We review in detail a number of approaches that have been adopted to try and explain the remarkable observation of our accelerating universe. In particular we discuss the arguments for and recent progress made towards understanding the nature of dark energy. We review the observational evidence for the current accelerated expansion of the universe and present a number of dark energy models in addition to the conventional cosmological constant, paying particular attention to scalar field models such as quintessence, K-essence, tachyon, phantom and dilatonic models. The importance of cosmological scaling solutions is emphasized when studying the dynamical system of scalar fields including coupled dark energy. We study the evolution of cosmological perturbations allowing us to confront them with the observation of the Cosmic Microwave Background and Large Scale Structure and demonstrate how it is possible in principle to reconstruct the equation of state of dark energy by also using Supernovae Ia observational data. We also discuss in detail the nature of tracking solutions in cosmology, particle physics and braneworld models of dark energy, the nature of possible future singularities, the effect of higher order curvature terms to avoid a Big Rip singularity, and approaches to modifying gravity which leads to a late-time accelerated expansion without recourse to a new form of dark energy.

Dynamics of dark energy

The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

Magnetoencephalography (MEG) is a noninvasive technique for investigating neuronal activity in the living human brain. The time resolution of the method is better than 1 ms and the spatial discrimination is, under favorable circumstances, 2-3 mm for sources in the cerebral cortex. In MEG studies, the weak 10 fT-1 pT magnetic fields produced by electric currents flowing in neurons are measured with multichannel SQUID (superconducting quantum interference device) gradiometers. The sites in the cerebral cortex that are activated by a stimulus can be found from the detected magnetic-field distribution, provided that appropriate assumptions about the source render the solution of the inverse problem unique. Many interesting properties of the working human brain can be studied, including spontaneous activity and signal processing following external stimuli. For clinical purposes, determination of the locations of epileptic foci is of interest. The authors begin with a general introduction and a short discussion of the neural basis of MEG. The mathematical theory of the method is then explained in detail, followed by a thorough description of MEG instrumentation, data analysis, and practical construction of multi-SQUID devices. Finally, several MEG experiments performed in the authors' laboratory are described, covering studies of evoked responses and of spontaneous activity in both healthy and diseased brains. Many MEG studies by other groups are discussed briefly as well.

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Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain

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

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.

Understanding the mechanism responsible for superconductivity in the high-temperature cuprates has been one of the major goals of condensed-matter physicists since the discovery of these exciting materials in 1986. Experimental evidence suggests that the pairing state may be unconventional, featuring an anisotropic order parameter for which a wide range of theoretical models for the superconducting pairing have been proposed. Recently, a new class of experiments has been presented that are sensitive to the phase of the superconducting order parameter, allowing an unambiguous determination of the symmetry of the pairing state. These experiments, based on the interference of the quantum-mechanical phases in Josephson tunnel junctions and dc SQUID devices, give strong evidence for pairing in a channel with $d$--wave symmetry in the most widely studied cuprate, $Y{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}x}$. Confirmation of this pairing state will focus efforts to develop a microscopic theory for the high-temperature superconductors and to apply them in power transmission and electronic device technologies.

Phase-sensitive tests of the symmetry of the pairing state in the high-temperature superconductors-Evidence for d x 2 -y 2 symmetry

We present an experiment designed to determine directly the symmetry of the pairing state in the cuprate superconductors. From the magnetic flux modulation of YBCO-Pb dc SQUIDs, we determine the spatial anisotropy of the phase of the order parameter in single crystals of YBCO. The experimental results are complicated by SQUID asymmetries and the trapping of magnetic vortices, but taken as a whole give rather strong evidence for a phase shift of \ensuremath{\pi} that is predicted for the ${\mathit{d}}_{\mathit{x}}^{2\mathrm{\ensuremath{-}}}$${\mathit{y}}^{2}$ pairing state. This is further corroborated by single junction modulation measurements.

https://link.aps.org/pdf/10.1103/PhysRevLett.71.2134

Experimental determination of the superconducting pairing state in YBCO from the phase coherence of YBCO-Pb dc SQUIDs.

We report on transport measurements of an InAs nanowire coupled to niobium nitride leads at high magnetic fields. We observe a zero-bias anomaly (ZBA) in the differential conductance of the nanowire for certain ranges of magnetic field and chemical potential. The ZBA can oscillate in width with either the magnetic field or chemical potential; it can even split and re-form. We discuss how our results relate to recent predictions of hybridizing Majorana fermions in semiconducting nanowires, while considering more mundane explanations.

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Anomalous modulation of a zero-bias peak in a hybrid nanowire-superconductor device.

We have measured the magnetic field modulation of the critical current of YBCO-Au-Pb Josephson junctions fabricated on the $a$ and $b$ edges and straddling the $a\ensuremath{-}b$ corners of YBCO single crystals. In corner junctions, the critical current drops sharply at zero applied field and increases as the field is increased in either direction, in sharp contrast to the Fraunhofer diffraction patterns observed for single edge junctions. This behavior signifies a sign change of the order parameter between the $a$ and $b$ directions, strong direct evidence that the superconducting pairing state of YBCO has ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry.

Evidence for dx2-y2 Pairing from the Magnetic Field Modulation of YBa2Cu3O7-Pb Josephson Junctions.

We have measured the spectral density of the 1/f voltage noise in current-biased resistively shunted Josephson tunnel junctions and dc SQUIDs. A theory in which fluctuations in the temperature give rise to fluctuations in the critical current and hence in the voltage predicts the magnitude of the noise quite accurately for junctions with areas of about 2 x 10/sup 4/ ..mu..m/sup 2/, but significantly overestimates the noise for junctions with areas of about 6 ..mu..m/sup 2/. DC SQUIDs fabricated from these two types of junctions exhibit substantially more 1/f voltage noise than would be predicted from a model in which the noise arises from critical current fluctuations in the junctions. This result was confirmed by an experiment involving two different bias current and flux modulation schemes, which demonstrated that the predominant 1/f voltage noise arises not from critical current fluctuations, but from some unknown source that can be regarded as an apparent 1/f flux noise. Measurements on five different configurations of dc SQUIDs fabricated with thin-film tunnel junctions and with widely varying areas, inductances, and junction capacitances show that the spectral density of the 1/f equivalent flux noise is roughtly constant, within a factor of three of (10/sup -10//f)phi/sup 2//submore » 0/Hz/sup -1/. It is emphasized that 1/f flux noise may not be the predominant source of 1/f noise in SQUIDS fabricated with other technologies.« less

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D. J. Van Harlingen

Papers

Phase-sensitive tests of the symmetry of the pairing state in the high-temperature superconductors-Evidence for d x 2 -y 2 symmetry

Experimental determination of the superconducting pairing state in YBCO from the phase coherence of YBCO-Pb dc SQUIDs.

Anomalous modulation of a zero-bias peak in a hybrid nanowire-superconductor device.

Evidence for dx2-y2 Pairing from the Magnetic Field Modulation of YBa2Cu3O7-Pb Josephson Junctions.

Flicker (1/f) noise in tunnel junction DC SQUIDS