Time-periodic Hamiltonian

The equation is written in velocity gauge. Atomic units are used. The particle has charge unity and mass m in units of the free electron mass. V is the constant potential energy appropriate for the interval under consideration. The vector potential is supposed to be spatially constant at the length scale of the structure. With such a vector potential, the A2 term contributes an irrelevant phase factor which can be omitted. For a one-mode field A(t) is written as Ao cos(ut). The associated electric field is 0 sin(ut), with 0 = uAq. px is the linear momentum i ld/dx. For such a time-periodic Hamiltonian, a scattering approach can be developped, with a well-defined initial energy, and time-independent transition probabilities for reflection and transmission. 

For time-periodic Hamiltonians Hit) = Hit + T), the Floqnet theorem allows a solution of the time-dependent Schrodinger equation (TDSE) in the following form ... 

It remains to investigate the zeros of Cg t) arising from having divided out by. The position and number of these zeros depend only weakly on G, but depends markedly on the fomi that the time-dependent Hamiltonian H(x, () has. It can be shown that (again due to the smallness of ci,C2,...) these zeros are near the real axis. If the Hamiltonian can be represented by a small number of sinusoidal terms, then the number of fundamental roots will be small. However, in the t plane these will recur with a period characteristic of the periodicity of the Hamiltonian. These are relatively long periods compared to the recurrence period of the roots of the previous kind, which is characteristically shorter by a factor of... 

The parameters e called quasienergies play the same role for periodic motion as the usual energies do for time-independent Hamiltonians. By using the definition (5.6) it is easy to obtain [Benderskii and Makarov 1992]... 

Formally the unperturbed Hamiltonian is equivalent to the Hamiltonian of the hydrogen atom in constant homogenious electric field. Chaotic dynamics of hydrogen atom in constant electric field under the influence of time-periodic field was treated earlier (Berman et. al, 1985 Stevens and Sundaraml987). To treat nonlinear dynamics of this system under the influence of periodic perturbations we need to rewrite (1) in action-angle variables. Action can be found using its standard definition ... 

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]. 

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]. 

Floquet theory provides a generalised form for the propagators of systems with periodically time dependent Hamiltonians [90,91]. The propagator for a doubly periodic Hamiltonian in the BMFT representation maybe written as [92, 93]... 

Define the periodic Hamiltonian in the Hilbert space and transform it to the time independent Fioquet Hamiltonian. 

The Average Hamiltonian Theory (AHT) [23,26] is applied to express the propagator of a periodic Hamiltonian T-Lmtit + krt) = Hintit) in terms of a time independent effective Hamiltonian %). When the Hamiltonian is also cyclic and Uint kTt) = Uint(0) = 1, the propagator gets the generalised form... 

In the semiclassical model the molecule is treated quantum mechanically whereas the held is represented classically. The held is an externally given function of time F that is not affected by any feedback from the interaction with the molecule. We consider the simplest case of a dipole coupling. The formalism is easily extended to other types of couplings. The time dependence of the periodic Hamiltonian is introduced through the time evolution of the initial phase F = F(0 + oat) = electric held and oa is its frequency. The semiclassical Hamiltonian can be, for example, written as... 

Figure 26. Experimental scheme of two-dimensional NMR spectroscopy, (a) The general scheme of two-dimensional spectroscopy. Here, t, and become two variables of two-dimensional response signals. H and are the Hamiltonians during t, and tj periods,

Suppose that we are now perturbing the system with a periodic external potential as explained above, the eigenstates and eigenvalues are solved by the time-dependent Hamiltonian ... 

Thus if the effect of a rf pulse or a free precession period on a spin system with the initial density matrix Ginitiai can be described by the time-independent Hamiltonian H[ then because ... 

The main question is what happens with these invariant surfaces when the streamfunction has a small time-periodic component, 0 < e 1. Are there any invariant surfaces preserved when the perturbation is small or they disappear for arbitrarily small perturbations and the orbits may wander anywhere in the phase space The answers are given by important theorems from the field of Hamiltonian dynamical systems. 

The methods axe applied to multielectron atoms represented by a one-electron potential (73-76). Ben-Tal, Moiseyev and coworkers also developed an approach combining complex-coordinates with the Floquet formalism and applied it to model Hamiltonians (77-82). They also considered non-periodic time-dependent Hamiltonians (81). 

It is useful to note that the case of two DoF Hamiltonian systems is special. In this case a classical result of Ref. [75] (see also Ref. [76]) gives convergence results for the classical Hamiltonian normal form in the neighborhood of a saddle-center equilibrium point. Recently, the first results on convergence of the QNF have appeared. In Ref. [77] convergence resulfs for a one and a half DoF system (i.e., time-periodically forced one DoF Hamiltonian system) have been given. It is not unreasonable that these results can be extended to the QNF in the neighborhood of a saddle-center equilibrium point of a fwo DoF system. 

The power of ultrafast MAS can easily be understood with AHT, as is explained in the seminal paper by Maricq and Waugh [52]. The solution to the periodic Hamiltonian problem H t) is obtained with a Magnus expansion that provides an effective Hamiltonian Hgg- acting on the spin system during a rotor period. This is relevant in the case of stroboscopic observation, that is, when a spectral window of or a sampling dwell time equal to Tr= 1/Vr is used HefF governs the shapes of the sidebands in the MAS spectrum and, indirectly, the resolution that can be achieved. On the other hand, the decay of the rotational echo is responsible for the shape of the spinning sideband pattern [52]. 

We summarize our findings. It is suggested the criterion of critical condition, that is, the extremal behavior of the reaction species concentration, may be applied to reveal the critical conditions of nonlinear chemical dynamic systems. This is with the changeover of different dynamic modes of the reactions, such as the quasi-periodic and chaotic oscillations of the intermediate concentrations, as well as the steady-state mode. At the same time the Hamiltonian formalism makes it possible not only to have a successful numerical identification of the critical reaction conditions, but also to specify the role of individual steps of the reaction mechanism under different conditions. 

T, it may undergo rapid time evolution over many adiabatic states due to the interaction with the external field. Such behavior of the Floquet state is sometimes described as a field-dressed state. Its importance in analysis in the field-induced dynamics is analogous to that of the energy eigenstate under a time-independent Hamiltonian. We will further show that even a small non-periodic term can be incorporated in this framework in an analogous manner as the nonadiabatic transitions in the field-free dynamics. 

Presumably the most straightforward approach to chemical dynamics in intense laser fields is to use the time-independent or time-dependent adiabatic states [352], which are the eigenstates of field-free or field-dependent Hamiltonian at given time points respectively, and solve the Schrodinger equation in a stepwise manner. However, when the laser field is approximately periodic, one can also use a set of field-dressed periodic states as an expansion basis. The set of quasi-static states in a periodic Hamiltonian is derived by a Floquet type analysis and is often referred to as the Floquet states [370]. Provided that the laser field is approximately periodic, advantages of using the latter basis set include (1) analysis and interpretation of the electron dynamics is clearer since the Floquet state population often vary slowly with the timescale of the pulse envelope and each Floquet state is characterized as a field-dressed quasi-stationary state, (2) under some moderate conditions, the nuclear dynamics can be approximated by mixed quantum-classical (MQC) nonadiabatic dynamics on the field-dressed PES. The latter point not only provides a powerful clue for interpretation of nuclear dynamics but also implies possible MQC formulation of intense field molecular dynamics. 

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... 

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