Hilbert space Floquet Hamiltonian

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... 

The Floquet Hamiltonian K, also called the quasienergy operator, is constructed as follows We define an enlarged Hilbert space... 

This space is generated by the orthonormal basis eM, k G Z (i.e., all integers). On the enlarged Hilbert space /f the Floquet Hamiltonian is defined as... 

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. 

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 Floquet Hamiltonian Kril acts on the enlarged Hilbert space Jf = 2(T. d()/2n), where Td is the d-dimensional unit torus. 

Hamiltonian in an extended space, the direct product of the usual molecular Hilbert space, and the space of periodic functions of f e [0,T]. This extension of the Hilbert space can be made somewhat more transparent by introducing a new time-like variable, to be distinguished from the actual time variable t. This new time variable can be defined through the arbitrary phase of the continuous (periodic) field, as done in Ref. [28, 29]. A variant of the idea is found in the (f, t ) method developed by Peskin and Moiseyev [30] and applied to the photodissociation of HJ [31, 32]. We will continue with the more traditional and simpler formulation of Floquet theory here, as this is sufficient to bring out ideas of laser-induced resonances in the dressed molecule picture. 

This space is spanned by the basis >, k eZ with 0 ) = We introduce a time-independent Hamiltonian, called the Floquet Hamiltonian, acting in the enlarged Hilbert space K = H defined as... 

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