  We propose a scheme to generate large amount of mechanical squeezing, far
beyond the $\rm{3dB}$ limit, which is based on synthetic magnetism in
optomechanical system that hosts a Backward Stimulated Brillouin Scattering
(BSBS) process. Our benchmark system consists of an acoustic mode coupled to
two optical modes through the BSBS process, and a Duffing mechanical oscillator
that couples to the same optical modes through the standard optomechanical
radiation pressure. The synthetic magnetism comes from the modulation of the
mechanical coupling between the acoustic and the mechanical mode. When there is
no synthetic magnetism, a given amount of mechanical squeezing is generated in
the system. This squeezing is mainly dependent on the BSBS process, and it is
fragile against thermal noise. By switching on the synthetic magnetism, the
degree of the generated squeezing is greatly enhanced and goes far beyond the
limit of the $\rm{3dB}$. This large magnetism induced squeezing persists even
when there is no BSBS process in the system. Moreover, this generated squeezing
is robust enough against thermal noise in comparison to the one induced when
the synthetic magnetism is off. Furthermore, both the mechanical variance
squeezing and effective phonon number exhibit series of peaks and dips
depending on the phase modulation of the mechanical coupling. This oscillatory
feature is reminiscent of a sudden death and revival of squeezing phenomenon,
which can be used to maintain a desired magnitude of squeezing by tuning this
phase. Our proposal provides a path toward a flexible scheme that generates
large amount of squeezing, far beyond the $\rm{3dB}$ limit. Such a generated
squeezed states can be used for quantum applications including quantum
information processing, quantum sensing and metrology, and quantum computing.
