The NV center contains a substitutional nitrogen ¹⁴N atom and a vacancy in an adjacent lattice site. Its ground state has a spin triplet S = 1 with a zero-field splitting D = 2.87 GHz between m_s = 0 and m_s = ±1 spin sublevels. In a sample with a natural abundance of ¹³C isotope (1.1%), a randomly placed ¹³C nucleus (spin I = 1/2) locates in the diamond lattice. The hyperfine splitting of ¹⁴N nuclear spin (spin I = 1) is a constant of −2.16 MHz,^[28] which is insensitive to the changes of external magnetic field. Therefore, we do not take into account the ¹⁴N nuclear spin temporarily, and only consider the hyperfine structure of NV electron spin coupling to the single ¹³C nuclear spin with corresponding Hamiltonian

Fig. 1.

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	Fig. 1. (color online) (a) The energy-level diagram of NV–13C center ground state. The hyperfine splitting of corresponds to ground-state manifolds due to the coupling to the nearest 13C nuclear spin. (b) Under the low MW power, six resonance peaks could be observed between energy levels due to hyperfine splitting of 14N nuclear spin.

Applying a microwave (MW) pulse causes transitions of the system between the electron spin levels and modulates the fluorescence intensity. The spin dynamics of the ground state are relevant to microwave power, which can particularly affect the quantization axis of sub-manifold m_s = 0. Concretely, in the case of relative low microwave power, the splitting of m_s = 0 corresponds to two eigenstates of and , between which the analytic form of the effective Larmor splitting obtained by second order perturbation theory

In the remaining cases of relative high MW power, the Larmor splitting of m_s = 0 corresponds to two linear superposition states of eigenstates and , whose quantization axis of nuclear spin are realigned to a new axis defined by nuclear spin states .^[33] can be approximated as and , where . In particular, the doublet transitions between and ( ) appear at the relatively low microwave power. When considering the hyperfine splitting of an ¹⁴N nucleus adjacent to the vacancy, previous double transitions could be split into six resonance peaks as demonstrated schematically in Fig. 1. In contrast, high MW power leads to a single peak transition, whose corresponding frequency is almost centered between the two transition frequencies of .

NV–¹³C was optically addressed at room temperature by using a confocal microscope combined with a photon-counting detection system. A permanent magnet near the bulk diamond was used to apply an unknown static vector magnetic field, whose direction relative to the NV symmetry axis (z-axis) is shown schematically in Fig. 2. The electron spin resonance (ESR) transitions are driven with a microwave field applied through a diameter copper wire directly spanned on the diamond surface.

Fig. 2.

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Fig. 2. (color online) The schematic of the experimental setup. A 532-nm laser was used for the initialization and readout of the NV electron spin. Control of the spin was realized through the resonant microwave pulse radiated from a copper wire mounted on the diamond. A static vector magnetic field is applied. NV frame of reference is defined by x, y, and z axes, where ϕ and θ represent the azimuth and polar angles with respect to the x and z axes respectively.

In this experiment, we used the NV–¹³C to implement the static vector magnetic field detection. As an identification of our chosen NV–¹³C sensor, in zero-field its doublet splitting should be about 130 MHz.^[34] For this purpose, the experimental optical detection magnetic resonance (ODMR) spectra was firstly applied in zero-field to determine the two resonance frequencies. As shown in Fig. 3, at room temperature the spectral lines of the chosen NV–¹³C defect exhibit about 126 MHz zero-field splitting, which conforms to the numerical simulation of 131(±9) MHz based on hyperfine tensors Table 1 and Hamiltonian Eq. (1). The deviation of zero-field splitting comes from of hyperfine components. Concrete analysis can be seen in Appendix B.

Fig. 3.

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	Fig. 3. (color online) An NV center with a nearest neighbor 13C nucleus was identified by the zero-field splitting value 126 MHz.

Table 1.

Table 1.

Table 1. Four experimental resonance frequencies and Larmor splitting Δ obtained by fitting experimental data. . Resonance frequency ν1/GHz ν2/GHz ν3/GHz ν4/GHz Δ/MHz Experimental data 2.974 2.909 2.847 2.784 5.3 FWHM of CW spectra 1.2 MHz

Table 1. Four experimental resonance frequencies and Larmor splitting Δ obtained by fitting experimental data. .

Under the detected vector magnetic field, experimental pulsed CW spectrums of NV–¹³C are demonstrated in Fig. 4. It should be noted that magnetic dipolar interactions with a bath of nuclear spin fundamentally affect the full width at half maximum (FWHM) Γ of CW spectrum, which is limited by the inhomogeneous dephasing rate. Furthermore, the FWHM could also be affected by power broadening, which is from the laser light used for polarizing electron spin and MW field used for spin rotation. Thus, the laser intensity and MW power should be appropriately reduced in experiment for achieving a narrower linewidth.^[35] Specifically, in Figs. 4(a1)–4(a4) MW power was reduced by −10 dB with relevant duration of π pulse 1000–1200 ns. Due to the coupling with the ¹³C and ¹⁴N nucleus, the resonance fluorescence spectrum in each one of Figs. 4(a1)–4(a4) shows six resonance peaks, for which Γ is about 1.2 MHz. It seems like each one of the Figs. 4(a1)–4(a4) has five peaks, because the middle two ones of these six resonance frequencies as schematically depicted in Fig. 1 are too close to each other. In addition, the fluorescence intensities of spectral lines as shown in Fig. 4 are asymmetric. The reason is that the transition probabilities between the two transition matrices of and are different. Based on the transition matrix, we could find that the relative fluorescence intensity of the doublet transition should depend on polar angle θ for a determined hyperfine tensor of an NV–¹³C system.

Fig. 4.

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Fig. 4. (color online) Experimental data demonstrate the hyperfine structure of NV–13C via a change in fluorescence detection with 100 sampling points. (a) At relatively high power, the chosen NV–13C indicates four resonance frequencies due to hyperfine coupling with a single 13C nucleus for a detected magnetic field configuration. (a1)–(a4) The relatively low microwave power reveals doublet transitions of each corresponding resonance line in panel (a) and hyperfine splitting induced by 14N nucleus. The hyperfine splitting of ground state induced by 13C nucleus is 5.3 MHz. The FWHM Γ of CW spectra is about 1.2 MHz.

The four transition frequencies are sensitive to the changes of polar angle θ, compared to its insensitivity of the changes of azimuth angle ϕ. However, the Larmor splitting is sensitive to the changes of ϕ. Thus, to precisely detect the vector magnetic field, both the two factors of and as listed in Table 1 should be taken into account to extract the information of magnetic strength and possible directions by a method akin to maximum likelihood estimation^[15] (seen also in Appendix C). Taking the spectral linewidth and known hyperfine component deviation into consideration by the numerical search program, we determined the eight possible directions of detected as listed in Table 2. Due to the lack of a calibrated magnetic field in our experimental apparatus, we only estimated the errors listed in Table 2. The spectral broadening determines the spatial resolution of the detected magnetic field as the simulation analysis in Appendix D. Finally, it should be noted that our scheme of detecting the vector field by observing CW spectrums is only suitable for magnetic field strength being less than 200 Gs. Otherwise, it will dramatically influence the fluorescence intensity of the NV center.

Table 2.

Table 2.

Table 2. Experimental result for the detected static vector magnetic field. . The parameters of Experimental results B0/Gs 19.3(±1.3) θ0/rad 1.03(±0.04), 2.11(±0.04) ϕ0/rad 1.25(±0.29), 1.89(±0.29) 4.39(±0.29), 5.03(±0.29)

Table 2. Experimental result for the detected static vector magnetic field. .