  Studying a single atomic ion confined in a time-dependent periodic anharmonic
potential, we find large amplitude trajectories stable for millions of
oscillation periods in the presence of stochastic laser cooling. The
competition between energy gain from the time-dependent drive and damping leads
to the stabilization of such stochastic limit cycles. Instead of converging to
the global minimum of the averaged potential, the steady-state phase-space
distribution develops multiple peaks in the regions of phase space where the
frequency of the motion is close to a multiple of the periodic drive. Such
distinct nonequilibrium behaviour can be observed in realistic radio-frequency
traps with laser-cooled ions, suggesting that Paul traps offer a
well-controlled test-bed for studying transport and dynamics of microscopically
driven systems.
