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Equilibration of observables in closed quantum systems that are described bya unitary time evolution is a meanwhile well-established phenomenon apart froma few equally well-established exceptions. Here we report the surprisingtheoretical observation that integrable as well as non-integrable spin ringswith nearest-neighbor or long-range isotropic Heisenberg interaction not onlyequilibrate but moreover also synchronize the directions of the expectationvalues of the individual spins. We highlight that this differs from spontaneoussynchronization in quantum dissipative systems. Here, we observe mutualsynchronization of local spin directions in closed systems under unitary timeevolution. Contrary to dissipative systems, this synchronization is independentof whether the interaction is ferro- or antiferromagnetic. In our numericalsimulations, we investigate the free induction decay (FID) of an ensemble of upto $N = 25$ quantum spins with $s = 1/2$ each by solving the time-dependentSchr\"odinger equation numerically exactly. Our findings are related to, butnot fully explained by conservation laws of the system. The synchronization isvery robust against for instance random fluctuations of the Heisenbergcouplings and inhomogeneous magnetic fields. Synchronization is not observedwith strong enough symmetry-breaking interactions such as the dipolarinteraction. We also compare our results to closed-system classical spindynamics which does not exhibit phase synchronization due to the lack ofentanglement. For classical spin systems the fixed magnitude of individualspins effectively acts like additional $N$ conservation laws.

Observation of phase synchronization and alignment during free induction decay of quantum spins with Heisenberg interactions