Bias-Field-Free Operation of Nitrogen-Vacancy Ensembles in Diamond for Accurate Vector Magnetometry

PRX Quantum 6, 040364 (2025) · DOI: 10.1103/zcdm-5qq3 · arXiv: 2505.24574

Paper license: CC BY 4.0. Data license: CC BY-SA 4.0.

Abstract

Accurate measurement of vector magnetic fields is critical for applications including navigation, geoscience, and space exploration. Nitrogen-vacancy (NV) center spin ensembles offer a promising solution for high-sensitivity vector magnetometry, as their different orientations in the diamond lattice measure different components of the magnetic field. However, the bias magnetic field typically used to separate signals from each NV orientation introduces inaccuracy from drifts in permanent magnets or coils. Here, we present a novel bias-field-free approach that labels the NV orientations via the direction of the microwave (MW) field in a variable-pulse-duration Ramsey sequence used to manipulate the spin ensemble. Numerical simulations demonstrate the possibility to isolate each orientation's signal with sub-nT accuracy in most terrestrial fields, even without precise MW field calibration, at only a moderate cost to sensitivity. We also provide proof-of-principle experimental validation, observing relevant features that evolve as expected with applied magnetic field. Looking forward, by removing a key source of drift, the proposed protocol lays the groundwork for future deployment of NV magnetometers in high-accuracy or long-duration missions.

Figures

26 panels with data across 8 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

• panel (d): Heatmap of the population in $m_s = 0$, $P_0(t,\tau)$, against evolution time $\tau$ ($2\pi/\omega_L$) and pulse duration $t$ ($2\pi/\Omega$), in the $\Omega/\omega_L \rightarrow \infty$ limit with constant microwave phase, no detuning, no dephasing and an axial field. The printed colorbar runs from 1 to 0; here it runs from 0 to 1. data.csv
• panel (e): Magnitude of the two-dimensional Fourier transform of the population in panel (d), pulse duration frequency $\nu$ ($\Omega$) against evolution frequency $\omega$ ($\omega_L$), each discrete frequency a dot of area proportional to $|a_{nm}|$. The printed size legend is replaced by a color bar and hover values. data.csv

Fig. 2

• panel (a): Discrete Fourier components of $\mathcal{P}_0(\nu, \omega)$ for $\Omega = 5\omega_L$, no detuning or dephasing and constant microwave phase, pulse duration frequency $\nu$ ($\Omega_{\mathrm{eff}}$) against free evolution frequency $\omega$ ($\omega_L$), dot area proportional to magnitude. The printed arrows at the four components traced in panel (b) are not drawn. data.csv
• panel (b): Fourier amplitude of the four positive-quadrant double-quantum components, $\nu = 2\Omega_{\mathrm{eff}}$, $3\Omega_{\mathrm{eff}}/2$, $\Omega_{\mathrm{eff}}$ and $\Omega_{\mathrm{eff}}/2$, against $\Omega/\omega_L$. The dotted line marks the value used in panel (a). The printed broken axis and its $\Omega/\omega_L \rightarrow \infty$ limit markers are not kept; the axis is continuous. data.csv

Fig. 3

• panel (a): Fourier transform magnitude (arb. units) of a simulated signal for an NV ensemble, pulse duration frequency (MHz) against free evolution frequency (MHz), positive-frequency quadrant. Colored lines for $\langle111\rangle$, $\langle1\bar{1}1\rangle$, $\langle11\bar{1}\rangle$ and $\langle\bar{1}11\rangle$ mark each orientation's Rabi frequency and harmonics; the printed arrowheads are not drawn. data.csv
• panel (b): Minimum Rabi harmonic separation (% $\Omega_{\mathrm{max}}$) against MW azimuthal and polar angle (deg). The white contour bounds the directions where one $\Omega_i < 0.5\,\Omega_{\mathrm{max}}$; the white marker, an arrow in the print, is the optimal MW orientation. data.csv

Fig. 4

• panel (a): Boxcar-windowed two-dimensional inner product $I(\nu, \omega)$ of a simulated data set, normalized amplitude against free evolution frequency $\omega$ (MHz) and pulse duration frequency $\nu$ (MHz). The white line is $\Omega_{\mathrm{eff},\langle\bar{1}11\rangle}$, along which the cut in a-inset-2 is taken. The two printed insets are hosted as a-inset-1 and a-inset-2. data.csv
• panel (a-inset-1): Right inset of printed panel (a): filter function $F(\nu)$ against pulse duration frequency $\nu$ (MHz) for the Boxcar and Blackman windows. The printed inset has frequency on the vertical axis; here it is on the horizontal axis. data.csv
• panel (a-inset-2): Lower inset of printed panel (a): the 2D inner product normalized amplitude along the white line of panel (a) with a Blackman window, against free evolution frequency $\omega$ (MHz), with its fit by three Lorentzians. data.csv
• panel (b): Blackman-windowed $\langle\bar{1}11\rangle$ Ramsey signal (arb. units) against free evolution time ($\mu$s) for the same simulated data set as panel (a) (markers), with the fit by three exponentially decaying sinusoids (line). data.csv

Fig. 5

• panel (1): Sub-map 1 of 4: inversion error (nT, log color scale) for the $\langle111\rangle$ orientation against $B_{\mathrm{dc}}$ azimuthal and polar angle (deg), for a $B_{\mathrm{dc}} = 50\,\mu$T field. The printed dotted boxes and in-panel labels are not drawn. data.csv
• panel (2): Sub-map 2 of 4: inversion error (nT, log color scale) for the $\langle1\bar{1}1\rangle$ orientation against $B_{\mathrm{dc}}$ azimuthal and polar angle (deg), for a $B_{\mathrm{dc}} = 50\,\mu$T field. The white dash-dot curve is the zero axial field contour. The printed dotted boxes and in-panel labels are not drawn. data.csv
• panel (3): Sub-map 3 of 4: inversion error (nT, log color scale) for the $\langle11\bar{1}\rangle$ orientation against $B_{\mathrm{dc}}$ azimuthal and polar angle (deg), for a $B_{\mathrm{dc}} = 50\,\mu$T field. The printed dotted boxes and in-panel labels are not drawn. data.csv
• panel (4): Sub-map 4 of 4: inversion error (nT, log color scale) for the $\langle\bar{1}11\rangle$ orientation against $B_{\mathrm{dc}}$ azimuthal and polar angle (deg), for a $B_{\mathrm{dc}} = 50\,\mu$T field. The cyan cross marks the field direction of Figs. 4 and 6. The printed dotted boxes and in-panel labels are not drawn. data.csv

Fig. 6

• panel (a): Inversion error (nT) against microwave amplitude $\Omega_{\mathrm{max}}/2\pi$ (MHz) for the four NV orientations $\langle111\rangle$, $\langle1\bar{1}1\rangle$, $\langle11\bar{1}\rangle$ and $\langle\bar{1}11\rangle$, with the vertical axis clipped to $\pm 2$ nT as printed. data.csv
• panel (b-1): Sub-map 1 of 4 of panel (b): inversion error (nT, log colors) for the $\langle111\rangle$ orientation against MW azimuthal and polar angle (deg), at $\Omega_{\mathrm{max}}/2\pi = 100$ MHz. The cyan cross is the nominal MW direction. The printed dotted boxes and in-panel labels are not drawn. data.csv
• panel (b-2): Sub-map 2 of 4 of panel (b): inversion error (nT, log colors) for the $\langle1\bar{1}1\rangle$ orientation against MW azimuthal and polar angle (deg), at $\Omega_{\mathrm{max}}/2\pi = 100$ MHz. The cyan cross is the nominal MW direction; the red line marks coincidence of $3\Omega_{\langle111\rangle}/2$ with this orientation's $\Omega_i$. The printed dotted boxes and in-panel labels are not drawn. data.csv
• panel (b-3): Sub-map 3 of 4 of panel (b): inversion error (nT, log colors) for the $\langle11\bar{1}\rangle$ orientation against MW azimuthal and polar angle (deg), at $\Omega_{\mathrm{max}}/2\pi = 100$ MHz. The cyan cross is the nominal MW direction; the red line marks coincidence of $3\Omega_{\langle111\rangle}/2$ with this orientation's $\Omega_i$. The printed dotted boxes and in-panel labels are not drawn. data.csv
• panel (b-4): Sub-map 4 of 4 of panel (b): inversion error (nT, log colors) for the $\langle\bar{1}11\rangle$ orientation against MW azimuthal and polar angle (deg), at $\Omega_{\mathrm{max}}/2\pi = 100$ MHz. The cyan cross is the nominal MW direction; the red line marks coincidence of $3\Omega_{\langle111\rangle}/2$ with this orientation's $\Omega_i$. The printed dotted boxes and in-panel labels are not drawn. data.csv

Fig. 7

• panel (b): Heatmap of the Rabi oscillation FFT (arb. units), Rabi frequency (MHz) against diamond rotation offset (deg), with no applied dc field. The white line marks the angle chosen for the experiments. data.csv
• panel (c): Example Fourier-domain data set at a coil voltage of 4 V: 2D inner product magnitude (arb. units) against free evolution and pulse duration frequency (MHz). For each of four orientations a cross marks the fitted double-quantum maximum and dots of the same color the expected locations of its other double-quantum peaks. data.csv
• panel (d): Inferred bare Rabi frequency (MHz) against coil voltage (V) for four distinct but unknown NV orientations, one symbol each. data.csv
• panel (e): Inferred field projection magnitude ($\mu$T) on each of four NV orientations' axes against coil voltage (V) (markers). Solid lines are the projections from a fit to a field that varies linearly with voltage. data.csv
• panel (f): Field components $B_x$, $B_y$ and $B_z$ ($\mu$T), in crystal coordinates, of the linearly varying field fitted to the projections of panel (e), against coil voltage (V). data.csv

Fig. 8

• panel (a): Monte Carlo estimates of $\eta_{\mathrm{VPDR}}(\tau_{\mathrm{opt}})/\eta_R(\tau_{\mathrm{opt}})$ against maximum pulse duration (ns), with Blackman (blue) and Boxcar (red) windows, each for $m_I = 0$ only (circles) and the HF triplet (crosses). The dashed and dotted lines are the hard-pulse limits with the Blackman and boxcar windows. data.csv
• panel (b): Monte Carlo ratio of fit to $\tau_{\mathrm{opt}}$ sensitivity against maximum free evolution time ($\mu$s), log scale, for VPDR and Ramsey without (blue) and with (green, VPDR HF and Ramsey HF) hyperfine structure. Dashed and dotted lines mark $\tau_{\mathrm{opt}}$ without and with hyperfine structure. data.csv

Cite

Lilian Childress, Vincent Halde, Kayla Johnson, Andrew Lowther, David Roy-Guay, Romain Ruhlmann, Adrian Solyom. Bias-Field-Free Operation of Nitrogen-Vacancy Ensembles in Diamond for Accurate Vector Magnetometry. PRX Quantum 6, 040364 (2025). https://doi.org/10.1103/zcdm-5qq3

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