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.

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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.

Recent experiments have shown several effects indicative of Bose-Einstein condensation in polaritons in GaAs-based microcavity structures when a harmonic potential trap for the two-dimensional polaritons is created by applied stress. These effects include both real-space and momentum-space narrowing, first-order coherence, and onset of linear polarization above a particle density threshold. Similar effects have been seen in systems without traps, raising the question of how important the role of the trap is in these experiments. In this paper we present results for both trapped conditions and resonant nontrapped conditions in the same sample. We find that the results are qualitatively different, with two distinct types of transitions. At low density in the trap, the polaritons remain in the strong-coupling regime while going through the threshold for onset of coherence; at higher density, there is a different threshold behavior which occurs with weak coupling and can be identified with lasing; this transition occurs both with and without a trap.

Role of the stress trap in the polariton quasiequilibrium condensation in GaAs microcavities

We extend our studies of microwave photoresistance of ultrahigh mobility two-dimensional electron systems (2DES) into the high-intensity, nonlinear regime employing both monochromatic and bichromatic radiation. Under high-intensity monochromatic radiation $\ensuremath{\omega}$ we observe zero-resistance states (ZRSs) that correspond to the rational values of $\ensuremath{\epsilon}=\ensuremath{\omega}∕{\ensuremath{\omega}}_{C}$ (${\ensuremath{\omega}}_{C}$ is the cyclotron frequency) and can be associated with multiphoton processes. Under bichromatic radiation ${\ensuremath{\omega}}_{1},{\ensuremath{\omega}}_{2}$ we discover a resistance minimum, possibly a precursor of bichromatic ZRSs, which seems to originate from a frequency mixing process, ${\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}$. These findings indicate that multiphoton processes play important roles in the physics of nonequilibrium transport of microwave-driven 2DES, and suggest new directions for theoretical and experimental studies.

Multiphoton processes in microwave photoresistance of two-dimensional electron systems

Low temperature near‐field scanning optical microscopy is used for spectroscopic studies of single, nanometer dimension, cleaved edge overgrown quantum wires. A direct experimental comparison between a two dimensional system and a single genuinely one dimensional quantum wire system, inaccessible to conventional far field optical spectroscopy, is enabled by the enhanced spatial resolution. We show that the photoluminescence of a single quantum wire is easily distinguished from that of the surrounding quantum well. Emission from localized centers is shown to dominate the photoluminescence from both wires and wells at low temperatures. A factor of 3 absorption enhancement for these wires compared to the wells is concluded from the photoluminescence excitation data.

Near‐field optical spectroscopy of single quantum wires

A robust technique for fabrication of metal wires with controlled widths substantially below 10 nm is presented. By etching a cleaved molecular-beam epitaxy grown substrate, a mechanical template is produced with surface relief of atomic lateral definition. Using metal deposition and directional ion etching of such a substrate, electrically continuous wires are formed from AuPd alloy with diameters as small as 3 nm and lengths greater than 1 μm. This technique can be used with a variety of materials and makes metallic nanostructures on a previously inaccessible size scale.

Fabrication of extremely narrow metal wires

We have used stress to create a harmonic potential for polaritons in GaAs microcavities and have previously reported that the polaritons undergo spontaneous coherence in the trap. In this paper we present results for both trapped conditions and resonant, nontrapped conditions in the same sample. We find that the results are qualitatively different with two distinct types of transitions. At low density in the trap, the polaritons remain in the strong coupling regime while going through the threshold for onset of coherence; at higher density, there is a different threshold behavior, which occurs with weak coupling and can be identified with lasing; this transition occurs both with and without a trap. The transition at lower density can therefore be identified as a type of nonequilibrium Bose–Einstein condensation.

L. N. Pfeiffer

Papers

Role of the stress trap in the polariton quasiequilibrium condensation in GaAs microcavities

Multiphoton processes in microwave photoresistance of two-dimensional electron systems

Near‐field optical spectroscopy of single quantum wires

Fabrication of extremely narrow metal wires

Lasing and polariton condensation: Two distinct transitions in GaAs microcavities with stress traps