  We propose and investigate a pump-probe spectroscopy scheme to unveil the
time-resolved dynamics of fermionic or bosonic impurities immersed in a
harmonically trapped Bose-Einstein condensate. In this scheme a pump pulse
initially transfers the impurities from a noninteracting to a resonantly
interacting spin-state and, after a finite time in which the system evolves
freely, the probe pulse reverses this transition. This directly allows to
monitor the nonequilibrium dynamics of the impurities as the dynamical
formation of coherent attractive or repulsive Bose polarons and signatures of
their induced-interactions are imprinted in the probe spectra. We show that for
interspecies repulsions exceeding the intraspecies ones a temporal
orthogonality catastrophe occurs, followed by enhanced energy redistribution
processes, independently of the impurity's flavor. This phenomenon takes place
over the characteristic trap timescales. For much longer timescales a steady
state is reached characterized by substantial losses of coherence of the
impurities. This steady state is related to eigenstate thermalization and it is
demonstrated to be independent of the system's characteristics.
