Quantum Geometric Tensor in the Wild: Resolving Stokes Phenomena via Floquet-Monodromy Spectroscopy

TL;DR

This paper introduces Floquet-Monodromy Spectroscopy, a novel method to extract hidden geometric data in quantum systems with singularities, enabling accurate topological classification beyond traditional invariants.

Contribution

The authors develop FMS to experimentally access Stokes phenomena, resolving failures of standard topological invariants in non-Hermitian and driven quantum systems.

Findings

FMS successfully extracts Stokes multipliers from superconducting qudits.

The Stokes Invariant acts as a new quantum number for phase classification.

Standard invariants fail in the presence of essential singularities.

Abstract

Standard topological invariants, such as the Chern number and Berry phase, form the bedrock of modern quantum matter classification. However, we demonstrate that this framework undergoes a \textbf{catastrophic failure} in the presence of essential singularities -- ubiquitous in open, driven, and non-Hermitian systems ("Wild" regime). In these settings, the local geometric tensor diverges, rendering standard invariants ill-defined and causing perturbative predictions to deviate from reality by order unity ($\sim 100\%$). We resolve this crisis by introducing the \textbf{Floquet-Monodromy Spectroscopy (FMS)} protocol, a pulse-level control sequence, which experimentally extracts the hidden \textit{Stokes Phenomenon} -- the "missing" geometric data that completes the topological description. By mapping the singularity's Stokes multipliers to time-domain observables, FMS provides a rigorous experimental bridge to \textbf{Resurgence Theory}, allowing for the exact reconstruction of non-perturbative physics from divergent asymptotic series. We validate this framework on a superconducting qudit model, demonstrating that the "Stokes Invariant" serves as the next-generation quantum number for classifying phases of matter beyond the reach of conventional topology.