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

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript addresses the solutions to the classical Mathieu-Hill (MH) equation in the context of trapped ions, specifically in electrodynamic (Paul) traps. The authors identify stable and bounded solutions for a harmonic oscillator (HO) levitated in these traps, focusing on the dynamics based on Floquet theory and various parameter regimes. The MH equation is well known for describing systems where the stability of periodic orbits is crucial, and the author looks into stability regions for both damped harmonic oscillators (HO) and parametric oscillators (PO). The author extends earlier work by analyzing small values of the q parameter and determining how the Floquet exponent influences stability. The manuscript is generally well written but needs substantial improvement by proofreading and providing a more complete exposure to the field, especially to those about the methods used (suggestions made below). With all the following edits and clarifications made satisfactorily, I recommend this work for publication in Mathematics. Detailed comments on the manuscript are provided below.  

SPECIFIC COMMENTS (MAJOR)  

1.     The introduction is well written but needs expanding to make it more straightforward to a broader audience. For example, the authors write about  “nonlinearity .” but do not explain in which context it is necessary and why, nor does it have a citation that could guide the reader to background information about nonlinearity.
For example, it might be nice to think about the idea of nonlinearity as a resource of quantumness., like https://doi.org/10.1103/PhysRevLett.99.040404 and references in terms of quantum sensing doi.org/10.1103/PhysRevResearch.5.013185

To make the presented results close to real physics, the authors should give appropriate experimental reports in the pioneering literature that all the chosen parameters are indeed realized in the domain and order of experimental works.

Author Response

Reply to Reviewer #1

 

Comment # 1

1.     The introduction is well written but needs expanding to make it more straightforward to a broader audience. For example, the authors write about  “nonlinearity.” but do not explain in which context it is necessary and why, nor does it have a citation that could guide the reader to background information about nonlinearity.

For example, it might be nice to think about the idea of nonlinearity as a resource of quantumness, like https://doi.org/10.1103/PhysRevLett.99.040404 and references in terms of quantum sensing doi.org/10.1103/PhysRevResearch.5.013185.

To make the presented results close to real physics, the authors should give appropriate experimental reports in the pioneering literature that all the chosen parameters are indeed realized in the domain and order of experimental works.

Author reply:

The suggestions made by the Reviewer # 1 have been implemented, both in terms of the papers suggested and with respect to nonlinearity (both in case of the Mathieu equation and as a source of quantumness) and quantum sensing. Two new paragraphs have been added, corresponding to lines 19-32 and 57-65, and extra bibliographical references. It has been emphasized that trapped ions are excellent candidates for quantum simulation and sensing. These aspects have been discussed in the paper published in Photonics 11 (2024).

Nonlinearity in the Mathieu equation is discussed thoroughly in the paper of Kovacic, Rand, and Sah, published in Appl. Mech. Rev. 70 (2018), which has been cited.

The issue of the experimental values of the Paul trap parameters has been discussed in the paper published in Photonics 11 (2024) and within many of the References in the Bibliography. The paper assumes that the values of the parameters |a|, |q| << 1, which is a hypothesis valid for any electrodynamic trap experiment. The scope of the paper is not to discuss these aspects (and thus avoid redundancy with the aforementioned paper), but to suggest an analytical model that enables identifying the solution of the Mathieu equation for systems of trapped ions and to discuss the physical realization of the HO and parametric oscillator for trapped ions, as well as the stability and instability regions of the Mathieu equation. And the discussion in Section Appendix B1 is relevant and self-consistent. Nevertheless, a new paragraph was added (lines 404-408) to discuss this issue and the experimental values of the Mathieu equation parameters.

Anharmonic corrections for nonlinear Paul traps are also discussed in Photonics 11 within the frame of perturbation theory, while the frontiers of the modified stability domains are determined as a function of the chosen perturbation parameter and it is demonstrated they are shifted towards negative values of the a parameter.

I would like to thank Reviewer #1 for his help in making the paper more accessible and easy to understand.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The author investigates the solutions of the classical Mathieu-Hill equation in order to describe the dynamics of trapped ions, based on the Floquet theory. It is shown that the corresponding equations of motion are equivalent to those of the harmonic oscillator.

The main purpose of the paper is to identify stable and bounded solutions for the dynamics of a harmonic oscillator levitated in an electrodynamic trap. 

The paper is well and clearly written and the obtained results are interesting and have the potential to find applications to electrodynamic traps employed in ultrahigh-resolution spectroscopy, mass spectrometry and in the domain of quantum technologies based on ultracold trapped ions. 

There are some minor points that I would recommend to the author: to check and correct some typos in the expressions in Eqs. (15), (A2), (A6), (A11), just to mention a few.

In conclusion, I recommend the publication of this manuscript in the journal Mathematics, after the mentioned corrections are performed. It is not necessary for me to see the manuscript after this minor revision is made.

Author Response

Comment # 1

The author investigates the solutions of the classical Mathieu-Hill equation in order to describe the dynamics of trapped ions, based on the Floquet theory. It is shown that the corresponding equations of motion are equivalent to those of the harmonic oscillator.

The main purpose of the paper is to identify stable and bounded solutions for the dynamics of a harmonic oscillator levitated in an electrodynamic trap. 

The paper is well and clearly written and the obtained results are interesting and have the potential to find applications to electrodynamic traps employed in ultrahigh-resolution spectroscopy, mass spectrometry and in the domain of quantum technologies based on ultracold trapped ions. 

There are some minor points that I would recommend to the author: to check and correct some typos in the expressions in Eqs. (15), (A2), (A6), (A11), just to mention a few.

In conclusion, I recommend the publication of this manuscript in the journal Mathematics, after the mentioned corrections are performed. It is not necessary for me to see the manuscript after this minor revision is made.

Author reply:

The author is really indebted to the reviewer # 2 for his kindness and great help in identifying the typos. The updated version of the paper has hopefully corrected all the typos.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript sets out to investigate solutions of the classical Mathieu-Hill equations for a linear oscillator in a time-dependent potential where the time-dependence is periodic. The main result of the paper seems to be a technique for determining the Fourier coefficients for the solutions (eq 60) in certain regimes, and a subsequent stability analysis of the results.

While I do not see any particular issues with the mathematics itself, I found this a very hard paper to read and understand. Specifically

(1) There is a very extended discussion particularly in the introduction of where these equations might be valid. Given that it is a driven harmonic oscillator, and this is almost textbook in its application in physics, this seemed excessive padding to boost the `importance' of the result.

(2) Inconsistent with the above extended discussion about physics application was the abrupt transition to the equations themselves, particularly when it is claimed that this is a quantum analysis. There is no particular attempt -- at least initially, despite the otherwise extended physics motivation in the introduction -- to connect the equation parameters with anything real for ions. Further,  as far as I can tell, the only reason we can call this 'quantum' is for the relatively trivial reason that for linear potentials, Ehrenfrest's theorem guarantees that the centroid of the quantum wavefunction follows the classical equations -- and this fact is not mentioned at all. There is no consideration or discussion about the more interesting behavior of spread variables, for example, as one might care about this for more quantum situations. I would also ask for a guidance to the (presumably target audience of) physicists by conneting things like the discussion of M,N to initial conditions to physical ideas about the trap and the ion, as well as when discussing averaging over `fast' versus `slow' time-scales, etc.

All of this adds up to what I think is an excessively padded and over-interpreted paper about some correct and probably interesting calculations and results, though the math itself is spelled out at almost pedagogical detail. I would strongly recommend shortening the introduction -- and the claims for application -- considerably and clarifying in the presentation the truly original -- and non-trivial parts of the calculations. The paper may well be of more use in that form, I believe.

Author Response

Reply to Reviewer #3

 

Comment # 1

This manuscript sets out to investigate solutions of the classical Mathieu-Hill equations for a linear oscillator in a time-dependent potential where the time-dependence is periodic. The main result of the paper seems to be a technique for determining the Fourier coefficients for the solutions (eq 60) in certain regimes, and a subsequent stability analysis of the results.

While I do not see any particular issues with the mathematics itself, I found this a very hard paper to read and understand. Specifically

(1) There is a very extended discussion particularly in the introduction of where these equations might be valid. Given that it is a driven harmonic oscillator, and this is almost textbook in its application in physics, this seemed excessive padding to boost the `importance' of the result.

Author reply:

The paper tries to present the original contributions and generally the entire layout in a coherent and self-consistent manner. Nevertheless, the reviewer states at the end of its comments that “the math itself is spelled out at almost pedagogical detail”. The Floquet theory is used and the solutions of the Mathieu equation for an ion confined in an electrodynamic (Paul) trap are introduced by taking into account both secular motion and micromotion. The mathematical apparatus used is far from complex and straightforward to follow. Eq. (15) in the revised version gives the system of equations used to determine the corresponding coefficients and implicitly the solution of the trapped ion dynamics, where the latter is treated as a HO. This is the first method suggested in the paper. The second method considers a Fourier series solution for the Mathieu equation. A recurrence relationship between these coefficients is inferred, which enables one to determine the solution. Again, mathematics is kept as simple as possible, the approach is classical. “Padding” has been hopefully discarded in the updated version of the paper that we attach.

Comment # 2

(2) Inconsistent with the above extended discussion about physics application was the abrupt transition to the equations themselves, particularly when it is claimed that this is a quantum analysis. There is no particular attempt -- at least initially, despite the otherwise extended physics motivation in the introduction -- to connect the equation parameters with anything real for ions. Further,  as far as I can tell, the only reason we can call this 'quantum' is for the relatively trivial reason that for linear potentials, Ehrenfrest's theorem guarantees that the centroid of the quantum wavefunction follows the classical equations -- and this fact is not mentioned at all. There is no consideration or discussion about the more interesting behavior of spread variables, for example, as one might care about this for more quantum situations. I would also ask for a guidance to the (presumably target audience of) physicists by connecting things like the discussion of M,N to initial conditions to physical ideas about the trap and the ion, as well as when discussing averaging over `fast' versus `slow' time-scales, etc.

All of this adds up to what I think is an excessively padded and over-interpreted paper about some correct and probably interesting calculations and results, though the math itself is spelled out at almost pedagogical detail. I would strongly recommend shortening the introduction -- and the claims for application -- considerably and clarifying in the presentation the truly original -- and non-trivial parts of the calculations. The paper may well be of more use in that form, I believe.

 

Reply:

I must emphasize that the paper does not perform a quantum approach with respect to the solutions of the Mathieu equation, the analysis is purely classical. This is why a discussion related to the spread variables is not necessary. Equation (1) in page 3 represents the most basic form of the Mathieu equation for an ion trapped confined within an electrodynamic trap, an aspect mentioned clearly in the text. In addition, to ease the understanding, the reader is explained the significance of the parameters in this equation, as they relate to the trapping voltages applied. The issue of choosing the operating point in the stability diagram depending on the values of the adimensional trap parameters a and q is extensively discussed both in the literature and in the paper published in Photonics 11, while the paper avoids redundancy and does not insist on this issue. In order to respond to the reviewer comments, we have introduced eqs. (2) that clearly specify the connection between the trap parameters and the Mathieu equation. All new text is marked as blue.

In line 261 of the paper we shortly discuss the values of the a and q parameters, but this issue has been approached in a detailed manner in the paper published in Photonics 11 (2024).

Averaging over slow or fast time scales has mainly to do with the pseudopotential approximation, where due to the fact that the micromotion takes time on a time scale which is considerably smaller than the time scale of the secular motion (in fact the micromotion performs an amplitude modulation of the secular motion), an average over the period of the micromotion leads to a pseudopotential and to an autonomous Hamilton function, which simplifies the physical system under investigation. The pseudopotential approximation explains well trapping of charged particles within a quadratic potential. This issue has been covered both in the Paper published in Photonics 11 and especially in the extended references list.

Along the guidelines suggested by the reviewer, the Discussion section has been renamed as Results. Its extent and content have been shortened, results should be better emphasized in its current form. The abstract of the paper has also been condensed into a more appropriate and coherent form, which better emphasizes the focus and progress of the paper. The introduction part has also been “compressed”, along with the structure of the paper, which was indeed too large in the initially submitted version.

Eq. (9) in the initial version submitted have been discarded, as it is well known. The discussion of M,N to initial conditions to physical ideas about the trap and the ion has been performed in the paper published in Appl. Sci. vol. 11 (7) 2928 (2021), where dynamical stability is investigated for a system of two ions confined in a Paul trap, assimilated with two coupled oscillators. The modes of oscillation are supplied and it is demonstrated that the weak coupling condition is inappropriate in practice, while for collective modes of motion (and strong coupling) only a peak of the mass can be detected. Phase portraits and power spectra are employed to illustrate how the trajectory executes quasiperiodic motion on the surface of torus, namely a Kolmogorov–Arnold–Moser (KAM) torus. Dynamical stability is described by means of an analytical model that characterizes critical points based on the Hessian matrix approach. The model is then applied to investigate quantum dynamics for many-body systems consisting of identical ions, levitated in 2D and 3D ion traps. Finally, the same model is applied to the case of a combined 3D Quadrupole Ion Trap (QIT) with axial symmetry, for which the associated Hamilton function is obtained. The equilibrium points for a 3D QIT are provided, and it is demonstrated that the configurations of minimum are exactly the regions where ion crystals are created.

Quantum dynamics of trapped ions is investigated in Ann. Phys. 388 100 (2018), where a dequantization algorithm is proposed, by which the classical Hamilton (energy) function associated to the system results as the expectation value of the quantum Hamiltonian on squeezed coherent states. The method is then applied to characterize the quantum Hamiltonian for both combined and RF nonlinear traps, that exhibit axial symmetry. combined and RF nonlinear traps, that exhibit axial symmetry. The classical Hamiltonian functions for the particular traps considered are also built and the classical equations of motion are inferred.

Quantum dynamics is also investigated in Ann. Phys. 442 169826 (2022). Time dependent Hamiltonians are characteristic for 3D Paul traps and they can be described by means of evolution operators associated with the symplectic group representation, applied in order to build coherent states. In addition, the expectation values of the quantum Hamilton function reduced through the evolution operators applied to such states, determine a classical Hamiltonian that exhibits a time periodic perturbative term. By averaging this Hamiltonian, an autonomous dynamical system results whose equilibrium configurations determine the family of ordered structures (ion crystals). In order that the trapped ion system reserves its stability, the associated quasienergy spectrum is required be discrete.

To summarize, the updated version of the paper discards most of the statements that are well known, hence, not necessary. The aim was to ease understanding and coherence of the paper.

 

Finally, the author would like to thank the reviewer for his time and for the useful comments uploaded, that have considered and implemented. I sincerely hope the replies clarify all the issues raised.

Author Response File: Author Response.pdf