  We perform numerical simulations of trapped $^{171}$Yb$^+$ ions that are
buffer gas cooled by a cold cloud of $^6$Li atoms. This species combination has
been suggested to be the most promising for reaching the quantum regime of
interacting atoms and ions in a Paul trap. Treating the atoms and ions
classically, we compute that the collision energy indeed reaches below the
quantum limit for a perfect linear Paul trap. We analyze the effect of
imperfections in the ion trap that cause excess micromotion. We find that the
suppression of excess micromotion required to reach the quantum limit should be
within experimental reach. Indeed, although the requirements are strong, they
are not excessive and lie within reported values in the literature. We analyze
the detection and suppression of excess micromotion in our experimental setup.
Using the obtained experimental parameters in our simulation, we calculate
collision energies that are a factor 2-11 larger than the quantum limit,
indicating that improvements in micromotion detection and compensation are
needed there. We also analyze the buffer-gas cooling of linear and
two-dimensional ion crystals. We find that the energy stored in the eigenmodes
of ion motion may reach 10-100 $\mu$K after buffer-gas cooling under realistic
experimental circumstances. Interestingly, not all eigenmodes are buffer-gas
cooled to the same energy. Our results show that with modest improvements of
our experiment, studying atom-ion mixtures in the quantum regime is in reach,
allowing for buffer-gas cooling of the trapped ion quantum platform and to
study the occurrence of atom-ion Feshbach resonances.
