Non-zero curvature in a waveguide leads to the appearance of an attractive quantum potential which crucially affects the dynamics in matter-wave circuits. Using methods of supersymmetric quantum mechanics, pairs of bent waveguides are found whose geometry-induced potentials share the same scattering properties. As a result, reflectionless waveguides, dual to the straight waveguide, are identified. Strictly isospectral waveguides are also found by modulating the depth of the trapping potential. Numerical simulations are used to demonstrate the efficiency of these approaches in tailoring and controlling curvature-induced quantum-mechanical effects.
Non-zero curvature in a waveguide leads to the appearance of an attractive quantum potential which crucially affects the dynamics in matter-wave circuits. Using methods of supersymmetric quantum mechanics, pairs of bent waveguides are found whose geometry-induced potentials share the same scattering properties. As a result, reflectionless waveguides, dual to the straight waveguide, are identified. Strictly isospectral waveguides are also found by modulating the depth of the trapping potential. Numerical simulations are used to demonstrate the efficiency of these approaches in tailoring and controlling curvature-induced quantum-mechanical effects.
The propagation of matter waves in curved geometry is relevant for ion transport, atomtronics and electrons in nanowires. Curvature effects are usually addressed within the adiabatic limit and treated via an effective potential acting on the manifold to which the particles are strongly confined. However, the strength of the confinements that can be achieved experimentally are limited in practice, and the adiabatic approximation often appears too restrictive for realistic guides. Here, we work out a design method for 2D sharply bent waveguides beyond this approximation using an exact inverse-engineering technique. The efficiency of the method is confirmed by the resolution of the 2D nonlinear Schrödinger equation in curved geometry. In this way, we realize reflectionless and ultrarobust curved guides, even in the presence of interactions. Here, the transverse stability is improved by several orders of magnitude when compared to circular guides of similar size.
Advances in quantum waveguide dynamics, with applications to physical and biological nanostructuresIAIN JOHN CLARK It is known [1,2] that as semiconductor microstructures get progressively smaller and more free of impurities, it becomes increasingly necessary to use quantum mechanics to describe accurately these systems, which may have features sized between a few and several hundred nanometers.Recent advances in fabrication technology [11] make it feasible to build such nanostructures or mesoscopic systems, as they are known.The term "quantum waveguide" [6] is used to denote a nanostructure which acts as a conducting wire, along which a wavefunction is guided to move by the boundary conditions.Unlike classical wires, curved quantum waveguides have curvature-induced bound states; see [6] for a recent review.This has implications for nanoelectronic circuit design, since the curvature-induced bound states will appear as resonances.However, the effect of torsion (where the waveguide twists out of a plane) and twisting (where the waveguide changes its orientation along its path) upon bound states has attracted little investigation, and is the motivation for the analysis in this work.This thesis develops a unified mathematical approach by which the energy spectrum of a quantum waveguide with curvature and torsion can be analysed, for any crosssectional geometry that remains uniform along the waveguide.Following the introduction and literature review of Chapter 1, a non-relativistic Hamiltonian H for a particle on a quantum waveguide is set up in Chapter 2. A system of curvilinear coordinates is used to express Schrodinger's equation using the Laplace-Beltrami operator V 2 .The metric tensor G is expressed as the product of a curvaturedependent matrix F and the metric tensor G for a "straightened-out" waveguide.The Hamiltonian H is then shown to be unitarily equivalent to the Hamiltonian Hy for a straightened-out waveguide plus interaction terms, Hy -H o + Ky + Vu, where H o is the Hamiltonian for a "straightened-out" waveguide, Ky is a kinetic energy perturbation term and Vu is an effective potential.In Chapter 3, the spectral theory needed to investigate the curvature-induced bound states is developed.The spectrum of a Hamiltonian is partitioned into the
A classical particle under spatial constraints is strictly confined to live on a specific space manifold or path, but this assumption is incompatible with the zero-point fluctuations of a quantum particle. One way to describe quantum mechanics under constraints is the confinement potential approach (CPA). For a non-relativistic particle, the CPA maps the problem onto the solution of a Schrödinger-type equation in an isometrically embedded Riemannian submanifold of Euclidean space while the motion along orthogonal directions are decoupled and spatially confined. This approach respects quantum uncertainty, and one of its key results is the appearance of geometry- and metric-induced potentials that affect the stationary states and the dynamics of the particle. For particles constrained to different spaces, such as structures hosting sharp bents, vertices, wedges, conical apices, tips, or self-intersections, a formalism beyond the CPA is needed. Here, a step towards a CPA extension for irregular spaces is presented. After classifying the possible geometric irregularities concerning the CPA formalism, the presentation is focused on a sharply bent quantum wire modeled as an embedded curve with singular (but absolute integrable) curvature. For a subclass fulfilling the additional requirement that the geometric potential is a distribution of first order, a solution scheme for the confined Schrödinger equation is presented based on singular Sturm-Liouville theory and operator theoretic methods. The analytical considerations and numerical simulations evidence the existence of curvature-induced bound states with non-differentiable wave functions localized around the singular point, with an extension well beyond the singularity. Furthermore, a multitude of scattering states appear that may affect the transport and optical properties of the system.
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.
The dynamics of bosons in curved geometries have recently attracted significant interest in quantum many-body physics. Leveraging recent experimental advances in tailored trapping landscapes, we investigate the quantum transport of weakly interacting bosons in two-dimensional bent trapping potentials, showing that geometry alone can serve as a precise control knob for tunneling dynamics. Using time-adaptive many-body simulations, complemented by mean-field analysis and exact diagonalization, we analyze both static and dynamical properties of bosons confined in the bent potential. We reveal how bending an initially straight channel induces a transition from density localization to delocalization and drives the buildup of correlations in the ground state. In the dynamics, the bend acts as a tunable barrier that enables controllable tunneling: weak curvature allows coherent tunneling across the bend, while a stronger bend suppresses transport and enhances self-trapping. The tunneling rate can be precisely tuned by geometric parameters, establishing bent traps as versatile platforms for geometry-controlled quantum transport.