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arXiv:cond-mat/0601594 · Physical Review B

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

Suk Bum Chung, Michael Stone

Published Jun 9, 2006• 27 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.73.245311

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To detect non-Abelian statistics in the \ensuremathν=12∕5\ensuremath{\nu}=12∕5\ensuremathν=12∕5 quantum Hall state through interferometry, we apply an analysis similar to the ones proposed for the non-Abelian \ensuremathν=5∕2\ensuremath{\nu}=5∕2\ensuremathν=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 \ensuremathν=5∕2\ensuremath{\nu}=5∕2\ensuremathν=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.
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.
Non-Abelian excitations are an interesting feature of many fractional quantum Hall phases, including those phases described by the Moore-Read (or Pfaffian) wave function. However, the detection of the non-Abelian quasiparticles is challenging. Here, we consider a system described by the Moore-Read wave function and assume that impurity particles bind to its quasiholes. Then, the angular momentum of the impurities, reflected also by the impurity density, provides a useful witness of the physics of the non-Abelian excitations. By demanding that the impurities are constrained to the lowest Landau level, we are able to write down the corresponding many-body wave function describing both the Moore-Read liquid and the impurities. Through Monte Carlo sampling, we determine the impurity angular momentum, and we show that it suggests a quantum-statistical parameter $\ensuremath{\alpha}=a\ensuremath{\nu}\ensuremath{-}b+P/2$ for the quasiholes, where $\ensuremath{\alpha}$ ranges from 0 for bosons to 1 for fermions. A reasonable agreement with the Monte Carlo results is obtained for a=1/4a=1/4a=1/4, b=1/8b=1/8b=1/8, and P=0,1P=0,1P=0,1 depending on the parity of the particle number in the Moore-Read liquid. This parity dependence of the angular momentum serves as an unambiguous demonstration of the non-Abelian nature of the excitations. In addition to the studies of excitations in the Moore-Read liquid, we also apply our scheme to Laughlin liquids, for which we focus on interacting bosonic impurities. With this, the impurities themselves form Laughlin states, which allows for a study of hierarchical fractional quantum Hall states.
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.
The tunneling current and shot noise of the current between two fractional quantum hall (FQH) edges in the $\ensuremath{\nu}=12/5$ FQH state in an electronic Mach--Zehnder interferometer are studied. It is shown that the tunneling current and shot noise can be used to probe the existence of k=3k=3k=3 parafermion statistics in the $\ensuremath{\nu}=12/5$ FQH state. More specifically, the dependence of the current on the Aharonov--Bohm flux in the Read--Rezayi state is asymmetric under the change in the sign of the applied voltage. This property is absent in the Abelian--Laughlin states. Moreover, the Fano factor can exceed 12.7 electron charges in the $\ensuremath{\nu}=12/5$ FQH state. This number well exceeds the maximum possible Fano factor in all Laughlin states and the $\ensuremath{\nu}=5/2$ Moore--Read state, which was previously shown to be eee and 3.2e3.2e3.2e, respectively.
Quasiholes in certain fractional quantum Hall states are promising candidates for the experimental realization of non-Abelian anyons. They are assumed to be localized excitations, and to display non-Abelian statistics when sufficiently separated, but these properties have not been explicitly demonstrated except for the Moore-Read state. In this work, we apply the newly developed matrix product state technique to examine these exotic excitations. For the Moore-Read and the Z_{3} Read-Rezayi states, we estimate the quasihole radii, and determine the correlation lengths associated with the exponential convergence of the braiding statistics. We provide the first microscopic verification for the Fibonacci nature of the Z_{3} Read-Rezayi quasiholes. We also present evidence for the failure of plasma screening in the nonunitary Gaffnian wave function.
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.