  Trapped ion in the Lamb-Dicke regime with the Lamb-Dicke parameter $\eta\ll1$
can be cooled down to its motional ground state using sideband cooling.
Standard sideband cooling works in the weak sideband coupling limit, where the
sideband coupling strength is small compared to the natural linewidth $\gamma$
of the internal excited state, with a cooling rate much less than $\gamma$.
Here we consider cooling schemes in the strong sideband coupling regime, where
the sideband coupling strength is comparable or even greater than $\gamma$. We
derive analytic expressions for the cooling rate and the average occupation of
the motional steady state in this regime, based on which we show that one can
reach a cooling rate which is proportional to $\gamma$, while at the same time
the steady state occupation increases by a correction term proportional to
$\eta^{2}$ compared to the weak sideband coupling limit. We demonstrate with
numerical simulations that our analytic expressions faithfully recover the
exact dynamics in the strong sideband coupling regime.
