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Guidelines for Designing Surface Ion Traps Using the Boundary Element Method Seokjun Hong 1 , Minjae Lee 1 , Hongjin Cheon 1 , Taehyun Kim 2, * and Dong-il “Dan” Cho 1, * 1 2

*

ISRC/ASRI, Department of Electrical and Computer Engineering, Seoul National University, Seoul 151-744, Korea; [email protected] (S.H.); [email protected] (M.L.); [email protected] (H.C.) Quantum Tech. Lab., SK Telecom, Seongnam-si, Gyeonggi-do 463-784, Korea Correspondence: [email protected] (T.K.); [email protected] (D.D.C.); Tel.: +82-31-710-5347 (T.K.); +82-2-880-6488 (D.D.C.)

Academic Editor: Vittorio M. N. Passaro Received: 24 February 2016; Accepted: 22 April 2016; Published: 28 April 2016

Abstract: Ion traps can provide both physical implementation of quantum information processing and direct observation of quantum systems. Recently, surface ion traps have been developed using microfabrication technologies and are considered to be a promising platform for scalable quantum devices. This paper presents detailed guidelines for designing the electrodes of surface ion traps. First, we define and explain the key specifications including trap depth, q-parameter, secular frequency, and ion height. Then, we present a numerical-simulation-based design procedure, which involves determining the basic assumptions, determining the shape and size of the chip, designing the dimensions of the radio frequency (RF) electrode, and analyzing the direct current (DC) control voltages. As an example of this design procedure, we present a case study with tutorial-like explanations. The proposed design procedure can provide a practical guideline for designing the electrodes of surface ion traps. Keywords: surface ion trap; electrode dimensions; design optimization; boundary element method; microfabrication

1. Introduction An ion trap is a device to trap charged particles in space using an electric or electromagnetic field. In particular, if atomic ions are trapped in an ultra-high vacuum environment, it can provide ideal isolation from the surroundings and allows individual manipulations of the trapped ions. These manipulations can be achieved using either a focused laser with beam-steering capability [1] or an optimized design of the microwave near field at trapped-ion location [2]. Because of these features, the ion trap method is considered a promising candidate for the physical implementation of quantum information processing [3] and a precise measurement system for fundamental physics applications such as the optical clock and mass spectroscopy [4–6]. Earlier ion traps were constructed using conventional machining and manual assembly. Following the proposal for a multi-zone ion trap with a scalable architecture for complex quantum information processing [7], microfabrication technologies have become methods of choice for realizing various ion traps [8]. In microfabrication technologies, the electrodes of conventional ion traps, which have a three-dimensional quadrupole configuration, are projected onto a plane. This allows for a lithography-based fabrication method that can significantly reduce alignment errors during manual assembly and provide scalability [9]. Microfabricated ion traps whose electrodes are laid on a plane are known as “surface ion traps” or “planar ion traps” (Figure 1). Currently, surface ion traps are widely used as platforms for quantum information processing [10–12]. In addition, the surface ion

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traps have potentiality in the realization of portable optical clocks, because the ion-trap system can be surface ion by traps have potentiality the realization miniaturized adopting the surfacein ion traps [13]. of portable optical clocks, because the ion-trap system can be miniaturized by adopting the surface ionion traps [13].several previous studies present With respect to the electrode design for surface traps, With respect to theHowever, electrodethese design for surface ionsimply traps,applied severalinprevious analytical models [14–18]. models cannot be practicalstudies designspresent due to models [14–18]. However, cannot bebeen simply applied inon practical designs due theanalytical complex structures of surface ion-trapthese chips.models There also have many studies designing electrode to the complex structures of surface ion-trap chips. There also have been many studies on designing dimensions using numerical simulations. Pearson et al. extracted specifications including trap depth electrode dimensions using boundary numericalelement simulations. Pearson al. extracted specifications including and secular frequency through method (BEM)etsimulations [19]. Brownnutt et al. used trap depth and secular frequency through boundary element method (BEM) simulations a simulation method that uses the sum of the electrical potentials generated by individual electrodes [19]. [20]. Brownnutt al. used a simulation method that uses the sum of the electrical potentials by Allcock et al. etused BEM simulations to calculate direct current (DC) voltage sets for generated shaping DC individual Allcock et presented al. used BEM simulations direct current (DC) potentials [21].electrodes Nizamani[20]. and Hensinger a geometric factorto in calculate the electrode dimensions that voltage sets for shaping DC potentials [21]. Nizamani and Hensinger presented a geometric factor in provides the maximum trap depth when the ion height is determined [22]. Siverns et al. presented the the electrode dimensions that provides the maximum trap depth when the ion height is determined relation between ion height and secular frequency in numerically simulated five-rail geometries [23]. [22].etSiverns et al. presented the relation between height and secular frequency numerically Doret al. proposed an iterative BEM method forion designing radio frequency (RF)inelectrodes of simulated five-rail geometries [23]. Doret et al. proposed an iterative BEM method for designing a complex shape that compensates for the distortion of the RF pseudopotential near the backside radio slot frequency (RF) electrodes a complex shape that compensates for RF theelectrodes distortionofofjunction the RF loading [24]. This iterative BEMofmethod can also be applied to design the pseudopotential near the backside loading slot [24]. This iterative BEM method can also be applied ion traps to decrease the amplitude of the pseudopotential barrier near the junction center [25–27]. to design the RF electrodes of junction ion traps to decrease the amplitude of the pseudopotential In addition, a few Ph.D. dissertations have proposed more detailed electrode design methods for barrier near the junction center [25–27]. In addition, a few Ph.D. dissertations have proposed more surface ion traps [28–31]. detailed electrode design methods for surface ion traps [28–31]. Table 1 summarizes the electrode dimensions and trap specifications of surface ion traps from Table 1 summarizes the electrode dimensions and trap specifications of surface ion traps from recent publications [12,21,24,25,27,32–34]. As shown in the table, most papers addressing electrode recent publications [12,21,24,25,27,32–34]. As shown in the table, most papers addressing electrode design for surface ion traps do not provide any characteristics of surface ion traps. Although several design for surface ion traps do not provide any characteristics of surface ion traps. Although several research groups have used numerical simulations to design electrodes, they have provided only partial research groups have used numerical simulations to design electrodes, they have provided only aspects of these simulations [35], and as yet, no comprehensive design procedure is available. partial aspects of these simulations [35], and as yet, no comprehensive design procedure is available. In this paper, we present a detailed design procedure for surface ion-trap chips. In Section 2, In this paper, we present a detailed design procedure for surface ion-trap chips. In Section 2, we we provide the background of surface ion traps and the specification requirements. In Section 3, we provide the background of surface ion traps and the specification requirements. In Section 3, we describe our step-by-step design methodology for surface ion-trap chips using numerical simulations. describe our step-by-step design methodology for surface ion-trap chips using numerical In simulations. Section 4, weInpresent a case study for our design methodology. We believe that the results of this Section 4, we present a case study for our design methodology. We believe that the study can provide an effective guideline for designing ion traps. results of this study can provide an effective guidelinesurface for designing surface ion traps.

DC electrodes Trapping zone

RF electrode

Figure 1. Scanning electron micrograph image of a surface ion trap fabricated by our group [11]. (The Figure 1. Scanning electron micrograph image of a surface ion trap fabricated by our group [11]. design layout of the ion trap uses the trap chip of Sandia National Laboratory [12] as the benchmark.) (The design layout of the ion trap uses the trap chip of Sandia National Laboratory [12] as the benchmark.)

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Table 1. Electrode dimensions and specifications of surface ion traps.

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Radial Secular Ion VRF ΩRF/2π Ion Trap Depth b q-Parameter Frequency Height (µm) (V) (MHz) Species (meV) of surface ion traps. Table 1. Electrode dimensions and specifications (MHz) (µm) 40Ca+ [12] 77 137 50–140 33 0.25–0.34 * 3–4 84 Ion 150 Radial Secular [21] 100 247 223 25.8 Ca+ 188 0.43 4.02 Ω /2π Ion Trap Depth Height63 q-Parameter Frequency Ref[24] No. a asymmetric (µm) b (µm) 100 VRF (V) 60 RF 40Ca+ 0.16–0.19 * 3.5–4 (MHz) Species (meV) (µm) (MHz) 24 + Mg 0.25 * 8 40 [25] asymmetric 51 90.7 40 Ca+ [12] 77 137 50–140 33 0.25–0.34 * 3–4 84 60 40 + Ca 0.05–0.12 * [27] 44 84 91 58.55 1–2.5 [21] 100 247 223 25.8 Ca+ 188 0.43 4.02 150 88Sr+ 25 0.15 *, 0.12 * 2.1, 1.7 79 [32] 75 95 155 40.6 40 Ca+ [24] asymmetric 100 60 0.16–0.19 * 3.5–4 63 43 + Ca + 59 0.3 4 75 [33] 45 136 72 38.7 24 [25] asymmetric 51 90.7 0.25 * 8 40 Mg 40Ca+ 75 - * [34] asymmetric 140 91 20.6 40 Ca+ [27] 44 84 58.55 0.05–0.12 1–2.560 230 88  . * These numbers were [32] 75 95 155 40.6 25 using 0.15 2.1, 1.7 79 Sr+ estimated q  *,2 0.12 2 *s 43 [33] 45 136 72 38.7 59 0.3  RF 4 75 Ca+ 40 Ca+ [34] asymmetric 140 20.6 75 230 ? * These numbers were estimated using q « 2 2 Ωωs . 2. Background Ref No.

a (µm)

RF

2. Background 2.1. Principle of Surface Ion Traps

Figure 2 shows schematic of a surface ion trap in a symmetric five-rail geometry [17]. By 2.1. Principle of Surface IonaTraps applying RF voltages to the two RF electrodes and maintaining the center and outer electrodes at RF Figure 2 shows a schematic of a surface ion trap in a symmetric five-rail geometry [17]. By applying ground, electric pseudopotentials are formed, which confine the trapped ions in radial directions. RF voltages to the two RF electrodes and maintaining the center and outer electrodes at RF ground, The outer electrodes are segmented to control the longitudinal electric potentials, which either electric pseudopotentials are formed, which confine the trapped ions in radial directions. The outer confine the trapped ions or shuttle them along the axial direction. The center electrode can be electrodes are segmented to control the longitudinal electric potentials, which either confine the trapped divided into two separate electrodes. The separated center electrodes allow for tilting of the ionsprincipal or shuttleaxes, themwhich, along in theturn, axialallows direction. The center electrode can be divided into two separate Doppler cooling of the trapped ions with a single laser [21]. electrodes. The separated center electrodes allow for tilting of the principal axes, which, in Note that the axis tilting can be achieved by applying asymmetric voltage on the turn, centerallows or outer Doppler cooling of the trapped ions with a single laser [21]. Note that the axis tilting can be achieved electrodes. by applying asymmetric voltage on the center or outer electrodes.

y z

DC RF DC

x

b a

lDC

RF DC

Figure 2. Schematic of a surface ion trap in a symmetric five-rail geometry. The red and blue rectangles Figure 2. Schematic of a surface ion trap in a symmetric five-rail geometry. The red and blue indicate the RF and DC electrodes, respectively. The curved arrows denote the direction of the electric rectangles indicate the RF and DC electrodes, respectively. The curved arrows denote the direction of field when the RF voltage is positive. The dashed black lines indicate the principal axes of ion motions, the electric field when the RF voltage is positive. The dashed black lines indicate the principal axes of which are tilted, for Doppler cooling with a single laser. ion motions, which are tilted, for Doppler cooling with a single laser.

2.2. Specifications of Surface Ion Traps 2.2. Specifications of Surface Ion Traps 2.2.1. Trap Depth 2.2.1. Trap Depth Trap depth is the difference in the total potential (sum of electric pseudopotential and DC potential) Trap depth is the difference in the total potential (sum of electric pseudopotential and DC between the RF null and the saddle point, which is a stationary point but not a local extremum potential) between the RF null and the saddle point, which is a stationary point but not a local (Figure 3). Deeper trap depths suppress the loss of ions induced by background gas collisions and extremum (Figure 3). Deeper trap depths suppress the loss of ions induced by background gas increase ion lifetime. Surface ion traps have an inherently shallower trap depth that is barely above collisions and increase ion lifetime. Surface ion traps have an inherently shallower trap depth that is

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barely above the energy mean kinetic energy of the evaporated neutral atoms [17]. Therefore,ofmaximization the mean kinetic of the evaporated neutral atoms [17]. Therefore, maximization trap depth is of trap depth is frequently considered a surface ion trap. frequently considered as a design goal as foraadesign surfacegoal ion for trap.

Saddle point

RF null point

Figure 3. Contour plot of total potential generated by a surface ion trap indicating the RF null point Figure 3. Contour plot figure of totalwas potential generated by a surface trap indicating the RF null point and saddle point. This obtained by boundary elemention method (BEM) simulations. and saddle point. This figure was obtained by boundary element method (BEM) simulations.

2.2.2. q-Parameter 2.2.2. q-Parameter The classical radial motions of ions trapped in hyperbolic electrodes can be described by the The classical motions of ions trapped in hyperbolic electrodes can be described by the standard Mathieu radial differential equation [36]: standard Mathieu differential equation [36]: d2 i

d 2 i2 ` pai ´ 2qi cos2τq i “ 0, i “ x, y dτ   ai  2 qi cos 2  i  0, i  x , y d 2

(1) (1)

where τ, ax , ay , qx , and qy are given below: where τ, ax, ay, qx, and qy are given below: Ωt 4eU 2eV τ “ t, a x “ ´ay “ 4eU , q “ ´q “ 2eV   2 , ax  ay  mr0 22Ω22 , qx x  qyy  mr0 22Ω22

2

mr0 

mr0 

(2) (2)

and Ω, t, e, m, r0 , U, and V indicate the RF voltage angular frequency, time, elementary charge, ion and Ω,ion–electrode t, e, m, r0, U, distance, and V indicate the RF voltage elementary charge, ion mass, offset voltage, and RFangular voltage frequency, amplitude, time, respectively. In addition, the mass, ion–electrode distance, offset voltage, and RF voltage amplitude, respectively. In addition, q-parameter is related to the secular frequencies and amplitudes of ion micromotion. The motionthe of q-parameter related to is the secular frequencies and amplitudes of ion micromotion. The motion of the ion in theisx-direction described as follows [36]: the ion in the x-direction is described´ as follows ¯ “ [36]: ‰ q β Ω Ω x “ x0 cos β x 2 t 1 ´ 2x cos pΩtq , ωs, x “ x2 q   x   b   x (3) x  x0 cos   x qx 2 t  1  ? cos ωs, x t   , s, x = a β x « ax ` 22 , qx « 2 2 2 Ω pwhen “ 0q x 2  (3) 2 s,secular qx amplitude of where x0 and ω s,x indicate the oscillation motion along the x-direction and the x ax  0   ax  respectively. , qx  2 2The ionwhen secular frequency along thex x-direction,  motion in the y-direction can be described 2 similarly. Moreover, these formulae are approximations. The ion motion comprises a secular motion where x0 andβωΩ/2 s,x indicate the oscillation amplitude of secular motion along the x-direction and the at frequency ! Ω and a micromotion at frequency Ω. Generally, a higher secular frequency x secular frequency x-direction, respectively. The ion motionq-parameter in the y-direction can be is preferred for thealong reasonthe described in Section 2.2.3, while a smaller is preferred to described similarly. Moreover, these formulae are approximations. The ion motion comprises minimize micromotion amplitude. However, they cannot be achieved simultaneously as evidenta secular motion(3), at unless frequency βxΩ/2 ≪ Ω and a ismicromotion at frequency Ω. Generally, a higher from Equation RF angular frequency increased. However, RF angular frequency is also secular frequency is preferred for the reason described in Section 2.2.3, while a smaller q-parameter limited by the maximum allowed RF voltage and the minimum desired trap depth as discussed later. is preferred to minimize micromotion amplitude. However, they cannot achieved Typically, the stable region usually used for hyperbolic electrode traps includes a and q, be where q < 0.7 simultaneously as evident from Equation (3), unless RF angular frequency is increased. However, 2 and |a/q | < 0.5 [16]. However, the stable region of a and q for a surface ion trap varies with the RF angular frequency is also limited by the maximum allowed RF voltage and the minimum desired trap depth as discussed later. Typically, the stable region usually used for hyperbolic





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tilted angle of the principal axes, which is dependent on the geometries of the DC electrodes and the arrangement of the DC voltage [37]. Because of the calculation complexity of the analytic derivation, an appropriate q-parameter value between 0.1 and 0.3 is generally considered as a starting point for design, and an optimal q-parameter can be determined as the experiment proceeds within stable ranges. This q-parameter range is consistent with other examples listed in Table 1. 2.2.3. Secular Frequency Secular frequency can be derived from the Mathieu parameters, as shown in Equation (3), and can be expressed as follows: e d2 φ ωs, i “ , i “ x, y, z (4) m di2 where φ is the total electric potential, which is the sum of the electric pseudopotential and static potentials. A higher secular frequency is preferred because it allows for tighter confinement, faster ion transportation [38], better cooling and less sensitivity to external noise [39]. Higher secular frequencies are also preferable for multi-qubit entangling gate operations utilizing motional modes [40]. Radial secular frequencies are strongly influenced by the electrode dimensions and are considered in the design of RF electrodes. The axial secular frequency is determined by adjusting both the DC voltages and the electrode geometry. When the peak voltage on the DC electrodes are limited by digital-to-analog converters, the maximum electric potential generated by a given electrode geometry can also be limited. Thus, occasionally, the designed electrode dimensions do not provide sufficiently high axial secular frequencies. Therefore, the effects of the DC voltage at the RF null point must be investigated after designing the RF electrodes. We explain this further in Section 3.4. 2.2.4. Ion Height As shown in Figure 2, the RF and DC electrodes of the surface ion trap are laid on the same plane and the RF null point is placed above the plane. The ion height is the vertical distance between the RF null point and the plane. Higher ion heights correspond to slower motional heating rates of the trapped ions [14]. However, a higher height also corresponds to a weaker trapping pseudopotential at the RF null point. The beam size of the lasers must also be considered when selecting the ion height. We describe the radius of a Gaussian beam at a distance d from the waist, w(d), as follows: d ˆ ˙2 πw0 2 d w pdq “ w0 1 ` , dR “ (5) dR λ where w0 is the beam waist, which is the radial size of the beam at its narrowest point, and λ is the wavelength [41]. Equation (5) indicates that a thinner beam waist can accompany a larger beam radius at the edge of the surface ion-trap chip. Figure 4 shows a schematic of a Gaussian beam injected parallel to the surface ion trap. An increase in the beam size induces beam clipping by the chip body, which can increase the laser scattering caused by the diffraction of the clipped laser. By setting the Rayleigh length of Gaussian beam (dR ) equal to half the traverse distance of the beam propagation above the chip as shown in Figure 4, the disturbance to beam propagation by the chip can be minimized. In addition, the ratio of ion height over beam size at the edge of the chip allows us to estimate the amount of potential laser scattering due to laser clipping.

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dR ૛w0

w(d)

d

w0

Figure4.4. Schematic Schematic of of aaGaussian Gaussianbeam beaminjected injectedparallel parallelto tothe thesurface surfaceion iontrap. trap.The Theblue bluelightning lightning Figure symbols indicate the scattering of the Gaussian beam by the surface ion-trap chip at the chip edges. symbols indicate the scattering of the Gaussian beam by the surface ion-trap chip at the chip edges.

2.3. Analytic Solutions 2.3. Analytic Solutions Based on a five-rail geometry, which assumes that the electrode surfaces are infinite and that Based on a five-rail geometry, which assumes that the electrode surfaces are infinite and that there there are no gaps between electrodes, the specifications of the surface ion trap described in the are no gaps between electrodes, the specifications of the surface ion trap described in the previous previous subsections are analytically solved as follows [15]: subsections are analytically solved as follows [15]:

eV 2 2 ( a  b) 2 p a ´ b q2 eV trapdepth  ˆ ˙ b trap depth “2 mΩ22 m  (πa2 pa`bb)q22 aa22`6ab6`ab 4b2abpa4`bq2ab( a  b)2 b2 `



8p b ´ a q eV ?  a) 2 mΩ2 π8( b abpa`bq

q-parametereV “ q-parameter  ? 2 ion height “ mab



(6) (6)

ab ( a  b)2

heightinFigure ab 2. The secular frequency can be calculated by combining where a and b are ion indicated Equations (3) and (6). However, these analytic solutions are not valid for the complex structures of where a and b are indicated in Figure 2. The secular frequency can be calculated by combining actual surface ion traps, including the loading slot, finite width of outer electrodes and segmentation, Equations (3) and (6). However, these analytic solutions are not valid for the complex structures of and multi-layered electrodes, because certain assumptions do not hold for these complex structures. actual surface ion traps, including the loading slot, finite width of outer electrodes and Nonetheless, these analytic solutions are very useful for making a rough first design of a surface ion segmentation, and multi-layered electrodes, because certain assumptions do not hold for these trap before applying the numerical optimization procedures outlined in the following sections. complex structures. Nonetheless, these analytic solutions are very useful for making a rough first design a surface ion trap before applying the numerical optimization procedures outlined in the 2.4. BEMofSimulations following sections. The application of advanced microfabrication technologies has allowed the development of complex trap structures such as loading slots, junctions, and even three-dimensional quadrupole 2.4. BEMion Simulations structures [24–27,42]. Complex ion trap structures are difficult to be analytically modeled; therefore, The application of advanced microfabrication technologies hasshown allowed the development of simulations and numerical analyses need to be used. BEMs have been to provide more precise complex ion trap structures such as loading slots, junctions, and even three-dimensional quadrupole and faster simulations of a surface ion trap than finite element methods [21,28]. In this paper, we use structures [24–27,42]. Complex ionParticle trap structures are difficult to be analytically modeled; therefore, the BEM-based software Charged Optics (CPO Ltd., Manchester, UK). We can simulate the simulations and numerical analyses need to be used. BEMs have been shown to provide more electrical potential and electric field induced by each electrode by applying 1 V to the electrode being precise and faster simulations a surface ion trap thanthe finite element methods In in this paper, measured while grounding theof other electrodes. Then, results are scaled and[21,28]. summed order to we use the BEM-based software Charged Particle Optics (CPO Ltd., Manchester, UK). We can calculate the total electric potential. The pseudopotential φpp (r) generated by the trap RF is defined by: simulate the electrical potential and electric field induced by each electrode by applying 1 V to the 2 electrode being measured while grounding the other electrodes. Then, the results are scaled and e2 |E prq| φpp prq “ (7) 2 summed in order to calculate the total electric potential. pseudopotential ϕpp(r) generated by the 4mΩThe trap RF is defined by: where r represents the position vector and E(r) represents the RF electric field amplitude generated by 2 2 applying voltage V on RF electrodes [14]. From the eestimated E  r  total electric potential, we can calculate (7) the specifications explained in Section 2.2,including q-parameter, secular frequencies, and pp  r   trap depth, 2 4 m  ion height. where represents the position vector and E(r) theelectric RF electric fieldgenerated amplitude To rdetermine the validity of the BEM tool, werepresents simulate the potential bygenerated a simple electrode structure with gaps, as shown in the upper-left inset of Figure 5. Thiswe figure by applying voltage V 1-µm on RFelectrode electrodes [14]. From the estimated total electric potential, can calculate the specifications explained in Section 2.2, including trap depth, q-parameter, secular frequencies, and ion height.

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of 16 the validity of the BEM tool, we simulate the electric potential generated 7by a simple electrode structure with 1-μm electrode gaps, as shown in the upper-left inset of Figure 5. This figure shows the analytic solution from(6) Equation and the simulated ionIfheights. use a shows the analytic solution from Equation and the(6) simulated ion heights. we useIfa we uniform 2 for both the RF and DC electrodes, the maximum error between uniform segment size of 3000 μm 2 segment size of 3000 µm for both the RF and DC electrodes, the maximum error between the analytic the and simulated values 1.62%. When the segment to size is adjusted to the 333357 μm2 for and analytic simulated values is 1.62%. Whenisthe segment size is adjusted 333´357 µm2 for RF electrodes 2 the electrodes μm for the electrodes, maximum error is 0.77%, 2 for68007400 andRF 6800´7400 µmand the DC electrodes, the DC maximum errorthe is reduced to 0.77%, asreduced shown intoFigure 5. as shown in Figure 5. The computational requirements for the two cases are 348 MB memory for 12 The computational requirements for the two cases are 348 MB memory for 12 min and 611 MB memory min 611 MB memory 50 min, respectively. perform the simulations usingCorporation, an i-7 4790 for 50and min, respectively. Wefor perform the simulationsWe using an i-7 4790 processor (Intel processor (Intel Corporation, Santa Clara, CA, USA). The adaptive mesh function of the tool Santa Clara, CA, USA). The adaptive mesh function of the BEM tool can also be used. InBEM this case, can also be used. In this case, the maximum error is 0.83% and the computational requirements are the maximum error is 0.83% and the computational requirements are 636 MB memory for 60 min. 636 MB memory for by 60 adjusting min. As described above, adjusting theand segment size boththe manually and As described above, the segment size by both manually automatically, differences automatically, the differences between the analytic solutions and the simulation results can be between the analytic solutions and the simulation results can be reduced for cases where a = 50 and reduced for bcases a =and 50 300 andµm. 100 Therefore, μm and bthe = 150, 200, 250, and using 300 μm. Therefore, the 100 µm and = 150,where 200, 250, simulation results the BEM tool seem simulation results using the BEM tool seem valid for estimating the electric potentials generated by valid for estimating the electric potentials generated by trap-chip electrodes. However, when a is fixed trap-chip electrodes. However, when a is fixed at 50 or 100 μm and b becomes very small or large, the at 50 or 100 µm and b becomes very small or large, the differences between the analytic solutions differences between theseem analytic solutionsThis and is simulation results seem increase. Thisregions is because and simulation results to increase. because the widths of to the simulation are the widths of the simulation regions are fixed at 1 mm in each simulation set, and that of the outer fixed at 1 mm in each simulation set, and that of the outer ground plane is decreased as b increases. ground is decreased as b increases. For such the simulation regions mustground also be adjusted For suchplane cases, the simulation regions must also cases, be adjusted to provide an outer plane of to provide ansize. outer ground planeresults of appropriate Thethat simulation results in Figure 5 showsolutions that the appropriate The simulation in Figuresize. 5 show the error between the analytic error between the analytic solutions and simulation results can be minimized when c ≈ 2b, where c is and simulation results can be minimized when c « 2b, where c is the width of the outer ground plane the width of the ground planeof(or the lengthtraps). of the DC electrodes of segmented traps). (or the length of outer the DC electrodes segmented

(a)

a

b

c

(b)

a

b

c

1V 1V

1V 1V (c)

-0.43% 0.08% 0.32% 0.77%

-0.69% -0.29% -0.02%

0.37% -33.40%

-31.05%

-31.90%

-31.36% -19.95%

-22.26%

-17.72%

-15.78%

Figure 5. 5. Simulation Simulationresults resultsofof heights for simple and complex electrode geometries. (a) a thethe ionion heights for simple and complex electrode geometries. (a) a simple simple electrode geometry with 1-μm electrode (b) a electrode complex electrode configuration electrode geometry with 1-µm electrode gaps; (b)gaps; a complex configuration with innerwith DC inner rails, aslot, loading and ground The of width the DC inner DCisrails 20regardless μm regardless rails, DC a loading andslot, ground planes. planes. The width the of inner rails 20 is µm of a. of The widths the loading for the = 50100 μmµm and μm and are 156 56 μm 156 μm, Thea.widths of the of loading slot forslot the case of acase = 50of µma and are100 56 µm µm,and respectively; respectively; simulation results of theThe ionblack heights. and red lines represent the analytic (c) simulation(c) results of the ion heights. andThe red black lines represent the analytic solution of the solution of the for thegeometries ideal five-rail geometries when = 50µm, μmrespectively. and 100 μm,The respectively. ion heights for ion the heights ideal five-rail when a = 50 µm anda100 numbers The near the bullets represent the errors between the analytic for thegeometries ideal five-rail nearnumbers the bullets represent the errors between the analytic solution for thesolution ideal five-rail and the simulation with the results same a with and bthe values. geometries andresults the simulation same a and b values.

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Next, we simulate the electric potential generated by a more complex electrode configuration shown in the upper-right inset in Figure 5, to determine the need for numerical simulations. The electrode structure includes two inner DC rails, an ion-loading slot between the rails, and ground planes. Moreover, the ground planes are placed 14 µm below the top electrode. The width of the inner DC rails is 20 µm regardless of a. The widths of the loading slot for the case of a = 50 µm and 100 µm are 56 µm and 156 µm, respectively. In this case, the differences between the analytic solutions and the BEM-simulated results are significant, as shown by the black and red triangular marks in Figure 5. Thus, using the BEM method is essential when designing surface ion traps with complex configurations. 3. Design Methodology for Surface Ion-Trap Chips The design procedure in this paper follows four steps: setting the basic assumptions, determining the shape and size of the chip, designing the dimensions of the RF electrodes, and investigating the effects of the DC voltages at the ion position. 3.1. Basic Assumptions First, we must establish some basic assumptions that are independent of the design layout. The ion species used in the ion-trap experiments determines the wavelengths of the lasers to be used. The ion mass is needed to calculate the trap specifications. The type of the ceramic chip package limits the maximum chip size and number of electrodes. In addition, to accurately simulate the electric potential, the thicknesses of the conducting films and the insulator materials must be defined. Next, we must determine the constraints of the design parameters including a, b, V, and Ω. The lengths a and b in Figure 2 determine the distance between the two RF electrodes and the distance of the outer DC electrodes from the ion position, respectively. Between the RF electrodes, we can position a loading slot and the center DC electrodes. Thus, we determine a by the widths of the loading slot and the center DC electrodes. The width of the loading slot can be designed with respect to the configuration of the components that supply neutral atoms, and are typically tens of micrometers. The role of the center DC electrodes is to generate an electric potential that can compensate for the asymmetry of the electric potential generated by the outer DC electrodes. If b increases, the electric field from the outer DC electrodes is reduced, which reduces the width of the center DC electrodes to keep the ion height constant. Essentially, the constraints for a and b are related. The optimal constraint dimensions for a and b cannot be determined in this step. Instead, the constraint dimensions are initially established by referring to previous work and must be optimized by repetitively designing the RF electrodes and DC voltage set. The maximum value of V is restricted to prohibit electric breakdown between the electrodes and the ground plane. The RF frequency Ω can be easily adjusted without many constraints. Both V and Ω affect the trap depth, q-parameter, and radial secular frequencies. The trap depth, radial secular frequencies, and q-parameter are proportional to V 2 /Ω2 , V/Ω, and V/Ω2 , respectively. Thus, V is assumed to be the maximum value and Ω is adjusted to achieve a low q-parameter and high trap depth and radial secular frequencies. We cannot calculate the RF breakdown voltage because the Paschen curve may not be applicable in an ultra-high vacuum environment [43]. However, it has been reported that an alternating current (AC) field of ~30 V µm´1 can induce electric breakdown between sharp electrodes in vacuum [44,45]. Because an electrode gap of ~10 µm is generally used in surface ion traps to prevent electric breakdown between the electrodes and considering the microfabrication constraints, the RF amplitude is restricted to ~300 V. In microfabrication, SiO2 and tetraethyl orthosilicate (TEOS) are typically used for thick dielectric layers. It is generally difficult to deposit SiO2 or TEOS at a thickness greater than 10 µm. It is also very difficult to pattern thick dielectric layers using UV-based lithography. Therefore, dielectric layers with a thickness of ~10 µm are widely used in surface ion traps.

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3.2. Shape and Size of the Chip As shown in Equation 3.2. Shape and Size of the Chip(5), the beam radius increases with distance from the waist. Therefore, a smaller chip size is advantageous for preventing laser clipping at the chip edges. The main factor As shown in Equation (5), the beam radius increases with distance from the waist. Therefore, a that restricts the chip miniaturization is the size and number of the DC electrodes. Wire bonding smaller chip size is advantageous for preventing laser clipping at the chip edges. The main factor that multiple electrodes to bonding pads requires sufficient lateral spaces on the electrode metal layer. To restricts the chip miniaturization is the size and number of the DC electrodes. Wire bonding multiple overcome this constraint, electrode routings using multi-layered metals and structures have been electrodes to bonding pads requires sufficient lateral spaces on the electrode metal layer. To overcome developed [46]. However, we do not consider this method in this study. As explained in Section 2.4, this constraint, electrode routings using multi-layered metals and structures have been developed [46]. as dimension b increases, the simulation regions must be expanded to provide an outer ground However, we do not consider this method in this study. As explained in Section 2.4, as dimension b plane of sufficient size. In the practical design of surface ion traps, the outer ground plane is increases, the simulation regions must be expanded to provide an outer ground plane of sufficient size. separated into DC electrodes and the simulation regions can be determined by the lengths of the In the practical design of surface ion traps, the outer ground plane is separated into DC electrodes and straight sections of the DC electrodes along the x-direction in Figure 2. To determine a sufficient the simulation regions can be determined by the lengths of the straight sections of the DC electrodes length for these straight sections, we simulate the ion height and trap depth with different lengths of along the x-direction in Figure 2. To determine a sufficient length for these straight sections, we simulate the DC electrodes. For these simulations, we fix the lengths a and b in Figure 2 at 50 μm and 200 μm, the ion height and trap depth with different lengths of the DC electrodes. For these simulations, we respectively. The simulation results shown in Figure 6 indicate that the variation rates of the trap fix the lengths a and b in Figure 2 at 50 µm and 200 µm, respectively. The simulation results shown in depth and ion height decrease as the length of the DC electrodes is increased. Thus, the accuracy of Figure 6 indicate that the variation rates of the trap depth and ion height decrease as the length of the the simulations and the length of the DC electrodes represent a trade-off problem. For example, the DC electrodes is increased. Thus, the accuracy of the simulations and the length of the DC electrodes errors of the trap depth and ion height in the cases using 150-μm and 200-μm lengths for the DC represent a trade-off problem. For example, the errors of the trap depth and ion height in the cases electrodes are less than 1%. Additionally, the errors in the cases with 400-μm and 450-μm lengths of using 150-µm and 200-µm lengths for the DC electrodes are less than 1%. Additionally, the errors in the DC electrodes are reduced to 0.2%. the cases with 400-µm and 450-µm lengths of the DC electrodes are reduced to 0.2%.

Figure 6. 6. Simulation Simulationresults resultsfor forthe thetrap trapdepth depthand andion ionheight heightas asaa function functionof of the the length length of of the the DC DC Figure electrodes. We b atb50 The ion species the simulations electrodes. We fix fixa and a and atµm 50 and μm 200 andµm, 200respectively. μm, respectively. The ionassumed species inassumed in the is 171 Yb+ . is 171Yb+. simulations

3.3. 3.3. Design DesignofofRF RFelectrodes electrodes Before Before designing designing the the RF RF electrodes, electrodes, we we need need to to approximately approximately calculate calculate the the constraint constraint for for the the 2 2 ion 1/e point” ion height. height. The The beam beam diameter diameter in in Equation Equation (5) (5) is based on the “half-width at 1/e point”definition, definition, 2 2times which the intensity intensityof ofthe thebeam beamtotobebe 1/e timesthe themaximum maximum intensity. However, to set which considers the 1/e intensity. However, to set the the proportion of clipped the clipped power at a specific the definitions beam proportion of the beambeam power at a specific value, value, we canwe usecan the use definitions of beam of intensity intensity and power described and power described below: below:  2`  2( x2y2 qy2 )  x ´2px Ipx, , P,“P  Ipx, I(xyq, y“) I0Iexp I(yqdxdy x, y)dxdy , 0 exp  w 2 2 0w „



0





(8) (8)

where I, I , and P represent beam intensity, beam intensity at the center of the beam, and beam power, where I, I00, and P represent beam intensity, beam intensity at the center of the beam, and beam respectively. Note that the coordinates of x and y in Equation (8) represent Cartesian coordinates power, respectively. Note that the coordinates of x and y in Equation (8) represent Cartesian perpendicular to the beam path. Using these definitions, we can calculate the proportion of the clipped coordinates perpendicular to the beam path. Using these definitions, we can calculate the proportion

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beam power as a function of ion height. Therefore, the constraint for the ion height can be calculated when we define a constraint for the proportion of the clipped beam power. In the design procedure for the RF electrodes, the amplitude of the RF voltage is generally assumed to be the maximum value available for prohibiting electric breakdown, as described in Section 3.1. The design of the RF electrodes has three inputs, a, b, and Ω, and four outputs, trap depth, q-parameter, radial secular frequencies, and ion height. To simplify the design problem, we fix one output specification at a constant value. The output specification can be selected from among the trap depth, q-parameter, and radial secular frequency because these values can be easily tuned to a constant by adjusting the RF frequency. After the output specification is fixed, two inputs and three outputs remain. Then, we can investigate the characteristics of the remaining output specifications. The appropriate constraints for each specification are given considering the purpose of a particular chip design, and the electrode dimensions a and b, the two remaining inputs, are determined to satisfy the given constraints. In Section 4, we explain in detail the design method of the RF electrodes and present a case study. 3.4. Investigation of Effects of DC Voltages After determining the lateral dimensions of the RF electrodes, we investigate the effects of the DC voltages at the ion position. This step is especially important because, generally, the output voltages of digital-to-analog converters are limited. As a first step, we must achieve an axial secular frequency greater than the constraint value. The constraint can be determined by considering the experimental goal and the available measurement systems. As the next step, we must determine the feasibility of tilting the principal axes near 45˝ . In order to tilt the principal axes of the ion’s secular motion, we apply DC voltages that are asymmetric along the z-axis to the outer DC electrodes. We can calculate the tilt angle by computing the Hessian matrix of the total trap potential and finding the eigenvalues. Finally, we must determine whether the electric potential from the center DC electrodes can compensate for the asymmetric electric potential of the outer DC electrodes. In other words, the total electric potential at the ion position generated by the DC electrodes must be zero. As long as the lateral dimensions of the RF electrodes satisfy the conditions described above, the constraint dimensions of a and b can be modified to improve the specifications of the surface ion trap described in Section 2. Essentially, the electrode dimensions can be optimized by repetitively designing RF electrodes with different constraint dimensions and then investigating the effects of the DC voltages. If the designed electrode dimensions cannot satisfy the requirements, the design procedure of the RF electrodes must be repeated with a larger constraint dimension of a and a smaller dimension of b than in the previous design. 4. Case Study of the Design Methodology In this section, we apply the design methodology described in the previous section to a particular trap-chip design. As a first step, we describe our basic assumptions. The surface ion-trap chip is designed to trap 171 Yb+ ions. Accordingly, lasers with wavelengths of 369.5 nm, 399 nm, 638 nm, and 935 nm must be used for Doppler cooling and state detection (2 S1/2 Ø 2 P1/2 ), photoionization, repumping from the 2 F7/2 state, and repumping from the 2 D3/2 state, respectively [47]. The chip is assumed to be mounted on the commercial chip package CPGA 10039 (Spectrum Semiconductor Materials, San Jose, CA, USA), with a mounting area of 1.2 cm ˆ 1.2 cm and a total pin number of 100. Figure 7 shows a cross-section schematic of the surface ion trap under consideration. The structure is designed with metalized sidewalls to shield the trapped ions from the inevitable charges built up on the dielectric sidewalls. We place two inner DC rails inside the RF rails to tilt the principal axes of the ion’s secular motion. The inner DC rails are placed on the lower layer to reduce the background scattering of the laser. The silicon substrate between the inner DC rails is penetrated for loading neutral atoms from the backside of the trap surface. The dimension a is initially constrained by the condition a ě 40 µm, which provides sufficient space for laying the inner DC rails and the loading slot between

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thecondition RF electrodes. b is constrained the condition 300 µm, which provides b ≤ 300The μm,dimension which provides the distancebybetween the ionsb ď and the outer electrodes. Wethe design between an asymmetric chip shape to reduce beam clipping from specific directions and fix the chip distance the ions and the outer electrodes. We design an asymmetric chip shape to reduce condition b ≤ 300 μm, which provides the distance between the ions and the outer electrodes. We design anassumed asymmetric chip shape reduce beam clipping specific directions and fix path theeach chip sizeclipping along the beam path to atand 2.26fixmm. The number of outer DC electrodes, a beam from specific directions the chip sizefrom along the assumed beam at with 2.26 mm. size along the assumed beam path at 2.26 mm. The number of outer DC electrodes, each with a width of 70ofμm, is 24. planar each dimensions based on µm, a previous study planar [12]. More details forare The number outer DCThese electrodes, with a are width of 70 is 24. These dimensions width 70 DC μm, electrode is 24. Thesecan planar dimensions are based on a previous study [12]. More details for the design ofofthe bedetails found in [48]. based on the a previous study [12]. More for the design of the DC electrode can be found in [48]. design of the DC electrode can be found in [48].

b a

b a

(a)

(b)

(b) in the case (a) Figure 7. (a) Three-dimensional schematic of the surface ion-trap structures assumed

Figure 7.study (a) Three-dimensional schematic of the surface ion-trap structures assumed in the case and (b) a cross-section schematic of the surface ion trap under consideration. The structure is Figure 7. (a) Three-dimensional schematic the surface ion-trap structures assumed in on the case is study and (b) a cross-section schematic of theof surface consideration. The designed with metalized sidewalls to shield the trappedion ionstrap fromunder the inevitable charges built up structure study and (b)metalized a cross-section schematic of the ionions trapfrom under The structure the dielectric sidewalls. designed with sidewalls to shield thesurface trapped theconsideration. inevitable charges built upison metalized sidewalls to shield the trapped ions from the inevitable charges built up on thedesigned dielectricwith sidewalls. Figure sidewalls. 8 shows the proportion of the clipped beam power as a function of ion height. In the the dielectric calcuation, we assume a laser with a wavelength of 369.5 nm, which is used in Doppler cooling and

171Yb+ ion. Figure shows the ofAs thedescribed clippedinbeam power as a the function of ion height. the state8 8 detection of proportion the the Section disturbance beam In In Figure shows the proportion of the clipped beam power 2.2.4, as a function of ion toheight. the 1/2 propagation by the chip body whenof the369.5 beamnm, waistwhich is equalistoused (lλ/π)in ,Doppler where l is cooling the calcuation, we assume a laser withisaminimized wavelength and calcuation, we assume a laser with a wavelength of 369.5 nm, which is used in Doppler cooling and 171 Yb distance between the+waist and the chip edge where beam clipping occurs. For example, assuming state detection of the ion. As described in the Section 2.2.4, the disturbance to beam propagation 171Yb+ ion. As described in the Section 2.2.4, the disturbance to beam state detection of thebetween that the distance the chip center and the edge is 2.26 mm and that1/2 the wavelength of the bypropagation the chip body is minimized the beam waist is equal tois(lλ/π) where is the ldistance 1/2 ,l where by the chip iswhen minimized when theis beam waist equal to ,(lλ/π) laser is 369.5 nm, the body proportion of the clipped beam minimized when the beam waist is 16.32 μm. is the between the waist and the chip edge where beam clipping occurs. For example, assuming distance the waist and the chip 8, edge where beam clipping For isexample, Forbetween the 369.5-nm laser shown in Figure the proportion of the clipped occurs. beam power calculatedassuming to that the −6 distance between the chip center and the edge is 2.26 mm and that the wavelength of the laser 10 whenbetween the beam waist is 16.32 μm. Therefore, we can theand constraint for the ion height of that thebedistance the chip center and the edge iscalculate 2.26 mm that the wavelength the is from Equation (8), and the calculated constraint value is 54.9 μm. We then use this value as a 369.5 nm, the proportion of the clipped is beam minimized when the beam 16.32 laser is 369.5 nm, the proportion of the beam clipped is minimized when the waist beam is waist is µm. 16.32For μm.the constraint when designing the RF electrodes. 369.5-nm shown inshown Figurein8,Figure the proportion of the clipped beam power is calculated to be 10 For the laser 369.5-nm laser 8, the proportion of the clipped beam power is calculated to´6 −6 when when the beamthe waist is waist 16.32 is µm. Therefore, we canwe calculate the constraint for the height from be 10 beam 16.32 μm. Therefore, can calculate the constraint for ion the ion height from Equation the calculated constraint is 54.9 thisasvalue as a Equation (8), and(8), the and calculated constraint value isvalue 54.9 µm. Weμm. thenWe usethen this use value a constraint constraint when designing the RF electrodes. when designing the RF electrodes.

Figure 8. Proportion of the clipped beam power fraction as a function of ion height. For the calculation, we assume a laser with a wavelength of 369.5 nm, which is used in Doppler cooling and state detection of the 171Yb+ ion.

Figure 8. Proportion of clipped the clipped fraction as a function of ion For height. For the Figure 8. Proportion of the beambeam powerpower fraction as a function of ion height. the calculation, calculation, we assume a laser with a wavelength of 369.5 nm, which is used in Doppler cooling and we assume a laser with a wavelength of 369.5 nm, which is used in Doppler cooling and state detection 171Yb+ ion. state detection of the 171 + of the Yb ion.

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Below, we describe an example design procedure for RF electrodes. As mentioned in Section 2.2.1,Below, the goal this design procedure is toprocedure maximizefor theRF trap depth. InAs this design scenario, we2.2.1, fix we of describe an example design electrodes. mentioned in Section the goal q-parameter at a constant valueisof and maximize trap depth. radial frequencies, s,x the of this design procedure to0.25 maximize the trapthe depth. In this The design scenario, we fixωthe 0.86 and ω s,y , which are the less stringent constraints, can be determined by the relation ω s,x /ω s,z > 0.73N q-parameter at a constant value of 0.25 and maximize the trap depth. The radial frequencies, ωs,x and 0.86, where N represents the number of ions in the ion string [49]. If arbitrary and ωs,y/ωs,zare > 0.73N ω the less stringent constraints, can be determined by the relation ωs,x /ωs,z > 0.73N0.86 s,y , which conditions and0.86 ωs,z, where = 551 kHz are considered as an example must and ωs,y /ωN 0.73N N represents the number of ions case, in thethe ionsecular string frequency [49]. If arbitrary s,z =>14 be higher than 4 MHz. Another less stringent constraint, the ion height, is restricted to be greater conditions N = 14 and ωs,z = 551 kHz are considered as an example case, the secular frequency must than 54.9 μm, in the previous paragraph. Figurethe 9 shows the simulation results with the be higher thanas4 explained MHz. Another less stringent constraint, ion height, is restricted to be greater q-parameter fixed at 0.25. We tune the q-parameter to 0.25 in each simulation by adjusting the RF than 54.9 µm, as explained in the previous paragraph. Figure 9 shows the simulation results with frequency. As shown in Figure 9a,b, small a values provide high trap depths and secular frequencies. the q-parameter fixed at 0.25. We tune the q-parameter to 0.25 in each simulation by adjusting the RF Based on these simulation results, wesmall selecta avalues value provide of a = 40 high μm. At this value,and thesecular secular frequencies. frequency frequency. As shown in Figure 9a,b, trap depths is larger than the design constraint value, 4 MHz, regardless of the b value. With regard to ion Based on these simulation results, we select a value of a = 40 µm. At this value, the secular frequency height, b must be smaller than 164 μm (Figure 9c). Thus, we select b = 164 μm to maximize the trap is larger than the design constraint value, 4 MHz, regardless of the b value. With regard to ion height, depth while satisfying the constraint on ion height. In conclusion, the expected values of the final b must be smaller than 164 µm (Figure 9c). Thus, we select b = 164 µm to maximize the trap depth while specifications are a trapon depth of 0.311IneV at an RF frequency of 56.48 secular frequency of satisfying the constraint ion height. conclusion, the expected valuesMHz, of thea final specifications are 4.80 MHz, and an ion height of 54.9 μm, when a = 40 μm and b = 164 μm are chosen a trap depth of 0.311 eV at an RF frequency of 56.48 MHz, a secular frequency of 4.80 MHz, andasanthe ion dimensions of the RF electrodes. height of 54.9 µm, when a = 40 µm and b = 164 µm are chosen as the dimensions of the RF electrodes.

(a)

(b)

ω = 4 MHz

(c)

4 h = 54.9 μm

+ ions, + ions, Figure 9. 9. Simulation results YbYb with the the q-parameter fixedfixed at 0.25, as Figure results for forthe thecase caseofoftrapping trapping171171 with q-parameter at 0.25, functions of of a and b. b.We as functions a and Wetune tunethe theq-parameters q-parameterstoto0.25 0.25inineach eachsimulation simulationset setbybyadjusting adjustingthe theRF RF frequency.The Theblue blueblocks blocks represent regions satisfy the constraints considered. (a) depth; trap frequency. represent thethe regions thatthat satisfy the constraints considered. (a) trap depth; (b)secular radial secular frequency; (c) ion height. (b) radial frequency; (c) ion height.

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Figure 10 10 shows shows aa DC DC voltage voltage set set for for trapping trappingions ionsand andthe theresult resultfor forthe thetilted tiltedprincipal principalaxes. axes. Figure When we use the voltage set in Figure 10a, the total electric potential at the ion position is zero. When we use the voltage set in Figure 10a, the total electric potential at the ion position is zero. In ˝ In addition, we obtain an axial secular frequency of 551 kHz and a principal axes tilt of 44.7 . Note that addition, we obtain an axial secular frequency of 551 kHz and a principal axes tilt of 44.7°. Note that the absolute absolute values values of of the the voltage voltage amplitudes amplitudes are are under under 10 10 V V (the limited by by digital-to-analog digital-to-analog the (the peak peak limited converters in our case). If the simulation results including the DC voltages do not satisfy necessary converters in our case). If the simulation results including the DC voltages do notthe satisfy the conditions of each experimental setup, the constraint dimensions for a and b must be adjusted and necessary conditions of each experimental setup, the constraint dimensions for a and b must the be design ofand the the RF electrodes must repeated.must be repeated. adjusted design of the RFbe electrodes

2 2 2 -8 -8 2 2 2 e2V2/4mΩ2 0.865 -0.484 2 e V2/4mΩ2 4 4 4 -6 -6 4 4 4 (a)

(b)

Figure of the the effects effectsof ofthe theDC DCvoltages voltagesatatthe the ion position. applied voltage Figure 10. 10. Investigation Investigation of ion position. (a)(a) applied DCDC voltage set; set; (b) axes tilt result of 44.7º. The tilt angle is achieved by computing the Hessian matrix of the ˝ (b) axes tilt result of 44.7 . The tilt angle is achieved by computing the Hessian matrix of the totaltotal trap trap potential finding the eigenvalues. potential and and finding the eigenvalues.

5. Conclusions 5. Conclusions In this paper, we presented a design methodology and a case study for designing the In this paper, we presented a design methodology and a case study for designing the electrodes electrodes of surface ion traps. We briefly presented the theory behind surface ion traps and of surface ion traps. We briefly presented the theory behind surface ion traps and explained their explained their specifications. We showed the accuracy and necessity of using the BEM tool for specifications. We showed the accuracy and necessity of using the BEM tool for designing an ion trap designing an ion trap by comparing analytic solutions with our simulation results. The design by comparing analytic solutions with our simulation results. The design methodology presented in methodology presented in this paper comprises the following four steps: setting the basic this paper comprises the following four steps: setting the basic assumptions, determining the shape assumptions, determining the shape and size of the chip, designing the dimensions of the RF and size of the chip, designing the dimensions of the RF electrodes, and investigating the effects of the electrodes, and investigating the effects of the DC voltages at the ion position. Based on the basic DC voltages at the ion position. Based on the basic assumptions and chip size, the lateral dimensions assumptions and chip size, the lateral dimensions of the RF electrodes can be designed. Then, we of the RF electrodes can be designed. Then, we investigated the validity of the RF and DC electrode investigated the validity of the RF and DC electrode designs by inspecting the axial secular designs by inspecting the axial secular frequency, the principal axes tilt, and the total electric potential frequency, the principal axes tilt, and the total electric potential at the ion position. The design at the ion position. The design process can be iteratively carried to optimize the required specifications process can be iteratively carried to optimize the required specifications and constraints. and constraints. We also explained the application of our design methodology by using a case study. The We also explained the application of our design methodology by using a case study. The length of length of the DC electrodes was set at 400 μm. To set the proportion of the clipped beam power ´ to the DC electrodes was set at 400 µm. To set the proportion of the clipped beam power to under 10 6 , 171Yb+ ion. The under 10−6, we calculated the required minimum ion height to be 54.9171 μm for the we calculated the required minimum ion height to be 54.9 µm for the Yb+ ion. The beam waist beam waist was assumed to be 16.32 μm to minimize the beam radius at the chip edge. We carried was assumed to be 16.32 µm to minimize the beam radius at the chip edge. We carried out the design out the design procedure with the following conditions: a q-parameter of 0.25, a secular frequency procedure with the following conditions: a q-parameter of 0.25, a secular frequency value greater value greater than 4 MHz, and an ion height of more than 54.9 μm. Using BEM simulations, an than 4 MHz, and an ion height of more than 54.9 µm. Using BEM simulations, an electrode geometry electrode geometry with a = 40 μm and b = 164 μm resulted in a trap depth of 0.311 eV at an RF with a = 40 µm and b = 164 µm resulted in a trap depth of 0.311 eV at an RF frequency of 56.48 MHz, frequency of 56.48 MHz, a secular frequency of 4.80 MHz, and an ion height of 54.9 μm. We˝ a secular frequency of 4.80 MHz, and an ion height of 54.9 µm. We achieved a principal axes tilt of 44.7 achieved a principal axes tilt of 44.7° with DC voltages of less than 10 V. For different experimental with DC voltages of less than 10 V. For different experimental objectives, an optimized design for each objectives, an optimized design for each case scenario can be obtained using the design case scenario can be obtained using the design methodology outlined in this paper. The developed methodology outlined in this paper. The developed design methodology can be applied to surface ion traps with more complex structures and those for trapping multiple species.

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design methodology can be applied to surface ion traps with more complex structures and those for trapping multiple species. Acknowledgments: This work was partly supported by the ITRC (Information Technology Research Center) support program of MSIP/IITP (IITP-2016-R0992-16-1017) and ICT R&D program of MSIP/IITP (10043464, Development of quantum repeater technology for the application to communication systems). Author Contributions: Seokjun Hong performed the BEM simulations and designed the variations of surface ion traps. Minjae Lee and Hongjin Cheon also performed the BEM simulations. Taehyun Kim determined the trap chip specifications and analyzed the numerical analysis results, Dong-il “Dan” Cho supervised the entire work, and they are the corresponding authors. All authors read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations The following abbreviations are used in this manuscript: AC BEM DC FEM RF SEM TEOS UV

Alternating current Boundary element method Direct current Finite element method Radio frequency Scanning electron micrograph Tetraethyl orthosilicate Ultra violet

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