Tunneling ionization time and position studied by backward propagation

We obtain tunneling ionization time and position with a novel backward-propagation method. In this method, the tunneling process is treated fully quantum-mechanically, and the resulting wave packet is then classically propagated backward in time to retrieve the initial configuration when the tunneling takes place. This method does not make any ad hoc assumptions or approximations. It is very powerful and gives highly differential information (correlated distribution of tunneling ionization time and position) regarding to the tunneling process. Using the backward-propagation method, we find that the tunneling ionization time is around pulse center, and the tunneling ionization position is close to but smaller than theoretical estimations.