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arXiv:1308.0681 · Physical Review A

Analytical model of overlapping Feshbach resonances

Krzysztof Jachymski, Paul S. Julienne

Published Nov 6, 2013• 37 citations
arXivHTMLPDF

Abstract

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 ${}^{7}$Li atoms or two Cs atoms.

DOI
10.1103/physreva.88.052701

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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 study atom-ion scattering in the ultracold regime. To this aim, an analytical model based on the multichannel quantum-defect formalism is developed and compared to close-coupled numerical calculations. We investigate the occurrence of magnetic Feshbach resonances, focusing on the specific 40Ca++Na^{40}\mathrm{Ca}^{+}+\mathrm{Na}40Ca++Na system. The presence of several resonances at experimentally accessible magnetic fields should allow the atom-ion interaction to be precisely tuned. A fully quantum-mechanical study of charge-exchange processes shows that charge-exchange rates should remain small even in the presence of resonance effects. Most of our results can be cast in a system-independent form and are important for the realization of charge-neutral ultracold systems.
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.
Starting from a multichannel quantum-defect theory, we derive analytic descriptions of a magnetic Feshbach resonance in an arbitrary partial wave lll and the atomic interactions around it. An analytic formula, applicable to both broad and narrow resonances of arbitrary lll, is presented for ultracold atomic scattering around a Feshbach resonance. Other related issues addressed include (a) the parametrization of a magnetic Feshbach resonance of arbitrary lll, (b) rigorous definitions of ``broad'' and ``narrow'' resonances of arbitrary lll and their different scattering characteristics, and (c) the tuning of the effective range and the generalized effective range by a magnetic field.
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 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.
We present analytic descriptions of atomic interaction at ultracold temperatures using both single-channel and multichannel quantum-defect theories. In the case of a single channel, addressed in this paper, the expansion of Gao [Phys. Rev. A 58, 4222 (1998)] is generalized to higher orders for angular momentum $l\ensuremath{\ge}2$ to give a more complete description of ultracold scattering, including an analytic description of ultracold shape resonances of arbitrary lll. We also introduce a generalized scattering length that is well defined and useful for all partial waves to replace the traditional definition that fails for $l\ensuremath{\ge}2$ due to the long-range van der Waals interaction. The results are used in a companion paper to derive analytic descriptions of atomic interaction around a magnetic Feshbach resonance of arbitrary angular momentum lll.