Non-dispersive wave packets in periodically driven quantum systems

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512 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 49. Electronic densities of the eigenstates of the upmost doublet states (top) of the n0 = 42 manifold of Fig. 46, and of their mirror states (bottom), shifted in energy by !=2 (Fig. 47), at driving eld phase !t = 0. The longitudinal localization on the Kepler ellipses (similar for all states) is apparent. On each ellipse, four di erent individual wave packets (or rather, due to azimuthal symmetry, two doughnut wave packets) can be distinguished, propagating along the Kepler ellipse. Notice the phase shift of in the temporal evolution on the two ellipses, implied by z-inversion. The microwave polarization axis along z is given by the vertical axis of the gure, with the nucleus at the center of the gure. longitudinally delocalized because of the time average. In fact, there are four such Floquet states displaying very similar electronic densities. These are the energetically highest doublet in the n0 =42 manifold, and the upmost doublet in the “mirror” manifold displayed in Fig. 47. Fig. 49 shows the electronic densities of these four Floquet eigenstates at phase !t = 0 of the driving eld: the four doughnuts are now clearly visible, as well as the orbital and radial localizations along the two elliptic trajectories. Very much in the same way as for a double well potential (or for the bouncer discussed in Section 5.2, compare Fig. 41), a linear combination of these four states allows for the selection of one single doughnut, localized along one single classical Kepler ellipse. This wave packet then evolves along this trajectory without dispersion, as demonstrated in Fig. 50. Note, however, that this single wave-packet is not a single Floquet state, and thus does not exactly repeat itself periodically. It slowly disappears at long times, for at least two reasons: rstly, because of longitudinal and transverse tunneling, the phases of the four Floquet eigenstates accumulate small di erences as time evolves, what induces complicated oscillations between the four possible locations of the wave-packet, and secondly, the ionization rates of the individual Floquet states lead to ionization and loss of phase coherence, especially if the ionization rates (see Section 7.1) of the four states are not equal.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 513 Fig. 50. Temporal evolution of a convenient linear combination of the four eigenstates of Fig. 49, for phases !t = 0 (top left), =2 (top center), (top right), 3 =2 (bottom left), 2 (bottom right) of the driving eld. Clearly, a single doughnut propagating along a single trajectory has been selected by the linear combination. This wave packet essentially repeats its periodic motion with period 2T =4 =!. It slowly disperses, because the four states it is composed of are not exactly degenerate (tunneling e ect), and because it ionizes (see Section 7.1). The microwave polarization axis along z is parallel to the vertical axis of the gure, with the nucleus at the center of the plot. 6. Alternative perspectives There are several known systems where an oscillating eld is used to stabilize a speci c mode of motion, such as particle accelerators [3], Paul traps [162] for ions, etc. In these cases, the stabilization is a completely classical phenomenon based on the notion of non-linear resonances. What distinguishes our concept of non-dispersive wave packets discussed in the preceding chapters from those situations is the necessity to use quantum (or semiclassical) mechanics to describe a given problem, due to relatively low quantum numbers. Still, the principle of localization remains the same, and consists in locking the motion of the system on the external drive. However, it is not essential that the drive be provided externally, it may well be supplied by a (large) part of the system to the (smaller) remainder. Note that, rather formally, also an atom exposed to a microwave eld can be understood as one large quantum system—a dressed atom, see Section 2—where the eld component provides the drive for the atomic part [18]. In the present section, we shall therefore brie y recollect a couple of related phase-locking phenomena in slightly more complicated quantum systems, which open additional perspectives for creating non-dispersive wave packets in the microscopic world. 6.1. Non-dispersive wave-packets in rotating molecules A situation closely related to atomic hydrogen exposed to CP microwaves (Section 3.4) is met when considering the dynamics of a single, highly excited Rydberg electron in a rotating molecule
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