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Quasiparticle Dynamics in NbN Superconducting Microwave Resonators at Single Photon Regime

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Pith's one-line read NbN superconducting microwave resonators at single-photon powers contain a non-equilibrium quasiparticle population that saturates near 50 per cubic micrometer at 120 mK, limiting their internal quality factor.

desk verdict Useful NbN resonator data, but the 50 µm^-3 saturation number is a residual against an unseen TLS model and should not be taken at face value. read the letter →

arxiv 2506.17816 v1 pith:EHPNCNAW submitted 2025-06-21 quant-ph cond-mat.supr-concs.SYeess.SYphysics.app-ph

Paniz Foshat , Shima Poorgholam-khanjari , Valentino Seferai , Hua Feng , Susan Johny , Oleg A. Mukhanov , Matthew Hutchings , Robert H. Hadfield
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Martin Weides Kaveh Delfanazari
This is my paper · ORCID
classification quant-phcond-mat.supr-concs.SYeess.SYphysics.app-ph
keywords NbNsuperconductingresonatorsquasiparticledensitynon-equilibriumquasiparticlesMattis-Bardeenconductivityinternalqualityfactorsingle-photonregimecoplanarwaveguideresonatormicrowaveloss
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 tries to establish that NbN superconducting coplanar waveguide resonators, operated at single-photon power, contain a measurable population of non-equilibrium quasiparticles at millikelvin temperatures—well below $T_c/10$, where thermal quasiparticles should be negligible. The evidence is a temperature sweep of the internal quality factor: measured $Q_i$ is lower than the sum of two-level-system loss and Mattis-Bardeen thermal quasiparticle loss, and the residual loss converts to a quasiparticle density that saturates near $50\,\mu\mathrm{m}^{-3}$ at 120 mK. If correct, this means millikelvin NbN quantum circuits have a decoherence floor set by non-equilibrium quasiparticles, and mitigation strategies such as quasiparticle traps, infrared shielding, or gap engineering would be needed to raise coherence times.

What carries the argument

The load-bearing object is the Mattis-Bardeen complex conductivity $\sigma(T)=\sigma_1(T)-j\sigma_2(T)$ of the NbN film, evaluated in the superconducting dirty limit where the mean free path is much shorter than the coherence and penetration lengths. The real part $\sigma_1$ is the resistive channel that lowers $Q_i$; the imaginary part $\sigma_2$ is the inductive channel that shifts the resonance frequency. The paper computes $\sigma_1/\sigma_N$ and $\sigma_2/\sigma_N$ from the standard Mattis-Bardeen formulas, feeds them into the surface-impedance expression to obtain the theoretical thermal quasiparticle loss $\delta_{qp,\mathrm{theory}}$, and defines the measured quasiparticle loss as $\delta_{qp,\mathrm{measured}} = 1/Q_{i,\mathrm{measured}} - 1/Q_{\mathrm{TLS}}$, with $Q_{\mathrm{TLS}}$ and the kinetic inductance fraction $\alpha$ taken from the prior study of the same resonators. Equation (7) then carries the argument from residual loss to quasiparticle density using the normal-state density of states $N_0$, the gap $\Delta(T)$, and $\alpha$.

What would settle it

Measure the same NbN resonator at 120 mK with additional infrared shielding or with a directly biased tunnel junction on the same film; if $Q_i$ rises and the inferred density drops well below $50\,\mu\mathrm{m}^{-3}$, or if the junction detects far fewer quasiparticles, the microwave-derived saturation is not an intrinsic quasiparticle density.

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Extended reading notes

Core claim

The paper's central claim is that the low-temperature internal quality factor of NbN coplanar waveguide resonators is limited by a non-equilibrium quasiparticle population. After subtracting the two-level-system loss $1/Q_{\mathrm{TLS}}$ from the measured $1/Q_i$, the remaining loss is attributed to quasiparticles; converting that residual loss through the Mattis-Bardeen relation yields a quasiparticle density that stops decreasing around $T \simeq 120\,$mK and saturates near $50\,\mu\mathrm{m}^{-3}$. Since thermal equilibrium theory predicts a negligible quasiparticle density in this temperature range, the paper concludes that the saturation is a non-equilibrium population acting as a decoherence source in NbN quantum circuits at millikelvin temperatures.

Load-bearing premise

The calculation assumes that all microwave loss not explained by the material's two-level-system defects and by thermally excited quasiparticles is caused by quasiparticles; if stray radiation, vortices, or another loss channel contributes at millikelvin temperatures, the reported saturation density is an artifact rather than a quasiparticle population.

Editorial extensions

If this is right

  • At temperatures below about 1 K, the internal quality factor of these NbN resonators is capped by quasiparticle loss rather than by two-level-system loss alone, so reducing TLS defects will not by itself raise $Q_i$ at base temperature.
  • The saturation density near $50\,\mu\mathrm{m}^{-3}$ at 120 mK means millikelvin NbN circuits have a decoherence floor tied to non-equilibrium quasiparticles, not to thermal excitation.
  • Mitigation strategies that reduce non-equilibrium quasiparticle populations—shielding stray infrared light, adding quasiparticle traps, or gap engineering—should translate directly into higher $Q_i$ and longer coherence times in NbN devices.
  • Mattis-Bardeen equilibrium theory is insufficient for NbN below $T_c/10$; a model including quasiparticle trapping, diffusion, and recombination is needed to describe the measured loss.

Reading between the lines

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

  • One implication the authors leave implicit: if the saturation is dominated by stray radiation rather than intrinsic material properties, then the same device with better infrared shielding should show a higher $Q_i$ and a lower inferred quasiparticle density, which is a testable separation between environmental and intrinsic contributions.
  • The resonance frequency shift observed between 1.6 K and 1.8 K is attributed to quasiparticle inductance, so a quantitative joint fit of $\Delta f/f_r$ and $Q_i$ would provide an independent cross-check of the inferred density.
  • The density conversion assumes a uniform quasiparticle distribution and a constant kinetic inductance fraction $\alpha$; if quasiparticles are concentrated near resonator edges, the reported $50\,\mu\mathrm{m}^{-3}$ is an effective average rather than a local density.
  • A direct test would be to fabricate a small tunnel junction on the same NbN film and measure its subgap current at 120 mK; the inferred quasiparticle density should match the microwave-derived saturation if the attribution is correct.
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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

3 major / 5 minor

Summary. The manuscript reports temperature-dependent microwave spectroscopy of NbN coplanar-waveguide resonators operated near single-photon powers, together with a Mattis-Bardeen calculation of the complex conductivity. The central claim is that the measured internal quality factor at millikelvin temperatures implies a quasiparticle density that saturates near 50 µm^-3 at T = 120 mK, well below T_c/10, indicating a non-equilibrium quasiparticle population that limits Q_i. The paper presents Q_i and frequency-shift data for two resonator modes, calculates theoretical TLS and quasiparticle losses, and attributes the low-temperature deviation between Q_i,measured and Q_i,theory to non-equilibrium quasiparticles.

Significance. If the central claim is correct, the work would provide useful evidence for a millikelvin loss floor in NbN resonators, a material of growing interest for hybrid superconducting circuits. The calculation framework is largely standard: the Mattis-Bardeen thermal conductivity expression is parameter-light, and the resonator parameter extraction follows a published circuit model. The authors are also transparent about the limitations of their interpretation, explicitly noting that future work and tunneling spectroscopy are needed to distinguish intrinsic from external contributions. However, the quantitative result at the heart of the paper depends on a residual subtraction against a TLS model imported from a prior study, with no restated parameters and no visible uncertainty propagation, so the 50 µm^-3 saturation should currently be read as a plausible hypothesis rather than an established measurement.

major comments (3)
  1. [Section IV-C, Eqs. (6)-(7)] The central result—n_qp,measured ≈ 50 µm^-3 at 120 mK—is defined as a residual, δ_qp,measured = 1/Q_i,measured − 1/Q_TLS, where Q_TLS(T) and α are taken from the authors' previous study [22] without restating their values, functional form, or fitting procedure. This is load-bearing: Fig. 6(a) indicates that Q_TLS and Q_i,measured are close near base temperature, so the inferred quasiparticle density is the difference of two comparable quantities, and a factor-of-order-two systematic error in the TLS model could make the residual consistent with zero. Please restate the TLS parameters, show the quality of the TLS fit for the specific resonators used here, and propagate uncertainties through Eq. (7).
  2. [Section IV-C, Figs. 4 and 6] No error bars or confidence intervals are shown for Q_i,measured(T), Δf/f_r, or the derived n_qp,measured, even though Fig. 2(a) demonstrates that such uncertainties exist for these measurements. Without an uncertainty estimate for the subtracted residual, the claimed saturation at 50 µm^-3 cannot be quantitatively assessed, and the word 'confirming' in the inset discussion is stronger than the displayed data support. Please include uncertainty propagation, for example from Monte Carlo resampling of the S21 fits, and show the resulting confidence band on n_qp,measured.
  3. [Section IV-C, concluding paragraph] The manuscript itself states that 'future works are needed to distinguish between intrinsic saturation and external non-equilibrium contributions' and that tunneling spectroscopy 'is critical' to validate the phenomenon, yet the same section concludes that a non-equilibrium quasiparticle density 'confirms a source of decoherence in quantum circuits at millikelvin temperature.' These statements are in tension. The data may demonstrate a low-temperature excess loss, but the specific attribution of that loss to quasiparticles—rather than to stray radiation, vortices, or an incomplete TLS model—is not uniquely established by the analysis presented. Please either add a discriminating measurement or temper the conclusion to an observed excess loss with quasiparticles as one plausible and well-motivated explanation.
minor comments (5)
  1. [Eq. (2)] The denominator of the expression for δ_qp,theory appears to have a parenthesis mismatch; it should read ω(Im(Z_s/ω) + L_g) rather than ω(Im(Z_s/ω)) + L_g).
  2. [Throughout] The notation is inconsistent: 'Q_qp,T heroy', 'Q_qp,theory', and 'δ_qp,T heory' all appear; please unify the notation, e.g., Q_qp,theory and δ_qp,theory, throughout the text and figures.
  3. [Section IV-B, Eqs. (3)-(4)] The approximation conditions ℏω ≪ Δ0 and k_B T ≪ Δ0 should be stated quantitatively; at T = 3 K, k_B T/Δ0 ≈ 0.16 for T_c = 10.7 K, which is not deep in the low-temperature limit, so the validity range of the simplified Mattis-Bardeen expressions deserves a brief comment.
  4. [Fig. 6(b) inset] The inset uses a different horizontal range from the main panel and the caption does not state explicitly that it is a zoomed view; please add axis labels and state in the caption that the inset magnifies the T < T_c/10 region.
  5. [References] Because the TLS model from reference [22] is central to the extraction, please provide its explicit equations and best-fit parameters in the main text or in an appendix so that the residual calculation is reproducible without consulting the prior paper.

Circularity Check

1 steps flagged · score 6.0 of 10

The 50 µm^-3 saturation at 120 mK is the residual of Eq. (6) after subtracting a TLS model and α imported from the authors' prior paper [22], so the quasiparticle-density claim is not independently derived.

  1. fitted input called prediction [Section IV-C, 'Thermal and Non-equilibrium Quasiparticle Density', Eqs. (6)-(7) and Fig. 6(b) inset]
    "δqp,measured(T) = 1/Qi,measured(T) − 1/QTLS(T) (6) nqp,measured(T)∼δqp,measured(T)N0∆(T)π/α sqrt(ℏω/2∆(T)) (7) Where nqp,measured(T) is quasiparticle density calculated from measurement results, N0∼1.86×10^28 (state/m3eV) is the density states at the Fermi level from [47,54], and α is the kinetic inductance ratio extracted from our previous study [22]. In the inset of Fig.6(b), quasiparticle density is saturating at 50µm^-3 at T=120 mK, confirming a source of decoherence in quantum circuits at millikelvin temperature."

    By Eq. (7), nqp,measured is a constant times δqp,measured, and by Eq. (6) δqp,measured is defined as 1/Qi,measured − 1/QTLS. QTLS(T) and α are taken from the authors' prior study [22] without restating or refitting them. The claimed 50 µm^-3 saturation at 120 mK is therefore, by construction, the residual after subtracting the imported TLS model; it is not an independent quasiparticle measurement. If QTLS from [22] already absorbs the low-temperature plateau, the residual and hence nqp,measured vanish. The paper acknowledges this: 'Future works are needed to distinguish between intrinsic saturation and external non-equilibrium contributions,' and says tunneling spectroscopy 'is critical' for validation—so attributing the residual to quasiparticles is an assumption, not a proven result.

full rationale

The derivation chain for the paper's headline claim is not self-contained. The direct measurement of Q_i(T) at single-photon power is an independent experimental result, and the low-temperature excess loss relative to a simple Mattis-Bardeen thermal model may well be real. However, the specific conclusion that this excess is a non-equilibrium quasiparticle density of about 50 µm^-3 at 120 mK is obtained by defining n_qp,measured through Eqs. (6) and (7) as a constant times [1/Q_i,measured − 1/Q_TLS], with Q_TLS(T) and α imported from the authors' previous paper [22] and not restated or refit. Thus the 50 µm^-3 saturation is, by construction, the residual after subtracting that imported TLS background; it is a transformed residual, not an independently predicted quasiparticle population. If the [22] TLS model had absorbed more of the low-temperature loss, the residual would be correspondingly smaller. The paper itself flags the underdetermination: it states that 'future works are needed to distinguish between intrinsic saturation and external non-equilibrium contributions' and that tunneling spectroscopy 'is critical' to validate the phenomenon. Figures 4(a) and 6 are shown without error bars, so the statistical significance of the residual cannot be independently assessed. This is a partial circularity (score 6), not a complete one: the Q_i data and the Mattis-Bardeen thermal calculation are external inputs, and the residual is not literally a renamed fit parameter in this paper, but the central claim reduces to a background subtraction whose background is a self-cited, unshown model.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a standard Mattis-Bardeen thermal model, a TLS model borrowed from prior work, and a residual-loss conversion. No new entities are introduced. The main burden is that the measured quasiparticle density is not directly observed: it is the leftover after subtracting a fitted TLS curve, scaled by a kinetic inductance fraction from an earlier paper. This makes the 50 per cubic micrometer saturation number dependent on several unstated or prior-fitted inputs.

free parameters (3)
  • Kinetic inductance ratio alpha = not stated in this paper, extracted from reference [22]
    Eq. 7 scales n_qp,measured inversely with alpha; without its value the quasiparticle density numbers cannot be reproduced.
  • Two-level system loss curve Q_TLS(T) = not stated; taken from prior fit in reference [22]
    Eqs. 5 and 6 subtract Q_TLS from measured Q_i; if this curve was fitted to similar NbN resonator data, the residual quasiparticle density is partly determined by that prior fit.
  • Geometrical inductance L_g = not stated
    Eq. 2 requires L_g to convert complex conductivity into quasiparticle loss delta_qp,theory, but no value or calculation is provided.
assumptions (5)
  • domain assumption NbN film is in the dirty limit, so surface impedance and Mattis-Bardeen formulas apply.
    Invoked in Section IV before Eqs. 1-4 via references [48,49]; no mean free path or coherence length data are given.
  • domain assumption BCS weak-coupling gap Delta0 = 1.76 k_B T_c describes NbN.
    Used in Eqs. 3-4 with T_c = 10.7 K; NbN can show strong-coupling corrections, but the paper does not test this.
  • domain assumption All non-TLS, non-thermal-quasiparticle loss is attributed to quasiparticles.
    Eq. 6 defines delta_qp,measured as the difference between measured loss and Q_TLS from [22]; stray radiation, vortices, and interface losses are not independently separated.
  • ad hoc to paper The TLS loss model from reference [22] applies unchanged to the resonators in this study.
    Q_TLS(T) enters Eq. 5; the paper does not re-fit or validate the model against the present devices.
  • domain assumption Photon number in the resonator is described by Eqs. 9-13 using the fitted Q_l and Q_c from the notch model.
    Appendix photon number calculation; assumes the resonator circuit model [51] and power budget are accurate.

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Pith. "Pith review of Quasiparticle Dynamics in NbN Superconducting Microwave Resonators at Single Photon Regime." pith.science (2026). https://pith.science/paper/EHPNCNAW

@misc{pith2026250617816,
  author       = {Pith},
  title        = {Pith review of: Quasiparticle Dynamics in NbN Superconducting Microwave Resonators at Single Photon Regime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHPNCNAW}},
  note         = {Machine review of arXiv:2506.17816}
}
read the original abstract

Exchanging energy below the superconducting gap introduces quasiparticle energy distributions in superconducting quantum circuits, which will be responsible for their decoherence. This study examines the impact of quasiparticle energy on the performance of NbN superconducting microwave coplanar waveguide resonators on silicon chips. We measured the resonance frequency and internal quality factor in response to temperature sweeps to evaluate the effect of quasiparticle dynamics. Moreover, by calculating the complex conductivity of the NbN film, we identified the contribution of quasiparticle density to the experimental results.

Figures

Figures reproduced from arXiv: 2506.17816 by the authors.

Figure 1
Figure 1. (a) Top view and cross-section illustrations of a 100 nm [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. (a) Qi in the photon number ranges from < nph > = 10−1 to 103 corresponding to the power ranges from Pin = -140 dBm to -100 dBm at T = 26 mK, and fr = 5.95 GHz, adapted from [22]. The inset plot shows the approximate < nph > concerning input power inside the transmission line in the CPWs. In this plot, the input power to reach the single photon regime is Pin ≃ -135 dBm. The vertical lines show Qi uncertainty range, … view at source ↗
Figure 3
Figure 3. (a) The XRD pattern of the NbN film. (b) Zoomed [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: (a) Qi at single photon regime versus temperature for fr = 5.95 GHz and fr = 4.45 GHz. (b) ∆f at single photon regime versus temperature for fr = 5.95 GHz and fr = 4.45 GHz. through the sample, an isolator was placed at the 4 K stage to prevent transmission signals fro…
Figure 5
Figure 5. Figure 5: (a) The real part of the normalized complex conduc [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: (a) Qi,measured and Qi,theory versus temperature at < nph > ∼ 1 fitted with the theoretical model of TLS and quasiparticle loss. Both plots are calculated and measured at fr = 5.95 GHz. (b) Theoretical and measured quasiparticle density of NbN CPW. The inset shows quas…

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Forward citations

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.