Effective Dirac equation for ultracold atoms in optical lattices: Role of the localization properties of the Wannier functions (original) (raw)
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Effective Dirac dynamics of ultracold atoms in bichromatic optical lattices
Physical Review A, 2011
We study the dynamics of ultracold atoms in tailored bichromatic optical lattices. By tuning the lattice parameters, one can readily engineer the band structure and realize a Dirac point, i.e. a true crossing of two Bloch bands. The dynamics in the vicinity of such a crossing is described by the one-dimensional Dirac equation, which is rigorously shown beyond the tight-binding approximation. Within this framework we analyze the effects of an external potential and demonstrate numerically that it is possible to demonstrate Klein tunneling with current experimental setups.
Tight-binding models for ultracold atoms in optical lattices: general formulation and applications
Science China Physics, Mechanics & Astronomy, 2016
Tight-binding models for ultracold atoms in optical lattices can be properly defined by using the concept of maximally localized Wannier functions for composite bands. The basic principles of this approach are reviewed here, along with different applications to lattice potentials with two minima per unit cell, in one and two spatial dimensions. Two independent methods for computing the tight-binding coefficients-one ab initio, based on the maximally localized Wannier functions, the other through analytic expressions in terms of the energy spectrum-are considered. In the one dimensional case, where the tight-binding coefficients can be obtained by designing a specific gauge transformation, we consider both the case of quasi resonance between the two lowest bands, and that between s and p orbitals. In the latter case, the role of the Wannier functions in the derivation of an effective Dirac equation is also reviewed. Then, we consider the case of a two dimensional honeycomb potential, with particular emphasis on the Haldane model, its phase diagram, and the breakdown of the Peierls substitution. Tunable honeycomb lattices, characterized by movable Dirac points, are also considered. Finally, general considerations for dealing with the interaction terms are presented.
New Journal of Physics, 2012
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 M and Vanderbilt D 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 to efficiently computing the tight-binding coefficients needed for mapping the continuous system to a discrete lattice model. Their behaviour 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.
Hubbard-like Hamiltonian for ultracold atoms in a 1D optical lattice
Physical Review A, 2005
Based on the standard many-fermion field theory, the authors construct models describing ultracold fermions in a 1D optical lattices by implementing a mode expansion of the fermionic field operator where modes, in addition to space localization, take into account the quantum numbers inherent in local fermion interactions. The resulting models are generalized Hubbard Hamiltonians whose interaction parameters are derived by a fully-analytical calculation. The special interest for this derivation resides in its model-generating capability and in the flexibility of the trapping techniques that allow the tuning of the Hamiltonian interaction parameters over a wide range of values. While the Hubbard Hamiltonian is recovered in a very low-density regime for a fermionic system, in general, far more complicated Hamiltonians characterise high-density regimes, revealing a rich scenario for both the phenomenology of interacting trapped fermions and the experimental realization of devices for quantum information processing. As a first example of the different situations that may arise beyond the models well known in the literature (the unpolarized-spin fermion model and the noninteracting spin-polarized fermion model), we derive a Rotational Hubbard Hamiltonian describing the local rotational activity of spin-polarized fermions. Based on a standard techniques we obtain the mean-field version of our model Hamiltonian and show how different dynamical algebras characterize the case of attractive and repulsive two-body potentials.
Hubbard-like Hamiltonian for ultracold atoms in a one-dimensional optical lattice
Physical Review A, 2005
The user has requested enhancement of the downloaded file. arXiv:cond-mat/0506316v2 [cond-mat.mes-hall] Abstract Based on the standard many-fermion field theory, we construct models describing ultracold fermions in a 1D optical lattice by implementing a mode expansion of the fermionic field operator where modes, in addition to space localisation, take into account the quantum numbers inherent in local fermion interactions. The resulting models are generalised Hubbard Hamiltonians whose interaction parameters are derived by a fully-analytical calculation. The special interest for this derivation resides in its model-generating capability and in the flexibility of the trapping techniques that allow the tuning of the Hamiltonian interaction parameters over a wide range of values. While the Hubbard Hamiltonian is recovered in a very low-density regime, in general, far more complicated Hamiltonians characterise high-density regimes, revealing a rich scenario for both the phenomenology of interacting trapped fermions and the experimental realization of devices for quantum information processing. As a first example of the different situations that may arise beyond the models well known in the literature (the unpolarised-spin fermion model and the noninteracting spin-polarised fermion model), we derive a Rotational Hubbard Hamiltonian describing the local rotational activity of spin-polarised fermions. Based on a standard techniques we obtain the mean-field version of our model Hamiltonian and show how different dynamical algebras characterize the case of attractive and repulsive two-body potentials.
Ultracold fermions in a graphene-type optical lattice
Physical Review A, 2009
Some important features of the graphene physics can be reproduced by loading ultracold fermionic atoms in a two-dimensional optical lattice with honeycomb symmetry and we address here its experimental feasibility. We analyze in great details the optical lattice generated by the coherent superposition of three coplanar running laser waves with respective angles 2π/3. The corresponding band structure displays Dirac cones located at the corners of the Brillouin zone and close to halffilling this system is well described by massless Dirac fermions. We characterize their properties by accurately deriving the nearest-neighbor hopping parameter t0 as a function of the optical lattice parameters. Our semi-classical instanton method proves in excellent agreement with an exact numerical diagonalization of the full Hamilton operator in the tight-binding regime. We conclude that the temperature range needed to access the Dirac fermions regime is within experimental reach. We also analyze imperfections in the laser configuration as they lead to optical lattice distortions which affect the Dirac fermions. We show that the Dirac cones do survive up to some critical intensity or angle mismatches which are easily controlled in actual experiments. In the tight-binding regime, we predict, and numerically confirm, that these critical mismatches are inversely proportional to the square-root of the optical potential strength. We also briefly discuss the interesting possibility of fine-tuning the mass of the Dirac fermions by controlling the laser phase in an optical lattice generated by the incoherent superposition of three coplanar independent standing waves with respective angles 2π/3.
Flat bands, Dirac cones, and atom dynamics in an optical lattice
Physical Review A, 2010
We study atoms trapped with a harmonic confinement in an optical lattice characterized by a flat band and Dirac cones. We show that such an optical lattice can be constructed which can be accurately described with the tight binding or Hubbard models. In the case of fermions the release of the harmonic confinement removes fast atoms occupying the Dirac cones while those occupying the flat band remain immobile. Using exact diagonalization and dynamics we demonstrate that a similar strong occupation of the flat band does not happen in bosonic case and furthermore that the mean field model is not capable for describing the dynamics of the boson cloud.
Hubbard-like Hamiltonian for ultracold atoms in a one-dimensional optical lattice (14 pages)
Phys Rev a, 2005
Based on the standard many-fermion field theory, we construct models describing ultracold fermions in a 1D optical lattice by implementing a mode expansion of the fermionic field operator where modes, in addition to space localisation, take into account the quantum numbers inherent in local fermion interactions. The resulting models are generalised Hubbard Hamiltonians whose interaction parameters are derived by a fully-analytical calculation. The special interest for this derivation resides in its model-generating capability and in the flexibility of the trapping techniques that allow the tuning of the Hamiltonian interaction parameters over a wide range of values. While the Hubbard Hamiltonian is recovered in a very low-density regime, in general, far more complicated Hamiltonians characterise high-density regimes, revealing a rich scenario for both the phenomenology of interacting trapped fermions and the experimental realization of devices for quantum information processing. As a first example of the different situations that may arise beyond the models well known in the literature (the unpolarised-spin fermion model and the noninteracting spin-polarised fermion model), we derive a Rotational Hubbard Hamiltonian describing the local rotational activity of spin-polarised fermions. Based on a standard techniques we obtain the mean-field version of our model Hamiltonian and show how different dynamical algebras characterize the case of attractive and repulsive two-body potentials.