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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542 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
9. Conclusions<br />
In this report, we have shown that novel and highly robust eigenstates of <strong>periodically</strong> <strong>driven</strong> <strong>quantum</strong><br />
<strong>systems</strong>—non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong>—are born out of classically mixed regular–chaotic dynamics.<br />
As much as a mixed phase space is generic for classical Hamiltonian <strong>systems</strong>, non-<strong>dispersive</strong><br />
<strong>wave</strong> <strong>packets</strong> are a generic manifestation thereof on the <strong>quantum</strong> level, given a su ciently high<br />
density of states (needed to resolve nite-size phase space structures). While we described their<br />
semiclassical properties and their experimental preparation, manipulation, and identi cation dur<strong>in</strong>g<br />
the largest part of this report for a speci c system—atomic Rydberg states <strong>driven</strong> by a micro<strong>wave</strong><br />
eld—it is clear from our approach that such “<strong>quantum</strong> particles” can be anchored to any non-l<strong>in</strong>ear<br />
resonance between a periodic drive and a periodic trajectory of a Hamiltonian system. As an alternative<br />
example, we have brie y touched upon the atomic realization of the gravitational bouncer,<br />
though many other realizations <strong>in</strong> simple <strong>quantum</strong> optical or atomic and molecular <strong>systems</strong> can be<br />
thought of. Let us only mention unharmonic traps for ions, atoms, or BEC condensates, <strong>periodically</strong><br />
kicked atoms [229], as well as molecular dynamics [230,231] on adiabatic potential surfaces (the<br />
<strong>driven</strong> frozen planet brie y discussed <strong>in</strong> Section 6.2 may be conceived as open<strong>in</strong>g a perspective<br />
<strong>in</strong> this direction). <strong>Non</strong>theless, atomic Rydberg states rema<strong>in</strong> arguably the best objects to study the<br />
fundamental properties of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> as the realization of Schrod<strong>in</strong>ger’s dream<br />
[2]. On one hand, they are microscopic realizations of the Keplerian motion and of Bohr’s orbitals<br />
us<strong>in</strong>g a well understood non-l<strong>in</strong>ear dynamical system. On the other hand, they possess the<br />
essential complication which open <strong>quantum</strong> <strong>systems</strong> add to bounded Hamiltonian dynamics—the<br />
driv<strong>in</strong>g-<strong>in</strong>duced, coherent coupl<strong>in</strong>g to the atomic cont<strong>in</strong>uum of free electronic states. On top of that,<br />
all these features can be controlled <strong>in</strong> real laboratory experiments, and we might actually dream<br />
of prob<strong>in</strong>g the characteristic properties of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> on s<strong>in</strong>gle, trapped atoms<br />
or ions, us<strong>in</strong>g novel experimental approaches yet to come. Let us nally dare to speculate on the<br />
potential use of non-<strong>dispersive</strong> <strong>wave</strong>-<strong>packets</strong> <strong>in</strong> coherent control: given their spectacular robustness—<br />
which we abundantly illustrated <strong>in</strong> this report—it is clear that they provide a means to store and to<br />
“ship” <strong>quantum</strong> probability densities <strong>in</strong> and across phase space, e.g., under adiabatic changes of the<br />
driv<strong>in</strong>g eld polarization and=or of the strength or orientation of additional static elds. Given the<br />
recent advances <strong>in</strong> coherent control of molecular reactions employ<strong>in</strong>g laser elds [230]—which so<br />
far do not explore the unique perspectives of non-l<strong>in</strong>ear dynamics—it looks like a promis<strong>in</strong>g (and<br />
challeng<strong>in</strong>g) program to systematically study non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> <strong>in</strong> molecular reaction<br />
dynamics.<br />
Acknowledgements<br />
It is a pleasure to acknowledge a longstand<strong>in</strong>g and fruitful collaboration with Robert Gebarowski,<br />
Benoˆt Gremaud, Klaus Hornberger, Andreas Krug, Romek Marc<strong>in</strong>ek, Krzysiek Sacha, Peter<br />
Schlagheck, and Sandro Wimberger on non-<strong>dispersive</strong> <strong>wave</strong>-<strong>packets</strong> and related topics over the<br />
past ve years.<br />
We acknowledge support of bilateral collaborations via programmes Procope (German–French) and<br />
Polonium (Polish–French). J.Z. acknowledges support by Polish Committee for Scienti c Research<br />
under grant 2P03B00915. Laboratoire Kastler Brossel is laboratoire de l’Universite Pierre et Marie