Version 1
: Received: 31 July 2024 / Approved: 31 July 2024 / Online: 8 August 2024 (17:32:42 CEST)
How to cite:
Mihalcea, B. M. Solutions of the Mathieu-Hill Equation for the Trapped Ion Harmonic Oscillator. Qualitative Discussion.. Preprints2024, 2024080604. https://doi.org/10.20944/preprints202408.0604.v1
Mihalcea, B. M. Solutions of the Mathieu-Hill Equation for the Trapped Ion Harmonic Oscillator. Qualitative Discussion.. Preprints 2024, 2024080604. https://doi.org/10.20944/preprints202408.0604.v1
Mihalcea, B. M. Solutions of the Mathieu-Hill Equation for the Trapped Ion Harmonic Oscillator. Qualitative Discussion.. Preprints2024, 2024080604. https://doi.org/10.20944/preprints202408.0604.v1
APA Style
Mihalcea, B. M. (2024). Solutions of the Mathieu-Hill Equation for the Trapped Ion Harmonic Oscillator. Qualitative Discussion.. Preprints. https://doi.org/10.20944/preprints202408.0604.v1
Chicago/Turabian Style
Mihalcea, B. M. 2024 "Solutions of the Mathieu-Hill Equation for the Trapped Ion Harmonic Oscillator. Qualitative Discussion." Preprints. https://doi.org/10.20944/preprints202408.0604.v1
Abstract
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 \cite{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.
Keywords
Mathieu-Hill equation; Floquet theory; Paul trap; stability diagram
Subject
Physical Sciences, Atomic and Molecular 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.