Microwave Cavity Modelling

TRANSDUCER DESIGN AND SIMULATION

4.4 Microwave Cavity Modelling

Table 4.3: Simulated optical cavity parameters

Parameter Value

Mirror Scattering Q,𝑄_𝑜,𝑠 10⁵ Mirror Reflection Q,𝑄_{𝑜,𝑖𝑛} 10⁴(10⁵)

Mode Volume 7.6𝜇𝑚³

FSR 350 GHz

𝐹_𝐺 0.0542

# of Ions,𝑁_𝑜 2.1·10⁸

𝑔_{𝑜,𝑚 𝑎𝑥 ,∥} =2𝜋 783 kHz

𝑔_{𝑜,𝑚 𝑎𝑥 ,⊥} =2𝜋 115 kHz

𝑔_{𝑜,𝑡 𝑜𝑡 ,∥} =2𝜋 1.04 GHz

𝑔_{𝑜,𝑡 𝑜𝑡 ,⊥} =2𝜋 153 MHz

The full optical cavity device is shown in Figure 4.12. Light can be coupled into the optical cavity from the grating coupler (labelled G1) such that we couple through the low reflectivity mirror. The second grating coupler, labelled G2, is to allow for room temperature transmission measurements, but is not used once the device within the dilution fridge. The waveguide bend radius is 10𝜇m to ensure the waveguide is low loss. The grating coupler, G1, is tilted at angle of 40 degrees from the cavity section such that it is horizontally aligned when mounted in the fridge. The whole chip will be mounted at an angle to accommodate the magnetic field orientation that needs to be generated on the chip.

Low Reflectivity Mirror High Reflectivity Mirror

Grating Couplers

G1 G2

Figure 4.12: CAD image of the full optical cavity device including the two photonic crystal mirrors with a 100𝜇m waveguide in between. Two grating couplers are used for coupling into the cavity.

Figure 4.13: CAD image of the microwave resonator including the inductive wire next to the optical circuit, the interdigitated capacitor and the coupling waveguide.

Table 4.4: Microwave cavity geometry parameters

Parameter Value

Inductive Wire Length, 𝐿_𝑖𝑛𝑑 100 𝜇m Inductive Wire Width,𝑊_𝑖𝑛𝑑 1 𝜇m Capacitor Finger Length,𝐿_{𝑐𝑎 𝑝} 485 𝜇m

Capacitor Finger Width,𝑊_{𝑐𝑎 𝑝} 10𝜇m Capacitor Finger Gap,𝐺_{𝑐𝑎 𝑝} 5 𝜇m Capacitor Finger Number,𝑁_{𝑐𝑎 𝑝} 20

Optical Gap Width,𝐺_𝑜,𝑤 235 𝜇m Optical Gap Height,𝐺_{𝑜, ℎ} 55𝜇m Waveguide Coupling Gap,𝐺_{𝑤 𝑔,𝑐𝑜𝑢} 4 𝜇m

Waveguide Width,𝑊_{𝑤 𝑔} 10𝜇m Waveguide Gap,𝐺_{𝑤 𝑔} 4.5𝜇m

The cavity geometry parameters are summarized in Table 4.4 and are shown in Figure 4.14. The length of the inductive wire length is 100𝜇m to match the length of the optical cavity and the inductive width is 1 𝜇m. In order to change the resonance frequency of the microwave cavity, the length of the capacitor fingers and the number of capacitor fingers was adjusted, but the resonator that was used for the main experiments has its parameters in the Table 4.4. On a given sample, we fabricated 10 resonators and tune their resonance frequencies∼200-300 MHz apart from each other, such that they would span∼4.5-7 GHz.

Figure 4.14: The geometric parameters that define the pattern of the microwave resonator including the parameters related to the inductive wire, the interdigitated capacitor, the coupling waveguide and the gap for the optical resonator.

The microwave cavity was modelled in COMSOL to determine the electromagnetic fields of its fundamental resonance mode. The normalized magnetic field distri- bution in log-scale is shown in Figure 4.15. In Figure 4.15a, the magnetic field is shown of the device plane, where the magnetic field is mostly confined to the induc- tive wire and the regions of the capacitor that are nearby. The maximum magnetic field is confined to the inductive wire (Figure 4.15b). The black line is a guide to the eye of the location of the optical resonator, where the magnetic field is near its peak value. We can also look at the cross-section of the mode in Figure 4.15c and see how the magnetic field strength decays away from the inductive wire.

From the magnetic field distribution, we can calculate a few different parameters.

First, we determine the magnetic field mode volume to be 156 𝜇m³. Next, we can simulate how the magnetic field decays as a function of the distance from the inductive wire (Figure 4.16). The optical resonator will reside next to the inductive wire, so we can see how much we can improve by decreasing the distance between the inductive wire and the optical resonator. We observe a 𝐵/𝐵_{𝑚 𝑎𝑥} ∼ 1/𝐷, where 𝐷is the distance from the inductive wire.

a) _{𝐵 / 𝐵𝑚𝑎𝑥} b)

c)

Figure 4.15: The normalized magnetic field distribution of the microwave resonator.

a) The in-plane magnetic field distribution in the plane of the niobium resonator itself. b) A closer look at the magnetic field distribution near the inductive wire.

The black line indicates the location of the optical resonator for reference. c) The cross-section of the microwave resonator magnetic field.

For our devices, we choose to center the optical cavity waveguide 1.5 𝜇m away from the inductor (i.e. the nearest edges of the optical waveguide and the supercon- ducting resonator is 1.2 𝜇m apart), which results in a value of𝐵/𝐵_{𝑚 𝑎𝑥} =0.043. In principle, we can decrease the gap and increase the microwave resonator magnetic field at the optical resonator, but we did not want to push this too extreme. If the distance between the two is too close, the niobium will induce optical losses on the optical resonator, the optical resonator losses will induce more quasi-particles in the superconductor and the fabrication process may start to get a bit trickier.

The next thing we can look at is the histogram of the different ion-cavity coupling rate to the 𝑍₁ ground state spin as shown in Figure 4.17. The coupling rate bin size is 0.0086 dB. The ions right at the surface of the superconducting have a maximum coupling rate of 𝑔_{𝜇,𝑚 𝑎𝑥}/2𝜋 = 5.8 kHz, while the ions at the position of the optical cavity have a coupling rate of𝑔_𝜇,𝑜/2𝜋 =255 Hz (as denoted by the red dotted line in the Figure 4.17). The excited state spins have a dipole moment that is𝜇_21,𝑌

1 =0.65𝜇_{21, 𝑍}

1 for the out-of-plane magnetic field component, so the excited state spin coupling rates can be scaled by that factor.

0 1 2 3 4 5 Distance from Inductor [ m]

0 0.1 0.2 0.3 0.4 0.5 0.6

B/B max

Figure 4.16: Microwave resonator normalized magnetic field strength as a function of the distance from the inductive wire.

Again, we define the number of ions as the ions with the largest coupling rates that make up 99.9% of the total ensemble coupling. This corresponds to∼ 1.64·10¹⁵ ions within the microwave cavity. Alternatively, if we define the number of ions as the 𝑁 ∼𝑉_𝜇 ·𝜌, we estimate ∼ 8.4·10⁸ions in the cavity. This suggests that there are a lot weakly coupled ions to the microwave cavity.

The ensemble coupling between the ground state spin to cavity is calculated to be 120 MHz, where we have included contributions for all magnetic field directions and the associated dipole moments of the spin along each direction. 83 MHz of this coupling is along the out-of-plane direction. We expect there to be∼2.1·10⁸ions within the optical cavity so the coupling of these spins to the microwave cavity is

∼3.6 MHz.

For the𝑌₁ excited state spin, we calculate a total ensemble coupling of 91 MHz with 53 MHz coming from the out-of-plane component. The contribution from ions within the optical cavity is∼2.3 MHz.

We can use the electromagnetic field distribution to determine the transduction figure of merit that relates to the microwave cavity as shown in Equation 4.1 and get 𝐹_𝜇 =3.55·10⁶m^−3/2.

10⁰ 10¹ 10² 10³ g [Hz]

10⁰ 10³ 10⁶ 10⁹ 10¹²

Number of Ions

Figure 4.17: Histogram of the single ion coupling rate between the ions and the microwave cavity. Each bin size is 0.0086 dB. The red line indicates the coupling rate for the ions that are also positioned within the optical cavity.

We can also define the more canonical circuit parameters of the lumped element microwave resonator. We can model the resonator as typical LC resonator, where the inductance has contributions from the inductive wire, 𝐿_{𝑤𝑖𝑟 𝑒}, and a parasitic inductance from the interdigitated capacitor, 𝐿𝑝 𝑎𝑟 𝑎 𝑠𝑖𝑡𝑖 𝑐, such that 𝐿 = 𝐿𝑝 𝑎𝑟 𝑎 𝑠𝑖𝑡𝑖 𝑐 + 𝐿_{𝑤𝑖𝑟 𝑒}. The circuit parameters were determined in Sonnet by adding ideal lumped element components into the model, measuring the shift in the resonance frequency and using𝜔_𝜇,0 = ^√1

𝐿𝐶, where𝜔_𝜇,0 is the resonance frequency,𝐿 is the inductance and𝐶 is the capacitance of the resonator.

From simulation, we determine that𝐿 =964 pH, 𝐿_{𝑤𝑖𝑟 𝑒} =208 pH,𝐿𝑝 𝑎𝑟 𝑎 𝑠𝑖𝑡𝑖 𝑐 =756 pH and𝐶 =839 pF. This results in a characteristic impedance of 𝑍₀ =

√︃

𝐿

𝐶 =33.9 Ω. The inductance fraction within the inductive wire (i.e. 𝐿_{𝑤𝑖𝑟 𝑒}/𝐿) is only 22% due to the parasitic inductance of the interdigitated capacitor. Reducing this parasitic capacitance would result in a smaller magnetic mode volume and increase the coupling rate of the spins to the microwave cavity.

Table 4.5: Simulated microwave cavity parameters

Parameter Value

Magnetic Mode Volume,𝑉_𝜇 156 𝜇m³ Spin-coupling rate at optical cavity𝑔_𝜇,𝑜/2𝜋 255 Hz (166 Hz)

𝑔_{𝜇,𝑚 𝑎𝑥}/2𝜋 5.6 kHz (3.6 kHz)

Total spin-coupling rate,𝑔_{𝜇,𝑡 𝑜𝑡} 120 MHz (91 MHz)

Inductance 964 pH

Capacitance 839 pF

Impedance 33.9Ω

𝐿_{𝑤𝑖𝑟 𝑒}/𝐿𝑝 𝑎𝑟 𝑎 𝑠𝑖𝑡𝑖 𝑐 0.22

Coupling Q,𝑄_{𝜇,𝑖𝑛} 10414

F𝜇 3.55·10⁶m^−3/2

In Sonnet, we also simulated the coupling quality factor (or coupling capacitance) of the coplanar waveguide. There is a ground plane in between the waveguide and resonator, 𝐺_{𝑤 𝑔,𝑐𝑜𝑢} = 4 𝜇m, to tune the coupling between the two. From simulation, we determine a coupling quality factor to be𝑄_{𝜇,𝑖𝑛} = 10414. The co- planar waveguide is a convenient way to couple to the resonators as we can frequency multiplex several resonators. However, this does reduce the device efficiency by up to 2x as the microwave cavity cannot exceed critical coupling.