# Asymmetric Lineshapes of Efimov Resonances in Mass-Imbalanced Ultracold Gases

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## Abstract

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## 1. Introduction

## 2. General Considerations and Methods

#### 2.1. The Adiabatic Hyperspherical Representation and the Semi-Classical Approach

#### 2.2. A Simplified Semi-Classical Model

## 3. Asymmetric Lineshapes in Three-Body Recombination Coefficients

## 4. Summary

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**An illustration of the lowest hyperspherical potential curves ${U}_{\nu}^{1/3}(R/{a}_{HH})$ with ${a}_{HH}>0$ and ${a}_{HL}<0$. The red (blue) line saturates at large hyperradii in the atom–dimer (three-body break-up) threshold. The quantities ${\Phi}_{L}^{U}$ and ${\Phi}_{L}^{U}$ indicate the JWKB phase accumulation in the upper potential curve. For the lower potential, the corresponding phase is denoted by ${\Phi}_{L}^{L}$. The vertical dashed line represents the hyperradius where the non-adiabatic coupling P-matrix element ${P}_{12}$ maximizes. The horizontal dotted line refers to the three-body collisional energy $\overline{E}$ in units of $\frac{{\u0127}^{2}}{{m}_{H}{a}_{HH}^{2}}$, and the three-body parameter, $\frac{{r}_{3b}}{{a}_{HH}}$, depicted by the blue region.

**Figure 3.**The degree of diabaticity p as a function of the scattering length ratio ${a}_{HH}/\left|{a}_{HL}\right|$ for different mass ratios ${m}_{H}/{m}_{L}$, covering the regime of strong-to-weak mass-imbalanced three-body systems.

**Figure 4.**The scaled $\frac{|{S}_{12}{|}^{2}}{{\left(k{a}_{HL}\right)}^{4}}$ matrix element versus the ratios $\frac{|{a}_{HL}|}{{a}_{HH}}$ and $\frac{{r}_{3b}}{{a}_{HH}}$ for the ${}^{6}\mathrm{Li}$-${}^{87}\mathrm{Rb}$-${}^{87}\mathrm{Rb}$ system at low energy $E=\frac{{\u0127}^{2}{k}^{2}}{2\mu}$. (

**a**) Semi-classical approach and (

**b**) R-matrix numerical calculations.

**Figure 6.**In the limit of $E\to 0$, the scaled recombination coefficient $\frac{{m}_{H}{K}_{3}}{\u0127{a}_{HL}^{4}}$ is shown as a function of $ln\frac{{r}_{3b}}{{a}_{HH}}$ for (

**a**) ${}^{6}\mathrm{Li}$-${}^{133}\mathrm{Cs}$-${}^{133}\mathrm{Cs}$ and (

**b**) ${}^{6}\mathrm{Li}$-${}^{87}\mathrm{Rb}$-${}^{87}\mathrm{Rb}$. The symbols refer to the corresponding calculations in the semi-classical approach. The solid lines indicate the fitting of Equation (15) using the universal parameters shown in Table 1.

**Figure 7.**A comparison of the scaled recombination coefficient obtained via the fitting of Equation (15) (points) and the Fano lineshape formula (solid lines) from Equation (16) for two HHL systems. The red points and lines correspond to ${}^{6}\mathrm{Li}$-${}^{133}\mathrm{Cs}$-${}^{133}\mathrm{Cs}$ for a scattering length ratio $\frac{|{a}_{\mathrm{CsLi}}|}{{a}_{\mathrm{CsCs}}}=67.8$. The black points and lines denote the ${}^{6}\mathrm{Li}$-${}^{87}\mathrm{Rb}$-${}^{87}\mathrm{Rb}$ system at $\frac{|{a}_{\mathrm{RbLi}}|}{{a}_{\mathrm{RbRb}}}=126.2$. Note that the total colliding energy is set to zero.

**Figure 8.**Panels (

**a**–

**d**) show the width of the Efimov resonance $\Gamma $ and the asymmetry parameter q versus the scattering length ratio $\frac{|{a}_{HL}|}{{a}_{HH}}$ for the ${}^{6}\mathrm{Li}$-${}^{133}\mathrm{Cs}$-${}^{133}\mathrm{Cs}$ (${}^{6}\mathrm{Li}$-${}^{87}\mathrm{Rb}$-${}^{87}\mathrm{Rb}$) system, respectively. Note that the total colliding energy is set to zero. Moreover, $\Gamma $ and q are obtained via Equations (17) and (18), respectively, using the universal parameters shown in Table 1.

**Table 1.**A summary of the universal parameters used in Equations (16)–(18) for the systems of ${}^{6}\mathrm{Li}$-${}^{133}\mathrm{Cs}$-${}^{133}\mathrm{Cs}$ and ${}^{6}\mathrm{Li}$-${}^{87}\mathrm{Rb}$-${}^{87}\mathrm{Rb}$. Note that the values of ${s}_{0}$ and ${s}_{0}^{*}$ are calculated in Ref. [51].

HHL System | ${\mathit{s}}_{0}$ | ${\mathit{s}}_{0}^{*}$ | $\mathit{\gamma}$ | ${\mathit{\psi}}_{1}$ | ${\mathit{\psi}}_{2}$ |
---|---|---|---|---|---|

${}^{6}\mathrm{Li}$-${}^{133}\mathrm{Cs}$-${}^{133}\mathrm{Cs}$ | 1.983 | 2.003 | 4.42 | 0.46 | 0.13 |

${}^{6}\mathrm{Li}$-${}^{87}\mathrm{Rb}$-${}^{87}\mathrm{Rb}$ | 1.633 | 1.682 | 3.13 | 0.8 | 0.4 |

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Giannakeas, P.; Greene, C.H.
Asymmetric Lineshapes of Efimov Resonances in Mass-Imbalanced Ultracold Gases. *Atoms* **2021**, *9*, 110.
https://doi.org/10.3390/atoms9040110

**AMA Style**

Giannakeas P, Greene CH.
Asymmetric Lineshapes of Efimov Resonances in Mass-Imbalanced Ultracold Gases. *Atoms*. 2021; 9(4):110.
https://doi.org/10.3390/atoms9040110

**Chicago/Turabian Style**

Giannakeas, Panagiotis, and Chris H. Greene.
2021. "Asymmetric Lineshapes of Efimov Resonances in Mass-Imbalanced Ultracold Gases" *Atoms* 9, no. 4: 110.
https://doi.org/10.3390/atoms9040110