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.
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
s-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
s-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.
Three-body collisions of ultracold identical Bose atoms under tight cylindrical confinement are analyzed. A Feshbach resonance in two-body collisions is described by a two-channel zero-range interaction. Elimination of the closed channel in the three-body problem reduces the interaction to a one-channel zero-range one with an energy-dependent strength. The related integrable Lieb-Liniger-McGuire (LLMG) model, with an energy-independent strength, forbids all chemical processes, such as three-atom association and diatom dissociation, as well as reflection in atom-diatom collisions. The resonant case is analyzed by a numerical solution of the Faddeev equations. The results demonstrate that as the internal symmetry of the LLMG model is lifted, reflection and chemical reactions become allowed and may be observed in experiments.