## 3 Results
### The Miscibility Phase Diagram
Figure 3: Normalized overlap as a function of the Feshbach field for different values of \(\eta\). For large \(\eta\), \(N_{\rm K}<<N_{\rm Na}\) (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 \(\eta=1\), \(N_{\rm K}=N_{\rm Na}\) (red stars), the normalized spatial overlap significantly increases only for \(B>111\) G remaining immiscible even if \(u_{12}^{2}<u_{11}u_{22}\) (with \(\delta<0\)). The miscible-immiscible phase diagram for the \({}^{23}\)Na-\({}^{39}\)K mixture under our experimental conditions is shown in Figure 5. The colormap represents the value of the normalized overlap for each combination of \(\eta\) and \(B\) ranging from zero to unit. The transition point for each \(\eta\) obtained from Figure 4b is displayed by the solid light gray curve with the shaded area covering its uncertainty. The black dashed line represents the transition point for the homogeneous case setting \(\delta>0\) to the left side of the curve and \(\delta<0\) to the right. Differently from earlier works performed with a balanced mixture of two distinct hyperfine states of a single species [29, 30, 31], in our case for \(\eta=1\), the system is more immiscible and the miscible phase only occurs for \(\delta\)\(\sim\)\(-0.5\). A deeper analysis of the \(\eta=1\) case is presented in Figure 6, where we show the normalized overlap for \(\eta=1\) under  
Figure 4: **(a)** Numerical second derivative of the normalized overlap, \(O_{\text{norm}}\). We identify the peak position of each curve as the magnetic field value, \(B_{\text{peak}}\), 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 \(B_{\text{peak}}\) as a function of \(\eta\) in a semilog scale which gives a logarithm dependence of \(B_{\text{peak}}\) 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 \(B_{0}=109.1\) G.  
different trapping conditions: real experimental conditions (star), without gravity (asterisk), considering equal trapping potentials with \(\vartheta_{1}=\vartheta_{2}\) (plus sign) and for the homogeneous case (gray solid curve) obtained setting the external potentials \(\vartheta_{1}(\mathbf{r})=\vartheta_{2}(\mathbf{r})=0\). The dashed vertical line in black represents the point for \(\delta=0\) which indeed matches the abrupt transition of \(O_{\text{norm}}\) from 0 to unity observed in the homogeneous case. The role of gravity and different trapping configurations for each species is clear in the data of Figure 6: setting \(g=0\) and \(\vartheta_{1}=\vartheta_{2}\) drifts the transition point to smaller magnetic fields approaching \(B_{0}\). However, due to the large difference between the intraspecies scattering lengths for sodium and potassium (\(a_{\text{Na}}=52a_{0}\) and \(a_{\text{K}}\)\(\sim\)7.59 - 8.73\(a_{0}\)), the system still behaves more immiscible than the homogeneous case. The identification of the miscibility regime of the Na-K mixture under realistic experimental conditions is important when defining the best parameters for studying different physical phenomena. In studies which the spatial overlap between the components of the mixture is important (i.e., coupled vortex dynamics [59; 60; 61], binary quantum turbulence [62], coupled superfluidity and excitations [63; 64], etc.