Overdamped dynamics of a Brownian particle levitated in a Paul trap

We study the dynamics of the center of mass of a Brownian particle levitated in a Paul trap. We mostly focus on the overdamped regime in the context of levitodynamics, comparing theory with numerical simulations and experimental data from our Paul trap. We provide an exact analytical solution to the stochastic equation of motion, expressions for the standard deviation of the motion, and thermalization times by using the WKB method under two different limits. Finally, we prove that the power spectral density of the motion can be approximated by that of an Ornstein-Uhlenbeck process and use the found expression to calibrate the motion of a trapped particle.