Transmission across non-Hermitian $\mathcal{PT}$-symmetric quantum dots and ladders

TL;DR

This paper investigates quantum transport through non-Hermitian $ ext{PT}$-symmetric quantum dots and ladders, revealing conditions for perfect transmission and the effects of $ ext{PT}$-symmetry breaking on conductance.

Contribution

It introduces a model for scattering in $ ext{PT}$-symmetric quantum dots and ladders, demonstrating current conservation and transmission properties related to symmetry phases.

Findings

Perfect transmission occurs in the unbroken phase with real eigenenergies.

Transmission peaks are affected by magnetic flux and symmetry breaking.

Transmission is suppressed in the broken phase of the ladder.

Abstract

A non-Hermitian region connected to semi-infinite Hermitian lattices acts either as a source or as a sink and the probability current is not conserved in a scattering typically. Even a $\mathcal{P}\mathcal{T}$-symmetric region that contains both a source and a sink does not lead to current conservation plainly. We propose a model and study the scattering across a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric two-level quantum dot~(QD) connected to two semi-infinite one-dimensional lattices in a special way so that the probability current is conserved. Aharonov-Bohm type phases are included in the model, which arise from magnetic fluxes ($\hbar\phi_{L} /e,~\hbar\phi_{R} /e$) through two loops in the system. We show that when $\phi_L=\phi_R$, the probability current is conserved. We find that the transmission across the QD can be perfect in the $\mathcal{P}\mathcal{T}$-unbroken phase (corresponding to real eigenenergies of the isolated QD) whereas the transmission is never perfect in the $\mathcal{P}\mathcal{T}$-broken phase (corresponding to purely imaginary eigenenergies of the QD). The two transmission peaks have the same width only for special values of the fluxes (being odd multiples of $\pi\hbar/2e$). In the broken phase, the transmission peak is surprisingly not at zero energy. We give an insight into this feature through a four-site toy model. We extend the model to a $\mathcal{P}\mathcal{T}$-symmetric ladder connected to two semi-infinite lattices. We show that the transmission is perfect in unbroken phase of the ladder due to Fabry-P\'erot type interference, that can be controlled by tuning the chemical potential. In the broken phase of the ladder, the transmission is substantially suppressed.