Emergent Structure and Coherence in Interacting Systems

Working Paper, June 2026

This research investigates the computational resolution of spectral singularities in non-linear transport systems. A matrix-valued Riccati transport framework is under development to map the evolution of Andreev Bound States (ABS) in systems characterized by strong spatial confinement and inhomogeneous potential landscapes.

Computational investigations explore simplified matrix-valued transport systems motivated by quasiclassical formulations of nonequilibrium superconductivity and interacting coherence dynamics. Numerical work focuses on iterative transport behavior, matrix-valued propagator structure, nonlinear operator stability, and self-consistent evolution in interacting systems.

Experimental investigations examine how coherence generates spatial structure within optical interference fields and how controlled perturbations alter, degrade, or reorganize that structure. Using interference visibility, spatial pattern evolution, and perturbation-sensitive coherence behavior as measurable observables, the work connects computational transport investigations with broader questions in photonics, wave dynamics, coherent sensing, and emergent organization in interacting systems.

Methodology & Technical Progress

The framework utilizes stabilized fixed-point propagation to solve the quasiclassical Eilenberger equations. Investigation has advanced from preliminary trajectory-based methods to an integrated computational framework that resolves ABS spectra through a stabilized Local Density of States (LDOS) diagnostic. Current efforts focus on the numerical conditioning of non-linear operator stability and the transition toward a Discontinuous Galerkin (DG) Finite Element Method (FEM) framework. This transition is essential for resolving the sharp spectral gradients observed at superconducting-normal interfaces, which trajectory ensembles often smooth out numerically.

The stability of the transport is monitored through the conditioning of the denominator matrix: D=I−γ^​γ~​^​ where I is the identity matrix, and γ^​ and γ~​^​ are the matrix-valued coherence amplitudes. The spectral stability of the system is evaluated by computing the minimum singular value of D. This follows the matrix-valued Riccati normalization required for stable quasiclassical transport.

Spectral Diagnostic: Local Density of States (LDOS)

The Local Density of States (LDOS) is utilized as a primary observable for spectral analysis. By analyzing the singular values (σmin​) of the denominator matrix, the research probes the transition between stable, coherent propagation and localized singularity formation.

Investigative Trajectory

The long-term objective is to establish a unified computational protocol for visualizing emergent coherence in many-body systems. By treating the experimental canvas as a physical resonator—and the numerical code as a high-resolution spectral analyzer—the project documents the resolution of microscopic chaos into macroscopic phase-stable equilibrium.

Current technical investigation focuses on three domains:

  • Numerical Stability: Development of convergence protocols for non-commutative matrix transport in FEM frameworks.

  • Topological Invariance: Characterization of the relationship between Riccati amplitude singularities and topological defects in confined fields.

  • Analogue Gravity: Assessment of whether these quasiclassical transport instabilities offer a valid computational model for non-linear field dynamics in curved spacetime.

This laboratory welcomes technical inquiry regarding the mapping of non-linear Riccati transport onto curved-space field dynamics.

 

Matrix-Valued Riccati Transport Systems and Coherence Dynamics in Nonequilibrium Superconductivity

Computational Prototype (June 2026)

This research investigates matrix-valued Riccati transport formulations in quasiclassical nonequilibrium superconductivity. Moving beyond initial proposals, the project now utilizes an integrated computational framework that resolves Andreev Bound State (ABS) spectra through a stabilized Local Density of States (LDOS) diagnostic. Current investigations focus on the numerical conditioning of non-linear operator stability and the transition toward a Discontinuous Galerkin (DG) FEM approach for complex geometries.

 

Coherence, Perturbation, and Emergent Spatial Structure in Optical Fields

Proposed Experimental Investigation, May 2026

This project investigates how coherence generates spatial structure and how controlled perturbations alter, degrade, or reorganize that structure within a coherent optical field. Using optical interference systems, the research examines perturbation-sensitive coherence behavior through fringe visibility, interference stability, and spatial pattern evolution.

The investigation establishes an experimentally accessible framework for studying coherence-driven structure formation relevant to photonics, wave dynamics, coherent sensing, and emerging quantum technologies. Future directions include computational modeling of coherence degradation, spectral analysis of interference fields, and extensions toward coupled optical and dynamical coherent systems.