  We investigate quantum control of a single atom in an optical tweezer trap
created by a tightly focused optical beam. We show that longitudinal
polarization components in the dipole trap arising from the breakdown of the
paraxial approximation give rise to significant internal-state decoherence. We
show that this effect can be mitigated by appropriate choice of magnetic bias
field, enabling Raman sideband cooling of a single atom close to its
three-dimensional ground state in an optical trap with a beam waist as small as
$w=900$ nm. We achieve vibrational occupation numbers of $\bar{n}_r = 0.01$ and
$\bar{n}_a = 8$ in the radial and axial directions of the trap, corresponding
to an rms size of the atomic wavepacket of 24 nm and 270 nm, respectively. This
represents a promising starting point for future hybrid quantum systems where
atoms are placed in close proximity to surfaces.
