Figure 2 - The Emergence of Chaos and the effect of interaction anisotropy on the chaotic level distribution in the weakly-bound dysprosium dimer.
Figure 1 - Coupled-channels simulation of Feshbach resonances of ultra cold collisions of erbium and dysprosium.
We study the resonant scattering of ultracold bosonic Er and Dy atoms in a magnetic field. This work was inspired by recent breakthroughs in the experi-mental realization of ultracold dipolar quantum gases of atoms with a large magnetic moment, Dy and Er, which have opened a new scientific playground for the study of strongly-correlated systems.
Our study has revealed unique features of colliding magnetic lanthanides that have not been observed in any other ultracold atomic system. These lanthanides have an unfilled 4f electron shell shielded by a closed 6s2 shell. They are cha-racterized by an exceptionally large electron orbital angular momentum, which leads to large anisotropic dispersion interactions between these atoms as well as the magnetic dipole-dipole interaction. Our theoretical estimate shows that for both Er and Dy the ratio of anisotropic to isotropic dispersion interaction is about 10% leading to significant splittings among the several tens of gerade short-range potentials of Er+Er and Dy+Dy.
We have performed coupled-channels and bound-state calculations with physically realistic angular-momentum couplings and interaction potentials for these atoms. The calculations were used to obtain a quantitative understanding of the observed chaotic distribution of Fano-Feshbach resonances as a function of magnetic field. The upper panel of Figure 1 shows an example of our theoretical near-threshold bound states and Feshbach resonances when their binding energy is zero. To ensure numerical convergence of the bound state energies as well as to get a better understanding of the density of magnetic Feshbach resonances we systematically increased the number of coupled channels. The results are shown in the bottom two images of Figure 1. For B=0 the total angular momentum J of a molecule is a good quantum number and, in fact, at most 49 and 82 Bose-symmetrized and parity-conserving channels are coupled for Er and Dy, respectively. For B>0 G the Zeeman interaction couples channels with different value of J. Its projection M remains conserved. We increase the number of channels by increasing Jmax, such that all channels with |M|< Jmax are included. I.e. we add 49 or 82 channels when Jmax is increased by one. Loosely speaking increasing Jmax also increases the number of coupled partial waves.
We investigate the role of interaction anisotropies on the level distribution of the most-weakly bound energy levels at zero magnetic field. There are two components to the anisotropy, the dispersion Vdisp (R) and magnetic dipole-dipole VMDD(R) contribution, and we define the total potential
Va (R)=Ldisp Vdisp(R) + LMDD VMDD(R)
with variable strength Ldisp and LMDD. We systematically increased the strengths Ldisp and LMDD from 0 to 1, where we recover the full physical strength. The results are shown in Figure 2.
We concluded that if we just consider the anisotropy from the magnetic dipole-dipole interaction our coupled-channel calculations indicate the binding energies of weakly-bound bound states are regular as a function of LMDD and there is no chaos in the level distribution. The strength of the dipole-dipole interaction is too small. On the other hand increasing the dispersion anisotropy ldisp leads to irregular binding energies and chaos is present. In addition, we have shown that the near-neighbours distributions for Dy and Er are very similar.
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Nature 507, 475 (2014);
Phys. Rev. Lett. 115, 203201 (2015);
Phys. Rev. X 5, 041029 (2015).