Analytical Study of the Non-Hermitian Semiclassical Rabi Model

The $\mathcal{PT}$ symmetric semiclassical Rabi model explores the fundamental interaction between a two-level atom and a classical field, revealing novel phenomena in open systems through the inclusion of non-Hermitian terms. We propose a single similarity transformation that yields an effective Hamiltonian in rotating-wave approximation, enabling an analytical solution. The phase boundary of the $\mathcal{PT}$-broken phase, derived from the analytical eigenvalues, closely matches the numerical exact one over a wide range of atomic frequencies, demonstrating the effectiveness of the analytical approach, especially at the main resonance. The Floquet parity operator is also introduced, providing a deeper physical understanding of the emergence of the $\mathcal{PT}$-broken phase. Furthermore, by analyzing the dynamics of excited-state population, we observe several stable oscillations in the Fourier spectrum, demonstrating the applicability of the analytical method beyond the single-photon resonance region. The Bloch-Siegert shift is also discussed and, surprisingly, resembles its Hermitian counterpart, except for the higher-order terms in the coupling strength. The present analytical treatment provides a concise and accurate description of the main physics of this non-Hermitian atom-field interaction system.