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

Abstract

The differential scalar polarizability $Δα_0(ω)$ 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 $Δα_0(ω)$, 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.

Figures (5)

• Figure 1: Fractional error due to the use of Eq. \ref{['eq:Model']} to represent $\Delta\alpha_0(\omega)$, as determined from atomic structure calculations. Solid: $\omega_0\approx 0.204\,580\,\mathrm{a.u.}$ (222.717 nm) minimizes the maximum absolute fractional error for wavelengths above 700 nm ($\omega \approx 0.065\,\mathrm{a.u.}$). Dashed: $\omega_0\approx 0.204\,696\,\mathrm{a.u.}$ (222.590 nm) gives agreement at $\omega=0$ between the model and the full calculation.
• Figure 2: (a) Relevant energy levels and transitions. (b) Front view of the trap showing the orientation of the Doppler cooling and detection lasers at 493 and 650 nm, and the repumping laser at 614 nm. (c) Top view of the trap showing the orientation of the clock laser at 1762 nm, which is aligned 34$^\circ$ to the magnetic field to allow driving the $|S_{1/2}, \pm 1/2\rangle \to |D_{5/2}, \pm M_J\rangle$ transitions for $M_J = 1/2, 3/2, 5/2$ efficiently; the 493 nm $\sigma^{\pm}$ beams, which are used for optical pumping into $|S_{1/2}, \pm 1/2\rangle$; and the Stark shifting laser at 481 nm. The magnetic field is aligned along the propagation directions of the 493 nm $\sigma^{\pm}$ beams. Linear polarization of the 481 nm laser lies either in the $xz$-plane (Config. I) or along the $y$-axis (Config. II).
• Figure 3: Ratio of scalar to tensor shifts $\delta_0(\omega)/\delta_2(\omega)$ as a function of frequency detuning relative to 623.570 THz for two polarization configurations of 481 nm. The black solid lines indicate linear $\chi^2$ fits, and the vertical black dashed lines indicate the zero crossings obtained from the fits.
• Figure 4: Summary of the measurements given in woods2010dipole and here. Solid black lines are as given in woods2010dipole with the inner ellipse giving the $1\sigma$ uncertainty. The dotted lines include the additional uncertainty from theory contributions and we give the corresponding $1\sigma$ and $2\sigma$ uncertainty boundaries. Boundaries are calculated assuming there is no correlation in the uncertainties defining the straight lines. The light grey region gives the ratio $R_0^2$ given woods2010dipole and the dark grey region the ratio reported here with the width in both cases indicating the $1\sigma$ uncertainty. Single points denote theoretical values from guet1991relativistic ($\blacklozenge$), iskrenova2008theoretical ($\bullet$), gopakumar2002electric ($\diamond$), safronova2011excitation ($\blacksquare$), and porsev2021role ($\circ$).
• Figure 5: Total fractional uncertainty in $\Delta\alpha_0(\omega)$. The dashed line shows the contribution arising from $c_{493}$. The remainder is primarily due to the uncertainty in $\omega_0$. Uncertainties from $\omega_{481}$ presented here and $\omega_{653}$ reported in chanu2020magic result in the divergences at those points with negligible contribution otherwise.