Edwards Mark, Krygier Michael, Seddiqi Hadayat, Benton Brandon, Clark Charles W
Department of Physics, Georgia Southern University, Statesboro, Georgia 30460-8031, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Nov;86(5 Pt 2):056710. doi: 10.1103/PhysRevE.86.056710. Epub 2012 Nov 26.
We present a method for approximating the solution of the three-dimensional, time-dependent Gross-Pitaevskii equation (GPE) for Bose-Einstein-condensate systems where the confinement in one dimension is much tighter than in the other two. This method employs a hybrid Lagrangian variational technique whose trial wave function is the product of a completely unspecified function of the coordinates in the plane of weak confinement and a Gaussian in the strongly confined direction having a time-dependent width and quadratic phase. The hybrid Lagrangian variational method produces equations of motion that consist of (1) a two-dimensional (2D) effective GPE whose nonlinear coefficient contains the width of the Gaussian and (2) an equation of motion for the width that depends on the integral of the fourth power of the solution of the 2D effective GPE. We apply this method to the dynamics of Bose-Einstein condensates confined in ring-shaped potentials and compare the approximate solution to the numerical solution of the full 3D GPE.
我们提出了一种方法，用于近似求解三维含时格罗斯-皮塔耶夫斯基方程（GPE），该方程用于描述玻色-爱因斯坦凝聚体系统，其中一维的约束比另外两维的约束要强得多。此方法采用了一种混合拉格朗日变分技术，其试探波函数是弱约束平面中坐标的完全未指定函数与强约束方向上具有随时间变化宽度和二次相位的高斯函数的乘积。混合拉格朗日变分方法产生的运动方程由以下两部分组成：（1）一个二维（2D）有效GPE，其非线性系数包含高斯函数的宽度；（2）一个宽度的运动方程，该方程取决于二维有效GPE解的四次方的积分。我们将此方法应用于限制在环形势中的玻色-爱因斯坦凝聚体的动力学，并将近似解与完整三维GPE的数值解进行比较。
