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arXiv:cond-mat/0512605 · Journal of Physics B Atomic Molecular and Optical Physics

Mean-field algebraic approach to the dynamics of fermions in a 1D optical lattice

Francesco Massel, Vittorio Penna

Published May 2, 2006• 1 citation
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Abstract

We consider a one-dimensional optical lattice of three-dimensional Harmonic Oscillators which are loaded with neutral fermionic atoms trapped into two hyperfine states. By means of a standard variational coherent-state procedure, we derive an effective Hamiltonian for this quantum model and the hamiltonian equations describing its evolution. To this end, we identify the algebra $\mathcal L$ of two-fermion operators --describing the relevant microscopic quantum processes of our model-- whereby the natural choice for the trial state appears to be a so(2r) coherent state. The coherent-state parameters, playing the role of dynamical variables for the effective Hamiltonian, are shown to identify with the $\mathcal L$-operator expectation values thus providing a clear physical interpretation of this algebraic mean-field picture.

DOI
10.1088/0953-4075/39/10/s14

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We consider a one-dimensional optical lattice of three-dimensional Harmonic Oscillators which are loaded with neutral fermionic atoms trapped into two hyperfine states. By means of a standard variational coherent-state procedure, we derive an effective Hamiltonian for this quantum model and the hamiltonian equations describing its evolution. To this end, we identify the algebra L\mathcal LL of two-fermion operators --describing the relevant microscopic quantum processes of our model-- whereby the natural choice for the trial state appears to be a so(2r) coherent state. The coherent-state parameters, playing the role of dynamical variables for the effective Hamiltonian, are shown to identify with the L\mathcal LL-operator expectation values thus providing a clear physical interpretation of this algebraic mean-field picture.
Based on the standard many-fermion field theory, we construct models describing ultracold fermions in a one-dimensional optical lattice 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 the very low-density regime, in general, far more complicated Hamiltonians characterize 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 standard techniques we obtain the mean-field version of our model Hamiltonian and show how different dynamical algebras characterize the cases of attractive and repulsive two-body potentials.
The mean-field pictures based on the standard time-dependent variational approach have widely been used in the study of nonlinear many-boson systems such as the Bose-Hubbard model. The mean-field schemes relevant to Gutzwiller-like trial states ∣F>|F>∣F>, number-preserving states ∣ξ>|ξ>∣ξ> and Glauber-like trial states ∣Z>|Z>∣Z> are compared to evidence the specific properties of such schemes. After deriving the Hamiltonian picture relevant to ∣Z>|Z>∣Z> from that based on ∣F>|F>∣F>, the latter is shown to exhibit a Poisson algebra equipped with a Weyl-Heisenberg subalgebra which preludes to the ∣Z>|Z>∣Z>-based picture. Then states ∣Z>|Z>∣Z> are shown to be a superposition of N\cal NN-boson states ∣ξ>|ξ>∣ξ> and the similarities/differences of the ∣Z>|Z>∣Z>-based and ∣ξ>|ξ>∣ξ>-based pictures are discussed. Finally, after proving that the simple, symmetric state ∣ξ>|ξ>∣ξ> indeed corresponds to a SU(M) coherent state, a dual version of states ∣Z>|Z>∣Z> and ∣ξ>|ξ>∣ξ> in terms of momentum-mode operators is discussed together with some applications.
We study collective behaviour of Fermi gases trapped in various external potentials, including optical lattices (OLs), in the framework of the mean-field (hydrodynamic) description. Using the variational method, we derive effective dynamical equations for the one- and two-dimensional (1D and 2D) settings from the general 3D mean-field equation. The respective confinement is provided by trapping potentials with the cylindrical and planar symmetries, respectively. The resulting equations are nonpolynomial Schrödinger equations coupled to equations for the local transverse size of the trapped states. Numerical simulations demonstrate close agreement of results produced by the underlying 3D equation and the effective low-dimensional ones. We consider the ground state in these settings. In particular, analytical solutions are obtained for the effectively 2D non-interacting Fermi gas. Differences between the 1D and 2D configurations are highlighted. Finally, we analyse the dependence of the 1D and 2D density patterns of the trapped gas, in the presence of the OL, on the strengths of the confining and OL potentials, and on the scattering length which determines the strength of interactions between non-identical fermions.
We discuss the theory of mixtures of bosonic and fermionic atoms in periodic potentials at zero temperature. We derive a general Bose-Fermi Hubbard Hamiltonian in a one-dimensional optical lattice with a superimposed harmonic trapping potential. We study the conditions for linear stability of the mixture and derive a mean-field criterion for the onset of a bosonic superfluid transition. We investigate the ground-state properties of the mixture in the Gutzwiller formulation of mean-field theory, and present numerical studies of finite systems. The bosonic and fermionic density distributions and the onset of quantum phase transitions to demixing and to a bosonic Mott-insulator are studied as a function of the lattice potential strength. The existence is predicted of a disordered phase for mixtures loaded in very deep lattices. Such a disordered phase possessing many degenerate or quasidegenerate ground states is related to a breaking of the mirror symmetry in the lattice.
We study collective behavior of Fermi gases trapped in various external potentials, including optical lattices (OLs), in the framework of the mean-field (hydrodynamic) description. Using the variational method, we derive effective dynamical equations for the one- and two-dimensional (1D and 2D) settings from the general 3D mean-field equation. The respective confinement is provided by trapping potentials with the cylindrical and planar symmetry, respectively. The resulting equations are nonpolynomial Schr% \"{o}dinger equations (NPSEs) coupled to equations for the local transverse size of the trapped states. Numerical simulations demonstrate close agreement of results produced by the underlying 3D equation and the effective low-dimensional ones. We consider the ground state in these settings. In particular, analytical solutions are obtained for the effectively 2D non-interacting Fermi gas. Differences between the 1D and 2D configurations are highlighted. Finally, we analyze the dependence of the 1D and 2D density patterns of the trapped gas, in the presence of the OL, on the strengths of the confining and OL potentials, and on the scattering length which determines the strength of interactions between non-identical fermions.
Abstract We studied the behavior of mixtures of 173Yb (with symmetry up to SU(6)) and 171Yb (up to SU(2)) fermionic isotopes loaded in one-dimensional (1D) optical lattices. To do so, we solved the Schrödinger equation describing different systems using a diffusion Monte Carlo technique. We considered continuous Hamiltonians in which the interactions between atoms of different species (isotopes and/or spins) were modeled by contact potentials with parameters derived from their experimental scattering lengths. This implies that we can find both attractive and repulsive interactions between fermion pairs in the same cluster. The strength of those interactions can be changed by varying the transverse confinement, leading to different cluster behaviors. Only balanced clusters, i.e. with the same number of 173Yb and 171Yb atoms were considered. We found that the standard state for these clusters is a metallic-like one with different populations of 173Yb–171Yb molecule-like pairs in each optical lattice potential well. However, for big enough clusters, insulator-like states are also possible.