Eur. Phys. J. D (2015) 69: 229
https://doi.org/10.1140/epjd/e2015-60389-7

Regular Article

Nearby states in non-Hermitian quantum systems I: Two states

¹ Department of Physics, McGill University, Montreal, H3A 2T8, Canada
² Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany

^a e-mail: rotter@pks.mpg.de

Received: 1 July 2015
Received in final form: 24 August 2015
Published online: 13 October 2015

Abstract

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 ℋ is sketched. Eigenvalues and eigenfunctions are parametrically controlled. Using a 2 × 2 model, we study the eigenfunctions of ℋ 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. Nonlinear terms in the Schrödinger equation appear nearby EPs which cause a mixing of the wavefunctions in a certain finite parameter range around the EP. The phases of the eigenfunctions jump by π at an EP. These results hold true for systems that can emit (“loss”) particles into the environment of scattering wavefunctions as well as for systems which can moreover absorb (“gain”) particles from the environment. In a parameter range far from an EP, open quantum systems are described well by a Hermitian Hamilton operator. The transition from this parameter range to that near to an EP occurs smoothly.