Zero crossings of the differential scalar polarizability of Ba$^+$ clock transition

The differential scalar polarizability $\Delta\alpha_0(\omega)$ of the Ba$^+$ S$_{1/2}$-to-D$_{5/2}$ clock transition has a zero crossing near 481nm, which is measured to be 623.603\,13(17)\,THz.

From this measurement, we infer a ratio of reduced matrix elements $\langle P_{3/2}\|r\|S_{1/2}\rangle/\langle P_{1/2}\|r\|S_{1/2}\rangle=1.411\,81(13)$, which provides a stringent test of atomic structure calculations and experimental determination of matrix elements.

Additionally, it enables the construction of an accurate approximation to $\Delta\alpha_0(\omega)$, valid for frequencies up to 450\,THz, with only one reduced matrix element, $\langle P_{1/2}\|r\|S_{1/2}\rangle$, appearing in the model's parameterization.

We discuss the achievable accuracy of the model, the application to the assessment of blackbody radiation (BBR) shifts in ion-based clocks, and the applicability of the approach to other alkaline-earth ions.