We study a non-Hermitian two-level system with square-wave modulated dissipation and coupling. Based on the Floquet theory, we achieve an effective Hamiltonian from which the boundaries of the $\mathcal{PT}$ phase diagram are captured exactly. Two kinds of $\mathcal{PT}$ symmetry broken phases are found whose effective Hamiltonians differ by a constant $\omega / 2$. For the time-periodic dissipation, a vanishingly small dissipation strength can lead to the $\mathcal{PT}$ symmetry breaking in the $(2k-1)$-photon resonance ($\Delta = (2k-1) \omega$), with $k=1,2,3\dots$ It is worth noting that such a phenomenon can also happen in $2k$-photon resonance ($\Delta = 2k \omega$), as long as the dissipation strengths or the driving times are imbalanced, namely $\gamma_0 \ne - \gamma_1$ or $T_0 \ne T_1$. For the time-periodic coupling, the weak dissipation induced $\mathcal{PT}$ symmetry breaking occurs at $\Delta_{\mathrm{eff}}=k\omega$, where $\Delta_{\mathrm{eff}}=\left(\Delta_0 T_0 + \Delta_1 T_1\right)/T$. In the high frequency limit, the phase boundary is given by a simple relation $\gamma_{\mathrm{eff}}=\pm\Delta_{\mathrm{eff}}$.

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${\mathscr{P}}{\mathscr{T}}$ 方波调制两电平系统的对称性

我们研究了具有方波调制耗散和耦合的非厄米两级系统。基于 Floquet 理论，我们实现了一个有效的哈密顿量，从中可以准确地捕获 $\mathcal{PT}$ 相图的边界。发现了两种 $\mathcal{PT}$ 对称破缺相，它们的有效哈密顿量相差一个常数 $\omega / 2$。对于时间周期耗散，极小的耗散强度会导致 $(2k-1)$-光子共振中的 $\mathcal{PT}$ 对称性破坏 ($\Delta = (2k-1) \omega$ )，其中 $k=1,2,3\dots$ 值得注意的是，这种现象也可以发生在 $2k$-光子共振 ($\Delta = 2k\omega$) 中，只要耗散强度或驱动时间不平衡，即 $\gamma_0 \ne - \gamma_1$ 或 $T_0 \ne T_1$。对于时间周期耦合，弱耗散引起的 $\mathcal{PT}$ 对称破坏发生在 $\Delta_{\mathrm{eff}}=k\omega$，其中 $\Delta_{\mathrm{eff}}=\left(\Delta_0 T_0 + \Delta_1 T_1\right)/T$。在高频极限中，相边界由简单的关系 $\gamma_{\mathrm{eff}}=\pm\Delta_{\mathrm{eff}}$ 给出。