Resonances coherent state superposition

Figure 3. Field-matter interactions for a pair of electronic states. The zero and first excited vibrational levels are shown for each state (A). The fields are resonant with the electronic transitions. A horizontal bar represents an eigenstate, and a solid (dashed) vertical arrow represents a single field-matter interaction on a ket (bra) state. (See Refs. 1 and 54 for more details.) A single field-matter interaction creates an electronic superposition (coherence) state (B) that decays by electronic dephasing. Two interactions with positive and negative frequencies create electronic populations (C) or vibrational coherences either in the excited (D) or in the ground ( ) electronic states. In the latter cases (D and E) the evolution of coherence is decoupled from electronic dephasing, and the coherences decay by the vibrational dephasing process.

We study the various superpositions of states that can be created by adiabatic passage in a robust way with respect to variations of the field amplitude, using the topological analysis with resonances of Section V.D (see also Section IV.B.3 for the case of one laser). We assume that one starts (at time t = ti) with a coherent state for the photon field and in the atomic state 1). We study here the A-system. Our results are easily extended to the other system (ladder and V), using the appropriate signs accompanying the field frequencies. We study the creation of a superposition of states at the final time t = tf. 

By preparing a coherent superposition of resonant states one could conceivably observe recurrences, section 4.4.1.1, which are damped by unimolecular decay. Consider the situation where there are only two resonances in the superposition state so that Eq. (8.10) becomes... 

The energy levels of a molecule placed in an off-resonant microwave field can be calculafed by diagonalizing fhe mafrix of fhe Floquef Hamiltonian in the basis of direct products y) ), where y) represents in the eigenstates of the molecule in the absence of the field and ) - fhe Fourier componenfs in Eq. (8.21). The states k) are equivalent to photon number states in the alternative formalism using the quantum representation of the field [11, 15, 26], The eigensfales of the Floquet Hamiltonian are the coherent superpositions... 

Conversely, a coherent superposition of continuum states with a population closely reproducing an isolated peak in the density of states, which corresponds to a resonance, can be built in such a way to give rise to a localized state. From this localized state, there will be an outward probability density flux, i.e., it will have a finite lifetime. In the limit of a resonance position far from any ionization threshold and a narrow energy width, the decay rate will be exponential with the rate constant T/ft. The decay is to all the available open channels, in proportion to their partial widths. 

The decay of the individual quasi-bound (metastable) resonance states follows an exponential law. The wave packet prepared by an ultrashort pulse can be represented as a (coherent) superposition of these states. The decay of the associated norm (i.e., population) follows a multi-exponential law with some superimposed oscillations due to quantum mechanical interference terms. The description given above is confirmed by experimental data. 

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]. 

Ducas, T.W., Littman, M.G. and Zimmerman, M.L. (1975). Observation of oscillations in resonance absorption from a coherent superposition of atomic states, Phys. Rev. Lett., 35, 1752-1754. 

Since VSFS is a coherent technique, and as Equation (25) implies, the oscillating electric field from each vibrational state involved in the generation of sum-frequency light can interfere with that from every other state and with that from the non-resonant response. This is quite different from IR spectroscopy where spectra are simple superpositions of intensity from individual vibrational modes. As such, VSFS leads to interesting line shapes that can be interpreted incorrectly if spectral intensities are compared visually without fitting the spectrum. 

Finally, we should mention the possibility of coherent excitation transfer when the donor-acceptor interaction is strong, but the coupling of the system to the thermal bath is weak. The resulting two-level weakly damped system lends itself to the time-dependent density-matrix approach [33], which is essentially identical to the familiar spin-1/2 treatment in magnetic resonance. Under certain circumstances, coherence effects can be important for singlet energy transfer, because the donor states are populated instantaneously by direct photoexcitation. With a sufficient band width of the excitation source (e.g., ultrashort femtosecond pulses), quantum superposition states can be prepared in a coherent fashion even in condensed media at room... 

D. Resonant Processes—Creation of Coherent Superposition of States—Half-Scrap... 

When the processes involve a zero-field resonance, one has to add the ingredient of lifting of degeneracy. This means that we have to consider the dynamics starting (or ending) near the conical intersection in a direction not parallel to the Cl = 0-plane. This can be seen in Fig. 6 where the surfaces of Fig. 2 have been redrawn for positive detunings (case of a one-photon resonance). When the dynamics starts this way, it is characterized by two adiabatic paths, one on each surface. They will lead in general to coherent superpositions of states. 

This process leading to a coherent superposition of states has been suggested in Ref. 63 and named half-scrap, since it is very similar to the scrap process except it starts (or ends) in resonance. 

Kosloff, 1994) have also been used to find the complex energies for compound-state resonances. A localized wave packet [i.e., a coherent superposition state, Eq. (4.7)], P(O) is initially placed in the bound region of the potential energy surface and propagated in time to give ( l (0) (r)), which is C t) in Eq. (4.16). If I (O) is a superposition of resonant states, it can be considered a zero-order state (see chapter 4) and can be written as... 

From magnetic resonance spectroscopy [49] it is well-known that IB effects are adequately circumvented by the tricks of a spin echo experiment. For instance, in a two-pulse echo experiment, IB effects are averaged out and one probes spin dephasing determined by time-dependent fluctuations characteristic of HB only (and not IB). More specifically, a nll-r-n microwave pulse sequence is applied, where the first nil pulse creates a coherent superposition state for which a la = 1 and the n pulse, applied at time r after the first pulse, generates a spin coherence (the echo) at time 2r after the initial pulse. The echo amplitude is traced with r. The echo amplitude decay time is characteristic of the pure dephasing dynamics. For phosphorescent triplet states it is possible to make the echo optically detectable by means of a final nil probe pulse applied at time f after the second pulse [44]. In Fig. 3b, the optically detected echo amplitude decay for the zero-field transition at 2320 MHz of... 

The Stokes laser generates a coherent superposition of the wavefunctions of levels 2) and 3). The states 2) and 3) are, however, not occupied before the pump pulse arrives. The wavefunction oscillates between levels 2) and 3) with the Rabi frequency which depends on the intensity of the Stokes pulse and its detuning from resonance. Now the pump pulse comes with a time delay At with respect to the Stokes pulse, where At is smaller than the width of the Stokes pulse, which means that the two pulses still overlap (Fig. 7.15b). This places the molecule at a coherent superposition of levels 1) and 2) and 2) and 3). If the delay At, the detuning A v and the intensities of the two lasers are correctly chosen, the population in level 11) can be completely transferred into level 13) without creating a population in level 2) (Fig. 7.15c). The coherently excited levels 1) and 2) are described by the wavefunction... 

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