Dynamic effect of Bloch electrons in time-periodic electric fields

The dynamic effect of Bloch electrons under the action of temporally periodic electric fields is studied. The explicit solutions for the quasienergies and the Floquet states are obtained exactly and generally, from which the localized or extended nature of Floquet states is demonstrated to be controlled by a reduced vector potential in one period of electric fields. When this reduced vector potential is an irrational, all the Floquet states are localized except the cases where the reduced vector potential is extremely well approximated by rationals (i.e., a Liouville number) and simultaneously the instersite hopping does not decay fast enough, for which the Floquet states are found to be more and more extended at large distances. When this reduced vector potential approaches an ordinary rational, all the Floquet states are extended. However, if the reduced vector potential has typical Diophantine properties, and if the intersite hopping decreases fast enough, the Floquet states cannot be extended.