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Open systems with gain, loss, or both, described by non-HermitianHamiltonians, have been a research frontier for the past decade. In particular,such Hamiltonians which possess parity-time ($\mathcal{PT}$) symmetry featuredynamically stable regimes of unbroken symmetry with completely realeigenspectra that are rendered into complex conjugate pairs as the strength ofthe non-Hermiticity increases. By subjecting a $\mathcal{PT}$-symmetric systemto a periodic (Floquet) driving, the regime of dynamical stability can bedramatically affected, leading to a frequency-dependent threshold for the$\mathcal{PT}$-symmetry breaking transition. We present a simple model of atime-dependent $\mathcal{PT}$-symmetric Hamiltonian which smoothly connects thestatic case, a $\mathcal{PT}$-symmetric Floquet case, and aneutral-$\mathcal{PT}$-symmetric case. We analytically and numerically analyzethe $\mathcal{PT}$ phase diagrams in each case, and show that slivers of$\mathcal{PT}$-broken ($\mathcal{PT}$-symmetric) phase extend deep into thenominally low (high) non-Hermiticity region.

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