Connecting active and passive mathcal{PT}-symmetric Floquet modulation models

Open systems with gain, loss, or both, described by non-Hermitian Hamiltonians, have been a research frontier for the past decade. In particular, such Hamiltonians which possess parity-time $mathcal{PT}$ symmetry feature dynamically stable regimes of unbroken symmetry with completely real eigenspectra that are rendered into complex conjugate pairs as the strength of the non-Hermiticity increases. By subjecting a mathcal{PT}-symmetric system to a periodic (Floquet) driving, the regime of dynamical stability can be dramatically affected, leading to a frequency-dependent threshold for the mathcal{PT}-symmetry breaking transition. We present a simple model of a time-dependent mathcal{PT}-symmetric Hamiltonian which smoothly connects the static case, a mathcal{PT}-symmetric Floquet case, and a neutral-mathcal{PT}-symmetric case. We analytically and numerically analyze the mathcal{PT} phase diagrams in each case, and show that slivers of mathcal{PT}-broken $mathcal{PT}$-symmetric phase extend deep into the nominally low (high) non-Hermiticity region.