Floquet operator

There are also important advances concerning the theoretical description of Rydberg atoms in strong radiation fields. Buchleitner et d. (1995) report on fully fiedged three-dimensional computations of the microwave ionization problem. They use the method of complex rotation discussed in Section 10.4.1, adapted to the computation of the resonances of the Floquet operator. The computed ionization probabilities are in good overall agreement with existing experimental data. 

The Fourier representation of the Floquet operator thus obtained is an infinite dimensional matrix in general. Fortunately, however, it often takes a simple form when the time dependence of the Hamiltonian is monochromatic, or composed of a single mode with frequency w in such case have nonzero values only for n — m = 0, 1. 

The Floquet theorem, when apphed to the quantum mechanics [370], implies the stationarity of Floquet states imder a perfectly periodic Hamiltonian. We define the electronic Floquet operator as 7ft = Hf — ihdt and the Floquet states as its periodic eigenstates which satisfy Tlt x t)) = A] A(f))- The above mentioned stationarity states that the solution of time-dependent Schrodinger equation ] t) can be expanded as... 

Substituting the parameterized Hamiltonian HsUH ) into Eq. (8.25), the Floquet-type operator becomes Hs TZ ) = Hs Tlt ) - ih- , which is now a true Floquet operator in the sense that it has the exact periodicity in the variable s. It thus follows that the eigenstates of this Floquet operator Ffs( 7 -f ) are also periodic in the variable s. We define parameterized Floquet states... 

To derive the Floquet operator associated with the Hamiltonian Eq. (6.77), we start from the TDSE... 

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. 

Floquet et al. (1994) applied SA to problems of separating a mixture of n components into pure products at minimal annual investment plus operating costs. The assumptions used were... 

The size of the matrix as it operates on the perturbation vector is directly related to the eigenvalues of J (or of B). The eigenvalues of J are known as the Floquet multipliers fit the eigenvalues of B are the Floquet exponents / ,. In general the former are easier to evaluate, although we should identify the parameter p2 introduced in chapter 5 with the Hopf bifurcation formula as a Floquet exponent for the emerging limit cycle (then P2 < 0 implies stability, P2 > 0 gives instability, and P2 = 0 corresponds to a bifurcation between these two cases). 

Floquet et al. (1985) proposed a tree searching algorithm in order to synthesize chemical processes involving reactor/separator/recycle systems interlinked with recycle streams. The reactor network of this approach is restricted to a single isothermal CSTR or PFR unit, and the separation units are considered to be simple distillation columns. The conversion of reactants into products, the temperature of the reactor, as well as the reflux ratio of the distillation columns were treated as parameters. Once the values of the parameters have been specified, the composition of the outlet stream of the reactor can be estimated and application of the tree searching algorithm on the alternative separation tasks provides the less costly distillation sequence. The problem is solved for several values of the parameters and conclusions are drawn for different regions of operation. 

Fig. 13a-d The number operators a N and b, and the ladder operators c and d F[ in the manifold of Floquet states nk) with n) the Fourier states corresponding to the spinning frequency ujr and fc) the Fourier states corresponding to the characteristic RF frequency ujc... 

Fourier and Floquet space. The definitions of the different block-elements are given in a for the Fourier representation and in b for the Floquet representation. The Hamiltonians retain the same general form, but the time-dependent elements in a become time-independent in b with the addition of the number operator elements on the diagonal... 

Before using these expressions and transforming the spin system to Floquet space, we should discuss some features of the evolution operator and... 

The choice of this form for the initial density matrix is in line with the above definitions of the Fourier and Floquet space, and maintains the form of all operators in Floquet space as in Eq. 28. 

The back transformation of operators from Floquet space to Hilbert space results in the expression originally derived by Chu et al. [92,93]. Using the expressions for and in Eq. 29 ... 

The Floquet Hamiltonian K, also called the quasienergy operator, is constructed as follows We define an enlarged Hilbert 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... 

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. 

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. 

The usual RWA consists in neglecting the 0-dependent operator V. The first term of V) (319) contains the counterrotating terms of the pump laser on the 1-2 transition and of the Stokes laser on the 2-3 transition. The next two terms correspond to the interactions of the pump laser on the 2-3 transition and of the Stokes laser on the 1-2 transition. Following the hypothesis (316), we neglect the first two terms and keep the last term, which becomes large (see Ref. 38 for details) when maxJ Oi(t), fi2(t) ] approaches or overcomes 8. The (approximate) effective one-mode Floquet Hamiltonian is thus... 

The use of van Vleck s contact transformation method for the study of time-dependent interactions in solid-state NMR by Floquet theory has been proposed. Floquet theory has been used for studying the spin dynamics of MAS NMR experiments. The contact transformation method is an operator method in time-independent perturbation theory and has been used to obtain effective Hamiltonians in molecular spectroscopy. This has been combined with Floquet theory to study the dynamics of a dipolar coupled spin (I = 1/2) system. 

A theoretical treatment of the DREAM adiabatic homonuclear recoupling experiment has been given using Floquet theory. An effective Hamiltonian has been derived analytically and the time evolution of the density operator in the adiabatic limit has been described. Shape cycles have been proposed and characterized experimentally. Application to spin-pair filtering as a mixing period in a 2D correlation experiment has been explored and the experimental results have been compared to theoretical predictions and exact numerical simulations. 

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