Quasiclassical quantum defect theory and the spectrum of highly excited rubidium atoms

We report on a significant discrepancy between recently published, highly accurate variational calculations and precise measurements of the spectrum of Rydberg states in ${}^{87}\mathrm{Rb}$ on the energy scale of fine splitting. Introducing a modified effective single-electron potential, we determine the spectrum of the outermost bound electron from a standard WKB approach. Overall very good agreement with precise spectroscopic data is obtained.

Figure 1

Reduced fine splitting in THz (a) for P states (cf. Table 1) and (b) for D states (cf. Table 2). The blue line corresponds to the theory from this work [see Eq. (17)]. The blue circles show the recent numerically calculated results from [7]. The red symbols denote experimental data from Refs. [11] (squares), [8] (crosses), and [10] (circles). The error bars for the experimental data are given in Tables 1 and 2.

Figure 2

The action integral $\nu \left(j,l,E\right)$ associated with the effective single-electron potential ${\tilde{V}}_{\mathrm{mod}}\left(r;j,l\right)$ vs scaled energy $\frac{1}{\sqrt{-E}}$ for $l=0$ (blue line), $l=1$ (red line), $l=2$ (green line), all for $j=l+\frac{1}{2}$. The curves for $j=l-\frac{1}{2}$ differ only by a tiny shift proportional to ${\alpha }^2$.

Figure 3

The quasiclassical momentum $\sqrt{-Q\left(r;j,l,E\right)}$ vs scaled distance $\frac{r}{r_c\left(l\right)}$ for $l=0$ (dashed black), $l=1$ (green), and $l=2$ (red) for $E=E_{n,j,l}$, corresponding to principal quantum number $n=57$ and $j=l+\frac{1}{2}$. The lower turning points $r^{\left(-\right)}\left(l\right)$ for the case of a pure Coulomb potential, for $l>0$ all being greater than $0.6\times r_c\left(l\right)$, are located well outside the core region. The main contribution to the quantum defect values in (24) thus originates from the inner-core region $r<r_c\left(l\right)$.