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arXiv:1506.08053 · Physical Review A

Interaction effects on Wannier functions for bosons in an optical lattice

Shaoqi Zhu, Biao Wu

Published Dec 30, 2015• 9 citations
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Abstract

We have numerically calculated the single-band Wannier functions for interacting Bose gases in optical lattices with a self-consistent approach. We find that the Wannier functions are broadened by repulsive interaction. The tunneling parameter $J$ and the on-site interaction $U$ computed with the broadened Wannier functions are found to change significantly with the number of atoms per site. Our theory can explain the nonuniform atomic clock shift observed in Campbell et al., Science 313, 649 (2006).

DOI
10.1103/physreva.92.063637

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We have numerically calculated the single-band Wannier functions for interacting Bose gases in optical lattices with a self-consistent approach. We find that the Wannier functions are broadened by repulsive interaction. The tunneling parameter JJJ and the on-site interaction UUU computed with the broadened Wannier functions are found to change significantly with the number of atoms per site. Our theory can explain the nonuniform atomic clock shift observed in Campbell et al., Science 313, 649 (2006).
We show that one can properly take into account of the interaction effects and construct a set of orthonormal Wannier functions for a Bose-Einstein condensate in an optical lattice. These interaction-dependent Wannier functions are used to compute the tunneling rate JJJ and the on-site repulsion UUU in the Bose-Hubbard model. Both parameters are found to be substantially different from ones calculated with the single-particle Wannier functions. Our numerical results of UUU are found in good agreement with the measured on-site energy in a recent experiment [Campbell {\it et al.} Science {\bf 314}, 281 (2006)].
We study the effects of repulsive on-site interactions on the broadening of the localized Wannier functions used for calculating the parameters to describe ultracold atoms in optical lattices. For this, we replace the common single-particle Wannier functions, which do not contain any information about the interactions, by two-particle Wannier functions obtained from an exact solution which takes the interactions into account. We then use these interaction-dependent basis functions to calculate the Bose-Hubbard model parameters, showing that they are substantially different both at low and high lattice depths from the ones calculated using single-particle Wannier functions. Our results suggest that density effects are not negligible for many parameter ranges and need to be taken into account in metrology experiments.
Superfluid to Mott-insulator transitions in atomic Bose–Einstein condensates (BEC) in optical lattices are investigated for the case of number of atoms per site larger than one. To account for mean field repulsion between the atoms in each well, we construct an orthogonal set of Wannier functions. The resulting hopping amplitude and on-site interaction may be substantially different from those calculated with single-atom Wannier functions. As illustrations of the approach, we consider lattices of various dimensionality and different mean occupations. We find that in three-dimensional optical lattices the correction to the critical lattice depth is significant enough to be measured experimentally even for a small number of atoms. Finally, we discuss the validity of the single-band model.
We construct the effective lowest-band Bose-Hubbard model incorporating interaction-induced on-site correlations. The model is based on ladder operators for local correlated states, which deviate from the usual Wannier creation and annihilation, allowing for a systematic construction of the most appropriate single-band low-energy description in the form of the extended Bose-Hubbard model. A formulation of this model in terms of ladder operators not only naturally contains the previously found effective multibody interactions, but also contains multibody-induced single-particle tunneling, pair tunneling, and nearest-neighbor interaction processes of higher orders. An alternative description of the same model can be formulated in terms of occupation-dependent Bose-Hubbard parameters. These multiparticle effects can be enhanced using Feshbach resonances, leading to corrections which are well within experimental reach and of significance to the phase diagram of ultracold bosonic atoms in an optical lattice. We analyze the energy-reduction mechanism of interacting atoms on a local lattice site and show that this cannot be explained only by a spatial broadening of Wannier orbitals on a single-particle level, which neglects correlations.
Recent experiments revealed the importance of higher-band effects for the Mott-insulator(MI)--superfluid transition(SF) of ultracold bosonic atoms or mixtures of bosons and fermions in deep optical lattices [Best et al., Phys. Rev. Lett. 102, 030408 (2009); Will et al., Nature (London) 465, 197 (2010)]. In the present work we derive an effective lowest-band Hamiltonian in three dimensions that generalizes the standard Bose-Fermi-Hubbard model taking these effects as well as nonlinear corrections of the tunneling amplitudes mediated by interspecies interactions into account. It is shown that a correct description of the lattice states in terms of the bare-lattice Wannier functions, rather than approximations such as harmonic-oscillator states, is essential. In contrast to self-consistent approaches based on effective Wannier functions, our approach captures the observed reduction of the superfluid phase for repulsive interspecies interactions.
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.