Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 469
Fig. 24. Temporal evolution of the electronic density of the extremal rotational quasienergy state |p =20〉 of the n0 =21
resonant manifold, for di erent phases !t = 0 (left), !t = =2 (center), !t = (right) of the driving eld, at amplitude
F0 =0:03, in cylindrical coordinates. Each box extends over ±700 Bohr radii, in both directions, (horizontal) and z
(vertical). The microwave polarization axis is oriented along z. Because of the azimuthal symmetry of the problem, the
actual 3D electronic density is obtained by rotating the gure around the vertical axis. The state represents a non-dispersive
wave packet shaped like a doughnut, moving periodically from the north to the south pole (and back) of a sphere. For
higher n0, the angular localization on the circular orbit should improve.
The eigenstates displayed here are localized along classical trajectories which are resonantly driven
by the external eld. Hence, we should expect them to exhibit wave packet like motion along these
trajectories, as the phase of the driving eld is changed. This is indeed the case as illustrated in
Fig. 24 for the state p = 20 with maximal angular momentum L=n0 1 [48,64,67]. Due to the
azimuthal symmetry of the problem, the actual 3D electronic density is obtained by rotating the
gure around the vertical axis. Thus, the wave packet is actually a doughnut moving periodically
from the north to the south pole (and back) of a sphere, slightly deformed along the eld direction.
The interference resulting from the contraction of this doughnut to a compact wave packet at the
poles is clearly visible at phases !t = 0 and in the plot. Note that the creation of unidirectional
wave-packet eigenstates moving along a circle in the plane containing the eld polarization axis is
not possible for the real 3D atom [67], as opposed to the reduced 2D problem studied in [48], due
to the above mentioned azimuthal symmetry (see also Section 3.5). For other states in the n0 =21
resonant manifold, the longitudinal localization along the periodic orbit is less visible. The reason is
that 1 is smaller than for the p=20 state, leading to a smaller resonance island in (Î; ˆ ) (see Fig. 20)
and, consequently, to less e cient localization. Proceeding to higher n0-values should improve the
situation.
Let us brie y discuss “excited” states in the resonance island, i.e., manifolds corresponding to
N¿0 in Eq. (78). Fig. 25 shows the exact level dynamics, with the semiclassical prediction for
N = 1 superimposed [67]. The states in this manifold originate from n0 = 22. We observe quite
good agreement between the quantum and semiclassical results for high lying states in the manifold
(for which the principal action island is large, see Fig. 20). For lower lying states the agreement is
improved for higher values of F0. IfF0 is too low, the states are not fully localized inside the resonance
island and, consequently, are badly reproduced by the resonant semiclassical approximation.
This is further exempli ed in Fig. 26, for N =2. Here, the agreement is worse than for smaller values
of N , and is observed only for large F0 and large p. This con rms the picture that the validity of
the semiclassical approach outlined here is directly related to the size, Eqs. (71) and (82), of the
resonance island in (Î; ˆ ) space (see also the discussion in Section 8.3).