Time Evolution of Two-States Non-Hermitian Systems

We study the time evolution of two-states non-Hermitian systems. The exact time-dependent wavefunctions are obtained with the evolution operates. It is shown that the non-Hermitian systems behave like the corresponding Hermitian systems, i.e., periodically oscillating between the two states, when the eigenenergy is real. But when the eigenenergy is imaginary, our analytical and numerical results show that the probabilities always grow with time for all parameter values in the two-states non-Hermitian systems (including the well-known PT -symmetry non-Hermitian system). We give a cursory explanation. Our exact results agree well with direct numerical simulations of the evolution equation, showing that our analytical results are reasonable and the numerical algorithm we adopted may be extended to many-states systems.