Semiclassical model Floquet Hamiltonian

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 in the semiclassical model the only dynamical variables are those of the molecule, and the extended Hilbert space. if = M M and the Floquet Hamiltonian K can be thought as only mathematically convenient techniques to analyze the dynamics, it was clear from the first work of Shirley [1] that the enlarged Hilbert space should be related to photons. This relation was made explicit by Bialynicki-Birula and co-workers [7,8] and completed in [9]. The construction starts with a quantized photon field in a cavity of finite volume in interaction with the molecule. The limit of infinite volume with constant photon density leads to the Floquet Hamiltonian, which describes the interaction of the molecule with a quantized laser field propagating in free space. The construction presented below is taken from Ref. 9, where further details and mathematical precisions can be found. 

The Schrodinger equation of the Floquet Hamiltonian in JT, where 9 is a dynamical variable, is equivalent, in an interaction representation, to the semiclassical Schrodinger equation in where 0 is considered as a parameter corresponding to the fixed initial phase. The dynamics of the two models are identical if the initial photon state in the Floquet model is a coherent state. 

The models we have discussed so far correspond to continuous (CW) lasers with a fixed sharp frequency and constant intensity. They can be easily adapted to the case of pulsed lasers that have slowly varying envelopes. They can furthermore have a chirped frequency—that is, a frequency that changes slowly with time. For periodic (or quasi-periodic) semiclassical Hamiltonians, the Floquet states are the stationary states of the problem. Processes controlled by chirped laser pulses include additional time-dependent parameters (the pulse... 

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