玻色-愛因斯坦凝聚態在一維光晶格中的動力學

Abstract

我們應用Gross-Pitaevskii equation 來描述在一維光晶格中的玻色-愛因斯坦疑聚態，基於光晶格位能的週期性，我們用Bloch function 來展開秩序參數(order parameter)可得到封包函數(envelope function)的一維非線性方程式。接著我們引用K• P 微擾近似的方法(或等效質量理論) 在特定k0展開能量E(k)將群速度和等效質量帶入我的討論中。在Thomas- Fermi 近似和流體力學(hydrodynamic)的方法可以解得密度變化量的方程式。解此方程式可得激發模組頻率大小，發現與等效質量有關和群速的大小無關，群速度只會影響密度變化的大小。 我們也解了在自由擴散中的隨時間變化的密度函數, 我們也展示在自由擴散時密度和流量密度的變化情形。自由擴散的這些推導是在由流體力學和非線性光學中的方法而得知的。

We applied the Gross-Pitaevskii equation for BEC condensate in 1D optical lattice. For the character of the periodic potential from optical lattice, the Bloch function is used to expand our order parameter to get a 1D nonlinear equation of the envelop function. We also introduce the K • P perturbation theory (effective mass theory) to expand the energy E to specific point k0 and bring the group velocity and effective mass into our discussion. To solve the nonlinear differential equation of envelope function we assume the Thomas Fermi approximation and use the hydrodynamic theory to get the equation for the density fluctuation. We solve the equation and get frequency of the excitation mode and find out that the frequency depends on the effective mass and is not influenced by the group velocity. The group velocity will influence only the change of density. The time dependent density of BEC in free expansion is also solved and we show how the density and the current density vary in free expansion. They are derived under the hydrodynamics with another method from nonlinear optics.