Floquet approach

The other way of describing the transitions is based on a Floquet approach.13-16 To illustrate the essential ideas we use a simple model to calculate the Rabi frequencies for the problem depicted in Fig. 10.10. There are two states 1, which has a linear Stark shift, and 2, which has no Stark shift. They are coupled by the... 

There are many forms in which the Floquet approach can be more rigorously implemented.13-16 One which has been used often is the infinite matrix approach of Shirley.14 It corresponds roughly to the infinite set of sidebands. An alternative more compact approach has been described by Sambe15 and Christiansen-Dalsgaard.16... 

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]. 

Although extending the Landau-Zener description to many cycles accurately describes the evolution from the non-resonant interaction with a single cycle to the resonant interaction with many cycles, this approach is not very convenient to use nor does it easily lead to any analytic predictions. Instead we use a Floquet approach [20-22]. We assume that we have two states, the (n -b 2)s state and the (n, 3) Stark state. The effective Hamiltonian,... 

We will discuss the Floquet approach from two different points of view. In the first one, discussed in Section II.A, the Floquet formalism is just a mathematically convenient tool that allows us to transform the Schrodinger equation with a time-dependent Hamiltonian into an equivalent equation with a time-independent Hamiltonian. This new equation is defined on an enlarged Hilbert space. The time dependence has been substituted by the introduction of one auxiliary dynamical variable for each laser frequency. The second point of... 

This Floquet approach provides a physical interpretation of the dynamics in terms of photons in interaction with the molecule, which is in close analogy to the theory of dressed states in a cavity (see Section II.D). 

An introduction to Floquet theory has been presented. The potential of this theoretical approach has been demonstrated using explicit calculations of the sideband patterns in MAS NMR. It has been shown that the Floquet theory works by expanding the periodic (due to sample spinning) Hamiltonian into a Fourier series, and that, regardless of the complexity of the time dependence of the Hamiltonian, the Floquet approach is the same. 

Equations (46) and (47) provide the required time dependence for an arbitrary number tt of periods of the field. Provided that the periodicity of the Hamiltonian is known, the Floquet approach outlined above is particularly useful for deriving accurate results for all the coherent mechanisms of multiphoton excitation outlined in Figure 2. In particular, direct multiphoton transitions in two-level models have been discussed by Shirley, using the Floquet approach. The Floquet-Liapounoff approach for the numerical treatment of many level molecular multiphoton excitation was introduced in Ref. 14. 

For the most general time dependent fields and also in the Floquet approach for one period of the field one needs numerical integration schemes. Without even making use of any matrix algebra, well known integration procedures for... 

A quite different approach from all other presented in this review has been recently proposed by Coffey [41]. This approach allows both the MFPT and the integral relaxation time to be exactly calculated irrespective of the number of degrees of freedom from the differential recurrence relations generated by the Floquet representation of the FPE. 

For the driven atom, we developed an accurate approach without any adjustable parameter, and with no other approximation than the confinement of the accessible configuration space to two dimensions. This method was successfully applied for the study of the near resonantly driven frozen planet configuration. Floquet states were found that are well localized in the associated phase space and propagate along near-... 

Wave propagation in periodic structures can be effieiently modeled using the eoncept of Bloeh (or Floquet-Bloch) modes . This approach is also applicable for the ealeulation of band diagrams of 1 -D and 2-D photonic crystals . Contrary to classical methods like the plane-wave expansion , the material dispersion ean be fully taken into aeeount without any additional effort. For brevity we present here only the basie prineiples of the method. 

E. This double -1 point is yet another codimension-two bifurcation, which will be discussed in detail later. Another period 1 Hopf curve extends from point F through points G and H. F is another double -1 point and, as one moves away from F along the Hopf curve, the angle at which the complex multipliers leave the unit circle decreases from it. The points G and H correspond to angles jt and ixr respectively and are hard resonances of the Hopf bifurcation because the Floquet multipliers leave the unit circle at third and fourth roots of unity, respectively. Points G and H are both important codimension-two bifurcation points and will be discussed in detail in the next section. The Hopf curves described above are for period 1 fixed points. Subharmonic solutions (fixed points of period greater than one) can also bifurcate to tori via Hopf bifurcations. Such a curve exists for period 2 and extends from point E to K, where it terminates on a period 2 saddle-node curve. The angle at which the complex Floquet multipliers leave the unit circle approaches zero at either point of the curve. 

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. 

A. Tikhonov, R.D. Coalson, Yu. Dahnovsky, Calculating electron transport in a tight binding model of a field-driven molecular wire Floquet theory approach, J. Chem. Phys. 116... 

The outcome of competition between two competitors depends on the stability properties of the single-competitor periodic solutions Ei and p2-It turns out that the stability of these solutions is determined, in each case, by a single Floquet exponent in a biologically intuitive way. Suppose that a chemostat is charged at / = 0 with only the competitor X[. According to Proposition 3.2, the concentration Xi very rapidly approaches the level... 

Rydberg atoms and microwave fields constitute an ideal system for the study of atom-strong field effects, and they have been used to explore the entire range of one electron phenomena [5]. Here we focus on an illustrative example, which has a clear parallel in laser experiments, a series of experiments which show that apparently non-resonant microwave ionization of nonhydronic atoms proceeds via a sequence of resonant microwave multiphoton transitions and that this process can be understood quantitatively using a Floquet, or dressed state approach. 

In the following section the experimental approach is briefly described. The initial observations of microwave ionization and the completely non-resonant picture initially used to describe it are then presented. Then microwave multiphoton transitions in a two level system analogous to the rate limiting step of microwave ionization are described both experimentally and theoretically. Experiments on this two level system with well controlled pulses of microwaves to show the applicability of an adiabatic Floquet theory to pulses are then described. We finally return to microwave ionization to see evidence for the resonant nature of the process. 

Evaluation of the response of the spin system to a time dependent Hamiltonian requires an appropriate mathematical framework. This framework must deal with Hamiltonians that are periodically time dependent with at least two characteristic frequencies, and Wc, that are not necessarily commensurate. We choose bimodal Floquet theory (BMFT) towards this, and in this Section we will set the basis of this theory. The approach is very similar to the single mode Floquet theory (SMFT) approach adapted by others to NMR spectroscopy [91]. 

To obtain the effective Hamiltonian we need to diagonalise the SMFT Hamiltonian in Floquet space. When this is not practical we should consider perturbation expansions. The van Vleck transformation [96] will be the most convenient approach in this case. The result will be an expansion of the effective Hamiltonian Heff in terms of higher-order terms with... 

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 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 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... 

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