Suppression of servo error uncertainty to 10^-18 level using double integrator algorithm in ion optical clock

A universal locking model for single ion optical clocks was built based on a simple integrator and a double integrator. Different integrator algorithm parameters have been analyzed in both numerical simulations and experiments. The frequency variation measured by the comparison of two optical clocks coincides well with the simulation results for different second integrator parameters. According to the experimental results, the sensitivity of the servo error influenced by laser frequency drift with the addition of a double integrator was suppressed by a factor of 107. In a week-long comparison of optical clocks, the relative uncertainty of the servo error is determined to be 1.9×10^-18, which is meaningful for the systematic uncertainty of the transportable single ⁴⁰Ca⁺ ion optical clock entering the 10^-18 level.