  We study a mechanical oscillator coupled to a two-level system driven by a
blue-detuned coherent source in the resolved sideband regime. For weak
mechanical damping, we find dynamical instabilities leading to limit cycles.
They are signaled by strong fluctuations in the number of emitted photons, with
a large Fano factor. The phonon-number fluctuations exhibit a strikingly
similar behavior. When the coupling strength becomes comparable to the
mechanical frequency, non-classical mechanical states appear. We discuss the
relation with cavity optomechanical systems. Candidates for observing these
effects include superconducting qubits, NV centers, and single molecules
coupled to oscillators.
