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.

Recent experiments have shown that two-dimensional electron systems with an externally applied magnetic field are an extremely rich ground for many-body physics. In particular, when only two of the Landau levels (LL) are filled an intricate magnetoresistance is found. This result stems from an interesting competition of electronic phases such as fractional quantum Hall liquids, reentrant integer Hall states, and unique quantized states at even denominator LL filling factors. We present a brief review of the transport properties of these electronic phases and discuss in detail the effects of an added in-plane magnetic field.

Exotic electronic phases in the second Landau level

Photoluminescence (PL) image size and angular resolved spectra were studied in a GaAs/AlAs planar microcavity (MC), with an embedded GaAs quantum well (QW) that contains a photogenerated, variable density, two-dimensional electron gas (2DEG). The PL was measured at T = 7 K, under nonresonant, cw excitation. The in-plane polariton migration from the photoexcitation spot (whose radius is ∼30 μm) is deduced from the PL image size. It is observed to be anisotropic in the MC plane, and for negative detuning energy values (δ = E C - E X ) it extends over in-plane distances of ∼300 μm. Introducing a 2DEG into the QW while maintaining the strong coupling between the excitons and the MC photon mode, reduces the migration range. On the other hand, the angular resolved PL spectra show that the emission from the k∥ ∼ 0 polariton states is enhanced as the 2DEG density is increased. From these observations we propose that the polaritons migrate while they occupy the long-lived states above the relaxation bottleneck in the lower polariton (LP) branch. The efficient electron-polariton scattering processes transfer the polaritons through the bottleneck into the radiative trap (k∥ ∼ 0) in the LP branch. These scattering processes occur at the point of photoexcitation in the MC plane, and therefore they decrease the migration range.

Anisotropic polariton migration and its effect on photoluminescence amplification in a GaAs/AlAs microcavity containing a two-dimensional electron gas

We report the results of tilted magnetic field experiments on three high quality wide GaAs quantum wells, with particular emphasis on the N ≥ 1 Landau levels. With an increasing component of in-plane magnetic field, B∥, we observe reentrant behavior for the odd filling factor quantum Hall states. This reentrance is not explained within a single-particle energy level crossing scheme and may be related to the collapse of interlayer tunneling.

OBSERVATIONS OF PARALLEL FIELD INDUCED REENTRANT QUANTUM HALL EFFECTS IN WIDE GaAs QUANTUM WELLS

Mixed neutral and negatively charged microcavity polaritons

Intense terahetz (THz) pulses induce a photoluminescence (PL) flash from undoped high-quality GaAs/AlGaAs quantum wells under continuous wave laser excitation The number of excitons increases 10000-fold from that of the steady state under only laser excitation The THz electric field dependence and the relaxation dynamics of the PL flash intensity suggest that the strong electric field of the THz pulse ionizes impurity states during the one-picosecond period of the THz pulse and release carriers from a giant reservoir containing impurity states in the AlGaAs layers

L. N. Pfeiffer

Papers

Exotic electronic phases in the second Landau level

Anisotropic polariton migration and its effect on photoluminescence amplification in a GaAs/AlAs microcavity containing a two-dimensional electron gas

OBSERVATIONS OF PARALLEL FIELD INDUCED REENTRANT QUANTUM HALL EFFECTS IN WIDE GaAs QUANTUM WELLS

Mixed neutral and negatively charged microcavity polaritons

Large Enhancement of the Photoluminescence Emission of Photoexcited Undoped GaAs Quantum Wells Induced by an Intense Single-Cycle Terahertz Pulse