Non-dispersive wave packets in periodically driven quantum systems

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510 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Energy (10 -4 a.u.) -2.822 -2.827- -2.832 -2.837 0 5 10 15 20 p 25 30 35 40 Fig. 47. Same as Fig. 46, but for the mirror manifold shifted in energy by !=2. While, for most states, the agreement with semiclassics is of the same quality as in Fig. 46, no quantum data are plotted at the bottom of the manifold. Indeed, at those energies, another Rydberg manifold strongly perturbs the spectrum due to close accidental degeneracy. Consequently, the unambiguous identi cation of individual states is very di cult. and quantum results is at most of the order of the spacing between adjacent levels. 32 Below the energy of the unstable xed points at L0 0:77; = =2; 3 =2, there are no more pairs of classical trajectories in the (L; ) plane corresponding to distinct classical dynamics related by z-parity. Hence, the doublet structure has to disappear, as con rmed by the exact quantum results shown in Fig. 46. On the other hand, there are two disconnected regions in the (L; ) plane which can give rise to quantized values of 2 (and, consequently, to quasienergies) within the same quasienergy range: the neighbourhood of the stable xed points (L0 =0; = =2; 3 =2), and the region close to L0 = 1. In the semiclassical quantization scheme, these regions are completely decoupled and induce two independent, non-degenerate series of quasi-energy levels. Consequently, the complete spectrum exhibits a rather complicated structure, caused by the interleaving of these two series. As discussed in Section 5.1, for a s:1 resonance, in a Floquet zone of width ! = s , there is not a single manifold of states, but rather a set of s di erent manifolds approximately identical and separated by . Deviations from the exact !=s periodicity are due to tunneling [42]. This tunneling process is however completely di erent from the “transverse” one in the (L; ) plane described above. It is a case of “longitudinal” tunneling, where the electron jumps from one location on a Kepler orbit to another, shifted along the same orbit. This longitudinal tunneling is similar in origin to the tunneling described in Section 5.2. It has to be stressed that it represents a general phenomenon in the vicinity of a s:1 resonance (with s ¿ 2), due to the phase space structure in the (Î; ˆ ) plane, see Section 5.1, in contrast to the “transverse” quasi-degeneracy (discussed in Fig. 46) due to the speci c form of 2. Inspecting the numerically exact quantum quasi-energy spectrum, we indeed nd the manifold shown in Fig. 47 (compared to the semiclassical prediction). Observe 32 A quantum approach based on the pendulum approximation and the Mathieu equation would give a much better prediction for such states.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 511 Fig. 48. Electronic density of the upmost eigenstate of the n0 = 42 manifold of Fig. 46, averaged over one microwave period. This state presents localization along a pair of Kepler ellipses oriented along the eld polarization axis. The box measures ±3500 Bohr radii in both and z directions, with the nucleus at the center. The microwave polarization axis along z is parallel to the vertical axis of the gure. The orientation and eccentricity of the ellipse are well predicted by the classical resonance analysis. that the agreement between quantum and semiclassical quasienergies is similar to that observed in Fig. 46, except for the low lying states. Here, incidentally, the states anchored to the resonance island are strongly perturbed by another Rydberg manifold; proper identi cation of the individual quantum states is very di cult in this region, and therefore no quantum data are shown at low energies. Finally, let us consider the localization properties of the wave-functions associated with the upmost states of the manifolds in Figs. 46 and 47. These wave functions should localize in the vicinity of stable trajectories of period 2, i.e., they should be strongly localized, both in angular and orbital coordinates, along an elliptic Kepler orbit of intermediate eccentricity. However, because of the longitudinal quasi-degeneracy, we expect the associated Floquet eigenstates to be composed of two wave packets on the ellipse, exchanging their positions with period T. Furthermore, due to the transverse quasi-degeneracy, we should have combinations of the elliptic orbits labeled by = 0 and . Altogether, this makes four individual wave packets represented by each Floquet state. Due to the azimuthal symmetry of the problem around the eld polarization axis, each wave packet actually is doughnut-shaped (compare Fig. 24 for the simpler s = 1 case). Exact quantum calculations fully con rm this prediction. Fig. 48 shows the electronic density of the upmost Floquet state in the n0 = 42 manifold (Fig. 46), averaged over one eld period. As expected, it is localized along two symmetric Kepler ellipses ( = 0 and , respectively), but
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Non-dispersive wave packets in periodically driven quantum systems

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