  Here we show that quantum states of a mechanical oscillator can be generated
in an optomechanical analogue of the micromaser, in absence of any atom-like
subsystem, thus exhibiting single-atom masing effects in a system composed
solely of oscillator components. In the regime where the single-photon coupling
strength is on the order of the cavity decay rate, a cavity mode with at most a
single-excitation present gives rise to sub-Poissonian oscillator limit-cycles
that generate quantum features in the steady state just above the renormalized
cavity resonance frequency and mechanical sidebands. The merger of multiple
stable limit-cycles markedly reduces these nonclassical signatures. Varying the
cavity-resonator coupling strength, corresponding to the micromaser pump
parameter, reveals transitions for the oscillator phonon number that are the
hallmark of a micromaser. The connection to the micromaser allows for a
physical understanding of how nonclassical states arise in this system, and how
best to maximize these signatures for experimental observation.
