{-7}$ asymptotic. For sufficiently large $R$, the oscillatory term whose magnitude falls off as $R^{-2}$ dominates the Wick-rotated term (the latter describes the retarded Casimir-Polder type of interaction).