  The properties and experimental identification of quantum spin liquids (QSL)
remains an important topic with many fundamental questions. Here, we explore
the dynamics of a single charge dopant (hole) in a $t$-$J_{1}$-$J_{2}$ model on
a square lattice, which realises a gapless $\mathbb{Z}_2$ QSL at half-filling.
Using a field theory approach based on the parton construction, which includes
an infinite number of scatterings between the low-energy quasiparticle
excitations of the QSL via a self-consistent Born approximation, we calculate
both the equilibrium and non-equilibrium properties of the hole for weak and
strong interactions. Quasiparticle branches as well as string-like excitations
of the hole are identified, and we furthermore explore the time-dependent
spreading of a hole throughout the QSL after it has been injected at a given
lattice site. The final ballistic expension speed is shown to exhibit a
non-monotonic behavior as a quantum phase transition between an
antiferromagnetic (AFM) and the QSL phase is crossed, which is caused by a
qualitative change in the fundamental kinematics of the interactions between
the hole and the surrounding spins. Our results demonstrate how charge dopants
can be used as a quantum probe for QSLs and are directly relevant to optical
lattice experiments with single site resolution, as well as the rapidly
increasingly number of new quanum materials such as transition metal
dichacogenides.
