  The Kuramoto model serves as a paradigm for describing spontaneous
synchronization in a system of classical interacting rotors. In this study, we
extend this model to the quantum domain by coupling quantum interacting rotors
to external baths following the Caldeira-Leggett approach. Studying the
mean-field model in the overdamped limit using Feynman-Vernon theory, we show
how quantum mechanics modifies the phase diagram. Specifically, we demonstrate
that quantum fluctuations hinder the emergence of synchronization, albeit not
entirely suppressing it. We examine the phase transition into the synchronized
phase at various temperatures, revealing that classical results are recovered
at high temperatures while a quantum phase transition occurs at zero
temperature. Additionally, we derive an analytical expression for the critical
coupling, highlighting its dependence on the model parameters, and examine the
differences between classical and quantum behavior.
