Probing the isolated vector magnetic field of structured laser beams by atoms

Electric and magnetic fields are inherently coupled in an electromagnetic wave. However, structured light beams enable their spatial separation. In particular, azimuthally polarized laser beams exhibit a localized magnetic field on-axis without the electric counterpart. Recent study by Martin-Domene et al. [App. Phys. Lett. 124, 211101 (2024)] has shown that combining these beams enables the generation of locally isolated magnetic fields with a controllable direction and phase. In the present paper we propose a method to probe and characterize such magnetic fields by studying their interaction with a single trapped atom. In order to theoretically investigate magnetic sublevel populations and their dependence on the relative orientation and phase -- i.e. the polarization state -- of the isolated magnetic field, we use a time-dependent density-matrix method based on the Liouville-von Neumann equation. As illustrative cases, we consider the $2s^2 2p^2 \, {}^3P_0 \, - \, 2s^2 2p^2 \, {}^3P_1$, the $1s^2 2s^2 \, {}^1S_0 \, - \, 1s^2 2s 2p \, {}^3P_2$, and the $2 s^2 2p \, {}^2 P_{1/2} \, - \, 2 s^2 2p \, {}^2 P_{3/2}$ transitions in ${}^{40}$Ca$^{14+}$, ${}^{10}$Be, and ${}^{38}$Ar$^{13+}$, respectively. Our results indicate that monitoring atomic populations serves as an effective tool for probing isolated vector magnetic fields, which opens avenues for studying laser-induced processes in atomic systems where electric field suppression is critical.