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We consider a coupled nonlinear Schr\"{o}dinger equations describing anatomic Bose-Einstein condensates coupled to a molecular condensates through thestimulated Raman adiabatic passage loaded in an external potential. Theexistence of dark state are investigated within the full parameter spaceaccounts for all the nonlinear collisions, together with the atom-moleculeconversion coupling and external potential. The results show that there are aclass of external potentials such as double-well, periodical and doubleperiodical, and harmonic potentials, where the exact dark solutions can beformed. Our results may raise the possibility of relative experiments for darkstate in the coupled atomic-molecular Bose-Einstein condensates.

Exact dark state solutions of the coupled atomic-molecular Bose-Einstein condensates in an external potential