#
Miscibility Regimes in a ^{23}Na–^{39}K Quantum Mixture

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Description of an Atomic Quantum Mixture

#### 2.2. Numerical Simulation of the Ground-State

## 3. Results

#### 3.1. Experimental Setup

#### 3.2. Ground-State of ${}^{23}$Na–${}^{39}$K Mixtures

#### 3.3. The Miscibility Phase Diagram

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Scattering lengths as a function of the magnetic field for the intra-species interactions of ${}^{23}$Na, ${a}_{\mathrm{Na}}$ (in blue), and ${}^{39}$K, ${a}_{\mathrm{K}}$ (in red), and for the inter-species interaction ${a}_{\mathrm{NaK}}$ (in green) considering both atoms in the $F=1,{m}_{F}=-1$ hyperfine state. The black dashed line given by ${\left({a}_{\mathrm{Na}}{a}_{\mathrm{K}}\right)}^{1/2}$ represents the value of ${a}_{\mathrm{NaK}}$ for which the system changes from immiscible to miscible with $\delta =0$. (

**b**) Miscibility parameter as a function of the magnetic field. At $B=109.1\phantom{\rule{3.33333pt}{0ex}}$ G with $\delta =0$ the system changes from immiscible to miscible when increasing B.

**Figure 2.**Density profiles along the z-axis of the simulated ground-state of ${}^{23}$Na (in blue) and ${}^{39}$K (in red): (

**a**) $B=100\phantom{\rule{3.33333pt}{0ex}}$ G with $\delta =4.32$, (

**b**) $B=108\phantom{\rule{3.33333pt}{0ex}}$ G with $\delta =0.22$ and (

**c**) $B=111\phantom{\rule{3.33333pt}{0ex}}$ G with $\delta =-0.50$. In each case, we display the results of three atom number ratio $\eta ={N}_{\mathrm{Na}}/{N}_{\mathrm{K}}$ equal to 50 (solid lines), 10 (dashed lines) and 5 (dotted lines).

**Figure 3.**Normalized overlap as a function of the Feshbach field for different values of $\eta $. For large $\eta $, ${N}_{\mathrm{K}}<<{N}_{\mathrm{Na}}$ (blue circles), the spatial overlap increases at earlier magnetic fields showing the transition to the miscible phase for ${u}_{12}^{2}>{u}_{11}{u}_{22}$ (with $\delta >0$). In the opposite scenario, for $\eta =1$, ${N}_{\mathrm{K}}={N}_{\mathrm{Na}}$ (red stars), the normalized spatial overlap significantly increases only for $B>111\phantom{\rule{3.33333pt}{0ex}}$ G remaining immiscible even if ${u}_{12}^{2}<{u}_{11}{u}_{22}$ (with $\delta <0$).

**Figure 4.**(

**a**) Numerical second derivative of the normalized overlap, ${O}_{\mathrm{norm}}$. We identify the peak position of each curve as the magnetic field value, ${B}_{\mathrm{peak}}$, for which the normalized overlap changes the most indicating the transition from immiscible to miscible. The dotted lines serve only as guide to the eyes. In (

**b**), we show ${B}_{\mathrm{peak}}$ as a function of $\eta $ in a semilog scale which gives a logarithm dependence of ${B}_{\mathrm{peak}}$ with the atom number ration. The red solid curve is a fit to the data points (see main text) and the black dashed line represents ${B}_{0}$ with $\delta =0$.

**Figure 5.**Phase diagram of the miscible–immiscible phase transition for the ${}^{23}$Na–${}^{39}$K mixture under our experimental conditions. The colormap represents the value of the normalized overlap for each combination of $\eta $ and B. The light gray line sets the phase transition point obtained from the second derivative of the normalized overlap. The condition for an homogeneous system is shown by the black dashed line at ${B}_{0}=109.1\phantom{\rule{3.33333pt}{0ex}}$ G.

**Figure 6.**Normalized overlap, ${O}_{\mathrm{norm}}$, as a function of the Feshbach field, B, for $\eta =1$ under different trapping conditions (see main text).

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**MDPI and ACS Style**

Gutierrez, E.M.; de Oliveira, G.A.; Farias, K.M.; Bagnato, V.S.; Castilho, P.C.M.
Miscibility Regimes in a ^{23}Na–^{39}K Quantum Mixture. *Appl. Sci.* **2021**, *11*, 9099.
https://doi.org/10.3390/app11199099

**AMA Style**

Gutierrez EM, de Oliveira GA, Farias KM, Bagnato VS, Castilho PCM.
Miscibility Regimes in a ^{23}Na–^{39}K Quantum Mixture. *Applied Sciences*. 2021; 11(19):9099.
https://doi.org/10.3390/app11199099

**Chicago/Turabian Style**

Gutierrez, Emmanuel Mercado, Gustavo Alves de Oliveira, Kilvia Mayre Farias, Vanderlei Salvador Bagnato, and Patricia Christina Marques Castilho.
2021. "Miscibility Regimes in a ^{23}Na–^{39}K Quantum Mixture" *Applied Sciences* 11, no. 19: 9099.
https://doi.org/10.3390/app11199099