The coupling of light and a mechanical resonator within an optomechanical setup can have significant effects on both the light field inside the cavity and the motion of the mechanical resonator. A prominent example is the cavity assisted side-band cooling of the mechanical motion, leading to low phonon occupation and thereby approaching the quantum regime [1,2]. However, while the preparation of highly nonclassical light fields like single photon states, entangled states or Schrödinger cat states is today an inherent part of the quantum optics toolbox, the preparation of nonclassical states for nanomechanical devices still constitutes a major challenge . Since the dynamics of purely harmonic oscillators will, even in the quantum regime, not differ from its classical analogue, a crucial requirement for observing quantum effects are effective nonlinearities which are typically provided by an ancilla system (ideally a qubit). Here we explore the alternative paradigm of exploiting the geometric nonlinearity inherent to doubly-clamped mechanical resonators. Thus, we consider the physics of a nanomechanical resonator with a Duffing nonlinearity of the form , coupled to different cavity modes that are each driven by a detuned laser. Here denotes the mechanical displacement and the strength of the nonlinearity. The system is coupled to its environment with decay rates κj for the cavity modes and γm for the mechanical mode.
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