Multiple frequencies of synchronization in classical and quantum networks.

Complex systems consisting of vector or matrix oscillators can synchronize to a common state characterized by a frequency matrix with distinct eigenvalues, leading to multiple frequencies of synchronization. In quantum networked systems the synchronized state is a linear combination of states corresponding to different energy levels. Suitable symmetry-breaking network interactions, however, allow only one or more such frequencies to appear. A specific example in three dimensions, where all trajectories lie on the 2-sphere, is a model of interacting spin-1 quantum angular momentum states, where synchronization to a nontrivial frequency occurs despite the presence of zero-frequency modes of oscillation.