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.

Excitation modes in the range $2/5\ensuremath{\ge}\ensuremath{\nu}\ensuremath{\ge}1/3$ of the fractional quantum Hall regime are observed by resonant inelastic light scattering. Spectra of spin-reversed excitations suggest a structure of lowest spin-split Landau levels of composite fermions that is similar to that of electrons. Spin-flip energies determined from spectra reveal significant composite fermion interactions. The filling factor dependence of mode energies displays an abrupt change in the middle of the range when there is partial population of a composite fermion level.

Evidence of Landau Levels and Interactions in Low-Lying Excitations of Composite Fermions at 1 / 3 ≤ ν ≤ 2 / 5

Electronic Mach–Zehnder interferometers in the quantum Hall (QH) regime are currently discussed for the realization of quantum information schemes. A recently proposed device architecture employs interference between two co-propagating edge channels. Here we demonstrate the precise control of individual edge-channel trajectories in quantum point contact devices in the QH regime. The biased tip of an atomic force microscope is used as a moveable local gate to pilot individual edge channels. Our results are discussed in light of the implementation of multi-edge interferometers.

Selective control of edge-channel trajectories by scanning gate microscopy

We present a summary of the experimental findings for a new dissipationless transport effect in high-mobility two-dimensional electron systems exposed to crossed millimeter wave and weak magnetic fields. These systems display zero-resistance states (ZRS) in Hall samples and zero-conductance states (ZCS) in Corbino geometry, when the electron mobility is sufficiently high. Samples of lower mobility show pronounced resistance oscillations which are periodic in 1/B. We survey the development from oscillatory resistance to ZRS as mobility is increased. Experimental data on temperature and microwave power dependence, as well as I–V characteristics in this effect, are presented. These results should help in distinguishing between competing theoretical models of the phenomena.

Dissipationless 2D electronic transport effect induced by microwaves

We present a method for all-optical switching using a microcavity exciton-polariton system. Unlike many other switching methods, we use a single wavelength for both the signal and gate, which means that this method could be used for cascading optical logic gates and amplification within an all-optical circuit. We resonantly pump the sample at an angle with a laser beam and probe the sample with another beam at normal incidence. Upon saturation of the exciton-polariton states, the normal-incidence resonance increases in energy to permit the probe beam to be transmitted through the sample. Experimental results demonstrate switching using a GaAs/AlAs microcavity. Switching times on the order of ten picoseconds and on/off ratios on the order of 10:1 have been observed, and we propose design options to improve upon these.

Single-wavelength, all-optical switching based on exciton-polaritons

Resonant inelastic light scattering experiments at $\ensuremath{\nu}=1/3$ reveal a novel splitting of the long-wavelength modes in the low energy spectrum of quasiparticle excitations in the charge degree of freedom. We find a single peak at small wave vectors that splits into two distinct modes at larger wave vectors. The evidence of well-defined dispersive behavior at small wave vectors indicates a coherence of the quantum fluid in the micron length scale. We evaluate interpretations of long-wavelength modes of the electron liquid.

L. N. Pfeiffer

Papers

Evidence of Landau Levels and Interactions in Low-Lying Excitations of Composite Fermions at 1 / 3 ≤ ν ≤ 2 / 5

Selective control of edge-channel trajectories by scanning gate microscopy

Dissipationless 2D electronic transport effect induced by microwaves

Single-wavelength, all-optical switching based on exciton-polaritons

Splitting of long-wavelength modes of the fractional quantum hall liquid at nu = 1/3.