Population Quantum Beats

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 7.12 Strong-laser-induced quantum interference. Quantum beats observed in the populations of the eigenstates v = 25, 27, and 29 as functions of the pump-NlR delay Tfjjjj. Each trace is an average of three repeated scans. The vertical scahngs of the traces for v = 25, 27, and 29 have been normahzed by their intensities averaged over Tpjjjj = —536 —283, —531 —278, and —535 —281 fs, respectively. Reproduced from Ref. [40] with permission from Nature Publishing Group.

From the above discussion one can see that due to <7 (At) the measurement of y(quantum beat the beat frequency is a>tt while the beat width is approximately dependent on its dephasing constant. From Eq. (4.53) it can be seen that the ultrafast dynamics of population and coherence appears in initial condition, created by the pumping laser. 

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... 

Figure 3.17 Time evolution of the ISe ISh exciton population as a function of the relaxation times and Coulomb coupling for the single and biexcitons. The rise time of the biexciton (formation time) and the presence of strong or damped quantum beating depends upon the relative values of the Wc, yi, and j2 as shown above. Source Shabaev et al. (2006).

Guo and Yang [53] have analyzed spontaneous decay from two atoms initially prepared in an entangled state. They have shown that the time evolution of the population inversion, which is proportional to the intensity (87), depends on the degree of entanglement of the initial state of the system. Ficek et al. [10] have shown that in the case of two nonidentical atoms, the time evolution of the intensity 7(R, t) can exhibit quantum beats that result from the presence of correlations between the symmetric and antisymmetric states. In fact, quantum beats are present only if initially the system is in a nonmaximally entangled state, and no quantum beats are predicted for maximally entangled as well as unentangled states. 

Ring, et al., (1998 and 1999) have used a time-dependent magnetic field and the combination of a static magnetic field in a direction perpendicular to that of a time-dependent field to create and manipulate novel coherences and to monitor the quantum beats associated with specifiable details of the time evolution of these coherences. The frequencies and decay rates of different classes of coherence (AMj = 2 and 1 polarization beats, AMj = 0 singlet triplet population beats) may be sampled and modified selectively. 

This suggests that a quantum beat spectrum of an AT-level system will, because it contains redundant information, be more complicated than the corresponding frequency domain spectrum. However, when the level spacings are approximately integer multiples of a common factor, such as 2B for upper-state A2F(J) = B (iJ + 2) rotational combination differences, then each upper state (J + 1, J — 1) pair of rotational levels coherently excited from all thermally populated lower-state J" levels contributes to a grand rephasing at tn = n [-gj ] (n = 1,2,...). This is Rotational Coherence Spectroscopy (RCS) (Felker and Zewail, 1987 and 1995 Felker, 1992). It provides upper state rotar tional constants without the need for a rotational analysis. 

Spectroscopic perturbations follow A J = 0, A Mj = 0 selection rules. Therefore perturbations are not a necessary feature of polarization quantum beats. However, such perturbations are essential to a major class of population quantum beats. 

The language of bright state and dark state is central to population quantum beats and also to the related polyatomic molecule radiationless decay processes (Bixon and Jortner, 1968 Rhodes, 1983), Intramolecular Vibrational Redistribution (IVR) (Parmenter, 1983 Nesbitt and Field, 1996 Wong and Gruebele, 1999 Keske and Pate, 2000), Inter-System Crossing (ISC), and Internal Conversion (IC), discussed in Section 9.4.15. 

The reason for the name population quantum beats is that the signal intensity (fluorescence, REMPI), integrated over all solid angles and polarization states, oscillates in time after the preparation pulse. It appears as if the population prepared in the excited state at t = 0 vanishes and returns periodically (see Fig. 9.5) (Felker and Zewail, 1984). In fact, the population does not oscillate, but the radiative capability of the time-evolving state prepared at t = 0, k(t), does oscillate. 

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.

When applying the quantum beats technique, it is essential that only the singlet-correlated pairs contribute to the oscillating component of spin evolution. Therefore, for systems with a single oscillation frequency, the singlet state population Ps t) is determined by... 

The purpose of this section is to show how to employ the density matrix method to study the population dynamics of a system. From the model shown in Fig. 4.2, we can see that due to the fact that there is only one system state, there is no system coherence (or phase). However, quantum beat may be observed under certain conditions. It should be noticed that the master equations of this model can be solved exactly and analytically. Likewise, its Schrodinger equation can also be solved exactly and analytically. 

Either two or more molecular levels of a molecule are excited coherently by a spectrally broad, short laser pulse (level-crossing and quantum-beat spectroscopy) or a whole ensemble of many atoms or molecules is coherently excited simultaneously into identical levels (photon-echo spectroscopy). This coherent excitation alters the spatial distribution or the time dependence of the total, emitted, or absorbed radiation amplitude, when compared with incoherent excitation. Whereas methods of incoherent spectroscopy measure only the total intensity, which is proportional to the population density and therefore to the square ir of the wave function iff, the coherent techniques, on the other hand, yield additional information on the amplitudes and phases of ir. 

Within the density-matrix formalism (Vol. 1, Sect. 2.9) the coherent techniques measure the off-diagonal elements pab of the density matrix, called the coherences, while incoherent spectroscopy only yields information about the diagonal elements, representing the time-dependent population densities. The off-diagonal elements describe the atomic dipoles induced by the radiation field, which oscillate at the field frequency radiation sources with the field amplitude Ak(r, t). Under coherent excitation the dipoles oscillate with definite phase relations, and the phase-sensitive superposition of the radiation amplitudes Ak results in measurable interference phenomena (quantum beats, photon echoes, free induction decay, etc.). 

The theoretical description of the quantum-beat structure in terms of oscillating population differences between Zeeman substates of different fine structure levels gives very satisfactory explanation for the appearance and form of the observed signal. 

A quantum-beat laser can be initialized by using lasing without population inversion. The mechanism for lasing without inversion essentially is CPT/EIT absorption is reduced while stimulated emission remains unaffected [50-56]. At the same time, the coherence leads to noise squeezing [57-61]. The fluctuations of the laser intensity are suppressed up to or more than 50% below the shot noise. We also note that noise squeezing is possible even when neither initial coherence nor external coherent driving is used. This is caused by dynamic noise reduction... 

A quantum-beat laser with external coherent driving [67,78]. Shown in Fig. 13 is a pumping and coupling scheme. The atoms are pumped from the ground state 10 ) to the excited states (/=1,2) to provide necessary population for the laser gain. An external coherent field of circular frequency coq is applied to the 2 )- 3 ) transitions to create atomic coherence, by which the system can operate without population inversion. Atoms emit photons into the laser modes o of circular frequencies a 12. In the dynamics, the atoms recycle throngh the snccessive channels... 

In terms of dressed states and the collective modes, the present system is similar to a single-mode system. As a consequence of quantum beats, modes 82,3 decouple and only mode 8 mediates into the interaction. Incoherent transfer of population is carried out through the channel... 

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... 

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