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.

中文翻译：

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连接有源和无源 PT 对称 Floquet 调制模型

非厄米哈密顿量所描述的具有增益、损失或两者兼有的开放系统，在过去十年中一直是研究前沿。特别是，这种拥有奇偶时间（$\mathcal{PT}$）对称性的哈密顿量具有动态稳定的不间断对称性，具有完全真实的特征谱，随着非厄米性强度的增加，这些特征谱被渲染成复共轭对。通过对 $\mathcal{PT}$ 对称系统进行周期性（Floquet）驱动，动态稳定性机制会受到显着影响，导致 $\mathcal{PT}$ 对称破坏的频率相关阈值过渡。我们提出了一个与时间相关的 $\mathcal{PT}$-对称哈密顿量的简单模型，它平滑地连接了静态情况、$\mathcal{PT}$-对称 Floquet 情况和中性-$\mathcal{PT} $-对称情况。