  We derive a self-consistent time-dependent harmonic approximation for the
quantum sine-Gordon model out of equilibrium and apply the method to the
dynamics of tunnel-coupled one-dimensional Bose gases. We determine the time
evolution of experimentally relevant observables and in particular derive
results for the probability distribution of subsystem phase fluctuations. We
investigate the regime of validity of the approximation by applying it to the
simpler case of a nonlinear harmonic oscillator, for which numerically exact
results are available. We complement our self-consistent harmonic approximation
by exact results at the free fermion point of the sine-Gordon model.
