We demonstrate that an ultracold many-body bosonic ensemble confined in a one-dimensional (1D) double-well (DW) potential can exhibit chaotic dynamics due to the presence of a single impurity. The non-equilibrium dynamics is triggered by a quench of the impurity-Bose interaction and is illustrated via the evolution of the population imbalance for the bosons between the two wells. While the increase of the post-quench interaction strength always facilitates the irregular motion for the bosonic population imbalance, it becomes regular again when the impurity is initially populated in the highly excited states. Such an integrability to chaos (ITC) transition is fully captured by the transient dynamics of the corresponding linear entanglement entropy, whose infinite-time averaged value additionally characterizes the edge of the chaos and implies the existence of an effective Bose-Bose attraction induced by the impurity. In order to elucidate the physical origin for the observed ITC transition, we perform a detailed spectral analysis for the mixture with respect to both the energy spectrum as well as the eigenstates. Specifically, two distinguished spectral behaviors upon a variation of the interspecies interaction strength are observed. While the avoided level-crossings take place in the low-energy spectrum, the energy levels in the high-energy spectrum possess a band-like structure and are equidistant within each band. This leads to a significant delocalization of the low-lying eigenvectors which, in turn, accounts for the chaotic nature of the bosonic dynamics. By contrast, those highly excited states bear a high resemblance to the non-interacting integrable basis, which explains for the recovery of the integrability for the bosonic species. Finally, we discuss the induced Bose-Bose attraction as well as its impact on the bosonic dynamics.
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