Exceptional points and dynamics of an asymmetric non-Hermitian two-level system

We investigated a damped two-level system interacting with a circularly polarized light as described by an asymmetric non-Hermitian Hamiltonian. This is a simple enough system to be studied analytically while complicated enough to exhibit a rich variety of behaviors. This system exhibits a ring of exceptional points in the parameter space of the real and imaginary dipole couplings where within the ring the energy eigenvalue of the system does not change. This leads to unstable regions inside the exceptional ring, which is shown using a linear stability analysis. These unstable regions are unique to gain-loss systems and have the surprising property that no matter how small the gain/loss ratio, the gain always prevails at long times. We also report on eigenvalue switching, phase rigidity and dynamics of the system around the exceptional points. We highlight that some of these properties are different from those in the widely studied case of symmetric non-Hermitian Hamiltonians.