Nonadiabatic transitions in non-Hermitian $\mathcal{PT}$-symmetric two-level systems

TL;DR

The paper tackles nonadiabatic transitions in generic non-Hermitian, PT-symmetric two-level systems with spin-dependent dissipation, focusing on how linearly sweeping the gap-control parameter $\eta(t)=\alpha t$ across exceptional points redistributes population. By reducing the dynamics to parabolic cylinder (Weber) equations, it derives analytic asymptotics and explicit final-state relations. A key finding is that equal redistribution of occupations in the slow-tuning limit occurs only when the underlying Hermitian part $H_0$ has a gap closing (i.e., $\delta_y=0$); for $\delta_y\neq 0$ (with $\delta_x=0$) the final-state ratio is $r_p=|(\,\gamma-\delta_y\,)/(\,\gamma+\delta_y\,)|$, independent of the initial state, while in the PT-broken/no-PT cases the equal redistribution generally disappears. The authors propose a dynamical protocol—driving across the PT-symmetry-breaking bubble and back—to identify gap closing in the Hermitian sector, offering a practical metrology-style tool, with numerical validation and experimental relevance across photonic, atomic, and circuit platforms.

Abstract

We systematically characterize the dynamical evolution of time-parity (PT )-symmetric two-level systems with spin-dependent dissipations. If the control parameters of the gap are linearly tuned with time, the dynamical evolution can be characterized with parabolic cylinder equations which can be analytically solved. We find that the asymptotic behaviors of particle probability on the two levels show initial-state-independent redistribution in the slow-tuning-speed limit as long as the system is nonadiabatically driven across exceptional points. Equal distributions appear when the nondissipative Hamiltonian shows gap closing. So long as the nondissipative Hamiltonian displays level anticrossing, the final distribution becomes unbalanced. The ratios between the occupation probabilities are given analytically. These results are confirmed with numerical simulations. The predicted equal distribution phenomenon may be used to identify the closing of the energy gap from anti-crossing between two energy bands.

Figures (3)

• Figure 1: Illustration of $\mathcal{PT}$-symmetry breaking in single-spin system $H_{0}$. (a)(b): The spectra (a) and spin structure (b) of a single-spin system with $\mathcal{PT}$-symmetry breaking. The red solid (blue dash-dotted) curves in (a) denote the real (imaginary) parts of the spectra. The variation of bubble size (horizontal diameter) with $\gamma$ is shown in the inset of (a).
• Figure 2: The illustration of instantaneous spectra (upper row) and projection properties (lower row) for different perturbation parameters in a cyclic time evolution, where $\eta(t)=-1+\alpha t$ when $t<t_{f}$ and $\eta(t)=1-\alpha t$ when $t>t_{f}$ with the one-way evolution time $t_{f}=15$ and tuning speed $\alpha=0.025$. The initial states are taken as $|\Psi(0)\rangle=\cos(\theta)|\Psi_{1}(0)\rangle+e^{i\varphi}\sin(\theta)|\Psi_{2}(0)\rangle$ with $\theta=\pi/3$ and $\varphi=\pi/6$, where $|\Psi_{1,2}(0)\rangle$ are the instantaneous eigenstates at moment $t=0$. The spectra of real-time Hamiltonian $H=\eta(t)\sigma_{z}+\delta_{x}\sigma_x+\delta_{y}\sigma_y+i\gamma\sigma_x$ (insets) and the projection probabilities of instantaneous states when $\gamma=\delta_{x}=\delta_{y}=0$ (a) [$\delta_{y}=\delta_{x}=0$, $\gamma=0.2$ (d)], $\gamma=\delta_{y}=0$, $\delta_{x}=0.15$ (b) [$\delta_{y}=0$, $\delta_{x}=0.15$, $\gamma=0.2$ (e)], and $\gamma=\delta_{x}=0$, $\delta_{y}=0.15$ (c) [$\delta_{x}=0$, $\delta_{y}=0.15$, $\gamma=0.2$ (e)], are shown. The red (blue) curves in the insets represent the real (imaginary) parts of the spectra. The solid (dotted) curves in the sub-figures show the projection probabilities for the corresponding levels in the insets. The projection probabilities $|C_{\xi=1,2}|^2$ are defined as $C_{1,2}=\langle\tilde{\Psi}_{1,2}(t)|\Psi(t)\rangle/|\langle\tilde{\Psi}_{1,2}|\Psi_{1,2}\rangle|$ with the instantaneous state $|\Psi(t)\rangle$ and the instantaneous right (left) eigenstates $|\Psi_{1,2}(t)\rangle$ ( $|\tilde{\Psi}_{1,2}(t)\rangle$) brody2013biorthogonal. We would like to note that the coefficients $C_{1,2}$ are not the spin components directly. However, because at $t=t_{f}$$\eta$ is far larger than other coefficients in the Hamiltonian, $\Psi_{1,2}$ approach the eigenstate of $\sigma_{z}$. Therefore, the asymptotic behaviors of $C_{1,2}$ approach those of $\psi_{1,2}$ given in the above section. The introduce of $C_{1,2}$ is just for the convenience of experimental observation.
• Figure 3: The difference between the probability amplitudes $|\Delta |C||=||C_{2}|-|C_{1}||$ for different initial states in cyclic evolution. The initial states are taken as $|\Psi(0)\rangle=\cos(\theta)|\Psi_{1}(0)\rangle+e^{i\varphi}\sin(\theta)|\Psi_{2}(0)\rangle$, where $\varphi$ is arbitrarily fixed at $\pi/6$ and $\theta$ is scanned. These sub-figures have the same parameters as the corresponding sub-figures in Fig. \ref{['fig:circle_evolution']}. The inset of (d) shows $|C_{1}|$ (blue dash-dotted line), $|C_{2}|$ (black dotted line) and their ratio $|C_{2}|/|C_{1}|$ (red solid line) at $t=t_{f}$. The ratio keeps constant except at around the zero point of $C$, which confirms the analytical result that the independence of initial states is preserved so long as the system crosses EPs.