  Interferometers operating at or close to quantum limits of precision have
found wide application in tabletop searches for physics beyond the standard
model, the study of fundamental forces and symmetries of nature and
foundational tests of quantum mechanics. The limits imposed by quantum
fluctuations and measurement backaction on conventional interferometers
($\delta \phi \sim 1/\sqrt{N}$) have spurred the development of schemes to
circumvent these limits through quantum interference, multiparticle
interactions and entanglement. A prominent example of such schemes, the
so-called $SU(1,1)$ interferometer, has been shown to be particularly robust
against particle loss and inefficient detection, and has been demonstrated with
photons and ultracold atoms. Here, we realize a $SU(1,1)$ interferometer in a
fundamentally new platform in which the interfering arms are distinct flexural
modes of a millimeter-scale mechanical resonator. We realize up to 15.4(3) dB
of noise squeezing and demonstrate the Heisenberg scaling of interferometric
sensitivity ($\delta \phi \sim 1/N$), corresponding to a 6-fold improvement in
measurement precision over a conventional interferometer. Our work extends the
optomechanical toolbox for the quantum manipulation of macroscopic mechanical
motion and presents new avenues for studies of optomechanical sensing and the
nonequilibrium dynamics of multimode optomechanical systems.
