  Backaction-evading (BAE) measurements of a mechanical resonator, by
continuously monitoring a single quadrature of motion, can achieve precision
below the zero-point uncertainty. When this happens, the measurement leaves the
resonator in a quantum squeezed state. The squeezed state so generated is
however conditional 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 achieve
deterministic manipulation of mechanical squeezing in an optomechanical system
subject to a continuous BAE measurement. We study in details two strategies,
direct (Markovian) and state-based (Bayesian) feedback. We show 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 friendly scenario) and an
imperfect BAE regime is considered, the ensuing non-optimal feedback may still
obtain significant amount of squeezing. In particular, we show that Bayesian
feedback control is nearly optimal for a wide range of sideband resolution. Our
analysis is of direct relevance for ultra-sensitive measurements and quantum
state engineering in state-of-the-art optomechanical devices.
