Precision measurement of light shifts in a single trapped Ba$^+$ ion as test of atomic theory

Using a single trapped barium ion we have developed an rf spectroscopy technique to measure the ratio of the off-resonant vector ac Stark effect (or light shift) in the $6S_{1/2}$ and $5D_{3/2}$ states to 0.1\% precision at the 514 nm argon-ion laser line. Our preliminary result\footnote{Sherman, J.A.\ et al., \emph{In preparation}.} of $R = -11.493(13)$ yields a new test of the theory of alkali-like atomic systems. Since the light shift ratio is expressible as sums of dipole matrix elements, our result also establishes a sum rule involving imprecisely known matrix elements, one of which is important for a proposed single-ion parity violation experiment\footnote{Koerber, T.W.\ et al., \emph{J.\ Phys.\ B} {\textbf{36}} (2003) 637--48.}. Along with these results we discuss analogous measurements at other wavelengths and future experimental plans.