Rabi frequency effective Hamiltonian

Processes that are resonant at zero held (i.e., with a atomic Bohr frequency that is an integer multiple of the laser frequency) can be investigated through an effective Hamiltonian of the model constructed from a multilevel atom driven by a quasi-resonant pulsed and chirped radiation held (referred to as a pump held). If one considers an w-photon process between the considered atomic states 1) and 2) (of respective energy E and Ef), one can construct an effective Hamiltonian with the two dressed states 11 0) (dressed with 0 photon) and 2 —n) (dressed with n photons) coupled by the w-photon Rabi frequency (2(f) (of order n with respect to the held amplitude and that we assume real and positive) and a dynamical Stark shift of the energies. It reads in the two-photon RWA [see Section III.E and the Hamiltonian (190)], where we assume 12 real and positive for simplicity,... 

Note that the Hamiltonian (237) is a good approximation for the one- and two-photon processes (as shown in Section III.E for the two-photon case), but that it is only a rough approximation for higher multiphoton processes, since the Stark shifts should contain additional terms of higher order to be consistent with the order of the effective Rabi frequency. 

The RWA is appropriate when the Rabi frequency is much smaller than the other frequencies in the problem, viz. the transition frequency and the detuning. The first of these conditions limits the intensity, and so ultimately RWA breaks down it then becomes necessary to include the effects of the counterrotating terms, and one returns to solving the time-dependent Schrodinger equation, with a Hamiltonian which is periodic in time this is done in a more general way by applying Floquet s theorem for differential equations with periodically varying coefficients. 

In order to induce strong dipole-dipole coupling we introduce a microwave field (x, t)ef with a frequency cof and Rabi-frequency tuned near resonance with the N = 0 N = I transition. The effective Hamiltonian acting on the lowest-energy states is obtained in second-order perturbation theory as... 

We note here that the TDSE associated with the effective RWA Hamiltonian of Eq. (6.26) is analytically solvable because an exactly resonant laser pulse is considered. A general, non-resonant two-level model, with an arbitrary expression for the Rabi frequency 2 (1) and the (time-dependent) detuning A(f), is not analytically solvable. However, there exists a number of models with specific expressions for S2(f) and A(f), for with analytical solutions of the TDSE are known. Examples of such models are the Rosen-Zener [2], Allen-Eberly [3] or the Demkov-Kunike [4] models, to cite only a few. 

Crowell discovered a variety of effects numerically, including modified Rabi flopping, which has an inverse frequency dependence similar to that observed in the solid state in reciprocal noise [73]. The latter is also explained by Crowell [17] using a non-Abelian model. A variety of other effects of RFR on the quantum electrodynamical level was also reported numerically [17]. The overall result is that the occurrence, classically, of the B V> field means that there is a quantum electrodynamical Hamiltonian generated by the classical term proportional to 3 2. This induces transitional behavior because it contributes to the dynamics of probability amplitudes [17]. The Hamiltonian is a quartic potential where the value of determines the value of the potential. The latter has two minima one where B = 0 and the other for a finite value of the B i) field, corresponding to states that are invariants of the Lagrangian but not of the vacuum. 

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