## 4 Dynamics
### Damped Monopole Oscillations  
Given the above picture, in the following we shall investigate the self evaporation dynamics for \(a_{12}=-82a_{0}\), where the droplet half-life is larger than 50 ms [see Figure 2b]. In particular, we consider two different configurations with \(N_{2}=4\times 10^{4}\) and \(N_{2}=1.2\times 10^{5}\), indicated by the yellow circles in Figures 1 and 2b, both lying in the self-evaporation regime. It is worth to remark that in the regime where the monopole mode is expected to be stable (see Figure 1) the droplet is affected by severe three-body losses (see Figure 2), which make the detection of this mode unfeasible. In order to distinguish between the droplet density distribution and the distribution of atoms that evaporate, we define the _droplet volume_ as that contained within a certain radius \(R_{d}\). Then, the various droplet properties can be easily computed by integrating within the droplet volume. In the present case, this radius can be conveniently fixed to \(R_{d}=8\) um, which is sufficiently large to include both the bulk and the droplet tails. The numerical simulations of the GP equations are performed on a computational box that is  
Figure 2: Heat map of \(t_{1/2}\) as a function of \(a_{12}\) and \(N_{2}(0)\). (**a**) Prediction of the analytical model, see Equation (18); (**b**) values extracted from the actual decay of \(N_{2}(t)\) obtained from the solution of the GP equations in (13). The color scale is saturated at \(t_{1/2}=100\) ms (dark blue). The dashed-dotted line corresponds to \(t_{1/2}=50\) ms. The two circles at \(a_{12}=-82a_{0}\) refer to the parameter configurations considered in the GP simulations of Section 4.2.  
at least one order of magnitude larger than \(R_{d}\). Accordingly, with \(N_{i}^{d}(t)\) we indicate the number of atoms of each species (\(i=1,2\)) within the droplet volume, at time \(t\). Then, we define a _running_ value of \(\widetilde{N}\) as [28]  
\[\widetilde{N}_{R}(t)\equiv\frac{1}{\xi^{3}}\frac{N_{1}^{d}(t)+N_{2}^{d}(t)}{N_ {1}+N_{2}}=k\sum_{i=1}^{2}N_{i}^{d}(t), \tag{19}\]  
where \(k\simeq 4.41\times 10^{-3}\) for the current values of the scattering lengths. The evolution of the system is shown in in Figure 3, where we plot the rms width \(\sigma(t)\) of the droplet, the running value of \([\widetilde{N}_{R}(t)-N_{c}]^{1/4}\), the ratio \(N_{1}^{d}(t)/N_{2}^{d}(t)\), and the fraction of evaporated atoms for each species \(N_{i}^{ev}(t)/N_{i}\), with and without three-body losses. In both cases the mixture is initially prepared in the ground state of a dipole trap of frequency \(\omega_{2}=2\pi\times 50\) Hz (we recall that \(\omega_{1}=\omega_{2}\sqrt{m_{2}/(\alpha m_{1})}\), see Section 3; in the present case \(\omega_{1}=2\pi\times 67.9\) Hz). Let us first focus on the clean case, in the absence of three-body losses. As expected, we find that in both the investigated cases the droplet width performs damped sinusoidal oscillations (corresponding to a damped monopole mode), and the system eventually relaxes to an equilibrium configuration [see Figure 3a, along with the top panels], corresponding to a smaller, stationary value of \(\widetilde{N}_{R}(t)\), see panels (b). This decrease in the number of particles is a consequence of the self-evaporation mechanism [1; 28], which takes place within the first 15 ms. Looking at panel (d), where we plot the fraction of atoms of each species that are lost by self-evaporation, it is clear that the values of \(N_{i}^{ev}(t)/N_{i}\) soon reach their asymptotic value (modulo small fluctuations). In addition, also the ratio between the atom numbers in  
Figure 3: Evolution of (**a**) the droplet width \(\sigma(t)\), (**b**) the running value of \(\widetilde{N}_{R}(t)\) (see text), (**c**) the ratio \(N_{1}^{d}(t)/N_{2}^{d}(t)\), and (**d**) the fraction of evaporated atoms for each species \(N_{i}^{ev}(t)/N_{i}\), after the release from a trap of frequency \(\omega_{2}=2\pi\times 50\) Hz, with and without three-body losses.