Correlated behavior of conductance and phase rigidity in the transition from the weak-coupling to the strong-coupling regime

TL;DR

This paper investigates how conductance and phase rigidity in small quantum systems are correlated during the transition from weak to strong coupling with leads, revealing collective effects and resonance trapping phenomena.

Contribution

It provides a detailed analysis of the correlation between conductance and phase rigidity, highlighting the role of resonance trapping and collective effects in the transition regime.

Findings

Conductance exhibits plateau-like enhancements and zero-transmission corridors.

Phase rigidity correlates with the inverse of the average conductance.

In strong coupling, only two resonance states dominate as traveling modes.

Abstract

We study the transmission through different small systems as a function of the coupling strength $v$ to the two attached leads. The leads are identical with only one propagating mode $\xi^E_C$ in each of them. Besides the conductance $G$, we calculate the phase rigidity $\rho$ of the scattering wave function $\Psi^E_C$ in the interior of the system. Most interesting results are obtained in the regime of strongly overlapping resonance states where the crossover from staying to traveling modes takes place. The crossover is characterized by collective effects. Here, the conductance is plateau-like enhanced in some energy regions of finite length while corridors with zero transmission (total reflection) appear in other energy regions. This transmission picture depends only weakly on the spectrum of the closed system. It is caused by the alignment of some resonance states of the system with the propagating modes $\xi^E_C$ in the leads. The alignment of resonance states takes place stepwise by resonance trapping, i.e. it is accompanied by the decoupling of other resonance states from the continuum of propagating modes. This process is quantitatively described by the phase rigidity $\rho$ of the scattering wave function. Averaged over energy in the considered energy window, $<G>$ is correlated with $1-<\rho>$. In the regime of strong coupling, only two short-lived resonance states survive each aligned with one of the channel wave functions $\xi^E_C$. They may be identified with traveling modes through the system. The remaining $M-2$ trapped narrow resonance states are well separated from one another.