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

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.

1. Introduction

Parametric excitation of second-order nonlinear systems [1,2] has been the subject of intense investigations [3,4], amongst which we mention the method of normal forms. Ref. [5] uses such a method to investigate the dynamical stability of a nonlinear system characterized by a nonlinear Mathieu–Duffing equation of motion [1,6,7,8,9,10,11] in case of parametric excitation [12]. It was demonstrated that the intrinsic nonlinear nature of the system induces a subharmonic region that was not previously reported for systems characterized by linear MH equations [8,13]. Normal modes analysis is also used in [14] to describe dynamical stability for trapped ion systems.

Nanoelectromechanical systems (NEMS) enable performing tests on the fundaments of quantum mechanics by studying the transition from classical to quantum behaviour of a driven nonlinear Duffing resonator [15]. The numerical solutions of the equations of motion associated to the resonator demonstrate that the quantum Wigner function slowly deviates from the corresponding classical phase-space probability density. Thus, nonlinearity is the cause of such differences which provide experimental evidence that NEMS resonators are excellent systems for quantum simulation and investigations of nonlinearity. On the other hand, ion traps are versatile tools that enable interdisciplinary investigations on nonlinearity [16,17,18,19].

Ref. [20] introduces a method to calculate the wave functions and energies of ground state and low-lying excited states of quantum multibody systems by employing the deep neural network and the unsupervised machine learning (ML) technique. A simple method of symmetrization for bosonic systems and antisymmetrization for fermionic systems is also proposed, in order to perform calculations in case of many-particle systems consisting of identical particles.

Investigations of the stability of physical systems that undergo periodic parametric driving [21], such as ions confined by oscillating electric fields (Paul traps) [22], is a subject of vivid scientific interest. The behaviour of these systems can be better explained based on an approach that employs the pseudopotential approximation, which explains well trapping of charged particles within a quadratic potential and resonances that arise out of parametric excitation [14]. In the pseudopotential approximation (for large values of the radiofrequency—RF), both in the classical and quantum cases, the time periodic Hamilton function that characterizes an ion confined within an electrodynamic trap is replaced by an autonomous (time-independent) Hamilton function. Furthermore, linear ion traps (LIT) that operate at two RF values represent a versatile tool to simultaneously confine two ion species of interest [23]. Parametric excitation is also employed to carry out mass-selective removal of ions out of an electrodynamic trap [24,25,26].

It is well established that a single trapped ion can be employed as a paradigm to test a choice of essential physical models realized as time-dependent (quantum) harmonic oscillators (QHO) [21,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]. A numerical modelling for a segmented Paul trap that exhibits four blade electrodes that generate the trapping field and a pair of two biasing rods, where the latter are employed to compensate micromotion [43], is performed in ref. [44]. Such an approach delivers enhanced optical access for both fluorescence spectroscopy and individual ion addressing, a mandatory requirement to achieve ion crystals or perform quantum engineering and manipulation [45] of trapped ion quantum states [46,47]. Applications span atomic clocks [48,49,50], which are excellent tools to search for physics Beyond the Standard Model (BSM) [51,52] or to test novel quantum technologies (QT) based on ultracold trapped ions [53,54]. Multipole linear Paul traps (LPTs) are also versatile tools to investigate cold ion-atom collisions and thus provide new experimental evidence of the phenomena involved [55,56,57,58].

A matter of utmost importance is the issue of quantum-enhanced measurements that exhibit potential to enhance high-precision sensing. Applications span areas such as ultraprecise optical clocks [50,59,60], realization of high-fidelity quantum logic gates [61], many-body quantum enhanced sensors [62,63] or measurements on the time-variations of the fundamental constants of nature at the cosmological scale [64]. Quantum sensors suffer from sensitivity loss in the presence of quantum noise [65]. To mitigate such an effect, different quantum error-correcting codes have been tested and implemented. Ultracold trapped ions demonstrate to be a promising platform for both quantum sensing and quantum error correction experiments [39,53,66].

Ref. [2] investigates Hamiltonian dynamics [67] of a single QHO [21,68,69] in the presence of dissipation and parametric driving, while it establishes that a time-dependent parametric frequency incessantly drives the system out of its dynamical equilibrium state. In addition, fine-tuning of the system parameters enables one to control the contention between dissipation and parametric driving in a similar way to which Ref. [16] investigates the competition between multipole anharmonicities of the trap electric potential and a periodic kicking term (laser field), driving the system from a regular motion regime towards chaotic dynamics [17]. To achieve control of a QHO, a key issue lies in characterizing and suppressing the inherent noise [70,71]. The intrinsic noise spectrum of a trapped ion can be characterized and experimental demonstrations of QHO control are recently reported in literature [39].

The paper is intended as a follow-up of the review paper recently published in Photonics [72]. Section 2 starts from the analytical model introduced in [73]. The paper investigates the issue of the solutions of the MH equation that explains ion dynamics within an electrodynamic ion trap (EIT). We show such an equation can be expressed as a HO equation and suggest two classes of complex solutions. It is well established that the MH equation solution can be expressed as a Hill series [74,75,76,77] and we analyse such cases. The first method we use yields a system of linear equations that enables one to determine the constants in the Fourier series solution. In case of the second method suggested, we derive a recursive relationship between these coefficients for a particular solution of the MH equation. Both approaches are original. Section 3 applies the results previously derived in Section 2 and supplies an analytical model to characterize ion dynamics as bound or unbound. By considering the residual interaction we supply the solutions of the MH equation. Finally, we demonstrate the MH equation can be regarded as a HO equation.

In Appendix A.1 we show a trapped ion can be assimilated with a kicked-damped HO model and discuss solutions depending on the discriminant of the equation. In Appendix B we analyse solutions of the MH equation for an ion confined within an electrodynamic (or RF) trap, assimilated with a parametric oscillator (PO).

The results in the paper apply to electrodynamic traps employed in ultra-high-resolution spectroscopy, mass spectrometry (MS) and in the domain of quantum technologies (QT) based on ultracold trapped ions [78,79,80,81], with an emphasis on quantum sensors and quantum metrology.

2. Solutions of the Mathieu–Hill Equation

An iconic paper that reports on an elaborate analysis and discussion of the transition of a few ion crystals into ion clouds is ref. [73]. The paper demonstrates how the crystalline phase prevails until the Mathieu stability limit is attained, emphasizing that the corresponding transition cannot be regarded as an order → chaos transition which occurs when a control parameter reaches a critical value. In addition, in the vicinity of this limit the system exhibits sensitivity to perturbations, which enables the experimenter to investigate crystal melting substantially forward the instability limit. The influence of the ion micromotion on ion crystal stability is also discussed in [73], where numerical modelling is used to characterize dynamical stability and discriminate between four possible regimes. Representations of large Coulomb crystals composed of trapped ions are reported in [55], where Molecular Dynamics (MD) numerical modelling is employed to account for cold ion–atom elastic collisions that occur between Coulomb crystals and very light virtual atoms.

A review of strongly coupled Coulomb systems consisting of ultracold trapped ions is performed in [82], whereas 2D crystals in zig-zag configurations and structural transitions are discussed in [83,84,85]. Recent approaches report direct observation of micromotion for a many-body Coulomb crystal consisting of ultracold trapped ions, while novel techniques to measure the micromotion amplitude are implemented [86]. A numerical modelling is employed to determine phase transitions in small-ion Coulomb crystals confined in an electrodynamic ion trap (EIT) [87], where the phase transition points are introduced as extremes of the interpolated functions employed. The novel conceptualization of fractal quasi-Coulomb crystals is introduced in [88], for the case of surface electrodynamic traps (SET) characterized by a Cantor Dust electrode configuration. New experimental evidence of orientational melting that arises in a 2D crystal consisting of up to 15 ions is reported in [89].

An analytical and numerical modelling focused on dynamical stability in the case of trapped ion systems is performed in [90] by extending the model introduced in [73,91]. Numerical modelling illustrates that the associated dynamics is either quasiperiodic or periodic, depending on the initial conditions. The evolution in time for a system of two coupled oscillators levitated in an RF trap is described using an analytical model that depends on the chosen control parameters, while ion dynamics is shown to be integrable only for discrete values of the ratio between the axial and the radial frequencies of the ion secular motion. A qualitative discussion of the system dynamical stability is then performed by employing the Morse theory. The results are further applied to many body strongly coupled Coulomb systems (trapped ions), locally investigated in the vicinity of equilibrium configurations that designate ordered structures. These equilibrium configurations exhibit a large interest for Coulomb ion crystals (atomic clocks) or to achieve quantum information processing (QIP).

For example, the Kibble–Zurek mechanism (KZM) generally describes nonequilibrium dynamics along with topological defect occurrence for physical systems that undergo second-order phase transitions. A generalized KZM that explains defect emergence in trapped ion systems is discussed in [92,93], where the analytical models proposed enable investigation of KZ physics in the case of inhomogeneous systems.

We consider the MH equation for an ion confined in an EIT [72,73]

$$\ddot{x}+f\left(\tau \right)x=0,f\left(\tau \right)=a+2qcos\left(2\tau \right).$$

(1)

Equation (1) is a linear differential equation with a periodic coefficient, and it is assumed that f is a periodic function $f\left(\tau +T_0\right)=f\left(\tau \right)$ with the particular case $T_0=\pi$. We introduce

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

(2)

where e is the elementary electric charge, m denotes the ion mass, $r^2=r_0^2+2z_0^2$ with $r_0$ and $z_0$ the trap semiaxes, while $U_0$ stands for the d.c. trapping voltage applied to the trap endcap electrodes. In addition, $a_x=-a_y$ and $q_x=-q_y$. Because we discuss a Paul (RF) trap $\tau =\Omega t/2$ is the dimensionless time [73,78], whilst $\Omega$ is the RF of the oscillating trapping voltage (denoted as $V_0$) supplied to the cylindrical electrodes in case of a 2D linear Paul trap, or to the ring electrode for a classical hyperbolic 3D geometry. Moreover, the function $f:\mathbb{R}\to \mathbb{R}$ is real and continuous for period $T_0>0$. We also analyse cases in which the function f is a Fourier series [72,74,94,95].

$$f\left(\tau \right)=A_0+\sum _{k>0}{A}_{k}cosk\tau +\sum _{k>0}{B}_{k}sink\tau ,$$

(3)

where $A_0$ is a known constant, whilst the $A_k$ and $B_k$ coefficients are considered as real

$$\sum _{k>0}\left(A_k^2+B_k^2\right)<\infty ,A_k,B_k\in \mathbb{R}.$$

(4)

The Floquet theory [77] explains the general solution (which is a complex function) can be cast as [73]

$$x_1=Q\left(\tau \right)\Phi \left(\tau \right),$$

(5)

where the function

$$Q\left(\tau \right)=e^{i\mu \tau },\mu \in \mathbb{C},$$

(6)

is associated with the slow (or secular) ion motion, while $\Phi \left(\tau \right)=\Phi \left(\tau +T_0\right)$ represents a complex, periodical and twice-differentiable function that characterizes the micromotion [72,96]. In the case of a MH equation ion dynamics is stable as long as the Floquet exponent is a complex number $\mu \in \mathbb{C}$ [97,98]. Hence, the solution is written as

$$x=e^{i\mu \tau }\Phi \left(\tau \right)+e^{-i\mu \tau }{\Phi }^{\ast }\left(\tau \right)\in \mathbb{R},$$

(7)

which depends on the coefficients $A_k$ and $B_k$, with $\left(k>0\right)$ and $A_0$. ${\Phi }^{\ast }\left(\tau \right)$ denotes the complex conjugate of the previously introduced $\Phi \left(\tau \right)$ function. One can also expand $\Phi \left(\tau \right)$ as a Fourier series

$$\Phi \left(\tau \right)=C_0+\sum _{k>0}{C}_{k}cosk\tau +\sum _{k>0}{D}_{k}sink\tau .$$

(8)

If

$$\sum _{k>0}\left(A_k^2+B_k^2\right)\ll A_0,$$

then Equation (1) can be expressed as the equation of a harmonic oscillator (HO) [40,41,42,99,100]

$$\ddot{x}+A_{0}x=0,$$

(9)

Further on, we illustrate two different methods to solve the MH equation:

• One chooses a solution expressed as $x=x_2+x_2^{\ast }$, where $x_2^{\ast }$ is the complex conjugate of $x_2$ so that

  $$x_2=e^{i\mu \tau }\left(C_0+C_{1}cos\tau +D_{1}sin\tau \right).$$

  (10)

  A double differentiation with respect to $\tau$ yields

  $${\dot{x}}_2=e^{i\mu \tau }\left[i\mu C_0+cos\tau \left(D_1+i\mu C_1\right)+sin\tau \left(i\mu D_1-C_1\right)\right],$$

  (11)
  $$\begin{array}{c}\hfill {\ddot{x}}_2=e^{i\mu \tau }\left\{-{\mu }^2{C}_0+cos\tau \left[2i\mu D_1-C_1\left(1+{\mu }^2\right)\right]-sin\tau \left[D_1\left(1+{\mu }^2\right)+2i\mu C_1\right]\right\}.\end{array}$$

  (12)

  We now revert to Equation (1) and introduce the expression $f\left(\tau \right)$ supplied by Equation (3)

  $${\ddot{x}}_2+\left(A_0+A_{1}cos\tau +B_{1}sin\tau \right)x_2=0.$$

  (13)

  Equation (13) is also verified for the complex conjugate $x_2^{\ast }$. Then, one uses Equations (11) and (12) along with Equation (13) to further derive

  $$\begin{array}{c}-{\mu }^2{C}_0+cos\tau \left[2i\mu D_1-C_1\left(1+{\mu }^2\right)\right]-sin\tau \left[D_1\left(1+{\mu }^2\right)+2i\mu C_1\right]\hfill \\ \hfill +\left(A_0+A_{1}cos\tau +B_{1}sin\tau \right)\left(C_0+C_{1}cos\tau +D_{1}sin\tau \right)=0.\end{array}$$

  (14)

  Equation (14) can also be cast as

  $$\begin{array}{c}-{\mu }^2{C}_0+A_0{C}_0+A_1{C}_1{cos}^2\tau +B_1{D}_1{sin}^2\tau +\left(A_1{D}_1+B_1{C}_1\right)sin\tau cos\tau \hfill \\ +cos\tau \left[A_0{C}_1+A_1{C}_0-C_1\left(1+{\mu }^2\right)+2i\mu D_1\right]\\ \hfill +sin\tau \left[A_0{D}_1+B_1{C}_0-2i\mu C_1-D_1\left(1+{\mu }^2\right)\right].\end{array}$$

  (15)

  One uses [101,102,103,104]

  $${sin}^2\tau ={\displaystyle \frac{1-cos2\tau }{2}},{cos}^2\tau ={\displaystyle \frac{1+cos2\tau }{2}},sin2\tau =2sin\tau cos\tau .$$

  (16)

  Finally, by performing the calculus in Equation (14) with $k>1$, one infers the following system of equations:

  $$\left\{\begin{array}{ccc}-{\mu }^2{C}_0+A_0{C}_0+{\displaystyle \frac{1}{2}}{A}_1{C}_1+{\displaystyle \frac{1}{2}}{B}_1{D}_1=0\hfill & \hfill & \\ B_1{C}_0+A_0{D}_1-D_1\left(1+{\mu }^2\right)-2i\mu C_1=0\hfill & \hfill & \\ A_1{C}_0+A_0{C}_1+2i\mu D_1-C_1\left(1+{\mu }^2\right)=0\hfill & \hfill \end{array}\right.$$

  (17)

  Using this system of equations one can determine the coefficients $C_0,C_1$ and $D_1$, respectively. Hence, one determines the solution $x_2$ from Equation (10) and implicitly the solution x in Equation (9). Hence, an ion confined within an electrodynamic trap can be treated as a HO and the method introduced above allows one to derive the MH equation that characterizes the associated dynamics.
• As demonstrated in [77] one chooses a solution of the form $x_2=e^{i\mu \tau }\Phi$. Therefore,

  $${\dot{x}}_2=e^{i\mu \tau }\left(i\mu \Phi +\dot{\Phi }\right)$$

  (18)

  $${\ddot{x}}_2=e^{i\mu \tau }\left(\ddot{\Phi }+2i\mu \dot{\Phi }-{\mu }^2\Phi \right)$$

  (19)

  We introduce the solution $x=x_2+x_2^{\ast }$ in the MH Equation (1) and derive

  $$\begin{array}{c}{e}^{i\mu \tau }\left(\ddot{\Phi }+2i\mu \dot{\Phi }-{\mu }^2\Phi \right)+e^{-i\mu \tau }\left(\ddot{\Phi }-2i\mu \dot{\Phi }-{\mu }^2\Phi \right)\hfill \\ +f\left(\tau \right)\left(e^{i\mu \tau }+e^{-i\mu \tau }\right)\Phi =0.\hfill \end{array}$$

  (20)

  By performing a Fourier series expansion [105] one can write

  $$\Phi =\sum _{k\in \mathbb{Z}}{c}_k{e}^{ik\tau },f=\sum _{k\in \mathbb{Z}}{a}_k{e}^{ik\tau },a_k^{\ast }=a_{-k},$$

  (21)

  where $a_k$ are arbitrary constants and $e^{i\mu \tau }=cos\mu \tau +isin\mu \tau$. Consequently, Equation (20) becomes

  $$\begin{array}{c}\sum _{k\in \mathbb{Z}}{c}_k\left[-k^2-2\mu k-{\mu }^2\right]e^{i\left(\mu +k\right)\tau }+\sum _{k\in \mathbb{Z}}\left(-k^2+2\mu k-{\mu }^2\right)e^{i\left(k-\mu \right)\tau }\hfill \\ +\sum _{p\in \mathbb{Z}}{a}_p\sum _{k\in \mathbb{Z}}{c}_k{e}^{i\left(k+\mu +p\right)}+\sum _{p\in \mathbb{Z}}{a}_p\sum _{k\in \mathbb{Z}}{c}_k{e}^{i\left(k+p-\mu \right)\tau }=0.\hfill \end{array}$$

  (22)

  It can be noticed that the equation above contains terms such as $e^{i\mu \tau }\left({\displaystyle \sum _{k+p=n}}{a}_p{c}_k\right)e^{in\tau }$. One denotes $b_n={\displaystyle \sum _{k+p=n}}{a}_p{c}_k={\displaystyle \sum _{p\in \mathbb{Z}}}{a}_p{c}_{n-p}$. As a result, Equation (22) can be recast as

  $$e^{i\mu \tau }E+e^{-i\mu \tau }{E}^{\ast }=0,$$

  (23)

  with

  $$\left\{\begin{array}{c}E={\displaystyle {\sum }_{k\in \mathbb{Z}}}\left[-c_k{\left(k+\mu \right)}^2+b_k\right]e^{ik\tau },\hfill \\ E^{\ast }={\displaystyle {\sum }_{k\in \mathbb{Z}}}\left[-c_k{\left(k+\mu \right)}^2+b_k\right]e^{-ik\tau }.\hfill \end{array}\right.$$

  (24)
  By using Equations (23) and (24) one obtains

  $$c_k{\left(k+\mu \right)}^2=b_k=\sum _{p\in \mathbb{Z}}{a}_{k-p}{c}_p,k\in \mathbb{Z},$$

  (25)

  which is amenable to

  $$\sum _{p\in \mathbb{Z}}\left[a_{k-p}-{\left(k+\mu \right)}^2{\delta }_{kp}\right]c_p=0,$$

  (26)

  where ${\delta }_{kp}$ stands for the Kronecker delta function. One further denotes

  $$A=\left[a_{k-p}-{\left(k+\mu \right)}^2{\delta }_{kp}\right].$$

  (27)
  Further on we introduce

  $$C=\left(\begin{array}{c}⋮\\ c_{-2}\\ c_{-1}\\ c_0\\ c_1\\ c_2\\ ⋮\end{array}\right).$$

  (28)
  Then, $A·C=0$ where C is the column matrix introduced above. One further infers

  $$detA=0,$$

  (29)

  which is exactly the MH equation that determines the Floquet exponent $\mu$ as a function of the $a_k$ coefficients (particularly as a function of the eigenvalue a and q parameter). It is assumed that $a_k=0$ for $\left|k\right|>2$ (Mathieu case) and $a_1=a_{-1}=0$. In such a case

  $$f\left(\tau \right)=a_0+a_2{e}^{2i\tau }+a_{-2}{e}^{-2i\tau }.$$

  (30)
  As explained above, one also supposes that $a_2=a_{-2}=q$ and $a_0=a$. Then

  $$f\left(\tau \right)=a+2qcos2\tau ,$$

  (31)

  which is exactly the classical MH equation. Based on Equation (25) one derives

  $$c_k{\left(k+\mu \right)}^2=a_0{c}_k+c_{k-2}{a}_2+a_{-2}{c}_{k+2}=a{c}_k+q\left(c_{k+2}+c_{k-2}\right),$$

  (32)

  with $q\ne 0$ and

  $$\left\{\begin{array}{c}{c}_{k+2}=\alpha c_k+\beta c_{k-2},\mathrm{if}k>0,\hfill \\ c_{k-2}=\gamma c_k+\zeta c_{k-2},\mathrm{if}k<0.\hfill \end{array}\right.$$

  (33)

  where $\alpha ,\beta ,\gamma$, and $\zeta$ are known coefficients. When $|c_k|\sim 0$ and $\left|k\right|>2$, the column matrix C [described by Equation (28)] can be recast as

  $$C=\left(\begin{array}{c}⋮\\ c_{-2}\\ 0\\ c_0\\ 0\\ c_2\\ ⋮\end{array}\right),$$

  (34)

  where it is assumed that $c_1=c_{-1}=0$ is an initial condition (hypothesis). In such case the function $\Phi$ [in Equation (21)] can be expressed as

  $$\Phi =c_0-2c_{2}cos2\tau ,$$

  (35)

  with $c_2=c_{-2}$. Then, for $k=0$ one finds

  $$c_0{\mu }^2=a{c}_0+2c_{2}q⟼{\mu }^2=a+2{\displaystyle \frac{c_2}{c_0}}q.$$

  (36)
  We consider the initial conditions

  $$\left\{\begin{array}{c}{c}_0=x_0,\hfill \\ 2{\displaystyle \frac{c_2}{c_0}}={\displaystyle \frac{q}{2}}.\hfill \end{array}\right.$$

  (37)
  Under these circumstances Equation (36) is recast as

  $${\mu }^2=a+{\displaystyle \frac{q^2}{2}}.$$

  (38)
  We return to Equation (1) and express f as

  $$f={\mu }^2+f_{res}.$$

  (39)
  Then, one uses Equation (38) and the residual interaction function $f_{res}$ is

  $$f_{res}=f-{\mu }^2,f_{res}=2qcos2\tau -{\displaystyle \frac{q^2}{2}},$$

  (40)

  which we discuss below.

Discussion

When $\left|k\right|>1$, then $a_k=0$. In case of the HO [6,8,41,72] $a_0>0$, and from Equation (32) one infers

$$c_k{\left(k+\mu \right)}^2=a_0{c}_k,$$

(41)

and we distinguish two distinct cases

(i) $k=0\Rightarrow a_0={\mu }^2$ with $c_0\ne 0$.

(ii) $c_k\ne 0\Rightarrow {\left(k+\mu \right)}^2={\mu }^2=a_0\Rightarrow k+\mu =\pm \mu$ and the two solutions of the equations are

$$\left\{\begin{array}{c}{k}_1=0,\hfill \\ k_2=-2\mu \notin \mathbb{Z},\hfill \end{array}\right.$$

(42)

where the latter case is discarded. Then, $\Phi =c_0$ and

$$x=e^{i\mu \tau }{c}_0+e^{-i\mu \tau }{c}_0^{\ast },$$

(43)

which stands for the equation of the HO for $\mu \in \mathbb{R}$, while $c_0^{\ast }$ stands for the complex conjugate of $c_0$. Hence, the MH equation turns into the equation of a classical HO

$$\ddot{x}+{\mu }^{2}x=0.$$

(44)

It is demonstrated that one can use the time-dependent variational principle (TDVP) [106,107,108] and then express the Hamilton equations of motion for the unidimensional case in order to derive the quantum equation of motion in the Husimi $\left(Q\right)$ representation, for a boson enclosed in a nonlinear ion trap [37,109]. This equation of motion is fully consistent with the one that portrays a perturbed classical oscillator. Hence, a phase-space formulation of quantum mechanics, e.g., the Husimi or Wigner representation, discloses the composition of the corresponding phase-space [36] and establishes a correlation between classical and quantum dynamics for this class of mesoscopic systems [110,111].

3. Stability Diagram of the MH Equation

The standard (canonical) form of the Mathieu equation for an ion confined within an electrodynamic trap, characterized by the parameters a and q, is [9,72,74,76,95,112,113]

$${\displaystyle \frac{d^{2}w}{d{\tau }^2}}+\left[a-2qcos2\tau \right]w=0.$$

(45)

We demonstrated in [72] that this is a Sturm–Liouville system [114] characterized by periodic boundary conditions, while the system exhibits solutions exclusively for particular values of the eigenvalue a, which depend upon q. It is also established that other solutions exist for every pair $(a,q)$, but we consider only periodic solutions [105].

In order to characterize ion dynamics, one of the possible approaches is to separate the ion micromotion from the secular motion [73]. The MH equation can be recast as (see Equation (1))

$$\ddot{x}+fx=0, \mathrm{where} \mathrm{one} \mathrm{introduces}f={\mu }^2+f_{res},$$

(46)

where $f_{res}$ describes the residual interaction that accounts for the ion micromotion. In addition,

$${\displaystyle \frac{1}{T_0}}{\int }_0^{T_0}{f}_{res}d\tau =0,T_0=\pi ,$$

(47)

so the micromotion term averages out during a period of the RF drive which leads us to the case of an autonomous Hamilton function for the trapped ion. By separating the micromotion Equation (46) changes accordingly

$$\ddot{x}+{\mu }^{2}x+f_{res}x=0,$$

(48)

which stands for the equation of the HO [8,21,36,37,100,115] of period $2\pi /\sqrt{\left|\mu \right|}$, with $\mu \in \mathbb{R},2\mu \notin \mathbb{Z}$. It can be observed that ion dynamics is decomposed into a slowly oscillating part (the secular motion) at frequency $\mu$ (see Equation (38)) and a fast parametric drive (the micromotion) $f_{res}$ at frequency $\Omega$. The secular ion motion (or the macromotion) is characterized by the equation

$$\ddot{x}+{\mu }^{2}x=0,$$

(49)

where $\mu$ is the Floquet exponent [74,77]. We now revert to Equation (25) and choose $k=0$. Consequently,

$${\mu }^2=a_0+{\displaystyle \frac{c_{-2}}{c_0}}{a}_2+{\displaystyle \frac{c_2}{c_0}}{a}_{-2}.$$

(50)

From Equation (46) one derives $f_{res}=f-{\mu }^2$, which we multiply by x. We consider the solution $x=x_2+x_2^{\ast }$ and use Equation (21). Hence,

$$f_{res}x=\left(\sum _k{a}_k{e}^{ik\tau }-{\mu }^2\right)\left(e^{i\mu \tau }\sum _p{c}_p{e}^{ip\tau }+e^{-i\mu \tau }\sum _p{c}_p{e}^{-ip\tau }\right).$$

(51)

One denotes

$$\varphi \left(\tau \right)=\sum _p{c}_p{e}^{ip\tau },{\varphi }^{\ast }\left(\tau \right)=\sum _p{c}_p{e}^{-ip\tau },$$

(52)

where ${\varphi }^{\ast }\left(\tau \right)$ stands for the complex conjugate of $\varphi \left(\tau \right)$. Because

$${\int }_0^{2\pi }{e}^{ik\tau }d\tau ={\displaystyle \frac{1}{ik}}{e}^{ik\tau }{|}_0^{2\pi }=0,\mathrm{when}k\ne 0,$$

(53)

$${\int }_0^{\pi }{e}^{ik\tau }d\tau ={\displaystyle \frac{1}{ik}}\left(e^{ik\pi }-1\right)=\left\{\begin{array}{cc}0,\hfill & k \mathrm{even}\hfill \\ 2i/k,\hfill & k \mathrm{odd}\hfill \end{array}\right.$$

(54)

We revert to Equation (47) and denote

$$\begin{array}{c}I=\underset{0}{\overset{\pi }{\int }}{f}_{res}xd\tau =\sum _{p,k}{a}_k{c}_p\left[{\displaystyle \frac{e^{i\left(k+p+\mu \right)\pi }-1}{i\left(\mu +k+p\right)}}-{\displaystyle \frac{e^{-i\left(k+p+\mu \right)\pi }-1}{i\left(\mu +k+p\right)}}\right]\hfill \\ -{\mu }^2\sum _p{c}_p{\displaystyle \frac{1}{i\left(\mu +p\right)}}\left[e^{i\left(\mu +p\right)\pi }-e^{-i\left(\mu +p\right)\pi }\right]\\ =\sum _{p,k}{a}_k{c}_p{\displaystyle \frac{2sin\left(\mu +k+p\right)\pi }{\mu +k+p}}-{\mu }^2\sum _p{\displaystyle \frac{2sin\left(\mu +p\right)\pi }{\mu +p}}.\hfill \end{array}$$

(55)

As demonstrated in [72], by choosing $a_0=a,a_2=a_{-2}=2q,\left|a_k\right|=0$ for $\left|k\right|>2$, one derives [73]

$$\begin{array}{c}I=\left\{\left(\sum _p{c}_p{\displaystyle \frac{2sin\left(\mu +p\right)\pi }{\mu +p}}\right)\left(a_0-{\mu }^2\right)+2q\sum _p{c}_p\left[{\displaystyle \frac{2sin\left(\mu +p\right)\pi }{\mu +2+p}}+{\displaystyle \frac{2sin\left(\mu +p\right)\pi }{\mu -2+p}}\right]\right\}\hfill \\ \hfill =2\sum _p{c}_{p}sin\left(\mu +p\right)\pi \left[{\displaystyle \frac{a_0}{\mu +p}}+{\displaystyle \frac{2q}{\mu +2+p}}+{\displaystyle \frac{2q}{\mu -2+p}}\right]=0.\end{array}$$

(56)

By assuming $c_1=c_{-1}=0;\left|c_k\right|=0$ for $\left|k\right|>2$ and $c_2=c_{-2}$ for values $p=-2,0,2$, and the micromotion term averages out (see [73])

$$\begin{array}{c}{c}_0\left[{\displaystyle \frac{a_0-{\mu }^2}{\mu }}+q\left({\displaystyle \frac{1}{\mu +2}}+{\displaystyle \frac{1}{\mu -2}}\right)\right]\hfill \\ +c_2\left[{\displaystyle \frac{a_0-{\mu }^2}{\mu }}+q\left({\displaystyle \frac{1}{\mu +4}}+{\displaystyle \frac{1}{\mu }}\right)\right]\hfill \\ +c_{-2}\left[{\displaystyle \frac{a_0-{\mu }^2}{\mu }}+q\left({\displaystyle \frac{1}{\mu -4}}+{\displaystyle \frac{1}{\mu }}\right)\right]=0,\hfill \end{array}$$

(57)

which is further amenable to

$${\displaystyle \frac{c_2}{c_0}}=-{\displaystyle \frac{2\left(a_0^2-\mu \right)+2q\left(2+\frac{\mu }{\mu +2}+\frac{\mu }{\mu -2}\right)}{a_0-{\mu }^2+2q\left(\frac{\mu }{\mu +2}+\frac{\mu }{\mu -2}\right)}}.$$

(58)

In cases when q is very small [72,73,95]

$$a_0={\mu }^2+{\displaystyle \frac{q^2}{2\left({\mu }^2-1\right)}}+{\displaystyle \frac{\left(5{\mu }^2+7\right)q^4}{32{\left({\mu }^2-1\right)}^3\left({\mu }^2-4\right)}}+\dots ,\mu \ne 1,2,3,$$

(59)

and the even solution can be expressed as

$${\displaystyle \frac{c_2}{c_0}}={\displaystyle \frac{q}{4\left(\mu +1\right)}}+{\displaystyle \frac{\left({\mu }^2+4\mu +7\right)q^3}{128{\left(\mu +1\right)}^3\left(\mu +2\right)\left(\mu -1\right)}}+\dots ,\mu \ne 1,2,$$

(60)

which completely determines the MH equation for an ion confined in a Paul trap, while it stands as the main result of the paper. In ref. [73] the values used are

$$a={\mu }^2-{\displaystyle \frac{1}{2}}{q}^2,{\displaystyle \frac{c_2}{c_0}}={\displaystyle \frac{q}{4}},$$

(61)

as the value of the Floquet exponent $\mu$ was chosen to be small, and implicitly the characteristic value (eigenvalue) a and parameter q in the MH equation as well. Both of them are real-valued, as previously emphasized in Section 1. Nevertheless, this hypothesis does not coincide with the values used in [73].

It was demonstrated in [72] that for ion trap operating points that lie within the first stability region of the MH equation and for small values of the eigenvalue a and q parameter, the coefficients $c_k$ rapidly converge to zero. Under such conditions higher harmonics become insignificant and the fundamental frequency prevails. It was also demonstrated that the solution of the MH equation is the sum between a shift that corresponds to the fundamental frequency and a correction due to the occurrence of higher harmonics associated with ion dynamics. The problem of evaluating the maximum amplitude of stable oscillations under given initial conditions was approached. The maximum is characterized by upper bounds determined by the relative position of the electrodes. For optimum trap operating parameters it is essential to consider all axes for which the equation of motion is of Mathieu type. A nonzero solution can be expressed as the combination of two linear dependent functions, which form a fundamental solution of the MH eq. Ion trajectories are investigated by building a family of ellipses, while the transfer matrix is supplied along with the maximum amplitude of stable oscillations [72].

The frontiers of the stability diagrams are defined by the equations [75,76,95,112]

$$a_0\left(q\right)=-{\displaystyle \frac{q^2}{2}}+{\displaystyle \frac{7q^4}{128}}-{\displaystyle \frac{29q^6}{2304}}+{\displaystyle \frac{68687q^8}{18874368}}+\dots ,$$

(62)

$$\begin{array}{c}{a}_1\left(-q\right)=b_1\left(q\right)=1-q-{\displaystyle \frac{q^2}{8}}+{\displaystyle \frac{q^3}{64}}-{\displaystyle \frac{q^4}{1536}}-{\displaystyle \frac{11q^5}{36864}}+{\displaystyle \frac{49q^6}{589824}}\hfill \\ -{\displaystyle \frac{55q^7}{9437184}}-{\displaystyle \frac{83q^8}{35389440}}+\dots ,\hfill \end{array}$$

(63)

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

(64)

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

(65)

When $a,\left|q\right|\ll 1$ and $r\ge 7$ (case when $a_r$ is approximately equal to $b_r$), the characteristic values of the frontiers of the stability region are described by the following power series approximation [8,76]

$$\begin{array}{c}\hfill a_r,b_r=r^2+{\displaystyle \frac{q^2}{2\left(r^2-1\right)}}+{\displaystyle \frac{\left(5r^2+7\right)q^4}{32{\left(r^2-1\right)}^3\left(r^2-4\right)}}+{\displaystyle \frac{\left(9r^4+58r^2+29\right)q^6}{64{\left(r^2-1\right)}^5\left(r^2-4\right)\left(r^2-9\right)}}+\dots .\end{array}$$

(66)

The analytical modelling performed in this paper is valid for all electrodynamic trap designs. It is essential to establish whether ion trajectories are stable or unstable depending on the experimental conditions, as well as whether the boundaries are stable and unstable regions associated with the ion dynamics. It is established that these limits correspond to combinations of cosine and sine elliptic series, as shown in [8,72,74,77,94,95]. Such value solutions of the MH equations are periodic but not bound, while they determine the point at which the ion trajectory becomes unbound.

4. Results

This paper suggests particular solutions of the classical MH equation with complex solutions for ions confined in electrodynamic ion traps (EITs) by employing the Floquet theory [74,77]. We analyse cases when the MH equation for a trapped ion can be cast as a Fourier series whose coefficients are determined by the analytical model introduced, which turns it into the equation of a HO. Thus, a system of linear equations is derived that enables one to determine the constants in the Fourier series solution.

The second method suggested delivers the solution of the MH equation for trapped ion systems. We derive a recursive relation that yields the Floquet coefficient as a function of the $a_k$ coefficients in the series expansions that we use. Ion dynamics within an electrodynamic (Paul) trap are shown to be consistent with the equation of motion for a classical HO. Finally, we apply these methods in Section 3 for an electrodynamic trap and we extend the model in [73]. Then, we derive the expression for $a_0$ and the coefficients of the even solution as a function of the Floquet coefficient for very low values of the q parameter. Hence, we completely determine the solution of the MH equation and the frontiers of the stability diagram.

The HO with damping is discussed in Appendix A, while the parametric HO is examined in Appendix B. It is demonstrated the solutions are linearly independent, while they also establish a fundamental system. We shortly review the Hill method to determine the Floquet coefficient [72,74,77,116,117] and supply the equation that defines the stability frontiers for the MH equation. A very short comment on EITs that operate under Standard Atmospheric Temperature and Pressure (SATP) conditions ends this paper.

The pseudopotential approximation can be safely used when $\Omega \gg 1$, which means the time scale of the secular motion is considerably larger than the time scale associated with the micromotion. In the case of the pseudopotential approximation for a Paul (RF) trap, the electric potential is described by a polynomial of rank 2. When considering the micromotion, stability regions result that are qualitatively similar to those of the Mathieu equation but with frontiers that are shifted depending on the anharmonicity parameter $\lambda \left(\beta \right)$, as shown in [72] and in the literature [118].

Applications of the methods used in this paper span MS, high-resolution spectroscopy, and trapping of ion crystals, where the latter is of extreme importance for optical atomic clocks, ultra-high-resolution spectroscopy, quantum metrology, and quantum information processing (QIP) with ultracold ions [42,46,60,119,120].