  Optical microtraps provide a strong spatial confinement for laser-cooled
atoms. They can, e.g., be realized with strongly focused trapping light beams
or the optical near fields of nano-scale waveguides and photonic
nanostructures. Atoms in such traps often experience strongly spatially varying
AC Stark shifts which are proportional to the magnetic quantum number of the
respective energy level. These inhomogeneous fictitious magnetic fields can
cause a displacement of the trapping potential that depends on the Zeeman
state. Hitherto, this effect was mainly perceived as detrimental. However, it
also provides a means to probe and to manipulate the motional state of the
atoms in the trap by driving transitions between Zeeman states. Furthermore, by
applying additional real or fictitious magnetic fields, the state-dependence of
the trapping potential can be controlled. Here, using laser-cooled atoms that
are confined in a nanofiber-based optical dipole trap, we employ this control
in order to tune the microwave coupling of motional quantum states. We record
corresponding microwave spectra which allow us to infer the trap parameters as
well as the temperature of the atoms. Finally, we reduce the mean number of
motional quanta in one spatial dimension to $\langle n\rangle=0.3 \pm 0.1$ by
microwave sideband cooling. Our work shows that the inherent fictitious
magnetic fields in optical microtraps expand the experimental toolbox for
interrogating and manipulating cold atoms.
