Phase projection errors in rf-driven optically pumped magnetometers

We investigate the phase relationship between the oscillating (rf) excitation field and the detected (light) power modulation in scalar rf-driven optically pumped magnetometers (OPMs), in particular in the $M_x$ configuration. While the static dependence of the demodulation phase on the direction of the external static magnetic field vector can be largely mitigated by aligning the oscillating rf field along the light propagation direction, we demonstrate that a dynamic (transient) phase response arises under magnetic field tilts. We analytically solve the corresponding modified Bloch equation and confirm agreement with experimental observations obtained using an $M_x$ magnetometer incorporating a paraffin-coated Cs vapor cell. The results reveal fundamental limitations of $M_x$ magnetometers regarding response time and accuracy, in particular when employed with active electronic feedback, such as a phase-locked loop. Therefore, this work is highly relevant to important magnetometry applications where the direction of the quasi-static magnetic field of interest is unknown \textit{a priori} or varies over time, or in measurements requiring a large detection bandwidth. Such conditions are encountered in applications such as geomagnetic surveying, particularly with mobile platforms.