Formation of dark state in coupled atomic-molecular Bose–Einstein condensates with an external potential

We consider three coupled nonlinear Schrödinger equations describing atomic Bose–Einstein condensates coupled to a molecular condensate through the stimulated Raman adiabatic passage loaded in an external potential. Using a combination of the homogeneous balance principle and sub-equation expansion technique, we obtain three families of exact stationary dark state solutions for this coupled system. The results show that there exist different trapping potentials, such as double-well, periodic and double-periodic, and harmonic potentials, where the exact dark solutions can be formed via a proper choice of the nonlinear parameters. Finally, the stability of the dark state solutions are checked by direct numerical simulations.