Systematic investigations on ion dynamics with noises in Paul trap

Ions confined in a Paul trap serve as crucial platforms in various research fields, including quantum computing and precision spectroscopy. However, the ion dynamics is inevitably influenced by different types of noise, which require accurate computations and general analytical analysis to facilitate diverse applications based on trapped ions with white or colored noise. In the present work, we investigate the motion of ions in a Paul trap via the Langevin equation using both analytical and numerical methods, systematically studying three different types of noise: the white noise, the colored noise via the Ornstein–Uhlenbeck process and the Wiener process. For the white noise of the case, we provide a recursion method to calculate ion motion for a wide range of parameters. Furthermore, we present an analytical solution to the more realistic stochastic process associated with the colored noise, verified by the Monte Carlo simulation. By comparing the results of the colored noise with those of the white noise, and additionally considering another limit of noise parameters corresponding to the Wiener process, we summarize the effects of different noise types on the ion dynamics.