Floquet Hamiltonian transformation

This effective Hamiltonian is again not unique but can be chosen such that its eigenvalue differences are smaller than l/2o t. Maricq [100-102] and others [14, 103] have demonstrated that the Magnus expansion of the effective Hamiltonian in AHT and the van Vleck transformation approach of the Floquet Hamiltonian are equivalent. At the time points krt the Floquet solution for the propagator in Eq. 24 has the form... 

For the block diagonalisation procedure we can use the van Vleck transformation. This procedure eliminates off-diagonal blocks of the Floquet Hamiltonian modifying the diagonal blocks. A first-order transformation removes... 

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... 

This transformed Floquet Hamiltonian is nonresonant for values of up to a certain amplitude > 2oo, and the KAM iteration based on Eq. (124) can be expected to converge. 

In this subsection we will combine the general ideas of the iterative perturbation algorithms by unitary transformations and the rotating wave transformation, to construct effective models. We first show that the preceding KAM iterative perturbation algorithms allow us to partition at a desired order operators in orthogonal Hilbert subspaces. Its relation with the standard adiabatic elimination is proved for the second order. We next apply this partitioning technique combined with RWT to construct effective dressed Hamiltonians from the Floquet Hamiltonian. This is illustrated in the next two Sections III.E and III.F for two-photon resonant processes in atoms and molecules. 

To extract from the effective Floquet Hamiltonian (185) an effective dressed Hamiltonian independent of 0, we can apply a contact transformation consisting... 

Transforming the polarizability into the laboratory frame and using 0 to denote the angle between the molecular axis and the polarization axis of the laser, we obtain the effective Floquet Hamiltonian in the ground electronic state of the linear molecule ... 

We will apply specific rotating wave transformations R that will allow us to identify resonant terms and to eliminate the nonresonant ones. We obtain an effective one-mode Floquet Hamiltonian of the form... 

Applying this transformation to the Floquet Hamiltonian (304) gives... 

To obtain the effective Floquet Hamiltonian, we apply the rotating wave transformation (RWT)... 

Acting with 2T-m from the left on (A 18) and with. %itn from the right, one obtains that C induces a unitary transformation of the Floquet Hamiltonian in the enlarged space JT ... 

The transformed Floquet Hamiltonian K 2 = S1K2S1 then reads... 

To obtain the effective Hamiltonian we need to diagonalise the SMFT Hamiltonian in Floquet space. When this is not practical we should consider perturbation expansions. The van Vleck transformation [96] will be the most convenient approach in this case. The result will be an expansion of the effective Hamiltonian Heff in terms of higher-order terms with... 

We will discuss the Floquet approach from two different points of view. In the first one, discussed in Section II.A, the Floquet formalism is just a mathematically convenient tool that allows us to transform the Schrodinger equation with a time-dependent Hamiltonian into an equivalent equation with a time-independent Hamiltonian. This new equation is defined on an enlarged Hilbert space. The time dependence has been substituted by the introduction of one auxiliary dynamical variable for each laser frequency. The second point of... 

Since in the Floquet representation the Hamiltonian K defined on the enlarged Hilbert space is time-independent, the analysis of the effect of perturbations (like, e.g., transition probabilities) can be done by stationary perturbation theory, instead of the usual time-dependent one. Here we will present a formulation of stationary perturbation theory based on the iteration of unitary transformations (called contact transformations or KAM transformations) constructed such that the form of the Hamiltonian gets simplified. It is referred to as the KAM technique. The results are not very different from the ones of Rayleigh-Schrodinger perturbation theory, but conceptually and in terms of speed of convergence they have some advantages. 

The use of van Vleck s contact transformation method for the study of time-dependent interactions in solid-state NMR by Floquet theory has been proposed. Floquet theory has been used for studying the spin dynamics of MAS NMR experiments. The contact transformation method is an operator method in time-independent perturbation theory and has been used to obtain effective Hamiltonians in molecular spectroscopy. This has been combined with Floquet theory to study the dynamics of a dipolar coupled spin (I = 1/2) system. 

For oscillating time-dependent elecbic fields with frequency (o we transform 77bo by Fourier expansion in o> to a Floquet picture, and thus a time-independent Hamiltonian, whose eigenvalues provide the dressed Bom-Oppenheimer potentials. 

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