用低密度变分法研究玻色凝聚气体的非线性动力学
Nonlinear dynamics of Bose-condensed gases by means of a low- to high-density variational approach
摘要：
本文提出了一种研究一维磁阱玻色凝聚气体时空动力学的变分方法。为此，我们采用了\emph{q} -高斯试探波函数，该函数在玻色凝聚气体的基态的低密度极限和高密度极限之间插值。我们的主要结果包括减少的Gross-Pitaevskii方程，一个非线性偏微分方程描述的T=0动力学的凝聚，一组只有三个方程：\emph{two coupled nonlinear ordinary differential equations}描述的相位和曲率的波函数和\emph{a separate algebraic equation}产生的广义宽度。我们的方程恢复那些通常的高斯变分法（在低密度政权），和流体动力学方程描述的高密度政权。最后，我们给出了我们的方程和原来的Gross-Pitaevskii方程的数值结果之间的详细比较。
英文摘要：
We propose a versatile variational method to investigate the spatio-temporal dynamics of one-dimensional magnetically-trapped Bose-condensed gases. To this end we employ a \emph{q}-Gaussian trial wave-function that interpolates between the low- and the high-density limit of the ground state of a Bose-condensed gas. Our main result consists of reducing the Gross-Pitaevskii equation, a nonlinear partial differential equation describing the T=0 dynamics of the condensate, to a set of only three equations: \emph{two coupled nonlinear ordinary differential equations} describing the phase and the curvature of the wave-function and \emph{a separate algebraic equation} yielding the generalized width. Our equations recover those of the usual Gaussian variational approach (in the low-density regime), and the hydrodynamic equations that describe the high-density regime. Finally, we show a detailed comparison between the numerical results of our equations and those of the original Gross-Pitaevskii equation.
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