${\mathscr{P}}{\mathscr{T}}$ Symmetry of a Square-Wave Modulated Two-Level System

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 $\gamma_{\mathrm{eff}}=\pm\Delta_{\mathrm{eff}}$.