  We propose nonreciprocal phonon lasing in a coupled cavity system composed of
an optomechanical and a spinning resonator. We show that the optical Sagnac
effect leads to significant modifications in both the mechanical gain and the
power threshold for phonon lasing. More importantly, the phonon lasing in this
system is unidirectional, that is the phonon lasing takes place when the
coupled system is driven in one direction but not the other. Our work
establishes the potential of spinning optomechanical devices for low-power
mechanical isolation and unidirectional amplification. This provides a new
route, well within the reach of current experimental abilities, to operate
cavity optomechanics devices for such a wide range of applications as
directional phonon switches, invisible sound sensing, and topological or chiral
acoustics.
