## 2 Methods  
### Description of an Atomic Quantum Mixture  
Consider a mixture of two different bosonic atoms, labeled 1 and 2, at \(T=0\) in the weakly interacting regime where interactions are treated as contact interactions. Let \(N_{1}\) and \(N_{2}\) be the number of particles and \(\phi_{1}(\mathbf{r})\) and \(\phi_{2}(\mathbf{r})\) be the corresponding normalized single-particle wave functions. In such a picture, and neglecting terms of the order of \(1/N_{1}\) and \(1/N_{2}\), the energy functional of the system [25; 26; 45] can be written as  
\[\begin{split} E=&\int d\mathbf{r}\left[\frac{\hbar^{ 2}}{2m_{1}}|\
abla\psi_{1}|^{2}+\phi_{1}(\mathbf{r})|\psi_{1}|^{2}+\frac{\hbar ^{2}}{2m_{2}}|\
abla\psi_{2}|^{2}+\phi_{2}(\mathbf{r})|\psi_{2}|^{2}\right.\\ &\left.+\frac{1}{2}u_{11}|\psi_{1}|^{4}+\frac{1}{2}u_{22}|\psi_{2 }|^{4}+u_{12}|\psi_{1}|^{2}|\psi_{2}|^{2}\right],\end{split} \tag{2}\]where \(m_{i}\) (with \(i=1,2\)) is the mass of atomic species \(i\), \(\phi_{i}(\mathbf{r})\) is the corresponding external potential, \(u_{ii}=4\pi\hbar^{2}a_{ii}/m_{i}\) are the intra-species interaction terms and \(u_{12}=2\pi\hbar^{2}a_{12}/m_{12}\) is the inter-species interaction term with \(m_{12}=m_{1}m_{2}/(m_{1}+m_{2})\), the reduced mass of the system. For all relations, \(a_{ii}\) is the associated two-body \(s\)-wave scattering length. The wave-functions \(\psi_{1}(\mathbf{r})\) and \(\psi_{2}(\mathbf{r})\) are the condensate wave-function of each atomic species, defined as  
\[\psi_{1}(\mathbf{r})=\sqrt{N_{1}}\phi_{1}(\mathbf{r})\;\;\text{and}\;\;\psi_{2 }(\mathbf{r})=\sqrt{N_{2}}\phi_{2}(\mathbf{r}). \tag{3}\]  
Minimizing the energy functional of Equation (2) under the constraint of fixed number of particles, \(N_{1}\) and \(N_{2}\), one obtains the time-independent coupled Gross-Pitaevskii equations  
\[\left[-\frac{\hbar^{2}}{2m_{1}}\
abla^{2}+\phi_{1}(\mathbf{r})+u_{11}|\psi_{1 }|^{2}+u_{12}|\psi_{2}|^{2}\right]\psi_{1}=\mu_{1}\psi_{1} \tag{4}\]  
\[\left[-\frac{\hbar^{2}}{2m_{2}}\
abla^{2}+\phi_{2}(\mathbf{r})+u_{22}|\psi_{2 }|^{2}+u_{12}|\psi_{1}|^{2}\right]\psi_{2}=\mu_{2}\psi_{2}, \tag{5}\]  
where \(\mu_{1}\) and \(\mu_{2}\) are the chemical potential of atomic species 1 and 2, respectively. If the interspecies interaction vanishes (\(u_{12}=0\)), Equations (4) and (5) are no longer coupled and each species behave as a single species atomic cloud. In this case, approximations such as the Thomas-Fermi approximation [25; 26; 45], for which the kinetic term of the GPE is neglected, can be used to find a solution for the ground-state of the system. On the other hand, when \(u_{12}\
eq 0\), the competition between inter- and intraspecies interactions gives rise to a phase transition from a miscible to an immiscible (phase-separated) phase when increasing the positive inter-species interaction strength. The existence of overlapping and non-overlapping regions between the atomic clouds dramatically changes the ground-state configuration of the system and it is not always possible to find analytical solutions for it, even relying on approximations [46]. A more powerful technique to obtain the ground-state of a trapped two-component BEC makes use of a numerical simulation with imaginary time evolution of the coupled GPEs.