  We study the motion of holes in a mixed-dimensional setup of an
antiferromagnetic ladder, featuring nearest neighbor hopping $t$ along the
ladders and Ising-type spin interactions along, $J_\parallel$, and across,
$J_\perp$, the ladder. We determine exact solutions for the low-energy one- and
two-hole eigenstates. The presence of the trans-leg spin coupling, $J_\perp$,
leads to a linear confining potential between the holes. As a result, holes on
separate legs feature a super-linear binding energy scaling as $(J_\perp /
t)^{2/3}$ in the strongly correlated regime of $J_\perp,J_\parallel \leq t$.
This behavior is linked to an emergent length scale $\lambda \propto
(t/J_\perp)^{1/3}$, stemming from the linear confining potential, and which
describes how the size of the two-hole molecular state diverges for
$J_\perp,J_\parallel \ll t$. On the contrary, holes on the same leg unbind at
sufficiently low spin couplings. This is a consequence of the altered
short-range boundary condition for holes on the same leg, yielding an effective
Pauli repulsion between them, limiting their kinetic energy and making binding
unfavorable. Finally, we determine the exact nonequilibrium quench dynamics
following the sudden immersion of initially localized nearest neigbhor holes.
The dynamics is characterized by a crossover from an initial ballistic quantum
walk to an aperiodic oscillatory motion around a finite average distance
between the holes due to the confining potential between them. In the strongly
correlated regime of low spin couplings, $J_\perp, J_\parallel \leq t$, we find
this asymptotic distance to diverge as $t / J_\perp$, showing a much stronger
scaling than the eigenstates. The predicted results should be amenable to
state-of-the-art quantum simulation experiments using currently implemented
experimental techniques.
