  Tunneling in a many-body system appears as one of the novel implications of
quantum physics, in which particles move in space under an otherwise
classically-forbidden potential barrier. Here, we theoretically describe the
quantum dynamics of the tunneling phenomenon of a few intricate bosonic clouds
in a closed system of a two-dimensional symmetric double-well potential. We
examine how the inclusion of the transverse direction, orthogonal to the
junction of the double-well, can intervene in the tunneling dynamics of bosonic
clouds. We use a well-known many-body numerical method, called the
multiconfigurational time-dependent Hartree for bosons (MCTDHB) method. MCTDHB
allows one to obtain accurately the time-dependent many-particle wavefunction
of the bosons which in principle entails all the information of interest about
the system under investigation. We analyze the tunneling dynamics by preparing
the initial state of the bosonic clouds in the left well of the double-well
either as the ground, longitudinally or transversely excited, or a vortex
state. We unravel the detailed mechanism of the tunneling process by analyzing
the evolution in time of the survival probability, depletion and fragmentation,
and the many-particle position, momentum, and angular-momentum expectation
values and their variances. As a general rule, all objects lose coherence while
tunneling through the barrier and the states which include transverse
excitations do so faster. Implications are briefly discussed.
