Scheme for steady-state preparation of a harmonic oscillator in the first excited state

We present a generic quantum master equation whose dissipative dynamics autonomously stabilizes a harmonic oscillator in the n=1 Fock state. A multi-mode optomechanical system is analyzed and shown to be an example of a physical system obeying this model. We show that the optomechanical setup enables preparation of a mechanical oscillator in a nonclassical steady state, and that this state indeed approaches a single phonon Fock state in the ideal parameter regime. The generic model may be useful in other settings, such as cavity or circuit quantum electrodynamics or trapped ion physics.