Evaluation of retardation energy shifts in a Rydberg helium atom

The microscopic system best suited to the high-precision confirmation of a retardation (or Casimir) effect—originating in the finiteness of the speed of light—would seem to be the Rydberg helium atom (a helium atom with one electron in its ground state and the other in a high-n state, preferably with l≊n). For the helium atom, calculation of the retardation energy shift Δ$E_{\mathrm{ret}(\mathrm{n}}$,l) that arises from the retardation effective potential $V_{\mathrm{ret}(\mathrm{r}}$) seen by the outer electron, where r is the separation of the nucleus and the outer electron, is not limited to the case of a Rydberg outer electron. We evaluate Δ$E_{\mathrm{ret}(\mathrm{n}}$,l) numerically in the dipole approximation for a range of values of n and l. The dynamic electric dipole polarizability of the ${\mathrm{He}}^{+}$ core is approximated using pseudostates (a finite number of effective excitation energies and oscillator strengths for the core). We also show how the evaluation can be performed analytically, again using pseudostates. Finally, we give $V_{\mathrm{ret}(\mathrm{r}}$) as an expansion in powers of 1/r; this often provides an easy means of estimating Δ$E_{\mathrm{ret}(\mathrm{n}}$,l) quickly and relatively accurately, and we present various results of use for the Rydberg helium atom, in tabular and graphical form. The effects of exchange, higher multipoles, and of the finite nuclear mass have not been included. As experimental capabilities improve, other systems, such as an ion composed of a nucleus with Z>2, a single core electron, and a Rydberg electron, may someday prove as useful as helium. The formalism for helium can readily be adapted to this case.