  Resolved sideband cooling is a standard technique for cooling trapped ions
below the Doppler limit to near their motional ground state. Yet, the most
common methods for sideband cooling implicitly rely on low Doppler-cooled
temperatures and tightly confined ions, and they cannot be optimized for
different experimental conditions. Here we introduce a framework which
calculates the fastest possible pulsed sideband cooling sequence for a given
number of pulses and set of experimental parameters, and we verify its
improvement compared to traditional methods using a trapped $^{171}$Yb$^+$ ion.
After extensive cooling, we find that the ion motional distribution is
distinctly non-thermal and thus not amenable to standard thermometry
techniques. We therefore develop and experimentally validate an improved method
to measure ion temperatures after sideband cooling. These techniques will
enable more efficient cooling and thermometry within trapped-ion systems,
especially those with high initial temperatures or spatially-extended ion
wavepackets.
