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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A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 469<br />
Fig. 24. Temporal evolution of the electronic density of the extremal rotational quasienergy state |p =20〉 of the n0 =21<br />
resonant manifold, for di erent phases !t = 0 (left), !t = =2 (center), !t = (right) of the driv<strong>in</strong>g eld, at amplitude<br />
F0 =0:03, <strong>in</strong> cyl<strong>in</strong>drical coord<strong>in</strong>ates. Each box extends over ±700 Bohr radii, <strong>in</strong> both directions, (horizontal) and z<br />
(vertical). The micro<strong>wave</strong> polarization axis is oriented along z. Because of the azimuthal symmetry of the problem, the<br />
actual 3D electronic density is obta<strong>in</strong>ed by rotat<strong>in</strong>g the gure around the vertical axis. The state represents a non-<strong>dispersive</strong><br />
<strong>wave</strong> packet shaped like a doughnut, mov<strong>in</strong>g <strong>periodically</strong> from the north to the south pole (and back) of a sphere. For<br />
higher n0, the angular localization on the circular orbit should improve.<br />
The eigenstates displayed here are localized along classical trajectories which are resonantly <strong>driven</strong><br />
by the external eld. Hence, we should expect them to exhibit <strong>wave</strong> packet like motion along these<br />
trajectories, as the phase of the driv<strong>in</strong>g eld is changed. This is <strong>in</strong>deed the case as illustrated <strong>in</strong><br />
Fig. 24 for the state p = 20 with maximal angular momentum L=n0 1 [48,64,67]. Due to the<br />
azimuthal symmetry of the problem, the actual 3D electronic density is obta<strong>in</strong>ed by rotat<strong>in</strong>g the<br />
gure around the vertical axis. Thus, the <strong>wave</strong> packet is actually a doughnut mov<strong>in</strong>g <strong>periodically</strong><br />
from the north to the south pole (and back) of a sphere, slightly deformed along the eld direction.<br />
The <strong>in</strong>terference result<strong>in</strong>g from the contraction of this doughnut to a compact <strong>wave</strong> packet at the<br />
poles is clearly visible at phases !t = 0 and <strong>in</strong> the plot. Note that the creation of unidirectional<br />
<strong>wave</strong>-packet eigenstates mov<strong>in</strong>g along a circle <strong>in</strong> the plane conta<strong>in</strong><strong>in</strong>g the eld polarization axis is<br />
not possible for the real 3D atom [67], as opposed to the reduced 2D problem studied <strong>in</strong> [48], due<br />
to the above mentioned azimuthal symmetry (see also Section 3.5). For other states <strong>in</strong> the n0 =21<br />
resonant manifold, the longitud<strong>in</strong>al localization along the periodic orbit is less visible. The reason is<br />
that 1 is smaller than for the p=20 state, lead<strong>in</strong>g to a smaller resonance island <strong>in</strong> (Î; ˆ ) (see Fig. 20)<br />
and, consequently, to less e cient localization. Proceed<strong>in</strong>g to higher n0-values should improve the<br />
situation.<br />
Let us brie y discuss “excited” states <strong>in</strong> the resonance island, i.e., manifolds correspond<strong>in</strong>g to<br />
N¿0 <strong>in</strong> Eq. (78). Fig. 25 shows the exact level dynamics, with the semiclassical prediction for<br />
N = 1 superimposed [67]. The states <strong>in</strong> this manifold orig<strong>in</strong>ate from n0 = 22. We observe quite<br />
good agreement between the <strong>quantum</strong> and semiclassical results for high ly<strong>in</strong>g states <strong>in</strong> the manifold<br />
(for which the pr<strong>in</strong>cipal action island is large, see Fig. 20). For lower ly<strong>in</strong>g states the agreement is<br />
improved for higher values of F0. IfF0 is too low, the states are not fully localized <strong>in</strong>side the resonance<br />
island and, consequently, are badly reproduced by the resonant semiclassical approximation.<br />
This is further exempli ed <strong>in</strong> Fig. 26, for N =2. Here, the agreement is worse than for smaller values<br />
of N , and is observed only for large F0 and large p. This con rms the picture that the validity of<br />
the semiclassical approach outl<strong>in</strong>ed here is directly related to the size, Eqs. (71) and (82), of the<br />
resonance island <strong>in</strong> (Î; ˆ ) space (see also the discussion <strong>in</strong> Section 8.3).