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Physical Review A

Feshbach resonance in atomic binary collisions in the Wigner threshold law regime

Maurice Raoult, Frederick H. Mies

Published Jul 27, 2004• 41 citations
Publisher

Abstract

We analyze, within the generalized multichannel quantum defect theory framework, atomic binary collision cross sections over the $1\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{K}$ to $10\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$ energy range just above the entrance channel threshold, a domain where the Wigner threshold law should apply. By adjusting the strength of a constant external magnetic field a Feshbach resonance is tuned at will over this energy range. In the threshold regime, the quasibound state interacts with an opening continuum whose wave function presents a strong energy dependence which reflects the breakdown of the WKB approximation. The effective discrete-continuum interactions become very sensitive to the energy. The consequences of these variations have been investigated by including threshold effects in the analysis of the Fano configuration-interaction theory [Phys. Rev. 124, 1866 (1961)] in terms of quantum defect theory quantities proposed by [Lecomte J. Phys. B 20, 3645 (1987)]. This analysis shows that the energy variations are in general so important that it becomes meaningless to associate a width with a Feshbach resonance. However, it is still possible to define the resonance energy, as long as the energy variation of the shift of the resonance, induced by the effective discrete-continuum interaction, remains linear over an energy range corresponding to the magnitude of the shift.

DOI
10.1103/physreva.70.012710

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We analyze, within the generalized multichannel quantum defect theory framework, atomic binary collision cross sections over the 1\ensuremathμK1\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{K}1\ensuremathμK to 10mK10\phantom{\rule{0.3em}{0ex}}\mathrm{mK}10mK energy range just above the entrance channel threshold, a domain where the Wigner threshold law should apply. By adjusting the strength of a constant external magnetic field a Feshbach resonance is tuned at will over this energy range. In the threshold regime, the quasibound state interacts with an opening continuum whose wave function presents a strong energy dependence which reflects the breakdown of the WKB approximation. The effective discrete-continuum interactions become very sensitive to the energy. The consequences of these variations have been investigated by including threshold effects in the analysis of the Fano configuration-interaction theory [Phys. Rev. 124, 1866 (1961)] in terms of quantum defect theory quantities proposed by [Lecomte J. Phys. B 20, 3645 (1987)]. This analysis shows that the energy variations are in general so important that it becomes meaningless to associate a width with a Feshbach resonance. However, it is still possible to define the resonance energy, as long as the energy variation of the shift of the resonance, induced by the effective discrete-continuum interaction, remains linear over an energy range corresponding to the magnitude of the shift.
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Compared to purely atomic collisions, ultracold collisions involving molecules have the potential to support a much larger number of Fano-Feshbach resonances due to the huge amount of ro-vibrational states available. In order to handle such ultracold atom-molecule collisions, we formulate a theory that incorporates the ro-vibrational Fano-Feshbach resonances in a statistical manner while treating the physics of the long-range scattering, which is sensitive to such things as hyperfine states, collision energy, and any applied electromagnetic fields, exactly within multichannel quantum defect theory. Uniting these two techniques, we can assess the influence of highly resonant scattering in the threshold regime, and in particular its dependence on the hyperfine state selected for the collision. This allows us to explore the onset of Ericson fluctuations in the regime of overlapping resonances, which are well known in nuclear physics but completely unexplored in the ultracold domain.
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 employ a quantum defect theory framework to provide a detailed analysis of the interplay between a magnetic Feshbach resonance and a shape resonance in cold collisions of ultracold 87Rb^{87}\mathrm{Rb}87Rb atoms as captured in recent experiments using a laser-based collider [M. Chilcott et al., Phys. Rev. Research 3, 033209 (2021)]. By exerting control over a parameter space spanned by both collision energy and magnetic field, the width of a Feshbach resonance can be tuned over several orders of magnitude. We apply a quantum defect theory specialized for ultracold atomic collisions to fully describe of the experimental observations. While the width of a Feshbach resonance generally increases with collision energy, its coincidence with a shape resonance leads to a significant additional boost. By conducting experiments at a collision energy matching the shape resonance and using the shape resonance as a magnifying lens, we demonstrate a feature broadening to a magnetic width of 8 G compared to a predicted Feshbach resonance width much less than 0.10.10.1 mG.
With the advent of evaporative cooling methods, atomic collisions can be studied at microkelvin and even sub-microkelvin temperatures. At these energies, resonance and threshold phenomena become very pronounced. Near zero-energy resonances lead to large variations in collision cross sections with energy. Magnetically tunable Feshbach resonances have been observed, which allow elastic collision cross sections and scattering lengths to be tuned over large ranges with small changes in an applied magnetic field. These resonance effects are extremely sensitive to small variations in atomic interaction parameters, including scattering lengths. The combined results of ultracold atomic collision and photoassociation spectroscopy experiments is leading to a complete and consistent picture of collisions of ultracold alkali 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.