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Solutions of the Mathieu–Hill Equation for a Trapped-Ion Harmonic Oscillator—A Qualitative Discussion

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  • Bogdan M. Mihalcea

    (National Institute for Laser, Plasma and Radiation Physics (INFLPR), Atomiştilor Str. Nr. 409, 077125 Măgurele, Romania)

Abstract

We investigate solutions of the classical Mathieu–Hill (MH) equation that characterizes the dynamics of trapped ions. The analytical model we introduce demonstrates the equations of motion are equivalent to those of a harmonic oscillator (HO). Two independent approaches are used, based on two classes of complex solutions of the MH equation. This paper addresses both a damped HO and parametric oscillator (PO) for an ion confined in an electrodynamic (Paul) trap, along with stability and instability regions for the associated periodic orbits.

Suggested Citation

  • Bogdan M. Mihalcea, 2024. "Solutions of the Mathieu–Hill Equation for a Trapped-Ion Harmonic Oscillator—A Qualitative Discussion," Mathematics, MDPI, vol. 12(19), pages 1-20, September.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:19:p:2963-:d:1484670
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    References listed on IDEAS

    as
    1. Karan K. Mehta & Chi Zhang & Maciej Malinowski & Thanh-Long Nguyen & Martin Stadler & Jonathan P. Home, 2020. "Integrated optical multi-ion quantum logic," Nature, Nature, vol. 586(7830), pages 533-537, October.
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