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arXiv:cond-mat/0308465 · Physical Review A

Feshbach resonance scattering under cylindrical harmonic confinement

Vladimir Yurovsky

Published Jan 19, 2005• 33 citations
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Abstract

A problem of collisions of atoms with two-channel zero-range interaction in an atomic waveguide is solved by using of a renormalization procedure. A matching of the solution to a solution of the related one-dimensional problem leads to relation between the one-dimensional and three-dimensional scattering parameters. The scattering amplitude and bound states for the confined system demonstrate differences from the related free and one-dimensional systems.

DOI
10.1103/physreva.71.012709

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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.

Cambridge University Press eBooks•Jan 30, 2021•0 citations
A problem of collisions of atoms with two-channel zero-range interaction in an atomic waveguide is solved by using of a renormalization procedure. A matching of the solution to a solution of the related one-dimensional problem leads to relation between the one-dimensional and three-dimensional scattering parameters. The scattering amplitude and bound states for the confined system demonstrate differences from the related free and one-dimensional systems.
Low-dimensional behavior of two ultracold atoms trapped in two- and one-dimensional waveguides is investigated in the vicinity of a magnetic Feshbach resonance. A quantitative two-channel model for the Feshbach mechanism is used, allowing an exhaustive analysis of low-dimensional resonant scattering behavior and of the confinement-induced bound states. The role of the different parameters of the resonance is depicted in this context. Results are compared with those of the zero-range approach. The relevance of the effective range approximation in low dimensions is studied. Examples of known resonances are used to illustrate the bound-state properties.
Bound states and collisions of atoms with two-channel two-body interactions in harmonic waveguides are analyzed. The closed-channel contributions to two-atom bound states become dominant in the case of a weak resonance. At low energies and values of the nonresonant scattering length the problem can be approximated by a one-dimensional resonant model. The three-body problem becomes nonintegrable and the properties of triatomic molecules become different from those predicted by the integrable Lieb-Liniger-McGuire model.
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.
It was recently predicted [Phys. Rev. Lett. 81, 938 (1998)]] that atom-atom scattering under transverse harmonic confinement is subject to a "confinement-induced resonance" where the effective one-dimensional coupling strength diverges at a particular ratio of the confinement and scattering lengths. As the initial prediction made use of the zero-range pseudopotential approximation, we now report numerical results for finite-range interaction potentials that corroborate this resonance. In addition, we now present a physical interpretation of this effect as a novel type of Feshbach resonance in which the transverse modes of the confining potential assume the roles of "open" and "closed" scattering channels.
A zero-range approach to atom-molecule coupling is developed in analogy to the Fermi-Huang pseudopotential approach to collisions. It is shown by explicit comparison to an exactly solvable finite-range model that replacing the molecular bound-state wave function with a regularized delta function can reproduce the exact scattering amplitude in the long-wavelength limit. Using this approach, we find an analytical solution to the two-channel Feshbach resonance problem for two atoms in a spherical harmonic trap, highlighting the strong dependence of the effective scattering length and bare-molecule population on the atom-molecule coupling strength.
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.