Bound states of two bosons in an optical lattice near an association resonance

We model two bosons in an optical lattice near a Feshbach or photoassociation resonance, focusing on the Bose-Hubbard model in one dimension. Whereas the usual atoms-only theory with a tunable scattering length yields one bound state for a molecular dimer for either an attractive or repulsive atom-atom interaction, for a sufficiently small direct background interaction between the atoms a two-channel atom-molecule theory may give two bound states that represent attractively and repulsively bound dimers occurring simultaneously. Such unusual molecular physics may be observable for an atom-molecule coupling strength comparable to the width of the dissociation continuum of the lattice dimer, which is the case, for instance, with narrow Feshbach resonances in Na, ${}^{87}\mathrm{Rb}$, and ${}^{133}\mathrm{Cs}$ or low-intensity photoassociation in ${}^{174}\mathrm{Yb}$.

Figure 1

The function $f(\omega ,L)$ [Eq. (10)] for $L=16$. The horizontal and positive-slope lines represent the right-hand sides of Eqs. (11) and (10), respectively, for $K=\Delta =\frac{1}{2}$ and $\Xi =1/\sqrt{2}$. The dashed vertical lines are the asymptotes of $f(\omega ,L)$ at the values of $\omega =-\cos \hspace{0.5em}q$ such that $f(\omega ,L)=\pm \infty$.