Quantum beat exponential decay

Fig. 23. The fluorescence decay of Cd vapor in a magnetic field, (a) Experimental data exhibiting the phenomenon of quantum beats, (b) The exponentially decaying component, (c) The decaying modulated component. This figure is reproduced from the work of Dodd, Kaul, and Warrington (158).

In summary, the dynamics of the electronic decay of inner-shell vacancies in a charged environment, such as created by interaction of a cluster with a high intensity FEL radiation, can be qualitatively different from the one induced by a low-intensity source. If the emitted electrons are slow enough to be trapped by the neighboring charges, the familiar exponential decay will be suppressed by quantum beats between the initial state and the quasi-continuum of discrete final states. Physically, the predicted oscillations correspond to creation of the initial vacancy due to the reflections of the emitted electron by the charged cluster potential and the subsequent inverse Auger transition. 

Fig. 4.20. Experimental signals of ground state quantum beats in Kat E+.v" = 1, J" = 73) (a) field switched off, B = 0 (b) B = 0.816 T (c) differential signal with compensation of exponential decay (d) independence of (jJj .

The coincidence measurements discussed in the previous section were concerned with the total coincidence signal, i.e. the signal obtained when the decay of a particular ensemble of states is integrated over. These states are produced in a very short time ( 10 s) in electron impact excitation, and can sometimes evolve in a complicated way. In the absence of internal fields (e.g. the n P states of helium) each of the fm) states decays with the same exponential time dependence exp(—yt), and the coincidence technique can be used to yield the decay constant y of the excited state (see Imhof and Read, 1977, and references therein). However, if the excited state is perturbed by an internal (or external) field before decay, then the exponential decay is modulated sinusoidally giving rise to the phenomenon of quantum beats (Blum, 1981). 

Using the former technique, the most significant result was the observation of quantum beats in the fluorescence decay of jet-cooled anthracene At low excess energies, the fluorescence and fluorescence excitation spectra of anthracene are very sharp, and the fluorescence decay of single vibronic levels is exponential. At an excess energy of 1400 cm however, clear quantum beats were seen, arising from the interference between the initially populated vibronic state, and a state produced... 

In time space cs(t) 2 is now a perfect exponential with decay rate (r r + Tr), a clear signature of a large molecule with a dense k manifold. Conversely, the observation of quantum beats is a signature of a small molecule. For a relatively large molecule we will have an intermediately dense manifold,... 

The case of N = 1 is trivial in a dynamics sense in that it corresponds to no IVR. A fluorescence spectrum belonging to this case consists entirely of vibrationally unrelaxed (u-type) bands. Each of these bands decays in the same manner. In most situations, these flecays are unmodulated, single exponentials, although quantum beats and multiexponential decays arising from couplings other than those associated with IVR are possible. 

This represents an exponential decay exp(-yt) superimposed by a modulation with the frequency energy separation A 2i of the two coherently excited levels (Fig. 7.9b). This modulation is called quantum beats, because it is caused by the interference of the time-dependent wave functions of the two coherently excited levels. 

The time-dependent fluorescence from these coherently excited states shows, besides the exponential decay cxp(—t/r), a beat period tqb = h/(Ea — Eb) due to the different frequencies coa and cob of the two fluorescence components (quantum beats, Sect. 12.2). 

In the large molecule limit, many (greater than 20) molecular (cluster) eigenstates are accessed by a laser pulse and therefore the zero order optical state contains many fourier components in its dephasing or quantum beat spectrum. The summation of these many fourier components leads to an exponential time dependence - an "IVR decay" or "dissipative IVR". IVR in this case can be treated as a relaxation process and rate constants for the "decays" can be measured by characterizing the rise and fall times of zero order molecular chromophore vibronic state emission. If VP does not take place then T2 = (IVR rate)-l and Ti = trad-... 

Because of the special properties of the exponential function the light decays with the same time constant r as the population decay. The light decay can be followed by a fast detector connected to fast, time-resolving electronics. If the excited state has a substructure, e.g. because of the Zeeman effect or hyperfine structure, and an abrupt, coherent excitation is made, oscillations (quantum beats) in the light intensity will be recorded. The oscillation frequencies correspond to the energy level separations and can be used for structure determinations. We will first discuss the generation of short optical pulses and measurement techniques for fast optical transients. 

We have already discussed quantum-beat spectroscopy (QBS) in connection with beam-foil excitation (Fig.6.6). There the case of abrupt excitation upon passage through a foil was discussed. Here we will consider the much more well-defined case of a pulsed optical excitation. If two close-lying levels are populated simultaneously by a short laser pulse, the time-resolved fluorescence intensity will decay exponentially with a superimposed modulation, as illustrated in Fig. 6.6. The modulation, or the quantum beat phenomenon, is due to interference between the transition amplitudes from these coherently excited states. Consider the simultaneous excitation, by a laser pulse, of two eigenstates, 1 and 2, from a common initial state i. In order to achieve coherent excitation of both states by a pulse of duration At, the Fourier-limited spectral bandwidth Au 1/At must be larger than the frequency separation ( - 2)/ = the pulsed excitation occurs at... 

In Zeeman quantum-beat measurements oscillations superimposed on an exponential decay of the fluorescence light intensity are observed. In experiments on ytterbium atoms with zero nuclear spin the beat frequency for the 6sl9d D2 state was 31.52 MHz for a flxed magnetic field, in which the beat frequency for the signal from the 6s6p P state (with known Qj value = 1.493) was 46.05 MHz. What is the gj value of the 6sl9d Z>2 state Is the result expected Discuss what can be learned from measurements of Lande gj factors. 

If two or more closely spaced molecular levels are simultaneously excited by a short laser pulse, the time-resolved total fluorescence intensity emitted from these coherently prepared levels shows a modulated exponential decay. The modulation pattern, known as quantwn beats is due to interference between the fluorescence amplitudes emitted from these coherently excited levels. Although a more thorough discussion of quantum beats demands the theoretical framework of quantum electrodynamics [11.33], it is possible to understand the basic principle by using more simple argumentation. 

Fig.11.23a,b. Quantum beat spectroscopy, (a) Level scheme illustrating coherent excitation of levels 1 and 2 with a short broad-band pulse, (b) Fluorescence intensity showing a modulation of the exponential decay... 

This shows that a modulation of the exponential decay is observed if both matrix elements for the transitions 1 - f and 2 - f are nonzero (Fig.11.23b). The measurement of the modulation frequency allows determination of the energy separation of the two levels, even if their splitting is less than the Doppler width. Quantum beat spectroscopy therefore allows Doppler-free resolution. 

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