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Motivated by the spin self-rephasing recently observed in an atomic clock, weintroduce a simple dynamical model to study the competition between dephasingand synchronization. Two spins $S$ are taken to be initially parallel and inthe plane perpendicular to an inhomogeneous magnetic field $\Delta$ that tendsto dephase them. In addition, the spins are coupled by exchange interaction $J$that tries to keep them locked. The analytical solution of the classicaldynamics shows that, there is a phase transition to a synchronized regime forsufficiently large exchange interaction $J>\Delta$ compared to theinhomogeneity. The quantum dynamics is solved analytically in four limits --large/small $J/\Delta$ and large/small $S$ -- and numerically in between. Insharp contrast to the classical case, the quantum solution features very rich$S$-dependent multiscale dynamics. For any finite $S$, there is nosynchronization but a crossover around $J=\Delta$ between two regimes. Thesynchronization transition is only recovered when $S\to \infty$, approachingthe classical solution in a non-trivial way. Quantum effects therefore suppressthe synchronization transition.

Quantum loss of synchronization in the dynamics of two spins