Effective nonlinear Schrödinger equations for cigar-shaped and disc-shaped Fermi superfluids at unitarity

In the case of tight transverse confinement (cigar-shaped trap), the three-dimensional (3D) nonlinear Schrödinger equation, describing superfluid Fermi atoms at unitarity (infinite scattering length | a |→ ∞ ), is reduced to an effective 1D form by averaging over the transverse coordinates. The resultant effective equation is a 1D nonpolynomial Schrödinger equation, which produces results in good agreement with the original 3D one. In the limit of small and large fermion numbers N , the nonlinearity is of simple power-law type. A similar reduction of the 3D theory to a 2D form is also performed for a tight axial confinement (disc-shaped trap). The resultant effective 2D nonpolynomial equation also produces results in agreement with the original 3D equation and has simple power-law nonlinearity for small and large N . For both cigar- and disc-shaped superfluids, our nonpolynomial Schrödinger equations are quite attractive for phenomenological applications.