  Both classical and quantum dynamics of the synchronization between two
nonlinear mechanical modes scattered from Bose-Einstein condensates (BECs) by
the standing-wave laser beam are comparatively investigated. As the ultra-low
dissipations of the momentum modes in the atomic BECs, the synchronized
dynamics are studied in a framework of closed-system theory in order to track
down both the classical and the quantum synchronizations from an angle of
quantum control. The classical synchronization and the relevant dynamics of
measure synchronization, the quantum synchronization and two different types of
measures proposed by Mari and estimated by mutual information based on
$Q$-function are studied respectively in order to reveal both the macroscopic
and the microscopic signatures of synchronized behaviors in a closed quantum
system. The results demonstrate that the "revival and collapse" of the quantum
fluctuations beyond the classical mean-value dynamics due to long-lasting mode
coherence discriminates the quantum synchronization from the classical one,
which not only excludes the possibilities of an exact synchronization and a
perfect density overlap in phase space, but also leads to upper limitations to
Mari measure and large unceasing fluctuations to mutual information between two
scattering modes. We reveal a close dynamic connection between Mari measure and
the mutual information of two nonlinear momentum modes in closed BEC systems by
demonstrating an opposite mean-value behavior but a similar fluctuation
variation with respect to their respective evolutionary scales.
