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

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 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 -operator expectation values, thus providing a clear physical interpretation of this algebraic mean-field picture.