Crossing scenario for a nonlinear non-Hermitian two-level system

The diabolic crossing scenario of two-state quantum systems can be generalized to a non-Hermitian case as well as to a nonlinear one. In the non-Hermitian case two different crossing types appear, distinguished according to the crossing or anticrossing of real parts or imaginary parts of the eigenvalues. In the nonlinear case additional stationary states can emerge, leading to looped structures in the eigenvalues. Here we discuss the basic properties of the most general situation, the combined nonlinear and non-Hermitian system. It is shown that the eigenvalues and eigenstates can be achieved from the real roots of a quartic equation. The corresponding crossing scenario is quite intricate and can be understood as a hybrid of the ones for the nonlinear Hermitian and the linear non-Hermitian systems. In addition, the implications of combined nonlinearity and non-Hermiticity on the system dynamics is studied in terms of a generalized Landau-Zener probability.