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 investigated the influence of the polarization of the incident radiation on the recently discovered microwave-induced resistance oscillations in state-of-the-art highest purity 2D electron systems. A quasi-optical setup allows us to tune in situ between different circular as well as linear polarization states. We find that the microwave-induced zero resistance and the resistance oscillations are notably immune to changes in the polarization. This observation is discrepant with a number of proposed theories. Deviations for different polarizations only occur for a bolometric contribution to the resistance associated with the resonant heating at the cyclotron resonance itself.

Circular-polarization-dependent study of the microwave-induced magneto-resistance oscillations in the 2D electron system

Recent optically pumped nuclear magnetic resonance (OPNMR) measurements in two different electron-doped multiple quantum well samples near the fractional quantum Hall ground state ν= 1 3 are reviewed. Below 0.5 K, the spectra provide evidence that the spin-reversed charged excitations of this ground state are localized over the NMR time scale of ∼40 μs. By varying NMR pulse parameters, the electron spin temperature (as measured by the Knight shift) could be driven above the lattice temperature. For Tlattice 1 3 .

Ultraslow electron spin dynamics in the fractional quantum Hall regime

Ballistic transport is nothing but an electron transport in the limit (/spl omega//spl tau//sub m/ > 1); the electron move without scattering between electric field cycles. Diffusive transport can be defined as the lower frequency range (/spl omega//spl tau//sub m/<1); in the diffusive limit there are many scattering events between electric field cycles. Here we measure for the first time the crossover from diffusive to ballistic transport as a function of frequency in a dc contacted 2DEG.

Crossover from diffusive to ballistic transport as a function of frequency in a two dimensional electron gas

Recent investigations of secondary emission from quantum well excitons following ultrafast resonant excitation have demonstrated an intricate interplay of coherent Rayleigh scattering and incoherent luminescence1,2,3,4. We have very recently demonstrated5 that it is possible to isolate and time resolve the coherent field associated with the Rayleigh component using ultrafast spectral interferometry6 or Tadpole7, thus, obtaining substantial and new information of the nature of resonant secondary emission.

L. N. Pfeiffer

Papers

Circular-polarization-dependent study of the microwave-induced magneto-resistance oscillations in the 2D electron system

Ultraslow electron spin dynamics in the fractional quantum Hall regime

Crossover from diffusive to ballistic transport as a function of frequency in a two dimensional electron gas

Ultrafast spectral interferometry of resonant secondary emission from quantum wells: From Rayleigh scattering to coherent emission from biexcitons

AlGaN/GaN HEMTs grown by Molecular Beam Epitaxy on sapphire, 6H-SiC, and HVPE-GaN templates