Key advances in the generation and shaping of spatially structured photonic fields both in the near and far field render possible the control of the duration, the phase, and the polarization state of the field distributions. For instance, optical vortices having a structured phase are nowadays routinely generated and exploited for a range of applications. While the light-matter interaction with optical vortices is meanwhile well studied, the distinctive features of the interaction of quantum matter with vector beams, meaning fields with spatially inhomogeneous polarization states, are still to be explored in full detail, which is done here. We analyze the response of atomic and low dimensional quantum structures to irradiation with radially or azimuthally polarized cylindrical vector beams. Striking differences to vortex beams are found: Radially polarized vector beams drive radially breathing charge-density oscillations via electric-type quantum transitions. Azimuthally polarized vector beams do not affect the charge at all but trigger, via a magnetic vector potential a dynamic Aharonov-Bohm effect, meaning a vector-potential driven oscillating magnetic moment. In contrast to vortex beams, no unidirectional currents are generated. Atoms driven by a radially polarized vector beam exhibit angular momentum conserving quadrupole transitions tunable by a static magnetic field, while when excited with azimuthally polarized beam different final-state magnetic sublevels can be accessed.
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