  Synchronization is a widespread phenomenon encountered in many natural and
engineered systems with nonlinear classical dynamics. How synchronization
concepts and mechanisms transfer to the quantum realm and whether features are
universal or platform specific are timely questions of fundamental interest.
Here, we present a new approach to model incoherently driven dissipative
quantum systems susceptible to synchronization within the framework of
Josephson photonics devices, where a dc-biased Josephson junction creates
(non-classical) light in a microwave cavity. The combined quantum compound
constitutes a self-sustained oscillator with a neutrally stable phase. Linking
current noise to the full counting statistics of photon emission allows us to
capture phase diffusion, but moreover permits phase locking to an ac-signal and
mutual synchronization of two such devices. Thereby one can observe phase
stabilization leading to a sharp emission spectrum as well as unique photon
emission statistics revealing shot noise induced phase slips. Two-time
perturbation theory is used to obtain a reduced description of the oscillators
phase dynamics in form of a Fokker-Planck equation in generalization of
classical synchronization theories.
