## 3 Results
### Ground-State of \({}^{23}\)Na-\({}^{39}\)K Mixtures  
The ground-state of \({}^{23}\)Na-\({}^{39}\)K mixtures was obtained with the numerical simulation method described in Section 2.2. In the simulations, we discretize the space with a three-dimensional grid of \(69\times 69\times 69\). The grid step size was chosen to be equal to 0.6 \(\upmu\)m resulting in a total volume of \(41\times 41\times 41\)\(\upmu\)m\({}^{3}\). The time interval for the simulations were \(\Delta t=50\times 10^{-6}\) in units of \(1/\overline{\omega}_{1}\), where \(\overline{\omega}_{1}=(\omega_{x}\omega_{y1}\omega_{z1})^{1/3}\) is the geometric mean of the trapping frequencies for species 1. We considered species 1 (2) as the potassium (sodium) atoms. We apply convergence methods based on the difference between the wave-functions of subsequent time intervals and monitor the total energy evolution in order to ensure the achievement of the ground-state configuration for both species. With these methods, typical integration times gave \(t_{\rm final}\)\(\sim\)3000. The number of sodium atoms was chosen \(N_{\rm Na}=5\times 10^{5}\) atoms in agreement with the numbers obtained in the experiment. The number of potassium atoms was varied with \(N_{\rm K}=1\times 10^{4}-5\times 10^{5}\) atoms setting \(\eta=N_{\rm Na}/N_{\rm K}=50-1\). The trapping frequencies were also set from the experimental values with \(\omega_{x,y}=2\pi\times 107(137)\) Hz and \(f_{\rm z}=2\pi\times 148(193)\) Hz for Na(K), respectively. The sodium scattering length was fixed to \(a_{\rm Na}=52\)\(a_{0}\), with \(a_{0}\) being the Born radius, while the scattering length of \({}^{39}\)K, \(a_{\rm K}\), and the interspecies scattering length, \(a_{\rm NaK}\), was varied according to the Feshbach resonances occuring at magnetic fields smaller than 300 G [14; 39]. In Figure 1, we show the values of the scattering lengths (\(a_{\rm Na}\), \(a_{\rm K}\) and \(a_{\rm NaK}\)) as a function of the magnetic field in the region with \(B=95\)-\(117.2\) G. In this region, both \(a_{\rm K}\) and \(a_{\rm NaK}\) are positive and the system changes its behaviour from immiscible to miscible with increasing the magnetic field. The predicted phase transition point for a homogeneous system (with \(\delta=0\)) occurs at \(B_{0}=109.1\) G [14]. The potassium scattering length was obtained with the simple relation:  
\[a(B)=a_{\rm bg}\Big{(}1-\frac{\Delta_{1}}{(B-B_{01})}-\frac{\Delta_{2}}{(B-B_ {02})}\Big{)}, \tag{10}\]  
where \(a_{\rm bg}=-19a_{0}\) is the background scattering length, \(B_{01}=32.6\) G and \(B_{02}=162.8\) G are the position of the first and second resonances for \({}^{39}\)K at the \(F=1,m_{F}=-1\) hyperfine state and \(\Delta_{1}=55\) G and \(\Delta_{2}=-37\) G are the corresponding resonance widths [39]. The \(a_{\rm NaK}\) curve displayed in Figure 1 was obtained from [14] via a private communication. Due to the presence of the gravitational force, each species suffers a different gravitation sag and the phase-separation at the immiscible phase occurs along the vertical direction (\(z\)-axis). In Figure 2, we show the simulated density profiles along the \(z\)-axis of  
Figure 1: **(a)** Scattering lengths as a function of the magnetic field for the intra-species interactions of \({}^{23}\)Na, \(a_{\rm Na}\) (in blue), and \({}^{39}\)K, \(a_{\rm K}\) (in red), and for the inter-species interaction \(a_{\rm NaK}\) (in green) considering both atoms in the \(F=1,m_{F}=-1\) hyperfine state. The black dashed line given by \((a_{\rm Na}a_{\rm K})^{1/2}\) represents the value of \(a_{\rm NaK}\) for which the system changes from immiscible to miscible with \(\delta=0\). **(b)** Miscibility parameter as a function of the magnetic field. At \(B=109.1\) G with \(\delta=0\) the system changes from immiscible to miscible when increasing \(B\).