Phase rigidity and avoided level crossings in the complex energy plane

Abstract

We consider the effective Hamiltonian of an open quantum system, its biorthogonal eigenfunctions ${\varphi }_{\lambda }$ and define the value $r_{\lambda }=({\varphi }_{\lambda }|{\varphi }_{\lambda })/<{\varphi }_{\lambda }|{\varphi }_{\lambda }>$ that characterizes the phase rigidity of the eigenfunctions ${\varphi }_{\lambda }$. In the scenario with avoided level crossings, $r_{\lambda }$ varies between 1 and 0 due to the mutual influence of neighboring resonances. The variation of $r_{\lambda }$ may be considered as an internal property of an {\it open} quantum system. In the literature, the phase rigidity $\rho$ of the scattering wave function ${\Psi }_C^E$ is considered. Since ${\Psi }_C^E$ can be represented in the interior of the system by the ${\varphi }_{\lambda }$, the phase rigidity $\rho$ of the ${\Psi }_C^E$ is related to the $r_{\lambda }$ and therefore also to the mutual influence of neighboring resonances. As a consequence, the reduction of the phase rigidity $\rho$ to values smaller than 1 should be considered, at least partly, as an internal property of an open quantum system in the overlapping regime. The relation to measurable values such as the transmission through a quantum dot, follows from the fact that the transmission is, in any case, resonant with respect to the effective Hamiltonian. We illustrate the relation between phase rigidity $\rho$ and transmission numerically for small open cavities.

6 pages, 3 figures

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