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Measuring intrinsic relaxation rates in superconductors using nonlinear response

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Pith's one-line read The paper shows that intrinsic quasiparticle relaxation times T1 and T2 can be extracted from the nonlinear terahertz response of superconductors, once the intrinsic power-law decay of the Higgs and current oscillations is separated out.

desk verdict A genuinely useful extension of Ref. [17] that gives experimentalists a practical THG/polarization-resolved route to T1 and T2 in s- and d-wave superconductors, but the central extraction map rests on a phenomenological damping ansatz that fixes 1/T2 = 1/(2T1) and sets pure dephasing to zero. read the letter →

arxiv 2510.07398 v2 pith:QALRKI6S submitted 2025-10-08 cond-mat.supr-con cond-mat.str-el

Wei-En Tseng , Rahul Nandkishore This is my paper · ORCID
classification cond-mat.supr-concond-mat.str-el
keywords AndersonpseudospinHiggsmodeT1/T2relaxationthirdharmonicgenerationterahertznonlinearresponsed-wavesuperconductorgapdynamicspolarizationcontrol
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 claims that the intrinsic quasiparticle relaxation times T1 (redistribution) and T2 (dephasing) of a superconductor can be measured directly from the nonlinear terahertz response. In the Anderson pseudospin formalism, the gap and the third-harmonic current each contain a universal power-law decay from inhomogeneous pseudospin dephasing (1/√t for s-wave, 1/t for d-wave) plus an exponential decay governed by the microscopic relaxation rates. The paper derives how to separate these contributions and extract T1 and T2 at specific Fermi-surface points, and shows that polarization control can isolate different irreducible representations in d-wave superconductors.

What carries the argument

The central object is the Anderson pseudospin, a two-level representation of each momentum state whose precession about a pseudomagnetic field encodes the superconducting dynamics. The paper adds a phenomenological spin-length-preserving damping (the effective field b_k^eff = b_k - γ_k bhat_k × s_k, with γ_k = 1/T2_k = 1/(2T1_k)) and then uses stationary-phase and Laplace methods to separate the intrinsic power-law decay from the exponential relaxation. Polarization enters through the momentum-dependent effective-mass tensor, which couples different irreducible representations (A1g, B1g, B2g) and enables mode-selective excitation and readout.

What would settle it

In an s-wave superconductor, measure the gap recovery and Higgs oscillation decay after an intense THz pump. If √t times the oscillation amplitude does not decay exponentially with a rate exactly half that of √t times the recovery, the spin-length-preserving relaxation model is ruled out. In a d-wave sample, the framework predicts J_xxxx ∝ e^{-t/T2_antinode}/t with T2 independent of pump intensity; a different time dependence would disprove the claim.

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Extended reading notes

Core claim

For s-wave superconductors with finite pseudospin damping at the Fermi surface, the Higgs amplitude oscillation decays as cos(2Δ∞ t + π/4) e^{-γ(0)t}/√t and the gap recovery as e^{-2γ(0)t}/√t, so 1/T2(ε=0)=γ(0) and 1/T1(ε=0)=2γ(0) can be read from the two channels. If the damping vanishes as |ε|^p near the Fermi surface, the recovery instead decays as t^{-1/p}, revealing the exponent p. For d-wave superconductors, the nonlinear current J_xxxx decays as e^{-t/T2,antinode}/t, giving T2 at the antinodes, while the gap recovery follows t^{-4/p}, giving the energy–momentum dependence p of the relaxation. These results provide a concrete recipe for extracting intrinsic relaxation rates from pump–p

Load-bearing premise

The entire extraction map assumes relaxation preserves each pseudospin's length and obeys γ = 1/T2 = 1/(2T1) with zero pure dephasing; if real damping includes pure dephasing or changes spin length, the measured decays no longer cleanly separate T1 and T2.

Editorial extensions

If this is right

  • In s-wave superconductors, correcting measured Higgs and gap-recovery time traces for the intrinsic 1/√t decay yields exponential decays whose rates directly give T2 and T1 at the Fermi surface.
  • In d-wave superconductors, T2 at the antinodes can be extracted from the nonlinear current J_xxxx ∝ e^{-t/T2}/t, while the gap recovery exponent t^{-4/p} encodes the energy–momentum scaling p of the relaxation.
  • Polarization control of pump and probe pulses permits selective excitation and readout of modes in different irreducible representations, each with its own relaxation rate.
  • If the damping vanishes at the Fermi surface (s-wave) or nodes (d-wave), the recovery becomes a pure power law, and that exponent reveals the energy dependence of the relaxation.

Reading between the lines

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

  • If the framework holds, the same two-channel extraction could be applied to amplitude modes in other ordered states (e.g., charge-density waves) that admit a pseudo-spin description.
  • A deviation between the measured recovery rate and twice the oscillation rate would signal pure dephasing or spin-length non-conservation, mechanisms the current model sets to zero.
  • The predicted d-wave t^{-4/p} recovery could be computed from microscopic electron-phonon or electron-electron scattering models to identify the physical origin of the damping.
  • The polarization-selective readout might be pushed further to map the full momentum dependence of relaxation across the Fermi surface, not just at nodes and antinodes.
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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 / 4 minor

Summary. The paper develops a theoretical protocol, based on Anderson pseudospin dynamics, for extracting intrinsic relaxation times from nonlinear (THz) pump-probe response in s- and d-wave superconductors. For s-wave, with a phenomenological spin-length-preserving damping (Eq. 37), the Higgs-mode oscillation decays as Δ_O(t) ∼ cos(2Δ∞t+π/4)e^{-γ(0)t}/√t while the gap recovery decays as Δ_R(t) ∼ e^{-2γ(0)t}/√t, so the authors propose extracting 1/T2(ε=0)=γ(0) and 1/T1(ε=0)=2γ(0) from the two channels. When the damping vanishes as |ε|^p, they predict Δ_R(t) ∝ t^{-1/p}. For d-wave, they find that the nonlinear current J_xxxx decays as e^{-t/T2,antinode}/t, allowing extraction of the antinodal T2, and that gap recovery follows Δ_R(t) ∝ t^{-4/p}, which is proposed as a way to measure the exponent p of the momentum-dependent damping. Polarization control is shown to selectively excite and read out A1g, B1g, and B2g pseudospin components.

Significance. If the proposed extraction map is correct, it provides a concrete, experimentally accessible route to measuring pseudospin T1 and T2 in superconductors, and it identifies polarization selection as a useful knob for isolating different irreducible representations. The s-wave analytic asymptotics and the identification of the Fermi-surface damping as the controlling rate are clear and internally consistent, and the numerical work is transparent. However, the central claims are conditional on a phenomenological damping ansatz whose microscopic origin is not derived, and two d-wave quantitative predictions are left as unexplained numerical observations. The paper is a useful contribution to the nonlinear-spectroscopy theory literature, but the extraction protocol is not as general as the Summary suggests.

major comments (3)
  1. [Section III.B, Eqs. (37)–(40); Summary] The extraction map is built on the spin-length-preserving damping b_eff = b − γ_k bhat_k × s_k, which gives 1/T2_k = γ_k = 1/(2T1_k) and 1/T2* = 0. Consequently the claimed independent extraction of 1/T1 and 1/T2 from Δ_R and Δ_O is really an extraction of γ_0 and 2γ_0; the two rates are not independent. If a pure-dephasing channel is present, the transverse decay rate becomes 1/T2 = 1/(2T1)+1/T2*, so Δ_O would decay with exponent γ+γ_φ while Δ_R still decays with 2γ, and the simple formulas in Eqs. (43) and (47) would no longer separate T1 and T2. The Summary's statement that the framework 'can be extended' to pure dephasing is not a derivation. Please derive that case or explicitly state that the T1/T2 extraction is conditional on 1/T2* = 0.
  2. [Section IV.A, Eq. (51) and Fig. 3(a)] The stationary-phase calculation predicts that the d-wave gap oscillation decays as 1/t, but the numerical inset of Fig. 3(a) shows a substantially faster power law, b ≈ 2.5. The text's 'perhaps because ... more complicated distribution or dynamics' is not an explanation. This matters because the same antinodal stationary-phase mechanism is stated to control both gap and current decay, yet only J_B1g follows the predicted 1/t behavior. The discrepancy should be resolved analytically or numerically, or the claim restricted to the nonlinear current.
  3. [Section IV.B, Eq. (54) and Fig. 4(c)] The recovery law Δ_R(t) ∝ t^{-4/p} is introduced with 'Numerically, we find' and is then used as the central d-wave extraction formula for p. The naive stationary-phase estimate would be t^{-2/p}; the difference is attributed to the pseudospins vanishing at the nodes, but no derivation is supplied. Since the extraction of p rests on this exponent, the -4/p law needs a supporting derivation or a systematic numerical study across p, band parameters, and pump conditions to establish its robustness. The caption of Fig. 4(c) also states that recovery is 'dominated by the antinodes,' which contradicts the text saying it is dominated by the nodes.
minor comments (4)
  1. [Section III.A, Eq. (36)] The displayed fitting form appears to be missing a division by √(Δ∞t): as written it shows a growing oscillation amplitude rather than the 1/√t decay described in the text.
  2. [Section III.B, Eqs. (42)–(43)] There is a typo '2∆tt' which should likely be '2Δt'; also check the phase expression involving tan^{-1} for consistency with the standard stationary-phase result.
  3. [Fig. 4(c) caption vs. text] The caption says the gap recovery is dominated by the antinodes, while the text (Section IV.B) says it is dominated by the nodal points. Please reconcile.
  4. [References] Reference [1] is incomplete (missing journal/volume/page/arXiv identifier); please complete all references in journal style.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper explicitly assumes the damping law and computes forward to macroscopic decay laws; extraction is an inversion of the stated model, and the numerical checks use known inputs.

full rationale

The derivation chain is self-contained and non-circular. The authors start from the standard Anderson-pseudospin equations of motion (Eqs. 3–5), add an explicitly stated phenomenological spin-length-preserving damping, b_eff = b − γ_k bhat_k × s_k (Eq. 37), and then derive, rather than assume, the late-time decay forms: from dθ/dt = −γ sin θ they obtain θ ∝ e^{−γ t} and hence |δs⊥|∝e^{−t/T2,k}, |δs∥|∝e^{−2γ t}=e^{−t/T1,k} (Eqs. 38–40). Inserting these into the momentum integrals for Δ_O(t) and Δ_R(t) and evaluating by stationary-phase/Laplace methods yields the s-wave results Δ_O∝e^{−γ(0)t}/√t and Δ_R∝e^{−2γ(0)t}/√t. The 'extraction' of 1/T2(ε=0)=γ0 and 1/T1(ε=0)=2γ0 is the inverse of this forward map; the numerical simulations are consistency checks in which γ0 is a known input, not a fit parameter being renamed as a prediction. The d-wave exponent b≈2.5 is explicitly reported as a numerical estimate, and if anything it is an acknowledged discrepancy with the analytic 1/t prediction, not a fitted value masquerading as a derivation. The paper also flags its own central limitation: the spin-length-preserving ansatz fixes 1/T2=1/(2T1) and sets pure dephasing 1/T2*=0, with the extension to pure dephasing stated but not derived. That is model dependence or underdetermination, which belongs to correctness risk, not circularity. No load-bearing step reduces by construction to its own input, and the self-citations in the introduction (spin-liquid spectroscopy examples) are not load-bearing for the central derivation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The analysis leans on standard BCS/pseudospin tools and adds a phenomenological damping ansatz; the extraction formulas are therefore model-dependent rather than first-principles. No new physical entities are introduced.

free parameters (4)
  • gamma_0 (overall damping strength) = 0.1 in numerics
    Overall amplitude of the phenomenological pseudospin damping; chosen by hand, not derived from microscopic theory.
  • p (damping exponent) = 1, 2, 3, 4 (cases)
    Exponent in gamma(eps)=gamma_0(omega/2)^p or gamma_0 Delta_0 |eps|^p; controls predicted decay laws and is varied by hand.
  • J' and J'' (further-neighbor hopping) = J' = 0.15J, J'' = 0.5J'
    Parameters in the d-wave band dispersion chosen to realize polarization-selective irreducible representations.
  • Pump peak amplitude A_0 = not stated in text
    Controls pump strength in the Gaussian vector potential; values are used numerically but not fully specified in the text.
assumptions (5)
  • domain assumption BCS mean-field and Anderson pseudospin mapping: H = 2 sum_k b_k . s_k with self-consistent gap equation Eq. (5)
    Underlies all dynamics; valid only within mean-field BCS and ignores beyond-mean-field fluctuations and quasiparticle non-conserving processes.
  • domain assumption Parity symmetry epsilon_k = epsilon_{-k} and q=0 light-matter coupling truncated at second order in A, Eq. (8)
    Used to eliminate linear coupling and justify quadratic drive; fails for systems with broken parity or finite-momentum light.
  • domain assumption Particle-hole symmetry near the Debye shell and decoupling of Higgs and phase modes
    Invoked in Section II.C to justify ignoring phase fluctuations and using the linearized equations (18).
  • ad hoc to paper Spin-length-preserving damping ansatz b_eff_k = b_k - gamma_k bhat_k x s_k with gamma_k = 1/T2_k = 1/(2T1_k)
    Phenomenological relaxation inserted by hand (Eq. 37, Section III.B); no microscopic derivation. The extraction formulas inherit this assumption.
  • standard math Smoothness of distribution f(eps, phi) and validity of stationary-phase/Laplace asymptotics at long times
    Used for the 1/sqrt(t), 1/t, e^{-gamma t}/sqrt(t) and t^{-1/p} asymptotics; assumes the relevant saddle point (Fermi surface, antinodes, or nodes) dominates.

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Pith. "Pith review of Measuring intrinsic relaxation rates in superconductors using nonlinear response." pith.science (2026). https://pith.science/paper/QALRKI6S

@misc{pith2026251007398,
  author       = {Pith},
  title        = {Pith review of: Measuring intrinsic relaxation rates in superconductors using nonlinear response},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QALRKI6S}},
  note         = {Machine review of arXiv:2510.07398}
}
abstract

We discuss intrinsic relaxation rates in superconductors, and how they may be measured using non-linear optical (terahertz) response. We consider both $s$ and $d$-wave superconductors, both with and without a phenomenological (energy dependent) damping. Intrinsic relaxation rates of interest include the Higgs mode decay rate, the quasiparticle redistribution rate ($1/T_1$) and the quasiparticle dephasing rate ($1/T_2$), where the latter two rates are zero in the pure BCS model, but non-zero in the presence of damping. Using the Anderson pseudospin formalism, we illustrate how these intrinsic relaxation rates are related to measurable quantities such as the time-dependent gap function and the non-linear current (a.k.a. third harmonic generation). Hence, we show how intrinsic relaxation rates may be experimentally extracted and discuss what one may thereby learn about the underlying damping. We also discuss the effects of polarization control (viz. non-linear response to light polarized in different directions), which offers a useful experimental knob, especially for $d$-wave superconductors, enabling selective excitation of modes in different irreducible representations (and readout of their corresponding relaxation rates).

Figures

Figures reproduced from arXiv: 2510.07398 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Gap dynamics in the collisionless limit (1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Gap dynamics with different pseudospin damping forms under [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Gap dynamics of d-wave superconductors under different light polarizations and light intensities [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Gap dynamics for d-wave superconductors with pseudospin damping forms [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Nonlinear current induced by an [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

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