Non-dispersive wave packets in periodically driven quantum systems

A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 463
• The time-scale of the “transverse” (or angular) motion along the (L; ), (M; ) variables. Because
these are constant for the unperturbed Coulomb system, the time derivatives like dL=dt and d =dt
generated by Eqs. (159)–(161) are proportional to F, and the resulting time-scale is proportional
to 1=F. More precisely, it is of the order of 1=F0 Kepler periods, i.e., once again, signi cantly
longer than the preceding time-scale.
From this separation of time scales, it follows that we can use the following, additional secular
approximation: for the motion in the (Î; ˆ ) plane, 1 and 1 are adiabatic invariants, which can be
considered as constant quantities. We then exactly recover the Hamiltonian discussed for the 1D
model of the atom, with a resonance island con ning trajectories with librational motion in the (Î; ˆ )
plane, and rotational motion outside the resonance island. The center of the island is located at
Î = Î 1 = ! −1=3 = n0; ˆ = − 1 : (162)
As already pointed out in Section 3.1.1, the size of the resonance island in phase space is determined
by the strength of the resonant coupling 1(Î; L; ). In the pendulum approximation, its extension in
Î scales as 1(Î 1;L; ), i.e., with 1 evaluated at the center, Eq. (162), of the island.
The last step is to consider the slow motion in the (L; ) plane. As usual, when a secular approximation
is employed, the slow motion is due to an e ective Hamiltonian which is obtained
by averaging of the secular Hamiltonian over the fast motion. Because the coupling 1(Î; L; ) exhibits
a simple scaling with Î (apart from a global Î 2 dependence, it depends on the scaled angular
variables L0 and only), the averaging over the fast motion results in an e ective Hamiltonian
for the (L0; ) motion which depends on 1(Î; L; ) only. We deduce that the slow motion follows
curves of constant 1(Î 1;L; ); at a velocity which depends on the average over the fast variables.
1(Î 1;L; ) is thus a constant of motion, both for the fast (Î; ˆ ) and the slow (L; ) motion. This
also implies that the order of the quantizations in the fast and slow variables can be interchanged:
using the dependence of 1(Î 1;L; )on(L; ); we obtain quantized values of 1 which in turn can
be used as constant values to quantize the (Î; ˆ ) fast motion. Note that the separation of time scales
is here essential. 16 Finally, the dynamics in (M; ) is trivial, since M is a constant of motion. In
the following, we will consider the case M = 0 for simplicity. Note that, when the eccentricity of
the classical ellipse tends to 1—i.e., L → 0 – and when → 0 Hamiltonian (158) coincides exactly
with the Hamiltonian of the 1D atom, Eq. (141). This is to be expected, as it corresponds to a
degenerate classical Kepler ellipse along the z-axis.
The adiabatic separation of the radial and of the angular motion allows the separate WKB quantization
of the various degrees of freedom. In addition to the quantization conditions in ( ˆPt;t) and
(Î; ˆ ), Eqs. (76)–(79), already formulated in our general description of the semiclassical approach
in Section 3.1.3, we additionally need to quantize the angular motion, according to:
1
L d = p +
2
1
˝ ; (163)
2
along a loop of constant 1 in the (L; ) plane.
16 If one considers non-hydrogenic atoms—with a core potential in addition to the Coulomb potential—the classical
unperturbed ellipse precesses, adding an additional time-scale, and the separation of time scales is much less obvious. See
also Section 8.2.