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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468 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
Fig. 23. Electronic densities of the extremal librational (p = 0, left), separatrix (p = 10, center), and extremal rotational<br />
(p = 20, right) quasi-energy states of the n0 = 21 manifold of a 3D hydrogen atom exposed to a resonant micro<strong>wave</strong> eld<br />
<strong>in</strong> cyl<strong>in</strong>drical coord<strong>in</strong>ates ( ; z), at di erent values of the driv<strong>in</strong>g eld amplitude F0 =Fn 4 0 =0:02 (top), 0:03 (middle), 0:04<br />
(bottom), averaged over one period of the driv<strong>in</strong>g eld. Note the clear localization along the classical orbits correspond<strong>in</strong>g<br />
to the respective contours <strong>in</strong> Fig. 19, for all eld amplitudes. The nodal l<strong>in</strong>es of the electronic densities clearly exhibit<br />
the direction of the underly<strong>in</strong>g classical motion. The eld-<strong>in</strong>duced nite decay rate of the eigenstates (see Section 7.1)<br />
is negligible on time scales shorter than approx. 10 6 Kepler periods. Each box extends over ±1000 Bohr radii, <strong>in</strong> both<br />
(horizontal) and z (vertical) directions, with the nucleus at the center of the plot. The micro<strong>wave</strong> polarization axis is<br />
oriented vertically along z.<br />
de ned by the stable or unstable xed po<strong>in</strong>ts of the (L; ) dynamics is obvious [64,67,87]. Note<br />
<strong>in</strong> particular the nodal structure of the state |p =10〉, associated with the unstable xed po<strong>in</strong>t:<br />
there are sharp nodal l<strong>in</strong>es perpendicular to the z-axis, re ect<strong>in</strong>g the dom<strong>in</strong>ant motion along the<br />
z-axis, but also nodal l<strong>in</strong>es of low visibility <strong>in</strong> the angular direction. They are a manifestation of the<br />
slow classical precession of the Kepler ellipse, i.e., the slow secular evolution <strong>in</strong> the (L; ) plane.<br />
The <strong>quantum</strong> state, however, dom<strong>in</strong>antly exhibits the motion along the z-axis, as a signature of the<br />
e ective separation of time scales of radial and angular motion. F<strong>in</strong>ally, it should be realized from a<br />
comparison of the top to the middle and bottom row of Fig. 23 that the quasi-classical localization<br />
properties of the eigenstates are essentially una ected as F rises, despite various avoided cross<strong>in</strong>gs<br />
which occur at <strong>in</strong>termediate eld values, see Fig. 22. Especially, the angular structure does not<br />
depend at all on F, as predicted by the secular approximation.