Abstract
Achieving and controlling non-reciprocity in engineered photonic structures is of fundamental interest in science and engineering. Here, we introduce a tunable, non-Hermitian, nonlinear microwave dimer designed to precisely implement phase-non-reciprocal hopping dynamics between two spatially separated cavities at room temperature. Our system incorporates simple components such as three-dimensional microwave cavities, unidirectional amplifiers, digital attenuators, and a digital phase shifter. By dividing the energy transfer into forward and backward paths, our platform enables precise control over the amplitude and phase of the propagating signals in each direction. Through a combination of theoretical and numerical analysis, we model the dynamics of the system under different operating conditions, including a parameter regime where the gain not only compensates for but significantly exceeds the inherent loss. Our model quantitatively reproduces the observed weak-drive transmission spectra, the amplitude and frequency of self-sustained limit cycles, and the phase locking synchronization effect between the limit cycle and an external microwave tone. Our results may have implications in areas ranging from sensing and synthetic photonic materials to neuromorphic computing and quantum networks, while providing new insight into the interplay between non-Hermitian and nonlinear dynamics.