Split-sideband spectroscopy in slowly modulated optomechanics

Optomechanical coupling between the motion of a mechanical oscillator and a cavity represents a new arena for experimental investigation of quantum effects on the mesoscopic and macroscopic scale.The motional sidebands of the output of a cavity offer ultra-sensitive probes of the dynamics. We introduce a scheme whereby these sidebands split asymmetrically and show how they may be used as experimental diagnostics and signatures of quantum noise limited dynamics. We show split-sidebands with controllable asymmetry occur by simultaneously modulating the light-mechanical coupling $g$ and $\omega_M$ - slowly and out of-phase. Such modulations are generic but already occur in optically trapped set-ups where the equilibrium point of the oscillator is varied cyclically. We analyse recently observed, but overlooked, experimental split-sideband asymmetries; although not yet in the quantum regime, the data suggests that split sideband structures are easily accessible to future experiments