Spectroscopic Search for Topological Protection in Open Quantum Hardware: The Dissipative Mixed Hodge Module Approach
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.
Summary
This paper addresses the limitations of standard spectroscopic protocols in characterizing open quantum systems, particularly near Exceptional Points (EPs) where the eigenbasis collapses. The authors introduce a novel geometric framework based on Dissipative Mixed Hodge Modules (DMHM) to resolve ambiguities in identifying topologically protected states amidst dissipation. They develop a "Weight Filtered Spectroscopy" (WFS) protocol that spatially separates decay channels based on the nilpotency rank of the Liouvillian, effectively acting as a "dissipative x-ray." This allows for the quantification of dissipative leakage and the certification of topological isolation, even when the spectral gap is closed. The approach involves lifting the system description from standard vector spaces to the DMHM framework, enabling the use of topological invariants to characterize open quantum systems. The WFS protocol leverages Laplace-domain filtration to resolve the algebraic structure of the Liouvillian, distinguishing topologically protected decay channels from trivial dissipation. The authors demonstrate the effectiveness of WFS in quantifying dissipative leakage in molecular polaritons and certifying topological isolation in Non-Hermitian Aharonov-Bohm rings. They introduce a "Dissipative Insulation" figure of merit (F_iso) based on the WFS cross-peak intensity, providing a metric for optimizing the design of robust quantum hardware. The paper also hints at future directions involving Floquet Monodromy Spectroscopy for dynamically engineering the "Singular Quantum Geometric Tensor."
Key Insights
- •Novel Geometric Framework: Introduces Dissipative Mixed Hodge Modules (DMHM) as a framework for analyzing open quantum systems, particularly near Exceptional Points (EPs).
- •Weight Filtered Spectroscopy (WFS): Develops WFS, a spectroscopic protocol that spatially separates decay channels based on the nilpotency rank of the Liouvillian, overcoming limitations of standard linewidth fitting.
- •Dissipative Insulation Figure of Merit (F_iso): Defines F_iso as the normalized inverse of the WFS cross-peak intensity, providing a quantifiable metric for certifying the algebraic decoupling of quantum states in the presence of strong bath interactions. F_iso is defined as `F_iso (∆) = (1 + ∫ | ̃ S(λ X ,λ C )|dλ ) −1`.
- •Hodge Filtered Spectroscopy (HFS): Introduces HFS as a complementary protocol to WFS. HFS uses an integral transform kernel derived from the Hodge grading to resolve a single, broad spectral feature into a grid of discrete Hodge components, distinguishing between "quantum" superpositions and "classical" mixtures.
- •Topological Protection Persists: Demonstrates that topological protection persists as an algebraic invariant even when the spectral gap is closed, challenging the conventional view that topological protection is solely dependent on a spectral gap.
- •Aharonov-Bohm Ring Robustness: Shows that Non-Hermitian Aharonov-Bohm rings exhibit exponentially suppressed leakage compared to the quadratic leakage of standard hybrid systems, confirming their suitability as decoherence-free subspaces.
- •Limitations: The numerical Laplace inversion in WFS can be ill-posed due to finite sampling and noise, necessitating robust Harmonic Inversion techniques like the Matrix Pencil Method or Padé Approximants.
Practical Implications
- •Design of Robust Quantum Hardware: The WFS protocol and the F_iso metric provide practical tools for engineers to design and optimize quantum hardware, particularly in hybrid quantum systems like molecular polaritons and Non-Hermitian Aharonov-Bohm rings.
- •Improved Spectroscopic Analysis: The DMHM framework and the WFS/HFS protocols offer a more rigorous and informative approach to spectroscopic analysis of open quantum systems, enabling researchers to extract deeper insights into the underlying dynamics and topological properties.
- •Material Synthesis Optimization: The Dissipative Insulation Figure of Merit (F_iso) can be used to guide the synthesis of organic polariton materials with long coherence times by optimizing the detuning parameter.
- •Future Research Directions: The paper opens up avenues for future research in Floquet Monodromy Spectroscopy, which aims to dynamically engineer the "Singular Quantum Geometric Tensor" and explore the renormalized geometry at Exceptional Points.
- •Application to Molecular Polariton Qubits: By certifying "Dissipative Insulation", this work provides a route to realize robust polariton qubits that are not limited by the fast decoherence of the excitonic component.