⇦Chinese Physics Letters, 2020, Vol. 37, No. 11, Article code 114203 Rapid Measurement and Control of Nitrogen-Vacancy Center-Axial Orientation in Diamond Particles Guobin Chen (陈国彬)1,2, Yang Hui (杨会)1, Junci Sun (孙军慈)1, Wenhao He (和文豪)2, and Guanxiang Du (杜关祥)2* Affiliations 1School of Mechanical and Electrical Engineering, Suqian College, Suqian 223800, China 2College of Telecommunication & Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210000, China Received 5 August 2020; accepted 9 September 2020; published online 8 November 2020 Supported by the National Key R&D Program of China (Grant No. 2017YFB0403602), the Nature Science Foundation of Jiangsu Province (Grant No. SBK2020041231), and the Suqian Sci&Tech Program (Grant No. K201912).
*Corresponding author. Email: duguanxiang@njupt.edu.cn Citation Text: Chen G B, Yang H, Sun J C, He W H and Du G X et al. 2020 Chin. Phys. Lett. 37 114203    Abstract Determination and control of nitrogen-vacancy (NV) centers play an important role in sensing the vector field by using their quantum information. To measure orientation of NV centers in a diamond particle attached to a tapered fiber rapidly, we propose a new method to establish the direction cosine matrix between the lab frame and the NV body frame. In this method, only four groups of the ODMR spectrum peaks shift data need to be collected, and the magnetic field along $\pm Z$ and $\pm Y$ in the lab frame is applied in the meantime. We can also control any NV axis to rotate to the $X$, $Y$, $Z$ axes in the lab frame according to the elements of this matrix. The demonstration of the DC and microwave magnetic field vector sensing is presented. Finally, the proposed method can help us to perform vector magnetic field sensing more conveniently and rapidly. DOI:10.1088/0256-307X/37/11/114203 PACS:42.50.Ex, 07.55.Ge, 03.65.Yz © 2020 Chinese Physics Society Article Text Negatively charged nitrogen-vacancy (NV$^{-}$) color centers in diamonds are a kind of spin defects with fluorescence characteristics, and each center consists of a substitutional nitrogen atom and an adjacent vacancy in the lattice of the diamond.[1] Because of its light stability, biocompatibility, chemical inertness, long spin coherence and relaxation time at atmospheric temperature, the NV center can be used for the solid-state quantum sensor in the application of sensing temperature,[2–4] rotation speed,[5,6] pressure,[7,8] magnetic field,[9–11] and electric field.[12] The orientations of NV centers known as the NV axis are aligned along four different axes according to the $C_{3\nu}$ symmetry of the diamond lattice, see Fig. 1(a). The spin state of the NV center can be read out via the optically detected magnetic resonance (ODMR). The fluorescence is affected by the components of the DC and microwave magnetic field along the orientation of the NV center, which makes it possible to sense the vector magnetic field.[13–16] Recent years, researchers have determined the orientation of the single NV center in diamond bulk or particle by analyzing the curve characteristics of ODMR spectral peak shifting with the external static magnetic field rotation.[17–19] However, they cannot meet the requirement of measuring the NV centers assembly orientation in the diamond particles. In addition, the orientation of the NV centers also needs to be adjusted to the assigned direction for the vector field sensing, which is not mentioned in these papers. The diamond particle with the NV centers assembly attached to the tapered fiber can be rotated to any direction in the space because of the fiber's flexibility compared with the diamond bulk. In addition, they can be used for sensing the field with the 3D high spatial resolution.[20–25] The orientations of NV assembly in the diamond particle are totally unknown and random when it located in a fixed position in the lab frame (LF). However, these methods cannot meet the requirement of orientating the NV assembly. Last year, we proposed an NV axis orientating method using an orthogonal 3D coil to determine the NV axis orientation in the LF.[26] However, it would take a long time to collect dozens group of the ODMR spectrum peaks shift data with different magnetic fields generated by the coils. In this Letter, we propose a rapid NV axis orientation measurement method so that only four group of the ODMR spectrum are needed. In addition, we also discuss how to rotate the NV axis to the $X$, $Y$, $Z$ axes in the LF, respectively. We construct the NV body frame (BF) by combining the four different NV axes in the diamond. Two steps are required to establish the transformation model between the NV BF and the lab frame (LF). The first step is to transform the NV BF to the triaxial orthogonal frame (TOF). The TOF is described as $Oxyz$, and the NV BF is described as $O\alpha \beta \gamma \delta$. The relative orientation setup of the frame $Oxyz$ and $O\alpha \beta \gamma \delta$ is shown in Fig. 1(a). We make $z$ overlap $\alpha$, and locate $\beta$ to $xOz$. Then, according to the diamond crystal structures, the angles among $\beta$, $\gamma$, $\delta$ and $xOy$ are the same, defined as $a$ ($a\,=\, 19.28^{\circ}$), the angles between the projection of $\gamma$, $\delta$ on $xOy$ and the projection of $\beta$ on $xOy$ are, respectively, defined as $b$ ($b\,=\,120^{\circ}$) and $c$ ($c\,=\,-120^{\circ}$). Therefore, transformation model between the NV BF and the TOF can be expressed as $$ \boldsymbol{B}_{\rm NV} ={K}_{O \rm N} \boldsymbol{B}_{O},~~ \tag {1} $$ where ${\boldsymbol B}_{\rm NV}\,=\,[B_{\alpha}, B_{\beta}, B_{\gamma}, B_{\delta} ]^{\rm T}$ is the magnetic-field vector in the NV BF and ${\boldsymbol B}_{O}\,=\,[B_{Ox}, B_{Oy}, B_{Oz}]^{\rm T}$ in the TOF. The transfer matrix $K_{O \rm N}$ can be calculated by $$\begin{align} {K}_{O \rm N} =\,&\begin{bmatrix} 0 \hfill & 0 \hfill & 1 \hfill \\ {\cos a} \hfill & 0 \hfill & {-\sin a} \hfill \\ {\cos b\cos a} \hfill & {\sin b\cos a} \hfill & {-\sin a} \hfill \\ {\cos c\cos a} \hfill & {\sin c\cos a} \hfill & {-\sin a} \hfill \\ \end{bmatrix}\\ =\,&\begin{bmatrix} 0 \hfill & 0 \hfill & 1 \hfill \\ {0.945} \hfill & 0 \hfill & {-0.33} \hfill \\ {-0.472} \hfill & {0.818} \hfill & {-0.33} \hfill \\ {-0.472} \hfill & {-0.818} \hfill & {-0.33} \hfill \\ \end{bmatrix}. \end{align} $$ In addition, Eq. (1) can be rewritten as $$ \boldsymbol{B}_{O} ={K}_{{\rm N}O} \boldsymbol{B}_{\rm NV},~~ \tag {2} $$ where ${K}_{{\rm N}O} =({{K}_{O \rm N}^{\rm T} {K}_{O \rm N}})^{-1}{K}_{O \rm N}^{\rm T}$. The second step is to transform the TOF to the LF. Here, the LF is described as OXYZ. As shown in Fig. 1(b), we can obtain TOF $Oxyz$ by rotating the LF OXYZ three times as follows: (1) Rotate the LF OXYZ through angle $\phi$ about $Z$ to the new frame $Ox_{1}y_{1}Z$. (2) Rotate the frame $Ox_{1}y_{1}Z$ through angle $\theta$ about $x_{1}$ to the new frame $Ox_{1}yz_{1}$. (3) Rotate the frame $Ox_{1}yz_{1}$ through angle $\omega$ about $y$ to the TOF $Oxyz$.
cpl-37-11-114203-fig1.png
Fig. 1. (a) The relative orientation between the NV body frame $O\alpha \beta \gamma \delta$ and the triaxial orthogonal frame $Oxyz$. (b) The rotation procedure from the lab frame OXYZ to the triaxial orthogonal frame $Oxyz$.
cpl-37-11-114203-fig2.png
Fig. 2. Rapid NV axis orientation measurement system
According to the theory of frame transformation, the transformation model between the TOF and LF the can be expressed as $$\begin{align} & B_{O} ={K}_{\omega } {K}_{\theta } {K}_{\phi } B_{e} \\ &=\begin{bmatrix} {\cos \omega } \hfill & 0 \hfill & {-\sin \omega } \hfill \\ 0 \hfill & 1 \hfill & 0 \hfill \\ {\sin \omega } \hfill & 0 \hfill & {\cos \omega } \hfill \\ \end{bmatrix}\begin{bmatrix} 1 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\cos \theta } \hfill & {\sin \theta } \hfill \\ 0 \hfill & {-\sin \theta } \hfill & {\cos \theta } \hfill \\ \end{bmatrix}\begin{bmatrix} {\cos \phi } \hfill & {\sin \phi } \hfill & 0 \hfill \\ {-\sin \phi } \hfill & {\cos \phi } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill \\ \end{bmatrix}\begin{bmatrix} B_{ex} \\ B_{ey} \\ B_{ez} \\ \end{bmatrix}\\ &=\begin{bmatrix} {\cos \phi \cos \omega -\sin \omega \sin \theta \sin \phi } \hfill & {\sin \phi \cos \omega +\cos \phi \sin \theta \sin \omega } \hfill & {-\cos \theta \sin \omega } \hfill \\ {-\sin \phi \cos \theta } \hfill & {\cos \phi \cos \theta } \hfill & {\sin \theta } \hfill \\ {\cos \phi \sin \omega +\sin \phi \sin \theta \cos \omega } \hfill & {\sin \phi \sin \omega -\cos \phi \sin \theta \cos \omega } \hfill & {\cos \theta \cos \omega } \hfill \\ \end{bmatrix}\begin{bmatrix} B_{ex} \\ B_{ey} \\ B_{ez} \\ \end{bmatrix}\\ &={K}_{eO} B_{e} .~~ \tag {3} \end{align} $$ We can establish the transformation model between the NV BF and the LF based on Eqs. (1) and (3) as follows: $$ \boldsymbol{B}_{\rm NV} ={K}_{e{\rm N}} {\boldsymbol B}_{e},~~ \tag {4} $$ where $K_{e{\rm N}}= K_{O{\rm N}}K_{eO}$ is a $4 \times 3$ the matrix. The matrix is unknown because the value of the angles $\phi$, $\theta$ and $\omega$ in matrix $K_{eO}$ is unknown with the random NV axis orientation in the LF. Therefore, the angles $\phi$, $\theta$ and $\omega$ need to be solved for measuring the NV axis orientation in the LF. To solving these angles, we need to find the components in the NV BF of the known external magnetic field vector firstly. The rapid NV axis orientation measurement system is shown in Fig. 2. We use the optical confocal system for the optical excitation and fluorescence collection of the NV center. The diamond particle is fixed to the tip of the tapered fiber and located in the uniformly distributed region of the magnetic field of the Helmholtz coils. The particle is also close to the place where the near magnetic field strength of the antenna is strongest. The position of the NdFeB magnet is set for making the ODMR spectrum split into four pairs of discrete peaks, as shown in Fig. 3(a). When $| B| \ll D/g_{e}\beta_{e }$ ($g_{e}$ is the electron Lander $g$-factor, $\beta_{e}$ is the Bohr magneton, $g_{e}\beta_{e} \approx 2.8$ MHz/G, $D$ is the zero field split 2.87 GHz at room temperature), the central-frequency difference of peaks of each pair ($f_{\alpha}$, $f_{\beta}$, $f_{\gamma}$ and $f_{\delta}$) is proportional to the strength of the magnetic field's component in each axis of the NV BF,[16,27] which can be expressed as $$ f_{\rm NV} \approx 2g_{e} \beta_{e} B_{\rm NV} =5.6B_{\rm NV}~~ ({{\rm NV~at~}\alpha,\beta,\gamma,\delta }).~~ \tag {5} $$
cpl-37-11-114203-fig3.png
Fig. 3. (a) The ODMR spectrum of the NV centers with four pairs of discrete peaks. (b) The ODMR spectrum peaks' shift difference when the magnetic field of Helmholtz coils is applied along $\pm Z$.
The Helmholtz coils are used to generate the known magnetic field vector in the $X$, $Y$ and $Z$ axes of the lab frame, which makes the central frequencies of the peaks shift. Figure 3(b) shows the ODMR spectrum peaks changing when the coils generate the magnetic fields ($\boldsymbol{B}_{e}^{Z+}=[{0~~0~~B_{z}}]^{\rm T}$, $\boldsymbol{B}_{e}^{Z-}=[{0~~0~~{-B}_{z}}]^{\rm T}$) with the same strength along the positive and negative directions of the $Z$ axis in the LF. The components of the magnetic field generated by the coils in the NV BF can be calculated by the central-frequency difference shift of each-pair peaks ($\Delta f_{\alpha}$, $\Delta f_{\beta}$, $\Delta f_{\gamma}$ and $\Delta f_{\delta}$) and Eq. (5). Then, these components can also be transformed to the components in the TOF ($\boldsymbol{B}_{O}^{Z+}$, $\boldsymbol{B}_{O}^{Z-}$) using the equations $$\begin{align} &{\boldsymbol B}_{O}^{Z+} -{\boldsymbol B}_{O}^{Z-} ={K}_{eO}({{\boldsymbol B}_{e}^{Z+} -{\boldsymbol B}_{e}^{Z-} }); \\ &B_{Ox}^{Z+} -B_{Ox}^{Z-} =-2B_{z} \cos \theta \sin \omega ,\\ &B_{Oy}^{Z+} -B_{Oy}^{Z-} =2B_{z} \sin \theta ,\\ &B_{Oz}^{Z+} -B_{Oz}^{Z-} =2B_{z} \cos \theta \cos \omega .~~ \tag {6} \end{align} $$ The angles $\phi$ and $\theta$ can be solved out using the equations $$\begin{align} &\omega =\arctan \Big({-\frac{B_{Ox}^{Z+} -B_{Ox}^{Z-} }{B_{Oz}^{Z+} -B_{Oz}^{Z-} }}\Big), \\ &\theta =\arctan \Big[{\frac{B_{Oy}^{Z+} -B_{Oy}^{Z-} }{({B_{Oz}^{Z+} -B_{Oz}^{Z-}})\cos \omega - ({B_{Ox}^{Z+} -B_{Ox}^{Z-}}\sin \omega)}}\Big].\\~~ \tag {7} \end{align} $$ Similarly, the Helmholtz coils are rotated 90$^{\circ}$ about the $X$ axis in the LB and then generate the magnetic fields ($\boldsymbol{B}_{e}^{Y+}=[0~~B_{Y}~~0]^{\rm T}$ and $\boldsymbol{B}_{e}^{Y-}=[0~~{-B}_{Y}~~0]^{\rm T}$) with the same strength along the positive and negative directions of the $Y$ axis. The angle $\omega$ can also be solved out using the calculated values of the angles $\phi$ and $\theta$ and the components in the TOF ($\boldsymbol{B}_{O}^{Y+}$, $\boldsymbol{B}_{O}^{Y-}$) and in the LB ($\boldsymbol{B}_{e}^{Y+}$, $\boldsymbol{B}_{e}^{Y-}$), $$\begin{alignat}{1} \phi =\arctan \Bigg[ {-\frac{\cos \omega ({B_{Ox}^{Y+} -B_{Ox}^{Y-}})+\sin \omega ({B_{Oz}^{Y+} -B_{Oz}^{Y-}})}{\cos \theta ({B_{Oy}^{Y+} -B_{Oy}^{Y-} })-\cos \omega \sin \theta ({B_{Oz}^{Y+}-B_{Oz}^{Y-}})+\sin \omega \sin \theta ({B_{Ox}^{Y+} -B_{Ox}^{Y-}})}} \Bigg].~~ \tag {8} \end{alignat} $$
cpl-37-11-114203-fig4.png
Fig. 4. The orientation of the NV $\gamma$ axis in the LF.
Therefore, we can know each element of the matrix $K_{eO}$, according Eq. (3). In addition, we can also know the transformation matrix $K_{e{\rm N}}$ from the LF to the NV BF, which is the result of multiplying the known matrix $K_{O{\rm N}}$ and $K_{eO}$. $K_{e{\rm N}}$ is the direction cosine matrix between LF and NV BF as follows: $$\begin{align} {K}_{e{\rm N}} =\begin{bmatrix} {\cos A_{aX} } \hfill & {\cos A_{aY} } \hfill & {\cos A_{aZ} } \hfill \\ {\cos A_{\beta X} } \hfill & {\cos A_{\beta Y} } \hfill & {\cos A_{\beta Z} } \hfill \\ {\cos A_{\gamma X} } \hfill & {\cos A_{\gamma Y} } \hfill & {\cos A_{\gamma Z} } \hfill \\ {\cos A_{\delta X} } \hfill & {\cos A_{\delta Y} } \hfill & {\cos A_{\delta Z} } \hfill \\ \end{bmatrix},~~ \tag {9} \end{align} $$ where $A_{ij}$ ($i =\alpha$, $\beta, \gamma$ and $\delta$; $j =X$, $Y$ and $Z$) represents the angles between the NV axes and the lab frame axes. We can know the orientation of each NV axis in the LF from Eq. (9). Figure 4 shows the orientation of the NV $\gamma$ axis in the LF as an example. A 2D-rotation stage and two rotation procedures are required to set the NV $\gamma$ axis orientation along the axes $X$, $Y$, $Z$ in the LF. For example, if we want to set the NV $\gamma$ along the axis $Z$, we need to rotate the NV $\gamma$ about the axis $X$ to the plate XOZ (NV $\gamma'$) firstly, as shown in Fig. 4. The rotation angle $R_{X}$ can be calculated using the element of the matrix $K_{e{\rm N}}$ as follows: $$ {R}_{X} =\arctan \Big({\frac{\cos A_{\gamma Y} }{\cos A_{\gamma Z} }} \Big).~~ \tag {10} $$ The second procedure is to make the NV $\gamma'$ rotate about the axis $Y$ to axis $X$, the rotation angle $R_{Y}\,=\,90^{\circ}-A_{\gamma X}$. In addition, we can set any NV axis in the diamond particle along the $X$, $Y$, $Z$ axes in the LF in a similar way as used earlier. To verify the rapid NV axis orientation measurement and control method, we remove the NdFeB magnet, make the Helmholtz coils generate the 15 Gs, 20 Gs and 25 Gs magnetic fields along the $Z$ and $X$ axes, and set the NV $\gamma$ axis along the $Z$ and $X$ axes, correspondingly. Figure 5 shows the ODMR spectra of the NV centers in the diamond particle in different coils' magnetic fields. We can see that there are always two pairs of peaks in the ODMR spectra with different applied magnetic fields because the orientations of the field vector and the NV $\gamma$ axis are the same, so that the components of the field in other three NV axes are the same, which makes the other three peaks overlap. In addition, the central frequencies of the peaks in Figs. 5(a) and 5(b) are nearly the same. This suggests that the orientation of the NV $\gamma$ axis is set along the $Z$ and $X$ axes. In addition, it can be seen that there are some splitting shifting errors between Figs. 5(a) and 5(b) because the axis orientation parameters may contain errors. The errors may be influenced by the factors such as diamond positioning, Helmholtz coil mechanical installation, and ODMR data fitting.
cpl-37-11-114203-fig5.png
Fig. 5. The ODMR spectra of the NV centers in the diamond particle in different coils' magnetic fields.
cpl-37-11-114203-fig6.png
Fig. 6. The distributions of the near magnetic field with the NV $\gamma$ axis along axes $Z$ (a) and $X$ (b).
We set the NV $\gamma$ axis along the $Z$ and $X$ axes, and correspondingly characterize the near magnetic field distributions on the plane 0.1 mm from the surface of CPW, as shown in Fig. 6. The working frequency of the CPW is 2.94 GHz. We can see that the characteristics of the near magnetic field distributions are different. This is because the fluorescence of the NV centers with the NV $\gamma$ axis is influenced by the near magnetic field components on the plane XOY and YOZ when the NV $\gamma$ axis along the $Z$ and $X$ axes. The characteristics of distributions of the near magnetic field on the plane XOY and YOZ are different. This also illustrates that the above method can make us measure and control the NV center axis orientation rapidly. In summary, we have presented a new rapid NV axis orientation measurement method. The direction cosine matrix between the LF and the NV BF is established by multiplying a known transformation matrix between the NV BF and the TOF and a transformation matrix with the unknown angles $\phi$, $\theta$ and $\omega$ between the TOF and the LF. The angles are solved by collecting four groups of the ODMR spectrum peak shift data when the magnetic field along $\pm Z$ and $\pm Y$ in the LF is applied. One of the four NV axes can be rotated to the $X$, $Y$, $Z$ axes in the LF by rotating twice, and the rotation angles can be calculated according to the elements in the direction cosine matrix. We demonstrate the rotated NV axis orientation by sensing the DC magnetic field vector and the microwave magnetic field near the surface of the CPW. This allows us to more conveniently sense the vector magnetic field using the NV center assembly in the diamond particles with a tapered fiber. References The nitrogen-vacancy colour centre in diamondTemperature shifts of the resonances of the NV − center in diamondAll-Optical Thermometry and Thermal Properties of the Optically Detected Spin Resonances of the NV – Center in NanodiamondHigh-Precision Nanoscale Temperature Sensing Using Single Defects in DiamondMeasurable Quantum Geometric Phase from a Rotating Single SpinQuantum measurement of a rapidly rotating spin qubit in diamondElectronic Properties and Metrology Applications of the Diamond NV − Center under PressureMeasurement of the full stress tensor in a crystal using photoluminescence from point defects: The example of nitrogen vacancy centers in diamondMagnetometry with nitrogen-vacancy defects in diamondHigh-resolution magnetic resonance spectroscopy using a solid-state spin sensorMicrowave Device Characterization Using a Widefield Diamond MicroscopeElectric-field sensing using single diamond spinsVector magnetic field sensing by a single nitrogen vacancy center in diamondHigh-resolution vector microwave magnetometry based on solid-state spins in diamondVector-magnetic-field sensing via multifrequency control of nitrogen-vacancy centers in diamondSimultaneous Broadband Vector Magnetometry Using Solid-State SpinsMeasuring the defect structure orientation of a single NV − centre in diamondComplete determination of the orientation of NV centers with radially polarized beamsIdentification of the orientation of a single NV center in a nanodiamond using a three-dimensionally controlled magnetic fieldRoom-temperature magnetic gradiometry with fiber-coupled nitrogen-vacancy centers in diamondConcept of a microscale vector magnetic field sensor based on nitrogen-vacancy centers in diamondHigh-resolution magnetic field imaging with a nitrogen-vacancy diamond sensor integrated with a photonic-crystal fiberA fiber based diamond RF B-field sensor and characterization of a small helical antennaUsing Diamond Quantum Magnetometer to Characterize Near-Field Distribution of Patch AntennaHigh Resolution Microwave B -Field Imaging Using a Micrometer-Sized Diamond Sensor *Nitrogen-Vacancy Axis Orientation Measurement in Diamond Micro-Crystal for Tunable RF Vectorial Field SensingFiber-optic magnetometry with randomly oriented spins
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