  Nanomechanical resonators have demonstrated great potential for use as
versatile tools in a number of emerging quantum technologies. For such
applications, the performance of these systems is restricted by the decoherence
of their fragile quantum states, necessitating a thorough understanding of
their dissipative coupling to the surrounding environment. In bulk amorphous
solids, these dissipation channels are dominated at low temperatures by
parasitic coupling to intrinsic two-level system (TLS) defects, however, there
remains a disconnect between theory and experiment on how this damping
manifests in dimensionally-reduced nanomechanical resonators. Here, we present
an optomechanically-mediated thermal ringdown technique, which we use to
perform simultaneous measurements of the dissipation in four mechanical modes
of a cryogenically-cooled silicon nanoresonator, with resonant frequencies
ranging from 3 - 19 MHz. Analyzing the device's mechanical damping rate at
fridge temperatures between 10 mK - 10 K, we demonstrate quantitative agreement
with the standard tunneling model for TLS ensembles confined to one dimension.
From these fits, we extract the defect density of states ($P_0 \sim$ 1 - 4
$\times$ 10$^{44}$ J$^{-1}$ m$^{-3}$) and deformation potentials ($\gamma \sim$
1 - 2 eV), showing that each mechanical mode couples on average to less than a
single thermally-active defect at 10 mK.
