  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 PT
symmetry, which can be broken when the drive is sufficiently strong. We derive
the boundaries of the 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 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.
