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arXiv:1010.0293 · Physical Review A

Scattering amplitude of ultracold atoms near the p -wave magnetic Feshbach resonance

Peng Zhang, Pascal Naidon, Masahito Ueda

Published Dec 23, 2010• 26 citations
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Abstract

Most of the current theories on the $p$-wave superfluid in cold atomic gases are based on the effective-range theory for the two-body scattering, where the low-energy $p$-wave scattering amplitude ${f}_{1}(k)$ is given by ${f}_{1}(k)=\ensuremath{-}1/[\mathit{ik}+1/(\mathcal{V}{k}^{2})+1/\mathcal{R}]$. Here $k$ is the incident momentum, $\mathcal{V}$ and $\mathcal{R}$ are the $k$-independent scattering volume and effective range, respectively. However, due to the long-range nature of the van der Waals interaction between two colliding ultracold atoms, the $p$-wave scattering amplitude of the two atoms is not described by the effective-range theory [J. Math. Phys. 4, 54 (1963); Phys. Rev. A 58, 4222 (1998)]. In this paper we provide an explicit calculation for the $p$-wave scattering of two ultracold atoms near the $p$-wave magnetic Feshbach resonance. We show that in this case the low-energy $p$-wave scattering amplitude ${f}_{1}(k)=\ensuremath{-}1/[\mathit{ik}+1/({\mathcal{V}}^{\mathrm{eff}}{k}^{2})+1/({\mathcal{S}}^{\mathrm{eff}}k)+1/{\mathcal{R}}^{\mathrm{eff}}]$ where ${\mathcal{V}}^{\mathrm{eff}},$ ${\mathcal{S}}^{\mathrm{eff}}$, and ${\mathcal{R}}^{\mathrm{eff}}$ are $k$-dependent parameters. Based on this result, we identify sufficient conditions for the effective-range theory to be a good approximation of the exact scattering amplitude. Using these conditions we show that the effective-range theory is a good approximation for the $p$-wave scattering in the ultracold gases of $^{6}\mathrm{Li}$ and $^{40}\mathrm{K}$ when the scattering volume is enhanced by the resonance.

DOI
10.1103/physreva.82.062712

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It is well-known that cold atoms near sss-wave Feshbach resonances have universal properties that are insensitive to the short-range details of the interaction. What is less known is that atoms near higher partial-wave Feshbach resonances also have remarkable universal properties. We illustrate this with a single atom interacting resonantly with two fixed static centers. At a Feshbach resonance point with orbital angular momentum $L\ensuremath{\ge}1$, we find 2L+12L+12L+1 shallow bound states whose energies behave like 1/R2L+11/{R}^{2L+1}1/R2L+1 when the distance RRR between the two centers is large. We then compute corrections to the binding energies due to other parameters in the effective range expansions. For completeness we also compute the binding energies near sss-wave Feshbach resonances, taking into account the corrections. Afterwards we turn to the bound states at large but finite scattering volumes. For ppp-wave and higher partial-wave resonances we derive a simple formula for the energies in terms of a parameter called ``proximity parameter.'' These results are applicable to a free atom interacting resonantly with two atoms that are localized to two lattice sites of an optical lattice, and to one light atom interacting with two heavy ones in free space. Modifications of the low energy physics due to the long range van der Waals potential are also discussed.
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.
Magnetically tunable scattering resonances have been used with great success for precise control of s-wave scattering lengths in ultracold atomic collisions. We describe relatively simple yet quite powerful analytic treatments of such resonances based on the analytic properties of the van der Waals long range potential. This theory can be used to characterize a number of properties of specific resonances that have been used successfully in various experiments with 87^{87}87Rb, 85^{85}85Rb, 40^{40}40K, and 6^{6}6Li. Optical Feshbach resonances are also possible and may be practical with narrow intercombination line photoassociative transitions in species like Sr and Yb.
We present analytic descriptions of atomic interaction at ultracold temperatures using both single-channel and multichannel quantum-defect theories. In the case of a single channel, addressed in this paper, the expansion of Gao [Phys. Rev. A 58, 4222 (1998)] is generalized to higher orders for angular momentum $l\ensuremath{\ge}2$ to give a more complete description of ultracold scattering, including an analytic description of ultracold shape resonances of arbitrary lll. We also introduce a generalized scattering length that is well defined and useful for all partial waves to replace the traditional definition that fails for $l\ensuremath{\ge}2$ due to the long-range van der Waals interaction. The results are used in a companion paper to derive analytic descriptions of atomic interaction around a magnetic Feshbach resonance of arbitrary angular momentum lll.
At long distances, interactions between neutral ground-state atoms can be described by the van der Waals potential. In the ultracold regime, atom-atom scattering is dominated by sss-waves phase shifts given by an effective range expansion in terms of the scattering length ${\ensuremath{\alpha}}_{0}$ and the effective range r0{r}_{0}r0​. We show that while the scattering length cannot be predicted for these potentials, the effective range is given by the universal low-energy theorem ${r}_{0}=A+B/{\ensuremath{\alpha}}_{0}+C/{\ensuremath{\alpha}}_{0}^{2}$, where AAA, BBB, and CCC depend on the dispersion coefficients Cn{C}_{n}Cn​ and the reduced diatom mass. We confront this formula to about 100 determinations of r0{r}_{0}r0​ and ${\ensuremath{\alpha}}_{0}$ and show why the result is dominated by the leading dispersion coefficient C6{C}_{6}C6​. Universality and scaling extend much beyond naive dimensional analysis estimates.
We revisit the basic properties of Fano-Feshbach resonances in two-body systems with van der Waals tail interactions, such as ultracold neutral atoms. Using a two-channel model and two different methods, we investigate the relationship between the width and shift of the resonances and their dependence on the low-energy parameters of the system. Unlike what was previously believed [C. Chin et al., Rev. Mod. Phys. 82, 1225 (2010)] for magnetic resonances, we find that the ratio between the width and the shift of a resonance does not depend only on the background scattering length, but also on a closed-channel scattering length. We obtain different limits corresponding to different cases of optical and magnetic resonances. Although the generalization of the theory to the multichannel case remains to be done, we found that our two-channel predictions are verified for a specific resonance of lithium-6.