The phenomenon of optical assisted transition in crystals in the case of a weak optical radiation has been experimentally studied in our quantum laboratory; both its mathematical modeling and numerical simulation are approached. The formalism of quantum physics has been used in order to describe the emission or absorption of the optical radiation in an atomic system, in the single photon regime. The Schrdinger equation in its operator format and state vectors has been used; the matrix format with the Dirac formalism was reserved exclusively for the calculations of scalar vector products. The atomic system is considered as a chosen system and the photon is considered as a perturbing system. The photon is considered a wavelet with nite spatial and temporal extension. The energy levels of the crystal are considered. The interaction between the two systems is studied on a basis similar with that one used in the information transmission theory. The structural algebraic formalism has been put in evidence for the description of the interaction of the two systems; the previous study of the authors has been reformulated and deepened. A quasi-complete analytic and numeric temporal analysis of the transition probability between two eigen states of a quantum system in interaction with a perturbation system, in this case an electromagnetic one, is performed. A general formalism has been elaborated based on the Hamiltonian of interaction concept and a rigorous treatment of both the terminology and the notations. In order to perform the temporal analysis of the transition probability, the rst order perturbation theory was used. The calculus relations were standardized. The complex spectral density (Fourier transform) for a perturbation with compact temporal support has been used in order to obtain the transition probability. An analytical study has been performed in order to validate the numerical simulation. Limit cases, speci c cases have been taken into account, graphical representation using MathCAD have been obtained, for numeric evaluation in SI (corresponding to the laser domain). Four cases of the temporal perturbation have been analyzed: monopuls, rectangular monopuls, sinusoidal monopuls modulated by a rectangular envelope, sinusoidal monopuls modulated by a sinusoidal envelope. A basis for the analysis, numerical calculus and simulations for emission or absorption of single photon has been created, useful for future analysis of two- and three- photonic processes.