  We analyze the heating of interacting bosonic atoms in an optical lattice due
to intensity fluctuations of the lasers forming the lattice. We focus in
particular on fluctuations at low frequencies below the band gap frequency,
such that the dynamics is restricted to the lowest band. We derive stochastic
equations of motion, and analyze the effects on different many-body states,
characterizing heating processes in both strongly and weakly interacting
regimes. In the limit where the noise spectrum is flat at low frequencies, we
can derive an effective Master equation describing the dynamics. We compute
heating rates and changes to characteristic correlation functions both in the
perturbation theory limit, and using a full time-dependent calculation of the
stochastic many-body dynamics in 1D based on time-dependent
density-matrix-renormalization-group methods.
