A saturation-absorption rubidium magnetometer with multilevel optical Bloch-equation modeling for intermediate-to-high fields

TL;DR

SASHMAG presents a rubidium-based magnetometer for intermediate-to-high magnetic fields ($0.2$–$0.4$ T) that operates in the hyperfine Paschen-Back regime. It combines saturated absorption spectroscopy in Faraday geometry with a multilevel optical Bloch-equation model solved in the uncoupled basis $|m_I,m_J angle$, plus a physics-constrained optimization to infer the magnetic field from sub-Doppler line centers. The approach is validated by reproducing power-broadened and Doppler-broadened spectral features and by achieving precise field estimates with MC-based uncertainty analysis, showing $ ext{B}$-estimates of $0.2602$–$0.4092$ T and a sensitivity of $0.42$ mT/$ ext{Hz}^{1/2}$. This framework enables high-field metrology with potential applications in MRI and fusion, and outlines a path toward autonomous sensing via synthetic data generated from the OBE model.

Abstract

We present SASHMAG (Saturated Absorption Spectroscopy High-field MAGnetometer), an atomic sensor designed for precision magnetic-field measurements in the intermediate-to-high field regime ($>0.2\,\text{T}$) using Rubidium-87 ($^{87}Rb$). The sensor operates in the hyperfine Paschen-Back regime, where the hyperfine and Zeeman interactions decouple, and utilizes counter-propagating pump-probe configuration in Faraday geometry to resolve isolated, Doppler-free Zeeman transitions. To interpret the resulting spectra in this strongly field-dependent regime, we developed a comprehensive multilevel optical Bloch-equation model solved explicitly in the uncoupled $\ket{m_I, m_J}$ basis, capturing state mixing and nonlinear saturation dynamics. This model reproduces measured spectra at sub-Doppler resolution and is consistent with analytical expectations for power broadening and thermal Doppler scaling. Magnetic field estimation is performed using a physics-constrained optimization routine that infers the magnetic field by minimizing the residual between experimentally extracted line centers and calculated transition frequencies from the field-dependent Hamiltonian. We demonstrate magnetic field retrieval from $0.2\,\text{T}$ to $0.4\,\text{T}$ with a precision of $\pm 0.0017 \,\text{T}$). Furthermore, the validated simulation establishes a foundation for generating synthetic training datasets, paving the way for autonomous, Machine Learning-enhanced magnetometry in applications ranging from MRI to fusion reactors.

Figures (5)

• Figure 1: (a) Experimental setup for saturated absorption spectroscopy (SAS) of the $^{87}$Rb. (b) Computational workflow for modeling high-field SAS using an optical Bloch equation framework.
• Figure 2: Optical schematic of the experimental setup for saturated absorption spectroscopy (SAS) of $^{87}\text{Rb}$. The light from the DBR Laser is split into two arms. The Reference Optics arm (top, orange section) provides coarse frequency markers using an Etalon and a stable, low-field ($0\text{ T}$) SAS lock signal using a separate ${}^{87}\text{Rb}$ cell and $\mathbf{PD2}$ for frequency positioning. The Main Experiment arm (bottom, blue section) performs the high magnetic field SAS measurement. This arm features the primary ${}^{87}\text{Rb}$ vapor cell ($\mathbf{VC}$) placed in a Faraday geometry between two permanent magnets ($\mathbf{PM}$) configured in a quasi-Helmholtz arrangement. The field-dependent spectrum is detected at $\mathbf{PD1}$. Key components include: OI (Optical Isolator), M (Mirror), $\lambda/2$ (Half Waveplate) and $\lambda/4$ (Quarter Waveplate), PBS (Polarizing Beam Splitter), BS (Beam Splitter), and PD (Photodiode).
• Figure 3: Frequency calibration and reference signals. The blue trace shows the Fabry–Perot transmission peaks (detected by PD3). The orange trace (top) is the saturated absorption spectrum (SAS) of the $^{87}\text{Rb}$$D_2$ line at zero field (PD2). The green trace (PD4) monitors the change in laser intensity during the ramping of the laser.
• Figure 4: (a) Stimulated saturated absorption spectra at magnetic field $B=0.4092$T (b) Experimental spectra obtained for the $^{87}\text{Rb}$$D_2$ transition at magnetic fields of $0.2602\,\text{T}$, $0.3263\,\text{T}$, and $0.4092\,\text{T}$ (from top to bottom). (c) Corresponding simulated spectra calculated using the Optical Bloch Equation (OBE) model at the same field strengths. All traces are vertically offset for clarity.
• Figure 5: (a) Simulated spectra showing power broadening at $I = 4.96\,\text{kW/m}^2$ and $27.1\,\text{kW/m}^2$. (b) Doppler-broadened profiles at $T = 300\,\text{K}$ (blue) and $600\,\text{K}$ (red). (c) Calculated $\sigma-$ transition detunings are shown as a function of the applied magnetic field $B$ (gray solid lines). Red circles indicate experimentally extracted resonance positions at $B = 0.2602$ T, $0.3263$ T, and $0.4092$ T, with vertical error bars representing the combined $1\sigma$ uncertainty from frequency calibration and peak fitting. Transitions 6, 7, and 8 belong to the $\ket{m_J=-1/2}\rightarrow\ket{m_J'=-3/2}$ manifold with $m_I = -1/2, +1/2,$ and $+3/2$, respectively..