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arXiv:cond-mat/0304527 · Physical Review Letters

Extended Molecules and Geometric Scattering Resonances in Optical Lattices

Peter O. Fedichev, M. Bijlsma, P. Zoller

Published Feb 27, 2004• 61 citations
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Abstract

We develop a theory describing neutral atom scattering at low energies in an optical lattice. We show that, for a repulsive interaction, as the microscopic scattering length increases the effective scattering amplitude approaches a limiting value which depends only on the lattice parameters. In the case of attractive interaction a geometric resonance occurs before reaching this limit. Close to the resonance, the effective interaction becomes repulsive and supports a weakly bound state, which can extend over several lattice sites.

DOI
10.1103/physrevlett.92.080401

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Strongly-interacting ultra-cold atoms in tight-binding optical lattice potentials provide an ideal platform to realize the fundamental Hubbard model. Here, after outlining the elementary single particle solution, we review and expand our recent work on complete characterization of the bound and scattering states of two and three bosonic atoms in a one-dimensional optical lattice. In the case of two atoms, there is a family of interaction-bound "dimer" states of co-localized particles that exists invariantly for either attractive or repulsive on-site interaction, with the energy below or above the two-particle scattering continuum, respectively. Adding then the third particle -- "monomer" -- we find that, apart from the simple strongly-bound "trimer" corresponding to all three particles occupying the same lattice site, there are two peculiar families of weakly-bound trimers with energies below and above the monomer-dimer scattering continuum, the corresponding binding mechanism being an effective particle exchange interaction.
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We study charged particles in three dimensions interacting via a short-range potential in addition to the Coulomb potential. When the Bohr radius and the scattering length are much larger than the potential range, low-energy physics of the system becomes independent from details of the short-range potential. We develop the zero-range theory to describe such universal physics in terms of the Bohr radius and the scattering length by generalizing the Bethe-Peierls boundary condition, which is then applied to two charged particles to reveal their bound states and resonances. Infinite resonances are found for a repulsive Coulomb potential, one of which turns into a bound state with increasing inverse scattering length, whereas infinite bound states exist for an attractive Coulomb potential with no resonances at any scattering length. The zero-range theory is also applied to three equally charged particles at infinite scattering length under the variational Born-Oppenheimer approximation. We find that the effective potential between two heavy particles induced by a light particle is an inverse-square attraction at distances shorter than the Bohr radius, leading to infinite deep bound states, whereas shallow ones successively turn into resonances with increasing Coulomb repulsion.