Non-perturbative electron dynamics in crossed fields

Intense AC electric fields on semiconductor structures have been studied in photon-assisted tunneling experiments with magnetic field applied either parallel (B_par) or perpendicular (B_per) to the interfaces. We examine here the electron dynamics in a double quantum well when intense AC electric fields F, and tilted magnetic fields are applied simultaneously. The problem is treated non-perturbatively by a time-dependent Hamiltonian in the effective mass approximation, and using a Floquet-Fourier formalism. For B_par=0, the quasi-energy spectra show two types of crossings: those related to different Landau levels, and those associated to dynamic localization (DL), where the electron is confined to one of the wells, despite the non-negligible tunneling between wells. B_par couples parallel and in-plane motions producing anti-crossings in the spectrum. However, since our approach is non-perturbative, we are able to explore the entire frequency range. For high frequencies, we reproduce the well known results of perfect DL given by zeroes of a Bessel function. We find also that the system exhibits DL at the same values of the field F, even as B_par non-zero, suggesting a hidden dynamical symmetry in the system which we identify with different parity operations. The return times for the electron at various values of field exhibit interesting and complex behavior which is also studied in detail. We find that smaller frequencies shifts the DL points to lower field F, and more importantly, yields poorer localization by the field. We analyze the explicit time evolution of the system, monitoring the elapsed time to return to a given well for each Landau level, and find non-monotonic behavior for decreasing frequencies.Comment: REVTEX4 + 11 eps figs, submitted to Phys. Rev.

The Almohad Mecca. Locating Igli and the Cave of Ibn Tūmart

Igli was the Mecca of the Almohads. In the Cave of Igli Muḥammad Ibn Tūmart proclaimed the beginning of the Almohad movement in North Africa. Despite the obvious importance of Igli, there was little knowledge about the precise location of Igli or the famous cave of Ibn Tūmart on the map. This paper details the discovery of the small village of Igli 30 kilometers East of Tarūdant. The location of this modem village corresponds with Almohad Igli. Primary sources, oral testimony and physical remains are all evidence that modem Igli was the birthplace of Ibn Tūmart.<br><br>Igli era La Meca de los Almohades. En la cueva de Igli Muḥammad Ibn Tūmart proclamó el principio del movimiento almohade en África del norte. A pesar de la importancia de Igli, no se conocía su localización exacta o de la cueva de Ibn Tūmart. Aquí se detalla el descubrimiento de la pequeña aldea de Igli 30 kilómetros al este de Tarūdant. La localización de esta aldea moderna parece corresponder con el Igli almohade. Las fuentes primarias, el testimonio oral y los restos físicos se aducen como evidencia de que el Igli moderno era el lugar de nacimiento de Ibn Tūmart