Density and Spectral-Density Matrices in Atomistic-Scale Models

Wen-ge Wang

For a schematic shell model, we show numerically that, contrary to the behaviour of eigenfunctions, the shapes of the so-called local spectral density of states become close to their forms at extremely strong perturbation (after rescaling) even when the perturbation is relatively weak. The same phenomenon is also found for the random version of the schematic shell model. We suggest that this property of the local spectral density of states may be common to models in which the Hamiltonian matrices in independent particle states have a banded and regular structure.

F. David Peat, R. J. C. Brown

Trevor Kling, Dong-Yeop Na, Mahdi Hosseini

We develop a model describing long-range atom-atom interactions in a two-dimensional periodic or a-periodic lattice of optical centers considering spectral and spatial broadening effects. Using both analytical and numerical Green's function techniques, we develop a mathematical framework to describe effective atom-atom interactions and collective behaviors in the presence of disorder. This framework is applicable to a broad range of quantum systems with arbitrary lattice geometries, including cold atoms, solid-state photonics, and superconducting platforms. The model can be used, for example, to scalably design quantum optical elements, e.g. a quantum lens, harnessing atomistic engineering (e.g. via ion implantation) of collective interactions in materials to enhance quantum properties at scale.

Shun-Ching Lee

This project (WP_007) establishes a mathematical and philosophical isomorphism between the African relational ontology of Ubuntu and quantum state non-separability. It proves that the Cartesian atomistic self is an epistemological illusion generated by the partial trace operation on a global density matrix ($\rho_A = \text{Tr}_B(\rho_{AB})$). The project formalizes love and empathy as macroscopic manifestations of global network phase-locking, offering operational "Relational De-Reification" and "Seriti Attunement" protocols for cognitive de-noising and emotional stabilization within a hyper-connected social manifold.

Karl Blum