Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 487
This can be nally rewritten as
Hsec = ˆPt − 1
2Î 2 − !Î + F 1(Î; M; ; ) cos( ˆ + 1) : (204)
Both, 1 and 1, depend on the shape and orientation of the electronic elliptical trajectory, as well
as on , and are given by
and
1(Î; M; ; )=
tan 1(M; ; )= U1
V2 1 + U2 1 ; (205)
V1
; (206)
which is once more the familiar form of a system with a resonance island in the (Î; ˆ ) plane. The
expressions obtained are in fact very similar to the ones we obtained for the three-dimensional
atom exposed to a circularly polarized microwave eld, Eqs. (169)–(171), in Section 3.4.2. This
is actually not surprising: the relevant parameter for the transverse dynamics is the magnitude of
the atomic dipole oscillating with the driving eld, i.e., the scalar product of the oscillating atomic
dipole with the polarization vector. The latter can be seen either as the projection of the oscillating
atomic dipole onto the polarization plane or as the projection of the polarization vector on the plane
of the atomic trajectory. If one considers a three-dimensional hydrogen atom in a circularly polarized
eld, the projection of the polarization vector onto the plane of the atomic trajectory is elliptically
polarized with ellipticity = L=M. This is another method to rediscover Hamiltonian (204) from
Hamiltonian (169).
To obtain a semiclassical estimation of the energies of the non-dispersive wave packets we proceed
precisely in the same way as for LP and CP. Since the radial motion in (Î; ˆ ) is much faster
than in the transverse=angular degree of freedom de ned by (M; ), we rst quantize the e ective
perturbation 1(Î; M; ; ) driving the angular motion. Fig. 33 shows 1=Î 2 , as a function of M0 =
M=Î and , for two di erent values of the driving eld ellipticity . Note that 1 becomes more
symmetric as → 0, since this limit de nes the LP case, where the dynamics cannot depend on
the rotational sense of the electronic motion around the nucleus. The four extrema of 1 de ne the
possible wave-packet eigenstates. Whereas the minima at = =2; 3 =2 correspond to elliptic orbits of
intermediate eccentricity 0 ¡e¡1 perpendicular to the driving eld major axis, the ( -independent)
maxima at M0 = ±1 de ne circular orbits which co- or contra-rotate with the driving eld. In the
limit → 0, the M0= maxima are associated with the same value of 1. Hence, the actual Floquet
eigenstates appear as tunneling doublets in the Floquet spectrum, each member of the doublet being
a superposition of the co- and contra-rotating wave packets.
The quantization of the fast motion in the (Î; ˆ ) plane is similar to the one already performed
in the LP and CP cases. As we already observed (see Fig. 20), the size of the resonance island
in the (Î; ˆ ) plane is proportional to √ 1. Correspondingly, also the localization properties of the
wave packet along the classical trajectory improve with increasing 1. We therefore conclude from
Fig. 33 that the eigenstates corresponding to the minima of 1 cannot be expected to exhibit strong
longitudinal localization, whereas the eigenstates localized along the circular orbits at the maxima
of 1 can.