  The intra-band tunneling of a Bose-Einstein condensate between three
degenerate high-symmetry X-points of the Brillouin zone of a cubic optical
lattice is studied in the quantum regime by reduction to a three-mode model.
The mean-field approximation of the deduced model is described. Compared to the
previously reported two-dimensional (2D) case [Phys. Rev. A 75, 063628 (2007)],
which is reducible to the two-mode model, in the case under consideration there
exist a number of new stable stationary atomic distributions between the
X-points and a new critical lattice parameter. The quantum collapses and
revivals of the atomic population dynamics are absent for the experimentally
realizable time span. The 2D stationary configurations, embedded into the 3D
lattice, turn out to be always unstable, while existence of a stable 1D
distribution, where all atoms populate only one X-state, may serve as a
starting point in the experimental study of the nonlinear tunneling in the 3D
lattice.
