Non-dispersive wave packets in periodically driven quantum systems

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488 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 33. The scaled e ective perturbation 1=Î 2 driving the transverse=angular motion of a two-dimensional hydrogen atom exposed to a resonant, elliptically polarized microwave eld, plotted as a function of the scaled angular momentum, M0 = M=Î, and of the angle between the Runge–Lenz vector and the major axis of the polarization ellipse. Left and right panels correspond to =0:1 and 0.6, respectively. Non-dispersive wave packets are localized around the maxima of this e ective potential, at M0 = ±1, and are circularly co- and contra-rotating (with respect to the microwave eld) around the nucleus (see Fig. 35). Fig. 34. Energy levels of the two-dimensional hydrogen atom driven by a resonant, elliptically polarized microwave eld of scaled amplitude F0 =0:03, for n0 = 21, as a function of the eld ellipticity (the resonant frequency of the microwave is ! =1=(21:5) 3 ). Full lines: semiclassical prediction; dotted lines: exact numerical result for the states originating from the n0 = 21 hydrogenic manifold. The non-dispersive wave packets are the states originating from the upper doublet at = 0. The ascending (resp. descending) energy level is associated with the wave packet co- (resp. contra-) rotating with the microwave eld. Fig. 34 compares the semiclassical prediction obtained by quantization of 1 and Hsec (following the lines already described in Section 3.3.2, for the ground state N = 0 in the resonance island) to the exact quasienergies (determined by numerical diagonalization of the Floquet Hamiltonian), for
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 489 Fig. 35. Non-dispersive wave packets of the two-dimensional hydrogen atom exposed to an elliptically polarized, resonant microwave eld. Scaled microwave amplitude F0 =0:03 (for resonant principal quantum number n0 = 21), and ellipticity =0:4. Top row: non-dispersive wave packet moving on a circular orbit corotating with the microwave eld, for phases !t =0; =4; =2 (from left to right). This wave packet evolves into the eigenstate represented in Fig. 27, under continuous increase of the ellipticity to = 1. Bottom row: non-dispersive wave packet launched along the same circular orbit, but contra-rotating with the driving eld (for the same phases). Note that, while the co-rotating wave packet almost preserves its shape during the temporal evolution, the contra-rotating one exhibits signi cant distortions during one eld cycle, as a direct consequence of its complicated level dynamics shown in Fig. 34. Still being an exactly time-periodic Floquet eigenstate, it regains its shape after every period of the microwave. The size of each box extends over ±800 Bohr radii, in both x and y directions, with the nucleus in the middle. The major axis of the polarization ellipse is along the horizontal x axis and the microwave eld points to the right at t =0: varying from LP to CP. The agreement is excellent, with slightly larger discrepancies between the semiclassical and the exact results for the states with smallest energy. For those states the resonance island is very small (small 1), what explains the discrepancy. The highest lying state in Fig. 34, ascending with , is a non-dispersive wave-packet state located on the circular orbit and corotating with the EP eld. It is shown in Fig. 35 for =0:4. As mentioned above, the corresponding counterrotating wave packet is energetically degenerate with the co-rotating one for = 0. Its energy decreases with (cf. Fig. 34). It is shown in the bottom row in Fig. 35 for =0:4. While the corotating wave packet preserves its shape for all values (except at isolated avoided crossings) the counter-rotating wave packet undergoes a series of strong avoided crossings for ¿0:42, progressively loosing its localized character. This is related to a strong decrease of the maximum of 1 at M0 = −1 with , clearly visible in Fig. 33. While we have discussed the two-dimensional case only, the CP situation (compare Section 3.4) indicates that for su ciently large , the important resonant motion occurs in the polarization plane, being stable vs. small deviations in the z direction. Thus the calculations presented above are also relevant for the real three-dimensional world, provided is not far from unity [69,73]. For arbitrary , a full 3D analysis is required. While this is clearly more involved, the general scenario of a wave packet anchored to a resonance island will certainly prevail.
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Non-dispersive wave packets in periodically driven quantum systems

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