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.