  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.
