Floquet theory

Schrodinger equation assumes a new and simpler form which can be solved for the functions, ip(r,t)  

The RWA is appropriate when the Rabi frequency is much smaller than the other frequencies in the problem, viz. the transition frequency and the detuning. The first of these conditions limits the intensity, and so ultimately RWA breaks down it then becomes necessary to include the effects of the counterrotating terms, and one returns to solving the time-dependent Schrodinger equation, with a Hamiltonian which is periodic in time this is done in a more general way by applying Floquet s theorem for differential equations with periodically varying coefficients. 

We now describe Floquet theory, as it has been applied to the strong 

The time-dependent Schrodinger equation for a one-electron atom in a coherent field of strength A and frequency u reads  

---

P. Kuchment, Floquet Theory for Partial Differential Equations, Birha, Boston, 1993. 

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... 

VII. Conclusions and Extensions Appendix A Elements of Floquet Theory Acknowledgments References... 

Floquet theory [60] represents a powerful method to analyze and describe NMR experiments including, in the present context, methods for dipolar recoupling. In particular the bimodal variants are promising in the sense that they may handle perturbations/interactions involving multiple time periodicities, as for example induced by combined sample spinning and rf irradiation in the case of dipolar... 

The quantum theory of molecular collisions in external fields described in this chapter is based on the solutions of the time-independent Schrodinger equation. The scattering formalism considered here can be used to calculate the collision properties of molecules in the presence of static electric or magnetic fields as well as in nonresonant AC fields. In the latter case, the time-dependent problem can be reduced to the time-independent one by means of the Floquet theory, discussed in the previous section. We will consider elastic or inelastic but chemically nonreac-tive collisions of molecules in an external field. The extension of the formalism to reactive scattering problems for molecules in external fields has been described in Ref. [12]. 

Fig. 15.10 Cross sections as a function of rf field strength for the first four orders of sideband resonances of the K 29s + K 27d radiative collisions in a 4 MHz rf field, (a) The zero-photon resonant collision cross section, (b) the +1 sideband resonance, (c) the —2 sideband resonance, and (d) the +3 sideband resonance. The solid line shows the experimental data, the bold line indicates the prediction the Floquet theory, and the dashed fine is the result of numerical integration of the transition probability (from ref. 17).

Although CPMAS is the most popular procedure in solid-state NMR, the physics of this phenomenon still requires more investigation. Using Floquet theory, a theoretical model for the description of CP experiments was proposed by Marks and Vega262 for spin-i systems and for CP between spin-) and spin-1... 

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... 

A standard reference for the material in this section is [CLJ. Here, the basic definitions and theorems are given but no proofs are presented. Floquet theory deals with the structure of linear systems of the form... 

This system is periodic and therefore the Floquet theory described in Section 4, Chapter 3, applies. Let 4>(/) be the fundamental matrix solution of (2.2). The Floquet multipliers of (2.2) are the eigenvalues of 4>(w) if /i is a Floquet multiplier and /i = e" then A is called a Floquet exponent. Only the real part of a Floquet exponent is uniquely defined. 

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. 

Keywords Solid-state NMR Bimodal Floquet theory... 

The efficient combination of RF pulses with MAS requires theoretical tools. While the initial multiple-pulse sequences were based on the average Hamiltonian theory (AHT) [14,25,26,32,33], some of the later schemes were more effectively formulated on the basis of Floquet theory [52], combinations of both AHT and Floquet theory [51], synchronisation arguments [42,43,46,55], and numerical methods [50]. A combination of RF pulses with MAS was also visualised in terms of certain symmetry conditions based on AHT [77]. An attempt to present all these in a unified picture is perhaps not out of place for a better understanding of the underlying phenomena. 

The bimodal Floquet theory representation of the time-dependent spin... 

To be consistent with our Floquet theory notation we denote the symmetry-adapted sequences from Levitt s group as CN and RN, instead of CN]( and RN](, as detailed later. 

In this section we introduce bimodal Floquet theory (BMFT), based on which we represent the spin Hamiltonians and analyse the various homonuclear dipolar decoupling sequences. 

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]. 

---