Non-dispersive wave packets in periodically driven quantum systems

466 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
Energy (10 -3 a.u.)
-1.120
-1.125
-1.130
-1.135
-1.115
-1.125
-1.135
-1.105
-1.115
-1.125
(a)
(b)
(c)
-1.135
0 5 10
p
15 20
Fig. 21. Comparison of the semiclassical quasienergies (circles; with N = 0, see Eq. (78)), originating from the unperturbed
n0 = 21 manifold, to the exact quantum result (crosses), at di erent values of the (scaled) driving eld amplitude
F0 = Fn 4 0 =0:02 (a), 0:03 (b), 0:04 (c). The agreement is excellent. The quantum number p =0:::20 labels the quantized
classical trajectories plotted in Fig. 19, starting from the librational state |p =0〉 at lowest energy, rising through the
separatrix states |p=10〉 and |p=11〉, up to the rotational state |p=20〉. The dashed line indicates the exact quasi-energy
of the corresponding wave packet eigenstate of the 1D model discussed in Section 3.3.1. The 1D dynamics is neatly
embedded in the spectrum of the real, driven 3D atom.
coupling vanishes identically ( 1 = 0), which shows that the semiclassical results obtained from our
rst-order approximation (in F) for the Hamiltonian may be quite inaccurate in the vicinity of this
orbit. Higher-order corrections may become important.
As discussed above, the classical motion in the (L; ) plane is slower than in the (Î; ˆ ) plane. In
the semiclassical approximation, the spacing between consecutive states corresponds to the frequency
of the classical motion (see also Eq. (40)). Hence, it is to be expected that states with the same
quantum number N , but with successive quantum numbers p, will lie at neighboring energies,
building well-separated manifolds associated with a single value of N . The energy spacing between
states in the same manifold should scale as F0, while the spacing between manifolds should scale
as √ F0 (remember that F01 in the case considered here). Accurate quantum calculations fully
con rm this prediction, with manifolds originating from the degenerate hydrogenic energy levels at
F0 = 0, as we shall demonstrate now. We rst concentrate on the N = 0 manifold, originating from
n0 = 21. Fig. 21 shows the comparison between the semiclassical and the quantum energies, for
di erent values of the scaled driving eld amplitude F0 = Fn 4 0
. The agreement is excellent, except
for the lowest lying states in the manifold for F0 =0:02. The lowest energy level (p=0) corresponds
to motion close to the stable xed point L =0, = =2 in Fig. 19; the highest energy level (p = 20)
corresponds to rotational motion L=n0 1. The levels with the smallest energy di erence (p=10; 11)
correspond to the librational and the rotational trajectories closest to the separatrix, respectively. The
narrowing of the level spacing in their vicinity is just a consequence of the slowing down of the
classical motion [60]. In the same gure, we also plot (as a dashed line) the corresponding exact
quasienergy level for the 1D model of the atom (see Section 3.3.1). As expected, it closely follows