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arXiv:2302.05642 · Physical Review A

Resonance-facilitated three-channel p -wave scattering

D. J. M. Ahmed-Braun, Paul S. Julienne, S. J. J. M. F. Kokkelmans

Published Jun 9, 2023• 1 citation
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Abstract

Feshbach resonances of arbitrary width are typically described in terms of two-channel models. Within these models, one usually considers a single dressed resonance, with the option to extend the analysis by including resonant open-channel features that can drastically change the observed threshold effects. For the strong $^{40}\mathrm{K} p$-wave resonance studied [Ahmed-Braun et al., Phys. Rev. Res. 3, 033269 (2021)], the interplay between an open-channel shape resonance and the Feshbach resonance could explain the unexpected nonlinear variation of the binding energy with magnetic field. However, the presented two-channel treatment relies on the introduction of two independent fitting parameters, whereas the typical Breit-Wigner expression would only account for one. This results in an effective magnetic moment that acquires a nonphysical value, which is an indication of a major shortcoming of the two-channel model treatment. In this study, we observe how the presence of a closed-channel shape resonance explains the physical mechanism behind the observations and demonstrates the need of a three-channel treatment. We introduce our model as resonance facilitated, where all coupling is mediated by the Feshbach state, while there is no direct coupling between the additional channel and the open channel. Notably, the resonance-facilitated structure greatly reduces the complexity of the full three-channel model. The typical Breit-Wigner form of the two-channel Feshbach formalism is retained and the full effect of the added channel can be captured by a single resonance dressing factor, which describes how the free propagation in the Feshbach state is dressed by the added channel.

DOI
10.1103/physreva.107.063304

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Feshbach resonances of arbitrary width are typically described in terms of two-channel models. Within these models, one usually considers a single dressed resonance, with the option to extend the analysis by including resonant open-channel features that can drastically change the observed threshold effects. For the strong 40Kp^{40}\mathrm{K} p40Kp-wave resonance studied [Ahmed-Braun et al., Phys. Rev. Res. 3, 033269 (2021)], the interplay between an open-channel shape resonance and the Feshbach resonance could explain the unexpected nonlinear variation of the binding energy with magnetic field. However, the presented two-channel treatment relies on the introduction of two independent fitting parameters, whereas the typical Breit-Wigner expression would only account for one. This results in an effective magnetic moment that acquires a nonphysical value, which is an indication of a major shortcoming of the two-channel model treatment. In this study, we observe how the presence of a closed-channel shape resonance explains the physical mechanism behind the observations and demonstrates the need of a three-channel treatment. We introduce our model as resonance facilitated, where all coupling is mediated by the Feshbach state, while there is no direct coupling between the additional channel and the open channel. Notably, the resonance-facilitated structure greatly reduces the complexity of the full three-channel model. The typical Breit-Wigner form of the two-channel Feshbach formalism is retained and the full effect of the added channel can be captured by a single resonance dressing factor, which describes how the free propagation in the Feshbach state is dressed by the added channel.
We study the near-threshold molecular and collisional physics of a strong 40K^{40}\mathrm{K}40K ppp-wave Feshbach resonance through a combination of measurements, numerical calculations, and modeling. Dimer spectroscopy employs both radio-frequency spin-flip association in the MHz band and resonant association in the kHz band. Systematic uncertainty in the measured binding energy is reduced by a model that includes both the Franck-Condon overlap amplitude and inhomogeneous broadening. Coupled-channels calculations based on mass-scaled 39K^{39}\mathrm{K}39K potentials compare well to the observed binding energies and also reveal a low-energy ppp-wave shape resonance in the open channel. Contrary to conventional expectation, we observe a nonlinear variation of the binding energy with magnetic field, and explain how this arises from the interplay of the closed-channel ramping state with the near-threshold shape resonance in the open channel. We develop an analytic two-channel model that includes both resonances as well as the dipole-dipole interactions which, we show, become important at low energy. Using this parametrization of the energy dependence of the scattering phase, we can classify the studied 40K^{40}\mathrm{K}40K resonance as broad. Throughout the paper, we compare to the well-understood sss-wave case and discuss the significant role played by van der Waals physics. The resulting understanding of the dimer physics of ppp-wave resonances provides a solid foundation for future exploration of few- and many-body orbital physics.
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).
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 We discuss s-wave scattering in an atomic binary collision with two coupled channels, tunable by an external magnetic field, one channel open and the other closed for the incident energies considered. The analysis is performed with a stylized model of square-well potentials. This simplification allows for a pedagogically thorough discussion of the different scattering resonances that appear in coupled channels. One of them, the potential resonances at vanishing energy, occurs as a bound state of the coupled system emerges, in turn, tuned at a very precise value of the external field. The other resonances, described by Feshbach theory, occur when the incident energy is near a bound state of the closed channel, as if it were decoupled from the open channel. These resonances exist for values of the external field above a particular threshold value. Besides the potential intrinsic value of this study in a quantum mechanics course, as the analysis can be performed with minor numerical calculations, it is also an aid for the understanding of current research advances in the exciting field of ultracold gases.
We discuss sss-wave scattering in an atomic binary collision with two coupled channels, tunable by an external magnetic field, one channel open and the other closed for the incident energies considered. The analysis is performed with a stylized model of square-well potentials. This simplification allows for a pedagogically thorough discussion of the different scattering resonances that appear in coupled channels. One of the them, the potential resonances at vanishing energy, occur as a bound state of the coupled system emerges, in turned tuned at a very precise value of the external field. The other resonances, described by Feshbach theory, occur when the incident energy is near a bound state of the closed channel, as if it were decoupled from the open channel. These resonances exist for values of the external field above a particular threshold value. Besides the potential intrinsic value of this study in a quantum mechanics course, as the analysis can be performed with minor numerical calculations, it is also an aid for the understanding of current research advances in the exciting field of ultracold gases.
We use the dressed potentials obtained in the adiabatic representation of two coupled channels to calculate sss-wave Feshbach resonances in a three-dimensional spherically symmetric potential with an open channel interacting with a closed channel. Analytic expressions for the sss-wave scattering length aaa and number of resonances are obtained for a piecewise constant model with a piecewise constant interaction of the open and closed channels near the origin. We show analytically and numerically that, for strong enough coupling strength, Feshbach resonances can exist even when the closed channel does not have a bound state.