A flexible optimization method for scaling surface-electrode ion traps

Abstract

Recent advances in scaling surface-electrode ion trap have demonstrated optimization of critical components, such as junctions and loading slots. In this study, a flexible method to systematically optimize the shape of radio frequency (rf) rails in different components was proposed. The rf rails were discretized in the electrostatic calculation; thus, the spatial fields of different components can be accumulated according to the superposition principle. The optimization process was accomplished by placing artificial control points along the edges of the rf rails, providing controllable degree of freedom. The locations of these control points were modified using ant colony optimization, which employs a proposed hybrid multi-objective function. The proposed method was verified with three kinds of components: an X junction, a Y junction, and a loading slot. Compared with the results obtained using non-optimized cases and the existing methods, the proposed method produced favorable results in maintaining the ion height and minimizing the axial pseudopotential barrier and rf noise heating. The proposed method can also be used to optimize other scaling components of surface-electrode ion traps.

1 Introduction

Trapped ion system exhibits exquisite coherence properties and high-fidelity manipulation [1–4]. It is thought to be one of the most promising systems for quantum information processing. Many studies have focused on scaling ion trap architectures for trapping and shuttling large numbers of ions. Surface-electrode ion trap (SEIT) is an efficient way used in achieving trapped ion scalability [4–6] because it enables fabrication of complex trap arrays leveraging advanced semiconductor and microfabrication techniques. SEITs require critical components, such as junctions and loading slots (LS). Junctions [8–10] are used in two-dimensional array ion trap architectures [7] to shuttle ions between different functional zones. LS [9, 11, 12] are used to alleviate neutral atom flux contaminating on the trap surface during ion loading and deterministic loading of a particular number or sequence of ions.

Although SEITs are widely used, their trap depth is generally lower than that of standard quadrupole traps of a similar size scale [6, 10]. Moreover, the presence of junctions and LS results in radio frequency (rf) pseudopotential deformation. The minimum rf pseudopotential value near a junction naively assembled from the intersection of linear sections is generally not zero at every point along the center of the pseudopotential tube, but pseudopotential barriers are formed instead [8]. Junctions also affect ion height. Loading slots necessarily lie directly below the pseudopotential null, causing variations in trap strength over the slot and pseudopotential barriers at the slot edges [11]. Loading slots also locally affect ion height. In addition, rf noise heating is proportional to the slope of the pseudopotential and significant in places near the pseudopotential barriers [13]. To address these issues, the magnitude and slope of the equilibrium pseudopotential must be minimized, particularly in the junction region and the loading zone, while maintaining a specified ion equilibrium height range.

Optimization of junctions (X, Y) and LS have recently been demonstrated (Table 1) [8–11]. However, existing studies have focused on specific cases and generalized analysis is unavailable. In the current study, a flexible method to optimize the shape of rf rails in different components is proposed. First, in the electrostatic calculation, the rf rails are meshed into rectangular elements and the field of each element is calculated and stored in a file. Such discretization process can be used to solve the spatial field of a wide variety of components based on the superposition principle. Second, the optimization is accomplished by placing artificial control points along the edges of the rf rails, providing controllable degree of freedom. The performance of each determined granularity can be evaluated. Then, optimal number and placement of the control points are obtained. Unlike in the previous studies [8, 10, 11], the number of control points is fixed, as the VarNum in Table 1 shows. Third, the locations of the control points are modified by ant colony optimization (ACO), which employs a proposed hybrid multi-objective function that facilitates weighting coefficients selection. The multi-objective function includes figures of merit for the ion height and the pseudopotential values and derivatives [13] along the equilibrium trap axis. To the best of our knowledge, the numerical value difference of the three metrics is considerable; thus, the weighting coefficients are difficult to determine in the traditional sum up manner [9].

The remainder of the paper is organized as follows. Section 2 presents the proposed method. The implementation of the proposed method is demonstrated in Sect. 3. The simulation setup and results comparison are introduced in Sect. 4. Finally, Sect. 5 provides the conclusion.

Table 1 Summary of the optimization studies of X, Y junctions, and loading slots (LS)

2 The proposed flexible optimization method

Searching for a modulated edge of the rf electrode rail with the target of minimize the ion height variation and the axial pseudopotential barrier and the pseudopotential slope, is similar to finding a “shortest path” in the foraging behavior of ants. ACO [14] is employed as the heuristic algorithm in the proposed method. Reference [15] claims that their ACO can be used in continuous domain optimization and ensuring convergence. Thus, the current study adopts the continuous ACO described in [15] to optimize the critical components. At the beginning of optimization, artificial ants initially explore the variables in a random manner. Each ant constructs a solution. At the end of each solution construction, the corresponding probability of each solution is updated. The amount of the probability depends on the quality of the solution, which biases subsequent ants toward the most promising regions of the solution space. Then, optimal solution can be obtained after a number of iterations. In order to abstract the optimization problem, a general model is established as follows.

Definition

A model $Q = (T,C,f)$ of the optimization problem consists of the following:

• (a) A solution space $T$ defined over a finite set of continuous decision variables. A solution determines a trial layout;

• (b) A set $C$ of constraints among the variables;

• (c) An objective function $f:T\rightarrow \mathbf {R}^{+}$ to be minimized. $f$ can be either the sub-objective or the hybrid multi-objective function;

• (d) The model $Q$ defines a class of optimization problems for scaling the SEIT.

In the rest of this section, the X junction is used as an example to introduce the controllable variable degree of freedom and its constraints in Sect. 2.1, the hybrid multi-objective function and the figures of merit in Sect. 2.2, and the electrostatics solver in Sect. 2.3. The flexibility of the proposed method is then generalized in Sect. 2.4. For the convenience of depiction, the initial layout is the rf electrode shape to be optimized, whereas the trial layout, a sub-geometry of the initial layout, is the candidate shape temporarily generated by ACO.

2.1 Solution space and the domain

Figure 1a shows a naive X junction. Two principal layout changes by carefully observing and comparing the final optimized layouts in the previous studies [8–10]: (1) introducing long rf rail “fingers” that extend into the junction; (2) the edges of the rf rails are perturbed during the transition between the junction and the linear region to maintain slow changing of the field. Our initial layout was set according to the changes (Fig. 1b). The X junction possesses a high degree of symmetry; thus , only one-eighth of the initial layout is schematically described (Fig. 1c). The left edge must be optimized, which can be accomplished by placing artificial control points along the left edge of the rf rail, providing controllable degree of freedom.

The solution space $T$ is defined as follows. Given that the number of variables is determined arbitrarily, the edges to be optimized are meshed into line elements with equal lengths, thereby fixing one coordinate of the control points at which the line elements connect. The other coordinate serves as variables for the trial layout generation. The left edge in Fig. 1c is equally meshed into $N_V+1$ line elements, i.e., the ordinate of the $N_V$ control points $z_{i \in \{ 1, 2, \ldots , N_{V} \} }$ is fixed; their abscissa ($x_{i}$) then acts as a solution to generate trial layouts. Such artificial division enhances the flexibility of the proposed optimization method because the meshing granularity is optionally determined. The performance of each chosen granularity can then be evaluated and compared, and the optimal variable number is determined.

The constraint set $C$ is the domain of $x_i$ determined by three aspects: the actual area of the initial electrode, the microfabrication capability, and the variable meshing granularity. The analytical expression (1) was used to determine the domain.

$$\begin{aligned} \left\{ \begin{array}{l} \frac{H_1}{V_1} z_i \le x_i \le \delta \cdot {{\rm min}}\left( H_2,\frac{H_2}{V_2} z_i\right) \\ z_i = \frac{V_1}{N_V+1} \cdot i\\ \end{array} \right. \end{aligned}$$

(1)

where $0 < \delta \le 1$. The parameter $\delta$ denotes the capability of the actual microfabrication equipment, and $\frac{V_1}{N_V+1}$ is the variable meshing granularity.

Fig. 1

a Naive rf electrode layout of an X junction. The DC control electrodes are not shown. The original point is located in the middle. The coordinate convention is: $z$ axis is on the surface-electrode plane and is parallel to the rf electrode rails. $x$ is also on the electrode plane and is perpendicular to the $z$ direction. $y$ is the direction normal to the electrode surface, i.e., the $y$ direction is perpendicular to the $x$–$z$ plane. b The initial layout to be meshed. c Rectangle mesh schematic for the initial X junction. $L_{x}$ and $L_{z}$ are the partition granularities in $x$ and $z$ directions, respectively. The diagonal vertex coordinates of a discrete rectangle, ($x_1,z_1$) and ($x_2,z_2$), are used to evaluate its field at specific spatial locations, such as $x$–$y$ or $z$–$y$ plane. The $N_V$ black points on the dotted line show the variables. The double arrows show the Horizontal manner of variable selection. All of the dimensions are used in Sect. 4

2.2 Hybrid multi-objective function

Based on the existing studies [8–11], the formulation of the cost function was improved in the following two aspects: (1) Two quantitative operations: the average (the integral operation) and the maximum value, were introduced to determine the slow variation of a physical quantity. The proposed sub-objective functions are summarized in Eq. (2). $f_{1,3,5}$ are the integrations of the ion height variation (${{\rm intIHV}}$), the pseudopotential (${{\rm intPP}}$), and its absolute first derivative (${{\rm intAFD}}$) [13]; $f_{2,4,6}$ are their corresponding maximum values (${{\rm maxIHV}}, {{\rm maxPP}}, {{\rm maxAFD}}$). It is worthy to note that the rf noise heating mechanism is proportional to the absolute first directive of the pseudopotential [13]. However, it only reveals the heating rate at individual locations along the transporting axis. In order to depict the total ion heating rate during a whole transporting, ${{\rm intAFD}}$ is proposed as an optimization criterion. (2) The hybrid multi-objective function can be solved in two phases. $f_{1,\ldots ,6}$ are determined in phase one and the hybrid multi-objective function $F$ is determined in phase two. The hybrid multi-objective function is proposed because it is difficult to determine the weighting coefficients that are used to trade the six sub-objective functions, given the considerable numerical value differences of the sub-objective functions.

After optimizing the six sub-objective functions, six optimal values of ${\mathbf{Op}}(f_{\tau})$ can be obtained. The hybrid multi-objective function $F$ can then be formulated using Eq. (2), where $\lambda _{\tau}$ are the weighting coefficients. According to Eq. (2), the proposed $F$ inherits the characteristics of the traditional weighted multi-objective functions and the normalized operation. The normalization makes the quantity of the sub-objective values rank the same order; thus, the selection of the weighting coefficients is simplified. For example, in order to treat the ion height more heavily, uniform coefficients $\lambda _{1,\ldots ,6}=1$ can be easily changed to $\lambda _{1,2}=1$ and $\lambda _{3,\ldots ,6}=0.2$ (or reasonable small values).

$$\begin{aligned}& {{\rm intIHV}}:\quad f_{1}=\int _{0}^{l_{{\rm max}}}|(y|_{{\rm min}({\varPsi })})-H|dl \nonumber \\&\text{maxIHV}:\quad f_{2}=\text{max}(y|_{{\rm min}({\varPsi })})-{\rm min}(y|_{{\rm min}({\varPsi })}) \nonumber \\&{\rm intPP} :\quad f_{3}=\int_{0}^{l_{\rm max}}{\varPsi }(y|_{{\rm min}({\varPsi })},l)dl\nonumber \\&{\rm maxPP} :\quad f_{4}={\rm max}({\varPsi}(y|_{{\rm min}({\varPsi })},l\in [0,l_{\rm max}]))\\&{\rm intAFD}:\quad f_{5}=\int_{0}^{l_{\rm max}}|\frac{\partial {\varPsi}(y|_{{\rm min}({\varPsi })},l)}{\partial l}|dl \nonumber\\&{\rm maxAFD}:\quad f_{6}={\rm max}|\frac{\partial{\varPsi }(y|_{{\rm min}({\varPsi })},l\in[0,l_{\rm max}])}{\partial l}| \nonumber \\&F=\sum _{\tau=1}^{6}\lambda _{\tau}\frac{f_{\tau}}{{\mathbf{Op}}(f_{\tau})},\quad \tau =1,\ldots ,6 \nonumber \end{aligned}$$

(2)

where $l$ is the equilibrium trap axis, $H$ is the ion height of the ion trap linear sections, and ${\varPsi }$ is the pseudopotential (Eq. (5)).

2.3 Electrostatics solver

According to Eq. (2), all of the objectives are indirect functions of the pseudopotential, which is also indirect function of the electrostatic potential. Thus, the electrostatic potential of a wide variety of trial layouts must be calculated in advance [16–21]. To enhance universality, an electrostatics solver that involves three steps was proposed: (1) as shown in Fig. 1c, the initial layout is meshed into rectangle elements and the spatial field of each rectangle element can be calculated [18, 19]; then, all of the fields are stored in a file; (2) The electrostatic field of each trial layout can be accumulated through file reading because the superposition principle and the trial layout are sub-geometries of the initial layout; (3) The pseudopotential of each trial layout can be calculated from the accumulated electrostatic field, according to the pseudopotential approximation [22].

As shown in Fig. 1c, the spatial field of each meshed rectangle electrode can be obtained using the differential operations described in [19] or the integral operations used in [18]. The field in the $y$ and $z$ directions is shown in Eqs. (3) and (4), where $p_t = y^2 + (x_t - x)^2 + (z_2 - z)^2, q_t = y^2 + (x_t - x)^2 + (z_1 - z)^2,$ and $t \in \{1, 2\}$. The coordinate system is already introduced in the caption of Fig. 1.

$$\begin{aligned} E_{y}&= \frac{V}{2\pi } \cdot \left\{ \frac{(x_1 - x)(z_2 - z)(p_1 + y^2)}{\sqrt{p_1}[y^2p_1+(x_1 - x)^2(z_2 - z)^2]}\right. \nonumber \\&\left.\quad - \frac{(x_2 - x)(z_2 - z)(p_2 + y^2)}{\sqrt{p_2}[y^2p_2+(x_2 - x)^2(z_2 - z)^2]} \right.\nonumber \\&\left.\quad + \frac{(x_2 - x)(z_1 - z)(q_2 + y^2)}{\sqrt{q_2}[y^2q_2+(x_2 - x)^2(z_1 - z)^2]}\right. \nonumber \\&\left.\quad - \frac{(x_1 - x)(z_1 - z)(q_1 + y^2)}{\sqrt{q_1}[y^2q_1+(x_1 - x)^2(z_1 - z)^2]} \right\} \end{aligned}$$

(3)

$$\begin{aligned} E_{z}&= \frac{V}{2\pi } \cdot \left\{ \frac{(x_1 - x)y^3 + (x_1 - x)^3y}{\sqrt{p_1}[y^2p_1+(x_1 - x)^2(z_2 - z)^2]}\right. \nonumber \\&\left.\quad - \frac{(x_2 - x)y^3 + (x_2 - x)^3y}{\sqrt{p_2}[y^2p_2+(x_2 - x)^2(z_2 - z)^2]}\right. \nonumber \\&\left.\quad + \frac{(x_2 - x)y^3 + (x_2 - x)^3y}{\sqrt{q_2}[y^2q_2+(x_2 - x)^2(z_1 - z)^2]}\right. \nonumber \\&\left.\quad - \frac{(x_1 - x)y^3 + (x_1 - x)^3y}{\sqrt{q_1}[y^2q_1+(x_1 - x)^2(z_1 - z)^2]} \right\} \end{aligned}$$

(4)

Fig. 2

a Naive layout (gray) and initial geometry of the Y junction marked with different colors. b Meshing and Horizontal manner variable selection of one sixth of the initial layout by the symmetrical geometry. $Y_x$ and $Y_z$ are the partition granularity in the $x$ and $z$ directions, respectively. The $y$ direction is perpendicular to the $x$–$z$ plane. c Naive (gray) and initial layout of LS. The original point is located at the middle. d Meshing and Horizontal manner variable selection of one quarter of the slot. $S_x$ and $S_z$ are the granularities. All of the dimensions are used in Sect. 4

Fig. 3

Rotated manner variable selection of the initial layout of the X junction (Fig. 1c) and the Y junction (Fig. 2b). Positions of the $N_V$ variables (black dots) are selected along the double arrows which are rotated by $45^\circ$ with respect to the $x$ direction. The corresponding variable constraint set $C$ must be modified according to edges of the initial layout. $\frac{V_1}{N_V+1}$ is the variable meshing granularity

Fig. 4

Implementation of the proposed method. Left frame is the flowchart of the variable meshing and the proposed electrostatic solver. Right pseudocode shows the metaheuristic of the model $Q$, i.e., the ACO algorithm. The file is accessed in real time and the constraint set $C$ limits the variable modification in the metaheuristic. The lhsdesign is a MATLAB function. $f$ denotes the sub-objective functions or the hybrid objective function $F$. The sort is an ascending operation. The ${{\varvec{FIRST}}}_{k}$ intercepts the first $k$ solutions in a sorted sequence. The length of the sequence is $k+M$

In addition, $E_{x} = 0$ in $y-z$ plane, in which the ion shuttling axis is located, because the layout of X junction is symmetrical to the $y-z$ plane (Fig. 1). Based on the superposition principle, the total $x$-direction electric field in $y$–$z$ plane of each trial layout is always 0. Thus, the pseudopotential approximation can be simplified as Eq. (5), where $V_{\mathrm{rf}}\cos ({\varOmega } _{\mathrm{rf}} t)$ is applied to the rf electrode and all other electrodes are grounded, and $e$ and $m$ are the charge and mass of the trapped ion, respectively. Such conclusion is applicable in the $x$–$y$ plane, i.e., $E_z$ in the $x$–$y$ plane is also always 0, and the pseudopotential approximation is modified as ${\varPsi } (x,y,z=0)$.

$$\begin{aligned} {\varPsi }(0,y,z) = \frac{eV_{\mathrm{rf}}^{2}}{4m{\varOmega } _{\mathrm{rf}}^{2}} |\nabla \psi _{\mathrm{rf}}|^{2} = \frac{eV_{\mathrm{rf}}^{2}}{4m{\varOmega } _{\mathrm{rf}}^{2}}(E_{y}^2 + E_{z}^2) \end{aligned}$$

(5)

Although calculating the field to be stored at the beginning stage remains time consuming, this is get done once and for ever. And the computer memory is relatively cheap. The rapidity advantage of the proposed solver is more prominent along with the increase in iteration numbers and optimization times. Large iteration number and optimization time are necessary in solving heuristic problems [14, 15]. Moreover, the electrostatics solver enhances the flexibility of the proposed optimization method by introducing the discretization and the superposition principle.

2.4 Model extension

Based on the definition of $T, C$, and $f$, the model $Q$ is capable of defining a class of problems because electrostatic field calculation and artificial variable determination are no longer limited to a particular layout optimization. Such flexibility of the proposed method can be proven by the following four aspects:

• 1. The right edge of the electrode (the bold line shown in Fig. 1c) can also be meshed by Eq. (1); hence, the solution space is enlarged and better results are obtained. However, this action is not implemented in the simulation setup section because the results should be fairly compared with those in existing studies [8–10];

• 2. The proposed model $Q$ is also suitable for Y junction and LS optimization (Fig. 2). The constraint of variable cross talk must be added to the set $C$ when in dual-edge optimization, which is described in the slot optimization in Sect. 4.3;

• 3. The solution selection scheme shown in Figs. 1 and 2 is called Horizontal manner. However, given the symmetry of the critical components, a set of axes, which are rotated by $45^\circ$ with respect to $x$ direction, is also a reasonable scheme for variable selection. As shown in Fig. 3, the $z$ position of the control points is multiply defined by moving the dots along the double arrows. Such extension is called Rotated manner and is verified in Sect. 4.1;

• 4. $Q$ can be used to model both the sub-objective optimization and the multi-objective problems.

3 Method implementation

The main aspects of the proposed method, including the variable meshing, objective formulation, and electrostatics field calculation, are described in detail in Sect. 2 and implemented in this section. As shown in Fig. 4, the naive rf electrode layout of the critical components and the microfabrication capacity are the starting points. The initial layout and the discretization-based frames determine the solution space (or the trial layout generation mechanism), domain (or the constraint set C), and electrostatics solver. The electric field of each discrete rectangle electrode of the initial layout is stored in a file. The file is accessed in real time for calculating the pseudopotential (Eq. (5)).

Equations (3) and (4) are used to evaluate the field of a rectangle electrode at fixed potential and surrounded by a ground plane, i.e., they are suitable for the X and Y junction optimization. However, these equations no longer work when used to optimize the rf electrode layout near LS because of the existence of a hole in the middle ground electrode. Field calculation of loading slots using physics simulation software (e.g., Maxwell [16] and COMSOL [17]) consumes less time than that of junctions because of the relatively simple structure. COMSOL is employed in this study.

The ACO part is supported by the left frames in two steps (Fig. 4): (1) Constraint set $C$ limits the variable modification in the metaheuristic; (2) the objectives solving by the superposition principle. The solution set $T$ tracks the $k$ solutions. Each solution consists of $N_V$ variables. Thus, $T$ is a $k$-row $N_V$-column table with element $x_{l}^{i}$ determined by Eq. (1) or Fig. 3, where $l \in \{1, \ldots , k\}$ and $i \in \{1, \ldots , N_V\}$. The $k$ solutions are ordered in the table depending on the quality of the objective function $f_{\tau}$ or $F$. In this paper, the minimum value ranks the top in $T$. $\omega _{l}$ is the vector of weights, $\mu ^{i}$ is the vector of means, $\sigma _{l}^{i}$ is the standard deviation, and $M$ is the ant numbers [15].

4 Simulation setup and results

In this section, the simulation setup used in evaluating the performance of the proposed method is presented. The flexibility characteristic is verified in terms of three different typical critical components: X junction, Y junction, and LS. These components are independently optimized in [8–11].

The following rules are used to guide the results comparison:

• (a) Adopting $^{40}\hbox {Ca}^{+}$ as the trapped ion, which influences the constants in Eq. (5). Most of the results of the existing studies are obtained for $^{40}\hbox {Ca}^{+}$ [9–11]. To achieve fair comparison, $^{40}\hbox {Ca}^{+}$ was also adopted, and the results of $^{24}\hbox {Mg}^{+}$ in [8] are then scaled accordingly.

• (b) The sub-objective and hybrid multi-objective functions, which are shown in Eq. (2), are emphasized. Less attention is given to the characteristics of the ACO algorithm, such as its convergence rate. The algorithm parameters (Table 2) are used in the following optimization processes. These parameters are empirically selected based on a number of tests.

Table 2 Algorithm parameters [15] used by our optimization

4.1 X junction optimization

In this subsection, the quality of the X junction optimized in [10] is compared with the proposed method. In accordance with [10], the rf electrode rails are set to 40 $\upmu$m wide and are separated by 80 $\upmu$m. The applied voltage is $V_{\mathrm{rf}}cos({\varOmega } _{\mathrm{rf}} t)$, where $V_{\mathrm{rf}} = 91$ V and ${\varOmega } _{\mathrm{rf}} = 58.55$ MHz. The optimization domain was set to $80\times 200\,\upmu \hbox {m}^{2}$ and the meshing granularity of the initial layout $L_{x} = L_{z} = 1\,\upmu \hbox {m}$. Thus, $(H_1, V_1) = (40, 200\,\upmu \hbox {m}), (H_2, V_2) = (80, 80\,\upmu \hbox {m})$, as shown in Fig. 1c. In the following simulations, the variable number $N_{V} \in \{3, 7, 9, 19, 39\}$ with Horizontal manner is simulated. Only one-eighth of the X junction needs to be optimized based on the symmetry. The data file introduced in Fig. 4 contains the $E_{y}$ and $E_{z}$ data of $y$–$z$ plane in Fig. 1(c).

The X junction hybrid multi-objective function is solved in two phases. Phase one involves the optimization of the six sub-objective functions. As shown in Fig. 5, separately guided by the six metrics in Eq. (2), six sets of optimal values are obtained with respect to $N_{V}$. The ion height $H$ of the linear trap region is 60 $\upmu$m (Fig. 5a). As shown in Fig. 5d–f, the obtained results of the pseudopotential barrier and its maximum derivative are better than those obtained in [10] for sub-objective optimization even when $N_V$ is set to seven. Meanwhile, increasing the number of variables does not improve quality. In addition, the variable partition granularity is compromisingly determined to satisfy fabrication capacities and to decrease the chance of arcing caused by sharp points. The following compromise values are employed: $N_{V} = 19$, $\mathbf{Op }(\mathrm{intIHV}) = 347\,\upmu \hbox {m}^2, \mathbf{Op }({\rm maxIHV}) = 13\,\upmu \hbox {m}$, $\mathbf{Op }(\mathrm{intPP}) = 0.0129\hbox { eV}\cdot \upmu \hbox {m}, \mathbf{Op }(\mathrm{maxPP}) = 0.128\,\hbox {meV}$, $\mathbf{Op }(\mathrm{intAFD}) = 0.824\times 10^{-3}\hbox { eV}$, $\mathbf{Op }(\mathrm{maxAFD}) = 0.95\times 10^{-5}\hbox { eV}/\upmu \hbox {m}$.

Fig. 5

Results of sub-objective optimization with 100 iterations for X junction. The dash dot lines show the quality quoted from [10] and the blue lines show the results obtained in the present study. a Optimal value of the integration of ion height variation ($\mathbf{Op }(\mathrm{intIHV})$) with respect to $N_{V}$. The axis integrated ranks from the junction center outward to 500 $\upmu$m in the linear trap region. b Optimal value of the integration of pseudopotential. c Optimal value of integration of the pseudopotential derivative along the trap axis. d Optimal maximum ion height variation. e Optimized maximum pseudopotential. f Optimal maximum absolute first derivative of pseudopotential in (e)

Fig. 6

Results of the proposed hybrid multi-objective optimization for X junction. a Convergence process of the optimization based on Eq. (2) and the parameters in Table 2. b Pseudopotential barrier of the naive X junction along the trap axis from the junction center to 500 $\upmu$m. c Derivative of the pseudopotential barrier in (b). d The solid blue line is the optimal ion height after 100 iterations. The dash dot green line is the ion height of the naive X junction. The dotted red line shows the ion height in [10]. e The solid blue line shows the trapping pseudopotential of the final junction in the current study. The six-pointed star is the location of the maximum barrier in [10]. f The solid blue line shows the derivative of the pseudopotential in (e). The six-pointed star indicates the location of the maximum derivative of [10]

Fig. 7

Results of the proposed hybrid multi-objective optimization for Y junction. a The convergence process. b The maximum pseudopotential barrier amplitude of the naive Y junction (non-opt), [8, 9], and the proposed method. c The detailed pseudopotential of the proposed method from the junction center to 500 $\upmu$m further. d The ion height variation of the non-opt Y junction, [8, 9], and the proposed method. The data on the top of the bars are the maximum ion height and the bottoms are the minimum ion height. e The detailed ion height of our final optimized junction. f The absolute derivative of the pseudopotential barrier in (c)

Phase two is the hybrid multi-objective optimization, which also follows the procedure in Fig. 4. Figure 6a shows the convergence process of $F$ in Eq. (2) with $\lambda _{\tau} = 1$. The proposed hybrid multi-objective function produces an efficient convergence. Figure 6d–f shows the optimized results. In [10], the ion height variation is 28 $\upmu$m and the maximum pseudopotential barrier is 0.21 meV, whereas in the present study, the ion height variation is 15 $\upmu$m and the barrier is 0.157 meV. However, the maximum derivative of our pseudopotential along the trap axis is $1.62\times 10^{-5}\hbox { eV}/\upmu \hbox {m}$ (Fig. 6f), which is larger than that of [10], but not coincident with that in Fig. 5(f). This phenomenon is attributed to the remarkably large solution search space of hybrid multi-objective optimization. The compromise solution does not provide optimal values for every single metrics. The layout of the final optimized X junction is shown in Fig. 8a.

Until now, the variable selection with Horizontal manner (Fig. 1c) is evaluated. In the following, two more extended situations are evaluated. (1) Rotated manner (Fig. 3) is utilized to guide the variable selection and the other parameters are unaltered. The optimized ion height, pseudopotential, and its slope are consistent with the trends shown in Fig. 6d–f. The specific values are summarized in the fifth column of Table 3. The layout of the X junction optimized with the Rotated manner is shown in Fig. 9a. (2) Larger $V_1$ in Fig. 1c means larger optimization area; thus, the optimization can be made beyond the edge shown in Fig. 1c with larger $V_1$. We predict that $z$ range of the initial layout is determined by the scope of the axis pseudopotential deformation caused by the naive X junction (Fig.1a). According to Fig. 6b, c, the axis pseudopotential deformation occurs primarily in the 200 $\upmu$m range. In order to determine whether the former initial area is acceptable or not, larger area of $80\times 300\,\upmu \hbox {m}^{2}$ is also evaluated (i.e., $V_1 = 300\,\upmu \hbox {m}$). Results are summarized in the sixth column of Table 3, which are only slightly different than the results obtained with $V_1 = 200\,\upmu \hbox {m}$ (the 4th and 5th columns of Table 3). The final optimized layout is shown in Fig. 9b.

Table 3 Summary of X junction optimization results

According to Table 3, the optimizations with Horizontal manner, Rotated manner, and larger optimization area possess analogous results. So, we conclude that the Horizontal manner and Rotated manner can be arbitrarily chosen. In addition, the scope of the axis pseudopotential deformation is an efficient indication of determining the optimization area. Compared with the method in [10], the proposed method in the present study produces better quality of all of the sub-objective functions. For multi-objective function, the proposed method also produces reasonable results. Compared with the naive non-optimized X junction, the optimized pseudopotential barrier is 63.7x smaller and the derivative is 18.75x smaller. In addition, each optimization can be achieved in about 3.35 hours running on a desktop with Intel Core i7 CPU and 32 GB RAM.

4.2 Y junction optimization

In this subsection, the quality of the Y junction optimized in [8, 9] is compared with the proposed method. The results of [8] were scaled by modifying the parameters $V_{\mathrm{rf}}, {\varOmega } _{\mathrm{rf}}$ and the mass of the trapped ion $m$ according to Eq. (5). The parameters are set in accordance with those in [9]. The rf electrode rails are 42.5 $\upmu$m wide, separated by 100.4 $\upmu$m, and applied with voltage $V_{\mathrm{rf}}cos({\varOmega } _{\mathrm{rf}} t)$, where $V_{\mathrm{rf}} = 120$ V (the maximum junction shuttling voltage used in [9]) and ${\varOmega }_{\mathrm{rf}} = 43$ MHz. The optimization domain is set to $92.7\times 200\,\upmu \hbox {m}^{2},$ and the partition granularity of the initial layout is set to $Y_{x} = Y_{z} = 1\,\upmu \hbox {m}$. As shown in Fig. 2b, $(H_1, V_1) = (50.2\,\upmu \hbox {m}, 200\,\upmu \hbox {m})$, $(H_2, V_2) = (92.7\,\upmu \hbox {m}, \frac{92.7}{\sqrt{3}}\,\upmu \hbox {m})$. Similar to the X junction optimization, only one-sixth of the Y junction is emphasized based on the symmetry.

In phase one, i.e., the sub-objective optimization, the quality for Horizontal manner was also evaluated with respect to $N_{V} \in \{3, 7, 9, 19, 39\}$. Similarly, the difference of the quality with respect to 9, 19, and 39 is small, consistent with the trends shown in Fig. 5. $N_{V} = 19$ was also compromisingly selected for hybrid multi-objective optimization. $\mathbf{Op }({\mathrm{intIHV}}) = 2,847\,\upmu \hbox {m}^2, \mathbf{Op }({\mathrm{maxIHV}}) = 9\,\upmu \hbox {m}, \mathbf{Op }({\mathrm{intPP}}) = 0.0435\hbox { eV}\cdot \upmu \hbox {m}$, $\mathbf{Op }({\mathrm{maxPP}}) = 0.44\,\hbox {meV}, \mathbf{Op }(\mathrm{intAFD}) = 3.015\times 10^{-2}\,\hbox { eV}$, $\mathbf{Op }({\mathrm{maxAFD}}) = 0.298\times 10^{-3}\,\hbox {eV}/\upmu \hbox {m}$ can then be obtained.

The results in Fig. 7 are obtained by employing $\lambda _{\tau} = 1$. As shown in Fig. 7b–c, the maximum pseudopotential barrier magnitude optimized by the proposed method is 9.7, 26.2, and 100.3 % than that of the non-optimized case, [9] and [8], respectively. As shown in Fig. 7d–e, the ion height variation of the proposed method is 8.1 $\upmu$m, which is a bit larger than that of [9] and is much smaller than that of [8] and the non-optimized junction. The absolute derivative of the pseudopotential barrier along the trap axis of the compared studies is unavailable, whereas the magnitude shown in Fig. 7f is the same with that in Fig. 6f.

Fig. 8

Horizontal manner optimization. a Optimized rf electrode layout of the X junction. b Optimized layout of the Y junction. Both images are shown in actual dimensions. The 19 dots denote the variables in Figs. 1c and F2b

Fig. 9

a Optimized rf electrode layout of the X junction with Rotated manner. b X junction Horizontal manner optimization with larger initial area

For the Y junction optimization in the proposed method, the range of the ion height variation, as well as the pseudopotential barrier and its derivative, is also significantly reduced. The specific optimized rf electrode layout of the Y junction is shown in Fig. 8b.

4.3 Loading slot optimization

In this subsection, the quality of the LS optimized in [9, 11] is compared with the proposed method. According to [9], the rf electrode rails are set to 42.5 $\upmu$m wide and are separated by 100.4 $\upmu$m and $V_{\mathrm{rf}} = 120$ V, ${\varOmega } _{\mathrm{rf}} = 43$ MHz. The dimension of the slot is $86\times 70\,\upmu \hbox {m}^2$. As shown in Fig. 2d, the partition granularity of the initial electrode layout $S_{x} = S_{z} = 1\,\upmu \hbox {m}$. The area of the blue shadow layout is $52.5\times 100\,\upmu \hbox {m}^{2}$. The range of each variables is 15 $\upmu$m, which can avoid the variable cross talk problem. After the sub-objective simulation, we only set $N_{V}$ to 10 because the electric field above the slot is less complex than that of the junctions, and less variables are needed. In addition, only the sub-objective functions $f_2, f_4,$ and $f_6$ are considered. $\mathbf{Op }(\mathrm{maxIHV}) = 3\,\upmu \hbox {m}$, $\mathbf{Op }(\mathrm{maxPP}) = 0.112$ meV, and $\mathbf{Op }(\mathrm{maxAFD}) = 28.56\,\hbox {eV}/\upmu \hbox {m}$ are then obtained.

Figure 10 shows the hybrid multi-objective optimization with $\lambda _{2,4,6} = 1$. As shown in Fig. 10b–c, the proposed pseudopotential barrier is much smaller than that of the non-optimized slot, and it is only $10\,\%$ of that in [9]. According to Fig. 10e–f, the proposed optimized ion height variation is also much smaller than that of the non-optimized slot and is comparable to that in [9] and [11]. The optimized layout and the variable locations are shown in Fig. 10d.

Fig. 10

Results of the LS optimization. a Convergence process of multi-objective optimization. b Amplitude of the pseudopotential barrier along the trap axis of the naive (non-opt) LS, [9] and the proposed slot. c Detailed optimized pseudopotential barrier of the proposed slot. d Optimized layout of the rf electrode layout (red) of the rectangle LS. The dots show the location of the variables after 100 iterations. e Ion height variation of the non-opt slot, [9, 11], and the proposed method. The data on the top of the bars are the maximum ion height, whereas those on the bottoms are the minimum ion height. f Detailed optimized ion height of the proposed method

5 Conclusion

The junction and LS optimization is necessary in scaling SEITs. A flexible uniform method was proposed to optimize different components. In particular, more sub-objective functions were adopted and a hybrid multi-objective function that reduces the difficulties in determining the weighting coefficients was proposed. The discretization used in selecting variable degree of freedom and calculating electric field can significantly enhance the flexibility of the proposed method. In addition, an X junction, a Y junction, and a LS were employed as testbench to evaluate the flexibility of the proposed method; favorable results were obtained. The proposed method can also be used to optimize other SEIT scaling components.