  A single $^{40}$Ca ion is confine in the harmonic potential of a Paul trap
and cooled to a temperature of a few mK, with a wave packet of sub-m spatial
and sub-m/s velocity uncertainty. Deterministically extracted from the Paul
trap, the single ion is propagating over a distance of 0.27 m and detected. By
engineering the ion extraction process on the initial wave packet,
theoretically modeling the ion trajectories, and studying experimentally the
time-of-flight distribution, we directly infer the state of the previously
trapped ion. This analysis allows for accurate remote sensing of the previous
motional excitation in the trap potential, both coherently or incoherently. Our
method paves a way to extract, manipulate and design quantum wave packets also
outside of the Paul trap.
