Abstract
Exceptional points (EPs) determine the dynamics of open quantum systems and cause also PT symmetry breaking in PT symmetric systems. From a mathematical point of view, this is caused by the fact that the phases of the wavefunctions (eigenfunctions of a non-Hermitian Hamiltonian) relative to one another are not rigid when an EP is approached. The system is therefore able to align with the environment to which it is coupled and, consequently, rigorous changes of the system properties may occur. We compare analytically as well as numerically the eigenvalues and eigenfunctions of a 2 × 2 matrix that is characteristic either of open quantum systems at high level density or of PT symmetric optical lattices. In both cases, the results show clearly the influence of the environment on the system in the neighborhood of EPs. Although the systems are very different from one another, the eigenvalues and eigenfunctions indicate the same characteristic features.
Highlights
Starting with paper [1], it has been shown that a wide class of PT symmetric non-Hermitian Hamilton operators provides entirely real spectra
In order to realize complex PT symmetric structures, the formal equivalence of the quantum mechanical Schrödinger equation to the optical wave equation in PT symmetric optical lattices [4] can be exploited by involving symmetric index guiding and an antisymmetric gain/loss profile
On the basis of 2 × 2 models, we have compared the influence of an Exceptional points (EPs) on the dynamics of an open quantum system with its influence on PT symmetry breaking in a PT symmetric system
Summary
Starting with paper [1], it has been shown that a wide class of PT symmetric non-Hermitian Hamilton operators provides entirely real spectra. Experimental results [5] have confirmed the expectations and have, demonstrated the onset of passive PT symmetry breaking within the context of optics This phase transition was found to lead to a loss-induced optical transparency in specially designed pseudo-Hermitian potentials. In another experiment [6], the wave propagation in an active PT symmetric coupled waveguide system is studied. Recent studies have shown the important role the singular points in the continuum play for the dynamics of open quantum systems, see, e.g., the review [10] These singular points are usually called exceptional points (EPs) after Kato, who studied their mathematical properties [11] many years ago.
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