The scope of the talk is to deliver recent insights on the mechanism of the tunneling process of a bosonic quantum many-body system through a potential barrier to open space (see http://arxiv.org/abs/1202.3447). The discussion of the physics will include the assessment of the dynamics of the coherence and condensation in the tunneling process and its manifestation in a very characteristic momentum distribution. The complete picture of the many-body process was obtained from numerically exact solutions of the underlying time-dependent Schroedinger equation with the multiconfigurational time-dependent Hartree for Bosons (MCTDHB) method -- the currently most efficient numerical algorithm for this purpose. The key ideas and implementation of the method will be introduced. | 
