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 $ϕ_λ$ and define the value $r_λ= (ϕ_λ|ϕ_λ)/<ϕ_λ|ϕ_λ>$ that characterizes the phase rigidity of the eigenfunctions $ϕ_λ$. In the scenario with avoided level crossings, $r_λ$ varies between 1 and 0 due to the mutual influence of neighboring resonances. The variation of $r_λ$ may be considered as an internal property of an {\it open} quantum system. In the literature, the phase rigidity $ρ$ of the scattering wave function $Ψ^E_C$ is considered. Since $Ψ^E_C$ can be represented in the interior of the system by the $ϕ_λ$, the phase rigidity $ρ$ of the $Ψ^E_C$ is related to the $r_λ$ and therefore also to the mutual influence of neighboring resonances. As a consequence, the reduction of the phase rigidity $ρ$ 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 $ρ$ and transmission numerically for small open cavities.