光电工程

NV色心矢量磁测中磁场投影等值与谱线重合的机制研究

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董士杰 , 赵一帆 , 刘艳龙 , 尚震 ^*^,^*

光电工程 | 科研论文 2026,53(7): 260115-1

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光电工程 |科研论文 2026 , 53 (7) : 260115 -1

NV色心矢量磁测中磁场投影等值与谱线重合的机制研究

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董士杰 (2004-)，男，中国海洋大学物理与光电工程学院物理学专业本科生。E-mail：2401311719@qq.com 2401311719@qq.com

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尚震 (1990-)，男，博士，硕士研究生导师，中国海洋大学讲师，主要研究方向为宽禁带半导体固态自旋色心物理性质研究与应用、量子传感等。E-mail：shangzhen@ouc.edu.cn shangzhen@ouc.edu.cn

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董士杰, 赵一帆, 刘艳龙, 尚震^*^,^*

作者信息

中国海洋大学物理与光电工程学院，山东青岛266100

通讯作者:

尚震，shangzhen@ouc.edu.cn

作者简介:

作者贡献

董士杰：研究资料收集、分析，实验设计与完成，论文构思与撰写；赵一帆：参与完成实验，论文修改；刘艳龙：参与完成实验；尚震：指导论文选题，论文审核与修改，提供课题实验条件与资源。

董士杰 (2004-)，男，中国海洋大学物理与光电工程学院物理学专业本科生。E-mail：2401311719@qq.com 2401311719@qq.com

Study on the mechanism of spectral line coincidence induced by equal magnetic field projections in NV-center vector magnetometry

Shijie Dong, Yifan Zhao, Yanlong Liu, Zhen Shang

Affiliations

College of Physics and Optoelectronic Engineering, Ocean University of China, Qingdao, Shandong 266100, China

Corresponding Author:

shangzhen@ouc.edu.cn

出版时间: 2026-07-25 doi: 10.12086/oee.2026.260115

文章导航

摘要

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针对金刚石NV色心矢量磁测中磁场投影等值引起的光探测磁共振‌(optically detected magnetic resonance‌, ODMR)谱线重合问题，本文开展了理论分析与实验验证。基于金刚石晶体对称性，建立了实验室坐标系与NV量子化轴之间的变换关系，推导得到三维磁场在各NV轴上的投影表达式及其等值判据。在此基础上，分析了投影等值对ODMR谱线分布的影响机制。实验中利用三维亥姆霍兹线圈施加可控偏置磁场，选取多个特征点进行测量。结果表明，当任意两个NV轴投影磁场相等时，ODMR谱呈现6个峰；当两对NV轴同时满足等值条件时，谱线进一步重合，仅表现为4个峰。实验结果与理论预测一致。该研究为理解谱线重合机制及优化磁场测量条件提供了理论依据。

关键词

NV色心 / 三维磁场传感 / 光探测磁共振

Abstract

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Objective

Nitrogen-vacancy (NV) centers in diamond are a robust platform for vector magnetometry with high sensitivity and nanoscale spatial resolution using optically detected magnetic resonance (ODMR). The Zeeman splitting of each NV center is proportional to the absolute magnetic field projection along its quantization axis. Because diamond hosts four possible NV orientations, full spectral separation of all eight resonances is required for unambiguous three-dimensional field reconstruction. In practice, an inappropriately chosen bias magnetic field can cause the projected field magnitudes on different NV axes to become equal, merging the corresponding resonance lines and creating ambiguity in field inversion. This work systematically identifies all conditions that lead to such projection equivalence, formulates rigorous mathematical criteria, and experimentally verifies their influence on ODMR spectra. The objective is to establish a quantitative framework for predicting and preventing spectral overlap, thereby improving the accuracy and reliability of NV-based vector magnetometry.

Methods

Taking into account the tetrahedral symmetry of diamond, a coordinate transformation was established between the laboratory frame and an internal frame aligned with the cubic crystal axes. The four NV quantization axes correspond to the body-diagonal directions of the cube. Using direction cosines, the projections of an arbitrary external magnetic field onto each NV axis were expressed as linear combinations of the laboratory field components. From these projection expressions, a complete set of eight independent equivalence conditions was derived algebraically; each condition describes the equality of absolute projected values on a specific pair of NV axes. Under fixed bias components, these conditions produced characteristic curves in the parameter space spanned by the variable field component and the azimuthal angle. An ODMR experimental setup was built with a 532 nm laser for spin initialization and readout, a microwave source for spin manipulation, and a three-axis Helmholtz coil system to generate precisely controlled bias magnetic fields. The coil currents were proportional to the field components. Two field-related currents were fixed, while the third current and the azimuthal orientation were systematically adjusted. Seven representative points were selected for measurement: one intersection of three equivalence curves, several points lying on a single equivalence curve, and points where two independent pairs of NV axes simultaneously satisfied the equivalence condition. At each point, ODMR spectra were recorded, and the number of distinct resonance peaks was identified as a direct indicator of spectral overlap.

Results and Discussion

The eight equivalence conditions, labeled B_X1 through B_X8, were classified by the NV-axis pairs involved. For example, B_X1 denoted equal projections on NV₁ and NV₄, B_X6 equal projections on NV₁ and NV₃, and B_X8 equal projections on NV₃ and NV₄. Under the fixed bias currents, these conditions produced curves in the parameter space defined by B_X and the azimuthal angle. When only one pairwise equality held, two resonance lines coincided, reducing the number of distinct peaks from eight to six. When two independent equalities occurred simultaneously— for instance, at the intersection of B_X1, B_X6, and B_X8, where NV₁, NV₃, and NV₄ shared identical projection magnitudes—only four peaks remained. The experimental intersection of these three curves yielded four peaks. Points on a single curve, such as B_X8 or B_X6, displayed six peaks. Points on B_X3 (where projections on NV₁ equaled NV₃ and on NV₂ equaled NV₄) and on B_X4 (where projections on NV₁ equaled NV₂ and on NV₃ equaled NV₄) both resulted in four peaks, confirming that two separate pairs could overlap simultaneously. All seven measured characteristic points matched the predictions without exception. These findings demonstrated that the equivalence of projected magnetic fields is the fundamental mechanism underlying ODMR spectral overlap. The degeneracies stem from the cubic symmetry of diamond combined with the linear Zeeman dependence on field projection. The derived criteria enable reliable identification of spectral degeneracies in experimental data. Moreover, they offer direct guidance for selecting bias field directions and magnitudes that avoid equivalence, ensuring complete peak separation essential for unambiguous vector magnetometry.

Conclusions

A systematic investigation of projection equivalence and its role in ODMR spectral overlap was conducted for NV-center-based vector magnetometry. An explicit coordinate transformation between the laboratory frame and the four NV quantization axes was formulated, yielding analytical magnetic field projection formulas. From these, all possible degeneracy conditions where absolute projections become equal were identified. These conditions were experimentally verified using a controlled ODMR setup with three-dimensional Helmholtz coils, and perfect agreement between theory and measurement was observed. The findings clarify the physical origin of spectral overlap and deliver a practical framework for optimizing experimental configurations. The criteria act as a quantitative tool for selecting bias fields that avoid resonance merging, thus enhancing the reliability of three-dimensional magnetic field measurements. This work advances both the fundamental understanding of NV quantum sensing and its application in high-precision magnetometry.

Key words

NV center / three-dimensional magnetic field sensing / optically detected magnetic resonance

引用本文

董士杰, 赵一帆, 刘艳龙, 尚震. NV色心矢量磁测中磁场投影等值与谱线重合的机制研究. 光电工程, 2026 , 53 (7) : 260115 -1 . DOI: 10.12086/oee.2026.260115

Shijie Dong, Yifan Zhao, Yanlong Liu, Zhen Shang. Study on the mechanism of spectral line coincidence induced by equal magnetic field projections in NV-center vector magnetometry[J]. Opto-Electronic Engineering, 2026 , 53 (7) : 260115 -1 . DOI: 10.12086/oee.2026.260115

正文

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1　引　言

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精密测量是基础物理理论验证与技术突破的核心支撑，随着研究尺度向微观延伸，磁场探测对灵敏度、空间分辨率与矢量感知能力的需求持续提升。金刚石氮-空位 (nitrogen-vacancy，NV)色心作为原子级量子传感器，兼具纳米级空间分辨率、室温工作与矢量磁场探测能力，是突破传统磁传感技术瓶颈的重要方案，在微观磁成像、基础物理检验等领域展现出巨大应用潜力。

NV色心是金刚石晶格中氮原子取代碳原子并伴随相邻空位形成的点缺陷，其电子自旋可通过光探测磁共振 (optically detected magnetic resonance, ODMR)技术实现光学极化与读出，结合微波调控可完成自旋态的精准操控^[1-4]。在外磁场作用下，NV色心自旋基态发生塞曼分裂，通过测量共振频率间隔即可反演对应轴向的磁场分量。由于金刚石单晶晶格的四面体结构，天然存在四种空间取向不同的NV主轴，因此单块金刚石可通过 ODMR 光谱同时获取磁场在四个轴向的投影，具备三维矢量磁场测量能力。

NV色心的自旋量子数

$ S=1 $

在外磁场的作用下，其轨道基态

$ {}^{3}{\mathrm{A}}_{2} $

会发生塞曼分裂^[5]。ODMR光谱技术可实现该自旋量子态的光制备与读出^[2,6]。同时，借助微波激发能够对自旋态进行共振调控，在外磁场中，自旋能级还会产生塞曼分裂^[7-8]。通过精准测量塞曼分裂的间隔，即可反演磁场大小。此外，金刚石晶格的四面体结构，令单晶体金刚石内存在四种取向互不相同的NV主轴。因此，完整的ODMR光谱可同时探测四个独立轴向的磁场分量，进而实现矢量磁场测量^[9-10]。

然而，基于NV色心的三维矢量磁场测量存在两个核心制约问题：其一，塞曼分裂频率仅与磁场在NV轴上的投影磁场的绝对值成正比，而由于NV色心的立方对称性导致空间中存在两组截然不同的三维磁场矢量，其在四个NV轴上的投影绝对值完全相等^[11]，造成磁场反演的二义性；其二，若实验中施加的偏置磁场方向与大小配置不当，会使不同取向NV色心的塞曼分裂间隔趋于一致，引发ODMR共振谱线重叠，无法区分各轴向的投影信号。上述问题直接影响测量结果的准确性与可靠性，而现有研究对投影磁场等值的完整条件、谱线重合的内在机制仍缺乏系统的理论判据。

针对这一问题，本研究建立了实验室坐标系与NV色心四个量子化轴的空间变换关系，系统推导了三维磁场测量中全部8种投影磁场等值条件，明确了不同NV取向下投影磁场等效与重合的数学判据，揭示了ODMR谱线重叠与三维磁场反演不唯一性的理论根源。该结果既可用于实验中谱线等值类型的快速判别，也可为偏置磁场方向选择与实验构型优化提供理论指导，对提升NV色心三维矢量磁场测量的准确性与稳定性具有重要意义。

2　实验设置与理论推导

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2.1　ODMR实验设置

实验中，采用532 nm的激光源，激光功率设置为375 mW，微波功率为20 dBm，微波频率范围为2.75~2.98 GHz，数据采集一次需115 s，金刚石NV色心浓度为4.5 × 10⁻⁶，亥姆霍兹线圈磁场标定精度为0.01 mT。通过三维亥姆霍兹线圈来施加磁场，线圈磁场与所施加电流成正比，即：

$ B=kI $

(k为单轴标定系数，经标定确定为8.82 × 10⁻⁴ T/A)。因此后续为方便计算，用施加的电流来表征磁场大小。

实验中将金刚石样品固定在三维亥姆霍兹线圈中心，随后对金刚石施加微波与激光并用光电探测器进行荧光计数，随后将数据传输到电脑中查看ODMR谱线。

2.2　理论推导

NV色心为自旋三重态点缺陷体系，其基态为轨道自旋三重态

$ {}^{3}{\rm{A}}_{2} $

，包含

$ {m}_{\rm{s}}=0 $

和

$ {m}_{\rm{s}}=\pm 1 $

三个磁子能级，零外磁场下，

$ {m}_{\rm{s}}=\pm 1 $

能级简并

$ {m}_{\rm{s}}=0 $

和

$ {m}_{\rm{s}}=\pm 1 $

之间存在零场分裂，分裂频率D≈2.87 GHz；激发态为三重态

$ {}^{3}{\mathrm{E}} $

，基态与激发态之间还存在单重态亚稳态

$ {}^{1}{\rm{A}}_{1} $

，完整能级结构与跃迁路径如图1(b)所示。

532 nm 激光的核心作用是实现自旋极化与光学读出：基态

$ {}^{3}{\rm{A}}_{2} $

上的电子吸收光子后跃迁至激发态

$ {}^{3}{\mathrm{E}} $

，随后通过两条自旋依赖的路径弛豫返回基态。

1)处于

$ {m}_{\rm{s}}=0 $

的激发态电子，主要通过辐射跃迁直接返回基态

$ {m}_{\rm{s}}=0 $

能级，同时辐射出红色荧光，该过程荧光量子效率高；

2)处于

$ {m}_{\rm{s}}=\pm 1 $

的激发态电子，会以较大概率通过系间窜越跃迁至亚稳态

$ {}^{1}{\rm{A}}_{1} $

，再经非辐射过程返回基态

$ {m}_{\rm{s}}=0 $

能级，该过程无荧光辐射。

在连续激光泵浦下，自旋布居会逐渐向能级

$ {m}_{\rm{s}}=0 $

聚集，实现自旋极化，稳态下体系荧光强度处于较高水平。此时若施加频率与基态塞曼分裂能级差匹配的微波场^[12-15]，基态

$ {m}_{\rm{s}}=0 $

与

$ {m}_{\rm{s}}=\pm 1 $

之间会发生受激跃迁，部分

$ {m}_{\rm{s}}=0 $

态的电子被转移至

$ {m}_{\rm{s}}=\pm 1 $

态，导致辐射跃迁占比下降、整体荧光强度降低。通过扫描微波频率并同步记录荧光强度，即可得到光探测磁共振 (ODMR)谱，谱线凹陷处对应的频率即为共振频率。

在外磁场的作用下NV色心能级会发生塞曼分裂：自旋态

$ {m}_{\rm{s}}=\pm 1 $

分裂为

$ {m}_{\rm{s}}=+1 $

与

$ {m}_{\rm{s}}=-1 $

两个自旋态，对应的ODMR谱峰同时产生了分裂，分裂量与磁场在NV轴上的投影幅值成正比，两个谱峰之间的共振频率差为

$ \Delta\omega=2g\mu\left|B_{\mathrm{NV}}\right|\;, $

(1)

其中：g是NV色心电子的g因子，μ是玻尔磁子，|B_NV|是外磁场沿着NV轴投影的绝对值。

因为金刚石具有立方对称性，结合图2(a)，可以对金刚石建立三维坐标系如图2(b)，通过分析金刚石内部的几何关系，可以得到NV轴投影磁场在金刚石晶体三维坐标系上的投影关系如式 (2)。其中所有磁场相关物理量均采用磁感应强度B作为基本符号，单位为特斯拉 (T)。

$ \left\{\begin{array}{l}B_{\mathrm{NV}_1}={\boldsymbol{B}}\cdot\left(-\dfrac{\sqrt{3}}{3},\dfrac{\sqrt{3}}{3},-\dfrac{\sqrt{3}}{3}\right) \\ B_{\mathrm{NV}_2}={\boldsymbol{B}}\cdot\left(\dfrac{\sqrt{3}}{3},\dfrac{\sqrt{3}}{3},\dfrac{\sqrt{3}}{3}\right) \\ B_{\mathrm{NV}_3}={\boldsymbol{B}}\cdot\left(-\dfrac{\sqrt{3}}{3},-\dfrac{\sqrt{3}}{3},\dfrac{\sqrt{3}}{3}\right) \\ B_{\mathrm{NV}_4}={\boldsymbol{B}}\cdot\left(\dfrac{\sqrt{3}}{3},-\dfrac{\sqrt{3}}{3},-\dfrac{\sqrt{3}}{3}\right)\end{array}\right.\; . $

(2)

随后对金刚石施加三维磁场，可以画出实验室坐标系三维磁场与金刚石晶体坐标系之间的关系如图2(c)，通过分析几何关系，可以得出式(3)，其中

$ {B}_{X} $

、

$ {B}_{Y} $

、

$ {B}_{Z} $

为实验室坐标系三维磁场，

$ B_x $

、

$ B_y $

、

$ B_z $

是金刚石晶体坐标系对应的磁场。

$ \left\{\begin{array}{l}B_X=B_y \sin \theta+B_x \cos \theta \\B_Y=B_y \cos \theta-B_x \sin \theta \\B_Z=B_z\end{array}\right. \;.$

(3)

联立式 (2)和式 (3)，可以将各个NV轴投影磁场与实验室坐标系三维磁场之间的关系表示出来：

$ \left\{ \begin{array}{l}\sqrt{3} B_{\mathrm{NV}_1}=B_Y (\cos \theta+\sin \theta)+B_X (\sin \theta-\cos \theta)-B_Z \\\sqrt{3} B_{\mathrm{NV}_2}=B_Y (\cos \theta-\sin \theta)+B_X (\sin \theta+\cos \theta)+B_Z \\\sqrt{3} B_{\mathrm{NV}_3}=B_Y (\sin \theta-\cos \theta)-B_X (\sin \theta+\cos \theta)+B_Z \\\sqrt{3} B_{\mathrm{NV}_4}=-B_Y (\sin \theta+\cos \theta)+B_X (\cos \theta-\sin \theta)-B_Z\end{array}\right. \;. $

(4)

实验中进行三维磁场测量时为确保NV轴的共振信号能被独立分辨，根据塞曼效应，需要满足金刚石中四个NV轴的投影磁场的绝对值各不相同，因此通过式 (4)，对不同NV轴上的磁场取绝对值并分别进行联立取不等号，可以得出：

$ \begin{aligned} & \left(\mathrm{NV}_1\right)\; \; \left(\mathrm{NV}_2\right)\; \; \left\{\begin{array}{l}B_Y\sin\theta\ne B_X\cos\theta+B_Z \\ \dfrac{B_X}{B_Y}\ne-\dfrac{1}{\tan\theta}\end{array}\right.\; , \\ & \left(\mathrm{NV}_1\right)\; \; \left(\mathrm{NV}_3\right)\; \; \left\{\begin{array}{l}B_Y\cos\theta\ne-B_X\sin\theta+B_Z \\ \dfrac{B_X}{B_Y}\ne\tan\theta\end{array}\right.\; , \\ & \left(\mathrm{NV}_1\right)\; \; \left(\mathrm{NV}_4\right)\; \; \left\{\begin{array}{l}\dfrac{B_X}{B_Y}\ne\dfrac{1+\tan\theta}{1-\tan\theta} \\ B_Z\ne0\end{array}\right.\; , \\ & \left(\mathrm{NV}_2\right)\; \; \left(\mathrm{NV}_3\right)\; \; \left\{\begin{array}{l}\dfrac{B_X}{B_Y}\ne\dfrac{\tan\theta-1}{1+\tan\theta} \\ B_Z\ne0\end{array}\right.\; , \\ & \left(\mathrm{NV}_2\right)\; \; \left(\mathrm{NV}_4\right)\; \; \left\{\begin{array}{l}B_Y\cos\theta\ne-B_X\sin\theta-B_Z \\ \dfrac{B_X}{B_Y}\ne\tan\theta\end{array}\right.\; , \\ & \left(\mathrm{NV}_3\right)\; \; \left(\mathrm{NV}_4\right)\; \; \left\{\begin{array}{l}B_Y\sin\theta\ne B_X\cos\theta-B_Z \\ \dfrac{B_X}{B_Y}\ne-\dfrac{1}{\tan\theta}\end{array}\right.\; ,\end{aligned} $

(5)

其中：公式左侧括号分别代表对应的NV轴，如 (NV₁) (NV₂)所对应右侧公式为式 (4)中NV₁轴投影磁场与NV₂轴投影磁场联立所得，若公式不等号成立，即NV₁轴投影磁场与NV₂轴投影磁场的绝对值不同。

随后对式 (5)通过变换，将

$ {B}_{X} $

单独提出放在不等号一边，将

$ {B}_{Y} $

、

$ {B}_{Z} $

与角度提到不等号另一边并进行整理归纳，可以得到

$ {B}_{X} $

与角度，

$ {B}_{Y} $

和

$ {B}_{Z} $

的关系式，其中

$ {B}_{Z} $

不为0。

$ \left\{\begin{array}{l}B_{X 1} \neq B_Y \cdot\left (\dfrac{1+\tan \theta}{1-\tan \theta}\right) \\B_{X 2} \neq B_Y \cdot\left (\dfrac{\tan \theta-1}{1+\tan \theta}\right) \\B_{X 3} \neq B_Y \cdot \tan \theta \\B_{X 4} \neq-\dfrac{B_Y}{\tan \theta} \\B_{X 5} \neq \dfrac{B_Y \sin \theta-B_Z}{\cos \theta} \\B_{X 6} \neq \dfrac{B_Z-B_Y \cos \theta}{\sin \theta} \\B_{X 7} \neq \dfrac{-B_Z-B_Y \cos \theta}{\sin \theta} \\B_{X 8} \neq \dfrac{B_Y \sin \theta+B_Z}{\cos \theta}\end{array}\right.\;, $

(6)

其中：条件

$ {B}_{X5} $

与

$ {B}_{X4} $

成立对应着NV₁与NV₂轴上的投影磁场的绝对值不同；条件

$ {B}_{X6} $

与

$ {B}_{X3} $

成立意味着NV₁与NV₃轴上的投影磁场的绝对值不同；条件

$ {B}_{X1} $

成立且

$ {B}_{Z} $

不为0意味着NV₁与NV₄轴上的投影磁场的绝对值不同；条件

$ {B}_{X2} $

成立且

$ {B}_{Z} $

不为0意味着NV₂与NV₃轴上的投影磁场的绝对值不同；条件

$ {B}_{X7} $

与

$ {B}_{X3} $

成立意味着NV₂与NV₄轴上的投影磁场的绝对值不同；条件

$ {B}_{X8} $

与

$ {B}_{X4} $

成立意味着NV₃与NV₄投影磁场的绝对值不同。

而在图中不等号的区域表示较为复杂，因此为了方便进行结果检验，后续将式 (6)的不等号改为等号，可以得到8个函数，若

$ {B}_{X} $

所对应的值落在了函数线上，即说明此时对应的两个NV轴投影磁场的绝对值相同，出现了ODMR谱峰重叠了现象。

若施加Z轴电流为1.5 A，Y轴电流为1 A的偏置磁场并将其带入式 (6)，可以得出对应的X轴向磁场对应的电流大小与角度之间的关系如图3。图中的每一条曲线都代表着两个NV轴投影磁场的绝对值相同，因此若X轴向磁场电流的大小落在了一条曲线上意味着此时发生了谱线重叠，此时ODMR谱线只存在六个峰，若落在两条曲线的交点处，则意味着至少存在三个NV轴上的投影磁场的绝对值相同，或是两两NV轴上的投影磁场对应相同，此时ODMR最多存在四个峰。

3　实验结果与讨论

收起

3.1　不同场景的共振峰数量

实验选取图3中的三线交点

$ \left ({{\text{π}}}/{11},1.873\right) $

；以及固定角度为

$ {{\text{π}} }/{6} $

，调整X轴电流大小，观察在图3中各个线上对应的ODMR峰的数量，结果如图4所示。

当外磁场仅满足一组 NV 轴的投影绝对值相等时，对应参数空间中一条等值曲线上的点位，此时仅一对共振峰发生重合，ODMR 谱线总数由 8 个减少为 6 个。

实验选取的图4(b, c, e, f) 对应的四组点位均属于该类构型：图4(b)的点位落在

$ {B}_{X8} $

等值曲线上，对应NV₃与 NV₄ 轴投影磁场相等；图4(c)的点位落在

$ {B}_{X6} $

等值曲线上，对应NV₁与NV₃轴投影磁场相等；图4(e)的点位对应NV₂与NV₃轴投影等值；图4(f)的点位对应NV₁与NV₂轴投影等值。实验测得上述点位的ODMR谱均呈现6个独立共振峰，与理论预测完全一致。从几何本质上看，该类重合对应磁场矢量位于某两个NV轴构成的角平分面上，是金刚石晶格镜面对称性的直接体现。

当磁场同时满足两组独立的投影等值条件时，两对共振峰同时发生重合，ODMR谱线总数进一步减少为4个。实验中观测到两种典型的双对等值场景：

第一种为三轴共简并场景，对应图4(a)的三线交点。该点为

$ {B}_{X1} $

、

$ {B}_{X6} $

、

$ {B}_{X8} $

三条等值曲线的交点，意味着NV₁、NV₃、NV₄三个轴的投影磁场绝对值完全相等，仅NV₂轴投影值独立，最终仅剩4个可分辨峰。该点位对应磁场矢量沿金刚石<111>型高对称晶向，是四面体对称性下的高度简并态。

第二种为两对独立简并场景，对应图4(d, g)的点位。其中，图4(d)的点位同时满足NV₁=NV₃、NV₂=NV₄两组等值条件，图4(g)的点位同时满足NV₁=NV₂、NV₃=NV₄ 两组等值条件，二者均形成两组两两重合的谱线结构，最终呈现 4 个独立峰。该类等值对应磁场矢量沿<001>、<110>等晶向，体现了立方晶体的旋转对称性。

综上，7组特征点位的实验结果均与理论判据一一对应，验证了投影等值条件的准确性：ODMR谱线重合的物理本质是不同NV轴上塞曼分裂间隔的简并，重合程度由磁场方向与晶体对称轴的相对空间关系直接决定。

3.2　晶体对称约束下磁场投影等值与谱线重合机理分析

本文观测到的投影等值与谱线重合，本质上源于金刚石立方晶格的Td点群对称性。NV色心的4个量子化轴对应立方体的4条体对角线<111>方向，在晶体对称操作下，不同NV取向可通过旋转、镜像操作互相映射。当外磁场方向落在某一对称元素上时，对称操作关联的两个NV轴上的磁场投影必然相等，对应谱线简并。

本文推导的8组等值条件，恰好对应Td群中全部对称元素约束的磁场方向：单对等值对应6个镜面 (两两 NV 轴构成的角平分面)，双对等值对应3个四重映轴与6个二重旋转轴，三轴简并对应4个三重旋转轴。这表明谱线重合是晶体对称性的必然结果，而非特定实验参数的偶然现象，只要采用体对角线取向的NV色心进行矢量磁测，该简并就必然存在于特定磁场方向上。

本文的等值判据基于理想均匀磁场、无应力、零温漂的理想条件推导，在实际测量场景中需结合环境因素进行修正与拓展，以下针对典型工况展开分析。

1)弱磁与零偏置磁场场景

零外磁场条件下，所有 NV 轴的投影磁场均为 0，塞曼分裂完全消失，ODMR 谱仅呈现对应零场分裂D值的一组共振峰，这是等值判据的极限简并情况。

当偏置磁场较弱、塞曼分裂量接近或小于 ODMR 本征线宽

$\varGamma $

时，即使投影磁场不满足严格相等的条件，不同 NV 轴的共振峰也会因线宽交叠而无法分辨，出现“等效重合”。此时需将严格等值判据修正为容差判据：

$ \left| {B}_{{{{\mathrm{NV}}}_{i}}}-{B}_{{{{\mathrm{NV}}}_{j}}}\right| \lt \dfrac{\varGamma }{{\gamma }_{{\mathrm{e}}}}\;, $

(7)

式中：

$ \varGamma $

是实际ODMR线宽，

$ {\gamma }_{{\mathrm{e}}} $

为电子旋磁比。

2)磁场噪声与空间不均匀性

实际测量中，三维亥姆霍兹线圈的磁场均匀度有限、环境杂散磁场 (地磁、工频噪声)、电流源噪声等因素，会导致 NV 色心感受到的投影磁场存在涨落与空间分布不均。当工作点靠近等值面时，微小的磁场噪声就可能导致谱线在 “重合-分离” 之间切换，造成共振峰展宽、幅值波动，降低测量信噪比。而本文选择的磁场远大于磁场噪声，以保证谱线稳定可分辨。

4　结　论

收起

本文针对金刚石NV色心三维矢量磁测中普遍存在的投影磁场等值与ODMR谱线重合问题，建立了实验室坐标系至NV量子化轴的磁场投影变换模型，系统推导了全部8组投影磁场绝对值等值的数学判据，结合三维亥姆霍兹线圈的连续波ODMR实验完成了多场景验证，并从晶体对称性层面揭示了谱线重合的内在物理机制。

研究表明，NV色心的谱线简并本质是金刚石Td点群对称性的必然结果：单对NV轴投影等值对应晶格的对角镜面，呈现6峰谱线构型；双对轴同时等值与三轴共简并分别对应晶格的旋转轴与三重对称轴，均呈现4峰谱线构型。基于三维亥姆霍兹线圈选取的7组特征点位实验结果与理论预测完全吻合，验证了等值判据的准确性与完备性。在此基础上，本文进一步将理想条件下的严格等值判据拓展至弱磁、噪声环境下的容差判据，给出了谱线可分辨的临界磁场差阈值，明确了判据的实际适用边界。

相较于现有研究多局限于谱线重合现象的定性描述，缺乏系统的等值条件推导与深层物理解释，本工作的核心创新在于：1)建立了完备的8组投影等值判据体系，实现了不同重合类型的定量预判与分类；2)从晶体点群对称性本源阐释了谱线简并的物理根源，建立了等值条件与对称元素的一一对应关系；3)拓展了实际工况下的容差修正方法，为偏置磁场的优化选型提供了可直接应用的理论依据。

该研究完善了NV色心矢量磁测的谱线简并理论，可为实验中规避谱线重叠、消除磁场反演歧义提供定量指导，对提升三维矢量磁测的准确性与鲁棒性具有重要的理论参考与工程应用价值。

参考文献

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2026年第53卷第7期

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doi: 10.12086/oee.2026.260115

接收时间：2026-04-30
首发时间：2026-10-22
出版时间：2026-07-25

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收稿日期：2026-04-30
修回日期：2026-07-14
录用日期：2026-07-15

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作者信息

中国海洋大学物理与光电工程学院，山东青岛266100

通讯作者:

尚震，shangzhen@ouc.edu.cn

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2种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

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