  We theoretically investigate the non-equilibrium dynamics in a quenched pair
of 1D Bose gases with density imbalance. We describe the system using its
low-energy effective theory, the Luttinger liquid model. In this framework the
system shows strictly integrable relaxation dynamics via dephasing of its
approximate many-body eigenstates. In the balanced case, this leads to the
well-known light-cone-like establishment of a prethermalized state, which can
be described by a generalized Gibbs ensemble. In the imbalanced case the
integrable dephasing leads to a state that, counter-intuitively, closely
resembles a thermal equilibrium state. The approach to this state is
characterized by two separate light-cone dynamics with distinct characteristic
velocities. This behavior is rooted in the fact that in the imbalanced case
observables are not aligned with the conserved quantities of the integrable
system. We discuss a concrete experimental realization to study this effect
using matterwave interferometry and many-body revivals on an atom chip.
