Non-adiabatic transitions in parabolic and super-parabolic $mathcal{PT}$-symmetric non-Hermitian systems in one-dimensional optical waveguides*

Exceptional points, which are spectral degeneracy points in the complex parameter space, are fundamental to non-Hermitian quantum systems. The dynamics of non-Hermitian systems in the presence of exceptional points differ significantly from those of Hermitian ones. Here we investigate non-adiabatic transitions in non-Hermitian $mathcal{P}mathcal{T}$-symmetric systems, in which the exceptional points are driven through at finite speeds which are quadratic or cubic functions of time. We identity different transmission dynamics separated by exceptional points, and derive analytical approximate formulas for the non-adiabatic transmission probabilities. We discuss possible experimental realizations with a $mathcal{P}mathcal{T}$-symmetric non-Hermitian one-dimensional tight-binding optical waveguide lattice, through non-Hermitian Bloch oscillations between different bands.