Driving field

As described at the end of section Al.6.1. in nonlinear spectroscopy a polarization is created in the material which depends in a nonlinear way on the strength of the electric field. As we shall now see, the microscopic description of this nonlinear polarization involves multiple interactions of the material with the electric field. The multiple interactions in principle contain infomiation on both the ground electronic state and excited electronic state dynamics, and for a molecule in the presence of solvent, infomiation on the molecule-solvent interactions. Excellent general introductions to nonlinear spectroscopy may be found in [35, 36 and 37]. Raman spectroscopy, described at the end of the previous section, is also a nonlinear spectroscopy, in the sense that it involves more than one interaction of light with the material, but it is a pathological example since the second interaction is tlirough spontaneous emission and therefore not proportional to a driving field... 

This solution is appropriate for the regime of a weak driving field E(t). If we now treat the material as a collection of non-hiteracting oscillators, we may write the induced polarization as a sum of the individual... 

We may then write the amplitude for the hannonically varying polarization as proportional to the corresponding quantity for the driving field, E(oi) ... 

Next we look for a substitution for the acceleration experienced by the charge. A convenient device for doing this originates from considering the oscillating dipole produced by the driving field. Since /a = aE, we can describe the periodic (subscript p) dipole moment of a molecule by... 

This dependence of the TS trajectory on the past and future driving is illustrated in Fig. 2, which shows the time dependence of the 5-functional 5[p, (t) for a fixed driving field t) and different values of the eigenvalue p. To demonstrate the properties of the 5-functional clearly, we have chosen a smooth driving field (t) that is given by Eq. (81) (with Ao = 1, co = 1, and N = 2). It is zero for f > 27i. 

Note that the 5-functionals are nonzero even in the time range when the field vanishes. For Rep < 0, as in the example of Fig. 2b, the 5-functional depends on the past of the driving fields. Consequently, it is zero before the onset of the pulse. After the end of the pulse, 5[—1, ] tends to zero... 

Figure 2. Illustration of the S-functional (23) for (a) the external driving field and the eigenvalues (b) jj, = —1, (c) (i = l, and (d) (j, = i. (The latter case is discussed in detail in Section IIIC.)...

Fig. 22 a, b. Orientation of the effective field B H and sense of rotation of the circularly polarized driving field for indue- m -.1 tion of nuclear transitions. S = 1/2, s 2... 

The nonresonant contributions pertain to electron cloud oscillations that oscillate at the anti-Stokes frequency but do not couple to the nuclear eigenfrequencies. These oscillatory motions follow the driving fields without retardation at all frequencies. The material response can, therefore, be described by a susceptibility that is purely real and does not depend on the frequencies of the driving fields. The resonant contributions, on the other hand, are induced by electron cloud oscillations that are enhanced by the presence of Raman active nuclear modes. The presence of nuclear oscillatory motion introduces retardation effects relative to the driving fields i.e., there is phase shift between the driving fields and the material oscillatory response. 

The driving fields in nonlinear coherent microscopy can be dressed with alternative phase profiles. One of the simplest phase masks is a one-dimensional r-phase step across the transverse Gaussian beam profile. The resulting phase pattern resembles... 

Protection of quantum states from the influence of noise is important. It has been shown that the alternating transport of a EEC generated by the fast-forward driving field suppresses the influence of a fluctuating random potential on the EEC [47], The EEC is kept undisturbed for a longer time than is characteristic of the simple trapping with a stationary potential because the effective potential, which the quanmm state feels, becomes uniform when the transport velocity is sufficiently large. 

Simple calculation leads to the form of the driving field [50]... 

So as to study the stability of the efficiency of the population transfer driven by a variable CDF, we represent the total driving field in the form... 

The results of the preceding analysis suggest a method that can be used to separate different ionic species. The idea is simple. The driving fields in Eq. (3.90) that are... 

Coherent excitation of quantum systems by external fields is a versatile and powerful tool for application in quantum control. In particular, adiabatic evolution has been widely used to produce population transfer between discrete quantum states. Eor two states the control is by means of a varying detuning (a chirp), while for three states the change is induced, for example, by a pair of pulses, offset in time, that implement stimulated Raman adiabatic passage (STIRAP) [1-3]. STIRAP produces complete population transfer between the two end states 11) and 3) of a chain linked by two fields. In the adiabatic limit, the process places no temporary population in the middle state 2), even though the two driving fields - pump and Stokes-may be on exact resonance with their respective transitions, 1) 2)and... 

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