  The similarity between matter waves in periodic potential and solid-state
physics processes has triggered the interest in quantum simulation using
Bose-Fermi ultracold gases in optical lattices. The present work evidences the
similarity between electrons moving under the application of oscillating
electromagnetic fields and matter waves experiencing an optical lattice
modulated by a frequency difference, equivalent to a spatially shaken periodic
potential. We demonstrate that the tunneling properties of a Bose-Einstein
condensate in shaken periodic potentials can be precisely controlled. We take
additional crucial steps towards future applications of this method by proving
that the strong shaking of the optical lattice preserves the coherence of the
matter wavefunction and that the shaking parameters can be changed
adiabatically, even in the presence of interactions. We induce reversibly the
quantum phase transition to the Mott insulator in a driven periodic potential.
