Floquet dynamics

Care must be exercised to distinguish the concept of adiabatic Floquet dynamics introduced here, which refers to an adiabatic time-evolution, or to the slow variations of the Floquet basis with time, from the concept of adiabatic representation defined in the previous section, which refers to the slow variations of the electronic Hamiltonian (Floquet or not) with respect to nuclear motions (i.e., the noncommutativity of the electronic Hamiltonian Hei and the nuclear KE operator Tjv). Where confusion is possible and to be avoided, we shall refer to this concept of adiabaticity related to the BO approximation as the R-adiabaticity, while adiabaticity in actual time evolution will be termed t-adiabaticity. Non-adiabatic effects in time evolution are due to a fast variation of the (Floquet) Hamiltonian with time, causing Floquet states to change rapidly in time, to the extent that in going from one time slice to another, a resonance may be projected onto many new resonances as well as diffusion (continuum) states [40], and the Floquet analysis breaks down completely. We will see in Section 5 how one can take advantage of such effects to image nuclear motions by an ultrafast pump-probe process. 

This dependence of the H+ KE on the XUV-IR delay in this case of the longer, 35 fs FWHM, IR pulse can be understood in terms of the adiabatic-ity of the Floquet dynamics underlying the dissociation processes, and the way that the IR intensity affects both the preparation and the propagation of the Floquet components of the wavepackets. More precisely, the IR probe pulse projects the various vibrational components of the wavepacket onto Floquet resonances, whose widths vary with the intensity of the IR pulse. We recall that these resonances are of two types Shape resonances supported by the lower adiabatic potential defined at the one-photon crossing between the dressed (g, n), (u, n ) channels and leading to efficient dissociation through the BS mechanism, or Feshbach resonances, vibrationally trapped in the upper adiabatic potential well. 

Here, pj denotes the nonadiabatic transition probability for one passage of the avoided crossing Xt, and / are the dynamical phases due to the nonadiabatic transition atX, is the kth adiabatic Floquet state, Xq = i and X3 = 2- The transition amplimde Eq. (152) can be explicitly expressed as... 

Floquet theory principles, 35—36 single-surface nuclear dynamics, vibronic multiplet ordering, 24—25 Barrow, Dixon, and Duxbury (BDD) method, Renner-Teller effect tetraatomic molecules, Hamiltonian equations, 626-628 triatomic molecules, 618-621 Basis functions ... 

Floppy molecules, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 701-711 Floquet theory, geometric phase theory principles of, 33-36... 

Quantum dynamical calculations on the IRMPE/D of 614, 615 O3. Quantum interference effects and discussion of the possibility of mode-selective excitation and reaction Confirmation that OCS does not undergo IRMPD at 616 high laser fluences (ca. 250 J cm ). Laser-induced dielectric breakdown in OCS, OCS-He, and OCS-Ar does lead to dissociation, giving CO + S Ab initio study of SO2 IRMPE using the most proba- 617 ble path approximation to select the most important paths within the semiclassical Floquet matrix. Conclude that collisionless MPD of SO2 will not occur at laser field strengths <20 GW cm ... 

A general review on distillation synthesis can be found in Westerberg (1985) and Floquet et ai. (1988). The synthesis of simple separation sequences based on heuristics is fairly well developed. Enumeration search methods (dynamic programming, branch-and-bound) have been proposed by Hendry and Hughes (1972) and Gomez-Munoz and Seader (1985), while evolutionary search procedures have been described by Stephanopoulos and Westerberg (1976), Seader and Westerberg (1977) and Nath and Motard (1981). 

CONTROL OF QUANTUM DYNAMICS BY LASER PULSES ADIABATIC FLOQUET THEORY... 

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... 

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). 

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. 

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. 

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. 

When considering the stationary Floquet Hamiltonian K ihmf - 11(0), 77(0) = Ho + F(0), describing the dynamics of a quantum (atomic or molecular) system Hq, illuminated by a strong photon field (of one frequency), we have to extend the preceding partitioning to the enlarged space JT = 0... 

More than one Floquet state can be involved in the dynamics—for example, if the initial condition is a linear combination of the instantaneous eigenvectors. These Floquet states span a subspace ff, and the adiabatic transport can be formulated in terms of eigenvectors ... 

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