K．P 理論於玻色-愛因斯坦凝聚態在光晶格之應用

Abstract

我們利用K．P(等效質量)理論探討具有吸引力或排斥力的玻色-愛因斯坦凝聚態在光晶格之特性。 玻色-愛因斯坦凝聚態的波函數可由Gross-Pitaevskii方程式描述。 當玻色-愛因斯坦凝聚態的能量接近能帶邊界(band edge)時，其波函數就可藉由等效質量理論得到解析解，即布拉格函數乘上由等效質量方程式得到之波包函數。 利用這樣的波函數，我們探討亮孤子及暗孤子在能帶結構中所表現之特性，並依其波函數之特性分類為布拉格反射型及全反射型的孤子。 在推導等效質量方程式過程中，我們保留能帶(band edge energy)這一項，並驗證此項對於描述玻色-愛因斯坦凝聚態在光晶格有重要意義。 此外，我們利用數值解Gross-Pitaevskii方程式來驗證由等效質量理論推得的解析解之準確性，並與Eiermann等人的實驗結果做比較。 我們證實玻色-愛因斯坦凝聚態在光晶格之特性不但可以 ”定性” 更可 ”定量” 地運用等效質量理論來描述。

We apply the K．P (effective mass) theory to study the dynamics of Bose-Einstein condensates (BECs) in optical lattices with either attractive or repulsive atom interactions. The macroscopic condensate wave function is described by Gross-Pitaevskii (G-P) equation. Near band edge, we obtain the analytic condensate wave function which is deduced from the effective mass theory and is found to be a Bloch function modulated by a soliton envelope function of the effective mass equation. We demonstrate that bright and dark solitons, corresponding to energy in band gap and energy within band, respectively, can exist for both attractive and repulsive atom interactions and can be categorized as Bragg reflection and internal reflection type solitons. In deriving the effective mass equation, we preserve the band edge energy term and we show this term is important to describe BECs in optical lattices. Numerically solving the G-P equation confirms the analytic results that agree reasonably well with simulations as well as compared with the experimental results reported by Eiermann et al. We demonstrate that BECs in optical lattices can be described, qualitatively and quantitatively, by the effective-mass theory.