Dressed Hamiltonians

It may be noticed that by virtue of being a dressed Hamiltonian including the effect of correlation, diagrams 37-39 account for some of the CIS(D)-type correlation corrections (22-24) to CIS (diagrams 35 and 36). There is no need to make ad hoc adjustments to diagrams to ensure size correctness because diagram 37 (unlinked) cancels exactly between the ground and excited states. In other words, P-EOM-MBPT(2) has the factorization approximation built in. 

D. Effective Dressed Hamiltonians by Partitioning of Floquet Hamiltonians... 

Effective Dressed Hamiltonians Partitioning in the Enlarged Space... 

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 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. 

The second step is the construction of an effective dressed Hamiltonian, independent of the 0-variable, inside the block connected to the initial condition. This can be done by the KAM iterations combined by the RWT techniques to treat the resonances. The second step depends on the specific problem that is treated. 

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

To determine an effective dressed Hamiltonian characterizing a molecule excited by strong laser fields, we have to apply the standard construction of the free effective Hamiltonian (such as the Born-Oppenheimer approximation), taking into account the interaction with the field nonperturbatively (if resonances occur). This leads to four different time scales in general (i) for the motion of the electrons, (ii) for the vibrations of the nuclei, (iii) for the rotation of the nuclei, and (iv) for the frequency of the interacting field. It is well known that it is a good strategy to take into account the time scales from the fastest to the slowest one. 

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 most general dressed Hamiltonian in the rotating wave approximation for these processes reads [69]... 

The derivation term is the relative number operator for pairs of photons, one pump-field photon minus one Stokes-field photon. The second term is the well-known RWA Hamiltonian (dressed Hamiltonian used in the usual STIRAP), and the third one can be viewed as a perturbation of this RWA Hamiltonian. 

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

Hence, the determination of the eigenelements of K in Jf is reduced to the determination of those of B in //. When such a transformation C(0) can be found, the operator B is called the dressed Hamiltonian. Although it acts only on the molecular Hilbert space M, it contains the information on the photons, which dress the molecule. The transformation C(0) can be interpreted as a change of representation. We remark that the transformation C(x, 0), and thus the dressed Hamiltonian B, is clearly not unique since C(x, 0) can be composed with any unitary transformation that acts inside Jf. 

If the cluster operator is connected, one can easily show that the dressed Hamiltonian and the matrix elements are also connected via multi-commutator expansion. Hence, the proof of the connectedness of the first term of Eq. (7) is quite... 

Since ij/Q is a CAS-type function, the first term and the dressed Hamiltonian of the second term of the left-hand side of Eq. (30) are manifestly connected involving H, T and via commutator expansion, and it is hence enough to show that the third term of the equation is connected. According to Baker-Campbell formula, the second term of the above equation can be written as follows ... 

We now introduce two effective operators. The first is a dressed Hamiltonian operator... 

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