We study the Landau damping of Bogoliubov excitations in two- and three-dimensional optical lattices at finite temperatures, extending our recent work on one-dimensional (1D) optical lattices. We use a Bose-Hubbard tight-binding model and the Popov approximation to calculate the temperature dependence of the number of condensate atoms
nc0(T) in each lattice well. As with 1D optical lattices, damping only occurs if the Bogoliubov excitations exhibit anomalous dispersion (i.e., the excitation energy bends upward at low momentum), analogous to the case of phonons in superfluid
4He. This leads to the disappearance of all damping processes in a
D-dimensional simple cubic optical lattice when $U{n}^{\mathrm{c}0}\ensuremath{\geqslant}6DJ$, where
U is the on-site interaction, and
J is the hopping matrix element.
Extending recent work to finite temperatures, we calculate the Landau damping of a Bogoliubov excitation in an optical lattice, due to the coupling to a thermal cloud of such excitations. For simplicity, we consider a one-dimensional Bose-Hubbard model and restrict ourselves to the first energy band. For energy conservation to be satisfied, the excitations in the collision processes must exhibit ``anomalous dispersion,'' analogous to phonons in superfluid
4He. This leads to the disappearance of all damping processes when $U{n}^{\mathrm{c}0}\ensuremath{\geqslant}6J$, where
U is the on-site interaction,
J is the hopping matrix element, and
nc0(T) is the number of condensate atoms at a lattice site. This phenomenon also occurs in two-dimensional and three-dimensional optical lattices. The disappearance of Beliaev damping above a threshold wave vector is noted.