  We study the time evolution of two coupled many-body quantum systems one of
which is assumed to be Bose condensed. Specifically, we consider two ultracold
atomic clouds populating each two localized single-particle states, i.e. a
two-component Bosonic Josephson junction. The cold atoms cloud can retain its
coherence when coupled to the condensate and displays synchronization with the
latter, differing from usual entrainment. We term this effect among the
ultracold and the condensed clouds as {\it hybrid synchronization}. The onset
of synchronization, which we observe in the evolution of average properties of
both gases when increasing their coupling, is found to be related to the
many-body properties of the quantum gas, e.g. condensed fraction, quantum
fluctuations of the particle number differences. We discuss the effects of
different initial preparations, the influence of unequal particle numbers for
the two clouds, and explore the dependence on the initial quantum state, e.g.
coherent state, squeezed state and Fock state, finding essentially the same
phenomenology in all cases.
