There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Topological quantum computation has emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations that are necessary for quantum computation are carried out by braiding quasiparticles and then measuring the multiquasiparticle states. The fault tolerance of a topological quantum computer arises from the nonlocal encoding of the quasiparticle states, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the $\ensuremath{\nu}=5∕2$ state, although several other prospective candidates have been proposed in systems as disparate as ultracold atoms in optical lattices and thin-film superconductors. In this review article, current research in this field is described, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. Both the mathematical underpinnings of topological quantum computation and the physics of the subject are addressed, using the $\ensuremath{\nu}=5∕2$ fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

Non-Abelian Anyons and Topological Quantum Computation

Coherent preparation by laser light of quantum states of atoms and molecules can lead to quantum interference in the amplitudes of optical transitions. In this way the optical properties of a medium can be dramatically modified, leading to electromagnetically induced transparency and related effects, which have placed gas-phase systems at the center of recent advances in the development of media with radically new optical properties. This article reviews these advances and the new possibilities they offer for nonlinear optics and quantum information science. As a basis for the theory of electromagnetically induced transparency the authors consider the atomic dynamics and the optical response of the medium to a continuous-wave laser. They then discuss pulse propagation and the adiabatic evolution of field-coupled states and show how coherently prepared media can be used to improve frequency conversion in nonlinear optical mixing experiments. The extension of these concepts to very weak optical fields in the few-photon limit is then examined. The review concludes with a discussion of future prospects and potential new applications.

/pdf/electromagnetically-induced-transparency-optics-in-coherent-2dhges92td.pdf

Electromagnetically induced transparency : Optics in coherent media

The so-called Klein paradox—unimpeded penetration of relativistic particles through high and wide potential barriers—is one of the most exotic and counterintuitive consequences of quantum electrodynamics. The phenomenon is discussed in many contexts in particle, nuclear and astro-physics but direct tests of the Klein paradox using elementary particles have so far proved impossible. Here we show that the effect can be tested in a conceptually simple condensed-matter experiment using electrostatic barriers in single- and bi-layer graphene. Owing to the chiral nature of their quasiparticles, quantum tunnelling in these materials becomes highly anisotropic, qualitatively different from the case of normal, non-relativistic electrons. Massless Dirac fermions in graphene allow a close realization of Klein’s gedanken experiment, whereas massive chiral fermions in bilayer graphene offer an interesting complementary system that elucidates the basic physics involved.

/pdf/chiral-tunnelling-and-the-klein-paradox-in-graphene-5g1fsh0dta.pdf

Chiral tunnelling and the Klein paradox in graphene

Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as {it Non-Abelian anyons}, meaning that they obey {it non-Abelian braiding statistics}. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding quasiparticles, and then measuring the multi-quasiparticle states. The fault-tolerance of a topological quantum computer arises from the non-local encoding of the states of the quasiparticles, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the nu=5/2 state, although several other prospective candidates have been proposed in systems as disparate as ultra-cold atoms in optical lattices and thin film superconductors. In this review article, we describe current research in this field, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. We address both the mathematical underpinnings of topological quantum computation and the physics of the subject using the nu=5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

J. H. Blokland,1 P. C. M. Christianen,1,* B. M. Ashkinadze,2 V. V. Rudenkov,1 L. N. Pfeiffer,3 and J. C. Maan1 1High Field Magnet Laboratory, Institute for Molecules and Materials, Radboud University Nijmegen, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands 2Solid State Institute, Technion-Israel Institute of Technology, Haifa 32000, Israel 3Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA Received 9 April 2010; published 12 May 2010

/pdf/optical-probing-of-electronic-fractional-quantum-hall-states-1lvir51jik.pdf

Optical probing of electronic fractional quantum Hall states

We present experimental results on the tunneling into the edge of a two dimensional electron gas (2DEG) obtained with GaAs/AlGaAs cleaved edge overgrown structures. The electronic properties of the edge of these systems can be described by a one-dimensional chiral Tomonaga–Luttinger liquid when the filling factor of the 2DEG is very small. Here we focus on the region where the Tomonaga–Luttinger liquid breaks down to form a standard Fermi liquid close to ν=1 and show that we recover a universal curve, which describes all existing data.

/pdf/experiments-on-the-fermi-to-tomonaga-luttinger-liquid-3hceh0navg.pdf

Experiments on the Fermi to Tomonaga–Luttinger Liquid Transition in Quasi-1D Systems

Low temperature electronic transports in the presence of a density gradient

While it has long been known that the ground state of spatially indirect excitons is optically inactive [1], its relevance to Bose-Einstein condensation (BEC) has been overlooked. In the past, signatures of excitonic BEC have been sought in the form of directional luminescence or coherence of the light emitted by excitons. In this contribution, we review our recent experiments on trapped excitons, and their relevance on excitonic BEC

Dark region of quantum well excitons in stress-induced potentials

The surface charge generated on an Al0.24Ga0.76As∕GaAs quantum well sample by electron bombardment was monitored by measuring the change in the conductivity of the channel. Upon turning off the electron bombardment the surface charge on adsorbed layers of xenon and water at 8K decays in room temperature darkness with a lifetime τ=0.30±0.02s. The average charging efficiency, μ0, defined as the ratio of the charge collected by the surface to the beam current times the charging time, is μ0≃0.001. Surface charging proves to be an effective method for contactless gating of field effect devices.

L. N. Pfeiffer

Papers

Optical probing of electronic fractional quantum Hall states

Experiments on the Fermi to Tomonaga–Luttinger Liquid Transition in Quasi-1D Systems

Low temperature electronic transports in the presence of a density gradient

Dark region of quantum well excitons in stress-induced potentials

Charging efficiency and lifetime of image-bound electrons on a dielectric surface