  We study the probability distribution of an atomic ion being laser-cooled in
a periodically-driven Paul trap using a Floquet approach to the semiclassical
photon scattering dynamics. We show that despite the microscopic nonequilibrium
forces, a stationary thermal-like exponential distribution can be obtained in
the Hamiltonian action, or equivalently in the number of quanta (phonons) of
the motion linearized about the zero of the potential. At the presence of
additional stray electric fields, the ion is pushed from the origin of the
potential and set into a large-amplitude driven oscillation, and above a
threshold amplitude of such "excess micromotion", the action distribution of
excitations about the driven oscillation broadens and becomes distinctly
nonthermal. We find that by a proper choice of the laser detuning the
distribution can be made exponential again, with a mean phonon number close to
that of the Doppler cooling limit. We derive a relation allowing to deduce just
from the experimentally observable photon scattering rate both the required
detuning for optimal cooling and the final mean phonon number. These results
are important for quantum information processing and other applications, and in
particular the derived approach can be applied to crystals of trapped ions in
planar configurations, where the driven motion of ions is unavoidable.
