A. E. Wallin

We report on a precision measurement of the differential static scalar polarizability $Δα_0$ of the ${}^{2}\!S_{1/2} \rightarrow {}^{2}\!D_{5/2}$ optical clock transition in the $^{88}$Sr$^{+}$ ion. The polarizability was determined from the 'magic' ion-trap drive frequency where the micromotion-induced second-order Doppler and quadratic Stark shifts cancel, using a single clock in an interleaved scheme by switching between minimized and large micromotion. By measuring at different Mathieu $q_z$ parameters, $Δα_0$ can be obtained without prior knowledge of the angle between the rf electric field and the trap axis, which would otherwise dominate the systematic uncertainty. For validation, measurements were carried out at different micromotion levels and by displacing the ion in opposite directions. The results show excellent consistency and our value, $Δα_0 = -4.8314(20)\times 10^{-40}\;\mathrm{J\, m^2/V^2} = -29.303(12)\;\mathrm{au}$, reduces the uncertainty by a factor of 3.5 compared to a previous measurement, while showing a discrepancy of $5 σ$. Our measurement reduces the polarizability-related uncertainty of the $^{88}$Sr$^{+}$ clock to $2.2\times 10^{-19}$ for a blackbody radiation temperature of 295 K -- a significant step towards total uncertainties ${<}1\times10^{-18}$. Using existing polarizability transfer schemes, the result can reduce the uncertainty also for other ion species.

The motion of an ion in a radiofrequency (rf) Paul trap is described by the Mathieu equation and the associated stability parameters that are proportional to the rf and dc electric field gradients. Here, a higher-order, iterative method to accurately solve the stability parameters from measured secular frequencies is presented. It is then used to characterize an endcap trap by showing that the trap's radial asymmetry is dominated by the dc field gradients and by measuring the relation between the applied voltages and the gradients. The results are shown to be in good agreement with an electrostatic finite-element-method simulation of the trap. Furthermore, a method to determine the direction of the radial trap axes using a 'tickler' voltage is presented and the temperature dependence of the rf voltage is discussed. As an application for optical ion clocks, the method is used to predict and minimize the electric quadrupole shift (EQS) using the applied dc voltages. Finally, a lower limit of 1070 for the cancellation factor of the Zeeman-averaging EQS cancellation method is determined in an interleaved low/high EQS clock measurement. This reduces the EQS uncertainty of our $^{88}$Sr$^+$ optical clock to ${\lesssim} 1\times 10^{-19}$ in fractional frequency units.