Spectral statistics of molecular resonances in erbium isotopes: How chaotic are they?

TL;DR

This study investigates the spectral statistics of molecular resonances in erbium isotopes to assess their chaotic behavior and the extent of missing resonances, using random matrix theory and experimental data analysis.

Contribution

It provides a comprehensive statistical analysis of molecular resonances in erbium isotopes, estimating the fraction of missing resonances and their impact on spectral chaos.

Findings

Chaoticity increases with magnetic field up to 30 G

Approximately 20-25% of resonances are missing

Resonance width distribution supports presence of small-width missing resonances

Abstract

We perform a comprehensive analysis of the spectral statistics of the molecular resonances in $^{166}$Er and $^{168}$Er observed in recent ultracold collision experiments [Frisch et al., Nature {\bf 507}, 475 (2014)] with the aim of determining the chaoticity of this system. We calculate different independent statistical properties to check their degree of agreement with random matrix theory (RMT), and analyze if they are consistent with the possibility of having missing resonances. The analysis of the short-range fluctuations as a function of the magnetic field points to a steady increase of chaoticity until $B \sim 30$ G. The repulsion parameter decreases for higher magnetic fields, an effect that can be interpreted as due to missing resonances. The analysis of long-range fluctuations allows us to be more quantitative and estimate a $20-25\%$ fraction of missing levels. Finally, a study of the distribution of resonance widths provides additional evidence supporting missing resonances of small width compared with the experimental magnetic field resolution. We conclude that further measurements with increased resolution will be necessary to give a final answer to the problem of missing resonances and the agreement with RMT.