  We explore the time evolution of two impurities in a trapped one-dimensional
Bose gas that follows a change of the boson-impurity interaction. We study the
induced impurity-impurity interactions and their effect on the quench dynamics.
In particular, we report on the size of the impurity cloud, the
impurity-impurity entanglement, and the impurity-impurity correlation function.
The presented numerical simulations are based upon the variational multilayer
multiconfiguration time-dependent Hartree method for bosons. To analyze and
quantify induced impurity-impurity correlations, we employ an effective
two-body Hamiltonian with a contact interaction. We show that the effective
model consistent with the mean-field attraction of two heavy impurities
explains qualitatively our results for weak interactions.
  Our findings suggest that the quench dynamics in cold-atom systems can be a
tool for studying impurity-impurity correlations.
