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

TL;DR

This paper reports on the microwave characterization of aluminum and niobium microstrip resonators within a 3D waveguide, achieving high quality factors and demonstrating their potential as building blocks for circuit QED architectures.

Contribution

It introduces a wire-bond-free, well-controlled environment for testing resonators, and shows their suitability for integration with transmon qubits in 3D waveguide circuit QED.

Findings

Internal quality factors up to one million at low temperature and power

Good agreement with models predicting conductive and TLS losses

Resonators serve as building blocks for 3D waveguide circuit QED

Abstract

Here we present the microwave characterization of microstrip resonators made from aluminum and niobium inside a 3D microwave waveguide. In the low temperature, low power limit internal quality factors of up to one million were reached. We found a good agreement to models predicting conductive losses and losses to two level systems for increasing temperature. The setup presented here is appealing for testing materials and structures, as it is free of wire bonds and offers a well controlled microwave environment. In combination with transmon qubits, these resonators serve as a building block for a novel circuit QED architecture inside a rectangular waveguide.

Figures

Tables

Table 1. Table 1: Fit results including fit errors for the measurement of the internal quality factor of the aluminum MSR, for stepwise increasing temperature. The results including the fits are shown in Fig. 3. The corresponding model is given in Eq.  S3 . Given that the the MSRs in the copper waveguide are not limited by TLS related losses, the fit values are not trustworthy. A similar conclusion can be drawn for the MSR in the aluminum waveguide, for Q other subscript 𝑄 other Q_{\text{other}} . The MSR seems to be either limited by TLS or surface impedance losses, so Q other subscript 𝑄 other Q_{\text{other}} is not seen in the measurement.

		$A$	$Q_{\text{other}}$	$k$
Al - cu	( )	$5.6\left(7\right)\text{×}{10}^{-4}$	$4\left(1\right)\text{×}{10}^5$	$1.2\left(6\right)\text{×}{10}^{-6}$
Al - cu	( )	$10.2\left(9\right)\text{×}{10}^{-4}$	$0.92\left(2\right)\text{×}{10}^5$	$4.46\text{×}{10}^{-1}$ $\pm$ $2.1$
Al - al	( )	$6.5\left(7\right)\text{×}{10}^{-4}$	-	$1.6\left(2\right)\text{×}{10}^{-6}$

Table 2. Table 2: Fit results including fit errors for the measurement of the internal quality factor of the niobium MSR, for stepwise increasing temperature. The measurement results including the fits are shown in Fig. 4. The corresponding model for the change of Q int subscript 𝑄 int Q_{\text{int}} is given in Eq. 2 and for the change of the resonance frequency in Eq. 3.

		fit to $Q_{\text{int}}\left(T\right)$		fit to $\Delta f_r\left(T\right)$
		$k$	$Q_{\text{other}}$	$k$
Nb - cu	( )	$10.0\left(2\right)\text{×}{10}^{-7}$	$2.53\left(4\right)\text{×}{10}^6$	$9.1\left(5\right)\text{×}{10}^{-7}$
Nb - al	( )	$7.5\left(4\right)\text{×}{10}^{-7}$	$4.7\left(5\right)\text{×}{10}^6$	$5.6\left(3\right)\text{×}{10}^{-7}$
Nb - al	( )	$7.9\left(7\right)\text{×}{10}^{-7}$	$7\left(2\right)\text{×}{10}^6$	$6.1\left(4\right)\text{×}{10}^{-7}$

Equations

$$\frac{1}{Q_{int}^{R_s}}=\frac{A}{T}exp\left(-\frac{\Delta }{k_{B}T}\right)+\frac{1}{Q_{\text{other}}}.$$

$$\frac{1}{Q_{\text{int}}^{\text{TLS}}}=k\tanh \left(\frac{h{f}_r(T)}{2k_{B}T}\right)+\frac{1}{Q_{\text{other}}^{\prime }}.$$

$$\Delta f_r(T)=f_r(0)\frac{k}{\pi }\times \left(\text{Re}\Psi \left(\frac{1}{2}+\frac{1}{2\pi i}\frac{h{f}_r(T)}{k_{B}T}\right)-\log \left(\frac{1}{2\pi }\frac{h{f}_r(T)}{k_{B}T}\right)\right).$$

$$S_{21}(f)=1-\frac{Q_l/|Q_c|e^{i{\varphi }_0}}{1+2i{Q}_l\frac{f-f_r}{f_r}}$$

$$S_{21}(f)=(a{e}^{i\alpha }{e}^{-2\pi if\tau })\left(1-\frac{Q_l/|Q_c|e^{i{\varphi }_0}}{1+2i{Q}_l\frac{f-f_r}{f_r}}\right)$$

$$\frac{1}{Q_{\text{int}}^{{\text{TLS + R}}_{\text{s}}}}=k\tanh \left(\frac{h{f}_r(T)}{2k_{B}T}\right)+\frac{A}{T}exp\left(-\frac{\Delta }{k_{B}T}\right)+Q_{\text{other}}$$

Full text

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Zoepfl et al. (2017) and may be found at https://aip.scitation.org/doi/full/10.1063/1.4992070.

Microwave resonators are an important building block for circuit QED systems where they are e.g. used for qubit readout Gambetta et al. (2008); Axline et al. (2016), to mediate coupling Majer et al. (2007) and for parametric amplifiers Bergeal et al. (2010). All of these applications require low intrinsic losses at low temperatures ($k_{B}T\ll hf_{r}$) 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) Pappas et al. (2011); Gao et al. (2008). These defects exist mainly in metal-air, metal-substrate and substrate-air interfaces as well as in bulk dielectrics Gao et al. (2008); Barends et al. (2010); Geerlings et al. (2012); Chu et al. (2016). 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 Gao et al. (2008); Reagor et al. (2013); Wang et al. (2015) or tries to improve the interfaces by a sophisticated fabrication process Bruno et al. (2015); Megrant et al. (2012). Reducing the participation ratio requires to decrease the electric field strength. This is typically done by increasing the size of the resonator Geerlings et al. (2012) or even implementing the resonator using three dimensional structures Reagor et al. (2013).

Our approach, a microstrip resonator (MSR) in a rectangular waveguide (Fig. 1), combines the advantages of three dimensional structures with a compact, planar design Axline et al. (2016); Paik et al. (2011). The U-shaped MSR (Fig. 1(a)) is effectively a capacitively shunted $\lambda/2$ resonator Pozar (2012). The sensitivity to interfaces is reduced, since the majority of the field is spread out over the waveguide, effectively reducing the participation ratio Wang et al. (2015). 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 environment Pozar (2012) without lossy seams Brecht et al. (2015) close to the MSR. As the MSR is capacitively coupled to the waveguide, no wirebonds Wenner et al. (2011) or airbridges Chen et al. (2014) 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 center Pozar (2012) (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 see .

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 $20\text{\,}\mathrm{m}\mathrm{K}$. The MSRs were analyzed regarding their resonance frequencies and quality factors by measuring $S_{21}$. We fit the measured data using a circle fit routine Probst et al. (2015) which utilizes the complex nature of the $S$-parameter see .

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 $1\text{\,}\mathrm{K}$ 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 Pappas et al. (2011). 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 $Q_{\text{int}}$ of more than 8 million see . Other experiments, using a more sophisticated fabrication process, report similar internal quality factors for planar NbTiN resonators on deep etched silicon Bruno et al. (2015) or for planar aluminum resonators on sapphire Megrant et al. (2012). Similar methods and materials might allow us to increase the single photon quality factor of the MSR.

The trend of increasing $Q_{\text{int}}$ 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 Song et al. (2009). We do not observe this effect for the niobium MSR, due to its higher critical field Janjušević et al. (2006). 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 $Q_{\text{int}}$, due to an increasing surface impedance. Considering a two fluid model Saito K. and Kneisel P. (1999), the following temperature dependence is found

[TABLE]

Here $T$ is the temperature, $\Delta$ the superconducting gap at zero temperature, $k_{B}$ the Boltzmann constant and $A$ a constant. An additional $Q_{\text{other}}$ accounts for other temperature independent losses. This model is expected to show good agreement until $T_{c}/2$. Gross and Marx (2014)

TLS saturate with increasing temperature, which leads to an increase in quality factor Pappas et al. (2011)

[TABLE]

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. $Q^{\prime}_{\text{other}}$ 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 see of TLS related losses (Eq. 2) and conductive losses (Eq. 1). Until about $200\text{\,}\mathrm{m}\mathrm{K}$ 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 $Q_{\text{int}}$ with temperature until $200\text{\,}\mathrm{m}\mathrm{K}$. Thus in this waveguide, TLS related losses most likely limit the quality factor of the MSR. Above $400\text{\,}\mathrm{m}\mathrm{K}$ all MSRs show a similar decrease in $Q_{\text{int}}$. This can be attributed to conductive losses, as the critical temperature of aluminum is around $1.19\text{\,}\mathrm{K}$ Gross and Marx (2014). 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 $9.2\text{\,}\mathrm{K}$ Gross and Marx (2014), 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 $Q_{\text{int}}$ up to $1\text{\,}\mathrm{K}$. For the MSR in the aluminum waveguide we measure a drop in the internal quality factor at $350\text{\,}\mathrm{m}\mathrm{K}$. 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 $1\text{\times}{10}^{6}\text{\,}$. Performing finite element simulations using the finite conductivity of the aluminum (Al5083 _co (2002)) waveguide wall we found a $Q_{\text{int}}$ of $1.16\text{\times}{10}^{6}\text{\,}$, which is consistent with our measurements.

TLS also lead to a shift in the resonance frequency Pappas et al. (2011)

[TABLE]

Here $\Psi$ 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 $Q_{\text{int}}$, off resonant TLS contribute to the frequency shift Pappas et al. (2011), which makes the resonance frequency independent of power see . 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 $350\text{\,}\mathrm{m}\mathrm{K}$. We attribute this again to the breakdown of superconductivity in the waveguide wall. Below $350\text{\,}\mathrm{m}\mathrm{K}$, 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 $k$ see 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 Pappas et al. (2011), which leads to a difference whether $Q_{int}$ or $\Delta f_{r}$ 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 $Q_{\text{int}}$ is given by $1/k$. Taking the $k$ value found fitting the change of $Q_{\text{int}}$ 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 Chen et al. (2014).

The presented setup, is an ideal platform for implementing interacting spin systems Viehmann, von Delft, and Marquardt (2013a, b); Dalmonte et al. (2015) 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 Reed et al. (2010); Sete, Martinis, and Korotkov (2015) 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 waveguide van Loo et al. (2013) 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 Diehl et al. (2008); Cho, Bose, and Kim (2011).

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 $20\text{\,}\mathrm{m}\mathrm{K}$. 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 structures Chu et al. (2016) or by improving the surface quality through deep reactive ion etching Bruno et al. (2015). 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 Chu et al. (2016).

Supplementary material

Technical details and further measurement results are shown in the supplementary material.

Acknowledgements

We want to thank our in-house workshop for the fabrication of the waveguides.

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement n*∘* 714235). MP, GK is supported by the Austrian Federal Ministry of Science, Research and Economy (BMWFW). CS is supported by the Austrian Science Fund FWF within the DK-ALM (W1259-N27).

Full setup

The full measurement setup is illustrated in Fig. S1. The VNA generates the microwave signal, indicated with ’VNA out’, to probe the sample, in this case the MSR. A DC block follows the VNA, which prevents DC currents. The black lines represent microwave lines. The signal enters the cryostat after the DC block and gets attenuated at $4\text{\,}\mathrm{K}$ by $20\text{\,}\mathrm{d}\mathrm{B}$ and further by $30\text{\,}\mathrm{d}\mathrm{B}$ at the base plate. The base plate is at a temperature of $20\text{\,}\mathrm{m}\mathrm{K}$. It then enters the waveguide, containing the MSR. The microwave propagates through the waveguide, interacts with the MSR, and leaves it at the other end. The sample is enclosed in a double layer cryoperm shield inside a completely closed copper can.

At the $4\text{\,}\mathrm{K}$ stage the signal is amplified by $40\text{\,}\mathrm{d}\mathrm{B}$, using a high electron mobility transistor (HEMT) amplifier. Two isolators, which are placed between the HEMT and the waveguide, protect the waveguide from HEMT noise. Another DC block after the HEMT prevents DC currents. Finally the VNA measures the microwave signal. The temperature sensor sits at the base plate of the fridge.

MSR and waveguide in detail

In this part we show photographs of the MSR on the silicon substrate, the MSR in the waveguide and the completely assembled waveguide.

In Fig. S2 we show a photograph of the MSR (a) and the MSR in the waveguide (b). In (c) the process of mounting the MSR is shown. We slide the MSR, which is assembled to a holder, from the top into the waveguide. Details about the MSR, including the fabrication, are discussed in the main article. Also the placement of the MSR inside the waveguide is discussed there.

Fig. S3 shows the fully assembled waveguide. The waveguide consists of three parts. At each end there is an identical coupler to receive and launch the microwave signals. The central part contains the samples. In this design no seams are present near the samples. It is possible to probe three samples, each in one of the three slots, which can be easily extended to more samples by using a different central section. One can see the individual slots for each sample, customized to the dimensions of our sample. In contrast to this waveguide, the copper waveguide only allows to probe a single sample at a time.

Example measurement and circle fit routine

The MSR in the waveguide represents a resonator in notch configuration Probst et al. (2015). For such a resonator, the $S_{21}$ parameter, which refers to a transmission measurement, follows like Probst et al. (2015):

[TABLE]

Here $Q_{l}$ is the total quality factor, $f_{r}$ is the resonance frequency and $Q_{c}$ is the coupling quality factor. In here $\phi_{0}$ accounts for an impedance mismatch in the transmission line before and after the resonator, which makes $Q_{c}$ a complex number ($Q_{c}=|Q_{c}|e^{-i\phi_{0}}$). The real part of the coupling quality factor determines the decay rate of the resonator, in our case the emission to the waveguide. The physical quantity is the decay rate, $\kappa$, which is inversely proportional to the quality factor Khalil et al. (2012) and therefore the real part is found as: $1/Q_{c}^{Re}=\text{Re}(1/Q_{c})=\cos{\phi_{0}}/Q_{c}$. Knowing $Q_{c}^{Re}$ and $Q_{l}$, the internal quality factor can be obtained, as $1/Q_{l}=1/Q_{c}^{Re}+1/Q_{\text{int}}$ Khalil et al. (2012). For simplification in all other chapters, $Q_{c}$ refers to the real part, $Q_{c}^{Re}$.

Plotting the imaginary versus the real part of $S_{21}$ forms a circle in the complex plane (in case of a resonance within the frequency range).

Equation S1 represents an isolated resonator, not taking effects from the environment into account. Including the environment, which arises by including the whole measurement setup before and after the MSR (Fig. S1), we have to modify Eq. S1 to Probst et al. (2015):

[TABLE]

Here $a$ and $\alpha$ are an additional attenuation and phase shift, independent of frequency. $\tau$ represents the phase delay of the microwave signal over the measurement setup, which has a linear dependence on frequency.

In Fig. S4 and Fig. S5 example measurement are shown including the circle fit, Eq. S2. Fig. S4 shows a measurement taken with high power and therefore a good signal to noise. The plot in the top left shows the unmodified measurement data. The imaginary part vs. real part of the $S_{21}$-parameter is plotted with frequency. On the top right and the bottom left, the magnitude and the phase of the measured data is seen. All plots also show the fit. In the bottom right, the data is shown free from the effects of the environment (see equation S1). Here we also see that the setup is near critically coupled. In Fig. S5 a measurement below the single photon limit, hence with a low signal to noise, is plotted. The panels show the same information as in Fig. S4. In the case of critically coupled setups, trustworthy results are achievable within a feasible measurement time of several hours for low powers.

Details on coupling between the waveguide and the MSR

As discussed in the main text, the coupling depends on the position of the MSR in the waveguide along the $x$-axis (Fig. 1). A critically coupled setup is inevitable to get trustworthy results of $Q_{\text{int}}$ and $Q_{c}$, in particular in the single photon limit, see Fig. S5. To accomplish such a setup, simulations were performed. After discussing those, the measurement results are compared to predictions from simulations.

Finite element simulations on the coupling

We ran simulations on the coupling between the MSR and the waveguide, using a finite element method.

Fig.  S6 illustrates the two considered cases. At first, the MSR was swept from the center towards the wall. Fig. S7(a) shows the results of the coupling. A critically coupled setup in the single photon limit requires a coupling quality factor on the order of $1\text{\times}{10}^{5}\text{\,}$ to $1\text{\times}{10}^{6}\text{\,}$, depending on the measured MSR (see Fig. 2). In the available waveguides, there are only discrete slots to place the sample. The first off-centered slot is around $3\text{\,}\mathrm{m}\mathrm{m}$ from the center, which leads to a coupling quality factor between $1\text{\times}{10}^{3}\text{\,}$ and $1\text{\times}{10}^{4}\text{\,}$, being around two magnitudes below critically coupled.

We decided to use a different approach, with the MSR in the center and an empty sapphire substrate in a neighbouring slot to displace the electric field, due to the higher $\epsilon_{r}$ of sapphire. This is illustrated in Fig. S6(b). We ran simulations with the MSR placed in the center and the empty substrate being shifted towards the wall (Fig. S7(b)). The substrate being one slot off center (neighbouring the MSR) leads to a coupling of around $1\text{\times}{10}^{5}\text{\,}$. For two slots off center we can reach the desired coupling quality factor of around $1\text{\times}{10}^{6}\text{\,}$. We should remark at this point, that for such high coupling quality factors, effects like the MSR having a slightly asymmetric leg length, or being placed off center on the chip or placed entirely off center, can have a big impact on the coupling. For instance simulations showed, that a displacement of $0.2\text{\,}\mathrm{m}\mathrm{m}$ off-center, can lead to a factor of 4 in the coupling quality factor.

Measurement results

Fig. S8 illustrates the actually measured configurations. In the copper waveguides (a) - (e), one sample was measured each time, in the aluminum waveguide, three samples could be measured at once, labelled (f1)-(f3) in the following. The three samples in the aluminum waveguide were put along the propagation direction, all in the same configuration. As they had to be apart in frequency, one MSR had longer legs (f3), leading to a nominally lower resonance frequency of around $0.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. We backed this MSR, as well as a second one (f2), having a resonance frequency of nominally $8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$, with an empty silicon substrate. This reduces the resonance frequency, due to the higher effective dielectric constant.

The results are plotted in Fig. S9. We observe overall agreement between the measurements and the simulation data. The weak dependence on the number of photons agrees well with the expected power-independence of the coupling. The closer the MSR is to the wall, the lower the coupling quality factor (a), (b), inline with simulations. With the additional empty substrate, the quality factors follow the predictions from simulations (c), (d). For the substrate further away from the MSR, (d)-(f), highest coupling quality factors are observed. We attribute the difference in the coupling between configurations (d), (e) and (f1), which should be similar (the only nominal difference is the waveguide width) to a slight displacement of the MSR as discussed before. The quality factors of (f2) and (f3) are higher, as they are already backed with an empty substrate. Thus the relative influence of the neighbouring substrate is reduced, leading to a higher $Q_{c}$.

Measurements results exceeding the main paper

Resonance frequency in dependence of photon number in the MSR

The measured resonance frequencies are shown in Fig. S10. The additional silicon substrate backing reduces the resonance frequencies of the $7.5\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$(f3) and $8\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$(f2) MSRs, plotted in (i). Except (f1), simulation results accurately predict the resonance frequencies. All the other setups have resonance frequencies in the same range (Fig. S10(ii)), which is predicted by simulations. There are several explanations for the $70\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ deviation of the resonance frequency, which is not seen in the simulation data. One possibility is, a variance in the chip dimension. This would lead to a different effective dielectric constant and thus a lower resonance frequency. Other possibilities include a slight difference between the MSRs or its placement on the substrate.

There is no dependence of the resonance frequency on the number of circulating photons.

Shift of the resonance frequency of the Al MSR with increasing temperature

The shift of the resonance frequency of the three measured aluminum MSRs for increasing base temperature is plotted in Fig. S11. A decrease of the resonance frequency is seen above $500\text{\,}\mathrm{m}\mathrm{K}$. The shift is similar for all three measured samples. The drop in resonance frequency can be explained with an increasing surface inductance over temperature, which originates from an increasing effective penetration depth Gao (2008). The increase of the penetration depth can be estimated with the Mattis Bardeen theory Mattis and Bardeen (1958).

The results are similar to the one found in Reagor et al. (2013); Gao (2008), where thin aluminum film resonators were measured. There, the frequency shift shows good agreement with the Mattis-Bardeen theory.

Resonance frequency shift of the niobium MSR with increasing temperature - comparison between low and high input powers

Fig. S12 compares the shift of the resonance frequency for input powers at the single photon limit to input powers six magnitudes greater. The main difference is the higher noise in the single photon limit leading to increasing uncertainties. Overall, the low and high power measurements show the same temperature dependence. Thus both can be taken to fit $\Delta f_{r}$ with the same results. Given the lower uncertainties, we took the high power measurements to perform the fit.

In Fig. S12 the measurement results are plotted until $1.4\text{\,}\mathrm{K}$. For the fit to the MSR in the copper waveguide, the data points above $0.8\text{\,}\mathrm{K}$ are omitted as the behavior above is not well described by the model anymore.

Internal quality factor of the niobium MSR for high excitation powers

Fig. S13 shows the internal quality factor of the niobium MSR over the whole measurement range. The best performing niobium MSR showed an internal quality factor of above eight million for high input powers. Due to attenuators in the measurement chain (Fig. S1) higher input powers were not possible. Given the tendency, we would expect an even higher quality factor for higher input powers. The difference between the two MSRs in the aluminum waveguide is probably related to TLS losses. We also find a slightly lower value for the combined loss parameter $k$ of the better performing MSR (Tab. 2). The finite conductivity of the copper is probably the reason for the lower quality factor measured for the MSR in the copper waveguide.

Fit results

In this section we provide the fit parameters to the fits, that were discussed in the main part.

The internal quality factor of the MSR changes with temperature. In case of the niobium MSR this is explained with the loss to two level systems (Eq. 2). We fit this model to the measurement data (Fig. 4(a)). TLS also lead to a shift of the resonance frequency, predicted by Eq. 3. The measurement results including the fits are shown in Fig. 4(b).

In case of the aluminum MSR, an increasing surface resistance also leads to an additional effect on $Q_{\text{int}}$, next to the TLS. Thus we use a combined model of the surface impedance (Eq. 1) and TLS (Eq. 2) for the fit:

[TABLE]

To fit this model to the change of $Q_{\text{int}}$, the inverse of Eq. S3 was taken. The fit parameters of all performed fits are listed here. In Tab. 1 the fit results of Eq. S3 to the measurements of the aluminum MSR (Fig. 3) are given.

In case of the aluminum MSR in the copper waveguide, $1/k$ can not be taken as a low energy low temperature limit for $Q_{\text{int}}$, as the MSR is limited by other losses. Only the MSR in the aluminum waveguide, being limited by TLS related losses, $1/k=$ $6(1)\text{\times}{10}^{5}\text{\,}$, is in agreement with the measurements. In turn, it was not possible to extract a useful value for $Q_{\text{other}}$, as the MSR was either limited by TLS effects or increasing conductive losses. For the MSRs in the copper waveguide $Q_{\text{other}}$ is in agreement with the measurements. The values obtained for $A$, which refers to the increasing surface impedance, are in the same range for all measurements.

Tab. 2 lists the fit results for the niobium MSR. We fitted both, the change of the internal quality factor with temperature (Fig. 4(a)) and the change of the resonance frequency with temperature (Fig. 4(b)).

For the niobium MSR the $k$ can be either determined by fitting the change of $Q_{\text{int}}$ or the change of the resonance frequency. In both cases the value we got are in the same range. Nevertheless, the value obtained fitting the resonance frequency is throughout $10\%-30\%$ higher, than fitting $Q_{\text{int}}$. The reason is explained in the main article and lies in the frequency distribution of the TLS Pappas et al. (2011). In addition, the $Q_{\text{int}}$ limit given through $k$ is also discussed in the main article. $Q_{\text{other}}$ gives an upper limit on the internal quality factor. The fit values are compatible with the measurements (Fig. S13).