Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

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

Bogdan M. Mihalcea
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Version 1 : Received: 31 July 2024 / Approved: 31 July 2024 / Online: 8 August 2024 (17:32:42 CEST)
Version 2 : Received: 27 August 2024 / Approved: 27 August 2024 / Online: 30 August 2024 (08:37:15 CEST)

A peer-reviewed article of this Preprint also exists.

Mihalcea, B.M. Solutions of the Mathieu–Hill Equation for a Trapped-Ion Harmonic Oscillator—A Qualitative Discussion. Mathematics 2024, 12, 2963. Mihalcea, B.M. Solutions of the Mathieu–Hill Equation for a Trapped-Ion Harmonic Oscillator—A Qualitative Discussion. Mathematics 2024, 12, 2963.

Abstract

We investigate solutions of the classical Mathieu-Hill (MH) equation which characterizes the dynamics of trapped ions, based on the Floquet theory and starting from a well known analytical model. The analytical model we introduce demonstrates that the equations of motion are equivalent to those of the harmonic oscillator (HO). Two independent approaches are used, based on two classes of complex solutions of the MH equation. The paper addresses both the damped HO and parametric oscillator (PO) for an ion confined in an electrodynamic (Paul) trap. Stability and instability regions for the periodic orbits associated to the MH equation which describes a HO (PO) confined in an electrodynamic trap are also discussed, depending on the values of the Floquet exponent, for very small values of the $q$ parameter. The main objective of the paper is to identify stable and bounded solutions for the dynamics of a HO levitated in an electrodynamic trap. This paper represents a follow-up of a recently published article in Photonics, 11 (6) 551.

Keywords

Mathieu-Hill equation; Floquet theory; Strum-Liouville theorem; electrodynamic trap; stability diagram

Subject

Physical Sciences, Mathematical Physics

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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