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arXiv:1609.00654 · Physical Review A

Wannier functions using a discrete variable representation for optical lattices

Saurabh Paul, Eite Tiesinga

Published Sep 7, 2016• 10 citations
arXivHTMLPDF

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 $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}, \stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}$, 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 $x$ 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.

DOI
10.1103/physreva.94.033606

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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 x\ifmmode^\elseˆ\fi,y\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}, \stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}x\ifmmode^\elseˆ\fi,y\ifmmode^\elseˆ\fi​, and z\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}z\ifmmode^\elseˆ\fi 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.
In systems with periodic potential fields, building relatively local Wannier functions can significantly simplify the Hamiltonian and enhance our understanding of the system's ground state and dynamic properties. In this work, we improve the current method of building the Wannier functions of ultracold atomic systems, including the case in the presence or absence of interactions. In noninteracting systems, we propose a method to directly obtain the real-valued maximally localized Wannier functions (MLWFs) by using real-valued eigenstates, and verify the effectiveness of this method in a two-dimensional (2D) degenerate system. In interacting systems, we obtain the effect of high-energy bands on the lowest-energy band by using the accurate calculation results of the two-particle system. In the two-particle system, we consider the effect of the entanglement between the particles and obtain the optimal two-particle Wannier functions. These Wannier functions are then further utilized to obtain the parameters of the extended Bose-Hubbard model. The effectiveness of the method is verified by taking a one-dimensional (1D) system with contact interaction as an example. In the three-particle and four-particle systems, compared calculation results with the original system and the unmodified two-band Bose-Hubbard model, we find that the effective Hamiltonian is more accurate than the unmodified two-band model. This verifies the effectiveness of our method, and the parameters obtained can reflect the original system well, which provides an effective method for accurate modeling of interacting systems.
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.
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 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.
Experiments with ultracold atoms in optical lattice have become a versatile testing ground to study diverse quantum many-body Hamiltonians. A single-band Bose-Hubbard (BH) Hamiltonian was first proposed to describe these systems in 1998 and its associated quantum phase-transition was subsequently observed in 2002. Over the years, there has been a rapid progress in experimental realizations of more complex lattice geometries, leading to more exotic BH Hamiltonians with contributions from excited bands, and modified tunneling and interaction energies. There has also been interesting theoretical insights and experimental studies on “un- conventional” Bose-Einstein condensates in optical lattices and predictions of rich orbital physics in higher bands. In this thesis, I present our results on several multi- band BH models and emergent quantum phenomena. In particular, I study optical lattices with two local minima per unit cell and show that the low energy states of a multi-band BH Hamiltonian with only pairwise interactions is equivalent to an effec- tive single-band Hamiltonian with strong three-body interactions. I also propose a second method to create three-body interactions in ultracold gases of bosonic atoms in a optical lattice. In this case, this is achieved by a careful cancellation of two contributions in the pair-wise interaction between the atoms, one proportional to the zero-energy scattering length and a second proportional to the effective range. I subsequently study the physics of Bose-Einstein condensation in the second band of a double-well 2D lattice and show that the collision aided decay rate of the con- densate to the ground band is smaller than the tunneling rate between neighboring unit cells. Finally, I propose a numerical method using the discrete variable repre- sentation for constructing real-valued Wannier functions localized in a unit cell for optical lattices. The developed numerical method is general and can be applied to a wide array of optical lattice geometries in one, two or three dimensions.
We study ultracold atoms in an optical lattice with two local minima per unit cell and show that the low-energy states of a multiband Bose-Hubbard (BH) Hamiltonian with only pairwise interactions are equivalent to an effective single-band Hamiltonian with strong three-body interactions. We focus on a double-well optical lattice with a symmetric double well along the xxx axis and single-well structure along the perpendicular directions. Tunneling and two-body interaction energies are obtained from an exact band-structure calculation and numerically constructed Wannier functions in order to construct a BH Hamiltonian spanning the lowest two bands. Our effective Hamiltonian is constructed from the ground state of the NNN-atom Hamiltonian for each unit cell obtained within the subspace spanned by the Wannier functions of the two lowest bands. The model includes hopping between ground states of neighboring unit cells. We show that such an effective Hamiltonian has strong three-body interactions that can be easily tuned by changing the lattice parameters. Finally, relying on numerical mean-field simulations, we show that the effective Hamiltonian is an excellent approximation of the two-band BH Hamiltonian over a wide range of lattice parameters, in both the superfluid and Mott insulator regions.