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arXiv:1105.5846 · Physical Review A

Analytic description of atomic interaction at ultracold temperatures. II. Scattering around a magnetic Feshbach resonance

Bo Gao

Published Aug 8, 2011• 41 citations
arXivHTMLPDF

Abstract

Starting from a multichannel quantum-defect theory, we derive analytic descriptions of a magnetic Feshbach resonance in an arbitrary partial wave $l$ and the atomic interactions around it. An analytic formula, applicable to both broad and narrow resonances of arbitrary $l$, 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 $l$, (b) rigorous definitions of ``broad'' and ``narrow'' resonances of arbitrary $l$ and their different scattering characteristics, and (c) the tuning of the effective range and the generalized effective range by a magnetic field.

DOI
10.1103/physreva.84.022706

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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 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.
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.
A low-energy effective theory based on a microscopic multichannel description of the atom-atom interaction is derived for the scattering of alkali-metal atoms in different hyperfine states. This theory describes all scattering properties, including medium effects, in terms of the singlet and triplet scattering lengths and the range of the atom-atom potential and provides a link between a microscopic description of Feshbach scattering and more phenomenological approaches. It permits the calculation of medium effects on the resonance coming from the occupation of closed channel states. The examination of such effects is demonstrated to be of particular relevance to an experimentally important Feshbach resonance for 40K^{40}\mathrm{K}40K atoms. We analyze a recent rethermalization rate experiment on 40K^{40}\mathrm{K}40K and demonstrate that a measurement of the temperature dependence of this rate can determine the magnetic moment of the Feshbach molecule. Finally, the energy dependence of the Feshbach interaction is shown to introduce a negative effective range inversely proportional to the width of the resonance. Since our theory is based on a microscopic multichannel picture, it allows the explicit calculation of corrections to commonly used approximations such as the neglect of the effective range and the treatment of the Feshbach molecule as a point boson.
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.
Studies of cold atom collisions and few-body interactions often require the energy dependence of the scattering phase shift, which is usually expressed in terms of an effective-range expansion.We use accurate coupled-channel calculations on 6 Li, 39 K, and 133 Cs to explore the behavior of the effective range in the vicinity of both broad and narrow Feshbach resonances.We show that commonly used expressions for the effective range break down dramatically for narrow resonances and near the zero crossings of broad resonances.We present an alternative parametrization of the effective range that is accurate through both the pole and the zero crossing for both broad and narrow resonances.However, the effective-range expansion can still fail at quite low collision energies, particularly around narrow resonances.We demonstrate that an analytical form of an energy and magnetic-field-dependent phase shift, based on multichannel quantum defect theory, gives accurate results for the energy-dependent scattering length.
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.