Wannier functions using a discrete variable representation for optical lattices

Abstract

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 26, 4269 (1982)], for a symmetric lattice with inversion symmetry, we construct Wannier functions as eigenstates of the position operators x̂, ŷ, and ẑ 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 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 model and show the dependence of these energies on lattice asymmetry.

Mott transition in a cavity-boson system: A quantitative comparison between theory and experiment

“…The alternative approaches are usually based on the mapping to the Bose-Hubbard model. Given the effective optical lattice potential, the Wannier functions can be estimated by different numerical methods, many of which are available for quantum optical systems [67][68][69][70]. The Wannier functions then allow the extraction of the Bose-Hubbard parameters t and U [cf.…”

Mott transition in a cavity-boson system: A quantitative comparison between theory and experiment

“…We describe here an alternative approach which is based on the mapping to the Bose-Hubbard model. Given the effective optical lattice potential, the Wannier functions can be estimated by different numerical methods, many of which are available for quantum optical systems [78][79][80][81]. The Wannier functions then allow the extraction of the Bose-Hubbard parameters t and U [cf.…”

Full reconstruction of acoustic wavefields by means of pointwise measurements

“…It can be done using the so-called discrete variable representation theory [33] (or shortly DVR), and the resulting set of functions χ j (z) can be referred to as DVR functions. DVR is extensively used in quantum mechanics in the context of multidimensional and/or many-body quantum calculations [33], [34], construction of atomic Wannier states in periodic potentials [35], and excitons in quantum dots [36]. Basically, the main idea of the DVR approach is to construct a basis in which the position operator (in our case z-coordinate operator) is diagonalized.…”