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) 409–547 515<br />
6.2. Driven Helium <strong>in</strong> a frozen planet con guration<br />
In the previous examples of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong>, the key po<strong>in</strong>t has been the generic<br />
appearance of a nonl<strong>in</strong>ear resonance for <strong>periodically</strong> <strong>driven</strong> <strong>quantum</strong> <strong>systems</strong> whose unperturbed dynamics<br />
is <strong>in</strong>tegrable. A natural question to ask is whether the concept of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong><br />
can be generalized to <strong>systems</strong> which exhibit mixed regular–chaotic dynamics even <strong>in</strong> the absence<br />
of the external perturbation. In the atomic realm, such a situation is realized for the helium atom,<br />
where electron–electron <strong>in</strong>teractions provide an additional source of non-l<strong>in</strong>earity. The correspond<strong>in</strong>g<br />
Hamiltonian writes <strong>in</strong> atomic units<br />
HHe = ˜p2 1<br />
2 + ˜p2 2 2<br />
− −<br />
2 r1<br />
2 1<br />
+ : (265)<br />
r2 |˜r1 −˜r2|<br />
As a matter of fact, the classical and <strong>quantum</strong> dynamics of the three-body Coulomb problem generated<br />
by Hamiltonian (265) has been a largely unexplored “terra <strong>in</strong>cognita” until very recently [164],<br />
s<strong>in</strong>ce the dimensionality of the phase space dynamics <strong>in</strong>creases from e ectively two to e ectively<br />
eight dimensions when a second electron is added to the familiar Kepler problem. Furthermore, the<br />
exact <strong>quantum</strong> mechanical treatment of the helium atom rema<strong>in</strong>s a formidable task s<strong>in</strong>ce the early<br />
days of <strong>quantum</strong> mechanics, and considerable advances could be achieved only very recently, with<br />
the advent of modern semiclassical and group theoretical methods [165–168]. Already the classical<br />
dynamics of this system exhibits a largely chaotic phase space structure, which typically leads to the<br />
rapid autoionization of the associated doubly excited <strong>quantum</strong> states of the atom. One of the major<br />
surprises <strong>in</strong> the analysis of the three-body Coulomb problem dur<strong>in</strong>g the last decades has therefore<br />
been the discovery of a new, highly correlated and classically globally stable electronic con guration,<br />
the “frozen planet” [169,170]. The appeal of this con guration resides <strong>in</strong> its counter<strong>in</strong>tuitive,<br />
asymmetric character where both electrons are located on the same side of the nucleus. Furthermore,<br />
this con guration turns out to be the most robust of all known doubly excited two-electron con gurations,<br />
<strong>in</strong> the sense that it occupies a large volume <strong>in</strong> phase space. Its stability is due to the strong<br />
coupl<strong>in</strong>g of the two electrons by the 1=|˜r1 − ˜r2| term <strong>in</strong> Hamiltonian (265), which enforces their<br />
highly correlated motion.<br />
The frozen planet is an ideal candidate to test the prevailance of the concept of non-<strong>dispersive</strong><br />
<strong>wave</strong> <strong>packets</strong> <strong>in</strong> <strong>systems</strong> with <strong>in</strong>tr<strong>in</strong>sically mixed dynamics. In a recent study [171,110,172,173] the<br />
response of this highly correlated two-electron con guration to a periodic force has been <strong>in</strong>vestigated<br />
from a classical and from a <strong>quantum</strong> mechanical po<strong>in</strong>t of view. The Hamiltonian for the <strong>driven</strong><br />
problem writes, <strong>in</strong> the length gauge,<br />
H = HHe + F cos(!t)(z1 + z2) : (266)<br />
Guided by the experience on non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> <strong>in</strong> one electron Rydberg states, the driv<strong>in</strong>g<br />
frequency ! was chosen near resonant with the natural frequency FP ≈ 0:3n −3<br />
i of the frozen planet,<br />
where ni denotes the pr<strong>in</strong>cipal <strong>quantum</strong> number of the <strong>in</strong>ner electron. It was found that, for a suitably<br />
chosen driv<strong>in</strong>g eld amplitude F, a non-l<strong>in</strong>ear resonance between the correlated electronic motion<br />
and the external drive can be <strong>in</strong>duced <strong>in</strong> the classical dynamics, at least for the coll<strong>in</strong>ear frozen<br />
planet where the three particles (two electrons and the nucleus) are aligned along the polarization<br />
axis of the driv<strong>in</strong>g eld.