Phonon amplification in two coupled cavities containing one mechanical resonator

We study a general theory of phonon lasing [I. S. Grudinin, Phys. Rev. Lett. 104, 083901 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.083901] in coupled optomechancial systems. We derive the dynamical equation of the phonon lasing using supermodes formed by two cavity modes. A general threshold condition for phonon lasing is obtained. We also show the differences between phonon lasing and photon lasing, generated by photonic supermodes and a two-level atomic ensemble, respectively. We find that the phonon lasing can be produced in a certain parameter regime near the threshold. The phase diagrams and degrees of second-order coherence for the phonon lasing are also studied to show some interesting phenomena that cannot be observed in the photon lasing generated by two-level atomic ensembles.