Strong polarization of individual nuclear spins weakly coupled to nitrogen-vacancy color centers in diamond

Nuclear spins, owing to their extremely long coherence times, have emerged as attractive candidates for solid-state quantum information processing and quantum-sensing applications [1–4]. Among the various physical platforms, nuclear spins interacting with nitrogen-vacancy (NV) center electron spins in diamond are of great interest as this spin system provides an unique method for measuring nuclear spin state through the measurement of NV centers' well-known spin dependent fluorescence and optically detected magnetic resonance [5].

Quantum manipulation using magnetic resonance plays a central role in this field of research [6]; and multi-spin-state initialization is a crucial step to achieve high-fidelity quantum manipulation such as entanglement generation [7] and error correction [8]. For the nuclear spins in diamond, an optically induced dynamic nuclear polarization method at the excited [9] or ground-state level anti-crossing [10] is conventionally used for high-fidelity polarization of the nitrogen nuclear spin inherent to the NV centers. On the other hand, for the spatially distributed ¹³C nuclear spins, the random magnitude of hyperfine tensor components and nuclear quantization axis differing from NV axis typically lead to much less polarization efficiency even at level anti-crossings [11]. Several strategies have been developed with widely varying degree of polarization and applicable external magnetic field range. Representative examples include dynamical-decoupling-based pulsed polarization [12], Hartmann–Hahn type cross polarization [13–15], and double resonance and double-π-pulse-based population transfer [16] method. Most of the existing polarization method, however, are targeted either for moderately or strongly coupled nuclear spins to the NV centers (hyperfine interaction >2 MHz) [11], often require long polarization sequence duration [16, 17], or targeted for large external magnetic field [9]. Since the hyperfine coupling strength of remote nuclear spins decreases rapidly as the distance between the two spins increases, there are limitations in expanding the spin system of the NV center via nuclear spins. By developing additional polarization methods to polarize even the weakly coupled nuclear spins, the size of the nuclear spin system accessible for quantum information or metrology can be expanded. Therefore, it is important to develop novel methods for efficient and fast polarization of weakly coupled nuclear spins in diamond.

In this study, we have demonstrated a novel ¹³C nuclear spin polarization method in diamond using energy-level selective microwave pump pulse and optical illumination. We exploited a tunable nuclear spin mixing parameter through the external magnetic field and showed that the interplay between the microwave pump rate and the optically induced relaxation rate leads to a population transfer-based nuclear polarization, which is higher than 85% at room temperature even for remote ¹³C nuclear spins exhibiting hyperfine interaction with NV centers of the order of 600 kHz. Moreover, we show that the method naturally enables selectivity on the target nuclear spin as well as controllable polarity with respect to the NV axis by adjusting the pump pulse frequency. A numerical simulation was performed using experimentally reasonable parameters to show the detailed spin dynamics and its dependence on the physical parameters such as hyperfine interaction dependent external magnetic field and polar coordination of ¹³C. With a room for further improvement and applicability to wide range of nuclear spins near NV center, we expect the present control method of nuclear-spin states can be used for high fidelity initialization of spin registers.

We studied a CVD-grown high-purity type-IIa bulk diamond with nitrogen concentration <10 ppb and natural abundance of ¹³C (∼1.1%), as shown in figure 1(a). The NV centers investigated in this study are the naturally formed color centers located roughly 2 μm below the {110}-oriented surface. We used a home-built confocal microscope to measure the optically detected magnetic resonance [18] and microfabricated a coplanar waveguide on the surface of the bulk diamond for applying the control pulses as shown in figure 1(a). Figure 1(b) shows the magnified confocal microscope image revealing the spatially resolved single NV centers near one arm of the waveguide. An external magnetic field B_z of ∼520 G was applied along the direction to the studied NV center axis with an alignment precision better than 0.1° [19]. At this field, the ¹⁴N nuclear spin intrinsic to the NV center is polarized to its magnetic quantum number m_n = +1 state with a degree higher than 98% through an optically induced nuclear spin mixing mechanism, which can be obtained by the excitation laser illumination during the readout process of each measurement sequence [9]. Thus, we do not consider the nitrogen nuclear spin degree of freedom in the present study. We used time-domain Ramsey spectroscopy to identify ¹³C nuclear spins weakly coupled to the NV center electron spin with the magnitude of hyperfine interaction parallel to the NV axis ∣A_zz∣ < 1 MHz.

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Figure 1(c) depicts the spin system considered in this study. In this two-spin system, the six eigen energy states are represented in terms of the superposition of the basis states of the electron spin quantum number m_s = 0, −1, 1 and the ¹³C nuclear spin quantum number m_c = $\uparrow ,\,\downarrow ,$ because the anisotropic hyperfine tensor component A_ani combined with external magnetic field leads to nonzero nuclear spin mixing. In particular, for m_s = +1 manifold the nuclear spin eigenstates are given by the following equations:

$$\begin{eqnarray}&&\left|+\right\rangle =\,\cos (\theta /2)\left|\uparrow \right\rangle +\,\sin (\theta /2){{\rm{e}}}^{{\rm{i}}\phi }\left|\downarrow \right\rangle ,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\left|-\right. \rangle =-\sin (\theta /2){{\rm{e}}}^{-{\rm{i}}\phi }\left|\uparrow \right. \rangle +\,\cos (\theta /2)\left|\downarrow \right. \rangle .\end{eqnarray} \tag{ 2 }$$

Here, the nuclear mixing parameter $\theta$ can be calculated as $\tan \,\theta ={A}_{{\rm{a}}{\rm{n}}{\rm{i}}}/({A}_{zz}+{\gamma }_{c}{B}_{z}),$ which is adjustable with the external magnetic field, where ${\gamma }_{c}\approx 1.07$ kHz G⁻¹ is the nuclear gyromagnetic ratio of ¹³C and $\phi$ is a phase set by the hyperfine tensor components (see appendix B for more details). In this experiment, we focused on a ¹³C nuclear spin that is weakly coupled to the NV center with ∣A_zz∣ ∼ 600 kHz ≫ A_ani, and A_zz < 0. With the application of B_z ∼ 520 G along the NV axis, we work in the regime where ${A}_{zz}+{\gamma }_{c}{B}_{z}\sim 0;$ thus $\theta \sim \pi /2$ so that the nuclear spin flipping transitions are equally allowed in the m_s = +1 manifold. On the other hand, for m_s = −1, the manifold mixing parameter $\theta ^{\prime}$ is set by $\tan \theta ^{\prime} ={A}_{{\rm{a}}{\rm{n}}{\rm{i}}}/(-{A}_{zz}+{\gamma }_{c}{B}_{z})\sim 0$ and the nuclear spin mixing is minimized. We point out that only the mixing parameter $\theta$ is involved in this polarization method, and it does not show explicit dependence on ∣A_zz∣ itself.

Figures 1(d) and (e) show the coherence property of the NV electron spin observed by Ramsey spectroscopy at B_z = 520 G for m_s = +1 (figure 1(d)) and m_s = −1 (figure 1(e)). With the inhomogeneous coherence time T₂* ∼ 2 μs, the Ramsey oscillations show beating due to the coexisting probability of the ¹³C nuclear spin initialized to m_c = ↑ or ↓ states. The beating has a frequency splitting of ∼600 kHz in both figures 1(d) and (e) (see fast Fourier transform (FFT) data as inset to figures 1(d) and (e), indicating γ_cB_z ∼ 600 kHz (from figure 1(d)) as well as ∣${A}_{zz}$∣ ∼ 600 kHz (from figure 1(e)); also see appendix C for more details). The FFT spectrum shows nearly equal amplitude for the two frequency components indicating that the initial nuclear spin is in a completely mixed state.

We now discuss the mechanism of the proposed nuclear spin polarization scheme focusing on the m_s = 0 and +1 manifolds: We consider an initial completely mixed state composed of $\left|{m}_{s},{m}_{c}\right\rangle =\left|0,\uparrow \right\rangle$ and $\left|0,\downarrow \right\rangle$ and applied selective microwave π pulse on this spin system, providing an initial nuclear spin dependent excitation. As mentioned earlier, we work in the $\theta \sim \pi /2$ regime, where the microwave π pulse excites the initial state to both nuclear spin projections, leading to efficient spin mixing at the m_s = +1 manifold, which is subsequently relaxed back to the m_s = 0 manifold by optical illumination, as shown in figure 2(a). Because of this unbalanced pumping and relaxation path, the system is polarized to some degree to one of the nuclear spin states after the repeated application of the sequence. We note that a similar eigenspace of the coupled NV electron and ¹³C nuclear spin system, along with double resonant microwaves, has been used to demonstrate coherent population trapping in microwave regime in diamond, preparing the system state to the dark state [20]. Likewise, we consider the proposed nuclear spin polarization method as a preparing the system in the dark state using a single resonant microwave driving starting from the statistical mixture.

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Generally, the limiting factor for nuclear spin polarization is the nuclear spin mixing that occurs at optically excited levels, exhibiting significant nuclear spin mixing induced by hyperfine interaction [21]. To minimize this unwanted additional spin mixing, we used a chopped laser pulse train with a pulse length of approximately 30 ns and a duty cycle ∼30%, where a single-pulse duration is shorter than the time scale for a typical nuclear spin mixing; however, repeated application ensures electron spin relaxation as theoretically and experimentally considered in recent studies [21]. After applying the polarization sequence with a given repetition number N, we measured the degree of polarization using Ramsey spectroscopy. The whole experimental pulse sequence including microwave and laser is shown in figure 2(b).

We performed Ramsey spectroscopy while sweeping the microwave frequency used in the π pulse of the polarization sequence, resulting the 2D plot shown in figure 2(c). The bottom panel of figure 2(c) is an example of a time domain result of the Ramsey sequence near the resonant condition showing only a single frequency component while maintaining the full optical contrast. Figures 2(d) and (e) show the results of a representative polarization when the microwave pump frequency f_mw is resonant with the transition frequencies f₁ (d) and f₂ (e), respectively (see figure 1(c) for definitions of f₁ and f₂). In this experiment, we used N = 6 and Rabi amplitude Ω of the microwave pump of ∼300 kHz.

Considering the polarization of ¹³C either to $\left|\uparrow \right\rangle$ or $\left|\downarrow \right\rangle ,$ we quantified the degree of polarization as follows:

$$\begin{eqnarray}&&P=\displaystyle \frac{{{\mathscr{P}}}_{\uparrow }-{{\mathscr{P}}}_{\downarrow }}{{{\mathscr{P}}}_{\uparrow }+{{\mathscr{P}}}_{\downarrow }},\end{eqnarray} \tag{ 3 }$$

where ${{\mathscr{P}}}_{\uparrow }$ (or ${{\mathscr{P}}}_{\downarrow }$) is the population of the $\left|\uparrow \right\rangle \left(\left|\downarrow \right\rangle \right)$ state of the ¹³C nuclear spin with respect to the NV axis, which we extracted from the experiment by fitting the FFT of the Ramsey data to two Lorentzian functions with different center frequencies. We observed that the value of P approaches ∼75% (93%) to the $\left|\uparrow \right\rangle \left(\left|\downarrow \right\rangle \right)$ orientation at the corresponding resonant conditions. The asymmetric maximum polarization values are likely because the experimental microwave sweep resolution was low in the $\left|\uparrow \right\rangle$ orientation range, and the error of fitting the FFT result cannot be ignored because of the lack of points used for fitting; however, we show below that generally symmetric maximum polarization to the $\left|\uparrow \right\rangle \left(\left|\downarrow \right\rangle \right)$ orientation is expected theoretically. Moreover, unlike previous studies [16, 17], our method does not require a slow rf nuclear Rabi pulse because the nuclear spin mixing is naturally allowed in the $\theta \sim \pi /2$ regime tuned by the external magnetic field. The results thus show a promising method for realizing a simple, target selectable, and high degree of polarization for weakly interacting nuclear spins in diamond.

To further support the detailed procedure of the nuclear spin polarization and examine the external magnetic field dependence, we performed time-dependent spin dynamics simulations using the python-based QuTIP package and the Hamiltonian in the rotating frame [22, 23]. In order to reflect the experimental conditions, we considered appropriate Lindblad operators providing optically induced relaxation rates; however, we did not consider the dephasing of the NV center during the polarization process as the light is illuminated in a chopped fashion (see appendix D for details and additional simulation results). In figure 3, we calculated the experimental polarization based on the time domain result of the Ramsey measurement sequence. We used two sine functions with decay to fit the two oscillations with different frequencies coming from different nuclear spin states, as shown in figure 2(c). The polarization was calculated and plotted using equation (3), where the fitted amplitude of each sine function represents ${{\mathscr{P}}}_{\uparrow }$ and ${{\mathscr{P}}}_{\downarrow }.$ Because of the fine time step used for the measurement, the time domain fitting result shows relatively small deviations for the fitting parameters, providing higher precision than the fitting of the FFT result. Figure 3(a) shows the simulation results of P as a function of f_mw at B_z = 520 G. The maximum experimental polarization value obtained from the time domain fitting differs from the values obtained from the FFT result fitting. Using the time domain fitting, we observe that the value of P approaches ∼90%(85%) to the $\left|\uparrow \right\rangle \left(\left|\downarrow \right\rangle \right)$ orientation. The results are in good agreement with our observations: the qualitative P versus f_mw trend is consistent with the theory and the quantitative maximum P at the resonant conditions matches extremely well with the theoretical value for the given hyperfine parameters obtained by the Ramsey experiment. Figure 3(b) shows the simulation result of P while varying the polarization sequence number N using the same hyperfine parameters as in figure 3(a), which is also in good agreement with the theory. Moreover, according to the simulation result for P versus B_z shown in figure 3(c), the maximum P occurs when ∣γ_cB_z∣=∣ A_zz∣, which is consistent with our interpretation. We further note that the maximum P occurs in a wide field range around the optimal value, demonstrating the robustness of our method. This robustness comes from nonzero A_ani, whose value allows for variations in B_z while maintaining high polarization, provided that the $\theta \sim \pi /2$ condition is met.

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Thus far, we demonstrated efficient spin polarization for a particular ¹³C nuclear spin near an NV center. An important question is whether the present protocol can be applied to widely varying locations of ¹³C with varying anisotropic tensor components. We turn to discuss this point with the aid of numerical simulation. Figure 4(a) shows the simulation result of P as a function of ${A}_{{\rm{a}}{\rm{n}}{\rm{i}}}$ and f_mw while all other parameters are fixed; and figure 4(b) is the plot of the maximum polarization value among various values of f_mw for each given A_ani value in figure 4(a). For a given A_zz, the dependence of P on A_ani reflects the dependence on the relative polar coordinate between the NV center and ¹³C resting at different lattice points. For this simulation, γ_cB_z− ∣A_zz∣ is fixed to a small nonzero value, and the result shows that while the resonant condition changes as A_ani is varied, near-unity polarization can be achieved for all A_ani values as long as the spin-mixing rate is significant set by the condition ∣A_zz∣ ∼ γ_cB_z, and provided A_ani > Ω. Thus, we conclude that the proposed polarization method is not limited to the ¹³C investigated in this study but can be generally applied to ¹³C spins at widely varying polar angles with respect to the NV direction under conditions that the hyperfine coupling strength of the ¹³C satisfies ∣A_zz∣ ∼ γ_cB_z, and that the selective microwave π pulse with microwave amplitude Ω satisfies ∣A_zz∣ ∼ γ_cB_z , A_ani > Ω.

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In conclusion, we have shown that the ground state spin sector of the NV centers coupled with a nearby ¹³C nuclear spin has eigenenergy space appropriate for the application of an efficient nuclear spin polarization sequence. Using selective resonant microwave pulses at optimized magnetic field for efficient spin-mixing excitation while using chopped laser pulses providing minimized spin-mixing relaxation, we experimentally demonstrated nuclear polarization, the magnitude and polarity of which can be controlled by the pump frequency. Moreover, we showed that the method is applicable to widely distributed ¹³C spins around the NV centers and that for most ¹³C spins that are weakly coupled with the NV centers, the maximum polarization can reach >85% when the nuclear quantization axis is perpendicular to the NV axis. These findings add to the existing nuclear polarization protocols a new method for selectively polarizing relatively weakly coupled nuclear spins in diamond, which is a crucial step toward the initialization of spin registers for quantum information applications. Moreover, as this method provides a means for controlling the environment state from a completely mixed state to pure states, our method can provide a unique opportunity for studying the open quantum system dynamics through controlled system–environment interaction.

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIT) (No. 2015R1A5A1037668 and No. 2018R1A2A3075438) and the Creative-Pioneering Researchers Program through Seoul National University (SNU).

Appendix. Confocal microscopy

Appendix. NV photon readout process

Appendix. Sample stage setup and magnet alignment

Appendix. Microwave setup

Appendix. Temperature tracking

We implemented a temperature feedback loop to stabilize the temperature of the permanent magnet along with the sample area within ∼30 mK fluctuation as shown in appendix figure A1. The resonance frequency fluctuation of the NV center due to this temperature fluctuation is within ∼50 kHz (N35 temperature coefficient of magnetization α₀ = −0.12%/°C), which is much smaller than the linewidth of the frequency spectrum of the Ramsey measurement.

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Appendix. Pulse sequence

The NV center was initialized to its m_s = 0 state by a 5 μs pulsed illumination of the 532 nm green laser. Then, a 2.2 μs rest time was allowed for the relaxation of the NV center electrons existing in the excited or singlet shelving state. After the rest time, the polarization pulse sequence described in the main text was applied: a chopped laser was created using the AOM with 30 ns on/60 ns off pulse sequence, which was repeated 17 times to complete the initialization of the NV center electron spin. After a 100 ns rest time, a 1.7 μs microwave π pulse was applied, followed by another 100 ns rest time. This polarization sequence comprising chopped laser and microwave application was repeated N = 6 times for the saturation of the polarization. Finally, a Ramsey pulse sequence containing 14.5 ns π/2 pulses was used to measure the order of polarization. A phase difference of π/2 was applied for the two π/2 pulses to provide robustness to the gate preparation error. The microwave frequency used in the π/2 pulses was detuned ∼5 MHz from the resonant frequency of the m_s = −1 manifold, which is smaller than the microwave power used for the pulses. The π gate used in the polarization sequence and π/2 gate for the Ramsey measurement were calibrated by measuring the Rabi oscillation frequency.

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In this study, the system of interest is a diamond crystal with a NV defect. In the neighborhood of this NV center, there exists ¹³C nuclear spin with 1.1% natural abundance. First, we consider the NV center electron spin and the ¹³C nuclear spin independently: The NV center electron spin has a zero-field splitting (D) of 2.87 GHz. ($\hslash =1$):

$$\begin{eqnarray}&&{H}_{ZF}=D{{S}_{z}}^{2},\end{eqnarray} \tag{ 4 }$$

where S_z is the z component of the spin-1 operator. Under an external magnetic field B_z, Zeeman splitting is applied to both the NV center and the ¹³C nuclear spin:

$$\begin{eqnarray}&&{H}_{z}={\gamma }_{e}{S}_{z}{B}_{z}+{\gamma }_{c}{I}_{z}{B}_{z},\end{eqnarray} \tag{ 5 }$$

where the gyromagnetic ratio of the NV center electron (or ¹³C nuclear spin) is ${\gamma }_{e}\approx 2.8$ MHz G⁻¹ (or ${\gamma }_{c}\approx 1.07$ kHz G⁻¹), and I_z is the Pauli spin operator. Next, we consider the dipole interaction of the two spins:

$$\begin{eqnarray}&&{H}_{{\rm{d}}{\rm{i}}{\rm{p}}}=-\displaystyle \frac{{\mu }_{0}}{4\pi {r}^{3}}\left(\displaystyle \frac{3}{{r}^{2}}(\mathop{S}\limits^{\longrightarrow}\cdot \mathop{r}\limits^{\longrightarrow})(\mathop{I}\limits^{\longrightarrow}\cdot \mathop{r}\limits^{\longrightarrow})-\mathop{S}\limits^{\longrightarrow}\cdot \mathop{I}\limits^{\longrightarrow}\right).\end{eqnarray} \tag{ 6 }$$

Here ${\mu }_{0}$ is the vacuum permeability and $\vec{r}$ is the relative position vector between the NV and the ¹³C spin. This can be rewritten as:

$$\begin{eqnarray}&&{H}_{{\rm{d}}{\rm{i}}{\rm{p}}}=\mathop{S}\limits^{\longrightarrow}\cdot \underline{\underline{A}}\cdot \mathop{I}\limits^{\longrightarrow},\end{eqnarray} \tag{ 7 }$$

where the tensor $\underline{\underline{A}}$ is defined as follows:

$$\begin{eqnarray}&&{A}_{i,j}=-\displaystyle \frac{{\mu }_{0}{\gamma }_{e}{\gamma }_{c}}{4\pi {r}^{3}}\left(\displaystyle \frac{3}{{r}^{2}}{r}_{i}{r}_{j}-{\delta }_{i,j}\right),i,j=x,y,z.\end{eqnarray} \tag{ 8 }$$

When the external magnetic field is aligned to the NV axis, considering $D-{\gamma }_{e}{B}_{z}\gg {A}_{zz},{A}_{{\rm{a}}{\rm{n}}{\rm{i}}},$ we can apply secular approximation to this system. The Hamiltonian of this system after using secular approximation is expressed as follows [11]:

$$\begin{eqnarray}&&{H}_{0}=D{{S}_{z}}^{2}+{\gamma }_{e}{S}_{z}{B}_{z}+{\gamma }_{c}{I}_{z}{B}_{z}+{S}_{z}{A}_{zz}{I}_{z}+\displaystyle \frac{{A}_{{\rm{a}}{\rm{n}}{\rm{i}}}}{2}{S}_{z}({I}_{+}{{\rm{e}}}^{-{\rm{i}}\phi }+{I}_{-}{{\rm{e}}}^{{\rm{i}}\phi }),\end{eqnarray} \tag{ 9 }$$

where is the azimuthal angle between the NV center axis and the position vector between the ¹³C nuclear spin and the NV center; and ${A}_{{\rm{a}}{\rm{n}}{\rm{i}}}=\sqrt{{{A}_{zx}}^{2}+{{A}_{zy}}^{2}},$ where ${A}_{ij}(i,j=x,y,z)$ is the $ij$ component of the hyperfine tensor; in particular, ${A}_{zz}$(or ${A}_{{\rm{a}}{\rm{n}}{\rm{i}}}$) is the position-dependent hyperfine tensor component of ¹³C parallel (or perpendicular) to the NV-axis. By diagonalizing this Hamiltonian, we derive the following eigenenergies: ${E}_{1,2}=\pm {\gamma }_{c}{B}_{z}/2,$ ${E}_{3,4}=D-{\gamma }_{e}{B}_{z}\mp \displaystyle \frac{1}{2}\sqrt{{{A}_{{\rm{a}}{\rm{n}}{\rm{i}}}}^{2}+{({A}_{zz}-{\gamma }_{c}{B}_{z})}^{2}},$ and ${E}_{5,6}=D-{\gamma }_{e}{B}_{z}\mp \displaystyle \frac{1}{2}\sqrt{{{A}_{{\rm{a}}{\rm{n}}{\rm{i}}}}^{2}+{({A}_{zz}+{\gamma }_{c}{B}_{z})}^{2}},$ where the corresponding eigenstates expressed in the basis of electron and nuclear spin states $\left|{m}_{s},{m}_{c}\right\rangle$ are:

$$\begin{eqnarray}&&\left|{\psi }_{1}\right\rangle =\left|0,\uparrow \right\rangle ,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\left|{\psi }_{2}\right\rangle =\left|0,\downarrow \right\rangle ,\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\left|{\psi }_{3}\right\rangle =\sin (\theta /2){{\rm{e}}}^{{\rm{i}}\phi }\left|+1,\uparrow \right\rangle +\,\cos (\theta /2)\left|+1,\downarrow \right\rangle ,\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\,\left|{\psi }_{4}\right\rangle =\cos (\theta /2)\left|+1,\uparrow \right\rangle -\,\sin (\theta /2){{\rm{e}}}^{-{\rm{i}}\phi }\left|+1,\downarrow \right\rangle ,\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\,\left|{\psi }_{5}\right\rangle =\cos (\theta ^{\prime} /2)\left|-1,\uparrow \right\rangle +\,\sin (\theta ^{\prime} /2){{\rm{e}}}^{{\rm{i}}\phi }\left|-1,\downarrow \right\rangle ,\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\,\left|{\psi }_{6}\right\rangle =-\sin (\theta ^{\prime} /2){{\rm{e}}}^{-{\rm{i}}\phi }\left|-1,\uparrow \right\rangle +\,\cos (\theta ^{\prime} /2)\left|-1,\downarrow \right\rangle ,\end{eqnarray} \tag{ 15 }$$

where $\theta$ and $\theta ^{\prime}$ represent the polar angle between the NV center axis and the position vector between the ¹³C nuclear spin and the NV center, which can be calculated as follows:

$$\begin{eqnarray}&&\tan \theta ^{\prime} =\displaystyle \frac{{A}_{{\rm{a}}{\rm{n}}{\rm{i}}}}{-{A}_{zz}+{\gamma }_{c}{B}_{z}},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\tan \,\theta =\displaystyle \frac{{A}_{{\rm{a}}{\rm{n}}{\rm{i}}}}{{A}_{zz}+{\gamma }_{c}{B}_{z}}.\end{eqnarray} \tag{ 17 }$$

By measuring the Ramsey oscillation using microwave π/2 pulses detuned by a few MHz from the resonant frequency of the transition m_s = 0 → −1, which is expressed as: Δ₍₋₎ = D − γ_eB, we can measure the energy spectrum of E₁, E₂, E₃, and E₄ using the four peak positions in frequency, ${p}_{1}^{\left(-\right)},$ ${p}_{2}^{\left(-\right)},$ ${p}_{3}^{\left(-\right)},$ and ${p}_{4}^{\left(-\right)},$ derived from its FFT result. The four frequencies that are derived from this measurement correspond to:

$$\begin{eqnarray}&&{{p}^{(-)}}_{1,2}={E}_{3,4}-{E}_{1}={{\rm{\Delta }}}_{(-)}\mp \displaystyle \frac{1}{2}\sqrt{{{A}_{{\rm{a}}{\rm{n}}{\rm{i}}}}^{2}+{({A}_{zz}-{\gamma }_{c}{B}_{z})}^{2}}-{\gamma }_{c}{B}_{z}/2,\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{{p}^{(-)}}_{3,4}={E}_{3,4}-{E}_{2}={{\rm{\Delta }}}_{(-)}\mp \displaystyle \frac{1}{2}\sqrt{{{A}_{{\rm{a}}{\rm{n}}{\rm{i}}}}^{2}+{({A}_{zz}-{\gamma }_{c}{B}_{z})}^{2}}+{\gamma }_{c}{B}_{z}/2.\end{eqnarray} \tag{ 19 }$$

However, under conditions where $\tan \theta ^{\prime} =\tfrac{{A}_{{\rm{a}}{\rm{n}}{\rm{i}}}}{-{A}_{zz}+{\gamma }_{c}{B}_{z}}\ll 1,$ which can be satisfied when A_zz < 0, the transitions $\left|{\psi }_{1}\right\rangle \leftrightarrow \left|{\psi }_{3}\right\rangle$ and $\left|{\psi }_{2}\right\rangle \leftrightarrow \left|{\psi }_{4}\right\rangle$ become forbidden as sin θ ∼ 0, cos θ ≈ 1 in equations (12) and (13), allowing only ${p}_{1}^{\left(-\right)}$ and ${p}_{4}^{\left(-\right)}$ to be visible. Moreover, when A_ani ≪ A_zz, the difference between the two peaks is equal to A_zz, which is measured to be ∼600 kHz, as shown in in figure 1(e) in the main text.

Similarly, for the m_s = +1 manifold, using microwaves detuned from Δ₍₊₎ = D + γ_eB_z for pulses, the energy-peak conversion relation is expressed as follows:

$$\begin{eqnarray}&&{{p}^{(+)}}_{1,2}={E}_{5,6}-{E}_{1}={{\rm{\Delta }}}_{\left(+\right)}\mp \displaystyle \frac{1}{2}\sqrt{{{A}_{{\rm{a}}{\rm{n}}{\rm{i}}}}^{2}+{({A}_{zz}+{\gamma }_{c}{B}_{z})}^{2}}-{\gamma }_{c}{B}_{z}/2,\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{{p}^{(+)}}_{3,4}={E}_{5,6}-{E}_{2}={{\rm{\Delta }}}_{\left(+\right)}\mp \displaystyle \frac{1}{2}\sqrt{{{A}_{{\rm{a}}{\rm{n}}{\rm{i}}}}^{2}+{({A}_{zz}+{\gamma }_{c}{B}_{z})}^{2}}+{\gamma }_{c}{B}_{z}/2.\end{eqnarray} \tag{ 21 }$$

Under the conditions: A_zz + γ_cB_z ∼ 0, A_ani ≪ A_zz, ${p}_{1}^{\left(+\right)}$ and ${p}_{2}^{\left(+\right)}$ (or ${p}_{3}^{\left(+\right)}$ and ${p}_{4}^{\left(+\right)}$ become degenerate, and only the following two frequencies can be observed during the Ramsey oscillation:

$$\begin{eqnarray}&&{p}_{1}^{\left(+\right)^{\prime} }={{\rm{\Delta }}}_{\left(+\right)}-{\gamma }_{c}{B}_{z}/2,\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{p}_{1}^{\left(+\right)^{\prime} }={{\rm{\Delta }}}_{\left(+\right)}+{\gamma }_{c}{B}_{z}/2.\end{eqnarray} \tag{ 23 }$$

Therefore, from the difference of the two frequencies, we can measure γ_cB_z, which is ∼600 kHz, as shown in figure 1(d) of the main text. This value is consistent with the magnetic field extracted from the frequency of the two peaks measured in the ESR measurement, which is 520 G. In this study, we used an external magnetic field of 520 G and considered A_zz ∼ −600 kHz.

In this study, we used the m_s = +1 manifold for the polarization sequence, and the m_s = −1 manifold was used for Ramsey sequence time-domain measurement to avoid the strong nuclear spin mixing that occurs at the m_s = +1 manifold. The different manifolds show different frequency behaviors, as shown in equations (18), (19), (22), and (23).

For the m_s = +1 state, the oscillation frequency originating from the m_c = ↑ state is lower than the frequency from m_c = ↓. On the other hand, for the m_s = −1 state, the oscillation frequency originating from the m_c = ↑ state is higher than the frequency from m_c = ↓, explaining the reversed frequency position of the dominant peak shown in figures 2(d) and (e) in the main text.

Appendix. Calculation of the degree of polarization P

Using the rotating wave approximation, the rotating frame Hamiltonian becomes:

$$\begin{eqnarray}&&{H}_{{\rm{r}}{\rm{o}}{\rm{t}}}={H}_{0}+{w}_{{\rm{m}}{\rm{w}}}{S}_{z}+{\rm{\Omega }}{S}_{x},\end{eqnarray} \tag{ 24 }$$

where ${H}_{0}$ is defined in equation (9), ${{w}}_{{\rm{m}}{\rm{w}}}$ is the applied microwave frequency, and ${\rm{\Omega }}$ is the Rabi amplitude. For the calculation, we used the following master equation in the Lindblad form:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\rho }{{\rm{d}}t}=-{\rm{i}}[{H}_{{\rm{r}}{\rm{o}}{\rm{t}}},\rho ]+\displaystyle \sum \left(L\rho {L}^{\dagger }-\displaystyle \frac{1}{2}\{{L}^{\dagger }L,\rho \}\right).\end{eqnarray} \tag{ 25 }$$

This equation contains the following Lindblad operators:

$$\begin{eqnarray}&&{L}_{{\rm{u}}{\rm{p}},\uparrow (\downarrow )}^{{\rm{g}}{\rm{l}}}=\sqrt{{{\rm{\Gamma }}}_{{\rm{g}}{\rm{l}}}}{n}_{{\rm{t}}{\rm{h}}}\left|-1,\uparrow (\downarrow )\right\rangle \left\langle 0,\uparrow (\downarrow )\right|,\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{L}_{{\rm{d}}{\rm{o}}{\rm{w}}{\rm{n}},\uparrow (\downarrow )}^{{\rm{g}}{\rm{l}}}=\sqrt{{{\rm{\Gamma }}}_{{\rm{g}}{\rm{l}}}}(1+{{\rm{n}}}_{{\rm{t}}{\rm{h}}})\left|0,\uparrow (\downarrow )\right\rangle \left\langle -1,\uparrow (\downarrow )\right|,\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&{L}_{i}^{D}=\sqrt{{{\rm{\Gamma }}}_{i}^{D}}\left|{\psi }_{i}\right\rangle \left\langle {\psi }_{i}\right|,i=1,2,3,4,\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{{L}}_{12(21)}^{{\rm{g}}{\rm{l}}}=\sqrt{{{\rm{\Gamma }}}_{n}^{{\rm{g}}{\rm{l}}}}\left|{\psi }_{1(2)}\right\rangle \left\langle {\psi }_{2(1)}\right|.\end{eqnarray} \tag{ 29 }$$

Each operator represents the decay rate caused by the green laser illumination (${{\rm{\Gamma }}}_{{\rm{g}}{\rm{l}}}$), transverse relaxation (${{\rm{\Gamma }}}_{i}^{D}$), and optically induced transverse relaxation (${{\rm{\Gamma }}}_{12(21)}^{{\rm{g}}{\rm{l}}}$) [21]. In order to capture realistic experimental results, we included in the simulation the time-dependent pulse sequence that was used to polarize our ¹³C nuclear spin. By solving the Lindblad master equation using this Hamiltonian in equation (24), we calculated the variation in the polarization result as a function of the various environmental factors of the NV center, such as the external magnetic field strength and the hyperfine coupling strength, as shown in figures 3 and 4 in the main text.

We calculated the solution of the equation (25) using the built-in mesolve function of the QuTIP (Quantum toolbox in Python) package. During the calculation, we considered the full time dependence of the pulse sequence (for example, switching on and off of Ω and ${{\rm{\Gamma }}}_{{\rm{g}}{\rm{l}}}$) reflecting the experimental condition.

The initial density matrix was set as a completely mixed state of $\left|{\psi }_{1}\right\rangle =\left|0,\uparrow \right\rangle$ and $\left|{\psi }_{2}\right\rangle =\left|0,\downarrow \right\rangle ,$ which is expressed as: ${\rho }_{0}={\left|0\right\rangle }_{e}{\left\langle 0\right|}_{e}\otimes \displaystyle \frac{1}{2}({\left|\uparrow \right\rangle }_{c}{\left\langle \uparrow \right|}_{c}+{\left|\downarrow \right\rangle }_{c}{\left\langle \downarrow \right|}_{c}).$ To calculate the order of polarization after the completion of the polarization process, we calculated $P=\tfrac{{{\mathscr{P}}}_{1}-{{\mathscr{P}}}_{2}}{{{\mathscr{P}}}_{1}+{{\mathscr{P}}}_{2}},$ where ${{\mathscr{P}}}_{1}=\left\langle {\psi }_{1}\right.\left|\rho ({t}_{f})\right|\left.{\psi }_{1}\right\rangle$ and ${{\mathscr{P}}}_{2}=\left\langle {\psi }_{2}\right.\left|\rho ({t}_{f})\right|\left.{\psi }_{2}\right\rangle .$ The calculated probability of each nuclear spin ${{\mathscr{P}}}_{1(2)}$ is different from the probability ${{\mathscr{P}}}_{\uparrow (\downarrow )}$ defined in the main text, but becomes equal at the end of each single polarization step where the state of the NV center spin is completely initialized to m_s = 0, which are the points where we can experimentally measure the order of polarization.

Appendix. Fitting P versus f_mw

The fitting of figure 3(a) in the main text was conducted using the curve_fit function built in SciPy. The fitting function was defined to solve the master equation containing pulse sequences with green laser time t_gl and microwave π pulse time t_mw. This pulse sequence was repeated N = 6 times; and t_gl, t_mw, and N match the condition used for the experiment. The fitting parameters for this function are f_rel, A_zz, and A_ani, where f_rel is defined to match the center frequency Δ = 0 between the experiment and simulation, and A_zz and A_ani are the hyperfine interaction strengths. The initial guess for the parameters of the curve_fit function was defined as: f_rel = 0, A_zz = 600 kHz, and A_ani = 100 kHz. The result of the curve_fit function is shown in figure 3(a), where the resulting fitting parameters are: A_zz = 687 kHz and A_ani = 215 kHz, which are close to the experimental values. The subsequent simulations share the hyperfine coefficients obtained from this process. Following figures A2–A4 are a summary of the additional simulation data and table A1 is the corresponding parameter table.

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Table A1.  Table of coefficients used for each simulation.

Coefficients	Fitting P versus fmw	P versus N	P versus Bz	P versus Bz and Aani
tgl	300 ns	300 ns	300 ns	300 ns
tmw	1.7 μs	1.7 μs	1.7 μs	20 μs
Ω	294.1176 kHz	294.1176 kHz	294.1176 kHz	25 kHz
Γgl	8 MHz	8 MHz	8 MHz	8 MHz
D	2.87 GHz	2.87 GHz	2.87 GHz	2.87 GHz
γe	2.8 MHz G−1	2.8 MHz G−1	2.8 MHz G−1	2.8 MHz G−1
γc	1.07 kHz G−1	1.07 kHz G−1	1.07 kHz G−1	1.07 kHz G−1
ΓiD	0	0	0	0
${{\rm{\Gamma }}}_{12(21)}^{{\rm{g}}{\rm{l}}}$	0	0	0	0
nth	0	0	0	0
Azz	—	686.5546 kHz	686.5546 kHz	625 kHz
Aani	—	215.3535 kHz	215.3535 kHz	—
Bz	520	520	—	520
N	6	—	6	10