Quantum chaos in ultracold collisions between $\text{Yb}\left({}^1{S}_0\right)$ and $\text{Yb}\left({}^3{P}_2\right)$

We calculate and analyze Feshbach resonance spectra for ultracold $\mathrm{Yb}\left({}^1{S}_0\right)+\mathrm{Yb}\left({}^3{P}_2\right)$ collisions as a function of an interatomic potential scaling factor $\lambda$ and external magnetic field. We show that, at zero field, the resonances are distributed randomly in $\lambda$, but that signatures of quantum chaos emerge as a field is applied. The random zero-field distribution arises from superposition of structured spectra associated with individual total angular momenta. In addition, we show that the resonances with respect to magnetic field in the experimentally accessible range of 400 to 2000 G are chaotically distributed, with strong level repulsion that is characteristic of quantum chaos.

Figure 1

Interatomic potentials for $\mathrm{Yb}\left({}^{1}S\right)+\mathrm{Yb}\left({}^{3}P\right)$. $\mathrm{\Sigma }$ and $\mathrm{\Pi }$ Born–Oppenheimer potentials calculated from Ref. [29] (red lines). Isotropic $V_0\left(R\right)$ and anisotropic $V_2\left(R\right)$ Legendre expansion coefficients, as described in the text (blue lines).

Figure 2

Statistical analysis of Feshbach resonance positions with respect to potential scaling factor $\lambda$. Panels (a), (c), and (e) show the NNS distributions $P\left(s\right)$: coupled-channel results (blue histograms); fitted Brody distributions (black lines, with the corresponding Brody parameters stated); Poisson and Wigner–Dyson distributions (green and red dashed lines, respectively). Panels (b), (d), and (f) show the corresponding number variances ${\mathrm{\Sigma }}^2\left(\mathrm{\Delta }\xi \right)$: coupled-channel results (blue solid lines); Poisson and GOE results (green and red dashed lines, respectively). (g) Calculated Brody parameter as a function of magnetic field. (h) NNS distribution for individual blocks of total angular momentum $J$ in the absence of magnetic field, averaged over $J=2,\cdots ,20$.

Figure 3

Statistical analysis of Feshbach resonance positions with respect to magnetic field for different isotopes of Yb. Upper panels show NNS distributions $P\left(s\right)$: coupled-channel calculations (blue histograms); fitted Brody distributions (black lines, with the corresponding Brody parameters stated); Poisson and Wigner–Dyson distributions (green and red dashed lines, respectively). Lower panels show the corresponding number variances ${\mathrm{\Sigma }}^2\left(\mathrm{\Delta }\xi \right)$: coupled-channel calculations (blue solid lines); Poisson and GOE results (green and red dashed lines, respectively).