The canonical example of a quantum-mechanical two-level system is spin. The simplest picture of spin is a magnetic moment pointing up or down. The full quantum properties of spin become apparent in phenomena such as superpositions of spin states, entanglement among spins, and quantum measurements. Many of these phenomena have been observed in experiments performed on ensembles of particles with spin. Only in recent years have systems been realized in which individual electrons can be trapped and their quantum properties can be studied, thus avoiding unnecessary ensemble averaging. This review describes experiments performed with quantum dots, which are nanometer-scale boxes defined in a semiconductor host material. Quantum dots can hold a precise but tunable number of electron spins starting with 0, 1, 2, etc. Electrical contacts can be made for charge transport measurements and electrostatic gates can be used for controlling the dot potential. This system provides virtually full control over individual electrons. This new, enabling technology is stimulating research on individual spins. This review describes the physics of spins in quantum dots containing one or two electrons, from an experimentalist’s viewpoint. Various methods for extracting spin properties from experiment are presented, restricted exclusively to electrical measurements. Furthermore, experimental techniques are discussed that allow for 1 the rotation of an electron spin into a superposition of up and down, 2 the measurement of the quantum state of an individual spin, and 3 the control of the interaction between two neighboring spins by the Heisenberg exchange interaction. Finally, the physics of the relevant relaxation and dephasing mechanisms is reviewed and experimental results are compared with theories for spin-orbit and hyperfine interactions. All these subjects are directly relevant for the fields of quantum information processing and spintronics with single spins i.e., single spintronics.

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Spins in few-electron quantum dots

Finite element methods for Maxwell's equations.

This article discusses finite element Galerkin schemes for a number of linear model problems in electromagnetism. The finite element schemes are introduced as discrete differential forms, matching the coordinate-independent statement of Maxwell's equations in the calculus of differential forms. The asymptotic convergence of discrete solutions is investigated theoretically. As discrete differential forms represent a genuine generalization of conventional Lagrangian finite elements, the analysis is based upon a judicious adaptation of established techniques in the theory of finite elements. Risks and difficulties haunting finite element schemes that do not fit the framework of discrete differential forms are highlighted.

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Finite elements in computational electromagnetism

Manipulation of single spins is essential for spin-based quantum information processing Electrical control instead of magnetic control is particularly appealing for this purpose, because electric fields are easy to generate locally on-chip We experimentally realized coherent control of a single-electron spin in a quantum dot using an oscillating electric field generated by a local gate The electric field induced coherent transitions (Rabi oscillations) between spin-up and spin-down with 90 degrees rotations as fast as approximately 55 nanoseconds Our analysis indicated that the electrically induced spin transitions were mediated by the spin-orbit interaction Taken together with the recently demonstrated coherent exchange of two neighboring spins, our results establish the feasibility of fully electrical manipulation of spin qubits

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Coherent Control of a Single Electron Spin with Electric Fields

Distance distributions between paramagnetic centers in the range of 1.8 to 6 nm in membrane proteins and up to 10 nm in deuterated soluble proteins can be measured by the DEER technique. The number of paramagnetic centers and their relative orientation can be characterized. DEER does not require crystallization and is not limited with respect to the size of the protein or protein complex. Diamagnetic proteins are accessible by site-directed spin labeling. To characterize structure or structural changes, experimental protocols were optimized and techniques for artifact suppression were introduced. Data analysis programs were developed, and it was realized that interpretation of the distance distributions must take into account the conformational distribution of spin labels. First methods have appeared for deriving structural models from a small number of distance constraints. The present scope and limitations of the technique are illustrated.

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DEER distance measurements on proteins

Computational tools for the design of accelerating structures are in use since decades. While highly accurate methods exist for quasi two dimensional cavities, fully three dimensional modeling with high precision is still a ‘big challenge’. The most widely used computer code in this area is MAFIA, basing on the Finite Integration Theory (FIT, [1,2]). While being well known for its robustness and reliability, MAFIA nevertheless suffers somewhat from a deficiency in being able to model very complicated 3Dcavities including curved boundaries with high precision. In this paper we present two recently developed algorithms, facing this challenge within FIT: the usage of generalized non-orthogonal computational grids (NOFIT), and the so-called Perfect Boundary Approximation (PBA) technique. Both methods represent consistent extensions of FIT, preserving all important properties as second order accuracy and stability of the transient solver. Especially the PBA technique reveals to be a highly efficient method, as it combines easy-to-use Cartesian grids with a perfect approximation of boundaries. We compare MAFIA with the PBA technique for typical accelerator components, and it turns out, that the PBA technique is more than one order of magnitude faster than the conventional method if many non Cartesian metallic boundaries appear inside the modeled structure.

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The Perfect Boundary Approximation Technique Facing the Big Challenge of High Precision Field Computation

Using numerical simulation techniques, the transmission and reflection coefficients, or S parameters, for left-handed metamaterials are calculated. Metamaterials consist of a lattice of conducting, nonmagnetic elements that can be described by an effective magnetic permeability μeff and an effective electrical permittivity eeff, both of which can exhibit values not found in naturally occurring materials. Because the electromagnetic fields in conducting metamaterials can be localized to regions much smaller than the incident wavelength, it can be difficult to perform accurate numerical simulations. The metamaterials simulated here, for example, are based on arrays of split ring resonators (SRRs), which produce enhanced and highly localized electric fields within the gaps of the elements in response to applied time dependent fields. To obtain greater numerical accuracy we utilize the newly developed commercially available code MICROWAVE STUDIO, which is based on the finite integration technique with the per...

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Ab initio numerical simulation of left-handed metamaterials: Comparison of calculations and experiments

A conformal finite-difference time-domain algorithm for the solution of electrodynamic problems in general perfectly conducting 3D geometries is presented. Unlike previous conformal approaches it has the second-order convergence without the need to reduce the maximal stable time step of conventional staircase approach. A novel proof for the local error rate for general geometries is given, and the method is verified and compared to other approaches by means of several numerical 2D examples. Copyright © 2003 John Wiley & Sons, Ltd.

A uniformly stable conformal FDTD-method in Cartesian grids

Effective electric permittivity and magnetic permeability for metamaterial structures are extracted from 3D field simulation data. The equivalent representation of the metamaterial is a homogeneous slab described with parameterized dispersive Drude and Lorentz models. The parameters of dispersive models are obtained by optimization. Proposed approach is applied to the extraction of effective material parameters for double negative metamaterial cells. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 285–288, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22105

Extraction of effective metamaterial parameters by parameter fitting of dispersive models

In this paper we present a reformulation of the FDTD algorithm on nonorthogonal grids, which was originally proposed by Holland in 1983. Based on the matrix-vector notation of the finite integration technique (FIT), the new formulation allows to study a special type of instability, which is due to the spatial discretization and independent of the choice of the timestep. It is shown, that this type of instability can be avoided by a symmetric evaluation of the metric coefficients of the nonorthogonal grid. Two numerical examples demonstrate the stability properties and the high accuracy of the new method.

Rolf Schuhmann

Papers

The Perfect Boundary Approximation Technique Facing the Big Challenge of High Precision Field Computation

Ab initio numerical simulation of left-handed metamaterials: Comparison of calculations and experiments

A uniformly stable conformal FDTD-method in Cartesian grids

Extraction of effective metamaterial parameters by parameter fitting of dispersive models

Stability of the FDTD algorithm on nonorthogonal grids related to the spatial interpolation scheme