Observation of synchronization between two quantum van der Pol oscillators in trapped ions

Abstract: Synchronization is a hallmark of nonlinear dynamics. It drives self-organized behavior across systems ranging from astronomy to chemistry. Among the simplest systems, the van der Pol oscillator captures the essence of limit-cycle behavior and forms the basis for diverse physical and biological models. While mutual synchronization has long been established in classical systems, it has yet to be experimentally observed in quantum limit-cycle oscillators, despite a decade of theoretical exploration. Here, we demonstrate synchronization between two quantum van der Pol oscillators using trapped ions. The synchronized state manifests itself as a stable relative phase, while the individual oscillator phases remain inaccessible. In view of potential sensing applications, we investigate the response of the synchronized system to an external field. Our results establish limit-cycle synchronization in the quantum regime and pave the way toward exploring complex synchronized dynamics in larger networks, where persistence of quantum features remains an open question.