Rabi frequency dynamics

For resonant excitation, 5 = 0, the splitting is determined only by the amplitude of the Rabi frequency, which is conveniently adjusted via the laser field amplitude. Finally, we obtain the population dynamics d t) = dJ(t)Y in the dressed state picture from the bare state amplitudes by the transformation d t) = V t)c t). 

The use of strong fields to drive the dynamics leads to somehow similar effects than those of ultrafast pulses. If the Rabi frequency or energy of the interaction is much larger than the energy spacing between adjacent vibrational states, a wave packet is formed during the laser action. The same laser can prepare and control the dynamics of the wave packet [2]. Both short time widths and large amplitudes can concur in the experiment. However, the precise manipulation of dynamic observables usually becomes more difficult as the duration of the pulses decreases. 

The dynamics of x(t) using intuitive sequences in APLIP is considerably different. It involves two intermediate steps at x(t) > x and x t) < x at initial and late times respectively, where the bond length is almost constant. The exact value of x(t) in these plateau regions is quite robust to the ratio of Rabi frequencies, but it can be controlled by the peak amplitude of the pulses. Therefore, the intuitive sequence provides a more robust scenario than the counterintuitive one for preparing the system at certain bond lengths. Additional control can be achieved by changing the time delays. 

The values of C (r) 2 and IC2WI2 obtained from (7.19) and (7.20) are compared in Figs. 9 and 10. The amplitudes and periods of the temporal evolution predicted by the two approaches to the system dynamics are seen to agree quite well. The differences seen in the amplitudes shown in Fig. 9 are a consequence of the replacement of the exact eigenfrequencies of the Rabi frequency matrix with a typical eigenfrequency from the P subspace. 

Figures 6a-c show the population dynamics encountered in a three-level system (see Fig. 4) interacting resonantly with two Fourier-transform-limited laser pulses with three different delay times between the two pulses. The calculation was done assuming that the chosen Rabi frequencies fulfill the relation > 1/pulse duration) in all three cases. This relation ensures that the typical time for a Rabi oscillation of the population in an isolated two-level system is shorter than the pulse duration. Ionization from level 2 was introduced as a fast laser intensity-dependent decay of level 2 [6, 60], and resonant laser frequencies were assumed.

In the upper parts of Figs. 6a-c the time-dependent Rabi frequencies of both laser pulses are shown for different delays. In all cases the dump laser pulse has a higher Rabi frequency than the pump laser pulse and twice its duration. Note that the Rabi frequency is proportional to the laser held strength and therefore to the square root of the pulse intensity. In the lower part the population dynamics for the three different pulse sequences is shown. The part of population transferred to the ionization continuum is indicated by a strong line. 

Figure 6. Coherent population dynamics calculated using the density matrix equation (3) for different delays (a-c) of the laser pulses. Upper part Time evolution of the Rabi frequencies of both laser pulses. Lower part Calculated time evolution of the level populations for three different delays.

Iplgure 11.8 Rates of association (Prec), back-dissociatioin (Pdiss), and total molecule Bti tion (P) vs. t in the counterintuitive scheme. Dashed lines are pulse intensity profile, spitted lines denote effective Rabi frequency d2(t)/% where is peak pulse intensity and (a) Initial wave packet width of SE = 10 3 cm-1 and other pulse parameters as fi Fig. 11.7. (b) Dynamics scaled by s— 10 [Eq. (19.81)] Initial wave packet width of 10 4 cm-1 both pulses lasting 85 ns pump pulse peaking at / =200 ns and dump... 

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,... 

The analysis of the dynamics consists of (i) the calculation of the dressed eigenenergy surfaces of the effective quasienergy operator as a function of the two Rabi frequencies flj and if, (ii) the analysis of their topology, and (iii) the application of adiabatic principles to determine the dynamics of processes in view of the topology of the surfaces. 

The path described above can be constructed by two smooth pulses, associated with the Rabi frequencies D, (t) and Q2(/j, with a time delay x. To a sequence of such pulses corresponds a closed loop in the parameter plane Oj and 02. Each of the two black curves [labeled (a) and (b)] correspond to a sequence of two smooth pulses of equal length T and equal peak Rabi frequencies Qmax = max2[fli(f)] = maxjf f)], separated by a delay such that the pulse 1 is switched on before the pulse 2. This path has been redrawn as a function of time on Fig. 16b, using sin2 envelopes of length T = 100/5 and a delay of x = T/3, shown in Fig. 16a. Details of this dynamics of bichromatic processes, in particular in relation with the initial condition for the photon field, are given and discussed in the next subsection. Path (c) needs two pulses with different peak amplitudes. 

Another property of atoms which is sensitive to the conditions in the outer reaches of the atomic field is of course orbital collapse, which can be controlled as described in section 5.23. This has led Golovinskiy et al. [480] to consider whether a strong laser field could be used to precipitate orbital collapse, and to propose an experiment in which dynamic collapse at the Rabi frequency could be detected by X-ray spectroscopy of the irradiated sample. 

As shown in Equations 8.25 and 8.27, the effect of the pump is equivalent to the production of a single coefficient heff that determines the multichannel dynamics. The source term for the production of heff is the (scaled) projection of the incoming vector ft ) on the multichannel (ffip vector. Alternatively, the source term can be viewed as the projection of the vector S2p ft ) on the vector of the dump Rabi frequencies S2p. Both the matrix S2p and the vector ffip are controlled by the pump and dump laser amplitudes. Therefore there is a large choice of possibilities to control the projective measurement of the incoming vector ft ) performed in the PA process. 

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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