Standard Quantum Limit for Probing Mechanical Energy Quantization

We derive a standard quantum limit for probing mechanical energy quantization in a class of systems with mechanical modes parametrically coupled to external degrees of freedom. To resolve a single mechanical quantum, it requires a strong-coupling regime-the decay rate of external degrees of freedom is smaller than the parametric coupling rate. In the case for cavity-assisted optomechanical systems, e.g., the one proposed by Thompson et al. [Nature (London) 452, 72 (2008)], zero-point motion of the mechanical oscillator needs to be comparable to the linear dynamical range of the optical system which is characterized by the optical wavelength divided by the cavity finesse.