1 A numerical model for light interaction with a two-level atom medium. T. Colin a and B. Nkonga a a Math´ ematiques Appliqu´ ees de Bordeaux, Universit´ e Bordeaux 1 et CNRS UMR 5466 351, Cours de la Liberation, 33405 Talence, mail: colin, nkonga@math.u-bordeaux.fr The aim of this work is to study the interaction of a laser pulse with a gas described by the Bloch equations. The physical model is derived from the Maxwell-Bloch equations under the assumption that the polarization and the electric field are both oriented in a single direction, and under the slowly varying envelope approximation. The Schr¨ odinger equation is formulated for a two-level atom medium under the rotating wave approxi- mation and dipole interaction. A rigorous asymptotic analysis of the simplified system is performed to point out the different mechanisms associated with the physical model. A numerical approach based on a splitting technique is proposed and successfully per- formed. Key Words: Nonlinear Optics, Two-level Atom, Asymptotic Analysis, Splitting Scheme, Lasers pulses. 1. Introduction Many topics of interest in nonlinear optical applications are related to the propagation of light and to the interaction of the electric field with material media (see for example [23,7,18] for numerical study). The regime of the laser sources (small wavelength) displays quantum mechanical coupling via the optical properties of the medium (see for example the classical textbooks [19,8,17]). The density of the material medium is either high [23] or low [16]. Then according to the intensity of the laser beams, we can obtain different levels of linear and nonlinear interactions. Modeling and computing those behaviors are difficult numerical challenges. The full description of the light-matter interaction involves the resolution of the Maxwell equation coupled to the atomic wave function which obeys the Schr¨ odinger equations [8,19]. The numerical approach of this model is only possible for simplified media with a small number of atoms. The nonlinear Maxwell-Bloch system is a macroscopic model taking into account the quantum mechanical coupling for a large class of media. It contains some essential numerical difficulties. In some cases, the full wave integration of this system is unavoidable and has been investigated in the finite difference time-domain (FDTD) context [22,15]. In general the computational time needed for those applications is very expensive (see [7,12,9] for general nonlinearity and [5,4,24,23,20] for Maxwell-Bloch systems). However, we are concerned with the propagation of a quasi- monochromatic optical field in a dilute atomic vapor. Atoms with two energy levels