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Injection locking can stabilize a source of radiation, leading to anefficient suppression of noise-induced spectral broadening and therefore, to anarrow spectrum. The technique is well established in laser physics, where aphenomenological description due to Adler is usually sufficient. Recently,locking experiments were performed in Josephson photonics devices, wheremicrowave radiation is created by inelastic Cooper pair tunneling across adc-biased Josephson junction connected in-series with a microwave resonator. Anin-depth theory of locking for such devices, accounting for the Josephsonnon-linearity and the specific engineered environments, is lacking.  Here, we study injection locking in a typical Josephson photonics devicewhere the environment consists of a single mode cavity, operated in theclassical regime. We show that an in-series resistance, however small, is animportant ingredient in describing self-sustained Josephson oscillations andenables the locking region. We derive a dynamical equation describing locking,similar to an Adler equation, from the specific circuit equations. The effectof noise on the locked Josephson phase is described in terms of phase slips ina modified washboard potential. For weak noise, the spectral broadening isreduced exponentially with the injection signal. When this signal is providedfrom a second Josephson device, the two devices synchronize. In the linearizedlimit, we recover the Kuramoto model of synchronized oscillators. The pictureof classical phase slips established here suggests a natural extension towardsa theory of locking in the quantum regime.

Injection locking and synchronization in Josephson photonics devices