Characterization of low loss microstrip resonators as a building block for circuit QED in a 3D waveguide

Microwave resonators are an important building block for circuit QED systems where they are e.g. used for qubit readout,^1,2 to mediate coupling³ and for parametric amplifiers.⁴ All of these applications require low intrinsic losses at low temperatures (⁠$kBT≪hfr$⁠) and single photon drive strength. In this low energy regime, the intrinsic quality factor is often limited by dissipation due to two level systems (TLS).^5,6 These defects exist mainly in metal-air, metal-substrate and substrate-air interfaces as well as in bulk dielectrics.^6–9 Two common approaches exist, to improve the intrinsic quality factor of resonators. Either one reduces the sensitivity to these loss mechanisms by reducing the participation ratio^6,10,11 or tries to improve the interfaces by a sophisticated fabrication process.^12,13 Reducing the participation ratio requires to decrease the electric field strength. This is typically done by increasing the size of the resonator⁸ or even implementing the resonator using three dimensional structures.¹⁰ 

Our approach, a microstrip resonator (MSR) in a rectangular waveguide (Fig. 1), combines the advantages of three dimensional structures with a compact, planar design.^2,14 The U-shaped MSR (Fig. 1(a)) is effectively a capacitively shunted $λ/2$ resonator.¹⁵ The sensitivity to interfaces is reduced, since the majority of the field is spread out over the waveguide, effectively reducing the participation ratio.¹¹ Moreover, the U-structure allows a tuneable coupling, by changing the position within the waveguide. Another advantage is, that the waveguide represents a clean and well controlled microwave environment¹⁵ without lossy seams¹⁶ close to the MSR. As the MSR is capacitively coupled to the waveguide, no wirebonds¹⁷ or airbridges¹⁸ are required, which can lead to dissipation or crosstalk.

To assess the performance of different materials, we investigate aluminum and niobium MSRs. The samples were fabricated using standard optical lithography techniques and sputter deposition of the metallic films. Structuring of the metal layer was done using a wet etching process for the aluminum samples and a reactive ion etching (RIE) process for niobium. After completely removing the photoresist, both samples were cleaned in an oxygen plasma.

For microwave transmission measurements, we place the MSR in a rectangular waveguide (Fig. 1(b)). The fundamental TE₁₀ mode, which has electric field components only along the y-axis, is the sole propagating mode at the resonance frequency of the MSR. Its field strength varies along the x-axis with a maximum in the center¹⁵ (dashed line in Fig. 1(b)). For the MSR placed off-center, the field strength is different on both legs, which leads to a capacitive coupling to the waveguide. Placed in the exact center of the waveguide, the field strength is equal on both legs of the MSR and the coupling vanishes. Instead of changing the position of the MSR in the waveguide, to change the coupling, we can also fabricate a MSR with legs of different length. To accurately predict the interaction of the MSR with the waveguide we performed simulations of the whole structure using a finite element solver (supplementary material).

We characterized the MSRs in waveguides fabricated from copper or aluminum. The waveguides were mounted to the baseplate of a dilution refrigerator and cooled down to 20 mK. The MSRs were analyzed regarding their resonance frequencies and quality factors by measuring S₂₁. We fit the measured data using a circle fit routine¹⁹ which utilizes the complex nature of the S-parameter (supplementary material).

Two sets of measurements were performed. First, we measured the MSRs under variation of input powers, ranging from below the single photon limit to several million photons circulating in the resonator. Second, we stepwise increased the base temperature to 1 K and performed measurements at single photon powers.

In Fig. 2(a) we show the dependence of the internal quality factor on the circulating photon number in the MSR. All measurements show a clear trend of an increasing quality factor with the number of photons. This indicates that the MSRs are limited by TLS losses, as they get saturated with increasing drive powers.⁵ We measure the highest single photon internal quality factor of one million for the two niobium MSRs placed in the aluminum waveguide. For high powers we even measure a $Qint$ of more than 8 million (supplementary material). Other experiments, using a more sophisticated fabrication process, report similar internal quality factors for planar NbTiN resonators on deep etched silicon¹² or for planar aluminum resonators on sapphire.¹³ Similar methods and materials might allow us to increase the single photon quality factor of the MSR.

The trend of increasing $Qint$ is weakest for the aluminum MSR in the copper waveguide, which indicates that this MSR is not limited by TLS. We rather believe that the normal conducting copper waveguide does not shield external fields. Thus vortices might limit the performance of the aluminum MSR in the copper waveguide.²⁰ We do not observe this effect for the niobium MSR, due to its higher critical field.²¹ The difference in quality factor of the niobium stripline in the copper and in the aluminum waveguide can be attributed to losses to the copper wall, as suggested by simulations.

We expect two effects on the internal quality factor, when raising the temperature of the MSRs. Approaching the critical temperature leads to a decrease of $Qint$⁠, due to an increasing surface impedance. Considering a two fluid model,²² the following temperature dependence is found

Here T is the temperature, $Δ$ the superconducting gap at zero temperature, k_B the Boltzmann constant and A a constant. An additional $Qother$ accounts for other temperature independent losses. This model is expected to show good agreement until T_c/2.²³ 

TLS saturate with increasing temperature, which leads to an increase in quality factor⁵ 

Where k is the loss parameter and hf_r(T) represents the energy of the TLS at the resonance frequency of the MSR for a given temperature. The resonance frequency barely changes with temperature (Fig. 3(b)), which allows us to fix the frequency of the TLS to the resonance frequency of the MSR in the low temperature limit. $Qother′$ is analogue to Eq. 1.

In Fig. 2(b) we plot the dependence of the internal quality factor of the aluminum MSRs on the base temperature of the dilution cryostat. We fit the data to a combined model (supplementary material) of TLS related losses (Eq. 2) and conductive losses (Eq. 1). Until about 200 mK the MSRs in the copper waveguide show a constant internal quality factor. This gives further evidence that dissipation due to TLS is not the dominant loss mechanism for the aluminum MSRs in the copper waveguide. In the aluminum waveguide we see an increase in $Qint$ with temperature until 200 mK. Thus in this waveguide, TLS related losses most likely limit the quality factor of the MSR. Above 400 mK all MSRs show a similar decrease in $Qint$⁠. This can be attributed to conductive losses, as the critical temperature of aluminum is around 1.19 K.²³ Near the critical temperature, we measure an internal quality factor slightly above 1000. This is close to the results of finite element simulations, which predict an internal quality factor of about 500.

In Fig. 3(a) we plot the temperature dependence of the internal quality factor of the niobium MSRs. Niobium has a critical temperature of about 9.2 K,²³ hence we do not expect to observe a breakdown of superconductivity. Thus, we only fit the data with the model describing TLS related losses (Eq. 2). The behavior of the MSR in the copper waveguide agrees well with predictions from theory throughout the whole measurement range. We observe an increase of $Qint$ up to 1 K. For the MSR in the aluminum waveguide we measure a drop in the internal quality factor at 350 mK. In this region we also see the breakdown of superconductivity for the aluminum MSRs (Fig. 2(b)). This indicates that the breakdown of superconductivity in the waveguide walls is the limiting factor here. For higher temperatures, the internal quality factor remains approximately constant around 1 $×$ 10⁶. Performing finite element simulations using the finite conductivity of the aluminum (Al5083²⁴) waveguide wall we found a $Qint$ of 1.16 $×$ 10⁶, which is consistent with our measurements.

TLS also lead to a shift in the resonance frequency⁵ 

Here $Ψ$ is the complex digamma function. Fig. 3(b) shows the frequency shift when increasing the temperature of the cryostat. In contrast to the effect on $Qint$⁠, off resonant TLS contribute to the frequency shift,⁵ which makes the resonance frequency independent of power (supplementary material). The only fit parameter is the combined loss parameter, k. For measurements of the Nb MSR in the aluminum waveguide, we observe a drop in the frequency shift above 350 mK. We attribute this again to the breakdown of superconductivity in the waveguide wall. Below 350 mK, the measurements are in good agreement with the model. In case of the niobium MSR in the copper waveguide, we observe agreement throughout the whole measurement range.

The values obtained for k (supplementary material) by fitting the shift of the resonance frequency are about 10% to 30% lower, than fitting the change of the internal quality factor (Fig. 3(a)). This can be attributed to a non-uniform frequency distribution of TLS,⁵ which leads to a difference whether Q_int or $Δfr$ is considered. The intrinsic quality factor depends on losses to TLS near the resonance frequency, whereas the shift of the resonance frequency depends on a wider frequency spectrum of TLS.

An approximate low power, low temperature limit on $Qint$ is given by 1/k. Taking the k value found fitting the change of $Qint$ gives a 20% to 30% higher limit, than found in the measurements. This suggests that the majority of losses happen to TLS, but there is also a second loss mechanism. According to simulations, the internal quality factor of the MSR in the copper waveguide could be limited by the wall conductivity. In the aluminum waveguide it could be attributed to bulk dielectric loss from the high-resistivity silicon, as the loss tangent is not very well known.¹⁸ 

The presented setup, is an ideal platform for implementing interacting spin systems^25–27 where we use the MSR for readout. In Fig. 4 we show a conceptual schematic for simulating spin chain physics. The orientation of the qubits relative to the waveguide allows us to control the coupling of the qubits to the waveguide mode. In Fig. 4 they are oriented along the axis of the waveguide which will lead to a large qubit-qubit interaction but negligible coupling to the waveguide. Three MSRs with different frequencies, all above the waveguide’s cutoff, are used to read out selected qubits. Another interesting aspect of this setup is the built-in protection from spontaneous emission due to the Purcell effect, similar to Refs. 28 and 29 but broadband. Even though the qubit is strongly coupled to the resonator it can not decay through the resonator, as the waveguide acts as a filter if the qubit frequency is below the cutoff.

This platform can also be used to investigate the interplay between short range direct interactions, long range photon mediated interaction via the waveguide³⁰ and dissipative coupling to an open system. It offers a new route to investigate non-equilibrium condensed matter problems and makes use of dissipative state engineering protocols to prepare many-body states and non-equilibrium phases.^31,32

In conclusion, we have presented a design for MSRs with a low interface participation ratio embedded in a rectangular waveguide. The MSRs show single photon intrinsic quality factors of up to one million at 20 mK. We find a strong dependence of the internal quality factor on the photon number and the temperature which indicates losses to two level systems. The presented setup is appealing for testing the material of the MSR, the substrate it is patterned on and for validating fabrication processes. The observed quality factors are expected to increase when more complex designs are used, such as suspended structures⁹ or by improving the surface quality through deep reactive ion etching.¹² Alternatively, switching to sapphire as a substrate is expected to improve quality factors, as interfaces on silicon generally show higher loss than those on sapphire.⁹