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arXiv:physics/0508060 · Physical Review A

Multichannel quantum-defect theory for slow atomic collisions

Bo Gao, Eite Tiesinga, Carl J. Williams, Paul S. Julienne

Published Oct 28, 2005• 98 citations
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Abstract

We present a multichannel quantum-defect theory for slow atomic collisions that takes advantages of the analytic solutions for the long-range potential and both the energy and angular momentum insensitivities of the short-range parameters. The theory provides an accurate and complete account of scattering processes, including shape and Feshbach resonances, in terms of a few parameters such as the singlet and triplet scattering lengths. As an example, results for $^{23}\mathrm{Na}\ensuremath{-}^{23}\mathrm{Na}$ scattering are presented and compared to close-coupling calculations.

DOI
10.1103/physreva.72.042719

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We present a multichannel quantum-defect theory for slow atomic collisions that takes advantages of the analytic solutions for the long-range potential and both the energy and angular momentum insensitivities of the short-range parameters. The theory provides an accurate and complete account of scattering processes, including shape and Feshbach resonances, in terms of a few parameters such as the singlet and triplet scattering lengths. As an example, results for 23Na\ensuremath−23Na^{23}\mathrm{Na}\ensuremath{-}^{23}\mathrm{Na}23Na\ensuremath−23Na scattering are presented and compared to close-coupling calculations.
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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.
We have developed a model of Feshbach resonances in gases of ultracold alkali-metal atoms using the ideas of multichannel quantum-defect theory. Our model requires just three parameters describing the interactions---the singlet and triplet scattering lengths, and the long-range van der Waals coefficient---in addition to known atomic properties. Without using any further details of the interactions, our approach can accurately predict the locations of resonances. It can also be used to find the singlet and triplet scattering lengths from measured resonance data. We apply our technique to 6Li-40K^{6}\text{L}\text{i-}^{40}\text{K}6Li-40K and $^{40}\text{K}\text{\ensuremath{-}}^{87}\text{R}\text{b}$ scattering, obtaining good agreement with experimental results, and with the more computationally intensive coupled channels technique.
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.
Multichannel quantum defect theory (MQDT) is shown to be capable of producing quantitatively accurate results for low-energy atom-molecule scattering calculations. With a suitable choice of reference potential and short-range matching distance, it is possible to define a matrix that encapsulates the short-range collision dynamics and is only weakly dependent on energy and magnetic field. Once this has been produced, calculations at additional energies and fields can be performed at a computational cost that is proportional to the number of channels NNN and not to N3{N}^{3}N3. MQDT thus provides a promising method for carrying out low-energy molecular scattering calculations on systems where full exploration of the energy dependence and the field dependence is currently impractical.
Multichannel Quantum Defect Theory (MQDT) is shown to be capable of producing quantitatively accurate results for low-energy atom-molecule scattering calculations. With a suitable choice of reference potential and short-range matching distance, it is possible to define a matrix that encapsulates the short-range collision dynamics. Multichannel quantum defect theory can provide an efficient alternative to full coupled-channel calculations for low-energy molecular collisions. However, the efficiency relies on interpolation of the Y matrix that encapsulates the short-range dynamics. It is shown how the phases of the MQDT reference functions may be chosen so as to remove such poles from the vicinity of a reference energy and dramatically increase the range of interpolation. For the test cases of Mg+NH and Li+NH, the resulting optimized Y matrix may be interpolated smoothly over an energy range of several Kelvin and a magnetic field range of over 1000G. Calculations at additional energies and fields can then be performed at a computational cost that is proportional to the number of channels N and not to N cubed. MQDT thus provides a promising method for carrying out low-energy molecular scattering calculations on systems where full exploration of the energy and the field dependence is currently impractical.