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.
