Non-dispersive wave packets in periodically driven quantum systems

484 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
Fig. 31. Di erence between the exact quantum energy of a non-dispersive wave packet of a 2D hydrogen atom in a
circularly polarized microwave eld, and two di erent semiclassical predictions, as a function of the scaled microwave
eld amplitude, F0. The thick lines are obtained from a harmonic approximation of the motion around the stable equilibrium
point (in the rotating frame), Eq. (199) and the thin lines use a quantum treatment (Mathieu approach, Section 3.1.4)
of the motion in the resonance island, combined with a semiclassical treatment of the secular motion. From top to
bottom ! =1=(30:5) 3 ; 1=(60:5) 3 ; 1=(90:5) 3 , corresponding to wave packets associated with Rydberg states of principal
quantum number n0 =30; 60; 90. The energy di erence is expressed in units of the mean level spacing, estimated by
n 4 0=2. The prediction of the Mathieu approach is consistently better for low F0, and the harmonic approximation becomes
clearly superior for larger F0: For larger and larger n0; the harmonic approximation is better and better. Note that both
approximations make it possible to estimate the energy of the non-dispersive wave packet with an accuracy better than
the mean level spacing, allowing for its simple and unambiguous extraction from exact numerical data.
as √ F, the harmonic approximation may become valid only for su ciently large microwave amplitudes,
when there is at least one state trapped in the island. As seen, however, in Fig. 31, the
harmonic approximation yields a satisfactory prediction for the wave packet energy (within few %
of the mean spacing) almost everywhere. For increasing n0, the harmonic approximation is better
and better and the Mathieu approach is superior only over a smaller and smaller range of F0 = Fn 4 0 ,
close to 0. Still, both approaches give very good predictions for the typical values of F0 used in
the following, say F0 0:03. The spikes visible in the gure are due to the many small avoided
crossings visible in Fig. 30.
While we have shown some exemplary wave packets for few values of n0 and F only, they
generally look very similar provided that
• n0 is su ciently large, say n0 ¿ 30. For smaller n0, the wave packet looks a bit distorted and
one observes some deviations from the harmonic approximation (for a more detailed discussion
of this point see Sections 7.1 and 7.2, where ionization and spontaneous emission of the wave
packets are discussed);
• F0 is su ciently large, say F0 ¿ 0:001, such that the resonance island can support at least one
state. For smaller F0; the wave packet becomes more extended in the angular coordinate, since
less atomic circular states are signi cantly coupled, see Fig. 27;