Rapid azimuthal rotation in the Hermitian and non-Hermitian Landau-Zener problem

In the limit of rapid passage in a time-dependent two-level system of the Landau-Zener type, the initial state undergoes a very simple yet non-trivial rotation. This effect takes place when the Hamiltonian reaches the stationary point of the system. We explain the origin of this rotation by simple means and extend it to open non-Hermitian (NH) systems where new features appear. In addition, we find that in contrast to the Hermitian case, the point at which this simple rotation takes place is not necessarily the point of minimal energy separation. Moreover, in NH Hamiltonian the stationary point may lie in the complex time plane, and then a surprisingly strong correction to the rotation angle appears. The NH aspects of this rotation can be observed in optical and quantum systems where decay rates can be controlled. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to Quantum physics with non-Hermitian operators.