  We study a one-dimensional interacting quantum liquid hosting a pair of
mobile impurities causing backscattering. We determine the effective retarded
interaction between the two impurities mediated by the liquid. We show that for
strong backscattering this interaction gives rise to resonances and
antiresonances in the finite-frequency mobility of the impurity pair. At the
antiresonances, the two impurities remain at rest even when driven by a (small)
external force. At the resonances, their synchronous motion follows the
external drive in phase and reaches maximum amplitude. Using a perturbative
renormalization group analysis in quantum tunneling across the impurities, we
study the range of validity of our model. We predict that these mechanical
antiresonances are observable in experiments on ultracold atom gases confined
to one dimension.
