# A Dual-Species Bose-Einstein Condensate with Attractive Interspecies Interactions

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## Abstract

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## 1. Introduction

## 2. Experiment

## 3. Tuning the Interspecies Interaction

## 4. Compensating for the Gravitational Sag

## 5. The Attractive Regime

#### 5.1. Phase Diagram: Free Space

#### 5.2. Observation of ${}^{41}$K-${}^{87}$Rb Droplets in Free Space

#### 5.3. Observation of ${}^{41}$K-${}^{87}$Rb Droplets in a Waveguide

## 6. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BEC | Bose–Einstein Condensate |

ODT | optical dipole trap |

TOF | time of flight |

MF | mean-field |

GP | Gross–Pitaevskii |

## References

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**Figure 1.**(

**a**) Schematics of the experiment, showing the optical dipole trap (ODT) beams, quadrupole and Feshbach coils. (

**b**) Absorption images of the dual-species Bose–Einstein condensate (BEC). Images are taken after 18 ms of TOF for ${}^{41}$K and 20.5 ms for ${}^{87}$Rb. ${N}_{\mathrm{K}}\sim 2\times {10}^{4}$ and ${N}_{\mathrm{Rb}}\sim 6\times {10}^{4}$. During TOF expansion, the Fesbach field is set at $77.5$ G, corresponding to ${a}_{12}\simeq 255{a}_{0}$.

**Figure 2.**(

**a**) Calculated interspecies scattering length ${a}_{12}$ for ${}^{41}$K and ${}^{87}$Rb in the $\left|F=1,{m}_{F}=1\right.\u232a$ state as a function of the magnetic field B. (

**b**) Absorption images of the dual-species BEC. First column: immiscible regime. During time-of-flight (TOF) expansion, the Feshbach field is set at $77.5$ G, corresponding to ${a}_{12}\simeq 255{a}_{0}$. Second column: miscible regime. During TOF expansion, the Feshbach field is set at 73 G, corresponding to ${a}_{12}\simeq 10{a}_{0}$.

**Figure 3.**Absorption images of the dual-species BEC for different values of ${b}_{z}$. During TOF expansion, the Feshbach field is set at $77.5$ G corresponding to ${a}_{12}\simeq 255{a}_{0}$.

**Figure 4.**Phase diagram of a ${}^{41}$K-${}^{87}$Rb attractive mixture, in free space, as a function of ${N}_{1}+{N}_{2}$ and ${a}_{12}$. For $\delta g>0$ (${a}_{12}>{a}_{12}^{\mathrm{c}}$) the gas is stable within the MF theory. For $\delta g<0\phantom{\rule{0.166667em}{0ex}}({a}_{12}<{a}_{12}^{\mathrm{c}})$, the MF energy is balanced by the Lee–Huang–Yang (LHY) term, and the system forms a LHY gas. For a sufficiently large atom number and sufficiently strong interactions, a self-bound droplet forms.

**Figure 5.**Calculated critical atom number ${N}_{\mathrm{c}}$ (top panel) and droplet density (bottom panel) in logarithmic scale as a function of $\delta a$ for the ${}^{41}$K-${}^{87}$Rb mixture (solid red lines) and the ${}^{39}$K spin mixture (dash-dotted black lines).

**Figure 6.**(

**a**) Absorption images (frame size: 190 × 190 $\mathsf{\mu}$m) of ${}^{41}$K and ${}^{87}$Rb for different values of TOF. For ${}^{41}$K from left to right: TOF = $17.5$, $19.5$, $21.5$, $23.5$, $25.5$ ms. For ${}^{87}$Rb from left to right: TOF = 20, 22, 24, 26, 28 ms. Top panels correspond to ${a}_{12}\simeq -18{a}_{0}$ ($\delta g>0$), and bottom panels to ${a}_{12}\simeq -85{a}_{0}$ ($\delta g<0$).

**Figure 7.**Absorption images (frame size: $34\times 760$ $\mathsf{\mu}$m) of ${}^{41}$K and ${}^{87}$Rb for different evolution times t in the waveguide. For ${}^{41}$K, from top to bottom: t = 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 ms. For ${}^{87}$Rb; from top to bottom: t = $17.5$, $22.5$, $27.5$, $32.5$, $37.5$, $42.5$, $47.5$, $52.5$, $57.5$, $62.5$ ms. Top panels correspond to ${a}_{12}\simeq -12{a}_{0}\phantom{\rule{0.166667em}{0ex}}(\delta g>0)$, and bottom panels to ${a}_{12}\simeq -83{a}_{0}$ ($\delta g<0$).

**Figure 8.**Simulated density distributions (frame size: 6 × 155 $\mathsf{\mu}$m) of ${}^{41}$K and ${}^{87}$Rb for ${a}_{12}=-12{a}_{0}$ ($\delta g>0$), at different evolution times t in the waveguide. For both species, from top to bottom $t=0,10,20,30,40$ ms; ${N}_{1}=1.5\times {10}^{3}$ and ${N}_{2}=4\times {N}_{1}$.

**Figure 9.**Simulated density distributions (frame size: 6 × 155 $\mathsf{\mu}$m) of ${}^{41}$K and ${}^{87}$Rb for ${a}_{12}=-83{a}_{0}$ ($\delta g<0$), at different evolution times t in the waveguide. For both species, from top to bottom $t=0,10,20,30,40$ ms; ${N}_{1}=1.5\times {10}^{3}$ and ${N}_{2}=4\times {N}_{1}$.

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**MDPI and ACS Style**

Burchianti, A.; D’Errico, C.; Prevedelli, M.; Salasnich, L.; Ancilotto, F.; Modugno, M.; Minardi, F.; Fort, C.
A Dual-Species Bose-Einstein Condensate with Attractive Interspecies Interactions. *Condens. Matter* **2020**, *5*, 21.
https://doi.org/10.3390/condmat5010021

**AMA Style**

Burchianti A, D’Errico C, Prevedelli M, Salasnich L, Ancilotto F, Modugno M, Minardi F, Fort C.
A Dual-Species Bose-Einstein Condensate with Attractive Interspecies Interactions. *Condensed Matter*. 2020; 5(1):21.
https://doi.org/10.3390/condmat5010021

**Chicago/Turabian Style**

Burchianti, Alessia, Chiara D’Errico, Marco Prevedelli, Luca Salasnich, Francesco Ancilotto, Michele Modugno, Francesco Minardi, and Chiara Fort.
2020. "A Dual-Species Bose-Einstein Condensate with Attractive Interspecies Interactions" *Condensed Matter* 5, no. 1: 21.
https://doi.org/10.3390/condmat5010021