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Intensities and pulse durations now attainable by lasers necessitate a semi-classical model to describe laser–matter interactions. Transient phenomena are very important and therefor a time-dependent model is needed. Interaction phenomena are all the larger as the frequency matches matter eigenfrequencies. The medium is then said to be resonant. To take these resonancies into account, modelling the impact of the medium on the wave (as is the case in classical models) is not sufficient and a quantum model is required to describe matter evolution.

In this context the most precise model is the Maxwell–Bloch model. This model does not assume a priori any shape for the solutions. According to the parameters of the medium and the wave, we can treat a large range of applications and combine many effects. However its applicability in scientific computing softwares is restricted to small experimental settings.

In some experimental settings much coarser models are used. Very often, a few frequencies are selected and with extra assumptions a nonlinear Schrödinger model is derived.

Many intermediate models may be found inthe literature (Lorentz model, rate equations,…). We address here the Maxwell–Bloch model. We first present the issues raised by the Bloch equations from the theoretical and numerical point of views. Then we treat the coupling with the Maxwell equation, leading to the Maxwell–Bloch model and applications.

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