Quantum graph as a quantum spectral filter

TL;DR

This paper investigates how a quantum spectral filter can be realized by attaching a graph to a line with a specific coupling, enabling spectral analysis and control of quantum transmission.

Contribution

It introduces a novel quantum spectral filtering method using scale-invariant couplings at a graph-line junction, with practical device designs for filtering and spectral separation.

Findings

Transmission probability approaches an indicator function of the graph's spectrum as coupling parameter increases.

The method enables spectral analysis and control of quantum transmission along a line.

Designs for a band-stop filter and spectral separator using special vertex couplings are proposed.

Abstract

We study the transmission of a quantum particle along a straight input--output line to which a graph $\Gamma$ is attached at a point. In the point of contact we impose a singularity represented by a certain properly chosen scale-invariant coupling with a coupling parameter $\alpha$. We show that the probability of transmission along the line as a function of the particle energy tends to the indicator function of the energy spectrum of $\Gamma$ as $\alpha\to\infty$. This effect can be used for a spectral analysis of the given graph $\Gamma$. Its applications include a control of a transmission along the line and spectral filtering. The result is illustrated with an example where $\Gamma$ is a loop exposed to a magnetic field. Two more quantum devices are designed using other special scale-invariant vertex couplings. They can serve as a band-stop filter and as a spectral separator, respectively.