Phase lapses in open quantum systems and the non-Hermitian Hamilton operator

Abstract

We study transmission through a system with $N=10$ states coupled to $K=2$ continua of scattering wave functions in the framework of the $S$ matrix theory by using the Feshbach projection operator formalism for open quantum systems. Due to the coupling of the system (being localized in space) to the (extended) continuum of scattering wave functions, the Hamilton operator ${H}_{\text{eff}}$ of the system is non-Hermitian. The numerical calculations are performed for different distributions of both the positions ${E}_{i}^{0}$ $(i=1,\dots{},N)$ of the states of the isolated (closed) system and the elements of the coupling vectors ${V}^{c}$ between system and continua $(c=1,\dots{},K)$. The overall coupling strength $\ensuremath{\alpha}$ simulating the degree of resonance overlapping, is used as a parameter. In all cases, the complex eigenvalues and eigenfunctions of ${H}_{\text{eff}}$ are controlled by $\ensuremath{\alpha}$. In the regime of overlapping resonances, the well-known spectroscopic reordering processes (resonance trapping) take place because the phases of the eigenfunctions of ${H}_{\text{eff}}$ are not rigid in the neighborhood of singular points (being crossing points of eigenvalue trajectories). Finally, width bifurcation generates $K=2$ short-lived and $N\ensuremath{-}K$ trapped long-lived states. Thus, narrow (Fano-like) resonances may appear in the transmission at high level density. They are similar to, but different from the Fano resonances in the scattering theory with $K=1$. Phase lapses are related to zeros in the transmission probability. Their number and position (in energy) are determined by the ${V}^{c}$ and ${E}_{i}^{0}$, but not by $\ensuremath{\alpha}$. However, number and position of the resonance states depend on $\ensuremath{\alpha}$ due to resonance trapping occurring in the regime of overlapping resonances. As a consequence, universal phase lapses between every two resonances may appear at high level density while the system will show mesoscopic features at low level density. The phase lapses are not a single phenomenon. Due to their link to singularities in the continuum, they are related also to other ``puzzling'' experimental results such as dephasing at low temperature.