Dressed Schrodinger equation

A. The Dressed Schrodinger Equation for Chirped Laser Pulses... 

We first derive the time-dependent dressed Schrodinger equation generated by the Floquet Hamiltonian, relevant for processes induced by chirped laser pulses (see Section IV.A). The adiabatic principles to solve this equation are next described in Section IV.B. 

The preceding analysis is well adapted when one considers slowly varying laser parameters. One can study the dressed Schrodinger equation invoking adiabatic principles by analyzing the Floquet Hamiltonian as a function of the slow parameters. 

It is convenient to consider explicitly the time scale in the slow parameters A (,v) and u err(.v), where s = t/r is a reduced time, x a characteristic time for the slow parameters, and t is the physical time. The slow parameters are gathered in a formal vector r( ) = [A (v), f err (-v)]- The dressed Schrodinger equation reads... 

Figure 21. From the dressed Schrodinger equation, with the same parameters as in Fig. 20. (a) Population histories P (t) for n = 1,2,3 (top frame), (c) photon histories (bottom frame), associated with the dressed spectrum in middle frame (b). The arrow characterizes the transfer eigenvector. Vertical lines indicate where the pump pulse starts and the Stokes pulse ends.

This is confirmed by the numerical solution of the dressed Schrodinger equation (308) with a number state as the initial condition for the photon field 11 0,0) It shows that the solution dressed state vector v /(t) (the transfer state, which in the bare basis is given by / /(0,0) mainly projects on the transfer eigenvector during the process. Additional data of the dressed solution during time are shown in Fig. 21a and 21c. Figure 21a displays the probabilities of being in the bare states 1, 2, and 3 ... 

Comparing Figs. 20a and 21a, we notice that, as expected, the solution of the dressed Schrodinger equation, with a number state as initial condition for the... 

To that effect, we choose the parameters 8 = 2flo and Qmax = 4.4 f>o, corresponding to the path (c) on the surfaces in Fig. 19. As shown in Fig. 22, the solution of the semiclassical Schrodinger equation (321) leads to nearly complete population transfer from state 11) to state 3). The analysis of the surfaces shows that the state 1 0,0) connects 3 —3,3). Thus the complete population transfer from the bare state 1) to the bare state 3) must be accompanied with absorption of three pump photons and emission of three Stokes photons at the end of the process. This is confirmed by the numerical solution of the dressed Schrodinger equation (308) with the initial state as a number state for the photon field 11 0,0), shown in Fig. 23a the dressed state vector /(f) approximately projects on the transfer eigenvectors during the process. It shows... 

In this Appendix we sketch an argument that leads to the adiabatic theorem for an A-level system with a Floquet Hamiltonian denoted Kr which generates the dressed Schrodinger equation [Eq. (229) of Section IV],... 

Discrete Fourier transform (DFT), non-adiabatic coupling, Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 153-155 Discrete variable representation (DVR) direct molecular dynamics, nuclear motion Schrodinger equation, 364-373 non-adiabatic coupling, quantum dressed classical mechanics, 177-183 formulation, 181-183... 

The dynamics of the dressed atom is determined by the effective one-mode time-dependent Schrodinger equation... 

The link between the simple propagator, g ° and the dressed propagator, g, is provided by the self-energy, Z, which is complex and energy-dependent and acts as a generalization of the ordinary potential energy in the conventional Schrodinger equation. 

We first consider an analysis of physical systems in periodic external fields using the Floquet theorem [134, 168]. As we shall see below, the theorem provides theoretical basis for the existence of field-dressed quasi-stationary state which expand the propagator. Earlier application of the Floquet theorem or related ideas to physical problems includes Refs. [224, 370], whereas the progress in this field is recently reviewed in Ref. [90]. Although it will later be extended to allow small non-periodic modulations, discussions in this section assumes perfect periodicity. The Schrodinger equation is given as... 

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 belief that computational chemists obtain molecular structures by solving Schrodinger s equation is often dressed up in so much jargon that the essential arguments are obscured. The general basis of the belief may be examined by considering the simplest possible molecule as a test case5. 

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