Schrodinger equation semiclassical formalism

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

Although this equation looks formally like the semiclassical Schrodinger equation (2), we emphasize that it is still different because it is defined in the enlarged Hilbert space X and the phase 0 does not have a definite value, since it is a dynamical variable on the same footing as x. In order to recover the semiclassical equation from Eq. (39), we have to reduce it to an equation defined in the Hilbert space Ji. From a mathematical point of view, this can be done by fixing a particular value of 0, as we did in Section II.A. Physically, this can be achieved, as we show in the following, by choosing the initial condition of the photon field as a coherent state. 

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