  Dynamical radiation pressure effects in cavity optomechanical systems give
rise to self-sustained oscillations or `phonon lasing' behavior, producing
stable oscillators up to GHz frequencies in nanoscale devices. Like in photonic
lasers, phonon lasing normally occurs in a single mechanical mode. We show here
that phase-locked, multimode phonon lasing can be established in a multimode
optomechanical system through Floquet dynamics induced by a temporally
modulated laser drive. We demonstrate this concept in a suitably engineered
silicon photonic nanocavity coupled to multiple GHz-frequency mechanical modes.
We find that the long-term frequency stability is significantly improved in the
multimode lasing state as a result of the phase locking. These results provide
a path towards highly stable ultra-compact oscillators, pulsed phonon lasing,
coherent waveform synthesis, and emergent many-mode phenomena in oscillator
arrays.
