The rotating wave approximation

A typical calculation based on equation (9.9) is shown in fig. 9.3, with parameters appropriate for three-photon ionisation of K I [472]. Note the Qn-reversal effect due to the influence of the perturber, displaced by the laser field into the energy range of the Rydberg manifold. 

For near-resonant excitation by a coherent light wave of frequency u , the frequency difference Aca = oj — ojo (where ojo is the frequency of the atomic transition) is very small, and indeed much less than the transition frequency. Thus, the maximum value of the interaction energy which induces the transition (p ) is expected to be much smaller than the transition energy  

Time-dependent perturbation theory proceeds by expanding a general time-dependent wavefunction 

The rotating wave approximation (RWA) is a useful simplification [465] which contains most of the features of coherent excitation it consists in assuming that only exp(iwt) terms are present (counter rotating terms are neglected), in which case the amplitudes Oi(t) and o/(t) experience only slow oscillations at the frequency Aw, according to the differential equations  

the populations are cycled or optically pumped by the coherent radiation field at a frequency Q, an effect which is not observed for incoherent excitation. 

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Here the oscillatory terms like e2,ta> and e 2lt<0 have been ignored, which is called the rotating wave approximation (RWA). Linear susceptibility yl et) is defined by... 

The theory has been generalized by us to finite temperatures and to qubits driven by an arbitrary time-dependent field, which may cause the failure of the rotating-wave approximation (RWA) [11]. It has also been extended to the analysis of multilevel systems, where quantum interference between the levels may either inhibit or accelerate the decay [19]. 

Eq. (6.26) is the TDSE in the Schrodinger picture. In general, it proves more convenient to discuss the time evolution of the driven system in a rotating frame, such as the frame rotating with the laser carrier frequency q- After transformation into the carrier frequency picture and application of the rotating wave approximation (RWA), the TDSE takes the form [92]... 

On the other hand, there are many instances when the rotating wave approximation cannot be used. For example, in order to find the energy levels of a molecule placed in a strong microwave field, it is necessary to diagonalize a large piece of the full Floquet matrix involving multiple n-states and multiple eigenstates of Hq, as discussed in Section 8.3.4. 

This reduces to (2.3) if one supposes (c) the rotating wave approximation terms with ana0 and 44 may be omitted. They are associated with the high frequencies Q + con and therefore negligible in most applications. For our present case this approximation was not indispensable, but simplified the algebra. 

Adopting the rotating-wave approximation (RWA) and introducing the detuning frequency Ao= Ujk-yjk and the Rabi frequency Mjk -we find... 

Figure 1- Liouville space diagram corresponding to the only term that contributes to the spontaneous light emission from a two-level system within the rotating-wave approximation [Eq. (2.7)]. Here ]g) and e) denote the ground and the excited states, respectively.

After transformation into the interaction picture and application of the rotating-wave approximation [46, SO, 54] the population dynamics can be calculated numerically by solving the time-dependent three-level Schrodinger equation or (if phenomenological relaxation rates are considered) by solving the density matrix equation (3) for the molecular system. The density matrix equation is given by... 

Let the laser field be expressed as s(t) — o(t) sin(coLt), where o(t) is the pulse envelope function, including polarization, and col is the central frequency. For simplicity, consider a 8 function excitation. In the rotating wave approximation, the field is expressed as e(t) = S(t) exp(—icoLt) with field strength sq. Integration over t in Eq. (25) gives... 

Moreover, within the rotating wave approximation, which is usual in the quantum theory of damping, the terms corresponding to double excitations or desexcita-tions in Eq. (104) are neglected. Thus, within this approximation, Eq. (104) reduces to... 

As a consequence, using the Bosons description and performing the rotating wave approximation lead us to write the effective Hamiltonians (102) and (103) describing the H-bond bridge coupled to the thermal bath as follows ... 

Consider then excitation of this mixed state with a Gaussian pulse, within the rotating wave approximation. The pulse is of the form... 

Equation (8.16), in the rotating wave approximation, is then given by... 

We now consider this scenario in detail by first writing the CPT (radiatiod + matter) Hamiltonian in the rotating wave approximation as... 

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... 

The theory of the saturable absorption effect in single-wall carbon nanotubes has been elaborated. The kinetic equations for density matrix of n-electrons in a single-wall carbon nanotube have been formulated and solved analytically within the rotating wave approximation. The dependence of the carbon nanotube absorption coefficient on the driving field intensity has been shown to be different from the absorption coefficient behavior predicted forthe case of two level systems. 

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. 

The most general dressed Hamiltonian in the rotating wave approximation for these processes reads [69]... 

We consider a two-level atom with excited and ground states e) and g) when in a photonic crystal coupled to the field of a discrete (or defect) mode and to the photonic band structure in the vacuum state. The hamiltonian of the system in the rotating-wave approximation assumes the form [Kofman 1994]... 

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... 

It is worth noticing that the problems go away if we apply the rotating-wave approximation [Stenholm 1994], Using the relations (33) we may then write... 

The adiabatic limit. In the rotating-wave approximation, the most general version of the Bloch equations, [Allen 1975], can be written as... 

Here, [in is the transition dipole moment for the transition between V22 and Vn. In the adiabatic approximation (see, e.g.. Ret. [Stenholm 1994]), the coupled bare potentials V22 and V33 can be replaced by uncoupled adiabatic potentials. Using the rotating-wave approximation, the dynamics of the system is then described by the Schrodinger equation... 

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