Spectroscopic Search for Topological Protection in Open Quantum Hardware: The Dissipative Mixed Hodge Module Approach

TL;DR

This paper introduces a novel topological spectroscopy method for open quantum systems that remains effective at Exceptional Points, revealing topological protection as an algebraic invariant despite spectral gap closures.

Contribution

It develops Dissipative Mixed Hodge Modules and Weight Filtered Spectroscopy to analyze dissipative quantum systems at Exceptional Points, overcoming limitations of traditional spectral methods.

Findings

WFS spatially separates decay channels based on nilpotency rank.

WFS quantifies dissipative leakage in molecular polaritons.

Topological protection persists as an algebraic invariant even at spectral gap closures.

Abstract

Standard spectroscopic protocols model the dynamics of open quantum systems as a superposition of isolated, exponentially decaying eigenmodes. This paradigm fails fundamentally at Exceptional Points, where the eigenbasis collapses and the response becomes dominated by non-diagonalizable Jordan blocks. We resolve this ambiguity by introducing a geometric framework based on \textit{Dissipative Mixed Hodge Modules} (DMHM). By replacing the scalar linewidth with a topological \textit{Weight Filtration}, we derive ``Weight Filtered Spectroscopy'' (WFS)--a protocol that spatially separates decay channels based on the nilpotency rank of the Liouvillian. We demonstrate that WFS acts as a dissipative x-ray, quantifying dissipative leakage in molecular polaritons and certifying topological isolation in Non-Hermitian Aharonov-Bohm rings. This establishes that topological protection persists as an algebraic invariant even when the spectral gap is closed.