Doppler quadratic

Chapter 3 is devoted to pressure transformation of the unresolved isotropic Raman scattering spectrum which consists of a single Q-branch much narrower than other branches (shaded in Fig. 0.2(a)). Therefore rotational collapse of the Q-branch is accomplished much earlier than that of the IR spectrum as a whole (e.g. in the gas phase). Attention is concentrated on the isotropic Q-branch of N2, which is significantly narrowed before the broadening produced by weak vibrational dephasing becomes dominant. It is remarkable that isotropic Q-branch collapse is indifferent to orientational relaxation. It is affected solely by rotational energy relaxation. This is an exceptional case of pure frequency modulation similar to the Dicke effect in atomic spectroscopy [13]. The only difference is that the frequency in the Q-branch is quadratic in J whereas in the Doppler contour it is linear in translational velocity v. Consequently the rotational frequency modulation is not Gaussian but is still Markovian and therefore subject to the impact theory. The Keilson-... 

Secondly, in the atomic frame, it comes with a motional electric field E = V x B hydrogen atom are specially sensitive to this field. As this electric field is proportional to v, the corresponding shift of the S level due to the interaction with a near P level, is quadratic with v. Moreover, since the nearest level (P1/2) is below the S one, this motional electric field can give a positive frequency shift of the transition Av able to compensate the negative shift due to the second order Doppler effect. 

The quadratic relationship of redshift to distance of the source is a complete departure from the conventionally assumed Doppler shift and Hubble s linear law. In order to test the quadratic model it is necessary (Segal, 1980) to eliminate the distance, which is not an observable quantity, using geometrical relations with parameters such as apparent luminosity or angular diameter. These relations were tested on data available for galaxies, quasars and radio sources. 

The evidence in favour of a quadratic dependence of redshift on distance cannot be ignored and the Doppler interpretation of cosmological redshifts needs revision. 

The first term represents the absorption frequency coq = Ek — Ei) of an atom at rest if the recoil of the absorbing atom is neglected. The second term describes the linear Doppler shift (first-order Doppler effect) caused by the motion of the atom at the time of absorption. The third term expresses the quadratic Doppler effect (second-order Doppler effect). Note that this term is independent of the direction of the velocity v. It is therefore not eliminated by the Doppler-free techniques described in Chaps. 2-5, which only overcome the linear Doppler effect. 

Example 9.1 A parallel beam of Ne ions accelerated by 10 keV moves with the velocity = 3 x 10 m/s. When the beam is crossed perpendicularly by a single-mode laser beam tuned to a transition with X = 500 nm, even ions with Vx =Vy=0 show a quadratic relativistic Doppler shift of Av/v = 5 X 10 , which yields an absolute shift of Av = 250 MHz. This should be compared with the linear Doppler shift of 600 GHz, which appears when the laser beam is parallel to the ion beam (Example 4.6). 

The velocities of very cold atoms are very small, i.e., the linear and the quadratic Doppler effects both become small and the recoil term becomes significant. It turns out that for cold Ca atoms at T = 10 pK, the recoil effect leads to large asymmetries in the Lamb dips of absorption spectra taken with short pulses [1119]. These are not found in experiments performed at room temperature, where the broad Doppler background masks these asymmetries, and they are based on the fundamental asymmetry between absorption and stimulated emission with short pulses. 

Although transit-time broadening is greatly reduced by the Ramsey technique, the quadratic Doppler effect is still present and may prevent the complete resolution of the recoil components. This may cause asymmetric line profiles where the central frequency cannot be determined with the desired accuracy. As was shown by Helmcke et al. [1269, 1277], one of the recoil components can be eliminated if the upper level Pi of the Ca transition is depopulated by optical pumping with a second laser. In Fig. 9.63 the relevant level scheme, the experimental setup, and the measured central Ramsey maximum of the remaining recoil component are shown. 

The visibility is then equal to the degree of coherence. Figure 2.28a depicts the visibility V of the fringe pattern in P as a function of the slit separation d, indicated in Fig. 2.24, when these slits are illuminated by monochromatic light from an extended uniform source with quadratic size bxb that appears from Si under the angle 0, Figure 2.28b illustrates the visibility as a function of path difference A5 in a Michelson interferometer which is illuminated with the Doppler-broadened line X = 632.8 nm from a neon discharge lamp. 

Note Equations (3.38) and (3.39) describe the linear Doppler shift. For higher accuracies, the quadratic Doppler effect must also be considered (Sect. 14.1). 

The measurement of absolute beam velocities, or the calibration of voltages, is already quite sensitive to the relativistic quadratic term in the Doppler shift formula (7). In fact, this transverse Doppler shift, caused by the time dilatation factor y = (1 - j8 )" , was first observed in the spectral lines of fast-moving hydrogen atoms from a 30-keV beam of H2 ions, viewed along and opposite the direction of propagation. Comparable accuracy in the percent range was also achieved in Mossbauer experi-ments, and more recently the time dilatation factor on the muon lifetime was determined to 1 x 10". ... 

Expanding the above expression into a series in powers ofv/c, we obtain, in addition to the linear shift proportional to uii/c, also a quadratic Doppler shift... 

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