Quantum dynamics of Bose-Hubbard Hamiltonians beyond Hartree-Fock-Bogoliubov: The Bogoliubov backreaction approximation

e 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 Phys. Rev. Lett. {\bf86}, 000568 (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.