Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
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520 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
|Shift| Distribution<br />
10 0<br />
10 -1<br />
10 -2<br />
10 -2<br />
10 -1<br />
10 0<br />
10 1<br />
10 -3<br />
(a) |AC Stark Shift| (b)<br />
Fig. 52. The distribution of energy shifts (a) and ionization widths (b) for the non-<strong>dispersive</strong> <strong>wave</strong> packet of a<br />
two-dimensional hydrogen atom <strong>in</strong> a circularly polarized eld, obta<strong>in</strong>ed by numerical diagonalization of the Hamiltonian<br />
(large b<strong>in</strong>s), compared to the random matrix model (small b<strong>in</strong>s). Both distributions are shown on a double logarithmic<br />
scale to better visualize the behavior over a large range of shift and width values. S<strong>in</strong>ce the energy shift may be positive<br />
or negative, we show the distribution of its modulus. The random matrix model ts very well the numerical results, with<br />
both distributions show<strong>in</strong>g regions of algebraic behavior followed by an exponential cut-o .<br />
were used for a s<strong>in</strong>gle t [65]. This allowed us to study the dependence of the parameters = ; =<br />
on n0 and F0. The dependence on n0 is shown <strong>in</strong> Fig. 53. Clearly, the tunnel<strong>in</strong>g rate = decreases<br />
exponentially with n0. S<strong>in</strong>ce n0 is the <strong>in</strong>verse of the e ective Planck constant <strong>in</strong> our problem (see<br />
the discussion <strong>in</strong> Section 3.3.1 and Eq. (147)), this shows that<br />
=<br />
<br />
˙ exp − S<br />
˝e<br />
<br />
; (267)<br />
where S, correspond<strong>in</strong>g to some e ective imag<strong>in</strong>ary action [188], is found to be given for our<br />
speci c choice of parameters by S 0:06 ± 0:01 (as tted from the plot). Such an exponential<br />
dependence is a hallmark of a tunnel<strong>in</strong>g process, thus con rm<strong>in</strong>g that the <strong>wave</strong> <strong>packets</strong> are strongly<br />
localized <strong>in</strong> the island and communicate with the outside world via tunnel<strong>in</strong>g. The n0 dependence<br />
of the dimensionless chaotic ionization rate = is very di erent: it shows a slow, algebraic <strong>in</strong>crease<br />
with n0. A simple analysis based on a Kepler map [131] description would yield a l<strong>in</strong>ear <strong>in</strong>crease<br />
with n0, whereas our data seem to suggest a quadratic function of n0. This discrepancy is not very<br />
surpris<strong>in</strong>g, bear<strong>in</strong>g <strong>in</strong> m<strong>in</strong>d the simplicity of the Kepler map approach.<br />
Similarly, we may study, for xed n0, the dependence of = and = on F0, i.e., on the micro<strong>wave</strong><br />
eld strength. Such studies, performed for both l<strong>in</strong>ear and circular polarizations, have <strong>in</strong>dicated that,<br />
not very surpris<strong>in</strong>gly, the chaotic ionization rate = <strong>in</strong>creases rather smoothly with the micro<strong>wave</strong><br />
amplitude F0. On the other hand, the tunnel<strong>in</strong>g rate = shows pronounced non-monotonic variations<br />
with F0, see Fig. 54. This unexpected behavior can nontheless be expla<strong>in</strong>ed [65]. The bumps <strong>in</strong> =<br />
occur at micro<strong>wave</strong> eld strengths where secondary nonl<strong>in</strong>ear resonances emerge with<strong>in</strong> the resonance<br />
island <strong>in</strong> classical phase space. For circular polarization, this corresponds to some resonance between<br />
two eigenfrequencies !+ and !− (see Section 3.4 and Figs. 30 and 59) of the dynamics <strong>in</strong> the<br />
classical resonance island. Such resonances strongly perturb the classical dynamics and necessarily<br />
a ect the <strong>quantum</strong> transport from the island.<br />
Let us stress nally that, even for rather strong micro<strong>wave</strong> elds (say F0=0:05), where most of the<br />
other Floquet states have lifetimes of few tens or hundreds of micro<strong>wave</strong> periods, and irrespective<br />
of the polarization of the driv<strong>in</strong>g eld or of the dimension of the accessible con guration space (1D,<br />
2D or 3D), the lifetime (modulo uctuations) of a non-<strong>dispersive</strong> <strong>wave</strong> packet is typically of the<br />
Width Distribution<br />
10 1<br />
10 -1<br />
10 -3<br />
10 -5<br />
10 -7<br />
10 -5<br />
10 -3<br />
10 -1<br />
Width<br />
10 1<br />
10 3