Vacuum Rabi splitting and quantum Fisher information of a non-Hermitian qubit in a single-mode cavity

A natural extension of the non-Hermitian qubit is to place it in a single-mode cavity. This setup corresponds to the quantum Rabi model (QRM) with a purely imaginary bias on the qubit, exhibiting parity-time ($\mathcal{P}\mathcal{T}$) symmetry. In this work, we first solve the $\mathcal{P}\mathcal{T}$-symmetric QRM using the Bogoliubov operator approach. We derive the transcendental function responsible for the exact solution, which can also be used to precisely identify exceptional points. The adiabatic approximation previously used can be easily formulated within this approach by considering transitions between the same manifolds in the space of Bogoliubov operators. By further considering transitions between the nearest-neighboring manifolds, we can analytically obtain more accurate eigensolutions. Moreover, these simple corrections can capture the main features of the dynamics, where the adiabatic approximation fails. Furthermore, the rich characteristics of the vacuum Rabi splitting in the emission spectrum are predicted. The width of the peaks increases with the coupling strength and the imaginary biases, reflecting the nature of open quantum systems. Additionally, we identify a quantum-criticality-enhanced effect by calculating the quantum Fisher information. Near the exceptional points, the quantum Fisher information in the $\mathcal{P}\mathcal{T}$-symmetric QRM is significantly higher than that of the non-Hermitian qubit component. This may open a new avenue for enhancing quantum sensitivity in non-Hermitian systems by incorporating coupling with an additional degree of freedom, enabling more precise parameter estimation.