Integrated Ion Traps using Surface Electrodes and Grating Couplers

Surface electrodes adjacent to grating couplers render an integrated ion trap. Here, we present a workflow, where we model surface electrodes using ANSYS Maxwell and grating couplers using ANSYS Lumerical . Surface electrodes have been used for long time to trap ions, whereas the combination with grating couplers only recently arise as a solution to get a fully integrated platform where optical and electric fields can interact with ions simultaneously. In this way, quantum computing may get closer to commercialization performed with a single, integrated device.

[[NOTE:]] Lumerical 2023 R1.0 edition and ANSYS Maxwell 2024 R1.02 editions are required for this workflow. Step 1 and 2 of this workflow are also presented as part of the PyANSYS for a fully automated workflow.

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Overview

Understand the simulation workflow and key results

Step 1: Electrostatic simulations of surface electrodes

We use ANSYS Maxwell to simulate the electrostatic response of the ion trap using the surface electrodes in a 3-rail surface electrode design.

Step 2: 2D optimization of an apodised grating coupler

Here, we feed the coordinates of the ion trap to the PS optimization algorithm with the goal to define the optimal two dimensional grating coupler design, which will focus the laser beam at the ion trap height. These two first steps, are also part of the PyANSYS workflow.

Step 3: 3D model of a single and 2x2 grating coupler array

We employ the optimized parameters of the grating as defined in step 2, in a three dimensional FDTD space, so to get the 3D field distribution. We also showcase how multiple grating couplers can provide a platform for more complex field distributions.

Step 4: Parameter sweep

We calculate the excited optical force on a 100nm polystyrene bead from a 2 by 2 grating coupler array and define the stability of the optical trap.

Run and results

Instructions for running the model and discussion of key results

Step 1: Electrostatic simulations of surface electrodes

Here, we calculate based on the finite element method solver, (FEM) the ion trap height. Below is shown a schematic of the electrodes design (upper left). The distribution of the electric field along the red line is calculated and the field is found to be zero at the y-coordinate 81.175 um (see bottom figure). This is the point where the ions feel minimal or no force, the so called nodal point, creating a null point. Ions are trapped near these nodal points, which act as potential minima where the ions' micromotion is minimized, allowing for stable trapping and precise control of their motion [1]. The ion trap height is dependent of the width ratio of the RF and DC electrodes. A parametric study on the impact of the ion trap height with the width ratio between the middle electrode and the side electrodes is presented on the table below and is accessible from the Project Manager>Optimetrics>w_dc_sweep. We choose the ratio value of 1.78 (annotated within yellow cells), between the RF and the DC electrode width, similar to the design parameters of the publication [2]. The design is a 2D design assuming the electrodes extending infinitely long in the out of plane direction. The ion trap is defined in the position where the field is zero at y=81.175 um, you may access the corresponding data; double-click on the Results>Plot_JNUXKH. A list of the distance over field will pop-up.

Step 2: 2D optimization of an apodised grating coupler

The grating used here is an apodised grating. Grating couplers consist of periodic refractive index change between the trenches and the un-etched region, allowing the guided light to be diffracted to free space. Here, we introduce a linear apodization on the grating to improve diffraction in free space [3]. By linearly varying the fill factor value along the grating the optical impedance matching between the waveguide and the grating section is improved and produces a self-focused output beam with a Gaussian-like shape.

The desired ion trap height is equal to 81.18 um and it is defined as the FOM for the optimization analysis. The actual focal point is at 85.61 um above the grating coupler, while the target distance was 81.18 um. This is a deviation is in the order of 3%. Further optimization can lead to even smaller discrepancies.

Users may want to change the geometry of the surface electrodes, therefore the FoM_beam analysis group has to change accordingly. Furthermore, modifications in the optical simulation are also possible, such as the wavelength of the laser beam or alternative materials.

Step 3: 3D model of a single and 2x2 grating coupler array

The side view and top view image of the light coming out of the 3D grating coupler is presented below. The top view of the E-field at the ion trap height is 81.75um.

Minor discrepancies between the 2D and 3D model can originate from the tapering of the waveguide to the grating coupler. Also, the focused beam has an oval shape, mainly because the grating coupler is not optimized to produce a rounded beam at focus.

Note: the fields  presented above are the interpolated fields. To plot the raw data, please uncomment the corresponding script lines.

A top view of the refractive index profile of the apodised grating is presented below. The color map below is plotted from the monitor named: index x_y.

Next, we also model the electric field distribution stemming from four counterpropagating laser beams intersecting at the focal point of 82.5 um, see schematic below.

In order to do this, we perform a three-fold rotation of the calculated far-field dataset produced from a single grating coupler and we add them coherently. The combined field distribution is symmetric and the beam waist is smaller (namely the FWHM is 1.8 um) than the beam waist produced from a single grating coupler. Next, we construct a rectilinear dataset with this data, and use it in the next step of the workflow as an imported source. Next we calculate the optical forces excited on a 50 nm radius polysterene nanobead.

Step 4: Calculation of force on a dielectric nanoparticle

In this final step, we consider a polystyrene nanobead of 50 nm in radius and a refractive index of 1.57, [5] which is illuminated by the light field calculated in the previous step of the workflow ( 4 grating couplers focusing the laser beam on the same focal point). The script uses a nested loop, where a 3D FDTD simulation runs repeatedly for the nanobead moving with a step of half micron across the x-y plane. Together with the nanobead, it is also moving an analysis box, that calculates the Maxwell Stress Tensor, see here for more details. After this process is complete, then we interpolate the fields so to get the final force colour maps, as shown below. Users can plot the raw data, if they uncomment the script lines 50 and 51. As we can see the Fx, and Fy distribution maps support negative slope of the dF/dx and dF/dy , therefore hold the conditions where the nanobead can be trapped. When the gradient of the force is positive the forces are repulsive and cannot keep the nanoobject in a fixed position. This is the case for the optical force at the (30,4) & (30,-4) position. The colour bars are normalized to the power of the source and have Newton units, where the maximum force is recorded to be up to 100fN.

This response should be studied in combination with the ion trap. As we see from the above distribution of force, there are several points where the nanobead can be trapped. These points are located in the periphery of the focal point of the combined beams rather than the center. If optical forces are not attractive in one axis, then the use of the electrostatic trap can act complimentary and form a stable ion trap. When the grating coupler design is satisfied for users application, it can be converted into GDS file using the instruction on this article.

Updating the model with your parameters

Instructions for updating the model based on your device parameters

Waveguide & grating coupler refractive index: The chosen refractive index values are representative of a silicon nitride waveguide embedded within silicon dioxide. users may want to modify this accordingly to the fabrication technology.

Operation wavelength: Here, we consider the telecomunication wavelength of 1550 nm as the light source, users may want to use other laser of higher power e.g. 1064 nm etc.

Source mode: Herem, we use TE mode as the propagating mode within the waveguide, users may want to modify to TM or other hybrid modes.

Combination of fields: Here, we combine 4 electromagnetic field datasets at the step 3 of the workflow. Users may want to use combination of two, a single or even more than 4 gratings, with the approriate modification of the script file 3D_4GC.lsf.

Important model settings

Description of important objects and settings used in this model

Pitch of the grating & etch depth

The low refractive index contrast between silicon nitride and silicon oxide, enables the tuning of the exit angle both to positive and negative angles using the pitch and etch depth as the main design parameters. Below we calculate the fields from the simulation file (step 2) for pitch size: 820 nm, etch depth: 300 nm (left) and pitch size: 1050 nm, etch depth: 150 nm, (right).

Nanoparticle refractive index

In the step 4 of this workflow, we use a low refractive index polystyrene nanobead, so to minimize the scattering from the nanoparticle. A particle made of higher refractive index e.g. Silicon, will immediately interact stronger with the laser beam, as it has a larger cross-section and therefore modify the gradient of the optical forces. Below are presented the colour maps of the E-fields applied to the nanobead of polysterene and Silicon, respectively. They are produced from the provided simulation file  by changing the refractive index of the nanobead.

Optimization figure of merit (FOM)

Since the purpose of the design is to have the best possible focus at the desired focal point, the optimization figure of merit is chosen to be the maximization of the electric field at the target height. This figure of merit is calculated by an analysis script in the “model” object, called FoM_beam.

Taking the model further

Information and tips for users that want to further customize the model

2D optimization : Here, we define a FoM that will steer the laser beam at the focal point calculated from the first step of the workflow. Another FoM can be the maximisation of the optical force at a given focal point. Or users may consider to design multiple traps, where more than one ions could be controlled, something very useful in quantum computing.

Taper optimization : In step 3 we use a simple non-adiabatic taper between the waveguide and the grating,[4] that may reduce considerably the optical power deliver to the ion trap. An optimization of the 3D FDTD design would help on this direction, however it may be time consuming and demand further memory resources.

Intersecting laser beams & grating combination : The combined fields from the gratings can result in several configuration. The intersection point of the laser beams, may offer an extra degree of freedom for this workflow where multiple optical traps can be designed.

Temperature dependent response : Typically ion integrated traps operate at low pressure and low temperature. Modification on the refractive index properties through the thermo-optic coefficient of the materials use can guide users on the correct grating design.

BEM based electronic simulations :With respect to the electrode design for surface ion traps, several studies present analytical models. However, these models cannot be simply applied in practical designs due to the complex structures of surface ion-trap chips. There also have been many studies on designing electrode dimensions using numerical simulations. Boundary element method (BEM) simulations can also be an alternative method. ANSYS support such solution using Q3D Extractor .

INTERCONNECT : Users may use this grating coupler as a compound element of the Photonic Integrated Circuit Simulator, INTERCONNECT with additional capabilities to model quantum effect using a dedicated module, the qInterconnect .

Additional resources

Additional documentation, examples and training material

Related publications

1. Integrated ion traps : Niffenegger, R.J., Stuart, J., Sorace-Agaskar, C. et al. Integrated multi-wavelength control of an ion qubit. Nature 586, 538–542 (2020).
2. Basic surface electrodes design : U. Tanaka, Kensuke Suziki, Yuki Ibaraki and Shinji Urabe, “Design of a surface electrode trap for parallel ion strings,” J. Phys, B: At, Mol. Opt, Phys, 47 (2014) 035301.
3. Apodized grating couplers design methods : Z. Zhao, S. Fan, Design ‘Principles of Apodized Grating Couplers,” J. Lightwave Technol. 2020, 38, 4435–4446.
4. Adiabatic and non-Adiabatic Tapering waveguides : Yunfei Fu, Tong Ye, Weijie Tang, and Tao Chu, "Efficient adiabatic silicon-on-insulator waveguide taper," Photon. Res. 2, A41-A44 (2014). "Efficient tapering to the fundamental quasi-TM mode in asymmetrical waveguides,"
   D Vermeulen, K Van Acoleyen, et. al. 15th European conference on Integrated Optics (ECIO 2010), 2010.
5. Using Mie Scattering to Determine the Wavelength-Dependent Refractive Index of Polystyrene Beads with Changing Temperature . Megan R. McGrory, Martin D. King, and Andrew D. Ward. The Journal of Physical Chemistry A 2020 124 (46), 9617-9625.
6. Surface Electrode design rules: Hong, Seokjun, Minjae Lee, Hongjin Cheon, Taehyun Kim, and Dong-il “Dan” Cho. 2016. "Guidelines for Designing Surface Ion Traps Using the Boundary Element Method" Sensors 16, no. 5: 616. https://doi.org/10.3390/s16050616

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