  Synchronization is a dynamical phenomenon found in complex systems ranging
from biological systems to human society and is characterized by the
constituents parts of a system locking their motion so that they have the same
phase and frequency. Recent intense efforts have focused on understanding
synchronization in quantum systems without clear semi-classical limits but no
comprehensive theory providing a systematic basis for the underlying physical
mechanisms has yet been found. Through a complete characterization of the
non-decaying persistently oscillating eigenmodes of smooth quantum evolutions
corresponding to quantum limit cycles, we provide a general algebraic theory of
quantum synchronization which provides detailed criteria for various types of
synchronization to occur. These results constitute the previously absent
framework within which to pursue a structured study of quantum synchronization.
This framework enables us to study both stable synchronization which lasts for
infinitely long times limit, and metastable synchronization which lasts for
long, but finite, times. Moreover, we give compact algebraic criteria that may
be checked in a given system to prove the absence of synchronization. We use
our theory to demonstrate the anti-synchronization of two spin-1/2's and
discuss synchronization in the Fermi-Hubbard model and its generalisations
which are relevant for various fermionic cold atom experiments, including
multi-species, multi-orbital and higher spin atoms.
