Quantum beat phase

First, it is possible to show by using the orthonormality of the eigenvector matrix C that each of the N band types arising from an N-level system has a decay with a unique quantum beat phase distribution. That is, each decay type is different from the others by virtue of its quantum beat phases. 

At this point, it may seem to the reader that the detailed consideration of quantum beat phase distributions is a somewhat abstract exercise bearing little relation to IVR. We would justify our attention to the problem of phases by noting that the proper interpretation of experimental results from picosecond-jet experiments on IVR relies on the ability to determine how closely one s experimental conditions correspond to one s theoretical model of the experiment. A particularly convenient way to do this is by comparing phase characteristics from experiment with those from theory. In addition, phase characteristics are useful in helping one assign the various bands in a fluorescence spectrum to band types. 

A series of NFS spectra of the spin-crossover complex [Fe(tpa)(NCS)2] were recorded over a wide temperature range [45]. A selection of spectra around the spin-crossover transition temperature is shown in Fig. 9.13. At 133 K, the regular quantum-beat structure reflects the quadrupole splitting from the pure high-spin (HS) phase, and the envelope of the spectrum represents the dynamical beating with a minimum around 200 ns. Below the transition, at 83 K, the QBs appear with lower frequency because of smaller AEq of the low-spin (LS) phase. Here the minima of... 

The events taking place in the RCs within the timescale of ps and sub-ps ranges usually involve vibrational relaxation, internal conversion, and photo-induced electron and energy transfers. It is important to note that in order to observe such ultrafast processes, ultrashort pulse laser spectroscopic techniques are often employed. In such cases, from the uncertainty principle AEAt Ti/2, one can see that a number of states can be coherently (or simultaneously) excited. In this case, the observed time-resolved spectra contain the information of the dynamics of both populations and coherences (or phases) of the system. Due to the dynamical contribution of coherences, the quantum beat is often observed in the fs time-resolved experiments. 

Figure 7.8 Pump-control-probe quantum beat signal obtained at various probe wavelengths from 378 (bottom) to 390 (top) nm. The delay between the pump and control pulses was set around timing. Each panel shows the different relative phase condition between the pump and control pulses. Reproduced with permission from the supplement of Ref. [39]. Copyright 2009 by the American Physical Society. (See color plate section for the color representation of this figure)...

Now, the eigenenergies of the Hamiltonian can be detected directly if the time dependence of the above average exhibits quantum beats. This will be the case if the spectrum is not too dense and the linewidths are smaller than the level spacings. From a Fourier transform of the autocorrelation function, we then obtain an expression of the form (2.26)-(2.27), which can be evaluated semiclassically in terms of periodic orbits and their quantum phases. 

Fig. 6.5. Numerical simulation of quantum beats measurements of DLL mutant RCs of Rb. capsulatus with 80 fsec pump pulse. Two vibrational frequencies are included in the simulation. The box with broken line indicates the time region in which the phase evolution of the vibrational quantum beams can be seen clearly.

Other possible types of resonance are discussed in [4, 100]. Non-linear parametric, phase and relaxation resonances of quantum beats and the possibilities of their observation in molecules are considered in [33, 37]. 

Auzinsh, M.P. (1990). Nonlinear phase resonance of quantum beats in the dimer ground state, Opt. Spectrosc. (USSR), 68, 750-752. 

The situation changed, however, with two advances. The first advance was the discovery that in the S, - S0 spectrum of jet-cooled anthracene a second band exists (at S, + 1420 cm-1), the excitation of which gives rise to quantum beat-modulated fluorescence decays.40 Besides indicating a somewhat more global importance to the beat phenomenon in anthracene, the characteristics of these new beats provided very strong evidence that they arose as a manifestation of IVR. In particular, the beats were shown to have phases and modulation depths dependent on the fluorescence band detected. Such behavior, which... 

Subsequent to the anthracene studies, picosecond-beam measurements of IVR in a number of other molecules have been made. These molecules include deuterated anthracenes,44 t-stilbene,45 and some alkyl anilines.46 One of the most significant results of these studies is that they have indicated that vibrational coherence30,40 (phase-shifted quantum beats) is a general phenomenon in molecules. Thus, it appears that an accurate understanding of IVR must rest firmly on an accurate understanding of vibrational coherence. 

The observation of novel quantum beats in the spectrally resolved fluorescence of anthracene21 forced one to consider, within the context of radiationless transition theory, the details of how IVR might be manifested in beat-modulated fluorescence decays. This work led to the concepts of phase-shifted quantum beats and restricted IVR,30a,4° and to a general set of results306 pertaining to the decays of spectrally resolved fluorescence in situations where an arbitrary number of vibrational levels, coupled by anharmonic coupling, participate in IVR. Moreover, three regimes of IVR have been identified no IVR, restricted (or coherent) IVR, and dissipative IVR.42... 

The detailed nature of the IVR in which the b -level participates is revealed by time-resolved results. One finds that the decays of individual bands in the 61 fluorescence spectrum are modulated by quantum beats, the phases and... 

Other excitation energies Other than the ones at S, + 1380 and S, + 1420 cm-, there are three prominent bands in the intermediate region of jet-cooled anthracene s excitation spectrum. Time- and frequency-resolved measurements subsequent to excitation of these bands have also been made. Without going into any detail concerning the results of these measurements, we do note that all three excitations give rise to quantum beat-modulated decays whose beat patterns (phases and modulation depths) depend on the fluorescence band detected.42 Figure 16 shows an example of this behavior for excitation to S, + 1514 cm-1. The two decays in the figure correspond to the detection of two different fluorescence bands in the S, + 1514 cm-1 fluorescence spectrum. 

The short-time spike in the decay, which can be attributed to the dephasing of many quantum beat terms (all with + 1 phases), represents the irreversible flow of vibrational energy out of the zero-order state prepared by the laser. The long-time component, although weakly modulated, represents an equilibration in the distribution of vibrational energy subsequent to the initial energy flow process. 

Figure 9.5 The phase of a vibrational quantum beat depends on whether the bright state for the excitation step is bright or dark in the fluorescence detection step, A molecular beam of anthracene, rotationally cooled to 3K, is excited by a 15 picosecond pulse at 1420 cm-1 above the Si <— So 0q origin band. Fluorescence is detected in a selected wavelength region through...

In particular, the laboratory frame orientation of the transition moment for spontaneous fluorescence evolves in time. The intensities of z— and (x,y) — polarized fluorescence are modulated 7t/2 out of phase, but the intensity of the total x + y + z polarized fluorescence is not modulated. This is the physical basis for polarization quantum beats (Aleksandrov, 1964 Dodd, et al., 1964) and Rotational Coherence Spectroscopy (Felker and Zewail, 1995). 

Figure 7a shows that there exists a longer time modulation (about 2 ps) in the chromophore population. The origin of this modulation can be directly traced to the i3 mode, the population of which displays the same evolution pattern as shown in Figure 7c. In this figure, we have compared the Aj and N3 average quantum numbers. It can be seen that these populations vary out of phase, indicating a resonance between the v, and v3 modes. From the quantum beat period, we expect a resonance between two states separated by 15 cm 1 as we noticed previously, the states with energies 16,169 cm 1 and 16,152 cm-1 have an important contribution from the 3 mode. Finally the N2, N4, and N6 populations, shown in Figure 7d, display a monotonic increase over the 15-ps interval, apart from modulations due to the N -N5 and Nt-N2 resonances. The overall relaxation from the 6v,)° mode is displayed in Figure 8.

Formally, a pattern of quantum beats can be characterized by the parameters such as the set of oscillation fi-equencies, oscillation decay time, the phase shift of oscillations, and, finally, their amplitude. Each parameter contains useful information about the processes in radiation spurs. The oscillation frequencies correspond to the splittings in the ESR spectrum of radical ions. The decay of oscillations contains information about spin relaxation times. The phase shift reflects the time delay of pair formation from its precursor. Finally, the amplitude of oscillating component is determined by the fraction of spin correlated pairs. 

The phase shifts of quantum beats have been studied in other alkanes as well [30], As expected, in linear alkanes the rate constants of hole trapping by acceptors, determined from the shifts, were close to the diffusion-controlled ones. In cyclic alkanes (cyclohexane, czs-decalin, and trans-decalin), the hole mobility is known to considerably exceed the mobility of molecular ions [31]. In these solvents the observed phase shifts had an intermediate value between that expected for the highly mobile holes and that assumed for molecular ions. Both types of ions are likely to take part in the formation of diphenylsulphide radical cation in cyclic alkanes. 

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