# A model of synchronization over quantum networks

## TL;DR

This paper extends the classical Kuramoto model to quantum networks, analyzing synchronization phenomena among quantum oscillators with nonlinear Schrödinger dynamics, and introduces a quantum order parameter to characterize synchronization.

## Contribution

It provides a comprehensive analysis of quantum synchronization in a non-Abelian Kuramoto model, including conditions for phase synchronization and the introduction of a quantum order parameter.

## Key findings

- Complete phase synchronization for identical oscillators.
- Synchronization of probability and current densities.
- Analysis based on correlations and quantum order parameter.

## Abstract

We investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by $N$ quantum oscillators ("nodes") connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schr\"odinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary $N>2$ identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the previous results, we obtain the synchronization of the probability and the current densities defined via the Madelung transformations.