  We report the detection of individual nuclear $\alpha$ decays through the
mechanical recoil of the entire micron-sized particle in which the decaying
nuclei are embedded. Momentum conservation ensures that such measurements are
sensitive to any particles emitted in the decay, including neutral particles
that may otherwise evade detection with existing techniques. Detection of the
minuscule recoil of an object more than $10^{12}$ times more massive than the
emitted particles is made possible by recently developed techniques in
levitated optomechanics, which enable high-precision optical control and
measurement of the mechanical motion of optically trapped particles.
Observation of a change in the net charge of the particle coincident with the
recoil allows decays to be identified with background levels at the
micro-Becquerel level. The techniques developed here may find use in fields
ranging from nuclear forensics to dark matter and neutrino physics.
