  The cross section of a given process fundamentally quantifies the probability
for that given process to occur. In the quantum regime of low energies, the
cross section can vary strongly with collision energy due to quantum effects.
Here, we report on a method to directly measure the atom-ion collisional cross
section in the energy range of 0.2-12 mK$\cdot$ k$_B$, by shuttling ultracold
atoms trapped in an optical-lattice across a radio-frequency trapped ion. In
this method, the average number of atom-ion collisions per experiment is below
one such that the energy resolution is not limited by the broad (power-law)
steady-state atom-ion energy distribution. Here, we estimate that the energy
resolution is below 200 $\mu$K$\cdot$k$_B$, limited by drifts in the ion's
excess micromotion compensation and can be reduced to the 10's
$\mu$K$\cdot$k$_B$ regime. This resolution is one order-of-magnitude better
than previous experiments measuring cold atom-ion collisional cross section
energy dependence. We used our method to measure the energy dependence of the
inelastic collision cross sections of a non-adiabatic
Electronic-Excitation-Exchange (EEE) and Spin-Orbit Change (SOC) processes. We
found that in the measured energy range, the EEE and SOC cross sections
statistically agree with the classical Langevin cross section. This method
allows for measuring the cross sections of various inelastic processes and
opens up possibilities to search for atom-ion quantum signatures such as
shape-resonances.
