Non-dispersive wave packets in periodically driven quantum systems

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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;
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 485 Fig. 32. Floquet eigenstate of the two-dimensional hydrogen atom in a circularly polarized microwave eld. This state is partially localized on the unstable equilibrium point (in the rotating frame), see Eqs. (185)–(188), for F0 =0:057 and n0 =60. This localization is of purely classical origin. The nucleus is at the center of the gure which extends over ±5000 Bohr radii. The microwave eld points to the right. • F0 is not too large, say smaller than F0 0:065. Our numerical data suggest that the upper limit is not given by the limiting value of q =8=9, for which the xed point is still stable. The limiting value appears to be rather linked to the 1 : 2 resonance between the !+ and !− modes, which occurs approx. at F0 0:065. • In particular, the value q =0:9562 (i.e., F0 0:04442), corresponding to optimal classical stability of the xed point, advertised in [34] as the optimal one, is by no means favored. A much broader range of microwave amplitudes is available (and equivalent as far as the “quality” of the wave packet is concerned). What is much more relevant, is the presence of some accidental avoided crossings with other Floquet states. Still, these are no very restrictive conditions, and we are left with a broad range of parameters favoring the existence of non-dispersive wave packets, a range which is experimentally fully accessible (see Section 8 for a more elaborate discussion of experimental aspects). Finally, in analogy with the LP case, we may consider Floquet states localized on the unstable xed point associated with the principal resonance island. From the discussion following Eq. (188), this point is located opposite to the stable xed point, on the other side of the nucleus. An example of such a state is shown in Fig. 32, for an amplitude of the microwave eld that ensures that most of the nearby Floquet states ionize rather rapidly. The eigenstate displayed in the gure lives much longer (several thousands of Kepler periods). The localization in the vicinity of an unstable xed point, in analogy to the LP case discussed previously, is of purely classical origin. As pointed out in [60], such a localization must not be confused with scarring [81]—a partial localization on
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Non-dispersive wave packets in periodically driven quantum systems

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