  We present experimental evidence of charge exchange between laser-cooled
potassium $^{39}$K atoms and calcium $^{40}$Ca$^+$ ions in a hybrid atom-ion
trap and give quantitative theoretical explanations for the observations. The
$^{39}$K atoms and $^{40}$Ca$^+$ ions are held in a magneto-optical (MOT) and a
linear Paul trap, respectively. Fluorescence detection and high resolution time
of flight mass spectra for both species are used to determine the remaining
number of $^{40}$Ca$^+$ ions, the increasing number of $^{39}$K$^+$ ions, and
$^{39}$K number density as functions of time. Simultaneous trap operation is
guaranteed by alternating periods of MOT and $^{40}$Ca$^+$ cooling lights, thus
avoiding direct ionization of $^{39}$K by the $^{40}$Ca$^+$ cooling light. We
show that the K-Ca$^+$ charge-exchange rate coefficient increases linearly from
zero with $^{39}$K number density and, surprisingly, the fraction of
$^{40}$Ca$^+$ ions in the 4p\,$^2$P$_{1/2}$ electronically-excited state.
Combined with our theoretical analysis, we conclude that these data can only be
explained by a process that starts with a potassium atom in its electronic
ground state and a calcium ion in its excited 4p\,$^2$P$_{1/2}$ state producing
ground-state $^{39}$K$^+$ ions and metastable, neutral Ca\,(3d4p$^3$P$_1$)
atoms, releasing only 150 cm$^{-1}$ equivalent relative kinetic energy.
Charge-exchange between either ground- or excited-state $^{39}$K and
ground-state $^{40}$Ca$^+$ is negligibly small as no energetically-favorable
product states are available. Our experimental and theoretical rate
coefficients of $9\times10^{-10}$ cm$^3$/s are in agreement given the
uncertainty budgets.
