Exceptional points and dynamics of a non-Hermitian two-level system without PT symmetry

In this work, taking the most general non-Hermitian Hamiltonian without parity-time symmetry and with non-Hermitian off-diagonal elements into account, we study the exceptional points of the model system. We find that there exists an exceptional ellipse in theory, compared with the exceptional ring studied in non-Hermitian symmetry Hamiltonian, in which the parameters in the model Hamiltonian take particular values. We also report on the dynamics of the model Hamiltonian and symmetry Hamiltonian. We find that they display different dynamical behaviors under different parameters. Moreover, there exists an unstable behavior of the system. The unstable behavior has a surprising property by which the gain always dominates at long times no matter how small the gain is. This phenomenon enables the compensation for losses in dissipative systems and opens a wide range of applications in quantum optics, plasmonics, and optoelectronics, where the loss is inevitable but can be compensated by the gain.