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.

At total Landau level filling factor nu(tot)=1 a double-layer two-dimensional electron system with small interlayer separation supports a collective state possessing spontaneous interlayer phase coherence This state exhibits the quantized Hall effect when equal electrical currents flow in parallel through the two layers In contrast, if the currents in the two layers are equal, but oppositely directed, both the longitudinal and Hall resistances of each layer vanish in the low-temperature limit This finding supports the prediction that the ground state at nu(tot)=1 is an excitonic superfluid

/pdf/vanishing-hall-resistance-at-high-magnetic-field-in-a-double-3neicelcf1.pdf

Vanishing Hall resistance at high magnetic field in a double-layer two-dimensional electron system.

Quasi-particles with fractional charge and statistics, as well as modified Coulomb interactions, exist in a two-dimensional electron system in the fractional quantum Hall (FQH) regime. Theoretical models of the FQH state at filling fraction $v={^{5}}/{_{2}}$ make the further prediction that the wave function can encode the interchange of two quasi-particles, making this state relevant for topological quantum computing. We show that bias-dependent tunneling across a narrow constriction at $v={^{5}}/{_{2}}$ exhibits temperature scaling and, from fits to the theoretical scaling form, extract values for the effective charge and the interaction parameter of the quasi-particles. Ranges of values obtained are consistent with those predicted by certain models of the ${^{5}}/{_{2}}$ state.

http://science.sciencemag.org/content/sci/320/5878/899.full.pdf

Quasi-particle properties from tunneling in the v = 5/2 fractional quantum Hall state.

In the high-magnetic-field low-disorder limit the ground state at v=1/5 Landau-level filling is an incompressible quantum limit. This is determined by observing vanishing resistivity p xx as the temperature T→0. At filling factors below v=1/5 as well as in a narrow region above v=1/5, p xx diverges exponentially as T→0. This contrasts the T dependence at any higher v. The quantum limit at v=1/5 is surrounded by a different phase. In as much as the exponential divergencies are indicative of an electron solid this solid phase is reentrant in a narrow region above v=1/5

Quantum liquid versus electron solid around nu =1/5 Landau-level filling.

In a $\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$ quantum well of density $1\ifmmode\times\else\texttimes\fi{}{10}^{11}\text{ }{\mathrm{c}\mathrm{m}}^{\ensuremath{-}2}$ we observed a fractional quantum Hall effect (FQHE) at $\ensuremath{\nu}=4/11$ and $5/13$, and weaker states at $\ensuremath{\nu}=6/17$, $4/13$, $5/17$, and $7/11$. These sequences of fractions do not fit into the standard series of integral quantum Hall effects of composite fermions (CF) at $\ensuremath{\nu}=p/(2mp\ifmmode\pm\else\textpm\fi{}1)$. They rather can be regarded as the FQHE of CFs attesting to residual interactions between these composite particles. In tilted magnetic fields the $\ensuremath{\nu}=4/11$ state remains unchanged, strongly suggesting it to be spin polarized. The weak $\ensuremath{\nu}=7/11$ state vanishes quickly with tilt.

https://arxiv.org/pdf/cond-mat/0303429.pdf

Fractional quantum Hall effect of composite fermions.

We compare the use of free-space electro-optic sampling (FSEOS) with photoconducting antennas to detect terahertz (THz) radiation in the range of 0.1–3 THz. For the same average THz power and low-frequency modulation, signal-to-noise ratio and sensitivity are better with antenna detection at frequencies smaller than 3 THz. When the modulation frequency is increased to more than 1 MHz in FSEOS, both detection schemes have comparable performance. Using a singular-electric-field THz emitter, we demonstrate the feasibility of a THz imaging system using real-time delay scanning in FSEOS and only 20 mW of laser power.

L. N. Pfeiffer

Papers

Vanishing Hall resistance at high magnetic field in a double-layer two-dimensional electron system.

Quasi-particle properties from tunneling in the v = 5/2 fractional quantum Hall state.

Quantum liquid versus electron solid around nu =1/5 Landau-level filling.

Fractional quantum Hall effect of composite fermions.

Coherent terahertz radiation detection: Direct comparison between free-space electro-optic sampling and antenna detection