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arXiv:2210.14200 · Physical Review A

Efimovian three-body potential from broad to narrow Feshbach resonances

J. van de Kraats, D. J. M. Ahmed-Braun, Jing-Lun Li, S. J. J. M. F. Kokkelmans

Published Feb 1, 2023• 11 citations
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Abstract

We analyze the change in the hyperradial Efimovian three-body potential as the two-body interaction is tuned from the broad to narrow Feshbach resonance regime. Here, it is known from both theory and experiment that the three-body dissociation scattering length ${a}_{\ensuremath{-}}$ shifts away from the universal value of $\ensuremath{-}9.7\phantom{\rule{4pt}{0ex}}{r}_{\mathrm{vdW}}$, with ${r}_{\mathrm{vdW}}=\frac{1}{2}{(m{C}_{6}/{\ensuremath{\hbar}}^{2})}^{1/4}$ the two-body van der Waals range. We model the three-body system using a separable two-body interaction that takes into account the full zero-energy behavior of the multichannel wave function. We find that the short-range repulsive barrier in the three-body potential characteristic for single-channel models remains universal for narrow resonances, while the change in the three-body parameter originates from a strong decrease in the potential depth. From an analysis of the underlying spin structure we further attribute this behavior to the dominance of the two-body interaction in the resonant channel compared to other nonresonant interactions.

DOI
10.1103/physreva.107.023301

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We analyze the change in the hyperradial Efimovian three-body potential as the two-body interaction is tuned from the broad to narrow Feshbach resonance regime. Here, it is known from both theory and experiment that the three-body dissociation scattering length a\ensuremath−{a}_{\ensuremath{-}}a\ensuremath−​ shifts away from the universal value of \ensuremath−9.7rvdW\ensuremath{-}9.7\phantom{\rule{4pt}{0ex}}{r}_{\mathrm{vdW}}\ensuremath−9.7rvdW​, with rvdW=12(mC6/\ensuremathℏ2)1/4{r}_{\mathrm{vdW}}=\frac{1}{2}{(m{C}_{6}/{\ensuremath{\hbar}}^{2})}^{1/4}rvdW​=21​(mC6​/\ensuremathℏ2)1/4 the two-body van der Waals range. We model the three-body system using a separable two-body interaction that takes into account the full zero-energy behavior of the multichannel wave function. We find that the short-range repulsive barrier in the three-body potential characteristic for single-channel models remains universal for narrow resonances, while the change in the three-body parameter originates from a strong decrease in the potential depth. From an analysis of the underlying spin structure we further attribute this behavior to the dominance of the two-body interaction in the resonant channel compared to other nonresonant interactions.
Here we present a four-channel model that incorporates a magnetically tunable Feshbach resonance in a system of three atoms that interact via pairwise van der Waals interactions. Our method is designed to model recent experiments where the tunability of the scattering length has been used to study three-body Efimov states, which appear in the limit of a diverging two-body scattering length. Using this model, we calculate three-body adiabatic and effective potential curves and study how the strength (or width) of the Feshbach resonance affects the three-body potential that is connected to the Efimov effect. We find that the position of the repulsive barrier, which has been used to explain the so-called van der Waals universality in broad resonances, is slightly shifted as the narrow resonance limit is approached and that this shift is correlated to the appearance of two avoided crossings in the adiabatic energy landscape. More importantly, the attractive well is markedly shifted upward in energy and is extremely shallow for the narrowest resonance. We argue that this behavior is connected to the breakdown of van der Waals universality for weak (narrow) resonances.
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.
We study Efimov physics of three identical bosons with pairwise multichannel interactions for Feshbach resonances of adjustable width in magnetic field. We find that the two-body multichannel realization of the interaction can affect the universal Efimov spectrum, especially for resonances of intermediate width. We analyze two scenarios that are equivalent on the two-body level but differ in their realization in three-body multichannel spin space. The deviations in the Efimov spectra of these scenarios are caused by trimer states in the closed interaction channels whose binding energy and coupling strength to the Efimov states depend on the spin realization. However, in the narrow resonance limit the Efimov spectrum is set by the resonance width parameter r∗{r}^{*}r∗ only and does not depend on the realization in spin space. We find this limit to be even independent of the interaction potential considered. In the broad resonance limit all excited Efimov trimer energies approach the ones from the corresponding single-channel system for the scenarios investigated.
We measure the critical scattering length for the appearance of the first three-body bound state, or Efimov three-body parameter, at seven different Feshbach resonances in ultracold ^{39}K atoms. We study both intermediate and narrow resonances, where the three-body spectrum is expected to be determined by the nonuniversal coupling of two scattering channels. Instead, our observed ratio of the three-body parameter with the van der Waals radius is approximately the same universal ratio as for broader resonances. This unexpected observation suggests the presence of a new regime for three-body scattering at narrow resonances.
1. Highlight “short-ranged,” “dilute,” and “low-energy” as three main features of interactions between ultracold atoms. 2. Introduce the important concept of the phase shift. 3. Introduce the sss-wave scattering length as a universal parameter describing the low-energy interaction between ultracold atoms. 4. Discuss the relation between divergent scattering length, low-energy bound state, and jump of phase shift. 5. Discuss the relation between the scattering length and the scattering amplitude. 6. Discuss under what condition a positive scattering length describes repulsive interaction. 7. Discuss the condition when an algebraically decayed potential can be treated as a finite range one. 8. Introduce two types of zero-range single-channel potentials to capture the universal low-energy sss-wave interaction between ultracold atoms. 9. Introduce the concept of renormalization condition and renormalizable theory. 10. Discuss how the spin rotational symmetry imposes constraints on interaction form for both alkali-metal and alkaline-earth-metal atoms. 11. Introduce Feshbach resonance as an important tool to tune scattering length. 12. Compare the two-channel Feshbach resonance with the single-channel shape resonance and compare the wide and the narrow resonances. 13. Introduce a zero-range two-channel model. 14. Introduce the confinement-induced resonance to tune interaction strength by an external potential. 15. Summarize three key conditions for a Feshbach resonance, and unify the optical Feshbach resonance, the orbital Feshbach resonance, and the confinement-induced resonance all in terms of these three conditions. 16. Introduce the Efimov effect as an important three-body effect at the vicinity of the two-body scattering resonance. 17. Highlight the symmetry aspect of the Efimov effect. 18. Discuss various connections between few-body and many-body physics. 19. Illustrate that few-body calculation can be used to determine properties of many-body systems by using high-temperature expansion as an example.
We provide an exact solution of the Efimov spectrum in ultracold gases within the standard two-channel model for Feshbach resonances. It is shown that the finite range in the Feshbach coupling makes the introduction of an adjustable three-body parameter obsolete. The solution explains the empirical relation between the scattering length a − where the first Efimov state appears at the atom threshold and the van der Waals length l vdw for open-channel dominated resonances. There is a continuous crossover to the closed-channel dominated limit, where the scale in the energy level diagram as a function of the inverse scattering length 1 / a is set by the intrinsic length r ⋆ associated with the Feshbach coupling. Our results provide a number of predictions for the deviations from universal scaling relations between energies and scattering lengths that can be tested in future experiments.