Optimal synchronization deep in the quantum regime: Resource and fundamental limit

We develop an analytical framework to study the synchronization of a quantum self-sustained oscillator to an external signal. Our unified description allows us to identify the resource on which quantum synchronization relies, and to compare quantitatively the synchronization behavior of different limit cycles and signals. We focus on the most elementary quantum system that is able to host a self-sustained oscillation, namely, a single spin 1. Despite the spin having no classical analog, we first show that it can realize the van der Pol limit cycle deep in the quantum regime, which allows us to provide an analytical understanding to recently reported numerical results. Moving on to the equatorial limit cycle, we then reveal the existence of an interference-based quantum synchronization blockade and extend the classical Arnold tongue to a snakelike split tongue. Finally, we derive the maximum synchronization that can be achieved in the spin-1 system, and construct a limit cycle that reaches this fundamental limit asymptotically.