标题： 用于光晶格的离散变量表示的Wannier函数 显示英文标题

标题： Wannier functions using a discrete variable representation for optical lattices

摘要： 我们提出一种数值方法，使用离散变量表示法（DVR）来构建在对称和非对称周期势中位于一个原胞内的实值Wannier函数。我们将这些结果应用于寻找被激光生成的光晶格束缚的超冷原子的Wannier函数。根据Kivelson\cite{kivelson_wannier_1982}，对于具有反演对称性的对称晶格，我们构造Wannier函数作为位置算符$\hat x$、$\hat y$和$\hat z$在属于一个或多个能带的单粒子布洛赫函数上的本征态。为了确保Wannier函数是实值的，我们使用均匀傅里叶网格DVR数值地获得能带结构和实值本征态。然后通过隧穿能量的比较，我们证明当使用双精度算术时，Wannier函数对于反演对称和非对称势的准确性优于十位有效数字。计算是在每个原胞内具有双势阱且沿$x$轴具有可调不对称性的光晶格以及沿垂直方向的单正弦势下进行的。每个原胞内的两个势能极小处的局域化函数同样被构造出来，但使用来自两个最低能带的单粒子解的叠加。最后，我们利用这些局域化基函数来确定玻色-哈伯德（BH）模型中的两体相互作用能，并展示这些能量与晶格不对称性的依赖关系。 显示英文摘要

摘要： We propose a numerical method using the discrete variable representation (DVR) for constructing real-valued Wannier functions localized in a unit cell for both symmetric and asymmetric periodic potentials. We apply these results to finding Wannier functions for ultracold atoms trapped in laser-generated optical lattices. Following Kivelson \cite{kivelson_wannier_1982}, for a symmetric lattice with inversion symmetry, we construct Wannier functions as eigen states of the position operators $\hat x$, $\hat y$ and $\hat z$ restricted to single-particle Bloch functions belonging to one or more bands. To ensure that the Wannier functions are real-valued, we numerically obtain the band structure and real-valued eigen states using a uniform Fourier grid DVR. We then show by a comparison of tunneling energies, that the Wannier functions are accurate for both inversion symmetric and asymmetric potentials to better than ten significant digits when using double-precision arithmetic. The calculations are performed for an optical lattice with double-wells per unit cell with tunable asymmetry along the $x$ axis and a single sinusoidal potential along the perpendicular directions. Localized functions at the two potential minima within each unit cell are similarly constructed, but using a superposition of single-particle solutions from the two lowest bands. We finally use these localized basis functions to determine the two-body interaction energies in the Bose-Hubbard (BH) model, and show the dependence of these energies on lattice asymmetry.