Microparticle driven by parametric and random forces: Theory and experiment

The confined motion of a charged microparticle within the Paul Trap (also known as the electrodynamic levitator trap) in an atmosphere near the standard temperature T and pressure Patm is studied both theoretically and experimentally. The suggested theoretical model is based on the Mathieu differential equation with damping term and stochastic source. This equation describes the damped microparticle motion subjected to the combined periodic parametric and random external excitations. To solve the equation in an experimentally investigated regime of extremely strong damping and periodic excitations, the singular perturbation theory (WKB theory) is applied. In order to compare experimental data obtained in the long-time imaging limit with an analytical solution obtained for the autocorrelation function, the last is averaged by employing the Bogoliubov general averaging principle. This comparison is performed in terms of the standard deviation of the microparticle confined stochastic motion. It results almost in the perfect agreement between the analytical result and the data obtained experimentally in an entire region of the investigated experimental parameters. The only theoretical restrictions imposed on the model parameters are 1/α1 and 4β/α21 (where α and β are the dimensionless drag and drive parameters). It is discovered both experimentally and theoretically that there is a minimum equal to [8kT/(mω2)]1/2 in the standard deviation of the microparticle confined stochastic motion (m is the microparticle mass and ω is the drive force frequency). The presence of this minimum, which takes place at β1.518α, reduces the thermal noise effects, providing unique opportunities for the spectroscopic studies. Comparison with the numerical simulation schemes developed in papers [Arnold, Folan, and Korn, J. Appl. Phys. 74, 4291 (1993); Blatt et al., Z. Phys. D 4, 121 (1986); Zerbe, Jung, and Hanggi, Phys. Rev. E 49, 3626 (1994)] is discussed.