  We analyze the amplitude and phase noise of limit-cycle oscillations in a
mechanical resonator coupled parametrically to an optical cavity driven above
its resonant frequency. At a given temperature the limit-cycle oscillations
have lower amplitude noise than states of the same average amplitude excited by
a pure harmonic drive; for sufficiently low thermal noise a sub-Poissonian
resonator state can be produced. We also calculate the linewidth narrowing that
occurs in the limit-cycle states, and show that while the minimum is set by
direct phase diffusion, diffusion due to the optical spring effect can dominate
if the cavity is not driven exactly at a side-band resonance.
