\mathcalPT 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 \mathcalPT phase diagram are captured exactly. Two kinds of \mathcalPT 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 \mathcalPT symmetry breaking in the (2k-1)-photon resonance (\varDelta = (2k-1) \omega), with k=1,2,3\dots It is worth noting that such a phenomenon can also happen in 2k-photon resonance (\varDelta = 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 \mathcalPT symmetry breaking occurs at \varDelta_\rm eff=k\omega, where \varDelta_\rm eff=(\varDelta_0 T_0 + \varDelta_1 T_1)/T. In the high frequency limit, the phase boundary is given by a simple relation \gamma_\rm eff=\pm\varDelta_\rm eff.