Non-dispersive wave packets in periodically driven quantum systems

492 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
Fig. 36. Contours of the e ective Hamiltonian, Eq. (212), in the (L0; ) plane (with L0=L=n0 the scaled angular momentum,
and the angle between the major axis of the elliptical trajectory and the eld axis). The potential surface generates
the slow evolution of the angular coordinates of the Kepler trajectory of a Rydberg electron exposed to collinear, static
and resonant microwave electric elds. Initial atomic principal quantum number n0 = 60; scaled microwave amplitude
F0 = Fn 4 0 =0:03, and scaled static eld amplitude Fs;0 =0:12F0 ¡Fs;c (a), Fs;0 =0:25F0 ¿Fs;c (b), with Fs;c the critical
static eld amplitude de ned in Eq. (213). The lighter the background, the higher the e ective energy. The contours are
plotted at the semiclassical energies which quantize He according to Eq. (163), and thus represent the 60 eigenenergies
shown in Fig. 37. Observe the motion of the stable island along the = line (corresponding to the energetically highest
state in the manifold), under changes of Fs.
Explicit evaluation of the ground state energy of the locally harmonic potential yields the e ective
Hamiltonian for the slow motion in the (L; ) plane:
He = −
(L; )
+ F 1(L; )+FsX0(L) cos ;
2
(212)
where all quantities are evaluated at Î =Î 1. For the determination of the angular localization properties
of the wave packet it is now su cient to inspect the extrema of He .
For Fs;0 = 0, we recover the pure LP case with a maximum along the line L0 = 1 (circular state),
and a minimum at L0 =0, = =2 (see Fig. 19), corresponding to a straight line orbit perpendicular
to the eld. For increasing Fs;0, the maximum moves towards lower values of L0, and contracts in
, whereas the minimum approaches = 0 for constant L0 = 0, see Fig. 36. It is easy to show that
there exists a critical value Fs;c of the static eld, depending on Î 1,
Fs;c = 2
3
F0J ′ 1(1) −
3F0J ′ 1 (1)
4Î 1
0:217F0 − 0:164
√ F0
Î 1
; (213)
above which both xed points reach L0 = 0. Then, in particular, the maximum at L0 =0, = ,
corresponds to a straight line orbit parallel to ˜Fs. Note that in the classical limit, Î 1 →∞, Eq. (213)
recovers the purely classical value [153] for angular stability of the straight line orbit along the
polarization axis, as it should. Therefore, by variation of Fs;0 ∈ [0;Fs;c], we are able to continuously
tune the position of the maximum in the (L0; ) plane. Consequently, application of an additional
static electric eld gives us control over the trajectory traced by the wave packet. This is further
illustrated in Fig. 37, through the semiclassical level dynamics of the resonantly driven manifold
originating from the n0 = 60 energy shell, as a function of Fs. In the limit Fs = 0, the spectrum