Atoms subjected to short pulses/'kicks' from optical lattices represent the quantum version of the standard paradigm of Hamiltonian classical chaos, the "Standard Map". There was a lot of excitement when in 1995, a Texas group demonstrated "Dynamical Localization" a phenomenon sometimes called the "quantum suppression of chaotic diffusion". The Figure alongside shows their results: the energy of cloud of atoms subjected to the kicks, initially grows linearly with time, as expected from the chaotic classical dynamics. However, for the quantum case this process stops at the "Break-time". The quantum momentum distribution (which should grow broader as the energy increases) freezes and assumes an exponential shape (the characteristic "triangular shape" using a log plot, as shown in the inset. The width of the distribution, its "localization length" L is proportional to 1/T where T is the kick period. |
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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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