  In this work we consider the non-equilibrium dynamics of two tunnel coupled
bosonic gases which are created from the coherent splitting of a
one-dimensional gas. The consequences of the tunneling both in the
non-stationary regime as well as at large time are investigated and compared
with equilibrium results. In particular, within a semiclassical approximation,
we compute correlation functions for the relative phase which are
experimentally measurable and we observe a transient regime displaying
oscillations as a function of the distance. The steady regime is very well
approximated by a thermal state with a temperature independent of the tunneling
strength.
