Multiphoton transition

Fig. 1.13. Energy diagram for driven flux qubits that supports high-order multiphoton transitions...

Meshulach, D., and Silberberg, Y. 1999. Coherent quantum control of multiphoton transitions by shaped ultrashort optical pulses. Phys. Rev. A 60 1287-92. 

Dudovich, N., Dayan, B., Eaeder, S. M. G., and SUberberg, Y. 2001. Transform-limited pulses are not optimal for resonant multiphoton transitions. Phys. Rev. Lett. 86 47-50. 

My question to Prof. B. Kohler (as representative of the group of K. R. Wilson) is whether he would agree with S. A. Rice s classification that puts the technique of K. R. Wilson et al. [8] into strategy (ii) What are the fundamental analogies and what are the differences between their approach [8] and the Tannor-Rice-Kosloff-Rabitz approach (see Refs. 2 and 3 and current chapter) Finally, I should like to point to another strategy (iii) of laser control by vibrationally mediated chemistry that is achieved by IR + UV continuous-wave (CW) multiphoton transitions (see the pioneering papers by Letokhov [9] and sequel theoretical developments [10] and experimental applications [11]). 

I. Concerning the first question of Prof. Kobayashi, I think that shifts of Raman lines by interaction with other levels than the investigated ones should be observable in the Raman process. A pronounced shift is expected, for example, if one of the laser frequencies is close to a resonant multiphoton transition to another rovibronic state. 

Sweeping the static field, which alters the energy spacing between the levels, while keeping the microwave field fixed leads to the observation of resonant multiphoton transitions. An example, shown in Fig. 10.9, is the set of the 18s—>(16,k) transitions observed by sweeping the static field with different strengths of the 10.35 GHz microwave field.8 The sequence of 18s—>(16,3) transitions 25 V/cm apart is quite evident. At the top of Fig. 9 there is a scale in terms of the number of 10.35 GHz photons required to drive the 18s—> (16,3) transition. 

The data of Fig. 10.9 show clearly that the K (n + 2)s— (n,k) transitions are multiphoton transitions. On the other hand, most of the data obtained by sweeping the microwave field agree qualitatively with the Landau-Zener description. To reconcile these apparently different descriptions we consider the problem shown in Fig. 10.10, in which there are two states 1 and 2, with a linear Stark shift and no Stark shift respectively.10 States 1 and 2 are coupled by V, the core... 

Fig. 10.19 The microwave frequency dependence of the n changing signals at low microwave power, where n changes up or down only by 1. Resonant multiphoton transitions are observed near the expected static field Stark shifted frequencies indicated. These resonances involve the absorption of four or five microwave photons. The down n changing atom production curve was obtained with the state analyzer field EA set at 50.0 V/cm, while up n changing was studied as n = 60 atom loss with EA = 45.5 V/cm. The locations of resonances for larger direct (not stepwise) changes in n are indicated along with...

We have considered in particular the case of multiphoton transitions, to be observed with the help of intense high frequency fields as produced by X-ray Lasers or Free-Electron Lasers (FEL). As a result of our analysis, we have shown that two-photon bound-bound transition amplitudes in high-Z hydrogenic systems are significantly affected by relativistic corrections, even for low values of the charge of the nucleus. For instance at Z = 20, the corrections amount to about 10%, a value much higher than what is observed for standard one-photon transitions in X-ray spectroscopy measurements for which the non-relativistic dipole (NRD) approximation agrees with the exact result to within 99% at comparable frequencies. 

In addition to the quantum approaches mentioned above, classical optimal control theories based on classical mechanics have also been developed [3-6], These methods control certain classical parameters of the system like the average nuclear coordinates and the momentum. The optimal laser held is given as an average of particular classical values with respect to the set of trajectories. The system of equations is solved iteratively using the gradient method. The classical OCT deals only with classical trajectories and thus incurs much lower computational costs compared to the quantum OCT. However, the effects of phase are not treated properly and the quantum mechanical states cannot be controlled appropriately. For instance, the selective excitation of coupled states cannot be controlled via the classical OCT and the spectrum of the controlling held does not contain the peaks that arise from one- and multiphoton transitions between quantum discrete states. 

An example for one of the first applications of the coupled-channel equations y with quantized fields for photodissociation problems is shown in Figure 12.1, where, the dissociation of the IBr molecule by a two-photon (visible + IR) process wasJS studied [388], The results of the calculations, shown in Figure 12.2, demonstrate 3 how the strong IR photon broadens the transition ( power broadening ) allowing th system to be dissociated even if the first photon is tuned substantially away from, 7 resonance. This illustrates how multiphoton transitions induced by strong fields [392] are less restricted insofar as they need not be very close to an intermediate T resonance, the situation described in Section 3.3. 

For the special case of non relativistic Hydrogen, the multiphoton transition rate can be obtained exactly using methods based on Green function techniques, which avoid summations over intermediate states. This approach was introduced in order to treat time independent problems, and later extended to time dependent ones [2]. In the Green function method, the evaluation of the infinite sums over intermediate states is reduced to the solution of a linear differential equation. For systems other than Hydrogen, this method can also be used, but the associated differential equation has to be integrated numerically. The two-photon transition rate can also be evaluated exactly by performing explicitly the summation over the intermediate states. 

Rydberg atoms and microwave fields constitute an ideal system for the study of atom-strong field effects, and they have been used to explore the entire range of one electron phenomena [5]. Here we focus on an illustrative example, which has a clear parallel in laser experiments, a series of experiments which show that apparently non-resonant microwave ionization of nonhydronic atoms proceeds via a sequence of resonant microwave multiphoton transitions and that this process can be understood quantitatively using a Floquet, or dressed state approach. 

In the following section the experimental approach is briefly described. The initial observations of microwave ionization and the completely non-resonant picture initially used to describe it are then presented. Then microwave multiphoton transitions in a two level system analogous to the rate limiting step of microwave ionization are described both experimentally and theoretically. Experiments on this two level system with well controlled pulses of microwaves to show the applicability of an adiabatic Floquet theory to pulses are then described. We finally return to microwave ionization to see evidence for the resonant nature of the process. 

The pair of levels 21s - (16,3) is exactly analogous to the extreme blue and red Na Stark states of n and + 1. The fact that only one has a permanent dipole moment is of no consequence it is only the difference in the permanent moments which is significant. Based on the single cycle Landau-Zener description of microwave ionization we expect that if atoms in the 18s state are exposed to a microwave field of amplitude equal to the crossing field, Eq = 753 V/cm, they would make transitions to the (16,3) state. On the other hand, if a static field is present as well as the microwave field it should be possible to see resonant microwave multiphoton transitions between these two bound states, and seeing the connection between these processes is part of our objective. 

Resonant and also non-resonant multiphoton transitions are well reproduced with the program. This was tested by changing the wavelength from 200 to 260 nm. 

The selection rules for multiphoton transitions are clearly different from the usual dipole selection rules, since each photon carries an angular momentum 1 Thus, for two-photon transitions, one rule is A J = 0 2, but further control can be exercised by selecting the polarisation of the light the A J = 0 transitions are only possible if the laser light is linearly polarised (i.e. contains both circular polarisation), while the choice of either circular polarisation results in an increase or a decrease of J. Detailed discussion of the selection rules for two-photon transitions can be found in several papers [453, 455, 459]. For multiphoton transitions, the same principles apply, and the role of polarisation is still more significant. A general reference is [460], in which selection rules are derived from first principles, and a list of selection rules for two-photon transitions is given in table 9.1. 

Although two-photon transitions were the first multiphoton transitions to be considered, there is no reason to limit studies of atomic response to just two optical waves, once the possibility of nonlinear coupling between the atoms and the radiation field has been recognised. Indeed, one of the most important processes involves the generation of a higher frequency lo by the superposition of three waves oq, ll>2 and 0J3, a process referred to as four-wave mixing. 

Fig. 9.11. (a) Schematic map of the quasienergies, plotted modulo laser photon energy, as a function of the laser intensity, showing avoided crossings, (b) The shape of the laser pulse as a function of time. Under appropriate conditions (as described in the text) a generalised form of Landau-Zener transition occurs which is what one commonly calls a multiphoton transition (after J.-P. Connerade et al. [482]). 

There are many variants of REMPI. In most REMPI experiments one or more of the photoexcitation steps is a multiphoton transition. The rotation and electronic selection rules and relative intensity factors are quite different for a one-photon... 

Unlike single photon absorption processes, the cross sections for multiphoton transitions are polarization dependent. 

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