Long-range atom-wall interactions and mixing terms: Metastable hydrogen

We investigate the interaction of metastable $2S$ hydrogen atoms with a perfectly conducting wall, including parity-breaking $S-P$ mixing terms (with full account of retardation). The neighboring $2P_{1/2}$ and $2P_{3/2}$ levels are found to have a profound effect on the transition from the short-range, nonrelativistic regime, to the retarded form of the Casimir-Polder interaction. The corresponding $P$ state admixtures to the metastable $2S$ state are calculated. We find the long-range asymptotics of the retarded Casimir-Polder potentials and mixing amplitudes for general excited states, including a fully quantum electrodynamic treatment of the dipole-quadrupole mixing term. The decay width of the metastable $2S$ state is roughly doubled even at a comparatively large distance of 918 a.u. (Bohr radii) from the perfect conductor. The magnitude of the calculated effects is compared to the unexplained Sokolov effect.

Figure 1

The modulus-squared admixtures to the coupled $|{\mathcal{S}}_{1/2}\rangle$ state are obtained from a diagonalization of the potential (11) in the basis of $|{\mathcal{S}}_{1/2}\rangle ,|{\mathcal{P}}_{1/2}\rangle$, and $|{\mathcal{P}}_{3/2}\rangle$ states, for close approach of the atom toward the wall. The subscript $j$ in Eq. (13) takes on the values $j=S$, as well as $j=1/2$ and $j=3/2,$ and denotes the state responsible for the admixture. As the $|{\mathcal{S}}_{1/2}\rangle$ state approaches the wall, the initially dominant $|S_{1/2}\rangle$ state contribution (solid curve, $j=S$) gradually fades and the $|P_{1/2}\rangle$ admixture (short-dashed curve, $j=1/2$) increases, while a significant admixture of the $|P_{3/2}\rangle$ state (long-dashed curve, $j=3/2$) is observed only for close approach. The atom-wall interaction energy becomes commensurate with the Lamb shift and fine structure at $\mathcal{Z}\approx 84$ and $\mathcal{Z}\approx 184$, respectively.