Beat resonance

Among other methods of creating and observing optical manifestations of coherent superpositions of non-degenerate magnetic sublevels (an elegant analysis may be found in [4]), the most widely used method is that of resonance of beats. The effect appears when the frequency fli of harmonic modulation of the excitation rate 

We will now try to present a comprehensible treatment of the phenomenon. To this end we turn again to Fig. 4.2. It is essential that the isotropy of angular momenta J distribution in the xy plane in Fig. 4.2(e) actually is the result of the superposition of shapes similar to those in 

The above is confirmed by the solution of Eq. (4.9). Assuming that in the ground (initial) state there is only the population ap ], the solution for bPg at i r 1 (steady process) is obtained in the form of the expression presented in [133]  

The first term in curly brackets does not depend on t and represents the usual Hanle effect. The second term oscillates at modulation frequency fii, with the modulation amplitude growing in resonance fashion if Qojj = fii holds (see Fig. 4.22). For u j T, when degeneracy is completely removed, we may write in the vicinity of resonance for Q 0  

Resonance-shaped growth of modulation amplitude manifests itself in the intensity of fluorescence, as expressed through 6Pq (t), according to (2.24). It may be pointed out that, owing to the harmonic factor exp(ifiit) the signal in (4.52) vanishes at registration averaged over the time span At fij-1. The position of resonance permits us to determine Qojj (see Fig. 4.22), i.e. the Lande factor, whilst the shape (width) of the envelope yields the constant Tk- 

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In spite of the apparent obviousness of the beat effect in optical radiation at pulsed excitation, it was only registered and studied comparatively recently. At the beginning of the 1960s Aleksandrov [3] and, independently, Dodd and coworkers [119] discovered beats in atomic emission. It may be pointed out that this, and the related phenomenon of beat resonance, was predicted by Podgoretskii [313], as well as by Dodd and Series [118]. The phenomenon was treated on the basis of well-known fundamental concepts on coherent superposition of states, and was named accordingly quantum beats. These ideas are amply expounded in reviews and monographs [4, 5, 6, 71, 96, 120, 146, 182, 188, 343, 348, 388]. 

The method of beat resonance has been used episodically to study the magnetism of excited states of molecules. One of the first papers of this kind appears to have been that by Lehmann et al. [89] on iodine I2(53n0+). Subsequently the same group [172, 173] used beat resonance for determining Lande factors for a number of rovibronic levels of... 

Fig. 4.23. Non-linear beat resonance signal of K2(X1E+,w" = 1, J" = 73) in the form of the dependence of the degree of polarization of radiation on modulation frequency fii in a constant magnetic field B = 0.589 T. The arrow indicates...

Auzinsh, M.P. and Ferber, R.S. (1984). Observation of a quantum-beat resonance between magnetic sublevels with AM = 4, JETP Lett., 39, 452-455. 

Auzinsh, M.P., Nasyrov, K.A., Tamanis, M.Ya., Ferber, R.S. and Shalagin, A.M. (1990). Determination of the ground-state Lande factor for diatomic molecules by a beat-resonance method, Chem. Phys. Lett., 167, 129-136. 

G. Alzetta, A. Gozzini, L. Moi, G. Orriols, Experimental-method for observation of rf transitions and laser beat resonances in oriented Na vapour, Nuovo Gimento Soc. Ital. Fis.B36 (1976)5. 

Alzetta, G., Gozzini, A., Moi, L., and Orriols, G. (1976). An experimental method for the observation of RF transitions and laser beat resonances in oriented Na vapor. Nuovo Cimento B, 36, 5-20. 

As already mentioned, electronically resonant, two-pulse impulsive Raman scattering (RISRS) has recently been perfonned on a number of dyes [124]. The main difference between resonant and nom-esonant ISRS is that the beats occur in the absorption of tlie probe rather than the spectral redistribution of the probe pulse energy [124]. These beats are out of phase with respect to the beats that occur in nonresonant ISRS (cosinelike rather tlian sinelike). RISRS has also been shown to have the phase of oscillation depend on the detuning from electronic resonance and it has been shown to be sensitive to the vibrational dynamics in both the ground and excited electronic states [122. 124]. 

Walmsiey I A, Wise F W and Tang C L 1989 On the difference between quantum beats in impulsive stimulated Raman scattering and resonance Raman scattering Chem. Phys. Lett. 154 315-20... 

Occurs when two or more machines mounted on common foundations operate at nearly the same speed Occasionally, a beat can develop in one machine if its operating speed is close to a structural component resonance... 

Second volume and choke-tube resonance frequency and beat frequencies... 

A beat frequency exists as the sum or difference of two frequencies. For a base of 10 cycles/sec and a resonant frequency of 3.48 cycles/sec, beat frequency = 10 — 3.48 = 6.52 cycles/sec, which is below the compressor base frequency of 10 cycles/sec and is acceptable. Beat frequencies of the harmonics of the base frequency will be filtered. 

It is important to investigate possible situations of harmful resonance. Although all pipes and pipe-volume combinations will resonate, it is necessary to investigate only the cases that might cause mechanical problems. This refers to a frequency that will given a objectionable beat frequency or to a length of pipe that is tuned to a stimulating frequency. 

A beat can occur as the sum or difference of two frequencies. The sum or difference of any resonant frequency and a compressor driving frequency must be equal to some low value, possibly only V2 cycle per sec or less. Therefore, only two frequencies or possible harmonics of these frequencies, which vary by a value of less than 2 cycles per sec, will be harmful as far as vibration is concerned for many, not all cases. It is important that risers be given close attention in... 

Equation (9.1) documents that quadmpole splittings A q exhibit quantum-beat spectra with period H/IuAEq superimposed over the time dependence of the nuclear decay exp(—f/t) with mean decay time t = 141 ns for Fe. In Fig. 9.2, quadmpole splittings A q = 0 and 2 mm s in the energy domain (conventional MS) are compared with those in the time domain (MS using synchrotron radiation) [7]. The QBs in the time domain spectmm for A q = 2 mm s are the result of the interference between the radiation scattered by different nuclear resonances. Consequently, their frequencies correspond to the energetic differences between these resonances. 

Variations of the msd of the resonant nuclei affect the dynamical beat pattern... 

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