The Stern-Gerlach Experiment Revisited

The Stern-Gerlach-Experiment (SGE) of 1922 is a seminal benchmark experiment of quantum physics providing evidence for several fundamental properties of quantum systems. Based on today's knowledge we illustrate the different benchmark results of the SGE for the development of modern quantum physics and chemistry. The SGE provided the first direct experimental evidence for angular momentum quantization in the quantum world and thus also for the existence of directional quantization of all angular momenta in the process of measurement. It measured for the first time a ground state property of an atom, it produced for the first time a `spin-polarized' atomic beam, it almost revealed the electron spin. The SGE was the first fully successful molecular beam experiment with high momentum-resolution by beam measurements in vacuum. This technique provided a new kinematic microscope with which inner atomic or nuclear properties could be investigated. The original SGE is described together with early attempts by Einstein, Ehrenfest, Heisenberg, and others to understand directional quantization in the SGE. Heisenberg's and Einstein's proposals of an improved multi-stage SGE are presented. The first realization of these proposals by Stern, Phipps, Frisch and Segr\`e is described. The set-up suggested by Einstein can be considered an anticipation of a Rabi-apparatus. Recent theoretical work is mentioned in which the directional quantization process and possible interference effects of the two different spin states are investigated. In full agreement with the results of the new quantum theory directional quantization appears as a general and universal feature of quantum measurements. One experimental example for such directional quantization in scattering processes is shown. Last not least, the early history of the `almost' discovery of the electron spin in the SGE is revisited.Comment: 50pp, 17 fig

Quantum dots and spin qubits in graphene

This is a review on graphene quantum dots and their use as a host for spin qubits. We discuss the advantages but also the challenges to use graphene quantum dots for spin qubits as compared to the more standard materials like GaAs. We start with an overview of this young and fascinating field and will then discuss gate-tunable quantum dots in detail. We calculate the bound states for three different quantum dot architectures where a bulk gap allows for confinement via electrostatic fields: (i) graphene nanoribbons with armchair boundary, (ii) a disc in single-layer graphene, and (iii) a disc in bilayer graphene. In order for graphene quantum dots to be useful in the context of spin qubits, one needs to find reliable ways to break the valley-degeneracy. This is achieved here, either by a specific termination of graphene in (i) or in (ii) and (iii) by a magnetic field, without the need of a specific boundary. We further discuss how to manipulate spin in these quantum dots and explain the mechanism of spin decoherence and relaxation caused by spin-orbit interaction in combination with electron-phonon coupling, and by hyperfine interaction with the nuclear spin system.Comment: 23 pages, 10 figures, topical review prepared for Nanotechnolog

Geometric phases and Bloch sphere constructions for SU(N), with a complete description of SU(4)

A two-sphere ("Bloch" or "Poincare") is familiar for describing the dynamics of a spin-1/2 particle or light polarization. Analogous objects are derived for unitary groups larger than SU(2) through an iterative procedure that constructs evolution operators for higher-dimensional SU in terms of lower-dimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation. For a general Hamiltonian of SU(4) with 15 parameters, and for Hamiltonians of its various sub-groups so that fewer parameters suffice, we derive Bloch-like rotation of unit vectors analogous to the one familiar for a single spin in a magnetic field. The unitary evolution of a quantal spin pair is thereby expressed as rotations of real vectors. Correspondingly, the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as a four-sphere for the SO(5) sub-group and an eight-dimensional Grassmannian manifold for the general SU(4). This latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectors moving on a five-sphere with an additional phase constraint.Comment: 9 page

Statistical signatures of critical behavior in small systems

The cluster distributions of different systems are examined to search for signatures of a continuous phase transition. In a system known to possess such a phase transition, both sensitive and insensitive signatures are present; while in systems known not to possess such a phase transition, only insensitive signatures are present. It is shown that nuclear multifragmentation results in cluster distributions belonging to the former category, suggesting that the fragments are the result of a continuous phase transition.Comment: 31 pages, two columns with 30 figure

Properties of Graphene: A Theoretical Perspective

In this review, we provide an in-depth description of the physics of monolayer and bilayer graphene from a theorist's perspective. We discuss the physical properties of graphene in an external magnetic field, reflecting the chiral nature of the quasiparticles near the Dirac point with a Landau level at zero energy. We address the unique integer quantum Hall effects, the role of electron correlations, and the recent observation of the fractional quantum Hall effect in the monolayer graphene. The quantum Hall effect in bilayer graphene is fundamentally different from that of a monolayer, reflecting the unique band structure of this system. The theory of transport in the absence of an external magnetic field is discussed in detail, along with the role of disorder studied in various theoretical models. We highlight the differences and similarities between monolayer and bilayer graphene, and focus on thermodynamic properties such as the compressibility, the plasmon spectra, the weak localization correction, quantum Hall effect, and optical properties. Confinement of electrons in graphene is nontrivial due to Klein tunneling. We review various theoretical and experimental studies of quantum confined structures made from graphene. The band structure of graphene nanoribbons and the role of the sublattice symmetry, edge geometry and the size of the nanoribbon on the electronic and magnetic properties are very active areas of research, and a detailed review of these topics is presented. Also, the effects of substrate interactions, adsorbed atoms, lattice defects and doping on the band structure of finite-sized graphene systems are discussed. We also include a brief description of graphane -- gapped material obtained from graphene by attaching hydrogen atoms to each carbon atom in the lattice.Comment: 189 pages. submitted in Advances in Physic

Hellmann-Feynman theorem and fluctuation-correlation analysis of the Calogero-Sutherland model

Exploiting the results of the exact solution for the ground state of the one-dimensional spinless quantum gas of Fermions and impenetrable Bosons with the mu/x_{ij}^2 particle-particle interaction, the Hellmann-Feynman theorem yields mutually compensating divergences of both the kinetic and the interaction energy in the limiting case mu to -1/4. These divergences result from the peculiar behavior of both the momentum distribution (for large momenta) and the pair density (for small inter-particle separation). The available analytical pair densities for mu=-1/4, 0, and 2 allow to analyze particle-number fluctuations. They are suppressed by repulsive interaction (mu>0), enhanced by attraction (mu<0), and may therefore measure the kind and strength of correlation. Other recently proposed purely quantum-kinematical measures of the correlation strength arise from the small-separation behavior of the pair density or - for Fermions - from the non-idempotency of the momentum distribution and its large-momenta behavior. They are compared with each other and with reference-free, short-range correlation-measuring ratios of the kinetic and potential energies.Comment: 30 pages, 9 figures, revised version, short version appeared as PRB 62, 15279-15282 (2000

Gigahertz quantized charge pumping in graphene quantum dots

Single electron pumps are set to revolutionize electrical metrology by enabling the ampere to be re-defined in terms of the elementary charge of an electron. Pumps based on lithographically-fixed tunnel barriers in mesoscopic metallic systems and normal/superconducting hybrid turnstiles can reach very small error rates, but only at MHz pumping speeds corresponding to small currents of the order 1 pA. Tunable barrier pumps in semiconductor structures have been operated at GHz frequencies, but the theoretical treatment of the error rate is more complex and only approximate predictions are available. Here, we present a monolithic, fixed barrier single electron pump made entirely from graphene. We demonstrate pump operation at frequencies up to 1.4 GHz, and predict the error rate to be as low as 0.01 parts per million at 90 MHz. Combined with the record-high accuracy of the quantum Hall effect and proximity induced Josephson junctions, accurate quantized current generation brings an all-graphene closure of the quantum metrological triangle within reach. Envisaged applications for graphene charge pumps outside quantum metrology include single photon generation via electron-hole recombination in electrostatically doped bilayer graphene reservoirs, and for readout of spin-based graphene qubits in quantum information processing.Comment: 13 pages, 11 figures, includes supplementary informatio