Measurement of light shifts at two off-resonant wavelengths in a single trappedBa+ion and the determination of atomic dipole matrix elements

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We have measured the ratio $R$ of the vector ac Stark effect (or light shift) in the $6{S}_{1∕2}$ and $5{D}_{3∕2}$ states of a single trapped barium ion to 0.2% accuracy at two different off-resonant wavelengths. We earlier found $R={\ensuremath{\Delta}}_{S}∕{\ensuremath{\Delta}}_{D}=\ensuremath{-}11.494(13)$ at $514.531\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ where ${\ensuremath{\Delta}}_{S,D}$ are the vector light shifts of the $6{S}_{1∕2}$ and $5{D}_{3∕2}$ $m=\ifmmode\pm\else\textpm\fi{}1∕2$ splittings due to circularly polarized light, and now we report the value at $1111.68\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, $R=+0.4176(8)$. These observations together yield a value of the $⟨5D\ensuremath{\Vert}er\ensuremath{\Vert}4F⟩$ matrix element. Also, comparison of our results with an ab initio calculation of dynamic polarizability would yield a test of atomic theory and improve the understanding of atomic structure needed to interpret a proposed atomic parity violation experiment.