Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 467
Fig. 22. Level dynamics of the numerically exact quasienergies (...) in the vicinity of the resonant manifold emerging
from n0 =21 (N =0), compared to the semiclassical prediction (—), for F0 =Fn 4 0 =0:::0:06. Note that the maximum eld
amplitude exceeds the typical ionization thresholds measured in current experiments for !n 3 0 1 [132–134]. Nontheless,
the semiclassical prediction accurately tracks the exact solution across a large number of avoided crossings with other
Rydberg manifolds.
the separatrix state p = 10. Such good agreement is a direct proof of the validity of the adiabatic
separation between the slow motion in (L; ), and the fast motion in (Î; ˆ ).
Fig. 22 shows a global comparison of the semiclassical prediction with the exact level dynamics
(energy levels vs. F0), in a range from F0 =0 to 0.06, which exceeds the typical ionization threshold
(F0 0:05) observed in current experiments [132–134]. We observe that the semiclassical prediction
tracks the exact quasienergies quite accurately, even for large F0-values, where the resonant n0 =21
manifold overlaps with other Rydberg manifolds, or with side bands of lower or higher lying Rydberg
states. The agreement becomes unsatisfactory only in the region of very small F0, where the size
of the resonance island in (Î; ˆ ) is very small. This is not unexpected, as semiclassics should fail
when the area of the resonance island is comparable to ˝, cf. Eq. (82). In this weak driving regime,
the pendulum approximation can be used to produce more accurate estimates of the energy levels.
The fast (Î; ˆ ) motion is essentially identical to the one of the 1D driven hydrogen atom: thus,
the Mathieu approach used in Section 3.3.1 can be trivially extended to the 3D case. The only
amendment is to replace the factor J ′ 1 (1)n2 0
in the expression of the Mathieu parameter q by the
various quantized values of 1 for 0 6 p 6 n0 − 1; and to use the same Eq. (149) for the energy
levels.
The semiclassical construction of the energy levels from classical orbits is—necessarily—re ected
in the localization properties of the associated eigenstates, as demonstrated by the electronic densities
of the states |p =0〉, |p =10〉, and |p =20〉 in Fig. 23, for the same eld amplitudes as in Fig. 21.
Note that, in this plot, the electronic densities are averaged over one eld cycle, hence display only
the angular localization properties of the eigenstates. Their localization along the classical orbits