  We investigate theoretically the many-emitter phonon laser based on optically
driven semiconductor quantum dots within an acoustic nanocavity. We map the
phonon laser Hamiltonian to a Tavis-Cummings type interaction with an
unexpected additional many-emitter energy shift. This many-emitter interaction
with the cavity mode results in a variety of resonances dependent on the number
of participating emitters. We show that the many-emitter phonon laser also
includes the single emitter resonance besides these collective phenomena.
However, we obtain a high quantum yield addressing these collective resonances.
We clearly demonstrate the best setup for maximal enhancement and show that the
output can be increased even more via lasing at the two phonon resonance.
