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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490 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
4. Manipulat<strong>in</strong>g the <strong>wave</strong> <strong>packets</strong><br />
We have shown <strong>in</strong> the previous sections that non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> are genu<strong>in</strong>e solutions<br />
of the Floquet eigenvalue problem, Eq. (75), under resonant driv<strong>in</strong>g, for arbitrary polarization<br />
of the driv<strong>in</strong>g eld. The semiclassical approximation used to guide our exact numerical approach<br />
directly demonstrates the localization of the electronic density <strong>in</strong> well de ned regions of phase<br />
space, which protect the atom aga<strong>in</strong>st ionization <strong>in</strong>duced by the external eld (see, however, Section<br />
7.1). We have also seen that classical phase space does not only undergo structural changes<br />
under changes of the driv<strong>in</strong>g eld amplitude (Figs. 10, 12 and 16), but also under changes of the<br />
driv<strong>in</strong>g eld ellipticity (Fig. 33). Therefore, the creation of non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong> can be<br />
conceived as an easy and e cient means of <strong>quantum</strong> control, which allows the manipulation and<br />
the controlled transfer of <strong>quantum</strong> population across phase space. In particular, one may imag<strong>in</strong>e the<br />
creation of a <strong>wave</strong> packet mov<strong>in</strong>g along the polarization axis of a l<strong>in</strong>early polarized micro<strong>wave</strong> eld.<br />
A subsequent, smooth change through elliptical to nally circular polarization allows to transfer the<br />
electron to a circular orbit.<br />
Add<strong>in</strong>g additional static elds to Hamiltonian (75) provides us with yet another handle to control<br />
the orientation and shape of highly excited Rydberg trajectories, and, hence, to manipulate the<br />
localization properties of non-<strong>dispersive</strong> <strong>wave</strong>-packet eigenstates <strong>in</strong> con guration and phase space.<br />
The key po<strong>in</strong>t is that trapp<strong>in</strong>g <strong>in</strong>side the non-l<strong>in</strong>ear resonance island is a robust mechanism which<br />
protects the non-<strong>dispersive</strong> <strong>wave</strong> packet very e ciently from imperfections. This allows to adjust<br />
the <strong>wave</strong>-packet’s properties at will, just by adiabatically chang<strong>in</strong>g the properties of the island itself.<br />
Moreover, when the strength of the external perturbation <strong>in</strong>creases, chaos generically <strong>in</strong>vades a large<br />
part of classical phase space, but the resonance islands most often survives. The reason is that<br />
the phase lock<strong>in</strong>g phenomenon <strong>in</strong>troduces various time scales <strong>in</strong> the system, which have di erent<br />
orders of magnitude. That makes the system quasi-<strong>in</strong>tegrable (for example through some adiabatic<br />
approximation a la Born–Oppenheimer) and—locally—more resistant to chaos.<br />
Hereafter, we discuss two possible alternatives of manipulat<strong>in</strong>g the <strong>wave</strong> <strong>packets</strong>. One is realized<br />
by add<strong>in</strong>g a static electric eld to the LP micro<strong>wave</strong> drive [72]. Alternatively, the addition of a<br />
static magnetic eld to CP driv<strong>in</strong>g enhances the region of classical stability, and extends the range<br />
of applicability of the harmonic approximation [30,44,46,54,62,144,151].<br />
4.1. Rydberg states <strong>in</strong> l<strong>in</strong>early polarized micro<strong>wave</strong> and static electric elds<br />
Let us rst consider a Rydberg electron <strong>driven</strong> by a resonant, l<strong>in</strong>early polarized micro<strong>wave</strong>, <strong>in</strong><br />
the presence of a static electric eld. We already realized (see the discussion <strong>in</strong> Section 3.3.2) that<br />
the classical 3D motion of the <strong>driven</strong> Rydberg electron is angularly unstable <strong>in</strong> a LP micro<strong>wave</strong><br />
eld. It turns out, however, that a stabilization of the angular motion is possible by the addition of<br />
a static electric eld Fs parallel to the micro<strong>wave</strong> polarization axis [72,152,153]. The correspond<strong>in</strong>g<br />
Hamiltonian reads<br />
H = p2 x + p 2 y + p 2 z<br />
2<br />
− 1<br />
r + Fz cos !t + Fsz ; (207)