Paul Saurabh, Tiesinga Eite
Joint Center for Quantum Information and Computer Science, Joint Quantum Institute and University of Maryland, Maryland 20742, USA.
Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, National Institute of Standards and Technology and University of Maryland, Gaithersburg, Maryland 20899, USA.
Phys Rev A (Coll Park). 2016 Sep;94(3). doi: 10.1103/PhysRevA.94.033606. Epub 2016 Sep 7.
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 S. Kivelson [Phys. Rev. B , 4269 (1982)], for a symmetric lattice with inversion symmetry, we construct Wannier functions as eigenstates of the position operators , , and 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 eigenstates 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 10 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 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 model and show the dependence of these energies on lattice asymmetry.
我们提出一种使用离散变量表示（DVR）的数值方法，用于构建在对称和非对称周期势的单位晶胞中局域化的实值万尼尔函数。我们将这些结果应用于寻找捕获在激光产生的光学晶格中的超冷原子的万尼尔函数。遵循S. Kivelson [《物理评论B》，4269（1982）]，对于具有反演对称性的对称晶格，我们将万尼尔函数构建为位置算符(x)、(y)和(z)的本征态，这些算符限制在属于一个或多个能带的单粒子布洛赫函数上。为确保万尼尔函数是实值的，我们使用均匀傅里叶网格DVR数值获得能带结构和实值本征态。然后，通过隧穿能量的比较，我们表明当使用双精度算法时，对于反演对称和非对称势，万尼尔函数的精度都优于10位有效数字。计算是针对每个单位晶胞具有双阱且沿(x)轴具有可调不对称性以及沿垂直方向具有单个正弦势的光学晶格进行的。每个单位晶胞内两个势阱处的局域函数以类似方式构建，但使用来自两个最低能带的单粒子解的叠加。我们最终使用这些局域基函数来确定玻色 - 哈伯德模型中的两体相互作用能，并展示这些能量对晶格不对称性的依赖性。
