  Synchronization is a ubiquitous phenomenon occurring in social, biological,
and technological systems when the internal rhythms of their constituents are
adapted to be in unison as a result of their coupling. This natural tendency
towards dynamical consensus has spurred a large body of theoretical and
experimental research in recent decades. The Kuramoto model constitutes the
most studied and paradigmatic framework in which to study synchronization. In
particular, it shows how synchronization appears as a phase transition from a
dynamically disordered state at some critical value for the coupling strength
between the interacting units. The critical properties of the synchronization
transition of this model have been widely studied and many variants of its
formulations have been considered to address different physical realizations.
However, the Kuramoto model has been studied only within the domain of
classical dynamics, thus neglecting its applications for the study of quantum
synchronization phenomena. Based on a system-bath approach and within the
Feynman path-integral formalism, we derive equations for the Kuramoto model by
taking into account the first quantum fluctuations. We also analyze its
critical properties, the main result being the derivation of the value for the
synchronization onset. This critical coupling increases its value as
quantumness increases, as a consequence of the possibility of tunneling that
quantum fluctuations provide.
