Material Loss Model Calibration for Tantalum Superconducting Resonators

Guy Moshel; Moshe Schechter; Sergei Masis; Shay Hacohen-Gourgy

REVIEW 2 major objections 3 minor 49 references

Material Loss Model Calibration for Tantalum Superconducting Resonators

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Tantalum resonator measurements show two-level-system loss rising linearly with frequency, implying an energy-dependent TLS density of states beyond the standard tunneling model.

desk verdict The paper's central claim of frequency-dependent TLS loss is very likely an artifact of an internal voltage-normalization inconsistency; the dataset and analytic model are still worth a serious look. read the letter →

arxiv 2507.12141 v1 pith:HZ232XQL submitted 2025-07-16 cond-mat.supr-con

Guy Moshel , Sergei Masis , Moshe Schechter , Shay Hacohen-Gourgy This is my paper · ORCID

classification cond-mat.supr-con

keywords superconductingresonatorstwo-levelsystemstantalumlosstangentstandardtunnelingmodelTLSdensityofstatesqualityfactorcoplanarwaveguide

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to turn routine quality-factor measurements of superconducting resonators into a quantitative calibration of two-level-system (TLS) loss, and to use that calibration on a family of α-tantalum resonators spanning 2.3 to 15.7 GHz. The authors build a fully analytical model that separates TLS loss from quasiparticle and other losses, so each resonator yields a small set of material parameters. The reason to care is that if the extracted TLS loss parameter $\delta_0^{\mathrm{MS}}$ really grows linearly with frequency as they find, the TLS density of states in these devices is not the energy-independent constant assumed by the standard tunneling model, and the dominant loss mechanism in tantalum circuits is not yet captured by that model. That would redirect both materials engineering and the theory of amorphous-surface losses in quantum circuits.

What carries the argument

The load-bearing object is the closed-form TLS loss expression $P_{\mathrm{MS}}$ for the metal–substrate interface, built from the quasi-static coplanar-waveguide electric field of Eq. 18, the thin-layer boundary-condition fields of Eq. 17, and an analytic integration over the resonator length using incomplete elliptic integrals. It reduces the TLS contribution to two material parameters, $A_0^{\mathrm{MS}}$ and $B_0^{\mathrm{MS}}$, which are fit along with a quasiparticle prefactor and a constant loss channel to the measured $Q_i(T,V_0)$ surfaces. The parameter $\delta_0^{\mathrm{MS}}$ extracted from $A_0^{\mathrm{MS}}$ is then read as proportional to the TLS density of states, which is what carries the paper's frequency-trend claim.

What would settle it

Fabricate two sets of identical resonators whose only difference is the metal–substrate interface, for example one set with a controlled 2 nm native oxide and one with a deliberately grown thicker oxide, and compare the extracted $\delta_0^{\mathrm{MS}}(f)$ slopes; the model predicts the slope scales with $t_{\mathrm{MS}}/\varepsilon_{\mathrm{MS}}$, so a slope that changes accordingly ties the trend to the metal–substrate layer, while a slope that is unchanged would mark the assignment of all loss to that layer as wrong.

Watch

Extended reading notes

Core claim

On the authors' own terms, the central result is a measured linear relation $\delta_0^{\mathrm{MS}} = a\,f + b$ with $a = 0.145\times10^{-2}\,\mathrm{ns}$ and $b = -0.151$ for the metal–substrate TLS loss parameter across resonators from 2.3 to 15.7 GHz. Since $\delta_0^{\mathrm{MS}}$ is proportional to the TLS density of states in the standard tunneling model, the linear growth contradicts that model's assumption of a constant density of states and instead supports a picture in which TLS–TLS interactions, possibly of electric-dipole origin, make the density of states rise with energy. The paper also claims that the underlying analytical model, which incorporates the spatially varying electric field of the coplanar-waveguide resonators without numerical simulation, describes every resonator's measured $Q_i$ over many orders of magnitude of readout power and up to the critical temperature, with the exception of one resonator whose gradual saturation is explained by a single strongly coupled, far-detuned TLS.

Load-bearing premise

The model assumes that essentially all two-level-system loss lives in a 2-nm-thick metal–substrate interface layer with dielectric constant 15, and that the metal–air and substrate–air interfaces contribute negligibly, so if the loss mostly sits elsewhere the extracted frequency trend would be an artifact.

Editorial extensions

If this is right

The same four-parameter fit can be applied to any thin-film superconducting resonator material, giving a direct, simulation-free way to compare TLS loss across fabrication runs.
For tantalum, the linear $\delta_0^{\mathrm{MS}}(f)$ trend means that low-frequency resonators should have comparatively less TLS loss per unit interface, so frequency planning and geometry can be used to suppress this loss channel.
The 3.53 GHz resonator's behavior shows that a single discrete TLS, not a continuous bath, can dominate the temperature dependence of $Q_i$; models that ignore TLS discreteness will misattribute such data to other loss channels.
The two-parameter reduction, with $t_{\mathrm{MS}}$ and $\varepsilon_{\mathrm{MS}}$ fixed from literature, makes the model easy to transfer to other materials, but it also concentrates all interface uncertainty into one parameter, so changes in fabrication that alter the real interface will show up directly in $\delta_0^{\mathrm{MS}}$.
If the frequency trend is confirmed, the standard tunneling model's constant density of states should be replaced, for these devices, by an energy-dependent density of states that theories of interacting TLSs are beginning to predict.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An extension the paper leaves implicit: the same measurement protocol could be run on resonators made from other high-coherence metals such as aluminum or niobium to see whether the slope $a$ is a universal feature of amorphous interfaces or specific to tantalum's native oxide.
Because the model assigns all interface loss to the metal–substrate layer, the linear $\delta_0^{\mathrm{MS}}$ trend could alternatively be explained by a metal–air loss that happens to scale with frequency; a test would be to measure resonators with a deliberately passivated or thickened metal–air interface and check whether the slope changes.
The strongly coupled TLS inferred for the 3.53 GHz resonator predicts a narrow avoided crossing or spectroscopic signature around 40 GHz; looking for such a discrete feature in two-tone spectroscopy would test that explanation directly.
If the TLS density of states grows linearly with energy, then TLS loss in higher-frequency qubits and resonators should be systematically worse, which would matter for choosing qubit transition frequencies in large processors.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me cut to the chase: the paper's central claim, a linear-in-frequency delta0_MS that contradicts the standard tunneling model, is very likely an artifact of a dimensional inconsistency in the voltage conversion. Equation (2) defines V0 = sqrt(2WZ0), while Eq. (4) gives W = lambda C V0^2 / 8. For a TEM line, lambda C Z0 = 1/f, so the two are consistent only at f = 0.25 Hz. The correct relation is V0^2 = 8 f Z0 W, a factor of 4f larger. Since the saturation term enters as V0^2, the fitted B0_MS absorbs this, and the extracted delta0_MS inherits a frequency-dependent factor. The reported trend delta0_MS = a f + b with b = -0.151 is itself a red flag: it goes negative over most of the measured range, which is unphysical for a loss tangent. The stress-test note is right.

Now the good parts. The dataset is genuinely new: six alpha-Ta resonators spanning 2.3-15.7 GHz, with Qi measured over temperature and power. The analytic integration of the saturated TLS loss over the resonator's spatial field profile is a real improvement over the empirical saturation model of Crowley et al., and it does so with no extra parameters. The fits look good visually. The paper is also honest about its limitations: it admits the fit is not good enough for chi-squared, it flags the single-TLS at 40 GHz as speculative, and it notes the model doesn't capture the low-power spike.

The soft spots, beyond the dimensional error: the model assumes all TLS loss is in the metal-substrate interface, with t_MS = 2 nm and epsilon_MS = 15 taken from the literature, so the extracted parameters are only as good as that assumption. The uncertainty estimates are ad hoc (5% square-error range). The single-TLS interpretation for the 3.53 GHz resonator uses a TLS at 40 GHz with g ~ 1.09 GHz, which is extreme and not independently verified.

So where does this leave us? The paper is a useful contribution to TLS-loss modeling in tantalum, but the headline result about energy-dependent TLS density of states is unsupported until the voltage normalization is corrected and the frequency trend reexamined. I'd send it to peer review, but any competent referee should catch the Eq. (2)/Eq. (4) problem. If the authors fix that, the paper might become a solid calibration study. As it stands, I would not cite the frequency trend.

Recommendation: accept for review, but expect heavy revision.

Referee Report

2 major / 3 minor

Summary. The manuscript presents an analytical loss model for α-Ta coplanar-waveguide resonators, combining a spatially resolved TLS saturation model based on the standard tunneling model with quasiparticle and residual-loss terms. The authors fit Qi(T, P) data for resonators spanning 2.3–15.7 GHz, extract per-resonator parameters A0_MS, B0_MS, Qqp0, and Qother, and report a linear frequency dependence of the TLS loss-tangent parameter δ0_MS, with slope a = 0.145×10^-2 ns and intercept b = −0.151. They interpret this trend as evidence for an energy-dependent TLS density of states beyond the STM. They also invoke a strongly coupled, far-detuned single TLS at 40 GHz to explain the anomalous temperature dependence of the 3.53 GHz resonator.

Significance. If correct, the paper would provide a useful analytical calibration framework for material loss in superconducting resonators and would add to the evidence for a frequency-dependent TLS density of states. The strengths include an analytical treatment without numerical simulation, measurements over a wide frequency range, and independent fits per resonator, so that the frequency trend is not imposed by the model. However, the central empirical claim rests on a voltage normalization that is dimensionally inconsistent with the model's own stored-energy expression; this issue must be corrected and the analysis redone before the physics claim can be evaluated.

major comments (2)

[II, Eq. (2); III, Eqs. (4), (25)–(27); Fig. 7] The conversion from stored energy to voltage is internally inconsistent. Eq. (2) defines V0 = sqrt(2WZ0), while Eq. (4) gives W = λ C V0^2/8. Using the TEM identity λ C Z0 = 1/f, substitution of Eq. (2) into Eq. (4) gives W = W/(4f), which is not an identity; moreover V0 from Eq. (2) has dimensions of V·s^{1/2}, not V. The correct relation from Eq. (4) is V0^2 = 8fZ0W. Because CT,V0 in Eq. (25) is proportional to V0^2 and the extracted δ0_MS is, through Eqs. (26)–(27), proportional to A0_MS B0_MS, a fit using the erroneous V0 can absorb the missing factor 4f into B0_MS. The reported linear trend δ0_MS = a f + b with a = 0.145×10^-2 ns and b = −0.151 could therefore be generated even for a frequency-independent TLS density of states; the negative intercept (δ0_MS < 0 below ≈104 GHz) further suggests an artifact. The authors must correct the normalization, refit all resonators, and test whether the trend survives.
[IV, Fig. 7 and uncertainty paragraph] The statistical support for the central trend is not quantified. The data are plotted without error bars, and parameter uncertainties are obtained from the range that increases the square error by 5% because, as the authors state, the fitting function does not describe the data well enough for a standard chi-squared analysis. Under these conditions the uncertainties on A0_MS and B0_MS are likely correlated and model-inadequacy dominated, yet no confidence interval is reported for the slope a and intercept b of the δ0_MS(f) line. A bootstrap or covariance-based analysis that propagates the per-resonator fit covariance and accounts for model discrepancy is needed to support the claim of a clear linear trend.

minor comments (3)

[III, after Eq. (17)] The model assigns all interface loss to the metal-substrate layer, with tMS = 2 nm and εMS = 15 taken from the literature; because the MS and MA field profiles are identical, this assumption is untestable with the present data. The extracted δ0_MS should be described as an effective interface loss parameter, and a sensitivity check to tMS and εMS should be reported.
[IV, Fig. 6 and single-TLS paragraph] The single-TLS model for the 3.53 GHz resonator adds several parameters, with 40 GHz chosen as the minimal frequency that reproduces the data; the paper should report parameter uncertainties and a fit-comparison statistic rather than the qualitative statement that the fit was dramatically improved.
[II and III, minor text] There are several typographical errors: 'Were we defined' after Eq. (12) should be 'where we defined'; 'hydroxid' should be 'hydroxide'; and the capitalization in 'the QP loss can be derived' should be fixed.

Circularity Check

1 steps flagged · score 8.0 of 10

The central δ0_MS(f) trend is forced by the paper's own voltage normalization, not by the data: Eq. (2) is inconsistent with Eq. (4) and injects the frequency dependence into the extracted TLS loss.

fitted input called prediction [Section II, Eqs. (2) and (4); Section III, Eqs. (7), (25)–(27); Section IV, Fig. 7 and following text]
"To convert this energy to voltage, which we used in the model described below, we used V0 = sqrt(2WZ0), ... The average stored energy in the resonator per cycle is W = ... = λCV0^2/8. ... A clear linear trend with frequency in δ0MS emerges, with parameters a = 0.145·10−2 ns and b = −0.151."

Equations (2) and (4) are mutually inconsistent: for a TEM line λCZ0 = 1/f, so replacing V0^2 by 2WZ0 in Eq. (4) gives W = W/(4f). The TLS saturation factor CT,V0 (Eq. 25) is proportional to V0^2, and in the low-power limit Eq. (26) gives PMS ∝ A0_MS B0_MS CT,V0, with A0_MS B0_MS independent of f (Eq. 27). Thus PMS ∝ δ0_MS W under Eq. (2). Since 1/QTLS = PMS/(ω0 W) (Eq. 7), the model forces 1/QTLS ∝ δ0_MS/ω0. All measured resonators have Qi ∼ 10^6, so a frequency-independent TLS density of states would already require δ0_MS ∝ f after this calibration. The reported line, with its negative intercept (unphysical below ~104 GHz), is the artifact of the missing factor 4f; the 'empirical frequency-dependent trend' is injected by the model's own voltage conversion, not measured.

full rationale

The per-resonator fits are genuine and the model is self-contained apart from the voltage conversion, so I do not treat the whole paper as circular. The single load-bearing circularity is the calibration of V0: Eq. (2) contradicts the paper's own Eq. (4), and the contradiction places a spurious 1/ω0 factor in the TLS loss channel (via Eq. 7) that the extracted δ0_MS must absorb. With Qi ∼ 10^6 across resonators, a constant TLS DOS would be reported as δ0_MS growing linearly with f, exactly the paper's central result; the negative intercept confirms the extracted parameter is a fitting artifact rather than a physical density of states. The appeal to the authors' Two-TLS model (refs. 47–48) is interpretive and therefore not scored separately, and the single-TLS fit for the 3.53 GHz resonator is post-hoc, not an independent prediction. Because the headline claim of 'mechanisms beyond the standard tunneling model' rests on this normalization artifact, the circularity score is high.

Assumptions & free parameters 7 free parameters · 6 assumptions · 1 invented entities

The central claim rests on a standard physical model plus four fitted parameters per resonator, and on one speculative entity (single TLS). The MS-dominance assumption is the largest unverified load.

free parameters (7)

A0_MS per resonator = not tabulated (see Fig. 7)
Material prefactor in Eq. 26 combining tMS, εMS, δ0_MS, and ζMS; fitted to Qi(T,V0) for each resonator.
B0_MS per resonator = not tabulated (see Fig. 7)
Material parameter in Eq. 26 involving ζMS and εMS; fitted to Qi(T,V0) for each resonator.
Qqp0 per resonator = not tabulated (see Fig. 7)
Quasiparticle loss prefactor in Eq. 12; fitted for each resonator.
Qother per resonator = not tabulated (see Fig. 7)
Constant background loss (radiation, packaging, non-equilibrium QPs) in Eq. 7; fitted for each resonator.
UTLS of single TLS (3.53 GHz resonator) = 40 GHz
Chosen by hand as the minimal TLS frequency that reproduces the anomalous turnover; not independently measured.
Q(1)_0, C0, D0 for single TLS = not tabulated
Additional parameters in Eq. 28 fitted for the 3.53 GHz resonator.
a and b (slope and intercept of δ0_MS vs f) = a = 0.145e-2 ns, b = -0.151
Linear fit to the six extracted δ0_MS values; the negative intercept is suspect.

assumptions (6)

domain assumption Only TLSs resonant with the resonator frequency contribute (U = ħω)
Invoked after Eq. 16 in Section III; standard for low-temperature TLS loss.
domain assumption Quasiparticle loss follows thermal-equilibrium Mattis-Bardeen form (Eq. 12)
The paper itself notes non-equilibrium QPs are possible and absorbs them into Qother.
ad hoc to paper TLS loss is dominated by the metal-substrate interface; metal-air and substrate-air losses are negligible
Section III reduction of free parameters; not directly verified.
domain assumption Interface layer parameters tMS = 2 nm and εMS = 15
Taken from refs. 41 and 42; used to reduce the model to two free material parameters.
ad hoc to paper Single strongly coupled TLS at 40 GHz exists for the 3.53 GHz resonator
Postulated in Section IV; frequency chosen to reproduce data.
standard math δ0_MS is proportional to the TLS density of states
Standard tunneling model relation, used to connect the fit trend to physics.

invented entities (1)

Far-detuned single TLS at 40 GHz with participation g ≈ 1.09 GHz
purpose: To explain the gradual turnover in Qi(T) for the 3.53 GHz resonator
No direct measurement; frequency chosen by hand and coupling derived from fit; the paper admits the coupling is extremely high.

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Cite this review

Pith. "Pith review of Material Loss Model Calibration for Tantalum Superconducting Resonators." pith.science (2026). https://pith.science/paper/HZ232XQL

@misc{pith2026250712141,
  author       = {Pith},
  title        = {Pith review of: Material Loss Model Calibration for Tantalum Superconducting Resonators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZ232XQL}},
  note         = {Machine review of arXiv:2507.12141}
}

abstract

Material research is a key frontier in advancing superconducting qubit and circuit performance. In this work, we develop a simple and broadly applicable framework for accurately characterizing two-level system (TLS) loss using internal quality factor measurements of superconducting transmission line resonators over a range of temperatures and readout powers. We applied this method to a series of $\alpha$-Ta resonators that span a wide frequency range, thus providing a methodology for probing the loss mechanisms in the fabrication process of this emerging material for superconducting quantum circuits. We introduce an analytical model that captures the loss behavior without relying on numerical simulations, enabling straightforward interpretation and calibration. Additionally, our measurements reveal empirical frequency-dependent trends in key parameters of the model, suggesting contributions from mechanisms beyond the standard tunneling model of TLSs.

Figures

Figures reproduced from arXiv: 2507.12141 by the authors.

**Figure 2.** FIG. 2. A SEM image of an edge of the etched tantalum, in the [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. TEM mode electric field of a CPW in the transverse [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. A typical lineshape and extracted Q-factors. This spe [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Measurements of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Fitting the 3.53GHz resonator with a model including [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. Best-fit parameters for the various resonators as a func [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

Discussion (0). Continue with ORCID to comment.

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