Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 489
Fig. 35. Non-dispersive wave packets of the two-dimensional hydrogen atom exposed to an elliptically polarized, resonant
microwave eld. Scaled microwave amplitude F0 =0:03 (for resonant principal quantum number n0 = 21), and ellipticity
=0:4. Top row: non-dispersive wave packet moving on a circular orbit corotating with the microwave eld, for phases
!t =0; =4; =2 (from left to right). This wave packet evolves into the eigenstate represented in Fig. 27, under continuous
increase of the ellipticity to = 1. Bottom row: non-dispersive wave packet launched along the same circular orbit, but
contra-rotating with the driving eld (for the same phases). Note that, while the co-rotating wave packet almost preserves
its shape during the temporal evolution, the contra-rotating one exhibits signi cant distortions during one eld cycle, as
a direct consequence of its complicated level dynamics shown in Fig. 34. Still being an exactly time-periodic Floquet
eigenstate, it regains its shape after every period of the microwave. The size of each box extends over ±800 Bohr radii, in
both x and y directions, with the nucleus in the middle. The major axis of the polarization ellipse is along the horizontal
x axis and the microwave eld points to the right at t =0:
varying from LP to CP. The agreement is excellent, with slightly larger discrepancies between the
semiclassical and the exact results for the states with smallest energy. For those states the resonance
island is very small (small 1), what explains the discrepancy.
The highest lying state in Fig. 34, ascending with , is a non-dispersive wave-packet state located
on the circular orbit and corotating with the EP eld. It is shown in Fig. 35 for =0:4. As mentioned
above, the corresponding counterrotating wave packet is energetically degenerate with the co-rotating
one for = 0. Its energy decreases with (cf. Fig. 34). It is shown in the bottom row in Fig. 35
for =0:4. While the corotating wave packet preserves its shape for all values (except at isolated
avoided crossings) the counter-rotating wave packet undergoes a series of strong avoided crossings
for ¿0:42, progressively loosing its localized character. This is related to a strong decrease of the
maximum of 1 at M0 = −1 with , clearly visible in Fig. 33.
While we have discussed the two-dimensional case only, the CP situation (compare Section 3.4)
indicates that for su ciently large , the important resonant motion occurs in the polarization plane,
being stable vs. small deviations in the z direction. Thus the calculations presented above are also
relevant for the real three-dimensional world, provided is not far from unity [69,73]. For arbitrary
, a full 3D analysis is required. While this is clearly more involved, the general scenario of a wave
packet anchored to a resonance island will certainly prevail.