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Hermitian Hamiltonians with time-periodic coefficients can be analyzed viaFloquet theory, and have been extensively used for engineering FloquetHamiltonians in standard quantum simulators. Generalized to non-HermitianHamiltonians, time-periodicity offers avenues to engineer the landscape ofFloquet quasi-energies across the complex plane. We investigate two-levelnon-Hermitian Hamiltonians with coefficients that have different periodicitiesusing Floquet theory. By analytical and numerical calculations, we obtain theirregions of stability, defined by real Floquet quasi-energies, and contours ofexceptional point (EP) degeneracies. We extend our analysis to study the phasesthat accompany the cyclic changes. Our results demonstrate that time-periodic,non-Hermitian Hamiltonians generate a rich landscape of stable and unstableregions.

Stability of time-periodic $\mathcal{PT}$ and anti-$\mathcal{PT}$-symmetric Hamiltonians with different periodicities