  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.
