Exploring electron spin dynamics in spin chains using defects as a quantum probe

Domain: biology
Published: 2026-03-17
Editorial outcome: major_revision

Abstract
We investigate the quantum dynamics of the electron spin resonance of topological defects (edge state) in dimerized chains. These objects are discontinuities of the spin chain protected by the properties of the global system leading to a quantum many-body multiplet protected from the environment decoherence. Despite recent achievements in the realization of isolated and finite spin chains,thepotentialimplementationinquantumdevicesneedstheknowledgeoftherelaxationanddecoherence sources. Our study reveals that electron spin lattice relaxation is governed at lowest temperatures by phonon-bottlenecked process and at high temperature by the chain dimerization gap. Weshowthattheinteredge-stateeffectivedipolarfieldisreducedbytheintrachainexchange coupling leading to a longer coherence time than isolated ions at eqSuivalent concentration. Ulti- mately, we demonstrate that the homogeneous broadening is governed by the intra-chain dipolar field, and we establish design principles for optimizing coherence in future materials.

Introduction
Topological quantum states in condensed matter systems represent a fascinating class of collective excitationswhosepropertiesaregovernedbyglobalsymmetries rather than local perturbations. A prime example is the S=1/2 topological defect, or ”edge state”, that emerges in gapped one-dimensional (1D) spin chains [1]. While their existence has been established for decades [2–5], a fundamentalquestionremainsunanswered: whatarethe intrinsic limits of their quantum coherence, and what physical mechanisms govern their interaction with the environment? Understanding these dynamics is essential bothforfundamentalphysics—torevealhowmany-body correlationsinfluencedecoherence—andforassessingthe potential of such states in future quantum applications.

Thispaperprovidesthefirstcomprehensiveexperimental and theoretical study of the relaxation and decoherence dynamics of these topological edge states.

Similar effects are predicted in systems undergoing a Spin-Peierls transition systems which have a nonmagnetic ground state because of a dimerization of the spin and/or lattice [6–8]. Furthermore, because the influence of the spins located at the regular sites of the chains is greatly reduced, a thorough study of the defect-induced magneticpropertiesispossiblewithsuchanon-magnetic singlet ground state. From a theoretical perspective, defects in spin chains have attracted considerable interest in thecontextofquantuminformationscience. SinceBose’s pioneering proposal in 2003 [9], numerous theoretical studies have explored the possibility that defects in spin chains could serve as qubits connected by spin-mediated interactions, potentiallyenablingquantumstatetransfer protocols [10–12]. While these proposals remain largely theoretical,recentexperimentsonartificialatomicchains [13]andquantumdots[14]havebeguntoexplorerelated physics. Acriticalopenquestionforanysuchapplication is: what limits the coherence time of embedded defects in real materials? Addressing this question requires a detailed understanding of the microscopic decoherence mechanisms, which has been lacking until now.

Magnetic properties of defects in cuprate spin chain like Sr2CuO3 [15] and CuGeO3 [16] have been reported but no coherence was observed so far. Recently, a wellknown class of systems, the organic spin chains with spin-Peierls ground state, has shown edge states with relatively long coherence times. Initially observed in (TMTTF)2PF6 [17], the absence of spin echo has prevented the study of decoherence mechanisms. Among the large family of organic spin chains, one of them is of particular interest: ortho-dimethyltetrathiafulvalene or o-DMTTF. The serie of (o-DMTTF)2X has been reported with centro-symmetric counter ions (X = Br,Cl and I) [18–20]. While these materials show some conductivity at room temperature, they become insulators below about 150 K and undergo a spin-Peierls transition around T_SP ≃ 50 K. Their magnetic properties can be described by the spin-Peierls hamiltonian:

H = J(1+δ)ΣS_{2i-1}·S_{2i} + J(1-δ)ΣS_{2i}·S_{2i+1} (1)

where J is the Heisenberg isotropic coupling, δ the dimerization parameter and S_i a spin S = 1/2 at the site i. At high temperature (i.e. T > 50 K), δ = 0 corresponds to the uniform spin chain while at low temperature (i.e. below the spin Peierls transition temperature, T_SP) δ ≠ 0 corresponds to the dimerized spin chain with a non-magnetic ground state. J as well as δ has been reported using continuous wave EPR [20] :

J_{X=Br} = 600 K, J_{X=Cl} = 600 K, J_{X=I} = 670 K ,
δ_{X=Br} =0.08, δ_{X=Cl} =0.083, and δ_{X=I} =0.093. Their highly one-dimensional character and the presence of a large spin gap in the bulk provide an ESR-silent background, allowing for an exceptionally clean probe of the in-gap edge states. Ourstudyisperformedbelow20K,so weconsiderthatthespin-Peierlsphaseisfullystabilized.

The presence of edge state is du to presence of imperfection in the crystal. In organic materials, this is mainly caused by electrochemical growth, which easily induces local breaks or stacking defects that cannot be corrected by annealing as in inorganic materials. Many numerical studies [4–8] have shown how these edge states can influence the overall behavior of the spin chain. In a finite chain, translational symmetry breaking will polarize the spins in the vicinity of the defect. Using computational methods such as Density Matrix Renormalization Group (DMRG),wecanshowthatthisspinpolarizationcreates aspinclusterwhosesize(i.e. correlationlength)depends on the dimerization parameter δ (see Figure S1).

For δ = 0.08 which is the estimated value for (o-DMTTF)2X at low temperature [20] the local magnetization of an open spin chain with odd or even number of spins is given in Figure 1. The end of the chain has the effect of polarising many spins forming a cluster of connected spins to the rest of the chain. This object has a ground state S=1/2 [21] and is denoted in the literature as of pinned soliton [16, 17, 22], end chain [6, 23] or edge state [24] and in the following it will be referred as quantum spin chain edge state (QSC-ES). Due to the topologicalprotectioninthismanybodysystem, intense effort have been devoted to create nanoscale individual QSC-ES [25–27] but the nature of decoherence and relaxation mechanism remain unclear.

In this work, we present a comprehensive study of relaxation and decoherence of QSC-ES of a quasi-1D spin chain embedded in 3D crystals. We have chosen the (o-DMTTF)2X familybecausethebulkmagneticproperties are well known [20, 21] with a large exchange J ∼600 K and a dimerization parameter δ ∼ 0.08 which is neither toosmallasin(TMTTF)2Xfamily(δ ∼0.03)[28]where very narrow homogeneous lines prevented echo observation in previous studies, nor too large like BEDT–TTF [29] which would lead to overly localized, trivial defects.

In the first part, we show that the spin lattice relaxation of QSD-EC is related the properties of the bulk spin chain. Then, the decoherence processes of QSD-EC are precisely studied. We demonstrate that the spin diffusion processes such as instantaneous and spectral diffusions are renormalized by the many body property of the topologicaly protected QSD-EC.

Results
Crystals description.
Single crystals of (o-DMTTF)2Br, (o-DMTTF)2Cl, and (o-DMTTF)2I were grown via electrocrystallization following the standard procedure described in Ref.[30]. Thestructural,electronicandmagneticcharacterizations werereportedinRefs.[18,30,31]. Theircrystallographic structures are shown in Figure 2. They all have tetragonal unit cell (space group I42d (no. 122) [30]) with cell parameters a = b = 16.93Å, 17.09Å, 17.40Å and c=7.040Å, 7.058Å, 7.098Å, respectively. The cell contains four pairs of molecules of o-DMTTF which stack alongtheaxisperpendiculartotheplaneofthemolecules (caxis). Eachpairofo-DMTTFmoleculeispartiallyoxidized and shares one hole and so, one spin. Each stack is rotated by 90◦ with respect of the neighboring stacks. Asaconsequence,theinteractionbetweenspinsalongthe stacking axis is very strong [18, 20, 30, 31] but the interactionbetweenspinchainsisveryweak. Thespin-Peierls transitionhasbeenreportedinallthreecompoundswith adimerizationtemperatureT_SP around50K.Withinthe temperature range examined in our study (<20 K) the dimerized phase is fully stabilized.

Spin Lattice Relaxation
The spin-lattice relaxation time, T1, represents the characteristic duration required for a spin to return to its equilibrium state after random flips along the spin quantizationaxis. LongitudinalrelaxationT1 imposesan intrinsic limit on the transverse relaxation T2, such that T2 ≤ 2T1. Therefore, understanding the mechanisms of spin-lattice relaxation is crucial for understanding spin decoherence. In magnetically diluted systems, such as rare earth ions, transition metal ions or nitrogen vacancy (NV) centers, which represent the majority of the studies on electron spin qubit, the spin-flip induced by the spin-phonon interaction is well understood and can be divided in three processes (Fig. 2): The direct absorptionoremissionofaphononresonatingbetweenthespin states |a⟩ and |b⟩. This process dominates at low temperature, when the density of phonon is mostly at the energy of the microwave hν =δ. When the temperature increases,thermallyassistedrelaxationmechanismusing two phonons is induced: The non resonant Raman process occurs by virtual absorption of a phonon of energy δ1 and is followed by emission of a phonon of energy δ2 and finally, the resonant Orbach process occurs through a third state |c⟩. This last mechanism exists only if the Debye temperature θD is larger than the energy of the third state |c⟩ . For a Kramers ion, the spin-lattice relaxation time as a function of temperature is given by [32, 33]:

1/T1 = α_D·T + α_R5·T5 + α_R7·T7 + α_R9·T9 + α_O·Δ3exp(-Δ0/kBT) (2)

where α_D is the direct absorption-emission process coefficient and is dependent to the external magnetic field as H4, α_Ri are the two-phonon Raman process coefficients such as α_R5 describes mostly molecular systems, α_R7 is proportional to H2, α_R9 is the principal coefficient expected in Kramers ions, α_O is the resonant Orbach coefficient[34]withΔ0 thegapwiththefirstexcited spin state.

In this section we study T1 as function of the temperature between T=5 K and 20 K and as function of the magneticfieldusingthreemicrowavefrequencies9.7GHz (0.35T),34GHz(1.21T)and94GHz(3.35T)usingthe inversionrecoverypulsesequence: Aπ pulseinversesthe populationofspins,andafterwaitingforatimet1 therecovered magnetization ⟨Sz⟩ is probed using a Hahn echo (π −t1 −π/2−τ −π −τ −echo). Examples of magnetization recovery at T = 7 K and 14.5 K are shown in Figure 3. While at high temperature, the recovery is clearly monoexponential, it deviates at low temperature and a stretched exponential is needed to fit the experimental values (Eq.(3)).

M0 - Mz(t) = exp(-(t/Tstr1)^β) (3)

Thevariationofthestretchedparameterβ asfunction oftemperature for the three frequencies used is shown for(o-DMTTF)2BrinFigure3(c). Athightemperature (T >13 K), β =1 indicates a uniform distribution of T1 while below (T < 13 K), β progressively decreases with aminimumof0.5atT =5K.Nosignificanteffectofthe microwave frequency, and consequently, of the magnetic field is observed.

The stretched exponential behavior of longitudinal magnetization has been observed in a wide variety of systems and research areas [35–37], and usually fits relaxation processes in disordered and quenched electronics. However the microscopic physical meaning of Tstr1 when β < 1 is debatable. Johnston [38] has shown that thestretchedexponentialistheLaplacetransformofthe probability density P(1/T1,β) of 1/T1. Knowing β it is then possible to reconstruct P(1/T1,β) (see Figure S12 for some examples). It is worth to notice that the maximum of P(1/T1,β) is not Tstr1 but Johnston has shown that in the case of 0.5 < β < 1, 1/Tstr1 can be described as the median of P(1/T1,β). Thus, while the temperature dependence of Tstr1 has a physical significance, extractingandinterpretingquantitativeprefactorscouldbe hazardous.

Now, we examine the temperature dependence of 1/Tstr1. Using the case of (o-DMTTF)2Br, Figure 4 illustratesthetemperaturedependenceof1/Tstr1 inlog-log scale for the three microwave frequencies available. The dashed line is the best fit for the low temperature part (T < 11 K) using the direct one-phonon process (α_D·T) which is assumed to be dominant in this range of temperature. Clearly the fit is poor and it is evident that the relaxation rates Tstr1 do not exhibit linear temperature dependence. This observation holds true for (o-DMTTF)2Cl (Fig. S5), and (o-DMTTF)2I (Fig. S6) as well. A much better fit was found using a square temperature dependence of 1/Tstr1 using α_B·T2 (black lines in Fig. 4).

The magnetic field dependence of the spin-lattice relaxation does not follow the usual behaviour of the direct regime. In fact, in a Kramers doublet,such as in our case, we would expect a dependence in H4 and therefore a variation of 4 orders of magnitude between the X-band and W-band measurements, which is clearly not the case. Here, the low-temperature behaviour is rather linear with the magnetic field. The linear field dependence and the quadratic temperature dependence of 1/Tstr1 suggest that the direct process is ”bottlenecked”. Indeed, in the direct-phonon process,itisassumedthatthephononemittedduringthe relaxationisrapidlyabsorbedbythethermalbath. However,ifthephononremainslongenoughinthecrystal,it mightbere-absorbedbythespinsystem,reducingtherelaxationefficiency. Thephonon-bottleneckwaspredicted by van Vleck [39] and observed in various systems. The relaxation rate in a phonon-bottlenecked direct process is approximated by [32, 40]:

1/T1phb = (3ω2Δω / 2π2v3nτph) * coth2(ħω / 2kBT) (4)

where v the velocity of sound in the crystal, Δω is the resonance linewidth, n is the concentration of spins, and τph isthelattice–bathrelaxationtime. AssumingkBT ≫ ħω, and the inhomogenous linewidth is field dependent we can simplify Eq.4 by

1/T1phb = α′_B·H0·T2 (5)

It has been observed that the phonon-bottleneck process of the spin lattice relaxation is accompanied by a non-exponential decay of the magnetization [41] (see Eq.(3)) whichisabehaviorwealsoobservedatlowtemperature. Wecanconcludethatthetemperaturedependenceofthe stretched parameter β is the signature of the phonon-bottleneck process that takes place below about 13 K (see Fig. 3(c)).

Having identified the low-temperature phonon-bottlenecked direct process, we now focus on the behavior at higher temperature. To study the two-phonon regimes, we have subtracted the low-temperature behavior from the total relaxation : 1/THT1 = 1/T1 - α_B·T2. Since the Orbach- and Raman-processes are highly non-linear, we present in Figure 5 the temperature dependence of the corrected relaxation either as (i) log(1/THT1) vs 1/T or (ii) log(1/THT1) vs log(T). In the first case (i) the Orbach (resonant) process is linear with respect of the inverse of temperature while in the second case (ii), the Raman (non-resonant) process is linear in the log-log scale with the slope corresponding to the exponent (see Eq. (2)).

The best fit using pure Orbach : 1/T1 = α_O·Δ3exp(-Δ0/kBT) and pure Raman : 1/T1 = α_R9·T9 shows two different groups of systems. (o-DMTTF)2Cl and (o-DMTTF)2Br exhibit an Orbach process while (o-DMTTF)2I has Raman-T9 process. We think that these two different relaxation behaviors are directly related to the relative value of the spin-triplet gap. In the cases of (o-DMTTF)2Cl and (o-DMTTF)2Br, the Orbach gaps Δ0 obtained from the best fit of the temperature dependence of T1 (Fig. 5) is coherent with the value of the spin-triplet gap Δ_SP. However, in the case of (o-DMTTF)2I, Δ_SP is large and no Orbach process is observedsuggestingthatnophononhastheenergytoreach thetripletstate(i.e. θD > Δ_SP).

Electronic spin coherence time
In this part, we focus on the electronic coherence time T2 andthemechanismsofdecoherenceof(o-DMTTF)2I, (o-DMTTF)2Br and (o-DMTTF)2Cl. Several phenomena contribute to the loss of coherence of electronic spins. Firstly, the spin-lattice relaxation resulting from spin-phonon interactions sets the physical limit for the coherence time, i.e., T2 < 2T1. We have discussed the various relaxation mechanisms in detail in section and identified which ones were involved in the relaxation of (o-DMTTF)2X. This limit is rarely reached because the excited spins are subject to random dephasing caused by spin-spin interactions. These dephasing processes contribute to reducing the coherence time T2 and are collectively represented by the term Γ_ϕ. The coherence time is then related to the relaxation time T1 by the relation:

1/T2 = Γ_ϕ + 1/(2T1) (6)

Let us note that the term Γ_ϕ is a sum of different contributions to the random phase shifts. Some contributions to the phase shifts are intrinsic, such as the flip-flop process,whileothersareduetotheenvironmentand may depend on the measurement method, such as spectral diffusion [42] and instantaneous spectral diffusion [43, 44]. Therefore, the coherence time value depends on the measurement method. These various decoherence mechanisms have been detailed in [45].

Weemploybothtwo-pulseHahnecho(2PE)andthree-pulsestimulatedecho(3PE)toclarifythedynamicinteractions that give rise to spin diffusion (SD) in QSC-ES ensembles. In this context, the 2PE involves flipping the spins into the transverse plane with a π/2 pulse. The spinsdefocusforatimeτ,andthenwerefocusthemwith a pulse corresponding to an angle θ2 (usually optimum for θ2 = π), the totalsequencebeing π/2−τ−θ2−τ−echo. The 3PE splits the refocusing π pulse 2 into two π pulses separatedbyatimeTw : π/2−τ−π/2−Tw−π/2−τ−echo.

The 2PE and 3PE sequences were first used by Mims et al. [42] and since then the underlying theory was improved using uncorrelated-sudden-jump [46, 47]:

S2(2τ+Tw) = V_ESEEM * exp(-2πΓ_effτ) * exp(-(2τ/T1)^β) (7)

Γ_eff = Γ0 + 2Γ_SD{Rτ + 1 - exp(-R*Tw)} (8)

where I0 is the echo amplitude at zero delay, and V_ESEEM represents to the electron spin echo envelope modulation (ESEEM) caused by the magnetic dipole-dipole interaction of the QSC-ED and the nuclear spins in (o-DMTTF)2X [48]. ESEEM from the host nuclear spinssuchas1H,13Candcounter-ionswasnotobserved. This absence of signal will be discussed after. T1 is the spin-lattice relaxation and we have added β to take into account the phonon-bottleneck effect presented before. Eq. (7) and (8) are valid when τ ≪ T1.

The effective decaying rate Γ_eff incorporates the coherence time Γ0, Γ_SD and the spin-flip rate R. The coefficient Γ0 represents the homogeneous broadening on the timescale of the measurements and also the instantaneous diffusion induced by the microwave excitation pulses. Γ_SD and R describe the spectral diffusion during the delays τ and Tw. Specifically, R denotes the average spin-flip rate, while Γ_SD represents the full width at half maximum (FWHM) contribution to the dynamic distribution of transition frequencies.

In the case of Tw =0 the sequence corresponds to the 2PE. This situation occurs when the microwave power is not sufficient to excite all spins constituting the inhomogeneous line.

I(2τ)=I0 exp(-2πΓ0τ - 2π Γ_SD*R*τ^2) (9)
=I0 exp(-(2τ/T2)^n) (10)

In the presence of pure spectral spin diffusion caused by non resonant spin bath (nuclear or electronic), the decay is expected to be a stretched exponential with n ranging between 2 and 3 [42, 43, 49].

Instantaneous diffusion
Instantaneous diffusion (ID) is a decoherence mechanism caused by the microwave excitations used to refocalize the spins that generate a spin echo. During a π refocusing pulse, spin populations are inverted, altering the quantum state m of the resonant spins, and the locally perceived field s for the spins is abruptly changed. The dephasing generated by ID is related to the coherence time by Eq.(11):

1/T2 = Γ0,ID + Γ_ID * ⟨sin2(θ2/2)⟩f (11)

wherethephaseshifttermΓ_ID ofEq.(12)isproportional to the spin concentration and Γ0,ID is the decoherence in absence of ID such as homogeneous broadening and spectral diffusion (SD).

Γ_ID = (μ0 * g^2 * μB^2 * C) / (9 * sqrt(3) * ħ) (12)

A key characteristic is the angle of rotation of the refocalisation pulse θ2 [51, 52] through the expression ⟨sin2(θ2/2)⟩f, which describes the average probability of inverting all spins. It depends on various parameters suchasthemicrowavefield(proportionaltotheRabifrequencyνR)andtheprofilef oftheESRline(alorentzian of half width at half maximum ΔH1/2 in our case):

⟨sin2(θ2/2)⟩f = ∫_{-∞}^{+∞} [νR^2 / (νR^2 + (η+Δ)^2)] * sin2(θ2/2) * f(η+Δ)dη (13)

When ⟨sin2(θ2/2)⟩f = 1, all spins involved in the ESR line are excited. However, in our case the effect of a partiallyexcitedlinehastobetakenintoaccount. Asanexamplelet’sconsider(o-DMTTF)2IwhereνR =10.5MHz and ΔH1/2 = 2.5 G (≡ 7 MHz). With our parameters, ⟨sin2(θ2/2)⟩f =0.56foratotalrefocusingpulseθ2 =180°.

To observe the effect of ID on the electronic coherence time, we measured the decay of the Hahn echo using the sequence : π/2 − τ − θ2 − τ − echo. Figure 6 shows the decay time T2^-1 of (o-DMTTF)2I at T =5.4 K. The inset displays two examples of Hahn echo decay with θ2 = π/4 (blue square) and θ2 = π (orange circles). In all our measurements using 2PE, the stretched parameter n in Eq.(10) was close to 1 showing that ID plays an important role in the decay time. It is also important to mention that the fits using Eq.(9) were not satisfying since the relative weight of Γ_SD*R was small compared to Γ0. The dependency of T2^-1 on θ2 and ⟨sin2(θ2/2)⟩f are presented in Figure 6(a) and (b) respectively. Using Eq.(11), we extract from the slope of Figure 6(b) πΓ_ID = 294 kHz, which leads to a concentration C = 3.5 × 1017 cm−3. The values of πΓ0,ID, πΓ_ID, and C for (o-DMTTF)2I, (o-DMTTF)2Br, and (o-DMTTF)2Cl are provided in Table II and comparedto values extracted fromcontinuous wave (cw) ESR measurements [20, 31].

For all compounds, the concentration obtained from ID is significantly lower than that determined by continuous wave electron spin resonance (cw-ESR). The concentrations from cw-ESR were derived by comparing the relative intensities of the spin chain and end chain signals, resulting in a very small error. In the case of ID, thefittingprocedureisdetailedintheSupplementaryInformation (SI) (Fig. S7-S10), with a cumulative error of less than 20%. The measurement error alone cannot accountforthediscrepancyinconcentrationresultingfrom the ID and cw-ESR studies. This discrepancy is a key finding of our work. For an ensemble of localized S=1/2 defects, the local spin concentration probed by dipolar interactions is expected to be equal than the average concentration (C_ID = C_cw). Our observation of C_ID < C_cw strongly contradicts the localized spin model. It is important to note that the concentration from cw-ESR is directly related to the number of S=1/2 spins, whereas the concentration from ID is extracted from the effective dipolar field. The nature of the defect in quantum spin chains is fundamentally different than that of isolated defects (such as metallic ions or free radicals). As explained in the introduction and in Ref. [4–8], the spin S=1/2 we observed is made of dozens of spins (see Figs. 1 and S1). It is not surprising that the strong correlation inside the spin chain might have an effect on the external couplings. To support this hypothesis, we performed DMRG calculation on two QSC-ES interacting though an Ising-likedipolarinteraction d*Sz1*Sz2. Wethencalculatedtheanistropicgap Δ_δ inducedbydformanyvalues of the dimerization factor δ and comparedthemwiththe anisotropic gap of 2 isolated spin Δ_iso. The details of calculation are presented in SI, and the main results are: For δ = 1 (i.e. chain of non interacting dimers with a gap Δ_δ=1 = Δ_iso), and Δ_δ=0 = 0 (ie. the uniform spin chain). In the particular cases of δ ranging from 0.08 to 0.096 for (o-DMTTF)2X [20], we obtain Δ_δ=0.08 = 0.29Δ_iso to Δ_δ=0.096 = 0.35Δ_iso which match with the reduction of the effective concentration observed by ID. Although this model is very simple, it allows us to semi quantitatively explain the reduction in the effective dipole field between 2 QSC-ES. This result can be interpreted as follows: QSC-ES are many-body objects and it is therefore expected that the strong internal correlations arising from the isotropic J exchange interaction reduce the effect of dipole interactions.

Spectral Diffusion
NowIDhasbeenquantifiedinall(o-DMTTF)2X samples,wefocusonothersourcesofdecoherencesuchashomogeneous broadening and spectral diffusion (SD). 2PE and 3PE are complementary methods to extract coherencecontributions. 2PEcannotseparatethelinewidthof the spectral diffusion Γ_SD and the spin-flip rate R (see. Eq.(9)). Moreover, 2PE sequence is accurate to extract spectraldiffusionwhenthestretchexponentninEq.(10) iscloseto2whileinourcaseitfallsintherange1to1.1. Asaconsequence,wehavefirststudiedthe3PEsequence at the lowest temperature. We measured the decay of the stimulated echo while increasing Tw at a constant τ value and we have then varied τ over the range of 200 ns to 4000 ns. This gives us a series of echo decays that we fitted simultaneously using Eq.(7) and Eq.(8). Selected stimulated echo decays for (o-DMTTF)2I are presented in Figure 7.

The extracted values of the spin-lattice relaxation T1^-1 are coherent with the ones obtained by inversion recovery T1,IR^-1. Thevaluesofthestretchedexponentsobtained from 3PE (Eq. (9)) and inversion recovery (Eq. (3)) are comparable,showingthatthephonon-bottleneckprocess is visible also in 3PE. The values of Γ_SD agree with the dipolar interactions between QSC-ES. Values of the flip-pingrateR areoneorderofmagnitudelargerthanwhat is expected for T1 spin flipping only. If we take into account the flip-flop process into the flipping rate (see S4) we obtain a total spin flip rate of R = 24 kHz close to our experimental value. Finally, we note that the homogeneouslinewidthΓ0 islargecomparedtovaluesreported inliteratureforequivalentspinconcentration[47,53,54].

Now, let’s compare the spin diffusion parameters obtained by 3PE with the temperature dependence of the 2PE decay. For this study, we performed a 2PE decay as function of temperature and we extracted the decoherence time T2-2PE using Eq. (10). We removed the ID contribution using the relation: T2-2PE,corr^-1 = T2-2PE^-1 - πΓ_ID*⟨sin2π/2⟩f. Corrected 2PE coherence time T2-2PE,corr^-1 and spin lattice relaxation time T1 are providedinFigure8for(o-DMTTF)2I,(o-DMTTF)2Cl and(o-DMTTF)2Br. The solid lines are the relaxation time using phonon bottleneck direct process and Raman or Orbach process as discussed previously. For the dashed lines we used the definition of the coherence time [43]:

T2 = (-a + sqrt(a^2 + 2b/π)) / b (14)

where a = Γ0, and b = Γ_SD*R/2 are taken from Table III. This equation is valid for T2 ≪ T1. To take into account the high temperature regime, we also include the limit due to spin lattice relaxation (see Eq.(6)). It is important to note that no free fit parameters were used to simulate the dashed lines in Fig. 8. In the absence of fit parameters, our data describe accurately the temperature dependence of the 2PE. However it appears that T2-2PE,corr is weakly sensitive to the diffusive part Γ_SD*R and the decoherence is mainly due to the homogeneous linewidth Γ0 which explains why we did not extract accurately the spin diffusion contribution from 2PE.

The microscopic origin of Γ0 remains an open question. The presence of nuclear spins in (o-DMTTF)2X is not negligible, and in similar molecular magnets with an S = 1/2 ground state, such as V15 [55] and Cr7Ni [56], they are the primary source of decoherence, often exhibiting a significant ESEEM signature. However, in our study, no ESEEM signal from the systems was detectable,andpreviousworkhasshownthatthehyperfine interactions in QSC-ED are significantly reduced by the exchange coupling within the spin chain [21].

Another possibility arises from the decoherence effect of the spins within the chain. Although we are probing QSC-ED, which display a low concentration, they are composed of dozens of spins (see Figs. 1, S1) strongly correlated to the spin chain. In such magnets, the homogeneous linewidth can be estimated using the theory of moments. Introduced by Anderson and Weiss [57] this method shows that the linewidth is proportional to sqrt(Σ M_i^2) where M_i is the second order moment of the interaction involved in the broadening such as electron-nuclear / electron-electron dipole interactions and anisotropic super-exchange interaction. Although this method is fairly simplistic regarding of the structure of QSC-ED, it does allow us to estimate the contribution of each interaction and their influence on the linewidth. To estimate the electro-nuclear dipolar field, DFT calculations were performed on a single molecule of(o-DMTTF)2Iandhaveshownaelectro-nucleardipolar field of M_e-n ∼ 10^13Hz^2 (see SI). The dipolar interaction caused by all the spins inside the chains give M_e-e ≃ 3×10^17Hz^2 and anisotropic exchange term although very small is estimated to M_ani ≃ 3×10^16Hz^2 (see S5). Finally, using the estimation of J ∼ 600 K obtained in [20, 31] we obtain an homogenous intrinsic linewidth Γ_h ∼ 25 kHz compatible with values reported in table III. Our method for estimating the sources of decoherence can be applied to other systems, in particular inorganic materials which were intensively studied in the 90s. As an example, let’s confront the method to CuGeO3 doped Zn [16]. While Me-n and Me-e remain weak, with Δg ∼ 0.1 and J ∼ 100 K we obtain M_ani ≃ 5×10^20Hz^2 andanhomogeneouscoherencetime of 1ns which explain why no coherence was reported in such material.

Methods
Crystals manipulation.
These crystals have a needle-like shape with the chain axis along the long length of the needle. The largest crystals were used in the low-temperature electron spin resonance (ESR) study, where defect signals were observed but remained weak due to the low concentration of strongly correlated defects. The typical dimensions were 0.2 × 0.2 × 3 mm3, the longest being the chain axis. To avoid the effects of temperature cycle history, a fresh sample was used for each series of measurements. The samples were glued to a suprasil quartz rod using a small amount of Apiezon grease on one side to minimize stress during temperature sweeps. The samples came from the same batches that the ones used for cw-ESR concentration obtained in [20] and this in order to avoid concentration difference between samples. Moreover, to be sure about the concentration we have re-performed a cw-ESRintensitymeasurementsrightaftertheIDstudy.

ESR spectrometer setup
Coherent spin dynamics were investigated using a pulsed X-band (9.4 GHz), Q-band (34 GHz) and W-band (94 GHz) ESR spectrometer (Bruker E580), equipped with an Oxford CF935 He-flow cryostat and a ColdEdge Stinger cryogen-free coldhead, allowing to reach a base temperature of 5.5 K at the sample position. The experimentswereconductedusingmicrowavecavitiesoptimized for pulsed operation. We performed inversion recoveryexperiments(π−t1−π/2−τ−π−τ−echo)forT1 measurements,andused2-pulseHahnecho(π/2−τ−π−τ−echo)and3-pulse(π/2−τ−π/2−Tw−π/2−τ−echo) stimulatedechosequencesforT2 measurements. Therepetition rate was set in order to let the spin system fully relax.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript reports ESR experiments on organic crystals of (o-DMTTF)X, with X=Br,Cl,I that contain S=1/2 spin chains with a modulation of the antiferromagnetic exchange on account of dimerization. The experiments focus on the spin relaxation and decoherence and are rationalized in terms of the S=1/2 that appear at the edges and defects of the spin chains of the dimerized chains. A systematic analysis of the temperature and field dependence (for 3 values of the field) is carried out and usign a multi-parameter formula, the contribution of diferente spin relaxation mechanisms is estimated. In addition, pulsed ESR techniques, such as two-pulse Hahn echo and 3-pulse stimulated echo is used to probe T2 as a function of temperatura. Both the T1 and the T2 experiments are carried out for 3 compounds (X=Cl,Br,I.). The echo experiments are also carried out as a function of the refocusing angle theta2\nThe work is supplemented with DMRG calculations, that reproduce results known in the literature (namely, the presence of S=1/2 moments localized at the edges that host a weak bond), with spin-phonon calculations as well as DFT calculations of the hyperfine interactions.\nI can not recommend the publication of this work in Nature Communications for the following reasons:\n1) The work is not sufficiently novel. The authors have carried out a related study, with cw ESR, on the same compounds, using the same underlying model (refs.14 and 20). I acknowledge that the present work is different, as it uses pulsed ESR-STM, that provides a better insight on the different spin relaxation and decoherence mechanisms.\n2) The interpretation of the data is not fully supported by experiments. The main claim is that ESR is probing the topological (edge) spin states of the chains. However, there are alternative explanations have not been ruled out (see below)\n3) The potential significance of the claims, if they were corroborated, is not particularly outstanding. ESR experiments of topological spin chains go back to the early 90 ( https://doi.org/10.1103/PhysRevLett.65.3181, https://doi.org/10.1103/PhysRevLett.65.1659) . It is not clear how the spin relaxation and decoherence mechanisms of the present system would be relevant for other one-dimensional spin chains with alternate exchange.\nComments/questions/recommendations\n• I could not find the conducting state of the compound. I guess it is an insulator, but in some previous paper it was mentioned that these compounds can be conductors (at higher temperatures). Of course, this would be relevant both for the origin of the resonating spins and for the spin relaxation mechanisms\n• Same comment for the magnetic state of the samples (magnetization vs field, magnetization vs temperature). Authors have reported these measurements in previous work, but they are not discussed here.\n• Specifically, an estimate of the density of resonating spins would help to substantiate the claim that they come from edge states. Specifically, the signal should increase with the area of the surface. Hence, a study of the signal vs surface area would help to substantiate the claim.\n• Some sort of magnetic imaging (perhaps NV magnetometry) may also help to map the increased magnetic density at the surface of the single-crystals\n• Introduction of some sort of non-magnetic doping that interrupts the chains, without adding free carriers would increase density of edges, and the ESR/magnetization signal, accordingly\n• In reference 20, authors show some sort of magnetic transition at around 50 Kelvin. However, in the present study, data is shown up to 20 Kelvin only. In reference 20 there is evidence of magnetic impurities, that is not discussed in the present work\n• I could not find the location of the magnetic moments in this compound. Perhaps DFT calculations could show that to the non-expert readers.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "The work of Bertaina and co-workers focuses on the electron spin dynamics of topological defects (edge states) in the so-called organic Spin-Peierls chains, namely the family of o-DMTTF. This system has been recently studied by the authors (ref.20) in terms of cw-EPR spectroscopy that was used to quantify the dimerization parameter δ=0.08, and pulse EPR methods (Rabi oscillations) that revealed the coexistence of single magnetic solitons with S=1/2 and exchanged-coupled solitons with S=1.\nThe present work employs well established pulse EPR methods to measure the relaxation and coherence times T and T 1 2 of three (o-DMTTF) X species (with X=Br, Cl, and I) which are key parameters for the implementation of such entities in 2 quantum devices. The work presents original research that is likely to attract the interest of various groups working in quantum technologies involving magnetic and paramagnetic materials. While some key challenges—such as scalability and fabrication—remain unresolved in the context of applying these many-body entities to quantum information processing, the study nonetheless represents a significant contribution to the fields of EPR spectroscopy and molecular magnetism.\nThe EPR methodology employed is robust, and the work adheres to high standards within the field. The claimed results of the work are summarized as follows: first, at low temperatures T is governed by the phonon bottleneck effect, and second, 1 due to the exchange coupling of the chain spins the dipolar fields (both electronic and nuclear) are reduced leading to increased T values as compared to those of isolated spins having the same concentration. While the findings are 2 noteworthy and the manuscript is generally well written, several key aspects require further clarification in order to fully substantiate the conclusions. The following points should be addressed as major revisions:\n1) From section IV it is concluded that instantaneous diffusion (ID) is the dominant decoherence mechanism in the studied systems. From the standard analysis of 1/T vs sin2θ2/2 a spin concentration of CID=3.5×1017 cm-3 is obtained which is about two times smaller than the concentration Ccw=8×1017 cm-3 measured with cw-EPR. This difference is interpreted by the authors as a clear evidence of reduction of the dipolar field as a consequence of the exchange coupling of the chain spins. According to them, this discrepancy cannot be ascribed to possible errors in the determination of CID and Ccw. However, although the good quality of ID measurements and the fit shown in Fig.6(b) justify an accurately determined CID value, the same is not clear for Ccw. The values of Ccw given in Table II refer to a previous study (ref.20) where the concentration is given in defects per atom, e.g. 4×10-4 at-1 (Table I of ref.20). How were these values transformed to cm-3? What are the transformation factors and what is their uncertainty? In line 405 it is mentioned that \"The concentrations from cw-ESR were derived by comparing the relative intensities of the spin chain and end chain signals, resulting in a very small error\", but it is not clear what this method includes and what is meant by \"small\". Please give details about the method used to calculate Ccw as well as an estimation of its error.\n2) Typically, the CID in Eq.12 refers to the local spin concentration while Ccw is the so-called average spin concentration. These two can differ considerably when a non uniform distribution is justified leading to CID≥Ccw. According to the authors, the opposite situation observed here is assigned to the strong correlation inside the spin chain and to support this hypothesis they performed DMRG calculations as described in line 425. Although they mention that these calculations are presented in SI, I could not find something in the version I have. Please give an explanation.\n3) The previous two points are particularly important as they directly relate to one of the central claims of this work. Despite the apparent accuracy in determining Ccw, one may question whether the comparison between CID and Ccw is practically meaningful, given that a new sample was used for each measurement (line 122). Can it be assumed that the defect concentration remains constant across different sample preparations? Additionally, were the samples used in Ref. 20 and those in the current study derived from the same preparation batch? Clarification on these points is essential to validate the comparison and its implications for the reported results.\nSome other points:\n4) line 81: \"which is neither too small nor too large:. Please specify small and large with respect to what. What is the influence of δ on the desired properties? Probably it is meant that the previously studied (TMTTF)2X family with δ=0.03 showed very narrow homogeneous lines preventing echo observation. Please add this here to help the reader understand the choice of this new family.\n5) Figure 2: please increase the font size of the axes labels.\n6) Table I: I could not find a reference to this table somewhere in the text, please add. Moreover, the parameter aB given in this table is not defined. If this is aD of Eq.2 please make a correction.\n7) Eq.11: please use a consistent symbol for the rotation angle of the refocusing pulse in both equations and text, θ2, as defined in line 385.\n8) It is not immediately clear how is θ2 changed; is it by changing ω1 or the pulse length Δtπ? What was the length of the first pulse, Δtπ/2? From the given data one can estimate 24 ns but please make this clear so that one can easily reproduce experiments.\n9) Eq.13: the symbol δ is used again here obviously as the offset frequency but it could be confused with the dimerization parameter used before. Please change the symbol and also define all parameters used in this equation. What is Δ? Moreover, it would help to replace θ2 with νRΔtπ.\n10) line 419: \"of free radicals\".\n11) line 427: what is d?\n12) Table II: concentration units should be in cm-3, not cm-1. Please correct.\n13) line 460: this paragraph describes the 3PE decays, therefore the corresponding data cannot be fitted with Eq.9 which occurs from Eq.7 and Eq.8 with TW=0. Please make the same correction to the figure caption of Fig.7.\n14) Table III: why are relaxation rates given as Γ and not as πΓ like in Table II? Please use one consistent unit for relaxation rates in Tables II and III, either kHz or ms-1.\n15) Eq.14: please explain the origin of this equation."}, "author_response": "Response to Referee #1 Comments\nComment 1: \"The work is not sufficiently novel. The authors have carried out a related study, with\ncw ESR, on the same compounds, using the same underlying model (refs.14 and 20). I acknowledge\nthat the present work is different, as it uses pulsed ESR-STM, that provides a better insight on the\ndifferent spin relaxation and decoherence mechanisms.\"\nResponse:\nWe thank the referee for this important point, which gives us the opportunity to clarify the specific advance\nthis work represents over our previous studies. As preliminary remark the present study was performed by\npulsed EPR and not EPR-STM.\nThe referee is correct that our prior works, (Ref.17) we have observed Rabi oscillation in TMTTF familly\nbut the absence of echo prevented the study of the nature of decoherence mechanism. We have add a\nsentence in this sens. Ref 20 introduced these (o-DMTTF)₂X compounds. However, that study was limited\nto continuous-wave (cw) ESR, which probes the static properties of the spin system. It allowed us to\nconfirm the spin-Peierls ground state and quantify the static dimerization parameter, δ. Indeed Rabi\noscillation was reported but only to confirm that the signal came from edge state.\nThe present manuscript addresses a fundamentally different and more advanced question: What are the\nquantum dynamics of the topological edge states that emerge in this system? To answer this, we\nemploy a suite of pulsed ESR techniques, which are necessary to measure spin-lattice relaxation (T₁) and,\ncrucially, spin decoherence (T₂). This is the first study to map out the complete landscape of relaxation and\ndecoherence mechanisms for these many-body topological defects. Our key novel finding is not just the\nmeasurement of T₁ and T₂, but the discovery of a coherence protection mechanism that is a direct\nconsequence of the edge state's delocalized, many-body nature. We demonstrate that the strong intrachain\ncorrelations renormalize and reduce the inter-spin dipolar fields, leading to a longer coherence time than\nwould be expected for isolated, point-like defects.\nTo make this distinction clearer, we have revised the introduction.\nComment 2: \"The interpretation of the data is not fully supported by experiments. The main claim is\nthat ESR is probing the topological (edge) spin states of the chains. However, there are alternative\nexplanations have not been ruled out (see below)... Specifically, an estimate of the density of\nresonating spins would help to substantiate the claim... a study of the signal vs surface area would\nhelp... magnetic imaging (perhaps NV magnetometry)... non-magnetic doping...\"\nResponse:\nThis is a critical point, and we thank the referee for these thoughtful suggestions. Our central claim that we\nare probing delocalized topological edge states, rather than simple point-like impurities, is supported by a\nkey piece of evidence from our instantaneous diffusion (ID) measurements, which we have now\nhighlighted and clarified in the revised manuscript.\nThe ID experiment allows us to measure an effective spin concentration (C_ID) based on the strength of the\ndipolar interactions between spins. We compare this to the absolute spin concentration (C_cw) determined\nby double integration of the cw-ESR signal. For a system of isolated, localized spins, one would expect\nC_ID = C_cw. Strikingly, we find the opposite: C_ID is 2 to 3 times smaller than C_cw (Table II).\nThis result is anomalous for localized spins, but it is the expected signature of delocalized, many-body\nobjects. Because the spin of an edge state is distributed over several lattice sites (as shown in our DMRG\nsimulations in Fig. 1), the dipolar field it exerts on a neighboring edge state is effectively \"smeared out\"\nand reduced. This leads to a weaker effective dipolar interaction, which our experiment measures as a\nlower effective concentration (C_ID). This is our strongest and most direct evidence.\nTo make this argument more prominent and readable, we have restructured Section IV of the manuscript.\nComment 3: \"The potential significance of the claims, if they were corroborated, is not particularly\noutstanding. ESR experiments of topological spin chains go back to the early 90s\n(https://doi.org/10.1103/PhysRevLett.65.3181, https://doi.org/10.1103/PhysRevLett.65.1659). It is not\nclear how the spin relaxation and decoherence mechanisms of the present system would be relevant\nfor other one-dimensional spin chains with alternate exchange.\"\nResponse:\nWe thank the referee for raising this important point about the significance of our work in the historical and\nbroader context of the field.\nThe referee correctly points to the pioneering ESR studies from the early 1990s. These foundational works\nwere instrumental in providing the first experimental evidence for the existence of topological S=1/2 edge\nstates in gapped spin chains, such as the Haldane system. They successfully used continuous-wave (cw)\nESR to probe the static properties of these states—confirming their existence and S=1/2 nature.\nHowever, the scientific landscape has evolved dramatically in the last 30 years. The significance of our\nwork lies in bridging the gap between this foundational discovery and the demands of modern quantum\nscience. Today, these topological objects are no longer just spectroscopic curiosities; they are being\nactively considered as potential building blocks for quantum technologies, such as robust qubits or\nquantum communication channels.\nThe main argument showing our study is a completly new approach in the study of quantum dynamics\ncomes from the fact that in topological defects presented in the 90s, no coherence has been reported.\nOur work provides the first comprehensive answer to the questions regarding relaxation and decoherence\nfor topological defects in a dimerized spin chain. By using a suite of advanced pulsed ESR techniques, we\nhave systematically measured T₁ and T₂, and, most importantly, we have dissected the various decoherence\nchannels. Our key finding—that the intrinsic many-body nature of the edge state actively suppresses\ndecoherence from dipolar fields—is a novel and non-trivial result with significant implications.\nRegarding the relevance to other systems, the physical mechanisms we identify are not unique to our\nspecific compound but are broadly applicable. We deeply thank the referee for this comment.\nIf we apply our analysis to topological defects in cupprate , like CuGeO3 the main source of decoherence\nis the anisotropic exchange interaction (See S5.1) with \\delta g=0.2 and J~100K we obtain M2= 4.10^20\nHz^2 and an homogenous coherence time of 1 ns which explain why coherence was not detected.\nTo ensure this context is clear to the reader, we have revised our manuscript to better frame our\ncontributions.\nWe thank the referee for these detailed questions and recommendations. They help us to clarify the context\nof our work and to better frame our central claims. We address each point below.\nComment: \"I could not find the conducting state of the compound. I guess it is an insulator, but in\nsome previous paper it was mentioned that these compounds can be conductors (at higher\ntemperatures). Of course, this would be relevant both for the origin of the resonating spins and for\nthe spin relaxation mechanisms.\"\nResponse:\nThe referee is correct; this is an important clarification. At room temperature, these organic charge-transfer\nsalts exhibit modest conductivity. However, upon cooling, they undergo a spin-Peierls transition at T_SP ≈\ntemperature regime of our study (T < 20 K).\nTherefore, the ESR signal we observe does not originate from conduction electrons. Instead, it arises from\nlocalized magnetic moments—the S=1/2 topological edge states—that exist within the spin gap of the\ninsulating bulk. The spin relaxation mechanisms we analyze are thus related to the coupling of these\nlocalized spins to lattice vibrations (phonons), not to charge carriers. We have clarified this in the\nmanuscript.\nMoreover the phase diagram was published in Ref 18\nComment: \"Same comment for the magnetic state of the samples (magnetization vs field,\nmagnetization vs temperature). Authors have reported these measurements in previous work, but\nthey are not discussed here.\"\nResponse:\nWe thank the referee for this point. Indeed, our previous work (Refs. 20) included detailed static magnetic\nsusceptibility measurements (SQUID magnetometry). These studies were crucial for establishing the\nfoundational properties of the system: they confirmed the presence of a spin gap (Δ_SP) below the spin-\nPeierls transition and revealed a small population of S=1/2 defects.\nThe current manuscript focuses on the quantum dynamics of these defects, which requires the use of\npulsed ESR techniques to measure T₁ and T₂. While the static magnetic measurements provide the essential\nbackdrop for this study, our focus here is on the dynamic relaxation and decoherence phenomena, which\nare distinct physical quantities.\nWe don’t think our manuscript needs a re-discussion on data already discussed previously\nComments on Substantiating the \"Edge State\" Claim (Combined response to the next three bullet\npoints):\n• \"Specifically, an estimate of the density of resonating spins would help to substantiate the\nclaim that they come from edge states. Specifically, the signal should increase with the area\nof the surface...\"\n• \"Some sort of magnetic imaging (perhaps NV magnetometry) may also help to map the\nincreased magnetic density at the surface of the single-crystals.\"\n• \"Introduction of some sort of non-magnetic doping that interrupts the chains... would\nincrease density of edges, and the ESR/magnetization signal, accordingly.\"\nResponse:\nWe are grateful to the referee for these excellent suggestions for future experiments. They highlight\npowerful techniques that could indeed provide complementary evidence for the nature of the defects.\nHowever, we believe our current study already provides strong, self-contained spectroscopic evidence for\ntheir topological origin, and these suggested experiments, while valuable, represent separate, major\nresearch efforts beyond the scope of this work.\n1. Surface Area Dependence: The defects arise from chain terminations essentially at internal\nboundaries/faults. These is caused by the method of growing which create deffect inside the crystal\nthat can’t be remouved by annealing. We added a sentence in the sense.\n2. NV Magnetometry: We agree that these would be fantastic follow-up studies. However the\ntopological defects are mainly inside the crystal and to our knowledge of NV magnetometry can’t\nbe observed by this method .\n3. Non-Magnetic Doping. In organic material the spin is distributed over the molecule and not on\nonly one atom like in inorganic ones. So the concept of « doping » is not possible.\nComment: \"In reference 20, authors show some sort of magnetic transition at around 50 Kelvin. However, in the present study, data is shown up to 20 Kelvin only. In reference 20 there is evidence of magnetic impurities, that is not discussed in the present work.\"\nResponse:\nThis point is well taken.\n• Temperature Range: The transition at 50 K is the spin-Peierls transition. But the coherence can\nbeen seen (and study) only up to 20K. This is caused by the relaxation time T1 (Orbach or Raman)\nwhich is quickly reduced when increasing temperature.\n• Magnetic Impurities: In Ref 20 we compared Squid susceptibility measurement to ESR\nsusceptibility. The conclusion was that in ESR thanks to the spectrocopic nature of the method we\nobseved only the esdge state while in Squid the signal contain edge state but also « extrinsic\nimpurity ». We don’t think it is uselfull to re-discuss the nature of the ESR signal since it has been\nproven in ref 20.\nResponse to Referee #2:\nWe thank Reviewer 2 for a very insightful and constructive review of our manuscript. We not that the\nreviewer acknowledges the quality of the experimental work. The comments have been invaluable in\nhelping us to significantly improve the paper's clarity, impact, and rigor. We appreciate that the referee\nrecognized the comprehensive and systematic nature of our ESR study. The main concerns regarding\nsignificance, the presentation of our central claim, and the statistical validity of our models are well-taken,\nand we have undertaken a major revision to address them directly. We believe the revised manuscript is\nnow a substantially stronger contribution that clearly articulates its novelty and significance.\nBelow, we provide a point-by-point response to each of the concerns raised.\nResponse to Referee #2: General Comments on Significance\nComment: \"However, I do not find this study significant enough. While the longer coherence time of\nthe delocalized edge mode is an interesting piece of physics, it is largely expected... it is also unclear\nwhat is the implication of this study. Therefore, I do not see a reason why this paper needs to be\npublished in Nature Communications.\"\nResponse:\nWe thank the referee for this frank assessment, which pushes us to better articulate the significance of our\nwork. While a longer coherence time for a delocalized state might be qualitatively expected, our work\nmoves far beyond this simple expectation to provide a quantitative and mechanistic understanding of\nwhy and how this protection occurs.\nThe key significance of our study is twofold:\n1. Mechanism of Coherence Protection: We are the first to experimentally demonstrate that the\ncoherence of a topological many-body state is actively protected by the renormalization of\ndipolar interactions due to internal correlations. We go beyond simply reporting a long T₂; we\ndissect the decoherence channels and pinpoint the physical mechanism responsible for its\nsuppression. This is a non-trivial result and provides a new, fundamental insight into the physics of\ndecoherence in correlated quantum systems.\n2. Implications for Quantum Science: Our findings have direct implications for the burgeoning\nfield of topological quantum matter. By identifying the dominant noise sources (dipolar fields) and\nthe intrinsic defense mechanism (delocalization), we provide a concrete design principle for\nengineering robust quantum states: enhancing the delocalization of a topological mode is a\npowerful, built-in strategy to shield it from environmental noise. This is a crucial lesson for\ndeveloping next-generation qubits and quantum sensors.\nRegarding Ref. 17, while it showed coherent dynamics in a spin-Peierls system, the absence of echo signal\n(only FID was detected) it did not provide a detailed decoherence study or identify the protection\nmechanism we report here. Our work is a significant step forward in understanding how to preserve and\ncontrol such quantum states.\nWe have substantially revised the Introduction and Conclusion to better frame our work in this modern\ncontext of quantum science and to make these points of significance and implication crystal clear.\nResponse to Referee #A: \"Major Concerns\"\nMajor Concern 1: \"My major concern is about the nature of the spin defects... whether they are\nactually measuring the edge mode... the discussion about this seems buried in the section on\ndephasing mechanisms and is also hardly readable... The authors need to provide clear evidence that\nthe mode they measure is indeed something non-trivial, presented in a readable way.\"\nResponse:\nThis is an excellent and absolutely crucial point. We agree that this central claim was not presented with\nthe clarity and prominence it deserves. The referee’s feedback has prompted us to completely restructure\nand rewrite Section IV to make this argument the clear, logical centerpiece of the paper.\nThe presence of the edge state was previously proven in Ref 20 but in the curent study we have found\nanother argument\nOur primary evidence is the anomalous discrepancy between 2 different kind of measurements of spin\nconcentrations:\n• C_cw: The absolute concentration measured by cw-ESR.\n• C_ID: The effective concentration measured by the strength of dipolar interactions via\ninstantaneous diffusion (ID).\nWe observe that C_ID is 2-3 times smaller than C_cw. This is a direct signature of a delocalized, many-\nbody state. A localized, point-like spin would result in C_ID = C_cw. The delocalization smears out the\nspin's magnetic field, weakening its interaction with neighbors, which our experiment correctly measures as\na lower effective concentration. This is the \"non-trivial\" evidence the referee is asking for.\nThe revised Section IV is now structured to walk the reader through this argument step-by-step, making it\nclear and compelling.\nMajor Concern 2: \"many conclusions in this paper are derived from fitting models with multiple\nparameters... I do not know the statistical significance of the fit. For example, in identifying the\nOrbach versus Raman mechanism, how much worse will the fit become when fitting with the wrong\nmodel? Overall, I would like to see more statistical tests...\"\nResponse:\nThis is a very important methodological criticism, and we thank the referee for demanding a higher level of\nstatistical rigor. We have now taken steps to validate our fitting models.\n1. Orbach vs. Raman Model Comparison: The purpose to present relaxation in figure 5 using two\nkind of scale was preciselly to prouve what mechanisms where involved : in log(1/T1)-1/T scale\n(left) Orbach process is lineare while in log(1/T1)- log(T) scale (right) Raman process is linear. To\ndirectly address the referee's example, we have performed a quantitative comparison of the Orbach\nand Raman models for the (o-DMTTF)₂Br compound. We have added a new figure to the\nSupplementary Information (Fig. S3) that shows the data fitted with both models. We include\nthe corresponding adjusted R² values, which demonstrate that the Orbach model for Br and Cl and\nRaman for I compounds provide a statistically superior description of the data, thus validating our\nphysical interpretation.\n2. Fit Errors : for all values extracted from fits ,the fit error is inside the last digit. We thank\nReviewer 2 to point that the information was missing and we have added it.\nResponse to Referee #A: \"Minor Concerns\"\nComment: \"Can the authors provide evidence of the spin-Peierls transition in their samples? This\nwould also provide evidence about the existence of the edge mode.\"\nResponse:\nThis evidence was published in Ref 18 and 20 using cw-ESR\nComment: \"Eq. (6) is well-known when beta=1, but is it true when beta is non-trivial?\"\nResponse:\nThis is a sharp observation. In the original model of spin diffusion probed by 3pulse stimulated echo, the\nspin lattice relaxation was concidered mono exponentiel and is accualy not a part of the derivation of the\nmodel but just a physical border. Since in our couse T1 is a stretched exponential it is logical to include this\nparameter in the 3-p stimulated echo fit. The fact that the beta found by stimulated echo and the one from\ninversion recovery tend to prove this hypothesis was right.\nComment: \"I do not see vertical error bars on many figures, such as Fig. 3(c), 4, and 5. Why?\"\nResponse:\nThe referee is correct to point this out. The error bars, which are derived from the standard error were\nsmaller than the size of the data markers especially when data are plotted in log scale. For clarity we didn’t\nincluded them but we have added a sentence.\nDear Referee #4,\nWe are extremely grateful for the optimistic and helpfull review. The detailed feedback has helped us to\nimprove the rigor and clarity of our manuscript. We appreciate that the referee recognizes the robustness of\nour methodology and the significance of our findings. We have carefully addressed every point raised, with\nparticular attention to the major revisions concerning the calculation of spin concentrations and the\ncomparison between different measurement techniques. We have added a new section to the\nSupplementary Information to provide these crucial details. We believe the revised manuscript is now a\nmuch more robust and convincing work.\nBelow is a point-by-point response to all comments.\nResponse to Referee #4: \"Major Revisions\"\nComment 1: Concerns the calculation of C_cw from Ref. 20. \"How were these values transformed to\ncm⁻³? What are the transformation factors and what is their uncertainty? ... Please give details about\nthe method used to calculate C_cw as well as an estimation of its error.\"\nResponse:\nThis is a critical point, and we thank the referee for requesting this essential clarification. We apologize for\nthe omission of these details. To address this, we have added a new section to the Supplementary\nInformation (Section S.3) dedicated to the calculation of the absolute spin concentration C_cw.\nIn this new section, we detail the step-by-step procedure:\n1. Starting Point: We start with the concentration in \"defects per formula unit\" from our previous\nwork (Ref. 20), which was determined by fitting the static magnetic susceptibility.\n2. Conversion: We convert this dimensionless value to a volumetric concentration (in cm⁻³) using\nthe molar mass of the (o-DMTTF)₂X compound and its crystallographic density, which is known\naccurately from X-ray diffraction data (Ref.). The formula used is: C_cw = (defects/formula_unit)\n* (ρ * N_A) / M, where ρ is the density, N_A is Avogadro's number, and M is the molar mass.\n3. Uncertainty Analysis: We have also performed an uncertainty analysis. The main source of error\ncomes from the fit of the magnetic susceptibility data, which we estimated to be around 10%. This\nis substantially smaller than the factor of 2-3 discrepancy we observe between C_cw and C_ID,\nconfirming that the difference is a real physical effect and not a calculation artifact.\nWe have also added the estimation of error in the ID annalysis.\nComment 2: Concerns the DMRG calculations. \"they performed DMRG calculations as described in\nline 425. Although they mention that these calculations are presented in SI, I could not find\nsomething in the version I have. Please give an explanation.\"\nResponse:\nWe sincerely apologize for this oversight; the referee is correct. The reference to the DMRG calculations\nsupporting our hypothesis was omited in the SI. It has been now added\nThe existing Supplementary Information section on DMRG (Section S1) has been updated.\nComment 3: Concerns the comparison of C_ID and C_cw from different samples/batches. \"Can it be\nassumed that the defect concentration remains constant across different sample preparations?\"\nResponse:\nThis is a very sharp and important question regarding experimental practice. The referee is right to question\nthis. The samples came fro mthe same batch that the sample used in ref 20 and we double checked the\nconcentration after having perfomed the ID analysis. We have added a sentance in this sens.\nResponse to Referee #4: \"Some Other Points\"\nResponse:\nWe thank the referee for this meticulous list of corrections, which have greatly improved the manuscript's\nquality and precision. We have implemented all of the following changes:\n4) line 81: \"neither too small nor too large\": Corrected. We have clarified this line.\nRevised Text: \"...a dimerization parameter δ ~ 0.08, which is neither too small, as in the (TMTTF)₂X\nfamily (δ ~ 0.03) where very narrow homogeneous lines prevented echo observation in previous studies,\nnor too large, which would lead to overly localized, trivial defects.\"\n5) Figure 2: Corrected. The font size of the axes labels has been increased.\n6) Table I: Corrected. We have added a reference to Table I in the main text. The parameter αβ has been\nexplicitly defined in the table caption as α_B, the coefficient for the phonon-bottleneck direct process (α_B\nT²).\n7) Eq. 11: Corrected. We have ensured that the symbol for the rotation angle of the refocusing pulse is\nconsistently θ₂ throughout the text, equations, and figures.\n8) How θ₂ is changed: Clarified. We have added a sentence in S3.\n9) Eq. 13: Corrected.\n10) line 419: Corrected.\n11) line 427: Corrected. The variable d is the dipolar field dS1zS2z as writen in line XXX.\n12) Table II: Corrected. The concentration units have been changed to cm⁻³.\n13) line 460 / Fig. 7 caption: Corrected. This is an excellent catch. The text and figure caption now\ncorrectly state that the data were fitted simultaneously using the full expression for the 3PE decay (Eqs. 7\nand 8), not the simplified 2PE case (Eq. 9).\n14) Table III: We undestrand the point of view but we would like to keep this notation. The reason is due\nto the property of the Fourier Transform. Gamma is the linewidth in the frequency domain (in Hz) while\npi*Gamma is inverte of T2 (so in ms-1) For reason of models and treatment we prefer to keep pi*Gamma\nfor ID and Gamma for 3p echo\n15) Eq. 14: This equation is the analytical solution for the decay time T₂ in the presence of pure spectral\ndiffusion, where the decay is given by exp[-(t/T₂)ⁿ] with n=2. It can be derived from the general model of\nKlauder and Anderson in the limit where T₂ ≪ T₁ and is valid for a Gaussian spectral diffusion process.\nWe have added the reference\nDear Editor,\nWe thank the Reviewers for their continued engagement with our manuscript and their\nacknowledgment that this is \"a solid piece of science.\" We respectfully disagree with the\nassessment that the significance is insufficient for Nature Communications. Below, we\nprovide a detailed response addressing both concerns and argue that our work represents a\nfoundational contribution to an emerging field at the intersection of topological quantum\nmatter and quantum information science.\nDetailed Response to Reviewer\nWe thank the Reviewers for their thoughtful evaluation. We address each concern below.\nConcern 1: Utility of the noise model for engineering longer coherence times\nReviewer's point: \"A good noise model should be able to support the engineering of a longer\ncoherence time by identifying and suppressing the dominant noise source. This has not yet\nbeen demonstrated.\"\nResponse:\nWe respectfully argue that our work does precisely this—it identifies the dominant noise\nsources and provides clear, actionable design principles for engineering longer coherence\ntimes. We acknowledge that we have not yet experimentally demonstrated improved\ncoherence in a new material, but the predictive framework we establish is itself a major\ncontribution. We elaborate on three key points:\n(1) Our model is predictive, not merely descriptive.\nThe central result of Figure 8 demonstrates that our theoretical prediction of coherence time\n(dashed lines) matches experimental data with no free fitting parameters. The parameters\nΓ₀, Γ_SD, and R were extracted independently from 3-pulse echo measurements (Table III),\nand these values successfully predict the temperature dependence of 2-pulse echo\ncoherence. This level of quantitative agreement demonstrates that the model captures the\nessential physics and can be used predictively.\n(2) We identify the key material parameter for engineering longer coherence: the\ndimerization factor \\delta.\nOur DMRG calculations (Figure S2) reveal that the effective dipolar field between edge states\nscales as d_eff/d ≈ 0.3 for \\delta ≈ 0.08–0.1, and approaches zero as \\delta → 0. This is a\nquantitative, predictive relationship that directly guides material design:\n• Prediction 1: Materials with smaller \\delta will exhibit longer coherence times due to\nreduced instantaneous diffusion. This explains the long-standing puzzle of why no\nspin echoes were observed in (TMTTF)₂X (\\delta ~ 0.03)—the spectral diffusion is so\nsuppressed that homogeneous broadening dominates, preventing echo formation\nwith standard techniques.\n• Prediction 2: The homogeneous linewidth \\Gamma_0 ~ 25 kHz arises from intra-\nchain dipolar interactions and is governed by J. Systems with larger J/\\delta ratios will\nexhibit narrower homogeneous lines.\n• Prediction 3: The Orbach relaxation pathway through the spin-Peierls gap (Table I)\nsuggests that materials with larger \\Delta_SP will have longer T₁ at intermediate\ntemperatures.\n(3) We provide a framework applicable to other topological spin systems.\nWe explicitly demonstrate the applicability of our approach by analyzing why CuGeO₃:Zn\nshows no coherence (Section IV, lines 573–580). Our method predicts \\Gamma_h ~ 1 ns⁻¹\ndue to the large anisotropic exchange term (\\Delta g ~ 0.1), which quantitatively explains the\nabsence of spin echoes in this extensively studied material. This validates our framework's\npredictive power across different material systems.\nIn summary, while we have not yet synthesized and measured a new optimized material\n(which would require a separate materials science effort), we have established the\ntheoretical and experimental framework that makes such optimization possible. This is\nanalogous to foundational works on NV center decoherence that identified nuclear spin\nbaths as limiting factors—those works did not immediately demonstrate isotopic purification\nbut provided the understanding that later enabled it.\nConcern 2: Overstated implications for quantum science\nReviewer's point: \"The implication for quantum science is overstated... no two-qubit gates\ndemonstrated... not optically addressable... I do not see how this physical setup might\nintroduce new capabilities.\"\nResponse:\nWe appreciate this critique and agree that some reframing is appropriate. We propose the\nfollowing perspective:\n(1) We do not claim to demonstrate a functional qubit or sensor.\nOur work is fundamentally about understanding decoherence mechanisms in a novel class of\nquantum objects—topologically protected many-body states. We agree that demonstrating\ntwo-qubit gates or competitive sensing is beyond the scope of this study. We will revise the\nmanuscript to moderate claims about immediate quantum device applications.\n(2) However, the significance lies in the fundamental physics, not device demonstration.\nThe key finding of our work is that quantum spin chain edge states are many-body objects\nwhose collective nature fundamentally modifies their interaction with the environment.\nSpecifically:\n• The effective dipolar coupling between edge states is renormalized by the intra-chain\nexchange interaction (C_ID < C_cw, Table II).\n• Nuclear hyperfine interactions are similarly reduced (no ESEEM observed, unlike in\nV15 or Cr7Ni molecular magnets with similar S=1/2 ground states).\n• The topological protection manifests not just in the existence of in-gap states, but in\ntheir dynamical properties.\nThis is a conceptually new result that has not been demonstrated in any previous study of\nspin chain defects. Prior work on Sr₂CuO₃ and CuGeO₃ established the existence of edge\nstates but could not probe their coherent dynamics.\n(3) This work addresses a timely question in the field.\nThe study of topological edge states in nanoscale spin chains is an active frontier, as\nevidenced by recent high-profile publications demonstrating individual edge states in atomic\nchains (Refs. 25–27, including 2024–2025 Nature Nanotechnology and Nature Synthesis\npapers). Our work provides the first quantitative understanding of what limits coherence in\nthese systems—a question that must be answered before any quantum application can be\nrealized. We quote from our conclusion (lines 615–618):\n\"This finding is important for the recent development of topological states in nano-objects in\nwhich the internal dipolar field could be the ultimate source of quantum decoherence.\""}, {"round": 2, "reviewer_comments": {"reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": ""}, {"round": 3, "reviewer_comments": {"reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_5": "Soriano et al report the electron spin dynamics in spin chain system by ESR spectra, the authors establish that the intrachain exchange coupling has positive effect on leading to a longer coherence time than isolated ions at equivalent concentration, this motives the optimizing coherence in future quantum spin materials. The authors have replied the comments from Referees well and improved their manuscripts. I recommended it to be published in Nat communication in present version.", "reviewer_6": "In this work the authors synthesized crystals of dimerized edge states and measured their spin relaxation properties. They perform a multifrequency and variable temperature examination of the spin-spin, spin-lattice relaxation times. In addition, they measure the effects of instantaneous and spectral diffusion due to the presences of imperfect pulses. These relaxation times are compared to known theory to elucidate determinants of relaxation leading to some predictive capabilities that can guide the design these systems. In particular the authors postulate that materials with weaker dimerization and larger exchange couplings may lead increased coherence times. The work is done with deep care and the analysis is thorough and for these reasons the work is topical and can be published as is, but the results are nevertheless specialized in nature and there is concern about the relevance to a general audience."}, "author_response": "The authors have convinced me that their model has predictive power. I’m also happy with their modified significance statement. Therefore, I can recommend the publication of this paper on nature communications."}]

Decision letter