  We study atomic Josephson junctions (AJJs) with one and two bosonic species
confined by a double-well potential. Proceeding from the second quantized
Hamiltonian, we show that it is possible to describe the zero-temperature AJJs
microscopic dynamics by means of extended Bose-Hubbard (EBH) models, which
include usually-neglected nonlinear terms. Within the mean-field approximation,
the Heisenberg equations derived from such two-mode models provide a
description of AJJs macroscopic dynamics in terms of ordinary differential
equations (ODEs). We discuss the possibility to distinguish the Rabi,
Josephson, and Fock regimes, in terms of the macroscopic parameters which
appear in the EBH Hamiltonians and, then, in the ODEs. We compare the
predictions for the relative populations of the Bose gases atoms in the two
wells obtained from the numerical solutions of the two-mode ODEs, with those
deriving from the direct numerical integration of the Gross-Pitaevskii
equations (GPEs). Our investigations shows that the nonlinear terms of the ODEs
are crucial to achieve a good agreement between ODEs and GPEs approaches, and
in particular to give quantitative predictions of the self-trapping regime.
