  We study the quantum dynamics of a BEC condensate trapped in a double-well
potential with a rising interwell barrier. We analytically find the
characteristic time scales of the splitting process and compare our results
with numerical analyses available in the literature. In first stage of the
dynamics, the condensate follows adiabatically the rising of the interwell
barrier. At a critical time $t_{ad}$, small amplitude fluctuations around the
average trajectory increase exponentially fast, signaling the break-down of
adiabaticity. We have found a highly non-trivial dependence of the dephasing
time $t_D$, defined by $\sigma_{\phi}(t_D)=1$, where $\sigma_{\phi}(t)$ is the
dynamical quantum phase spreading, on $t_{ad}$ and on the ramping time of the
interwell barrier.
