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Loss tangent fluctuations due to two-level systems in superconducting microwave resonators

T0 review · 1 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Low-power quality factors of superconducting microwave resonators fluctuate by 13% over 12–16 hours, and the paper traces these fluctuations to time-varying two-level-system loss.

desk verdict Solid, well-controlled study of long-timescale Qi fluctuations in superconducting resonators, but the attribution to TLS loss-tangent fluctuations leaves an unaddressed frequency-noise contamination channel. read the letter →

arxiv 2412.05482 v2 pith:TDHLR2XL submitted 2024-12-07 quant-ph

André Vallières , Megan E. Russell , Xinyuan You , David A. Garcia-Wetten , Dominic P. Goronzy , Mitchell J. Walker , Michael J. Bedzyk , Mark C. Hersam
show 5 more authors
Alexander Romanenko Yao Lu Anna Grassellino Jens Koch Corey Rae H. McRa
This is my paper · ORCID
classification quant-ph
keywords two-levelsystemslosstangentinternalqualityfactorsuperconductingmicrowaveresonatorsTLSsaturationfluctuationsqubitcoherencelow-frequencynoise
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

Superconducting microwave resonators, which are used as stand-ins for qubits when benchmarking materials loss, show slow, large wandering of their internal quality factor at low probe power: over 12–16 hours the relative standard deviation is about 13%, and single excursions reach 37% from the mean. The paper argues that this wandering is not measurement noise or coupling drift but real variation in the loss tangent set by two-level systems (TLS) in the device materials. The evidence is that fluctuations shrink as power and temperature increase, exactly the behaviour of saturable TLS, and that interleaved low- and medium-power measurements fluctuate together while high-power measurements do not. If true, standard single-sweep extractions of the TLS loss tangent capture only a snapshot of a distribution, and the same fluctuating loss could contribute to the time-varying relaxation times seen in superconducting qubits.

What carries the argument

The load-bearing object is the effective TLS loss tangent $F\delta^{0}_{\mathrm{TLS}}$, the filling-factor-weighted intrinsic TLS loss tangent. The paper tracks its time dependence through the TLS saturation model $1/Q_i = F\delta^{0}_{\mathrm{TLS}}\tanh(\hbar\omega_r/2k_BT)(1+\langle n\rangle/n_c)^{-\beta} + 1/Q_{\mathrm{PI}}$ and, in the two-plateau limit, the simple subtraction $F\delta^{0}_{\mathrm{TLS}}\simeq 1/Q_{\mathrm{LP}} - 1/Q_{\mathrm{HP}}$. A second supporting mechanism is the admittance model with a uniform TLS density of states and fluctuating couplings, which predicts the observed linear relation $\sigma_{Q_i}\propto Q_i$. The diagnostic toolkit is low-frequency noise spectroscopy (spectral densities $S_{Q_i}$) and signal coherence between interleaved traces, which distinguish genuine TLS-driven fluctuations from amplifier noise or common-mode drift.

What would settle it

Measure low-power $Q_i$ for 16 hours on two identical resonators, one with its TLS-rich surface layer altered by a different surface treatment, while continuously tracking a high-power reference: if the 13% fluctuations are TLS loss-tangent variations, the fractional fluctuation amplitude should scale with the extracted $F\delta^{0}_{\mathrm{TLS}}$ and nearly vanish when the TLS contribution is suppressed, whereas a fridge or amplifier drift would survive unchanged.

Watch

Extended reading notes

Core claim

On the paper's own terms, the finding is that the internal quality factor $Q_i$ of distributed-element superconducting microwave resonators is not a fixed number at low power: it wanders with a relative standard deviation of about 13% over 12–16 hours, and the wandering is caused by temporal fluctuations of the effective TLS loss tangent $F\delta^{0}_{\mathrm{TLS}}$. This is established by showing that the fluctuation amplitude falls by up to four orders of magnitude as input power is raised through the TLS saturation region, and by roughly an order of magnitude as temperature is raised toward the quasiparticle-dominated regime. In 16-hour interleaved runs, the low-power and medium-power traces are strongly coherent while neither correlates with the high-power trace, and the high-power quality factor stays roughly fixed while the low-power value swings between $3.3\times10^5$ and $1.0\times10^6$. The effective TLS loss tangent extracted as $F\delta^{0}_{\mathrm{TLS}}\simeq 1/Q_{\mathrm{LP}} - 1/Q_{\mathrm{HP}}$ follows a log-normal distribution with mean $(9.0\pm2.2)\times10^{-7}$, and its fluctuations explain the spread in $Q_i$. Across many resonators, chips, and cooldowns the fluctuation size scales linearly with $Q_i$, giving $\sigma_{Q_i}/Q_i\simeq13\%$ at low power and $0.5\%$ at high power.

Load-bearing premise

The whole attribution rests on the TLS saturation model of Eq. (2) and on the assumption that the high-power quality factor $Q_{\mathrm{HP}}$ sits on a stable, power-independent plateau; if that reference drifts with time or the saturation model misdescribes the power dependence, the inferred TLS loss-tangent fluctuations would be contaminated by other loss channels.

Editorial extensions

If this is right

  • Because a single low-power sweep can miss the instantaneous TLS loss tangent by tens of percent, one-shot extractions should be replaced or supplemented by repeated measurements over at least a few hours to recover the mean.
  • Time-averaging a long record captures the mean TLS loss tangent but not the width of its distribution; that width is itself reproducible information about the material environment.
  • The effect is generic across resonators, chips, and cooldowns with $\sigma_{Q_i}/Q_i\simeq13\%$ at low power, so any low-power resonator benchmark that ignores the fluctuations is incomplete.
  • If the same loss-tangent wandering occurs in superconducting qubits, then background $T_1$ fluctuations need not require individual near-resonant TLS; a time-fluctuating TLS ensemble would produce similar statistics.
  • The proportionality $\sigma_{Q_i}\propto Q_i$ with a power-dependent constant (13% low, 0.5% high) gives a quantitative expectation for quality-factor noise at a chosen operating point.

Reading between the lines

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

  • If the fluctuating-coupling model is correct, surface or interface treatments that lower the TLS density should reduce the fractional fluctuation amplitude $\sigma_{Q_i}/Q_i$, not merely raise the mean $Q_i$; reporting both would make a sharper materials benchmark.
  • The log-normal distribution of $F\delta^{0}_{\mathrm{TLS}}$ and its gradual symmetrization with averaging time imply that qubit $T_1$ statistics depend on sampling rate: faster sampling should expose a skewed tail of low-$T_1$ events that long averages smooth away.
  • Because the $Q_i$ noise spectra are roughly $1/f$, the underlying fluctuators are slow; correlating $Q_i$ wandering with independent charge-noise or TLS-population measurements on the same chip could identify the physical fluctuators.
  • The interleaved low/medium/high-power coherence test could be reused as a general diagnostic to separate TLS-dominated loss from other loss channels in any resonator, including qubit readout resonators.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 3 minor

Summary. The paper reports large temporal fluctuations in the internal quality factor Qi of superconducting Nb-on-Si coplanar waveguide resonators at low power and low temperature, with relative standard deviation sigma_Qi/Qi of about 13% over 12 to 16 hours. The authors attribute these fluctuations to variations in the TLS loss tangent, supported by the decrease of fluctuations at higher power and temperature, the correlation structure in interleaved low- and medium-power measurements, and the stability of the coupling quality factor. They further track the effective TLS loss tangent F delta0_TLS, characterize its log-normal distribution and its dependence on averaging time, and hypothesize that such loss-tangent fluctuations contribute to background T1 fluctuations in superconducting qubits.

Significance. If the attribution to TLS loss-tangent fluctuations holds, the results are significant for resonator-based loss characterization and for understanding long-time noise in superconducting qubits. The paper's strengths include a well-controlled measurement campaign with multiple diagnostics: an explicit JPA on/off comparison (Appendix D), interleaved two-resonator coherence checks (Appendix E), Qc stability analysis (Appendix I), and reporting of fitting uncertainties. The inclusion of data from several devices, chips, and cooldowns, together with comparison to literature values, makes the universality claim credible. The central quantitative claim, however, depends on the fitted Qi being an unbiased instantaneous loss measurement, and that point is not yet fully established.

major comments (1)
  1. [Appendix C / Eq. (1)] The attribution of the observed Qi fluctuations to TLS loss-tangent variations rests on the assumption that each fitted Qi value is an unbiased instantaneous loss measurement. However, Appendix C and Fig. A.3 document substantial low-frequency resonance-frequency noise S_fr that increases at low power, and the paper never checks the impact of this frequency noise on the static Lorentzian fit of Eq. (1). If the resonance frequency shifts by a significant fraction of the linewidth during a single VNA sweep, the circle fit will see a time-averaged, broadened line and report a biased Qi that fluctuates with the frequency noise. Such an artifact would (i) decrease at higher power and temperature because S_fr decreases, (ii) produce correlations between LP and MP fluctuations, and (iii) leave Qc stable, as observed in Fig. A.9. The paper convincingly rules out amplifier noise and Qc instability, but it does not report the sweep duration, compute the expected broadening from the measured S_fr, or examine the cross-correlation between Qi and fr fluctuations. For these reasons, the quantitative extraction of F delta0_TLS in Fig. 3 and the central claim of TLS loss-tangent fluctuations are not yet fully supported. I ask the authors to add at least one of the following: (a) report the VNA frequency span, number of points, IF bandwidth, and sweep time, and estimate the in-sweep frequency jitter from the measured S_fr; (b) compute the signal coherence or cross-spectral density between Qi and fr and show that the two are not strongly correlated at relevant timescales; or (c) repeat a subset of measurements with substantially different sweep durations and demonstrate that the inferred sigma_Qi is independent of sweep time.
minor comments (3)
  1. [Main text / Fig. 1] The paper states sigma_Qi/Qi = 13% but does not provide an uncertainty estimate for this relative standard deviation, which is a key quantitative result.
  2. [Fig. 4 / Appendix H] The fits to the linear model sigma_Qi proportional to Qi in Fig. 4 and Fig. A.8 are shown without reporting the fitted slopes, confidence intervals, or goodness-of-fit; reporting these would help the reader judge the claimed universality.
  3. [Data Availability Statement] The data availability statement limits access to 'available from the corresponding author upon reasonable request'; making the raw time traces and fit parameters publicly available would strengthen reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the TLS attribution, Fδ0_TLS extraction, and σQi∝Qi scaling are empirical or independently modeled, not reductions to fitted inputs.

full rationale

The paper's central claim is that measured Qi fluctuations are attributable to TLS loss-tangent variations. This is supported by three independent empirical observations: fluctuations decrease with power and temperature, interleaved LP-MP traces correlate while HP traces do not, and the low-power Qi distribution is log-normal with a large spread. None of these observations is defined in terms of the conclusion; they are measured time-series statistics. The Fδ0_TLS extraction uses the standard TLS saturation model of Eq. (2), but the model is cited from external literature (Gao, Burnett, and others), and the extracted quantity 1/QLP − 1/QHP is a direct algebraic combination of two independently measured quality factors, not a fitted parameter relabeled as a prediction. The authors explicitly note that QHP is roughly constant and discuss the small offset that would arise if the HP point is not on the true plateau, which is a transparent assumption rather than a circular input. The σQi ∝ Qi relation in Fig. 4 is a fit to data that includes independent points from Refs. 29 and 33, and the proportionality constant is obtained from the data, not from the model; the model only motivates the linear form. The paper includes self-citations (e.g., Refs. 9 and 11 include author McRae), but these are background references for established loss-separation techniques and are not load-bearing for the fluctuation claim. The only substantive caveat, discussed in the skeptical reading, is that low-frequency resonance-frequency noise (Appendix C, Fig. A.3) could in principle bias fitted Qi if a resonance shifts during a VNA sweep; the paper rules out amplifier noise and Qc instability but does not explicitly test sweep-time dependence. That is a potential systematic-error concern, not a circularity, because it does not identify any input that is equivalent to the output by construction. Overall, the derivation chain is self-contained against external benchmarks and independent data, so no circular step is exhibited.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

Central claim rests on standard literature models; no new entities are introduced. The main quantitative outputs are fitted parameters, not derived constants.

free parameters (3)
  • Effective TLS loss tangent Fdelta0_TLS = 9.0 +/- 2.2 x 10^-7 (one resonator)
    Computed as 1/QLP - 1/QHP from interleaved measurements; its temporal distribution is the paper's main quantitative output.
  • TLS saturation model parameters (Fdelta0_TLS, nc, beta, 1/QPI) = Values not all reported; used in dashed fits in Figs. 1b, 1e and 3b
    Fit to Eq. (2) to establish operating points and to guide interpolation; central to the interpretation that fluctuations saturate at high power/temperature.
  • Relative fluctuation constant sigmaQi/Qi = 13% at low power; 0.5% at high power
    Slope of sigmaQi vs Qi linear fit in Figs. 4 and A.8; used to claim universality and power dependence.
assumptions (4)
  • domain assumption TLS saturation model Eq. (2): 1/Qi = Fdelta0_TLS tanh(hbar omega/2kB T) (1 + <n>/nc)^(-beta) + 1/QPI
    Standard model from Refs 10,11,19,30; used to interpret Qi(P,T) and to justify TLS as the source of fluctuations.
  • standard math Circle-fit model Eq. (1) for S21 transmission
    Used to extract Qi and Qc; from Refs 26,27.
  • domain assumption Admittance model with uniform TLS density of states and fluctuating couplings (Ref 28)
    Motivates linear sigmaQi vs Qi relation in Fig. 4; not independently verified in this paper.
  • domain assumption Approximation Fdelta0_TLS = 1/QLP - 1/QHP
    Assumes QHP is on the power-independent plateau and stable; authors verify QHP roughly constant.

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Cite this review

Pith. "Pith review of Loss tangent fluctuations due to two-level systems in superconducting microwave resonators." pith.science (2026). https://pith.science/paper/TDHLR2XL

@misc{pith2026241205482,
  author       = {Pith},
  title        = {Pith review of: Loss tangent fluctuations due to two-level systems in superconducting microwave resonators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TDHLR2XL}},
  note         = {Machine review of arXiv:2412.05482}
}
abstract

Superconducting microwave resonators are critical to quantum computing and sensing technologies. Additionally, they are common proxies for superconducting qubits when determining the effects of performance-limiting loss mechanisms such as from two-level systems (TLS). The extraction of these loss mechanisms is often performed by measuring the internal quality factor $Q_i$ as a function of power or temperature. In this work, we investigate large temporal fluctuations of $Q_i$ at low powers over periods of 12 to 16 hours (relative standard deviation $\sigma_{Q_i}/Q_i = 13\%$). These fluctuations are ubiquitous across multiple resonators, chips and cooldowns. We are able to attribute these fluctuations to variations in the TLS loss tangent due to two main indicators. First, measured fluctuations decrease as power and temperature increase. Second, for interleaved measurements, we observe correlations between low- and medium-power $Q_i$ fluctuations and an absence of correlations with high-power fluctuations. Agreement with the TLS loss tangent mean is obtained by performing measurements over a time span of a few hours. We hypothesize that, in addition to decoherence due to coupling to individual near-resonant TLS, superconducting qubits are affected by these observed TLS loss tangent fluctuations.

Figures

Figures reproduced from arXiv: 2412.05482 by the authors.

Figure 1
Figure 1. FIG. 1. Internal quality factor fluctuations with varying power (a-c) and temperature (d-f). Measurements for (a-c) were performed at base [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Interleaved power measurements of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Standard deviation of the internal quality factor [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (1 more)
Figure 3
Figure 3. Figure 3: FIG. 3. Fluctuations of the effective TLS loss tangent [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.