By Christian J. Bordé (auth.), F. T. Arecchi, F. Strumia, H. Walther (eds.)

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Numerical methods have been used with success in the case of strong fields witharbitrary spatial or temporal dependence. However, perturbative calculations which provide closed-form expressions bring a lot of insight and are preferred in many cases for a detailed physical discussion. It is well-known that diagrams are a useful tool to keep track of the various terms in perturbative expansions. This is illustrated by the wide use of Feynman diagrams. Furthermore besides the interest of diagrams in book-keeping terms, diagrammatic rules can be of great help to write down the solutions automatically without having to constantly refer to the basic equations.

The physics can be understood by inspection of the diagrams. The constant term corresponds to both one-photon processes occurring in the same field zone,whereas the oscillating term comes from an interference of two one-photon processes occurring in two different zones. On the diagram, we can also see that, at the first vertex in the first field zone, an optical dipole (off-diagonal matrix element or optical coherence) is created and precesses freely in the dark zone for the time duration d/v. When this dipole is projected on the field at the second interaction vertex ,in the second zone,the phase difference,between the internal atomic clock and'the laser field oscillation during the time d/v ,gives rise to the interference term in the upper state population or in the absorbed power.

5. a a a' a a a a' a' a a a 13' Figure 5 a (3' c. J. BORDE 34 With these new diagrams we have the choice between two possibilities : I) Use second-quantization for the electromagnetic field as well as for molecules. It is then necessary to introduce a propagator for the radiation density matrix elements. 2) Use the semiclassical approximation in which the classical electromagnetic field enters at each vertex with the corresponding space-time argument. ' s ,whereas the exponential factor exp i0 oscillates rapidly ex~ept when the phase is stationary,and the greatest contribution to the velocity and.