Quantum dynamics of Bose-Hubbard Hamiltonians beyond the Hartree-Fock-Bogoliubov approximation: The Bogoliubov back-reaction approximation

We formulate a method for studying the quantum field dynamics of ultracold Bose gases confined within optical lattice potentials, within the lowest Bloch-band Bose-Hubbard model. Our formalism extends the two-sites results of Vardi and Anglin [Phys. Rev. Lett. 86, 568 (2001)] to the general case of M lattice sites. The methodology is based on mapping the Bose-Hubbard Hamiltonian to an SU(M) pseudospin problem and truncating the resulting hierarchy of dynamical equations for correlation functions, up to pair correlations between SU(M) generators. Agreement with few-site exact many-particle calculations is consistently better than the corresponding Hartree-Fock-Bogoliubov approximation. Moreover, our approximation compares favorably with a more elaborate two-particle irreducible effective action formalism, at a fraction of the analytic and numerical effort.