$\mathcal{PT}$-Symmetric Spin--Boson Model with a Continuous Bosonic Spectrum: Exceptional Points and Dynamics

TL;DR

This work investigates a PT-symmetric non-Hermitian spin-boson model with a continuous bosonic environment. By combining a projection method based on a generalized Silbey–Harris ansatz with Dirac–Frenkel TDVP dynamics, it reveals that the continuum bath supports a single exceptional point and only one real eigenvalue below it, contrasting with finite-mode systems. Dynamically, the PT-unbroken phase shows sustained oscillations with suppressed decoherence, while the PT-broken phase experiences enhanced dissipation and rapid relaxation, with coupling strength and bias tuning the phase boundaries. The results illuminate how PT symmetry protects coherent light–matter interactions in open quantum systems and highlight avenues for improving variational treatments with higher-order coherent-state expansions.

Abstract

This work studies a $\mathcal{PT}$-symmetric non-Hermitian spin--boson model, consisting of a non-Hermitian two-level system coupled to a continuous bosonic bath. The static properties of the system are analyzed through a projection method derived from the displacement operator. We find that only a single exceptional point (EP) emerges, in contrast to non-Hermitian spin--boson models with finite modes, which typically exhibit multiple EPs. Notably, only a single real eigenvalue is found before the EP, which differs markedly from typical non-Hermitian systems where a pair of real eigenvalues precedes the EP. The time evolution of observables is further investigated via the Dirac--Frenkel time-dependent variational principle. Compared to its Hermitian counterpart, the non-Hermitian model exhibits distinct dynamical signatures, most notably the emergence of oscillations with periodic amplified amplitude. In the $\mathcal{PT}$-unbroken phase, the system exhibits sustained oscillatory dynamics with suppressed decoherence, whereas in the $\mathcal{PT}$-broken phase, additional dissipative channels accelerate decoherence and drive rapid convergence toward a stable steady state. These results shed light on how $\mathcal{PT}$ symmetry protects coherent light--matter interactions in non-Hermitian quantum systems.

Figures (7)

• Figure 1: Eigenvalue spectrum of the $\mathcal{PT}$-symmetric non-Hermitian spin-boson model (a) and the corresponding quantum Rabi model (b). Parameters: $\Delta = 0.3$ and $\epsilon = 0.1$.
• Figure 2: Eigenvalue spectrum of the $\mathcal{PT}$-symmetric non‑Hermitian spin‑boson model versus the bias strength $\epsilon$. Other parameters are set as (a) $\Delta = 0.1,\lambda = 0.01$; (b) $\Delta = 0.1,\lambda = 0.1$; (c) $\Delta = 0.3,\lambda = 0.1$; (d) $\Delta = 0.3,\lambda = 0.3$.
• Figure 3: Time evolution of the average spin component $\langle s_{z} \rangle$ under different parameter sets: (a) $\Delta = 0.1$, $\lambda = 0.01$, $\epsilon = 0.05$ (black), $0.1$ (red); (b) $\Delta = 0.1$, $\lambda = 0.1$, $\epsilon = 0.05$ (black), $0.1$ (red); (c) $\Delta = 0.3$, $\lambda = 0.1$, $\epsilon = 0.1$ (black), $0.3$ (red); (d) $\Delta = 0.3$, $\lambda = 0.3$, $\epsilon = 0.1$ (black), $0.3$ (red).
• Figure 4: (a) Time evolution of the average photon occupation $\langle n_{\text{b}}\rangle$ and (b) dynamics of the normalization factor $\mathcal{N}$. The inset in (b) displays the same data on a exponential $y$-axis. Parameters: (a) $\Delta = 0.1$, $\lambda = 0.01$, $\epsilon = 0.05$ (black), $0.1$ (red); (b) $\Delta = 0.3$, $\lambda = 0.1$, $\epsilon = 0.1$ (black), $0.3$ (red).
• Figure 5: Lowest two eigenvalues of the non‑Hermitian quantum Rabi model. (a) Real parts; (b) Imaginary parts. Parameters: $\omega_0=1$, $\Delta = 0.5$, $\epsilon = 0.1$.