  Mechanical transduction of torque has been key to probing a number of
physical phenomena, such as gravity, the angular momentum of light, the Casimir
effect, magnetism, and quantum oscillations. Following similar trends as mass
and force sensing, mechanical torque sensitivity can be dramatically improved
by scaling down the physical dimensions, and therefore moment of inertia, of a
torsional spring. Yet now, through precision nanofabrication and sub-wavelength
cavity optomechanics, we have reached a point where geometric optimization can
only provide marginal improvements to torque sensitivity. Instead, nanoscale
optomechanical measurements of torque are overwhelmingly hindered by thermal
noise. Here we present cryogenic measurements of a cavity-optomechanical
torsional resonator cooled in a dilution refrigerator to a temperature of 25
mK, corresponding to an average phonon occupation of <n> = 35, that demonstrate
a record-breaking torque sensitivity of 2.9 yNm/Hz^{1/2}. This a 270-fold
improvement over previous optomechanical torque sensors and just over an order
of magnitude from its standard quantum limit. Furthermore, we demonstrate that
mesoscopic test samples, such as micron-scale superconducting disks, can be
integrated with our cryogenic optomechanical torque sensing platform, in
contrast to other cryogenic optomechanical devices, opening the door for
mechanical torque spectroscopy of intrinsically quantum systems.
