Microparticle Driven by
Parametric and Random Forces: Theory and Experiment

A.F. Izmailov, S. Arnold, S. Holler, and A.S. Myerson

Phys.Rev.E **52**, 1325-1332(1995)

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 Matieu
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 Bogliubov 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/a << 1 and 4b/a2 << 1 (where a and b and
the dimensionless drag and drive parameters). It is discovered both
experimentally and theoretically that there is a minimum equal to [8kT/(mw2)]1/2
in the standard deviation of the microparticle confined stochastic motion (m is
the microparticle mass and w is the drive force frequency). The presence of this
minimum which take place at b » 1.518a, reduces the thermal noise effects,
providing unique opportunities for the spectroscopic studies. Comparison with
numerical simulation schemes develpoed 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.