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) 409–547 481<br />
Fig. 29. Isovalue plots (at 30% of the maximum value) of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> <strong>in</strong> the three-dimensional hydrogen<br />
atom <strong>driven</strong> by a circularly polarized micro<strong>wave</strong> eld. Frequency of the driv<strong>in</strong>g ! =1=60 3 , amplitude F0 =0:04442. In<br />
the laboratory frame, the <strong>wave</strong> <strong>packets</strong> propagate—without chang<strong>in</strong>g their shapes—along a circular trajectory centered<br />
around the nucleus <strong>in</strong>dicated by a cross. The cube edges measure 10 000 Bohr radii. The micro<strong>wave</strong> polarization plane<br />
is horizontal with the eld po<strong>in</strong>t<strong>in</strong>g to the right. The four <strong>wave</strong> <strong>packets</strong> shown represent the ground state <strong>wave</strong> packet<br />
|0; 0; 0〉 (top left), and the excited states |1; 0; 0〉 (bottom left), |0; 1; 0〉 (top right), and |0; 0; 1〉 (bottom right). Eventually,<br />
the micro<strong>wave</strong> eld will ionize such states, but their lifetimes are extremely long, of the order of thousands to millions<br />
of Kepler periods.<br />
Gaussians and Hermite polynomials of the position coord<strong>in</strong>ates, but the Hermite polynomials have<br />
to be evaluated for complex values. This results <strong>in</strong> the unusual pattern of the probability density<br />
displayed <strong>in</strong> Fig. 28. An improvement over the harmonic approximation is possible, by bend<strong>in</strong>g the<br />
axis <strong>in</strong> the spirit of [47], <strong>in</strong> order to account for the spherical symmetry of the dom<strong>in</strong>ant Coulomb<br />
potential. With this improvement, the probability density, shown <strong>in</strong> the middle row of Fig. 28, is<br />
almost <strong>in</strong>dist<strong>in</strong>guishable from the exact result. Let us repeat that this bend<strong>in</strong>g—and consequently the<br />
deviation from Gaussian character of the <strong>wave</strong> function—does not a ect at all the non-<strong>dispersive</strong><br />
character of the <strong>wave</strong> packet.<br />
Let us now turn to the realistic 3D model of the atom. Fig. 29 shows an isovalue contour<br />
of several non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> for the hydrogen atom <strong>driven</strong> by a micro<strong>wave</strong> eld with<br />
frequency ! =1=60 3 , i.e., roughly resonant with the n0 =60 → 59; 61 transitions (see Eqs. (40),<br />
(65) and (119)). The best localized <strong>wave</strong> packet is the ground state |0; 0; 0〉, while the three other