Surveying the Multicomponent Scattering Matrix: Unitarity and Symmetries

TL;DR

This paper introduces a structured unitarity condition for multicomponent scattering matrices, ensuring unitarity preservation in complex quantum transport systems with multiple interacting components, even with nonzero incoming amplitudes.

Contribution

It develops a robust theoretical framework for unitarity in multicomponent scattering, extending standard properties to systems with multiple interacting channels within the envelope function approximation.

Findings

Derived a structured unitarity condition applicable to multicomponent systems.

Recovered standard unitarity properties for single-component cases.

Provided tools for more accurate tunneling threshold analysis.

Abstract

Multicomponent-multiband fluxes of spim-charge carriers, whose components propagate mixed and synchronously, with \emph{a priori} nonzero incoming amplitudes, do not obey the standard unitarity condition on the scattering matrix for an arbitrary basis set. For such cases, we have derived a robust theoretical procedure, which is fundamental in quantum-transport problems for unitarity preservation and we have named after \emph{structured unitarity condition}. Our approach deals with $(N \times N)$ interacting components (for $N \geq 2$), within the envelope function approximation (EFA), and yet the standard unitary properties of the ($N = 1$) scattering matrix are recovered. Rather arbitrary conditions to the basis-set and/or to the output scattering coefficients, are not longer required, if the \emph{eigen}-functions are orthonormalized in both the configuration and the spinorial spaces. We expect the present model to be workable, for different kind of multiband-multicomponent physical systems described by Hermitian Hamiltonians within the EFA, with small transformations if any. We foretell the interplay for the state-vector transfer matrix, together with the large values of its condition number, as a novel complementary tools for a more accurate definition of the threshold for tunnelling channels in a scattering experiment.