Non-dispersive wave packets in periodically driven quantum systems

526 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
SQUARED DIPOLE
1.0
0.5
0.0
ω−ω +
ω
ω+ ω−
0.0 0.5 1.0 1.5 2.0
Fig. 56. Square of the dipole matrix element (scaled w.r.t. n0, i.e., divided by n 2 0) connecting the |0; 0; 0〉 non-dispersive
wave packet of a three-dimensional hydrogen atom in a circularly polarized microwave eld with other Floquet states,
as a function of the energy di erence between the two states. The stick spectrum is the exact result obtained from a
numerical diagonalization with ! =1=60 3 , corresponding to a principal quantum number n0 = 60, and scaled amplitude
F0 =0:04446; in natural units, the microwave frequency !=2 is 30:48 GHz, and the microwave amplitude 17:6 V=cm.
The crosses represent the analytic prediction within the harmonic approximation [68]. There are three dominant lines
( + polarized) discussed in the text, other transitions (as well as transitions with − or polarizations) are negligible,
what proves the validity of the harmonic approximation. If a weak probe eld (in the microwave domain) is applied to the
system in addition to the driving eld, its absorption spectrum should therefore show the three dominant lines, allowing
an unambiguous characterization of the non-dispersive wave packet.
and the semiclassical energies observed in Figs. 30 and 59). This is not completely surprising as
the energy levels themselves are well reproduced by this harmonic approximation, see Section 3.4.4.
However, the photoabsorption spectrum probes the wave functions themselves (through the overlaps)
which are well known to be much more sensitive than the energy levels. The good agreement for
both the energy spectrum and the matrix elements is a clear-cut proof of the reliability of the
harmonic approximation for physically accessible principal quantum numbers, say n0 ¡ 100; in fact,
it is good down to n0 30, and the non-dispersive wave packet exists even for lower n0 values
(e.g. n0 = 15 in [49]) although the harmonic approximation is not too good at such low quantum
numbers. There were repeated claims in the literature [30,44,46,54,62,144] that the stability island as
well as the e ective potential are necessarily unharmonic in the vicinity of the equilibrium point, and
that the unharmonic terms will destroy the stability of the non-dispersive wave packets. The present
results prove that these claims are doubly wrong: rstly, as explained in Section 3.4.4, harmonicity
is not a requirement for non-dispersive wave-packets to exist (the only condition is the existence of
a su ciently large resonance island); secondly, the harmonic approximation is clearly a very good
approximation even for moderate values of n0.
Multiplication by the free space density of states transforms Fig. 56 in Fig. 57, which shows
that the corresponding spontaneous decay rates are very low, of the order of 100 Hz at most. They
are few orders of magnitude smaller than the ionization rates and thus may be di cult to observe.
With increasing n0, the spontaneous rate decreases algebraically while the ionization rate decreases
exponentially, see Section 7.2.4. Thus for large n0, the spontaneous emission may be the dominant
ωp