Quantum-coherent phase oscillations in synchronization

Recently, synchronization of quantum systems has attracted increasing attention. So far, in these studies the synchronization dynamics itself remained overdamped and thus necessarily incoherent. We study the generic model of a quantum Van der Pol oscillator synchronized to an external drive and show that regimes of underdamped and even quantum-coherent phase motion exist. To this end, we derive an effective quantum model which allows us to quantify the quality of quantum coherence. We identify the quantum-coherent regime and illustrate the long-lived coherence by showing that initial negativities of a Wigner density can persist many oscillations of the system dynamics. Possible experimental implementations can be envisioned with optomechanical systems, trapped ions, and microwave circuits.