Quantum Synchronization of Perturbed Oscillating Coherences

Quantum mutual synchronization has recently been explored through the persistent oscillation of local observables that arises from undamped eigenmodes of dissipative dynamics. However, these oscillating modes require strictly fine-tuning the system to satisfy algebraic constraints. Here, we investigate the robustness of synchronization against generic perturbations that break these constraints. We identify conditions under which the steady state of the perturbed system exhibits correlations that indicate mutual synchronization, even as the oscillations decay. That synchronization persists as imprints in the time-independent, asymptotic steady state directly bridges the dynamical notion of synchronization with the steady-state notion, which have so far been treated as distinct phenomena. Moreover, we discover in a spin-1 model that the resulting steady-state synchronization is manifested in unexpected geometries of locked phases that are multiples of $\pi/3$. Our work establishes a link between the two primary paradigms of quantum synchronization while demonstrating its inherent robustness against generic perturbations.