Tunnelling current through fractional quantum Hall interferometers

TL;DR

This paper derives a general formula for tunnelling current in fractional quantum Hall interferometers, accounting for multiple edge channels and asymmetries, and proposes a method to measure edge velocities via current oscillations.

Contribution

It introduces a comprehensive theoretical framework for tunnelling currents in fractional quantum Hall edges with multiple channels and asymmetries, extending previous models.

Findings

Current exhibits multiple oscillations as a function of edge length.

Edge velocities can be extracted from the Fourier spectrum of the edge current.

The numerical scheme for the Carlson's R function facilitates analysis of various edge states.

Abstract

We calculate the tunnelling current through a Fabry-P\'{e}rot interferometer in the fractional quantum Hall regime. Within linear response theory (weak tunnelling but arbitrary source-drain voltage) we find a general expression for the current due to tunnelling of quasiparticles in terms of Carlson's $R$ function. Our result is valid for fractional quantum Hall states with an edge theory consisting of a charged channel and any number of neutral channels, with possibly different edge velocities and different chiralities. We analyse the case with a single neutral channel in detail, which applies for instance to the edge of the Moore-Read state. In addition we consider an asymmetric interferometer with different edge lengths between the point contacts on opposite edges, and we study the behaviour of the current as a function of varying edge length. Recent experiments attempted to measure the Aharanov-Bohm effect by changing the area inside the interferometer using a plunger gate. Theoretical analyses of these experiments have so far not taken into account the accompanying change in the edge lengths. We show that the tunnelling current exhibits multiple osculations as a function of this edge length, with frequencies proportional to the injected edge current and inversely proportional to the edge velocities. In particular the edge velocities can be measured by looking at the Fourier spectrum of the edge current. We provide a numerical scheme to calculate and plot the $R$ function, and include sample plots for a variety of edge states with parameter values which are experimentally relevant.