The Mechanism of Electron Spin Decoherence in a Partially Deuterated Glassy Matrix

Abstract

Long electron spin coherence lifetimes are essential for applications in quantum information science and electron paramagnetic resonance, for instance for nanoscale distance measurements in biomolecular systems using double electron–electron resonance. We experimentally investigate the decoherence dynamics under the Hahn echo sequence of the organic radical d₁₈-TEMPO in a variably deuterated frozen water:glycerol matrix. The coherence time (phase memory time) T_M scales with proton concentration as [¹H]^−0.65. For selectively deuterated matrices, decoherence is accelerated in the presence of proton clustering, that is, substantial short-range density in the proton–proton radial distribution functions (< 3 Å). Simulations using molecular dynamics and many-body spin quantum dynamics show excellent agreement with experiment and show that geminal proton pairs such as CH₂ and OH₂ groups are major decoherence drivers. This provides a predictive tool for designing molecular systems with long electron spin coherence times.

Graphical Abstract

In pulse EPR (Electron Paramagnetic Resonance) spectroscopy and in some quantum sensing applications, coherent pulse excitation schemes of electron spins are used to obtain information about the local environment of unpaired electrons. In these schemes, electron spin coherence is generated at the beginning of the pulse sequence, and it is refocused and detected at the end in the form of a spin echo. An essential requirement is an electron spin coherence time long enough that the generated coherence survives until detection. The presence of nearby magnets (other electrons and magnetic nuclei) modulates the echo amplitude, and if these modulations are resolved, they provide valuable structural insight.^1–3 However, many unresolved modulations lead to irreversible decoherence and loss of echo amplitude.^4,5 Similar balancing problems also occur when using electron spins as qubits; complex calculations require coupling multiple qubits together, but other undesired couplings cause decoherence, limiting the time available for computations. ⁶

At temperatures low enough for most motional degrees of freedom to have frozen out, and at sufficiently low electron spin concentrations, the primary decoherence pathway for electron spins in non-T₁-limited systems is via the surrounding bath of coupled nuclear spins, communicated to the electron by hyperfine couplings. ^5,7–9 The decoherence timescale depends on the system and the pulse sequence.^5,10,11 For dilute nitroxides in a water:glycerol matrix at 20 K under a Hahn echo sequence, the coherence time (phase memory time) T_M is about 5 μs.

Decoherence from bath spins can be strongly reduced by using approximately spin-free or spin-dilute environments. A prime example are nitrogen–vacancy centers in diamond. The low 1.1 % natural abundance for ¹³C provides long T_Ms,^12,13 and reducing the ¹³C concentration further increases the coherence time.¹⁴ For molecular systems, hydrogen-free molecules in matrices such as CS₂ can be used.¹⁵ Another approach is the use of deuterated molecules and solvents.^16,17 This replaces the large magnetic moments of the protons for the smaller magnetic moments of deuterons. The smaller hyperfine couplings, combined with the detuning effect of deuteron electric quadrupole couplings, prolongs the coherence timescale.^18,19

It is important to understand and be able to predict a priori the decoherence dynamics of electron spins. Such an improved mechanistic understanding is practically relevant, as it provides guidance for avoiding systems with short electron spin coherence times that limit resolution in pulse EPR experiments, sensitivity in quantum sensing, and the number of operations in quantum algorithms.

Here, we investigate the electron spin decoherence in a system where T_M is limited by proton spins as a function of proton concentration and distribution, using a frozen dilute solution of the nitroxide radical d₁₈-TEMPO (Figure 1) in 1:1 mass:mass water:glycerol. We measure the electron spin decoherence dynamics under the Hahn echo pulse sequence. We find that the coherence time scales with the proton concentration as [¹H]^−0.65, but also depends significantly on the proton–proton radial distribution function.

Figure 1:

Top: Measuring the Hahn echo amplitude as a function of total pulse sequence length, 2τ, provides the simplest way to observe decoherence effects. Bottom: The system under study is d₁₈-TEMPO (unpaired electron shown in green) solvated in a frozen 1:1 mass:mass (≈ 5:1 mole:mole) water:glycerol solution, with solvent exchangeable hydrons (blue), solvent non-exchangeable hydrons (yellow) and TEMPO’s deuterons (red) contributing to decoherence.

To rationalize the experimental observations and to develop a predictive model, we employ in silico modeling. There are a number of theoretical methods developed to model nuclear-spin-driven electron spin decoherence, from semi-classical approaches that model the nuclear spin bath as a source of stochastic magnetic noise (requiring the choice of a characteristic timescale)^4,20–23 to many-body quantum dynamics models with the advantage that the timescale emerges from the model.^24–35 Here, we employ the ensemble cluster correlation expansion (CCE) approach^28,29 in combination with molecular dynamics (MD). We have successfully employed this approach on proton-only systems in previous publications.^9,36

We measured Hahn echo decays as a function of solvent proton concentration, utilizing appropriate mixtures of D₂O:d₈-glycerol and natural-abundance H₂O:h₈-glycerol. To minimize instantaneous diffusion, the d₁₈-TEMPO concentration was kept low at 5 μM. At this concentration, the predicted instantaneous diffusion timescale is T_ID = 1.6 ms,³⁷ much longer than any decoherence time scale observed in this work. To eliminate decoherence contributions from thermal motions and spin–lattice relaxation, echo decays were measured at 20 K. Decoherence contributions from the TEMPO methyl groups are minimal due to methyl perdeuteration.

The experimental Hahn echo decays are shown in Figure 2 (top). The observed decays are of stretched-exponential form. The initial 8 and 16 MHz modulation is independent of solvent proton concentration and is therefore attributed to ESEEM (electron spin echo envelope modulation) from TEMPO deuterons. As deuteration is increased and the proton concentration decreases, the decay timescale increases from about 5 μs at 111.4 M protons (natural abundance) to 70 μs at 1 M protons.

Figure 2:

Experimental Hahn echo decays (upper part; 20 K, 1.2 T, 33.8 GHz) and simulated decays (lower part; reflected) are shown as a function of proton fraction (color-coded) for 5 μM d₁₈-TEMPO in a 1:1 water:glycerol matrix where the proton concentration was varied between natural abundance and 1 M via the ratio of H₂O:h₈-glycerol to D₂O:d₈-glycerol.

Figure 2 (bottom) shows explicit quantum spin dynamics simulations based on a structure of the solvated radical obtained by molecular dynamics. The resulting simulated decays are in good agreement with experiment. This is remarkable, since the many-body quantum simulation is exclusively based on the MD structure and does not include any adjustable parameters. The simulations include the electron spin and about 1200 surrounding protons (via periodic boundary conditions; see SI) and use a modified CCE approach. While full CCE converges to the exact result, truncation at a maximum cluster size can give reasonably good approximations. Here, we choose 2-CCE as a compromise between numerical accuracy and computational cost. This leaves some $𝒪\left({\tau }^3\right)$ terms unaccounted for.^26,28 The 2-CCE level appears to get within 20% of the correct timescale.

Excluding the ESEEM-affected initial parts, the experimental and simulated decays of Figure 2 are well fit by stretched exponentials of the form

$$V\left(2\tau \right)=V_0\cdot {\text{e}}^{-{\left(2\tau /T_{\text{M}}\right)}^{\xi }}$$ (1)

with coherence time or phase memory time T_M, initial amplitude V₀, and stretching exponent ξ. Figure 3 plots the fitted T_M and ξ against the proton concentration. (The fits are shown in the SI.) It reveals good agreement between experiment and computational prediction for both T_M and ξ, although the simulations overestimate T_M by about 20 %. To assess the contribution of deuterons to T_M, Figure 3 also compares simulations with and without the deuterons in the system (see SI). The results show that T_M is noticeably affected by deuterons only at [¹H] < 3 M. At higher proton concentrations, the effect of solvent deuterons, or their absence, on decoherence is negligible.

Figure 3:

Coherence time (top) and stretching exponent (bottom) are plotted versus proton concentration for the experimental and simulated data from Figure 2. The 95 % confidence intervals are smaller than the circle size. The dashed black line is a fit of Eq. (2) to the experimental data with T_M,1M = 93 ± 10 μs and p = 0.65 ± 0.05 (R² = 0.99). Small black dots indicate fits to the computational predictions, either including deuterons (red line) or excluding them (green line).

Figure 3 shows that ξ is concentration dependent, varying between 1.5 and 2.6. This indicates that it is not strictly possible to infer the decoherence mechanism from the value of ξ, as sometimes done in the literature, where values less than 2 have been associated with methyl rotation or instantaneous diffusion.^18,38 Figure 3 also shows that the experimental T_M decreases almost linearly with increasing concentration in this log–log plot, indicating a

$$T_{\text{M}}=T_{\text{M},1\text{M}}\cdot {\left({[}^1\text{H}]/\text{M}\right)}^{-p}$$ (2)

relationship, where T_M,1M is the coherence time at 1 M proton concentration, [¹H]/M is the proton concentration divided by a reference 1 molar concentration, and the exponent p is the negative slope in the log–log plot, that is, p = –d lnT_M/d ln([¹H]/M). The fit to the experimental T_M data yields p = 0.65(5) (R² = 0.99), and the simulated data are fit with p = 0.639(5) (R² = 0.999), again showing that the simulations recover the observations well.

Expectations for p are mixed. On one hand, p < 1 is consistent with our earlier work on 3-maleimido-proxyl, where we observed a similarly shallow dependence in the refocused Hahn echo.³⁶ Experimental T_M data on P donors in Si crystals show a p ≈ 0.85 dependence on the ²⁹Si mole fraction between 4.7 % and 99.2 %.³⁹ On the other hand, a [¹H]⁻¹ (p = 1) fit has previously worked for data where [¹H] was varied by solvent type.¹⁸ A p = 1 dependence is also expected from analogy with electron spin decoherence driven by electrons (experimentally, ⁴⁰ and from simulations⁴¹) or ¹³C simulations.^42,43

A simple analytical model for the coherence decay V (2τ ) can be derived under the assumptions that the electron–proton radial distribution function (rdf) g(R) and the proton–proton radial distribution function g(r) are concentration-independent (see SI). By comparison with Eq. (1), the model predicts p ≈ 2/ξ. Taking the simple model to the low-concentration limit, where both radial distribution functions can be further approximated as scale invariant (g(R) = g(r) = 1), the model predicts p = 1.

Both the p ≈ 2/ξ and the p = 1 predictions are inconsistent with the observations, as shown in Figure 4. To further explore the connection between our observations and the model predictions, Figure 4 extends the deuteron-free simulations to proton concentrations lower than achievable experimentally. (Even if such low proton concentrations were feasible, the deuterons would dominate the coherence time in this regime.) As the proton concentration decreases, p approaches 1 (dashed lines), as predicted by the simple model. The figure also shows simulations with the assumption that all hydron sites are individually exchangeable. This is helpful for illustrating the importance of glycerol protons exchanging in groups of five. This yields the p ≈ 1 low-concentration limit over a larger concentration range. Figure 4 also compares the ξ values from fits with Eq. (1) (solid lines), with the model prediction of 2/p (dashed lines). There is a clear discrepancy, except for the all-exchangeable series. This indicates that the assumptions underlying the analytical model, that is, the concentration independence of g(r), are not adequate for these samples.

Figure 4:

Plot of p (left axis, dashed lines) and ξ (right axis, solid lines) vs. [¹H] shows three cases, the ξ fitted from experiment (red points) and p = 0.65 (red dashed), p and ξ for the simulations series without deuterons (green), and p and ξ of another simulation (again without deuterons), where all hydrons are treated as exchangeable (blue). The simple model prediction ξ ≈ 2/p is used to relate the p and the ξ axis. Note that only the fully exchangeable simulations match the simple model’s prediction.

The above analysis suggests that the ¹H–¹H rdf g(r) is important in determining the decoherence dynamics. To experimentally investigate the effects of g(r) on T_M, we selectively varied [¹H] in either the pool of exchangeable hydrons (OH of water and glycerol; using d₅-glycerol/H₂O with d₈-glycerol/D₂O) or the pool of non-exchangeable hydrons (CH₅ in glycerol; using d₃-glycerol/D₂O with d₈-glycerol/D₂O), keeping the other pool maximally (99 %) deuterated. Hydrons disperse evenly across all exchangeable OH sites with a rate constant of 10⁹ to 10¹⁰ M⁻¹s⁻¹,^44,45 whereas hydrons within the CH₅ pool do not exchange, so that the CH₅ clusters with five protons in d₃-glycerol are preserved. The proton concentrations was kept below 10 M.

Figure 5A shows the T_M values extracted from experimental echo decays of these selectively isotope-labeled samples. (The echo decays are shown in the SI.) A clear dependence on pool type is observable. For the same total proton concentration, the system with CH₅ proton clusters has a shorter coherence time than the system with OH protons, and the system with a mixture of both CH₅ proton clusters and OH protons falls in between. The simulations, also shown, correctly reproduce the observations. When the simulations are modified to only include proton pairs that are separated by more than 4 Å, decoherence slows down substantially and the differences between the three series vanish. This suggests that close proton pairs are responsible for decoherence and for the observed difference in decoherence between the different samples. This is consistent with You et al.’s studies on malonic acid crystals: their simulations identify geminal protons as the largest single contributor to central spin decoherence. ³⁵

Figure 5:

A: Coherence time versus proton concentration, both experimental (circles, diamonds, and squares) and simulated (dots), for all-H (red circles), CH₅ (yellow squares) and OH protiation (blue diamonds) are shown. The upper part of the plot shows simulations where only proton pairs with more than 4 Å separation are included. B: The ¹H–¹H radial distribution functions, g(r), from the MD-simulated structure of solvated TEMPO, for the C¹H₅ and O¹H samples at [¹H] = 4 M (96.5 % deuteration), plotted against ¹H–¹H distance r. The peaks are labeled by the proton pair type that is primarily responsible at that separation. C: Simulated decoherence time are shown versus maximum proton pair separation distance, r_max, for the same samples as in B. Since the coherence decays are not stretched exponentials for small r_max (see SI), T_M here is taken as the 1/e decay time.

To investigate this further, Figure 5B shows the proton–proton rdfs for both O¹H and C¹H₅ samples with [¹H] = 4 M (96.5 % deuteration), as determined numerically from MD snapshots. (Note that the rdf for a sample with uniform 4 M deuteration differs from these two.) Both rdfs show multiple peaks at distances characteristic for various intramolecular proton pairs (OH₂ at ≈1.5 Å, CH₂ at ≈ 1.8 Å, etc.). The peak for inter-molecular OH–OH overlaps with the HCCH peak. The rdf of the O¹H sample shows peaks at distances corresponding to CH₂ and HCCH. This is due to the 1 % protons in the CH pool. Except at the OH₂ distance, the rdf for the C¹H₅ sample is of larger magnitude than the one for the O¹H sample. The reason for this is that within a CH₅ proton cluster in d₃-glycerol, the relative probability of a second proton near a particular proton is high even at low bulk proton concentrations. In other words, the local CH proton concentration around a CH proton is more independent of the bulk concentration than that of the local OH concentration around a OH proton. Therefore, the C¹H₅ sample has a higher concentration of close proton pairs.

These close proton pairs are the main drivers of decoherence and the reason that decoherence is faster in the CH₅-protiated matrix than in the OH-protiated matrix. This is evident from Figure 5C, which shows T_Ms obtained from a series of simulations for both O¹H and C¹H₅ matrices again at [¹H] = 4 M (96.5 % deuterated). For T_M at a particular r_max, only proton pairs with a separation smaller than r_max were included. The calculated T_M of the C¹H₅ substitution sample shows a large drop at r_max corresponding to CH₂, demonstrating that these geminal proton pairs are the dominant decoherence drivers. In contrast, for the O¹H sample, protons pairs over a larger range of separations contribute to T_M, in part due to the fact that isotope scrambling reduces the number of geminal O¹H pairs. Therefore, in the O¹H sample, the contribution from the geminal OH₂ pairs is small, whereas the contributions from the CH₂ and HCCH pairs are comparable, but cumulatively smaller than in the CH₅ sample. This indicates that even in the O¹H sample, the 1 % non-exchangeable clustered protons are the dominant decoherence drivers.

The balance of magnetic dipolar couplings provides an explanation of how close proton pairs such as CH₂ drive decoherence. The analytical model of the Hahn echo modulation in an electron–proton–proton system²⁵ shows that the echo modulation depth from a proton pair (m, n) is maximal if the magnetic dipolar coupling strength among the protons $\left|b_{m,n}\right|$ matches the difference in electron–proton hyperfine couplings $\left|A_m-A_n\right|$, while the modulation frequency is proportional to their Pythagorean sum. The most significant proton pairs can be expected to be the ones with large enough $\left|b_{m,n}\right|\approx \left|A_m-A_n\right|$ for the modulations from similar clusters with slightly different modulation frequencies to become phase shifted with respect to each other within the timescale of the Hahn echo, and that occur in sufficient numbers. Clustering the protons in groups of five on the glycerols better achieves these conditions than uniformly scattering the protons throughout the sample.

In conclusion, the influence of matrix hydrogen isotope composition upon the electron spin decoherence dynamics of an organic radical in a frozen solid glassy matrix can be accurately predicted computationally and reveals several effects. Protons are the main decoherence drivers. Replacing protons with deuterons prolongs the coherence timescale. Large coherence time gains are only achieved at high levels of deuteration, > 90% ([¹H] < 10 M). Even in this regime, the coherence timescale is still primarily limited by the remaining protons rather than the numerically dominant deuterons. Furthermore, for a given concentration of protons, clustered protons are much stronger decoherence drivers than dispersed protons. Clustering leads to coherence times that do not follow an inverse-concentration dependence. Among proton clusters, those with separations < 2.5 Å are mainly responsible for the coherence loss. This means that both in EPR experiments and in molecular and solid-state spin qubit architectures where long coherence times are required, special attention should be given to potential proton inclusions, in particular to geminal protons pairs, such as CH₂. In molecular systems, where protons are mostly unavoidable, declustering protons increases the coherence time, even while leaving the overall proton concentration the same. Similar conclusions should hold for other spin–1/2 nuclei (e.g. ¹³C or ²⁹Si) in matrices where they dominate decoherence, although timescales will be different because of the different gyromagnetic ratios.