Mathieu–Hill Equation Stability Analysis for Trapped Ions: Anharmonic Corrections for Nonlinear Electrodynamic Traps

Abstract

The stability properties of the Hill equation are discussed, especially those of the Mathieu equation that characterize ion motion in electrodynamic traps. The solutions of the Mathieu-Hill equation for a trapped ion are characterized by employing the Floquet theory and Hill’s method solution, which yields an infinite system of linear and homogeneous equations whose coefficients are recursively determined. Stability is discussed for parameters a and q that are real. Characteristic curves are introduced naturally by the Sturm–Liouville problem for the well-known even and odd Mathieu equations $c{e}_m(z,q)$ and $s{e}_m(z,q)$. In the case of a Paul trap, the stable solution corresponds to a superposition of harmonic motions. The maximum amplitude of stable oscillations for ideal conditions (taken into consideration) is derived. We illustrate the stability diagram for a combined (Paul and Penning) trap and represent the frontiers of the stability domains for both axial and radial motion, where the former is described by the canonical Mathieu equation. Anharmonic corrections for nonlinear Paul traps are discussed 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 we demonstrate they are shifted towards negative values of the a parameter. The applications of the results include but are not restricted to 2D and 3D ion traps used for different applications such as mass spectrometry (including nanoparticles), high resolution atomic spectroscopy and quantum engineering applications, among which we mention optical atomic clocks and quantum frequency metrology.

1. Introduction

Linear differential equations (LDEs) with variable (periodic) coefficients are ubiquitous in both physics and engineering, but their solutions are generally identified by means of numerical simulations. In an effort to identify a solution for such a system, it is essential to infer the so-called Floquet or characteristic exponents that define a fundamental matrix associated to the system [1]. One of the most prevalent approaches used in case of LDEs with periodic coefficients is based on a truncated Fourier series [2] whose coefficients are derived by means of the Harmonic Balance (HB) approximate method [3,4], which is employed to investigate nonlinear oscillating systems described by ordinary nonlinear differential equations (NLDEs) [5].

Mathieu functions of period $\pi$ or $2\pi$, also known as elliptic cylinder functions, were introduced in 1868 by Mathieu [6] together with the so-called modified Mathieu functions [7] in order to characterize the vibrations of an elastic membrane placed in a fixed elliptical hoop [8,9]. As the Mathieu equation does not possess a closed-form analytic solution, its applications are affected by analytical [10] and numerical approximation schemes [11] along with the nonlinear analysis of the associated stability charts [12,13]. Therefore, analytical solutions of Mathieu–Hill (MH) or Duffing equations are generally investigated by means of various perturbation techniques [14,15,16], while other approaches also use non-perturbative techniques to characterize the parametric damping nonlinear Mathieu–Duffing oscillator [17].

Analytic periodic approximations for differential equations of Hill type are investigated in Ref. [3], where two different methods are employed in the straightforward case of a Mathieu equation. The first method is triggered by the HB method [4,18], with an aim to identify analytic approximations for the critical values and focus on periodic solutions of the Mathieu equation. It is demonstrated that these kinds of solutions are valid for any values of the parameter q. The second method exploits truncations of the Fourier series [2].

In the last decades, Mathieu-type linear or nonlinear parametric differential equations have been the subject of large scientific interest because of their frequent use in applied mathematics [19,20], quantum physics [21,22], quantum optics [23], engineering [12], analytical chemistry and mechanics [13,14], along with general relativity [24] and astrophysics [25].

The advent of ion traps (ITs) has produced remarkable breakthroughs in modern physics, as they enable the trapping of electrically charged particles—usually in ultra-high vacuum (UHV)—which creates an almost interaction-free environment [26,27,28,29,30] that delivers long coherence times and narrow atomic lines. The particles are localized in a sharply defined region in space under conditions of dynamical equilibrium [31,32]. Hence, an IT is a versatile tool to perform high-resolution atomic spectroscopy. Ion dynamics in a Paul (or RF) trap [26,33] is characterized by a MH-type equation [21,34]. The Paul trap apparatus [in the case of both 2D linear ion traps (LITs) and 3D versions] has been developed and refined for high-finesse quantum engineering experiments, high-precision spectroscopy [35,36], classical mass spectrometry (MS) [37,38,39,40,41,42] and chemical analysis [43], including the detection of aerosols and chemical warfare [44,45,46,47,48,49,50,51,52].

Apart from that, ITs also enable exceptional control in preparing and manipulating atomic quantum states [53,54,55,56,57,58] because they make possible applications such as quantum logic [59,60,61,62,63], quantum sensing [64,65,66,67], quantum metrology [68,69] and even time fractals [70] or time crystals [71]. To these, one adds high-accuracy optical frequency standards [72,73,74,75], which are amongst the most sensitive quantum sensors [76,77] used to perform searches for physics beyond the Standard Model (BSM) [78,79,80] or to disseminate atomic time scales and redefine the SI unit of time, the second [81,82,83,84,85]. The sensitivity of optical atomic clocks [72,86] is limited by the Standard Quantum Limit (SQL) imposed by the inherent projection noise of a quantum measurement [87]. Recent developments in atomic physics have enabled quantum engineering of many-body entangled states that boost the performance of quantum sensors beyond the SQL [55,66,88,89]. Furthermore, it is anticipated that use of compound atomic clocks can enhance the stability of single ion clocks with long clock transition lifetimes to levels comparable to those of optical lattice clocks [75,90]. For ion species characterized by shorter lifetimes, the stability can be improved directly by increasing the number of ions. Nevertheless, this approach requires special care in the selection of the atomic transition and offers the potential for a stability beyond the SQL, which could represent a viable approach to further improve the stability of optical clocks [91] and provide quantum-limited optical time transfer [82,92], with an aim to achieve intercontinental clock comparisons through a common-view node in geostationary orbit (GEO). Several missions such as NASA’s Deep Space Atomic Clock (DSAC) or the Atomic Clock Ensemble in Space (ACES), along with ongoing or future European Space Agency (ESA) missions are based on deploying optical clocks in space.

An electrodynamic (Paul) trap is highly susceptible to geometric imperfections and electrode misalignment [39,93,94], which results in the occurrence of local minima in the trapping electric potential and nonlinear resonances in ion dynamics [42,95,96,97,98]. In the case of 2D traps, motional coupling between axial and radial directions is reported [99,100]. Trap potential deviations with respect to an ideal quadrupole lead to parasitic effects [101] which impose limitations in an explicit manner on the number of ions that can be confined, and implicitly on the signal to noise ratio (SNR) [65,102,103]. The effects produced by higher-order anharmonic terms [60] in the electric trapping potential (for linear strings of trapped ions) is investigated in [104]. As the ratio between the anharmonic and harmonic terms typically increases when the ion–electrode distance is reduced, it is demonstrated that anharmonic effects become more critical by lessening the ion trap scale. This requires special care, particularly in the case of single-ion [90,105,106] and multi-ion optical clocks [74,91,107,108] or in quantum sensing [109] and quantum computing experiments with trapped ions [65,110]. On the other hand, it is demonstrated that an increased dodecapole potential generated in an asymmetric LIT [99] opens new pathways for exciting applications in mass spectrometry or Coulomb crystals. Such a setup also enables dark ion (inappropriate confined species) elimination from a trapped ion chain consisting of physical qubits [42], or performing searches for time and parity violations that could constrain sources of new physics BSM [78,111,112].

The solution of a weakly nonlinear Mathieu equation (NME) is employed in [113] to characterize ion dynamics in the neighborhood of the stability boundary of ideal ITs (for which higher-order terms of the trap electric potential are discarded). The HB method [4] is employed in [114] to explore the coupling effects of hexapole and octopole fields on ion dynamics in a quadrupole ion trap (QIT). Ion motion characteristics, e.g., motion center displacement, secular frequency shift, nonlinear resonance curve and buffer gas damping effects have been investigated. It turns out that hexapole fields have a larger impact on ion motion center displacement, while octopole fields are mainly responsible for the ion secular frequency shift. In addition, the nonlinear features induced by hexapole and octopole fields may enhance or cancel each other. Ion dynamics in nonlinear Paul traps is investigated in [115] using the theoretical HB method, which allows one to derive the analytical ion trajectory and ion motion frequency in the superimposed octopole field by solving the NME [101]. The HB method is then validated by means of the numerical fourth-order Runge–Kutta (4th RK) method.

Amongst other issues, the response of a Duffing oscillator (DO) [15,116] to a harmonic excitation, in presence of (viscous) damping, is known to exhibit hysteretic and chaotic behaviors [1,13,100,117,118]. It is demonstrated that in particular cases, the damping force may induce instability in the ion dynamics [119,120]. For instance, Ref. [121] explores the role of field inhomogeneities in altering the stability boundaries in nonlinear Paul traps, taking into account higher-order terms in the equation of motion. The Poincaré–Lighthill–Kuo (PLK) method is employed in [122] to derive an analytical expression on the stability boundary and characterize the ion trajectory within a nonlinear Paul trap. The paper shows that a multipole superposition model (which essentially involves the octopole component) explains quite well how the field inhomogeneities shift the stable trapping region.

Particle dynamics in a nonlinear quadrupole Paul trap with octopole anharmonicity (which is a well-known dissipative system [32,123]) is described by an NME [4,124], which includes all perturbing contributions such as damping, multipole terms of the potential and a harmonic excitation force (periodic kicking). To complicate the picture even more, the ion also undergoes interaction with a laser field [125], in presence of contributions from the hexapole and octopole fields that superpose over the harmonic trap potential. Hence, it was demonstrated that a damped parametric oscillator (PO) levitated in a Paul trap exhibits fractal properties and complex chaotic orbits, along with the emergence of strange attractors (which are often called fractal sets) [125,126]. Ion motion on the strange attractor exhibits sensible dependence on the initial conditions. Several recent papers also demonstrate that ion dynamics in a Paul trap can be assimilated with the equation that describes a damped driven DO [100,101]. The equation of motion for such system is similar to a perturbed Duffing-type equation [15], which is a generalization of the linear differential equation that describes damped and forced harmonic motion [127,128].

Ref. [129] explores classical and quantum integrability for a trapped ion Hamiltonian in 3D space, under conditions of a non-quadratic potential with superpositions of the hexapole and octopole terms in the series expansion of the electric potential [130]. The Painlevé analysis is employed to determine new integrable cases. It is also demonstrated that the 3D perturbed Hamiltonian is completely integrable in the sense of Liouville [129]. Recent results also show the effective potential of an ion in 3D multipole fields in the mixed-mode is characterized by multiple isolated local minima. In addition, the specific number and spatial positions of the stable quasi-equilibrium points depend on the $a/q$ parameters ratio and the polarity of the electrodes’ DC voltage component [131]. An iterative method is employed to derive the stability parameters from measured secular frequencies, which is presented in Ref. [105].

Structure of the Paper

Section 1 is a review of ion traps and their deep impact for modern physics and quantum technologies. Section 2 and Section 3 investigate the properties of the Mathieu–Hill (MH) equations based on Floquet’s theorem, Hill’s Method solution and Sträng’s technique [132]. We demonstrate how the recurrence relationship that supplies the Floquet exponent is derived. The stability properties of the solutions of the MH equation for a trapped ion are discussed for real parameters a and q. The issue of bounded or unbounded ion dynamics is also discussed, pointing out that the $(a,q)$ plan is divided into stability and instability regions. A stable solution of motion corresponds to a superposition of harmonic motions and we find the frequency spectrum in such case. The stability diagram for an electrodynamic (Paul) trap is illustrated as a function of the Mathieu function eigenvalues. We also illustrate the stability diagram for a combined trap, where both axial and radial dynamics are considered, and we investigate the damped PO which generally exhibits fractal properties, chaotic complex orbits and exotic attractors [125,126,133]. Ion trajectories in the phase space are explored and we infer the transfer matrix along with the maximum amplitude of stable oscillations for ions confined in a quadrupole Paul trap.

We demonstrate that in case of small values of the trap operating parameters $(a,q)$, located within the first stability region, higher harmonics are practically insignificant and the fundamental frequency prevails. We identify the solution of the MH equation in such a case and we further apply the pseudopotential method to infer the shift in the solution, along with the first correction.

Section 4 approaches an issue of large interest by investigating multipole anharmonicities of the trap electric potential. We employ the perturbation theory and supply the modified frontiers of the stability diagram in case of an electrodynamic trap with a nonlinear octopole term of the electric potential, as a function of the perturbation parameter we choose. Annexes are intended to help the reader acquire a better grasp of the mathematics involved, including the perturbation theory technique, with a focus on simplicity.

2. Mathieu–Hill Equations

The Hill differential equation is similar to Mathieu’s equation but exhibits a more general nature. It arises in Hill’s method [134,135] to establish the motion of the Lunar Perigee and it represents a generalization of the Mathieu equation [19,136,137]. The MH equation is a homogeneous, second-order ordinary differential equation (ODE), with variable (periodic) coefficients [4,13]. It characterizes dynamical systems that exhibit intrinsic periodicity and parametric behaviour, such as the modulation of radio carrier waves, transverse vibrations of a tense elastic membrane, the stability of periodic motion in a nonlinear system, as well as the focus and defocus of beams in particle accelerators [138,139], and it is also employed to explore planetary dynamics [25]. The Hill equation takes the canonical or standard form of a reduced linear equation of the second order [8,10,20,140]:

$$\frac{d^{2}w}{d{\tau }^2}+J\left(\tau \right)w=0$$

(1)

where J stands for a continuous and even function that is periodic in $\tau$. The period is usually taken equal to $\pi$ for historical reasons [140]. The differential Equation (1) was discussed by Mathieu in 1868 in connection with the problem of vibrations of an elliptic membrane [6]. It is also assumed that the function $J\left(\tau \right)$ can be expressed as a Fourier series [137]

$$J\left(\tau \right)={\theta }_0+\sum _{s=1}^{\infty }2{\theta }_{s}cos2\tau$$

(2)

which converges within an infinite band in the $\tau$ plan that includes the real axis. If ${\theta }_s=0$ for $s\ge 2$, then Equation (1) turns into the Mathieu equation. Furthermore, Equation (1) can also be arranged as follows:

$$\frac{d^{2}w}{d{\tau }^2}+\left[a-2q\psi \left(\tau \right)\right]w=0$$

(3)

where $\psi \left(\tau \right)$ represents a continuous and even function of period $\pi$, while a and q denote adimensional parameters [141]. The parameter a is often called the characteristic value or eigenvalue, and it is generally determined by means of given boundary conditions, while the $-2q$ parameter represents a fixed datum. The solution $w\left(\tau \right)$ for a given value of a is usually called the eigenfunction. As a rule, one seeks out periodic solutions of Equation (3) with boundary conditions defined at two finite fixed points, for example, 0 and p, provided in the form $w\left(0\right)=w\left(p\right)$ and $\dot{w}\left(0\right)=\dot{w}\left(p\right)$, where p is the period. This represents a particular case of a Sturm–Liouville problem [3,141]. In the particular case of Mathieu’s general equation or Hill’s equation, a fundamental system of solutions consists of $e^{\mu \tau }\varphi \left(\tau \right)$ and $e^{-\mu \tau }\varphi \left(-\tau \right)$, as the equation is invariable to the change $-\tau \to \tau$. Consequently, Floquet demonstrated that the complete solution of Mathieu’s general Equation (1) can be expressed as follows [136,137,140,142,143]:

$$w=A{e}^{\mu \tau }\varphi \left(\tau \right)+B{e}^{-\mu \tau }\varphi \left(-\tau \right)$$

(4)

where $\mu \in \mathbb{C}$ is called the Floquet or characteristic exponent and it is a definite function of the a and q parameters, $\varphi$ denotes a periodic class $C^2$ function (twice continuously differentiable) [144], while A and B stand for arbitrary constants. Then, based on the Floquet’s theorem and according to Hill’s method one may assume a series solution:

$$w=e^{\mu \tau }\sum _{s=-\infty }^{\infty }{c}_{2s}{e}^{2is\tau }=\sum _{s=-\infty }^{\infty }{c}_{2s}{e}^{\left(\mu +2is\right)\tau }$$

(5)

By introducing Equation (5) into Equation (1), one infers a recurrence relation:

$${\left(\mu +2is\right)}^2{c}_{2s}+\sum _{m=-\infty }^{\infty }{\theta }_{2m}{c}_{2s+m}=0,s\in \mathbb{Z},$$

(6)

where $\mathbb{Z}$ is the set of integer numbers, with ${\theta }_{-2m}={\theta }_{2m}$. If one tries to eliminate the term $c_{2s}$ in Equation (6), then a non-convergent infinite determinant [137] would result. To avoid such an outcome, every expression in Equation (6) is divided to its central term ${\left(\mu +2is\right)}^2+{\theta }_0$. To determine the characteristic exponent, one multiplies the system matrix described by Equation (6) with a diagonal matrix. Then, a matrix results whose diagonal entries are each equal to unity, assuming that none of the terms ${\left(\mu +2is\right)}^2+{\theta }_0$ vanish. By denoting the determinant of this matrix as $▵\left(i\mu \right)$, the equation that determines the Floquet exponent $\mu$ is [137]

$$▵\left(i\mu \right)=0$$

(7)

Then, Equation (7) can be further arranged as

$$cosh\pi \mu =1-2▵\left(0\right){sin}^2\frac{1}{2}\pi {\sqrt{\theta }}_0$$

(8)

When $\mu$ has been determined, the $c_{2s}$ coefficients can be inferred in terms of $c_0$ and co-factors of $▵\left(i\mu \right)=0$. Thus, the solution of the Hill differential equation is complete. In the case of a fairly rapid convergence of the determinant $▵\left(i\mu \right)$, Equation (7) can be used in its algebraic as well as recursive and explicit form, which represents Hill’s original method [9,137]. The evaluation of the determinant $▵\left(0\right)$ and the use of Equation (8) represents an alternative method when this determinant converges quite well. A practical method to solve the Hill equation was suggested by Brillouin [140] based on the expression

$${sinh}^2\left(\frac{1}{2}\pi \mu \right)={\dot{w}}_1\left(\frac{\pi }{2}\right)w_2\left(\frac{\pi }{2}\right)$$

(9)

where $w_1$ and $w_2$ stand for two fundamental solutions, and ${\dot{w}}_1$ stands for the time derivative. If a periodic fundamental solution exists for the Mathieu equation, then the existence of the other fundamental solution is forbidden and the characteristic exponent of the periodic solution corresponds to the a and q parameters of the characteristic curves that separate the stability domains. In the case of the Hill equation, such a property is not generally satisfied. Equation (9) is useful to compare with Whittaker’s theory on the Hill equation [137]. The advantage of applying such method lies in the fact that it works with periodic $J\left(\tau \right)$ functions that exhibit a finite number of discontinuities.

For example, Ref. [145] investigates the modes which diagonalize the dynamical problem for linearly coupled Mathieu equations, which leads to the Floquet–Lyapunov transformation, where the motion is associated with decoupled linear oscillators. The method is then used to solve the Heisenberg equations of the corresponding quantum–mechanical problem and to determine the quantum wavefunctions for stable oscillations in the configuration (Hilbert) space. Such a transformation and solution can be applied to more generic linear systems with periodic coefficients, such as coupled Hill equations and periodically driven parametric oscillators.

3. Stability of the Solutions of the Mathieu–Hill Equation for a Trapped Ion

The equation of motion for an ion confined within a Paul (electrodynamic) trap exhibits the standard form of the Mathieu equation [19,146,147,148]:

$$\frac{d^{2}w}{d{\tau }^2}+\left[a-2qcos\left(2\tau \right)\right]w=0$$

(10)

with $\tau =\Omega t/2$ being dimensionless, where $\Omega$ is the RF of the trapping voltage $V_0$ applied between the electrodes. We introduce

$$\left\{\begin{array}{cc}{a}_z=-2a_x=-2a_y=-\frac{8Q{U}_0}{m{\Omega }^2\left(r_0^2+2z_0^2\right)}\hfill & \\ q_z=-2q_x=-2q_y=\frac{4Q{V}_0}{m{\Omega }^2\left(r_0^2+2z_0^2\right)}\hfill & \hfill \end{array}\right.$$

(11)

where Q is the ion electric charge, m denotes the ion mass, $r^2=r_0^2+2z_0^2$ with $r_o$ and $z_0$ being the trap semiaxes, and $U_0$ stands for the DC trapping voltage applied to the trap (endcap) electrodes. In addition, $a_x=-a_y$ and $q_x=-q_y$. To investigate the stability of ion trajectories, one uses the stability properties of the Mathieu equation solutions. As shown in Equations (4) and (5), the solution of Equation (10) can be expressed as a Hill series

$$w\left(\tau \right)=A{e}^{\mu \tau }\sum _{s=-\infty }^{\infty }{c}_{2s}{e}^{2is\tau }+B{e}^{-\mu \tau }\sum _{s=-\infty }^{\infty }{c}_{2s}{e}^{-2is\tau }$$

(12)

where the A and B constants are determined from the initial conditions. If $i\mu \notin \mathbb{Z}$, then using Equation (12), one infers a fundamental system of solutions associated to Equation (10) for $A=1,B=0$ and $A=0,B=1$ (the Floquet theorem [19,137]). The characteristic exponent $\mu$ and the $c_{2s}$ coefficients are functions of the parameters a and q [146].

By introducing the solution supplied by Equation (5) in the Mathieu Equation (10), the latter is cast into

$$\sum _{s=-\infty }^{\infty }{c}_{2s}\left[{\left(\mu +2is\right)}^2+a-2q\frac{e^{2i\tau }+e^{-2i\tau }}{2}\right]e^{\left(\mu +2is\right)\tau }=0$$

(13)

which, after matching the terms according to the powers of s, results in a recurrence relation

$$-q{c}_{2s-2}+\left[{\left(\mu +2is\right)}^2+a\right]-q{c}_{2s+2}=0$$

(14)

Further on, one multiplies with $i^2=-1$ and then divides by the middle term in Equation (14), which leads to an infinite system of linear equations which are homogeneous in the $c_{2s}$ coefficients [142]:

$$\frac{q}{{\left(2s-\mu i\right)}^2-a}{c}_{2s-2}+c_{2s}+\frac{q}{{\left(2s-\mu i\right)}^2-a}{c}_{2s+2}=0$$

(15)

which can be expressed as

$${\gamma }_{2s}{c}_{2s-2}+c_{2s}+{\gamma }_{2s}{c}_{2s+2}=0,s=0,\pm 1,\pm 2,\dots$$

(16)

with

$${\gamma }_{2s}=q{\left[{\left(2s-i\mu \right)}^2-a\right]}^{-1}$$

(17)

The system of Equation (16) admits a nontrivial solution if and only if the corresponding infinite determinant $▵\left(i\mu \right)$ vanishes for s noninfinite [137,149]

$$△\left(i\mu \right)=\left|\begin{array}{ccccc}\dots & & & & \\ & {\gamma }_{-2}& 1& {\gamma }_{-2}& & \\ & & {\gamma }_0& 1& {\gamma }_0& \\ & & & {\gamma }_2& 1& {\gamma }_2\\ & & & & & \dots \end{array}\right|$$

(18)

Due to the symmetry of $△\left(0\right)$, ${\gamma }_{-j}={\gamma }_j$. Solving this determinant is not a simple issue, which is why one uses the approach of Whittaker [137]. The method is described in Appendix A and it was introduced by Sträng [132,149]. The equation

$$▵\left(\mu \right)=0$$

(19)

supplies the characteristic exponent $\mu$, while Equation (16) enable one to recursively determine the coefficients $c_{2s}$. The infinite-order determinant $▵\left(\mu \right)$ is absolutely convergent and it represents a meromorphic function [150,151] of $\mu$, with simple poles for $\mu =\pm i\left(\sqrt{a}+2s\right);s=0,\pm 1,\dots$. Hence, Equation (18) is equivalent to [19]

$$cosh\left(\pi \mu \right)=1+2▵\left(0\right){\left[sin\left(\pi \sqrt{a}/2\right)\right]}^2$$

(20)

Hereinafter, we discuss the stability of the solutions of Equation (10) for $\tau ,a,q\in \mathbb{R}$. The solution of Equation (12) is stable or unstable if $\left|w\left(\tau \right)\right|$ is bounded or unbounded along the $\tau >0$ semi-axis, respectively. From Equation (20), one infers $cosh\left(\pi \mu \right)\in \mathbb{R}$ which renders the solution of Equation (12) as stable when $\left|cosh\left(\pi \mu \right)\right|<1$, in other words when $i\mu$ is a real non-integer. If $\left|cosh\left(\pi \mu \right)\right|>1$ then $\alpha =\Re e\mu \ne 0$, while $\left|w\left(\tau \right)\right|$ is bounded along the $\tau$ axis. In such a case, $\left|w\left(\tau \right)\right|$ is bounded along the $\tau >0$ semiaxis only when $A=0,\alpha >0$ or $B=0,\alpha <0$. As an outcome, the $\left(a,q\right)$ plan is divided into stability regions with $\left|cosh\left(\pi \mu \right)\right|<1$ and instability regions characterized by $\left|cosh\left(\pi \mu \right)\right|>1$, separated by the curve $\left|cosh\left(\pi \mu \right)\right|=1$ for which a stable and periodic solution of Equation (10) exists, although the general solution is unbounded. The curves characterized by integer values of $i\mu$ are called characteristic curves. They are naturally introduced by means of the Sturm–Liouville (eigenvalue) problem [137,141,152] for the Mathieu functions $c{e}_m\left(z,q\right)$ and $s{e}_m\left(z,q\right)$ [20,148], which are treated as characteristic functions of Equation (10) with the limit conditions

$$\frac{dw\left(0\right)}{d\tau }=\frac{dw\left(\pi \right)}{d\tau }=0,w\left(0\right)=w\left(\pi \right)=0$$

(21)

The $c{e}_m\left(z,q\right)$ and $s{e}_m\left(z,q\right)$ functions [7,153] are even and odd, respectively, known up to a constant factor, with period $\pi$ when m is even and with period $2\pi$ when m is odd [19,137]. The characteristic curves $a_m\left(q\right)$ and $b_m\left(q\right)$ correspond to the Mathieu functions $c{e}_m\left(z,q\right)$ with $m=0,1,\dots$ and $s{e}_m\left(z,q\right)$ with $m=1,\dots$, respectively. In addition, the characteristic curves are analytical in q, while being characterized by the subsequent properties [137]:

$$a_{2m}\left(q\right)=a_{2m}\left(-q\right),b_{2m+2}\left(q\right)=b_{2m+2}\left(-q\right),a_{2m+1}\left(q\right)=a_{2m+1}\left(-q\right)$$

(22)

$$a_{2m+1}\left(-q\right)=b_{2m+1}\left(q\right),a_m\left(q\right)<b_{m+1}\left(q\right)<a_{m+1}\left(q\right)$$

(23)

with $m=0,1,\dots$ and $q>0$.

The family of curves characterized by $\left|cosh\left(\pi \mu \right)\right|=1$ coincides with the family made up by the characteristic curves $a_m\left(q\right)$ with $m=0,1,\dots$ and $b_m\left(q\right)$ with $m=1,\dots$, which divide the $\left(a,q\right)$ plan into stability regions [ranging from $a_m\left(q\right)$ to $b_{m+1}\left(q\right)$ for $q\ge 0$ and from $a_m\left(q\right)$ to $a_{m+1}\left(q\right)$ or from $b_m\left(q\right)$ to $b_{m+1}\left(q\right)$ for $q\le 0$] and instability regions located below $a_0\left(q\right)$ or ranging from $b_m\left(q\right)$ to $a_m\left(q\right)$.

Ref. [154] shows that in case of microparticles levitated within a 2D linear Paul trap (LPT) operating under standard temperature and pressure (STP) conditions (in air), the Mathieu equations describing the trapping process are homogeneous along the x and y axes, with the well-known solutions and stability domains. On the other hand, the z axis motion is described by an inhomogeneous Mathieu equation. In this case, stability is obtained for (a) $i\mu \in \mathbb{R}$, (b) $\mu \in \mathbb{R}$ and $\mu <\Lambda$ and (c) $\mu -i\in \mathbb{R}$ and $|\mu -i|<\Lambda$, where $\mu$ is the Floquet exponent, $\Lambda =K/m\Omega$, K stands for the coefficient which describes the drag aerodynamic force and m represents the particle mass. Hence, stability regions for solutions of the inhomogeneous equations of motion in the presence of drag forces include both the stability regions of the homogeneous equations along with a part of the instability regions, which extends them considerably.

The Mathieu Equation (10) exhibits $\pi -$ and $2\pi -$ periodic solutions on continuous stability curves $a=a\left(q\right)$, starting from points $a=n^2,n=1,2,3,\dots$ Furthermore, the periodic solutions and the boundaries between stability and instability regions in the $(a,q)$ parameter plane can be found by means of the Lindstedt–Poincaré method [16]. A cosine elliptic $c{e}_n$ and a sine elliptic $s{e}_n$ function are associated to any value of n, where each one of these functions has its own characteristic number $a_{c{e}_n}$ and $a_{s{e}_n}$. The Mathieu functions and their eigenvalues (characteristic numbers) in the power series of q for $a,\left|q\right|\ll 1\left(q\simeq 0\right)$ are expressed as follows:

$$a_0\left(q\right)=a_{c{e}_0}\left(q\right)=-\frac{q^2}{2}+\frac{7q^4}{128}-\frac{29q^6}{2304}+\frac{68687q^8}{18874368}+\frac{123707q^{10}}{104857600}+\dots$$

(24)

$$\begin{array}{c}{a}_1\left(q\right)=a_{s{e}_1}\left(q\right)=1+q-\frac{q^2}{8}-\frac{q^3}{64}-\frac{q^4}{1536}+\frac{11q^5}{36864}+\frac{49q^6}{589824}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{55q^7}{9437184}-\frac{83q^8}{35389440}-\frac{12121q^9}{15099494400}+\dots \hfill \end{array}$$

(25)

$$\begin{array}{c}{b}_1\left(q\right)=a_{c{e}_1}\left(q\right)=1-q-\frac{q^2}{8}+\frac{q^3}{64}-\frac{q^4}{1536}-\frac{11q^5}{36864}+\frac{49q^6}{589824}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{55q^7}{9437184}-\frac{83q^8}{35389440}+\frac{12121q^9}{15099494400}+\dots \hfill \end{array}$$

(26)

$$a_2\left(q\right)=a_{s{e}_2}\left(q\right)=4+\frac{5q^2}{12}-\frac{763q^4}{13824}+\frac{1002401q^6}{79626240}-\frac{1669068401q^8}{458647142400}+\dots$$

(27)

$$b_2\left(q\right)=a_{c{e}_2}\left(q\right)=4-\frac{q^2}{12}+\frac{5q^4}{13824}-\frac{289q^6}{79626240}+\frac{21391q^8}{458647142400}+\dots$$

(28)

The eigenvalue associated with the even solutions of the Mathieu functions, $c{e}_k(z,q)$, is labelled by $a_k\left(q\right),k=0,1,2,\dots$, while the one associated with the odd Mathieu function, $s{e}_k(z,q)$, is denoted as $b_k\left(q\right),k=1,2,\dots$, as illutrated in Figure 1. The higher rank terms that describe the frontiers of the stability regions are given in Appendix B. A method to generate stability plots for the Mathieu equation in the case of a toroidal ion trap mass analyzer is presented in Ref. [155].

To graphically illustrate the stability diagram (associated to the Mathieu equation) that characterizes the dynamics of an ion confined within a combined quadrupole trap (a combination between a Penning and a Paul trap), we have used Equations (24)–(28) and Equations (A20)–(A28). The result is shown in Figure 2.

In case of a quadrupole Paul trap, the stable solution of the equation of motion (10) (for $\mu =i\beta ,\beta \in \mathbb{R}$) corresponds to a superposition of harmonic motions [21,29,103]

$$w\left(t\right)={\alpha }_1\sum _{s=-\infty }^{\infty }{c}_{2s}cos\left[\left(s+\frac{\beta }{2}\right)\Omega t\right]+{\alpha }_2\sum _{s=-\infty }^{\infty }{c}_{2s}sin\left[\left(s+\frac{\beta }{2}\right)\Omega t\right]$$

(29)

with the frequency spectrum

$${\nu }_s=\left(2s\pm \beta \right)\frac{\Omega }{4\pi },s=0,1,\dots$$

(30)

In Figure 3, we have illustrated the stability diagram for a combined (Paul and Penning) trap. One notices that $a_0,b_1,a_1,b_2,a_2$ represent the frontiers of the stability diagram associated with the canonical Mathieu equation that describes axial motion. We also introduce

$$\begin{array}{c}{c}_0\left(q\right)=c-a_0\left(-q/2\right)/2=c+\frac{1}{16}{q}^2-\frac{7}{4096}{q}^4+\frac{29}{2304}\frac{q^6}{64}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{68687}{18874368}\frac{q^8}{512}-\frac{123707}{104857600}\frac{q^{10}}{2048}\hfill \end{array}$$

(31)

$$\begin{array}{c}{c}_1\left(q\right)=c-a_1\left(-q/2\right)/2=c-\frac{1}{2}+\frac{q}{4}+\frac{q^2}{64}-\frac{q^3}{512}+\frac{1}{1536}\frac{q^4}{32}+\frac{11}{36864}\frac{q^5}{64}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{49}{589824}\frac{q^6}{128}+\frac{55}{9437184}\frac{q^7}{256}+\frac{83}{35389440}\frac{q^8}{512}-\frac{12121}{15099494400}\frac{q^9}{1024}\hfill \end{array}$$

(32)

$$\begin{array}{c}{d}_1\left(q\right)=c-b_1\left(-q/2\right)/2=c-\frac{1}{2}-\frac{q}{4}+\frac{q^2}{64}+\frac{q^3}{1024}+\frac{1}{1536}\frac{q^4}{32}-\frac{11}{36864}\frac{q^5}{64}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{49}{589824}\frac{q^6}{128}-\frac{55}{9437184}\frac{q^7}{256}+\frac{83}{35389440}\frac{q^8}{512}+\frac{12121}{15099494400}\frac{q^9}{1024}\hfill \end{array}$$

(33)

Further on, we consider trap operating points of $\left(a,q\right)$ that lie within the first stability region, with $0<\beta <1$ and $q\ge 0$. By minimizing the parameters a and $q\left(a/q\ll 1,q<0.4\right)$, the coefficients $c_{2s}$ (with $s\ne 0$) rapidly converge to zero, in such a way that higher harmonics are practically insignificant and the fundamental frequency ${\nu }_0=\beta \Omega /\left(4\pi \right)$ prevails [156]. In such an event, $w\left(t\right)=w_0\left(t\right)+w_1\left(t\right)$, where $w_0\left(t\right)$ stands for the average of the $w\left(t\right)$ shift during the interval $\left[t,t+2\pi /\Omega \right]$ where $\left|w_1\left(t\right)\right|\ll \left|w_0\left(t\right)\right|$ and $\left|d{w}_1/dt\right|\gg \left|d{w}_0/dt\right|$. Under such circumstances, the solution of the Mathieu equation can be approximated as the solution of the following equation:

$$\frac{d^{2}w}{d{t}^2}=\frac{{\Omega }^2}{4}\left[-a+q\left(\frac{a}{2}+2\right)cos\Omega t-q^2{cos}^2\Omega t\right]w$$

(34)

By averaging Equation (34) over a period of the RF voltage (which is exactly the case of the pseudopotential approximation, when the electric forces that are time-averaged over an RF period generate a harmonic potential [21,29,157,158]), one derives

$$\frac{d^2{w}_0}{d{t}^2}=-{\beta }^2{w}_0,\beta =\sqrt{a+\frac{q^2}{2}}$$

(35)

The shift $w_0$ in Equation (35) corresponds to the harmonic motion of fundamental frequency ${\nu }_0$, while the correction $w_1$ is determined by the higher harmonics $\left(2-\beta \right)\Omega /4\pi$, $\left(2+\beta \right)\Omega /4\pi ,\dots$ The axial frequencies are twice the radial frequencies, $a_z=-2a_r$ and $q_z=-2q_r$, while the trapped ions describe approximate Lissajoux trajectories when the ratio between the frequencies along the axes is 1:1:2. The initial conditions have no effect whatsoever on the stability of ion trajectories, but they establish their position with respect to the area located between the trap electrodes for every operating point $\left(a,q\right)$ fixed within a stability domain. In this regard, the problem of evaluating the maximum amplitude of stable oscillations under given initial conditions is approached. The maximum admits upper bounds determined by the relative position of the electrodes. For optimum operating parameters, it is mandatory to take into account all axes for whom the equation of motion is of Mathieu type.

One considers the stable solution of Equation (29) expressed as

$$w\left(\tau \right)={\alpha }_1{w}_1\left(\tau \right)+{\alpha }_2{w}_2\left(\tau \right)$$

(36)

with

$$w_1\left(\tau \right)=\sum _{s=-\infty }^{\infty }{c}_{2s}cos\left(2s+\beta \right)\tau ,w_2\left(\tau \right)=\sum _{s=-\infty }^{\infty }{c}_{2s}sin\left(2s+\beta \right)\tau$$

(37)

where both $w_1\left(\tau \right)$ and $w_2\left(\tau \right)$ are differentiable functions. If $w\left(\tau \right)\ne 0$ (which also implies that the coefficients ${\alpha }_1,{\alpha }_2\ne 0$) then $w_1$ and $w_2$ are linearly dependent functions. Hence, $w_1$ and $w_2$ form a fundamental system of solutions of the Mathieu equation (10) and the Wronskian determinant [7] is

$$W=w_1\left(\tau \right){\dot{w}}_2\left(\tau \right)-{\dot{w}}_1\left(\tau \right)w_2\left(\tau \right)\ne 0$$

(38)

Furthermore, the derivative of the Wronskian determinant is

$$\dot{W}=w_1\left(\tau \right){\ddot{w}}_2\left(\tau \right)-{\ddot{w}}_1\left(\tau \right)w_2\left(\tau \right)\ne 0$$

(39)

The modulo of the solution of Equation (36) admits the following upper limit:

$$\Lambda =\sum _{-\infty }^{\infty }\left|c_{2s}\right|\sqrt{{\alpha }_1^2+{\alpha }_2^2}$$

(40)

One denotes

$$d\left(\tau \right)=-W^{-1}\left[w_1\left(\tau \right){\dot{w}}_1\left(\tau \right)+w_2\left(\tau \right){\dot{w}}_2\left(\tau \right)\right]$$

(41)

$$b\left(\tau \right)=W^{-1}\left[w_1^2\left(\tau \right)+w_2^2\left(\tau \right)\right],c\left(\tau \right)=W^{-1}\left[{\dot{w}}_1^2\left(\tau \right)+{\dot{w}}_2^2\left(\tau \right)\right]$$

(42)

$$\epsilon =W{\Lambda }^2{\left[\sum _{s=-\infty }^{\infty }\left|c_{2s}\right|\right]}^{-2}$$

(43)

By eliminating the parameters ${\alpha }_1$ and ${\alpha }_2$ in Equations (36) and (40), it follows that

$$c\left(\tau \right)w^2\left(\tau \right)+2d\left(\tau \right)w\left(\tau \right)\dot{w}\left(\tau \right)+b\left(\tau \right){\dot{w}}^2\left(\tau \right)=\epsilon$$

(44)

Out of Equations (41)–(43) one infers

$$b\left(\tau \right)c\left(\tau \right)-d^2\left(\tau \right)=1$$

(45)

Hence, for an initial given phase $\tau ={\tau }_0$, Equation (44) represents the equation of an ellipse of area $\pi \epsilon$ in the plan $\left(w,\dot{w}\right)$. For initial conditions given such that the point $\left(w\left({\tau }_0\right),\dot{w}\left({\tau }_0\right)\right)$ is located on the ellipse described by Equation (44) or within it, one finds $\left|w\left(\tau \right)\right|<M$ at any moment in time. The ions are not captured at the electrodes if all points w (with $\left|w\right|<M$) belong to the projection along the w axis of the region located within the electrodes.

To investigate ion trajectories in the phase space, it will suffice to build the family of ellipses described by Equation (44) for $\tau$ ranging between $0\le \tau <\pi$. Indeed

$$\left(\begin{array}{c}w\left(\tau +\pi \right)\\ \dot{w}\left(\tau +\pi \right)\end{array}\right)=M\left(\tau \right)\left(\begin{array}{c}w\left(\tau \right)\\ \dot{w}\left(\tau \right)\end{array}\right)$$

(46)

with the transfer matrix expressed as

$$M\left(\tau \right)=\left(\begin{array}{cc}cos\pi \theta +d\left(\tau \right)sin\pi \theta & b\left(\tau \right)sin\pi \theta \\ -c\left(\tau \right)sin\pi \theta & cos\pi \theta -d\left(\tau \right)sin\pi \theta \end{array}\right)$$

(47)

The maximum amplitude of stable oscillations for trapped ions results from Equation (44)

$$w_{max}=\underset{0\le \tau \le \pi }{max}\sqrt{\epsilon b\left(\tau \right)}.$$

(48)

The Kicked Damped Parametric Oscillator

One considers the differential equation (DE) which describes a damped, kicked parametric oscillator (PO)

$$\ddot{u}+f\left(\tau \right)\dot{u}+g\left(\tau \right)u=h\left(\tau \right)$$

(49)

where $f,g$ and h are continuous functions of $\tau$, while $h\left(\tau \right)$ stands for the kicking term (external forcing), which is usually periodic. The homogeneous equation

$$\ddot{u}+f\left(\tau \right)\dot{u}+g\left(\tau \right)u=0$$

(50)

exhibits the fundamental solutions, which we denote as ${\phi }_1$ and ${\phi }_2$. Then, the general solution of Equation (49) is expressed as

$$\begin{array}{c}u\left(\tau \right)=c_1{\phi }_1\left(\tau \right)+c_2{\phi }_2\left(\tau \right)+{\phi }_2\left(\tau \right)\underset{{\tau }_0}{\overset{\tau }{\int }}\frac{{\phi }_1\left({\tau }^{\prime }\right)h\left({\tau }^{\prime }\right)}{W\left({\tau }^{\prime }\right)}d{\tau }^{\prime }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\phi }_1\left(\tau \right)\underset{{\tau }_0}{\overset{\tau }{\int }}\frac{{\phi }_2\left({\tau }^{\prime }\right)h\left({\tau }^{\prime }\right)}{W\left({\tau }^{\prime }\right)}d{\tau }^{\prime }\hfill \end{array}$$

(51)

where $W\left(\tau \right)={\phi }_1\left(\tau \right){\dot{\phi }}_2\left(\tau \right)-{\dot{\phi }}_1\left(\tau \right){\phi }_2\left(\tau \right)\ne 0$ is the Wronskian determinant. Equation (50) changes into the normal form

$$\ddot{w}+J\left(\tau \right)w=0$$

(52)

with

$$w\left(\tau \right)=u\left(\tau \right)exp\frac{1}{2}\underset{{\tau }_0}{\overset{\tau }{\int }}f\left({\tau }^{\prime }\right)d{\tau }^{\prime }$$

(53)

In particular, if the J function is periodic and continuous, the normal form described by Equation (52) is a Hill equation. In case when f is a constant function $f\left(\tau \right)=\lambda >0$ one derives

$$u\left(\tau \right)=w\left(\tau \right)exp\left[-\frac{\lambda }{2}\left(\tau -{\tau }_0\right)\right],J\left(\tau \right)=g\left(\tau \right)-\frac{{\lambda }^2}{4}$$

(54)

It can be noticed that the solution of Equation (49) is stable if $2\left|\Re e\mu \right|<\lambda$, with the Floquet exponent $\mu$ given by Equation (4). If w is stable then u is also definitely stable. Nevertheless, there exist unstable solutions w for which the corresponding solutions u are stable. The frontiers of the stability domains are supplied by the equation $2\left|\Re e\mu \right|=\lambda$.

4. Anharmonic Corrections for Electrodynamic (Paul) Traps: Perturbation Method Analysis

An issue of large interest lies in exploring the competition between the ion micromotion and the multipole anharmonicities of the trap electric potential, in order to discriminate between stable and unstable (chaotic) dynamics [133,159,160,161]. Ref. [125] reports on the dynamics of an ion confined in a nonlinear Paul trap [56,121,162], assimilated with a time-periodic differential dynamical system. As such systems are characterized by low dissipation, exotic phenomena can be observed such as strange attractors, limit cycles, period doubling bifurcation and fractal basin boundaries [70,126,133]. We investigate the following equation of motion, which describes the axial dynamics for a particle of mass M and electric charge Q, confined within a quadrupole Paul trap, with anharmonicity derived from an octopole (quartic) electric potential $\lambda z^4/4$ [125,163]:

$$\frac{d^{2}z}{d{\tau }^2}+\left[a-2qcos\left(2\tau \right)\right]z+\lambda z^3=0$$

(55)

where the adimensional parameters a and q are defined by Equation (11), while $r_0$ and $z_0$ stand for the Paul trap radial and axial dimensions. Generally, in the case of a typical 3D Paul trap, $a\le 0.05$ and $q\le 0.3$. The frequency of the applied AC voltage (micromotion) is denoted by $\Omega$, and $U_0$ and $V_0$ stand for the static and time-varying trapping voltages, respectively. The analytical modeling of the dimensionless Equation (55) is based on employing techniques that are characteristic to the global bifurcation theory [14,127]. Numerical modelling is used to explore the associated dynamics and discuss chaos in such a nonlinear system [133,159,160,161].

Further on, we introduce the perturbation parameter $\epsilon =2q$ and choose $\lambda =\epsilon \beta$, with the anharmonicity parameter $\beta$ fixed. Then, one can perform a series expansion of the a parameter and z coordinate as a function of the $\epsilon$ parameter:

$$a=\sum _{k=0}^{\infty }{\epsilon }^k{a}_k,z=\sum _{k=0}^{\infty }{\epsilon }^k{z}_k$$

(56)

4.1. Solutions of the Mathieu Equation

The frontiers of the stability domains in the plan of the control parameters $q-a$ are determined by the periodic solutions of Equation (55) [105,125]. There is a requirement that z is a periodic solution of period $2\pi$ and known parity. Within the limit $\epsilon \to 0$, the z solution is expressed as $cosn\tau$ or $sinn\tau$, with $n\in \mathbb{Z}$. One considers the case when $\epsilon =0(\epsilon =2q,\lambda =\epsilon \beta )$, where $\beta$ denotes an anharmonicity parameter. Then, Equation (55) is cast as

$$\frac{d^{2}z}{d{\tau }^2}+az=0$$

(57)

whose solution can be expressed as $z=cosn\tau$. Then, $\ddot{z}=-n^{2}cosn\tau$ and Equation (57) changes into

$$-n^2+a=0\Rightarrow a_0=n^2$$

(58)

From Equation (56), one infers

$$\ddot{z}=\sum _{k=0}^{\infty }{\epsilon }^k{\ddot{z}}_k$$

(59)

$$z^3=\sum _{k=0}^{\infty }\sum _{k^{\prime }=0}^{\infty }\sum _{k^{″}=0}^{\infty }{\epsilon }^{k+k^{\prime }+k^{″}}{z}_k{z}_{k^{\prime }}{z}_{k^{″}}=\sum _{p=0}^{\infty }{\epsilon }^p\sum _{\genfrac{}{}{0pt}{}{k,k^{\prime }\ge 0}{k+k^{\prime }\le p}}{z}_k{z}_{k^{\prime }}{z}_{p-k-k^{\prime }},p=k+k^{\prime }+k^{″}$$

(60)

We use the expression for $z^3$ derived above and revert to Equation (55)

$$\sum _{k=0}^{\infty }{\epsilon }^k{\ddot{z}}_k+a\sum _{k=0}^{\infty }{\epsilon }^k{z}_k-cos\left(2\tau \right)\sum _{k=0}^{\infty }{\epsilon }^{k+1}{z}_k+\beta \sum _{p=0}^{\infty }{\epsilon }^{p+1}\sum _{\genfrac{}{}{0pt}{}{k,k^{\prime }\ge 0}{k+k^{\prime }\le p}}{z}_k{z}_{k^{\prime }}{z}_{p-k-k^{\prime }}=0$$

(61)

Then, Equation (61) can be cast into

$$\sum _{p=0}^{\infty }{\epsilon }^p{\ddot{z}}_p+a\sum _{p=0}^{\infty }{\epsilon }^p{z}_p-cos\left(2\tau \right)\sum _{p=0}^{\infty }{\epsilon }^p{z}_{p-1}+\beta \sum _{p=0}^{\infty }{\epsilon }^p\sum _{\genfrac{}{}{0pt}{}{k,k^{\prime }\ge 0}{k+k^{\prime }\le p-1}}{z}_k{z}_{k^{\prime }}{z}_{p-k-k^{\prime }-1}=0$$

(62)

Therefore, by employing Equations (55) and (56), after identifying the rank of the $\epsilon$ parameter, one infers a system of differential equations that recursively determines the $a_k$ parameters and $z_k$ coordinates. Equation (62) can be expressed as

$$\frac{d^2{z}_p}{d{\tau }^2}+a_0{z}_p=z_{p-1}cos\left(2\tau \right)-\beta \sum _{\genfrac{}{}{0pt}{}{k,k^{\prime }\ge 0}{k+k^{\prime }\le p-1}}{z}_k{z}_{k^{\prime }}{z}_{p-k-k^{\prime }-1}-\sum _{k=0}^{p-1}{a}_{p-k}{z}_k,p\ge 1$$

(63)

where $z_0=cos\left(n\tau \right)$ or $z_0=sin\left(n\tau \right)$, with $a_0=n^2,n=0,1,\dots$ Therefore, the frontiers of the Mathieu stability diagram are either even functions $a_{+}^{\left(k\right)}$ or odd functions $a_{-}^{\left(j\right)}$ that depend on the parameter q, which within the limit $q\to 0$ approaches $k^2$ and $j^2$, respectively. The k and j indexes are integers, with $k\ge 0$ and $j>0$. By performing a series expansion of these functions depending on the q parameter $\left(\left|q\right|\ll 1\right)$, one derives the first frontiers of the stability diagram [164]

$$a_0\left(q\right)=\epsilon +2q\beta -\frac{q^2}{2}\left(1+4\beta \right)-\frac{q^3}{8}\beta \left(21+16\beta \right)+\dots$$

(64)

$$\begin{array}{c}{b}_1\left(q\right)=a_{-}^{\left(1\right)}=1+\epsilon -\frac{q}{2}\left(2+3\beta \right)-\frac{q^2}{32}\left(4+20\beta +9{\beta }^2\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{q^3}{256}\left(12+46\beta +60{\beta }^2+21{\beta }^3\right)+\dots \hfill \end{array}$$

(65)

$$\begin{array}{c}{a}_1\left(q\right)=a_{-}^{\left(1\right)}=1+\epsilon +\frac{q}{2}\left(2-3\beta \right)-\frac{q^2}{32}\left(4+4\beta -3{\beta }^2\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{q^3}{256}\left(4+20\beta -11{\beta }^2\right)+\dots \hfill \end{array}$$

(66)

$$\begin{array}{c}{b}_2\left(q\right)=4+\epsilon -\frac{3}{2}q\beta -q^2\left(\frac{1}{12}+\frac{1}{4}\beta +\frac{9}{128}{\beta }^2\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{q^3}{64}\beta \left(\frac{19}{6}+\frac{61}{20}\beta +\frac{15}{32}{\beta }^2\right)+\dots \hfill \end{array}$$

(67)

where $\beta =\left(\chi -\epsilon /6\right)/\left(2q\right)$, and $\chi$ is a constant.

Hence, the frontiers of the stability diagrams for the Mathieu equation with anharmonic perturbation are characterized by Equations (64)–(67). It is evident how the frontiers of the stability diagram are shifted towards negative values along the a axis in the control parameters plan $a-q$. Figure 4 and Figure 5 illustrate the stability diagram for the dynamics of an ion in the anharmonic trap we have investigated.

It can be noticed that the first stability region is bounded by the curves $a_0\left(q\right)$ and $b_1\left(q\right)$, while the second stability region is bounded by the curves $a_1\left(q\right)$ and $b_2\left(q\right)$. As opposed to the pseudopotential approximation when the electric RF potential is described by a polynomial of rank 2 in $z^2$ [21,28,29,115,143], considering the micromotion results in stability regions that are qualitatively similar to those of the Mathieu equation for low enough values of c, but generally with frontiers sensibly altered as a function of the anharmonicity parameter $\beta$. Hence, explicit analytic calculus which is numerically illustrated, enables one to establish the differences between the stability domains of the nonlinear dynamical system that is periodic in time (described by Equation (5)) and those of the autonomous dynamical system that is associated by means of the pseudopotential approximation [158].

4.2. The Frontiers of the Stability Diagram for the Mathieu Equation with Nonlinear Term

This section of the paper presents the technique we have developed to derive $a_0$. The frontiers of the stability diagrams of the Mathieu equation with nonlinear terms (anharmonic perturbation) are given by Equations (64) –(67), which correspond to the following limit solutions for $q=0$:

$$a_0=0;a_1=1\mathrm{and}{z}_0=cos\tau ;a_1=-1\mathrm{and}{z}_0=sin\tau$$

(68)

The motion of a single charged particle in a nonlinear Paul trap in the presence of the damping force is theoretically investigated in [119] and the modified stability diagrams in the parameter space are calculated. The results show that the stable regions in the $a-q$ parameter plane are not only enlarged but also shifted. Our results also show that the frontiers of the stability diagrams for the Mathieu equation are shifted towards negative values of the a parameter (within the plan defined by the control parameters $a-q$). In disagreement with the pseudopotential approximation for a Paul (RF) trap [115,158], when the electric potential is described by a polynomial of rank 2 in $z^2$, by taking into account the micromotion (when not in the pseudopotential approximation), stability regions result that are qualitatively similar to those of the Mathieu equation but with frontiers that are shifted depending on the anharmonicity parameter $\lambda \left(\beta \right)$. Further on, we express Equation (63) in case $p=1$ as

$$\frac{d^2{z}_1}{d{\tau }^2}+a_0{z}_1=z_{0}cos\left(2\tau \right)-a_1{z}_0-\beta z_0^3$$

(69)

and we compute the right term for $z_0=cos\tau$. Then, the right hand side of Equation (69) is cast into

$$cos\left(2\tau \right)cos\tau -a_{1}cos\tau -\beta {cos}^3\tau$$

(70)

which can be further expressed as (see Appendix C)

$$\left(2{cos}^2\tau -1\right)cos\tau -a_{1}cos\tau -\beta {cos}^3\tau$$

(71)

One writes

$$cos3\tau =cos2\tau cos\tau -sin2\tau sin\tau =2{cos}^3\tau -cos\tau -2\left(1-{cos}^2\tau \right)cos\tau$$

(72)

as

$$cosx=\frac{e^{ix}+e^{-ix}}{2}$$

(73)

Then, one derives

$$\begin{array}{c}{\left(cosx\right)}^n=\frac{1}{2^n}\sum _{k=0}^n{C}_n^k{e}^{ix\left(n-k\right)}{e}^{-ixk}=\frac{1}{2^n}\sum _{k=0}^n{C}_n^k{e}^{ix\left(n-2k\right)}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{2^n}\sum _{k=0}^n{C}_n^k\left[cos\left(n-2k\right)x+isin\left(n-2k\right)x\right]\hfill \end{array}$$

(74)

Hence, Equation (74) becomes

$${\left(cosx\right)}^n=\frac{1}{2^n}\sum _{k=0}^n{C}_n^{k}cos\left(n-2k\right)x$$

(75)

Using Equation (75), we infer

$$cos3\tau =4{cos}^3\tau -3cos\tau \mapsto co{s}^3\tau =\frac{1}{4}\left(cos3\tau +3cos\tau \right)$$

(76)

Then, Equation (69) becomes

$$\frac{d^2{z}_1}{d{\tau }^2}+a_0{z}_1=\left(2-\beta \right){cos}^3\tau -\left(a_1+1\right)cos\tau$$

(77)

By using Equation (76), Equation (77) changes accordingly to

$$\frac{d^2{z}_1}{d{\tau }^2}+a_0{z}_1=\left(2-\beta \right)\frac{1}{4}cos3\tau +\left(\frac{1}{2}-\frac{3}{4}\beta -a_1\right)cos\tau$$

(78)

The general solution of the harmonic oscillator (HO) writes as

$$z_{1g}=\alpha cos\tau +\beta sin\tau$$

(79)

We choose an even solution, such as $z_{1g}=\alpha cos\tau$. Then, one tries a particular solution such as

$$z_{1part}=Acos3\tau +Bcos\tau$$

(80)

with

$${\ddot{z}}_{1part}=-9Acos3\tau -Bcos\tau$$

(81)

Then, Equation (78) changes into

$$-8Acos3\tau =\frac{1}{4}\left(2-\beta \right)cos3\tau +\left(\frac{1}{2}-\frac{3}{4}\beta -a_1\right)cos\tau$$

(82)

which enables us to infer

$$a_1=\frac{1}{4}\left(2-3\beta \right),A=\frac{1}{32}\left(\beta -2\right)\Rightarrow z_1=cos\tau +\frac{1}{32}\left(\beta -2\right)cos3\tau$$

(83)

Hence, the solution of the Mathieu equation can be expressed as

$$z=z_0+z_1+\dots =1+cos\tau +\frac{1}{32}\left(\beta -2\right)cos3\tau +\dots$$

(84)

More explicit details with respect to solving the Mathieu equation by means of the perturbation theory and determining approximations to all other Mathieu functions and eigenvalues are supplied in Appendix C.

5. Discussion

An ideal IT exhibits a pure 3D quadrupole field, while the associated dynamics is characterized by the exact analytical solution of the MH equation [21,29,158]. In a quadrupole ideal IT the equations of motion are linear and uncoupled. As a matter of fact, the situation is entirely different in the case of real IT where the trapping field is far from linear due to geometric imperfections [39], such as the truncation of electrodes or geometric misalignment [165], stray electric charges, etc. These limitations superimpose weak multipole fields such as hexapole, octopole, decapole and higher-order electric fields [103,120,166,167,168]. The superposition of higher multipole fields significantly alters ion dynamics with respect to the case of a pure quadrupole trap [169,170]. The equation that characterizes ion motion in an anharmonic, real trap [56,125,163], is the nonlinear Mathieu equation which has no analytical solutions [115,129]. Furthermore, electrode misalignment results in symmetry breaking and leads to extra local minima in the trapping potential [98,165]. Trapped ion trajectories in the quadrupole, hexapole, octopole and dodecapole RF traps are investigated in [170]. A detailed analysis of the motion of trapped ions as a function of the amplitude, phase and stability of the ion’s motion is used to evaluate the experimental prospects for such traps.

This paper approaches the issues of the Hill equation solution completeness and those of periodic fundamental solutions of the Mathieu equation. We use the Floquet theorem, and based on the MH equation we find the recurrence relationship that supplies the coefficients and the characteristic exponent. We show the coefficients $c_s$ exhibit non-trivial solutions if the infinite determinant satisfies $△\left(i\mu \right)=0$ (vanishes) for noninfinite s. Then, a holomorphic function with a single pole (similar to the determinant) is introduced, and we use the Liouville theorem for complex calculus and derive the expression of the Floquet characteristic exponent. It is assumed the Floquet characteristic exponent $\mu$ is chosen to satisfy the prerequisite such that the determinant described by Equation (18) vanishes. The solution will be unbounded unless the Floquet exponent is imaginary. We supply this solution as a function of $△\left(0\right)$. To calculate this last determinant, one uses the Sträng recursion formula (method) [132], which is described in Appendix A.1.

Then, we discuss the stability of the MH equation for a trapped ion in the case of trap operating points $a,q\in \mathbb{R}$ that lie within the first stability region, pointing out when the associated dynamics are bounded or unbounded. By minimizing the parameters a and q, the coefficients $c_{2s}$ (with $s\ne 0$) of the Hill series solution rapidly converge to zero, in such a way that higher harmonics are practically insignificant and only the fundamental frequency prevails. We infer the solution of the Hill equation in such a case and find the solution by means of the pseudopotential approximation. We find both the shift in particle position and the first correction. We demonstrate that the initial conditions have no effect whatsoever on the stability of ion trajectories, but they establish their position with respect to the area located between the trap electrodes for every operating point located within a stability domain. We show the characteristic curves (which are analytical in q) are naturally introduced by means of the Sturm–Liouville (eigenvalue) problem [137,152] for the Mathieu functions $c{e}_m\left(z,q\right)$ (even) and $s{e}_m\left(z,q\right)$ (odd) [20], which are treated as characteristic functions of Equation (10) with limit conditions. These characteristic functions (curves) separate the $(a,q)$ plane into stability and instability regions, separated by the curve $\left|cosh\left(\pi \mu \right)\right|=1$ for which a stable and periodic solution of Equation (10) exists, although the general solution is unbounded. The curves characterized by integer values of $i\mu$ are called characteristic curves. We also show the periodic solutions as well as the boundaries between stability and instability regions in the $\left(a,q\right)$ parameter plane can be found by means of the Lindstedt–Poincaré method. Then, we illustrate the stability diagram for the Mathieu equation in case of an electrodynamic trap as a function of the associated eigenvalues and show the stable solution to be a superposition of harmonic motions.

In this regard, the problem of evaluating the maximum amplitude of stable oscillations under given initial conditions is approached. The maximum admits upper bounds determined by the relative position of the electrodes. To investigate ion trajectories in the phase space, we build a family of ellipses while we supply the transfer matrix and find the maximum amplitude of stable oscillations. We exemplify our analytical model for the case of a damped, kicked PO, for which we supply the general solution and discuss the frontiers of the stability domains for the homogeneous equation.

The stability diagram (associated to the Mathieu equation) that characterizes the dynamics of an ion confined within a combined quadrupole trap (a combination between a Penning and a Paul trap) is included. We illustrate the frontiers of the stability diagram for the canonical Mathieu equation for axial motion, as well as the frontiers of the stability domains for the radial trap motion, which are shown to also depend on the cyclotronic frequency.

Different ion trap geometries and electrode space arrangements are generally used (especially hyperbolic and cylindrical traps) in order to achieve a harmonic (electric) trapping potential around the trap center. Whilst hyperbolic geometries surround the trap center almost entirely, cylindrical ones usually exhibit open-endcap designs that allow better axial access to the center and enhanced interaction between levitated particles and laser beams used for cooling or manipulation [171], which makes them suited for MS, quantum optics, optical frequency metrology and quantum information processing (QIP) applications. For this reason, classical (hyperbolic) endcap electrode geometries are frequently employed when building single-ion optical clocks, as they provide certain benefits such as a saddle-shaped trapping potential which is essentially quadrupole, along with an open electrode structure that enables effective fluorescence detection [90,105]. Furthermore, a hyperbolic trap potential exhibits true azimuthal angle independence due to the electrode symmetry [172]. An open endcap trap design also exhibits a smaller quadrupole field component in the series expansion of the trap potential, while higher order terms must be considered when calculating the potential apart from the origin. Consequently, besides compensation of the effects of higher-order terms of the electric potential, some of which are approached in this paper, a careful trap design must consider and minimize issues such as potential frequency shifts and systematic errors in the atomic clock operation [72,73,173,174].

Axial stability for ion dynamics in a nonlinear Paul trap is investigated. In the case of the octopole trap (with an anharmonicity of order 4) considered, the frontiers of the stability diagram have been explicitly determined as a series expansion of the control parameters chosen. We illustrate graphically that the frontiers of the stability diagrams for the Mathieu equation with anharmonic perturbation are shifted towards negative values along the a axis, in the control parameters plan $a-q$, in agreement with [120,121]. Moreover, in the case of an octopole trap, the equations of motion are nonlinear and coupled which means that ion trajectories exhibit strong sensitivity with respect to the initial conditions. As an outcome of such a fact, it is not possible to define absolute stability conditions based only on the trap operating parameters in the stability diagram ($a,q$). The issue of dynamical stability for multiple ions, among other topics, is investigated in [31]. It is our intention to investigate the stability of ion (Coulomb) crystals used for optical clocks in the near future, taking into consideration externally forced Duffing PO as a natural model for ITs. Classical and quantum chaos is also expected to occur for specific parameters, used in numerical simulations or in the experiments performed, as well as strange attractors and fractal basin boundaries.

Both the paper submitted and the scientific literature demonstrate that geometrical imperfections play an important role in altering trap pseudo-potential, but they do not recommend corrections of the nominal trap geometry that could result in quantifiable improvements in the trapping efficiency [90]. We consider the results to be of interest for 2D and 3D ion traps used for different applications in ultrahigh-resolution spectroscopy, MS, and especially in the domain of quantum technologies (QT) based on ion traps, among which we evidence optical clocks (as extremely high-precision quantum sensors) and quantum metrology experiments.