Vacuum Rabi Splitting and Quantum Fisher Information of a Non-Hermitian Qubit in a Single-Mode Cavity

TL;DR

This work analyzes a non-Hermitian qubit in a single-mode cavity, i.e., a $\mathcal{PT}$-symmetric extension of the quantum Rabi model, and develops a Bogoliubov-operator framework that produces a transcendental $G(E)$ function whose zeros give the exact spectrum while enabling EP detection via $G(E)=G'(E)=0$. It further introduces a symmetric adiabatic correction (CAA) that includes transitions between nearest-neighbor manifolds, yielding eigenstates and dynamics that closely match exact results across coupling regimes and revealing rich vacuum Rabi splitting behavior. The study also evaluates quantum Fisher information (QFI) and finds pronounced enhancement near EPs for the PTQRM, indicating improved parameter sensitivity compared to Hermitian counterparts and to the non-Hermitian two-level system. Together, these results bridge Hermitian and non-Hermitian light-matter coupling, offer analytical tools for non-Hermitian spectra, and point to practical routes for enhanced quantum sensing using PT-symmetric open systems.

Abstract

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.

Figures (9)

• Figure 1: ln$\lvert G\rvert^{2}$, where G is calculated from $G$ function (\ref{['Gge']}), in the complex $E$ plane. Open circles mark the zeros of both the real and imaginary parts of the $G$ function, which coincide exactly with the eigenvalues from the ED. The parameters are $\Delta =0.5$, $\epsilon =0.2$, and $g=0.25$.
• Figure 2: (a) $G$ curves from Eq. ( \ref{['Gfunc']}) (blue line) at $g=0.1$ in the real eigenvalue regime, with circles representing ED results. (b) The real part of eigenvalues as a function of g from the ED, with the dashed lines marking the real part of eigenvalues in the $\mathcal{P}\mathcal{T}$-broken region. Crossing values between the vertical dotted lines at $g=0.1,0.5$ and $0.6828$ and real eigenvalues (solid line) in the $\mathcal{P}\mathcal{T}$-unbroken region are confirmed in the $G$ functions in (a) and the lower panel. $G$ curves from Eq. ( \ref{['Gfunc']}) at $g=0.5,0.6828$ and $0.7$ are shown in (c), (d), and (e) with blue lines, respectively. The parameters are$\Delta=0.5$ and$\epsilon = 0.2$.
• Figure 3: The real part (left) of the first four pairs and the imaginary part (right) of only the fourth pair of eigenvalues (for clarity) as a function of coupling strength $g$ at resonance $\Delta = 1$ for $\epsilon = 0.1$ (upper panel) and $\epsilon = 0.5$ (lower panel). The black lines represent the ED results, the blue dots show the AA results, and the red dots indicate the CAA results. The eigenvalues and corresponding eigenstates along the dashed vertical line will be used to calculate the dynamics shown in Fig. \ref{['Dynamic_ultra']}, \ref{['Dynamic_near_deep']} and \ref{['Dynamic_deep']}.
• Figure 4: Time evolution of $<\sigma_{z}>$ (left) and the corresponding emission spectrum (right) at resonance ($\Delta = 1$) for the ultrastrong coupling ($g=0.2$) with $\epsilon = 0.1$ (upper panel) and $\epsilon = 0.5$ (lower panel). The peak positions are also shown in the left-hand panel.
• Figure 5: Time evolution of $<\sigma_{z}>$ (left) and the corresponding emission spectrum (right) at resonance ($\Delta = 1$) for near deep strong coupling ($g=0.75$) with $\epsilon = 0.1$ (upper panel) and $\epsilon = 0.5$ (lower panel).