Wavepacket interference

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.

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]. 

By making use of classical or quantum-mechanical interferences, one can use light to control the temporal evolution of nuclear wavepackets in crystals. An appropriately timed sequence of femtosecond light pulses can selectively excite a vibrational mode. The ultimate goal of such optical control is to prepare an extremely nonequilibrium vibrational state in crystals and to drive it into a novel structural and electromagnetic phase. 

The needle-shaped wavepacket solutions of the individual photon model, which agree at least sometimes with the dot-shaped marks on a screen in interference experiments... 

In many cases the photon can be represented by the two alternative models of a plane wave and a particle-like wavepacket. This should also apply to interference phenomena with individual photons [21]. For a given point at the screen of an experiment with two apertures, the resulting interference pattern obtained from individual photon impacts could thus be interpreted in two alternative ways ... 

The photon is as an axisymmetric wavepacket divided into two parts that pass the apertures and interfere with each other when ending up at the screen to form a common dot-shaped mark. 

Irrespective of whether the photon is considered as a plane wave or a wavepacket of narrow radial extension, it must thus be divided into two parts that pass each aperture. In both cases interference occurs at a particular point on the screen. When leading to total cancellation by interference at such a point, for both models one would be faced with the apparently paradoxical result that the photon then destroys itself and its energy hv. A way out of this contradiction is to interpret the dark parts of the interference pattern as regions of forbidden transitions, as determined by the conservation of energy and related to zero probability of the quantum-mechanical wavefunction. 

These questions appear to be understandable in terms of both photon models. The wavepacket axisymmetric model has, however, an advantage of being more reconcilable with the dot-shaped marks finally formed by an individual photon impact on the screen of an interference experiment. If the photon would have been a plane wave just before the impact, it would then have to convert itself during the flight into a wavepacket of small radial dimensions, and this becomes a less understandable behavior from a simple physical point of view. Then it is also difficult to conceive how a single photon with angular momentum (spin) could be a plane wave, without spin and with the energy hv spread over an infinite volume. Moreover, with the plane-wave concept, each individual photon would be expected to create a continuous but weak interference pattern that is spread all over the screen, and not a pattern of dot-shaped impacts. 

The explanation for the absence of interference comes naturally from the causal model of a particle whose undulatory part is described by a finite localized wavelet. In this situation, the limitless spreading of matter wavepackets originates from the fact that in the initial burst, coming from the source, each individual localized quantum particle travels at a different velocity. Therefore, as the time increases the distance among them also increases, as shown in Fig. 24. 

We have employed this phase-sensitive pump-probe technique to further investigate the multiphoton ionization of Na2 with 618-nm femtosecond pulses as discussed in the previous paragraph and have observed the interference of the A E and 2 Tlg wavepackets created by the first pulse and those created by the second pulse in the Na2+ signal. The amplitude of the high-frequency oscillations in the Na2+ signal was obtained as a function of pump-probe delay by filtering the transient with the laser frequency. It is shown in Fig. 8 (top). Below the averaged Na2+ transient of Fig. 4 is... 

Figure 8. Frequency-filtered Na2+ pump-probe signal in comparison to the averaged signal of Fig, 4. The filtered signal measures by how much the Na2+ signal is modulated with the laser frequency. Such modulations occur when there is interference between excitation by the probe pulse and the wavepackets formed by the pump laser pulse. This interference effect causes both the A EJ and the 2 1 Ilg state wavepacket motion to be observable in the signal.

On the other hand, additional spectroscopic information can be obtained by making use of this technique The Fourier transform of the frequency-filtered transient (inset in Fig. 8) shows that the time-dependent modulations occur with the vibrational frequencies of the A E and the 2 IIg state. In the averaged Na2+ transient there was only a vanishingly small contribution from the 2 IIg state, because in the absence of interference at the inner turning point ionization out of the 2 IIg state is independent of intemuclear distance, and this wavepacket motion was more difficult to detect. In addition, by filtering the Na2+ signal obtained for a slowly varying pump-probe delay with different multiples of the laser frequency, excitation processes of different order may be resolved. This application is, however, outside the scope of this contribution and will be published elsewhere. 

For a three-photon and a two-photon process we have shown that vibrational wavepacket propagation excited by an ultrashort laser pulse can be used to drive a molecule to a nuclear configuration where the desired product formation by a second probe pulse is favored (Tannor-Kosloff-Rice scheme). In both cases the relative fragmentation and ionization yield of Na2 was controlled as a function of pump-probe delay. By varying the delay between pump and probe pulses very slowly and therefore controlling the phase relation between the two pulses, additional interference effects could be detected. 

The peak shift data in Fig. 17 show oscillatory character, as is our first two examples (I2 and LH1). This arises from vibrational wavepacket motion. In addition, the very fast drop in peak shift to about 65% of the initial value in -20 fs results from the interference between the wavepackets created in different intramoleculear modes. This conclusion follows directly from obtaining the frequencies and relative coupling strengths of the intramolecular modes from transient grating studies of IR144, carried out in the same solvents (data not shown). Thus, by visual inspection of Fig. 17, an answer to a long-standing question—What fraction of the spectral width arises from intra- and intermolecular motion —is immediately apparent. 

Thus a wavepacket initiated in well A passes to well B by a curve crossing. Prof. Fleming showed an interesting case of persistent coherence in such a situation, despite the erratic pattern of the eigenvalue separations. An alternative, possibly more revealing approach, is to employ Stuckelberg-Landau-Zener theory, which relates the interference (i.e., coherence) in the two different wells via the area shown in Fig. 2. A variety of applications to time-independent problems may be found in the literature [1]. 

Note that the calculation of the optimal pulse shape is a double-ended boundary-value problem tp is known at t = 0 while x is known at t = T. This aspect of the calculation of the optimal pulse shape mirrors the considerations advanced concerning the competition between spreading of the wavepacket as it moves on the potential-energy surface and the use of interference between pump and dump fields to counteract that spreading. 

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