CPW Resonator for Circuit Quantum Electrodynamics

Developments in the last decade have made circuit quantum electrodynamics (cQED) the leading architecture for quantum computation. cQED is the solid-state version of cavity QED, which studies the basic light-matter interactions at the quantum level. This new architecture for quantum hardware has two main components: superconducting qubits and transmission line resonators. Superconducting qubits are the artificial meta-atoms that serve as a two-level quantum system and the transmission-line resonators are high-quality superconducting oscillators that play the role of cavities.

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 Plank’s constant, and f01 is the operating frequency of quantum qubits. This frequency is typically in the range of 4–8 GHz and also equal to the frequency of the microwave pulse used to coherently excite the qubits. Just like atoms, superconducting quantum qubits interact with microwave photons at quanta levels.

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

In this model, one of the main component of cQEDs, transmission-line resonators, are demonstrated. This resonator can be built from CPW transmission lines terminated with a combination of open and short ends. Open-short ends form nodes or antinodes at the end of the CPW which results in standing wave patterns within the transmission line and the transmission line serves as a resonator. Figure 1 illustrates the CPW resonator terminated with an open and a shorted end, also called a quarter-wave resonator, coupled to a CPW feeding line used in the model.

Model Definition

Figure 2 shows the schematic cross section of a CPW line that is used for the resonator and the feeding line. The impedance of the CPW is basically related to the ratio between center conductor width and gap width and the dielectric constants of the substrate. The conductive regions are simulated as perfect conductors for the sake of simplicity to mimic superconductors. A more realistic temperature-dependent superconductor model can be employed easily. Silicon is used as a substrate with relative permittivity 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 employed to excite and terminate the feeding CPW line. Therefore the boundary conditions on those surfaces are the corresponding mode fields. Radiation boundary conditions are used on the remaining boundaries. Since the loss is very small, the quality factor of the system is too high, and the resonance is very narrow. To make the simulation computationally efficient adaptive frequency sweep is employed.

In such a high-quality factor component, to capture the resonance behavior, a very fine frequency sweep is required in the vicinity of the resonance. By default, COMSOL stores field values for all the frequencies, within the 3D computational domain. If we proceed with the default settings, generated file which stores field distribution would be quite large. Since we are mainly interested in the field distribution on the CPW filter surface, to reduce the size of the file, we may only choose to store field distribution on that surface. For this purpose, geometric selections are created, and in the study settings, field values for that geometric selection are stored.

Results and Discussion

Figure 3 illustrates S parameters of the system which demonstrate resonance behavior. At the resonant frequency, all the energy is efficiently coupled to the resonator. At off resonances, electromagnetic energy does not interact with the resonator at all.

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.

Figure 4 demonstrates the standing wave formation in the resonator at the resonance frequency. Nodes and antinodes can be observed at the short and open ends of the resonator.

Notes About the COMSOL Implementation

Since the CPW resonator is a very high-quality factor system, it is a challenging structure to simulate. It is highly mesh-sensitive and a mesh-refinement study is necessary to make sure that the results are reliable. To show a simple modeling workflow, this model only focuses on the simulation of a CPW resonator and the mesh-refinement study is not included in the example. It requires ~20 GB of memory with the settings used here. To get accurate results, it is highly recommended to follow all the steps in this document; ignoring some of the settings may result in inaccuracies such as resonance shift. Even though this shift could be very small, since the system is very high Q, it could be challenging to find the resonance.

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

For the sake of simplicity, import geometry feature is used since the geometry is complicated and is used for . The CPW resonator contains high-aspect-ratio features, and it’s also highly mesh-dependent due to its very narrow bandwidth. It is important to use the to reproduce the results shown in this example. Otherwise, minor variations in the mesh structure may result in discrepancies. The could be used, however, a mesh refinement process should be performed accordingly. In general, the choice of the kernel does not matter. But, this is a very high-quality factor device, and special care should be taken.

Geometry 1

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 .

There is no analytical solution to define mode field of the CPW. Use numeric ports and perform boundary mode analyses. The field distribution obtained from the boundary mode analysis will be used for 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 .

Materials

In the toolbar, click

and choose .

Add Material

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

As mentioned in the port section, there is no analytical solution to define mode field of the CPW. Perform . The field distribution obtained from the boundary mode analysis will be used for 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 the field values within the computational domain, for each frequency in the study step. 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 omit the field values that we are not interested. For this purpose, a geometric selection could be generated and field values only within the geometric selection could be saved.

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 presmoother in the settings of to get a faster convergence and reduce computational time. The usage is limited in this model.

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. From the list, choose .

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

In the window, expand the node.

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 .

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.