Four-order power reduction in nanoscale electron-nuclear double resonance with a nitrogen-vacancy center in diamond
Zhiyi Hu Fengjian Jiang Jingyan He Yulin Dai Ya Wang Nanyang Xu Jiangfeng Du





§ ABSTRACT 
Detecting nuclear spins using single Nitrogen-Vacancy (NV) centers is of particular importance in nano-scale science and engineering, but often suffers from the heating effect of microwave fields for spin manipulation, especially under high magnetic fields. Here, we realize an energy-efficient nano-scale nuclear-spin detection using a phase-modulation electron-nuclear double resonance scheme. The microwave field can be reduced to 1/250 of previous requirements and the corresponding power is over four orders lower. Meanwhile, the microwave-induced broadening to the line-width of the spectroscopy is significantly canceled and we achieve a nuclear-spin spectrum with a resolution down to 2.1 kHz under a magnetic field at 1840 Gs. The spectral resolution can be further improved by upgrading the experimental control precision. This scheme can also be used in sensing microwave fields and extended to a wide range of applications in the future.





Nuclear magnetic resonance (NMR) spectroscopy is one of the most important analytical technique that is widely used in physics, chemistry, biology and medical sciences <cit.>. It resolves the chemical specificity of elements from the resonance frequency of nuclear spins in order to obtain molecular structure or image of materials non-destructively. Conventional NMR, which often suffers from low detection efficiency, relies primarily on a large amount of sample molecules for signal accumulation <cit.>. Many efforts have been provided to improve the sensitivity for example the hyper-polarization approaches <cit.>, the magnetic resonance force microscopy<cit.> and optically-detected NMR <cit.>. Recently, the Nitrogen-Vacancy (NV) center in diamond has shown its outstanding capability as a nano-scale sensor for nuclear-spin detection that associates with a significantly-improved sensitivity even at the single-molecule level <cit.>.



In the field of NMR, the resonance frequency of nuclear spins is determined by the Larmor frequency and its local environment, i.e., the neighboring nuclear spins and surrounding electron distribution, which generates an additional field and changes the resonance frequency. The relative difference between the resonance frequency and the Larmor frequency also called the chemical shift, reveals fruitful knowledge of the host molecule <cit.>. Therefore, a high spectral resolution is necessary for the NMR spectroscopy applications. Generally, the resolution is improved by increasing the external magnetic field which can enlarge the chemical shift. Besides, it also brings crucial benefits to the decoherence time and polarization of the sample spins under a high magnetic field.


NV-based NMR often utilizes the standard pulsed dynamical decoupling (pulsed-DD) sequences <cit.>, such as XY-N <cit.> or Carr-Purcell-Meiboom-Gill-N <cit.>, as band-pass noise filters to detect the nuclear spins by matching their resonance frequency, as well as their extensions with a quantum(classical) memory to realize a high(arbitrary)-resolution signal detection<cit.>. But the pulsed-DD tends to work under low magnetic fields because the maximal band-pass frequency (limited by square root of the pulse power) has practical upper limitation and cannot increase as the magnetic field increases. An alternative scheme under high magnetic fields is the continuous-wave dynamical decoupling (CWDD) sequence <cit.>. It also requires the continuous microwave driving to match the resonance frequency that is linearly-scaled with the magnetic field. However, the microwave power has practical limitations and will induce various issues including power broadening of line-width and sample heating<cit.>. A nanoscale NMR spectroscopy that can largely reduce the power requirements is highly benefited for high-field NMR and the related applications including bio-sensing applications <cit.>.  

Recently, the phase-modulation (PM) technique is introduced into NV-based quantum sensing, which works as a mixer to down-convert high-frequency microwave fields to a proper range <cit.>. Also, a theoretical method combining the PM and CWDD schemes has been proposed to reduce the power requirement for nuclear-spin detection under high magnetic fields <cit.>. In this work, we experimentally examine the PM-based CWDD scheme and realize the new scheme to detect in vivo nuclear spin under a field up to 3015 Gs. The microwave power is about four orders of magnitude lower than the standard DD schemes. We show that the power-induced broadening of the spectrum is also suppressed, where four weakly-coupled ^13C spins are distinguished from the spin bath with a minimal line-width of 2.1 kHz under 1840 Gs. Besides, we also contribute to the theoretical scheme by extending the resonant condition to both sidebands of the mixer, enabling its future applications in zero-field situations. This scheme can be easily extended to sensing of in vitro nuclear spins or classical magnetic fields in general.



The  NV center in diamond has emerged as an excellent solid-state platform for quantum information processing <cit.>. The center consists of a nitrogen atom with a neighboring vacancy and its negatively charged state NV^- forms a spin triplet in the orbital ground state. The electron spin with nearby nuclear spins  form a hybrid system that has been used as quantum registers <cit.> for computation <cit.> and simulation <cit.> tasks, or interacting nodes of quantum network <cit.>. NV center is also used as a nano-scale sensor to detect magnetic field <cit.>, electric field <cit.>, temperature <cit.> or other spins <cit.>. Under this situation, the Hamiltonian of the system is often formulated as

    H_0/2π =DS_z^2 -γ_e B_z S_z - ∑_jγ^(j)_nB_zI^(j)_z+∑_jS_z𝐀^(j)·𝐈^(j)

where D≈ 2870MHz is the zero-field splitting, S_i (I_i^(j)) the electron (j-th nuclear) spin operator, B_z the static magnetic field applied along the NV axis, γ_e (γ_n^(j)) the gyro-magnetic ratio of electron (j-th nuclear) spin and 𝐀^(j)=(A^(j)_z x, A^(j)_z y, A^(j)_z z) the interaction vector between electron and nuclear spin. In our experiment, only the subspace of |m_s=0⟩ and |m_s= 1⟩ of electron spin is utilized. A resonant microwave field with frequency ω is applied to drive the transition between |0⟩ and |1⟩ with a controlling Hamiltonian H_c=√(2)Ωcos(ω t+ϕ)S^(0,1)_x, where Ω is the Rabi frequency, ϕ the phase and S^(0,1)_x=(|0⟩⟨ 1|+| 1⟩⟨ 0|)/√(2) the transition operator. In the following discussion, we consider a simple system that contains only the NV electron spin and a single ^13C nuclear spin (the notation j is ignored in the following description).

Conventionally, the pulsed-DD scheme (e.g., XY-N) utilizes a basic operation sequence like τ-π-2τ-π-τ, where τ is the interval time and π pulse flips the electron spin between |0⟩ and |1⟩.  During this process, the extra phase generated by the nuclear spin is accumulated on the electron spin if its precessing speed (2γ_n B_z+A_zz) matches with τ. This protocol requires the microwave to be as strong as possible to ensure π pulse is much shorter than τ. While in CWDD scheme (Fig. <ref>d), the electron spin is instead prepared on one of the dressed states |±⟩≡(|0⟩± |1⟩)/√(2) and experiences a spin-locking process driven by another microwave. During this process, the spin would stay on the dressed state except for a global phase if no perturbation from the nuclear spin exists. Otherwise, when the electron-spin Rabi frequency Ω matches with the energy gap between nuclear-spin states, i.e., the Hartmann-Hahn (HH) condition is satisfied, the double resonance (DR) happens between states |+↑⟩ to |-↓⟩, where |↑⟩ and |↓⟩ are the nuclear-spin eigenstates. In the above protocols, the microwave field (power) is scaled linearly (quadratically) with the magnetic field B_z. As shown in Fig. <ref>f, this makes them extremely hard to be applied under high magnetic fields. 





Recently, new schemes based on HHDR are proposed where the phase or amplitude of the microwave field is modulated in a regular <cit.> or arbitrary<cit.> way to compensate for the mismatch between Rabi frequency and nuclear-spin energy gap. To elaborate this new protocol, we first explain the original HHDR and show its schematic in Fig. <ref>b. In electron-spin m_s=0 subspace, the hyperfine interaction takes no effect and the nuclear spin precesses under magnetic field B_z with frequency γ_nB_z only. But in m_s=1 subspace (Fig. <ref>c) it generates an extra precessing A_∥ along the z axis and an effective field A_⊥/γ_n in the x-y plane on nuclear spin, where A_∥=A_zz and A_⊥=√(A_zx^2 + A_zy^2). As a result, when electron spin rotates along the x-axis during the spin-locking, the nuclear spin experience a z-axis precessing with frequency γ_n B_z-1/2 A_∥ in average and an effective microwave field in x-y plane with frequency Ω. Once the HH condition is satisfied, i.e.,  Ω=γ_n B_z-1/2 A_∥, the nuclear-spin resonance happens and the electron-spin state is changed by the hyperfine interaction as well.


In the phase-modulated (PM-) HHDR scheme (Fig. <ref>e) , the driven field is replaced with 

    H_c= √(2)Ω_+ S^(0,1)_x cos (ω t)+√(2)Ω_- S^(0,1)_x cos (ω t-ϕ),

where the phase ϕ switches periodically between the values 0 and π with a modulation frequency ν. The total effect of this modulation is to change the electron-spin rotating speed between (Ω_++Ω_-) and (Ω_+-Ω_-) in the same frequency ν. In experiment, we make  Ω^' =Ω_+=Ω_- to ensure a best signal contrast (see Supporting Information).
 The effective magnetic field on the nuclear spin is also modulated to generate two sidebands with frequency Ω^'+ν and Ω^'-ν. Then the resonant condition in the PM-HHDR scheme is changed to

    |Ω^'±ν|=|γ_n B_z-A_∥/2|.

and the signal in the resonance ν is described by cos^2( A_⊥ J_1(4 Ω^' / πν) t_f/ 4 ), where Ω^' is defined as effective Rabi frequency, t_f is integration time, J_1 is the first kind of Bessel function. This new condition brings a possibility to make the microwave power significantly reduced when we drive the electron spin to achieve the double resonance under high magnetic fields (Fig. <ref>f). Note that J_1 associates with a relatively small value in experiment, which weakens the effective coupling between the sensor and target spins, thus reduces the sensitivity of this scheme. It is worth pointing out that the theoretical proposal <cit.> only consider the single-side resonant situation Ω^' + ν=γ_n B_z-A_∥/2, which is different with Eq.<ref> here (see Supporting Information).


Here, we utilize single NV centers (NV1 and NV2) in diamond to detect intrinsic ^13C nuclear spins at 1840 Gs as an example. The PM-HHDR spectrum is obtained by sweeping the modulation frequency ν with Ω^' fixed. The result is shown in Fig.<ref>, where we can observe two resonance dips symmetrical about γ_nB_z, i.e., the Larmor frequency of ^13C nuclear spin. Additionally, with decreasing Ω^' the positions of resonance moves towards the Larmor frequency. The relation between the resonance position and Ω^' is highlighted in the Fig. <ref>, which matches well with the condition defined in Eq.<ref>. Most importantly, the effective Rabi frequency Ω^' is reduced to around 1/20 (1/250) of the microwave field used in conventional HHDR (XY-N) where the modulation frequency ν is chosen much larger than Ω^' (see Supporting Information). Noted that A_∥ is averaged for different ^13C nuclear spins, thus it is not apparently shown in the relation. 

Since the resolution of the HHDR spectrum is mainly determined by the noise in microwave amplitude, low-power control can also improve the spectrum in principle <cit.>. However, this also weaken the decoupling efficiency of the CWDD sequence that protects the electron spin against (z-axis) magnetic fluctuations. In order to improve the performance, a single-spin lock-in detection <cit.> is realized to monitor the electron-spin resonance in real-time together with a PID-based temperature control of the setup within ±5 mK.




Meanwhile, we optimize the experimental parameters, and finally the single resonance dip is split into five narrow lines as shown in Fig. <ref>a (the upper line). By fitting the spectrum, four weakly-coupled individual ^13C nuclear spins and a spin-bath signal are assigned to each dip. The calculated coupling parameter is listed in Tab.<ref>. Because of the reduction in microwave noise, a minimal (average) line-width of 2.1 (3.5) kHz is achieved. For comparison, we realize conventional HHDR spectroscopy under the same field in Fig. <ref>a (the lower line). Since the microwave amplitude is around 20 times higher than that is used in PM-HHDR, the line-width is over 100 kHz and none of the weakly-coupled nuclear spins can be observed (see Supporting Information).










To verify this result, we perform a standard XY-32 experiment on the same NV center. As shown in Fig. <ref>b (the upper line), a spectrum with five distinguishable dips is observed under a field near 500 Gs and the minimal (average) line-width in the spectrum is 2.3 (3.0) kHz respectively. The extracted coupling information (also shown in Tab.<ref>) matches well with the result from PM-HHDR. The experiment under a higher magnetic field (the same field as PM-HHDR) is also realized, where the spectral lines become overlapped and no splitting can be observed anymore (the lower line in Fig. <ref>b). For even higher magnetic fields, the scheme works well and the results are shown in Fig. <ref>. Due to the maximal sampling frequency of the microwave pulse generator, the frequency-modulation resolution is increased to 2 kHz and we have to boost the Rabi frequency Ω^' to broaden the line-width accordingly (see Supporting Information).



Here we demonstrate a nano-scale nuclear-spin detection spectroscopy under high magnetic fields and achieve an spectral line-width down to 3.5 kHz at about 1840 Gs. The microwave power is about two orders below previous double resonance schemes, which associates with a detection bandwidth over 100 kHz (see Supporting Information). The performance is mainly limited by the minimal configuration precision of the modulation frequency ν in our case. High-resolved pulse control based on time-delay lines can further improve the line-width to around 10 Hz. 
In general cases, the ultimate limit is defined by the rotating-frame relaxation time T_1^ρ of the electron spin, where T_1^ρ is measured to be over mili-second in the experiment thus the spectral line-width is possibly pushed down to the sub-kHz level in the future. Although our work is focused on the electron-nuclear hyperfine interactions, this method can also applied to more general spin-based quantum sensing fields, for example, detection of electron-electron spin interaction <cit.>, chemical shift analysis <cit.> and spin manipulation in two-dimensional materials<cit.>. Finally, we have extended the resonant condition to both sidebands towards the real resonance <cit.>, which could potentially be utilized in the ultra-low or zero-field cases, i.e., to up-convert the required Rabi frequency to a proper range in the future <cit.>. 





The authors thank Bing Chen and Ying Dong for helpful discussion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 92265114, 92265204), the Fundamental Research Funds for the Central Universities (Grant No. 226-2023-00139),the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302200).





  
  * Supporting Information: Details of the setup and experimental scheme, and analysis of the performance. (PDF).







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subitem(mcitesubitemcount)

  
  

[Aue et al.(1976)Aue, Bartholdi, and Ernst]ref1
Aue, W. P.; Bartholdi, E.; Ernst, R. R.; Two-dimensional spectroscopy
  Application to nuclear magnetic resonance. The Journal of Chemical
  Physics 1976, 64, 2229–2246
[Oschkinat et al.(1988)Oschkinat, Griesinger, Kraulis,
  Sørensen, Ernst, Gronenborn, and Clore]ref2
Oschkinat, H.; Griesinger, C.; Kraulis, P. J.; Sørensen, O. W.; Ernst, R. R.;
  Gronenborn, A. M.; Clore, G. M. Three-dimensional NMR spectroscopy of a
  protein in solution. Nature 1988, 332, 374–376
[Wüthrich(2001)]ref3
Wüthrich, K. The way to NMR structures of proteins. Nature Structural
  Biology 2001, 8, 923–925
[Bucher et al.(2020)Bucher, Glenn, Park, Lukin, and
  Walsworth]ref47
Bucher, D. B.; Glenn, D. R.; Park, H.; Lukin, M. D.; Walsworth, R. L.
  Hyperpolarization-Enhanced NMR Spectroscopy with Femtomole Sensitivity Using
  Quantum Defects in Diamond. Physical Review X 2020,
  10, 021053
[Pfender et al.(2019)Pfender, Wang, Sumiya, Onoda, Yang,
  Dasari, Neumann, Pan, Isoya, Liu, and Wrachtrup]ref48
Pfender, M.; Wang, P.; Sumiya, H.; Onoda, S.; Yang, W.; Dasari, D. B. R.;
  Neumann, P.; Pan, X.-Y.; Isoya, J.; Liu, R.-B.; Wrachtrup, J. High-resolution
  spectroscopy of single nuclear spins via sequential weak measurements.
  Nature Communications 2019, 10, 594
[Lovchinsky et al.(2017)Lovchinsky, Sanchez-Yamagishi, Urbach,
  Choi, Fang, Andersen, Watanabe, Taniguchi, Bylinskii, Kaxiras, Kim, Park, and
  Lukin]ref6
Lovchinsky, I.; Sanchez-Yamagishi, J. D.; Urbach, E. K.; Choi, S.; Fang, S.;
  Andersen, T. I.; Watanabe, K.; Taniguchi, T.; Bylinskii, A.; Kaxiras, E.;
  Kim, P.; Park, H.; Lukin, M. D. Magnetic resonance spectroscopy of an
  atomically thin material using a single-spin qubit. Science
  2017, 355, 503–507
[Shi et al.(2014)Shi, Kong, Wang, Kong, Zhao, Liu, and
  Du]ref57
Shi, F.; Kong, X.; Wang, P.; Kong, F.; Zhao, N.; Liu, R.B.; Du, J. Sensing and
  atomic-scale structure analysis of single nuclear-spin clusters in diamond.
  Nature Physics 2014, 10, 21–25
[Fratila and Velders(2011)Fratila, and Velders]ref78
Fratila, R. M.; Velders, A. H. Small-Volume Nuclear Magnetic Resonance
  Spectroscopy. Annual Review of Analytical Chemistry 2011,
  4, 227–249
[Fernéndez-Acebal et al.(2018)Fernéndez-Acebal, Rosolio,
  Scheuer, Müller, Müller, Schmitt, McGuinness, Schwarz, Chen, Retzker,
  Naydenov, Jelezko, and Plenio]hyperpolarization1
Fernéndez-Acebal, P.; Rosolio, O.; Scheuer, J.; Müller, C.; Müller, S.;
  Schmitt, S.; McGuinness, L. P.; Schwarz, I.; Chen, Q.; Retzker, A.;
  Naydenov, B.; Jelezko, F.; Plenio, M. B. Toward Hyperpolarization of Oil
  Molecules via Single Nitrogen Vacancy Centers in Diamond. Nano Letters
  2018, 18, 1882–1887
[King et al.(2015)King, Jeong, Vassiliou, Shin, Page, Avalos,
  Wang, and Pines]hyperpolarization
King, J. P.; Jeong, K.; Vassiliou, C. C.; Shin, C. S.; Page, R. H.;
  Avalos, C. E.; Wang, H.-J.; Pines, A. Room-temperature in situ nuclear spin
  hyperpolarization from optically pumped nitrogen vacancy centres in diamond.
  Nature Communications 2015, 6, 8965
[Abrams et al.(2014)Abrams, Trusheim, Englund, Shattuck, and
  Meriles]hyperpolarization2
Abrams, D.; Trusheim, M. E.; Englund, D. R.; Shattuck, M. D.; Meriles, C. A.
  Dynamic Nuclear Spin Polarization of Liquids and Gases in Contact with
  Nanostructured Diamond. Nano Letters 2014, 14,
  2471–2478
[Degen et al.(2009)Degen, Poggio, Mamin, Rettner, and
  Rugar]mrfm
Degen, C. L.; Poggio, M.; Mamin, H. J.; Rettner, C. T.; Rugar, D. Nanoscale
  magnetic resonance imaging. Proceedings of the National Academy of
  Sciences 2009, 106, 1313–1317
[Balasubramanian et al.(2008)Balasubramanian, Chan, Kolesov,
  Al-Hmoud, Tisler, Shin, Kim, Wojcik, Hemmer, Krueger, Hanke, Leitenstorfer,
  Bratschitsch, Jelezko, and Wrachtrup]odnmr
Balasubramanian, G.; Chan, I. Y.; Kolesov, R.; Al-Hmoud, M.; Tisler, J.;
  Shin, C.; Kim, C.; Wojcik, A.; Hemmer, P. R.; Krueger, A.; Hanke, T.;
  Leitenstorfer, A.; Bratschitsch, R.; Jelezko, F.; Wrachtrup, J. Nanoscale
  imaging magnetometry with diamond spins under ambient conditions.
  Nature 2008, 455, 648–651
[Maze et al.(2008)Maze, Stanwix, Hodges, Hong, Taylor,
  Cappellaro, Jiang, Dutt, Togan, Zibrov, Yacoby, Walsworth, and Lukin]odnmr1
Maze, J. R.; Stanwix, P. L.; Hodges, J. S.; Hong, S.; Taylor, J. M.;
  Cappellaro, P.; Jiang, L.; Dutt, M. V. G.; Togan, E.; Zibrov, A. S.;
  Yacoby, A.; Walsworth, R. L.; Lukin, M. D. Nanoscale magnetic sensing with an
  individual electronic spin in diamond. Nature 2008,
  455, 644–647
[Doherty et al.(2013)Doherty, Manson, Delaney, Jelezko,
  Wrachtrup, and Hollenberg]ref61
Doherty, M. W.; Manson, N. B.; Delaney, P.; Jelezko, F.; Wrachtrup, J.;
  Hollenberg, L. C. L. The nitrogen-vacancy colour centre in diamond.
  Physics Reports 2013, 528, 1–45
[Xie et al.(2021)Xie, Zhao, Kong, Ma, Wang, Ye, Yu, Yang, Xu,
  Wang, Wang, Shi, and Du]ref38
Xie, T.; Zhao, Z.; Kong, X.; Ma, W.; Wang, M.; Ye, X.; Yu, P.; Yang, Z.;
  Xu, S.; Wang, P.; Wang, Y.; Shi, F.; Du, J. Beating the standard quantum
  limit under ambient conditions with solid-state spins. Science
  advances 2021, 7, 9204
[Holger Försterling(2010)]ref24
Holger Försterling, F. Spin dynamics: Basics of Nuclear Magnetic Resonance,
  Second Edition. Medical Physics 2010, 37,
  406–407
[Kong et al.(2015)Kong, Stark, Du, McGuinness, and
  Jelezko]ref54
Kong, X.; Stark, A.; Du, J.; McGuinness, L.; Jelezko, F. Towards Chemical
  Structure Resolution with Nanoscale Nuclear Magnetic Resonance Spectroscopy.
  Physical Review Applied 2015, 4, 024004
[de Lange et al.(2010)de Lange, Wang, Risté, Dobrovitski, and
  Hanson]ref20
de Lange, G.; Wang, Z. H.; Risté, D.; Dobrovitski, V. V.; Hanson, R. Universal
  Dynamical Decoupling of a Single Solid-State Spin from a Spin Bath.
  Science 2010, 330, 60–63
[Hegde et al.(2020)Hegde, Zhang, and Suter]ref46
Hegde, S. S.; Zhang, J.; Suter, D. Efficient Quantum Gates for Individual
  Nuclear Spin Qubits by Indirect Control. Physical Review Letters
  2020, 124, 220501
[Lang et al.(2019)Lang, Madhavan, Tetienne, Broadway, Hall,
  Teraji, Monteiro, Stacey, and Hollenberg]ref49
Lang, J. E.; Madhavan, T.; Tetienne, J. P.; Broadway, D. A.; Hall, L. T.;
  Teraji, T.; Monteiro, T. S.; Stacey, A.; Hollenberg, L. C. L. Nonvanishing
  effect of detuning errors in dynamical-decoupling-based quantum sensing
  experiments. Physical Review A 2019, 99, 012110
[Zhao et al.(2012)Zhao, Honert, Schmid, Klas, Isoya, Markham,
  Twitchen, Jelezko, Liu, Fedder, and Wrachtrup]ref63
Zhao, N.; Honert, J.; Schmid, B.; Klas, M.; Isoya, J.; Markham, M.;
  Twitchen, D.; Jelezko, F.; Liu, R.-B.; Fedder, H.; Wrachtrup, J. Sensing
  single remote nuclear spins. Nature Nanotechnology 2012,
  7, 657–662
[Taminiau et al.(2012)Taminiau, Wagenaar, van der Sar, Jelezko,
  Dobrovitski, and Hanson]ref17
Taminiau, T. H.; Wagenaar, J. J. T.; van der Sar, T.; Jelezko, F.;
  Dobrovitski, V. V.; Hanson, R. Detection and Control of Individual Nuclear
  Spins Using a Weakly Coupled Electron Spin. Physical Review Letters
  2012, 109, 137602
[Casanova et al.(2015)Casanova, Wang, Haase, and Plenio]ref55
Casanova, J.; Wang, Z. Y.; Haase, J. F.; Plenio, M. B. Robust dynamical
  decoupling sequences for individual-nuclear-spin addressing. Physical
  Review A 2015, 92, 042304
[Carr and Purcell(1954)Carr, and Purcell]ref18
Carr, H. Y.; Purcell, E. M. Effects of Diffusion on Free Precession in Nuclear
  Magnetic Resonance Experiments. Physical Review 1954,
  94, 630–638
[Meiboom and Gill(1958)Meiboom, and Gill]ref19
Meiboom, S.; Gill, D. Modified Spin Echo Method for Measuring Nuclear
  Relaxation Times. Review of Scientific Instruments 1958,
  29, 688–691
[Liu et al.(2017)Liu, Xing, Ma, Wang, Li, Po, Zhang, Fan, Liu,
  and Pan]ref52
Liu, G.-Q.; Xing, J.; Ma, W.-L.; Wang, P.; Li, C.-H.; Po, H. C.; Zhang, Y.-R.;
  Fan, H.; Liu, R.-B.; Pan, X.-Y. Single-Shot Readout of a Nuclear Spin Weakly
  Coupled to a Nitrogen-Vacancy Center at Room Temperature. Physical
  Review Letters 2017, 118, 150504
[Boss et al.(2017)Boss, Cujia, Zopes, and
  Degen]arb_resolution
Boss, J. M.; Cujia, K. S.; Zopes, J.; Degen, C. L. Quantum sensing with
  arbitrary frequency resolution. Science 2017, 356,
  837–840
[Schmitt et al.(2017)Schmitt, Gefen, StÃŒrner, Unden, Wolff,
  MÃŒller, Scheuer, Naydenov, Markham, Pezzagna, Meijer, Schwarz, Plenio,
  Retzker, McGuinness, and Jelezko]RN222
Schmitt, S. et al.  Submillihertz magnetic spectroscopy performed with
  a nanoscale quantum sensor. Science 2017, 356,
  832–837
[Cai et al.(2012)Cai, Naydenov, Pfeiffer, McGuinness, Jahnke,
  Jelezko, Plenio, and Retzker]ref74
Cai, J. M.; Naydenov, B.; Pfeiffer, R.; McGuinness, L. P.; Jahnke, K. D.;
  Jelezko, F.; Plenio, M. B.; Retzker, A. Robust dynamical decoupling with
  concatenated continuous driving. New Journal of Physics 2012,
  14, 113023
[Zhou et al.(2014)Zhou, Huang, Zhang, Wang, Tan, Xu, Shi, Rong,
  Ashhab, and Du]zhouPRL124
Zhou, J.; Huang, P.; Zhang, Q.; Wang, Z.; Tan, T.; Xu, X.; Shi, F.; Rong, X.;
  Ashhab, S.; Du, J. Observation of Time-Domain Rabi Oscillations in the
  Landau-Zener Regime with a Single Electronic Spin. Physical Review
  Letters 2014, 112, 010503
[Li et al.(2020)Li, Kong, Zhao, Cheng, Qin, Wang, Zhang, Wang,
  Wang, Shi, and Du]kongPRL112
Li, R.; Kong, F.; Zhao, P.; Cheng, Z.; Qin, Z.; Wang, M.; Zhang, Q.; Wang, P.;
  Wang, Y.; Shi, F.; Du, J. Nanoscale Electrometry Based on a
  Magnetic-Field-Resistant Spin Sensor. Physical Review Letters
  2020, 124, 247701
[London et al.(2013)London, Scheuer, Cai, Schwarz, Retzker,
  Plenio, Katagiri, Teraji, Koizumi, Isoya, Fischer, McGuinness, Naydenov, and
  Jelezko]ref25
London, P.; Scheuer, J.; Cai, J. M.; Schwarz, I.; Retzker, A.; Plenio, M. B.;
  Katagiri, M.; Teraji, T.; Koizumi, S.; Isoya, J.; Fischer, R.;
  McGuinness, L. P.; Naydenov, B.; Jelezko, F. Detecting and Polarizing Nuclear
  Spins with Double Resonance on a Single Electron Spin. Physical Review
  Letters 2013, 111, 067601
[Kucsko et al.(2013)Kucsko, Maurer, Yao, Kubo, Noh, Lo, Park,
  and Lukin]ref79
Kucsko, G.; Maurer, P. C.; Yao, N. Y.; Kubo, M.; Noh, H. J.; Lo, P. K.;
  Park, H.; Lukin, M. D. Nanometre-scale thermometry in a living cell.
  Nature 2013, 500, 54–58
[McGuinness et al.(2011)McGuinness, Yan, Stacey, Simpson, Hall,
  Maclaurin, Prawer, Mulvaney, Wrachtrup, Caruso, Scholten, and
  Hollenberg]ref80
McGuinness, L. P.; Yan, Y.; Stacey, A.; Simpson, D. A.; Hall, L. T.;
  Maclaurin, D.; Prawer, S.; Mulvaney, P.; Wrachtrup, J.; Caruso, F.;
  Scholten, R. E.; Hollenberg, L. C. L. Quantum measurement and orientation
  tracking of fluorescent nanodiamonds inside living cells. Nature
  Nanotechnology 2011, 6, 358–363
[Le Sage et al.(2013)Le Sage, Arai, Glenn, DeVience, Pham,
  Rahn-Lee, Lukin, Yacoby, Komeili, and Walsworth]ref81
Le Sage, D.; Arai, K.; Glenn, D. R.; DeVience, S. J.; Pham, L. M.;
  Rahn-Lee, L.; Lukin, M. D.; Yacoby, A.; Komeili, A.; Walsworth, R. L. Optical
  magnetic imaging of living cells. Nature 2013, 496,
  486–489
[Wang et al.(2022)Wang, Kong, Zhao, Huang, Yu, Wang, Shi, and
  Du]mixer1
Wang, Z.; Kong, F.; Zhao, P.; Huang, Z.; Yu, P.; Wang, Y.; Shi, F.; Du, J.
  Picotesla magnetometry of microwave fields with diamond sensors.
  Science Advances 2022, 8, eabq8158
[Wang et al.(2022)Wang, Liu, Schloss, Alsid, Braje, and
  Cappellaro]mixer2
Wang, G.; Liu, Y.-X.; Schloss, J. M.; Alsid, S. T.; Braje, D. A.;
  Cappellaro, P. Sensing of Arbitrary-Frequency Fields Using a Quantum Mixer.
  Phys. Rev. X 2022, 12, 021061
[Casanova et al.(2019)Casanova, Torrontegui, Plenio,
  Garcí­a-Ripoll, and Solano]ref27
Casanova, J.; Torrontegui, E.; Plenio, M.; García-Ripoll, J.; Solano, E.
  Modulated Continuous Wave Control for Energy-Efficient Electron-Nuclear Spin
  Coupling. Physical Review Letters 2019, 122,
  010407
[Aharon et al.(2019)Aharon, Schwartz, and Retzker]power_limit
Aharon, N.; Schwartz, I.; Retzker, A. Quantum Control and Sensing of Nuclear
  Spins by Electron Spins under Power Limitations. Physical Review
  Letters 2019, 122, 120403
[Ge et al.(2022)Ge, Chen, Wang, Zhou, Zheng, Yang, Qian, and
  Xu]omega
Ge, F.; Chen, B.; Wang, Y.; Zhou, F.; Zheng, R.; Yang, X.; Qian, P.; Xu, N. A
  wideband balun-based microwave device for quantum information processing with
  nitrogen vacancy centers in diamond. Journal of Lightwave Technology
  2022, 40, 7572–7577
[Gruber et al.(1997)Gruber, Dräbenstedt, Tietz, Ludovic,
  Wrachtrup, and Borczyskowski]ref34
Gruber, A.; Dräbenstedt, A.; Tietz, C.; Ludovic, F.; Wrachtrup, J.;
  Borczyskowski, C. Scanning Confocal Optical Microscopy and Magnetic Resonance
  on Single Defect Centers. Science 1997, 276,
  2012–2014
[Staudacher et al.(2013)Staudacher, Shi, Pezzagna, Meijer, Du,
  Meriles, Reinhard, and Wrachtrup]ref5
Staudacher, T.; Shi, F.; Pezzagna, S.; Meijer, J.; Du, J.; Meriles, C. A.;
  Reinhard, F.; Wrachtrup, J. Nuclear Magnetic Resonance Spectroscopy on a
  (5-Nanometer)3 Sample Volume. Science 2013,
  339, 561–563
[Childress et al.(2006)Childress, Dutt, Taylor, Zibrov,
  Jelezko, Wrachtrup, Hemmer, and Lukin]ref36
Childress, L.; Dutt, M. V. G.; Taylor, J. M.; Zibrov, A. S.; Jelezko, F.;
  Wrachtrup, J.; Hemmer, P. R.; Lukin, M. D. Coherent Dynamics of Coupled
  Electron and Nuclear Spin Qubits in Diamond. Science 2006,
  314, 281–285
[Dutt et al.(2007)Dutt, Childress, Jiang, Togan, Maze, Jelezko,
  Zibrov, Hemmer, and Lukin]register1
Dutt, M. V. G.; Childress, L.; Jiang, L.; Togan, E.; Maze, J.; Jelezko, F.;
  Zibrov, A. S.; Hemmer, P. R.; Lukin, M. D. Quantum Register Based on
  Individual Electronic and Nuclear Spin Qubits in Diamond. Science
  2007, 316, 1312–1316
[Fuchs et al.(2011)Fuchs, Burkard, Klimov, and
  Awschalom]register2
Fuchs, G. D.; Burkard, G.; Klimov, P. V.; Awschalom, D. D. A quantum memory
  intrinsic to single nitrogen-vacancy centres in diamond. Nature
  Physics 2011, 7, 789–793
[Zhang et al.(2021)Zhang, Guo, Ji, Wang, Yin, Kong, Lin, Yin,
  Shi, Wang, and Du]register3
Zhang, Q.; Guo, Y.; Ji, W.; Wang, M.; Yin, J.; Kong, F.; Lin, Y.; Yin, C.;
  Shi, F.; Wang, Y.; Du, J. High-fidelity single-shot readout of single
  electron spin in diamond with spin-to-charge conversion. Nature
  Communications 2021, 12, 1529
[Zu et al.(2014)Zu, Wang, He, Zhang, Dai, Wang, and
  Duan]computation1
Zu, C.; Wang, W. B.; He, L.; Zhang, W. G.; Dai, C. Y.; Wang, F.; Duan, L. M.
  Experimental realization of universal geometric quantum gates with
  solid-state spins. Nature 2014, 514, 72–75
[van der Sar et al.(2012)van der Sar, Wang, Blok, Bernien,
  Taminiau, Toyli, Lidar, Awschalom, Hanson, and Dobrovitski]computation2
van der Sar, T.; Wang, Z. H.; Blok, M. S.; Bernien, H.; Taminiau, T. H.;
  Toyli, D. M.; Lidar, D. A.; Awschalom, D. D.; Hanson, R.; Dobrovitski, V. V.
  Decoherence-protected quantum gates for a hybrid solid-state spin register.
  Nature 2012, 484, 82–86
[Taminiau et al.(2014)Taminiau, Cramer, van der Sar,
  Dobrovitski, and Hanson]computation3
Taminiau, T. H.; Cramer, J.; van der Sar, T.; Dobrovitski, V. V.; Hanson, R.
  Universal control and error correction in multi-qubit spin registers in
  diamond. Nature Nanotechnology 2014, 9, 171–176
[Shi et al.(2010)Shi, Rong, Xu, Wang, Wu, Chong, Peng,
  Kniepert, Schoenfeld, Harneit, Feng, and Du]computation
Shi, F.; Rong, X.; Xu, N.; Wang, Y.; Wu, J.; Chong, B.; Peng, X.; Kniepert, J.;
  Schoenfeld, R.-S.; Harneit, W.; Feng, M.; Du, J. Room-Temperature
  Implementation of the Deutsch-Jozsa Algorithm with a Single Electronic Spin
  in Diamond. Physical Review Letters 2010, 105,
  040504
[Childress et al.(2006)Childress, Taylor, Sørensen, and
  Lukin]computation5
Childress, L.; Taylor, J. M.; Sørensen, A. S.; Lukin, M. D. Fault-Tolerant
  Quantum Communication Based on Solid-State Photon Emitters. Physical
  Review Letters 2006, 96, 070504
[Cai et al.(2013)Cai, Retzker, Jelezko, and Plenio]simulation
Cai, J.; Retzker, A.; Jelezko, F.; Plenio, M. B. A large-scale quantum
  simulator on a diamond surface at room temperature. Nature Physics
  2013, 9, 168–173
[Ji et al.(2020)Ji, Zhang, Wang, Zhang, Guo, Chai, Rong, Shi,
  Liu, Wang, and Du]simulation1
Ji, W.; Zhang, L.; Wang, M.; Zhang, L.; Guo, Y.; Chai, Z.; Rong, X.; Shi, F.;
  Liu, X.; Wang, Y.; Du, J. Quantum Simulation for Three-Dimensional Chiral
  Topological Insulator. Physical Review Letters 2020,
  125, 020504
[Reiserer et al.(2016)Reiserer, Kalb, Blok, van Bemmelen,
  Taminiau, Hanson, Twitchen, and Markham]network1
Reiserer, A.; Kalb, N.; Blok, M. S.; van Bemmelen, K. J. â.; Taminiau, T. H.;
  Hanson, R.; Twitchen, D. J.; Markham, M. Robust Quantum-Network Memory Using
  Decoherence-Protected Subspaces of Nuclear Spins. Physical Review X
  2016, 6, 021040
[Hastie et al.(2017)Hastie, Zandonatti, Kleinfelter, Heinrich,
  Rowland, Chandran, Branco, Robinson, Garry, and Saphire]network2
Hastie, K. M.; Zandonatti, M. A.; Kleinfelter, L. M.; Heinrich, M. L.;
  Rowland, M. M.; Chandran, K.; Branco, L. M.; Robinson, J. E.; Garry, R. F.;
  Saphire, E. O. Structural basis for antibody-mediated neutralization of Lassa
  virus. Science 2017, 356, 923–928
[Humphreys et al.(2018)Humphreys, Kalb, Morits, Schouten,
  Vermeulen, Twitchen, Markham, and Hanson]network3
Humphreys, P. C.; Kalb, N.; Morits, J. P. J.; Schouten, R. N.; Vermeulen, R.
  F. L.; Twitchen, D. J.; Markham, M.; Hanson, R. Deterministic delivery of
  remote entanglement on a quantum network. Nature 2018,
  558, 268–273
[Pompili et al.(2021)Pompili, Hermans, Baier, Beukers,
  Humphreys, Schouten, Vermeulen, Tiggelman, Martins, Dirkse, Wehner, and
  Hanson]network4
Pompili, M.; Hermans, S. L. N.; Baier, S.; Beukers, H. K. C.; Humphreys, P. C.;
  Schouten, R. N.; Vermeulen, R. F. L.; Tiggelman, M. J.; Martins, L. d. S.;
  Dirkse, B.; Wehner, S.; Hanson, R. Realization of a multinode quantum network
  of remote solid-state qubits. Science 2021, 372,
  259–264
[Casola et al.(2018)Casola, van der Sar, and Yacoby]ref7
Casola, F.; van der Sar, T.; Yacoby, A. Probing condensed matter physics with
  magnetometry based on nitrogen-vacancy centres in diamond. Nature
  Reviews Materials 2018, 3, 17088
[Schmitt et al.(2017)Schmitt, Gefen, StÃŒrner, Unden, Wolff,
  MÃŒller, Scheuer, Naydenov, Markham, Pezzagna, Meijer, Schwarz, Plenio,
  Retzker, McGuinness, and Jelezko]ref8
Schmitt, S. et al.  Submillihertz magnetic spectroscopy performed with
  a nanoscale quantum sensor. Science 2017, 356,
  832–837
[Mamin et al.(2013)Mamin, Kim, Sherwood, Rettner, Ohno,
  Awschalom, and Rugar]ref58
Mamin, H. J.; Kim, M.; Sherwood, M. H.; Rettner, C. T.; Ohno, K.;
  Awschalom, D. D.; Rugar, D. Nanoscale Nuclear Magnetic Resonance with a
  Nitrogen-Vacancy Spin Sensor. Science 2013, 339,
  557–560
[Chen et al.(2020)Chen, Hou, Ge, Zhang, Ji, Li, Qian, Wang, Xu,
  and Du]lab2
Chen, B.; Hou, X.; Ge, F.; Zhang, X.; Ji, Y.; Li, H.; Qian, P.; Wang, Y.;
  Xu, N.; Du, J. Calibration-Free Vector Magnetometry Using Nitrogen-Vacancy
  Center in Diamond Integrated with Optical Vortex Beam. Nano Letters
  2020, 20, 8267–8272
[Bian et al.(2021)Bian, Zheng, Zeng, Chen, Stöhr, Denisenko,
  Yang, Wrachtrup, and Jiang]ref9
Bian, K.; Zheng, W.; Zeng, X.; Chen, X.; Stöhr, R.; Denisenko, A.; Yang, S.;
  Wrachtrup, J.; Jiang, Y. Nanoscale electric-field imaging based on a quantum
  sensor and its charge-state control under ambient condition. Nature
  Communications 2021, 12, 2457
[Doherty et al.(2014)Doherty, Struzhkin, Simpson, McGuinness,
  Meng, Stacey, Karle, Hemley, Manson, Hollenberg, and Prawer]ref10
Doherty, M. W.; Struzhkin, V. V.; Simpson, D. A.; McGuinness, L. P.; Meng, Y.;
  Stacey, A.; Karle, T. J.; Hemley, R. J.; Manson, N. B.; Hollenberg, L. C.;
  Prawer, S. Electronic properties and metrology applications of the diamond
  NV- center under pressure. Phys Rev Lett 2014, 112,
  047601
[Dolde et al.(2011)Dolde, Fedder, Doherty, Nöbauer, Rempp,
  Balasubramanian, Wolf, Reinhard, Hollenberg, Jelezko, and Wrachtrup]ref65
Dolde, F.; Fedder, H.; Doherty, M. W.; Nöbauer, T.; Rempp, F.;
  Balasubramanian, G.; Wolf, T.; Reinhard, F.; Hollenberg, L. C. L.;
  Jelezko, F.; Wrachtrup, J. Electric-field sensing using single diamond spins.
  Nature Physics 2011, 7, 459–463
[Michl et al.(2019)Michl, Steiner, Denisenko, Bülau,
  Zimmermann, Nakamura, Sumiya, Onoda, Neumann, Isoya, and Wrachtrup]ref67
Michl, J.; Steiner, J.; Denisenko, A.; Bülau, A.; Zimmermann, A.;
  Nakamura, K.; Sumiya, H.; Onoda, S.; Neumann, P.; Isoya, J.; Wrachtrup, J.
  Robust and Accurate Electric Field Sensing with Solid State Spin Ensembles.
  Nano Letters 2019, 19, 4904–4910
[Fujiwara and Shikano(2021)Fujiwara, and Shikano]ref11
Fujiwara, M.; Shikano, Y. Diamond quantum thermometry: from foundations to
  applications. Nanotechnology 2021, 32, 482002
[Singam et al.(2019)Singam, Nesladek, and Goovaerts]ref12
Singam, S. K. R.; Nesladek, M.; Goovaerts, E. Nitrogen-vacancy nanodiamond
  based local thermometry using frequency-jump modulation.
  Nanotechnology 2019, 31, 105501
[Neumann et al.(2013)Neumann, Jakobi, Dolde, Burk, Reuter,
  Waldherr, Honert, Wolf, Brunner, Shim, Suter, Sumiya, Isoya, and
  Wrachtrup]ref71
Neumann, P.; Jakobi, I.; Dolde, F.; Burk, C.; Reuter, R.; Waldherr, G.;
  Honert, J.; Wolf, T.; Brunner, A.; Shim, J. H.; Suter, D.; Sumiya, H.;
  Isoya, J.; Wrachtrup, J. High-Precision Nanoscale Temperature Sensing Using
  Single Defects in Diamond. Nano Letters 2013, 13,
  2738–2742
[Plakhotnik et al.(2014)Plakhotnik, Doherty, Cole, Chapman, and
  Manson]ref72
Plakhotnik, T.; Doherty, M. W.; Cole, J. H.; Chapman, R.; Manson, N. B.
  All-Optical Thermometry and Thermal Properties of the Optically Detected Spin
  Resonances of the NV-Center in Nanodiamond. Nano Letters
  2014, 14, 4989–4996
[Liu et al.(2022)Liu, Henning, Heindl, Allert, Bartl, Sharp,
  Rizzato, and Bucher]ref15
Liu, K. S.; Henning, A.; Heindl, M. W.; Allert, R. D.; Bartl, J. D.;
  Sharp, I. D.; Rizzato, R.; Bucher, D. B. Surface NMR using quantum sensors in
  diamond. Proceedings of the National Academy of Sciences
  2022, 119, e2111607119
[Soshenko et al.(2021)Soshenko, Bolshedvorskii, Rubinas,
  Sorokin, Smolyaninov, Vorobyov, and Akimov]ref39
Soshenko, V. V.; Bolshedvorskii, S. V.; Rubinas, O.; Sorokin, V. N.;
  Smolyaninov, A. N.; Vorobyov, V. V.; Akimov, A. V. Nuclear Spin Gyroscope
  based on the Nitrogen Vacancy Center in Diamond. Physical Review
  Letters 2021, 126, 197702
[Sushkov et al.(2014)Sushkov, Lovchinsky, Chisholm, Walsworth,
  Park, and Lukin]ref56
Sushkov, A.; Lovchinsky, I.; Chisholm, N.; Walsworth, R.; Park, H.;
  Lukin, M. Magnetic Resonance Detection of Individual Proton Spins Using
  Quantum Reporters. Physical Review Letters 2014, 113,
  197601
[Loretz et al.(2013)Loretz, Rosskopf, and Degen]ref59
Loretz, M.; Rosskopf, T.; Degen, C. L. Radio-Frequency Magnetometry Using a
  Single Electron Spin. Physical Review Letters 2013,
  110, 017602
[Xu et al.(2023)Xu, Zhou, Ye, Lin, Chen, Zhang, Yue, Chen,
  Wang, and Du]2201.06002
Xu, N.; Zhou, F.; Ye, X.; Lin, X.; Chen, B.; Zhang, T.; Yue, F.; Chen, B.;
  Wang, Y.; Du, J. Noise Prediction and Reduction of Single Electron Spin by
  Deep-Learning-Enhanced Feedforward Control. Nano Letters
  2023, 23, 2460–2466 
[Zhou et al.(2021)Zhou, Song, Deng, Zhang, Chen, and Xu]ref33
Zhou, F.; Song, S.; Deng, Y.; Zhang, T.; Chen, B.; Xu, N. Mixed-signal data
  acquisition system for optically detected magnetic resonance of solid-state
  spins. Review of Scientific Instruments 2021, 92,
  114702
[Belthangady et al.(2013)Belthangady, Bar-Gill, Pham, Arai, Le Sage, Cappellaro and Walsworth]EE
Belthangady, C.; Bar-Gill, N.; Pham, L. M; Arai, K.; Le Sage, D.; Cappellaro, P.; Walsworth, R. L. Dressed-State Resonant Coupling between Bright and Dark Spins in Diamond. Physical Review Letters
  2013, 110, 157601 


[Aslam et al.(2017)Aslam, Pfender, Neumann, Reuter, Zappe,
  Fávaro de Oliveira, Denisenko, Sumiya, Onoda, Isoya, and Wrachtrup]RN221
Aslam, N.; Pfender, M.; Neumann, P.; Reuter, R.; Zappe, A.; Fávaro de
  Oliveira, F.; Denisenko, A.; Sumiya, H.; Onoda, S.; Isoya, J.; Wrachtrup, J.
  Nanoscale nuclear magnetic resonance with chemical resolution. Science
  2017, 357, 67–71
[Zhou et al.(2023)Jiang, Liang, Ru, Bettiol, and
  Gao]ref101
Zhou, F.; Jiang, Z.; Liang, H.; Ru, S.; Bettiol, A.; Gao, W. DC Magnetic Field Sensitivity Optimization of Spin Defects in Hexagonal Boron Nitride.
  Nano Letters 2013, 23, 6209–6215
[Kong et al.(2018)Kong, Zhao, Ye, Wang, Qin, Yu, Su, Shi, and
  Du]ref23
Kong, F.; Zhao, P.; Ye, X.; Wang, Z.; Qin, Z.; Yu, P.; Su, J.; Shi, F.; Du, J.
  Nanoscale zero-field electron spin resonance spectroscopy. Nature
  Communications 2018, 9, 1563





