## 3 Preparation of the Initial State  
Following Ref. [14], we initially prepare the atomic mixture in the ground state of an optical dipole trap. In order to simplify the discussion, and motivated by the fact that quantum droplets at equilibrium are spherically symmetric objects, we assume (i) that the two condensates are initially prepared in a spherically symmetric potential \(U_{j}^{d}(r)\) (\(j=1,2\)), (ii) that the differential vertical gravitational sag (due to the different masses of the two atomic species) can be exactly compensated, and (iii) that equilibrium droplets can be smoothly prepared without any appreciable effect of three body losses [14]. With these assumptions, which will be maintained throughout this work for easiness of calculations and conceptual clarity, both the ground state and the dynamical evolution can be obtained by solving spherically symmetric equations. We remark that this approach restricts the analysis to the excitations of the monopole mode only. Indeed, the excitation of surface modes (with angular momentum \(\ell\
eq 0\)), which arise when the condensate is prepared in an asymmetric trap, is ruled out by the assumption of spherical symmetry. However, the presence of these modes is not expected to modify qualitatively the picture resulting from the following discussion. The ground state of the system is obtained by minimizing the energy functional in Equation (1) by means of a steepest-descent algorithm [32]. Formally, this corresponds to solve the following set of stationary Gross-Pitaevskii (GP) equations for two wave functions \(\psi_{j}\)[33]  
\[\left\{ \begin{aligned} &\left[-\frac{\hbar^{2}}{2m_{1}}\
abla_{r}^{2}+U_{1 }^{d}(r)+\mu_{1}(n_{1},n_{2})\right]\psi_{1}=\mu_{1}\psi_{1}\\ &\left[-\frac{\hbar^{2}}{2m_{2}}\
abla_{r}^{2}+U_{2}^{d}(r)+\mu_ {2}(n_{1},n_{2})\right]\psi_{2}=\mu_{2}\psi_{2},\end{aligned} \right. \tag{9}\]  
where \(\
abla_{r}^{2}\) represents the radial component of the Laplacian:  
\[\
abla_{r}^{2}=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{ \partial}{\partial r}\right)=\frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r} \frac{\partial}{\partial r}. \tag{10}\]  
The local chemical potentials \(\mu_{i}(n_{1},n_{2})\) include both the usual mean-field term and the Lee-Huang-Yang (LHY) correction [1], namely  
\[\mu_{i}(n_{1},n_{2})=g_{ii}n_{i}+g_{12}n_{j}+\frac{\delta E_{LHY}}{\delta n_{ i}},\quad i\
eq j, \tag{11}\]  
with \(n_{i}(r)\equiv|\psi_{i}(r)|^{2}\). We recall that droplets form in the regime of parameters where the mean-filed term alone would lead to the collapse of the mixture, that is instead avoided thanks to the repulsive effect of quantum fluctuations. As for the optical dipole potential experienced by the two atomic species, it can be written as \(U_{j}^{d}(r)=U_{0j}I(r)\), where \(U_{0j}\) is species-dependent (\(j=1,2\)) and it depends on the atomic species polarizability and the wavelength of the optical potential, and \(I(r)\) represents the intensity of the laser beam. For the atomic mixture \({}^{41}\)K-\({}^{87}\)Rb and a wavelength \(\lambda=1064\) nm [14] one has \(\alpha\equiv U_{02}/U_{01}\simeq 1.15\). Then, in the harmonic approximation, we have  
\[U_{j}^{d}(r)=\frac{1}{2}m_{j}\omega_{j}^{2}r^{2}, \tag{12}\]  
with \(\omega_{1}=\omega_{2}\sqrt{m_{2}/(\alpha m_{1})}\). The value of the trap frequency can be tuned in order to produce the desired amount of initial excitations. For low enough \(\omega_{j}\), the droplet will be prepared close to the equilibrium configuration in free space, whereas a tight confinement obviously corresponds to the excitation of compressional modes. In this paper we shall use \(\omega_{2}=2\pi\times 50\) Hz, which produces a not too large excitation of the monopole mode (see later on). The other free parameters of the system are the inter-species scattering length \(a_{12}\), that we will consider in the range \(a_{12}\in[-95,-73.6]a_{0}\), below to the mean-field collapse threshold, and the number of atoms of the two species.