# Optical signatures of the superconducting Goldstone mode in granular aluminum: experiments and theory

## TL;DR

This study combines THz experiments and theoretical modeling to reveal optically-active Goldstone modes in granular aluminum, showing how spatial inhomogeneities enable their optical detection and providing insights into collective superconducting excitations.

## Contribution

The paper demonstrates the optical visibility of Goldstone modes in granular aluminum and models the inhomogeneities as a disordered Josephson array, linking experimental data with theory.

## Key findings

- Observation of temperature-dependent sub-gap optical absorption.
- Identification of Goldstone modes as the source of absorption.
- Quantitative agreement between theory and experiment.

## Abstract

Recent advances in the experimental growth and control of disordered thin films, heterostructures, and interfaces provide a fertile ground for the observation and characterisation of the collective superconducting excitations emerging below $T_c$ after breaking the $U(1)$ gauge symmetry. Here we combine THz experiments in a nano-structured granular Al thin film and theoretical calculations to demonstrate the existence of optically-active phase modes, which represent the Goldstone excitations of the broken gauge symmetry. By measuring the complex transmission trough the sample we identify a sizeable and temperature-dependent optical sub-gap absorption, which cannot be ascribed to quasiparticle excitations. A quantitative modelling of this material as a disordered Josephson array of nano-grains allows us to determine, with no free parameters, the structure of the spatial inhomogeneities induced by shell effects. Besides being responsible for the enhancement of the critical temperature with respect to bulk Al, already observed in the past, this spatial inhomogeneity provides a mechanism for the optical visibility of the Goldstone mode. By computing explicitly the optical spectrum of the superconducting phase fluctuations we obtain a good quantitative description of the experimental data. Our results demonstrate that nanograins arrays are a promising setting to study and control the collective superconducting excitations via optical means.