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arXiv:1303.2213 · Physical Review A

Ab initio derivation of Hubbard models for cold atoms in optical lattices

R. Walters, G. Cotugno, T. H. Johnson, Stephen R. L. Clark, Dieter Jaksch

Published Apr 10, 2013• 58 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.87.043613

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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.
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.
Optical lattices provide ideal experimental tools for simulating a wide variety of physical systems. They exhibit excellent coherence properties, their microscopic dynamics are well-understood, and they can be dynamically controlled with great precision. Recent advances in cooling and trapping gases of dipolar molecules and Rydberg atoms have further increased the versatility of optical lattices as quantum simulators, particularly for solid state systems possessing long-range interactions. In this thesis we further examine the microscopic properties of cold atoms trapped in an optical lattice, going beyond the case where the lattice axes are orthogonal. We present a transformation of the single-particle energy eigenstates to a complete, orthonormal basis of so-called 'generalised Wannier functions' that are localised to each lattice site. Generalised Wannier functions, as opposed to ordinary Wannier functions, are formed by mixing states between bands and are required when the bands are degenerate. We modify an algorithm devised by Marzari and Vanderbilt (Phys. Rev. B 56(20) 12847) for calculating maximally localised generalised Wannier functions, specifically to determine these functions for optical lattice systems of arbitrary geometry, including those with degenerate lower bands. Moreover, we overcome an issue with the initialisation of the original algorithm to find the maximally localised set without fail. We then present results for a variety of optical lattice systems in one- and two-dimensions, including hexagonal and Kagomé geometries. In each case we use the generalised Wannier functions to calculate the hopping and interaction parameters, and thus determine a reduced form of the system's Hamiltonian that contains only local interactions. We also study two spin-1/2 systems with long-range interactions as paradigm examples of the models that can be simulated using optical lattices loaded with dipolar molecules. We first examine the critical properties of a 1D spin-1/2 Ising chain possessing exponentially decaying interactions as the range of the interactions is increased. Using the infinite density matrix renormalisation group method, we calculate the location of the critical point and observe a smooth approach to the classical mean-field result in the limit of infinite-ranged interactions. We go on to consider the quantum dynamics of a single spin-1/2 coupled to a bath of interacting spins as a model for decoherence in solid state quantum memories. The large degree of symmetry of the bath, described by the infinite-ranged Lipkin-Meshkov-Glick model, allows us to find parameter regimes where the initial qubit state is revived at well-defined times after the qubit preparation. We observe that these times may become independent of the bath size for large baths and thus enable faithful qubit storage even in the presence of strong coupling to the bath.
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 new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.
We introduce a method for quantum simulation of U(1) lattice gauge theories coupled to matter, utilizing alkaline-earth(-like) atoms in state-dependent optical lattices.The proposal enables the study of both gauge and fermionic matter fields without integrating out one of them in one and two dimensions.We focus on a realistic and robust implementation that utilizes the long-lived metastable clock state available in alkaline-earth(-like) atomic species.Starting from an ab initio modeling of the experimental setting, we systematically carry out a derivation of the target U(1) gauge theory.This approach allows us to identify and address conceptual and practical challenges for the implementation of lattice gauge theories that-while pivotal for a successful implementation-have never been rigorously addressed in the literature: those include the specific engineering of lattice potentials to achieve the desired structure of Wannier functions and the subtleties involved in realizing the proper separation of energy scales to enable gauge-invariant dynamics.We discuss realistic experiments that can be carried out within such a platform using the fermionic isotope 173 Yb, addressing via simulations all key sources of imperfections, and provide concrete parameter estimates for relevant energy scales in both one-and two-dimensional settings.
We consider the physical implementation of a 2D optical lattice with schemes involving 3 and 4 light fields. We illustrate the wide range of geometries available to the 3 beam lattice, and compare the general potential properties of the two lattice schemes. Numerically calculating the band structure we obtain the Wannier states and evaluate the parameters of the Bose-Hubbard models relevant to these lattices. Using these results we demonstrate lattices that realize Bose-Hubbard models with 2, 4, or 6 nearest neighbors, and quantify the extent that these different lattices effect the superfluid to Mott-insulator transition.