INTRODUCTION

Since the first experiments with trapped ions were conducted in the late 1950s [1–3], systems with stored ions became very popular in frequency standards and quantum simulation applications [4–8]. Precise frequencies are being used for accurate definition of physical units and timekeeping. Enough to mention, the resolution of global navigation satellite systems (GNSS) like GPS, GLONASS, and GALILEO rely on the stability of the onboard atomic clocks [9, 10].

One of the ways to increase the stability of atomic clocks is to shift the operating frequency of the standard from microwave (1010–1011 Hz) to the optical range (1014–1015 Hz) as the Q-factor of an oscillator is \(f{\text{/}}\delta f\) [10].

There are two most advanced optical frequency standards types based on elements used as atomic references: systems with trapped neutral atoms and stored ions. Each approach has its advantages and disadvantages. In systems with neutral atoms, the number of atoms N served as atomic reference counted in 106, that improves the signal-to-noise ratio as \(\sqrt N \) [10]. However, systems with trapped ions are more robust because of huge trap depths of several electronvolts leading to almost infinite lifetime of ions in a trap [10].

The further development of optical frequency standards for practical applications as (sat and inertial navigation, gravimetry, etc.) requires a significant miniaturization of all key components of existing standards by a factor of 10 [9]. As an example, it is possible to create very compact and transportable systems for space and airborne timekeeping applications [9]. The most promising architecture for a compact Paul trap typically used for frequency standard applications is a chip-based surface trap with electrodes located in one plane. Such architecture also has the advantage of optical access to the stored ion because ions are trapped above the surface of the trap. It allows guiding radiation from different laser systems into the ion, detecting it, and collecting light with a higher probability [11]. Also such architecture can be used with integrated photonic circuits which can help to minimize the sizes of the whole system and make it more convenient to use.

First ion traps with electrodes in one plane were tested at NIST in 2006 [8]. They used a gold metallization on a quartz dielectric substrate to create a trap. Electrodes were made with photolithography technology.

A vast amount of surface traps has been built since 2006. The fabrication process and trap performance were improved using different materials for substrate and electrodes [12, 13]. Finally, the fabrication of surface traps was combined with well-known silicon-pattering technologies [14]. It significantly helped to improve the resolution of narrow regions on a trap and made the process of planar trap production much more convenient.

METHODS

We chose a surface-electrode trap design in which all electrodes were aligned in the same plane. The model of our trap is based on the approach described in [11, 15, 16] and is shown in Fig. 1. A stripe of a ground (GND) electrode is located between two stripe-shaped radio frequency (RF) electrodes. The direct current (DC) voltages are applied to the segmented electrodes on both sides of the RF electrodes which confine the ions in the axial direction and compensate the secular motions. All electrodes are located on top of the dielectric substrate. An example of the pseudopotential well to store an ion in such a trap is demonstrated in Fig. 2.

Fig. 1.
figure 1

(Color online) Planar ion trap design serve as equivalent to the rod electrodes. (a) Side and (b) top views of the surface electrodes and (c) the surface trap with the hole in the central electrode.

Full size image
Fig. 2.
figure 2

(Color online) Example of the pseudopotential in symmetric planar trap.

Full size image

It is possible to rotate the trapping axis of the ion trap by adjusting the relative sizes of the RF electrodes. Such a feature, as demonstrated in [15, 16], enables the cooling of the trapped ion in all directions with laser beams parallel to the trap surface. This method helps to avoid scattering of the laser light from the electrode surface, which potentially leads to extra heating of the trapped ion. To have an additional optical access for the laser light to the stored ions and/or enable additional optical elements such as the photonic integrated circuits into the design of ion traps, it is possible to introduce a hole in the central electrode [11] under the stored ion, as shown in Fig. 1c.

As for the main tap parameters which we want to calculate, we chose the depth of the trap (d), the secular frequency (\({{\omega }_{{{\text{sec}}}}}\)), and the height of the ion above the surface of the electrodes (h). According to the linear Paul trap theory, which is described in [16], the depth of the trap can be defined as the highest value of a pseudopotential which the ion needs to overcome on its way to infinity. The pseudopotential can be written as:

$$\Psi (x,y,z) = \frac{Q}{{4M{{\Omega }^{2}}}} \times {{({\text{|}}\nabla \Phi (x,y,z){\text{|}})}^{2}},$$
(1)

where Q is the charge of the ion, M is the mass of the ion, and Ω is the frequency of the RF field. From the simulations of Ψ, it is clear that in the \(xy\) \(plane\) maximum of the Ψ is located on the y \(axis\) (perpendicular to the trap \(xz\) \(plane\)). So d can be calculated as:

$$d = \Psi ({{h}_{{\max }}}) - \Psi (h),$$
(2)

where \({{h}_{{\max }}}\) is the coordinate (altitude) where the highest value of a pseudopotential on a y-\(axis\) above the ion is achieved. To estimate the secular frequency, we use the following equation from the linear Paul trap theory [16]:

$${{\omega }_{{{\text{sec}}}}}(x,y) = \frac{{{{q}_{0}}\Omega }}{{2\sqrt 2 }},$$
(3)

where \({{q}_{0}} = q\varepsilon \) is a stability parameter (q-parameter) of a non-hyperbolic trap, ε is an efficiency parameter (typically 0.2−0.3 for the surface trap as shown in [16]) and q is a q-parameter for a hyperbolic trap:

$$q = \frac{{2Q{{V}_{{{\text{rf}}}}}}}{{mR_{0}^{2}{{\Omega }^{2}}}},$$
(4)

where Vrf is an amplitude of the RF voltage. To be in the experimental conditions convenient for the laser focusing, we have to consider \(h > 150\) μm.

RESULT AND DISCUSSION

For trap simulations we used the Electrostatics module of the COMSOL Multiphysics software package. Figure 3 illustrates an example of the pseudopotential calculated in dependence on the distance from the center of the trap surface. To enhance the visualization, the pseudopotential axis is presented on the logarithmic scale.

Fig. 3.
figure 3

(Color online) Calculated pseudopotential in dependence on the distance from the center of the trap surface.

Full size image

The point 1 in Fig. 3 corresponds to the pseudopotential minimum \(\Psi (h)\) and represents the position of the stored ion. The point 2 corresponds to the local maximum of the pseudopotential \({{\Psi }_{{\max }}}\) and represents the highest value of a pseudopotential which ion needs to overcome on its way to infinity. For the optimization of the trap parameters, we were using the “Live Link to MATLAB.” It helped to vary different trap parameters and run simulations automatically.

The following parameters were being changed during the optimization process: \(wc\)—the width of the central electrode, \(wr\)—the width of RF electrodes, and \(hc\)—the width of the central hole (for the trap with the hole in the central electrode). The following parameters were being calculated in our simulations: d—the trap depth, h—the height of the stored ion above trap surface, and \({{\omega }_{{{\text{sec}}}}}\)—the secular frequency of the stored ion. The frequency of the RF field (Ω) in our simulations was chosen to be 2π × 10 MHz and an amplitude of the RF voltage (Vrf) was chosen to be 100 V as a suitable value for electrodes situated at a distance of 5–10 μm to prevent electrical breakdown in the vacuum between them.

For the surface trap, the parameters were changed in the following ranges: \(wc \in [200;600]\) μm, \(wr \in \) [100; 400] μm. For the case of a trap with a hole in the central electrode, the parameter ranges were slightly different: \(wc \in [70;100]\) μm, \(wr \in [70;100]\) μm, \(hc \in [20;50]\) μm. The range of each parameter was roughly estimated during the previous simulations so \(h > 150\) μm for the regular trap and \(h \in [80;150]\) μm for the trap with the hole in the central electrode. By varying these parameters, it is possible to achieve a trap width of about 1 mm, which will make it possible to direct the cooling laser beams at a significant angle for better cooling of the ion.

In order to speed-up the optimization process, it is more efficient to locate the minimum and maximum of the pseudopotential on a one-dimensional line (Fig. 3) rather than a two-dimensional plane (Fig. 2). The location of the line with the minimum of the pseudopotential will shift as the geometrical dimensions of the asymmetric trap are changed. To ensure a consistent optimization process, for our optimizations we used a symmetrical trap configuration where the width of both RF electrodes was kept equal.

By varying the geometrical parameters of the trap, including \(wc\), \(wr\), and \(hc\), we were able to obtain the trap depth, the secular frequency, and the distance between the trapped ion and the electrode surface. The simulation results indicate that by decreasing the geometrical parameters of the trap we have an increase in the trap depth and the secular frequency, while the height of the stored ion is decreased. There are no significant changes in these parameters with the variation of the width of the RF electrodes, particularly in comparison to changes resulting from the varying of other geometrical parameters. These findings could relate to the fact that the stored ion is located at a significant distance from the furthest edges of the RF electrodes.

The most interesting Figs. 4–6 demonstrate the correlations between the calculated parameters d, h, and \({{\omega }_{{{\text{sec}}}}}\). These figures are particularly valuable in defining the geometrical parameters required to produce a trap with the specific characteristics and for the specific applications.

Fig. 4.
figure 4

(Color online) Correlation between the trap depth and the ion height for the trap (top) without and (bottom) with the hole in the central electrode at \(\Omega = 2\pi \times 10\) MHz, Vrf = 100 V, and \(hc \in [20;50]\) μm for (top) \(wc \in [200;\) 600] μm, \(wr \in [100;400]\) μm and (bottom) \(wc \in [70;\) 100] μm, \(wr \in [70;100]\) μm.

Full size image
Fig. 5.
figure 5

(Color online) Correlation between the secular frequency and the ion height for the trap (top) without and (bottom) with the hole in the central electrode at \(\Omega = 2\pi \times 10\) MHz, Vrf = 100 V, and \(hc \in [20;50]\) μm for (top) \(wc \in [200;600]\) μm, \(wr \in [100;400]\) μm and (bottom) \(wc \in [70;100]\) μm, \(wr \in [70;100]\) μm. The secular frequency given with the trap efficiency parameter.

Full size image
Fig. 6.
figure 6

(Color online) Correlation between the secular frequency and the trap depth for the trap (top) without and (bottom) with the hole in the central electrode at \(\Omega = 2\pi \times 10\) MHz, Vrf = 100 V, and \(hc \in [20;50]\) μm for (top) \(wc \in [200;600]\) μm, \(wr \in [100;400]\) μm and (bottom) \(wc \in [70;100]\) μm, \(wr \in [70;100]\) μm.

Full size image

The graphs clearly show the distinct groups of dots due to the discrete changes in the parameters of the simulation process. This effect is particularly noticeable in the results obtained from the simulations of the trap without a hole in the central electrode, as only two parameters (\(wc\) and \(wr\)) were changed during the calculations. Based on the calculated data presented in Figs. 4–6, one can select appropriate parameters and fabricate the ion trap with customized characteristics to suit a specific range of applications.

Using these results, we have identified several designs for our future surface ion trap. The parameters of these designs chosen to meet the specific requirements and based on the outcomes of this study.

CONCLUSIONS

Our study demonstrates the development process of the chip-based surface ion traps with the electrodes located in a plane. Such an architecture is very suitable for applications such as the frequency standards and the scalable quantum computers. This construction design offers several advantages over other types of the traps, including enhanced optical access to stored ions and the ability to use the elements of photonic integrated circuits to deliver and collect the radiation to or from the ion.

In this research, we have presented the development process for the class of the segmented planar ion traps that is used for various applications. Our focus was primarily on the determination of the trap depth, the secular frequency, and the ion height above the surface. Our results can be used to design the traps with the designed characteristics and for the specific purposes. Furthermore, the scalability of the trap design allows the storage of extended chains of ions, making it a promising candidate for the quantum computation.