Non-Hermitian time evolution: From static to parametric instability

Eigenmode coalescence imparts remarkable properties to non-Hermitian time evolution, culminating in a purely non-Hermitian spectral degeneracy known as an exceptional point (EP). Here, we revisit time evolution around EPs, looking at both static and periodically modulated non-Hermitian Hamiltonians. We connect a Möbius group classification of two-level non-Hermitian Hamiltonians with the theory of Hill's equation, which unlocks a large class of analytical solutions. Together with the classification, this allows us to investigate the impact of the shape of the temporal modulation on the long-term dynamics of the system. In particular, we find that EP encircling does not predict the temporal class, and that the elaborate interplay between non-Hermitian and modulation instabilities is better understood through the lens of parametric resonance. Finally, we identify specific signatures of complex parametric resonance by exhibiting stability diagrams with features that cannot occur in traditional parametric resonance.