Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 465
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(a) (b) (c)
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−1.0 −0.5 0.0 0.5
−1.0 −0.5 0.0 0.5 −1.0 −0.5 0.0 0.5 1.0
(θ+δ 1)/π
Fig. 20. Isovalue curves of the secular Hamiltonian Hsec, Eq. (159), generating the (Î; ˆ ) motion of a Rydberg electron
in a resonant microwave eld. Î and ˆ correspond to the atomic principal quantum number, and to the polar angle of the
electron on the Kepler ellipse, respectively. The scaled microwave amplitude is xed at F0 =0:03. Since the isovalues
of Hsec depend on the transverse motion in (L; ) via the constant value of 1, Eq. (160), contours (—) are shown for
three characteristic values of 1, corresponding to xed quantum numbers p =0; 10; 20, Eq. (163), of the angular motion
for the n0 = 21 resonant manifold. Only the “ground state” orbit satisfying Eq. (78) with N = 0 is shown, together with
the separatrix (- - -) between librational and rotational motion in the (Î; ˆ ) plane. The separatrix encloses the principal
resonance island in phase space, see also Eqs. (66) and (71). Panel (a) corresponds to the orbit with L0 = L=n0 1
(rotational orbit, p = 20), panel (b) to the orbit close to the separatrix of the angular motion (p = 10), panel (c) to the
librational orbit close to the stable xed point L0 =0, = =2. Note that the resonance island is smallest for librational,
largest for rotational, and of intermediate size for separatrix modes of the angular motion.
actually correspond to the same physics. The oscillating dipole clearly vanishes there, resulting
in a global minimum of 1(L =0; = =2; 3 =2) = 0. This stable xed point is surrounded by
“librational” trajectories in the (L; ) plane.
• (L =0; =0; ). This corresponds to a degenerate, straight line trajectory along the microwave
eld, i.e., the situation already considered in the 1D model of the atom. = 0 and correspond
to the two orbits pointing up and down, which are of course equivalent. This is a saddle point of
1(L =0; =0; )=J ′ 2
1 (1)Î 1: Hence, it is an unstable equilibrium point. As an implication, in the
real 3D world, the motion along the microwave axis, with the phase of the radial motion locked
on the external drive, is angularly unstable (see also Section 4.1). This leads to a slow precession
of the initially degenerate Kepler ellipse o the axis, and will manifest itself in the localization
properties of the 3D analog of the non-dispersive wave packet displayed in Fig. 13. This motion
takes place along the separatrix between librational and rotational motion.
Once the quantized values of 1 (represented by the trajectories in Fig. 19) have been determined,
we can quantize the (Î; ˆ ) motion with these values xed. Fig. 20 shows the equipotential lines of
Hsec, for the three values of 1 corresponding to the p = 0, 10 and 20 states, see Eq. (163), of
the n0 = 21 manifold. In each case, the contour for the lowest state N = 0 has been drawn, together
with the separatrix between the librational and rotational (Î; ˆ ) modes. The separatrix determines
the size of the principal resonance island for the di erent substates of the transverse motion. Note
that the principal resonance is largest for the p = 20 state, localized closest to the stable circular
orbit (hence associated with the maximum value of 1), whereas the smallest resonance island is
obtained for the p = 0 state, localized in the vicinity of (though not precisely at) the straight line
orbit perpendicular to the eld axis (minimum value of 1). For the latter orbit itself, the rst-order
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