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arXiv:0808.0120 · Physical Review A

Scattering and binding of different atomic species in a one-dimensional optical lattice

Rune Piil, Nicolai Gayle Nygaard, Klaus Mølmer

Published Sep 15, 2008• 20 citations
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Abstract

The theory of scattering of atom pairs in a periodic potential is presented for the case of different atoms. When the scattering dynamics is restricted to the lowest Bloch band of the periodic potential, a separation in relative and average discrete coordinates applies and makes the problem analytically tractable. We present a number of results and features, which differ from the case of identical atoms.

DOI
10.1103/physreva.78.033611

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The theory of scattering of atom pairs in a periodic potential is presented for the case of different atoms. When the scattering dynamics is restricted to the lowest Bloch band of the periodic potential, a separation in relative and average discrete coordinates applies and makes the problem analytically tractable. We present a number of results and features, which differ from the case of identical atoms.
We consider a two-particle Schrödinger difference operator with a periodic potential perturbed by an exponentially decaying interaction potential for particles on a one-dimensional lattice. We obtain rigorous results for the two-particle scattering problem in the case of a small interaction and low velocities. Here, as in other quasi-one-dimensional models, small interactions can significantly affect the scattering pattern. In particular, we find the probability that the velocities of two particles in a periodic medium (e.g., they can be ultracold atoms in a one-dimensional optical lattice) change their signs during a collision. This probability increases as the relative velocity decreases and also as the absolute value of the matrix element between single-particle unperturbed Bloch states increases.
We present the theory of a pair of atoms in a one-dimensional optical lattice interacting via a narrow Feshbach resonance. Using a two-channel description of the resonance, we derive analytic results for the scattering states inside the continuum band and the discrete bound states outside the band. We identify a Fano resonance profile, and the survival probability of a molecule when swept through the Bloch band of scattering states by varying an applied magnetic field. We discuss how these results may be used to investigate the importance of the structured nature of the continuum in experiments.
Optical lattices are formed when cold atoms get trapped in the periodic array of optical potential wells created by a number of traveling light waves. Exact periodicity is guaranteed when the number of traveling waves exceeds the number of dimensions by no more than one. We present a few simple exactly solvable models for the transport of atoms over a lattice in the tight-binding limit. One model concerns the spreading of an atom by tunneling through the barriers and by momentum recoil induced by optical pumping. Another model gives results for the amplitude of Bloch oscillations, which occur when a uniform is added to the periodic potential. Finally the dissipative spreading of the quasimomentum due to optical pumping is modeled.
This study concerns the two-body scattering of particles in a one-dimensional periodic potential. A convenient ansatz allows for the separation of center-of-mass and relative motion, leading to a discrete Schr\"odinger equation in the relative motion that resembles a tight-binding model. A lattice Green's function is used to develop the Lippmann-Schwinger equation, and ultimately derive a multiband scattering KKK matrix which is described in detail in the two-band approximation. Two distinct scattering lengths are defined according to the limits of zero relative quasimomentum at the top and bottom edges of the two-body collision band. Scattering resonances occur in the collision band when the energy is coincident with a bound state attached to another higher or lower band. Notably, repulsive on-site interactions in an energetically closed lower band lead to collision resonances in an excited band.
The quantum-mechanical transport properties of cold atoms in an optical lattice are described in terms of an analytically tractable model. The atoms are cooled down to the lowest energy band in the periodic optical potential, in the tight- binding limit. The transport mechanisms are coherent tunneling through potential barriers, and momentum recoil at optical pumping. Expressions are obtained for the time-dependent population distributions over the energy eigenstates and over localized states. These expressions are modified in the presence of a dark atomic state, that is uncoupled to the field.