on the competition between intra- and interspecies interactions: if \(u_{12}\) overcomes the intraspecies interaction terms (\(\delta>0\)), the fluids strongly repel each other making the system immiscible. On the contrary, if \(u_{12}\) is smaller than the intraspecies interactions (\(\delta<0\)), the fluids overlap and the system is miscible. In such a picture, the miscibility regime of a two-component quantum gas can be controlled by varying the interaction coupling terms, which can be experimentally realized with the use of Feshbach resonances [27].  
However, until very recently [28], homogeneous atomic BECs were not experimentally produced. Instead, trapped atomic BECs exhibit an inhomogeneous density distribution as a result of the confinement. The increased role of the kinetic energy in such systems contributes to a more miscible mixture where phase-separation occurs for larger \(u_{12}\) than the condition set by Equation (1). The shift at the miscible-immiscible critical point has been obtained in the case of mixtures composed of distinct hyperfine states of the same atomic species [29; 30; 31]. In the broader scenario of unbalanced mixtures of different atomic species, the atom number ratio [32], the mass imbalance and the difference in trapping configurations between the components were also shown to affect the boundary of the miscibility phase transition [33; 34; 35; 36]. The contribution of gravity, relevant for all real experiments due to the induced gravitational sag [37; 38], is rarely taken into account in numerical simulations.  
In this work, we perform numerical simulations of the ground-state of a two-component quantum mixture of \({}^{23}\)Na and \({}^{39}\)K atoms for different interaction strengths, according to the relevant Feshbach resonances for magnetic fields in the range of 95-117 G [14; 39], in order to show the realistic miscibility regimes accessible in the experimental setup being developed in our laboratory [40] in the presence of gravity. We explore the effect of changing the number of atoms of the minority species (\({}^{39}\)K), thereby changing the atom number ratio \(\eta\), and calculating the spatial overlap between the atomic clouds as a quantity able to characterize the change in the miscibility regime of the system. The numerical simulations are performed at zero temperature, which satisfactorily reproduces the experimental results for the case of strongly degenerate atomic mixtures [41], although theoretical works at finite temperature have shown a change of the miscibility condition of the system favoring phase separation [42; 43; 44].  
The article is organized as follows. In Section 2, we describe the two-component quantum gas at zero temperature in terms of a pair of coupled Gross-Pitaevskii equations (GPEs) (Section 2.1) and the numerical simulation method used to obtain the ground-state of the system (Section 2.2). In Section 3, we first present our experimental system producing the \({}^{23}\)Na-\({}^{39}\)K atomic mixtures (Section 3.1), followed by the results of the numerical simulation performed with realistic experimental parameters (Section 3.2) and the construction of the phase diagram of the miscible-immiscible transition for such a mixture (Section 3.3). Finally, in Section 4, we highlight our main findings and discuss some future perspectives for identifying the miscibility regime of a quantum mixture comparing with the results presented in this article.