Unconditional mechanical squeezing via backaction-evading measurements and nonoptimal feedback control

Backaction-evading (BAE) measurements of a mechanical resonator, by continuously monitoring a singlequadrature of motion, can achieve precision below the zero-point uncertainty. When this happens, the mea-surement leaves the resonator in a quantum squeezed state. The squeezed state so generated is however con-ditional on the measurement outcomes, while for most applications it is desirable to have a deterministic, i.e.,unconditional, squeezed state with the desired properties. In this work we apply feedback control to achievedeterministic manipulation of mechanical squeezing in an optomechanical system subject to a continuous BAEmeasurement. We study in details two strategies, direct (Markovian) and state-based (Bayesian) feedback. Weshow that both are capable to achieve optimal performances, i.e., a vanishing noise added by the feedback loop.Moreover, even when the feedback is restricted to be a time-varying mechanical force (experimentally friendlyscenario) and an imperfect BAE regime is considered, the ensuing non-optimal feedback may still obtain signif-icant amount of squeezing. In particular, we show that Bayesian feedback control is nearly optimal for a widerange of sideband resolution. Our analysis is of direct relevance for ultra-sensitive measurements and quantumstate engineering in state-of-the-art optomechanical devices