Synchronization and asynchronization in non-Hermitian systems

Synchronization and asynchronization are ubiquitous occurrences in a wide range of natural and artificial systems. The underlying mechanisms responsible for these events are typically complex, presenting a considerable difficulty to grasp the crucial elements that cause synchronization or asynchronization. Moreover, real-world systems frequently encounter energy exchanges with their surrounding environment; synchronization and asynchronization in such non-Hermitian systems have not been fully understood. In this work, we study the synchronization and asynchronization phenomena in a simple non-Hermitian system that involves two coupled bosonic modes. We find that such an open system evolves towards a state of lower dissipation as it synchronizes, whereby the existence of particular symmetries, such as anti-parity-time (anti-$\mathcal{PT}$) symmetry and $\mathcal{PT}$ symmetry, impact synchronization. Peculiarly, we show that when the anti-$\mathcal{PT}$ symmetry is broken or the $\mathcal{PT}$ symmetry is unbroken, a state of asynchronization occurs as the result of the degeneracy of imaginary parts of the eigenvalues. We designate this occurrence as ``degeneracy-induced asynchronization.'' Our findings shed light on generating and controlling synchronization and asynchronization in non-Hermitian systems.