Extending and measuring dephasing times of nuclear spins in NV centers of diamond

Dieter Suter; Fedor Jelezko; Jingfu Zhang; Swathi S. Hegde

REVIEW 3 major objections 4 minor 26 references

Extending and measuring dephasing times of nuclear spins in NV centers of diamond

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read By using a SWAP gate to move nuclear-spin coherence onto the electron spin and a Hahn echo to refocus it, the paper measures 13C dephasing times of 8.66 ms and 14.10 ms in NV centers—longer than the electron spin's 5.5 ms $T_1$.

desk verdict Nice SWAP-readout scheme, but the claim that nuclear T2 exceeds the electron T1 is not secure until the mS=+1 subspace is modeled. read the letter →

arxiv 2505.23512 v1 pith:47GL4OE3 submitted 2025-05-29 quant-ph

Jingfu Zhang , Swathi S. Hegde , Fedor Jelezko , Dieter Suter This is my paper

classification quant-ph

keywords NVcentersnuclearspincoherencedephasingtimeHahnechoSWAPgatediamondquantummemory13Celectronrelaxation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper targets a built-in ceiling on nuclear spin coherence in NV centers: because the nuclear spin is initialized and read out through the electron spin, its measured dephasing time has appeared capped by the electron spin's longitudinal relaxation time $T_{1}^{e}$. It proposes two changes—a SWAP gate that transfers nuclear coherence into electron coherence just before readout, and a Hahn echo applied during the storage period—and tests them on a strongly coupled 13C nuclear spin in a single NV center. The measured free-induction dephasing time is $T_2^* = 8.66$ ms and the Hahn-echo time is $T_2 = 14.10$ ms, exceeding the electron $T_{1}^{e}=5.5$ ms by factors of 1.6 and 2.6. If correct, the nuclear spin can act as a quantum memory that outlives the electron spin that controls it.

What carries the argument

The load-bearing piece is a SWAP gate between the electron spin qubit and the 13C nuclear spin qubit, implemented through indirect control without radio-frequency pulses. It transfers the nuclear spin coherence into electron spin coherence, which is then converted to an observable population difference by a $90^\circ$ microwave pulse and read out by laser fluorescence. Around the storage interval, a Hahn echo ($180^\circ$ pulse on the nuclear spin at half the evolution time) partially refocuses the dephasing. The readout phase is swept with time, $\varphi = 2\pi\nu_d\tau + \varphi_0$, so the signal oscillates at a known frequency and its decaying amplitude is fitted with $c_{\mathrm{FID}}(\tau)=e^{-\tau/T_2^*}$ and $c_{\mathrm{Hahn}}(\tau)=e^{-\tau/T_2}$.

What would settle it

Run the Hahn-echo sequence with the electron held in the $m_S=-1$ subspace for the full evolution and with the nuclear spin prepared in a superposition that is not an eigenstate of the effective Hamiltonian in that subspace: if the 14.10 ms single-exponential decay is not reproduced, or an extra oscillation appears, the eigenstate assumption that makes the echo a genuine nuclear coherence measurement fails.

Watch

Extended reading notes

Core claim

The central claim is that the apparent dephasing of a 13C nuclear spin in a single NV center is not intrinsically bounded by the electron spin lifetime. With a two-qubit SWAP operation at the end of the evolution to convert the nuclear coherence into electron coherence, and a Hahn echo on the nuclear spin at mid-evolution, the measured coherence decays exponentially with $T_2^* = 8.66$ ms for free precession and $T_2 = 14.10$ ms with one echo, compared with the electron $T_{1}^{e} = 5.5$ ms. The echo extends the dephasing time by 60% relative to the free-induction decay. The paper explains the echo result by noting that in the electron $m_S=-1$ subspace the nuclear spin is approximately an eigenstate of the effective Hamiltonian (roughly $I_x$) and therefore does not evolve during the echo, while in the $m_S=0$ subspace the echo refocuses the nuclear spin evolution.

Load-bearing premise

The reported times assume the nuclear coherence decays as a single exponential and that, during the echo, the nuclear spin is effectively frozen when the electron is in its $-1$ internal state; if either assumption fails, the 14.10 ms value is not a clean nuclear spin coherence time.

Editorial extensions

If this is right

Nuclear spin coherence can outlast the electron spin longitudinal relaxation in a strongly coupled NV–13C system, so the nuclear spin can serve as a quantum memory that survives the electron spin used to initialize and read it.
A single Hahn echo extends the measured nuclear dephasing time by 60%, and the paper expects that multiple echoes would extend it further toward the minute-scale coherence seen at low temperature.
Because readout no longer requires the electron to remain polarized at the moment of measurement, the apparent nuclear dephasing time is no longer capped by $T_{1}^{e}$ in the same way as in earlier readout schemes.
The method is not tied to NV centers specifically; the paper states it should transfer to other hybrid electron–nuclear systems such as malonic acid radicals and other solid-state defects.

Reading between the lines

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

If the eigenstate approximation in the $m_S=-1$ subspace is exact, the one-echo value of $T_2 = 14.10$ ms may still underestimate the intrinsic nuclear spin coherence, since a single echo leaves residual dephasing from the $m_S=0$ subspace uncorrected.
The SWAP-readout scheme turns electron $T_1$ noise into a slowly varying background rather than a fast coherence killer, so the same protocol could measure nuclear dephasing in NV centers with much shorter electron $T_1$, where earlier readout would fail.
The observed ratio $T_2/T_{1}^{e} \approx 2.6$ is a quantitative statement about how much electron-spin noise a single echo removes; comparing this ratio across centers with different $T_{1}^{e}$ would test whether the residual decay is set by the nuclear spin bath or by incomplete refocusing of electron flips.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a competent experimental letter with a genuinely new measurement scheme—SWAP-based readout plus one Hahn echo on a strongly coupled NV-13C system—but the headline claim that nuclear dephasing times exceed the electron T1 is not as secure as the paper suggests. The FID value (T2*=8.66 ms) is plausible, but the Hahn-echo T2=14.10 ms is vulnerable to a systematic that isn't modeled.

What's new and good: refs [8,9] used electron-assisted readout and saw apparent times capped by T_e1. This group replaces that with a SWAP gate that transfers nuclear coherence to the electron, plus a Hahn echo during storage. That's a sensible extension, implemented with known indirect control gates, and the fits in Fig. 3 look reasonable. The population model uses a separately measured T_e1=5.5 ms, which is the right way to account for T1 in the amplitude.

Now the soft spots. The main one is the treatment of electron spin flips to mS=+1. The paper factors out the population decay [P0+P-1]/2 in the Hahn echo, but that only removes the loss of population from the (0,-1) subspace. It does not account for the phase a 13C spin acquires during a sojourn in mS=+1, where the Hamiltonian H_1/(2π)=-0.310 I_z+0.110 I_x is far from the I_x eigenstate that makes the echo work. The nuclear 180x pulse does not refocus that evolution. With T_e1=5.5 ms, the rate kappa=1/(16.5 ms) is comparable to the quoted T2=14.1 ms, so this contamination could easily explain a large part of the measured decay. The eigenstate approximation in mS=-1 (overlap 0.999) is fine; that's not the problem. But the mS=+1 subspace is. The same history effect also affects the FID, though less severely.

Two smaller issues: no error bars on T2* or T2, and the assumed single-exponential decay c=e^{-τ/T2} is asserted without justification. With no error bars and a systematic that lands in the same range, the numbers in Table I should be treated as upper bounds until the mS=+1 effect is modeled.

Bottom line: good experimental scheme, worth publishing after a serious revision that adds the mS=+1 term to the model, reports uncertainties, and either confirms or lowers the T2 values. For a referee, I'd accept, but the report should push hard on the systematic. I wouldn't cite these numbers as-is in my own work.

Referee Report

3 major / 4 minor

Summary. The manuscript reports measurements of the dephasing time of a 13C nuclear spin strongly coupled to a single NV center in diamond. The authors introduce a SWAP-based readout that transfers nuclear-spin coherence to the electron spin, and combine it with a Hahn-echo refocusing sequence during the storage period. From fits to the measured oscillation amplitude they obtain T2* = 8.66 ms for free precession and T2 = 14.10 ms with one echo, both exceeding the electron spin longitudinal relaxation time Te1 = 5.5 ms. The analysis models electron T1 relaxation as a three-level population dynamics and assumes an exponential decay of the nuclear-spin coherence.

Significance. If the reported values are correct, the work provides a useful demonstration that a strongly coupled 13C nuclear spin can retain coherence for times exceeding the electron spin T1, and the SWAP-based detection scheme is a sensible extension of earlier readout methods. The protocol uses hyperfine parameters obtained in prior work by the same group, and the experiment addresses a regime of strong coupling without ancillary qubits. However, the central quantitative claim depends on the validity of a single-exponential model and on the treatment of electron T1 jumps into the mS=1 subspace, for which the manuscript currently gives only a qualitative argument and no error bars.

major comments (3)

[Experimental protocol and results, Eqs. (12)-(13), Table I] The Hahn-echo T2 interpretation does not account for T1 jumps into the mS=1 subspace. During such a jump the nuclear spin evolves under H_1/(2π) = -0.310 MHz Iz + 0.110 MHz Ix, which is not refocused by the 180x pulse because the Ix term does not change sign under conjugation by that pulse. The associated coherence-loss timescale is of order 1/κ ≈ 16.5 ms, close to the reported T2 = 14.10 ms. With a single-exponential fit and no error bars, the extracted T2 cannot be distinguished from this relaxation-induced floor. The paper itself acknowledges the mS=1 issue in the Conclusion, but the central claim that the measured T2 represents nuclear-spin coherence beyond Te1 requires a quantitative model of mS=1 excursions or an experiment that suppresses them.
[Table I and Fig. 3] The reported values T2* = 8.66 ms and T2 = 14.10 ms are obtained from fits of c_FID(τ) = c0 exp(-τ/T2*) and c_Hahn(τ) = c0 exp(-τ/T2), but the manuscript reports no uncertainties, goodness-of-fit measures, or residuals for these fits. Because the headline claim is that these times exceed Te1 by factors of 1.6 and 2.6, the absence of error bars is load-bearing. Please report standard errors on all fitted parameters and, if possible, confidence intervals for the ratios T2*/Te1 and T2/Te1.
[Conclusion, H_-1 eigenstate approximation] The argument that T1 jumps from |0> to |-1> are harmless relies on the statement that the 13C state |+> is approximately an eigenstate of the effective Hamiltonian in mS = -1. With the quoted parameters, H_-1/(2π) = -0.006 MHz Iz - 0.110 MHz Ix, so |+> has an admixture of the excited eigenstate with amplitude of order 0.03. The manuscript does not quantify the effect of this admixture on the echo signal over the 14 ms timescale. Although the effect appears small, it should be estimated numerically or bounded experimentally to justify the interpretation of Eq. (12).

minor comments (4)

[Introduction, last paragraph] The word 'introudced' should be 'introduced', and the caption of Table I contains a misplaced comma in 'Table, I'.
[Conclusion] The sentence 'where indirect control in this subspace' is incomplete and should be rephrased to describe how the echo is applied in the |0>,|-1> subspace.
[Experimental protocol and results, Eqs. (10)-(12)] The detuning frequencies νd^a and νd^b are introduced with different sign conventions; please clarify why using νd^b = -0.342 MHz in Eq. (12) yields the same oscillation frequency as the FID signal.
[Eqs. (5)-(7)] The population model defines Te1 = 1/(3κ) but does not specify the individual transition rates between the electron spin levels; given the relevance of mS=1 jumps to the Hahn-echo interpretation, this should be stated explicitly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the dephasing times are experimental fit outputs, and the model inputs (T_e1 and hyperfine couplings) come from independent measurements.

full rationale

The paper's central quantities, T2* = 8.66 ms and T2 = 14.10 ms, are obtained by fitting measured oscillation amplitudes to explicitly assumed exponential decay functions, c_FID(τ)=e^{-τ/T2*} and c_Hahn(τ)=e^{-τ/T2} (Eqs. 11 and 13). These fitted times are the output of the experiment, not inputs used to derive a prediction. The amplitude correction uses the electron-spin populations P0(τ) and P−1(τ), which depend on the separately measured T_e1 = 5.5 ms and the initial polarization s1 = 0.80; these are independent inputs, not quantities derived from the target dephasing times. The hyperfine parameters νC, Azz, and Azx are cited from the authors' prior measurement [14], but that is an external empirical determination used as input, not a self-referential derivation of the reported coherence times. The exponential-decay model and the mS = −1 eigenstate approximation stated in the Conclusion are modeling assumptions; if they fail, the extracted values could be misinterpreted, but that is a correctness risk, not circularity. No equation reduces by construction to the target result, and no uniqueness theorem or ansatz is smuggled in via self-citation. The result is a self-contained experimental measurement with standard fitting, so it receives a low circularity score.

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

The central claim depends on a small number of stated assumptions: exponential coherence decay, a rate-equation population model, and the eigenstate approximation for the mS=-1 subspace. No invented entities are introduced. The fitted amplitudes and decay times are the experimental outputs.

free parameters (5)

c0,FID = 0.80
Initial amplitude of nuclear coherence in FID fit, Eq. (11).
c0,Hahn = 0.76
Initial amplitude in Hahn-echo fit, Eq. (13).
T2* (FID) = 8.66 ms
Fitted nuclear dephasing time in FID.
T2 (Hahn echo) = 14.10 ms
Fitted nuclear dephasing time in Hahn echo.
d0 = 0.086
Offset in background fit for FID/Hahn signals.

assumptions (4)

domain assumption Coherence decays exponentially: c_FID(τ)=e^{-τ/T2*} and c_Hahn(τ)=e^{-τ/T2}
Stated in main text: 'In the following, we assume an exponential decay of the coherence'. Used to fit T2 values.
domain assumption Nuclear spin in the electron mS=-1 subspace is approximated as an eigenstate of the effective Hamiltonian (~Ix) and does not evolve during the echo
Conclusion and outlook: needed to reason that electron flips between |0> and |-1> do not dephase the nuclear spin in the echo sequence.
domain assumption Electron spin populations follow the rate-equation model in Eqs. (5)-(7) with a single relaxation rate κ=1/(3T1)
Used to separate population decay from coherence decay; details in SM Section I.
domain assumption The 14N nuclear spin remains in mN=1 throughout the experiment
Stated in System section as safe because T_N1 >> T_e1.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Extending and measuring dephasing times of nuclear spins in NV centers of diamond." pith.science (2026). https://pith.science/paper/47GL4OE3

@misc{pith2026250523512,
  author       = {Pith},
  title        = {Pith review of: Extending and measuring dephasing times of nuclear spins in NV centers of diamond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/47GL4OE3}},
  note         = {Machine review of arXiv:2505.23512}
}

abstract

Long coherence times rank among the most important performance measures for many different types of quantum technology. In NV centers of diamond, the nuclear spins provide particularly long dephasing times. However, since initialization and readout require assistance from the electron spin, the apparent dephasing times can be reduced by the electron spin lifetime. Here we propose and implement schemes for measuring and extending the dephasing times of nuclear spins, resulting in dephasing times that are longer than the longitudinal relaxation time $T_{1}$ of the electron spin.

Figures

Figures reproduced from arXiv: 2505.23512 by the authors.

**Figure 3.** FIG. 3: Experimental results for the background [ [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 2.** FIG. 2: Measured signals with the fitted curves for the FID [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

26 extracted references · 23 canonical work pages

[1]

C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017)

2017
[2]

The state of the electron qubit can be detected by laser pulses, as shown in Fig. 1 [5]. Applying the Hadamard gate H to the 13C spin generates the superposition state |+⟩ = ( | ↑⟩ + | ↓⟩)/ √ 2 from | ↑⟩. Therefore ρe,C ini shown in Eq. (2) is transformed to ρe,C sup = s1|0⟩⟨0| ⊗ |+⟩⟨+| + (1 − s1)|1 ↓⟩⟨1 ↓ |, (3) where the state of the 13C spin in the sub...

work page 2020
[3]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)

2000
[4]

Stolze and D

J. Stolze and D. Suter, Quantum Computing: A Short Course from Theory to Experiment (Wiley-VCH, Berlin, 2nd edition, 2008)

work page 2008
[5]

T. D. Ladd, F. Jelezko, R. Laﬂamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Nature 464, 45 (2010)

work page 2010
[6]

Suter and F

D. Suter and F. Jelezko, Progress in Nuclear Magnetic Resonance Spectroscopy 98-99, 50 (2017)

work page 2017
[7]

Kurizki, P

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Pet- rosyan, P. Rabl, and J. Schmiedmayer, Proceedings of the National Academy of Sciences 112, 3866 (2015)

work page 2015
[8]

C. E. Bradley, J. Randall, M. H. Abobeih, R. C. Berrevoets, M. J. Degen, M. A. Bakker, M. Markham, D. J. Twitchen, and T. H. Taminiau, Phys. Rev. X 9, 031045 (2019)

2019

Show all 26 references

[9]

van der Sar, Z

T. van der Sar, Z. H.Wang, M. S. Blok, H. Bernien, T. H. Taminiau, D. M. Toyli, D. A. Lidar, D. D. Awschalom, R. Hanson, and V. V. Dobrovitski, Nature 484, 82 (2012)

2012
[10]

J. H. Shim, I. Niemeyer, J. Zhang, and D. Suter, Phys. Rev. A 87, 012301 (2013)

2013
[11]

N. Kalb, P. C. Humphreys, J. J. Slim, and R. Hanson, Phys. Rev. A 97, 062330 (2018)

2018
[12]

P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, et al., Science 336, 1283 (2012)

2012
[13]

Pfender, N

M. Pfender, N. Aslam, H. Sumiya, S. Onoda, P. Neu- mann, J. Isoya, C. A. Meriles, and J. Wrachtrup, Nature Communications 8, 834 (2017)

2017
[14]

H. P. Bartling, M. H. Abobeih, B. Pingault, M. J. Degen, S. J. H. Loenen, C. E. Bradley, J. Randall, M. Markham, D. J. Twitchen, and T. H. Taminiau, Phys. Rev. X 12, 011048 (2022),

2022
[15]

Zhang, S

J. Zhang, S. S. Hegde, and D. Suter, Phys. Rev. Applied 12, 064047 (2019)

2019
[16]

S. S. Hegde, J. Zhang, and D. Suter, Phys. Rev. Lett. 124, 220501 (2020)

2020
[17]

Soshenko, I

V. Soshenko, I. Cojocaru, S. Bolshedvorskii, O. Rubina s, A. Smolyaninov, V. Vorobyov, V. Sorokin, and A. Aki- mov, Quantum Electronics 51, 1144 (2021)

2021
[18]

Khaneja, Phys

N. Khaneja, Phys. Rev. A 76, 032326 (2007)

2007
[19]

J. S. Hodges, J. C. Yang, C. Ramanathan, and D. G. Cory, Phys. Rev. A 78, 010303 (2008),

2008
[20]

Zhang, S

J. Zhang, S. S. Hegde, and D. Suter, Phys. Rev. Lett. 130, 090801 (2023)

2023
[21]

Suter and G

D. Suter and G. A. ´Alvarez, Rev. Mod. Phys. 88, 041001 (2016)

2016
[22]

A. M. Souza, G. A. ´Alvarez, and D. Suter, Phil. Trans. R. Soc. A 370, 4748 (2012)

2012
[23]

Unden, P

T. Unden, P. Balasubramanian, D. Louzon, Y. Vinkler, M. B. Plenio, M. Markham, D. Twitchen, A. Stacey, I. Lovchinsky, A. O. Sushkov, et al., Phys. Rev. Lett. 116, 230502 (2016)

2016
[24]

Zaiser, T

S. Zaiser, T. Rendler, I. Jakobi, T. Wolf, S.-Y. Lee, S. Wagner, V. Bergholm, T. Schulte-Herbrueggen, P. Neu- mann, and J. Wrachtrup, Nature Communications 7, 12279 (2016). 5

2016
[25]

Zhang, C

Y. Zhang, C. A. Ryan, R. Laﬂamme, and J. Baugh, Phys. Rev. Lett. 107, 170503 (2011)

2011
[26]

Senkalla, G

K. Senkalla, G. Genov, M. H. Metsch, P. Siyushev, and F. Jelezko, Phys. Rev. Lett. 132, 026901 (2024)

2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017)

2017

[2] [2]

The state of the electron qubit can be detected by laser pulses, as shown in Fig. 1 [5]. Applying the Hadamard gate H to the 13C spin generates the superposition state |+⟩ = ( | ↑⟩ + | ↓⟩)/ √ 2 from | ↑⟩. Therefore ρe,C ini shown in Eq. (2) is transformed to ρe,C sup = s1|0⟩⟨0| ⊗ |+⟩⟨+| + (1 − s1)|1 ↓⟩⟨1 ↓ |, (3) where the state of the 13C spin in the sub...

work page 2020

[3] [3]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)

2000

[4] [4]

Stolze and D

J. Stolze and D. Suter, Quantum Computing: A Short Course from Theory to Experiment (Wiley-VCH, Berlin, 2nd edition, 2008)

work page 2008

[5] [5]

T. D. Ladd, F. Jelezko, R. Laﬂamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Nature 464, 45 (2010)

work page 2010

[6] [6]

Suter and F

D. Suter and F. Jelezko, Progress in Nuclear Magnetic Resonance Spectroscopy 98-99, 50 (2017)

work page 2017

[7] [7]

Kurizki, P

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Pet- rosyan, P. Rabl, and J. Schmiedmayer, Proceedings of the National Academy of Sciences 112, 3866 (2015)

work page 2015

[8] [8]

C. E. Bradley, J. Randall, M. H. Abobeih, R. C. Berrevoets, M. J. Degen, M. A. Bakker, M. Markham, D. J. Twitchen, and T. H. Taminiau, Phys. Rev. X 9, 031045 (2019)

2019