PT -symmetric Rabi model: Perturbation theory

We study a non-Hermitian version of the Rabi model, where a two-level system is periodically driven with an imaginary-valued drive strength, leading to alternating gain and loss. In the Floquet picture, the model exhibits $\mathcal{PT}$ symmetry, which can be broken when the drive is sufficiently strong. We derive the boundaries of the $\mathcal{PT}$ phase diagram for the different resonances by doing perturbation theory beyond the rotating-wave approximation. For the main resonance, we show that the non-Hermitian analog of the Bloch-Siegert shift corresponds to maximal $\mathcal{PT}$ breaking. For the higher-order resonances, we capture the boundaries to lowest order. We also solve the regime of high frequency by mapping to the Wannier-Stark ladder. Our model can be experimentally realized in waveguides with spatially modulated loss or in atoms with time-modulated spontaneous decay.