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