Approach the fluctuation of eigenstates around the avoided level crossing by two-level dynamics

Abstract

From level dynamics, we show that when two levels come to an avoided level crossing, violent fluctuations of the corresponding eigenstates are similar to a two level process: rapid mixing of the two eigenstates. This can be checked sensitively by the expectation values of a smooth observable and its derivative with respect to the strength parameter of perturbation. In Husimi distribution, the mixing can be viewed as that the two eigenstates exchange parts of their distributions in a simple way.