Eigenvalues dressed Hamiltonian

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... 

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

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]. 

The polaron transformation, executed on the Hamiltonian (12.8)-( 12.10) was seen to yield a new Hamiltonian, Eq. (12.15), in which the interstate coupling is renormalized or dressed by an operator that shifts the position coordinates associated with the boson field. This transformation is well known in the solid-state physics literature, however in much of the chemical literature a similar end is achieved via a different route based on the Bom-Oppenheimer (BO) theory of molecular vibronic stmcture (Section 2.5). In the BO approximation, molecular vibronic states are of the form (/) (r,R)x ,v(R) where r and R denote electronic and nuclear coordinates, respectively, R) are eigenfunctions of the electronic Hamiltonian (with corresponding eigenvalues E r ) ) obtained at fixed nuclear coordinates R and... 

We derive a set of Born-Oppenheimer potentials by first separating Equation 12.19 into center-of-mass and relative coordinates, and diagonalizing the Hamiltonian //rei for the relative motion for fixed molecular positions. Within an adiabatic approximation, the corresponding eigenvalues play the role of an effective three-dimensional interaction potential in a given state manifold dressed by the external field. 

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. 

Kato and Kono [4] were the first to develop an approximate method for computing the TDPES in an intense laser field. In their approach, the TDPES was defined in terms of instantaneous eigenvalues of the field-dressed electronic Hamiltonian. In other words, in the Kato-Kono approach, the TDPES is basically the adiabatic potential energy that is computed by including the instantaneous value of electron-field interactions in the Hamiltonian. This method has proved to be extremely helpful in understanding many intense field phenomena in realistic molecular systems. However, this method lacks the exact dynamical features of the intense laser-molecule interactions, and therefore, a need still exists for a method that is not computationally as expensive as the exact methods but captures the dynamical features of electrons/molecules in strong laser fields. 

---