Non-dispersive wave packets in periodically driven quantum systems

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.