Bent waveguides for matter-waves: supersymmetric potentials and reflectionless geometries

Quantum mechanics of particles constrained to spiral curves with application to polyene chains

CONTEXT

Due to advances in synthesizing lower-dimensional materials, there is the challenge of finding the wave equation that effectively describes quantum particles moving on 1D and 2D domains. Jensen and Koppe and Da Costa independently introduced a confining potential formalism showing that the effective constrained dynamics is subjected to a scalar geometry-induced potential; for the confinement to a curve, the potential depends on the curve's curvature function.

METHOD

To characterize the π electrons in polyenes, we follow two approaches. First, we utilize a weakened Coulomb potential associated with a spiral curve. The solution to the Schrödinger equation with Dirichlet boundary conditions yields Bessel functions, and the spectrum is obtained analytically. We employ the particle-in-a-box model in the second approach, incorporating effective mass corrections. The π -π ∗ transitions of polyenes were calculated in good experimental agreement with both approaches, although with different wave functions.

Quantum Control beyond the Adiabatic Regime in 2D Curved Matter-Wave Guides

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.