The evaluation of a weak magnetic field with high spatial resolution has many potential applications, ranging from material science to biomedical science.^[1–5] Consequently, a variety of magnetometers based on the techniques such as superconduction quantum interference devices (SQUIDs)^[6] or Hall effect in semiconductors^[7] have been developed in the past few decades. Alternatively, electronic spin associated with nitrogen–vacancy (NV) color center in diamond can be well addressed, initialized and manipulated, even at ambient temperature. Due to its small size and long coherence time, of milliseconds,^[8] the NV center has been demonstrated to be an excellent solid-state quantum sensor that is able to detect magnetic fields with nanoscale resolution and high sensitivity.^[8–22] To extract the information of detected magnetic field, the idea behind the use of NV electron is that the Zeeman shift of the electron spin sublevels is monitored through ODMR spectrums or a measurement is taken based on a quantum phase evolution.^{[9, 10, 23, 24]} However, due to the symmetry of the defect, a single NV electron spin as a magnetometer can only determine the polar angle relative to NV axis and the strength B of detected static magnetic field.^[10] To reconstruct the complete information of a static vector magnetic field , the azimuth angle should be determined simultaneously. Various methods have been developed to address this issue, including the high-spin system theoretically,^[25] the NV center with a single nearest neighbor ¹³C system,^{[26, 27]} and the techniques of ensemble imaging.^{[28, 29]}

In our work, we demonstrated a proof-of-principle experiment to reconstruct a static vector magnetic field by three NV sensing spin centers with different crystal axes. By observing their respective continuous wave (CW) spectrums, we were able to reconstruct the complete information of a detected static vector magnetic field.

The negatively charged NV center consists of a vacancy defect and an adjacent substitutional nitrogen atom. The corresponding ground state of the NV center is a spin S = 1 system with a zero field splitting at 2.87 GHz, separated by the lower-energy level and levels. An external magnetic field can be applied to lift the degeneracy between the levels, as shown in Fig. 1.

Fig. 1.

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	Fig. 1. The energy level scheme of NV electron spin ground states. The zero-field splitting value is D=2.87 GHz. For the split-energy levels of correspond to doublet resonance frequencies and , respectively.

The corresponding Hamiltonian of the ground state is

A 532-nm green laser was used for the initialization and readout of the electronic spin state. The fluorescence signal was detected by an avalanche photo diode. Coherent control of the electron spin was realized through the resonant microwave pulse radiated from a 50- copper line mounted on the diamond. Three different axial NV centers could be addressed using scanning confocal microscopy at room temperature.

The distance between each chosen NV center is less than , as shown in Fig. 2. By measuring the corresponding ODMR frequencies of the NV centers with different crystallographic orientations, whose distinct CW spectrums are demonstrated under an applied static vector magnetic field, as demonstrated in Fig. 3. However, the limitation of our implementation is that the resonance frequencies measured by monitoring the CW spectrum is only applicable to detect the magnetic field strength below 0.02 T, otherwise the fluorescence of the NV center will be dramatically influenced. The resonance frequencies (shown in table 1) could be measured by fitting experimental data. The polar angle θ and field strength B could be calculated directly from the resonance frequencies and the relative energy level shift associated with the detected field.

Fig. 2.

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	Fig. 2. A schematic diagram of the experimental setup. (a) A 532-nm laser was used for the initialization and readout of the NV electron spin. The sample was imaged with an oil immersion objective lens. (b) Zoom-in view of the diamond sample for panel (a) shows respective symmetry axes labeled by red arrows of the chosen three NV centers, the distance between them is less than .

Fig. 3.

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	Fig. 3. Experimental data are obtained by monitoring the photoluminescence intensity. The CW spectrums of fitting curves with blue, green and red lines correspond to NV1, NV2, and NV3 centers, respectively. The marked correspond to their respective resonance frequencies.

Table 1.

Table 1.

Table 1. Measured six resonance frequencies of CW spectrums. . Frequency /MHz /MHz /MHz /MHz /MHz /MHz Experiment data 2819.7 2921.3 2836.8 2906.6 2860.6 2883.3 FWHM of CW spectra 16.7 11.6 9.9 9.7 8.0 8.2 Standard deviation 0.20 0.14 0.14 0.14 0.15 0.17

Table 1. Measured six resonance frequencies of CW spectrums. .

The polar angles corresponding to the three NV centers could be calculated according to

Table 2.

Table 2.

Table 2. The polar angles relative to their respective symmetry axes and the strengths of static magnetic field B corresponding to the three NV centers. . NV1 NV2 NV3 θ1,2,3/rad (0.55 or π-0.55) ±0.03 (1.01 or π-1.01)±0.005 (1.38 or π-1.38)±0.003 B/T (21.2±0.35)×10−4 (23.4±0.30)×10−4 (22.2±0.38)×10−4

Table 2. The polar angles relative to their respective symmetry axes and the strengths of static magnetic field B corresponding to the three NV centers. .

To reconstruct a static vector magnetic field , the corresponding vector direction of magnetic field was projected onto Cartesian components , , in the laboratory frame. Alternatively, the vector field can be projected onto each of the NV axes corresponds to different NV reference frames. Due to the fixed crystallographic axes inherent to the NV solid-state system, each of the tetrahedral diamond axes of NV centers demonstrates different CW spectrum features. By using a similar method as shown in Ref. ^[32] that reconstructed an AC vector magnetic field, a static vector magnetic field could be entirely determined by at least three of four possible NV axes aligned along [111], , , and .

Concretely, by using a geometric arguments to transform the tetrahedral components into a Cartesian coordinate (see Appendix B), as shown schematically in Fig. 4, the transformation between the two sets of coordinate system is given by

Fig. 4.

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	Fig. 4. Four inherent NV symmetry axes in the diamond’s tetrahedral structure are demonstrated by . A bias static vector magnetic field is applied. The corresponding polar angles taken with respect to each of NV axes can be represented by respectively.

Fig. 5.

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	Fig. 5. The transformation of laboratory and NV1 frame of reference are linked by a rotation around the ( ) axis. It is noted that the two reference frames share a same axis.

With a method akin to maximum likelihood estimation,^{[26, 32]} our numerical calculations implemented a search procedure to match the target experimental resonance frequencies , , and according to the formula

Table 3.

Table 3.

Table 3. Experimental results of detected static vector magnetic field with respect to the laboratory frame. . α0/rad β0/rad B0/T /MHz Result 1 1.34 ± 0.01 1.15 ± 0.01 (21.1 ± 0.1)×10−4 0.68 Result 2 1.80 ± 0.01 4.29 ± 0.01 (21.1 ± 0.1)×10−4 0.76

Table 3. Experimental results of detected static vector magnetic field with respect to the laboratory frame. .

In this paper, we have provided an experimental study of the detection of a weak static vector magnetic field by using three NV centers with different symmetry axes under ambient conditions. By measuring the resonance frequencies of the ODMR spectrums, the corresponding information of direction and strength of the detected field could be reconstructed. In future work, it may be possible to further improve our spatial resolution by reduction of spectral broadening and determine the real vector direction of the detected magnetic field by installing the calibration of the magnetic field.

The spin Hamiltonian of the ground state of an NV center is

The corresponding two resonance frequencies are given by

The transformation of laboratory and NV₁ frame of reference is shown in Fig. B1. For example, the transformation between laboratory and NV₁ frame of reference is given by a 3 × 3 orthogonal rotation matrix

The vector direction of can be represented by a unit vector

Via the rotation matrix of Eq. (B1), the transformed vector direction of in the NV₁ frame of reference can be expressed as . Due to the symmetry of the NV center, the main concern is the z component of parallel to the NV₁ axis as the following matrix multiplication

In the same way, , , and can be given by the transformation taken with respect to each NV reference frame.

We thank Peng-Fei Wang and Hong-Wei Chen for their helpful discussions.