For $a_{12}<-73.6a_0$, the mixture is stabilized by the LHY term and turns into a quantum droplet above a critical atom number [5].

The ratio of ${}^{41}\mathrm{K}$ and ${}^{87}\mathrm{Rb}$ atom numbers in the droplet is approximately constant and close to one [9]. We have typically $N_1\sim 0.4N_2$, this implies that all ${}^{41}\mathrm{K}$ is bound while a residual fraction of ${}^{87}\mathrm{Rb}$ remains unbound.

The system’s center of mass moves along the $x$ axis, mainly due to a shift between the initial droplet position and the center of the magnetic curvature. The images in Fig. 1 are recentered on the center of mass.

The critical ramping time for the onset of instability depends on $a_{12}$ and $N_1$. In the experiment, we have observed that a 25 ms interaction ramp allows us to explore the droplet breakup dynamics over a wide range of interactions.

For a filament along the $x$ axis, the classical PR instability assumes a perturbation of the form $R(x,t)=R_0+\delta e^{\omega (t-t_0)}\cos (kx)$, with $k=2\pi /\lambda$ the wave vector, $\delta$ the amplitude of the perturbation, and $\omega (k)$ the growth rate.

In the range of parameters used in the simulations, if the length of the filament is smaller than ${\lambda }_c$ before expanding, then we find $t^{*}\in [-2,0.5]\mathrm{ms}$, else we set $t^{*}=0\mathrm{ms}$.