Article révisé par les pairs
Résumé : In this work, we analyze the dynamical properties of periodically driven band models. Focusing on the caseof Bose-Einstein condensates, and using a mean-field approach to treat interparticle collisions, we identify theorigin of dynamical instabilities arising from the interplay between the external drive and interactions. Wepresent a widely applicable generic numerical method to extract instability rates and link parametricinstabilities to uncontrolled energy absorption at short times. Based on the existence of parametric resonances,we then develop an analytical approach within Bogoliubov theory, which quantitatively captures theinstability rates of the system and provides an intuitive picture of the relevant physical processes, including anunderstanding of how transverse modes affect the formation of parametric instabilities. Importantly, ourcalculations demonstrate an agreement between the instability rates determined from numerical simulationsand those predicted by theory. To determine the validity regime of the mean-field analysis, we compare thelatter to the weakly coupled conserving approximation. The tools developed and the results obtained in thiswork are directly relevant to present-day ultracold-atom experiments based on shaken optical lattices and areexpected to provide an insightful guidance in the quest for Floquet engineering.