Generalized Donkor model with induced dipole-dipole forbidden transitions using Maple

Abstract

ABSTRACT The idea of this work was to generalize the Donkor model (Quantum Information and Computation IX, edited by Eric Donkor, Andrew R. Pirich, Howard E. Brandt, Proc. of SPIE Vol. 8057, 80570O · © 2011 SPIE · CCC code: 0277-786X/11/$18 · doi: 10.1117/12.884501) about the application of induced dipole-dipole forbidden transitions to quantum computation. Using computer algebra we were able to reproduce the original Donkor model. Then we applied some modifications to this model and obtained the respective solutions. It is expected that this model has applications for quantum computation. Keywords: Donkor Model, Generalized Donkor model, forbidden transition, entropy, computer algebra, Bessel functions. 1. INTRODUCTION In a previous work [1], Professor Erik Donkor presented a model describing forbidden transitions induced by external force in rare-earth elements. According to his conclusions, the model in high Z elements can lead to quantum information processing which can have a lower decoherence rate. Specifically, the Donkor model considers the possible prospects for quantum computation when transitions in the 4f shell of the rare-earth elemen ts are considered. Concretely, the Donkor model is centered on the element Ytterbium Yb and for this element some interesting quantum magnitudes are calculated: the selection rule, the estimate decoherence time for N-qubit syst em created by optically induced spin-orbit coupling The original Donkor model [1] was formulat ed and solved in the case of ideal laser beam with ideal atoms and with ideal interactions between laser and atoms. It is interesting to study the Donkor model in the case with more realistic laser beams and atoms. In this work, we formulate and solve a generalized Donkor model for which the behavior of the laser beams and atoms is deviated respect to the ideal behavi or. Specifically we consider th e case when the electric field of the laser or the electric dipole of the atoms or the interaction laser-atoms d ecays exponentially with the time. Such exponentially decay is relevant fo r long periods of exposition of the atoms to the radiation. Initially the original Donkor model, with ideal laser and atoms, is analytically solved using computer algebra software [2] (specifically Maple [3]) and the original results of the Donkor model are reproduced and embedded inside a powerful computational platform. At second place, the generalized Donkor model, with more realistic laser beams and atoms, is formulated and analytically solved using again Maple and the new results are also embedded inside the previously mentioned powerful computation platform.