  A Bose-Einstein condensate in an optical lattice exhibits parametric
resonances when the intensity of the lattice is periodically modulated in time.
These resonances correspond to an exponential growth of the population of
counter-propagating Bogoliubov excitations. A suitable linearization of the
Gross-Pitaevskii (GP) equation is used to calculate the stability diagram and
the growth rates of the unstable modes. The results agree with the ones
extracted from time-dependent GP simulations, supporting our previous claim (M.
Kraemer et al., Phys. Rev. A (2005) in press) concerning the key role of
parametric resonances in the response observed by Stoeferle et al. (Phys. Rev.
Lett. 92, 130403 (2004)) in the superfluid regime. The role of the seed
excitations required to trigger the parametric amplification is discussed. The
possible amplification of the quantum fluctuations present in the quasiparticle
vacuum, beyond GP theory, is also addressed, finding interesting analogies with
similar processes in nonlinear quantum optics and with the dynamic Casimir
effect. Our results can be used in exploiting parametric instabilities for the
purpose of spectroscopy, selective amplification of a particular excitation
mode and for establishing a new type of thermometry.
