High-accuracy determination of Paul-trap stability parameters for electric-quadrupole-shift prediction

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