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

We perform a comprehensive analysis of the spectral statistics of the molecular resonances in ${}^{166}\mathrm{Er}$ and ${}^{168}\mathrm{Er}$ observed in recent ultracold collision experiments [Frisch et al., Nature (London) 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.

Figure 1

Comparison of the accumulated number of levels as a function of the magnetic field $N\left(B\right)$ for ${}^{166}\mathrm{Er}$. Symbols represent data from Ref. [17] while lines are linear (dashed) and polynomial (full) fits. Similar results hold for ${}^{168}\mathrm{Er}$ (not shown).

Figure 2

Brody parameter vs magnetic field from fits of the nearest-neighbor spacing distribution $P\left(s\right)$ on a moving window of 60 levels for ${}^{166}\mathrm{Er}$ (empty red circles) and ${}^{168}\mathrm{Er}$ (filled blue squares). The dashed horizontal lines indicate the limiting values corresponding to Poisson and WD distributions. The vertical error bars indicate the errors in the fit.

Figure 3

Power spectrum of the ${\delta }_q$ for ${}^{166}\mathrm{Er}$ (top) and ${}^{168}\mathrm{Er}$ (bottom). Symbols stand for data from experiments, while lines correspond to the Poisson prediction, a fit to Eq. (3), and the GOE prediction (from top to bottom in each panel).

Figure 4

(Top) Observed fraction estimated from the PT distribution (4) if only widths $\mathrm{\Delta }>{\mathrm{\Delta }}_{\min }$ are observed. The full line is the analytical estimate (5), the symbol stands for the estimate of the situation corresponding to the experiments in Ref. [17] (see text for details). (Bottom) Probability density of widths $P\left(\mathrm{\Delta }\right)$ for ${}^{168}\mathrm{Er}$. Probabilities derived from experimental data are shown as shaded bins while lines correspond to best fits to the PT (4) (dashed line, ${\chi }_{\mathrm{red}}^2=5.634$) and MPT (6) (solid line, ${\chi }_{\mathrm{red}}^2=3.116$) distributions. The hashed rectangle illustrates the range of minimum experimental magnetic field resolution $\delta B_{\mathrm{res}}=5–10$ mG [17]. The inset shows analogous results for ${}^{166}\mathrm{Er}$ with ${\chi }_{\mathrm{red}}^2\left(\mathrm{PT}\right)=6.842, {\chi }_{\mathrm{red}}^2\left(\mathrm{MPT}\right)=3.572$.

Figure 5

Short-range correlations from three finite samples drawn from a Wigner-Dyson distribution from which $20\%$ of levels have been dropped. Symbols stand for the corresponding data, while lines stand for the Poisson prediction $P\left(s\right)=exp(-s)$ (dashed line), the Wigner surmise (dotted-dashed line), and fits to Eq. (1) (long-dashed blue line) and to the model in Ref. [19] (solid orange line). The colored regions show the uncertainty stemming from the fitted values. The top panels (a)–(c) correspond to samples with $L=120$ levels. The bottom panels correspond to samples with $L=3000$ (d) and 5000 (e) levels, respectively. Fit parameters are (a) ${\beta }_{\mathrm{fit}}=0.90\pm 0.15, {\varphi }_{\mathrm{fit}}=0.95\pm 0.10$; (b) ${\beta }_{\mathrm{fit}}=0.81\pm 0.11, {\varphi }_{\mathrm{fit}}=0.78\pm 0.07$; (c) ${\beta }_{\mathrm{fit}}=0.43\pm 0.16, {\varphi }_{\mathrm{fit}}=0.66\pm 0.10$; (d) ${\beta }_{\mathrm{fit}}=0.62\pm 0.05, {\varphi }_{\mathrm{fit}}=0.34\pm 0.04$; (e) ${\beta }_{\mathrm{fit}}=0.79\pm 0.03, {\varphi }_{\mathrm{fit}}=0.15\pm 0.03$. Note how the uncertainties in the fitted parameters for the small samples (a)–(c) notably underestimate their actual spread.

Figure 6

Power spectrum for a typical example of a finite sample with $L=135$ levels taken out of a GOE matrix with a fraction of observed levels $\varphi =0.8$. The data (symbols) are compared to the fitted result $\varphi =0.76$ and the GOE and Poisson predictions.