玻色愛因斯坦凝聚於光晶格中靠近零色散點的行為

Abstract

我們使用一個方法去探討玻色-愛因斯坦凝聚態在光晶格中靠近零色散點附近的行為。玻色-愛因斯坦凝聚態可由一維的Gross-Pitaevski 方程式來描述其動力學行為，並且可以應用等效質量法將之以非線性薛丁格方程來描述。當考慮零色散點附近的行為時即第二階色散效應趨近於零,必須考慮第三階的色散效應。此時玻色-愛因斯坦凝聚態將由廣義的非線性薛丁格方程來描述，我們利用由逆散射法之非線性薛丁格方程的解在小振幅的限制下找出一個假設解解出 廣義的非線性薛丁格方程(即考慮第三階色散效應)之進似解析解，我們得到一個可以存在大部份區域之暗孤子以及在一個特殊區域時暗孤子將轉成光孤粒子的解,即所謂的在背景上的孤立子。此外，我們利用直接的模擬數數值解來觀察其解析解的存在性。同時,在數值解我們可以觀察到當在較大的振幅時將產生一個輻射衰退的效應,此效應被認為是由於第三階色散的影響。

We demonstrate a method to analytically study the effective-mass method of Bose-Einstein Condensates (BEC) in optical lattices near the zero dispersion(Z-D) point where the effect of the second-order dispersion is zero. We use one dimensional Gross-Pitaevskii (G-P) equation describes the dynamic behavior of BEC to the optical lattices. By using effective-mass theory to our system in theneighborhood of the Z-D point we need to consider the third-order dispersion term to our equation. That is, our system is described by the generalized NLS equation.We take the dark-soliton solution form of the NLS equation solved by inverse scattering method in the small amplitude limit as anassumed solution to substitute into our equation. We obtain dark solitons solution may exist near the Z-D point and we also show a region near the Z-D point where a special solitary wave form, the so-called soliton on the constant background, may be observed. We use directly numerical simulations of the full generalized NLS equation which includes the third-order dispersion term to observe that the existence of the new solitary wave form. Numerical computation also shows that a radiation emission exits near the Z-D point in the larger amplitude, which is regarded as the effect of three-order dispersion