  We report on the injection locking of an optically levitated nanomechanical
oscillator (a silica nanosphere) to resonant intensity modulations of an
external optical signal. We explore the characteristic features of injection
locking in this system, e.g. the phase pull-in effect and the injection-induced
reduction of the oscillation linewidth. Our measurements are in good agreement
with theoretical predictions and deepen the analogy of injection locking in
levitated optomechanical systems to that in optical systems (lasers). By
measuring the force noise of our feedback cooled free-running oscillator, we
attain a force sensitivity of $\sim23~\rm{zN}/\sqrt{\rm{Hz}}$. This can readily
allow, in fairly short integration times, for tests of violations of Newtonian
gravity and searching for new small-scale forces. As a proof of concept, we
show that the injection locking can be exploited to measure the forces
optically induced on levitated nanoparticles, with potential applications in
explorations of optical binding and entanglement between optically coupled
nanomechanical oscillators.
