Two-body problem in periodic potentials

TL;DR

This paper analyzes how two atoms interact in a deep optical lattice, deriving analytical results for bound states and scattering, revealing a dimensional crossover influenced by tunneling effects.

Contribution

It provides analytical expressions for two-body bound states and scattering in optical lattices of arbitrary dimension, highlighting the dimensional crossover and tunneling effects.

Findings

Dimensional crossover from confined to 3D regime due to tunneling

Logarithmic dependence of critical scattering length for D=1

Good agreement between analytical results and numerical simulations for D=1

Abstract

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.