Parametric excitation of a Bose-Einstein condensate in a 1D optical lattice

We study the response of a Bose-Einstein condensate to a periodic modulation of the depth of an optical lattice. Using Gross-Pitaevskii theory, we show that a modulation at frequency $\omega_{\rm m}$ drives the parametric excitation of Bogoliubov modes with frequency $\omega_{\rm m}/2$. The ensuing nonlinear dynamics leads to a rapid broadening of the momentum distribution and of the condensate size after free expansion. Our results reproduce the main features of the spectrum measured in the superfluid phase by St\"oferle {\it et al.}, Phys. Rev. Lett. {\bf 92}, 130403 (2004), and show that the presence of a large condensate depletion is not essential to explain the observed broad response.