Quantum loss of synchronization in the dynamics of two spins

Motivated by the spin self-rephasing recently observed in an atomic clock, we introduce a simple dynamical model to study the competition between dephasing and synchronization. Two spins $S$ are taken to be initially parallel and in the plane perpendicular to an inhomogeneous magnetic field $\Delta$ that tends to dephase them. In addition, the spins are coupled by exchange interaction $J$ that tries to keep them locked. The analytical solution of the classical dynamics shows that, there is a phase transition to a synchronized regime for sufficiently large exchange interaction $J>\Delta$ compared to the inhomogeneity. The quantum dynamics is solved analytically in four limits -- large/small $J/\Delta$ and large/small $S$ -- and numerically in between. In sharp contrast to the classical case, the quantum solution features very rich $S$-dependent multiscale dynamics. For any finite $S$, there is no synchronization but a crossover around $J=\Delta$ between two regimes. The synchronization transition is only recovered when $S\to \infty$, approaching the classical solution in a non-trivial way. Quantum effects therefore suppress the synchronization transition.