  In a sequence of single spins interacting longitudinally with a mechanical
oscillator, and using the micromaser model with random injection, we show that
after an appropriate post-selection of each spin, a phonon laser analog with
Poisson statistics is created with nearly perfect coherence, evidenced by the
second order coherence function that goes asymptotically to one. The non-linear
gain of the system depends crucially on the properly post-selected spin state
as well as the pump. Our model and results suggest that the mechanism of
interaction followed by a post-selected state or partial trace (common in
laser/maser theory) of the spins may create the coherent vibrational radiation.
However, for situations where the mechanical losses are high and it is
imposible to decrease these, then the heralded post-selection can be the only
resource to get phonon lasing if compared to the partial trace operation. These
ideas and results may be useful for further theoretical and technical
developments.
