Half-integer vs. integer effects in quantum synchronization of spin systems

We study the quantum synchronization of a single spin driven by an external semiclassical signal for spin numbers larger than $S = 1$, the smallest system to host a quantum self-sustained oscillator. The occurrence of interference-based quantum synchronization blockade is found to be qualitatively different for integer vs. half-integer spin number $S$. We explain this phenomenon as the interplay between the external signal and the structure of the limit cycle in the generation of coherence in the system. Moreover, we show that the same dissipative limit-cycle stabilization mechanism leads to very different levels of quantum synchronization for integer vs. half-integer $S$. However, by choosing an appropriate limit cycle for each spin number, comparable levels of quantum synchronization can be achieved for both integer and half-integer spin systems.