Dynamics in non-Hermitian systems with nonreciprocal coupling

We reveal that 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 non-orthogonal eigenstates. Demonstrating two extreme cases of Hamiltonians, namely one having complex eigenenergies with orthogonal eigenstates and one having real eigenenergies with non-orthogonal eigenstates, we elucidate the differences between the amplifications from complex eigenenergies and from non-orthogonal eigenstates.