Non-dispersive wave packets in periodically driven quantum systems

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