Phonon amplification in two coupled cavities containing one mechanical   resonator
  We study a general theory of phonon lasing [I. S. Grudinin et al., Phys. Rev. Lett. 104, 083901 (2010)] 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 two-level atomic systems, respectively. We find that the phonon lasing can be realized in certain parameter regime near the threshold. The phase diagram and second-order correlation function of the phonon lasing are also studied to show some interesting phenomena that cannot be observed in the common photon lasing with the two-level systems. 
Hui Wang, Zhixin Wang, Jing Zhang, Sahin Kaya Ozdemir, Lan Yang, Yu-xi   Liu
quant-ph