Eigenfrequencies periodicity

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. 

Because it contains sin2 ka, the function co = co(k) is periodic. All possible eigenfrequencies are obtained when in Eq. (II.4) sin2ka is varied only in the range 0 < ka < it 12. The magnitude of the wave vector is thus... 

We now describe how the periodic orbit theory of Section 3.6, which relates the energy levels with the poles of the spectral function g(E), can be extended to two dimensions. For simplicity we shall illustrate this extension by the simplest model, in which the total PES is constructed of two paraboloids that cross at some dividing plane. Each paraboloid is characterized by two eigenfrequencies, dividing plane (symmetric case) or with respect to a point (antisymmetric case). Our discussion draws on Benderskii et al. [1992b]. 

Fourier transformation over an incremented Hartmann-Hahn evolution period yields the eigenfrequencies of the (effective) Hartmann-Hahn Hamiltonian. In solid samples with resolved heteronuclear dipolar couplings (Muller et al., 1974), this approach yields heteronuclear dipolar oscillation spectra (Hester et al., 1975) if the heteronuclear spins are Hartmann-Hahn matched during the evolution period of the experiment. In liquid state NMR, Fourier transformation over incremented homonu-clear Hartmann-Hahn transfer periods yields so-called coherence-transfer... 

Similar to smectic membranes, periodic electric fields (although in this case the field is parallel to the fiber axis) can induce mechanical vibration of the strings. Studying the eigenfrequencies of the vibrations of the strings under... 

If the laser pulse applied to the sample molecules is sufficiently long and intense, a molecule (represented by a two-level system) will be driven back and forth between the two levels at the Rabi flopping frequency (2.134). The time-dependent probability amplitudes a (t) and a (t) are now periodic functions of time and we have the situation depicted in Fig.2.30. Since the laser beam is alternatively absorbed (induced absorption E ) and amplified (induced emission E2-> E ), the intensity of the transmitted beam will display an oscillation. Because of relaxation effects this oscillation is damped and the transmitted intensity reaches a steady state determined by the ratio of induced to relaxation transitions. According to (2.133) the Rabi frequency depends on the laser intensity and on the detuning molecular eigenfrequency o) 2 laser frequency co. This detuning can... 

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