Hubbard-like Hamiltonian for ultracold atoms in a 1D optical lattice

Abstract

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.