$\mathcal{PT}$ 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 $ω/ 2$. For the time-periodic dissipation, a vanishingly small dissipation strength can lead to the $\mathcal{PT}$ symmetry breaking in the $(2k-1)$-photon resonance ($Δ= (2k-1) ω$), with $k=1,2,3\dots$ It is worth noting that such a phenomenon can also happen in $2k$-photon resonance ($Δ= 2k ω$), as long as the dissipation strengths or the driving times are imbalanced, namely $γ_0 \ne - γ_1$ or $T_0 \ne T_1$. For the time-periodic coupling, the weak dissipation induced $\mathcal{PT}$ symmetry breaking occurs at $Δ_{\mathrm{eff}}=kω$, where $Δ_{\mathrm{eff}}=\left(Δ_0 T_0 + Δ_1 T_1\right)/T$. In the high frequency limit, the phase boundary is given by a simple relation $γ_{\mathrm{eff}}=\pmΔ_{\mathrm{eff}}$.

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