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Physical Review A

Prediction of Feshbach resonances using an analytical quantum-defect matrix

Gao‐Ren Wang, Ting Xie, Wei Zhang, Yin Hui Huang, Shu‐Lin Cong

Published Mar 14, 2012• 10 citations
Publisher

Abstract

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 $1/{R}^{6}$ 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 ${}^{6}$Li-${}^{40}$K scattering. The theoretical results agree quite well with experimental values.

DOI
10.1103/physreva.85.032706

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We present a simple theoretical model for describing ultracold-atomic collision dynamics and Feshbach resonances (FRs) without onerous numerical computation. An effective quantum-defect phase shift ${\ensuremath{\delta}}_{\mathrm{Y}}^{\mathrm{eff}}$ is derived and the reference potentials are shown acting as an assistant tool in the calculation. With the help of singlet and triplet scattering lengths aS{a}^{\mathrm{S}}aS and aT{a}^{\mathrm{T}}aT, as well as an adjustable parameter ${\ensuremath{\gamma}}_{\mathrm{sr}}$, this model can be used to reliably predict FRs. An application example of $^{6}\mathrm{Li}\ensuremath{-}^{40}\mathrm{K}$ collision shows that the results given by the simple model agree quite well with both the relevant experimental data and the coupled-channel calculations.
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 an analytical model for ultracold atom–ion collisions using the multichannel quantum-defect formalism. The model is based on analytical solutions of the r −4 long-range potential and on the application of a frame transformation between asymptotic and molecular bases. This approach allows the description of atom–ion interaction in the ultracold domain in terms of only three parameters: the singlet and triplet scattering lengths, assumed to be independent of the relative motion angular momentum, and the lead dispersion coefficient of the asymptotic potential. We also introduce corrections to the scattering lengths that improve the accuracy of our quantum-defect model for higher-order partial waves, a particularly important result for an accurate description of shape and Feshbach resonances at finite temperature. The theory is applied to the system composed of a 40 Ca + ion and a Na atom, and compared with numerical coupled-channel calculations carried out using ab initio potentials. For this particular system, we investigate the spectrum of bound states, the rate of charge-transfer processes and the collision rates in the presence of magnetic Feshbach resonances at zero and finite temperature.
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.
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