Analytical study of the non-Hermitian semiclassical Rabi model

The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi mathvariant="script">PT</a:mi></a:math>-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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi mathvariant="script">PT</c:mi></c:math>-broken phase, derived from the analytical eigenvalues, closely matches the numerically 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi mathvariant="script">PT</e:mi></e:math>-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.