Non-dispersive wave packets in periodically driven quantum systems

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