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We study atomic Josephson junctions (AJJs) with one and two bosonic speciesconfined by a double-well potential. Proceeding from the second quantizedHamiltonian, we show that it is possible to describe the zero-temperature AJJsmicroscopic dynamics by means of extended Bose-Hubbard (EBH) models, whichinclude usually-neglected nonlinear terms. Within the mean-field approximation,the Heisenberg equations derived from such two-mode models provide adescription of AJJs macroscopic dynamics in terms of ordinary differentialequations (ODEs). We discuss the possibility to distinguish the Rabi,Josephson, and Fock regimes, in terms of the macroscopic parameters whichappear in the EBH Hamiltonians and, then, in the ODEs. We compare thepredictions for the relative populations of the Bose gases atoms in the twowells obtained from the numerical solutions of the two-mode ODEs, with thosederiving from the direct numerical integration of the Gross-Pitaevskiiequations (GPEs). Our investigations shows that the nonlinear terms of the ODEsare crucial to achieve a good agreement between ODEs and GPEs approaches, andin particular to give quantitative predictions of the self-trapping regime.

Nonlinear quantum model for atomic Josephson junctions with one and two bosonic species