Floquet Hamiltonian

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]. 

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

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... 

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

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

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... 

Combining the KAM techniques with the RWT will allow us to construct effective Hamiltonians in a systematic way and to estimate the order of the neglected terms (see Section III.D). We show that the KAM technique allows us to partition at a desired order operators in orthogonal Hilbert subspaces. We adapt this partitioning technique to treat Floquet Hamiltonians. Its connection with the standard adiabatic elimination is shown at a second-order approximation. 

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... 

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