Wavepackets

This section begins with a brief description of the basic light-molecule interaction. As already indicated, coherent light pulses excite coherent superpositions of molecular eigenstates, known as wavepackets , and we will give a description of their motion, their coherence properties, and their interplay with the light. Then we will turn to linear and nonlinear spectroscopy, and, finally, to a brief account of coherent control of molecular motion. 

A1.6.2.1 WAVEPACKETS SOLUTIONS OF THE TIME-DEPENDENT SCHRODINGER EQUATION... 

Figure Al.6.1. Gaussian wavepacket in a hannonic oscillator. Note tliat the average position and momentum change according to the classical equations of motion (adapted from [6]).

Much of the previous section dealt with two-level systems. Real molecules, however, are not two-level systems for many purposes there are only two electronic states that participate, but each of these electronic states has many states corresponding to different quantum levels for vibration and rotation. A coherent femtosecond pulse has a bandwidth which may span many vibrational levels when the pulse impinges on the molecule it excites a coherent superposition of all tliese vibrational states—a vibrational wavepacket. In this section we deal with excitation by one or two femtosecond optical pulses, as well as continuous wave excitation in section A 1.6.4 we will use the concepts developed here to understand nonlinear molecular electronic spectroscopy. 

The pioneering use of wavepackets for describing absorption, photodissociation and resonance Raman spectra is due to Heller [12, 13,14,15 and 16]- The application to pulsed excitation, coherent control and nonlinear spectroscopy was initiated by Taimor and Rice ([17] and references therein). 

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. 

Figure Al.6.7. Schematic diagram illustrating the different possibilities of interference between a pair of wavepackets, as described in the text. The diagram illustrates the role of phase ((a) and (c)), as well as the role of time delay (b). These cases provide the interpretation for the experimental results shown in figure Al.6.8. Reprinted from [22],...

Figure Al.6.8. Wavepacket interferometry. The interference contribution to the exeited-state fluoreseenee of I2 as a fiinotion of the time delay between a pair of ultrashort pulses. The interferenee eontribution is isolated by heterodyne deteetion. Note that the stnieture in the interferogram oeeurs only at multiples of 300 fs, the exeited-state vibrational period of f. it is only at these times that the wavepaeket promoted by the first pulse is baek in the Franek-Condon region. For a phase shift of 0 between the pulses the returning wavepaeket and the newly promoted wavepaeket are in phase, leading to eonstnietive interferenee (upper traee), while for a phase shift of n the two wavepaekets are out of phase, and interfere destnietively (lower traee). Reprinted from Seherer N F et 0/1991 J. Chem. Phys. 95 1487.

Second-order effects include experiments designed to clock chemical reactions, pioneered by Zewail and coworkers [25]. The experiments are shown schematically in figure Al.6.10. An initial 100-150 fs pulse moves population from the bound ground state to the dissociative first excited state in ICN. A second pulse, time delayed from the first then moves population from the first excited state to the second excited state, which is also dissociative. By noting the frequency of light absorbed from tlie second pulse, Zewail can estimate the distance between the two excited-state surfaces and thus infer the motion of the initially prepared wavepacket on the first excited state (figure Al.6.10 ). 

CN] —> I + CN. Wavepacket moves and spreads in time, with its centre evolving about 5 A in 200 fs. Wavepacket dynamics refers to motion on the intennediate potential energy surface B. Reprinted from Williams S O and lime D G 1988 J. Phys. Chem.. 92 6648. (c) Calculated FTS signal (total fluorescence from state C) as a fiinction of the time delay between the first excitation pulse (A B) and the second excitation pulse (B -> C). Reprinted from Williams S O and Imre D G, as above. 

We now proceed to some examples of this Fourier transfonn view of optical spectroscopy. Consider, for example, the UV absorption spectnun of CO2, shown in figure Al.6.11. The spectnuu is seen to have a long progression of vibrational features, each with fairly unifonu shape and width. Wliat is the physical interpretation of tliis vibrational progression and what is the origin of the width of the features The goal is to come up with a dynamical model that leads to a wavepacket autocorrelation fiinction whose Fourier transfonn... 

Figure Al.6.14. Schematic diagram showing the promotion of the initial wavepacket to the excited electronic state, followed by free evolution. Cross-correlation fiinctions with the excited vibrational states of the ground-state surface (shown in the inset) detennine the resonance Raman amplitude to those final states (adapted from [14].

The central dynamical object that enters mto the polarization are the coherences of the fonn ( / t) p vj/ i(t)) and etc. These quantities are overlaps between wavepackets moving on different potential... 

As discussed above, the nonlinear material response, P f) is the most connnonly encountered nonlinear tenn since vanishes in an isotropic medium. Because of the special importance of P we will discuss it in some detail. We will now focus on a few examples ofP spectroscopy where just one or two of the 48 double-sided Feymnan diagrams are important, and will stress the dynamical interpretation of the signal. A pictorial interpretation of all the different resonant diagrams in temis of wavepacket dynamics is given in [41]. 

Figure Al.6.20. (Left) Level scheme and nomenclature used in (a) single time-delay CARS, (b) Two-time delay CARS ((TD) CARS). The wavepacket is excited by cOp, then transferred back to the ground state by with Raman shift oij. Its evolution is then monitored by tOp (after [44])- (Right) Relevant potential energy surfaces for the iodine molecule. The creation of the wavepacket in the excited state is done by oip. The transfer to the final state is shown by the dashed arrows according to the state one wants to populate (after [44]).

Figure Al.6.21. Bra and ket wavepacket dynamics which detennine the coherence overlap, (( ) ( ) ). Vertical arrows mark the transitions between electronic states and horizontal arrows indicate free propagation on the potential surface. Full curves are used for the ket wavepacket, while dashed curves indicate the bra wavepacket. (a) Stimulated emission, (b) Excited state (transient) absorption (from [41]).

Figure Al.6.24. Schematic representation of a photon echo in an isolated, multilevel molecule, (a) The initial pulse prepares a superposition of ground- and excited-state amplitude, (b) The subsequent motion on the ground and excited electronic states. The ground-state amplitude is shown as stationary (which in general it will not be for strong pulses), while the excited-state amplitude is non-stationary. (c) The second pulse exchanges ground- and excited-state amplitude, (d) Subsequent evolution of the wavepackets on the ground and excited electronic states. Wlien they overlap, an echo occurs (after [40]).

Figure Al.6.26. Stereoscopic view of ground- and excited-state potential energy surfaces for a model collinear ABC system with the masses of HHD. The ground-state surface has a minimum, corresponding to the stable ABC molecule. This minimum is separated by saddle points from two distmct exit chaimels, one leading to AB + C the other to A + BC. The object is to use optical excitation and stimulated emission between the two surfaces to steer the wavepacket selectively out of one of the exit chaimels (reprinted from [54]).

Pollard W T, Lee S-Y and Mathies R A 1990 Wavepacket theory of dynamic absorption spectra in femtosecond pump-probe experiments J. Chem. Phys. 92 4012... 

Knopp G, Pinkas I and Prior Y 2000 Two-dimensional time-delayed coherent anti-Stokes Raman spectroscopy and wavepacket dynamics of high ground-state vibrations J. Raman Spectrosc. 31 51... 

Kosloff R, Rice S A, Gaspard P, Tersigni S and Tannor D J 1989 Wavepacket dancing achieving chemical selectivity by shaping light pulses Chem. Phys. 139 201-20... 

A beautiful, easy-to-read introduction to wavepackets and their use in interpreting molecular absorption and resonance Raman spectra. 

A comprehensive discussion of wavepackets, classical-quantum correspondence, optical spectroscopy, coherent control and reactive scattering from a unified, time dependent perspective. 

In order to localize the particle, it is necessary to superimpose wavefiinctions i with different momenta k. A very general way to do this is to construct a wavepacket, defined tlirough the integral... 

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