An Accurate Vector Magnetometer via Zeeman Rabi Oscillations

Abstract

Accurate magnetic field direction sensing in compact platforms is critical in applications spanning magnetic navigation, space science, and biomedical imaging. We demonstrate a single-optical-axis vector optically pumped magnetometer based on Rabi oscillations between Zeeman sublevels driven by a series of resonant radiofrequency (RF) polarization ellipses (PEs). A calibration protocol based on controlled rotations of the DC magnetic field determines the spatial orientation of each PE. We develop a detailed theoretical model describing the angular dependence of the Rabi frequencies, incorporating key systematics including RF Stark shifts and Bloch-Siegert shifts. We also account for an RF-based heading-error systematic affecting Rabi-frequency measurements arising from the nonlinear Zeeman effect. Simultaneous Larmor measurements yield the magnitude of the magnetic field, enabling integrated vector-scalar measurements. The magnetometer achieves deadzone-free vector operation with 80 $μ$rad mean angular accuracy and angular noise densities as low as 8 $μ$rad$/\sqrt{\mathrm{Hz}}$, offering a pathway towards miniaturized sensors without requiring 3D optical access or sensor rotations.

Figures (10)

• Figure 1: Vector magnetometry using Zeeman Rabi oscillations driven by resonant RF PEs. (a) Level diagram of $^{87}$Rb in a 50 $\upmu$T DC field, showing Larmor spin precession (blue arrows) at frequency $\nu_L$. Dotted lines depict dressed states in the rotating frame, while red arrows denote Rabi oscillations under a resonant RF PE, $\bm{\mathcal{B}}_\mathrm{{RF}}$. (b) Angular dependence of $\Omega_{\sigma+}$ is shown as a function of $\mathbf{B}_{\mathrm{DC}}$ direction, $\left(\alpha,\beta\right)$ in the orthogonalized laboratory X--Y--Z coordinate system. $\langle\mathbf{F}\rangle$ denotes the initial optically pumped spin polarization. (c) $\hat{x}'_B- \hat{y}'_B-\hat{z}_B$ denotes the rotating frame coordinate system (rotating at RF frequency, $\nu_{RF} = \omega_{RF}/2\pi$), with $\hat{z}_B$ aligned along the DC field direction. In this frame, Rabi oscillation is visualized as spin precession driven by $\left(\bm{\mathcal{B}}_\mathrm{RF}\right)_\mathrm{\sigma+}$. Bloch--Siegert shifts tilt the effective precession axis by $\Phi$. (d) Faraday rotation, $\theta_f$ measurement showing Zeeman Rabi oscillations. (e) Angular dependence of the Rabi frequency, $\Omega_{\sigma+}$ for the PEs used. Inset highlights a zoomed region of PE 4, showing variations in vector sensitivity determined by the angular gradient of the Rabi frequency.
• Figure 2: Experimental schematic showing co--propagating pump and probe lasers through a microfabricated $^{87}$Rb vapor cell, RF and DC coil systems, polarizing beam splitter (PBS) and photodetectors (PDs).
• Figure 3: Systematic effects in Rabi frequency measurements and comparison with theory. (a) Residuals between measured Rabi frequency and predictions from a RWA model, $\Delta\Omega_{\sigma+,\mathrm{RWA}}$ as a function of magnetic field direction. Bottom and left panels show residuals from the RWA model and a Floquet--based model, $\Delta\Omega_{\sigma+,\mathrm{Flq}}$ along the horizontal and vertical cuts at fixed $\beta$ = 28.65$^\circ$ and $\alpha = 80.21^\circ$, respectively. Measurement uncertainties are smaller than the markers. (b) Variation of $\Omega_{\sigma+}$ with the applied RF phase, $\phi$ for three polar angles. $\alpha=0$ for all three directions. Results of the model (solid lines) accounting for dynamic heading errors in the PE, $\bm{\mathcal{B}}_\mathrm{RF}\sim 9.1 \cos(\omega_{RF}t+\phi_0+\phi) \hat{x}\; \upmu \mathrm{T}$ with technical offset phase, $\phi_0 \sim 232^\circ$.
• Figure 4: Schematic of the measurement protocol illustrating PE calibration and vector magnetometry sequences. (a) Timing diagram illustrating the sequence of the applied 50 $\upmu$T DC magnetic field orientations. During calibration, Rabi and Larmor frequencies are measured at thirty random but predefined magnetic field directions, $\left(\alpha_{cj},\beta_{cj}\right)$, followed by measurements at random test directions, $\left(\alpha_{tj},\beta_{tj}\right)$, each repeated 10 times. Each DC field change is followed by a 0.85 s delay to mitigate transient effects from the DC current controller. (b) For each DC field orientation, atoms are interrogated using six PEs generated by distinct permutations of RF pulses applied to the RF coil system. $\mathrm{B}_{\mathrm{RF,i}}, i\in \{x,y,z\}$ denotes the RF signals applied to the corresponding coil pair. To reduce technical noise, each Rabi measurement is repeated and averaged 20 times, while each Larmor measurement is averaged over 5 repetitions.
• Figure 5: Evaluation of vector magnetometer performance. (a) Transverse component errors, $\overline{\updelta\mathrm{B}}_\mathrm{x}$, $\overline{\updelta\mathrm{B}}_\mathrm{y}$ and (b) transverse component noise densities $\mathrm{S_{B_x},S_{B_y}}$ for more than 340 applied magnetic field orientations. Each point corresponds to a distinct field direction and is color coded by polar angle, $\beta$. Gray contours indicate boundaries of constant angular accuracy, $\delta\theta$ and angular noise density, $S_{\theta}$, and dashed lines mark the mean angular accuracy (80 $\upmu$rad) and noise density (22 $\upmu$rad$/\sqrt{\mathrm{Hz}}$) over the test directions in (a) and (b), respectively. (c) Histogram shows the relative angular error, $\mathrm{\Delta\uptheta}$ between vector measurements obtained using disjoint sets of PEs. (d) Angular noise density, $S_{\theta}$ binned as a function of the polar angle, $\beta$, illustrating the variation of noise with magnetic field orientation.