Floquet Hamiltonian eigenvalues

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... 

The eigenvectors of K are the same ones as those for K, and the eigenvalues just have to be multiplied by Ha. The difference with the preceding discussion is that here Ho and V are both of order e. Thus we take as the unperturbed Floquet Hamiltonian just... 

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. 

The Structure of Eigenvectors and Eigenvalues of Floquet Hamiltonians—The Concept of Dressed Hamiltonian... 

This structure of the eigenvectors and eigenvalues of Floquet Hamiltonians can be understood by considering an alternative interpretation of the Floquet eigenvalue problem. 

Using the fact that the Floquet Hamiltonian is time-independent, the solution of the TDSE in T L Eq. (6.57) can be directly expressed as a function of the eigenstates and eigenvalues obtained through... 

We label these two continuous branches by the instantaneous Floquet states v and Y ] The two eigenvalues 7.1 can be deduced from an effective local dressed Hamiltonian... 

This shows that the time evolution is exactly like that of a stationary state of a fime-independenf Hamiltonian, provided the probing is limited to T, or any multiple of T. Since, (0)) is equal fo 4> (0)), Eq. (27) also shows fhaf exp ( - iEiT/h) is an eigenvalue of fhe evolution operator over one period of the field. Suppose now thaf we wish to follow fhe developmenf in fime of an arbitrary initial wavepacket rj 0). We can expand it over the complete set of Floquet eigenfunctions of a given Brillouin zone af fime f = 0 ... 

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. 

In each case, we first studied the laser driven dynamics of the system in the framework of the Floquet formalism, described in Sect. 6.5 of Chap. 6, which provides a geometrical interpretation of the laser driven dynamics and its dependence on the frequency and amplitude of the laser field, through the analysis of the eigenvalues of the Floquet operator, called quasienergies. Various effective models were used for that purpose. This analysis allowed us to explain the shape of the relevant quasienergy curves as a function of the laser parameters, and to obtain the parameters of the laser field that induce the CDT. We then used the MCTDH method to solve the TDSE for the molecule in interaction with the laser field and compare these results with those obtained from the effective Hamiltonian described in Sect. 8.2.3 above. 

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