Bias-field-free operation of nitrogen-vacancy ensembles in diamond for accurate vector magnetometry

TL;DR

This work presents a bias-field-free method for vector magnetometry with NV ensembles by labeling NV orientations through the angle of a single microwave drive in a variable-pulse-duration Ramsey sequence. The VPDR protocol yields orientation-specific, cross-talk-free signals by exploiting a phase-cancelled double-quantum Ramsey component that is robust to detuning and temperature fluctuations, with orientation separation achieved via distinct Rabi frequencies. Theory, numerical simulations, and a proof-of-principle experiment show sub-nanotesla axial-field accuracy across most terrestrial field directions, along with resilience to MW amplitude and direction drifts. The approach eliminates large bias magnets, supports long-duration operation, and offers a practical path toward high-accuracy NV vector magnetometry in compact setups, albeit with dead zones and a need for sign determination of projections.

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 (9)

• Figure 1: (a) Illustration of the four possible directions of the NV symmetry axis (N-to-V axis) in the conventional diamond crystal unit cell. (b) Variable-pulse-duration Ramsey (VPDR) sequence with two microwave (MW) pulses of duration $t$ separated by free evolution time $\tau$. (c) Illustration of dynamics in the bright-dark basis. (d) Population in $|0\rangle$ ($P_0(t,\tau)$) after application of the pulse sequence in (b) in the $\Omega/\omega_L \rightarrow \infty$ limit, with constant MW phase, no detuning, no dephasing, and an axial magnetic field. The pulse duration $t$ is given in units of the Rabi period and the evolution time $\tau$ is given in units of the Larmor period. (e) Magnitude of the 2D Fourier transform of the population in (d), with discrete frequencies illustrated with dots of area proportional to their magnitude. The quantitative values given in the legend are $|a_{nm}|$ in the Fourier expansion $P_0(t,\tau) = \sum_n \sum_m a_{nm}e^{i \nu_n t}e^{i\omega_m \tau}$. The frequency associated with the pulse duration (free evolution time) is given in units of the Rabi frequency $\Omega$ (Larmor frequency $\omega_L$).
• Figure 2: (a) $\mathcal{P}_0(\nu, \omega)$ for $\Omega = 5 \omega_L$, no detuning or dephasing, and constant MW phase. The area of the dots is proportional to the magnitude of the discrete Fourier components as in Fig. \ref{['fig1']}e. The pulse duration frequency $\nu$ is given in units of the effective Rabi frequency $\Omega_{\mathrm{eff}} = \sqrt{\Omega^2 + 4\omega_L^2}$ while the free evolution frequency is given in units of the Larmor frequency $\omega_L$. (b) Amplitude of the positive-quadrant $DQ$ Fourier components, corresponding to the four components indicated with arrows in (a), as a function of $\Omega/\omega_L$. The dotted line indicates the value of $\Omega/\omega_L$ for which (a) is evaluated.
• Figure 3: (a) The discrete Fourier transform (positive-frequency quadrant only) of a simulated VPDR Ramsey signal with SQ cancellation for an NV ensemble. $\Omega_\text{max}/2\pi =$ 100 MHz, $T_2^* = 2~\upmu$s, $\mathrm{B} = (185, 204, 223)~\upmu$T in the conventional diamond crystal coordinates, and $\mathbf{B_\text{MW}} \parallel (0.2054, .1188, .9714)$ corresponding to the locally optimal orientation indicated in (b). Thick arrow-tipped lines indicate each orientation's bare Rabi frequency $\Omega_i$, while thin arrow-tipped lines indicate $\Omega_i/2, 3\Omega_i /2$ and $2\Omega_i$. (b) Optimization of the MW magnetic field direction. The minimum separation between $\Omega_i$ and $n\Omega_j$ for NV orientations $i\neq j$ and $n \in \{\frac{1}{2},1,\frac{3}{2},2\}$ is given as a function of the direction of the MW magnetic field $\mathbf{B_\text{MW}}$ relative to the conventional diamond unit cell. For directions inside the white contour, one of the $\Omega_i$ is less than $\Omega_\text{max}/2$; the white arrow indicates an optimal orientation outside the contour.
• Figure 4: Inner-product-based inversion. All plots in this figure analyze the same simulated $SQ$-cancelled data set $S_\text{VPDR}(t_j, \tau_k)$ performed for $\mathbf{B_\text{DC}} = (-38.4, 25.7, 19.1)~\upmu$T in conventional diamond unit cell coordinates, at the optimum MW angle from Fig. \ref{['fig3']}, with $\Omega_\text{max} = 100$ MHz, $T_2^* = 2\upmu$s, a maximum pulse duration $t_j$ of 397.5 ns and a pulse step size of 2.5 ns. a) Boxcar-windowed two-dimensional inner product $I(\nu, \omega)$ as defined in Eqs. \ref{['eq:I']}-\ref{['eq:f']}. (right inset) Filter functions $F(\nu)\propto\sum_j W(t_j)\cos{\nu t_j}$ for boxcar (blue) and Blackman (orange) windows. (lower inset) $I(\nu, \omega)$ evaluated along the white line in (a), with a Blackman window, fit with three Lorentzians. (b) Blackman-windowed $\tau-$domain Ramsey signal $f(\tau_k) \propto \sum_j S_\text{VPDR}(t_j,\tau_k)W(t_j)\cos{(\Omega_{\langle\bar{1}11\rangle} t_j)}$. The signal is fit with the sum of three decaying sinusoids with commensurate decay times, frequencies constrained to $\Delta\omega = 2|\omega_0 + m_I A|$ for $m_I \in \{-1, 0, 1\}$, and variable phase (due to an inner product at fixed $\Omega$ rather than $\Omega_\text{eff}$).
• Figure 5: Systematic inversion error for a $B_\text{DC} = 50~\upmu$T field with varying direction relative to diamond crystalline axes. Simulations are run with $\Omega_\text{max}=100$ MHz, $T_2^* = 2\upmu$s, MW direction as shown in Fig. \ref{['fig3']}b, pulse durations up to 797.5 ns in 2.5 ns steps, and free evolution times up to 2.98 $\upmu$s in 20 ns steps. The time-domain Ramsey signal $f(\tau, \Omega)$ is calculated for each orientation using a Blackman window and fit to extract the $m_s = \pm 1$ transition frequency of the highest-frequency hyperfine line. The error compared to the exact eigenvalue difference is expressed as equivalent error in the axial field, with plots for each orientation labeled by their crystallographic direction and approximate Rabi frequency $\Omega$. The dashed-dotted white line shows the region of vanishing axial field for the $\langle 1\bar{1}1\rangle$ orientation. Dotted boxes indicate regions in which the indicated rms errors are calculated; the cyan $\times$ marks the field and orientation for which Fig. \ref{['fig4']} is calculated.