Non-dispersive wave packets in periodically driven quantum systems

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468 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 23. Electronic densities of the extremal librational (p = 0, left), separatrix (p = 10, center), and extremal rotational (p = 20, right) quasi-energy states of the n0 = 21 manifold of a 3D hydrogen atom exposed to a resonant microwave eld in cylindrical coordinates ( ; z), at di erent values of the driving eld amplitude F0 =Fn 4 0 =0:02 (top), 0:03 (middle), 0:04 (bottom), averaged over one period of the driving eld. Note the clear localization along the classical orbits corresponding to the respective contours in Fig. 19, for all eld amplitudes. The nodal lines of the electronic densities clearly exhibit the direction of the underlying classical motion. The eld-induced nite decay rate of the eigenstates (see Section 7.1) is negligible on time scales shorter than approx. 10 6 Kepler periods. Each box extends over ±1000 Bohr radii, in both (horizontal) and z (vertical) directions, with the nucleus at the center of the plot. The microwave polarization axis is oriented vertically along z. de ned by the stable or unstable xed points of the (L; ) dynamics is obvious [64,67,87]. Note in particular the nodal structure of the state |p =10〉, associated with the unstable xed point: there are sharp nodal lines perpendicular to the z-axis, re ecting the dominant motion along the z-axis, but also nodal lines of low visibility in the angular direction. They are a manifestation of the slow classical precession of the Kepler ellipse, i.e., the slow secular evolution in the (L; ) plane. The quantum state, however, dominantly exhibits the motion along the z-axis, as a signature of the e ective separation of time scales of radial and angular motion. Finally, it should be realized from a comparison of the top to the middle and bottom row of Fig. 23 that the quasi-classical localization properties of the eigenstates are essentially una ected as F rises, despite various avoided crossings which occur at intermediate eld values, see Fig. 22. Especially, the angular structure does not depend at all on F, as predicted by the secular approximation.
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 469 Fig. 24. Temporal evolution of the electronic density of the extremal rotational quasienergy state |p =20〉 of the n0 =21 resonant manifold, for di erent phases !t = 0 (left), !t = =2 (center), !t = (right) of the driving eld, at amplitude F0 =0:03, in cylindrical coordinates. Each box extends over ±700 Bohr radii, in both directions, (horizontal) and z (vertical). The microwave polarization axis is oriented along z. Because of the azimuthal symmetry of the problem, the actual 3D electronic density is obtained by rotating the gure around the vertical axis. The state represents a non-dispersive wave packet shaped like a doughnut, moving periodically from the north to the south pole (and back) of a sphere. For higher n0, the angular localization on the circular orbit should improve. The eigenstates displayed here are localized along classical trajectories which are resonantly driven by the external eld. Hence, we should expect them to exhibit wave packet like motion along these trajectories, as the phase of the driving eld is changed. This is indeed the case as illustrated in Fig. 24 for the state p = 20 with maximal angular momentum L=n0 1 [48,64,67]. Due to the azimuthal symmetry of the problem, the actual 3D electronic density is obtained by rotating the gure around the vertical axis. Thus, the wave packet is actually a doughnut moving periodically from the north to the south pole (and back) of a sphere, slightly deformed along the eld direction. The interference resulting from the contraction of this doughnut to a compact wave packet at the poles is clearly visible at phases !t = 0 and in the plot. Note that the creation of unidirectional wave-packet eigenstates moving along a circle in the plane containing the eld polarization axis is not possible for the real 3D atom [67], as opposed to the reduced 2D problem studied in [48], due to the above mentioned azimuthal symmetry (see also Section 3.5). For other states in the n0 =21 resonant manifold, the longitudinal localization along the periodic orbit is less visible. The reason is that 1 is smaller than for the p=20 state, leading to a smaller resonance island in (Î; ˆ ) (see Fig. 20) and, consequently, to less e cient localization. Proceeding to higher n0-values should improve the situation. Let us brie y discuss “excited” states in the resonance island, i.e., manifolds corresponding to N¿0 in Eq. (78). Fig. 25 shows the exact level dynamics, with the semiclassical prediction for N = 1 superimposed [67]. The states in this manifold originate from n0 = 22. We observe quite good agreement between the quantum and semiclassical results for high lying states in the manifold (for which the principal action island is large, see Fig. 20). For lower lying states the agreement is improved for higher values of F0. IfF0 is too low, the states are not fully localized inside the resonance island and, consequently, are badly reproduced by the resonant semiclassical approximation. This is further exempli ed in Fig. 26, for N =2. Here, the agreement is worse than for smaller values of N , and is observed only for large F0 and large p. This con rms the picture that the validity of the semiclassical approach outlined here is directly related to the size, Eqs. (71) and (82), of the resonance island in (Î; ˆ ) space (see also the discussion in Section 8.3).
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Non-dispersive wave packets in periodically driven quantum systems

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