Curved Quantum Waveguides Research Articles

Quantum Eigenstates of Curved and Varying Cross-Sectional Waveguides

A simple one-dimensional differential equation in the centerline coordinate of an arbitrarily curved quantum waveguide with a varying cross section is derived using a combination of differential geometry and perturbation theory. The model can tackle curved quantum waveguides with a cross-sectional shape and dimensions that vary along the axis. The present analysis generalizes previous models that are restricted to either straight waveguides with a varying cross-section or curved waveguides, where the shape and dimensions of the cross section are fixed. We carry out full 2D wave simulations on a number of complex waveguide geometries and demonstrate excellent agreement with the eigenstates and energies obtained using our present 1D model. It is shown that the computational benefit in using the present 1D model to calculate both 2D and 3D wave solutions is significant and allows for the fast optimization of complex quantum waveguide design. The derived 1D model renders direct access as to how quantum waveguide eigenstates depend on varying cross-sectional dimensions, the waveguide curvature, and rotation of the cross-sectional frame. In particular, a gauge transformation reveals that the individual effects of curvature, thickness variation, and frame rotation correspond to separate terms in a geometric potential only. Generalization of the present formalism to electromagnetics and acoustics, accounting appropriately for the relevant boundary conditions, is anticipated.

Various manifestations of Wood anomalies in locally curved quantum waveguides

Various manifestations of Wood anomalies in locally curved quantum waveguides

Stark resonances in a quantum waveguide with analytic curvature

We investigate the influence of an electric field on trapped modes arising in a two-dimensional curved quantum waveguide i.e. bound states of the corresponding Laplace operator . Here the curvature of the guide is supposed to have an analytic extension in some complex domain around the real axis satisfying as . We show that under conditions on the electric field , has resonances near the discrete eigenvalues of .

Stark Resonances in 2-Dimensional Curved Quantum Waveguides

In this paper we study the influence of an electric field on a two-dimensional waveguide. We show that bound states that occur under a geometrical deformation of the guide turn into resonances when we apply an electric field of small intensity having a nonzero component on the longitudinal direction of the system.

Nonadiabatic couplings and gauge-theoretical structure of curved quantum waveguides

We investigate the quantum mechanics of a single particle constrained to move along an arbitrary smooth reference curve by a confinement that is allowed to vary along the waveguide. The Schr\"odinger equation is evaluated in the adapted coordinate frame and a transverse mode decomposition is performed, taking into account both curvature and torsion effects and the possibility of a cross-section potential that changes along the curve in an arbitrary way. We discuss the adiabatic structure of the problem, and examine nonadiabatic couplings that arise due to the curved geometry, the varying transverse profile and their interplay. The exact multi-mode matrix Hamiltonian is taken as the natural starting point for few-mode approximations. Such approximate equations are provided, and it is worked out how these recover known results for twisting waveguides and can be applied to other types of waveguide designs. The quantum waveguide Hamiltonian is recast into a form that clearly illustrates how it generalizes the Born-Oppenheimer Hamiltonian encountered in molecular physics. In analogy to the latter, we explore the local gauge structure inherent to the quantum waveguide problem and suggest the usefulness of diabatic states, giving an explicit construction of the adiabatic-to-diabatic basis transformation.

Magnetic Effects in Curved Quantum Waveguides

The interplay among the spectrum, geometry and magnetic field in tubular neighbourhoods of curves in Euclidean spaces is investigated in the limit when the cross section shrinks to a point. Proving a norm resolvent convergence, we derive effective, lower-dimensional models which depend on the intensity of the magnetic field and curvatures. The results are used to establish complete asymptotic expansions for eigenvalues. Spectral stability properties based on Hardy-type inequalities induced by magnetic fields are also analysed.

Electron transport through curved two-dimensional quantum waveguide

Abstract This paper details the study of electron transport through a two-dimensional curved quantum waveguide. The curvature of the quantum waveguide is found to greatly affect the transport properties. It is shown that no evanescent mode exists when the inner radius of the curved part of the waveguide is zero. Also, this paper details how localized modes with energy below the Fermi energy can be formed in the curved part due to curvilinear effects. A series of Fano-resonance type dips of the transmission coefficient appear because of the presence of localized modes. Due to the quantum effect, the transmission coefficient and the traversal time increase nonlinearly with the curved angle θ 0 . This nonlinear property is destroyed as the radius increases.

Resonance phenomena in curved quantum waveguides coupled via windows

A system of two curved waveguides mutually coupled via a small window is considered by solving a boundary value problem with the Dirichlet conditions at the external boundaries and the Neumann conditions at an interface with the window. Asymptotic expansions (with respect to the window size) for a resonance related to the window and for a bound state related to the waveguide curvature are obtained using the method of matching of the asymptotic expansions of solutions of the boundary-value problem.

Curved quantum waveguides in uniform magnetic fields

A theoretical study of a planar electronic waveguide with a uniformly curved section in the perpendicular homogeneous magnetic field $\mathbf{B}$ is presented within the envelope function approximation. Utilizing analytical solutions in each part of the waveguide, exact expressions are derived for the scattering and reflection matrices and for the transcendental equation defining bound-state energies. It is shown that in the magnetic field a propagation threshold in the continuously curved channel is always smaller than its counterpart for the straight arm which means that bound states in the uniform magnetic field always exist. Their energies do not depend on the direction of the field, and at high magnetic intensities they approach the lowest Landau level. For the transport in the fundamental mode an interaction of a quasibound level split off from the higher-lying threshold as a result of the bend, with its degenerate continuum counterpart, causes a dip in the transmission. In the magnetic field, contrary to the field-free case, conductance in the minimum ${G}_{\mathit{min}}$, generally, ceases to be zero. It is shown that growing magnetic fields cause ${G}_{\mathit{min}}$ to saturate to $2{e}^{2}∕h$ which means that a quasibound level formed as a result of the bend is completely dissolved by the increasing $\mathbf{B}$; however, this transformation is very different for the different bend angles and radii. In particular, quasibound states of the fundamental propagation mode survive stronger fields for the smaller bend angles which is explained by the larger total magnetic flux through the curved section where these levels are formed. Since a magnetic length ${l}_{B}={(\ensuremath{\hbar}∕eB)}^{1∕2}$ is inversely proportional to the square root of $B$, states for the waveguide with a smaller radius also survive stronger fields, and their asymptotic approach to the dissolution possesses nonmonotonic ${G}_{\mathit{min}}$ dependence on the magnetic field with minimum conductance again reaching zero for some special values of $B$. Vortex structure of the currents flowing in the waveguide near the resonance is strongly affected by the field. In particular, small magnetic intensities change zero-field vortices in the straight arms into the magnetic antivortices which correspond to the interacting with each other surface currents flowing along opposite walls of the channel. Increasing the magnetic field suppresses the formation of the vortices pushing currents to the outer (inner) walls in the straight (bent) section. For fields larger than the saturation magnetic intensity, the only consequence of the bend is a strong surface current near the convex wall of the bend, and the electronic flow along the junctions between straight and curved parts.

On the Spectrum of Curved Planar Waveguides

The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound to the distance between the essential spectrum and the spectral threshold for locally curved strips. The paper is also intended as an overview of some new and old results on spectral properties of curved quantum waveguides.

The nature of the essential spectrum in curved quantum waveguides

We study the nature of the essential spectrum of the Dirichlet Laplacian in tubes about infinite curves embedded in Euclidean spaces. Under suitable assumptions about the decay of curvatures at infinity, we prove the absence of singular continuous spectrum and state properties of possible embedded eigenvalues. The argument is based on Mourre conjugate operator method developed for acoustic multistratified domains by Benbernou and Dermenjian et al. As a technical preliminary, we carry out a spectral analysis for Schrodinger-type operators in straight Dirichlet tubes. We also apply the result to the strips embedded in abstract surfaces.

Asymptotics of resonances and bound states for laterally coupled curved quantum waveguides

A system of two curved waveguides coupled laterally through small window is considered. The asymptotics (in the width of window) of eigenvalue close to the threshold and curvature-induced resonance are obtained. The technique is matching of asymptotic expansions of the solutions.

On the number of particles that a curved quantum waveguide can bind

We discuss the discrete spectrum of N particles in a curved planar waveguide. If they are neutral fermions, the maximum number of particles that the waveguide can bind is given by a one-particle Birman–Schwinger bound in combination with the Pauli principle. On the other hand, if they are charged, e.g., electrons in a bent quantum wire, the Coulomb repulsion plays a crucial role. We prove a sufficient condition under which the discrete spectrum of such a system is empty.

Band Gap of the Spectrum in Periodically Curved Quantum Waveguides

In this paper we study the spectral gap of the Dirichlet Laplacian on a periodically curved strip. We begin with the definition of the planar strip and the Dirichlet Laplacian. For γ ∈ C(R; R), let $$ {k_\gamma }:R \ni s \mapsto \left( {{a_\gamma }\left( s \right),{b_\gamma }\left( s \right)} \right) \ni {R^2}$$ be a curve parameterized by its arc length whose curvature at s is γ(s). For d > 0, we define $${\Omega _{d,\gamma }} \equiv \left\{ {\left( {{a_\gamma }\left( s \right) - u\frac{d}{{ds}}{b_\gamma }\left( s \right),{b_\gamma }\left( s \right) + u\frac{d}{{ds}}{a_\gamma }\left( s \right)} \right) \in {R^2};s \in R, - d < u < d} \right\}.$$

A quantum pipette

Curved quantum waveguides are known to bind particles. We show that a number of charged fermions in such a trap can be tuned by an external electrostatic field; if the latter is slowly increased, the bent duct can serve as a single particle ejector up to the spin degeneracy.

Curvature vs. thickness in quantum waveguides

It is known that there is at least one bound state in a curved quantum waveguide provided it is sufficiently thin. In this paper we investigate the critical thickness of two-dimensional waveguides above which the discrete spectrum becomes void. We have found an expression for it in the case of a small bending angle. In the general case, its asymptotic behaviour with respect to the bending angle is shown to be governed by local smoothness properties of the boundary. Uniqueness of the critical thickness is also discussed.

Bound states in curved quantum waveguides

A free quantum particle living on a curved planar strip Ω of a fixed width d with Dirichlet boundary conditions is studied. It can serve as a model for electrons in thin films on a cylinder-type substrate, or in a curved quantum wire. Assuming that the boundary of Ω is infinitely smooth and its curvature decays fast enough at infinity, it is proved that a bound state with energy below the first transversal mode exists for all sufficiently small d. A lower bound on the critical width is obtained using the Birman–Schwinger technique.