# Probing localization and quantum geometry by spectroscopy

## TL;DR

This paper introduces a spectroscopy-based method to probe localization, quantum fluctuations, and geometry in quantum systems, applicable to ultracold gases and many-body systems, revealing new insights into quantum matter properties.

## Contribution

It proposes a novel spectroscopy technique to measure localization and quantum geometry, including the quantum metric tensor, in engineered quantum systems.

## Key findings

- Demonstrates spectroscopy as a tool to detect Anderson localization and topological edge modes.
- Shows how to extract the quantum metric tensor from excitation rate measurements.
- Provides a unified framework linking localization, quantum fluctuations, and geometry.

## Abstract

The spatial localization of quantum states plays a central role in condensed-matter phenomena, ranging from many-body localization to topological matter. Building on the dissipation-fluctuation theorem, we propose that the localization properties of a quantum-engineered system can be probed by spectroscopy, namely, by measuring its excitation rate upon a periodic drive. We apply this method to various examples that are of direct experimental relevance in ultracold atomic gases, including Anderson localization, topological edge modes, and interacting particles in a harmonic trap. Moreover, inspired by a relation between quantum fluctuations and the quantum metric, we describe how our scheme can be generalized in view of extracting the full quantum-geometric tensor of many-body systems. Our approach opens an avenue for probing localization, as well as quantum fluctuations, geometry and entanglement, in synthetic quantum matter.