Dynamics of $2\times2$ matrix non-Hermitian quantum systems on Bloch sphere

By casting evolution to Bloch sphere, the dynamics of $2\times2$ matrix non-Hermitian systems is investigated in detail. It shows that there are four kinds of dynamical mode for such systems. The different modes are classified by different kinds of fixed points, namely, the elliptic point, spiral point, critical node, and degenerate point. The Hermitian systems and the unbroken $\mathcal{PT}$ non-Hermitian cases belong to the category with elliptic points. The degenerate point just corresponds to the systems with exceptional point (EP). The topological properties of the fixed point are also discussed. It is interesting that the topological charge for degenerate point is $2$ while the others are $1$.