The Resonant Wave Approximation

In this subsection, we introduce the resonant wave approximation (RWA). This approximation, which is valid when the pulsation of the electric field is close to the resonance o 2 - wi - w <derive effective Hamiltonians that are easier to manipulate than the exact Hamiltonian. We first rewrite the Hamiltonian of Eq. (6.7) in a slightly different form... 

The results of two parameter correlations AE m and AE"m (A 1/2=E /2X-H /2H) with constants of the substituents CTjCTr, a ak", FR and CTjCTr+ (Table 3.49) show that the substituent influence on the first half-wave potential follows both induction and resonance mechanisms, the ratio of contributions of these effects being approximately equal and, practically, independent of a choice of the substituent parameters. The correlation results between AE"1/2 and substituent parameters indicate that the substituent influence is mainly achieved by the resonance mechanism (approximately 80%) (Table 3.49) [991],... 

We can use a simple two state model in the rotating wave approximation (RWA) [20] to explain the main features of Fig. 7. Consider a six photon resonance in a pulse such that I peak ires- The ground and resonance states are... 

For very small field amplitudes, the multiphoton resonances can be treated by time-dependent perturbation theory combined with the rotating wave approximation (RWA) [10]. In a strong field, all types of resonances can be treated by the concept of the rotating wave transformation, combined with an additional stationary perturbation theory (such as the KAM techniques explained above). It will allow us to construct an effective Hamiltonian in a subspace spanned by the resonant dressed states, degenerate at zero field. 

As we stressed above, the rotating wave approximation consists of taking into account only the terms varying with time at approximately the same frequency, that is, according to the resonance condition (9.11). Then we neglect in eqn (9.18) all interaction terms except A B oc exp[—i(w6 — lo0)L and in eqn (9.19) all interaction terms except A2 oc exp(— 2iujat). As a result, we arrive at the following simple system of equations... 

The second terms in Eqs. (1.167) and (1.168) are rapidly vibrating non-resonant terms having minor contribution on average, so they are neglected in further calculations. This is called the rotating wave approximation. By solving these simultaneous equations with the initial condition of T0 = T, for / = 0, we obtain ... 

This is only natural, for otherwise the interaction of the field with the quantum system would no longer be resonant. In this case, all the time-dependent terms can be greatly simplified by treating them only by reference to fast motions of the type With what is known as the rotating-wave approximation (Feynman et al. 1957), the probability amplitudes ai and oscillate only at the slow frequency A. Under these conditions (which generally hold true), we have, instead of eqn (2.33), the following set of simple equations for the probability amplitudes ... 

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