Phase projection errors in rf-driven optically pumped magnetometers

We study the phase 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 overcome by aligning the oscillating rf field along the light propagation direction, we show that a dynamic (transient) phase response can appear under tilts of the magnetic field. We analytically solve the corresponding modified Bloch equation and obtain agreement with experimental observations in an $M_x$ magnetometer setup using a paraffin-coated Cs vapor cell. The results reveal fundamental limitations of $M_x$ magnetometers in terms of response time and accuracy, in particular when operated with active electronic feedback, e.g., using a phase-locked loop. Thus, the work is highly relevant in important magnetometry applications, where the direction of the quasi-static magnetic field of interest is not known \textit{a priori} and/or not constant over time, or in measurements, in which a large detection bandwidth is paramount. Such conditions are encountered, e.g., in geomagnetic surveying, in particular with mobile platforms.