Wavepacket component

Using the wavepacket propagation methodology outlined in Section 3.2, the wavepacket components are generated at a given time f, from some initial state defined at f = 0, which will be specified later. Various observables can be calculated from the wavepackets at the current time, among which the total instantaneous probabilities for the ion to remain in bound vibrational states of fhe ground electronic state, i.e.. 

Figure 2.2 illusfrafes the dynamics of W and of the associated nuclear probability distributions. For 5 = 0 the field is at its peak intensity at f = 0, when the initial wavepacket is prepared on the inner repulsive edge of fhe affracfive potential (close to Cg in this region). Only its tail penetrates the gap region (R 4 a.u.) which, at that time, is widely open between W+ and W. At f = T/4, the wavepacket components reach the gap region with a gap now closed due to the vanishing field amplitude, preventing thus any... 

To numerically absorb the wavepacket components at the boundaries, which are set beforehand to prevent the packet from flowing out to the asymptotic ranges, a damping function e is introduced in the Chebyshev recursion scheme such that... 

Interpretation of spectra Wavepacket components and ionization channels... 

The excited wavepacket formed by the pump pulse evolves into several components as seen in Figs. 5.21 and 5.22 (1) part of the initial wavefunction that was not electronically excited by the pump pulse, (2) the bulk of the excited wavepacket formed by the pump pulse, (3) two symmetrically equivalent components formed on electronic state 1 at the time of the first passage through the conical intersection (P in Fig. 5.21, snapshot shown in the lower panel at 8 fs in Fig. 5.22), and (4) another set of symmetrically equivalent components formed on electronic state 2 at the time of the second passage through the conical intersection (Pj in Fig. 5.21, or the additional components shown in the upper panel at 20 fs in Fig. 5.22). These wavepacket components are probed by the four ionization channels depicted in Fig. 5.15. 

When affected by the control pulse, there is much greater transfer between the diabatic states before the wavepacket reaches the conical intersection geometry. Thus the wavepacket component on state 1 is formed both just before and after first passage through the conical intersection region. 

At long times (> 400 fs) LiF on diabatic state 2 would dissociate, handled by an optical potential eliminating the wavepacket component going out of the spatial grid in the computation. The dissociating population is kept track of and the figures do not show the decrease due to dissociation. 

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. 

Gaussian wavepackets multiple spawning, 402 propagation, 380-381 molecular systems, component amplitude analysis, wave packet construction, 229-230... 

Under even more intense photoexcitation ( 10mJ/cm2), the coherent A g and Eg phonons of Bi and Sb exhibit a collapse-revival in their amplitudes (Fig. 2.10) [42,43], This phenomenon has a clear threshold in the pump density, which is common for the two phonon modes but depends on temperature and the crystal (Bi or Sb). At first glance, the amplitude collapse-revival appears to be analogous to the fractional revival in nuclear wavepackets in molecules [44,45]. However, the pump power dependence may be an indication of a polarization, not quantum, beating between different spatial components of the coherent response within the laser spot [46],... 

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