Dimensional reduction of a binary Bose-Einstein condensate in mixed dimensions

We present effective reduced equations for the study of a binary Bose-Einstein condensate (BEC), where the confining potentials of the two BEC components have distinct asymmetry so that the components belong to different space dimensions as in a recent experiment [ G. Lamporesi et al. Phys. Rev. Lett. 104 153202 (2010)]. Starting from a binary three-dimensional (3D) Gross-Pitaevskii equation (GPE) and using a Lagrangian variational approach we derive a binary effective nonlinear Schrödinger equation with components in different reduced dimensions, for example, the first component in one dimension and the second in two dimensions as appropriate to represent a cigar-shaped BEC coupled to a disk-shaped BEC. We demonstrate that the effective reduced binary equation, which depends on the geometry of the system, is quite reliable when compared with the binary 3D GPE and can be efficiently used to perform numerical simulation and analytical calculation for the investigation of static and dynamic properties of a binary BEC in mixed dimensions.