  We present a systematic study on the effects of dynamical transfer and
steady-state synchronization of quantum states in a hybrid optomechanical
network, consisting of two cavities with atoms inside and interacting via a
common moving mirror (i.e. mechanical oscillator), are studied. It is found
that high fidelity transfer of Schr\"{o}dinger's cat and squeezed states
between the cavities modes is possible. Additionally, we show the effect of
synchronization of cavity modes in a steady squeezed states at high fidelity
realizable by the mechanical oscillator which intermediates the generation,
transfer and stabilization of the squeezing. In this framework, we also have
studied the generation and evolution of bipartite and tripartite entanglement
and found its interconnection to the effects of transfer and synchronization.
Particularly, when the transfer occurs at the maximal fidelity, at this instant
any entanglement is almost zero, so the modes are disentangled. On the other
hand, when the two bosonic modes are synchronized in a squeezed stationary
state, then these modes are also entangled. The results found in this study may
find their applicability in quantum information and computation technologies,
as well in metrology setups, where the squeezed states are essential.
