标题： 一维和二维对简并费米气体平均场描述的约化 显示英文标题

标题： One- and two-dimensional reductions of the mean-field description of degenerate Fermi gases

摘要： 我们研究在各种外部势场中（包括光学晶格）费米气体的集体行为，在平均场（流体动力学）描述框架下。 使用变分法，我们从一般的三维平均场方程中推导出一维和二维（1D 和 2D）情况的有效动力学方程。 相应的限制分别由具有圆柱对称性和平面对称性的势场提供。 所得方程是与束缚态局部横向尺寸方程耦合的非多项式薛定谔方程（NPSEs）。 数值模拟表明，由原始三维方程和有效低维方程产生的结果之间有密切的一致性。 我们在这些情况下考虑基态。 特别是，得到了有效二维非相互作用费米气体的解析解。 强调了一维和二维配置之间的差异。 最后，我们分析了在存在光学晶格的情况下，一维和二维束缚气体密度图案对限制势和光学晶格势强度以及决定非相同费米子之间相互作用强度的散射长度的依赖关系。 显示英文摘要

摘要： We study collective behavior of Fermi gases trapped in various external potentials, including optical lattices (OLs), in the framework of the mean-field (hydrodynamic) description. Using the variational method, we derive effective dynamical equations for the one- and two-dimensional (1D and 2D) settings from the general 3D mean-field equation. The respective confinement is provided by trapping potentials with the cylindrical and planar symmetry, respectively. The resulting equations are nonpolynomial Schr% \"{o}dinger equations (NPSEs) coupled to equations for the local transverse size of the trapped states. Numerical simulations demonstrate close agreement of results produced by the underlying 3D equation and the effective low-dimensional ones. We consider the ground state in these settings. In particular, analytical solutions are obtained for the effectively 2D non-interacting Fermi gas. Differences between the 1D and 2D configurations are highlighted. Finally, we analyze the dependence of the 1D and 2D density patterns of the trapped gas, in the presence of the OL, on the strengths of the confining and OL potentials, and on the scattering length which determines the strength of interactions between non-identical fermions.