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

TL;DR

This paper develops a mean-field algebraic framework using coherent states to analyze the dynamics of fermions in a one-dimensional optical lattice, providing a new approach to understanding their quantum evolution.

Contribution

It introduces an algebraic mean-field method based on coherent states and the algebra of two-fermion operators for modeling fermionic dynamics in optical lattices.

Findings

Derived an effective Hamiltonian using a variational coherent-state approach.

Identified the algebra of two-fermion operators as key to the model.

Connected the coherent-state parameters to physical expectation values.

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.