High-Accuracy Determination of Paul-Trap Stability Parameters for Electric-Quadrupole-Shift Prediction

TL;DR

This paper introduces a higher-order iterative method to accurately determine Paul-trap stability parameters from measured secular frequencies, enabling precise prediction and minimization of electric quadrupole shifts in optical ion clocks.

Contribution

A novel iterative approach for accurately calculating stability parameters from experimental data, improving electric quadrupole shift prediction in ion trapping.

Findings

The method accurately characterizes trap asymmetry and electric field gradients.

Good agreement with electrostatic finite-element simulations.

Achieved a fractional frequency uncertainty of less than 1×10^{-19} in Sr+ optical clocks.

Abstract

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.