Hubbard model for ultracold bosonic atoms interacting via zero-point-energy–induced three-body interactions

We show that, for ultracold neutral bosonic atoms held in a three-dimensional periodic potential or optical lattice, a Hubbard model with dominant, attractive three-body interactions can be generated. In fact, we derive that the effect of pairwise interactions can be made small or zero starting from the realization that collisions occur at the zero-point energy of an optical lattice site and the strength of the interactions is energy dependent from effective-range contributions. We determine the strength of the two- and three-body interactions for scattering from van der Waals potentials and near Fano-Feshbach resonances. For van der Waals potentials, which for example describe scattering of alkaline-earth atoms, we find that the pairwise interaction can only be turned off for species with a small negative scattering length, leaving the ${}^{88}\mathrm{Sr}$ isotope a possible candidate. Interestingly, for collisional magnetic Feshbach resonances this restriction does not apply and there often exist magnetic fields where the two-body interaction is small. We illustrate this result for several known narrow resonances between alkali-metal atoms as well as chromium atoms. Finally, we compare the size of the three-body interaction with hopping rates and describe limits due to three-body recombination.

Figure 1

(a) Zero-point energy, (b) tunneling energies, and (c) the scaled first-order two-body interaction strength $U_2/\left(k_{L}a\right)$ with no effective-range correction as a function of lattice depth $V_0$ for a cubic, three-dimensional optical lattice. Solid red curves are based on exact band-structure calculations and exact Wannier functions. Dashed blue lines are based on oscillator solutions of the isotropic harmonic approximation around the lattice minima [43].

Figure 2

The effective range volume $r_e{a}^2/2$ (solid red curve) as a function of scattering length $a$ for a van der Waals potential. All lengths are expressed in units of the mean scattering length $\overline{a}$. The dashed blue curves correspond to $-2a{\ell }^2/3$ for two values of $\ell$. At intersections of $r_e{a}^2/2$ and $-2a{\ell }^2/3$ the effective two-body interaction is tuned to zero.

Figure 3

Effective volume $r_e{a}^2/2$ of the Feshbach resonances listed in Table 1 as a function of scattering length $a$ with lengths in units of $\overline{a}$. Solid lines correspond to volumes based on Eq. (9) with $\alpha ,\beta \ne 0$ or equivalently the coupled-channels calculation of [58]. Dashed lines follow from Eq. (9) with $\alpha =\beta =0$.

Figure 4

(a) Two-body interaction strength $U_2$ and (b) minus one times the three-body strength $-U_3$ in a harmonic trap with $\omega /\left(2\pi \right)=50$ kHz as a function of the scattering length $a$ for a narrow ${}^{23}\mathrm{Na}$ (black lines), ${}^{39}\mathrm{K}$ (red lines), ${}^{52}\mathrm{Cr}$ (orange lines), and ${}^{133}\mathrm{Cs}$ (green lines) Feshbach resonance tabulated in Table 1. Solid circles in both panels and arrows in (b) indicate where $U_2=0$.