Effective Nonlinear Schrödinger Equations for Cigar-Shaped and Disk-Shaped Fermi Superfluids at Unitarity

In the case of tight transverse confinement (cigar-shaped trap) the threedimensional (3D) nonlinear Schrödinger equation, describing superfluid Fermi atoms at unitarity (infinite scattering length |a| → ∞), is reduced to an effective one-dimensional form by averaging over the transverse coordinates. The resultant effective equation is a 1D nonpolynomial Schrodinger equation, which produces results in good agreement with the original 3D one. In the limit of small and large fermion number N the nonlinearity is of simple power-law type. A similar reduction of the 3D theory to a two-dimensional form is also performed for a tight axial confinement (disk-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 cigarand disk-shaped superfluids our nonpolynomial Schrödinger equations are quite attractive for phenomenological application. PACS numbers: 05.30.Fk,71.10.Ay, 03.75.Ss,67.85.Lm ‡ [email protected]; URL: www.ift.unesp.br/users/adhikari § [email protected]; URL: www.padova.infm.it/salasnich Effective Nonlinear Schrödinger Equations at Unitarity 2