  We present an experimental study on the non-equilibrium tunnel dynamics of
two coupled one-dimensional Bose-Einstein quasi-condensates deep in the
Josephson regime. Josephson oscillations are initiated by splitting a single
one-dimensional condensate and imprinting a relative phase between the
superfluids. Regardless of the initial state and experimental parameters, the
dynamics of the relative phase and atom number imbalance shows a relaxation to
a phase-locked steady state. The latter is characterized by a high phase
coherence and reduced fluctuations with respect to the initial state. We
propose an empirical model based on the analogy with the anharmonic oscillator
to describe the effect of various experimental parameters. A microscopic theory
compatible with our observations is still missing.
