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arXiv:1112.2845 · New Journal of Physics

Maximally localized Wannier functions for ultracold atoms in one-dimensional double-well periodic potentials

Michele Modugno, Giulio Pettini

Published May 1, 2012• 22 citations
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Abstract

We discuss a method for constructing generalized Wannier functions that are maximally localized at the minima of a one-dimensional periodic potential with a double well per unit cell. By following the approach of Marzari and Vanderbilt (1997 Phys. Rev. B 56 12847), we consider a set of band-mixing Wannier functions with minimal spread and design a specific two-step gauge transformation of the Bloch functions for a composite two-band system. This method is suited for efficiently computing the tight-binding coefficients needed for mapping the continuous system to a discrete lattice model. The behaviour of the tight-binding coefficients is analyzed here as a function of the symmetry properties of the double well (including the possibility of parity-breaking), in a range of feasible experimental parameters.

DOI
10.1088/1367-2630/14/5/055004

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We discuss a method for constructing generalized Wannier functions that are maximally localized at the minima of a one-dimensional periodic potential with a double well per unit cell. By following the approach of Marzari and Vanderbilt (1997 Phys. Rev. B 56 12847), we consider a set of band-mixing Wannier functions with minimal spread and design a specific two-step gauge transformation of the Bloch functions for a composite two-band system. This method is suited for efficiently computing the tight-binding coefficients needed for mapping the continuous system to a discrete lattice model. The behaviour of the tight-binding coefficients is analyzed here as a function of the symmetry properties of the double well (including the possibility of parity-breaking), in a range of feasible experimental parameters.
We discuss how to construct tight-binding models for ultracold atoms in honeycomb potentials, by means of the maximally localized Wannier functions (MLWFs) for composite bands introduced by Marzari and Vanderbilt [Phys. Rev. B 56, 12847 (1997)]. In particular, we work out the model with up to third-nearest neighbors, and provide explicit calculations of the MLWFs and of the tunneling coefficients for the graphenelike potential with two degenerate minima per unit cell. Finally, we discuss the degree of accuracy in reproducing the exact Bloch spectrum of different tight-binding approximations, in a range of typical experimental parameters.
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 $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}, \stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}$, and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ 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 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 xxx 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.
Abstract We obtain ab initio the Hubbard parameters for Rydberg-dressed atoms in a one-dimensional (1D) sinusoidal optical lattice on the basis of maximally-localized Wannier states. Finite range, soft-core interatomic interactions become the trait of Rydberg admixed atoms, which can be extended over many neighboring lattice sites. In contrast to dipolar gases, where the interactions follow an inverse cubic law, the key feature of Rydberg-dressed interactions is the possibility of making neighboring couplings to the same magnitude as that of the onsite ones. The maximally-localized Wannier functions (MLWFs) are typically calculated via a spread-minimization procedure (Marzari N and Vanderbilt D 1997 Phys. Rev. B 56 12847 ) and are always found to be real functions apart from a trivial global phase when an isolated set of Bloch bands are considered. For an isolated single Bloch band, the above procedure reduces to a simple quasi-momentum-dependent unitary phase transformation. Here, instead of minimizing the spread, we employ a diagonal phase transformation which eliminates the imaginary part of the Wannier functions. The resulting Wannier states are found to be maximally localized and in exact agreement with those obtained via a spread-minimization procedure. Using these findings, we calculate the Hubbard couplings from the Rydberg admixed interactions, including dominant density-assisted tunneling (DAT) coefficients. Finally, we provide realistic lattice parameters for the state-of-the-art experimental Rydberg-dressed rubidium setup.
We review the derivation of the effective Dirac equation for ultracold atoms in one-dimensional bichromatic optical lattices, following the proposal by Witthaut et al. [Phys. Rev. A 84, 033601 (2011)]. We discuss how such a derivation---based on a suitable rotation of the Bloch basis and on a coarse-graining approximation---is affected by the choice of the Wannier functions entering the coarsening procedure. We show that in general the Wannier functions obtained by rotating the maximally localized Wannier functions for the original Bloch bands can be sufficiently localized for justifying the coarse-graining approximation. We also comment on the relation between the rotation needed to achieve the Dirac form and the standard Foldy-Wouthuysen transformation. Our results provide a solid ground for the interpretation of the experimental results by Salger et al. [Phys. Rev. Lett. 107, 240401 (2011)] in terms of an effective Dirac dynamics.
We investigate the localization properties of gapped periodic quantum systems, modeled by a periodic or covariant family of projectors, as e.g. the orthogonal projectors on the occupied orbitals at fixed crystal momentum for a gas of non-interacting electrons. We prove a general localization dichotomy for dimension d≤3d\leq 3d≤3: either the system is topologically trivial i.e. all the Chern numbers vanish, or any arbitrary choice of composite Wannier functions yields an infinite expectation value for the squared position operator. Equivalently, in the topologically non-trivial phase, the localization functional introduced by Marzari and Vanderbilt diverges, as already noticed in the case of the Haldane model. Our result is formulated by using only the relevant symmetries of the system, and it is thus largely model-independent. Possible applications include both tight-binding and continuous models of crystalline solids, cold gases in optical lattices as well as flat band superconductivity.
We derive ab initio local Hubbard models for several optical-lattice potentials of current interest, including the honeycomb and kagome lattices, verifying their accuracy on each occasion by comparing the interpolated band structures against the originals. To achieve this, we calculate the maximally localized generalized Wannier basis by implementing the steepest-descent algorithm of Marzari and Vanderbilt [Phys. Rev. B 56, 12847 (1997)] directly in one and two dimensions. To avoid local minima we develop an initialization procedure that is both robust and requires no prior knowledge of the optimal Wannier basis.