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

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.