Rotating-wave model

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... 

In this subsection we will combine the general ideas of the iterative perturbation algorithms by unitary transformations and the rotating wave transformation, to construct effective models. We first show that the preceding KAM iterative perturbation algorithms allow us to partition at a desired order operators in orthogonal Hilbert subspaces. Its relation with the standard adiabatic elimination is proved for the second order. We next apply this partitioning technique combined with RWT to construct effective dressed Hamiltonians from the Floquet Hamiltonian. This is illustrated in the next two Sections III.E and III.F for two-photon resonant processes in atoms and molecules. 

Here we discuss in detail a model for measurement-induced decay modification in a multilevel system. The system with energies frwn, 1 < n < N, is coupled to a zero-temperature bath of harmonic oscillators with frequencies uj. The corresponding Hamiltonian, in the rotating-wave approximation, is... 

These results leave several basic questions open How to derive a non-Markovian master equation (ME) for arbitrary time-dependent driving and modulation of a thermally relaxing two-level system Would the two-level system (TLS) model hold at all for modulation rates, that are comparable to the TLS transition frequency u)a (between its states e) and g)) which may invalidate the standard rotating-wave approximation (RWA), [to hen-Tannoudji 1992] Would temperature effects, which are known to incur upward g) —> e) transitions, [Lifshitz 1980], further complicate the dynamics and perhaps hinder the suppression of decay How to control decay in an efficient, optimal fashion We address these questions by outlining the derivation of a ME of a TLS that is coupled to an arbitrary bath and is driven by an arbitrary time-dependent field. 

Finally, it is also interesting to compare the result (9.65) to the result (8.106) ofthe very different semiclassical formalism presented in Section (8.3.3). If we identify y of the present treatment with the factor Zgoi/< Eq. (8.96) the two results are identical for e = hco ksT. The rotating wave approximation used in the model (9.44) cannot reproduce the correct result in the opposite, classical, limit. Most studies of vibrational relaxation in molecular systems are done at temperatures considerably lower than s/ks, where both approaches predict temperature-independent relaxation. We will see in Chapter 13 that temperature-dependent rates that are often observed experimentally are associated with anhannonic interactions that often dominate molecular vibrational relaxation. 

This rate has two remarkable properties First, it does not depend on the temperature and second, it is proportional to the bath density of modes g(ct>) and therefore vanishes when the oscillator frequency is larger than the bath cutoff frequency (Debye frequency). Both features were already encountered (Eq. (9.57)) in a somewhat simpler vibrational relaxation model based on bilinear coupling and the rotating wave approximation. Note that temperature independence is a property of the energy relaxation rate obtained in this model. The inter-level transition rate, Eq. (13.19), satisfies (cf. Eq, (13.26)) k = k (l — and does depend on temperature. 

We have not yet implemented the fully adiabatic theory represented by Equation 5. That theory bears some resemblance to the bending-corrected rotating linear model (BCRLM)(16-18). In this model a partial wave hamiltonian is given by... 

Another model of the detector, which has only two energy levels, has been considered [188]. The most significant features can be described, in the rotating-wave approximation (RWA), in the framework of the following generalization of the Jaynes-Cummings (JC) Hamiltonian ... 

In order to treat the reaction in the laser field, we introduce an idea of "laser dressed" states [7]. Photoabsorption and photoemission processes are then modeled as the nonadiabatic transitions between the dressed ground and excited states. Under the assumptions of (1) two-state model, (2) single-photon excitation, and (3) rotating wave approximation, the laser dressed adiabatic PESs are given for two diabatic PESs W- and and the laser frequency as [7],... 

This contribution reviews the basic tools which are currently employed for interpreting ESR and NMR observables in condensed phases, with an emphasis on stochastic modeling as key for the prediction of continuous-wave ESR (cw-ESR) lineshapes and NMR relaxation times of proteins. Section 12.2 is therefore devoted to the definition of reduced (effective) magnetic Hamiltonians and the stochastic (Liouville) approach to spin/molecular dynamics in order to clarify the basic stochastic approach to cw-ESR observables. Section 12.3 provides a short overview of rotational stochastic models for the evaluation of relaxation NMR data in biomolecules. Conclusions are briefly summarized in Section 12.4. 

Distorted wave Born approximation Schatz, Miller, Tang and coworkers [81] Bending-corrected rotating linear model Bowman, Clary, Hayes, Walker and coworkers [69] ... 

The model considered is the simplest system of ODEs which (i) has the symmetries important for the spiral dynamics (rotations, reflections, and translations), and which (ii) has a supercritical Hopf bifurcation from a rotating wave solution. The model equations are ... 

The state = ( = 0 is a steady state of Equations (11) for all parameter values, and by the choice of the constant terms in expansions for / and g, it is linearly stable for all values of a and a2. For the full system (8) this state corresponds to v = iv = 0, p = po= constant. The trivial steady state coexists with rotating-wave and modulated-wave solutions discussed next. Hence it plays much the same role in the ODE model as the homogeneous steady state plays in excitable media. That is, in all excitable media, there is a homogeneous steady state (e.g. u — v = O the reaction-diffusion model), which is linearly stable for all parameter values, and which coexists with rotating waves and modulated waves when they exist. The trivial steady state in the ODE model has the same character as this homogeneous state in excitable media. 

Throughout this chapter I have taken the point of view that the meandering of spiral waves in excitable media can and should be examined from the perspective of bifurcation theory. With this approach, it has been possible to show that the organizing center for spiral dynamics is a particular codimension-two bifurcation resulting from the interaction of a Hopf bifurcation from rotating waves with symmetries of the plane. From this observation has followed a simple ordinary-differential-equation model for spiral meandering. 

This holds as long as hco > ksT. In the opposite, classical, limit the rotating wave approximation invoked here cannot be used. This can be seen by comparing Eq. (9.65) to Eqs (8.104) and (8.106). Polyatomic solids have of course high frequencies associated with their intramolecular motions. Indeed, both represent, in their corresponding models, the zero temperature transition rate from level = 1 to level = 0 of the harmonic oscillator. 

---