  We unravel the correlated quantum quench dynamics of a single impurity
immersed in a bosonic environment confined in an one-dimensional double-well
potential. A particular emphasis is placed on the structure of the time-evolved
many-body wave function by relying on a Schmidt decomposition whose
coefficients directly quantify the number of configurations that are
macroscopically populated. For a non-interacting bosonic bath and weak
postquench impurity-bath interactions, we observe the dynamical formation of a
two-fold fragmented many-body state which is related to intra-band excitation
processes of the impurity and manifests as a two-body phase separation
(clustering) between the two species for repulsive (attractive) interactions.
Increasing the postquench impurity-bath coupling strength leads to the
destruction of the two-fold fragmentation since the impurity undergoes
additional inter-band excitation dynamics. By contrast, a weakly interacting
bath suppresses excitations of the bath particles and consequently the system
attains a weakly fragmented many-body state. Our results explicate the
interplay of intra- and inter-band impurity excitations for the dynamical
generation of fragmented many-body states in multi-well traps and for designing
specific entangled impurity states.
