Efficient methods for extracting superconducting resonator loss in the single-photon regime

Superconducting resonators are essential circuit elements in microwave photonics and quantum computing applications.¹ The most popular superconducting qubit design, the transmon qubit,^2,3 uses a Josephson junction as a non-linear inductor in an LC tank circuit to form an anharmonic oscillator with the ground and first excited states comprising the qubit. It is known that a large fraction of the loss is associated with the electric fields of the qubit penetrating the shunt capacitor and the ground plane outside the junction.⁴ The primary benefit of using resonators as opposed to qubits to extract loss is the general ease of design and fabrication. For example, planar linear resonator fabrication can be simplified to a one-layer deposition and patterning process without a junction, bandage, or integration layer.⁵ 

At millikelvin temperatures and single-photon excitation powers, superconducting resonator quality factors are ultimately limited by two-level systems (TLSs) residing in the amorphous dielectric materials at the interfaces between the air, superconductor, and substrate.^6,7 At high power, TLSs are saturated^8,9 and become less saturated as power decreases leading to a power-dependent resonator internal quality factor $Q i$ (or internal loss $1/ Q i$⁠). When measured as a function of photon number $⟨ n ph⟩$ or internal field strength, this power-dependent relationship can be exploited to differentiate TLS-related loss from power-independent losses due to quasiparticles, radiation, trapped flux, etc.^10–13 

The power-dependent losses are commonly measured using a vector network analyzer (VNA) to obtain a set of scattering parameters, $S 21(f)$⁠, as a function of $⟨ n ph⟩$ within a few linewidths around the resonance frequency $f r$⁠. The quality factor and other resonator parameters can be extracted from independent fits to the complex $S 21$ for each power. Determining the TLS-induced loss this way often requires $⟨ n ph⟩≲1$ so one must decrease the VNA power to very low levels. In this regime, the measurement signal-to-noise ratio (SNR) is poor and limited by the cryogenic amplifier noise in typical setups. The low SNR reduces the fit quality and often produces unreliable $Q i$ values with large uncertainties.¹⁴ To increase SNR, one must inevitably increase acquisition time, but this, in turn, makes the measurement susceptible to long term drift from TLS and the amplifiers. For these reasons, accurately determining resonator $Q i$ at single-photon power levels remains challenging and time consuming.

In this paper, we address this challenge by improving the measurement and data analysis methodologies. Instead of fitting parameters independently at each power as is conventionally done, our strategy is to extract a subset of the resonator fitting parameters (such as the coupling $Q c$ and the impedance mismatch angle $ϕ$⁠) from high power data and fix these parameters in the analysis of low power data. We use this method of partial parameter extraction to analyze experimental data acquired from Al and Nb resonators and show that great improvements have been achieved in terms of accuracy and robustness of the results.

The diameter correction method works well for obtaining resonator parameters at large microwave powers but begins to produce poor fits at lower powers where the SNR is low. We introduce an updated version of DCM called parameter-constrained DCM (PC-DCM) that improves the quality and reliability of the fitting procedures at all powers. The main motivation behind PC-DCM is that many parameters listed in Eq. (1) are constant for any number of photons as long as the system response is linear (free from nonlinear effects such as amplifier saturation and nonlinear kinetic inductance). These parameters can be determined with much higher accuracy at higher powers, and then reused to fit data at lower powers. Parameters such as $ϕ,a,α,τ$ describe circuit properties of the signal chain rather than the resonator itself. Meanwhile, $Q c$ is fixed by design. The two remaining parameters, $f r$ and $Q i$⁠, can depend on power (e.g., due to TLS saturation and nonlinear kinetic inductance effect¹⁹). For low power resonator loss measurements on resonators with a low kinetic inductance fraction, the power dependence of $f r$ is typically much weaker than $Q i$⁠.¹⁶ 

The PC-DCM procedure has four steps. First, accurately determine the cable delay $τ$ from a wideband frequency sweep (same as in DCM). Second, use DCM to fit all parameters using high power data. The values obtained from the fit, such as $a,α,ϕ$ and $Q c$⁠, are now treated as constant values just like $τ$⁠. Third, perform a parameter-constrained circle fit, phase fit and an optional refining fit on the lower power data using the previously determined $a,α,ϕ, Q c,τ$ as constant values. This reduces the fitting problem in Eq. (1) down to a two-parameter fit for $f r$ and $Q l$⁠. Finally, extract $Q i$ from Eq. (3).

We measured a 100 nm Al resonator patterned on a Si substrate [Fig. 1(a)], to show how PC-DCM can improve the fitting quality over DCM. Figure 2(a) shows the measured $z$-data of the resonator at $⟨ n ph⟩∼2$⁠. This data set spanned 8 linewidths is fit using DCM (green circle). A subset of the data spanning 1 linewidth enclosed by the dashed box is fit using PC-DCM (red circle) with the constrained parameters $z ∞$⁠, $Q c$⁠, and $ϕ$ predetermined at a higher power. We extract $1/ Q i$ to be $7.5±3.2× 10 − 8$ for DCM and $2.5±0.23× 10 − 7$ for PC-DCM. The reason for this discrepancy can be traced to back to the phase fit in Fig. 2(b) which shows that the DCM fit converged to a sub-optimal result. The corresponding reduced $χ ν 2$ for the DCM fit is 5.01, whereas it is 1.07 for the PC-DCM fit. This highlights the importance of constraining the parameter space to improve fitting reliability in the presence of noise and reduce the number of erroneous fits.

In this section, we describe a new method for analyzing resonators based on a conformal map of the $S 21$ data. We show how the unwrapped data $z(f)$ can be mapped from the $z$-plane to a new $w 2$-plane with $w 2=1/ Q i+2δf/ f r$⁠. Conformal mapping is a powerful mathematical tool that has broad applications in microwave engineering.²⁰ It is often used to map orthogonal curvilinear coordinates into cartesian coordinates. In our method, the two orthogonal quadratures, often referred to as the frequency quadrature (tangential to the circle) and the dissipation quadrature (perpendicular to circle),²¹ are mapped to the imaginary part and real part in the new $w 2$-plane, respectively. This is possible because Eq. (1) has the general form of a bilinear (or Mobius) transformation:²²  $ξ=(aη+b)/(cη+d)$⁠, which is a special class of conformal maps that map lines and circles in the $η$-plane into lines and circles in the $ξ$-plane. The two following subsections illustrate how to use this new fitting method to determine resonator parameters at high power as well as establish the conformal map to analyze data taken at low power.

The conformal map provides a straightforward transformation of $S 21$ data into $1/ Q i$ and $2δf/ f r$ values. According to Eq. (4), it is most efficient to measure $S 21$ at a single frequency $f= f r$ at lower powers, where $f r$ is the resonance frequency determined at high power in the linear regime, instead of performing a frequency sweep around $f r$ as is done in the DCM and PC-DCM. This is similar to other time-domain resonator measurement applications, such as kinetic inductance detectors and qubit readout, where the probe tone is fixed at the resonance frequency. The time-dependent single-frequency $S 21(t,f= f r)$ data taken for a period of time $T$ at each lower power are directly mapped to $w 2$-plane using Eq. (11). We plot these data as the colored dots in Figs. 3(a) and 3(c) alongside the high power data obtained from a frequency sweep. In Fig. 3(d), we plot extracted $1/ Q i$ values against $⟨ n ph⟩$ generated from the mean of $Re[ w 2]$ at each power with the error bar quoting the standard error of the mean. We hereafter refer to this method of directly mapping the single frequency $S 21(t)$ data to the $w 2$-plane to calculate $1/ Q i$ as the parameter-constrained conformal mapping (PC-CM) method. The ability to process lower power data without the need for any fitting is a unique feature of the PC-CM method and makes it advantageous over non-linear fitting because it is insensitive to choice of initial conditions or weighting of points.

We report comparison of the DCM, PC-DCM, CM, and PC-CM methods by analyzing experimental data on 100 nm evaporated Al and 100 nm sputtered Nb resonators on Si substrates. Each resonator chip has 12 $λ/4$ CPW resonators with designed resonance frequencies between 5.2 and 6.2 GHz and coupling $Q c=2× 10 6$⁠. An electron micrograph of the fabricated resonator is shown in Fig. 1(a). The devices were cooled down in a dilution refrigerator or an adiabatic demagnetization refrigerator (ADR). Measurements were carried out using a VNA in a conventional cryogenic setup⁴ shown in Fig. 4.

In the following discussions, unless otherwise noted, we always used the fitting results of DCM at the highest power to constrain the parameters in PC-DCM (⁠ $Q c$⁠, $z ∞$ and $ϕ$⁠) and CM (⁠ $z ∞$ only). As for the PC-CM method, the lower power data were mapped to $1/ Q i$ using the conformal map established by a CM fit at the highest power. In addition, all error bars are quoted at the 68% confidence level (⁠ $1σ$⁠).

The methods (PC-DCM, PC-CM) introduced here depend on the independence of the off resonance point $z ∞$⁠, impedance mismatch angle $ϕ$⁠, and coupling $Q c$ from the photon number $⟨ n ph⟩$⁠. Here we show that these three parameters are unchanged across a wide range of $⟨ n ph⟩$ using data from a 100 nm Nb resonator measured in the ADR; we also show that these parameters are unchanged as a function of temperature for temperatures well below the critical temperature (9.2 K). We plot in Fig. 5(a) the power-dependent resonance curves measured at 300 mK and $⟨ n ph⟩=1− 10 4$⁠. In Fig. 5(b), we plot the temperature-dependent resonance curves measured between 100 mK and 1.5 K at $⟨ n ph⟩∼ 10 4$⁠. All measurements used the same number of points, acquisition time, and frequency span.

We begin with an analysis of the power-dependent data. We determined the center of each resonance circle (marked by “ $×$”) with a circle fit. Next, we fit the center points to a straight line (dashed lines) with a slope of $−0.8273±0.0098$ and an $R 2$ goodness-of-fit value of 0.998.²³ The fact that all the circle centers aligned along this common line indicates that the angle $ϕ$ remained unchanged at different $⟨ n ph⟩$⁠. We then used the intersection of this line with the circle fits to determine the off resonance points (marked by “ $⋆$”) which we show in a close-up plot in Fig. 5(d). The large overlap of the points suggests that the off-resonance points also remained unchanged at different $⟨ n ph⟩$⁠. Finally, we used these values of $ϕ$ and $z ∞$ to perform independent PC-DCM fits of all the resonance circles while allowing $Q c$ to vary. We plot the extracted $Q c$ values in Fig. 5(e). This analysis was repeated on the temperature-dependent resonance circles. The straight line fit of the circle centers in this case yielded a slope of $−0.8293±0.0048$ and an $R 2$ goodness-of-fit value of 0.999. We plot the corresponding off resonance points and $Q c$ values in Figs. 5(d) and 5(e) which overlap well with the power-dependent results. Our results confirm that the resonator parameters $z ∞$⁠, $ϕ$⁠, and $Q c$ were unchanged when the internal loss changed in a wide range (whether due to TLS saturation or thermal quasi-particles). We note that it is good practice to carry out this validation procedure to verify the power-independence assumption prior to using the parameter-constrained methods. Once confirmed, it is valid to treat these parameters as constant values, determine them from resonance circles measured at high power, and reuse them to fit data measured at lower powers.

In Fig. 6, we present data measured on a 100 nm Al resonator in the DR and compare the extracted $1/ Q i$ from each of the four methods as a function of $⟨ n ph⟩$⁠. We took three data sets in total. Each data set contained 100 data points with one set spanning six linewidths for DCM, another spanning one linewidth for PC-DCM and CM, and lastly a single frequency measurement for PC-CM, all centered around the resonance frequency. The same integration time per point was used for each data set.

We plot the extracted $1/ Q i$ against $⟨ n ph⟩$ in Fig. 6(a). The error bars for DCM and PC-DCM were calculated from the covariance matrix of the fit while CM and PC-CM use the standard error of the mean as stated previously. The four methods give overall consistent curves that show $1/ Q i$ decreases with $⟨ n ph⟩$ due to the TLS saturation effect. At $⟨ n ph⟩≫1$ the $1/ Q i$ values agree well with each other to within 1% showing that they give equivalent results when the SNR is high. This implies that either DCM or CM can be used to determine the power-independent parameters at high power. At intermediate powers of $10<⟨ n ph⟩< 10 5$⁠, the results from all four methods agree to within $6%$⁠. Finally in the single photon regime (⁠ $⟨ n ph⟩∼1$⁠), we begin to see large deviations over $10%$⁠. There are two main contributions that factor into the deviations: the fitting uncertainty from the HEMT (high electron mobility transistor) noise and the TLS-induced fluctuations. Superconducting CPW resonators are known to fluctuate in quality factor as a result of two-level systems. Changes of up to 30% within tens of minutes have been observed in the literature.^24–26 In our case, the third single frequency data set was taken several minutes after the first data set. Also, the error bars reported in Fig. 6(a) only account for the measurement white noise and do not factor in TLS-induced fluctuations.

We further explored these TLS-induced fluctuations by measuring single frequency data spanning a period of 1000 s at three power levels and processed the data using the PC-CM method. We plot the resulting $1/ Q i$ against time in Fig. 6(c). Each data point is the average of 575 $1/ Q i$ values measured in 5.75 s with the error bars quoting the standard error of the mean. We confirm that the internal loss is capable of changing by 30% or more within minutes at $⟨ n ph⟩∼1$⁠. In addition, we also found the magnitude of fluctuations decreased with increasing $⟨ n ph⟩$ as expected from TLS saturation. While a discussion of the rich physics behind these observations is beyond the scope of this methodology-focused paper, we have demonstrated that the presented non-conventional single-frequency measurement and conformal mapping data analysis techniques are useful tools to study these phenomena.

An additional feature we see in Fig. 6 is that the error bars at any $⟨ n ph⟩$ are different and in a fixed order (largest for DCM and smallest for PC-CM). Here we take a closer look at the errors (uncertainty) in the extracted $1/ Q i$ by measuring a 100 nm Nb resonator in the ADR and compare the errors reported by each of the fitting methods. Four data sets were measured in total. We took one frequency sweep that spanned 3 linewidths for DCM, one that spanned 1 linewidth for PC-DCM and CM, and a single frequency measurement for PC-CM. The fourth data set was also a single frequency measurement that was taken off resonance (2 MHz de-tuned from the resonance) to assess the measurement noise.

We plot the extracted $1/ Q i$ vs $⟨ n ph⟩$ in Fig. 7(a). The results from the four methods agree with each other with a maximum difference of $1.1%$ occurring at the lowest $⟨ n ph⟩∼1000$⁠. Fit quality was evaluated by calculating the reduced chi-square $χ ν 2$ statistic and we plot the results in Fig. 7(c). We find an average $⟨ χ ν 2⟩∼1.28$ across all $⟨ n ph⟩$⁠. However, the different methods yielded different standard errors in $1/ Q i$ as shown in Fig. 7(b). Two clear trends can be observed in the data. First, the errors all scale inversely with the photon number as expected from the reduced SNR at lower powers. Second, at any photon number the errors follow the order $σ DCM> σ PC − DCM≈ σ CM> σ PC − CM$⁠, showing that PC-DCM, CM, and PC-CM achieve lower error than the conventional DCM with PC-CM showing the lowest error. This second trend is consistent with our intuition that data taken closer to (further away from) the resonance contains more (less) information about the properties of the resonator and should result in better (worse) estimates of $1/ Q i$⁠. Therefore, it is unsurprising that the different frequency spans used by the four methods resulted in different fitting uncertainties.

The ratios of uncertainties in $1/ Q i$ from the four methods at each $⟨ n ph⟩$ [Fig. 7(b)] is found to be $σ DCM: σ PC − DCM( σ CM):$ $σ PC − CM≈3.5:1.5(1.5):1$⁠. Because the measurement time to achieve a given uncertainty level in $1/ Q i$ scales as $1/ σ 2$⁠, we expect that the PC-CM and CM (PC-DCM) methods can reduce the measurement time by a factor of 10 and 5 (5), respectively, as compared to the conventional DCM. To demonstrate this measurement time reduction advantage, we measured additional data sets at $⟨ n ph⟩=1000$ with the same 200 data points but used different VNA IF bandwidths (IFBW) from 2000 Hz down to 100 Hz. This corresponded to measurement times from 0.1 to 2 s. The errors in $1/ Q i$ against IFBW are plotted in Fig. 7(d). We find that the standard error achieved by PC-CM (CM and PC-DCM) at an IFBW of 1000 Hz (500 Hz) is comparable to the one given by DCM at an IFBW of 100 Hz (dashed line), which indeed translates into a factor 10 (5) reduction in measurement time for the PC-CM (CM and PC-DCM) from the conventional DCM. In addition, we find that the error in Fig. 7(d) for each method scales as the square-root of the IFBW. This suggests the measurement is dominated by the white noise from our HEMT which has an input-referred noise temperature of 5 K. This showcases that the new methods are able to improve measurement uncertainty or efficiency over the conventional diameter correction method in the presence of white noise. In addition, the decreased measurement times of the PC-DCM and PC-CM should also help to reduce fitting errors due to TLS-induced fluctuations.

As we have discussed in the previous section, there are two main noise sources that contribute to the uncertainty in the extracted $1/ Q i$ measured in the single photon regime: amplifier noise and the intrinsic TLS-induced fluctuations. Our proposed methods (PC-DCM and PC-CM) address the first contribution and improve the fitting uncertainty by focusing the measurement to be at or very close to the resonance frequency where the $S 21$ is most sensitive to the underlying parameter $1/ Q i$⁠. We remark that this is only possible by constraining parameters because the conventional unconstrained DCM requires data away from resonance by necessity in order to fit the resonance circle and extract values for $z ∞,ϕ$⁠, and $Q c$⁠. We also note that these new methods are equally applicable to both HEMT or parametric amplifiers with quantum limited noise. Our new methods do not address the noise contribution from the intrinsic TLS-induced fluctuations but provide a useful tool to study this phenomenon. In particular, PC-CM can be used to measure changes in the resonator frequency by reading the fractional frequency shift from the imaginary part of $w 2$⁠.

Two additional considerations should be noted when applying the new analysis techniques. First, the single measurement frequency $f$ used in the PC-CM method should be set as close as possible to the resonance frequency $f r$⁠, but a small detuning or drift in $f r$ is tolerable. This is because the conformal map defined in Eq. (11) is valid for any frequency on or off resonance, assuming Eq. (1) accurately describes the $S 21$ response of the resonator. The conformal map separates changes in $f r$ and $1/ Q i$ into two orthogonal directions so that drift in $f r$ corresponds to movement only along the imaginary axis in the $w 2$ plane without any changes in the $1/ Q i$ value; this also suggests that the $1/ Q i$ value extracted using PC-CM is insensitive to frequency noise. As a rule of thumb, we recommend to always keep $f$ to within one linewidth of $f r$ at all times when applying PC-CM. Second, the new analysis techniques have been successfully applied to Nb and Al resonators with low kinetic inductance fractions, but special care should be taken when applying our technique to resonators fabricated from materials with high kinetic inductance, such as TiN or NbTiN. One should avoid using these techniques in the high power, non-linear regime where the resonance circles are distorted and Eq. (1) is no longer valid in describing the $S 21$ response. Any fitting or mapping should be done in the linear regime, which applies in general to measurement near the single-photon level.

In conclusion, we presented new measurement and data analysis methods for resonator internal loss that are more accurate, efficient, and robust than the conventional diameter correction method. Extracting a subset of power-independent resonator parameters from high power enabled us to spend measurement time more efficiently on resonance to reduce the fitting uncertainty as we have shown with experimental data. In particular, the conformal mapping method provides an efficient way to analyze transmission data gathered on resonators by directly mapping $S 21$ data into $1/ Q i$ and $2δf/ f r$ values. All together, these new resonator loss measurements extend the toolbox available to experimentalists for studying the physical origins of dissipation, noise and decoherence in quantum circuits as well as to characterize materials used for quantum devices.

We thank the staff from across the AWS Center for Quantum Computing that enabled this project. We also thank Simone Severini, Bill Vass, James Hamilton, Nafea Bshara, and Peter DeSantis at AWS for their involvement and support of the research activities at the AWS Center for Quantum Computing.

The authors have no conflicts to disclose.

  C.C. and D.P. contributed equally to this work.

Cliff Chen: Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). David Perello: Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Shahriar Aghaeimeibodi: Methodology (supporting); Software (supporting). Guillaume Marcaud: Methodology (supporting). Ignace Jarrige: Methodology (supporting). Hanho Lee: Investigation (supporting). Warren Fon: Investigation (supporting). Matt Matheny: Conceptualization (equal); Project administration (equal); Resources (equal); Writing – review & editing (equal). Jiansong Gao: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.