Author Correction: Decoherence of nitrogen-vacancy spin ensembles in a nitrogen electron-nuclear spin bath in diamond

Correction to: npj Quantum Information https://doi.org/10.1038/s41534-022-00605-4, published online 12 August 2022

The authors became aware that a previous version of the manuscript slightly different from the correct version was incorrectly used for publication. As a result of this error, the following changes have been made to the original version of this Article.

The second sentence in the Abstract originally read “While such applications require the long spin lifetimes of the NV centers…”. In the corrected version, “spin lifetimes” is replaced by “spin coherence lifetimes”.

In the second and last sentences of the first paragraph under “Introduction”, “spin lifetimes” is replaced by “spin coherence lifetimes”.

The second last sentence of the second paragraph under “Introduction” originally read “It has been shown that the NV-spin lifetime is much more…”. In the corrected version, “NV-spin lifetime” is replaced by “NV spin coherence”.

The third sentence of the fourth paragraph under “Introduction” originally read “showing a linear dependence of T₂ on [P1] on a log scale with a stretched exponential parameter of ~1.37”. In the corrected version, it reads “showing a linear dependence of T₂ on [P1] on a log scale with a stretched exponential parameter (n) of ~1.37, which describes the exponential decay of the NV coherence as $\exp \left( { - \left( {\frac{{{{\mathrm{t}}}}}{{{{{\mathrm{T}}}}_2}}} \right)^{{{\mathrm{n}}}}} \right)$.”.

The last sentence of the last paragraph under “Introduction” originally read “which can provide a key reference for developing diamond samples with optimized NV-spin lifetimes for different applications”. In the corrected version, “NV-spin lifetime” is replaced by “NV spin coherence”.

In “System and model” under “Results”, the second sentence of the second paragraph originally stated “An external magnetic field of 500 G is applied in the same direction as the symmetry axis of the NV spin.” In the corrected version, this sentence is removed.

In “System and model” under “Results”, the fourth sentence of the second paragraph originally stated “…and we consider various [P1] from 1 to 100 ppm”. In the corrected version, this sentence is expanded and several references are added, and it reads: “…and we consider various [P1] from 1 ppm to 100 ppm, which are commonly found in diamond samples used in magnetometry and quantum simulation applications ^{29,34,36,41-44}.”

Before the second last sentence “The P1 center accompanies…” of the second paragraph in “System and model” under “Results”, a new sentence is added, which reads “To compute the maximum T₂ time of the NV ensembles imposed by the P1 bath, we applied an external magnetic field of 500 G in the same direction as the symmetry axis of the NV spin, over which the NV decoherence is mainly driven by flip-flop transitions between P1 spins. We note that the cross-relaxation effect between the NV and P1 spins, which occurs near B = 514 G^45,46 was not included.”

In “System and model” under “Results”, the third sentence of the third paragraph originally stated “We compared our DFT calculations with experimental results…”. In the corrected version, “DFT calculations” is replaced by “density functional theory (DFT) calculations”.

After the last paragraph ending with “… can be found in the Methods section and Supplementary Notes 3 and 4.” in “System and model” under “Results”, new sentences are added, which read “It is worth noting that our NV decoherence model do not include the spin-lattice relaxation (T₁) effects of the P1 center^27,56 and the NV center^27,45, which may be a valid assumption to model the P1-driven NV decoherence at room temperature. At room temperature and below, the T₁ time of the P1 center is longer than ms, meaning that thermally driven random flips of the P1 spins are unlikely in the time scale of the NV spin echo decoherence. We note, however, that the spin-lattice relaxation could significantly contribute to the NV decoherence if temperature is much higher than the room temperature as the T₁ time of the P1 center significantly decreases as the temperature increases^27,45,56.”

The second sentence of the third paragraph in “The coherence time of the NV center as a function of [P1]” under “Results” originally read “…in a P1 bath is examined and results indicate that…”. In the corrected version, it reads “…in a P1 bath is examined³⁰ (see Supplementary Note 5). Interestingly, our results indicate that…”.

At the conjunction of second and third sentences of the third paragraph in “The coherence time of the NV center as a function of [P1]” under “Results”, it originally read “…when the concentration of the extra spins ([Ex]) is similar to [P1]. Figure 3a, b show the…”. In the corrected version, it reads “…when the concentration of the extra spins ([Ex]) is similar to [P1], which indicates a close relationship between [Ex] and [P1]. Fig. 3(a) and (b) show the…”.

In Figure 3 captions, a new sentence is appended at the end after “…as a function of [Ex] for [P1] = 5 ppm”, which reads “In (c, e), the black dotted line shows an experimental T₂ time of 65 μs and 30 μs for [P1] = 1.5 ppm and 6 ppm, respectively, taken from the reference [29].”

The second sentence of the first paragraph under “Discussion” originally read “Our results show a clear linear dependence of the T₂ time on [P1] on a log scale with a slope of -1.07”. In the corrected version, it reads “Our results show a clear linear dependence of the T₂ time on [P1] on a log scale with a slope of -1.06, which is consistent with a slope of -1.07 used to describe T₂ vs [P1] in the log scale in the previous study²⁹”.

After the first paragraph ending with “…suppressing the electron spin flip-flop transitions” under “Discussion”, a new paragraph is added, which reads “It is worth commenting that the relation between T₂ and [N_s⁰] derived in this study could be applied to estimate the P1 concentration and the N to NV^- conversion ratio, which are crucial information in NV-based magnetometry and quantum simulation applications since varying $\left[ {{{{\mathrm{P}}}}1} \right]$ can vastly change the ensemble spin dynamics. Our result shows that ${{{\mathrm{T}}}}_2 = 416.65 \times \left[ {{{{\mathrm{P}}}}1} \right]^{ - 1.06}\mu {{{\mathrm{s}}}}$ for $\left[ {{{{\mathrm{P}}}}1} \right]$ > 1 ppm, assuming the P1 spins to be the dominant spin bath. By relating measured ${{{\mathrm{T}}}}_2$ to the CCE derived relation, one can extract the lower-bound for $\left[ {{{{\mathrm{P}}}}1} \right]$ without any destructive measurements. In addition, the sensitivity of echo-based AC magnetometry can be expressed as⁷ $\eta \propto \frac{1}{{\sqrt {{{{\mathrm{NT}}}}_2} }} \cdot \sqrt {\frac{{{{{\mathrm{T}}}}_2 + \tau _{{{\mathrm{I}}}} + \tau _{{{\mathrm{R}}}}}}{{{{{\mathrm{T}}}}_2}}}$, where ${{{\mathrm{N}}}}$ is the total number of sensor spins, ${{{\mathrm{T}}}}_2$ is the Hahn echo spin coherence time, $\tau _{{{\mathrm{I}}}}$ is the initialization time, and $\tau _{{{\mathrm{R}}}}$ is the readout time. Typical values for $\tau _{{{\mathrm{I}}}} + \tau _{{{\mathrm{R}}}} \sim 5\,\mu s$ for NV ensemble sensor. Since ${{{\mathrm{N}}}} \propto \left[ {{{{\mathrm{NV}}}}^ - } \right] = {{{\mathrm{c}}}} \cdot \left[ {{{{\mathrm{P}}}}1} \right]$, where ${{{\mathrm{c}}}}$ is the $[{{{\mathrm{P}}}}1]$ to $[{{{\mathrm{NV}}}}^ - ]$ conversion ratio, the sensitivity as a function of $\left[ {{{{\mathrm{P}}}}1} \right]$ can be simplified as $\eta \propto \sqrt {\left[ {{{{\mathrm{P}}}}1} \right]/{{{\mathrm{c}}}}}$. By measuring NV ensemble sensor’s echo AC magnetometer noise floor and relating that to estimated $\left[ {{{{\mathrm{P}}}}1} \right]$, one can calculate the conversion ratio ${{{\mathrm{c}}}}$”.

The second paragraph originally starting with “Additionally, the computed T2 times are consistently larger than…” under “Discussion” is rewritten and it reads “The fact that the computed T₂ times are consistently larger than previous experimental results indicates the importance of other decoherence sources in describing the NV spin decoherence in a P1 spin bath. In our study, we considered extra electron spins as an additional source of decoherence inspired by recent studies³⁰. Notably, we found that the calculated T₂ time becomes similar to the experimental T₂ time when the concentration of the extra spins is similar to the P1 concentration, which may indicate a close relationship between the concentration of extra parasitic spins and the P1 concentration in diamond.

“It should be noted, however, that other paramagnetic defects or complexes could be considered as possible extra spins to better understand the source of the NV decoherence⁵⁸. During the CVD growth or ion-implantation to create NV centers in diamond, many vacancies or defect complexes could be easily formed, acting as parasitic spins in the diamond lattice. Such examples may include NV⁰ (S=1/2), NVH^- (S=1/2), VH⁰ (S=1/2), VH^- (S=1), N₂⁺ (S=1/2), V^- (S=3/2), V⁺ (S=1/2), VV⁰ (S=1), to name a few, which are paramagnetic defects commonly found in diamond⁷. However, their contribution to the NV decoherence is largely unknown. We remark that the methods developed in this work could be applied to study these defects, which we planned as a separate future study. Further systematic investigation on those paramagnetic spins will help to better understand NV spin decoherence in practical applications.”

A new sentence is appended at the end of the Acknowledgments section, which reads “This work was supported by the National Supercomputing Center with supercomputing resources including technical support (KSC-2019-CRE-0156)”.

Finally, new references [41-46], [56], [58] are inserted in the References section associated to the above changes in the main text. They are:

41. Cujia, K. S., Herb, K., Zopes, J., Abendroth, J. M., & Degen, C. L. Parallel detection and spatial mapping of large nuclear spin clusters. Nat. Commun. 13, 1260 (2022).

42. Pezzagna, S., Naydenov, B., Jelezko, F., Wrachtrup, J., & Meijer, J. Creation efficiency of nitrogen-vacancy centres in diamond. New J. Phys. 12, 065017 (2010).

43. Ziem, F., Garsi, M., Fedder, H., & Wrachtrup, J. Quantitative nanoscale MRI with a wide field of view. Sci. Rep. 9, 12166 (2019).

44. Ku, M. J. H. et al. Imaging viscous flow of the Dirac fluid in graphene. Nature 583, 537-541 (2020).

45. Jarmola, A., Acosta, V. M., Jensen, K., Chemerisov, S., & Budker, D. Temperature- and Magnetic-Field-Dependent Longitudinal Spin Relaxation in Nitrogen-Vacancy Ensembles in Diamond. Phys. Rev. Lett. 108, 197601 (2012).

46. Wang, H.-J. et al. Optically detected cross-relaxation spectroscopy of electron spins in diamond. Nat. Commun. 5, 4135 (2014).

56. Reynhardt, E. C., High, G. L., & Wyk, J. A. v. Temperature dependence of spin-spin and spin-lattice relaxation times of paramagnetic nitrogen defects in diamond. J. Chem. Phys. 109, 8471-8477 (1998).

58. Pellet-Mary, C., Huillery, P., Perdriat, M., Tallaire, A., & Hétet, G. Optical detection of paramagnetic defects in diamond grown by chemical vapor deposition. Phys. Rev. B 103, L100411 (2021).

As a consequence of these inserted references, current citations in the text to references [47–55], [57], [59–63] were previously incorrectly numbered [41–49], [50], [51–55] respectively.

This has now been corrected in both the PDF and HTML versions of the Article.