Two-body problem in periodic potentials

We investigate the problem of two atoms interacting via a short range $s$-wave potential in the presence of a deep optical lattice of arbitrary dimension $D$. Using a tight-binding approach, we derive analytical results for the properties of the bound state and the scattering amplitude. We show that the tunneling through the barriers induces a dimensional crossover from a confined regime at high energy to an anisotropic three-dimensional regime at low energy. The critical value of the scattering length needed to form a two-body bound state shows a logaritmic dependence on the tunneling rate for $D=1$ and a power law for $D>1$. For the special case $D=1$, we also compare our analytical predictions with exact numerics, finding remarkably good agreement.

Figure 1

Binding energy as a function of the inverse scattering length for different values of the laser intensity: the analytical prediction (solid line) is compared with the exact numerical data (squares). Also shown are the position of the bandwidth (dashed line) and the asymptotic behaviors for the binding energy (dotted lines) in the two regimes [see Eqs. (21, 22)].

Figure 2

Critical value of the inverse scattering length as a function of the laser intensity for quasimomentum $Q=0$ and $Q=q_B$. The analytical prediction (solid line) is shown with the exact numerical data (circles).

Figure 3

The inverse effective mass of the molecule as a function of the inverse scattering length for $s=5$: analytical prediction (solid line) and exact numerical data (squares).

Figure 4

Molecular bandwidth as a function of the inverse scattering length for different values of the laser intensity: analytical prediction (solid line) and exact numerical data (squares). The atomic bandwidth is $w_{\mathit{at}}=0.264E_R$ for $s=5$ and $w_{\mathit{at}}=0.077E_R$ for $s=10$.