  We propose a minimal model to study the real-time dynamics of a
$\mathbb{Z}_2$ lattice gauge theory coupled to fermionic matter in a cold atom
quantum simulator setup. We show that dynamical correlators of the gauge fields
can be measured in experiments studying the time-evolution of two pairs of
impurities, and suggest the protocol for implementing the model in cold atom
experiments. Further, we discuss a number of unexpected features found in the
integrable limit of the model, as well as its extensions to a non-integrable
case. A potential experimental implementation of our model in the latter regime
would allow one to simulate strongly-interacting lattice gauge theories beyond
current capabilities of classical computers.
