The core-shell state is obtained from a rotationally invariant initial state. During the imaginary time evolution, it always preserves $SO(3)$ rotational symmetry due to the same symmetry of Hamiltonian. In comparison, the dimer state is from a symmetry-broken initial state where (1, 2) and (2, 3) droplets are arranged side by side.

Should the system be unstable against any density fluctuation, its associated mode will develop an imaginary part such that the fluctuation grows exponentially with time to destabilize the system. For all of the atom numbers considered in our Letter, we do not observe such instability.

Note that a field gradient on component-2 can be gauged away in the moving frame of this component, so only the field gradients on two of the components (here 1 and 3) are independent.

Mode matching requires that the densities of initial (in a trap) and final (without a trap) states match each other as much as possible. If this requirement is violated, there will be a huge amount of internal energy to release during the dynamics, leading to complex dynamical outcomes. In [2], the initial and final states are substantially different, as seen from the switched core and shell components therein. This may be due to the combined effect of tight trapping potential and large $N_{1,2}$ taken in [2], such that (1, 2) prefers to stay in the trap center (core) to avoid large potential energy and accordingly (2,3) becomes the shell.