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.

We describe the construction and operation of a two-channel noise detection system for measuring power and cross spectral densities of current fluctuations near 2 MHz in electronic devices at low temperatures. The system employs cryogenic amplification and fast-Fourier-transform based spectral measurement. The gain and electron temperature are calibrated using Johnson noise thermometry. Full shot noise of 100 pA can be resolved with an integration time of 10 s. We report a demonstration measurement of bias-dependent current noise in a gate defined GaAs/AlGaAs quantum point contact.

https://arxiv.org/pdf/cond-mat/0604018.pdf

A system for measuring auto- and cross-correlation of current noise at low temperatures

Temperature dependent weak localization is measured in metallic nanowires in a previously unexplored size regime down to width w = 5 nm. The dephasing time, tau(phi), shows a low temperature T dependence close to quasi-1D theoretical expectations (tau(phi) approximately T(-2/3)) in the narrowest wires, but exhibits a relative saturation as T-->0 for wide samples of the same material, as observed previously. As only sample geometry is varied to exhibit both suppression and divergence of tau(phi), this finding provides a new constraint on models of dephasing phenomena.

http://www.ruf.rice.edu/~rau/phys600/PRL001821.pdf

Geometry-Dependent Dephasing in Small Metallic Wires

We report the observation of a colossal, narrow resistance peak that arises in ultraclean (mobility ~3 × 10 7 cm 2 /Vs) GaAs/AlGaAs quantum wells (QWs) under millimeter wave irradiation and a weak magnetic field. Such a spike is superposed on the 2nd harmonic microwave-induced resistance oscillations (MIRO) but having an amplitude >300% of the MIRO, and a typical FWHM ~50 mK, comparable with the Landau level width. Systematic studies show a correlation between the spike and a pronounced negative magnetoresistance in these QWs, suggesting a mechanism based on the interplay of strong scatterers and smooth disorder. Alternatively, the spike may be interpreted as a manifestation of quantum interference between the quadrupole resonance and the higher-order cyclotron transition in well-separated Landau levels.

Observation of a cyclotron harmonic spike in microwave-induced resistances in ultraclean GaAs/AlGaAs quantum wells.

Microwave radiation has a dramatic effect on the magneto-resistance of two-dimensional electron systems, even reducing it below zero. It is thought that this is the result of the formation of distinct current domains. Direct experimental evidence for these domains is now presented for the first time.

Random telegraph photosignals in a microwave-exposed two-dimensional electron system

We report direct observations of vortices in a square superconducting wire grid imaged using scanning Hall probe microscopy. Real space images of vortex configurations are obtained as a function of the flux per unit cell f by measuring the local magnetic field just above the sample. At f=1/2 we observe domains of the checkerboard ground state. As f is reduced from 1/2 to 1/3 vacancies first penetrate the grain boundaries and then the checkerboard domains. Near f=-1/3 we observe domains of the 1/3 staircase ground state. Heating the sample close to T c produces correlated vortex hopping

L. N. Pfeiffer

Papers

A system for measuring auto- and cross-correlation of current noise at low temperatures

Geometry-Dependent Dephasing in Small Metallic Wires

Observation of a cyclotron harmonic spike in microwave-induced resistances in ultraclean GaAs/AlGaAs quantum wells.

Random telegraph photosignals in a microwave-exposed two-dimensional electron system

Direct spatial imaging of vortices in a superconducting wire network.