Position measurements in the de Broglie–Bohm interpretation of quantum mechanics

The de Broglie–Bohm interpretation of quantum mechanics assigns positions and trajectories to particles. We analyze the validity of a formula for the velocities of Bohmian particles which makes the analysis of these trajectories particularly simple. We apply it to particle detectors of four different types and show that the detectors of three of these types lead to “surrealistic trajectories”, i.e., leave a trace where the Bohmian particle was not present.     

Ballistic Aharonov–Bohm quantum bits and quantum gates

We propose an architecture to perform quantum computation, using ballistic electrons as qubits and coupled quantum rings as quantum gates. In the proposed architecture two adjacent one-dimensional wires, creating a single qubit, are connected to two coupled quantum rings, where the required magnetic flux is provided by enclosed nano-sized magnets. The phase modulation of the wave function of the ballistic electrons under the Aharonov–Bohm effect is carefully designed to facilitate reprogrammable and dynamically controllable quantum gates. Arbitrary single-qubit quantum gates with high fidelity can be constructed on the basis of this architecture.     

Geometric phase of a classical Aharonov–Bohm Hamiltonian

We present a gauge-invariant approach for associating a geometric phase with the phase space trajectory of a classical dynamical system. As an application, we consider the classical analog of the quantum Aharonov–Bohm (AB) Hamiltonian for a charged particle orbiting around a current carrying long thin solenoid. We compute the classical geometric phase of a closed phase space trajectory, and also determine its dependence on the magnetic flux enclosed by the orbit. We study the similarities and differences between this classical geometric phase and the AB phase acquired by the wave function of the quantum AB Hamiltonian. We suggest an experiment to measure the geometric phase for the classical AB system, by using an appropriate optical fiber ring interferometer.     

Geometry of the Aharonov–Bohm Effect

We show that the connection responsible for any Abelian or non-Abelian Aharonov–Bohm effect with n parallel “magnetic” flux lines in ℝ³, lies in a trivial G-principal bundle P→M, i.e. P is isomorphic to the product M×G, where G is any path connected topological group; in particular a connected Lie group. We also show that two other bundles are involved: the universal covering space , where path integrals are computed, and the associated bundle P× _G ℂ ^m →M, where the wave function and its covariant derivative are sections.     

Two-dimensional quantum ring in a graphene layer in the presence of a Aharonov–Bohm flux

In this paper we study the relativistic quantum dynamics of a massless fermion confined in a quantum ring. We use a model of confining potential and introduce the interaction via Dirac oscillator coupling, which provides ring confinement for massless Dirac fermions. The energy levels and corresponding eigenfunctions for this model in graphene layer in the presence of Aharonov–Bohm flux in the centre of the ring and the expression for persistent current in this model are derived. We also investigate the model for quantum ring in graphene layer in the presence of a disclination and a magnetic flux. The energy spectrum and wave function are obtained exactly for this case. We see that the persistent current depends on parameters characterizing the topological defect.     

Aharonov–Bohm effect in the ghost interference

Chaotic systems demonstrate complex behaviour in their state variables and their parameters, which generate some challenges and consequences. This paper presents a new synchronisation scheme based on integral sliding mode control (ISMC) method on a class of complex chaotic systems with complex unknown parameters. Synchronisation between corresponding states of a class of complex chaotic systems and also convergence of the errors of the system parameters to zero point are studied. The designed feedback control vector and complex unknown parameter vector are analytically achieved based on the Lyapunov stability theory. Moreover, the effectiveness of the proposed methodology is verified by synchronisation of the Chen complex system and the Lorenz complex systems as the leader and the follower chaotic systems, respectively. In conclusion, some numerical simulations related to the synchronisation methodology is given to illustrate the effectiveness of the theoretical discussions.     

On the Aharonov–Bohm Hamiltonian

Using the theory of self-adjoint extensions, we construct all the possible Hamiltonians describing the nonrelativistic Aharonov–Bohm effect. In general, the resulting Hamiltonians are not rotationally invariant so that the angular momentum is not a constant of motion. Using an explicit formula for the resolvent, we describe the spectrum and compute the generalized eigenfunctions and the scattering amplitude.     

Aharonov–Bohm effects in magnetohydrodynamics

It is shown that an Aharonov–Bohm (AB) effect exists in magnetohydrodynamics (MHD). This effect is best described in terms of the MHD variational variables (Kats, 2004; Yahalom and Lynden-Bell, 2008; Yahalom, 2010) [1], [10], [12]. If a MHD flow has a non-trivial topology some of the functions appearing in the MHD Lagrangian are non-single-valued. These functions have properties similar to the phases in the AB celebrated effect (Aharonov and Bohm, 1959; van Oudenaarden et al., 1998) [2], [3]. While the manifestation of the quantum AB effect is in interference fringe patterns (Tonomura et al., 1982) [4], the manifestation of the MHD Aharonov–Bohm effects are through new dynamical conservation laws.     

Aharonov–Bohm effect in non-uniform crater-like quantum dot with double rim

We study electron energies in a double concentric quantum ring with anisotropy in the rims heights in the presence of the external magnetic field applied along the symmetry axis. To this end, we consider a model in which the thickness grows linearly from the axis up to the inner rim with a slope different from one between the inner and the outer rims. The anisotropy in the rims heights originated by the presence in the structure of various valleys we simulate by periodic dependence of the slope on the radial direction. We show that the wave functions of the electron confined in such structure can be found analytically if the slopes in all radial directions are the same, and by using a simple exact diagonalization procedure otherwise. The behavior of the electron energies as functions of the magnetic field, rings radii and rims heights, as well as the number of the valleys and their depths is consistently described with our formalism. The entanglement of the states with different radial and orbital quantum numbers, the period and the amplitude of the Aharonov–Bohm oscillations are very sensible to any variations of the rims heights.     

An investigation of the gate dependence in an electronic Aharonov–Bohm interferometer

In this study we have obtained the dependence of the electronic distribution on the gate shape and the applied gate voltage in a quantum Hall effect based Aharonov–Bohm interferometer using a method presented in our previous studies. We have discussed the relation between the distribution of incompressible strips and observation of Aharonov–Bohm oscillations. We have obtained the distributions of the incompressible strips for various gate voltages and have shown that a gate potential sweep and a magnetic field sweep would be equivalent. Our calculations also predict that for wider gate separations it is possible observe a silent region while sweeping the magnetic field or the gate voltage.     

Geometric model of a fullerene molecule in the presence of Aharonov–Bohm flux

In this study, we describe a geometric model of a fullerene molecule with Ih symmetry. We combine the well known non-Abelian monopole approach and the geometric theory of defects, where every topological defect is associated with curvature and torsion, to describe a fullerene molecule. The geometric theory of defects in solids is used to consider the topological defects that allow this molecule to form and we apply a continuum formulation to describe this spherical geometry in the presence of an external Aharonov–Bohm flux. We solve a Dirac equation for this model and obtain the eigenvalues and eigenfunction of the Hamiltonian, and we obtain the persistent current for this model and show that it depends on the geometrical and topological properties of the fullerene.     

Detection of Majorana fermions in an Aharonov–Bohm interferometer

By using the non-equilibrium Green's function technique, we investigate the electronic transport properties in an Aharonov–Bohm interferometer coupling with Majorana fermions. We find a fixed unit conductance peak which is independent of the other factors when the topological superconductor is grounded. Especially, an additional phase appears when the topological superconductor is in the strong Coulomb regime, which induces a new conductance resonant peak compared with the structure of replacing the topological superconductor by a quantum dot, and the conductance oscillation with the magnetic flux reveals a 2π phase shift by raising(lowering) a charge on the capacitor.     

Impurity-modulated Aharonov–Bohm oscillations and intraband optical absorption in quantum dot–ring nanostructures

In this work we study the electronic states in quantum dot–ring complex nanostructures with an on-center hydrogenic impurity. The influence of the impurity on Aharonov–Bohm energy spectra oscillations and intraband optical absorption is investigated. It is shown that in the presence of a hydrogenic donor impurity the Aharonov–Bohm oscillations in quantum dot–ring structures become highly tunable. Furthermore, the presence of the impurity drastically changes the intraband absorption spectra due to the strong controllability of the electron localization type.     

Does the Aharonov–Bohm Effect Exist?

We draw a distinction between the Aharonov–Bohm phase shift and the Aharonov–Bohm effect. Although the Aharonov–Bohm phase shift occurring when an electron beam passes around a magnetic solenoid is well-verified experimentally, it is not clear whether this phase shift occurs because of classical forces or because of a topological effect occurring in the absence of classical forces as claimed by Aharonov and Bohm. The mathematics of the Schroedinger equation itself does not reveal the physical basis for the effect. However, the experimentally observed Aharonov–Bohm phase shift is of the same form as the shift observed due to electrostatic forces for which the consensus view accepts the role of the classical forces. The Aharonov–Bohm phase shift may well arise from classical electromagnetic forces which are simply more subtle in the magnetic case since they involve relativistic effects of the order v ² /c ² . Here we first review the experimentally observable differences between phenomena arising from classical forces and phenomena arising from the quantum topological effect suggested by Aharonov and Bohm. Second we point out that most discussions of the classical electromagnetic forces involved when a charged particle passes a solenoid are inaccurate because they omit the Faraday induction terms. The subtleties of the relativisitic v ² /c ² classical electromagnetic forces between a point charge and a solenoid have been explored by Coleman and Van Vleck in their analysis of the Shockley–James paradox; indeed, we point out that an analysis exactly parallel to that of Coleman and Van Vleck suggests that the Aharonov–Bohm phase shift is actually due to classical electromagnetic forces. Finally we note that electromagnetic velocity fields penetrate even excellent conductors in a form which is unfamiliar to many physicists. An ohmic conductor surrounding a solenoid does not screen out the magnetic field of the passing charge, but rather the time-integral of the magnetic field is an invariant; this time integral is precisely what is involved in the classical explanation of the Aharonov–Bohm phase shift. Thus the persistence of the Aharonov–Bohm phase shift when the solenoid is surrounded by a conductor does not exclude a classical force-based explanation for the phase shift. At present there is no experimental evidence for the Aharonov–Bohm effect.     

On a relation of the angular frequency to the Aharonov–Casher geometric phase in a quantum dot

By analysing the behaviour of a neutral particle with permanent magnetic dipole moment confined to a quantum dot in the presence of a radial electric field, Coulomb-type and linear confining potentials, then, an Aharonov–Bohm-type effect for bound states and a dependence of the angular frequency of the system on the Aharonov–Casher geometric phase and the quantum numbers associated with the radial modes, the angular momentum and the spin are obtained. In particular, the possible values of the angular frequency and the persistent spin currents associated with the ground state are investigated in two different cases.     

The Aharonov–Bohm effect in scattering theory

The Aharonov–Bohm effect is considered as a scattering event with nonrelativistic charged particles of the wavelength which is less than the transverse size of an impenetrable magnetic vortex. The quasiclassical WKB method is shown to be efficient in solving this scattering problem. We find that the scattering cross section consists of two terms, one describing the classical phenomenon of elastic reflection and another one describing the quantum phenomenon of diffraction; the Aharonov–Bohm effect is manifested as a fringe shift in the diffraction pattern. Both the classical and the quantum phenomena are independent of the choice of a boundary condition at the vortex edge, providing that probability is conserved. We show that a propagation of charged particles can be controlled by altering the flux of a magnetic vortex placed on their way.     

The Aharonov–Bohm effect in graphene rings

This is a review of electronic quantum interference in mesoscopic ring structures based on graphene, with a focus on the interplay between the Aharonov–Bohm effect and the peculiar electronic and transport properties of this material. We first present an overview on recent developments of this topic, both from the experimental as well as the theoretical side. We then review our recent work on signatures of two prominent graphene-specific features in the Aharonov–Bohm conductance oscillations, namely Klein tunneling and specular Andreev reflection. We close with an assessment of experimental and theoretical development in the field and highlight open questions as well as potential directions of the developments in future work.     

Observations of the Ramsauer–Townsend effect in quaternionic quantum mechanics

In this article, one of the well-known effects in quantum mechanics is addressed and also the extended form of quantum mechanics which is based on quaternions is presented. In the presence of this version of quantum mechanics the Ramsauer–Townsend effect has been investigated and the existence of this phenomenon is studied according to quaternionic calculations; results are presented by graphs.     

Quantum motion of a point particle in the presence of the Aharonov–Bohm potential in curved space

The nonrelativistic quantum dynamics of a spinless charged particle in the presence of the Aharonov–Bohm potential in curved space is considered. We chose the surface as being a cone defined by a line element in polar coordinates. The geometry of this line element establishes that the motion of the particle can occur on the surface of a cone or an anti-cone. As a consequence of the nontrivial topology of the cone and also because of two-dimensional confinement, the geometric potential should be taken into account. At first, we establish the conditions for the particle describing a circular path in such a context. Because of the presence of the geometric potential, which contains a singular term, we use the self-adjoint extension method in order to describe the dynamics in all space including the singularity. Expressions are obtained for the bound state energies and wave functions.