CPW Resonator for Circuit Quantum Electrodynamics

Developments in the last decade have led to circuit quantum electrodynamics (cQED) becoming the leading architecture candidate for quantum computation. The cQED architecture for quantum hardware has three main components: superconducting qubits, transmission lines, and transmission line resonators. Superconducting qubits are the artificial meta-atoms that serve as a two-level quantum system, transmission line resonators are high-quality superconducting oscillators that play the role of cavities, and transmission lines route energy through the architecture.

The energy difference between the quantum states of superconducting qubits is given by E01 = hf01, wherein a two-level quantum system E01 is the energy difference between the ground state and the excited state, h is Planck’s constant, and f01 is the transition frequency between the states. This frequency is typically designed to be in the range of 4–8 GHz so that it is above the thermal energy of dilution refrigerators (~10 mK) but also well below the superconducting gap of any constituent materials, like aluminum (82 GHz). Just like atoms, superconducting quantum qubits interact with microwave photons at quanta levels.

Figure 1: A CPW resonator coupled to a CPW transmission line. The air domains are removed for a better view.

In this model, one of the main components of cQEDs, a transmission line resonator, is demonstrated. This resonator can be built from CPW transmission lines terminated with a combination of open and short ends. These ends create a resonator out of a CPW, with the open and short ends functioning as zero current and zero voltage boundary conditions, respectively. Figure 1 illustrates a quarter-wave resonator, which is formed from a CPW terminated with an open and a shorted end, and shows how the quarter-wave resonator can be coupled to a CPW feeding line.

Model Definition

Figure 2 shows the schematic cross section of the CPW line used for the resonator and the feed line. The impedance of CPW is related to the dielectric constant of the substrate and the ratio between the center conductor and gap widths. The conductive regions are treated as perfect conductors to capture the lossless behavior of the superconducting metal, and these regions are also treated as 2D layer because they are much thinner than any other relevant length scales in the model. The substrate is silicon with a relative permittivity of 11.7.

Figure 2: Schematic of the CPW cross section where w/d=7/4 and the characteristic impedance is 50 Ω.

Numeric ports are used to excite and terminate the feeding CPW line. Therefore, the boundary conditions on those surfaces are the corresponding mode fields. Scattering boundary conditions are used on the remaining boundaries, although due to the highly confined mode structure of CPW this results in very little loss in the system and an extremely high the quality. To model this system in a computationally efficient manner, an adaptive frequency sweep is used. This is because very fine frequency resolution is required to capture the narrow bandwidth of the resonance, and an adaptive frequency sweep allows evaluation of many frequency points without explicit calculation of each and every one.

To further reduce the model size, the field distribution is only stored on the CPW filter surface. By default, COMSOL stores the electromagnetic field values for the entire computational domain for each frequency evaluated, but because of the large number of frequencies accounted for in the adaptive frequency sweep this can result in a massive file size. Since we are mainly interested in the field distribution on the CPW filter surface, we choose to only store field for that surface, and this can be done through the use of geometric selections in the study settings.

Results and Discussion

Figure 3 shows the S-parameters of the system. The behavior of the quarter-wave resonator can be seen in the strong resonant reflectivity in S11dB, whereas there feed line is highly transmissive away from the resonance frequency. This is further illustrated in Figure 4, which shows the standing wave formation in the CPW resonator. There is a large electric field enhancement at the open end and zero field at the shorted end, consistent with the anticipated behavior of the boundary conditions.

Figure 3: The S-parameters plot demonstrates a very narrow resonance behavior.

Figure 4: Illustration of standing wave pattern formed within the resonator. The height distribution corresponds to the total electric field. Antinode and node can be observed at the open and short ends.

Notes About the COMSOL Implementation

In this model we conduct two studies. The first is an eigenfrequency simulation to find the resonant frequency of the structure. The second performs a frequency sweep +/- 3 MHz around this point to calculate the S-Parameters for the feed line.

Since the CPW resonator is a very high-quality factor system, it can be a challenging structure to simulate. High-Q systems can be extremely mesh sensitive, and a mesh refinement study is necessary to ensure reliable results. Very fine meshes in turn require more memory, which may require fine tuning of the solver settings for large models. For this demonstration, you can obtain a reasonable mesh using the feature and a size setting close to the dielectric gap width. In your own modeling, we recommend performing a mesh refinement investigation. Inherent in a mesh refinement study is a tradeoff between computational resources and accuracy, and so an important question to consider is exactly how accurately the resonance frequency needs to be known. Slight changes to the mesh used here can result in shifts on the order of ~1 MHz, and the model takes ~20 GB of memory to solve. As a result of this large model size, the Eigenfrequency solver settings require slight modifications to converge.

RF_Module/Filters/cpw_resonator

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

To focus on the model physics and implementation, the import geometry feature is used to build the structure. Because the structure is highly sensitive to the mesh, and the mesh is in turn sensitive to the geometry, it is important to use the to exactly reproduce the results in this example. The could be used as well, however a mesh refinement process should be performed accordingly as the resonance may shift slightly despite identical mesh settings.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Import 1 (imp1)

In the toolbar, click

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file cpw_resonator.mphbin.

Click

Click the

button in the toolbar.

Electromagnetic Waves, Frequency Domain (emw)

Perfect Electric Conductor 2

In the window, under right-click and choose .

Select Boundaries 9, 15, and 17 only.

Scattering Boundary Condition 1

In the toolbar, click

and choose .

Select Boundaries 1, 3, 4, 7, 10, and 19–21 only.

Port 1

In the toolbar, click

and choose .

To excite the CPW we need the mode profile of the electromagnetic fields. Since there is no analytical equation to define these fields, we use numeric ports and calculate them as a preprocessing step using two Boundary Mode Analysis study steps. The field distributions obtained are used for both the eigenfrequency and frequency domain analysis. Since a quasi-TEM wave is propagating on a CPW, use the option and define .

Select Boundaries 2, 5, and 8 only.

In the window for , locate the section.

From the list, choose .

Select the check box.

Integration Line for Voltage 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Select Edge 47 only.

Port 2

In the toolbar, click

and choose .

Select Boundaries 11–13 only.

In the window for , locate the section.

From the list, choose .

Select the check box.

Integration Line for Voltage 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Select Edge 48 only.

Locate the section. Click .

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Materials

Silicon (mat2)

Select Domain 2 only.

Field is confined in the close vicinity of the CPW gaps. Use to refine the mesh in the vicinity of CPW gap.

Mesh 1

In the window, under click .

In the window for , locate the section.

Select the check box.

From the list, choose .

In the text field, type 5[um].

Click

To see the mesh structure on the CPW surface, Use .

Definitions

Hide for Physics 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 7, 8, and 10 only.

Mesh 1

Here we perform the to calculate the mode profiles. The field distributions obtained will be used for both the and .

Study 1

Boundary Mode Analysis

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 5[GHz].

In the text field, type 2.5217.

Step 2: Boundary Mode Analysis 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 2.

Eigenfrequency

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 4.94[GHz].

Select the check box. In the associated text field, type 1.

From the list, choose .

By default, COMSOL stores all field values for every spatial location and frequency considered in the model. For a densely meshed problem with a fine frequency sweep, the size of the automatically generated result file could be extremely large. To reduce the file size, we can select a spatial subset of the model the store only the values of interested. For this purpose, we create a geometric selection and to save only the values within that selection.

Definitions

Explicit 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Click

In the dialog box, type 2, 5, 8, 9, 11-18 in the text field.

Click .

Study 1

Step 3: Eigenfrequency

In the window, under click .

In the window for , click to expand the section.

Find the subsection. From the list, choose .

Under , click

In the dialog box, select in the list.

Click .

Solution 1 (sol1)

In the toolbar, click

For this specific example using the combination of boundary mode analysis and eigenfrequency, one can take the advantage Vanka presmoother in the settings of Eigenvalue Solver to get a faster convergence and reduce computational time.

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type 1.0E-5.

In the window, expand the node.

Right-click and choose .

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the text field, type 1.

Under , click

In the dialog box, select in the list.

Click .

In the window for , locate the section.

From the list, choose .

In the text field, type 1.

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type 1.

In the text field, type 0.5.

In the toolbar, click

Results

Multislice

In the window, expand the node, then click .

In the window for , locate the section.

Find the subsection. In the text field, type 0.

Locate the section. Click

In the dialog box, select in the tree.

Click .

Deformation 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.

In the text field, type emw.normE.

In the toolbar, click

Surface 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Study 1

Step 1: Boundary Mode Analysis, Step 2: Boundary Mode Analysis 1

In the window, under , Ctrl-click to select and .

Right-click and choose .

Study 2

Step 1: Boundary Mode Analysis

In the window, right-click and choose .

In the window for , locate the section.

Clear the check box.

Adaptive Frequency Sweep

In the toolbar, click

and choose .

Use the evaluated resonant frequency for frequency sweep study settings that can be directly copied from the . Right click on the frequency value in the to access the context menu and choose for use later.

In the input field, sweep +-3[MHz] around the resonant frequency with 0.1[MHz] step. For instance, if the computed resonant frequency is 5[GHz], then type range(5[GHz]-3[MHz],0.1[MHz],5[GHz]+3[MHz]).

Since the CPW resonator has a very sharp resonance, the can be utilized to reduce computational time. A good choice for the increases the efficiency of adaptive frequency sweep. Magnitude of S11 is a suitable choice for this problem to decrease computational cost.