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Experiments by the cold atoms group at UCL (ref[1]) tried instead to
apply pairs of closely spaced kicks. The results were
surprisingly different from the standard single-kick case. Even in the chaotic
regime, the classical phase-space space has a `cellular' structure, with momentum regions partly separated
by porous regions where diffusion is slow. The quantum momentum probability distributions
have a novel 'staircase' structure superposed on the exponential of the
Dynamical localization. The diffusion had a completely different character
to the usual Standard Map, because of the "trapping regions" which divide the
cells. At first we conjectured that they might be cantori: these represent a sort of fractal
dust which arises as the regular trajectories (tori) break up when a Hamiltonian system becomes chaotic.
A very nice result of chaos theory shows that the tori which are the most resistant
to chaos (the last to break up) are the "golden cantori". These correspond
to momenta P/(2*pi)= golden ratio~0.618). As it happens, these would occur in the vicinity of
the trapping regions, so at the outset, this seemed a reasonable explanation.
The top figure on the right shows that the trapping regions of the DKR2
(middle two figures) coincide with the golden cantori (which surround the
period 3 island chain on the Surface of Section for the Standard Map on the
far right).
The quantum momentum probability distributions
have a novel 'staircase' structure superposed on the exponential of the
Dynamical localization. Examples of the staircase distribution are shown in the
bottom figure on the right .
The width ('Localization Length') of the staircase
scales with a fractional power of (-2/3)rds of the
period, unlike the usual kicked system, which scales with an integer power of (-1).
This too pointed to golden cantori as they have been associated with ratios
of (2/3) and (3/4). However, repeating the calculations in regimes
with no cantori whatsoever, made no difference to the fractional
scaling ratio of (2/3). A closer analysis however found a completely
different explanation for this ratio. Its origin resides in a
function called an Airy function, which is involved in the mathematical
description of the dynamics in the trapping regions. |
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