Solutions of the Mathieu-Hill Equation for the Trapped Ion Harmonic Oscillator. Qualitative Discussion.

We investigate solutions of the classical Mathieu-Hill (MH) equation which describes the dynamics of trapped ions, based on the Floquet theory and the analytical model introduced in {Blu89}. We show the equations of motion are equivalent to those of the harmonic oscillator (HO) and demonstrate methods to determine the solutions of the MH equation. The regions of stability and instability for the MH equation in case of a trapped particle are also discussed. What is more, we address both the damped HO and parametric oscillator (PO) for an ion confined in a Paul trap. The paper is a follow-up of a recently published paper in Photonics, 11 (6) 551.