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.

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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.

We have explored the behavior of a two-dimensional hole system in the regime of very low densities and hence large r s . The electronic phases at the largest magnetic fields have surprising behavior. We found that with decreasing density the reentrant insulating phase weakens until it completely disappears. We also found that at the collapse of the reentrant insulator the nearby ν = 1 3 fractional quantum Hall liquid is unexpectedly suppressed. Both of these properties can be understood as stemming from quantum fluctuations of the insulating electronic solid.

The influence of quantum fluctuations on quantum hall phases at large rs

A tunable coherent far infrared radiation source from a biased semiconductor microcavity is demonstrated. The radiation is emitted after resonant excitation of the cavity polaritons by a short laser pulse and its energy is tuned by applying different bias fields.

Tunable coherent far infrared radiation emission from biased semiconductor microcavities

Terahertz photoconductivity of 2D electrons in top-gated semiconductor structures

We report on a detailed study of the free electron scattering effects on the (e1:hh1)1S and (e1:lh1)1S excitons in a GaAs quantum well with a variable density two-dimensional electron gas, that is either bare or embedded in a GaAs/AlGaAs microcavity. These effects are studied by measuring the reflection line width of the bare excitons or cavity polaritons as a function of photoexcitation intensity, temperature (2 K ) and a perpendicularly applied magnetic field (0 T ). This field induces the formation of charged polaritons at temperatures higher than the range of charged exciton existence without a magnetic field. In order to interpret the line-width data observed at high temperatures, when no charged polaritons exist, we developed a theoretical model that is based on calculating the exciton–electron direct and exchange interaction matrix elements, from which we derive the scattering rates of bare excitons. The model accounts well for the polariton line-widths dependence on microcavity-photon detuning energy and the electron density.

Magnetic field effect on the free electron–exciton scattering in GaAs/AlGaAs bare quantum wells and in microcavities

We report a study of the integer quantum Hall plateau-to-plateau transitions in the two-dimensional electron system in Al x Ga 1 - x As – Al 0.32 Ga 0.68 As heterostructures. For two-dimensional electron systems with disorder dominated by short-ranged alloy potential fluctuations, a power-law temperature scaling is established with the critical exponent obtained to be κ = 0.42 ± 0.01 , disregarding the density of the samples.

L. N. Pfeiffer

Papers

The influence of quantum fluctuations on quantum hall phases at large rs

Tunable coherent far infrared radiation emission from biased semiconductor microcavities

Terahertz photoconductivity of 2D electrons in top-gated semiconductor structures

Magnetic field effect on the free electron–exciton scattering in GaAs/AlGaAs bare quantum wells and in microcavities

Plateau-to-plateau transitions in random-alloy scattering dominated 2DEGs of different densities