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arXiv:quant-ph/0608119 · Physical Review Letters

Decoherence of Anyonic Charge in Interferometry Measurements

Parsa Bonderson, Kirill Shtengel, Johannes K. Slingerland

Published Feb 12, 2007• 25 citations
arXivHTMLPDF

Abstract

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'.

DOI
10.1103/physrevlett.98.070401

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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'.
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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.
Two distinct methods for measuring topological charge in a nonabelian anyonic system have been discussed in the literature: projective measurement of a single point-like quasiparticle and interferometric measurement of the total topological charge of a group of quasiparticles. Projective measurement by definition is only applied near a point and will project to a topological charge sector near that point. Thus, if it is to be applied to a group of anyons to project to a total charge, then the anyons must first be fused one by one to obtain a single anyon carrying the collective charge. We show that interferometric measurement is strictly stronger: Any protocol involving projective measurement can be simulated at low overhead by another protocol involving only interferometric measurement.
The manipulation of non-Abelian anyons is a foundational ingredient for topological quantum computing and underlies topologically protected quantum memories. Here we analyze a quantum-impurity-based protocol for braiding non-Abelian anyons. Exploiting mobile impurities with addressable internal spin states, we provide a concrete protocol to experimentally measure the matrix elements of the RRR and FFF matrices that characterize the non-Abelian statistics of anyons. In particular, focusing on the case of Ising and Fibonacci anyons, we illustrate how the anyonic nature of the excitations emerges directly in the Ramsey interferometer we propose. Building on this framework, we further introduce a conventional-to-topological qubit encoder to transfer quantum information from conventional flying qubits into a topologically protected quantum memory.
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.
Three manifold topology is used to analyze the effect of anyonic interferometers in which the probe anyons' path along an arm crosses itself, leading to a "twisted" or braided space-time trajectory for the probe anyons. In the case of Ising non-Abelian anyons, twisted interferometry is shown to be able to generate a topologically protected π/8π/8π/8-phase gate, which cannot be generated from quasiparticle braiding.