I am learning about Quantum synchronization and read [1,2] on Sunday, which are about many-body bosonic systems. I understand that what happens when the system synchronizes is that all the particles in the system start oscillating in phase. In practice this happens when the destruction and creation operators $a$ and $a^\dagger$ acquire a finite expectation value, and the system spontaneously breaks $U(1)$ symmetry. This happens in a driven-dissipative set-up.

On the other hand systems that are periodically driven can become time crystals and spontaneously break the discrete time-translation symmetry. I expect from the literature that this can not happen if time-translation symmetry is not already broken to a discrete one because nobody talks about it, but am not sure. Does the equilibrium argument apply to a nonequilibrium (driven-dissipative) set-up?

Both phenomena (quantum synchronization and time crystals) share the same idea of an emergent periodic time dependence, but the terminology (and literature) seem very different to me. I can see one difference between them in the fact that time-translation is explicitly broken (to a discrete symmetry) for time crystals but not for quantum synchronization, but what are the similarities and are there other differences between the two phenomena?

[1] https://iopscience.iop.org/article/10.1088/1367-2630/aae947

[2] https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.234101



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