  We analyze charge-$e/4$ quasiparticle tunneling between the edges of a point
contact in a non-Abelian model of the $\nu=5/2$ quantum Hall state. We map this
problem to resonant tunneling between attractive Luttinger liquids and use the
time-dependent density-matrix renormalization group (DMRG) method to compute
the current through the point contact in the presence of a {\it finite voltage
difference} between the two edges. We confirm that, as the voltage is
decreases, the system is broken into two pieces coupled by electron hopping. In
the limits of small and large voltage, we recover the results expected from
perturbation theory about the infrared and ultraviolet fixed points. We test
our methods by finding the analogous non-equilibrium current through a point
contact in a $\nu=1/3$ quantum Hall state, confirming the Bethe ansatz solution
of the problem.
