Mathieu–Hill Equation Stability Analysis for Trapped Ions: Anharmonic Corrections for Nonlinear Electrodynamic Traps
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsIn this work,
the author studied the stability of the Mathieu-Hill (MH) equations for the specific case of electrodynamics trapped ions, considering the anharmonic perturbations from non-linear effects in electrodynamics.
The analysis performed uses well known methods developed by Hill and the Floquet theorems.
The author focused on the case of Paul's traps showing several stability diagrams for radial and axial trajectories of ions.
I think the paper is well written and the analysis looks correct; it has certainly possible implications in physics of ion traps and non-linear electrodynamics. However, I suggest to the author to stress what are the new results compared to the large literature in both mathematics and physics sides. Indeed, MH equations as well as Floquet methods are very well known and extensively studied in many contexts, so it is important to show the novelty of these new stability analysis which can be relevant for improving technical methods to stabilise the ions in traps. After the revision suggested, I will reconsider the paper for the publication.
Comments on the Quality of English LanguageMinor revision of English is demanded.
Author Response
I have performed the suggested changes, please read the attached document as well as the replies to the other 2 reviewers. Thank you very much.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThe article discusses the stability of the Mathieu-Hill equations, particularly the Mathieu equations that describe ion motion in electrodynamic traps. Using Floquet theory and Hill's method solution, the article explores the solutions of these equations and analyzes the impact of various parameters on the stability of ion motion. Additionally, the article addresses anharmonic corrections in nonlinear electrodynamic traps and their effects on the stability. Overall, although there has been extensive research on the Mathieu-Hill equations and ion trap technology, this article makes new contributions in the areas of anharmonic corrections and numerical simulations. I believe this article has a certain degree of novelty. However, I have some minor suggestions that I hope the authors can address before resubmitting the article.
1. The introduction section is somewhat lengthy. The introduction can be made more concise by avoiding excessive historical background and focusing on the innovative aspects of the current research.
2. In Chapter 2, "Mathieu-Hill equations," the mathematical derivation of the Hill equation solution, particularly the matrix description, is somewhat complex. It is recommended to add illustrations to aid readers' understanding.
3. The article involves a large number of formula derivations. The typesetting of the mathematical formulas in the article needs to be optimized to ensure consistency and make the formulas clearer and more readable.
4. The structure of the summary section is somewhat disorganized, with multiple topics mixed together, which may confuse the reader. It is recommended to organize the summary in the order of research objectives, innovative methods, and significance to make it more logical and coherent. Additionally, the summary section does not mention the limitations of the research. It is suggested to include a discussion on the limitations and propose future research directions and improvement suggestions to comprehensively present the strengths and weaknesses of the research.
Author Response
I have performed most of the suggested changes, please read the attached document as well as the replies to the other 2 reviewers. Thank you very much.
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThe article "Stability analysis of the Mathieu-Hill equation for trapped ions. Anharmonic corrections for nonlinear electrodynamic traps", is devoted to an actual problem that is of interest to a wide range of scientists. Unfortunately, it is presented as an “article“, but the structure and style correspond to the style of the "review". Therefore, it can be published after major revision. In my opinion, the author should focus on solving the problem indicated in the title, rather than giving an overview in the introduction of all the main results obtained over the past 150 years. The introduction should be reduced and completely rewritten.
Despite the fact that a formal mathematical analysis has been carried out in the manuscript, the parameters of the Mathieu-Hill equation should be discussed for various, but well-defined physical problems. In particular, the parameters of the problems of classical and quantum physics will have different meanings, while the equation may be the same. The author discusses the application of the equation to solve problems related to classical systems. Therefore, in particular, it is necessary to discuss the results obtained for application in quantum technologies, since this is stated in the abstract.
In addition, there are some technical remarks.
1. In Eq. (10), the parameters a and q are not discussed, but in the following they are associated with some spatial parameters (coordinates of the particle) of the system, except of Eq.(55). The author need to show how in the scalar equation one can obtain both axial and radial motion simultaneously.
2. In line 417, the fundamental frequency ν0 is not defined.
3. Derivatives are indicated by both a stroke in Eq. (9) and a dot in the following equations (see Equation(37)).
4. Eq. (48) is written for a damped, kicked parametric oscillator, but only a homogeneous equation is analyzed.
Author Response
I have performed the suggested changes, please read the attached document as well as the replies to the other 2 reviewers. Thank you very much.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
Comments and Suggestions for AuthorsThe author took into account the most part of the comments.
