Wavepacket phase coherent

D. J. Tannor I would like to point out that the Scherer-Fleming wavepacket interferometry experiment is very different from the Tannor-Rice pump-dump scheme, in that it exploits optical phase coherence of the laser light (optical phase coherence translates into electronic phase coherence between the wavepackets on different potential surfaces). However, there was a paragraph in the first paper of Tannor and Rice [7. Chem. Phys. 83, 5013 (1985), paragraph above Eq. (11)] that did in fact discuss the role of optical phase and suggested the possibility of experiments of the type performed by Scherer and Fleming. 

When a femtosecond laser pulse passes through nearly any medium, coherent vibrational excitation (in general, initiation of coherent wavepacket propagation) is likely [33, 34]. One- or two-photon absorption of a visible or ultraviolet pulse into an electronic excited state can result in phase-coherent motion in the excited-state potential [35]. Impulsive stimulated Raman scattering can initiate phase-coherent vibrational motion in the electronic... 

Things are not quite as simple as they seem. In order for the constructive interference, which is at the core of wavepacket interferometry, to occur, not only must (t + At) = (t), but also the phases of apump and aprobe> which depend on the optical phase of the femtosecond laser rather than the molecular phase, must match. A rigorous treatment of the phase coherent pump/probe scheme using optically phase-locked pulse pairs is presented by Scherer, et al., [1990, 1991, 1992] and refined by Albrecht, et al., (1999), who discuss the distinction between and consequences of pulse envelope delays vs. carrier wave phase shifts (see Fig. 9.6). A simplified treatment, valid only for weak optical pulses is presented here. 

For 900 nm radiation, the path difference between pump and probe is stepped in 0.9 /j,m increments (or integer multiples), which corresponds to At, = 3 fs. This crucial modification is called phase coherent wavepacket interferometry (Scherer, et al, 1990, 1991, 1992) and it results in a profound simplification of... 

In the phase-coherent, one-color pump/probe scheme (see Section 9.1.9) the wavepacket is detected when the center of the wavepacket returns to its to position, (x)to+nT — (x)to, after an integer number of vibrational periods. The pump pulse creates the wavepacket. The probe pulse creates another identical wavepacket, which may add constructively or destructively to all or part of the original pump-produced wavepacket. If the envelope delay and optical phase of the probe pulse (Albrecht, et al, 1999) are both chosen correctly, near perfect constructive or destructive interference occurs and the total spontaneous fluorescence intensity (detected after the pump and probe pulses have traversed the sample) is either quadrupled (relative to that produced by the pump pulse alone) or nulled. As discussed in Section 9.1.9, the probe pulse is delayed, relative to the pump pulse, in discrete steps of At = x/ojl- 10l is selected by the experimentalist from within the range (ljl) 1/At (At is the temporal FWHM of the pulse) to define the optical phase of the probe pulse relative to that of the pump pulse and the average excitation frequency. However, [(E) — Ev ]/K is selected by the molecule in accord with the classical Franck-Condon principle (Tellinghuisen, 1984), also within the (ojl) 1/At range. When the envelope delay is chosen so that the probe pulse arrives simultaneously with the return of the center of the vibrational wavepacket to its position at to, a relative maximum (optical phase at ojl delayed by 2mr) or minimum (optical phase at u>l delayed by (2n + l)7r) in the fluorescence intensity is observed. 

An alternative perspective is as follows. A 5-frmction pulse in time has an infinitely broad frequency range. Thus, the pulse promotes transitions to all the excited-state vibrational eigenstates having good overlap (Franck-Condon factors) with the initial vibrational state. The pulse, by virtue of its coherence, in fact prepares a coherent superposition of all these excited-state vibrational eigenstates. From the earlier sections, we know that each of these eigenstates evolves with a different time-dependent phase factor, leading to coherent spatial translation of the wavepacket. 

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

Coherent states and diverse semiclassical approximations to molecular wavepackets are essentially dependent on the relative phases between the wave components. Due to the need to keep this chapter to a reasonable size, we can mention here only a sample of original works (e.g., [202-205]) and some summaries [206-208]. In these, the reader will come across the Maslov index [209], which we pause to mention here, since it links up in a natural way to the modulus-phase relations described in Section III and with the phase-fiacing method in Section IV. The Maslov index relates to the phase acquired when the semiclassical wave function haverses a zero (or a singularity, if there be one) and it (and, particularly, its sign) is the consequence of the analytic behavior of the wave function in the complex time plane. 

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

How well do these quantum-semiclassical methods work in describing the dynamics of non-adiabatic systems There are two sources of errors, one due to the approximations in the methods themselves, and the other due to errors in their application, for example, lack of convergence. For example, an obvious source of error in surface hopping and Ehrenfest dynamics is that coherence effects due to the phases of the nuclear wavepackets on the different surfaces are not included. This information is important for the description of short-time (few femtoseconds) quantum mechanical effects. For longer timescales, however, this loss of information should be less of a problem as dephasing washes out this information. Note that surface hopping should be run in an adiabatic representation, whereas the other methods show no preference for diabatic or adiabatic. 

Unlike the case of simple diatomic molecules, the reaction coordinate in polyatomic molecules does not simply correspond to the change of a particular chemical bond. Therefore, it is not yet clear for polyatomic molecules how the observed wavepacket motion is related to the reaction coordinate. Study of such a coherent vibration in ultrafast reacting system is expected to give us a clue to reveal its significance in chemical reactions. In this study, we employed two-color pump-probe spectroscopy with ultrashort pulses in the 10-fs regime, and investigated the coherent nuclear motion of solution-phase molecules that undergo photodissociation and intramolecular proton transfer in the excited state. 

A. H. Zewail Prof. Chetgui s observations are very interesting. As discussed in this conference, coherent wavepacket motion has been observed in condensed phases in many laboratories. In Prof. Chergui s experiment it is now possible to study the time scale for bubble formation. Molecular dynamics should tell us the nature of forces which maintain any coherence in such systems. 

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