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arXiv:cond-mat/0402278 · Physical Review A

Feshbach resonances with large background scattering length: Interplay with open-channel resonances

Bout Marcelis, Eric van Kempen, B. Verhaar, S. J. J. M. F. Kokkelmans

Published Jul 1, 2004• 76 citations
arXivHTMLPDF

Abstract

Feshbach resonances are commonly described by a single-resonance Feshbach model, and open-channel resonances are not taken into account explicitly. However, an open-channel resonance near threshold limits the range of validity of this model. Such a situation exists when the background scattering length is much larger than the range of the interatomic potential. The open-channel resonance introduces strong threshold effects not included in the single-resonance description. We derive an easy-to-use analytical model that takes into account both the Feshbach resonance and the open-channel resonance. We apply our model to $^{85}\mathrm{Rb}$, which has a large background scattering length, and show that the agreement with coupled-channel calculations is excellent. The model can be readily applied to other atomic systems with a large background scattering length, such as $^{6}\mathrm{Li}$ and $^{133}\mathrm{Cs}$. Our approach provides full insight into the underlying physics of the interplay between open-channel (or potential) resonances and Feshbach resonances.

DOI
10.1103/physreva.70.012701

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Feshbach resonances are commonly described by a single-resonance Feshbach model, and open-channel resonances are not taken into account explicitly. However, an open-channel resonance near threshold limits the range of validity of this model. Such a situation exists when the background scattering length is much larger than the range of the interatomic potential. The open-channel resonance introduces strong threshold effects not included in the single-resonance description. We derive an easy-to-use analytical model that takes into account both the Feshbach resonance and the open-channel resonance. We apply our model to 85Rb^{85}\mathrm{Rb}85Rb, which has a large background scattering length, and show that the agreement with coupled-channel calculations is excellent. The model can be readily applied to other atomic systems with a large background scattering length, such as 6Li^{6}\mathrm{Li}6Li and 133Cs^{133}\mathrm{Cs}133Cs. Our approach provides full insight into the underlying physics of the interplay between open-channel (or potential) resonances and Feshbach resonances.
Abstract Efimov physics in the vicinity of two overlapping narrow Feshbach resonances can be explored within a framework of a three-channel model where a non-interacting open channel is coupled to two closed molecular channels. Here, we determine how it compares to the extended two-channel model, which includes an open channel with finite background scattering and a single molecular channel. We identify the parameter range in which the three-channel model surpasses the extended two-channel model. Furthermore, the three-channel model is extended to include background scattering, and then both models are applied to the experimentally relevant system of bosonic lithium atoms polarized on two different energy levels, with an isolated and two overlapping narrow Feshbach resonances, respectively. We confirm, in agreement with previous studies, that being small, the background scattering length in lithium has a negligible effect on the Efimov features in the case of isolated resonance. However, in the case of overlapping Feshbach resonances, the inclusion of background scattering improves the performance of the theory with respect to the experimentally measured position of the Efimov resonance.
The physics of Feshbach resonance is analyzed using an analytic expression for the s-wave scattering phase shift and the scattering length a which we derive within a two-channel tight-binding model. Employing a unified treatment of bound states and resonances in terms of the Jost function, it is shown that, for strong interchannel coupling, Feshbach resonance can occur even when the closed channel does not have a bound state. This may extend the range of ultracold atomic systems that can be manipulated by Feshbach resonance. The dependence of the sign of a on the coupling strength in the unitary limit is elucidated. As a by-product, analytic expressions are derived for the background scattering length, the external magnetic field at which resonance occurs, and the energy shift ε-ε(B), where ε is the scattering energy and ε(B) is the bound-state energy in the closed channel (when there is one).
We study the influence of a single Feshbach resonance on the near-threshold bound and continuum states in interatomic potentials with attractive tails. We present an energy-independent parametrization of the Feshbach resonance to reveal its influence on the bound state spectrum as well as on the scattering properties. The applicability of this parametrization is confirmed by comparison with numerical coupled-channel calculations. A formula for the scattering length is derived, which---in contrast to previous parametrizations---transparently exposes the dependence on the characteristic quantities of the uncoupled channels and on channel-coupling effects and is valid for all deep potentials falling off faster than $\ensuremath{-}1/{r}^{3}$ asymptotically.
We calculate the spectrum of three-body Efimov bound states near a Feshbach resonance within a model which accounts both for the finite range of interactions and the presence of background scattering. The latter may be due to direct interactions in an open channel or a second overlapping Feshbach resonance. It is found that background scattering gives rise to substantial changes in the trimer spectrum as a function of the detuning away from a Feshbach resonance, in particular in the regime where the background channel supports Efimov states on its own. Compared to the situation with negligible background scattering, the regime where van der Waals universality applies is shifted to larger values of the resonance strength if the background scattering length is positive. For negative background scattering lengths, in turn, van der Waals universality extends to even small values of the resonance strength parameter, consistent with experimental results on Efimov states in 39K^{39}\mathrm{K}39K. Within a simple model, we show that short-range three-body forces do not affect van der Waals universality significantly. Repulsive three-body forces may, however, explain the observed variation between around $\ensuremath{-}8$ and $\ensuremath{-}10$ of the ratio between the scattering length where the first Efimov trimer appears and the van der Waals length.
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 Feshbach resonance arises in cold atom scattering due to the complex interplay between several coupled channels. However, the essential physics of the resonance may be encapsulated in a simplified model consisting of just two coupled channels. In this paper we describe in detail how such an effective Feshbach model can be constructed from knowledge of a few key parameters, characterizing the atomic Born-Oppenheimer potentials and the low energy scattering near the resonance. These parameters may be obtained either from experiment or full coupled-channel calculations. Using RRR-matrix theory we analyze the bound state spectrum and the scattering properties of the two-channel model, and find it to be in good agreement with exact calculations.