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Bulletin of the American Physical Society

Analytic models of ultracold atomic collisions for application to confinement-induced resonances

Satyan Bhongale, Ivan Deutsch, S. J. J. M. F. Kokkelmans

Published May 21, 2009• 0 citations
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Abstract

We construct simple analytic models of the S matrix, accounting for both scattering resonances and smooth background contributions for collisions that occur below the s-wave threshold. Such models are important for studying confinement-induced resonances such as those occurring in cold collisions of 133Cs atoms in separated sites of a polarization-gradient optical lattice. Because these resonances occur at negative energy with respect to the s-wave threshold, they cannot be studied easily using direct numerical solutions of the Schrödinger equation. Using our analytic model, we extend previous studies of negative-energy scattering to the multichannel case, accounting for the interplay of Feshbach resonances, large background scattering lengths, and inelastic processes.

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We construct simple analytic models of the S matrix, accounting for both scattering resonances and smooth background contributions for collisions that occur below the s-wave threshold. Such models are important for studying confinement-induced resonances such as those occurring in cold collisions of 133Cs atoms in separated sites of a polarization-gradient optical lattice. Because these resonances occur at negative energy with respect to the s-wave threshold, they cannot be studied easily using direct numerical solutions of the Schrödinger equation. Using our analytic model, we extend previous studies of negative-energy scattering to the multichannel case, accounting for the interplay of Feshbach resonances, large background scattering lengths, and inelastic processes.
We construct simple analytic models of the SSS matrix, accounting for both scattering resonances and smooth background contributions for collisions that occur below the sss-wave threshold. Such models are important for studying confinement-induced resonances such as those occurring in cold collisions of 133Cs^{133}\text{C}\text{s}133Cs atoms in separated sites of a polarization-gradient optical lattice. Because these resonances occur at negative energy with respect to the sss-wave threshold, they cannot be studied easily using direct numerical solutions of the Schr\"odinger equation. Using our analytic model, we extend previous studies of negative-energy scattering to the multichannel case, accounting for the interplay of Feshbach resonances, large background scattering lengths, and inelastic processes.
Feshbach resonances in ultracold collisions often result from an interplay between many collision channels. Simple two-channel models can be introduced to capture the basic features, but cannot fully reproduce the situation when several resonances from different closed channels contribute to the scattering process. Using the formalism of multichannel quantum defect theory we develop an analytical model of overlapping Feshbach resonances. We find a general formula for the variation of the scattering length with magnetic field in the vicinity of an arbitrary number of resonances, characterized by simple parameters. Our formula is in excellent agreement with numerical coupled-channel calculations for several cases of overlapping resonances in the collisions of two 7{}^{7}7Li atoms or two Cs atoms.
We develop a grid method for multichannel scattering of atoms in a waveguide with harmonic confinement. This approach is employed to extensively analyze the transverse excitations and deexcitations as well as resonant scattering processes. Collisions of identical bosonic and fermionic as well as distinguishable atoms in harmonic traps with a single frequency $\ensuremath{\omega}$ permitting the center-of-mass (c.m.) separation are explored in depth. In the zero-energy limit and single mode regime we reproduce the well-known confinement-induced resonances (CIRs) for bosonic, fermionic, and heteronuclear collisions. In the case of the multimode regime up to four open transverse channels are considered. Previously obtained analytical results are extended significantly here. Series of Feshbach resonances in the transmission behavior are identified and analyzed. The behavior of the transmission with varying energy and scattering lengths is discussed in detail. The dual CIR leading to a complete quantum suppression of atomic scattering is revealed in multichannel scattering processes. Possible applications include, e.g., cold and ultracold atom-atom collisions in atomic waveguides and electron-impurity scattering in quantum wires.
A detailed study of the anharmonicity-induced resonances caused by the coupling of center-of-mass and relative motion is presented for a system of two ultracold atoms in single-well potentials. As has been confirmed experimentally, these inelastic confinement-induced resonances are of interest since they can lead to coherent molecule formation, losses, and heating in ultracold atomic gases. A perturbative model is introduced to describe the resonance positions and the coupling strengths. The validity of the model and the behavior of the resonances for different confinement geometries are analyzed in comparison with exact numerical ab initio calculations. While such resonances have so far only been detected for large positive values of the s-waves\text{-wave}s-wave scattering length, it is found that they are present also for negative s-waves\text{-wave}s-wave scattering lengths, i.e., for attractive interactions. The possibility to coherently tune the resonances by a variation of the external confinement geometry might pave the way for coherent molecule association where magnetic Feshbach resonances are inaccessible.
We consider the feasibility of observing a trap-induced resonance [Stock et al., Phys. Rev. Lett. 91, 183201 (2003)] for the case of two 133Cs^{133}\mathrm{Cs}133Cs atoms, trapped in separated wells of a polarization-gradient optical lattice, and interacting through a multichannel scattering process. Due to the anomalously large scattering length of cesium dimers, a strong coupling can occur between vibrational states of the trap and a weakly bound molecular state that is made resonant by the ac Stark shift of the lattice. We calculate the energy spectrum of the two-atom system as a function of the distance between two potential wells by connecting the solutions of the Schr\"odinger equation for the short-range molecular potential to that of the long-range trap in a self-consistent manner. The short-range potential is treated through a multichannel pseudopotential, parametrized by the KKK matrix, calculated numerically for atoms in free space in a close-coupling approximation. This captures both the bound molecular spectrum as well as the energy-dependent scattering for all partial waves. We establish realistic operating conditions under which the trap-induced resonance could be observed and show that this strong and coherent interaction could be used as a basis for high-fidelity two-qubit quantum logic operations.
We present a theoretical model for describing ultracold atomic collision based on the multichannel quantum-defect theory. An analytical expression for the quantum-defect matrix Y is derived for the truncated 1/R61/{R}^{6}1/R6 model potential, and threshold behaviors of the quantum-defect parameters are discussed. The model allows us to predict the Feshbach resonances when the singlet and triplet scattering lengths are known or to fit the scattering lengths when a couple of Feshbach resonances is located experimentally. As an illustrative example, we investigate the 6{}^{6}6Li-40{}^{40}40K scattering. The theoretical results agree quite well with experimental values.