Non-Abelian Anyons and Interferometry

Parsa Bonderson

Published Jun 4, 2007• 169 citations

Abstract

This thesis is primarily a study of the measurement theory of non-Abelian anyons through interference experiments. We give an introduction to the theory of anyon models, providing all the formalism necessary to apply standard quantum measurement theory to such systems. This formalism is then applied to give a detailed analysis of a Mach-Zehnder interferometer for arbitrary anyon models. In this treatment, we find that the collapse behavior exhibited by a target anyon in a superposition of states is determined by the monodromy of the probe anyons with the target. Such measurements may also be used to gain knowledge that would help to properly identify the anyon model describing an unknown system. The techniques used and results obtained from this model interferometer have general applicability, and we use them to also describe the interferometry measurements in a two point-contact interferometer proposed for non-Abelian fractional quantum Hall states. Additionally, we give the complete description of a number of important examples of anyon models, as well as their corresponding quantities that are relevant for interferometry. Finally, we give a partial classification of anyon models with small numbers of particle types.

DOI: 10.7907/5ndz-w890

Related papers

Decoherence of Anyonic Charge in Interferometry Measurements

Parsa Bonderson, Kirill Shtengel, Johannes K. Slingerland

We examine interferometric measurements of the topological charge of (non-Abelian) anyons. The target's topological charge is measured from its effect on the interference of probe particles sent through the interferometer. We find that superpositions of distinct anyonic charges a and a' in the target decohere (exponentially in the number of probes particles used) when the probes have nontrivial monodromy with the charges that may be fused with a to give a'.

arXiv:quant-ph/0608119Physical Review LettersFeb 12, 200725 citations

Non-Abelian Quantum Hall States and Fractional Statistics

Waheb Bishara

The discovery of the fractional quantum Hall effect stimulated the investigation of anyons, particles with fractional statistics which are neither bosons nor fermions. This thesis focuses on the study of quantum Hall states which may support non-Abelian anyons. We first address the validity of assumptions used in the numerical study of such states, and then proceed with analyzing different experiments which can detect non-Abelian fractional statistics. We quantitatively analyze the two-point contact interferometer experiment, which is hoped to display clear-cut, direct evidence of non-Abelian fractional statistics. We calculate the temperature and voltage dependence of the interference experiment outcome, and the signal attenuation due to finite temperature loss of coherence. We then analyze the edge theory of a family of non-Abelian quantum Hall states in the second Landau level, and examine the tunneling between these states and a quantum dot. This tunneling problem maps onto the multi-channel Kondo problem, and will allow distinguishing between different quantum Hall states. Finally, we use the same theoretical methods for analyzing Sagnac interference in the conductance of a carbon nanotube loop, a one-dimensional system.

May 28, 20091 citation

Anyon Interferometry to Detect Braiding Statistics of Neutral Modes

Cheolhee Han, June-Young M. Lee, H.-S. Sim

On the edge of certain fractional quantum Hall states, e.g., at 2/3 and 5/2 filling, a local fractional excitation, occurring by anyon tunneling at a quantum point contact, is further fractionalized into counterpropagating charge and neutral (Abelian or non-Abelian) anyonic excitations. We propose a scheme to detect the braiding statistics of the charge and neutral anyons separately. It is the injection of a dilute beam of a target (charge or neutral) anyon to a Fabry-Perot interferometer. The monodromy of the target anyon is obtained by comparing the amplitude and phase of the interference current with a reference signal of the same setup but without the injection. Our proposal relies on braiding between anyons on the edge, and applies even in the presence of bulk-edge couplings.

Physical Review LettersOct 29, 20240 citations

Proposal for reading out anyon qubits in non-Abelian ν=12∕5 quantum Hall state

Suk Bum Chung, Michael Stone

To detect non-Abelian statistics in the $\ensuremath{\nu}=12∕5$ quantum Hall state through interferometry, we apply an analysis similar to the ones proposed for the non-Abelian $\ensuremath{\nu}=5∕2$ quantum Hall state. The result is that the amplitude of the Aharonov-Bohm oscillation of this interference is dependent on the internal states of quasiholes, but, in contrast to the $\ensuremath{\nu}=5∕2$ quantum Hall state, independent of the number of quasiholes. However, if the quasiholes are in a superposition state, it is necessary for the interferometer to have certain additional features to obtain the coefficients.

arXiv:cond-mat/0601594Physical Review BJun 9, 200627 citations

Inequivalent classes of interference experiments with non-Abelian anyons

B. J. Overbosch, F.A. Bais

We present a theoretical analysis of inequivalent classes of interference experiments with non-Abelian anyons using an idealized Mach-Zender-type interferometer. Because of the non-Abelian nature of the braid group action one has to distinguish the different possibilities in which the experiment can be repeated, which lead to different interference patterns. We show that each setup will, after repeated measurement, lead to a situation where the two-particle (or multiparticle) state gets locked into an eigenstate of some well-defined operator. Also, the probability to end up in such an eigenstate is calculated. Some representative examples are worked out in detail.

arXiv:quant-ph/0105015Physical Review ANov 19, 200131 citations

Anyon braiding statistics： From fundamental physics to scalable quantum computing

HE Jianhong, GAO Jie

Topological quantum computation aims to encode quantum information in the nonlocal degrees of freedom of non-Abelian anyons and manipulate it through braiding and fusion operations， offering a promising pathway to fault-tolerant quantum computation. Non-Abelian anyons， with their nontrivial braiding statistics and intrinsic topological fault tolerance， are regarded as a key building block for realizing scalable and decoherence-resistant quantum computing. This review systematically surveys the key developments in anyon physics over the past four decades， from the early theoretical prediction of fractional statistics to direct experimental verification of anyon braiding in fractional quantum Hall states. The physical principles and key findings of essential experimental schemes， including Fabry-Pérot interferometers， Mach-Zehnder interferometers， and anyon colliders， are comprehensively covered. Based on these foundations， the characterization of topological order in even-denominator fractional quantum Hall states， exemplified by the ν=5/2 state， and transport and interferometric probes of non-Abelian statistics are discussed in detail. Furthermore， recent advances in topological quantum states in novel material platforms， including graphene and twisted transition metal dichalcogenides， are reviewed， together with artificial quantum platforms that simulate non-Abelian braiding using superconducting quantum processors. Finally， the key challenges and future directions on the path from anyon physics to scalable topological quantum computing prototype devices are outlined.

Sep 1, 20260 citations