The Quantum Kuramoto Model

Abstract

Synchronization is an ubiquitous phenomenon occurring in social, biological and technological systems when the internal rhythms of a large number of units evolve coupled. This natural tendency towards dynamical consensus has spurred a large body of theoretical and experimental research during the last decades. The Kuramoto model constitutes the most studied and paradigmatic framework to study synchronization. In particular, it shows how synchronization shows up 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 has been considered to address different physical realizations. However, the Kuramoto model has been only studied within the domain of classical dynamics, thus neglecting its applications for the study of quantum synchronization phenomena. Here we provide with the quantization of the Kuramoto model. Based on a system-bath approach and within the Feynman path-integral formalism, we derive the equations for the Kuramoto model by taking into account the first quantum fluctuations. We also analyze its critical properties being the main result the derivation of the value for the synchronization onset. This critical coupling turns up to increase its value as quantumness increases, as a consequence of the possibility of tunneling that quantum fluctuations provide.