{"id":"3dea0e30-41ad-4053-ad47-3d167640fce2","arxiv_id":"2507.12141","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Fitting a TLS loss model to tantalum resonators from 2.3 to 15.7 GHz yields a TLS loss strength that increases roughly linearly with frequency.","lead":"This paper builds an analytical model for two-level system losses in superconducting tantalum resonators, using quality factor measurements over temperature and readout power. It reports a fitted TLS loss parameter that rises with frequency, evidence that the standard tunneling model of TLSs does not fully describe tantalum.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central δ0_MS(f) trend is likely an artifact of Eq. (2): V0 = sqrt(2WZ0) is dimensionally inconsistent with Eq. (4), omitting a frequency factor that the extracted linear-in-f δ0_MS could absorb.","rationale":"The reader's conditional verdict targeted the interface-loss attribution and fixed layer parameters, but the more serious and more specific problem is the dimensional inconsistency in Eq. (2). The central result rests entirely on the δ0_MS(f) trend, and that trend is extracted through a fitting procedure that uses V0 from Eq. (2). Because the stored-energy expression Eq. (4) and the transmission-line identity λ C Z0 = 1/f imply V0^2 = 8 f Z0 W, the paper's V0^2 = 2 W Z0 is wrong by a frequency-dependent factor of 4f. This error enters the saturation term and propagates into the fitted B0_MS and hence into δ0_MS ∝ A0_MS B0_MS. The observed linear dependence on f is precisely the kind of artefact such a missing factor would produce. The negative intercept reinforces the concern: the reported line gives negative δ0_MS over most of the measured band, which cannot represent a TLS loss tangent. A direct refit with the corrected normalization would settle the matter. If the authors can show that their code used the correct formula and that the trend persists, the paper could become conditionally acceptable; as written, the central claim is unsupported by the stated equations.","tokens_in":15660,"tokens_out":18050,"duration_ms":214841,"concrete_test":"Re-derive V0 from Eq. (4) using the correct distributed-line relation V0^2 = 8 f Z0 W (equivalently, V0^2 = 4 ω0 Z0 W / π) and refit all six resonators' Qi(T, P) data with this corrected normalization, keeping the same model. Then examine the extracted δ0_MS versus f: if the slope collapses to zero (or becomes statistically insignificant) and the values become positive, the claimed linear trend and beyond-STM interpretation are artifacts of the missing frequency factor; if a positive linear trend persists, the concern is resolved.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central claim is the linear frequency dependence of the extracted TLS parameter δ0_MS. The most load-bearing weakness is not the interface assignment but the voltage normalization in Eq. (2). The authors define V0 = sqrt(2WZ0), while Eq. (4) gives W = λ C V0^2/8. For a TEM transmission line, λ C Z0 = 1/f, so substituting Eq. (2) into Eq. (4) yields W = W/(4f), which is self-consistent only at f = 0.25 Hz. The correct conversion from Eq. (4) is V0^2 = 8 f Z0 W, a factor of 4f larger than the value used in the paper. Since the saturation term CT,V0 (Eq. 25) enters as V0^2 and the fit parameter B0_MS is adjusted to compensate, the product A0_MS B0_MS — and therefore δ0_MS — inherits an extra factor proportional to f. The reported trend δ0_MS = a f + b with a = 0.145e-2 ns could thus be generated even for a frequency-independent TLS density of states. The negative intercept b = -0.151 additionally implies δ0_MS < 0 below ~104 GHz, which is unphysical for a loss tangent and suggests the extracted parameter is a fitting artifact. This internal inconsistency must be resolved before any conclusion beyond the standard tunneling model can be drawn.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16073,"tokens_out":13137,"duration_ms":157764,"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":[{"comment":"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.","section":"II, Eq. (2); III, Eqs. (4), (25)–(27); Fig. 7"},{"comment":"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.","section":"IV, Fig. 7 and uncertainty paragraph"}],"minor_comments":[{"comment":"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.","section":"III, after Eq. (17)"},{"comment":"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.","section":"IV, Fig. 6 and single-TLS paragraph"},{"comment":"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.","section":"II and III, minor text"}],"recommendation":"major_revision","confidential_remarks":"The voltage-normalization problem in Eq. (2) appears to be a likely origin of the paper's headline frequency trend. Before inviting a revision, the editor may wish to ask the authors to confirm, with the corrected relation V0^2 = 8fZ0W, whether the δ0_MS(f) trend survives the reanalysis. If it does not, the paper would be reduced to a calibration study without the claimed beyond-STM result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nNow 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.\n\nThe 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.\n\nSo 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.\n\nRecommendation: accept for review, but expect heavy revision.","headline":"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.","tokens_in":16574,"tokens_out":6558,"would_cite":false,"duration_ms":64130,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["superconducting resonators","two-level systems","tantalum","loss tangent","standard tunneling model","TLS density of states","quality factor","coplanar waveguide"],"falsifier":"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.","tokens_in":15448,"feed_emoji":"🔬","tokens_out":5962,"duration_ms":67543,"temperature":0.7,"pith_summary":"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.","feed_headline":"Measured tantalum resonator loss rises with frequency","feed_subtitle":"A wide-frequency study of α-Ta resonators finds two-level-system loss contradicting the standard tunneling model.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the earlier tantalum TLS saturation model with empirical parameters that this work expands by incorporating the spatial field variation.","marker":"[20]"},{"why":"Provides the three-interface decomposition of dielectric loss and the boundary-condition treatment used to write the interface fields.","marker":"[36]"},{"why":"Gives the standard TLS loss tangent expression that the model's power-saturation and temperature dependence are built on.","marker":"[31]"},{"why":"Gives the analytic quasi-static coplanar-waveguide electric field used for the explicit loss integral.","marker":"[39]"},{"why":"Establishes tantalum as a record-coherence material and supplies the fabrication context that motivates this loss calibration.","marker":"[8]"},{"why":"Shows theoretically that TLS–TLS interactions can produce an energy-dependent TLS density of states, which the paper cites to interpret the linear frequency trend.","marker":"[47]"},{"why":"Provides the Two-TLS model with dominant electric-dipole interactions, cited as a possible origin of the energy-dependent density of states.","marker":"[48]"},{"why":"Reports prior experimental evidence of a strong frequency dependence of TLS density of states in resonators, supporting that the observed trend is physical.","marker":"[46]"}],"fun_headline_variants":["Tantalum TLS loss scales with frequency, defying standard model","Measured TLS loss in Ta rises linearly with frequency","Linear rise in Ta resonator loss challenges TLS model","Analytical model reveals frequency-dependent TLS loss in Ta"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Tantalum TLS loss scales with frequency, defying standard model","Measured TLS loss in Ta rises linearly with frequency","Linear rise in Ta resonator loss challenges TLS model","Analytical model reveals frequency-dependent TLS loss in Ta"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00087,"raw_usage":{"total_tokens":3738,"prompt_tokens":888,"completion_tokens":2850,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":504,"completion_tokens_details":{"reasoning_tokens":2785}},"tokens_in":504,"tokens_out":2850,"duration_ms":22410,"temperature":1.0,"reasoning_tokens":2785,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:53:38.836410+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the earlier tantalum TLS saturation model with empirical parameters that this work expands by incorporating the spatial field variation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the three-interface decomposition of dielectric loss and the boundary-condition treatment used to write the interface fields."},{"cited_title":"V on Schickfus and S","cited_arxiv_id":null,"evidence_quote":"Gives the standard TLS loss tangent expression that the model's power-saturation and temperature dependence are built on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the analytic quasi-static coplanar-waveguide electric field used for the explicit loss integral."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes tantalum as a record-coherence material and supplies the fabrication context that motivates this loss calibration."},{"cited_title":"Churkin, S","cited_arxiv_id":null,"evidence_quote":"Shows theoretically that TLS–TLS interactions can produce an energy-dependent TLS density of states, which the paper cites to interpret the linear frequency trend."},{"cited_title":"Schechter and P","cited_arxiv_id":null,"evidence_quote":"Provides the Two-TLS model with dominant electric-dipole interactions, cited as a possible origin of the energy-dependent density of states."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports prior experimental evidence of a strong frequency dependence of TLS density of states in resonators, supporting that the observed trend is physical."}],"review_version":1}