Abstract: The evaluation of the second-order Doppler frequency shift (SODFS) of trapped ions in Paul traps has been studied in this paper.The motion of confined ions in a Paul trap consists of three components:secular motion,micromotion,and excess micromotion.This paper focused on the ion excess micromotion,which is rarely discussed when evaluating the SODFS of ions.Based on the hypothesis that the ion density is uniformly distributed in the radial direction,a model was proposed to evaluate the total SODFS for an ion ensemble with temperature less than 100 mK in this paper,which consists of three contributions of ion motion.The effectiveness of the model has been verified by molecular dynamics simulations and experiments.The model in this paper offers advantages in accurately evaluating the SODFS for the ion trap,which is appliable for optical clocks based on ions and microwave clocks based on ions,especially those of laser-cooled ion microwave clocks based on large ion clouds.

Key words: Doppler frequency shift, Molecular dynamics, Ion trap