Revealing spin-flip two-level systems using ultra-thin film superconducting resonators

Bao-Chuan Wang; Gang Cao; Guang-Can Guo; Guo-Ping Guo; Hai-Ou Li; Shu-Kun Ye; Tian-Yi Jiang; Tian-Yue Hao; Xiang-Xiang Song; Yong-Qiang Xu

REVIEW 3 major objections 4 minor 64 references

Revealing spin-flip two-level systems using ultra-thin film superconducting resonators

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Spin-flip two-level systems explain the anomalous resonator frequency rise at low magnetic fields.

desk verdict A reproducible and striking experimental anomaly in ultra-thin TiN resonators, but the central quantitative model has a load-bearing derivation gap: the stated θ distribution gives a temperature-dependent prefactor absent from Eq. (4), so the claimed temperature check does not follow as written. read the letter →

arxiv 2412.15856 v1 pith:CBEC3PPC submitted 2024-12-20 cond-mat.mes-hall quant-ph

Zi-Qing Huang , Shu-Kun Ye , Yong-Qiang Xu , Tian-Yi Jiang , Tian-Yue Hao , Bao-Chuan Wang , Xiang-Xiang Song , Hai-Ou Li

show 3 more authors

Guang-Can Guo Gang Cao Guo-Ping Guo

This is my paper · ORCID

classification cond-mat.mes-hallquant-ph

keywords spin-fliptwo-levelsystemsmagneticdefectssuperconductingresonatorsTiNthinfilmseffectivespin-orbitcouplingfrequencyshiftquantumdevicenoisesystemsaturation

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

Ultra-thin titanium nitride superconducting resonators show an unexpected increase in resonant frequency when a small in-plane magnetic field is applied, in contrast to the quadratic decrease predicted by BCS theory. The paper attributes this anomaly to spin-flip two-level systems (sTLSs), in which an electron tunneling between two sites also flips its spin because the local magnetic field from nearby magnetic defects points differently at the two sites. A model with an effective spin-orbit coupling yields a closed-form expression for the sTLS contribution to the resonator frequency shift, and the resulting formula quantitatively reproduces the measured frequency-field curves and their temperature dependence. If correct, this connects the long-known magnetic-defect noise in quantum devices to the TLS losses that limit superconducting circuits, and suggests magnetic fields could be used to engineer those losses.

What carries the argument

The load-bearing object is the spin-flip TLS Hamiltonian $H_{\mathrm{sys}} = \frac{1}{2}(m\sigma_z + 2\Delta_0 \tau_x) + b(|L\rangle\langle L|\sigma_x + |R\rangle\langle R|\sigma_\theta)$, where $b$ is the spin-flip rate induced by the perpendicular component of the local magnetic field at each well, $\theta$ is the relative angle between the local fields in the two wells, $\Delta_0$ is the usual tunneling amplitude, and $m = g\mu_B B$ is the Zeeman energy. This Hamiltonian converts the field-dependent spin-flip rate into a field-dependent energy splitting $\varepsilon = \sqrt{m^2 + 4b^2}$ for $\theta \approx 0$, and into an electric-dipole matrix element $p_{\mathrm{eff}} = p \sin\Gamma$ with $\sin\Gamma \approx (b \sin(\theta/2))/(\Delta_0 - \varepsilon^2/4\Delta_0)$. Substituting the assumed distributions $P_{\Delta_0} \sim n_{\mathrm{sTLS}}/\Delta_0$, $P_{\cos\phi}$ uniform, $P_b = \delta(b-b_0)$, and $P_\theta$ from a negative-$J$ angular distribution into the susceptibility integral yields the closed form of Eq. (4), which then enters the observed frequency shift through Eq. (5). The key step is that the spin-flip rate $b$, not the tunnel splitting $\Delta_0$, sets the low-energy scale of the defect.

What would settle it

Measure the frequency-field curve on an ultra-thin TiN resonator before and after a surface treatment that removes magnetic defects: if the low-field frequency rise survives the treatment, the sTLS attribution fails. Alternatively, compare the extracted spin-flip rate $b_0 \approx 10\,\mu\mathrm{eV}$ with electron-spin-resonance spectra; the amplitude of the anomaly should track the density of the specific spin species, and a material with negligible surface magnetism should show no rise at all.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the anomalous low-field rise in frequency is not a quasiparticle effect, but the signature of a new class of two-level defects. In each sTLS, an electron tunnels between two wells while the perpendicular component of the local field from magnetic defects, oriented differently at the two wells, induces a spin flip; the inhomogeneous field thereby creates an effective spin-orbit coupling that mixes orbital and spin states. As a result, the sTLS energy takes the field-dependent form $\varepsilon = \sqrt{(g\mu_B B)^2 + 4b^2}$ rather than the field-independent tunnel splitting of a conventional TLS, and its electric-dipole coupling is reduced by the spin-orbit mixing angle $\Gamma$. Integrating the susceptibility of such systems over parameter distributions gives the closed form of Eq. (4), and adding the BCS quasiparticle term, $\Delta f_r/f_r = kB^2 + h(B,T)$, fits the measured curves at 10 mK and at higher temperatures with only three parameters. The extracted zero-field sTLS energies lie between 300 and 600 mK and the implied magnetic defect density is around $10^{17}\,\mathrm{m}^{-2}$, placing these defects squarely in the surface layer of the TiN film.

Load-bearing premise

The calculation replaces the unknown distribution of spin-flip rates by a single constant $b_0$, treating $P_b = \delta(b-b_0)$, and assumes the relative local-field angle $\theta$ is near zero because the defect-defect coupling $J$ is negative, so that $\varepsilon = \sqrt{m^2 + 4b^2}$; if the real distributions of $b$ or $\theta$ are broad, the predicted field and temperature dependence would differ from the fitted form.

Editorial extensions

If this is right

The sTLS zero-field energies (300–600 mK, i.e., 6–12 GHz) fall directly in the operating band of superconducting qubits and microwave resonators, so these defects contribute to loss and noise at device-relevant frequencies.
Because sTLSs couple to the electric field through the reduced dipole $p \sin\Gamma$, they saturate at higher drive power than conventional TLSs, which can explain the stronger loss suppression with magnetic field observed at intermediate power.
Polarizing defect spins with a magnetic field reduces the inhomogeneity of the local field and thereby weakens the effective spin-orbit coupling, suggesting a path to magnetically decouple sTLSs from the device.
An optimal magnetic field should exist that minimizes decoherence, balancing sTLS suppression against quasiparticle generation and other field-induced losses.
The estimated defect density of about $10^{17}\,\mathrm{m}^{-2}$ supports the view that surface magnetic defects, not bulk TLSs, are responsible for the anomaly, which is consistent with earlier surface-treatment experiments.

Reading between the lines

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

The delta-function replacement for $P_b$ is a strong simplification; a direct way to test it would be to measure the frequency shift at fields well beyond the fitted range and check that the functional form of $h(B,T)$ still holds.
The model predicts a specific dependence on the direction of the in-plane field relative to the local disorder axes; rotating the field in the plane while keeping its magnitude fixed could reveal an anisotropic component of the anomaly that the current data do not address.
Because the sTLS susceptibility is nonresonant (the photon energy $hf_r$ is far from $\varepsilon$), the same mechanism should also produce a small shift in the resonator's internal quality factor; measuring $Q_i$ as a function of $B$ and $T$ would provide an independent check.
The sTLS picture could be extended to other disordered superconductors that show surface magnetism; the model predicts that their sTLS spectra, and hence their optimal operating fields, depend on the specific defect species.

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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

First thing you should know: the experiment is genuinely interesting. A reproducible low-field increase in resonant frequency in 12 ultra-thin TiN resonators, absent in perpendicular field, is not in the prior literature, and the authors have done the basic sanity checks (field direction, vortex pinning, hysteresis, reproducibility). The idea that spin-flip TLSs from magnetic defects couple to the resonator is physically plausible and connects to existing glass-model work. I'd want this on the record.

The soft spot is the theory. The step from Eq. (3) to Eq. (4) is not shown, and when you try to do it with the distributions stated, it does not work. With Pθ as given and J ~ -1 meV, θ is peaked near zero, and the dipole factor (b sin θ/2)^2 averages to roughly kB T/(2|J|). That is a temperature-dependent prefactor that grows about 60x between 10 mK and 600 mK for t ~ 0.5 K, while the tanh in Eq. (4) falls only ~2.5x. So the sTLS susceptibility should grow with temperature, not saturate as the data show. Eq. (4) has no such prefactor. Unless the authors can exhibit the integration, the claimed quantitative temperature dependence does not follow.

There are also smaller issues: the b distribution is replaced by a delta function after admitting it cannot be defined, the fits use the same data that define the anomaly, and the extracted shifts in Fig. 2B are small enough that missing error bars matter. The consistency of the fitted BCS coefficient k with the 600 mK value is encouraging, and the extracted t ~ 300–600 mK is sensible for GHz-frequency TLSs.

Net: this is a valuable experimental paper wrapped in a theory that is not yet established. The observation deserves a serious referee. I'd send it out, but with a request for the full derivation of Eq. (4), a numerical check of the θ integration, and error bars on the fractional shifts. If the derivation cannot be produced, the paper should be reframed as an experimental report with a speculative model.

Referee Report

3 major / 4 minor

Summary. The manuscript reports an anomalous increase in resonant frequency at low in-plane magnetic fields in 7.9-nm TiN superconducting resonators, in contrast to the quadratic decrease expected from quasiparticle generation. It attributes the anomaly to a proposed 'spin-flip two-level system' (sTLS), in which an electron tunneling between two wells is coupled to magnetic defects whose inhomogeneous local fields mix charge tunneling and spin flips. A closed-form susceptibility, Eq. (4), is proposed and used together with a BCS quasiparticle term to fit frequency-field and frequency-temperature data, claiming quantitative reproduction with three parameters k, W, and t.

Significance. If correct, the sTLS mechanism would connect magnetic surface defects to TLS losses and would suggest magnetic-field engineering of decoherence in quantum devices. The experimental part has strengths: the anomaly is reported as reproducible across 12 resonators, hysteresis and vortex effects are discussed, and the high-temperature BCS baseline is consistent with known behavior. However, the central theoretical claim is compromised by the unshown reduction of Eq. (3) to Eq. (4) and by a temperature dependence that appears to contradict the stated θ-distribution. The model as presented is therefore not an independent prediction, and the quantitative statement is not supported in its current form.

major comments (3)

[§II, Eqs. (3)–(4)] The passage from the integral in Eq. (3) to the closed form in Eq. (4) is not derived. With the stated distribution Pθ for J < 0 and |J| >> kBT, the θ integral of the dipole factor (b sin(θ/2))² gives ⟨sin²(θ/2)⟩ ≈ kBT/(2|J|), a factor linear in T that is absent from the T-independent coefficient A in Eq. (4). Since ε ≈ sqrt((vB)² + t²) and the tanh factor saturates at low temperature, the resulting sTLS contribution would grow with temperature, whereas the data show the anomaly disappearing by 600 mK. As written, Eq. (4) is not the integral of Eq. (3), so the claimed quantitative reproduction and the temperature check do not follow. Please provide the full integration or revise the model so that the predicted temperature dependence matches the data, and redo the fits accordingly.
[§II, Pb definition] The replacement Pb = δ(b − b0) is asserted with only a reference to unshown numerical calculations ('we have verified through numerical calculations'), and the text concedes that Pb cannot be accurately defined. Because b enters both the sTLS energy and the electric-dipole matrix element, a broad Pb would change the field and temperature dependence of h(B,T). Without a derivation, a sensitivity analysis, or the actual numerical check, the three-parameter fit cannot discriminate the sTLS model from a generic empirical curve.
[§III, Fig. 3C and 3D] The central claim of quantitative reproduction rests on fitting k, W, and t to the same 10-mK field sweep that defines the anomaly, and then refitting W and t for each temperature in Fig. 3D. There is no out-of-sample or parameter-free prediction; the statement that the temperature dependence is 'reproduced' is therefore overstated. A stronger test would be to fix W and t from one condition and predict the other curves, or to report residuals and parameter uncertainties for all 12 resonators.

minor comments (4)

[Eq. (4)] The coefficient A is said to be 'positively related to nsTLS', but its exact relation to the microscopic parameters and to the integrals over Δ0 and θ is not given; please define it explicitly.
[§II, Pb paragraph] The numerical verification of the delta-function assumption for Pb is not shown; please include it in the Supplementary Material so that the assumption can be checked.
[§III, Fig. 3D] The sTLS contribution in Fig. 3D is obtained by subtracting the BCS fit at 600 mK from data at all temperatures; please clarify why the modified formula's own quasiparticle contribution at each temperature is not used for this subtraction.
[References] Some references are cited indirectly, notably [42] for the θ-distribution and [46] for the sign of J; please cite primary sources for these assumptions so that the reader can verify them.

Circularity Check

2 steps flagged · score 5.0 of 10

Partial circularity: the field anomaly is fitted with W and t on the same data that define it, and the temperature check re-fits the same parameters; Eq. (4) is asserted as the integral of Eq. (3) although the stated θ distribution would introduce a T-dependent prefactor.

fitted input called prediction [Section after Eq. (5), Fig. 3C/D, and Fig. 4]
"We use this formula to fit the experimental results with three fitting parameters, k, W = GA/(2ǫm), and t. As shown in the upper panel of Fig. 3C, the data measured at 10 mK, which cannot be well-fitted using the pure BCS model (red curve), are quantitatively reproduced by our modified formula (blue curve). ... All these behaviors are well-fitted by Eq. (5) with fitting parameters W and t."

The central anomaly is defined by the 10 mK frequency-field curve, and W and t are fitted to exactly that curve; the 'quantitative reproduction' of the B-field dependence is therefore a fit, not an independent prediction. The temperature dependence is then checked by subtracting a BCS background and re-fitting Eq. (5) with W and t to the extracted sTLS residuals (Fig. 3D and Fig. 4). Because the same flexible parameters are re-optimized, the temperature check does not test a parameter-free prediction; the model's functional shape still provides some non-tautological content, so the circularity is partial.
other [Equations (3)-(4), paragraph beginning 'Substituting these distributions into Eq. (3)']
"The distribution of θ is related to the defect–defect interaction J, and can be expressed as Pθ = [2kBT sinh(J/kBT)/J]−1sin(θ)e−Jcosθ/kBT ... Therefore, θ should be close to zero, which makes ε ≈ sqrt(m²+4b²). Substituting these distributions into Eq. (3), we can obtain the final expression of ǫsTLS: ǫsTLS(B,T) = At²√((vB)²+t²)/((vB)²+t²−(hfr/kB)²) tanh(√((vB)²+t²)/2T)."

This step is presented as a derivation, but the stated inputs do not produce the displayed output. In Eq. (3) the integrand contains the electric-dipole factor (b sin θ/2)², and with the quoted Pθ peaked near θ ≈ 0 (negative J), the θ-integral yields a T-dependent average scale proportional to kBT/|J|. That prefactor is absent from Eq. (4), where A and t are T-independent. The thermal-saturation dependence in Eq. (4) is therefore not obtained from the model's stated distributions; it is effectively inserted by the chosen closed form. The temperature check then tests that inserted formula rather than the physical model as specified.

full rationale

There is no load-bearing self-citation chain or imported uniqueness theorem; the sTLS model is an original construction. The main circularity is statistical rather than logical: the B-field anomaly that the paper claims to reproduce is the same data set used to fix W and t, and the temperature evolution is re-fitted with the same parameters rather than compared with a parameter-free prediction. The paper itself concedes Pb cannot be accurately defined and replaces it with a delta function, and the passage from Eq. (3) to Eq. (4) omits the T-dependent prefactor implied by the stated Pθ distribution; this makes the claimed derivation of Eq. (4) an ansatz. Still, the model is not a tautology: it makes specific functional predictions involving √((vB)²+t²) and a tanh saturation that can, in principle, be falsified, and the qualitative disappearance of the anomaly at 600 mK is a nontrivial qualitative check. The circularity is therefore partial, warranting a score of 5 rather than 0 or 10.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central claim rests on three fitted parameters (k, W, t) and on several domain assumptions inherited from standard TLS and BCS theory. The most fragile additions are the delta-function distribution for b and the θ≈0 assumption, which are introduced to obtain the analytics of Eq. (4). No code or data are shipped, and the supplementary material that would document the numerical checks and full fits is not included in the provided text.

free parameters (3)

k (BCS quasiparticle quadratic coefficient) = (-0.94 ± 0.03) × 10^-9 mT^-2 at 10 mK; (-1.03 ± 0.01) × 10^-9 mT^-2 at 600 mK
Quadratic coefficient for the quasiparticle contribution to the frequency shift in the fit to Δfr/fr versus B.
W (sTLS amplitude, W = GA/(2ε_m)) = Not stated numerically in the main text; 'almost of the same order' across field and temperature fits
Overall amplitude of the sTLS-induced frequency shift, absorbing the filling factor, sTLS density, and the angular coupling factors.
t (average zero-field sTLS energy in temperature units, t = 2b0/kB) = 300-600 mK across fits
Controls the B and T dependence of the sTLS susceptibility through sqrt((vB)²+t²); fitted to the field and temperature curves.

assumptions (5)

domain assumption Standard BCS quasiparticle theory gives a quadratic frequency shift with magnetic field, Δfr^qp/fr = kB², with k independent of temperature.
Used to separate the quasiparticle background from the sTLS contribution; the consistency of k at 10 mK and k' at 600 mK is presented as support.
domain assumption The standard TLS susceptibility formalism from Gao's thesis (ref 34) applies to spin-flip TLSs, so the frequency shift is -G ε_sTLS/(2 ε_m).
Carried over from conventional TLS theory without adaptation to the spin-flip case.
domain assumption The magnetic defects are concentrated at the TiN-vacuum interface and have a negative exchange coupling J of order meV, forcing the local field angle θ near zero.
Basis for approximating ε ≈ sqrt(m²+4b²) and for the closed form of Eq. (4); rooted in cited room-temperature ferromagnetism in TiO2 and TiN.
ad hoc to paper The distribution of spin-flip rates b can be replaced by a delta function Pb = δ(b−b0).
The paper states 'we cannot accurately define it' and then asserts a numerical verification that b can be treated as a constant; this simplification makes the integral in Eq. (3) analytically tractable and is not independently justified in the text.
domain assumption The dipole orientation angle cosφ is uniformly distributed and PΔ0 ~ nsTLS/Δ0.
Imported from standard TLS models to evaluate Eq. (3).

invented entities (1)

Spin-flip two-level system (sTLS): an electron tunneling between two potential wells with spin flips induced by local magnetic fields from magnetic defects
purpose: Explains the anomalous increase in resonator frequency with in-plane magnetic field and its temperature dependence.
The sTLS is not directly observed; its energy scale t and coupling W are extracted from the same frequency shifts it is meant to explain. The temperature dependence is a weak independent check, but the fitting parameters are re-adjusted.

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Cite this review

Pith. "Pith review of Revealing spin-flip two-level systems using ultra-thin film superconducting resonators." pith.science (2026). https://pith.science/paper/CBEC3PPC

@misc{pith2026241215856,
  author       = {Pith},
  title        = {Pith review of: Revealing spin-flip two-level systems using ultra-thin film superconducting resonators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CBEC3PPC}},
  note         = {Machine review of arXiv:2412.15856}
}

read the original abstract

Material disorders are one of the major sources of noise and loss in solid-state quantum devices, whose behaviors are often modeled as two-level systems (TLSs) formed by charge tunneling between neighboring sites. However, the role of their spins in tunneling and its impact on device performance remain highly unexplored. In this work, employing ultra-thin TiN superconducting resonators, we reveal anomalous TLS behaviors by demonstrating an unexpected increase in resonant frequency at low magnetic fields. Furthermore, a spin-flip TLS model is proposed, in which an effective spin-orbit coupling is generated by inhomogeneous local magnetic fields from defect spins. This mechanism mixes charge tunnelings and spin flips, quantitatively reproducing the observed frequency-field relationship and its temperature dependence. This work deepens the understanding of spin-dependent TLS behaviors, offering the possibility of magnetically engineering noise and loss in solid-state quantum devices.

Figures

Figures reproduced from arXiv: 2412.15856 by the authors.

**Figure 2.** shows the measured fr of four typical resonators as a function of magnetic field B (results from the other eight resonators can be found in SM). Under a perpendicular magnetic field B⊥, fr decreases rapidly. This is because the magnetic field breaks the Cooper pairs to generate qp, and increases the kinetic inductance of the resonator [15, 20]. We use the fractional frequency shift ∆fr/fr to quantify the frequency… view at source ↗

**Figure 3.** FIG. 3. (A) Schematic illustration of the sTLS model with [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** FIG. 4. Measured ∆ [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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