The Floquet Hamiltonian

To finalise the discussion let us have a closer look at the Floquet Hamiltonian of our spin system. Its general form is  

The schematic of Floquet energy levels is shown in Fig. 16. For certain characteristic frequencies these energy differences can become zero and a Floquet energy level crossing occurs  

of course, is not always the case, and when Eq. 47 fails, an exact diagonalisation should be performed. At the non-crossing conditions perturbation approaches are justified when oJagn,admi. and ojabGn,abdmli of the CSA and the DD/iomo interaction tensor elements are relatively small. For example, for the elimination of the homo influence on the effective Hamiltonian we should require at least 

The conditions for CSA terms are more problematic because in many instances the CSA frequency is larger than the spinning frequency, uja and 

With the constraints of Eq. 47 we can, in general, use perturbation theory to find the approximate eigenvalues oiT-Lp. In the coming sections we will do so by using van Vleck perturbation theory, but only after discussing Floquet energy level crossings. 

---

The states an) are the Floquet states and is the Floquet Hamiltonian. The diagonal matrix elements of Hp in the Floquet basis are... 

The sums over n in the Fourier expansions (8.16) and (8.17) extend to infinity. Therefore, the matrix of the Floquet Hamiltonian in Eq. (8.20) is infinitely large. 

If the basis of molecular states is restricted only to two states a and / , the matrix of the Floquet Hamiltonian has the following form ... 

In the presence of the field, the molecular states are coherent superpositions of the states AM ) A ). In principle, the basis set must include an infinite number of states A ). However, the Floquet Hamiltonian matrix is block-diagonal and the diagonal matrix elements of the Floquet matrix separate in values SiS k-k increases. This suggests that it may be possible to include in the basis set a finite number of states from - max to max seek convergence with respect to In other words, the eigenstates of the Floquet Hamiltonian... 

The energy levels of a molecule placed in an off-resonant microwave field can be calculafed by diagonalizing fhe mafrix of fhe Floquef Hamiltonian in the basis of direct products y) ), where y) represents in the eigenstates of the molecule in the absence of the field and ) - fhe Fourier componenfs in Eq. (8.21). The states k) are equivalent to photon number states in the alternative formalism using the quantum representation of the field [11, 15, 26], The eigensfales of the Floquet Hamiltonian are the coherent superpositions... 

Complex rotation can be usefully applied also to the case of the interaction of an atom with a time-dependent perturbation. With the Floquet formalism by Shirley [41], it was shown that, for a time-periodic field, the dressed states of the combined atom-field system can be characterized non-perturbatively by the eigenstates of a time-independent, infinite-dimensional matrix. The combination of the Floquet approach with complex rotation, proposed by Chu, Reinhardt, and coworkers [37, 42, 43], permits to account for the field-induced coupling to the continuum in an efficient way. As in the time-independent case, this results in complex eigenvalues (this time to the Floquet Hamiltonian matrix) and again the imaginary parts give the transition rate to the continuum. This combination has since then been successfully used to examine various strong field phenomena a review can be found in Ref. [44]. 

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 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... 

In this expression, H(Q) is just the semiclassical Hamiltonian (1) but with the phase 0(f) taken at the (fixed) initial value 0 corresponding to t = 0. The usefulness of the Floquet Hamiltonian comes from the fact that it is time-independent and that the dynamics it defines on is essentially equivalent with the one of Eq. (2). This can be formulated as follows. The Floquet Hamiltonian K defines a time evolution in through the equation... 

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. 

We will establish a precise relation between dressed states in a cavity and the Floquet formalism. We show that the Floquet Hamiltonian K can be obtained exactly from the dressed Hamiltonian in a cavity in the limit of infinite cavity volume and large number of photons K represents the Hamiltonian of the molecule interacting in free space with a field containing a large number of photons. We establish the physical interpretation of the operator... 

In this limit, the Hamiltonian //ml is identical, up to an additive constant, to the Floquet Hamiltonian K... 

The formal hypothesis (31) must be interpreted in relation to the functions on which — /S/00 acts. The statement is that if all the states elk()) that are relevant in the dynamics are such that k exchanged between light and matter compared to the average photon number n contained in the laser field — then the coupled Hamiltonian //[ () can be identified with the Floquet Hamiltonian K. 

From the formulation of the Floquet formalism given above, we can establish the precise connection between the dynamics in the enlarged space C/f defined by the Floquet Hamiltonian K, and the one defined by the semiclassical Hamiltonian in with a classical description of the electric field ... 

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. 

Interaction Representation The Schrodinger equation of the Floquet Hamiltonian in... 

As we have stated, the Floquet Hamiltonian (113) has no terms that are resonant if we take small enough e, and the iteration of the KAM procedure converges. However, if we take e large enough, we encounter new resonances that are not present at zero or small fields that is, they are not related to degeneracies of the unperturbed eigenvalues of Kq that lead to the zero-field resonances we have discussed in the previous subsection. These new resonances are related to degeneracies of the new effective unperturbed operator K 0(e), which appear at some specific finite values of e. These are the dynamical resonances. 

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. 

We next partition the Floquet Hamiltonian with respect to these atomic blocks. We obtain effective Floquet Hamiltonians inside each block. 

We partition the Floquet Hamiltonian such that the states a) and b) span the Hilbert subspace and the other atomic states 11),..., N) span the Hilbert subspace. ... 

We remark that this effective Hamiltonian (190) constructed by the combination of a partitioning of the Floquet Hamiltonian, a two-photon RWT, and a final 0-averaging can be seen as a two-photon RWA, which extends the usual (one-photon) RWA [39,40], We have thus rederived a well-known result, using stationary techniques that allow us to estimate easily the order of the neglected terms. This method allows us also to calculate higher order corrections. We apply it in the next subsection to calculate effective Hamiltonians for molecules illuminated by strong laser fields. 

The Born-Oppenheimer approximation allows us to decouple the electronic and nuclear motions of the free molecule of the Hamiltonian Hq. Solving the Schrodinger equation //o l = with respect to the electron coordinates r = r[, O, gives rise to the electronic states (r, R) = (r n(R)), n = 0,..., Ne, of respective energies En (R) as functions of the nuclear coordinates R, with the electronic scalar product defined as (n(R) n (R))r = j dr rj( r. R) T,-(r, R). We assume Ne bound electronic states. The Floquet Hamiltonian of the molecule perturbed by a field (of frequency co, of amplitude 8, and of linear polarization e), in the dipole coupling approximation, and in a coordinate system of origin at the center of mass of the molecule can be written as... 

For a process with two lasers, the Floquet Hamiltonian is (see Section II.E)... 

We first derive the time-dependent dressed Schrodinger equation generated by the Floquet Hamiltonian, relevant for processes induced by chirped laser pulses (see Section IV.A). The adiabatic principles to solve this equation are next described in Section IV.B. 

We derive the Floquet Hamiltonian K associated to this semiclassical Hamiltonian by starting with the following definition of the corresponding propagator, which is the natural generalization of (13)... 

The above result is proved in Appendix A. We point out the appearance of an effective instantaneous frequency in the Floquet Hamiltonian (227). 

The preceding analysis is well adapted when one considers slowly varying laser parameters. One can study the dressed Schrodinger equation invoking adiabatic principles by analyzing the Floquet Hamiltonian as a function of the slow parameters. 

A sketch of an argument that leads to the adiabatic theorem for the Floquet Hamiltonian of an iV-level system is given in Appendix C. 

---