  Effective transition between the population-inverted optical eigenmodes of
two coupled microcavities carrying mechanical oscillation realizes a phonon
analogue of optical two-level laser. By providing an approach that linearizes
the dynamical equations of weak nonlinear systems without relying on their
steady states, we study such phonon laser action as a realistic dynamical
process, which exhibits time-dependent stimulated phonon field amplification
especially when one of the cavities is added with optical gain medium. The
approach we present explicitly gives the conditions for the optimum phonon
lasing, and thermal noise is found to be capable of facilitating the phonon
laser action significantly.
