Dynamics in non-Hermitian systems with nonreciprocal coupling

Non-Hermitian Hamiltonians with nonreciprocal coupling can achieve amplification of initial states without external gain due to a kind of inherent source. We discuss the source and its effect on time evolution in terms of complex eigenenergies and nonorthogonal eigenstates. Demonstrating two extreme cases of Hamiltonians, namely one having complex eigenenergies with orthogonal eigenstates and one having real eigenenergies with nonorthogonal eigenstates, we elucidate the differences between the amplifications from complex eigenenergies and from nonorthogonal eigenstates.