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Moscow University Physics Bulletin

Scattering problem in a quantum waveguide in a tight-binding model

Ivan A. Ado

Published Oct 1, 2010• 1 citation
Publisher

Abstract

The scattering problem in a quantum waveguide is considered within a light-binding approach. The relationship between the stationary and time-dependent definition of an S-matrix is established.
DOI
10.3103/s0027134910050036

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The exact expression for the transition rate for quantum-mechanical scattering problems is derived formally without recourse to adiabatically switching on the interaction or averaging the initial state in time. The equivalence is shown of the stationary definition of the SSS matrix and an appropriate definition by the time-dependent method.
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.
Many advancements in optics have relied on the tight-binding approximation, which simplifies the description and prediction of complex system behaviors. This approximation describes the dynamics of the total light field by examining the coupling between the guided modes of individual single-mode substructures -- also known as coupled mode theory. However, the underlying assumption, that the guided modes of individual waveguides form an orthogonal basis, breaks down when waveguides are brought into close proximity or when larger arrays are considered. In this work, we systematically analyze the consequences of this non-orthogonality and show that it leads to a generalized eigenvalue problem involving an overlap matrix, causing a fundamental mismatch between the standard TB model and solutions of the paraxial wave equation. To resolve this issue, we introduce a modified TB framework based on the Löwdin orthogonalization, which constructs an orthonormal basis from the non-orthogonal guided modes while minimally altering their physical shape and preserving their symmetry properties. The resulting Löwdin-TB method restores the standard eigenvalue problem and yields excellent agreement with exact beam propagation simulations across a wide range of system sizes and waveguide separations. Furthermore, it captures important physical effects, such as enhanced long-range coupling and nontrivial hopping phases, that are absent in the standard approach.