Anharmonic modulation

Fig. 10.18 Experimental coherence spectrum obtained with two anharmonic modulation frequencies I ] = 2.0 and I 2 = 2.7. Additional major peaks in the spectrum correspond to beat frequency (0.7 Hz), first harmonic of I ) (4.0 Hz), frequency sum (4.7 Hz), and first harmonic of / 2 (5.4 Hz). The red line represents the same experiment but with 10% of white noise added (Kikas et al., 2001b)...

More general anharmonic modulations can be considered by extending the sum in Equation (29) to a higher number of waves, say D in such a case the D q vectors are linear combinations of the basic (rationally independent) d q vectors d < D). In the following we considered only the harmonic model of dimension d. 

Planck s constant as given in the appendix of Ch. 1, a value of 0.06 A is found for 8Qq. This value is not at all small and emphasizes the important geometrical consequence of the anharmonic modulation of the freqnency of by intermonomer modes (4). It gives a value of 0.017 A for 8Qq -8Qq (eq. (7.1)), a value close to 0.02 A, measured by electron diffraction. It shows that the theoretical description of the anharmonic coupling of and intermonomer modes of H-bonds, presented above, not only quite precisely reproduces spectra, but also reproduces such small geometrical effects, which are measured by completely independent methods. The experimentally verified correlation this theoretical description establishes between IR bandshapes and isotopic length variations 8Qq - 8Q° of H-bonds constitutes a strong snpport of its validity. Values of 8 - 5 can be in principle also measured by X-ray... 

Replacing H-bonds by D-bonds has no effect on the electronic structure of H-bonds when nuclei are held at fixed positions. It nevertheless has a small, indirect effect on thermodynamic properties, because quantities such as enthalpies of formation of H- or D-bonds not only depend on their electronic structure, but also incorporate a significant vibrational component that is sensitive to this substitution. The result is a difference of enthalpies of formation of H- and D-bonds that amounts to only some percent of the total enthalpy of formation and remains consequently weak with respect to kT at room temperature for weak and intermediate strength H-bonds. It means that at room temperature this difference is not really apparent. Such a substitution consequently conveys almost no information for calorimetric methods. The geometry of the H-bond is also little affected by such a substitution that results in a small lengthening of the X- Y equilibrium distance in X-D- Y as compared to X-H---Y. It is an indirect, or second-order consequence of the strong and characteristic anharmonic modulation of the stretching vibration of the H-atom by the X- Y distance. It implies a weak effect with little information. 

Figure 9.45(a) shows fhe resulting fluorescence intensify as a function of time. This is dominated by oscillations wifh a period of abouf 300 fs buf shows an amplifude modulation wifh a period of abouf f 0 ps. The 300 fs period is fhaf of a vibration in fhe u = 7 — f 2 range and fhe modulation is due to fwo neighbouring levels having slighfly differenf frequencies due to fhe vibration being anharmonic. 

The cornerstone of the strong anharmonic coupling theory relies on the assumption of a modulation of the fast mode frequency by the intermonomer distance. This behavior is correlated by many experimental observations, and it is undoubtly one of the main mechanisms that take place in a hydrogen bond. Because the intermonomer distance is, in the quantum model, represented by the dimensionless position coordinate Q of the slow mode, the effective angular frequency of the fast mode may be written [52,53]... 

Although this potential is written within the harmonic approximation, it is extrinsically anharmonic when one assumes the modulation hypothesis (6) and (7). Then, U expands into... 

Melinger JS, McMorrow D, Hillegas C, Warren WS. Selective excitation of vibrational overtones in an anharmonic ladder with frequency- and amplitude-modulated laser pulses. Phys Rev A 1995 51 3366-3369. 

But we will later see a violation of a simple energy gap behavior such as the above. It is critical to appreciate that this frequency effect is modulated by the factors that depend on the intramolecular potential it suffices to recall that in an harmonic approximation only the transition (001) -> (000) would be allowed. To place this issue in perspective, we write the rate constant [Equation (5)] for transitions out of the anharmonic... 

The experimental data was fitted, as shown in Fig. 5.10, to a convolution of this response function with the instrument response function. As the result, the decay time T-2/2 was estimated to be 1.1 0.1 ps. Recently, the population lifetime Ti of G-phonons was measured by incoherent time-resolved anti-Stokes Raman scattering and the lifetime was found to be 1.1-1.2 ps in semiconducting SWNTs [57]. Therefore, one can reasonably assume ipu Ti at room temperature. This result is consistent with the conventional Raman line width of semiconducting SWNTs [58]. The observed short lifetime of the G-phonons implies anharmonic mode coupling between G-phonons and RBM-phonons [59]. In fact, a frequency modulation of the G mode by the RBM has been reported, suggesting the anharmonic coupling between these vibrations [56]. 

Apart from the bilinear connection between acoustic and optical vibrations, the usual phonon anharmonicity takes place in the chain. The acoustic vibrations modulate the distances between A atoms and thus influence the frequency of proton vibrations on hydrogen bonds. The proton vibration frequency can be presented as... 

It seems intuitively clear that such an achievement is due to the control of quantum dissipative dynamics through the application of a suitably tailored, time-modulated driving field. Indeed, some interesting examples of suppression of quantum decoherence by the modulation of system parameters have been considered in [Viola 1998 Vitali 1999], An improvement of sub-Poissonian statistics of an anharmonic oscillator by the application of amplitude-modulated pump field have been demonstrated in [Kryuchkyan 2002 Kryuch-kyan 2003],... 

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