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) 409–547 459<br />
Fig. 15. Husimi representation of the <strong>wave</strong>-packet eigenstate of Fig. 13 <strong>in</strong> classical phase space, for the same phases !t<br />
and scales (0 6 6 2 ; 30 6 I 6 90) as employed for the classical surface of section <strong>in</strong> Fig. 12. Clearly, the <strong>quantum</strong><br />
mechanical eigenstate of the atom <strong>in</strong> the eld follows the classical evolution without dispersion, except for its transient<br />
spread<strong>in</strong>g when re ected o the nucleus (at !t = , bottom left), due to the divergence of the classical velocity at that<br />
position.<br />
electronic density, without a ect<strong>in</strong>g its shape or localization properties, after 10 6 classical Kepler<br />
periods.<br />
3.3.2. Realistic three-dimensional atom<br />
Extend<strong>in</strong>g our previous analysis to the three-dimensional hydrogen atom <strong>driven</strong> by a l<strong>in</strong>early<br />
polarized micro<strong>wave</strong> eld, we essentially expand the accessible phase space. S<strong>in</strong>ce the Hamiltonian<br />
HLP = ˜p2 1<br />
− + Fz cos(!t) (151)<br />
2 r<br />
is <strong>in</strong>variant under rotations around the eld polarization axis, the projection of the angular momentum<br />
is a conserved quantity and gives rise to a good <strong>quantum</strong> number M. Hence, only two dimensions<br />
of con guration space are left, which, together with the explicit, periodic time dependence, span a<br />
ve-dimensional phase space.<br />
In the 1D situation described previously, the key <strong>in</strong>gredient for the existence of non-<strong>dispersive</strong><br />
<strong>wave</strong> <strong>packets</strong> was the phase lock<strong>in</strong>g of the <strong>in</strong>ternal degree of freedom on the external drive. In the<br />
3D situation, there rema<strong>in</strong>s one s<strong>in</strong>gle drive, but there are several <strong>in</strong>ternal degrees of freedom. In<br />
the generic case, not all <strong>in</strong>ternal degrees of freedom can be simultaneously locked on the external<br />
drive, and one can expect only partial phase lock<strong>in</strong>g, i.e., only partially localized <strong>wave</strong> <strong>packets</strong>.<br />
The non-trivial task is to understand how the phase lock<strong>in</strong>g of one degree of freedom modi es the