Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 509
Energy (10 -4 a.u.)
-2.822
-2.827
-2.832
-2.837
0 5 10 15 20
p
25 30 35 40
Fig. 46. Comparison of numerically exact quasi-energies originating from the n0 = 42 manifold (depicted by pluses) to
the semiclassical prediction (open symbols) based on the quantization of the s = 2 resonance island (microwave frequency
=2× Kepler frequency), for scaled microwave eld F0 =0:04. Circles correspond to doubly degenerate states localized
in the vicinity of maxima of 2, around the elliptic xed points at (L0 0:65; =0; ) in Fig. 45. Triangles correspond
to almost circular states in the vicinity of the stable minimum at (L0 =1; arbitrary), while diamonds correspond to
states localized around the stable minima at L0 =0; = =2; 3 =2. The agreement between the semiclassical and quantum
energies is very good, provided the size of the resonance island in the (Î; ˆ ) plane is su ciently large (high lying states
in the manifold). For low lying states in the manifold, the discrepancies between quantum and semiclassical results are
signi cant, due to the insu cient size of the island.
with a large resonance island in the (Î; ˆ ) plane. The motion in their vicinity is strongly con ned,
both in the angular (L0; ) and in the (Î; ˆ ) coordinates: the corresponding eigenstates can be
characterized as non-dispersive wave packets, localized both longitudinally along the orbit (locked
on the microwave phase), and in the transverse direction.
In order to separate quantum states localized in di erent regions of the (L0; ) space, we show
in Fig. 46 a comparison between the semiclassical prediction and the numerically exact Floquet
energies (obtained as in Section 3.3.2 for the s = 1 resonance) originating from this manifold, with
N = 0 in Eqs. (78) and (79), at F0 =0:04. Observe that the 16 upmost states appear in eight
quasi-degenerate pairs di ering by parity. Exact degeneracy does not happen because of tunneling
e ects: the lower the doublet in energy, the larger its tunneling splitting. The tunneling process
involved here is a “transverse” tunneling in the (L; ) plane, where the electron jumps from the
elliptic (L0 0:65; = 0) Kepler trajectory to its image under z-parity, the (L0 0:65; = )
trajectory (compare Fig. 45). This tunneling process is entirely due to the speci c form of 2, with
two distinct maxima.
The energetically highest doublet in Fig. 46 corresponds to states localized as close as possible
to the xed points L0 0:65; =0; . For these states (large resonance island in the (Î; ˆ ) plane),
semiclassical quantization nicely agrees with the quantum results. On the other hand, the agreement
between quantum and semiclassical results progressively degrades for lower energies, as the size
of the island in the (Î; ˆ ) plane becomes smaller. Still, the disagreement between semiclassical