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) 409–547 485<br />
Fig. 32. Floquet eigenstate of the two-dimensional hydrogen atom <strong>in</strong> a circularly polarized micro<strong>wave</strong> eld. This state is<br />
partially localized on the unstable equilibrium po<strong>in</strong>t (<strong>in</strong> the rotat<strong>in</strong>g frame), see Eqs. (185)–(188), for F0 =0:057 and<br />
n0 =60. This localization is of purely classical orig<strong>in</strong>. The nucleus is at the center of the gure which extends over ±5000<br />
Bohr radii. The micro<strong>wave</strong> eld po<strong>in</strong>ts to the right.<br />
• F0 is not too large, say smaller than F0 0:065. Our numerical data suggest that the upper limit<br />
is not given by the limit<strong>in</strong>g value of q =8=9, for which the xed po<strong>in</strong>t is still stable. The limit<strong>in</strong>g<br />
value appears to be rather l<strong>in</strong>ked to the 1 : 2 resonance between the !+ and !− modes, which<br />
occurs approx. at F0 0:065.<br />
• In particular, the value q =0:9562 (i.e., F0 0:04442), correspond<strong>in</strong>g to optimal classical stability<br />
of the xed po<strong>in</strong>t, advertised <strong>in</strong> [34] as the optimal one, is by no means favored. A much broader<br />
range of micro<strong>wave</strong> amplitudes is available (and equivalent as far as the “quality” of the <strong>wave</strong><br />
packet is concerned). What is much more relevant, is the presence of some accidental avoided<br />
cross<strong>in</strong>gs with other Floquet states.<br />
Still, these are no very restrictive conditions, and we are left with a broad range of parameters favor<strong>in</strong>g<br />
the existence of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong>, a range which is experimentally fully accessible<br />
(see Section 8 for a more elaborate discussion of experimental aspects).<br />
F<strong>in</strong>ally, <strong>in</strong> analogy with the LP case, we may consider Floquet states localized on the unstable<br />
xed po<strong>in</strong>t associated with the pr<strong>in</strong>cipal resonance island. From the discussion follow<strong>in</strong>g Eq. (188),<br />
this po<strong>in</strong>t is located opposite to the stable xed po<strong>in</strong>t, on the other side of the nucleus. An example<br />
of such a state is shown <strong>in</strong> Fig. 32, for an amplitude of the micro<strong>wave</strong> eld that ensures that<br />
most of the nearby Floquet states ionize rather rapidly. The eigenstate displayed <strong>in</strong> the gure lives<br />
much longer (several thousands of Kepler periods). The localization <strong>in</strong> the vic<strong>in</strong>ity of an unstable<br />
xed po<strong>in</strong>t, <strong>in</strong> analogy to the LP case discussed previously, is of purely classical orig<strong>in</strong>. As po<strong>in</strong>ted<br />
out <strong>in</strong> [60], such a localization must not be confused with scarr<strong>in</strong>g [81]—a partial localization on