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Multi-cavity strong coupling to an electron spin ensemble: spectral and dark-state signatures

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper reports that a spin ensemble on one resonator can strongly couple to two hybridized cavity modes, and that a shared transmission line can make the central mode radiatively dark.

desk verdict Solid experiment and clean spectra, but the dark-state claim hinges on a phase in Eq. (18) that their own input–output derivation contradicts. read the letter →

arxiv 2608.05765 v1 pith:EFF6EN2A submitted 2026-08-06 quant-ph cond-mat.mtrl-sci

P. Oehrl , B. Pérez González , A. Dunaev , M. Althammer , T. S. Parvini , F. Piazza , M. Benito , H. Huebl This is my paper · ORCID
classification quant-phcond-mat.mtrl-sci PACS 42.50.Pq76.30.-v03.67.-a
keywords strongcouplingdarkstatespinensemblesuperconductingmicrowaveresonatorsinput-outputformalismwaveguidequantumelectrodynamicselectronresonancetripartitehybridization
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 aims to show that a single spin ensemble can enter strong coupling with two photonic modes at the same time even though it touches only one resonator, because a direct coupling between two near-degenerate resonators hybridizes them and distributes the spin interaction over both branches. It also aims to show that the same device can host a stationary dark state — a hybridized mode whose radiation into a given channel is cancelled by destructive interference — because the shared transmission line provides a common radiation environment. Using an input-output model with complex waveguide couplings and propagation phases, the authors reproduce the measured cwESR spectra and extract the couplings, linewidths, and cooperativities of both hybrid branches. The fitted parameters make the central eigenstate satisfy the condition for vanishing radiation into the transmitted channel, which is why its absorption feature disappears near 210 mT. If these claims hold, spin ensembles in multi-cavity circuits could be loaded and read out through a bright configuration and then switched by magnetic field into a radiatively protected configuration, without extra control drives.

What carries the argument

The load-bearing object is the common transmission line treated as a shared continuum, with each resonator coupled to it through a complex rate $\kappa_j=|\kappa_j|e^{-i\phi_j}$ and a propagation phase $\theta_j$ between coupling points. The paper defines radiative jump operators $\hat{L}_+=\sqrt{\kappa_3}\,\hat{a}_3+e^{i\Theta}\sqrt{\kappa_2}\,\hat{a}_2$ and $\hat{L}_-=\sqrt{\kappa_3}\,\hat{a}_3+e^{-i\Theta}\sqrt{\kappa_2}\,\hat{a}_2$ for the right- and left-moving field channels, and an eigenstate is dark when its radiation vanishes, $\hat{L}_\pm|\psi\rangle=0$. These operators yield the hybridized-mode linewidths, the dark-state amplitude condition $\tan(\varphi/2)=\sqrt{|\kappa_2|/|\kappa_3|}$, and the phase condition $\pm\Theta+(\phi_3-\phi_2)/2=\pm(2n-1)\pi$. The mixing angle $\tan\varphi=2\zeta/(\Omega_2-\Omega_3)$ redistributes the spin coupling between branches, so the inter-resonator coupling $\zeta$ and detuning $\Omega_-$ act as control knobs for strong coupling, cooperativity, and darkness.

What would settle it

Measure both transmission and reflection around $B_0\approx 211$ mT and $5.87$ GHz: the model predicts a transmission dip with a non-vanishing reflected signal at the same field, because the fitted phases make the central state dark only to the transmitted channel. If the reflection also shows a simultaneous dip, or if the field position of the transmission dip does not shift when the propagation phase $\Theta$ (for example, the resonator spacing) is changed as Eqs. (14)–(15) predict, the radiative-interference dark-state explanation would be falsified.

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Extended reading notes

Core claim

The paper's central claim is that the observed avoided crossings are not a spin coupling to one bare resonator but a tripartite hybridization between the spin ensemble and the two normal modes $|\tilde{\psi}_+\rangle$ and $|\tilde{\psi}_-\rangle$ of the coupled resonator pair. In that basis the spin–photon couplings are $g\sin(\varphi/2)$ and $g\cos(\varphi/2)$ with $\tan\varphi=2\zeta/(\Omega_2-\Omega_3)$, and for the fitted parameters $g/2\pi=20$ MHz, $\gamma/2\pi=12$ MHz, $\zeta/2\pi=6$ MHz the effective linewidths of the two branches are $\tilde{\kappa}_+/2\pi\approx 2.54$ MHz and $\tilde{\kappa}_-/2\pi\approx 1.98$ MHz, so both satisfy $g_\pm>(\tilde{\kappa}_\pm+\gamma)/2$; the extracted cooperativities are $C_+\approx 41.95$ and $C_-\approx 13.47$. Using the radiative jump operators $\hat{L}_\pm$ for the two propagation directions, the paper derives the dark-state condition $\hat{L}_\pm|\psi\rangle=0$ and shows that at $B_0\approx 211$ mT the central eigenstate satisfies this condition for the transmitted channel, matching the measured suppression of the central resonance. Because the fitted complex phases $\phi_j$ break the symmetry, the state is dark to transmission but retains a reflective signal, making it a subradiant rather than fully dark mode.

Load-bearing premise

The load-bearing premise is that the fitted complex coupling phases $\phi_2$ and $\phi_3$, the propagation phase $\theta_3$, and the point-like coupling model correctly describe how the fields radiated by resonators 2 and 3 interfere in the shared transmission line; if those phases are not physical, the observed central dip could arise from a different mechanism.

Editorial extensions

If this is right

  • Both hybridized branches satisfy strong coupling, so one spin ensemble can act as a shared quantum interface for two resonator modes; the ratio $g_+/g_-$ is set by $\zeta$ and $\Omega_2-\Omega_3$, giving a design knob for distributing the spin's oscillator strength.
  • A hybridized eigenmode can be switched between bright and dark by the magnetic field: bright for loading and readout through the waveguide, dark for storage with suppressed radiative decay, with no additional control drive required.
  • Dark states survive away from the symmetric point: unequal linewidths, complex coupling phases, or fabrication detunings can be compensated by tuning $\zeta$ or the resonator detuning, so the protection mechanism is not limited to ideally matched resonators.
  • For zero inter-resonator coupling, darkness requires exact degeneracy and equal waveguide couplings; finite $\zeta$ relaxes this to an amplitude condition plus a phase condition, providing a concrete design rule for future devices.
  • The same shared-continuum mechanism suggests an engineering route to dissipation control in multi-cavity circuits, such as selectively suppressing radiative loss of a chosen hybridized excitation.

Reading between the lines

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

  • The dark-state condition is effectively phase-sensitive: because the dip position ties together $\Theta$, $\phi_2$, and $\phi_3$, sweeping the resonator spacing or frequency and tracking the dip would test the fitted phases and could calibrate them without a separate background measurement.
  • The interference mechanism is independent of the spin's microscopic nature, so replacing the spin ensemble with a transmon, magnon mode, or mechanical resonator should still yield a transmission-dark eigenstate, as long as the linear multi-mode coupling to the shared continuum is preserved.
  • A two-port measurement at the dark point could distinguish interference darkness from absorption-based explanations: the model predicts residual reflected power at the same field and frequency, while an alternative damping or asymmetric-line-shape mechanism would produce a different relation between the dips in transmission and reflection.
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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 reports cwESR transmission measurements on a circuit with three superconducting resonators coupled to a common transmission line and a DPPH spin ensemble coupled to only one resonator. Using input-output theory, the authors reproduce the measured spectra and extract coupling parameters. They identify two avoided crossings and claim that both hybridized photonic branches are in the strong-coupling regime, i.e., a tripartite spin-resonator-resonator strong coupling. They further interpret a pronounced suppression of the central transmission feature at B0 ≈ 211 mT as a stationary dark state of the tripartite system, arising from destructive interference between the fields radiated by resonators 2 and 3 into the common feedline. The central claim is that for the fitted parameters the corresponding hybridized eigenstate satisfies f_+(v)=0, making it dark to transmission.

Significance. If established, this would be a valuable experimental demonstration of multi-mode spin-resonator strong coupling and of a radiatively dark state in a circuit-QED architecture, with potential applications to protected spin-ensemble storage. The input-output model is standard and the paper is transparent about the derivation; the two-resonator dark-state analysis in Appendix D is instructive, and the comparison between exact pole positions and approximate linewidths in Fig. 7 is a useful methodological check. The experimental data show clear avoided crossings and a striking central dip. However, the dark-state interpretation currently rests on an internal phase inconsistency and on parameters that were fitted to the very spectrum containing the dip, so the central claim is not yet established.

major comments (3)
  1. [§IV, Eq. (18); cf. Appendix C, Eq. (C10)] The phase multiplying √κ2 in the right-moving source operator is θ3, not Θ=θ2+θ3. From Eq. (C10), d_out,+,3 = √κ3 a3 + e^{iθ3}√κ2 a2 + e^{i(θ2+θ3)}√κ1 a1 + e^{i(θ2+θ3)} d_in,+,1. With a1=0, the radiative source operator for the right-moving channel is L_+ = √κ3 a3 + e^{iθ3}√κ2 a2, and the total phase Θ multiplies only the input field, which vanishes for a zero-input stationary eigenstate. Eq. (18) therefore over-rotates the √κ2 term by e^{iθ2}; the same error appears in Eqs. (13), (16), and (E6). The two-resonator derivation in Appendix D.2 consistently uses θ3 (Eqs. (D19) and (D22)), so the substitution of Θ in the tripartite condition is unjustified. The numerical dark-state check in Fig. 5 and the statement that Θ≈2.2 rad is in excellent agreement with the fitted phases are therefore based on a mis-specified phase. The authors should replace Θ by θ3 in the dark-state condition and re-evaluate. The issue is compounded by Appendix G, which reports that the fitted θ3=1.8 rad deviates from the geometric estimate of about 0.45 rad by a factor of four, so the phase used in the model is not independently verified.
  2. [Table I; §III; Fig. 5] No uncertainties are given for any fitted parameter, and the dark-state condition is evaluated with parameters obtained from the same two-dimensional fit that reproduces the central dip. This makes the 'prediction' that f_+(v)=0 at B0≈211 mT a restatement of the fit rather than an independent test. To support the dark-state interpretation, the authors should report parameter uncertainties or confidence intervals and test the dark-state condition using parameters constrained by spectral regions away from the dip (for example, B0<209 mT and B0>213 mT), so that the vanishing of f_+ is a genuine prediction. The same uncertainty issue affects the quantitative strong-coupling claim: with fitted values g_-/2π≈8.9 MHz and (κ~_-+γ)/2≈6.99 MHz, the inequality is not robust without error bars.
  3. [§IV, Eq. (10) and following text] The labels g_+ and g_- appear to be interchanged. In Eq. (10), the |ψ_+> row has coupling g sin(φ/2) and the |ψ_-> row has g cos(φ/2), but the text reports g_+/2π≈17.9 MHz and g_-/2π≈8.9 MHz, which for the fitted φ correspond to g cos(φ/2) and g sin(φ/2), respectively. Please use consistent definitions, since the strong-coupling conditions and cooperativities C_± depend on which coupling belongs to which branch.
minor comments (4)
  1. [Fig. 5] The 'dashed medium-blue curve' referred to in the text is not clearly identifiable in the printed panels; please add a legend or use a distinct linestyle so the total-decay curve can be distinguished from the transmission and reflection curves.
  2. [After Eq. (18)] The parameter vector is written with 'θ' rather than 'Θ'; define this symbol and use it consistently, especially since the distinction between θ3 and Θ is central to the dark-state condition.
  3. [Appendix B, Eq. (B2)] The linear background correction would be clearer if m and n were explicitly identified as real fitting constants and their fitted values were reported.
  4. [Appendix E] The dark-state condition in Eq. (E6) uses e^{±iθ} without defining θ; this should be θ3 and should be stated explicitly to avoid further confusion with the total propagation phase Θ.

Circularity Check

2 steps flagged · score 6.0 of 10

The central dark-state claim reduces to the same fitted transmission dip it is supposed to explain; the phase used in Eq. (18) also disagrees with the paper's own input-output source operator.

  1. fitted input called prediction [Section III (fit over B0=205–215 mT) and Section IV / Fig. 5 (dark-state evaluation), after Eq. (18)]
    "we extract the collective spin–cavity coupling and the spin relaxation rate from a two-dimensional fit to the hybridized region (B0 = 205 mT to 215 mT). ... Our numerical results, shown in Fig. 5, confirm the presence of a transmission-dark state in the central mode for the fitted parameters of Table I and Θ≈2.2 rad."

    The global input-output fit was performed on the field window B0 = 205–215 mT, which contains the central dip at ≈211 mT that is subsequently presented as a dark-state signature. The parameters entering f_+(v) — θ3, φ2, φ3, κ2, κ3, and the eigenstate amplitudes α, β — are outputs of that same fit. Evaluating f_+(v)=0 at B0≈211 mT therefore restates, in operator form, the transmission dip the fit already reproduced; it is not an out-of-sample prediction. Appendix G also admits that the fitted θ3=1.8 rad deviates from the geometric estimate ≈0.45 rad, so the phase controlling the predicted cancellation is a free fit parameter, making the claimed 'excellent agreement' of Θ self-referential.

  2. other [Main-text Eq. (18) vs. Appendix C Eq. (C10) and Appendix D.2 Eq. (D18)]
    "√κ3 [α sin(φ/2)+β cos(φ/2)] + e^{±iΘ}√κ2 [α cos(φ/2) − β sin(φ/2)] = 0, (Eq. 18) ... d̂in_{+,j} = e^{iθ_j} d̂out_{+,j−1} = e^{iθ_j} √κ_{j−1} â_{j−1} + e^{iθ_j} d̂in_{+,j−1}. (Eq. C10)"

    The paper's own input-output recursion places the inter-resonator phase θ3 on the √κ2 term in the right-moving radiative source operator (L_+ = √κ3 a3 + e^{iθ3}√κ2 a2, Eq. D18; cf. Eq. C10). Eq. (18) instead multiplies √κ2 by e^{±iΘ}, with Θ = θ2 + θ3, inserting an extra e^{±iθ2} rotation. The dark-state zero at Θ≈2.2 rad is therefore not a consequence of the derived model but follows from the phase chosen in the dark-state condition. The agreement between the fitted Θ and the fitted θ2+θ3 is built into the evaluation, so the claimed transmission-dark state is an artifact of the Θ definition rather than a predicted signature.

full rationale

The paper contains two independent claims: multi-mode strong coupling and dark-state formation. The strong-coupling analysis is largely self-contained: it combines the coherent two-resonator hybridization (Eqs. 7–10) with non-Hermitian linewidth expressions (Eq. 11) and fitted g, γ, κ values to verify g± > (κ̃± + γ)/2; this does not reduce to the fitted spectra in a circular way. The dark-state claim, however, is substantially circular and, on the paper's own equations, mis-specified. The dark-state condition f_+(v)=0 is evaluated with Table I parameters obtained from a fit whose field window (B0 = 205–215 mT) includes the very dip at ≈211 mT that the condition is used to explain; the 'confirmation' is thus a post-hoc restatement of the fit. Additionally, Eq. (18) uses a total phase Θ = θ2 + θ3 on the √κ2 amplitude, whereas the input-output recursion in Appendix C and the two-resonator expression in Eq. (D18) put the inter-resonator phase θ3 on that term. The predicted destructive interference is therefore tied to a phase convention that the paper's own derivation does not support, and Appendix G admits the decisive θ3 is not independently verified. I do not see load-bearing self-citation or imported-uniqueness issues here; the circularity is not about references but about the fitted parameter set being renamed as a prediction. Hence score 6: the strong-coupling part stands, while the central dark-state signature partially reduces to the fit and to a phase artifact.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

The central claims rest on a set of fitted coupling parameters (g, gamma, zeta, Omega2, Omega3, kappa2, kappa3, theta2, theta3, phi2, phi3) and on the domain assumption that a single collective Tavis-Cummings mode with negligible inhomogeneous broadening describes the spin ensemble. No new entities are introduced; the dark-state phenomenon is entirely due to interference of known photonic and spin fields.

free parameters (8)
  • collective spin-photon coupling g = 20.0 MHz
    Fitted to the avoided crossings in the cwESR spectrum; sets the spin-photon coupling strength used for the strong-coupling and dark-state analysis.
  • spin dephasing rate gamma = 12.0 MHz
    Fitted spin linewidth; used in the strong-coupling criterion g > (kappa+gamma)/2 and in the cooperativity.
  • inter-resonator coupling zeta = 6.0 MHz
    Fitted from the lowest-field spectrum and the full 2D fit; controls the hybridized mode composition via the mixing angle phi.
  • resonator frequencies Omega2, Omega3 = 5.864 GHz, 5.873 GHz
    Fitted cavity frequencies; their detuning enters the mixing angle phi and the hybridized mode energies.
  • external coupling rates kappa2, kappa3 = 0.211 MHz, 0.297 MHz
    Fitted resonator-to-transmission-line coupling amplitudes; enter the dark-state condition Eq. (18).
  • coupling phases phi2, phi3 = -0.92 rad, 0.95 rad
    Fitted phases of the complex external couplings; they set the interference condition for the dark state and are not independently verified.
  • propagation phases theta2, theta3 = 0.4 rad, 1.8 rad
    Fitted phases between resonator coupling points; theta3 deviates strongly from the geometric estimate (0.45 rad) and controls the dark-state condition.
  • magnetic field offset DeltaB0 = 7.223 mT
    Fitted offset used to align the spin resonance to the DPPH g-factor; uncertainty in this offset affects all field positions, including the claimed dark state.
assumptions (5)
  • domain assumption Single collective Tavis-Cummings mode with negligible inhomogeneous broadening describes the DPPH spin ensemble in the single-excitation limit.
    Invoked in Section II; if the ensemble linewidth were dominated by inhomogeneous broadening, the effective coupling and eigenstate composition would change, affecting both the strong-coupling and dark-state claims.
  • domain assumption The resonators couple point-like to the transmission line with Markovian input-output relations.
    Used throughout Appendix C; Appendix G acknowledges finite resonator width makes this approximate, and the fitted theta3 differs from the geometric estimate.
  • domain assumption Only the inter-resonator coupling zeta23 between resonators 2 and 3 is finite; couplings to resonator 1 are negligible.
    Stated in Section II based on detuning; if zeta12 or zeta13 were non-negligible, the dark-state condition would change.
  • standard math A stationary eigenstate is dark when the jump operator of the relevant channel annihilates it, L_p |Psi> = 0.
    Standard criterion in the input-output formalism (Refs. [61-65]); used to derive Eqs. (13) and (18).
  • domain assumption The spin ensemble couples only to resonator 3.
    Design statement in Appendix A; if DPPH couples to other resonators, the model would need extra terms.

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Cite this review

Pith. "Pith review of Multi-cavity strong coupling to an electron spin ensemble: spectral and dark-state signatures." pith.science (2026). https://pith.science/paper/EFF6EN2A

@misc{pith2026260805765,
  author       = {Pith},
  title        = {Pith review of: Multi-cavity strong coupling to an electron spin ensemble: spectral and dark-state signatures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EFF6EN2A}},
  note         = {Machine review of arXiv:2608.05765}
}
read the original abstract

Spin ensembles are considered as potential candidates for quantum memory and quantum enhanced sensing applications. Here, we explore the controlled coupling of multiple superconducting microwave cavities to a spin ensemble, which shows signatures of strong coupling and, due to the multi-mode character, the formation of dark states. In particular, the latter are of interest, as they provide a potential pathway to enhance memory times and enable protected storage of non-classical states in spin ensembles due to the suppressed coupling to the circuit environment. We model the spin multi-cavity hybrid to reproduce the spectra and extract characteristic coupling strengths using the input-output formalism.

Figures

Figures reproduced from arXiv: 2608.05765 by the authors.

Figure 1
Figure 1. Conceptual depiction of the setup. (a) The superconducting microwave circuit consists of three hanger￾type, lumped-element resonators of frequencies Ω1, Ω2, Ω3, which are coupled to a common transmission line with cou￾pling strengths |κi|e −iϕi . One of the resonators is strongly coupled, with coupling strength g, to the spin ensemble with resonance frequency ∆. We investigate the complex transmis￾sion parameter S21… view at source ↗
Figure 2
Figure 2. Measured and modeled cwESR spectra. (a) Measured (upper panel) and simulated (lower panel) microwave absorption amplitude |S21| as a function of the applied magnetic field B0 and the probe frequency ωprobe. The modeled microwave absorption amplitude is based on Eq. 1 and the coupling rates summarized in Tab.I. Vertical lines (1)−(5) represent B0−fixed cuts represented in panel (b). (b) Comparison of the measured (so… view at source ↗
Figure 3
Figure 3. Simulated energy spectrum for the tripartite system. Calculated energies of the effective three-component system formed by the spin ensemble and resonators 2 and 3, as a function of B0, for the fitted parameters in Table I, com￾paring (a) ζ/2π = 0, (b) ζ/2π = 6 MHz. The dotted lines indicate the bare energies of the modes that are hybridizing: (a) ∆, Ω2, and Ω3, (b) ∆, Ω˜ +, and Ω˜ −. expressed entirely in terms of … view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Dark and directionally dark modes in the two-coupled-resonator system. Dark state condition f ± 2r/max[f ± 2r] for |ψ˜+⟩, as a function of the propagation phase Θ, for complex κi = |κi|e −iϕi , and for both propagation chan￾nels Lˆ±. The diagnostic function F2r/max[F2r…
Figure 5
Figure 5. Figure 5: Dark state in the tripartite system. (a) Dark state condition for the central eigenstate of the Hamil￾tonian in Eq. (10), evaluated through f±(⃗v)/ max(f±(⃗v)), using the parameters ⃗v = (κ2, κ3, α, β, φ, θ) obtained from the fitting in Table I. The observed dark state…
Figure 6
Figure 6. Figure 6: Background correction. Dashed lines: Frequency-dependent microwave transmission measured at 1 T. Comparison of the raw measured microwave transmis￾sion (left panels) and corrected microwave transmission (right panels) for magnetic field strengths of (a) B0 = 198 mT, (b…
Figure 7
Figure 7. Figure 7: Comparison of mode frequency and linewidth obtained from different methods. (a) Energy mode Ω˜′ ± = Re(˜ω±) from Eq. (D8), versus Ω˜ ±, obtained in Eq. (9) from diagonalizing the Hermitian Hamiltonian ma￾trix. (b) Effective linewidth of the hybridized modes, ob￾tained …
Figure 8
Figure 8. Figure 8: Transmission coefficient |S21| as a function of different parameter choices, given fixed κ2 = κ3 ∈ R (ϕ2 = ϕ3 = 0). All parameters are expressed in the same arbitrary units of energy. (a) |S21| as a function of the input frequency ω, for a fixed resonator detuning Ω− =…
Figure 9
Figure 9. Figure 9: Dark state condition for the lowest￾and highest-energy mode. Dark state condition for the (a) lowest-energy eigenstate, (b) highest-energy eigen￾state, of the Hamiltonian in Eq. (10), evaluated through f±(⃗v)/ max(f±(⃗v)), using the parameters ⃗v = (κ2, κ3, α, φ, θ) ob…

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    Decay rates for hybridized photonic modes We first write the EOM for the interacting resonators, following the same procedure as for Eqs. (C7a)-(C7c). We get h −i(ω−Ω 2) + κcav,2 2 i ˆa2 =− √κ2 ( ˆdin +,2 + ˆdin −,2)−iζˆa3, (D1) and h −i(ω−Ω 3) + κcav,3 2 i ˆa3 =− √κ3 ( ˆdin +,3 + ˆdin −,3)−iζˆa2. (D2) The input-output relations for this system, ˆdout +,j...

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