Role of coherence and degeneracies in quantum synchronization

Progress on the study of synchronisation in quantum systems has been largely driven by specific examples which resulted several frequency entrainment as well mutual synchronisation. Here we utilising Liouville space perturbation theory. We begin clarifying role centers, symmetries and oscillating coherences context then analyse eigenspectrum superoperator generating dynamics system determine conditions under arises. apply our framework to derive a powerful relationship between energy conservation, degeneracies systems. Finally, demonstrate approach analysing two mutually coupled thermal machines close thermodynamic quantities.