  We study the out-of-equilibrium dynamics of two interacting atoms in a
one-dimensional harmonic trap after a quench by a tightly pinned impurity atom.
We make use of an approximate variational calculation called the Lagrange-mesh
method to solve the Schr\"odinger equation as a function of inter-particle
interaction and impurity quench strength. We investigate the out-of-equilibrium
dynamics by calculating the Loschmidt echo which quantifies the irreversibility
of the system following the quench, while its probability distribution after
long times can be used to identify distinct dynamical regimes. These quantities
are related to the spectral function which describes the full dynamical
spectrum, and we show through a thorough examination of the parameter space the
existence of distinct scattering states and collective oscillations. This work
demonstrates how these dynamics are strongly dependent on the interaction
strength between the atoms and may be tuned to reach the orthogonality
catastrophe in few-body systems.
