We consider different properties of small open quantum systems coupled to an environment and described by a non-Hermitian Hamilton operator. Of special interest is the non-analytical behavior of the eigenvalues in the vicinity of singular points, the
so-called exceptional points (EPs), at which the eigenvalues of two states coalesce and the corresponding eigenfunctions are linearly dependent from one another. The phases of the eigenfunctions are not rigid in approaching an EP and providing therewith the possibility to put information from the environment into the system. All characteristic properties of non-Hermitian quantum systems hold true not only for natural open quantum systems that suffer loss due to their embedding into the continuum of scattering wavefunctions. They appear also in systems coupled to different layers some of which provide gain to the system. Thereby gain and loss, respectively, may be fixed inside every layer, i.e. characteristic of it.
In part I, the formalism for the description of open quantum systems (that are embedded into a common well-defined environment) by means of a non-Hermitian Hamilton operator $ch$ is sketched. Eigenvalues and eigenfunctions are parametrically controll
ed. Using a 2$times$2 model, we study the eigenfunctions of $ch$ at and near to the singular exceptional points (EPs) at which two eigenvalues coalesce and the corresponding eigenfunctions differ from one another by only a phase. In part II, we provide the results of an analytical study for the eigenvalues of three crossing states. These crossing points are of measure zero. Then we show numerical results for the influence of a nearby (third) state onto an EP. Since the wavefunctions of the two crossing states are mixed in a finite parameter range around an EP, three states of a physical system will never cross in one point. Instead, the wavefunctions of all three states are mixed in a finite parameter range in which the ranges of the influence of different EPs overlap. We may relate these results to dynamical phase transitions observed recently in different experimental studies. The states on both sides of the phase transition are non-analytically connected.
We present a model describing the use of ultra-short strong pulses to control the population of the excited level of a two-level quantum system. In particular, we study an off-resonance excitation with a few cycles pulse which presents a smooth phase
jump i.e. a change of the pulses phase which is not step-like, but happens over a finite time interval. A numerical solution is given for the time-dependent probability amplitude of the excited level. The control of the excited levels population is obtained acting on the shape of the phase transient, and other parameters of the excitation pulse.
Using a recently developed technique to solve Schrodinger equation for constant mass, we studied the regime in which mass varies with position i.e position dependent mass Schrodinger equation(PDMSE). We obtained an analytical solution for the PDMSE a
nd applied our approach to study a position dependent mass $m(x)$ particle scattered by a potential $mathcal{V}(x)$. We also studied the structural analogy between PDMSE and two-level atomic system interacting with a classical field.
