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.

In an ultraclean 2D electron system (2DES) subjected to crossed millimeterwave (30--150 GHz) and weak ($Bl2\text{ }\mathrm{k}\mathrm{G}$) magnetic fields, a series of apparently dissipationless states emerges as the system is detuned from cyclotron resonances. Such states are characterized by an exponentially vanishing low-temperature diagonal resistance and a classical Hall resistance. The activation energies associated with such states exceed the Landau level spacing by an order of magnitude. Our findings are likely indicative of a collective ground state previously unknown for 2DES.

Evidence for a new dissipationless effect in 2D electronic transport.

The tunneling conductance between two parallel 2D electron systems has been measured in a regime of strong interlayer Coulomb correlations. At total Landau level filling νT = 1 the tunnel spectrum changes qualitatively when the boundary separating the compressible phase from the ferromagnetic quantized Hall state is crossed. A huge resonant enhancement replaces the strongly suppressed equilibrium tunneling characteristic of weakly coupled layers. The possible relationship of this enhancement to the Goldstone mode of the broken symmetry ground state is discussed.

/pdf/resonantly-enhanced-tunneling-in-a-double-layer-quantum-hall-5fwvb3wp8d.pdf

Resonantly enhanced tunneling in a double layer quantum hall ferromagnet.

The sign of the interference (constructive or destructive) between quantum-mechanical paths depends on the phase difference between the paths. In the Fano effect1 two optical paths from the ground state of a system — one direct and one mediated by a resonance — to a state in an energy continuum interfere to produce an asymmetric absorption spectrum that falls to zero near the absorption maximum. Zero absorption occurs as the wavelength is scanned across the resonance, at a photon energy corresponding to a 180 ° phase difference between the paths. Similar interference effects occur when two absorption paths are mediated by two different states, and they provide the basis for lasers that operate without a population inversion2,3,4,5,6,7. Here we report the control, by quantum mechanical tunnelling, of interference in optical absorption. The two intermediate states are resonances that arise from the mixing of the states in two adjacent semiconductors quantum wells, which are broadened by tunnelling into the same energy continuum through an ultra-thin potential-energy barrier. Inverting the direction of tunnelling by reversing the position of the barrier with respect to the two quantum wells changes the interference from destructive to constructive, as predicted theoretically. This effect might provide a way to make semiconductor lasers without population inversion8.

Controlling the sign of quantum interference by tunnelling from quantum wells

The reverse bias leakage current in macroscopic GaN Schottky diodes is found to be insensitive to barrier height. Using a scanning current–voltage microscope, we show that the reverse bias current occurs at small isolated regions, while most of the sample is insulating. By comparing the current maps to topographic images and transmission electron microscopy results, we conclude that reverse bias leakage occurs primarily at dislocations with a screw component. Furthermore, for a fixed dislocation density, the V/III ratio during the molecular beam epitaxial growth strongly affects reverse leakage, indicating complex dislocation electrical behavior that is sensitive to the local structural and/or chemical changes.

http://manfragroup.org/wp-content/uploads/2014/12/Inhomogeneous-spatial-distribution-of-reverse-bias-leakage-in-GaN-Schottky-diodes.pdf

Inhomogeneous spatial distribution of reverse bias leakage in GaN Schottky diodes

During the past ten years, coupled quantum wells have emerged as a promising system for experiments on Bose condensation of excitons, with numerous theoretical1,2,3,4,5,6 and experimental7,8,9,10,11,12 studies aimed at the demonstration of this effect. One of the issues driving these studies is the possibility of long-range coherent transport of excitons. Excitons in quantum wells typically diffuse only a few micrometres from the spot where they are generated by a laser pulse; their diffusion is limited by their lifetime (typically a few nanoseconds) and by scattering due to disorder in the well structure. Here we report photoluminescence measurements of InGaAs quantum wells and the observation of an effect by which luminescence from excitons appears hundreds of micrometres away from the laser excitation spot. This luminescence appears as a ring around the laser spot; almost none appears in the region between the laser spot and the ring. This implies that the excitons must travel in a dark state until they reach some critical distance, at which they collectively revert to luminescing states. It is unclear whether this effect is related to macroscopic coherence caused by Bose condensation of excitons.

L. N. Pfeiffer

Papers

Evidence for a new dissipationless effect in 2D electronic transport.

Resonantly enhanced tunneling in a double layer quantum hall ferromagnet.

Controlling the sign of quantum interference by tunnelling from quantum wells

Inhomogeneous spatial distribution of reverse bias leakage in GaN Schottky diodes

Long-range transport in excitonic dark states in coupled quantum wells