Amplitude mode in a multi-gap superconductor MgB$_2$ investigated by terahertz two-dimensional coherent spectroscopy

Jiahao Liang; Ke Chen; Kota Katsumi; N. P. Armitage; Ralph Romero III; Xiaoxing Xi

REVIEW 3 major objections 6 minor 2 cited by

Amplitude mode in a multi-gap superconductor MgB$_2$ investigated by terahertz two-dimensional coherent spectroscopy

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The multigap superconductor MgB2's amplitude mode is overdamped, with a fitted damping rate of 0.55 THz that the authors attribute to interband coupling.

desk verdict The normalized FH result is a real new observation, but the overdamped pi-band amplitude mode with delta = 0.55 THz is a plausible hypothesis that needs more than a single-oscillator fit to support it. read the letter →

arxiv 2411.10852 v2 pith:P2EW654W submitted 2024-11-16 cond-mat.supr-con

Kota Katsumi , Jiahao Liang , Ralph Romero III , Ke Chen , Xiaoxing Xi , N. P. Armitage This is my paper · ORCID

classification cond-mat.supr-con

keywords MgB2multigapsuperconductoramplitudemodeterahertztwo-dimensionalcoherentspectroscopyfirst-harmonicnonlinearresponseinterbandcouplingoverdampedcollectivesuperconductinggapdynamics

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 the collective amplitude mode of the superconducting order parameter in the multigap superconductor MgB2 is strongly damped, and that the damping comes from coupling between the two superconducting bands. Using terahertz two-dimensional coherent spectroscopy with both broadband and narrowband driving fields, the authors find a nonlinear response at twice the lower gap energy at low temperature, but the normalized first-harmonic signal shows a monotonic increase on cooling rather than the resonant enhancement seen in single-gap NbN. The data are fit with a damped-oscillator model giving a π-band amplitude-mode damping rate δ=0.55 THz, roughly 4.5 times larger than NbN's. If correct, this shows that interband coupling qualitatively changes collective excitations in multigap superconductors and sets a benchmark for how amplitude modes decay in such systems.

What carries the argument

The central object is the superconducting amplitude mode—the collective oscillation of the magnitude of the order parameter—in the lower-gap π band of MgB2. The argument is carried by the first-harmonic (FH) intensity of the THz 2DCS nonlinear signal as a function of temperature, normalized by the sixth power of the transmitted drive field to remove screening effects. The fitting machinery is the damped-oscillator resonance model I(ω,T)=I0 Δπ(T)^2 / ((ω+iδ)^2-(2Δπ(T))^2), used with ω=Ω for the FH signal and ω=2Ω for the third-harmonic signal, where δ is the amplitude-mode damping rate. A resonant enhancement is expected when the drive frequency matches twice the gap; its absence in the normalized FH data is read as overdamping.

What would settle it

Measure the normalized first-harmonic intensity with narrow-band drives at several frequencies sweeping Ω through 2Δπ(T) at low temperature and fit each temperature scan with the damped-oscillator model leaving δ free; a resonant peak at Ω=2Δπ with δ near 0.12 THz would refute the overdamped/interband-coupling claim, as would the same monotonic FH behavior in a single-gap superconductor.

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Extended reading notes

Core claim

The paper reports that in the multigap superconductor MgB2, terahertz two-dimensional coherent spectroscopy with narrow-band drive at Ω/2π=0.63 THz produces first- and third-harmonic nonlinear signals whose normalized first-harmonic intensity rises monotonically as temperature decreases, with no resonant enhancement at the temperature where Ω=2Δπ(T). This contrasts with NbN, where the normalized FH signal peaks resonantly at that matching condition, and is interpreted as evidence that the π-band amplitude mode in MgB2 is strongly overdamped. Fitting the temperature dependence to a single damped oscillator gives damping rate δ=0.55 THz, about 4.5 times the δ=0.12 THz found in NbN. The paper attributes this overdamping to interband coupling between the π and σ bands, which theory says suppresses the π-band amplitude mode. A well-defined amplitude-mode signal is seen only at the lowest temperatures, in a broadband nonlinear peak at 2Δπ≈1 THz.

Load-bearing premise

The load-bearing assumption is that the measured first-harmonic signal is dominated by the π-band amplitude mode and behaves like a single damped oscillator; if ordinary quasiparticle or diamagnetic nonlinearity dominates instead, the fitted 0.55 THz damping rate is not the mode's linewidth and the interband-coupling conclusion collapses.

Editorial extensions

If this is right

If the π-band amplitude mode is overdamped by interband coupling, the first-harmonic channel of THz 2DCS will not show a clean amplitude-mode resonance in MgB2 at Ω≈2Δπ; searches for the mode must rely on the third harmonic or on lower temperatures where the mode is better defined.
The normalization procedure—dividing raw nonlinear intensity by the sixth power of the transmitted drive field—becomes mandatory for comparing superconductors whose optical conductivity changes strongly with temperature; raw FH peaks near Tc are artifacts of screening.
The fitted δ=0.55 THz predicts that the amplitude-mode line is broadened to a degree that should be observable in other MgB2 nonlinear or Raman experiments, providing a cross-check.
Because theory associates the suppression with interband coupling strength, the result supports the view that materials with weaker interband coupling should exhibit sharper amplitude-mode resonances than MgB2.

Reading between the lines

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

The paper leaves implicit that a direct test would be to tune the narrow-band drive across a range of frequencies near 2Δπ at fixed low temperature; an underdamped mode would show a Lorentzian peak in the normalized FH intensity as a function of drive frequency, whereas pure overdamping would show only a monotonic increase.
If interband coupling is the cause, MgB2 films with modified σ/π coupling, for example through disorder or doping, should show a systematically different damping rate; measuring δ across such samples would separate interband from intrinsic lifetime effects.
The 2D rephasing and non-rephasing spectra, which the paper notes do not show clean diagonal broadening, might still separate amplitude-mode and quasiparticle contributions if analyzed with a two-band model; that is a natural next step not pursued here.

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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

Here's my take on the MgB2 THz 2DCS paper. The headline: the normalized FH temperature dependence is a genuine new experimental result, and the paper is candid about the interpretive step, but that step—from a monotonic FH increase to an overdamped pi-band amplitude mode with delta = 0.55 THz—is load-bearing and not yet supported by the model fit alone.

What's new and good: this is the first THz 2DCS applied to a multigap superconductor. The broad-band data show a nonlinear peak at 2Delta_pi at low T, and the narrow-band measurements separate FH and TH responses, both scaling as E^6 as expected for third-order nonlinearities. The field-normalization procedure is thoughtful: the raw FH peak near Tc disappears once you divide by the sixth power of the transmitted field, and you get a monotonic increase with decreasing T. The pump-probe controls showing no condensate depletion at 12-30 kV/cm are solid, and the two-band BCS gap solution tracks the measured 2Delta_pi well. These are careful experiments.

The soft spots are in the interpretation, not the data. Eq. (1) is a single damped oscillator for the pi-band amplitude mode, but the drive frequency (0.63 THz) only matches 2Delta_pi near Tc. Over most of the temperature range the response is off-resonant, and in that regime the non-resonant diamagnetic or quasiparticle nonlinearity, which grows with superfluid density as T drops, will give a monotonic FH increase with no amplitude mode needed. The kF l argument from the NbN work was established at resonance; it does not transfer automatically off-resonance. There is also a tension with the paper's own cited theory: Ref. [32] says the pi-band amplitude mode is strongly suppressed when interband coupling is finite—the authors acknowledge this and then assume the pi mode dominates because it is closer to the drive. That may be true, but it needs a two-band calculation, not an assertion. Finally, delta = 0.55 THz comes from a fit with no error bars, and the 30 kV/cm data require an ad hoc reduction of 2Delta_pi by 0.81 to keep the model alive.

To be fair, the paper signals these limitations itself ('it is reasonable to assume,' 'likely indicates'). The raw observation stands; the interband-coupling conclusion is a plausible hypothesis, not an established result. This is exactly what peer review is for. I'd send it to a referee who can push for a quantitative estimate of the diamagnetic contribution and a two-band linewidth calculation, or at minimum error bars on delta.

Referee Report

3 major / 6 minor

Summary. The paper reports THz two-dimensional coherent spectroscopy (2DCS) measurements on a MgB2 thin film (Tc = 38 K). Using broadband THz pulses, the authors observe a nonlinear signal peaked near 2Δπ ≈ 1 THz at low temperature. Using narrowband pulses at Ω/2π = 0.63 THz, they observe first-harmonic (FH) and third-harmonic (TH) responses that scale as E^6, consistent with third-order nonlinearity. After normalizing the FH intensity by the sixth power of the transmitted drive field, they find a monotonic increase with decreasing temperature, in contrast to the resonant enhancement previously observed in NbN at Ω = 2Δ. The authors interpret this as evidence for a strongly overdamped π-band amplitude mode (δ = 0.55 THz, about 4.5 times larger than in NbN) and attribute the large damping to interband coupling.

Significance. If the interpretation is correct, the paper would provide the first THz 2DCS study of a multigap superconductor and would demonstrate that interband coupling qualitatively changes the amplitude-mode response, a prediction of recent two-band theories (Ref. [32]). The raw data are of good quality: the nonlinear signal is clearly resolved, the E^6 scaling is verified, and the THz pump-probe controls in the SM show that the narrowband drive does not deplete the superconducting condensate. The paper also carefully addresses the normalization issue that affects the temperature dependence of the nonlinear response. However, the central interpretation rests on an unverified single-band model assumption and does not exclude a non-resonant diamagnetic contribution, which would also produce a monotonic FH increase with decreasing temperature. The quantitative claim about the damping rate therefore requires additional analysis or a more cautious framing.

major comments (3)

[§4 (normalized FH signal) and Eq. (1)] The assignment of the monotonic normalized FH signal to an overdamped π-band amplitude mode is not uniquely supported because the drive frequency Ω/2π = 0.63 THz lies below 2Δπ ≈ 1 THz for all measured temperatures; in this off-resonant regime, a non-resonant diamagnetic nonlinearity that grows with superfluid density would also produce a monotonic increase with decreasing temperature. The argument from Ref. [20] that the amplitude-mode paramagnetic coupling dominates for kF l ≈ 13.5 was established for resonant conditions in NbN and does not automatically carry over to the off-resonant case here. A quantitative estimate separating the paramagnetic (amplitude-mode) and diamagnetic/quasiparticle contributions to the FH response, or a measurement at a drive frequency matched to 2Δπ, is needed to support the attribution.
[§4, Eq. (1) and discussion of Ref. [32]] The single-band damped-oscillator model of Eq. (1) is applied to the π band of a two-gap superconductor without a derivation or two-band calculation, and the manuscript itself notes that for realistic MgB2 parameters the π-band amplitude mode is strongly suppressed by interband coupling. This internal tension undermines the premise that the π mode dominates the FH response; the authors should either provide a two-band calculation justifying the dominance or explicitly present the fit as a phenomenological characterization rather than as evidence for the amplitude-mode damping rate.
[§4, Eq. (1) and fitting procedure] The reported damping rate δ = 0.55 THz is obtained from a fit with free parameters I0 and δ, and an additional ad hoc factor of 0.81 reduction of 2Δπ at 30 kV/cm, with no reported uncertainties, goodness-of-fit, or justification for fixing δ across field strengths while reducing the gap. Given that this fit is the core quantitative evidence for the claim that the damping is about 4.5 times larger than in NbN, the paper must report error bars and fit residuals, and should test whether the data actually constrain δ or whether a similar-quality fit is possible with very different damping values.

minor comments (6)

[Abstract and §1] The phrase 'distinct from the single-gap superconductor NbN' would be more accurate as 'distinct from observations in the single-gap superconductor NbN', since the comparison is to one specific film and measurement.
[§2, paragraph 2] The word 'intrisic' in 'To obtain the intrisic response' appears to be a typo for 'intrinsic'.
[Fig. 1 caption] In the caption of Fig. 1(a), 'pules' should be 'pulses'.
[§2, Fig. S3 reference] The sentence 'For fields larger than 21 kV/cm (Fig. S3.)' should be rephrased to specify that the long-lived component is observed in the THz pump-probe data shown in Fig. S3, and the reference should include the relevant subfigure.
[§3 and §4 (normalization)] The physical meaning of the normalized quantity INL/EA^6 should be stated explicitly: it is proportional to the magnitude squared of the effective third-order susceptibility, assuming a purely electronic third-order process. This would help the reader understand why the sixth power is used.
[§2, Fig. 2 caption and text] The orange dashed curve in Fig. 2(b) is described as 2Δπ from a two-band BCS calculation, but the parameters of that calculation are only given in the SM; a brief mention in the main text would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central data are independent of the model, and the fitted damping rate is not an input to the measurement.

full rationale

The paper's central claim is that the normalized first-harmonic (FH) response of MgB2 increases monotonically with decreasing temperature, unlike NbN, and that this can be fit by an overdamped amplitude-mode model with a damping rate δ=0.55 THz. The FH and TH intensities are raw experimental data; the driving fields, transmission corrections, and the two-band BCS gap curves are determined independently from equilibrium THz conductivity, STM values, and solution of the two-band gap equations. Equation (1) is a damped-oscillator model taken from Ref. [20], but δ is a free parameter fitted to the MgB2 data, not a predetermined input. The comparison with NbN and the interpretation of a larger damping due to interband coupling is an inference, not a derivation that reduces to its own assumptions. While Ref. [20] is from the same group and supplies the model and normalization procedure, that prior result is an externally established, independently falsifiable study on a different material (NbN), so citing it does not make the MgB2 conclusion circular. The paper itself acknowledges that incorporating interband coupling in Eq. (1) is challenging and that the dominance of the π-band amplitude mode is assumed rather than derived from a two-band calculation. That is a correctness or scope concern, not a circularity: no equation in the paper is equivalent by construction to an input, and no fitted quantity is renamed as a prediction. Therefore the circularity score is 0.

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

The central claim rests on a single-band amplitude-mode model applied to the π band, on a fitted damping rate, and on a normalization procedure inherited from the same group's NbN work. No new physical entities are postulated, but the interpretation depends on several domain assumptions taken from prior literature.

free parameters (4)

δ (amplitude-mode damping rate) = 0.55 THz
Fitted to simultaneously reproduce the FH and TH temperature dependence at 12 kV/cm; reused for 21 and 30 kV/cm. The central inference of overdamping depends on this fitted value.
I0 (amplitude prefactor in Eq. 1) = not reported
Overall scaling constant in the resonance model, adjusted to match the absolute measured FH and TH intensities; one per drive field.
2Δπ scaling factor at 30 kV/cm = 0.81
Multiplies the BCS 2Δπ(T) curve at 30 kV/cm to fit the data, accounting for partial suppression of superconductivity by the stronger drive.
2Δπ(0) (π-band gap at T=0) = 1.1 THz (2Δπ(0)/2π)
Input from the Mattis-Bardeen fit to the measured equilibrium optical conductivity, used to set the resonance condition and the BCS curve; taken similar to Ref. [5], but effectively fit to this film's data.

assumptions (4)

domain assumption Two-band BCS gap equations (SM Eq. S1) with λ11=0.28, λ22=0.96, λ12=λ21=0.19, N2/N1=0.73, ωc=1.78 THz describe the temperature-dependent gaps of MgB2.
These parameters come from literature, not measured here; they set the 2Δπ(T) and 2Δσ(T) curves used for all comparisons and fits.
ad hoc to paper Equation (1), a single-band damped oscillator model from Ref. [20], applies to the π band of a multi-gap superconductor with a single effective damping rate.
The paper applies a model calibrated for single-gap NbN to the π band of MgB2, assuming interband coupling modifies only the damping and not the form of the response.
domain assumption The THz nonlinearity in this parameter range is dominated by paramagnetic coupling to the amplitude mode rather than diamagnetic or quasiparticle terms.
Borrowed from Ref. [20] for NbN and dirty-limit theories in Refs. [31,32]; not directly verified for this MgB2 film.
domain assumption Normalizing the nonlinear intensity by the sixth power of the transmitted drive field removes the screening effect and exposes the intrinsic nonlinear susceptibility.
The procedure follows Ref. [20] and is cross-checked with A and B pulses in SM, but it assumes the transmitted field is the correct internal drive and that no other temperature-dependent factor enters.

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Cite this review

Pith. "Pith review of Amplitude mode in a multi-gap superconductor MgB$_2$ investigated by terahertz two-dimensional coherent spectroscopy." pith.science (2026). https://pith.science/paper/P2EW654W

@misc{pith2026241110852,
  author       = {Pith},
  title        = {Pith review of: Amplitude mode in a multi-gap superconductor MgB$_2$ investigated by terahertz two-dimensional coherent spectroscopy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P2EW654W}},
  note         = {Machine review of arXiv:2411.10852}
}

abstract

We have investigated the terahertz (THz) nonlinear response of the multigap superconductor MgB$_2$, using THz two-dimensional coherent spectroscopy (THz 2DCS). With broadband THz drive fields, we identified a nonlinear response at twice the lower superconducting gap energy $2\Delta_{\pi}$ at the lowest temperatures. Using narrow-band THz driving pulses, we observed first (FH) and third harmonic responses. The FH intensity shows a monotonic increase with decreasing temperature when properly normalized by the driving field strength. This is distinct from the single-gap superconductor NbN, where the FH signal exhibited a resonant enhancement at temperatures when twice the gap energy $2\Delta$ was resonant with the driving photon energy, which was interpreted to originate from the superconducting amplitude mode. Our results in MgB$_2$ are consistent with a well-defined amplitude mode only at the lowest temperatures and indicate strong damping as temperature increases. This likely indicates the importance of interband coupling in MgB$_2$ and its influence on the nature of the amplitude mode and its damping.

Figures

Figures reproduced from arXiv: 2411.10852 by the authors.

**Figure 4.** FIG. 4. Temperature dependence of the frequency-integrated [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

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Forward citations

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