Injection locking and parametric locking in a superconducting circuit

When a signal is injected in a parametric oscillator close enough to its resonance, the oscillator frequency and phase are locked to those of the injected signal. Here we demonstrate two frequency-locking schemes using a Josephson mixer in the parametric down-conversion regime, pumped beyond the parametric oscillation threshold. The circuit then emits radiation out of two spectrally and spatially separated resonators at frequencies determined by the locking schemes that we choose. When we inject the signal close to a resonance, it locks the oscillator emission to the signal frequency by injection locking. When we inject the signal close to the difference of resonances, it locks the oscillator emission by parametric locking. We compare both schemes and investigate the dependence of the parametric locking range on the pump and the injected-signal power. Our results can be interpreted using Adler's theory for lasers, which provides a link between laser physics and superconducting circuits that could enable better understanding of pumped circuits for quantum-information applications such as error correction, circulators, and photon-number detectors.