  Results are reported from a search for a class of composite dark matter
models with feeble, long-range interactions with normal matter. We search for
impulses arising from passing dark matter particles by monitoring the
mechanical motion of an optically levitated nanogram mass over the course of
several days. Assuming such particles constitute the dominant component of dark
matter, this search places upper limits on their interaction with neutrons of
$\alpha_n \leq 1.2 \times 10^{-7}$ at 95\% confidence for dark matter masses
between 1--10 TeV and mediator masses $m_\phi \leq 0.1$ eV. Due to the large
enhancement of the cross-section for dark matter to coherently scatter from a
nanogram mass ($\sim 10^{29}$ times that for a single neutron) and the ability
to detect momentum transfers as small as $\sim$200 MeV/c, these results provide
sensitivity to certain classes of composite dark matter models that
substantially exceeds existing searches, including those employing kg-scale or
ton-scale targets. Extensions of these techniques can enable
directionally-sensitive searches for a broad class of previously inaccessible
heavy dark matter candidates.
