  Motivated by a recent experiment with optical lattices that has realized a
ladder geometry with an effective magnetic field (Atala et al., Nature Physics
\textbf{10}, 588 (2014)), we study the dynamics of bosons on a tight-binding
two-leg ladder with complex hopping amplitudes. This system displays a quantum
phase transition even without interactions. We study the non-equilibrium
dynamics without and with interactions, in the presence of a harmonic trapping
potential. In particular we consider dynamics induced by quenches of the
trapping potential and of the magnitude of the rung hopping. We present a
striking "slowing down" effect in the collective mode dynamics near the phase
transition. This manifestation of a slowing down phenomenon near a quantum
phase transition can be visualized unusually directly: the collective mode
dynamics can be followed experimentally in real time and real space by imaging
the atomic cloud.
